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abstract: 'By means of white noise analysis, we prove some limit theorems for nonlinear functionals of a given Volterra process. In particular, our results apply to fractional Brownian motion (fBm) and should be compared with the classical convergence results of the 1980s due to Breuer, Dobrushin, Giraitis, Major, Surgailis and Taqqu, as well as the recent advances concerning the construction of a Lévy area for fBm due to Coutin, Qian and Unterberger.'
address:
- 'Université Aix-Marseille I, 39 rue Joliot Curie, 13453 Marseille Cedex 13, France.\'
- 'Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie (Paris VI), Boîte Courrier 188, 4 place Jussieu, 75252 Paris Cedex 05, France. '
- 'Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA.\'
author:
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title: Limit theorems for nonlinear functionals of Volterra processes via white noise analysis
---
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Introduction {#intro}
============
Fix $T>0$ and let $B=(B_t)_{t\geq 0}$ be a fractional Brownian motion with Hurst index $H\in(0,1)$, defined on some probability space $(\Omega,\mathcal{B},P)$. Assume that $\mathcal{B}$ is the completed $\sigma$-field generated by $B$. Fix an integer $k\geq 2$ and, for $\e>0$, consider $$\label{geps}
G_\e = \e^{-k(1-H)}\int_0^T h_k
\biggl(\frac{B_{u+\e}-B_u}{\e^{H}}\biggr)\,\mathrm{d}u.$$ Here, and in the rest of this paper, $$\label{herm-pol}
h_k(x)=(-1)^k \mathrm{e}^{x^2/2}\frac{\mathrm{d}^k}{\mathrm{d}x^k} (\mathrm{e}^{-x^2/2} )$$ stands for the $k$th Hermite polynomial. We have $h_2(x)=x^2-1$, $h_3(x)=x^3-3x$ and so on.
Since the seminal works [@BM; @DM; @GS; @taqqu75; @taqqu79] by Breuer, Dobrushin, Giraitis, Major, Surgailis and Taqqu, the following three convergence results are classical:
- if $H<1- \frac 1{2k}$, then $$\label{cv<}
\bigl((B_t)_{t\in[0,T]}
,\e^{k(1-H) -1/2} G_{\e} \bigr)
\displaystyle\mathop{\stackrel{\mathrm{Law}}{\longrightarrow}}_{\e\to 0}
\bigl((B_t)_{t\in[0,T]}, N \bigr),$$ where $N\sim\mathscr{N} (0,T
\times k!\int_0^T\rho^k(x)\,\mathrm{d}x )$ is independent of $B$, with $\rho(x)=\frac12 (|x+1|^{2H}+|x-1|^{2H}-2|x|^{2H} )$;
- if $H=1- \frac 1{2k}$, then $$\label{cv=}
\biggl((B_t)_{t\in[0,T]}
, \frac{G_{\e}}
{
\sqrt{\log(1/\e)}
} \biggr) \displaystyle\mathop{\stackrel{ \mathrm{Law}}{\longrightarrow}}_{\e\to 0 }
\bigl((B_t)_{t\in[0,T]}, N \bigr),$$ where $N\sim\mathscr{N} (0,T
\times 2k!(1-\frac1{2k})^k(1-\frac1k)^k )$ is independent of $B$;
- if $H>1-\frac1{2k}$, then $$\label{cv>}
G_{\e} \displaystyle\mathop{\stackrel{L^2(\Omega)}{\longrightarrow}}_{\e\to 0}
Z_T^{(k)},$$ where $Z^{(k)}_T$ denotes the Hermite random variable of order $k$; see Section \[sec31\] for its definition.
Combining (\[cv<\]) with the fact that $\sup_{0<\e\leq 1}E [|\e^{k(1-H)-1/2}G_\e|^p ]<\infty$ for all $p\geq 1$ (use the boundedness of $\operatorname{Var}(\e^{k(1-H)-1/2}G_\e)$ and a classical hypercontractivity argument), we have, for all $\eta\in
L^2(\Omega )$ and if $H<1-\frac1{2k}$, that $$\e^{k(1-H)-1/2}E[\eta G_\e]
\mathop{\longrightarrow}_{\e\to 0}
E(\eta N)=E(\eta)E(N)=0$$ (a similar statement holds in the critical case $H=1-\frac1{2k}$). This means that $\e^{k(1-H)-1/2} G_\e$ converges *weakly* in $L^2(\Omega)$ to zero. The following question then arises. Is there a normalization of $G_\e$ ensuring that it converges *weakly* towards a *non-zero* limit when $H\leq 1-\frac1{2k}$? If so, then what can be said about the limit? The first purpose of the present paper is to provide an answer to this question in the framework of *white noise analysis*.
In [@nualartwhite], it is shown that for all $H\in(0,1)$, the time derivative $\dot{B}$ (called the *fractional white noise*) is a distribution in the sense of Hida. We also refer to Bender [@bender], Biagini *et al.* [@BOSW] and references therein for further works on the fractional white noise.
Since we have $E(B_{u+\e}-B_u)^2=\e^{2H}$, observe that $G_\e$ defined in (\[geps\]) can be rewritten as $$G_{\e}=\int_0^T \biggl(\frac{B_{u+\e}-B_u}{\e} \biggr)^{\diamond k}\,\mathrm{d}u, \label{e1bis}$$ where $(\ldots)^{\diamond k}$ denotes the $k$th Wick product. In Proposition \[thm-fbm\] below, we will show that for all $H\in (\frac12-\frac1k,1 )$, $$\label{cvwick}
\lim_{\e\to 0}\int_0^T
\biggl(\frac{B_{u+\e}-B_u}{\e} \biggr)^{\diamond k}\,\mathrm{d}u
=\int_0^T \dot{B}_u^{\diamond k}\,\mathrm{d}u,$$ where the limit is in the $(\mathcal{S})^*$ sense.
In particular, we observe two different types of asymptotic results for $G_{\e}$ when $H\in (\frac12-\frac1k,1- \frac 1{2k} )$: convergence (\[cvwick\]) in $(\mathcal{S})^*$ to a Hida distribution, and convergence (\[cv<\]) in law to a normal law, with rate $\e^{ 1/2 - k(1-H) }$. On the other hand, when $H\in (1-\frac1{2k},1 )$, we obtain from (\[cv>\]) that the Hida distribution $\int_0^T \dot{B}_s^{\diamond k}\,\mathrm{d}s$ turns out to be the square-integrable random variable $Z_T ^{(k)}$, which is an interesting result in its own right.
In Proposition \[CVS\*\], the convergence (\[cvwick\]) in $(\mathcal{S})^*$ is proved for a general class of Volterra processes of the form $$\label{def-vol}
\int_0^t K(t,s)\, \mathrm{d}W_s,\qquad t\geq 0,$$ where $W$ stands for a standard Brownian motion, provided the kernel $K$ satisfies some suitable conditions; see Section \[volterra\].
We also provide a new proof of the convergence (\[cv<\]) based on the recent general criterion for the convergence in distribution to a normal law of a sequence of multiple stochastic integrals established by Nualart and Peccati [@NP] and by Peccati and Tudor [@PT], which avoids the classical method of moments.
In two recent papers [@MR1; @MR2], Marcus and Rosen have obtained central and non-central limit theorems for a functional of the form (\[geps\]), where $B$ is a mean zero Gaussian process with stationary increments such that the covariance function of $B$, defined by $\sigma^2(|t-s|)=\operatorname{Var}(B_t-B_s)$, is either convex (plus some additional regularity conditions), concave or given by $\sigma^2(h)=h^r$ with $1<r<2$. These theorems include the convergence (\[cv<\]) and, unlike our simple proof, are based on the method of moments.
In the second part of the paper, we develop a similar analysis for functionals of two independent fractional Brownian motions (or, more generally, Volterra processes) related to the Lévy area. More precisely, consider two *independent* fractional Brownian motions $B^{(1)}$ and $B^{(2)}$ with (for simplicity) the same Hurst index $H\in(0,1)$. We are interested in the convergence, as $\e\to
0$, of $$\label{tilde}
\widetilde{G}_\e:=\int_0^T B^{(1)}_u \frac{B^{(2)}_{u+\e}-B^{(2)}_u}{\e}\, \mathrm{d}u$$ and $$\label{breve}
\breve{G}_\e:=\int_0^T \biggl(\int_0^u \frac{B^{(1)}_{v+\e}-B^{(1)}_v}{\e}\, \mathrm{d}v \biggr)
\frac{B^{(2)}_{u+\e}-B^{(2)}_u}{\e} \,\mathrm{d}u .$$ Note that $\widetilde{G}_\e$ coincides with the $\e$-integral associated with the forward Russo–Vallois integral $\int_0^T B^{(1)}\,\mathrm{d}^-B^{(2)}$; see, for example, [@RVLN] and references therein. Over the last decade, the convergences of $\widetilde{G}_\e$ and $\breve{G}_\e$ (or of related families of random variables) have been intensively studied. Since $\e^{-1}\int_0^u (B^{(1)}_{v+\e}-B^{(1)}_v)\,\mathrm{d}v$ converges pointwise to $B^{(1)}_u$ for any $u$, we could think that the asymptotic behaviors of $\widetilde{G}_\e$ and $\breve{G}_\e$ are very close as $\e\to 0$. Surprisingly, this is not the case. Actually, only the result for $\breve{G}_\e$ agrees with the seminal result of Coutin and Qian [@CQ] (that is, we have convergence of $\breve{G}_\e$ in $L^2(\Omega)$ if and only if $H>1/4$) and with the recent result by Unterberger [@unterbergertcl] (that is, adequately renormalized, $\breve{G}_\e$ converges in law if $H<1/4$). More precisely:
- if $H<1/4$, then $$\label{star10}
\bigl( \bigl(B^{(1)}_t,B^{(2)}_t\bigr) _{t\in [0,T]}, \e^{1/2-2H} \breve{G}_{\e} \bigr)
\displaystyle\mathop{\stackrel{ \mathrm{Law}}{\longrightarrow}}_{\e\to 0}\hspace*{1pt}
\bigl( \bigl(B^{(1)}_t,B^{(2)}_t\bigr) _{t\in [0,T]}, N \bigr),$$ where $N \sim \mathscr{N} (0,T\breve{\sigma}^2_H )$ is independent of $(B^{(1)},B^{(2)})$ and $$\begin{aligned}
&&\breve{\sigma}^2_H=\frac{1}{4(2H+1)(2H+2)}
\int_{\mathbb{R}}(|x+1|^{2H}+|x-1|^{2H}-2|x|^{2H} )\\
&&\hspace*{119pt}{}\times (2|x|^{2H+2}-|x+1|^{2H+2}-|x-1|^{2H+2} )\,\mathrm{d}x;\end{aligned}$$
- if $H=1/4$, then $$\label{star10crit}
\biggl( \bigl(B^{(1)}_t,B^{(2)}_t\bigr) _{t\in [0,T]},
\frac{ \breve{G}_{\e}}{\sqrt{\log(1/\e)}} \biggr)
\displaystyle\mathop{\stackrel{ \mathrm{Law}}{\longrightarrow}}_{\e\to 0}\hspace*{1.5pt}
\bigl( \bigl(B^{(1)}_t,B^{(2)}_t\bigr) _{t\in [0,T]}, N \bigr),$$ where $N \sim \mathscr{N} (0,T/8 )$ is independent of $(B^{(1)},B^{(2)})$;
- if $H> 1/4$, then $$\label{star11}
\breve{G}_{\e} \displaystyle\mathop{\stackrel{ L^2(\Omega) }{\longrightarrow}}_{\e\to 0}
\int_0^T B^{(1)}_u\diamond \dot{B}^{(2)}_u \,\mathrm{d}u=\int_0^T B^{(1)}_u\,\mathrm{d}B^{(2)}_u;$$
- for all $H\in(0,1)$, we have $$\label{star12}
\breve{G}_{\e} \displaystyle\mathop{\stackrel{ (\mathcal{S})^*}{\longrightarrow}}_{\e\to 0}
\int_0^T B^{(1)}_u\diamond \dot{B}^{(2)}_u \,\mathrm{d}u.$$
However, for $\widetilde{G}_\e$, we have, in contrast:
- if $H<1/2$, then $$\label{star1}
\bigl( \bigl(B^{(1)}_t,B^{(2)}_t\bigr) _{t\in [0,T]}, \e^{1/2-H} \widetilde{G}_{\e} \bigr)
\displaystyle\mathop{\stackrel{ \mathrm{Law}}{\longrightarrow}}_{\e\to 0 }\hspace*{1.5pt}
\bigl( \bigl(B^{(1)}_t,B^{(2)}_t\bigr) _{t\in [0,T]}, N\times S \bigr),$$ where $$S=\sqrt{\int_0^\infty (|x+1|^{2H}+|x-1|^{2H}-2|x|^{2H} )\,\mathrm{d}x\times\int_0^T \bigl(B^{(1)}_u\bigr)^2\,\mathrm{d}u}$$ and $N\sim\mathscr{N}(0,1)$, independent of $(B^{(1)},B^{(2)})$;
- if $H\geq 1/2$, then $$\label{star3}
\widetilde{G}_{\e}
\displaystyle\mathop{\stackrel{ L^2(\Omega) }{\longrightarrow}}_{\e\to 0}
\int_0^T B^{(1)}_u\diamond \dot{B}^{(2)}_u \,\mathrm{d}u=\int_0^T B^{(1)}_u\,\mathrm{d}B^{(2)}_u;$$
- for all $H\in(0,1)$, we have $$\label{star4}
\widetilde{G}_{\e} \displaystyle\mathop{\stackrel{ (\mathcal{S})^*}{\longrightarrow}}_{\e\to 0}
\int_0^T B^{(1)}_u\diamond \dot{B}^{(2)}_u \,\mathrm{d}u.$$
Finally, we study the convergence, as $\e\to 0$, of the so-called $\e$*-covariation* (following the terminology of [@RVLN]) defined by $$\widehat{G}_\e:=
\int_0^T \frac{B^{(1)}_{u+\e}-B^{(1)}_u}{\e}\times\frac{B^{(2)}_{u+\e}-B^{(2)}_u}{\e} \,\mathrm{d}u\label{hat}$$ and we get:
- if $H<3/4$, then $$\label{star5}
\bigl( \bigl(B^{(1)}_t,B^{(2)}_t\bigr) _{t\in [0,T]}, \e^{3/2-2H} \widehat{G}_{\e} \bigr)
\displaystyle\mathop{\stackrel{ \mathrm{Law}}{\longrightarrow}}_{\e\to 0 }\hspace*{1pt}
\bigl( \bigl(B^{(1)}_t,B^{(2)}_t\bigr) _{t\in [0,T]}, N \bigr)$$ with $N \sim \mathscr{N}(0,T\widehat{\sigma}^2_H)$ independent of $(B^{(1)},B^{(2)})$ and $$\widehat{\sigma}^2_H=\frac14
\int_{\mathbb{R}}(|x+1|^{2H}+|x-1|^{2H}-2|x|^{2H} )^2\,\mathrm{d}x;$$
- if $H=3/4$, then $$\label{star6}
\biggl( \bigl(B^{(1)}_t,B^{(2)}_t\bigr) _{t\in [0,T]},\frac{\widehat{G}_{\e}}
{ \sqrt{\log(1/\e)} } \biggr) \displaystyle\mathop{\stackrel{ \mathrm{Law}}{\longrightarrow}}_{\e\to 0 }\hspace*{1.5pt}
\bigl( \bigl(B^{(1)}_t,B^{(2)}_t\bigr) _{t\in [0,T]}, N \bigr)\\$$ with $N \sim \mathscr{N}(0,9T/32)$ independent of $(B^{(1)},B^{(2)})$;
- if $H>3/4$, then $$\label{star7}
\widehat{G}_{\e} \displaystyle\mathop{\stackrel{ L^2(\Omega) }{\longrightarrow}}_{\e\to 0}
\int_0^T \dot{B}^{(1)}_u\diamond \dot{B}^{(2)}_u \,\mathrm{d}u;$$
- for all $H\in(0,1)$, we have $$\label{star8}
\widehat{G}_{\e} \displaystyle\mathop{\stackrel{ (\mathcal{S})^* }{\longrightarrow}}_{\e\to 0}
\int_0^T \dot{B}^{(1)}_u\diamond \dot{B}^{(2)}_u \,\mathrm{d}u.$$
The paper is organized as follows. In Section \[sec2\], we introduce some preliminaries on white noise analysis. Section \[volterra\] is devoted to the study, using the language and tools of the previous section, of the asymptotic behaviors of $G_\e$, $\widetilde{G}_\e$ and $\widehat{G}_\e$ in the (more general) context where $B$ is a Volterra process. Section \[sec4\] is concerned with the fractional Brownian motion case. In Section \[sec5\] (resp., Section \[sec6\]), we prove (\[cv<\]) and (\[cv=\]) (resp., (\[star10\]), (\[star10crit\]), (\[star1\]), (\[star5\]) and (\[star6\])).
White noise functionals {#sec2}
=======================
In this section, we present some preliminaries on white noise analysis. The classical approach to white noise distribution theory is to endow the space of tempered distributions $\mathcal{S}'({\mathbb{R}})$ with a Gaussian measure ${\mathbb{P}}$ such that, for any rapidly decreasing function $\eta\in \mathcal{S}({\mathbb{R}})$, $$\int_{\mathcal{S}'({\mathbb{R}})}\mathrm{e}^{\mathrm{i}\la x,\eta\ra}{\mathbb{P}}(\mathrm{d}x)=\mathrm{e}^{-\sfrac{|\eta|_0^2}{2}}.$$ Here, $|\cdot|_0$ denotes the norm in $L^2({\mathbb{R}})$ and $\langle \cdot, \cdot \rangle$ the dual pairing between $\mathcal{S}'({\mathbb{R}})$ and $\mathcal{S}({\mathbb{R}})$. The existence of such a measure is ensured by Minlos’ theorem [@kuo].
In this way, we can consider the probability space $(\Omega,\mathcal{B},{\mathbb{P}})$, where $\Omega=\mathcal{S}'({\mathbb{R}})$. The pairing $\langle x, \xi \rangle$ can be extended, using the norm of $L^2(\Omega)$, to any function $\xi \in L^2({\mathbb{R}})$. Then, $W_t=\la\cdot,\1_{[0,t]}\ra$ is a two-sided Brownian motion (with the convention that $ \1_{[0,t]}= -\1_{[t,0]}$ if $t<0$) and for any $\xi \in L^2({\mathbb{R}})$, $$\la\cdot,\xi \ra= \int_{-\infty}^\infty \xi\, \mathrm{d}W=I_1(\xi)$$ is the Wiener integral of $\xi$.
Let $\Phi\in L^2(\Omega)$. The classical Wiener chaos expansion of $\Phi$ says that there exists a sequence of symmetric square-integrable functions $\phi_n \in L^2({\mathbb{R}}^n)$ such that $$\label{wiener}
\Phi= \sum_{n=0} ^\infty I_n(\phi_n),$$ where $I_n$ denotes the multiple stochastic integral.
The space of Hida distributions
-------------------------------
Let us recall some basic facts concerning tempered distributions. Let $(\xi_n)_{n=0}^\infty$ be the orthonormal basis of $L^2({\mathbb{R}})$ formed by the Hermite functions given by $$\label{xi_n}
\xi_n(x)=\curpi^{-1/4}(2^nn!)^{-1/2}\mathrm{e}^{-x^2/2}h_n(x), \qquad x\in{\mathbb{R}},$$ where $h_n$ are the Hermite polynomials defined in (\[herm-pol\]). The following two facts can immediately be checked: (a) there exists a constant $K_1>0$ such that $\|\xi_n\|_\infty\leq K_1(n+1)^{-1/12}$; (b) since $\xi'_n=\sqrt{\frac{n}2}\xi_{n-1}-\sqrt{\frac{n+1}{2}}\xi_{n+1}$, there exists a constant $K_2>0$ such that $\|\xi'_n\|_\infty\leq K_2 n^{5/12}$.
Consider the positive self-adjoint operator $A$ (whose inverse is Hilbert–Schmidt) given by $A=-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+(1+x^2)$. We have $A\xi_n=(2n+2)\xi_n$.
For any $p\ge 0$, define the space $\mathcal{S}_p({\mathbb{R}})$ to be the domain of the closure of $A^p$. Endowed with the norm $|\xi|_p:=|A^p\xi|_0$, it is a Hilbert space. Note that the norm $|\cdot|_p$ can be expressed as follows, if one uses the orthonormal basis $(\xi_n)$: $$|\xi|_p^2=\sum_{n=0}^\infty \langle \xi,\xi_n\rangle^2(2n+2)^{2p}.$$ We denote by $\mathcal{S}'_p({\mathbb{R}})$ the dual of $\mathcal{S}_p({\mathbb{R}})$. The norm in $S'_p({\mathbb{R}})$ is given by (see [@ob], Lemma 1.2.8) $$|\xi|^2_{-p}=\sum_{n=0} ^\infty |\la \xi,A^{-p}\xi_n\ra |^2
=\sum_{n=0}^\infty \langle \xi,\xi_n\rangle^2(2n+2)^{-2p}$$ for any $\xi\in \mathcal{S}'_p({\mathbb{R}})$. One can show that the projective limit of the spaces $\mathcal{S}_{p}({\mathbb{R}})$, $p\geq 0$, is $\mathcal{S}({\mathbb{R}})$, that the inductive limit of the spaces $ \mathcal{S}_p({\mathbb{R}})'$, $p\ge 0$, is $\mathcal{S}'({\mathbb{R}})$ and that $$\mathcal{S}({\mathbb{R}})\subset L^2({\mathbb{R}})\subset \mathcal{S}'({\mathbb{R}})$$ is a Gel’fand triple.
We can now introduce the Gel’fand triple $$(\mathcal{S})\subset L^2(\Omega)\subset(\mathcal{S})^*,$$ via the second quantization operator $\Gamma(A)$. This is an unbounded and densely defined operator on $L^2(\Omega)$ given by $$\Gamma(A)\Phi= \sum_{n=0} ^\infty I_n(A^{\otimes n}\phi_n),$$ where $\Phi$ has the Wiener chaos expansion (\[wiener\]). If $p\geq 0$, then we denote by $(\mathcal{S})_{p}$ the space of random variables $\Phi \in L^{2}(\Omega)$ with Wiener chaos expansion (\[wiener\]) such that $$\| \Phi \| ^p_{p}:=E [ | \Gamma(A)^p \Phi |^2 ] =
\sum_{n=0} ^\infty n! |\phi_n|_p^2 <\infty.$$ In the above formula, $|\phi_n|_p$ denotes the norm in $\mathcal{S}_p({\mathbb{R}}) ^{\otimes n}$. The projective limit of the spaces $(\mathcal{S})_{p}$, $p\geq 0$, is called the space of test functions and is denoted by $(\mathcal{S})$. The inductive limit of the spaces $(\mathcal{S})_{-p}$, $p\geq 0$, is called the space of Hida distributions and is denoted by $(\mathcal{S})^{\ast }$. The elements of $(\mathcal{S})^*$ are called *Hida distributions*. The main example is the time derivative of the Brownian motion, defined as $\dot{W}_t=\langle \cdot,\delta_t\rangle$. One can show that $|\delta_t|_{-p}<\infty$ for some $p>0$.
We denote by $\langle\!\langle\Phi, \Psi\rangle\!\rangle$ the dual pairing associated with the spaces $(\mathcal{S})$ and $(\mathcal{S})^*$. On the other hand (see [@ob], Theorem 3.1.6), for any $\Phi\in (\mathcal{S})^*$, there exist $\phi_n\in \mathcal{S}({\mathbb{R}}^n)'$ such that $$\langle\!\langle \Phi,\Psi \rangle\!\rangle=\sum_{n=0}^\infty n!\langle \phi_n,\psi_n\rangle,$$ where $\Psi=\sum_{n=0}^\infty I_n(\psi_n)\in (\mathcal{S})$. Moreover, there exists $p>0$ such that $$\|\Phi\|_{-p}^2=\sum_{n=0}^\infty n!|\phi_n|_{-p}^2.$$ Then, with a convenient abuse of notation, we say that $\Phi$ has a generalized Wiener chaos expansion of the form (\[wiener\]).
The $S$-transform
-----------------
A useful tool to characterize elements in $(\mathcal{S})^*$ is the $S$-transform. The Wick exponential of a Wiener integral $I_1(\eta)$, $\eta \in L^2({\mathbb{R}})$, is defined by $$: \mathrm{e}^{I_1(\eta)} :\ = \mathrm{e}^{ I_1(\eta) - |\eta|_0^2/2}.$$ The $S$-transform of an element $\Phi\in (\mathcal{S})^*$ is then defined by $$S(\Phi)(\xi)=\big\langle\big\langle\Phi,:\mathrm{e}^{I_1(\xi)}:\big\rangle\big\rangle,$$ where $\xi\in \mathcal{S}({\mathbb{R}})$. One can easily see that the $S$-transform is injective on $(\mathcal{S})^*$.
If $\Phi\in L^2(\Omega)$, then $S(\Phi)(\xi)=E[\Phi:\mathrm{e}^{I_1(\xi)}:]$. For instance, the $S$-transform of the Wick exponential is $$S\bigl(:\mathrm{e}^{I_1(\eta)}:\bigr)(\xi)=\mathrm{e}^{\langle\eta,\xi\rangle}.$$ Also, $S(W_t)(\xi)=\int_0^t\xi(s)\,\mathrm{d}s$ and $S(\dot{W}_t)(\xi)=\xi(t)$.
Suppose that $\Phi\in (\mathcal{S})^*$ has a generalized Wiener chaos expansion of the form (\[wiener\]). Then, for any $\xi\in \mathcal{S}({\mathbb{R}})$, $$S(\Phi)(\xi)= \sum_{n=0}^\infty \la \phi_n, \xi^{\otimes n} \ra,$$ where the series converges absolutely (see [@ob], Lemma 3.3.5).
The Wick product of two functionals $\Psi=\sum_{n=0}^\infty I_n(\psi_n)$ and $\Phi=\sum_{n=0}^\infty I_n(\phi_n)$ belonging to $(\mathcal{S})^*$ is defined as $$\Psi\diamond\Phi =\sum_{n,m=0}^\infty I_{n+m}(\psi_n \otimes \phi_m).$$ It can be proven that $\Psi\diamond\Phi \in (\mathcal{S})^*$. The following is an important property of the $S$-transform: $$\label{prod}
S(\Phi \diamond \Phi)(\xi) =S(\Phi)(\xi)S(\Psi)(\xi).$$ If $\Psi$, $\Phi$ and $\Psi\diamond\Phi $ belong to $L^2(\Omega)$, then we have $E[\Psi\diamond\Phi]=E[\Psi] E[\Phi]$.
The following is a useful characterization theorem.
\[F\] A function $F$ is the $S$-transform of an element $\Phi\in (\mathcal{S})^*$ if and only if the following conditions are satisfied:
(1) for any $\xi,\eta\in \mathcal{S}$, $z\mapsto F(z\xi+\eta)$ is holomorphic on ${\mathbb{C}}$;
(2) there exist non-negative numbers $K,a$ and $p$ such that for all $\xi\in \mathcal{S}$, $$|F(\xi)|\leq K \exp(a|\xi|_p^{2}).$$
See [@kuo], Theorems 8.2 and 8.10.
In order to study the convergence of a sequence in $(\mathcal{S})^*$, we can use its $S$-transform, by virtue of the following theorem.
\[th\_kuo\] Let $\Phi_n \in (\mathcal{S})^*$ and $S_n=S(\Phi_n)$. Then, $\Phi_n$ converges in $(\mathcal{S})^*$ if and only if the following conditions are satisfied:
1. $\lim_{n\to \infty}S_n(\xi)$ exists for each $\xi\in \mathcal{S}$;
2. there exist non-negative numbers $K,a$ and $p$ such that for all $n\in {\mathbb{N}}$, $\xi\in (\mathcal{S})$, $$|S_n(\xi)|\leq K \exp(a|\xi|_p^{2}).$$
See [@kuo], Theorem 8.6.
Limit theorems for Volterra processes {#volterra}
=====================================
One-dimensional case
--------------------
Consider a Volterra process $B=( B_t )_{t\geq 0}$ of the form $$\label{star}
B_t=\int_0^t K(t,s)\,\mathrm{d}W_s,$$ where $K(t,s)$ satisfies $\int_0^tK(t,s)^2 \,\mathrm{d}s<\infty$ for all $t>0$ and $W$ is the Brownian motion defined on the white noise probability space introduced in the last section. Note that the $S$-transform of the random variable $B_t$ is given by $$\label{e1aa}
S(B_t)(\xi)=\int_0^{t}K(t,s)\xi(s)\,\mathrm{d}s$$ for any $\xi\in \mathcal{S}({\mathbb{R}})$. We introduce the following assumptions on the kernel $K$:
1. $K$ is continuously differentiable on $\{0<s<t<\infty \}$ and for any $ t>0$, we have $$\int_0^t \bigg|\frac{\partial K}{\partial t}(t,s) \bigg|(t-s)\,\mathrm{d}s <\infty;$$
2. $k(t)=\int_0^{t}K(t,s)\,\mathrm{d}s $ is continuously differentiable on $(0,\infty)$.
Consider the operator $K_+$ defined by $$K_+\xi(t)=k'(t)\xi(t)+\int_0^t \frac{\partial K}{\partial t}(t,r)\bigl(\xi(r)-\xi(t)\bigr)\,\mathrm{d}r,$$ where $t>0$ and $\xi\in \mathcal{S}({\mathbb{R}})$. From Theorem \[F\], it follows that the linear mapping $\xi\rightarrow K_+\xi(t)$ is the $S$-transform of a Hida distribution. More precisely, according to [@nualartwhite], define the function $$\label{C}
C(t)=|k'(t)|+\int_0^t \bigg|\frac{\partial K}{\partial t}(t,r) \bigg|(t-r)\,\mathrm{d}r, \qquad t\geq 0,$$ and observe that the following estimates hold (recall the definition (\[xi\_n\]) of $\xi_n$): $$\begin{aligned}
\label{e11}
|K_+\xi (t)|
& \leq & C(t) (\|\xi\|_\infty+\|\xi'\|_\infty) \nonumber \\[1pt]
& \leq & C(t) \sum_{n=0}^\infty|\la\xi,\xi_n\ra| (\|\xi_n\|_\infty+\|\xi'_n\|_\infty) \nonumber\\[1pt]
&\leq& C(t)M\sum_{n=0}^\infty|\la \xi,\xi_n\ra|(n+1)^{5/12}\\[1pt]
& \leq & C(t) M \sqrt{ \sum_{n=0}^\infty|\la\xi,\xi_n\ra| ^2 (2n+2) ^{17/6} }\sqrt{\sum_{n=0}^\infty(n+1)^{-2}}\nonumber\\[1pt]
&= & C(t) M |\xi|_{17/12}\nonumber\end{aligned}$$ for some constants $M>0$ whose values are not always the same from one line to the next.
We have the following preliminary result.
\[def-int\] Fix an integer $k\geq 1$. Let $B$ be a Volterra process with kernel $K$ satisfying the conditions $(\mathrm{H_1})$ and $(\mathrm{H_2})$. Assume, moreover, that $C$ defined by (\[C\]) belongs to $L^k([0,T])$. The function $\xi\mapsto \int_0^T (K_+\xi(s))^k\,\mathrm{d}s$ is then the $S$-transform of an element of $(\mathcal{S})^*$. This element is denoted by $\int_0^T \dot{B}_u^{\diamond k}\,\mathrm{d}u$.
We use Theorem \[F\]. Condition (1) therein is immediately checked, while for condition (2), we just write, using (\[e11\]), $$\bigg| \int_0^T (K_+\xi(s))^k\, \mathrm{d}s \bigg|\leq \int_0^T | K_+\xi(s)|^k\, \mathrm{d}s\leq M|\xi|_{17/12}\int_0^T C^k(s)\,\mathrm{d}s.$$
Fix an integer $k\ge 1$ and consider the following, additional, condition.
1. The maximal function $D(t)=\sup_{0<\e\leq \e_0}\frac1\e\int_t^{t+\e}C(s)\,\mathrm{d}s$ belongs to $L^k([0,T])$ for any $T>0$ and for some $\e_0>0$.
We can now state the main result of this section.
\[CVS\*\] Fix an integer $k\geq 1$. Let $B$ be a Volterra process with kernel $K$ satisfying the conditions $\mathrm{(H_1)}$, $\mathrm{(H_2)}$ and $(\mathrm{H}_3^{k})$. The following convergence then holds: $$\int_0^T
\biggl(\frac{B_{u+\e}-B_u}{\e} \biggr)^{\diamond k}\,\mathrm{d}u \displaystyle\mathop{\stackrel{ (\mathcal{S})^*}{\longrightarrow}}_{\e\to 0}
\int_0^T \dot{B}_u^{\diamond k}\,\mathrm{d}u.$$
Fix $\xi\in{\mathcal{S}}({\mathbb{R}})$ and set $$S_\e(\xi)= S \biggl(\int_0^T \biggl(\frac{B_{u+\e}-B_u}{\e} \biggr)^{\diamond k}\,\mathrm{d}u \biggr)(\xi).$$ From linearity and property (\[prod\]) of the $S$-transform, we obtain $$\label{e8}
S_\e(\xi)=\int_0^T \frac{(S(B_{u+\e}-B_u)(\xi))^{k}}{\e^k}\,\mathrm{d}u.$$ Equation (\[e1aa\]) yields $$\label{e314}
S(B_{u+\e}-B_u)(\xi)= \int_0^{u+\e} K(u+\e,r) \xi(r)\,\mathrm{d}r -\int_0^{u} K(u,r) \xi(r)\,\mathrm{d}r.$$ We claim that $$\label{e7}
\int_0^{u+\e} K(u+\e,r) \xi(r)\,\mathrm{d}r -\int_0^{u} K(u,r) \xi(r)\,\mathrm{d}r
=\int_u^{u+\e}K_+\xi(s) \,\mathrm{d}s.$$ Indeed, we can write $$\begin{aligned}
\label{e6}
\hspace*{-10pt}\nonumber \int_u^{u+\e}K_+\xi(s) \,\mathrm{d}s
&=&\int_u^{u+\e} k'(s) \xi(s) \,\mathrm{d}s
+ \int_u^{u+\e}\biggl( \int_0^s \frac{\partial K}{\partial s}(s,r)\bigl(\xi(r)-\xi(s)\bigr) \,\mathrm{d}r \biggr) \,\mathrm{d}s \\[-8pt]\\[-8pt]
&=& A^{(1)}_u +A^{(2)}_u.\nonumber\end{aligned}$$ We have, using Fubini’s theorem, that $$\begin{aligned}
A^{(2)}_u&=& - \int_u^{u+\e} \mathrm{d}s \nonumber
\int_0^s \mathrm{d}r \,\frac{\partial K}{\partial s}(s,r) \int_r^s \mathrm{d}\theta \, \xi'(\theta) \\[-8pt]\\[-8pt]
&=& -\int_0^{u+\e} \mathrm{d}\theta\, \xi'(\theta)\int_0^\theta
\mathrm{d}r\,
\bigl( K(u+\e, r)- K(\theta \vee u,r) \bigr).\nonumber\end{aligned}$$ This can be rewritten as $$\begin{aligned}
\label{g1}
A^{(2)}_u&=& -\int_0^u \bigl( K(u+\e,r) -K(u,r) \bigr) \bigl(\xi(u)-\xi(r)\bigr) \,\mathrm{d}r \nonumber \\[-8pt]\\[-8pt]
&&{}-\int_u^{u+\e} \mathrm{d}\theta\, \xi'(\theta) \int_0^\theta \mathrm{d}r \, \bigl( K(u+\e,r) -K(\theta,r) \bigr).\nonumber\end{aligned}$$ On the other hand, integration by parts yields $$\begin{aligned}
\label{e3}
A^{(1)}_u&=&\xi(u+\e) \int_0^{u+\e} K(u+\e,r) \,\mathrm{d}r \nonumber\\[-8pt]\\[-8pt]
&&{} -\xi(u) \int_0^{u} K(u,r) \,\mathrm{d}r
-\int_u^{u+\e}\,\mathrm{d}s\, \xi'(s) \int_0^s \,\mathrm{d}r \, K(s,r).\nonumber\end{aligned}$$ Therefore, adding (\[e3\]) and (\[g1\]) yields $$\begin{aligned}
\label{g3}
A^{(1)}_u +A^{(2)}_u
&=& \xi(u+\e) \int_0^{u+\e} K(u+\e,r) \,\mathrm{d}r
- \xi(u ) \int_0^{u } K(u,r) \,\mathrm{d}r \nonumber \\
&&{} -\int_0^u \bigl( K(u+\e,r) -K(u,r) \bigr) \bigl(\xi(u)-\xi(r)\bigr) \,\mathrm{d}r \\
&&{} -\int_u^{u+\e} \mathrm{d}\theta\, \xi'(\theta) \int_0^\theta K(u+\e,r) \,\mathrm{d}r.\nonumber\end{aligned}$$ Note that, by integrating by parts, we have $$\begin{aligned}
\label{g2}
&&-\int_u^{u+\e} \mathrm{d}\theta\, \xi'(\theta) \int_0^\theta K(u+\e,r)\, \mathrm{d}r\nonumber
\\
&&\quad
=-\xi(u+\e) \int_0^{u+\e} K(u+\e,r)\, \mathrm{d}r
+ \xi(u ) \int_0^{u } K(u+\e,r) \, \mathrm{d}r
\\
&&\qquad{}+ \int_u^{u+\e} K(u+\e,r) \xi(r) \, \mathrm{d}r.\nonumber\end{aligned}$$ Thus, substituting (\[g2\]) into (\[g3\]), we obtain $$A^{(1)}_u +A^{(2)}_u = \int_0^{u+\e} K(u+\e,r) \xi(r) \, \mathrm{d}r- \int_0^{u } K(u,r) \xi(r) \, \mathrm{d}r,$$ which completes the proof of (\[e7\]). As a consequence, from (\[e8\])–(\[e7\]), we obtain $$S_\e (\xi)=\int_0^{T} \biggl( \frac 1 \e \int_u^{u+\e} K_+\xi (s)\, \mathrm{d}s \biggr)^k\, \mathrm{d}u.$$ On the other hand, using (\[e11\]) and the definition of the maximal function $D$, we get $$\begin{aligned}
\label{v12}
\sup_{0< \e\leq \e_0} \bigg| \frac 1 \e \int_u^{u+\e} K_+\xi(s)\, \mathrm{d}s \bigg |^k
&\leq& M^k |\xi|^k_{17/12} \sup_{0<\e\leq \e_0} \biggl(\frac1\e\int_u^{u+\e}C(s)\, \mathrm{d}s \biggr)^k\nonumber\\[-8pt]\\[-8pt]
&=&M^k|\xi|^k_{17/12}D^k(u).\nonumber\end{aligned}$$ Therefore, using hypothesis $(\mathrm{H}_3^{k})$ and the dominated convergence theorem, we have $$\lim_{\e\to0} S_\e(\xi)=\int_0^{T} (K_+\xi (s))^k\,\mathrm{d}s. \label{e12}$$ Moreover, since $|S_\e(\xi)|\leq M^k|\xi|^k_{17/12}\int_0^T D^k(u)\,\mathrm{d}u$ for all $0<\e\leq \e_0$ (see (\[v12\])), conditions (1) and (2) in Proposition \[CVS\*\] are fulfilled. Consequently, $\e^{-k}\int_0^T (B_{u+\e}-B_u)^{\diamond k}\,\mathrm{d}u$ converges in $({\mathcal{S}}^*)$ as $\e\to 0$.
To complete the proof, it suffices to observe that the right-hand side of (\[e12\]) is, by definition (see Lemma \[def-int\]), the $S$-transform of $\int_0^T \dot{B}_s^{\diamond k}\,\mathrm{d}s$.
In [@nualartwhite], it is proved that under some additional hypotheses, the mapping $t\rightarrow B_t$ is differentiable from $(0,\infty)$ to $(\mathcal{S})^*$ and that its derivative, denoted by $\dot B_t$, is a Hida distribution whose $S$-transform is $K_+\xi(t)$.
Bidimensional case {#3.2}
------------------
Let $W=(W_t)_{t\in {\mathbb{R}}}$ be a two-sided Brownian motion defined in the white noise probability space $(\mathcal{S}'({\mathbb{R}}), \mathcal{B}, {\mathbb{P}})$. We can consider two independent standard Brownian motions as follows: for $t\geq 0$, we set $W^{(1)}_t= W_t$ and $W^{(2)}_t = W_{-t}$.
In this section, we consider a bidimensional process $B=(B^{(1)}_t,B^{(2)}_t)_{t\geq 0}$, where $B^{(1)}$ and $B^{(2)}$ are independent Volterra processes of the form $$B^{(i)}_t=\int_0^t K(t,s)\,\mathrm{d}W^{(i)}_s, \qquad t\geq 0, i=1,2. \label{B1B2}$$ For simplicity only, we work with the same kernel $K$ for the two components.
First, using exactly the same lines of reasoning as in the proof of Lemma \[def-int\], we get the following result.
\[def-int-bis\] Let $B=(B^{(1)}_t,B^{(2)}_t)_{t\geq 0}$ be given as above, with a kernel $K$ satisfying the conditions $\mathrm{(H_1)}$ and $\mathrm{(H_2)}$. Assume, moreover, that $C$ defined by (\[C\]) belongs to $L^2([0,T])$ for any $T>0$. We then have the following results:
(1) the function $\xi\mapsto \int_0^T (\int_0^u K_+\xi (-y)\,\mathrm{d}y ) K_+\xi (u) \, \mathrm{d}u$ is the $S$-transform of an element of $(\mathcal{S})^*$, denoted by $\int_0^T B^{(1)}_u\diamond \dot{B}^{(2)}_u\,\mathrm{d}u$;
(2) the function $\xi\mapsto \int_0^T K_+\xi (-u) K_+\xi (u) \,\mathrm{d}u$ is the $S$-transform of an element of $(\mathcal{S})^*$, denoted by $\int_0^T \dot{B}^{(1)}_u\diamond \dot{B}^{(2)}_u\,\mathrm{d}u$.
We can now state the following result.
\[CVS\*-bid\] Let $B=(B^{(1)}_t,B^{(2)}_t)_{t\geq 0}$ be given as above, with a kernel $K$ satisfying the conditions $\mathrm{(H_1)}$, $\mathrm{(H_2)}$ and $(\mathrm{H}_3^2)$. The following convergences then hold: $$\begin{aligned}
\int_0^T B^{(1)}_u \frac{B^{(2)}_{u+\e}-B^{(2)}_u}{\e} \,\mathrm{d}u
&\displaystyle\mathop{\stackrel{ (\mathcal{S})^*}{\longrightarrow}}_{\e\to 0} &
\int_0^T B^{(1)}_u\diamond \dot{B}^{(2)}_u \,\mathrm{d}u, \\
\int_0^T \biggl(\int_0^u \frac{B^{(1)}_{v+\e}-B^{(1)}_v}{\e}\,\mathrm{d}v \biggr)\frac{B^{(2)}_{u+\e}-B^{(2)}_u}{\e} \,\mathrm{d}u
&\displaystyle\mathop{\stackrel{ (\mathcal{S})^* }{\longrightarrow}}_{\e\to 0}&
\int_0^T B^{(1)}_u\diamond \dot{B}^{(2)}_u \,\mathrm{d}u, \\
\int_0^T \frac{B^{(1)}_{u+\e}-B^{(1)}_u}{\e}\times\frac{B^{(2)}_{u+\e}-B^{(2)}_u}{\e} \,\mathrm{d}u
&\displaystyle\mathop{\stackrel{ (\mathcal{S})^*}{\longrightarrow}}_{\e\to 0}&
\int_0^T \dot{B}^{(1)}_u\diamond\dot{B}^{(2)}_u \,\mathrm{d}u.\end{aligned}$$
Set $$\widetilde{G}_\e=
\int_0^T B^{(1)}_u \frac{B^{(2)}_{u+\e}-B^{(2)}_u}{\e} \,\mathrm{d}u=
\int_0^T B^{(1)}_u\diamond\frac{B^{(2)}_{u+\e}-B^{(2)}_u}{\e} \,\mathrm{d}u.$$ From linearity and property (\[prod\]) of the $S$-transform, we have $$S (\widetilde{G}_\e)(\xi)=\frac1\e \int_0^T S\bigl(B^{(1)}_u\bigr)(\xi )
S\bigl(B^{(2)}_{u+\e} - B^{(2)}_u\bigr) (\xi ) \,\mathrm{d}u$$ so that $$S (\widetilde{G}_\e)(\xi)= \int_0^T \biggl( \int_0^{u} K_+\xi (-y)\,\mathrm{d}y \biggr)
\biggl( \frac1\e\int_u^{u+\e} K_+\xi (x)\,\mathrm{d}x \biggr) \,\mathrm{d}u.$$ Therefore, using (\[e11\]) and (\[v12\]), we can write $$\begin{aligned}
|S (\widetilde{G}_\e)(\xi)| &\leq& M^2 |\xi|^2_{17/12} \int_0^T \biggl(\int_0^u C(t)\,\mathrm{d}t \biggr) D(u)\,\mathrm{d}u\\
&\leq& M^2 |\xi|^2_{17/12} \int_0^T \biggl(\int_0^u D(t)\,\mathrm{d}t \biggr)D(u)\,\mathrm{d}u\\
&=& \frac12 M^2 |\xi|^2_{17/12} \biggl( \int_0^T D(u)\,\mathrm{d}u \biggr)^2\\
&\leq& \frac T2 M^2 |\xi|^2_{17/12} \int_0^T D^2(u)\,\mathrm{d}u.\end{aligned}$$ Hence, by the dominated convergence theorem, we get $$\label{e12bis}
\lim_{\e\to 0} S (\widetilde{G}_\e)(\xi)=
\int_0^T \biggl(\int_0^u K_+\xi (-y)\,\mathrm{d}y \biggr) K_+\xi (u) \,\mathrm{d}u.$$ The right-hand side of (\[e12bis\]) is the $S$-transform of $\int_0^T B^{(1)}_u \diamond \dot{B}^{(2)}_u\, \mathrm{d}u$, due to Lemma \[def-int-bis\]. Therefore, by Theorem \[th\_kuo\], we obtain the desired result in point (1).
The proofs of the other two convergences follow exactly the same lines of reasoning and are therefore left to the reader.
Fractional Brownian motion case {#sec4}
===============================
One-dimensional case {#sec31}
--------------------
Consider a (one-dimensional) fractional Brownian motion (fBm) $B=( B_t )_{t\geq 0}$ of Hurst index $H\in(0,1)$. This means that $B$ is a zero mean Gaussian process with covariance function $$R_H(t,s)=E(B_tB_s) = \tfrac 12(t^{2H} + s^{2H} - |t-s| ^{2H}).$$ It is well known that $B$ is a Volterra process. More precisely (see [@DU]), $B$ has the form (\[star\]) with the kernel $K(t,s)=K_{H}(t,s)$ given by $$K_{H}(t,s)=c_{H} \biggl[ \biggl( \frac{t(t-s)}{s} \biggr) ^{H-1/2}
-\biggl(H-\frac{1}{2}\biggr)s^{1/2-H}\int_{s}^{t}u^{H-3/2}(u-s)^{H-1/2}\,\mathrm{d}u \biggr].$$ Here, $c_H$ is a constant depending only on $H$. Observe that $$\label{b1}
\frac{\partial K_{H}}{\partial t}(t,s)=c_{H}\biggl(H-\frac{1}{2}\biggr)(t-s)^{H-\sfrac{3}{2}} \biggl( \frac{s}{t} \biggr) ^{\sfrac{1}{2}-H}\qquad \mbox{for $t>s> 0$.}$$
Denote by $\mathscr{E}$ the set of all ${\mathbb{R}}$-valued step functions defined on $[0,\infty)$. Consider the Hilbert space $\HH$ obtained by closing $\mathscr{E}$ with respect to the inner product $$\big\langle \mathbf{1}_{[0,u]},\mathbf{1}_{[0,v]}\big\rangle_\HH=E(B_uB_v).$$ The mapping $\mathbf{1}_{[0,t]}\mapsto B_t$ can be extended to an isometry $\varphi\mapsto B(\varphi)$ between $\HH$ and the Gaussian space $\mathcal{H}_1$ associated with $B$. Also, write $\HH^{\otimes k}$ to indicate the $k$th tensor product of $\HH$. When $H>1/2$, the inner product in the space $\HH$ can be written as follows, for any $\varphi$, $\psi\in \mathscr{E}$: $$\langle \phi, \psi \rangle _{\HH}=H(2H-1)\int_0^\infty\!\! \int_0^\infty \phi(s) \psi(s') |s-s'|^{2H-2}\,\mathrm{d}s\,\mathrm{d}s'.$$ By approximation, this extends immediately to any $\varphi$, $\psi\in {\mathcal{S}}({\mathbb{R}})\cup\mathscr{E}$.
We will make use of the multiple integrals with respect to $B$ (we refer to [@nualart] for a detailed account on the properties of these integrals). For every $k\geq 1$, let $\mathcal{H}_{k}$ be the $k$th Wiener chaos of $B$, that is, the closed linear subspace of $L^{2}(\Omega)$ generated by the random variables $\{h_{k}(B(\varphi)), \varphi\in \mathfrak{H}, \|
\varphi \| _{\mathfrak{H}}=1\}$, where $h_{k}$ is the $k$th Hermite polynomial (\[herm-pol\]). For any $k\geq 1$, the mapping $I_{k}(\varphi^{\otimes k})=h_{k}(B(\varphi))$ provides a linear isometry between the *symmetric* tensor product $\mathfrak{H}^{\odot k}$ (equipped with the modified norm $\sqrt{k!} \| \cdot \| _{\mathfrak{H}^{\otimes k}}$) and the $k$th Wiener chaos $\mathcal{H}_{k}$.=1
Following [@NNT], let us now introduce the Hermite random variable $Z^{(k)}_T$ mentioned in (\[cv>\]). Fix $T>0$ and let $k\geq 1$ be an integer. The family $(\varphi_\e)_{\e>0}$, defined by $$\label{phieps}
\varphi _{\e}=\e^{-k}\int_0^T \mathbf{1}_{[u,u+\e]}^{\otimes k}\,\mathrm{d}u,$$ satisfies $$\begin{aligned}
\label{variance}
\hspace*{-20pt}\lim_{\e,\eta\rightarrow 0 } \langle \varphi_\e,
\varphi _{\eta} \rangle _{\mathfrak{H}^{\otimes k}}
=
H^k(2H-1)^{k}
\int_{[0,T]^2}|s-s'|^{(2H-2)k}\,\mathrm{d}s\,\mathrm{d}s'
=c_{k,H} T^{(2H-2)k+2}\end{aligned}$$ with $c_{k,H}=\frac{H^k(2H-1)^{k}}{(Hk-k+1)(2Hk-2k+1) }$. This implies that $\varphi_\e$ converges, as $\e$ tends to zero, to an element of $\mathfrak{H}^{\otimes k}$. The limit, denoted by $\pi^k_{\mathbf{1}_{[0,T]}}$, can be characterized as follows. For any $\xi_i\in {\mathcal{S}}({\mathbb{R}})$, $i=1,\ldots,k$, we have $$\begin{aligned}
&&\big\langle\pi^k_{\mathbf{1}_{[0,T]}},\xi_1\otimes \cdots\otimes \xi_k\big\rangle_{\mathfrak{H}^{\otimes k}}\\
&&\quad=\lim_{\e\to 0} \langle \varphi_\e,\xi_1\otimes \cdots\otimes \xi_k\rangle_{\mathfrak{H}^{\otimes k}}\\
&&\quad=\lim_{\e\to 0} \e^{-k}\int_0^T \mathrm{d}u \,\prod_{i=1}^k \big\langle \mathbf{1}_{[u,u+\e]},\xi_i\big\rangle_{\mathfrak{H}}\\
&&\quad=\lim_{\e\to 0} \e^{-k}H^k(2H-1)^k\int_0^T \mathrm{d}u\, \prod_{i=1}^k\int_{u}^{u+\e} \mathrm{d}s\, \int_0^T
\mathrm{d}r\,
|s-r|^{2H-2}\xi_i(r)\\
&&\quad=
H^k(2H-1)^k \int_0^T \mathrm{d}u\, \prod_{i=1}^k \int_0^T \mathrm{d}r\,|u-r|^{2H-2}\xi_i(r).\end{aligned}$$ We define the $k$th Hermite random variable by $Z^{(k)}_T=
I_{k}(\pi^k_{\mathbf{1}_{[0,T]}})$. Note that, by using the isometry formula for multiple integrals and since $G_\e=I_k(\varphi_\e)$, the convergence (\[cv>\]) is just a corollary of our construction of $Z^{(k)}_T$. Moreover, by (\[variance\]), we have $$E \bigl[\bigl(Z_T^{(k)}\bigr)^2 \bigr]=c_{k,H}\times t^{(2H-2)k+2}.$$
We will need the following preliminary result.
\[lmlm\]
(1) The fBm $B$ verifies the assumptions $\mathrm{(H_1)}$, $\mathrm{(H_2)}$ and $(\mathrm{H}_3^{k})$ if and only if $H\in (\frac12-\frac1k,1 )$.
(2) If $H\in (\frac12-\frac1k,1 )$, then $\int_0^T \dot{B}_u^{\diamond k}\,\mathrm{d}u$ is a well-defined element of $(\mathcal{S})^*$ (in the sense of Lemma \[def-int\]).
(3) If we assume that $H>\frac12$, then $\int_0^T \dot{B}_u^{\diamond k}\,\mathrm{d}u$ belongs to $L^2(\Omega)$ if and only if $H>1-\frac1{2k}$.
\(1) Since $$\label{k'}
k'(t)=k'_H(t)=\bigl(H+ \tfrac 12 \bigr)c_1(H)t^{H-\sfrac 12 }$$ and $$\label{terme2}
\int_0^t \bigg|\frac{\partial K_H}{\partial t}(t,s) \bigg|(t-s)\,\mathrm{d}s=
\bigg|\int_0^t \frac{\partial K_H}{\partial t}(t,s)(t-s)\,\mathrm{d}s \bigg|=
c_2(H)t^{H+ \sfrac 12}$$ for some constants $c_1(H)$ and $c_2(H)$, we immediately see that assumptions $\mathrm{(H_1)}$ and $\mathrm{(H_2)}$ are satisfied for all $H\in(0,1)$. It therefore remains to focus on assumption $(\mathrm{H}_3^{k})$. For all $H\in(0,1)$, we have $$\label{$1}
\sup_{0<\e\leq \e_0}\frac1\e\int_{t}^{t+\e} s^{H-1/2}\,\mathrm{d}s\leq t^{H-\sfrac 12} \vee
(t+\e_0)^{H-\sfrac 12}$$ and $$\label{$3}
\sup_{0<\e\leq \e_0}\frac1\e\int_{t}^{t+\e} s^{H+1/2}\,\mathrm{d}s\leq (t+\e_0)^{H+1/2}.$$ Consequently, since $\int_0^T t^{kH-k/2}\,\mathrm{d}t$ is finite when $H>\frac12-\frac1k$, we deduce from (\[k’\])–() that $(\mathrm{H}_{3}^{k})$ holds in this case. Now, assume that $H\leq \frac12-\frac1k$. Using the fact that $D(t) \ge C(t)$, we obtain $$\int_0^T D^k(t)\,\mathrm{d}t \ge \int_0^T C^k(t)\,\mathrm{d}t
= \biggl(H+\frac 12\biggr)^kc_1(H)^k \int_0^T t^{kH-\sfrac k2}\,\mathrm{d}t=\infty.$$ Therefore, in this case, assumption $(\mathrm{H}_{3}^{k})$ is not verified.
\(2) This fact can be proven immediately: simply combine the previous point with Lemma \[def-int\].
\(3) By definition of $\int_0^T \dot{B}_u^{\diamond k}\, \mathrm{d}u$ (see Lemma \[def-int\]), it is equivalent to show that the distribution $\tau^{k}_{\1_{[0,T]}}$, defined via the identity $\int_0^t \dot{B}_s^{\diamond k}\,\mathrm{d}s= I_k(\tau^{k}_{\1_{[0,t]}})$, can be represented as a function belonging to $L^2([0,T]^k)$. We can write $$\begin{aligned}
\big\langle \tau^{k}_{\1_{[0,T]}}, \xi_1\otimes\cdots \otimes \xi_k \big\rangle & =&
\int_0^T K_+\xi_1(s)\cdots K_+\xi_k(s)\,\mathrm{d}s\\
& =& \int_0^T \mathrm{d}s\,\prod_{i=1}^{k} \int_0^s \frac{\partial K_H}{\partial s}(s,r)\xi_i(r)\,\mathrm{d}r\end{aligned}$$ for any $\xi_1,\ldots, \xi_k\in \mathcal{S}({\mathbb{R}})$. Observe that $K_+\xi(s)=\int_0^s \frac{\partial K_H}{\partial s}(s,r)\xi(r)\,\mathrm{d}r$ because $K_H(s,s)=0$ for $H>1/2$. Using Fubini’s theorem, we deduce that the distribution $\tau^{k}_{\1_{[0,T]}}$ can be represented as the function $$\begin{aligned}
\tau _{\mathbf{1}_{[0,T]}}^{k}(x_{1},\ldots,x_{k})=\mathbf{1}_{[0,T]^{k}}(x_{1},\ldots,x_{k})
\int_{\max (x_{1},\ldots,x_{k})}^{T}\frac{\partial K_H}{\partial s}(s,x_{1})\cdots \frac{\partial K_H}{\partial s}
(s,x_{k})\,\mathrm{d}s.\end{aligned}$$ We then obtain $$\begin{aligned}
&&\big\| \tau _{\mathbf{1}_{[0,T]}}^{k}\big \| _{L^2([0,T]^k)}^{2}
\\
&&\quad=\int_{[0,T]^{k}}\int_{\max (x_{1},\ldots,x_{k})}^{T}\int_{\max (x_{1},\ldots,x_{k})}^{T}\frac{\partial K_H}
{\partial s}(s,x_{1})\cdots \frac{\partial K_H}{\partial s}(s,x_{k}) \\
&&\qquad\hspace*{130pt}{}\times \frac{\partial K_H}{\partial s}(r,x_{1})\cdots \frac{\partial K_H}{\partial s}(r,x_{k})
\,\mathrm{d}s\,\mathrm{d}r\,\mathrm{d}x_{1}\cdots \,\mathrm{d}x_{k} \\
&&\quad=\int_{[0,T]^{2}} \biggl( \int_{0}^{r\wedge s}\frac{\partial K_H}{\partial s}(s,x)\frac{\partial K_H}{\partial s}(r,x)\,\mathrm{d}x \biggr)^k
\,\mathrm{d}r\,\mathrm{d}s.\end{aligned}$$ Using the equality (\[b1\]) and the same computations as in [@nualart], page 278, we obtain, for $s<r$, $$\label{etunetunetunzero}
\int_{0}^{s}\frac{\partial K_H}{\partial s}(s,x)\frac{\partial K_H}{\partial r} (r,x)\,\mathrm{d}x=
H(2H-1) (r-s)^{2H-2}.$$ Therefore, $$\big\|\tau^k_{\mathbf{1}_{[0,T]}}\big \|^2_{L^2([0,T]^k)}=
\bigl(H(2H-1)\bigr)^k \int_0^T\int_0^T |r-s|^{2Hk-2k} \,\mathrm{d}r\,\mathrm{d}s.$$ We immediately check that $\|\tau^k_{\mathbf{1}_{[0,T]}} \|^2_{L^2([0,T]^k)}<\infty$ if and only if $2Hk-2k>-1$, that is, $H>1-\frac{1}{2k}$. Thus, in this case, the Hida distribution $\int_0^T \dot{B}_s^{\diamond k}\,\mathrm{d}s$ is a square-integrable random variable with $$\begin{aligned}
E \biggl[ \biggl(\int_0^T \dot{B}_s^{\diamond k}\,\mathrm{d}s \biggr)^2 \biggr] =
\big\|\tau^k_{\mathbf{1}_{[0,T]}}\big\|^2_{L^2([0,T]^k)}
=c_{k,H}\times T^{2Hk-2k+2}.\end{aligned}$$
According to our result, the two distributions $\tau^k_{\mathbf{1}_{[0,T]}}$ and $\pi^k_{\mathbf{1}_{[0,T]}}$ should coincide when $H>1/2$. We can check this fact by means of elementary arguments. Let $\xi_i\in {\mathcal{S}}({\mathbb{R}})$, $i=1,\ldots,k$. From (\[e7\]), we deduce that $$\big\langle \mathbf{1}_{[u,u+\e]},\xi_i\big\rangle_{\mathfrak{H}}=\int_u^{u+\e}K_+\xi_i(s) \,\mathrm{d}s$$ and then $$\lim_{\e\to 0} \frac{1}{\e}\big\langle \mathbf{1}_{[u,u+\e]},\xi_i\big\rangle_{\mathfrak{H}}= K_+\xi_i(u).$$ Using (\[v12\]) with $k=1$ for each $\xi_i$ and applying the dominated convergence theorem, since the fractional Brownian motion satisfies the assumption $(\mathrm{H}_3^k)$ when $H\in (\frac12-\frac1k,1 )$, we get, for $\varphi_\e$ defined in (\[phieps\]), $$\lim_{\e\to 0} \langle \varphi_\e,\xi_1\otimes \cdots\otimes \xi_k\rangle_{\mathfrak{H}^{\otimes k}}
= \int_0^T K_+\xi_1(u)\cdots K_+\xi_k(u)\,\mathrm{d}u,$$ which yields $\tau^k_{\mathbf{1}_{[0,T]}}=\pi^k_{\mathbf{1}_{[0,T]}}$.
We can now state the main result of this section.
\[thm-fbm\] Let $k\geq 2$ be an integer. If $H>\frac 12- \frac 1k$ (note that this condition is immaterial for $k=2$), the random variable $$G_\e=\int_0^T \biggl(\frac{B_{u+\e}-B_u}{\e} \biggr)^{\diamond k}\,\mathrm{d}u
=\e^{-k(1-H)}\int_0^T h_k
\biggl(\frac{B_{u+\e}-B_u}{\e^{H}}\biggr)\,\mathrm{d}u$$ converges in $(\mathcal{S}^*)$, as $\e\to 0$, to the Hida distribution $\int_0^T \dot{B}_u^{\diamond k}\,\mathrm{d}u$. Moreover, $G_\e$ converges in $L^2(\Omega)$ if and only if $H>1-\frac{1}{2k}$. In this case, the limit is $\int_0^T \dot{B}_u^{\diamond k}\,\mathrm{d}u=Z_T^{(k)}$.
The first point follows directly from Proposition \[CVS\*\] and Lemma \[lmlm\] (point 1). On the other hand, we already know (see (\[cv>\])) that $G_\e$ converges in $L^2(\Omega)$ to $Z_T^{(k)}$ when $H>1-\frac1{2k}$. This implies that when $H>1-\frac1{2k}$, $\int_0^T \dot{B}_s^{\diamond k}\,\mathrm{d}s$ must be a square-integrable random variable equal to $Z_T^{(k)}$. Assume, now, that $H\leq 1-\frac1{2k}$. From the proof of (\[cv<\]) and (\[cv=\]) below, it follows that $E(G^2_\e)$ tends to $+\infty$ as $\e$ tends to zero, so $G_\e$ does not converge in $L^2(\Omega)$.
Bidimensional case {#sec32}
------------------
Let $B^{(1)}$ and $B^{(2)}$ denote two independent fractional Brownian motions with (the same) Hurst index $H\in(0,1)$, defined by the stochastic integral representation (\[B1B2\]), as in Section \[3.2\].
By combining Lemma \[lmlm\] (point 1 with $k=2$) and Lemma \[def-int-bis\], we have the following preliminary result.
\[lmlmlm\] For all $H\in(0,1)$, the Hida distributions $\int_0^T B^{(1)}_u \diamond \dot{B}^{(2)}_u\,\mathrm{d}u$ and $\int_0^T \dot{B}^{(1)}_u \diamond \dot{B}^{(2)}_u\,\mathrm{d}u$ are well-defined elements of $(\mathcal{S})^*$ (in the sense of Lemma \[def-int-bis\]).
We can now state the following result.
\[thm-fbm2\]
(1) For all $H\in(0,1)$, $\widetilde{G}_\e$ defined by (\[tilde\]) converges in $(\mathcal{S}^*)$, as $\e\to 0$, to the Hida distribution $\int_0^T B^{(1)}_u\diamond \dot{B}^{(2)}_u \,\mathrm{d}u$. Moreover, $\widetilde{G}_\e$ converges in $L^2(\Omega)$ if and only if $H\geq 1/2$.
(2) For all $H\in(0,1)$, $\breve{G}_\e$ defined by (\[breve\]) converges in $(\mathcal{S}^*)$, as $\e\to 0$, to the Hida distribution $\int_0^T B^{(1)}_u\diamond \dot{B}^{(2)}_u \,\mathrm{d}u$. Moreover, $\breve{G}_\e$ converges in $L^2(\Omega)$ if and only if $H> 1/4$.
(3) For all $H\in(0,1)$, $\hat{G}_\e$ defined by (\[hat\]) converges in $(\mathcal{S}^*)$, as $\e\to 0$, to the Hida distribution $\int_0^T \dot{B}^{(1)}_u\diamond \dot{B}^{(2)}_u \,\mathrm{d}u$. Moreover, $\widehat{G}_\e$ converges in $L^2(\Omega)$ if and only if $H>3/4$.
\(1) The first point follows directly from Proposition \[CVS\*-bid\] and Lemma \[lmlm\] (point 1 with $k=2$). Assume that $H<1/2$. From the proof of Theorem \[cvgausslevyarea\] below, it follows that $E(\widetilde{G}^2_\e)\to \infty$ as $\e$ tends to zero, so $\widetilde{G}_\e$ does not converge in $L^2(\Omega)$. Assume that $H=1/2$. By a classical result of Russo and Vallois (see, for example, the survey [@RVLN]) and since we are, in this case, in a martingale setting, we have that $\widetilde{G}_\e$ converges in $L^2(\Omega)$ to the Itô integral $\int_0^T B^{(1)}_u \,\mathrm{d}B^{(2)}_u$. Finally, assume that $H> 1/2$. For $\e,\eta>0$, we have $$E( \widetilde{G}_\e \widetilde{G}_\eta)
= \frac 1{\e \eta} \int_{[0,T]^2} \rho_{\e,\eta}(u- u' ) R_H(u,u' ) \,\mathrm{d}u\,\mathrm{d}u',$$ where $$\label{rhoeps}
\rho_{\e,\eta}(x )= \tfrac 12 [ |x+\e|^{2H} + |x-\eta|^{2H} - |x|^{2H}
-|x+\e-\eta |^{2H} ].$$ Note that as $\e$ and $\eta$ tend to zero, the quantity $(\e\eta)^{-1} \rho_{\e,\eta}(u- u' )$ converges pointwise to (and is bounded by) $H(2H-1) |u-u' |^{2H-2}$. Then, by the dominated convergence theorem, it follows that $E( \widetilde{G}_\e \widetilde{G}_\eta)$ converges to $$H(2H-1) \int_{[0,T]^2} |u-u' |^{2H-2}R_H(u,u' ) \,\mathrm{d}u\,\mathrm{d}u'$$ as $\e,\eta\to 0$, with $\int_{[0,T]^2} |u-u' |^{2H-2}|R_H(u,u' )| \,\mathrm{d}u\,\mathrm{d}u' <\infty$, since $H> 1/2$. Hence, $\widetilde{G}_\e$ converges in $L^2(\Omega)$.
\(2) The first point follows directly from Proposition \[CVS\*-bid\] and Lemma \[lmlm\] (point 1 with $k=2$). Assume that $H\leq 1/4$. From the proof of Theorem \[cvgausslevyareabis\] below, it follows that $E(\breve{G}^2_\e)\to \infty$ as $\e$ tends to zero, so $\breve{G}_\e$ does not converge in $L^2(\Omega)$. Assume that $H>1/4$. For $\e,\eta>0$, we have $$E( \breve{G}_\e \breve{G}_\eta)
= \frac 1{\e^2 \eta^2} \int_{[0,T]^2}\,\mathrm{d}u\,\mathrm{d}u' \rho_{\e,\eta}(u- u' )
\int_0^u\mathrm{d}s\int_0^{u' }\mathrm{d}s'\, \rho_{\e,\eta}(s-s')$$ with $\rho_{\e,\eta}$ given by (\[rhoeps\]). Note that, as $\e$ and $\eta$ tend to zero, the quantity $(\e\eta)^{-1} \rho_{\e,\eta}(u- u' )$ converges pointwise to $H(2H-1) |u-u' |^{2H-2}$, whereas $ (\e\eta)^{-1}\int_0^u\mathrm{d}s\int_0^{u' }\mathrm{d}s'\,\rho_{\e,\eta}(s-s')$ converges pointwise to $R_H(u,u' )$. It then follows that $E(\breve{G}_\e \breve{G}_\eta)$ converges to $$-\frac{H}2(2H-1) \int_{[0,T]^2}|u-u' |^{4H-2}\,\mathrm{d}u\,\mathrm{d}u' +H\int_0^T u^{2H}\bigl(u^{2H-1}+(T-u)^{2H-1}\bigr)\,\mathrm{d}u$$ as $\e,\eta\to 0$ and each integral is finite since $H> 1/4$. Hence, $\breve{G}_\e$ converges in $L^2(\Omega)$.
\(3) Once again, the first point follows from Proposition \[CVS\*-bid\] and Lemma \[lmlm\] (point 1 with $k=2$). Assume that $H\leq 3/4$. From the proof of Theorem \[cvgausscovariation\] below, it follows that $E(\widehat{G}^2_\e)\to \infty$ as $\e$ tends to zero, so $\widehat{G}_\e$ does not converge in $L^2(\Omega)$. Assume, now, that $H>3/4$. For $\e,\eta>0$, we have $$E( \widehat{G}_\e \widehat{G}_\eta)
= \frac 1{\e^2 \eta^2} \int_{[0,T]^2} \rho_{\e,\eta}(u- u' )^2 \,\mathrm{d}u\,\mathrm{d}u'$$ with $\rho_{\e,\eta}$ given by (\[rhoeps\]). Since the quantity $(\e\eta)^{-1} \rho_{\e,\eta}(u- u' )$ converges pointwise to (and is bounded by) $H(2H-1) |u-u' |^{2H-2}$, we have, by the dominated convergence theorem, that $E( \widehat{G}_\e \widehat{G}_\eta)$ converges to $$H^2(2H-1)^2 \int_{[0,T]^2} |u-u' |^{4H-4}\,\mathrm{d}u\,\mathrm{d}u'$$ as $\e,\eta\to 0$, with $\int_{[0,T]^2} |u-u' |^{4H-4}\,\mathrm{d}u\,\mathrm{d}u' <\infty$, since $H>3/4$. Hence, $\widehat{G}_\e$ converges in $L^2(\Omega)$.
Proof of the convergences (\[cv<\]) and (\[cv=\]) {#sec5}
====================================================
In this section, we provide a new proof of these convergences by means of a recent criterion for the weak convergence of sequences of multiple stochastic integrals established in [@NP] and [@PT]. We refer to [@MR1] for a proof in the case of more general Gaussian processes, using different kind of tools.
Let us first recall the aforementioned criterion. We continue to use the notation introduced in Section \[sec31\]. Also, let $\{e_{i}, i\geq 1\}$ denote a complete orthonormal system in $\mathfrak{H}$. Given $f\in \mathfrak{H}^{\odot k}$ and $g\in \mathfrak{H}^{\odot l}$, for every $r=0,\ldots,k\wedge l$, the *contraction* of $f$ and $g$ of order $r$ is the element of $\mathfrak{H}^{\otimes (k+l-2r)}$ defined by $$f\otimes _{r}g=\sum_{i_{1},\ldots,i_{r}=1}^{\infty }\langle
f,e_{i_{1}}\otimes \cdots \otimes e_{i_{r}}\rangle _{\mathfrak{H}^{\otimes
r}}\otimes \langle g,e_{i_{1}}\otimes \cdots \otimes e_{i_{r}}\rangle _{\mathfrak{H}^{\otimes r}}.$$ (Note that $f\otimes_{0}g=f\otimes g$ equals the tensor product of $f$ and $g$ while, for $k=l$, $f\otimes _{k}g=\langle f,g\rangle _{\mathfrak{H}^{\otimes k}}$.) Fix $k\geq 2$ and let $(F_\e)_{\e>0}$ be a family of the form $F_\e=I_k(\phi_\e)$ for some $\phi_\e\in\HH^{\odot k}$. Assume that the variance of $F_\e$ converges as $\e\to 0$ (to $\sigma^2$, say). The criterion of Nualart and Peccati [@NP] asserts that $F_\e\stackrel{\mathrm{Law}}{\longrightarrow}N\sim\mathscr{N}(0,\sigma^2)$ if and only if $ \| \phi_\e \otimes_r \phi_\e\|_{\HH^{\otimes (2k-2r)} } \to 0$ for any $r=1,\ldots, k-1$. In this case, due to the result proved by Peccati and Tudor [@PT], we automatically have that $$(B_{t_1},\ldots,B_{t_k},F_\e)\stackrel{\mathrm{Law}}{\longrightarrow}(B_{t_1},\ldots,B_{t_k},N)$$ for all $t_k> \cdots> t_1>0$, with $N\sim\mathscr{N}(0,\sigma^2)$ *independent* of $B$.
For $x\in{\mathbb{R}}$, set $$\label{rho}
\rho(x)= \tfrac 12 ( |x+1|^{2H} + |x-1|^{2H} - 2|x|^{2H} ),$$ and note that $\rho(u-v)=E [(B_{u+1}-B_u)(B_{v+1}-B_v) ]$ for all $u,v\geq 0$ and that $\int_{\mathbb{R}}|\rho(x)|^k \,\mathrm{d}x$ is finite if and only if $H<1-\frac1{2k}$ (since $\rho(x)\sim H(2H-1)|x|^{2H-2}$ as $|x|\to\infty$).
We now proceed with the proof of (\[cv<\]). The proof of (\[cv=\]) would follow similar arguments.
[Proof of (\[cv<\])]{} Because $\e^{k(1-H)-\sfrac12}G_\e$ can be expressed as a $k$th multiple Wiener integral, we can use the criterion of Nualart and Peccati. By the scaling property of the fBm, it is actually equivalent to considering the family of random variables $(F_\e)_{\e>0}$, where $$F_\e = \sqrt{\e} \int_0^{T/\e} h_k(B_{u+1} - B_u) \,\mathrm{d}u.$$
*Step . Convergence of the variance*. We can write $$\begin{aligned}
E(F^2_{\e})&=& \e k!
\int_0^{T/\e} \mathrm{d}u \int_0^{T/\e} \mathrm{d}s\, \rho(u-s)^k
\\
&=& \e k! \int_{-T/\e}^{T/\e} \rho(x)^{k}(T/\e - |x|)\,\mathrm{d}x,\end{aligned}$$ where the function $\rho$ is defined in (\[rho\]). Therefore, by the dominated convergence theorem, $$\lim_{\e \downarrow 0} E(F^2_{\e})
=T k! \int_{{\mathbb{R}}} \rho(x)^{k}\,\mathrm{d}x.$$
*Step . Convergence of the contractions*. Observe that the random variable $h_{k }(B_{u+1} -B_u)$ coincides with the multiple stochastic integral $I_k( \mathbf{1}_{[u,u+1]} ^{\otimes k })$. Therefore, $
F_\e = I_k(\phi_\e),
$ where $ \phi_\e = \sqrt{\e} \int_0^{T/\e} \mathbf{1}_{[u,u+1]} ^{\otimes k }\,\mathrm{d}u.$ Let $r\in\{1,\ldots,k-1\}$. We have $$\phi_\e \otimes_r \phi_\e= \e \int_0^{T/\e} \int_0^{T/\e} \bigl( \mathbf{1}_{[u,u+1]} ^{\otimes (k-r) }
\otimes \mathbf{1}_{[s,s+1]} ^{\otimes ( k-r) } \bigr) \rho(u-s)^r \,\mathrm{d}u\,\mathrm{d}s.$$ As a consequence, $\| \phi_\e \otimes_r \phi_\e\|_{\HH^{\otimes (2k-2r)} }^2$ equals $$\e^2 \int_{[0,{T/\e}]^4} \rho(u-s)^r \rho(u' -s')^r
\rho(u-u' )^{k-r} \rho(s-s')^{k-r} \,\mathrm{d}s\,\mathrm{d}s'\,\mathrm{d}u\,\mathrm{d}u'.$$ Making the changes of variables $x=u-s $, $y=u' -s'$ and $z=u-u' $, we obtain that $ \| \phi_\e \otimes_r \phi_\e\|_{\HH^{\otimes (2k-2r)} }^2$ is less than $$A_\e= \e \int_{D_\e} |\rho(x)|^r |\rho(y)|^r
|\rho(z)|^{k-r} |\rho(y+z-x)|^{k-r} \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z,$$ where $D_\e= [-T/\e,{T/\e}]^3$. Consider the decomposition $$\begin{aligned}
A_\e&=& \e \int_{D_\e \cap \{|x|\vee |y| \vee |z| \leq K\}} |\rho(x)|^r |\rho(y)|^r
|\rho(z)|^{k-r} |\rho(y+z-x)|^{k-r} \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z \\
&&{} +\e \int_{D_\e \cap \{|x|\vee |y| \vee |z| > K\}} |\rho(x)|^r |\rho(y)|^r
|\rho(z)|^{k-r} |\rho(y+z-x)|^{k-r} \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z \\
&=& B_{\e,K} + C_{\e,K}.\end{aligned}$$ Clearly, for any fixed $K>0$, the term $B_{\e,K}$ tends to zero because $\rho$ is a bounded function. On the other hand, we have $$D_\e \cap \{|x|\vee |y| \vee |z| > K\}\subset
D_{\e,K,x}\cup D_{\e,K,y}\cup D_{\e,K,z},$$ where $D_{\e,K,x}=\{|x|>K\}\cap \{|y|\leq T/\e\}\cap \{|z|\leq T/\e\}$ ($D_{\e,K,y}$ and $D_{\e,K,z}$ being defined similarly). Set $$C_{\e,K,x}
=\e \int_{D_{\e,K,x}} |\rho(x)|^r |\rho(y)|^r
|\rho(z)|^{k-r} |\rho(y+z-x)|^{k-r} \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z.$$ By Hölder’s inequality, we have $$\begin{aligned}
C_{\e,K,x} &\leq& \e \biggl( \int_{D_{\e,K,x}} |\rho(x)|^k|\rho(y)|^k \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z \biggr)^{\sfrac rk}\\
&&{}\times
\biggl( \int_{D_{\e,K,x}} |\rho(z)|^k|\rho(y+z-x)|^k \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z \biggr)^{1-\sfrac{r}k}\\
&\leq&
2T \biggl(\int_{\mathbb{R}}|\rho(t)|^k \,\mathrm{d}t \biggr)^{2-\sfrac rk} \biggl(\int_{|x|>K}|\rho(x)|^k \,\mathrm{d}x \biggr)^{\sfrac rk}
\mathop{\longrightarrow}_{K\to\infty} 0.\end{aligned}$$ Similarly, we prove that $C_{\e,K,y}\to 0$ and $C_{\e,K,z}\to 0$ as $K\to\infty$. Finally, it suffices to choose $K$ large enough in order to get the desired result, that is, $\| \phi_\e \otimes_r \phi_\e\|_{\HH^{\otimes (2k-2r)} }\to 0$ as $\e\to 0$.
*Step . Proof of the first point.* By step 1, the family $$\bigl((B_t)_{t\in[0,T]},\e^{\sfrac12-2H}G_\e \bigr)$$ is tight in $C([0,T])\times \mathbb{R}$. By step 2, we also have the convergence of the finite-dimensional distributions, as a by-product of the criteria of Nualart and Peccati [@NP] and Peccati and Tudor [@PT] (see the preliminaries at the beginning of this section). Hence, the proof of the first point is complete.
Convergences in law for some functionals related to the Lévy area of the fractional Brownian motion {#sec6}
===================================================================================================
Let $B^{(1)}$ and $B^{(2)}$ denote two independent fractional Brownian motions with Hurst index $H\in(0,1)$. Recall the definition (\[tilde\]) of $\widetilde{G}_\e$: $$\widetilde{G}_\e=\int_0^T B^{(1)}_u \frac{B^{(2)}_{u+\e}-B^{(2)}_u}{\e}\, \mathrm{d}u.$$
\[cvgausslevyarea\] Convergence in law (\[star1\]) holds.
We fix $H<1/2$. The proof is divided into several steps.
*Step . Computing the variance of $\varepsilon ^{\sfrac{1}{2}-H}\widetilde{G}_\e$*.
By using the scaling properties of the fBm, first observe that $\varepsilon ^{\sfrac{1}{2}-H}\widetilde{G}_\e$ has the same law as $$\label{gg}
\widetilde{F}_{\varepsilon }=\e^{1/2+H} \int_{0}^{T/\varepsilon
}B_{u}^{(1)} \bigl( B_{u+1}^{(2)}-B_{u}^{(2)} \bigr)\, \mathrm{d}u.$$ For $\rho(x)=\frac12 (|x+1|^{2H}+|x-1|^{2H}-2|x|^{2H} )$, we have $$\begin{aligned}
E(\widetilde{F}_{\varepsilon }^{2})
&=&\varepsilon^{1+2H} \int_{0}^{T/\varepsilon}\mathrm{d}u\int_{0}^{T/\varepsilon }\mathrm{d}s\, R_{H}(u,s)\rho (u-s)
\\
&=&\alpha_{\varepsilon }-\beta_{\varepsilon },\end{aligned}$$ where $$\begin{aligned}
\alpha_{\varepsilon }
&=& \varepsilon^{1+2H} \int_{0}^{T/\varepsilon}\mathrm{d}u\, u^{2H}\int_{0}^{T/\e}\mathrm{d}s\, \rho (u-s),
\\
\beta_{\varepsilon }
&=&\varepsilon^{1+2H} \int_{0}^{T/\varepsilon}\mathrm{d}u\,\int_{0}^{u}\mathrm{d}s\, (u-s)^{2H}\rho (u-s).\end{aligned}$$ Let us first study $\beta_{\varepsilon }$. We can write $$\beta_{\varepsilon }= \e^{2H}\int_{0}^{T/\varepsilon }x^{2H}\rho (x)(T-\varepsilon x)
\,\mathrm{d}x.$$ The integral $\int_{0}^{\infty }x^{2H}\rho (x)\,\mathrm{d}x$ is convergent for $H<1/4$, while $\int_{0}^{T/\varepsilon }x^{2H}\rho (x)\,\mathrm{d}x$ diverges as $-\frac18\log(1/\e)$ for $H=1/4$ and as $H(2H-1)T^{4H-1}\e^{1-4H}$ for $1/4<H<1/2$. The integral $\int_{0}^{T/\varepsilon }x^{2H+1}\rho (x)\,\mathrm{d}x$ diverges as $H(2H-1)T^{4H}
\varepsilon^{-4H}$. Therefore, $$\lim_{\varepsilon \rightarrow 0}\beta_{\varepsilon }=0.$$ Second, let us write $\alpha_\e$ as $$\begin{aligned}
\alpha_{\varepsilon } &=& \varepsilon^{1+2H} \int_{0}^{T/\varepsilon
}\mathrm{d}u\, u^{2H}\int_{0}^{T/\e}\mathrm{d}s\, \rho (u-s)\\
&=& \varepsilon^{1+2H} \biggl( \int_{0}^{T/\varepsilon
}\mathrm{d}u\, u^{2H}\int_{0}^{u}\mathrm{d}s\, \rho (u-s) + \int_{0}^{T/\varepsilon
}\mathrm{d}u\, u^{2H}\int_{u}^{T/\e}\mathrm{d}s\, \rho (u-s) \biggr)\\
&=& \frac{1}{2H+1} \int_{0}^{T/\varepsilon
}\rho(x) \bigl(T^{2H+1}-(\e x)^{2H+1}+(T-\e x)^{2H+1} \bigr)\,\mathrm{d}x.\end{aligned}$$ Hence, by the dominated convergence theorem, we have $$\lim_{\varepsilon \rightarrow 0}\alpha_{\varepsilon }=\frac{2T^{2H+1}}{2H+1}\int_0^\infty \rho(x)\,\mathrm{d}x$$ so that $$\lim_{\varepsilon \rightarrow 0}\e^{1-2H}E[\widetilde{G}_{\varepsilon }^2]
=\lim_{\varepsilon \rightarrow 0}E[\widetilde{F}_{\varepsilon }^2]=
\frac{2T^{2H+1}}{2H+1}\int_0^\infty \rho(x)\,\mathrm{d}x.$$
*Step . Showing the convergence in law in* (\[star1\]).
By the previous step, the distributions of the family $$\bigl( \bigl(B^{(1)}_t,B^{(2)}_t\bigr) _{t\in [0,T]}, \e^{\sfrac 12-H} \widetilde{G}_{\e} \bigr)_{\e>0}$$ are tight in $C([0,T]^2) \times \mathbb{R}$ and it suffices to show the convergence of the finite-dimensional distributions. We need to show that for any $\lambda \in \mathbb{R}$, any $0<t_1\leq\cdots\leq t_k$, any $\theta_1,\ldots,\theta_k\in{\mathbb{R}}$ and any $\mu_1,\ldots,\mu_k\in{\mathbb{R}}$, we have $$\begin{aligned}
\label{stardesstars}
&&\lim_{\varepsilon \downarrow 0} E \bigl[ \mathrm{e}^{\mathrm{i}\sum_{j=1}^k\theta_jB_{t_j}^{(1)}}
\mathrm{e}^{\mathrm{i}\sum_{j=1}^k \mu_j B^{(2)}_{t_j}} \mathrm{e}^{\mathrm{i}\lambda \e^{ \sfrac 12 - H} \widetilde{G}_{\varepsilon }} \bigr] \nonumber
\\[-8pt]\\[-8pt]
&&\quad=
E \bigl[
\mathrm{e}^{-(\sfrac12)\operatorname{Var} (\sum_{j=1}^k \mu_j B^{(2)}_{t_j} )}
\bigr] E \bigl[ \mathrm{e}^{\mathrm{i}\sum_{j=1}^k\theta_jB_{t_j}^{(1)}}
\mathrm{e}^{- \sfrac {\lambda^2 S^2} 2} \bigr],\nonumber\end{aligned}$$ where $S=\sqrt{2\int_0^\infty \rho(x)\,\mathrm{d}x\int_0^T( B^{(1)}_u)^2 \,\mathrm{d}u}$. We can write $$\begin{aligned}
&&E \bigl[
\mathrm{e}^{\mathrm{i}\sum_{j=1}^k\theta_jB_{t_j}^{(1)}}
\mathrm{e}^{\mathrm{i}\sum_{j=1}^k \mu_j B^{(2)}_{t_j}} \mathrm{e}^{\mathrm{i}\lambda \e^{ \sfrac 12 - H} \widetilde{G}_{\varepsilon }} \bigr]\\
&&\quad = E \bigl[ \mathrm{e}^{\mathrm{i}\sum_{j=1}^k\theta_jB_{t_j}^{(1)}} E\bigl[ \mathrm{e}^{\mathrm{i}\sum_{j=1}^k \mu_j B^{(2)}_{t_j}}
\mathrm{e}^{\mathrm{i}\lambda \e^{ \sfrac 12 - H} \widetilde{G}_{\varepsilon }} \big| B^{(1)}\bigr] \bigr]\\
&&\quad = E \bigl[ \mathrm{e}^{\mathrm{i}\sum_{j=1}^k\theta_jB_{t_j}^{(1)}}
\mathrm{e}^{-\lambda \e^{\sfrac12-H}\sum_{j=1}^k\mu_j\int_0^T B^{(1)}_u E (B_{t_j}^{(2)}\times
\sfrac{B^{((2)}_{u+\e}-B^{(2)}_u)}{\e} )\,\mathrm{d}u}
\\
&&\qquad\hspace*{9pt}{}\times \mathrm{e}^{-(\sfrac{\lambda^2}2) \e^{1-2H}\int_{[0,T]^2} B^{(1)}_uB^{(1)}_v \rho_\e(u-v)\,\mathrm{d}u\,\mathrm{d}v}
\mathrm{e}^{-(\sfrac12)\operatorname{Var} (\sum_{j=1}^k \mu_j B^{(2)}_{t_j} )}\bigr]\end{aligned}$$ with $\rho_\e(x)=\frac12 (|x+\e|^{2H}+|x-\e|^{2H}-2|x|^{2H} )$. Observe that $$\int_{[0,T]^2} B^{(1)}_uB^{(1)}_v\rho_\e(u-v)\,\mathrm{d}u\,\mathrm{d}v\geq 0$$ since $\rho_\e(u-v)=E [(B^{(2)}_{u+\e}-B^{(2)}_u)(B^{(2)}_{v+\e}-B^{(2)}_v) ]$ is a covariance function. Moreover, for any fixed $t\geq 0$, we have $$\begin{aligned}
&&\int_0^T B^{(1)}_u E \biggl(B_{t}^{(2)}\times
\frac{B^{(2)}_{u+\e}-B^{(2)}_u}{\e} \biggr)
\,\mathrm{d}u\\
&&\quad=
\frac1{2}\int_0^T B^{(1)}_u
\biggl(\frac{(u+\e)^{2H}-u^{2H}}{\e}+\frac{|t-u|^{2H}-|t-u-\e|^{2H}}{\e}\biggr)\,\mathrm{d}u\\
&&\quad\stackrel{\mathrm{a.s.}}{\mathop{\longrightarrow}_{\e\to 0}}
H\int_0^T B^{(1)}_u (u^{2H-1}-|t-u|^{2H-1} )\,\mathrm{d}u.\end{aligned}$$ Since $H<1/2$, this implies that $$\mathrm{e}^{-\lambda \e^{\sfrac12-H}\sum_{j=1}^k\mu_j\int_0^T B^{(1)}_u E
(B_{t_j}^{(2)}\times \afrac{B^{(2)}_{u+\e}-B^{(2)}_u}{\e} )
\,\mathrm{d}u} \stackrel{\mathrm{a.s.}}{\mathop{\longrightarrow}_{\e\to 0}} 1.$$ Hence, to get (\[stardesstars\]), it suffices to show that $$\label{stardes}
E
\bigl[
\mathrm{e}^{\mathrm{i}\sum_{j=1}^k\theta_jB_{t_j}^{(1)}}
\mathrm{e}^{-(\sfrac{\lambda^2}2) \e^{1-2H}\int_{[0,T]^2} B^{(1)}_uB^{(1)}_v \rho_\e(u-v)\,\mathrm{d}u\,\mathrm{d}v}
\bigr]
\mathop{\longrightarrow}_{\e\to 0}E \bigl[
\mathrm{e}^{\mathrm{i}\sum_{j=1}^k\theta_jB_{t_j}^{(1)}} \mathrm{e}^{-(\sfrac{\lambda^2}2)S^2} \bigr].$$ We have $$\begin{aligned}
C_\e&:=&E
\Biggl[\exp
\Biggl(\mathrm{i}\sum_{j=1}^k\theta_jB_{t_j}^{(1)}
-\frac{\lambda^2}2 \e^{1-2H}\int_{[0,T]^2} B^{(1)}_uB^{(1)}_v \rho_\e(u-v)\,\mathrm{d}u\,\mathrm{d}v \Biggr)
\Biggr]\\
&=&E
\Biggl[\exp
\Biggl(\mathrm{i}\sum_{j=1}^k\theta_jB_{t_j}^{(1)}
-\lambda^2 \e^{1-2H}\int_0^T B^{(1)}_u \biggl(\int_0^u B^{(1)}_{u-x} \rho_\e(x)\,\mathrm{d}x \biggr)\,\mathrm{d}u \Biggr)
\Biggr]\\
&=&E
\Biggl[\exp
\Biggl(\mathrm{i}\sum_{j=1}^k\theta_jB_{t_j}^{(1)}
-\lambda^2 \e^{1-2H}\int_0^T \rho_\e(x) \biggl(\int_x^T B^{(1)}_uB^{(1)}_{u-x} \,\mathrm{d}u \biggr)\,\mathrm{d}x \Biggr)
\Biggr]\\
&=&E
\Biggl[\exp
\Biggl(\mathrm{i}\sum_{j=1}^k\theta_jB_{t_j}^{(1)}
-\lambda^2 \int_0^{T/\e} \rho(x) \biggl(\int_{\e x}^T B^{(1)}_uB^{(1)}_{u-\e x} \,\mathrm{d}u \biggr)\,\mathrm{d}x \Biggr)
\Biggr],\end{aligned}$$ the last inequality following from the relation $\rho_\e(x)=\e^{2H}\rho(x/\e)$. By the dominated convergence theorem, we obtain $$\begin{aligned}
C_\e&\displaystyle\mathop{\longrightarrow}_{\e\to 0} &
E
\Biggl[\exp
\Biggl(\mathrm{i}\sum_{j=1}^k\theta_jB_{t_j}^{(1)}
-\lambda^2 \int_0^{\infty} \rho(x)\,\mathrm{d}x\times \int_{0}^T \bigl(B^{(1)}_u\bigr)^2 \,\mathrm{d}u \Biggr)
\Biggr]\\
&=&E
\Biggl[\exp
\Biggl(\mathrm{i}\sum_{j=1}^k\theta_jB_{t_j}^{(1)}
-\frac{\lambda^2}2 S^2 \Biggr)
\Biggr],\end{aligned}$$ that is, (\[stardes\]). The proof of the theorem is thus completed.
Recall the definition (\[rho\]) of $\rho$ and the definition of $\breve{G}_\e$: $$\breve{G}_\e=
\int_0^T \biggl(\int_0^u \frac{B^{(1)}_{v+\e}-B^{(1)}_v}{\e} \,\mathrm{d}v \biggr)\frac{B^{(2)}_{u+\e}-B^{(2)}_u}{\e} \,\mathrm{d}u.$$
\[cvgausslevyareabis\] Convergences in law (\[star10\]) and (\[star10crit\]) hold.
We only show the first convergence, the proof of the second one being very similar. By using the scaling properties of the fBm, first observe that $\varepsilon ^{\sfrac{1}{2}-2H}\breve{G}_\e$ has the same law as $$\breve{F}_{\varepsilon }=\sqrt{\e} \int_{0}^{T/\varepsilon}
\biggl(\int_0^u
\bigl(B_{v+1}^{(1)}-B^{(1)}_v \bigr)\,\mathrm{d}v\biggr)
\bigl( B_{u+1}^{(2)}-B_{u}^{(2)} \bigr) \,\mathrm{d}u.$$
We now fix $H<1/4$ and the proof is divided into several steps.
*Step . Computing the variance of $\breve{F}_{\varepsilon }$*. We can write $$\begin{aligned}
E(\breve{F}_{\varepsilon }^{2}) &=&\varepsilon \int_{[0,T/\varepsilon]^2}
\mathrm{d}u\,\mathrm{d}u' \,\rho(u-u' )\int_{0}^{u }\mathrm{d}v\int_0^{u' }\mathrm{d}v'\,\rho(v-v')\end{aligned}$$ with $\rho(x)=\frac12 (|x+1|^{2H}+|x-1|^{2H}-2|x|^{2H} )$. We have $$\int_0^u\mathrm{d}v\int_0^{u' }\mathrm{d}v'\,\rho(v-v')=\frac{\Psi(u-u' )-\Psi(u)-\Psi(u'
)+2}{2(2H+1)(2H+2)},$$ where $$\label{PPSSII}
\Psi(x)=2|x|^{2H+2}-|x+1|^{2H+2}-|x-1|^{2H+2}.$$ Consider first the contribution of the term $\Psi(u-u' )$. We have $$\lim_{\e\to 0}\e\int_{[0,T/\e]^2}\rho(u-u' )\Psi(u-u' )\,\mathrm{d}u\,\mathrm{d}u' =T\int_{\mathbb{R}}\rho(x)\Psi(x)\,\mathrm{d}x.$$ Note that $\rho(x)\sim H(2H-1)|x|^{2H-2}$ and $\Psi(x)\sim -(2H+2)(2H+1)|x|^{2H}$ as $|x|\to\infty$ so that $\int_{\mathbb{R}}|\rho(x)\Psi(x)|\,\mathrm{d}x<\infty$ because $H<1/4$. On the other hand, we have $$\e\int_{[0,T/\e]^2} \rho(u-u' )\Psi(u)\,\mathrm{d}u\,\mathrm{d}u'
=\e\int_0^{T/\e}\mathrm{d}u\,\Psi(u)\int_{u-T/\e}^u\mathrm{d}x\,\rho(x)$$ and this converges to zero as $\e\to 0$. Indeed, since $\rho(x)\sim H(2H-1)x^{2H-2}$ as $x\to\infty$, we have $\int_u^\infty \rho(x)\,\mathrm{d}x \sim Hu^{2H-1}$ as $u\to\infty$; hence, since $\int_{\mathbb{R}}\rho(x)\,\mathrm{d}x=0$, $H<1/4$ and $\Psi(u)\sim -(2H+2)(2H+1)u^{2H}$ as $u\to\infty$, we have $$\lim_{u\to\infty}\Psi(u)\int_{-\infty}^u\rho(x)\,\mathrm{d}x
=-\lim_{u\to\infty}\Psi(u)\int_{u}^\infty\rho(x)\,\mathrm{d}x
=0.$$ Also, we have $$\lim_{\e\to 0}\e\int_{[0,T/\e]^2}\rho(u-u' )\,\mathrm{d}u\,\mathrm{d}u' =\int_{\mathbb{R}}\rho(x)\,\mathrm{d}x=0.$$ Therefore, $\lim_{\e\to 0}E(\breve{F}_\e^2)=\breve{\sigma}_H^2$.
*Step . Showing the convergence in law* (\[star10\]). We first remark that by step 1, the laws of the family $ ((B^{(1)}_t,B^{(2)}_t) _{t\in [0,T]},\e^{\sfrac12-2H}\breve{G}_{\varepsilon } ) _{\e>0}$ are tight. Therefore, we only have to prove the convergence of the finite-dimensional laws. Moreover, by the main result of Peccati and Tudor [@PT], it suffices to prove that $$\label{felow}
\e^{\sfrac12-2H}\breve{G}_\e\stackrel{\mathrm{Law}}{=}\breve{F}_{\varepsilon }\stackrel{\mathrm{Law}}{\longrightarrow}
\mathscr{N}(0,T\breve{\sigma}^2_H) \qquad\mbox{as $\e\to 0$}.$$ We have $$\begin{aligned}
E(\mathrm{e}^{\mathrm{i}\lambda \breve{F}_\e})
&=&E
\biggl(\exp \biggl\{-\frac{\lambda^2\e}2\int_{\lbrack 0,T/\varepsilon]^{2}}
\bigl(B^{(2)}_{u+1}-B^{(2)}_{u}\bigr)\bigl(B^{(2)}_{u' +1}-B^{(2)}_{u' }\bigr)
\\
&&\hspace*{93pt}{}\times
\biggl( \int_{0}^{u}\int_{0}^{u'}\rho
(v-v^{\prime })\,\mathrm{d}v\,\mathrm{d}v^{\prime } \biggr) \,\mathrm{d}u\,\mathrm{d}u' \biggr\}
\biggr).\end{aligned}$$ Since $\rho(v-v')=E [(B^{(1)}_{v+1}-B^{(1)}_v)(B^{(1)}_{v'+1}-B^{(1)}_v) ]$ is a covariance function, observe that the quantity inside the exponential in the right-hand side of the previous identity is negative. Hence, since $x\mapsto\exp (-\frac{\lambda^2}2x_+ )$ is continuous and bounded by 1 on ${\mathbb{R}}$, (\[felow\]) will be a consequence of the convergence $$\label{toshow2}
A_\e \stackrel{\rm law}{\longrightarrow}T\breve{\sigma}_H^2 \qquad\mbox{as $\e\to 0$}$$ with $$A_{\varepsilon}:=\varepsilon \int_{\lbrack 0,T/\varepsilon
]^{2}}(B_{u+1}-B_{u})(B_{u' +1}-B_{u' }) \biggl( \int_{0}^{u }\int_{0}^{u'}\rho
(v-v^{\prime })\,\mathrm{d}v\,\mathrm{d}v^{\prime } \biggr) \,\mathrm{d}u\,\mathrm{d}u',$$ $B$ denoting a fractional Brownian motion with Hurst index $H$. The proof of (\[toshow2\]) will be achieved by showing that the expectation (resp., the variance) of $A_\e$ tends to $T\breve{\sigma}_H^2$ (resp., zero). By step 1, observe that $$E(A_\e)=E(\breve{F}_\e^2)\to T\breve{\sigma}^2_H$$ as $\e\to 0$. We now want to show that the variance of $A_{\varepsilon }$ converges to zero. Performing the changes of variables $s=u\varepsilon$ and $t=u' \varepsilon$ yields $$A_{\varepsilon }=\varepsilon ^{-1}\int_{[0,T]^{2}}(B_{s/\varepsilon
+1}-B_{s/\varepsilon })(B_{t/\varepsilon +1}-B_{t/\varepsilon})
\biggl(\int_{0}^{s/\varepsilon }\int_{0}^{t/\varepsilon }\rho (v-v^{\prime })\,\mathrm{d}v\,\mathrm{d}v^{\prime } \biggr) \,\mathrm{d}s\,\mathrm{d}t,$$ which has the same distribution as $$\begin{aligned}
C_{\varepsilon } &=&\varepsilon ^{-1-2H}\int_{[0,T]^{2}}(B_{s+\varepsilon
}-B_{s})(B_{t+\varepsilon }-B_{t}) \biggl( \int_{0}^{s/\varepsilon
}\int_{0}^{t/\varepsilon }\rho (u-u' )\,\mathrm{d}u\,\mathrm{d}u' \biggr) \,\mathrm{d}s\,\mathrm{d}t \\
&=&\varepsilon ^{-1-2H}\int_{[0,T]^{2}}(B_{s+\varepsilon
}-B_{s})(B_{t+\varepsilon }-B_{t})\Lambda _{\varepsilon }(s,t)\,\mathrm{d}s\,\mathrm{d}t,\end{aligned}$$ where $\Lambda _{\varepsilon }(s,t)=\int_{0}^{s/\varepsilon
}\int_{0}^{t/\varepsilon }\rho (u-u' )\,\mathrm{d}u\,\mathrm{d}u'.$ This can be written as $$C_{\varepsilon }=\varepsilon ^{-1-2H}\int_{\mathbb{R}^{2}}B_{s}B_{t}\Sigma_{\varepsilon }(s,t)\,\mathrm{d}s\,\mathrm{d}t,$$ where $$\begin{aligned}
\label{e1}
\hspace*{-15pt}\Sigma _{\varepsilon }(s,t) &=&\mathbf{1}_{[\varepsilon,T+\varepsilon ]}(s)
\mathbf{1}_{[\varepsilon,T+\varepsilon ]}(t)\Lambda _{\varepsilon}(s-\varepsilon,t-\varepsilon )
-\mathbf{1}_{[0,T]}(s)\mathbf{1}_{[\varepsilon,T+\varepsilon ]}(t)\Lambda _{\varepsilon }(s,t-\varepsilon )
\nonumber \\[-8pt]\\[-8pt]
&&{}-\mathbf{1}_{[\varepsilon,T+\varepsilon ]}(s)\mathbf{1}_{[0,T]}(t)\Lambda_{\varepsilon }
(s-\varepsilon,t)+\mathbf{1}_{[0,T]}(s)\mathbf{1}_{[0,T]}(t)\Lambda _{\varepsilon }(s,t).\nonumber\end{aligned}$$ Moreover, $$C_{\varepsilon }-E(C_{\varepsilon })=\varepsilon ^{-1-2H}I_2 \biggl(\int_{\mathbb{R}^{2}}
\mathbf{1}_{[0,s]}\otimes \mathbf{1}_{[0,t]} \Sigma_{\varepsilon }(s,t)\,\mathrm{d}s\,\mathrm{d}t \biggr),$$ where $I_{2}$ is the double stochastic integral with respect to $B$. Therefore, $$\begin{aligned}
\operatorname{Var}(C_{\varepsilon }) &=&2\varepsilon ^{-2-4H}
\bigg\| \int_{\mathbb{R}^{2}} \mathbf{1}_{[0,s]}\otimes \mathbf{1}_{[0,t]}\Sigma _{\varepsilon }(s,t)\,\mathrm{d}s\,\mathrm{d}t\bigg\|^2_{\HH^{\otimes 2}} \\
&=&2\varepsilon ^{-2-4H}\int_{\mathbb{R}^{4}}R_{H}(s,s^{\prime
})R_{H}(t,t^{\prime })\Sigma _{\varepsilon }(s,t)\Sigma _{\varepsilon
}(s^{\prime },t^{\prime })\,\mathrm{d}s\,\mathrm{d}t\,\mathrm{d}s^{\prime }\,\mathrm{d}t^{\prime }.\end{aligned}$$ Taking into account that the partial derivatives $\frac{\partial R_{H}}{\partial s}$ and $\frac{\partial R_{H}}{\partial t}$ are integrable, we can write $$\begin{aligned}
&& \operatorname{Var}(C_{\varepsilon })=2\varepsilon ^{-2-4H}\int_{\mathbb{R}^{4}}
\biggl( \int_{0}^{s}\frac{\partial R_{H}}{\partial\sigma }(\sigma,s^{\prime})\,\mathrm{d}\sigma \biggr)
\biggl( \int_{0}^{t^{\prime }}\frac{\partial R_{H}}{\partial \tau }(t,\tau )\,\mathrm{d}\tau \biggr) \\
&&\hspace*{96pt} {} \times \Sigma _{\varepsilon }(s,t)\Sigma_{\varepsilon }(s^{\prime },t^{\prime })
\,\mathrm{d}s\,\mathrm{d}t\,\mathrm{d}s^{\prime }\,\mathrm{d}t^{\prime}.\end{aligned}$$ Hence, by integrating by parts, we get $$\begin{aligned}
&& \operatorname{Var}(C_{\varepsilon })=2\varepsilon ^{-2-4H}\int_{\mathbb{R}^{4}}\frac{\partial R_{H}}{\partial s}(s,s^{\prime })
\frac{\partial R_{H}}{\partial t^{\prime }}(t,t^{\prime }) \\
&&\hspace*{97pt}{} \times \biggl( \int_{0}^{s}
\times \Sigma _{\varepsilon
}(\sigma,t)\,\mathrm{d}\sigma \biggr) \biggl( \int_{0}^{t^{\prime }}\Sigma_{\varepsilon }(s^{\prime },\tau )\,\mathrm{d}\tau \biggr)
\,\mathrm{d}s\,\mathrm{d}t\,\mathrm{d}s^{\prime }\,\mathrm{d}t^{\prime}.\end{aligned}$$ From (\[e1\]), we obtain $$\int_{0}^{s}\Sigma _{\varepsilon }(\sigma,t)\,\mathrm{d}\sigma =\mathbf{1}_{[0,T]}(s) \bigl( \mathbf{1}_{[0,\varepsilon ]}(t)-
\mathbf{1}_{[T,T+\varepsilon ]}(t) \bigr) \int_{s-\varepsilon }^{s}\Lambda_{\varepsilon }(\sigma,t-\varepsilon )\,\mathrm{d}\sigma.$$ In the same way, $$\int_{0}^{t^{\prime }}\Sigma _{\varepsilon }(s^{\prime },\tau )\,\mathrm{d}\tau =
\mathbf{1}_{[0,T]}(t^{\prime }) \bigl( \mathbf{1}_{[0,\varepsilon
]}(s^{\prime })-\mathbf{1}_{[T,T+\varepsilon ]}(s^{\prime }) \bigr)
\int_{t^{\prime }-\varepsilon }^{t^{\prime }}\Lambda _{\varepsilon
}(s^{\prime }-\varepsilon,\tau )\,\mathrm{d}\tau.$$ As a consequence, $$\begin{aligned}
\operatorname{Var}(C_{\varepsilon }) &=&2\varepsilon ^{-2-4H}\int_{\mathbb{R}^{4}}
\frac{\partial R_{H}}{\partial s}(s,s^{\prime })\frac{\partial R_{H}}
{\partial t^{\prime }}(t,t^{\prime }) \biggl( \int_{s-\varepsilon }^{s}\Lambda_{\varepsilon }(\sigma,t-\varepsilon )\,\mathrm{d}\sigma \biggr) \\
&&\hspace*{53pt}{}\times \biggl( \int_{t^{\prime }-\varepsilon }^{t^{\prime }}
\Lambda_{\varepsilon }(s^{\prime }-\varepsilon,\tau )\,\mathrm{d}\tau \biggr) \mathbf{1}_{[0,T]}(s)
\bigl( \mathbf{1}_{[0,\varepsilon ]}(t)-\mathbf{1}_{[T,T+\varepsilon ]}(t) \bigr) \\
&&\hspace*{53pt}{}\times \mathbf{1}_{[0,T]}(t^{\prime }) \bigl( \mathbf{1}_{[0,\varepsilon]}(s^{\prime })-\mathbf{1}_{[T,T+\varepsilon ]}(s^{\prime }) \bigr)
\,\mathrm{d}s\,\mathrm{d}t\,\mathrm{d}s^{\prime }\,\mathrm{d}t^{\prime }
=\sum_{i=1}^{4}H_{\varepsilon }^{i},\end{aligned}$$ where $$\begin{aligned}
H_{\varepsilon }^{1} &=&\int_{0}^{T}\int_{0}^{\varepsilon
}\int_{0}^{\varepsilon }\int_{0}^{T}G_{\varepsilon }(s,t,s^{\prime
},t^{\prime })\,\mathrm{d}s\,\mathrm{d}t\,\mathrm{d}s^{\prime }\,\mathrm{d}t^{\prime}, \\
H_{\varepsilon }^{2} &=&-\int_{0}^{T}\int_{T}^{T+\varepsilon
}\int_{0}^{\varepsilon }\int_{0}^{T}G_{\varepsilon }(s,t,s^{\prime
},t^{\prime })\,\mathrm{d}s\,\mathrm{d}t\,\mathrm{d}s^{\prime }\,\mathrm{d}t^{\prime}, \\
H_{\varepsilon }^{3} &=&-\int_{0}^{T}\int_{0}^{\varepsilon
}\int_{T}^{T+\varepsilon }\int_{0}^{T}G_{\varepsilon }(s,t,s^{\prime
},t^{\prime })\,\mathrm{d}s\,\mathrm{d}t\,\mathrm{d}s^{\prime }\,\mathrm{d}t^{\prime}, \\
H_{\varepsilon }^{4} &=&\int_{0}^{T}\int_{0}^{T+\varepsilon
}\int_{0}^{T+\varepsilon }\int_{0}^{T}G_{\varepsilon }(s,t,s^{\prime
},t^{\prime })\,\mathrm{d}s\,\mathrm{d}t\,\mathrm{d}s^{\prime }\,\mathrm{d}t^{\prime}\end{aligned}$$ and $$\begin{aligned}
G_{\varepsilon }(s,t,s^{\prime },t^{\prime })&=&2\varepsilon ^{-2-4H}
\frac{\partial R_{H}}{\partial s}(s,s^{\prime })\frac{\partial R_{H}}{\partial
t^{\prime }}(t,t^{\prime }) \\
&&{} \times \biggl( \int_{s-\varepsilon }^{s}\Lambda_{\varepsilon }(\sigma,t-\varepsilon )\,\mathrm{d}\sigma \biggr)
\biggl(\int_{t^{\prime }-\varepsilon }^{t^{\prime }}\Lambda _{\varepsilon
}(s^{\prime }-\varepsilon,\tau )\,\mathrm{d}\tau \biggr).\end{aligned}$$ We only consider the term $H_{\varepsilon }^{1}$ because the others can be handled in the same way. We have, with $\Psi$ given by (\[PPSSII\]), $$\Lambda _{\varepsilon }(s,t)=\int_{0}^{s/\varepsilon
}\int_{0}^{t/\varepsilon }\rho (u-u')\,\mathrm{d}u\,\mathrm{d}u'=\frac{\Psi (\sfrac{(s-t)}{\varepsilon })
-\Psi (\sfrac{s}{\varepsilon })-\Psi (\sfrac{t}{\varepsilon })+2}{2(2H+1)(2H+2)}.$$ Note that $$\begin{aligned}
\bigg| \Psi \biggl(\frac{s-t}{\varepsilon }\biggr) \bigg| &\le &\varepsilon
^{-2H-2} | 2|s-t|^{2H+2}-|s-t+\varepsilon |^{2H+2}-|s-t-\varepsilon
|^{2H+2} | \\
&\leq &C\varepsilon ^{-2H}\end{aligned}$$ for any $s,t\in \lbrack 0,T]$. Therefore, $ | \Lambda _{\varepsilon
}(s,t) | \leq C\varepsilon ^{-2H}$ and we obtain the estimate $$| G_{\varepsilon }(s,t,s^{\prime },t^{\prime }) | \leq
C\varepsilon ^{-8H} ( s^{2H-1}+|s-s^{\prime }|^{2H-1} )
(t^{\prime 2H-1}+|t-t^{\prime }|^{2H-1} ).$$ As a consequence, $$\begin{aligned}
| H_{\varepsilon }^{1} | &\leq &\int_{0}^{T}\int_{0}^{\varepsilon
}\int_{0}^{\varepsilon }\int_{0}^{T} | G_{\varepsilon }(s,t,s^{\prime
},t^{\prime }) | \,\mathrm{d}s\,\mathrm{d}t\,\mathrm{d}s^{\prime }\,\mathrm{d}t^{\prime } \\
&\leq &C\varepsilon ^{-8H}\int_{0}^{T}\int_{0}^{\varepsilon
}\int_{0}^{\varepsilon }\int_{0}^{T} ( s^{2H-1}+|s-s^{\prime}|^{2H-1} ) \\
&&{}\hspace*{96pt} \times ( t^{\prime 2H-1}+|t-t^{\prime }|^{2H-1} )
\,\mathrm{d}s\,\mathrm{d}t\,\mathrm{d}s^{\prime }\,\mathrm{d}t^{\prime } \\
&\leq &C\varepsilon ^{2-8H},\end{aligned}$$ which converges to zero because $H<\frac{1}{4}$.
Recall the definition (\[hat\]) of $\widehat{G}_\e$, $$\widehat{G}_\e=
\int_0^T \frac{B^{(1)}_{u+\e}-B^{(1)}_u}{\e}\times\frac{B^{(2)}_{u+\e}-B^{(2)}_u}{\e} \,\mathrm{d}u.$$ We have the following result.
\[cvgausscovariation\] Convergences (\[star5\]) and (\[star6\]) hold.
We use the same trick as in [@nourdinjfa], Remark 1.3, point 4. Let $\beta$ and $\widetilde{\beta}$ be two independent one-dimensional fractional Brownian motions with index $H$. Set $B^{(1)}=(\beta+\widetilde{\beta})/\sqrt{2}$ and $B^{(2)}=(\beta-\widetilde{\beta})/\sqrt{2}$. It is easily checked that $B^{(1)}$ and $B^{(2)}$ are also two independent fractional Brownian motions with index $H$. Moreover, we have $$\begin{aligned}
\label{dollar}
\hspace*{-25pt}\e^{\sfrac32-2H}\widehat{G}_\e
&=&\frac12 \e^{\sfrac32-2H}\int_0^T \biggl(\frac{\beta_{u+\e}-\beta_u}{\e} \biggr)^2\,\mathrm{d}u
-\frac12 \e^{\sfrac32-2H}\int_0^T \biggl(\frac{\widetilde{\beta}_{u+\e}-\widetilde{\beta}_u}{\e} \biggr)^2\,\mathrm{d}u\nonumber\\
&=&\frac1{2\sqrt{\e}}\int_0^T \biggl(\frac{\beta_{u+\e}-\beta_u}{\e^H} \biggr)^2\,\mathrm{d}u
-\frac1{2\sqrt{\e}}\int_0^T \biggl(\frac{\widetilde{\beta}_{u+\e}-\widetilde{\beta}_u}{\e^H} \biggr)^2\,\mathrm{d}u\\
&=&\frac1{2\sqrt{\e}}\int_0^T h_2 \biggl(\frac{\beta_{u+\e}-\beta_u}{\e^H} \biggr)\,\mathrm{d}u
-\frac1{2\sqrt{\e}}\int_0^T h_2 \biggl(\frac{\widetilde{\beta}_{u+\e}-\widetilde{\beta}_u}{\e^H} \biggr)\,\mathrm{d}u.\nonumber\end{aligned}$$ The proofs of the desired convergences in law are now direct consequences of the convergence (\[cv<\]) with $k=2$, taking into account that $\beta$ and $\widetilde{\beta}$ are independent.
As a by-product of the decomposition (\[dollar\]), and taking into account (\[cv>\]) for $k=2$, we get that $\int_0^T \dot{B}^{(1)}_u\diamond \dot{B}^{(2)}_u\, \mathrm{d}u$ and $ (Z_T^{(2)}-\widetilde{Z}_T^{(2)} )/2$ have the same law when $H>3/4$, where $\widetilde{Z}_T^{(2)}$ stands for an independent copy of the Hermite random variable $Z_T^{(2)}$.
Acknowedgements {#acknowedgements .unnumbered}
===============
The research of I. Nourdin was supported in part by the ANR project ‘Exploration des Chemins Rugueux’. The research of D. Nualart was supported by NSF Grant DMS-0904538.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We discuss an open driven-dissipative many-body system, in which the competition of unitary Hamiltonian and dissipative Liouvillian dynamics leads to a nonequilibrium phase transition. It shares features of a quantum phase transition in that it is interaction driven, and of a classical phase transition, in that the ordered phase is continuously connected to a thermal state. Within a generalized Gutzwiller approach which includes the description of mixed state density matrices, we characterize the complete phase diagram and the critical behavior at the phase transition approached as a function of time. We find a novel fluctuation induced dynamical instability, which occurs at long wavelength as a consequence of a subtle dissipative renormalization effect on the speed of sound.'
author:
- Sebastian Diehl
- Andrea Tomadin
- Andrea Micheli
- Rosario Fazio
- Peter Zoller
title: 'Dynamical Phase Transitions and Instabilities in Open Atomic Many-Body Systems'
---
Experiments with cold atoms provide a unique setting to study nonequilibrium phenomena and dynamics, both in closed systems but also for (driven) open quantum dynamics. This relies on the ability to control the many-body dynamics and to prepare initial states far from the ground state. For closed systems we have seen a plethora of studies of quench dynamics [@QuenchTh; @QuenchExp], thermalization [@ThermalizationTh; @ThermalizationExp], and transport [@Transport], and also dynamical studies of crossing in a finite time quantum critical points in the spirit of the Kibble-Zurek mechanism [@KibbleZurekTh; @KibbleZurekExp]. On the other hand, systems of cold atoms can be driven by external (light) fields and coupled to dissipative baths, thus realizing driven open quantum systems. As familiar e.g. from the quantum optics of the laser, the steady state of such a system (if it exists) is characterized by a dynamical equilibrium between pumping and dissipation, and can exhibit various nonequilibrium phases and phase transitions [@ExcitonPolariton; @Dalla09] as function of external control parameters. In the present work we will study such scenarios for quantum degenerate gases. Our emphasis is on understanding quantum phases and dynamical phase transitions of cold atoms as an interacting many-body condensed matter system far from equilibrium.
For a many-body system in thermodynamic equilibrium the competition of two noncommuting parts of a microscopic Hamiltonian $H=H_{1}+gH_{2}$ manifests itself as a quantum phase transition (QPT), if the ground states for $g\ll g_{c}$ and $g\gg g_{c}$ have different symmetries [@SachdevBook]. For temperature $T=0$ the critical value $g_{c}$ then separates two distinct quantum phases, while for finite temperature this defines a quantum critical region around $g_{c}$ in a $T$ vs. $g$ phase diagram. A seminal example in the context of cold atoms in optical lattices is the superfluid–Mott insulator transition in the Bose-Hubbard (BH) model, with Hamiltonian $$\label{eq:BHHamil}
H=-J\sum_{\langle \ell,\ell'\rangle}b_{\ell}^{\dagger}b_{\ell'} - \mu\sum_{\ell}\hat{n}_{\ell} +\tfrac{1}{2}U\sum_{\ell}\hat{n}_{\ell}(\hat{n}_{\ell}-1)~,$$ with $b_{\ell}$ bosonic operators annihilating a particle on site $\ell$, $\hat{n}_{\ell}=b_{\ell}^{\dagger}b_{\ell}$ number operators, $J$ the hopping amplitude, and $U$ the onsite interaction strength. For a given chemical potential $\mu$, chosen to fix a mean particle density $n$, the critical coupling strength $g_c = (U/Jz)_{c}$ separates a superfluid $Jz \gg U$ from a Mott insulator regime $Jz \ll
U$ ($z$ the lattice coordination number).
In contrast, we consider a nonequilibrium situation in which the competition of microscopic quantum mechanical operators results from an interplay of unitary (Hamiltonian) and dissipative (Liouvillian) dynamics. We study a cold atom evolution described by a master equation for the many-body density operator $$\begin{aligned}
\partial_{t}\rho&=& -i[H,\rho]+{\cal L}[\rho]~,\\\nonumber
\mathcal{L}[\rho] & = & \frac{1}{2}\kappa\sum_{\langle \ell,\ell' \rangle}\left(2 c_{\ell\ell'}\rho c_{\ell\ell'}^{\dagger}- c_{\ell\ell'}^{\dagger}c_{\ell\ell'}\rho- \rho c_{\ell\ell'}^{\dagger}c_{\ell\ell'}\right)~,
\label{mastereq}\end{aligned}$$ where $c_{\ell\ell'}=(b_{\ell}^{\dagger}+b_{\ell'}^{\dagger})(b_{\ell}-b_{\ell'})$ are Lindblad “jump operators” acting on adjacent sites $\langle\ell,\ell'\rangle$. The energy scale $\kappa$ is the dissipative rate. As shown in [@Diehl08], such dissipative reservoir couplings are obtained in a setup where laser driven atoms are coupled to a phonon bath provided by a second condensate. For no interaction $U=0$ this dissipation drives the system to a dynamical equilibrium independent of the initial state [@Diehl08] given by the [*pure many body state*]{} $\rho_{ss}=|{\rm BEC}\rangle\langle
{\rm BEC}|$ representing a Bose Einstein condensate. From an atomic physics point of view this is remarkable, as typical decoherence mechanisms, such as spontaneous emission acting locally on lattice sites, will destroy long range order, whereas here the bath coupling is engineered to suppress phase fluctuations. This can be easily understood in momentum space, where the annihilation part of $c_{\ell\ell'}$ reads $ \sum_\lambda (1 - \exp(\mathrm i
\textbf{q}_\lambda a))b_\textbf{q}$, with $\lambda$ the reciprocal lattice directions and $a$ the lattice constant. $c_{\ell\ell'}$ thus feature a (unique) dissipative zero mode at ${\bf q} = 0$ – a many-body “dark state” $|\mathrm{BEC}\rangle\sim
b_{\textbf{q}=0}^{\dagger\, N}|{\rm vac}\rangle$ decoupled from the bath, into which the system is consequently driven for long wait times. The dynamics behind Eq. (\[mastereq\]) can thus be understood as a “dark state laser cooling" [@LaserCooling] into a condensate, although in a many-body context.
$|\mathrm{BEC}\rangle$ is also an eigenstate of kinetic energy. In contrast, turning on an interaction measured by $u = U/(4\kappa z)$ provides a Hamiltonian term in (\[mastereq\]) which is incompatible with kinetic energy and dissipation. This competition leads to novel dynamical equilibria which cannot be understood as thermodynamic equilibrium states found from minimizing a free energy. They are summarized in the steady state phase diagram in Fig. \[fig:phasediagram\]. Most prominently, it features a strong coupling phase transition as a function of $u$. A first hallmark of the nonequilibrium nature of the system is this: The transition shares features of a QPT in that it is interaction driven, and of a classical phase transition in that the ordered phase terminates in a mixed state. This contrasts e.g. the well-known dissipation induced phase transition to a superconductor in Josephson junction arrays [@JJarrays], in which detailed balance guarantees that the system’s state remains pure despite the suppression of phase fluctuations via the coupling to a zero temperature bath.
Furthermore, we show the existence of a novel dynamical instability that covers an extensive domain of the phase diagram. Again, this is a nonequilibrium effect, since in equilibrium, finite momentum excitations carry positive kinetic energy ruling out dynamical instabilities. It persists at arbitrarily weak interaction parameters $Un$ due to its fluctuation induced nature elucidated below. This is in marked contrast to the “classical” dynamical instabilities of condensates in boosted lattices [@DynInstabTh; @DynInstabExp] or in exciton-polariton systems [@Carusotto10], which are induced by external tuning of parameters beyond finite critical values.
*Nonlinear mean field master equation*.—To solve the master equation we developed a generalized Gutzwiller approach, expected to hold in sufficiently high spatial dimension, which allows to include density matrices corresponding to mixed states. This is implemented by a product ansatz $\rho = \bigotimes_{\ell}\rho_{\ell}$, with the reduced local density operators $\rho_{\ell} = \mathrm{Tr}_{\ne
\ell}\,\rho$. The equation of motion (EoM) reads $$\label{eq:redmasterequation}
\partial_{t}\rho_{\ell} = -i [h_{\ell},\rho_{\ell}] +{\cal L}_{\ell}[\rho_{\ell}]~,$$ with the local Hamiltonian $h_{\ell} = - J \sum_{\langle \ell' | \ell
\rangle} (\langle b_{\ell'} \rangle b_{\ell}^{\dag} + \langle
b_{\ell'}^{\dag}\rangle b_{\ell} ) -\mu \hat{n}_{\ell} +\frac{1}{2} U
\hat{n}_{\ell}(\hat{n}_{\ell} - 1)$ reproducing the standard form of the Gutzwiller mean field approximation and a Liouvillian of the form $ {\cal L}_{\ell}[\rho_{\ell}] = \kappa \sum_{\langle \ell'| \ell
\rangle} \sum_{r,s=1}^{4} \Gamma_{\ell'}^{rs}[2 A_{\ell}^{r}
\rho_{\ell} A_{\ell}^{s\dag} - A_{\ell}^{s\dag} A_{\ell}^{r}
\rho_{\ell} - \rho_{\ell} A_{\ell}^{s\dag} A_{\ell}^{r}]$. The Liouvillian is constructed with the vector of operators ${\bf A}_\ell
= (1, b_{\ell}^{\dag}, b_{\ell}, \hat{n}_{\ell})$ and the matrix of correlation functions $\Gamma_{\ell}^{r,s} = \sigma^{r} \sigma^{s}
{\rm Tr}_{\ell} A_{\ell}^{(5-s)\dag} A_{\ell}^{(5-r)}\rho_{\ell}$, for $\sigma = (-1,-1,1,1)$. The $\rho$-dependent correlation matrix makes the master equation *nonlinear* in $\rho_\ell$.
![\[fig:phasediagram\] (color online) Nonequilibrium phase diagram for the model in Eq. (\[eq:redmasterequation\]). The solid lines indicate the border of the dynamical quantum phase transition from a condensed to a homogeneous thermal steady state. The dashed lines delimit the region where the condensed state is stable with respect to spatial fluctuations. The black (blue) lines are the numerical results corresponding to average density $n=1.0$ ($n=0.1$). The red line corresponds to the analytical results for $n=0.1$. ](fig1){width="\linewidth"}
*Dynamical quantum phase transition*.—At $U=0$ a steady state solution of Eq. (\[eq:redmasterequation\]) is given by the pure state $\rho^{\rm (c)}_{\ell} = |\Psi\rangle\langle\Psi|$ for any $\ell$ together with the choice $\mu = - Jz$, where $|\Psi\rangle$ is a coherent state of parameter $n e^{i\theta}$ for any phase $\theta$ [@GardinerZoller]. In order to understand the effect of a finite interaction $U$, we apply the rotating-frame transformation $\hat{V}(U) = \exp[i U \hat{n}_{\ell}(\hat{n}_{\ell} -1)t]$ to Eq. . This removes the interaction term from the unitary evolution, but the annihilation operators become $\hat{V} b_{\ell} \hat{V}^{-1} = \sum_{m} \exp(i m U t)
|m\rangle_{\ell}\langle m| b_{\ell}$. The effect of a finite $U$ is thus to rotate the phase of each Fock state differently, leading to dephasing of the coherent state $\rho_{\ell}^{(c)}$. Hence, for strong enough $U$, off-diagonal order is suppressed completely and the density matrix becomes diagonal. In this case Eq. (\[eq:redmasterequation\]) reduces precisely to the master equation for a system of bosons coupled to a thermal reservoir with occupation $n$ [@GardinerZoller], whose solution is a mixed diagonal thermal state $\rho^{\rm (t)}$. Interestingly, this state is thermal-like; however the role of the thermal bath is played by the system itself, being provided by the mean occupation of neighbouring sites.
We substantiate the discussion above with the numerical integration of the EoM (\[eq:redmasterequation\]) for a homogeneous system (we drop the index $\ell$). The system is initially in the coherent state and the condensate fraction $|\psi|^{2}/n$, where $\psi = \langle b
\rangle$, decreases in time depending on the value of the interaction strength $U$. The result is a continuous transition from the coherent state $\rho^{\rm (c)}$ to the thermal state $\rho^{\rm (t)}$, shown in Fig. \[fig:timeconvergence\] for some typical parameters. The boundary between the thermal and the condensed phase with varying $J,n$ is shown in Fig. \[fig:phasediagram\] with solid lines.
The transition is a smooth crossover for any finite time, but for $t\to \infty$ a sharp nonanalytic point indicating a second order phase transition develops. In the universal vicinity of the critical point, $1/\kappa t$ may be viewed as an irrelevant coupling in the sense of the renormalization group. We may use this attractive irrelevant direction to extract the critical exponent $\alpha$ for the order parameter from the scaling solution $|\psi(t)| \propto (\kappa
t)^{-\alpha}$. In the inset of Fig. \[fig:timeconvergence\] we plot $\alpha(t) = d \log (\psi) / d \log (1/t)$ and read off the critical exponent $\alpha = 0.5$ in the scaling regime, which is an expected result given the mean field nature of the Gutzwiller ansatz. We emphasize that following the relaxation dynamics of the condensate fraction for critical system parameters gives an experimental handle for the measurement of $\alpha$.
*Low-density limit*.—In the low density limit $n\ll 1$ we obtain an analytical understanding of the time evolution based on the observation that the six correlation functions $\psi$, $\langle
b_{\ell}^{2} \rangle$, $\langle b_{\ell}^{\dag} b^{2}_\ell \rangle$, and complex conjugates, form a closed (nonlinear) subset which decouples from the *a priori* infinite hierarchy of normal ordered correlation functions $\langle b_{\ell}^{\dag n} b_{\ell}^{m}
\rangle$. We first use this result to obtain analytically the critical exponent $\alpha$ discussed above. For a homogeneous system with $J=0$ the EoMs read $$\begin{aligned}
\label{eq:corrfunccrit}
\partial_{t} \psi & = & i\mu \psi + (-iU + 4\kappa) \langle b^{\dag} b^{2} \rangle - 4 \kappa \psi^{\ast} \langle b^{2} \rangle ~, \nonumber \\
\partial_{t} \langle b^{\dag} b^{2} \rangle & = & 8 n \kappa \psi + (-iU + i \mu - 8 \kappa) \langle b^{\dag} b^{2} \rangle ~,\nonumber \\
\partial_{t} \langle b^{2} \rangle & = & (-iU + 2 i \mu - 8 \kappa) \langle b^{2} \rangle + 8 \kappa \psi^{2}~.\end{aligned}$$ The structure of the equations suggest that $\langle b^{2} \rangle$ decays much faster than the other correlations for $U=U_\text{c}$, so that we may take $\partial_{t} \langle b^{2} \rangle = 0 $ and hence $\langle b^{2} \rangle \propto \psi^{2}$. At the critical point the two linear contributions to $\partial_t \psi$ vanish due to the zero mass eigenvalue at criticality and $\partial_{t} \psi \propto \kappa
\psi^{2}\psi^{\ast}$. It follows that $|\psi| \simeq 1 / (4
\sqrt{\kappa t})$ in agreement with the numerical result in Fig. \[fig:timeconvergence\].
![\[fig:timeconvergence\] Stroboscopic plot of the time evolution of the condensate fraction as a function of the interaction strength $U$, for $J = 1.5\,\kappa$ and $n=1$. For large times it converges to the lower thick solid line. The critical point is $U_{\rm c} \simeq 4.5\, \kappa z$. Inset: Near critical evolution reflected by the logarithmic derivative of the order parameter $\psi(t)$, for $J=0$, $n=1$, and $U\lesssim U_{\rm c}$. The early exponential decay ($\times$) is followed by a scaling regime ($\circ$) with exponent $\alpha \simeq 0.5$. The final exponential runaway ($+$) is due to a small deviation from the critical point. ](fig2){width="\linewidth"}
To study the interaction induced depletion of the condensate fraction, it is convenient to use “connected” correlation functions, built with the fluctuation operator $\delta b = b - \psi_{0}$. Here $\psi_{0}$ is the constant value of the order parameter in the steady state, and $\langle \delta b \rangle = 0$. From (\[eq:corrfunccrit\]) we obtain a closed *linear* system of EoMs, if $\psi_0$ is considered as a parameter, determined self-consistently from the identity $n = \langle \delta b^{\dag}
\delta b \rangle + |\psi_{0}|^{2}$. The value of the chemical potential is fixed to remove the driving terms in the equations for $\langle \delta b \rangle $, leading to $\mu = n U$. This is an equilibrium condition similar to the vanishing of the mass of the Goldstone mode in a thermodynamic equilibrium system with spontaneous symmetry breaking. The solution of the equations in steady state yields the condensate fraction $$\label{eq:Depletion}
\frac{|\psi_0|^{2}}{n} = 1 - \frac{2 u^2 \left(1+(j + u)^2\right)}{1+u^2 + j( 8 u +6 j \left(1+2 u^2\right) +24 j^2 u + 8 j^3)}~,$$ with dimensionless variable $j = J/(4\kappa)$. Eq. reduces to the simple quadratic expression $1
- 2u^2$ in the limit of zero hopping, with the critical point $U_{c}(J=0) = 4 \kappa z/\sqrt{2}$. The phase boundary, obtained by setting $\psi_{0} = 0$ in Eq. , reads $u_c =
j+\sqrt{ 1 / 2 + 2 j^2}$. Fig. \[fig:phasediagram\] shows that these compact analytical results (solid red line) match the full numerics for small densities (solid blue line), and also explain the qualitative features of the phase boundary for large densities. We note the absence of distinct commensurability effects for e.g. $n=1$, tied to the fact that the interaction also plays the role of heating.
*Dynamical instability*.—Numerically integrating the full EoM (\[eq:redmasterequation\]) with site-dependence (in one dimension for simplicity), we observe a dynamical instability, manifesting itself at late times in a long wavelength density wave with growing amplitude. Numerical linearization of Eq. (\[eq:redmasterequation\]) around the homogeneous steady state allows to draw a phase border for the unstable phase (see Fig. \[fig:phasediagram\]). The instability is cured by the increase of hopping $J$, which is associated to an operator compatible with dissipation $\kappa$. Furthermore, we note that the thermal state is always dynamically stable against long wavelength perturbations.
![\[fig:spectrum\] Real (dissipative) part of the spectrum $\gamma_\textbf{q}$ from the analytical low density limit for $J=0$, $n=0.1$, and $U = 1.0\,\kappa$. The inset magnifies the parameter region with unstable modes (red solid line). The black solid line is the bare dissipative spectrum $\kappa_{\textbf{q}}$. ](fig3){width="\linewidth"}
The origin of this instability is intriguing and we discuss it analytically within the low-density limit introduced above. We linearize in time the EoM (\[eq:redmasterequation\]), writing the generic connected correlation function as $\langle \hat{\cal O}_{\ell}
\rangle(t) = \langle \hat{\cal O}_{\ell} \rangle_{0} + \delta \langle
\hat{\cal O}_{\ell} \rangle(t)$, where $\langle \hat{\cal O}_{\ell}
\rangle_{0}$ is evaluated on the homogeneous steady state of the system. The EoM for the time and space dependent fluctuations is then Fourier transformed, resulting in a $7\times7$ matrix evolution equation $\partial_{t} \delta \Phi_{\textbf{q}} = M \delta
\Phi_{\textbf{q}}$ for the correlation functions $\Phi_{\textbf{q}} =
(\langle \delta b\rangle_{\textbf{q}},$$ \langle \delta
b^{\dag}\rangle_{\textbf{q}}, \langle \delta b^{\dag} \delta b
\rangle_{\textbf{q}},$$ \langle \delta b^{2} \rangle_{\textbf{q}},$$
\langle \delta b^{\dag 2} \rangle_{\textbf{q}}, \langle \delta
b^{\dag} \delta b^{2} \rangle_{\textbf{q}},$$ \langle \delta b^{\dag
2} \delta b \rangle_{\textbf{q}} )$. We note that the fluctuation $\delta \langle \delta b \rangle_{\textbf{q}}$ ($\delta \langle \delta
b^{\dag} \rangle_{\textbf{q}}$) coincides with the fluctuation of the order parameter $\delta \psi_{\textbf{q}}$ ($\delta
\psi_{-\textbf{q}}^{\ast}$). The full matrix $M$ can be easily diagonalized numerically revealing the spectrum in Fig. \[fig:spectrum\] (we display only the relevant real part $\gamma$ corresponding to damping). The lowest-lying branch gives $\gamma_\textbf{q}< 0$ in a small interval around the origin $\textbf{q}=0$. This means that the correlation functions grow exponentially $\propto e^{\gamma t}$ in a range of low momenta, resulting e.g. in a long wavelength density wave.
Due to the scale separation for $\textbf{q} \to 0$ in the matrix $M$ apparent from Fig. \[fig:spectrum\], we can apply second order perturbation theory twice in a row to integrate out the fast modes $\gamma \propto \kappa$ and $ \propto \kappa n$. We then obtain an effective low energy EoM for the fluctuations of the order parameter $(\delta \psi_{\textbf{q}} , \delta \psi_{-\textbf{q}}^{\ast})$, governed by a $2\times 2$ matrix $$\begin{aligned}
\label{eq:linear2x2}
M_{\text{eff}} = \left(
\begin{array}{cc}
Un + \epsilon_{\textbf{q}} - i \kappa_{\textbf{q}} & Un + 9 u n \kappa_{\textbf{q}} \\
- Un - 9 u n \kappa_{\textbf{q}} & - Un - \epsilon_{\textbf{q}} - i \kappa_{\textbf{q}}
\end{array}
\right)~,\end{aligned}$$ where $\epsilon_{\textbf{q}}=J\textbf{q}^{2}$ represents the kinetic contribution and $\kappa_{\textbf{q}}=2(2n+1)\kappa \textbf{q}^{2}$ is the bare dissipative spectrum. The form of the EoM reflects the structure of the spatial fluctuations which are included in our approach, that may be understood as scattering off the mean fields in opposite directions. We note that a naive *a priori* restriction to the $2\times2$ set corresponding to the subset $(\delta\psi_{\ell},\delta\psi_\ell^{\ast})$ would be inconsistent, for example destroying the dark state property present in the correct solution $M_{\text{eff}}$. On the other hand, factorizing the correlation functions in the Liouvillian ${\cal L}_{\ell}$ yields a dissipative Gross-Pitaevski equation but its linearization in time produces a matrix $M_{\text{eff}}$ *without* the fluctuation induced term $\sim u$ and fails to describe the dynamical instability. Thus, in order to correctly capture the physics of the instability at long wavelength $\textbf{q}\to 0$, the onsite quantum correlations renormalizing $M_{\text{eff}}$ have to be properly taken into account.
We can make the nature of the instability even more transparent from calculating the lowest eigenvalue of $M_{\text{eff}}$, $\gamma_\textbf{q} \simeq \mathrm i c |\textbf{q}| +
\kappa_{\textbf{q}}$, with speed of sound $c=\sqrt{2 U n[J - 9 U n /
(2z) ]}$. If the hopping amplitude is smaller than the critical value $J_{\rm c} = 9Un / (2z)$ the speed of sound turns imaginary and contributes to the dissipative real part of $\gamma_\textbf{q}$. The nonanalytic renormalization contribution $\sim |\textbf{q}|$ always dominates the bare quadratic piece for low momenta, explaining the shape in the inset of Fig. \[fig:spectrum\] and rendering the system unstable. The linear slope of the stability border for small $J$ and $U$ is clearly visible from the numerical results in Fig. \[fig:phasediagram\]. In summary, the origin of the instability is traced back to a subtle renormalization effect of the speed of sound at low energies, which in turn is due to an interplay of short time quantum and long wavelength classical fluctuations.
*Conclusion*.—We have discussed the steady state phase diagram resulting from a competition of unitary Bose-Hubbard and dissipative dynamics with dark state. The features found in the present model are expected to be generic and representative for a whole class of nonequilibrium models discussed recently in the context of reservoir engineering and dissipative preparation of given long range ordered entangled states of qubits or spins on a lattice [@Verstraete09; @Weimer10] and paired fermions [@Diehl08; @Yi10]. In particular, we emphasize the importance of a compatible energy term for the achievement of stability of driven-dissipative many-body systems in future experiments.
We thank M. Hayn, A. Pelster, S. Kehrein, M. Möckel, and J. V. Porto for interesting discussions. This work was supported by the Austrian Science Foundation through SFB FOQUS, SCALA and by EU Networks.
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---
abstract: 'New types of maximal symplectic partial spreads are constructed.'
address: 'U. of Oregon, Eugene, OR 97403 and Northeastern U., Boston, MA 02115'
author:
- 'W. M. Kantor'
title: On maximal symplectic partial spreads
---
Introduction {#Introduction}
=============
Since very few papers concern maximal symplectic partial spreads in dimension $>4$ [@Grassl], this paper will focus on those dimensions. The largest and most obvious type of maximal partial spread of a $2n$-dimensional symplectic ${\hbox{\Bbb F}}_q$-space is a spread, of size $q^n+1$, which we will not consider here. (However, there are relatively few known types of symplectic spreads, see [@Ka3] for a survey as of 2012.)
On the other hand, when $n$ is even Grassl [@Grassl] initially conjectured that the smallest possible size of a maximal symplectic partial spread is $q^{n/2} +1$, and he provided examples of this size for all even $q$ and $n$. However, when $2n=8$ the conjecture is not correct [@Grassl]. Families of counterexamples using Suzuki-Tits ovoids are in [Section \[Grassl’s Conjecture\]]{}. It still seems plausible that Grassl’s conjecture may be correct if $2n>8$ or if $q$ is odd. Thus far all counterexamples to this conjecture have size greater than $q^{n/2} /2$.
Most of our examples are based on standard properties of orthogonal and symplectic spaces, involving either orthogonal spreads or the standard method for obtaining them (Sections \[Using orthogonal spreads\], \[More maximal symplectic partial spreads\] and \[Projections\]), or partial $\O^+(8,q)$-ovoids and triality (Section \[$8$-dimensional partial spreads\]). Approximately half of this paper is concerned with spaces of dimension 4 or 8, where we can use points as crutches: the Klein correspondence in dimension 4 [@Taylor p. 196] and triality in dimension 8 [@Ti] turn sets of points into sets of subspaces (of dimension 2 or 4). In dimension $>4$ our results are summarized in Table \[Dimension at least 6\]; the pairs of dimensions of the form $4n,$ $4n-2$ arise from orthogonal partial spreads and are explained in [Section \[Projections\]]{}.
Maximal symplectic partial spreads have a straightforward use in Quantum Physics for finding sets of mutually unbiased bases [@MBGW; @Grassl] (e.g., by plugging into [@Ka3 Eqs. (3.2) or (3.4)] in order to obtain sets of complex vectors).
There are tables of computer-generated sizes of maximal symplectic partial spreads in ${\hbox{\Bbb F}}_q^{2n}$ for very small $n$ and $q$ [@CDFS; @Grassl]. A few of these are special cases of constructions given here. However, since these tables contain integer intervals that consist of sizes of these partial spreads, it is clear that new types of construction techniques are needed in all dimensions.
Background {#Background}
==========
The letter $q$ will always denote a prime power, while $n,m,k,s$ and $i$ will be integers.
\[h\]
dimensions parity of $q$ Size Restrictions Theorems
-------------------- --------------- --------------------------- ----------------------- -------------------------------------------------- -- -- -- -- -- -- -- -- -- -- --
$ 4m$ arbitrary $\!q^{2m}-q^m+(2,q-1) \!$ \[using transversals\]
$\! 4mk , 4mk-2\!$ even $ q^{2 m k-k}+1 $ $ m > (k+1)/2 $ \[nk\], \[project k\]
$ 4k$,$ 4k-2$ even $q^k+1\,$ \[Grassl example\], \[project symplectic1\]
$ 4k$ even $2q^k+1$ \[Grassl example\]
even $q^3 -q^2+1 $ $q \ge4$ \[triality to spread\], \[Sp6 corollary\]
even $ n_s$ $1\le s\le q/ 5 $ \[orthovoids\], \[Sp6 corollary\]
even $n_4-1$ $q\ge16 $ \[orthovoids\], \[Sp6 corollary\]
even $q^2+1$ \[partial ovoid from spread\], \[Sp6 corollary\]
even $2q^2+1$ \[2q2+1\], \[Sp6 corollary\]
even $q^2+q+1\ $ $q=2^{2e+1}>2$ \[q2+q+1\], \[Sp6 corollary\]
even $q^2-q+1 $ $q=2^{2e+1}>2$ \[easiest Grassl\], \[Sp6 corollary\]
even $ q^2-sq+2s-1 $ $q=2^{2e+1}>2$ \[smaller Grassl\], \[Sp6 corollary\]
$1 < s\le 2^{e} -1$
$ 6$ arbitrary $q^3-q^{2}+1$ \[group\]
[This corresponds to the excluded possibility $m=1$ in dimensions $4mk,$ $4mk-2$.]{}
[$n_s=q^3-sq^2+(s-1)(q+2)+\binom{s}{2}(q-2)+1 $]{}
See [@Taylor] for the standard properties of the symplectic and orthogonal vector spaces used here. We name geometries using their isometry groups. We will be concerned with singular vectors and totally singular (t.s.) subspaces of orthogonal spaces, and totally isotropic (t.i.) subspaces of symplectic spaces. A subspace of an orthogonal space is *anisotropic* if it contains no nonzero singular vector – and hence has dimension $\le2$. In characteristic 2, an orthogonal vector space is also a symplectic space, and t.s. subspaces are also t.i. subspaces.
The $n$-dimensional t.s. subspaces of an $\O^+(2n,q)$-space are of two types, with two such subspaces of the same type if and only if their intersection has dimension $\equiv n$ (mod 2). Each t.s. $n -1$-space is contained in one member of each type. Since we will be concerned with subspaces intersecting in 0, $n$ will be even.
A triality map for an $\O^+(8,q)$-space [@Ti] permutes the t.s. subspaces, sending singular points to a type of t.s. 4-spaces and non-perpendicular pairs of points to pairs of 4-spaces having zero intersection.
A *partial ovoid* of an orthogonal space is a set $\Omega$ of t.s. points such that each maximal t.s. subspace contains at most one point in the set; $\Omega$ is an *ovoid* if it meets every such subspace. A *partial spread* in a $2n$-dimensional vector space $V$ is a set $\Sigma$ of $n$-spaces any two of which have only 0 in common; $\Sigma$ is a *spread* if every vector is in a member of $\Sigma$. If $V$ is a $2n$-dimensional symplectic or orthogonal vector space, a *symplectic or orthogonal partial spread* $\Sigma$ is a partial spread consisting of t.i. or t.s. $n$-spaces; $\Sigma$ is a *symplectic or orthogonal spread* if every vector or every singular vector is in a member of $\Sigma$. This note concerns *maximal* symplectic or orthogonal partial spreads: maximal with respect to inclusion. In some situations we will even obtain symplectic maximal partial spreads: maximal partial spreads that happen to be symplectic.
Two symplectic partial spreads are *equivalent* if there is a semilinear automorphism of the symplectic geometry sending one partial spread to the other. If $\Sigma $ is a set of subspaces of an $\Sp(2n,q)$-space, then $\Sp(2n,q)_\Sigma$ is its set-stabilizer in the symplectic group $\Sp(2n,q)$. There are similar definitions for orthogonal spaces and for the automorphism group of a symplectic or orthogonal partial spread.
Maximal partial $\Sp(4m,q)$-spreads {#Maximal partial $Sp(4m,q)$-spreads}
===================================
Our most general result is the following
\[using transversals\] For any $q $ and $m\ge1,$ an $\Sp(4m,q)$-space has a maximal symplectic partial spread of size $q^{2m}-q^m+(2,q-1)$.
We begin with notation. Let $F={\hbox{\Bbb F}}_{q^{2 m}}\supset E={\hbox{\Bbb F}}_{q^{m}} \supset K={\hbox{\Bbb F}}_q$, with trace map $T\col F\to K$, so that $T(xy)$ is a nondegenerate symmetric $K$-bilinear form on $F$. By dimensions, $\{x\in F\mid T(x E)=0\} = \theta E$ for some $\theta\in F$.
Equip the $K$-space $V =F^2 $ with the nondegenerate alternating $K$-bilinear form $f\big ((x,y),(x',y') \big):=T(xy')-T(x'y)$.
Let $\Sigma$ be the desarguesian symplectic spread of $V $ consisting of the t.i. $2$-spaces $[x=0]$ and $[y=ax]$ for $a\in F$. Let $Z_\star$ be the t.i. $2m$-space $(E,\theta E) = E\oplus \theta E$ (t.i. since $T(E\theta E)=0$).
Let $\Sigma_\star \subset \Sigma$ consist of the members of $\Sigma$ met nontrivially by $Z_\star$ (namely, the $2m$-spaces $[x=0]$ and $[y=a\theta x]$ for $a\in E).$ We need information concerning some transversals of $\Sigma_\star$:
\[one transversal\] There are exactly $(2,q-1)$ t.i. $2m$-spaces of the $\Sp(4m,q)$-space $V$ that meet each member of $\Sigma_\star$ in an $m$-space. If there are two such subspaces then they intersect in $0$.
If $Z$ is such a subspace let $Z\cap [y=0]= (U,0)$ and $Z\cap [x=0]= (0,W)$ for $m$-dimensional $K$-subspaces $U$ and $W$ of $F$. Since $Z=(U,0) + (0,W)$ is t.i. we have $T(UW)=0$.
Since $Z\cap [y=a\theta x] $ (for $a\in E$) consists of the vectors $(u,a\theta u)$ with $u\in U$, we see that $W=\theta U$ (using $a=1$) and $W$ is closed under multiplication by elements of $E$. Then $W$ is an $E$-subspace of $F$. Let $U=\a E,$ $ \a\in F^*$, so that $W=\theta \a E$. Then $0=T(UW ) = T( \a \theta \a E ) $, so that $\a^2\theta \in \theta E$. Thus, there are $(2,q-1)$ choices for the coset $\a F^*\in F^*/E^*$, and hence also $(2,q-1)$ choices for $Z=(U, W)=(\a E, \theta \a E)$.
This argument reverses: if the coset $\a E^*$ has order at most 2, then $(\a E, \theta \a E)$ is a t.i. $2m$-space that meets each member of $\Sigma_\star$ in an $m$-space.
Finally, if there are two subspaces $( E, \theta E)$ and $ (\a E, \theta \a E)$, then $\a\notin E$ and these have intersection $0$.
Let $\Sigma $ and $\Sigma_\star$ be as above. By the lemma, there are t.i. $2m$-spaces $Z$ (if $q$ is even) or $Z,Z'$ (if $q$ is odd) such that $\Sigma_\star$ is the set of elements of $\Sigma$ met nontrivially by either of these $2m$-spaces. Then $$\Sigma^\bullet:=
\begin{cases}
(\Sigma - \Sigma_\star )\cup \{Z\} &\mbox{if $q$ is even}
\vspace{2pt}
\\
(\Sigma - \Sigma_\star )\cup \{Z,Z'\} &\mbox{if $q$ is odd}
\vspace{-2pt}
\end{cases}$$ is a symplectic partial spread of size $ q^{2m}-q^m+(2,q-1)$.
*Maximality*: Suppose that $X$ is a t.i. $2m$-space meeting each member of $\Sigma^\bullet$ in zero. Since $\Sigma $ is a spread, the set $\Sigma_X$ of members of $\Sigma$ meeting $X$ nontrivially must be contained in $\Sigma_\star$. If $(\star) $$\,\Sigma_X = \Sigma_\star $ *and $\dim X\cap Y=0$ or $m$ for each* $Y \in \Sigma, $ then $X=Z$ or $Z'$ by [Lemma \[one transversal\]]{}, which contradicts the fact that $X\notin \Sigma^\bullet$.
We count in order to prove $(\star) $. Let $a_i$ be the number of $Y\in \Sigma$ such that $\dim X\cap Y=i$, where $1\le i\le 2m-1$. Since the intersections $X\cap Y$ produce a partition of $X-\{0\}$, $$\begin{array}{rll}
\displaystyle \sum_1^{2m-1}a_i(q^i-1) &\hspace{-6pt}=\hspace{-6pt}& q^{2m} -1
\vspace{-4pt}
\\
\displaystyle\sum_1^{2m-1}a_i &\hspace{-6pt}=\hspace{-6pt}& |\Sigma_X| \le |\Sigma_\star|=q^m+1 .
\end{array}
\vspace{-2pt}$$
There cannot be two subspaces of $X$ of dimension $>m$ and $\ge m$ having zero intersection. Thus, if $a_k\ne0$ for some $k>m$ then $a_k=1$ and $a_i=0$ whenever $m\le i\le 2m-1 ,$ $ i\ne k$. This produces the contradiction $q^{2m}-1=(q^k-1)+ \sum_1^{m-1}a_i (q^i-1) \le
\vspace{1pt}
(q^k-1)+\sum_1^{m-1}a_i (q^{m-1}-1)
\le (q^k-1)+ (q^m +1-1) (q^{m-1}-1).
$
Thus, $a_k=0$ for $k>m$, and $q^{2m}-1= \sum_1^{m }a_i (q^i-1) \le
\sum_1^{m }a_i (q^m-1)
\le (q^m+1) (q^{m }-1).$ Then $a_i=0$ for $i<m$ and $a_m=q^m+1$, as required.
When $2m=4$ the theorem is a special case of [@CDFS; @ThK] and [Theorem \[conics\]]{}, which suggests the question: *Can more than one subset like $\Sigma_\star$ be removed in* [Theorem \[using transversals\]]{}?
The last part of the proof showed that *a partition of the nonzero vectors of ${\hbox{\Bbb F}}_q^{2m}$ induced by a set of proper subspaces has size at least $q^m+1,$ with equality if and only if the subspaces all have dimension $m$.*
Orthogonal spreads {#Using orthogonal spreads}
==================
Let $V$ be an $\O^+(4m ,q )$-space (for even $q$ and $4m\ge8$) with quadratic form $Q$. Then $V$ has an orthogonal spread $\Sigma$, and $|\Sigma|=q^{2m-1}+1$ (first proved in [@Dillon], then rediscovered in [@Dye]; cf. [@Ka2; @KaW]). This leads to our simplest examples:
\[orthogonal spread\] $\Sigma$ is a maximal partial spread of size $q^{2m-1} + 1,$ and is symplectic.
For even $q$, t.s. subspaces are also t.i., so $\Sigma $ is symplectic. [*Maximality*]{}: since $2m > 2$, the quadratic form induced by $Q$ on any $2m$-space has a nontrivial zero. Thus, every $2m$-space has nonzero intersection with some member of $\Sigma $. \[1/2\] If $d=2^{2 m}$ and $q=2$ then $| \Sigma|= \frac{1}{2}d +1$. Finding maximal symplectic partial spreads of size $\frac{1}{2}d +1$ appears to be a goal of [@MBGW].
\[nonzero singular\] Let $ E={\hbox{\Bbb F}}_q\subseteq F={\hbox{\Bbb F}}_{q^k}\! $ with $q$ even. If $X$ is an $E$-subspace of an orthogonal $F$-space and $|X| > q^{k^2+k},$ then $X$ contains a nonzero $F$-singular vector.
We are given an $F$-space $V$ equipped with a quadratic form $Q$ and associated bilinear form $f(~,~)$; both forms have values in $F$ not in $E$. The symbol $\perp$ will refer to the $F$-space $V$, while $\<\ ~ \ \>_L$ refers to spanning an $L$-subspace for $L=E$ or $F$.
For $i=1,\dots, k+1,$ we will construct $E$-linearly independent vectors $x_1,\dots,x_{i} \in X$ and an $E$-subspace $X_i$ such that $\<x_1,\dots,x_{i}\>_E\le X_i \le \<x_1,\dots,x_{i}\>_F^\perp\cap X$ and $|X_i|\ge|X|/|F|^{i}$. (In particular, $ x_1,\dots,x_{k+1}\in \<x_1,\dots,x_{k+1}\>_F^\perp\cap X$.)
Let $0\ne x_1\in X$ and $X_1:=\<x_1\>_F^\perp\cap X$. Then $x_1\in X_1$ (since $q$ is even and hence $V$ is symplectic) and $|X_1|=|\<x_1\>_F^\perp|| X|/| {\<x_1\>_F^\perp+X} | \ge
|\<x_1\>_F^\perp|| X|/|V | =| X |/|F|$.
For induction, let $1\le i\le k$ and assume that we have $x_1,\dots,x_{i}$ and $X_i$. Then $|X_i| \! \ge\! |X|/|F|^i\! >\! q^{k^2+k}/(q^k)^k \ge | \<x_1,\dots,x_{i}\>_E|$. Let $x_{i+1}\in X_i- \<x_1,\dots,x_{i}\>_E$ and $X_{i+1}:=\<x_{i+1}\>_F^\perp \cap X_i$. Then $x _{i+1}\in X_{i+1} \le \<x _{i+1}\>_F^\perp\cap \<x_1,\dots,x_{i}\>_F^\perp\cap X$ and $|X_{i+1}| \! =
\! { |\<x_{i+1}\>_F^\perp || X_i|/| \<x_{i+1}\>_F^\perp\! +\! X |}\! \ge \! (|X |/|F|^i)/|F|$, as needed for induction.
Since $\<x_1,\dots,x_{k+1}\> _E $ is in $ X\cap \<x_1,\dots,x_{k+1}\> _E^\perp$ and has size $q^{k+1}>|F| $, the additive map $Q\col
\< x_1,\dots,x_{k+1}\>_E \to F$ has a nonzero kernel. The preceding argument did not require anything about the nature of the quadratic form, which could even have a large radical.
Although the argument used the fact that all vectors are isotropic, it can still be used for unitary spaces and orthogonal spaces of odd characteristic. One minor difference is that we need to know that $X_i$ has an isotropic vector $x_{i+1}\in X_i- \<x_1,\dots,x_{i}\>_E$. This is clear if $X_i$ is the span of its isotropic vectors; and that holds unless $X_i/\operatorname{rad}X_i$ is anisotropic, hence of dimension 1 or (in the orthogonal case) 2. Thus, there is a choice $x_{i+1}$ for each $i$ if we replace the condition $|X| > q^{k^2+k}$ by $|X| > (q^2)^{k^2+k+1}$ for unitary spaces and by $|X| > q^{k^2+k+2}$ for orthogonal spaces.
These observations do not, however, lead to useful unitary or odd characteristic orthogonal analogues of the next theorem: unfortunately, there is no unitary spread in dimension $\ge6$ [@Thas1990] and no known odd characteristic orthogonal spread in dimension $>8$.
\[nk\] If $q$ is even and $ m > (k+1)/2 ,$ then ${\hbox{\Bbb F}}_q^{4 m k}$ has a maximal partial spread of size $ q^{2 m k-k}+1 $ that is orthogonal and hence also symplectic.
Let $V$ be an $\O^+(4 m ,q^k)$-space with quadratic form $Q$, and let $\Sigma$ be an orthogonal spread in $V$. Let $T\col {\hbox{\Bbb F}}_{q^k}\to {\hbox{\Bbb F}}_q$ be the trace map. Then $Q'(v):=T(Q(v))$ is a quadratic form that turns $V$ into an $\O^+(4 m k,q)$-space. Moreover, $\Sigma$ is still an orthogonal partial spread in this space, of size $(q^k)^{2 m-1}+1$.
[*Maximality*]{}: If $X$ is an ${\hbox{\Bbb F}}_q$-subspace of $V$ of dimension $2 m k$, then $|X|=q^{2 m k}
>q^{k^2+k }$. By [Lemma \[nonzero singular\]]{}, $X$ contains a nonzero ${\hbox{\Bbb F}}_{q^k}$-singular vector that must lie in some member of the $\O^+(4 m ,q^k)$-spread $\Sigma$. Thus, $X$ has nonzero intersection with a member of $\Sigma$.
\[conjectures\] Is every $\,\O^+(4 m ,q^k)$-spread also a maximal orthogonal partial spread in an $\O^+(4 mk ,q )$-space? This seems plausible since it is correct when either $ m > (k+1)/2~$ ([Theorem \[nk\]]{}) or $m=2$ [@Grassl] (cf. [Theorem \[Grassl example\]]{}(i)).
If $q=2$ and $d=2^{2 mk}$ with $ m > (k+1)/2 $, then the maximal symplectic partial spreads in [Theorem \[nk\]]{} have size $\frac{1}{2^k}d +1$, resembling [Remark \[1/2\]]{}. Grassl’s computer data [@Grassl] suggests much more: *for even $q$ there appears to be a maximal symplectic partial spread of size $2^i+1$ in $\Sp(2n,q)$-space whenever $q^{n/2}+1\le 2^i+1\le q^{n}+1.$*
We will need the following elementary observation several times: \[orthogonal implies symplectic\] If $\Sigma$ is a maximal orthogonal partial spread of an $\O^+(4m,q)$-space with $q$ even and $m\ge 2,$ then it is also a maximal symplectic partial spread.
Suppose that $Y\notin \Sigma $ is a t.i. $2m$-space such that $\Sigma\cup \{Y\}$ is a symplectic partial spread. The quadratic form on $V$ restricts to a semilinear map on the t.i. subspace $Y$; the kernel is a t.s subspace $Y_0$ of dimension $\ge 2m-1$. If $ \dim \, Y_0 =2m$ then $ Y= Y_0 $ must have the same type as the members of $\Sigma $ (cf. [Section \[Background\]]{}).
In any case let $W$ be the t.s. $2m$-space containing $Y_0$ having the same type as the members of $\Sigma$. By maximality, $\Sigma\cup \{W\}$ is not an orthogonal partial spread, so that $W\cap X\ne 0$ for some $X\in \Sigma$. Since $\dim \, W\cap X \ \equiv 2m$ (mod 2) we have $\dim \, W\cap X \ge2$. Since $ Y_0,$ $ W\cap X \le W$ and $\dim \, Y_0\ge 2m-1$, it follows that $Y_0\cap (W\cap X) \ne0$ and hence that $Y\cap X \ne0. $ This contradicts the fact that $\Sigma\cup \{Y\}$ is a partial spread.
$\O^+(4,q^k)$-space {#More maximal symplectic partial spreads}
===================
\[Folklore\] *If $q$ is even then an $\Sp(4,q)$-space has a maximal symplectic partial spread of size $ q+1 $ that is also a maximal orthogonal partial spread.*
An $\O^+(4,q)$-space has $(q+1)^2$ singular points partitioned by exactly two orthogonal spreads $\Sigma,$ $
\Sigma^\dagger $, arising from the two types of t.s. 2-spaces (cf. [Section \[Background\]]{}); each member of $\Sigma $ and each member of $\Sigma^\dagger $ meet nontrivially. Possibly the most elementary (and most opaque) way to see that these are maximal symplectic spreads is to count the number of t.i. lines containing at least one singular point. There are $2|\Sigma| +(q+1)^2(q-1)
=(q^2+1)(q+1)$ such lines, which is exactly the total number of t.i. lines.
\[Grassl example\] Let $q$ be even and $k\ge1$.
- An $\Sp(4k,q)$-space has a maximal symplectic partial spread of size $q^k+1$ that is also a maximal orthogonal partial spread.
- An $\Sp(4k,q)$-space has a maximal symplectic partial spread of size $2q^k+1$.
\(i) The preceding example produces a maximal symplectic partial spread $\Sigma$ of an $\Sp(4,q^k)$-space $V$ that is also a maximal orthogonal partial spread. Viewed over ${\hbox{\Bbb F}}_{q}$ (using a trace map as in the proof of [Theorem \[nk\]]{}) the set $\Sigma$ again is an orthogonal partial spread. It is a maximal symplectic partial spread by [@Grassl], and hence also a maximal orthogonal partial spread.
We include slightly more detail: in [@Grassl] the ${\hbox{\Bbb F}}_q$-space $({\hbox{\Bbb F}}_{q^k}^2)^2$ is equipped with the alternating bilinear form $\big ((u,v),(u',v') \big) :=T ( u\cdot v'-u'\cdot v)$ using the trace map $T\col {\hbox{\Bbb F}}_{q^k}\to {\hbox{\Bbb F}}_q $. The partial spread $\Sigma$ consists of the t.i. subspaces $\{(0,0,y_1,y_2)\mid y_1,y_2\in {\hbox{\Bbb F}}_{q^k}\}$ and $\{(x_1,x_2, x_2\a,x_1\a)\mid x_1,x_2\in {\hbox{\Bbb F}}_{q^k}\}$ for each $\a\in {\hbox{\Bbb F}}_{q^k}$. These are t.s. $2k$-spaces for the quadratic form $Q(u,v)=T(u\cdot v)$. In the preceding example, $\Sigma^\dagger $ is $ \Sigma^j$, where $j\col (x_1,x_2, y_1,y_2 )\mapsto
(x_1,y_1, x_2,y_2)$.
\(ii) Choose any $Z\in \Sigma$. Obtain a new symplectic partial spread $ \Sigma ^\bullet$ by removing $Z$ and then, for each 1-dimensional ${\hbox{\Bbb F}}_{q^k}$-subspace $W$ of $Z$, adjoining one 2-dimensional t.i. ${\hbox{\Bbb F}}_{q^k}$-subspace that contains $W$ and is different from both $Z$ and the member of $\Sigma^\dagger$ containing $W$. This produces a maximal symplectic partial spread of the ${\hbox{\Bbb F}}_{q^k}$-space $V$ .
In fact [*$ \Sigma ^\bullet$ is also a maximal symplectic partial spread of the ${\hbox{\Bbb F}}_{q}$-space $V$.*]{} For, let $X$ be a t.i. $2k$-dimensional ${\hbox{\Bbb F}}_{q}$-subspace of $V$ having zero intersection with all members of $\Sigma^\bullet$. By (i), $X$ has nonzero intersection with some member of $\Sigma$, which therefore must be $Z$. Then $X$ has nonzero intersection with some ${\hbox{\Bbb F}}_{q^k}$-point $W$ of $Z$ and hence with the adjoined ${\hbox{\Bbb F}}_{q^k}$-space in $\Sigma ^\bullet$ containing $W$, which is a contradiction.
Part (i) contains [Theorem \[partial ovoid from spread\]]{}(i) as a special case, and amounts to the case $m=1$ not dealt with in [Theorem \[nk\]]{}.
The proof of (i) in [@Grassl] uses a neat computational idea. Unsupported optimism suggests that there should also be a nice geometric proof.
\[intersect in spread\] [Proposition \[Grassl example\]]{}(i) points to a general construction (compare Remarks \[use ovoid\]). Let $V={\hbox{\Bbb F}}_q^{4m}$ be an orthogonal, symplectic or unitary space. Let $X$ and $Y$ be t.i./t.s. $2m$-spaces with zero intersection, and let $\Sigma_X$ be a partial spread (of $m$-spaces) of $X$. Each $A\in \Sigma_X$ determines another $m$-space $A':=A^\perp \cap Y$, and $A+A'$ is a t.i./t.s. $2m$-space. Then [*$\Sigma:=\{A+ A'\mid A\in \Sigma_X\}$ is a partial spread of the same type as the underlying space $V$*]{}. (If $A\ne B\in \Sigma_X$ then $V=X\oplus Y = (A\oplus B)\oplus ( A'\oplus B')$, so that $A\oplus A'$ and $B\oplus B'$ have zero intersection.)
When $\Sigma_X$ is a maximal partial spread (or even a spread), some of these partial spreads may be maximal orthogonal, symplectic or unitary partial spreads of size $q^{m}+1$ (as in [Proposition \[Grassl example\]]{}(i) and [Theorem \[partial ovoid from spread\]]{}), but we do not see how to prove that. (See Question \[spanning ovoid\] for instances of such symplectic partial spreads that are [*not*]{} maximal. As noted earlier, there is no unitary spread in dimension $\ge6$ [@Thas1990].)
Projections {#Projections}
===========
Let $q$ be even. A key ingredient of [@Ka2; @Ka3; @KaW] is the fact that there is a natural transition between $\O^+(4m,q)$-spreads and $\Sp(4m-2,q)$-spreads. This uses any nonsingular point $z$ of an $\O^+(4m,q)$-space and projects into the symplectic space $z^\perp/z$. This procedure also applies to orthogonal and symplectic partial spreads:
\[orthogonal to symplectic\] Let $z$ be a nonsingular point of an $\O^+(4m,q)$-space $V$.
- If $\Sigma $ is a maximal orthogonal partial spread of $V, $ then $\Sigma/z:=\break
\{\<z^\perp\cap X,z\>/z \mid X\in \Sigma \}$ is a maximal symplectic partial spread of the $\Sp(4m-2,q)$-space $z^\perp/z$.
- If $\Sigma'$ is a maximal symplectic partial spread of $z^\perp/z,$ then there is a maximal orthogonal partial spread $\Sigma$ of $\,V\!$ such that $\Sigma'=\Sigma/z$. Moreover$,$ $\Sigma$ is a maximal symplectic partial spread.
- If $\Sigma_1 $ is a maximal orthogonal partial spread of $V$ and $z_1$ is a nonsingular point of $V,$ then $\Sigma/z$ and $\Sigma_1/z_1$ are equivalent symplectic partial spreads if and only if $\Sigma_1 $ is the image of $\Sigma $ under an automorphism of the orthogonal geometry of $V$ that sends $z$ to $z_1$.
\(i) *$\Sigma/z$ is a symplectic partial spread*: If $X$ and $Y$ are distinct members of $\Sigma$ and $\<z^\perp\cap X,z\>\cap \<z^\perp\cap Y ,z\>\ne z$, then $z\in \<x,y\>$ for some points $x\in z^\perp\cap X,$ $ y\in z^\perp\cap Y$. Then $x$ and $y$ are perpendicular to $z$ and hence to one another, so that $\<x,y\>$ is t.s. whereas $z$ is nonsingular.
*Maximality*: Suppose that $(\Sigma/z ) \cup \{U/z\}$ is a larger symplectic partial spread for a t.i. $2m$-space $U$ of $V$ containing $z$. Let $U'$ be the hyperplane of $U$ consisting of singular vectors (i.e., $U'$ is the kernel of the semilinear map $U\to {\hbox{\Bbb F}}_q$ induced by the quadratic form on $V$). The members of $\Sigma$ all have the same type (cf. [Section \[Background\]]{}). Let $\hat U$ be the t.s. $2m$-space of that type containing $U'$. Then $\hat U$ meets each $X\in \Sigma$ in at most a 1-space and hence only in 0 (by [Section \[Background\]]{}, $1\ge \dim\, ( \hat U \cap X )\ \equiv 2m$ (mod 2) and hence $\hat U\cap X=0$). Thus, $\Sigma\cup \{\hat U\}$ is an orthogonal partial spread properly containing $\Sigma$, whereas $\Sigma$ is assumed to be maximal.
\(ii) Choose a type of t.s. $2m$-space of $V$. If $U/z\in \Sigma'$ let $U'$ be the hyperplane of singular vectors of the t.i. $2m$-space $U$, and let $ \hat U$ be the t.s. $2m$-space containing $U'$ of the chosen type. Then the set $\Sigma$ consisting of these subspaces $ \hat U$ is an orthogonal partial spread: distinct members of $\Sigma $ meet in at most a 1-space and hence have intersection 0 since members of $\Sigma $ have the same type. Clearly $\Sigma'=\Sigma/z$.
*Maximality*: If $\Sigma^+$ is an orthogonal partial spread properly containing $\Sigma$, then $\Sigma^+/z$ properly contains $\Sigma/z \! = \!\Sigma',$ whereas $\Sigma'$ is maximal.
The final statement follows from [Lemma \[orthogonal implies symplectic\]]{}.
\(iii) As a consequence of Witt’s Lemma [@Taylor p. 57], an equivalence from $\Sigma /z $ to $\Sigma_1 /z _1$ lifts first to $z^\perp\to z_1 ^\perp$ and then to an automorphism of the orthogonal geometry on $V$ sending $z\to z_1$ and $\Sigma \to \Sigma_1$. The converse is clear.
By (iii), a maximal orthogonal partial spread $\Sigma$ produces many inequivalent maximal symplectic partial spreads for different choices of $z$, where the number of inequivalent ones requires knowledge of the automorphism group of $\Sigma$. This was crucial in [@Ka2; @Ka3; @KaW].
\[project symplectic1\] If $k \ge2$ then there is a maximal partial $\Sp(4k-2,q)$-spread of size $q^{k}+1$.
Use [Lemma \[orthogonal to symplectic\]]{}(i) and [Theorem \[Grassl example\]]{}(i).
\[project k\] If $ m > (k+1)/2 $ then there is a maximal partial $\Sp(4mk-2,q)$-spread of size $q^{2mk-k}+1$.
Use [Lemma \[orthogonal to symplectic\]]{}(i) and [Theorem \[nk\]]{}.
$8$-dimensional partial spreads {#$8$-dimensional partial spreads}
===============================
In $\O^+(8,q)$-spaces, triality [@Ti] allows us to use more easily visualized points and partial ovoids in place of partial spreads: a triality map sends orthogonal (partial) ovoids to orthogonal (partial) spreads. This produces maximal partial $\Sp(8,q)$-spreads when $q$ is even.
$8$-dimensional ovoids
----------------------
Spreads and ovoids are known in $\O^+(8,q)$-spaces when $q$ is prime, a power of 2 or 3, or $\equiv 2$ (mod 3) (some of these ovoids are described in [@Ka1]). They have size $q^3+1$.
\[use orthogonal ovoid\] Let $\Omega$ be an ovoid in an $\O^+(8,q)$-space $V,$ where $q>2$. Let $a\notin \Omega$ be a singular point that is the only singular point in $ \<a^\perp\cap\Omega\> ^\perp$. [(Examples appear below in Appendix \[appendix A\] for all even $q>2$.)]{} Then $\Omega^\bullet:= \big ( \Omega - (a^\perp\cap \Omega) \big ) \cup \{a\} $ is a maximal orthogonal partial ovoid of size $q^3 -q^2+1 $.
Clearly $\Omega ^\bullet$ is an orthogonal partial ovoid. If $b$ is a singular point not perpendicular to any member of $\Omega^\bullet$ then $b^\perp\cap \Omega\subseteq a^\perp\cap \Omega$. Since both of these sets have size $q^2+1$ (e.g., by [@Ka1 p. 1196]), we obtain the contradiction that both $a$ and $b$ are the singular point in $ \<a^\perp\cap\Omega\>^\perp$.
\[triality to spread\] Applying triality $\tau $ to the preceding lemma produces a maximal orthogonal partial spread $\Sigma^\bullet := \Omega^\bullet{}^\tau$ of size $q^3 -q^2+1 $ in an $\O^+(8,q)$-space when $q>2$. If $q$ is even then $\Sigma^\bullet $ is a maximal symplectic partial spread.
In particular$,$ such maximal symplectic partial spreads exist for all even $q$.
By the previous lemma, $\Sigma ^\bullet$ is a maximal orthogonal partial spread. If $q$ is even use [Lemma \[orthogonal implies symplectic\]]{}.
We can imitate the preceding result and remove several sets $a^\perp\cap \Omega$ by using a specific type of ovoid. \[orthovoids\] If $q>2$ is even and $1\le s \le q/ 5,$ then an $\O^+(8,q)$-space has maximal orthogonal partial spreads of size $\, n_s=q^3-s q^2+(s-1)(q+2)+\binom{s}{2}(q-2)+1 .
$ There is also a maximal orthogonal partial spread of size $n_4-1$ if $q\ge 16$.
These are also maximal symplectic partial spreads.
As in [Theorem \[triality to spread\]]{} we will construct maximal orthogonal ovoids. Since this the only part of this paper involving detailed computations, those computations have been postponed to Appendix \[appendix A\].
For the ovoid $\Omega$ in Appendix \[appendix A\], [Example \[examples\]]{}(i) provides us with many sets ${\mathcal P}$ of $s$ singular points disjoint from $\Omega$ together with the sizes $ |\bigcap_{p\in {\mathcal P} ' } p ^\perp \cap \Omega | $ for all ${\mathcal P} '\subseteq {\mathcal P}$. Then $$\Omega_s^\bullet:=
\big (\Omega-\bigcup _{p\in {\mathcal P}}(p^\perp \cap \Omega)\big )\cup {\mathcal P}$$ is an orthogonal partial ovoid of size $(q^3+1)-s(q^2+1)+\binom{s}{2}(2q) -
\vspace{1pt}
{\binom{s}{3}(q+2)} + \cdots
\pm \binom{s}{s}(q+2) +s =
\vspace{2pt}
(q^3+1)-s(q^2+1)+\binom{s}{2}(2q) -(q+2)
+ s(q+2) -\binom{s}{2}(q+2) +(1- 1)^s(q+2)+s.
$
[*Maximality of $\Omega_s^\bullet$*]{}: Suppose that $b$ is a singular point not perpendicular to every member of $\Omega_{s}^\bullet$. Since $\Omega$ is an orthogonal ovoid, $b^\perp \cap \Omega$ must be contained in $\bigcup_{p\in {\mathcal P}}(p^\perp \cap \Omega ) $. By [Lemma \[non-perp\]]{}, $s (5q- 5)\ge \sum _{p\in {\mathcal P}} |b^\perp\cap p ^\perp \cap \Omega |
\ge |b^\perp \cap \Omega|=q^2+1$, which contradicts our assumption that $ s \le q/5$.
The same argument can be used for [Example \[examples\]]{}(ii), producing the stated additional maximal orthogonal partial spreads.
Use [Lemma \[orthogonal implies symplectic\]]{} for the final assertion.
The preceding proof should be compared to the proofs of Theorem \[smaller Grassl\] and the more elementary Theorem \[conics\]. In those proofs the needed intersection sizes are known for simple geometric reasons. Here there does not seem to be a geometric explanation for the various intersection sizes occurring in Appendix \[appendix A\].
$4$- and $5$-dimensional orthogonal ovoids
------------------------------------------
The next 8-dimensional partial spreads (in [Theorem \[partial ovoid from spread\]]{}) arise from small-dimensional ovoids.
\[elliptic\] If $\Omega$ is an $\O^-(4,q)$-ovoid [(i.e., an elliptic quadric)]{} in an $\O^-(4,q)$-space $W$ inside a nondegenerate orthogonal ${\hbox{\Bbb F}}_q$-space $V,$ then $\Omega$ is a maximal orthogonal partial ovoid of $V$.
If $x$ is any point of $V$ then $x^\perp\cap W$ contains a hyperplane of $W$ and hence contains either $p^\perp\cap W$ for a singular point $p$ of $W$ or $n^\perp\cap W$ a nonsingular point $n$ of $W$. Each such hyperplane of $W$ contains a singular point of $W$, and hence meets $\Omega$ nontrivially.
A more general version of this example is a simple consequence of 5-dimensional results [@Bagchi-Sastry; @Ball] (also see [Lemma \[hyperplane intersections\]]{}):
\[partial ovoid from ovoid\] If $\Omega$ is an ovoid in an $\O(5,q)$-subspace of a nondegenerate orthogonal ${\hbox{\Bbb F}}_q$-space $V,$ then it is a maximal orthogonal partial ovoid of $V$.
Once again we will show that [*each point $x$ of $V$ is perpendicular to some point in $\Omega$.*]{} We may assume that $U:=\<\Omega\>\not\leq x^\perp$, so that $H:=x^\perp \cap U$ is a hyperplane of $U$. By the preceding example, we may also assume that $U$ is not of type $\O^-(4,q)$.
If $H$ has type $\O^+(4,q)$ then $H$ contains a t.s. line, and each t.s. line of $U$ meets each ovoid of $U$ (by definition; see [Section \[Background\]]{}).
If $H$ has type $\O^-(4,q)$ then its set $\Lambda$ of singular points is a classical quadric. Then $ \Lambda \cap \Omega \ne \emptyset$ [@Bagchi-Sastry; @Ball].
Thus, $H$ is degenerate. If there is a singular point $y$ in its radical $\operatorname{rad}\, H$, then every t.s. line of $U$ on $y$ meets $\Omega$ at a point perpendicular to $y$.
Finally, if $\operatorname{rad}\, H $ is a nonsingular point then $q$ is even and the radical $r$ of $U$ is in $H$ (since all hyperplanes of $U$ not containing its radical are nonsingular). Let “bar” denote the projection map $U\to U / r$. Then $\overline H $ is a tangent or secant plane of the ovoid $\overline \Omega$ in the 4-space $\overline U$, so that $\overline H$ contains 1 or $q+1$ points of $\overline\Omega$. If $T/r$ is one of these points, then the line $T$ has a unique singular point, and this lies in both $H\le x^\perp$ and $\Omega$. *\[spanning ovoid\] Which ovoids in orthogonal spaces are partial ovoids in all larger-dimensional orthogonal spaces over the same field$\,?$ This requires that all hyperplanes of the smaller orthogonal space meet the ovoid. *Perhaps* this does not hold for any ovoids of $\O^+(6,q)$-spaces that span the underlying 6-space (and there are, indeed, many such ovoids for which this requirement does not hold). However, this requirement does hold for some of the known $\O^+(8,q)$ ovoids, as in [@Coop Theorem 3.9] and for the ovoids in [@Ka1 Sec. 7] and Appendix \[appendix A\].*
\[partial ovoid from spread\] Let $q$ be a prime power.
- There are [inequivalent]{} maximal partial $\O^+(8,q)$-spreads $\Sigma$ of size $ q^{2 }+1 $$:$
- One for which $\Sp(8,q)_\Sigma$ has a subgroup $\SL(2,q^2) $ acting $2$-transitively on $\Sigma;$ and
- One occurring when $q$ is odd but not prime and for which $\Sp(8,q)_\Sigma$ is intransitive on $\Sigma.$
- If $q=2^e$ then there are [inequivalent]{} maximal partial $\Sp(8,q)$-spreads $\Sigma$ of size $ q^{2 }+1 $ that are orthogonal partial spreads$:$
- One for which $\Sp(8,q)_\Sigma$ has a subgroup $\SL(2,q^2) $ acting $2$-transitively on $\Sigma;$ and
- One occurring when $e>1$ is odd and for which $\Sp(8,q)_\Sigma$ has a subgroup $\Sz(q ) $ acting $2$-transitively on $\Sigma.$
Let $\tau $ be a triality map for an $\O^+(8,q)$-space $V$. For $\Omega$ in the preceding example or lemma, $\Sigma =\Omega^\tau $ is a maximal orthogonal partial spread of $V$.
For (ia) use an elliptic quadric, whose group of isometries produces the last part. For (ib) there are other choices for $\Omega$ in [Lemma \[partial ovoid from ovoid\]]{}, such as those in [@Ka1 Sec. 5].
If $q$ is even then [Lemma \[partial ovoid from ovoid\]]{} applies, where the only known choices for $\Omega$ are an elliptic quadric ([Example \[elliptic\]]{}) and a Suzuki-Tits ovoid (see Appendix \[Background on Suzuki-Tits ovoids\]). The stated groups arise from subgroups of $\Omega^+(8,q)$ acting on $\Omega $.
The various partial spreads are inequivalent as orthogonal partial spreads, since the corresponding maximal orthogonal partial ovoids $\Sigma ^{\tau^{-1}}=\Omega $ are inequivalent. However, this is orthogonal inequivalence, which is not the same as symplectic inequivalence in (ii).
Nevertheless, the symplectic partial spreads in (iia) and (iib) are inequivalent. This can be proved using the groups appearing in (iia) and (iib), but a geometric proof is simpler. The construction of the orthogonal partial spread in (iia) provides us with $q^2+1$ t.i. 4-spaces (in fact, t.s. 4-spaces in $\Omega^\perp{}^\tau$) meeting each of its members. If the symplectic partial spreads are equivalent then (iib) has $q^2 +1$ t.i. 4-spaces $U$ meeting each of its members. Then $U$ is t.s. since it is spanned by singular vectors. Now $(\Sigma\cup \{U\})^{\tau^{-1}} =\Omega\cup \{U^{\tau^{-1}}\} $ for a singular point $ U ^{\tau^{-1}}$ of $ \Omega ^\perp$. This contradicts the fact that, for (iib), the 3-space $ \Omega ^\perp$ contains only $q+1$ singular points.
\[use ovoid\] We excluded $q=2 $ in (iib) since that produces the same partial spread as in (iia). Part (iia) is a very special case of a result in [@Grassl] (cf. [Theorem \[Grassl example\]]{}). Is there an analogous generalization of (ii)?
Note that $\Sp(8,q)_\Sigma$ even contains subgroups $\SL(2,q^2) \times \SL(2,q^2) $ in (i) and $\Sz(q) \times \O(3,q) \cong
\Sz(q) \times \SL(2,q) $ in (iib).
In both (i) and (ii) there are t.s. 4-spaces $X,Y$ such that the members of $ \Sigma$ meet $X$ and $Y$ in spreads of each (cf. [Example \[intersect in spread\]]{}).
See [@PW] for a survey of $\O(5,q)$-ovoids.
Extending a partial spread {#Extending a partial spread}
--------------------------
*How can one search for maximal symplectic partial spreads*? One obvious answer is to start with a symplectic or orthogonal partial spread and try to extend it to a maximal one (this was the computational method used to produce the table in [@Grassl]). The instances considered below may have extensions to maximal ones other than the ones we provide.
Once again, points are easier to deal with than subspaces.
### $\O^-(4,q)$-ovoids {#$O-(4,q)$-ovoids}
A simple example of an orthogonal partial ovoid is $(\Omega -\{p\})\cup\{x\}$, where $p$ is a point in the set $\Omega$ of singular points of an $\O^-(4,q)$-space $U$ and $x\notin U$ is a singular point in $ (p^\perp\cap U)^\perp -U^\perp$.
\[extend to maximal partial ovoid\] For any $q$ an $\O^+(8,q)$-space has a maximal orthogonal partial ovoid of size $2q^2+1$.
In an $\O^+(8,q)$-space $V$ consider anisotropic 2-spaces $A,A'$ and a totally singular 2-space $\<p,p'\>$ such that $\<A,A',p,p'\>= A\perp A' \perp \<p,p'\>$. Let $E= \<A, p \> $ and $ E '= \<A', p' \>, $ and let $ x$ be a point of $ \<p,p'\>-\{p,p'\}$.
Let $U $ and $U' $ be non-perpendicular $\O^-(4,q)$-subspaces of $V$ such that $E'^ \perp > U> E $ and $E^ \perp > U'> E'$. (In order to construct these, note that $p$ and $p'$ are in t.s. lines $\ne \<p,p'\> $ of the $\O^+(4,q)$-space $ ( A\perp A')^\perp$. Choose singular points $u,u' \in ( A\perp A')^\perp - \<p,p'\>$ perpendicular to $p'$ and $p$, respectively, but not to each other. Then $U:= A\perp \<p,u\>=\<E ,u\>$ and $U':= A'\perp \<p',u'\>=\<E' ,u' \>$ are non-perpendicular $\O^-(4,q)$-subspaces such that $U=\<A,p,u \><\<A' ,p ' \>^\perp=E'{} ^\perp$ and $U' < E ^\perp$ behave as required.)
If $\Omega$ and $\Omega'$ are the sets of singular points of $U$ and $U '$, respectively, we claim that $$\Omega^\bullet:= ( \Omega -\{p\} )\cup
( \Omega' -\{p' \} )\cup \{x\}$$ behaves as stated in the lemma. Clearly, $| \Omega^\bullet|=q^2+q^2+1$.
*Orthogonal partial ovoid*: $x^\perp \cap U=p^\perp \cap U=E$ has only one singular point $p$, and $p\notin
\Omega^\bullet$. Suppose that there are perpendicular singular points $y\in \Omega-E $ and $y'\in \Omega ' -E ' $. Since $y\in U < E'^\perp $ and $y' <E^\perp $, while $E$ and $E'$ are perpendicular, we obtain to the contradiction that $\<y,E\>=U$ and $\<y',E'\>=U'$ are perpendicular.
*Maximality*: Suppose that $h$ is a singular point such that $ h^\perp \cap \Omega^\bullet =\emptyset$. Then $h^\perp\cap U$ is a hyperplane of $U$ and hence contains a singular point, which must be $p$. Then $h^\perp\cap U = p^\perp\cap U =E $. Also $h^\perp\cap U '= E'$. Now $h\in \<E,E'\>^\perp = \<p,p'\>$, which contradicts the assumption that $h$ is not perpendicular to $x\in \Omega^\bullet$.
\[2q2+1\] For any $q$ an $\O^+(8,q)$-space has a maximal orthogonal partial spread $\Sigma$ of size $2q^2+1$. If $q$ is even then $\Sigma$ is symplectic.
Applying triality to the lemma proves the first part, while [Lemma \[orthogonal implies symplectic\]]{} implies the second part.
When $q$ is even, [Theorem \[Grassl example\]]{}(ii) contains another maximal symplectic partial spread of size $2q^2+1$ that need not be orthogonal.
Note that these examples, and others earlier in this section, would have been awkward to describe using t.s. 4-spaces instead of points.
### Suzuki-Tits ovoids {#Using a Suzuki-Tits ovoid}
Another example of an orthogonal partial ovoid is $(\Omega -\{p\})\cup\{x \}$, where $p$ is a point of a Suzuki-Tits ovoid $\Omega$ in an $\O(5,q)$-space $U$ and $x \notin U$ is a singular point in $ (p^\perp\cap U)^\perp -U^\perp$ (see Appendix \[Background on Suzuki-Tits ovoids\]). This time it is easier to extend this to a maximal orthogonal partial ovoid of an $O^+(8,q)$-space. In the next section we will see further advantages of $\Omega$ over an elliptic quadric.
\[q2+q+1\] If $q=2^{2e+1}>2$ then an $\O^+(8,q)$-space has a maximal orthogonal partial spread $\Sigma$ of size $q^2+q+1$ that is symplectic.
By triality and [Lemma \[orthogonal implies symplectic\]]{}, we need to construct a maximal orthogonal partial ovoid of the stated size in an $\O^+(8,q)$-space $V$ containing $U$. The radical $r$ of $U$ is also the radical of the 3-space $U^\perp $, and $(p^\perp\cap U)^\perp = \<p,U^\perp \> = p\perp U^\perp $ for $p\in \Omega$. Each singular point in the 4-space $(p^\perp\cap U)^\perp$ lies on a t.s. line containing $p$ and meeting $U^\perp $ in one of its $q+1$ singular points.
For each singular point $x _0$ in $ U^\perp $ let $x$ be any point in $ \<p,x_0\> - \{ p,x_0\}$. Let $X$ be the resulting set of $q+1$ points $x$. We claim that $$\Omega^\bullet:= ( \Omega -\{p\} )\cup X\vspace{-4pt}$$ behaves as required. Clearly, $| \Omega^\bullet|=q^2+q+1$.
*Orthogonal partial ovoid*: $x^\perp \cap U=p^\perp \cap U $ since $x_0^\perp\ge U$. Then $x^\perp \cap\Omega=\{p\} .$ No two members of $X$ are perpendicular since no two singular points in $U^\perp$ are.
*Maximality*: Suppose that $h$ is a singular point such that $ h^\perp \cap \Omega^\bullet =\emptyset$. Then $h^\perp\cap U$ is a hyperplane of $U$ that cannot contain a point of $\Omega -\{p\}$. By [Lemma \[hyperplane intersections\]]{}, $h^\perp\ge h^\perp\cap U = p^\perp\cap U$. Then $h\in (p^\perp\cap U) ^\perp = \<p,U^\perp \>$, so that $h$ lies on one of the above lines $\<p,x_0\>$, whereas $ h^\perp \cap X =\emptyset$.
Instead of using a single pair $(p,x)$ for replacement what happens if several such pairs are used*?* Can [[Section \[$O-(4,q)$-ovoids\]]{}]{} be handled better than at present in order to use several replacement pairs*?*
Small maximal partial spreads {#Grassl's Conjecture}
------------------------------
We will describe counterexamples to Grassl’s conjecture, which was stated in the Introduction. Grassl [@Grassl] has also found counterexamples to his conjecture in an $\Sp(8,8)$-space by a computer search.
\[easiest Grassl\] If $q=2^{2e+1}>2$ then there is a maximal partial $\O^+(8,q)$-spread of size $q^2-q+1;$ this is also a maximal partial $\Sp(8,q)$-spread.
In view of triality and [Lemma \[orthogonal implies symplectic\]]{}, it suffices to construct a maximal partial $\O^+(8,q)$-ovoid of size $q^2-q+1.$ We use the notation in [Section \[Using a Suzuki-Tits ovoid\]]{} and Appendix \[Background on Suzuki-Tits ovoids\].
Let $\Omega$ be a Suzuki-Tits ovoid in an $\O(5,q)$-space $U$. Embed $U$ into an $\O^+(8,q)$-space $V$.
Let $\Omega^\bullet:=
{\big (\Omega-(x^\perp \cap \Omega) \big )\cup\{x\}}$ for a singular point $x$ of $U$ not in $\Omega$ (this uses $\dim \,U>4$). Then $|\Omega^\bullet|=q^2-q+1$ and $ \Omega^\bullet $ is an orthogonal partial ovoid of $U$ and hence of $V$.
*Maximality*: Suppose that $h$ is a singular point of $V$ such that $h^\perp\cap\Omega^\bullet =\emptyset$. We will consider the possibilities for the hyperplane $h^\perp \cap U$ of $U$ in [Lemma \[hyperplane intersections\]]{}. We have $ h^\perp \cap\Omega
\subseteq x^\perp \cap \Omega $ since $h^\perp\cap\Omega^\bullet =\emptyset$. Also, $\Omega^\perp =U^\perp < x^\perp$ since $x \in U=\< \Omega\>$.
Case 1. $h^\perp \cap \Omega =\{p\}$ for some $p\in x^\perp \cap \Omega.$ Then $ h^\perp \cap U = p^\perp \cap U $ since [Lemma \[hyperplane intersections\]]{} implies that $p^\perp \cap U $ is the only hyperplane of $U$ meeting $\Omega$ just in $p$. Then $h^\perp\ge h^\perp \cap U = p^\perp \cap U $, so that $h\in \<p , U^\perp\> \le x^\perp$, whereas $h$ is assumed not to be perpendicular to $x\in \Omega^\bullet$.
Case 2. $|h^\perp \cap \Omega|=q+1.$ Since $h^\perp \cap\Omega \subseteq x^\perp \cap \Omega $ for sets of size $q+1$, we have $h^\perp\ge \<h^\perp \cap\Omega\>=
\<x^\perp \cap \Omega\> $, where $\<x^\perp \cap \Omega\>=
x^\perp \cap U $ by the end of [Lemma \[hyperplane intersections\]]{}(ii). Then $h\in \<x , U^\perp\> \le x^\perp$, which produces the same contradiction as before. (This is where an elliptic quadric $\Omega$ would not suffice: we would only have $\<x^\perp \cap \Omega\> <x^\perp \cap U $ since $\dim \, U=5$.)
Case 3. $1< |h^\perp \cap \Omega | <q+1$. Since $ h^\perp \cap \Omega$ lies in a set $x^\perp \cap \Omega $ that projects into a plane of $U/ r$, this contradicts the irreducibility in [Lemma \[hyperplane intersections\]]{}(iii).
We can go further (mimicking the proofs of Theorems \[orthovoids\] and \[conics\]):
\[smaller Grassl\] An $\O^+(8,q)$-space has a maximal orthogonal partial spread of size $q^2-sq+2s-1$ whenever $q=2^{2e+1}>2$ and $1 < s\le \sqrt{q/2} -1 $. Each of these is a maximal partial symplectic spread.
In particular$,$ there is a maximal partial $\Sp(8,q)$-spread of size $q^2-\sqrt{q^3/2} +q+\sqrt{2q}-3.$
Once again we will construct maximal partial $\O^+(8,q)$-ovoids. Let $\Omega$, $U$, $r$ and $V$ be as before. Choose distinct $a,b \in \Omega$. Then $\{a,b\}^\perp \cap U$ is a nondegenerate plane containing $r$. There are $q+1$ singular points $x\in \{a,b\}^\perp $. These produce $q+1$ subspaces $\<x,a,b,r\> =x^\perp \cap U$ that induce a partition of $\Omega-\{a,b\}$ using the $q+1$ [circles]{} $\Omega_x:=x^\perp \cap \Omega$ of the inversive plane $\I(\Omega)$ determined by $\Omega$ [@Demb Sec. 6.4].
Let $\mathcal X$ be any set of $s $ singular points $x\in \{a,b\}^\perp $. We will show that *$$\Omega ^\bullet :=
\big (\Omega- \bigcup_{x\in\mathcal X } \Omega_x \big )\cup \mathcal X
\vspace{-4pt}$$ is a maximal partial ovoid of the stated size.*
1\. $|\Omega ^\bullet |=
(q^2+1) -2 - |\mathcal X| (q-1) + |\mathcal X| $.
2\. [*Partial ovoid*]{}: If $x\in \mathcal X$ then $\Omega_x= x ^\perp\cap \Omega $ was replaced by $ x $, and $\mathcal X$ lies in a conic of $\{a,b\}^\perp \cap U$.
3\. [*Maximality*]{}: Suppose that $h$ is a singular point of $V$ such that $h^\perp\cap \Omega^\bullet=\emptyset.$ Then $h^\perp \cap\Omega
\subseteq \cup_{x\in \mathcal X}\Omega_x $. We will consider the various possibilities in [Lemma \[hyperplane intersections\]]{} for the hyperplane $h^\perp \cap U$ of $U$.
Case 1. $h^\perp \cap \Omega =\{p\}$ for some $p\in \Omega.$ Then $p\in \Omega_x $ for some $x\in \mathcal X \subset \Omega^\bullet$. By [Lemma \[hyperplane intersections\]]{}, $h^\perp \cap U = p^\perp \cap U $. This is inside $h^\perp$, so that $h\in \<p , U^\perp\> \le x^\perp$, whereas $h$ is assumed not to be perpendicular to $x\in \Omega^\bullet$.
Case 2. $h^\perp \cap \Omega$ is a circle. If $h^\perp \cap \Omega$ contains $\{a,b\}$ then $h^\perp \cap \Omega= \Omega_x\subset x^\perp$ for some $x\in \mathcal X$ since the circles $ \Omega_y,$ $ y\in \{a,b\}^\perp$, induce a partition of $\Omega-\{a,b\}$. Then $h^\perp $ contains $\<\Omega_x\cap U\> =x^\perp \cap U$ by [Lemma \[hyperplane intersections\]]{}(ii), which again produces the contradiction $h\in
\<x,U^\perp\> \le x^\perp$.
If $h^\perp \cap \Omega$ does not contain $\{a,b\}$ then it meets each circle $\Omega_x$, $x\in \mathcal X$, in at most two points. This produces the contradiction $q+1=|h^\perp \cap \Omega|\le 2| \mathcal X |=2s$.
Case 3. $h^\perp \cap \Omega$ is an orbit of a cyclic group $T<G$ of order $|h^\perp \cap \Omega|=q\pm \sqrt{2q}+1$ ([Lemma \[hyperplane intersections\]]{}(iii)). We replace the argument used in [Theorem \[easiest Grassl\]]{} by counting helped by $T$. Note that $ |T |$ divides $ q^2+1$ and hence is relatively prime to $q(q-1)$, the order of the stabilizer in $G$ of a circle [@Suz Theorem 9]. Thus, given circles $C_1$ and $C_2$, at most one element of $T$ can send $C_1$ to $C_2$.
For each $t\in T$ we have $h^\perp \cap \Omega=(h^\perp \cap \Omega)^t \subseteq \cup_{x\in \mathcal X}\Omega_x^t$, involving two sets of $s$ circles: $ \{\Omega_x \mid x\in \mathcal X\}$ and $ \{\Omega_x^t \mid x\in \mathcal X \}$. For an ordered pair $x,y$ of distinct elements of $\mathcal X$ there is at most one such $t\ne 1$ with $\Omega_x^t=\Omega_y$. Thus, if we choose $t$ to be one of at least $|T|-1-s(s-1) \ge q - \sqrt{2q}-s(s-1) >0$ elements of $T$ that do not behave this way for all $x,y$, then we will have two disjoint sets of $s$ circles, with the union of each set containing $h^\perp \cap \Omega$. Since distinct circles meet in at most two points, $q\pm \sqrt{2q}+1 =| h^\perp \cap \Omega | \le s\cdot s\cdot2$, which is not the case.
Case 4. $\Lambda:= h^\perp \cap \Omega$ has size $q+1$, and its stabilizer in $G$ has a cyclic subgroup $T$ of order $q-1$ having orbit-lengths $1,1,q-1$ on $\Lambda$ ([Lemma \[hyperplane intersections\]]{}(iv)). There are $q+1$ orbits of $T$ of size $q-1$; every nontrivial element of $T$ fixes just two points.
If $a$ and $b$ are not the two points fixed by $T$ then it follows that every nontrivial element of $T$ moves every $\Omega_x$. Thus, the argument in Case 3 can be repeated (this time with $|T|-1-s(s-1) = q-s(s-1) >0$ and the contradiction $q +1 \le s\cdot s\cdot2$).
If $T$ fixes $a$ and $b$ then $\Lambda$ is not one of the circles $\Omega_y$ by [Lemma \[hyperplane intersections\]]{}(iv). Note that $T$ fixes two circles containing $ \{a,b\}$ and is transitive on the remaining $q -1 $ circles containing $ \{a,b\}$. Thus, if a nontrivial element of $T$ fixes $\Omega_y$ then $\Omega_y-\{a,b\}$ is an orbit of $T$, while $\Lambda-\{a,b\}$ is a different orbit, so that $\Omega_y$ can be deleted in our union (of $\Omega_x$, $x\in \mathcal X$) that contains $\Lambda$. Since $T$ is transitive on the set of $q-1$ circles $\Omega_x$ that it does not fix, we obtain the contradiction $s=|\mathcal X|\ge |T|=q-1$. (Alternatively, the argument in Case 3 can be repeated again.)
We have proved, more generally, that *$\Omega^\bullet$ is a maximal partial ovoid of any nonsingular orthogonal ${\hbox{\Bbb F}}_q$-space containing $U$* since every hyperplane of $U$ has nonempty intersection with $\Omega^\bullet$ (cf. Question \[spanning ovoid\]).
$\Sp(6,q)$-space consequences {#$Sp(6,q)$-space consequences}
-----------------------------
\[Sp6 corollary\] For even $q>2, $ an $\Sp(6,q)$-space has maximal symplectic partial spreads of size
- $n_1 = q^3-q^2+1 ,$
- $q^2+1,$
- $2q^2+1,$
- $q^2+q+1$ if $q=2^{2e+1},$
- $q^2-q+1$ if $q=2^{2e+1},$
- $q^2-sq+2s-1$ if $q=2^{2e+1} $ and $1\le s\le \sqrt{q/2} -1 ,$
- $n_r$ if $1\le r\le q/ 5 $ [(where $n_r$ is defined in [Theorem \[orthovoids\]]{}),]{} and
- $n_4-1$ if $q\ge16$.
Use [Lemma \[orthogonal to symplectic\]]{}(i) together with Theorems \[triality to spread\], \[partial ovoid from spread\], \[2q2+1\], \[q2+q+1\], \[easiest Grassl\], \[smaller Grassl\] and \[orthovoids\].
By [Lemma \[orthogonal to symplectic\]]{}(ii), the set of sizes of maximal partial $\Sp(6,4)$-spreads is contained in the set of sizes of maximal partial $\Sp(8,4)$-spreads. This can be compared with the list in [@Grassl].
$6$-dimensional partial spreads {#Using groups}
===============================
We again consider arbitrary characteristic. In characteristic 2 the examples in the next theorem already appear in [Theorem \[Sp6 corollary\]]{}(i) but using an entirely different method to prove maximality.
\[group\] If $q$ is a prime power then an $\Sp(6,q)$-space has a maximal symplectic partial spread of size $q^3-q^{2}+1$.
In an $\Sp(6,q)$-space let $\Sigma$ be a desarguesian spread preserved by $G= \SL(2,q^3)=\Sp (2,q^3) <\Sp(6,q)$. Let $X\in \Sigma$. Let $U$ be a t.i. 3-space such that $U\cap X=L$ is a line. If $\Sigma_U$ is the set of members of $\Sigma$ met nontrivially by $U$, then we will show that *$\Sigma^\bullet := (\Sigma -\Sigma_U)\cup\{U\}$ is a maximal symplectic partial spread of size $q^3-q^2+1$.*
If $U$ meets $Y\in \Sigma-\{X\}$ nontrivially then $U\cap Y$ must meet $U\cap X=L$ trivially and hence is a point; the number of such points is the number $q^{2}$ of points in $U$ not in $L$. Thus, $|\Sigma_U |= q^2+1$ and $\Sigma^\bullet $ is a symplectic partial spread of size $q^3-q^2+1$.
The set-stabilizer $G_X$ of $X$ has order $q^3(q^3-1)$, with an abelian normal subgroup $Q$ of order $q^3$ inducing $1$ on $X $ and a cyclic subgroup $S$ of order $q^3-1$ that is transitive on both $X-\{0\}$ and the $q^2+q+1$ lines $L$ of $X$. Then $|G_L|=q^3(q-1)$. Since $Q$ is transitive on the $q$ t.i. 3-spaces $\ne X$ containing $L$, $|G_ U |=q^2(q-1)$ and $Q_U$ fixes each of those 3-spaces.
Since $S$ is transitive on both the $q^2+q+1$ lines $L$ of $X$ the $G_X$-conjugates of $Q_U$, we obtain a bijection $L\mapsto Q_U$ between these sets.
Also $Q$ is transitive on the $q^3$ points in $\{ L^\perp\cap Y \mid Y\in \Sigma-\{X\} \}$. Since $U$ contains $q^{2}$ of these points, and each such point and $L$ generate $U$, it follows that $Q_U$ is transitive on these $q^{2}$ points. Since $Q_U$ fixes each t.i. 3-space containing $L$, the $q$ orbits of $Q_U$ partitioning $ \Sigma-\{X\}$ correspond to the $q$ t.i. 3-spaces $\ne X$ containing $L$. *Maximality*: Assume that $W\notin \Sigma^\bullet$ is a t.i. 3-space such that $\Sigma^\bullet \cup\{W\}$ is a symplectic partial spread. Then $\Sigma_W\subseteq \Sigma_U$ since $\Sigma$ is a spread. Clearly, $W$ meets each member of $\Sigma$ in 0, a point or a line. Since $|\Sigma_W|\le |\Sigma_U|={q^2+1}$, some intersection is a line, and it is unique (since two lines of $W$ would meet nontrivially). Thus, $W$ arises in the same manner as $U$, and $G_W$ acts on ${\Sigma_W=\Sigma_U}$.
We cannot have $ L=X\cap U = X \cap W $ in view of the above orbit partition, and we cannot have $ G_U = G_W $ in view of the above bijection. Then $\< G_U,G_W \>$ is generated by distinct subgroups of $G$ of order $q^2(q-1)$, and hence has a subgroup of order $q^3$ with an orbit on $\Sigma$ of size $q^3$. (This is a very special case of 115-year-old group theory summarized in [@Dic Ch. XII].) This contradicts the fact that $|\Sigma_U|<q^3$.
$4$-dimensional partial spreads {#Generalized quadrangles}
===============================
Finally, we survey families of maximal partial spreads of $\Sp(4,q)$-spaces. See [@CDFS; @Grassl] for lists and tables of known families. As suggested in [Section \[Introduction\]]{}, we can use more easily visualized points in $\O(5,q)$-space instead of lines in $\Sp(4,q)$-space due to the Klein correspondence [@Taylor p. 196].
\[conics\]
- For odd $q$ an $\Sp(4,q)$-space has a maximal partial spread of size $q^2-s q+ 3 s-1 $ whenever $1\le s<(q + 1)/2$.
- For even $q$ an $\Sp(4,q)$-space has a maximal partial spread of size $q^2- sq+ 2 s -1 $ whenever $1\le s<(q + 1)/2$.
We will construct maximal partial $\O(5,q)$-ovoids. Start with an $\O^-(4,q)$ ovoid $\Omega$ in a 4-dimensional subspace $U$. Choose distinct $a,b \in \Omega$. If $y$ is a singular point not in $\Omega,$ then $y^\perp\cap \Omega$ is an oval (more precisely, a conic in $\<y^\perp\cap \Omega\>$).
\(i) The planes $E$ of $U$ containing $\{a,b\}$ fall into two sets $\Pi_k$, $k=0$ or 2, each of size $(q+1)/2$, such that the nonsingular line $E^\perp $ has exactly $k$ singular points. Let $\Omega_E:=E\cap \Omega $. With each $E\in \Pi_2$ is an associated $E'\in \Pi_2$ (for an involution $E\mapsto E'$ without fixed points) such that there are exactly two singular points $x_{E },x_{E' } $ in $ E\cap \{a,b\}^\perp $ and $(*)\ \Omega_E =
x_{E } ^\perp\cap \Omega =
x_{E '} ^\perp\cap \Omega =\Omega _{E '}$. Then $\{x_{E },x_{E' } \mid E\in \Pi_2 \} $ lies in a conic in the plane $\{a,b\}^\perp $. The members of $\Pi_ 0\cup \Pi_ 2$ induce a partition of $\Omega-\{a,b\}$.
Let $\mathcal S$ be any set of $s < (q+1)/2$ planes $E \in \Pi_ 2 $ such that the conics $\Omega_E$,$\,E\in\mathcal S$, are distinct (cf. $(*)$). We claim that *$\Omega ^\bullet :=
{ \big (\Omega- \bigcup_{E\in\mathcal S } \Omega_E \big )\cup \bigcup_{E\in\mathcal S}\{x_{E},x_{E' }\} }$ is a maximal partial $\O(5,q)$-ovoid of the stated size.*
1\. $|\Omega ^\bullet |=
(q^2+1) -2 - |\mathcal S| (q-1) +2 |\mathcal S| $.
2\. [*Orthogonal partial ovoid*]{}: If $E\in \mathcal S$ then $\Omega_E= x_{E} ^\perp\cap \Omega
= x_{E' } ^\perp\cap \Omega $ was replaced by $\{x_{E},x_{E' }\}$, lying in a conic of $\{a,b\}^\perp$.
3\. [*Maximality*]{}: Every point of $\Omega$ is either in $\Omega ^\bullet$ or is perpendicular to $\{x_{E},x_{E' }\}$ for some $ E \in\mathcal S$. Suppose that $h\notin \Omega$ is a singular point such that $h^\perp\cap \Omega ^\bullet =\emptyset$. Then $h^\perp\cap \Omega\subseteq \bigcup_{E\in\mathcal S} \Omega_E$ and $h^\perp\cap \Omega$ is either a point or a circle of the inversive plane $\I(\Omega)$ determined by $\Omega$ [@Demb Sec. 6.4] (compare [Example \[elliptic\]]{}).
If $h^\perp\cap \Omega$ is a point $p$ then $h^\perp\cap U$ is the tangent plane to $\Omega$ at $p$ in $U$. Then $h\in (h^\perp\cap U)^\perp = (p^\perp\cap U)^\perp =\<p,U^\perp\>$, which has just one singular point $p$, whereas $h\notin \Omega$. Thus, $h^\perp\cap \Omega \subseteq \bigcup_{E\in\mathcal S} \Omega_E$ is a circle.
If $h^\perp\cap \Omega=\Omega_E$ with $E\in\mathcal S$, then $h\in (\Omega_E)^\perp = E^\perp = \<x_{E},x_{E' }\>$, whereas $h\notin \{x_{E},x_{E' }\}$. Thus, $h^\perp\cap \Omega$ is a circle lying in the union of $s$ other circles, each of which it meets at most twice. This produces the contradiction $q+1=| h^\perp\cap \Omega|\le 2s$. (ii) This is proved as above but is simpler: ${(E^\perp\cap \Omega)^\perp}$ contains just one singular point for each plane $E$ of $U$ containing $\{a,b\}$; no permutation $E\mapsto E'$ is involved.
It is tempting to hope that the above argument only used an $\O(5,q)$-ovoid $\Omega$. However, when $q$ is odd the intersections $x^\perp \cap \Omega$ are not sufficiently well-behaved. When $q$ is even [Section \[Grassl’s Conjecture\]]{} contains versions of the preceding argument in $\O^+(8,q)$-space (also compare Theorem \[orthovoids\]).
\[3q-1\] A maximal partial ovoid of size $3q-1$ in $\Sp(4,q)$-space, $q\ge4,$ is constructed in . The proof in that paper shows that this is a maximal partial ovoid in $\Sp(2m,q)$-space for all $m\ge2$.
This partial ovoid is the set of points in $\bigcup_1^3\big (\<x_i,y_{i+1}\>-\{x_i,y_{i+1}\}\big)\cup \{x,y\}$ (subscripts mod 3), where $x_1, x_2, x_3, x$ are four points of $X$ and $y_1, y_2, y _3, y$ are four points of $Y$ for t.i. 2-spaces $X,Y$ intersecting in 0, with each pair $x_i,y_i$ perpendicular and $x,y$ not perpendicular.
Dualizing [@Taylor p. 196] produces a maximal symplectic partial spread of size $3q-1$ in $\Sp(4,q)$-space for even $q\ge4$.
For arbitrary $q$, [@RS; @PRS] has integer intervals that consist of sizes of maximal partial $\Sp(4,q)$-spreads.
There is a maximal partial spread of size $q^2-1$ in $\Sp(4,q)$-space for $q\in\{3,5,7 ,11 \}.$ This is constructed using a subgroup of $\Sp(2,q)=\SL(2,q)$ sharply transitive on ${\hbox{\Bbb F}}_q^2-\{0\}\,$ [@Pe; @DH; @CDS]. It is contained in the non-symplectic spread of ${\hbox{\Bbb F}}_q^4$ corresponding to the associated affine irregular nearfield plane.
Concluding remarks
==================
The preceding examples makes it clear that there are rather few known types of maximal symplectic partial spreads. There are amazingly few known types in odd characteristic, especially in view of the tables in [@CDFS; @Grassl]. We mentioned a number of symplectic partial spreads whose maximality has yet to be decided.
We have not yet considered most inequivalence questions for given dimension and field size. Suppose that $q$ is even. The number of inequivalent orthogonal spreads in $\O^+(4m ,q )$-spaces is not bounded above by any polynomial in $q^m$ [@KaW]; these produce inequivalent maximal symplectic partial spreads in [Proposition \[orthogonal spread\]]{}, \[nk\] and \[project k\]. In addition, there are at least $q^{q^k}/q^{4k^2}$ inequivalent examples in [Theorem \[Grassl example\]]{}(ii), $\binom{ q-1 }{s}/q^{30 }$ for each pair $q,s$ in [Theorem \[orthovoids\]]{}, $( q-1 )^{q+1}/q^{30 }$ in [Theorem \[q2+q+1\]]{}, $\binom{ q+1 }{s}/q^{30 }$ for each pair $q,s$ in [Theorem \[smaller Grassl\]]{}, $\binom{q+1 }{s}/q^{11 }$ for each pair $q, s$ in [Theorem \[conics\]]{} and $(q-2)(q-3)/6\log q$ in [Example \[3q-1\]]{}.
I am grateful to Markus Grassl for stimulating my interest in maximal symplectic partial spreads by pointing out the scarcity of examples in dimension $>4$. This research was supported in part by a grant from the Simons Foundation.
[BGWW]{}
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The desarguesian ovoid in $\O^+(8,q)$-space {#appendix A}
===========================================
In order to prove [Theorem \[orthovoids\]]{} we will consider a specific orthogonal ovoid in an $\O^+(8,q)$-space with [$q>2$ even.]{} Let $F={\hbox{\Bbb F}}_{q^3}\supset K={\hbox{\Bbb F}}_q$, with trace map $T\col F\to K$ and norm $N\col F\to K$. Then $Q(a,\b,{\gamma },d):=ad+T(\b{\gamma })$ turns $V:=K\oplus F \oplus F \oplus K$ into an $\O^+(8,q)$-space.
The $q^3+1 $ points $\<(0,0,0,1 ) \>$ and $\<(1,t,t^{q+q^2}, N(t )\>$, $t\in F$, form an ovoid $\Omega$ on which $G:=\SL(2,q^3)$ acting 3-transitively. In [@Ka1 p. 1204] this is called a [*desarguesian ovoid*]{} (since it arises from a desarguesian spread of an $\Sp(6,q) $-space using [Lemma \[orthogonal to symplectic\]]{}(ii) and triality), and it is observed that $G$ has exactly two orbits of singular points of $V$, one of which is $\Omega$. If $q>2$ and $p$ is any singular point not in $\Omega$, then $\<p^\perp\cap \Omega\>=p^\perp$ [@Ka1 p. 1204], as required in [Lemma \[use orthogonal ovoid\]]{}. \[dagger\]Let $\pi\in F$ with $T(\pi)=0\ne T(\pi^{{1+q}} )$. Use the nondegenerate symmetric $K$-bilinear form $T(xy)$ on $F$ to see that $\pi^{q} \notin\{t\in F\mid T(\pi t)=0\}= K+K\pi.$ \[non-perp\] If $p_1$ and $p_2$ are distinct singular points not in $\Omega ,$ then $|p_1^\perp \cap p_2^\perp \cap \Omega|\le 5q-5$.
By the transitivity of $G$ we may assume that $p_1=\<(0,0,\pi,0)\>$ and $p_2=\<(a,\b,{\gamma },d)\>$ for some $a,\b,{\gamma },d$. We need to estimate the number of solutions $t$ to the equations $$T(t\pi)= 0=aN(t)+d+T(\b t^{q+q^2} +{\gamma }t)$$ corresponding to points $\<(1,t,t^{q+q^2},t^{1+q+q^2})\>$. By we can write $t=u+v\pi$ with $u,v\in K$. Then the second equation is $$aN(u+v\pi)+d+T(\b[u+v\pi]^{q+q^2} +{\gamma }[u+v\pi]) = 0,$$ which expands to $$\label{main equation2}
\begin{array}{lll}
a\{u^3+uv^2T( \pi^{q+q^2}) + v^3N(\pi)\} +d
\vspace{2pt}
\\
\hspace{23pt}
+~u^2T(\b ) +uvT(\b\pi) +v^2 T(\b \pi^{q+q^2}) +uT({\gamma }) +v T({\gamma }\pi)
= 0.
\end{array}$$ For each $u$ this is a $K$-polynomial in $v$ of degree at most three, and hence has at most three roots $v\in K$ if it is not the zero polynomial. Let $B$ be the number of “bad” $u$ for which this polynomial in $v$ is the zero polynomial. Then $|p_1^\perp \cap p_2^\perp \cap \Omega|\le
(q-B)3+Bq +1$ (the last term occurs since $\<(0,0,0,1)\>$ may be in the intersection). We will show that $B\le2$, which produces the bound in the lemma.
The coefficients of our polynomial show that, for a “bad” $u$, we must have $a=0$, then $T(\b \pi^{q+q^2}) =0$,$\,u T(\b\pi )+T({\gamma }\pi)=0 $ and $ u^2T(\b ) + uT({\gamma })+ d =0$. If $T(\b\pi )\ne0$ then there is one “bad” $u$, and if $ T(\b\pi )=T({\gamma }\pi)=0 $ then there are at most two “bad” $u$ unless $ T(\b ) = T({\gamma })= d =0$.
Thus, we must show that ${ T(\b \pi^{q+q^2}) =T(\b\pi )=T({\gamma }\pi)= T(\b ) = T({\gamma })=0 }$ cannot all occur. Since $ T(\b ) =T(\b\pi ) =0$, by we have $\b=x\pi$ with ${x\in K}$. Then $0=T(\b \pi^{q+q^2}) =xT(N(\pi))$, so that $x=0$. Similarly, $ T({\gamma }) =T({\gamma }\pi ) =0$ implies that ${\gamma }=y\pi$ with $y\in K$. Now $p_2=\<(0,0,y\pi,0)\>=p_1$, which is not the case.
Let $\Omega_0\subset \Omega$ consist of $\<(0,0,0,1 ) \>$ and $\<(1,t,t^{q+q^2},t^{1+q+q^2})\>$, $t\in K$. There are $(q+1)^2$ singular points in $\Omega_0^\perp,$ all having the form $\<(0,\b,{\gamma },0)\>$ with $T(\b)=T({\gamma })=T(\b{\gamma })=0 $. The sets $\Omega_0$ and $\Omega_0^\perp$ are acted on by a naturally embedded subgroup $G_0 = \SL(2,q)$ of $G$ containing the transformations $$\begin{array}{llll}
\hspace{-5.5pt}u_s\col (a,\b,{\gamma },d)\mapsto
(a,\b+sa,{\gamma }+ as^2 + \b^qs +\b^{q^2}\! s,d+as^3 + T(\b)s^2 + T({\gamma })s)
\vspace{2pt}
\\
j\col (a,\b,{\gamma },d)\mapsto (d,{\gamma }, \b,a) .
\vspace{-2pt}
\end{array}$$ with $ s\in K$. These act on each of the $q+1$ lines $\<(0,\b,0, 0), (0,0,\b, 0) \>$ with $T(\b) =0\ne \b $ that partition the $(q+1)^2$ singular points in $\Omega_0^\perp$, sending $$\label{generators}
\begin{array}{llll}
\hspace{-5.5pt}u_s\col (0,\b,{\gamma },0)\mapsto
(0,\b,{\gamma }+ \b s ,0)
\vspace{2pt}
\\
j\col (0,\b,{\gamma },0)\mapsto (0,{\gamma }, \b,0) . \hspace{185pt}
\end{array}\vspace{-1pt}$$ An [*ordinary*]{} point is a singular point in $\Omega_0^\perp$ of the form $\< (0,\b, {\gamma },0)\>$ such that either $\b=0$ and $T({\gamma }^{1+q})\ne 0$, or $T(\b^{1+q})\ne 0$ (recall that $ T(\b)=T({\gamma })= T(\b{\gamma }) = 0 $). Since any $\b\in F^*$ has characteristic polynomial $x^3+T(\b)x^2+T(\b^{1+q})+N(\b)$, the ordinary requirement can fail for some $\b,{\gamma }$ if and only if $q\equiv 1$ (mod 3). Moreover, if $\b\in F-K$ then ${\b ^q \in \b K}\iff~{ \b^{q-1}\in K } \iff \b^{(q-1,q^2+q+1)}\in K \iff \b^{3}\in K
\iff T(\b^{1+q})= 0$.
For $\pi$ in , since $T((a\pi+ \pi^q)(a\pi + \pi^{q }) ^q )=(a^2+a+1)T(\pi^{1+q})$ the points of the line $\<(0,a\pi+ \pi^q ,0 ,0), (0,0 , a\pi+ \pi^q ,0) \>,$ $a\in K,$ are ordinary if and only if $a^2+a+1\ne 0$, so that all points are ordinary if $q\equiv 2$ (mod 3), but there are two lines of this form all of whose points are not ordinary when $q\equiv 1$ (mod 3).
The significance of ordinary points is the following
\[move to ordinary\] If $p$ is an ordinary point then
- $p$ has the form $ \<(0,0,{\gamma }, 0)\> $ with $T({\gamma })=0$ or $\<(0,\b,a \b, 0)\> $ with $T(\b)=0$ and $a\in K,$ and
- $p^g=\<(0,0,\pi',0)\>$ for some $g\in G_0,$ where $\pi'$ behaves as $\pi $ does in $:$ $T(\pi')=0\ne T(\pi'{}^{{1+q}})$.
We may assume that $p=\<(0,\b,{\gamma }, 0)\> $ with $\b\ne0$.
\(i) Since $p$ is ordinary, $\b^q\notin K\b $, so that $\b$ and $\b^q$ span $\ker T $. Write ${\gamma }=k\b+b\b^q$ with $k,b\in K$. Then $0=T(\b{\gamma })=bT(\b^{{1+q}})$ implies that $b=0$.
\(ii) By , $p^{u_k j}=\<(0,0,\b,0)\>$ behaves as stated.
\[new intersection sizes\] If $p_1,p_2,$ and $p_3$ are pairwise non-perpendicular ordinary points$,$ then
- $ |p_1^\perp \cap p_2^\perp \cap \Omega | =2q$ and
- $ |p_1^\perp \cap p_2^\perp \cap p_3^\perp \cap \Omega | =q+2. $
By [Lemma \[move to ordinary\]]{}(ii) we may assume that $p_1$ has the form $\<(0,0,\pi,0)\>$ and $p_2 = \<(0,\b,{\gamma },0)\>$, where $T(\b)=T({\gamma })=T(\b{\gamma })=0$. Also $ T(\b\pi)\ne0$ since $p_1$ and $p_2$ are not perpendicular. All $(0,0,0,1) $ and $(1,t,t^{q+q^2},N(t)),t \in K$, are in each of the stated intersections, so we will focus on vectors $(1,t,t^{q+q^2},N(t))$ with $t=u+v\pi \notin K$ in the intersections.
\(i) Here states that $$\label{to massage}
uv T(\b\pi ) +v^2T(\b\pi^{q+q^2})+vT({\gamma }\pi)=0.$$ Since $T(\b\pi)\ne0,$ each $v\ne0$ determines a unique $u$. This argument reverses: the intersection size is $ (q+1)+(q-1)$.
Before continuing we massage . By [Lemma \[move to ordinary\]]{}(i), ${\gamma }=k\b$ for some $k\in K$. Since $\dim \ker T =2$ we can write $\b=x\pi +y\pi^q$ with $x,y\in K$. Since $0\ne T(\b\pi) =yT(\pi^{1+q} )$ we have $y\ne0$ and $\b\in ((x/y)\pi + \pi^q)K$. We may assume that $\b = a\pi + \pi^q $ with $a\in K$. Then $$\label{p2}
p_2 = \<(0,a\pi + \pi^q ,k (a\pi + \pi^q ),0)\>,$$ so that $T(\b\pi)=T(\pi^{{1+q}})$ and becomes $$\label{mn eq}
uT(\pi^{{1+q}}) + v[aN(\pi)+ T(\pi^{2q+q^2})]+kT(\pi^{{1+q}}) =0.$$
\(ii) We may assume that $p_3=\<(0,\b',{\gamma }',0\>)$ with ${\gamma }'=k'\b'$ and $\b' =a'\pi + \pi^q $ for some $k',a'\in K$. Then $(a+a')(k+k')T(\pi\pi^q)= T(\b{\gamma }'+{\gamma }\b')\ne0$. The two versions of imply that $$\label{mn}
v=\frac{k+k'}{a+a'}\frac{T(\pi^{{1+q}})}{N(\pi)}, \ \
u=k+ \frac{k+k'}{a+a'} \Big(a+\frac{T(\pi^{2q+q^2}) }{ N(\pi)}\Big),$$ which proves (ii).
\[examples\] [(i)]{} If $ {\mathcal P}\subseteq
\{\<(0,0,\pi,0)\>, \<(0,a\pi+ \pi^q ,a^2\pi+a\pi^q ,0)\> \mid {a \in K,}~
a^2+a+1\ne0\},$ then $$\displaystyle \Big |\bigcap_{p\in {\mathcal P} } p ^\perp \cap \Omega \Big | =
\begin{cases}
{q^2+1} &\mbox{if} \ \ |{\mathcal P} | =1 \\
{2q} &\mbox{if} \ \ |{\mathcal P} |=2 \\
{q+2} &\mbox{if} \ \ |{\mathcal P} |\ge 3 .
\end{cases}$$ [(ii)]{} If ${\mathcal P} \subseteq
\{\<(0,0,\pi,0)\>, \
\<(0, \pi^q ,0 ,0)\>,\
\<(0, \pi+ \pi^q , \pi+ \pi^q ,0)\>,
\<(0,a\pi+ \pi^q ,a^3\pi+a^2\pi^q ,0)\>
\} $ for an arbitrary $a\in K-\{0,1\} $ such that $ a^2+a+1\ne0,$ then $$\displaystyle \Big |\bigcap_{p\in {\mathcal P} } p ^\perp \cap \Omega \Big | =
\begin{cases}
{q^2+1} &\mbox{if} \ \ |{\mathcal P} |=1 \\
{2q} &\mbox{if} \ \ |{\mathcal P} |=2 \\
{q+2} &\mbox{if} \ \ |{\mathcal P} |=3 \\
{q+1} &\mbox{if} \ \ |{\mathcal P} |=4 .
\end{cases} \vspace{-2pt}$$
All of the stated points are ordinary.
\(i) In , $k = a$ for all listed points other than $\<(0,0,\pi,0)\>$. By , $t =
\frac{T(\pi^{2q+q^2}) }{ N(\pi)\raisebox{1.5ex}{\hspace{-1pt}}\raisebox{-.4ex}{\hspace{-1pt}}}
+ \frac{T(\pi^{{1+q}})}{ N(\pi)\raisebox{1.5ex}{\hspace{-1pt}}\raisebox{-.4ex}{\hspace{-1pt}}}\pi
$ is in every intersection (which is easily checked directly); so is $\Omega_0$, so that every intersection has size $\geq q+2$. Since any intersection of three sets $p^\perp\cap \Omega$ has size $q+2$ (by [Lemma \[new intersection sizes\]]{}(ii)), so does any intersection of at least four such sets.
\(ii) The last three of these four ordinary points correspond to the pairs $(a,k)=(0,0),\,(1,1), \,(a,a^2)$ in . Then and different 3-sets in $\mathcal P$ produce different values of $v$, so that $ { |\bigcap_{p\in {\mathcal P}} p ^\perp \cap \Omega | }=q+1$ if $| {\mathcal P}|=4$. The remaining sizes are given in [Lemma \[new intersection sizes\]]{}.
Suzuki-Tits ovoids: background {#Background on Suzuki-Tits ovoids}
==============================
We will need information concerning a Suzuki-Tits ovoid $\Omega$ in an $\O(5,q)$-space $U$ with radical $r$, where $q=2^{2e+1}$. The standard view of these ovoids is in symplectic space. For our purposes, the view from an $\O(5,q)$-space has advantages, such as lying in an $\O^+(8,q)$-space.
Let $\bar\Omega$ denote a standard Suzuki-Tits ovoid in the symplectic 4-space $U/r$ [@Ti2]. If $\<x,r\>/r\in \bar\Omega$ then the line $\<x,r\>$ has a unique singular point. Thus, there is a set $\Omega $ of $q^2+1$ singular points of $U$ that projects onto $\bar\Omega$. The group $ \Sz(q) $ lifts from a subgroup of $\Sp(4,q)$ to a group $G < \O(5,q)$ preserving $ \Omega $. See [@Demb Sec. 6.4] for information concerning the inversive plane $\I(\Omega)$ produced by $ \Omega$.
We will assume that $q>2$. Then $U=\<\Omega\>$ since $G$ does not act on an $\O^{\pm}(4,q)$-space. (If $q=2$ then $\Omega$ spans an $\O^-(4,2)$-space.)
\[hyperplane intersections\] Every hyperplane meets $\Omega$. More precisely$,$ there are exactly five types of hyperplanes $H$ of $U$$:$
- Tangent hyperplanes $p^\perp$ for $p\in \Omega,$ with $r\in H$ and $H\cap \Omega=\{p\};$
- Secant hyperplanes $x^\perp=H$ containing $r,$ where $x$ is a singular point$,$ $x^\perp\cap \Omega$ is a circle of $\I(\Omega)$ and $\<x^\perp\cap \Omega\>=x^\perp;$
- $\O^-(4,q)$-hyperplanes for which $H\cap \Omega$ is an orbit of a cyclic subgroup of $G$ of order $|H\cap \Omega|=q\pm\sqrt{2q}+1$ acting irreducibly on $U/ r;$ and
- $\O^+(4,q)$-hyperplanes for which $ H\cap \Omega $ contains an orbit of a cyclic subgroup of $G$ of order $|H\cap \Omega|-2=q -1$ that fixes two points of $H\cap \Omega .$ Moreover$, $ $ H\cap \Omega $ is not one of the circles in [(ii)]{}.
\(i) Projecting mod $r$ shows that each point of $\Omega$ behaves as stated.
\(ii) If $x$ is a singular point not in $\Omega$ then each of the $q+1$ t.s. lines on $x$ meets $\Omega$ since $\Omega$ is an ovoid, so that $|x^\perp\cap \Omega|=q+1$. Also, $\dim\<x^\perp\cap \Omega\>=4$, as otherwise its set of singular points would project into a plane of $U/r$, and hence be contained in a conic, which is not the case since $q>2$ [@Ti3 pp. 51-52]. Since $\<x^\perp\cap \Omega\>$ lies in the 4-space $x^\perp$, these subspaces coincide
\(iii) This is [@Bagchi-Sastry2 Theorem 1(a)].
\(iv) The set of singular points of $H$ is partitioned by $q+1$ t.s. lines, and each t.s. line of $U$ meets $\Omega$ since $\Omega$ is an ovoid. Thus, $|H\cap \Omega|=q+1$. We use the orbits of $G$ to find $G_H$. There are exactly two point-orbits on $U/r$: $\bar\Omega$ and the remaining $q(q^2+1)$ points. There is a subgroup of $G$ of order $q-1$ that fixes four points of $U/r$ and induces all scalars on each of these 1-spaces [@HB p. 183]. Since each line containing $r$ has a unique singular point, the two point-orbits on $U/r$ produce four point-orbits on $U-\{r\}$.
Since $G$ has five point-orbits it also has five hyperplane-orbits, so that all $q^2(q^2+1)/2$ hyperplanes $H$ in (iv) lie in an orbit. Then $|G_H|= |G| /[q^2(q^2+1)/2]=2(q-1)$, so that $G_H$ is dihedral of order $2(q-1)$, with orbits of size $2$ and $q- 1$ on $\Omega$ [@Suz Theorem 9].
For the final assertion, if $H\cap \Omega$ lies in two hyperplanes then it is in a plane, and hence is a conic, which is not the case [@Ti3 pp. 51-52].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider random walks (RWs) and self-avoiding walks (SAWs) on disordered lattices directly at the percolation threshold. Applying numerical simulations, we study the scaling behavior of the models on the incipient percolation cluster in space dimensions $d=2, 3, 4$. Our analysis yields estimates of universal exponents, governing the scaling laws for configurational properties of RWs and SAWs.'
address:
- '$^1$ Institut für Theoretische Physik and Centre for Theoretical Sciences (NTZ),Universität Leipzig, Postfach 100920, D-04009 Leipzig, Germany'
- '$^2$ Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, UA–79011 Lviv, Ukraine'
author:
- 'V Blavatska$^{1,2}$ and W Janke$^1$'
title: 'Walking on fractals: diffusion and self-avoiding walks on percolation clusters'
---
Introduction
============
The model of a random walk (RW) provides a good description of diffusion processes, such as for example encountered for electrons in metals or colloidal particles in solution [@RWbook]. The averaged mean square displacement of the diffusive particle at time $t$ (or, equivalently, after $t$ steps on a lattice) scales as $$\langle R^2 \rangle \sim t^{2\nu_{{\rm RW}}}, \label{diff}$$ where in a non-disordered medium $\nu_{{\rm RW}}=1/2$, independently of the space dimension $d$. A RW is a fractal object, with fractal dimension $d_{{\rm RW}}=1/\nu_{RW}$. The number of all possible trajectories $C_t$ for a randomly walking particle of $t$ steps can be found exactly: $C_t=z_0^t$, where $z_0$ is the coordination number of the corresponding lattice.
Forbidding the trajectory of a random walk to cross itself, we obtain a self-avoiding walk (SAW), which is one of the most successful in describing the universal configurational properties of a long, flexible single polymer chain in good solvent [@desCloizeaux90]. The average squared end-to-end distance $\langle R^2\rangle$ and the number of configurations $ C_N $ of SAWs with $N$ steps on the underlying lattice obey the scaling laws: $$\label{scaling}
\langle R^2 \rangle
\sim N^{2\nu_{{\rm SAW}}},\mbox{\hspace{3em}}
C_N \sim z^{N}
N^{\gamma_{{\rm SAW}}-1},$$ where $\nu_{{\rm SAW}}, \gamma_{{\rm SAW}}$ are universal exponents that only depend on the space dimensionality $d$, and $z$ is a non-universal fugacity, counting the average number of accessible nearest-neighbor sites. The properties of SAWs on a regular lattice have been studied in detail both in analytical approaches [@Guillou80; @Nienhuis82; @Guillou85; @Guida98] and computer simulations [@Rosenbluth55; @Madras88; @MacDonald92; @MacDonald00; @Li95; @Caracciolo98]. For example, in the space dimension $d{=}3$ one finds within the frame of the field-theoretical renormalization group approach $\nu_{\rm SAW}{=}0.5882\pm 0.0011$ [@Guida98] and Monte Carlo simulations give $\nu_{\rm SAW}{=}0.5877\pm0.0006$ [@Li95]. For space dimensions $d$ above the upper critical dimension $d_{\rm up}{=}4$, the effect of self-avoidance becomes irrelevant and SAWs behave effectively as random walks with exponents $\nu_{{\rm RW}}=1/2$, $\gamma_{{\rm RW}}=1$.
The problem of random walks in disordered media is of great interest since it is connected with a large amount of physical phenomena: transport properties in fractures and porous rocks, the anomalous density of states in randomly diluted magnetic systems, silica aerogels and in glassy ionic systems, diffusion-controlled fusion of excitations in porous membrane films etc. (see, e.g., Ref. [@Havlin87] for a review). Similarly, SAWs on randomly diluted lattices may serve as a model of linear polymers in a porous medium.
Much of our understanding of disordered systems comes from percolation theory [@Stauffer]. A disordered medium can be modelled as randomly diluted lattice, with a given concentration $p$ of lattice sites allowed for walking. Most interesting is the case, when $p$ equals the critical concentration $p_{c}$, the site-percolation threshold (see Table \[dim\]), and an incipient percolation cluster can be found in the system. Studying properties of percolative lattices, one encounters two possible statistical averages. In the first, one considers only percolation clusters with linear size much larger than the typical length of the physical phenomenon under discussion. The other statistical ensemble includes all the clusters which can be found in a percolative lattice. For the latter ensemble of all clusters, the walks can start on any of the clusters, and for an $N$-step walk, performed on the $i$th cluster, we have $\langle R^2 \rangle \sim l_i^2$, where $l_i$ is the averaged size of the $i$th cluster. In what follows, we will be interested in the former case, when trajectories of walks reside only on the percolation cluster. In this regime, the scaling laws (\[diff\]), (\[scaling\]) hold with new exponents $\nu_{{\rm RW}}^{p_c}\neq \nu_{{\rm RW}}$ [@Sahimi83; @Majid84; @Pandey83; @Alexander82; @Avraham82; @Havlin83; @Argyrakis84; @McCarthy88; @Lee00; @Bug86; @Hong84; @Mastorakos93; @Webman81; @Gefen83; @Rammal83; @Mukherjee95], $\nu_{{\rm SAW}}^{p_c}\neq\nu_{{\rm SAW}},\gamma_{{\rm SAW}}^{p_c}\neq\gamma_{{\rm SAW}} $ [@Kremer81; @Lee89; @Kim90; @Woo91; @Grassberger93; @Lee96; @Meir89; @Lam90; @Nakanishi92; @Rintoul94; @Ordemann00; @Nakanishi91; @Barat91; @Sahimi84; @Rammal84; @Kim87; @Roy90; @Roy87; @Aharony89; @Lam84; @Blavatska04; @Janssen07; @Blavatska08]. A hint to the physical understanding of this phenomenon is given by the fact that weak disorder does not change the dimension of a lattice, whereas the percolation cluster itself is a fractal object with fractal dimension $d_{p_c}^F$ dependent on $d$ (see Table \[dim\]). In this way, scaling law exponents of residing walks change with the dimension $d_{p_c}^F$ of the (fractal) lattice on which the walk resides. Since $d_{\rm up}{=}6$ for percolation [@Stauffer], the exponents $\nu_{{\rm SAW}}^{p_c}(d\geq 6){=}1/2$, $\gamma_{{\rm SAW}}^{p_c}(d\geq 6){=}1$.
Our present paper aims to supplement the studies of random and self-avoiding walks on percolative lattices by obtaining numerical values for exponents, governing the scaling behavior of the models, up to $d=4$ by computer simulations. The layout of the paper is as follows: in the next section, we will present in detail the procedure of extracting the percolation cluster and its backbone on disordered lattices at the percolation threshold. In section III we describe the pruned-enriched Rosenbluth algorithm, applied to study the scaling of self-avoiding walks, and present the results obtained. In the next section we consider the method for studying random walks on percolation clusters. In Section V, we end up by giving conclusions and an outlook.
Construction of percolation cluster
===================================
We consider site percolation on a regular lattice of edge length $L=400,200,50$ in dimensions $d=2,3,4$, respectively. Each site of the lattice is assigned to be occupied with probability $p_c$ (values of critical concentration in different dimensions are given in the Table \[dim\]), and empty otherwise. To describe the procedure of extracting the percolation cluster, let us consider schematically the two-dimensional case. We apply an algorithm based on the one proposed by Hoshen and Kopelman [@Hoshen76]. As a first step, a label is prescribed to each of the occupied sites. Such a labeling process is regulated, we start, for example, from the first “column" of the lattice, label the occupied sites upwards, and then turn to the next “column", as shown in Fig. \[gratka\], left. Next, we start the procedure of burning the occupied sites. Again, in the same order, starting from the bottom of the first “column" of the lattice, for each of the labeled sites (say, $i$), we check whether its nearest neighbors are also occupied or not. If yes, two possibilities appear: 1) The label of the neighbor is larger than the label of site $i$. In this case, we change the label of the neighbor to that of site $i$. 2) The label of the neighbor is smaller than that of $i$. Then, we change the label of site $i$ to that of the neighbor.
![\[gratka\]Procedure of site labeling and extracting the percolation cluster.](gratka1.eps "fig:"){width="4cm"} ![\[gratka\]Procedure of site labeling and extracting the percolation cluster.](gratka2.eps "fig:"){width="3.8cm"}
Such a burning procedure is applied until no more change of site labels is needed. As a result, we end up with groups of clusters of occupied sites with the same labels (Fig. \[gratka\], right). Finally, we check, whether there exists a cluster, that wraps around the lattice in all $d$ directions. If yes, we have found the percolation cluster (Fig. \[perc\]). If not, this disordered lattice is rejected and a new one is constructed. Note, that on finite lattices the definition of spanning clusters is not unique (e.g., one could consider clusters connecting only two opposite borders), but all definitions are characterized by the same fractal dimension and are thus equally legitimate. The here employed definition has the advantage of yielding the most isotropic clusters. Note also that directly at $p=p_c$ more than one spanning cluster can be found in the system, and the probability $P(k)$ for at least $k$ separated clusters grows with the space dimension as $P(k)\sim \exp(-\alpha k^{ d/(d-1)})$ [@Aizenman97]. In our study, we take into account only one cluster per each disordered lattice constructed, in order to avoid presumable correlations of the data.
![\[perc\]Percolation cluster on a $d=2$-dimensional regular lattice of edge length $L=50$.](perc.eps){width="4cm"}
Aiming to investigate the scaling of SAWs on percolative lattices, we are interested in the backbone of percolation clusters, which can be defined as follows. Assume that each bond (or site) of the cluster is a resistor and that an external potential drop is applied at two ends of the cluster. The backbone is the subset of the cluster consisting of all bonds (or sites) through which the current flows; i.e., it is the structure left when all “dangling ends" are eliminated from the cluster. The SAWs can be trapped in “dangling ends", therefore infinitely long chains can only exist on the backbone of the cluster. The algorithm for extracting the backbone of obtained percolation clusters was first introduced in Ref. [@Herrmann84] and improved in Ref. [@Porto97]. We choose the starting point – “seed” – at the center of the cluster, and find the chemical distance $l$ of all the sites belonging to the percolation cluster to this starting point. In Fig. \[himia\], the starting point is numbered with $0$, and the chemical distance of all the other sites are depicted. The burning algorithm is divided into two parts. First, we start from some site of the edge of the lattice belonging to the percolation cluster and consider it as burning site. At the next step, if the nearest neighbor of this site has the chemical distance smaller than the burning site itself, the nearest neighbor site is burnt. Such a procedure ends when the “seed" site is reached. All the thus obtained burnt sites are located along the shortest path between the “seed" and the given site at the edge of the percolation cluster, as is shown in Fig. \[himia\]. The same algorithm is applied successively to all the edge sites. As a result, we obtain the so-called skeleton or elastic backbone [@Havlin84], shown in Fig. \[geom\], left. In the second part of the algorithm, we consider successively each site of the elastic backbone, and check, whether a “loop" starts from this site. “Loop" is a path of sites, belonging to the percolation cluster, which is connected with the elastic backbone by no less than two sites. All sites of the elastic backbone together with the sites of “loops" form finally the geometric backbone of the cluster (see Fig. \[geom\], right).
![\[himia\] For all sites of a percolation cluster the chemical distance from the starting site is calculated. The minimal paths from all the sites on the edge of the percolation cluster to this starting point are found, which form the elastic backbone of the percolation cluster.](himia.eps){width="4cm"}
![\[geom\]Elastic and geometrical backbones of the percolation cluster depicted in Fig. \[perc\].](elas.eps "fig:"){width="4cm"} ![\[geom\]Elastic and geometrical backbones of the percolation cluster depicted in Fig. \[perc\].](geom.eps "fig:"){width="4cm"}
The results for fractal dimensions of the percolation cluster and its geometrical backbone in $d{=}2,3,4$ are compiled in Table \[dim\].
Self-avoiding walks on percolation clusters
===========================================
The method
----------
For the sampling of SAWs, we use the pruned-enriched Rosenbluth method (PERM), proposed in the work of Grassberger [@Grassberger97]. The algorithm is based on ideas from the very first days of Monte Carlo simulations, the Rosenbluth-Rosenbluth (RR) method [@Rosenbluth55] and enrichment [@Wall59]. Let us consider the growing polymer chain, i.e. the $n$th monomer is placed at a randomly chosen neighbor site of the last placed $(n-1)$th monomer ($n\leq N$, where $N$ is total length of polymer). The growth is stopped, if the total length $N$ of the chain is reached, then the next chain is started to grow from the same starting point. In order to obtain correct statistics, if this new site is occupied, any attempt to place a monomer at it results in discarding the entire chain. This leads to an exponentional “attrition", the number of discarded chains grows exponentially with the chain length, which makes the method useless for long chains. In the RR method, occupied neighbors are avoided without discarding the chain, but the bias is corrected by means of giving a weight $W_n\sim (\prod_{l=1}^n m_l)$ to each sample configuration at the $n$th step , where $m_l$ is the number of free lattice sites to place the $l$th monomer. When the chain of total length $N$ is constructed, the new one starts from the same starting point, until the desired number of chain configurations are obtained. The configurational averaging, e.g., for the end-to-end distance $r(N)\equiv \sqrt{R^2(N)}$, then has the form: $$\begin{aligned}
&&\langle r (N) \rangle=\frac{1}{Z_N}{\sum_{{\rm conf}}W_N^{{\rm conf}}(\vec{r}_N-\vec{r}_0)^2},
\,\,\,\,Z_N=\sum_{{\rm conf}} W_N^{{\rm conf}} \label {R},\end{aligned}$$ where $\vec{r}_0$ is the position of the starting point of the growing chains, $\vec{r}_k$ is the position of the $k$th monomer, and $Z_N$ is the partition sum.
The Rosenbluth method, however, also suffers from attrition: if all next neighbors at some step ($n<N$) are occupied, i.e., the chain is running into a “dead end", the complete chain has to be discarded and the growth process has to be restarted. Combining the chain growth algorithm with population control, such as PERM (pruned-enriched Rosenbluth method) [@Grassberger97] leads to a considerable improvement of the efficiency by increasing the number of successfully generated chains. The weight fluctuations of the growing chain are suppressed in PERM by pruning configurations with too small weights, and by enriching the sample with copies of high-weight configurations [@Grassberger97]. These copies are made while the chain is growing, and continue to grow independently of each other. Pruning and enrichment are performed by choosing thresholds $W_n^{<}$ and $W_n^{>}$ depending on the estimate of the partition sum for the $n$-monomer chain. These thresholds are continuously updated as the simulation progresses. The zeroes iteration is a pure chain-growth algorithm without reweighting. After the first chain of full length has been obtained, we switch to $W_n^{<}$, $W_n^{>}$. If the current weight $W_n$ of an $n$-monomer chain is less than $W_n^{<}$, a random number $r={0,1}$ is chosen; if $r=0$, the chain is discarded, otherwise it is kept and its weight is doubled. Thus, low-weight chains are pruned with probability $1/2$. If $W_n$ exceeds $W_n^{>}$, the configuration is doubled and the weight of each identical copy is taken as half the original weight. For a value of the weight lying between the thresholds, the chain is simply continued without enriching or pruning the sample. For updating the threshold values we apply similar rules as in [@Hsu03; @Bachmann03]: $W_n^{>}=C(Z_n/Z_1)(c_n/c_1)^2$ and $W_n^{<}=0.2W_n^{>}$, where $c_n$ denotes the number of created chains having length $n$, and the parameter $C$ controls the pruning-enrichment statistics. After a certain number of chains of total length $N$ is produced, the given tour is finished and a new one starts. We adjust the pruning-enrichment control parameter such that on average 10 chains of total length $N$ are generated per each tour [@Bachmann03]. Also, what is even more important for efficiency, in almost all iterations at least one such a chain was created. The number of different trajectories of SAWs with $N$ steps can be then estimated as averaged weight: $$C_N =\frac{1}{t}\sum_{{\rm conf}}W_N^{{\rm conf}},
\label{number}$$ where $t$ is the number of successful tours. PERM has been applied to a wide class of problems, in particular study of $\Theta$-transition in homopolymers [@Grassberger97], trapping of random walkers on absorbing lattices [@Mehra02], study of protein folding [@Frauenkron98; @Bachmann03] etc.
Results
-------
To study scaling properties of SAWs on percolating lattices, we have to perform two types of averaging: the first average is performed over all SAW configurations on a single percolation cluster according to (\[R\]); the second average is carried out over different realizations of disorder, i.e. over all percolation clusters constructed: $$\begin{aligned}
&&\overline{\langle r \rangle}{=}\frac{1}{M}\sum_{i{=}1}^M \langle r \rangle_i,\label{av}\\
&&\overline{ C_N }{=}\frac{1}{M}\sum_{i{=}1}^M C_{N,i},\end{aligned}$$ where $M$ is the number of different clusters and the index $i$ means that a given quantity is calculated on the cluster $i$.
![Disorder averaged end-to-end distance vs number of steps in double logarithmic scale for SAWs on the backbone of percolation clusters in $d{=}2$ (pluses), $d{=}3$ (stars), $d{=}4$ (squares). Lines represent linear fitting, statistical error bars are of the size of symbols.[]{data-label="sawr"}](sawr.eps){width="7cm"}
[@r l l l ]{} $\nu_{{\rm SAW}}^{p_c} \setminus d$ & 2 & 3 & 4\
FL Eq. (\[kremer\]) & 7/9 & 0.665& 0.594\
[@Sahimi84] & 0.778 & 0.662 & 0.593\
[@Roy90] & 0.77& 0.66 & 0.62\
[@Roy87] & 0.770 & 0.656 & 0.57\
[@Aharony89] & 0.76 & 0.65 & 0.58\
EE [@Lam90] &0.745(10)& 0.635(10)&\
[@Rintoul94]&0.770(5)& 0.660(5)&\
[@Ordemann00]&0.778(15)& 0.66(1)&\
RS [@Sahimi84] & 0.778&0.724&\
[@Lam84] & 0.767 & &\
RG [@Blavatska04] & 0.785 & 0.678& 0.595\
[@Janssen07] & 0.796 & 0.669& 0.587\
MC [@Woo91] & 0.77(1) & &\
[@Grassberger93] & 0.783(3) & &\
[@Lee96] & & 0.62–0.63 &0.56–0.57\
[@Ordemann00]&0.787(10)& 0.662(6)&\
our results & $ 0.782\pm 0.003$ & $0.667\pm 0.003$ & $0.586\pm 0.003$\
![Disorder averaged distribution function $r\overline{P(r,N)}$ vs the scaling variable $r/N^{\nu_{{\rm SAW}}^{p_c}}$ in $d{{=}}2$ dimensions. Lattice size $L{{=}}400$, number of SAW steps $N{{=}}140$ (squares), $N{{=}}120$ (triangles), $N{{=}}100$ (crosses).[]{data-label="prsaw2"}](rozpR3d.eps){width="6.8cm"}
![Disorder averaged distribution function $r\overline{P(r,N)}$ vs the scaling variable $r/N^{\nu_{{\rm SAW}}^{p_c}}$ in $d{{=}}3$ dimensions. Lattice size $L{{=}}200$, number of SAW steps $N{{=}}80$ (squares), $N{{=}}60$ (triangles), $N{{=}}50$ (crosses).[]{data-label="prsaw3"}](rozpR2d.eps){width="7cm"}
![Disorder averaged distribution function $r\overline{P(r,N)}$ vs the scaling variable $r/N^{\nu_{{\rm SAW}}^{p_c}}$ in $d{{=}}4$ dimensions. Lattice size $L{{=}}50$, number of SAW steps $N{{=}}30$ (squares), $N{{=}}20$ (triangles), $N{{=}}15$ (crosses).[]{data-label="prsaw4"}](rozpR4d.eps){width="6.6cm"}
The SAW statistics crucially depends on the type of disorder averaging, namely, whether the disorder is considered to be “annealed" (positions of defects are in thermodynamical equilibrium with the system) or “quenched" (positions of defects are out of equilibrium). As it was pointed out in Ref. [@Doussal91], the correctness of results, obtained in the picture of “quenched" disorder, depends on whether the location of the starting point of a SAW is fixed while the configurational averaging is performed, or not. In the latter case, one has to average over all locations and effectively this corresponds to the case of annealed disorder.
An interesting question arises: what is the difference in statistics between SAWs walking on percolation clusters and the backbone of percolation clusters, after eliminating all the “dead ends"? First Kim [@Kim90] claimed, based on a scaling argument, that the critical behavior on the percolation cluster is the same as that on the backbone, namely $\nu_{{\rm SAW}}^{p_c}=\nu_{{\rm SAW}}^{B}$. This equality was also assumed by Rammal [@Rammal84] in deriving a Flory formula for SAWs on fractal substrates. This can be easily explained: since the walks, which visit the dead ends are eventually terminated after a certain number of steps, the walks that survive in the limit $N\to \infty$ are those confined within the backbone. However, in a numerical study [@Woo91] it was found, that $\nu_{{\rm SAW}}^B>\nu_{{\rm SAW}}^{p_c}$, and, moreover, that $\nu_{{\rm SAW}}^{p_c}$ almost equals the value for SAWs on pure lattices. It was argued, that the averaged end-to-end distance of SAWs on the backbone is significantly enhanced in comparison with averaging on the full percolation cluster. We have checked this, comparing results obtained by us for the averaged end-to-end distance $\overline{\langle r(N) \rangle}$ on percolation clusters and the backbone of percolation clusters. We conclude, that there is practically no difference in scaling for SAWs on both types of clusters, the SAW statistics is determined by the backbone of percolation clusters.
To study the scaling properties of SAWs on the backbone of percolation clusters, we choose as the starting point the “seed" of the cluster, and apply the PERM algorithm, taking into account, that a SAW can have its steps only on the sites belonging to the backbone of the percolation cluster. We use lattices of size up to $L_{{\rm max}}{=}400, 200, 50$ in $d{=}2,3,4$, respectively, and performed averages over 1000 percolation clusters in each case. Estimates for the critical exponents $\nu_{{\rm SAW}}^{p_c}$ were obtained by linear least-square fitting (see Tables \[2d\], \[3d\], \[4d\] in the Appendix). Note, that since we can construct lattices only of a finite size $L$, it is not possible to perform very long SAWs on it. For each $L$, the scaling for $\overline{\langle r(N) \rangle}$ holds only up to some “marginal" number of SAWs steps $N_{{\rm marg}}\sim L^{1/\nu_{{\rm SAW}}^{p_c}}$ [@Blavatska08]. We take this into account when analyzing the data obtained; for each lattice size we are interested only in values of $N<N_{{\rm marg}}$, thus avoiding distortions, caused by finite-size effects. Our results for the scaling exponent $\nu_{{\rm SAW}}^{p_c}$ for SAWs on the backbone of percolation clusters [@Blavatska08] are given in Table \[allnu\] and compared with previous estimates obtained by a variety of different techniques. We see that the value of $\nu_{{\rm SAW}}^{p_c}$ is larger than the corresponding exponent on the pure lattice; presence of disorder leads to stretching of the trajectory of self-avoiding walks.
A simple modified Flory formula for the exponent of a SAW on a percolation cluster, proposed a long time ago by Kremer [@Kremer81], $$\nu_{\rm SAW}^{p_c}=3/(d_{p_c}^F+2),
\label{kremer}$$ gives numbers in an astonishingly good agreement with our numerical data (see Table \[allnu\]). For the estimates we have used the values of the fractal dimension of percolation clusters from Table \[dim\]. Since $d_{\rm up}=6$ for percolation and $d_{p_c}^F(d\geq6)=4$ [@Stauffer], we receive from Eq. (\[kremer\]) that the exponent $\nu_{\rm SAW}^{p_c}(d\geq 6)=1/2$. Note, that there exists a whole family of more sophisticated Flory-like theories [@Kim90; @Sahimi84; @Rammal84; @Kim87; @Roy90; @Roy87; @Aharony89].
The disorder averaged distribution function, defined via $$\overline{\langle r \rangle}=\sum_{r}r {\overline {P(r,N)}}
\label{prob}$$ can be written in terms of the scaled variables $r/\overline{\langle r\rangle}$ as $$r\overline{P(r,N)}\sim f(r/\overline{\langle r\rangle})\sim f(r/N^{\nu_{{\rm SAW}}^{p_c}}).$$ The distribution function is normalized according to $\sum_{r}{\overline {P(r,N)}}{{=}}1$. The numerical results for the distribution function in $d=2$,$3$, and $4$ are shown in Figs. \[prsaw2\], \[prsaw3\], and \[prsaw4\] for different $N$. When plotted against the scaling variable $r/N^{\nu_{{\rm SAW}}^{p_c}}$, the data are indeed found to nicely collapse onto a single curve, using our values for the exponent $\nu_{{\rm SAW}}^{p_c}$ reported in Table \[allnu\].
[@r lll ]{} $z^{p_c}\setminus d$ & 2 & 3 & 4\
SS [@Barat91] & $1.31\pm0.03$ &&\
EE [@Lam90] &$1.53\pm 0.05$ & &\
[@Ordemann00] & $ 1.565\pm0.005 $& $1.462\pm0.005$ &\
MC [@Woo91] & $1.459\pm0.003$ & &\
[@Ordemann00] & $1.456\pm0.005$ & $1.317 \pm 0.005$ &\
our results & $1.566\pm0.005$ & $1.459\pm0.005 $ & $1.340\pm0.005$\
$z\cdot p_c$ &1.564&1.460&1.333\
\[allmu\]
[@ r l l l ]{} $\gamma_{{\rm SAW}}^{p_c} \setminus d$ & 2 & 3 & 4\
FL [@Roy87] & 1.384 & 1.379& 1.27\
EE [@Lam90] & $1.3\pm0.1 $ & &\
[@Ordemann00] & $1.34\pm0.05$ & $1.29\pm0.05$ &\
MC [@Lee89] & $1.31\pm0.03$ & $1.40 \pm 0.02$ &\
[@Ordemann00] & $1.26\pm0.05$ & $1.19\pm 0.05$ &\
our results & $ 1.350\pm 0.008$ & $1.269 \pm 0.008 $ & $1.250\pm0.008 $\
![Averaged connectivity constant for SAWs on the backbone of percolation clusters in $d{=}2$ (triangles), $d{=}3$ (squares), $d{=}4$ (stars).[]{data-label="fugpc"}](fugpc.eps){width="8.2cm"}
![Disorder averaged number of SAWs configurations vs number of steps for SAWs on the backbone of percolation clusters in $d{=}2$ (triangles), $d{=}3$ (squares), $d{=}4$ (stars).[]{data-label="gammapc"}](gammapc.eps){width="8cm"}
Let us now turn our attention to estimating the number of different possible SAW configurations ${\overline{ C_N }}$, defined by Eq. (\[scaling\]). First, let us note, that the fugacity or connectivity constant $z$ is obviously affected by introducing disorder into the lattice. For the case, when the SAW is not confined only to the percolation cluster, namely when averaging over all the clusters is performed, then $z^{p_c}{=}p_c z$ exactly. In Table \[allmu\] we present results of this estimate, taking values for $p_c$ from Table \[dim\]. However, since each existing bond on the infinite percolation cluster has a non-trivial (correlated) probability of occurrence, a similar argument cannot be applied to the SAWs confined to the infinite percolation cluster only. However, enumeration estimates [@Chakrabarti83] suggested $z^{p_c}{\simeq}p_c z$ up to $p_c$ for SAWs on the percolation cluster. It turns out, that this difference from linear dependence for incipient cluster is subtle and could hardly be detected. We have estimated $z_p$ as the averaged number of possibilities to perform the next step in the PERM procedure for SAWs on the backbone of percolation clusters (see Fig. \[fugpc\]); results are presented in Table \[allmu\].
In the analytical study of Ref. [@Lyklema84], it was argued that the exponent $\gamma$, governing the scaling of the number of SAW configurations, is not changed by the presence of disorder even at $p=p_c$. This was supported by an exact enumeration study [@Lam90]. However, this argument disagrees with results of studies [@Roy87; @Ordemann00; @Lee89], where averaging over single percolation clusters was performed and different values for $\gamma_{{\rm SAW}}^{p_c}$ were found. In Ref. [@Roy87] it was proven, using scaling arguments, that at $p=p_c$ the exponents $\gamma_{{\rm SAW}}$ will crossover to $\gamma_{{\rm SAW}}^{p_c}=\gamma_{{\rm SAW}}+d(\nu_{\rm SAW}^{p_c}-\nu_{{\rm SAW}})$ at $p=p_c$; the estimates based on this equality are given in the first row of Table \[allgamma\].
We obtained numerical estimates for $\gamma_{{\rm SAW}}^p$, studying the behavior of the quantity $$\ln \overline{ C_N }/N=\ln(A)/N+\ln(z^{p_c})+(\gamma_{{\rm SAW}}^{p_c}-1)\frac{\ln N}{N},
\label{ggg}$$ where $A$ is a constant. Figure \[gammapc\] shows these values for the backbone of percolation clusters in $d=2,3,4$. Estimates for $\gamma_{{\rm SAW}}^{p_c}$ are obtained by linear least-square fits (see Tables \[g2d\], \[g3d\], \[g4d\] in the Appendix). Our final results are presented in Table \[allgamma\].
Random walks on percolation clusters
====================================
![\[3drw\] Averaged end-to-end distance vs number of steps in a double logarithmic scale of RWs on percolation clusters (crosses) and the backbone of percolation clusters (pluses) in $d=2$.](3drw.eps){width="7.8cm"}
To simulate the diffusion process in a disordered medium, the picture of the “ant in the labyrinth", proposed by de Gennes [@deGennes76] is traditionally used. Here the walker (an “ant") starts at an arbitrary point on the diluted lattice and tries to move randomly to the nearest site. If the randomly chosen direction leads to an empty site, it moves and the steps increment by 1. If the chosen site is occupied by a defect (in our case, does not belong to the percolation cluster) it stays at the current position for this time step. The growth is stopped, if the total number of steps $N$ is performed, than the next trajectory is started to grow.
After averaging the end-to-end distance over RW configurations on a single percolation cluster, the disorder average is carried out as in Eq. (\[av\]) over all constructed percolation clusters. Let us note that, in contrast to the SAW problem, discussed previously, the scaling behavior of RWs on a percolation cluster is different from that on its backbone. Let us remind, that statistics of long SAWs on percolation clusters is nevertheless determined by its backbone, since the walks, which visit the “dead ends" are eventually terminated after a certain number of steps. Simple random walks cannot be trapped in “dead ends", however, visiting these parts of a cluster will lead to some “slowing down" of the diffusion process in comparison with the behavior on the backbone where all the dead ends are removed. This is really confirmed by analyzing our results for the averaged end-to-end distance of random walks on a percolation cluster and its backbone (see Fig. \[3drw\]).
![Disorder averaged distribution function $r\overline{P(r,N)}$ vs the scaling variable $r/\overline{\langle r\rangle}$ in $d=2$ dimensions, left: percolation cluster, right: backbone of percolation cluster. Lattice size $L=400$, number of RW steps $N{{=}}180$ (squares), $N{{=}}160$ (triangles), $N{{=}}140$ (crosses).[]{data-label="prrw2"}](rozrwpc2d.eps "fig:"){width="6.4cm"} ![Disorder averaged distribution function $r\overline{P(r,N)}$ vs the scaling variable $r/\overline{\langle r\rangle}$ in $d=2$ dimensions, left: percolation cluster, right: backbone of percolation cluster. Lattice size $L=400$, number of RW steps $N{{=}}180$ (squares), $N{{=}}160$ (triangles), $N{{=}}140$ (crosses).[]{data-label="prrw2"}](rozrwback2d.eps "fig:"){width="6.8cm"}
![Disorder averaged distribution function $r\overline{P(r,N)}$ vs the scaling variable $r/\overline{\langle r\rangle}$ in $d=3$ dimensions, left: percolation cluster, right: backbone of percolation cluster. Lattice size $L=200$, number of RW steps $N{{=}}100$ (squares), $N{{=}}90$ (triangles), $N{{=}}80$ (crosses).[]{data-label="prrw3"}](rozrwpc3d.eps "fig:"){width="7.1cm"} ![Disorder averaged distribution function $r\overline{P(r,N)}$ vs the scaling variable $r/\overline{\langle r\rangle}$ in $d=3$ dimensions, left: percolation cluster, right: backbone of percolation cluster. Lattice size $L=200$, number of RW steps $N{{=}}100$ (squares), $N{{=}}90$ (triangles), $N{{=}}80$ (crosses).[]{data-label="prrw3"}](rozrwback3d.eps "fig:"){width="6.58cm"}
![Disorder averaged distribution function $r\overline{P(r,N)}$ vs the scaling variable $r/\overline{\langle r\rangle}$ in $d=4$ dimensions, left: percolation cluster, right: backbone of percolation cluster. Lattice size $L=50$, number of RW steps $N{{=}}60$ (squares), $N{{=}}50$ (triangles), $N{{=}}40$ (crosses).[]{data-label="prrw4"}](rozrwpc4d.eps "fig:"){width="6.8cm"} ![Disorder averaged distribution function $r\overline{P(r,N)}$ vs the scaling variable $r/\overline{\langle r\rangle}$ in $d=4$ dimensions, left: percolation cluster, right: backbone of percolation cluster. Lattice size $L=50$, number of RW steps $N{{=}}60$ (squares), $N{{=}}50$ (triangles), $N{{=}}40$ (crosses).[]{data-label="prrw4"}](rozrwback4d.eps "fig:"){width="7.1cm"}
We have studied RWs both on the percolation cluster and its backbone, performing $10^7$ realizations on each cluster and average over $1000$ clusters in each space dimensions $d=2,3,4$. Estimates of scaling exponents $\nu_{{\rm RW}}^{p_c}$ and $\nu_{{\rm RW}}^{B}$, describing scaling of walks on percolation cluster and backbone, respectively, are obtained by linear least-square fitting and given in Table \[nurw\]. One can see, that the inequality $\nu_{{\rm RW}}^{p_c} < \nu_{{\rm RW}}^{B}$ holds, and that the quantitative difference between these two values grows with increasing the space dimension $d$. On the other hand, both values are [*smaller*]{} than the corresponding exponent $\nu_{{\rm RW}}=1/2$, governing scaling of random walks on the pure lattice: presence of disorder slows down the diffusion process. The reason for this subdiffusive behavior is intuitively clear: due to the presence of defects, the randomly walking particle returns back to already visited sites more often, thus its walking distance is shorter than on the pure lattice. This has also been observed in recent studies of less disordered deterministic fractals such as two-dimensional Sierpinski carpet composites [@Anh07].
The disorder averaged distribution function, defined in Eq. (\[prob\]), and rewritten in terms of the scaled variables $r/\overline{\langle r\rangle}$ as: $$r\overline{P(r,N)}\sim f(r/\overline{\langle r\rangle})\sim f(r/N^{\nu^{p_c}})$$ is shown in Figs. \[prrw2\], \[prrw3\] and \[prrw4\] for $d=2,3,4$, both for the cases of percolation cluster and backbone. When plotted against the scaling variable $r/N^{\nu^{p_c}}$, the data are indeed found to nicely collapse onto a single curve, using our values for the exponent $\nu_{{\rm RW}}^{p_c}, \nu_{{\rm RW}}^{B}$ reported in Table \[nurw\].
[@ r l l l l ]{} $\nu_{{\rm RW}}^{p_c}\setminus d$ & 2 & 3 & 4\
RS [@Sahimi83]& 0.356 & 0.285 &\
EE [@Majid84] & 0.349$\pm0.002$ &&\
[@Pandey83] & & 0.266$\pm0.01$ &\
analytic [@Alexander82] & 0.352 & 0.268 &\
MC [@Avraham82] & $0.352\pm0.006$ & $0.271\pm0.004$ &\
[@Havlin83] &$0.352\pm0.006$ &$0.271\pm0.004$&\
[@Argyrakis84] & $0.392\pm 0.007$ & $0.282\pm0.003$ &\
[@McCarthy88] & $0.348\pm0.009$ & $0.274\pm0.007$ &\
[@Lee00]& & & $0.222\pm0.007$\
our results &$ 0.353\pm 0.003 $&$ 0.273\pm 0.003$&$ 0.231 \pm 0.003$\
$\nu_{{\rm RW}}^{B}\setminus d$ &2&3&4\
analytic [@Bug86] & $0.371\pm0.001$ &&\
EE [@Hong84] & $0.372\pm0.005$ &&\
MC [@Mastorakos93] &$0.370\pm0.003$ &&\
our results &$0.372\pm 0.002 $&$ 0.306\pm0.002$ & $ 0.262\pm0.002$\
Conclusions
===========
We studied the scaling behavior of simple random walks and self-avoiding walks on disordered lattices. Both models are of great interest: RWs provide a good description of diffusion processes, SAWs are successful in describing the universal properties of long flexible polymer macromolecules in a good solvent.
We consider the case, when concentration $p$ of lattice sites allowed for walking equals the critical concentration $p_{c}$ and the incipient percolation cluster can be found in the system. Studying properties of percolative lattices, one encounters two possible statistical averages: in the first, one considers only percolation clusters with linear size much larger than the typical length of the physical phenomenon under discussion, the other statistical ensemble includes all the clusters, which can be found on a percolative lattice. In our study, we considered only the first case, being interested in random and self-avoiding walks on a percolation cluster, which has a fractal structure. In this regime, the critical behavior of both models belongs to a new universality class: scaling law exponents change with the dimension of the (fractal) lattice on which the walk resides.
We performed numerical simulations of random and self-avoiding walks on percolation clusters and the backbone of percolation clusters on lattice sizes $L=400, 200, 50$ in space dimensions $d=2,3,4$, respectively. Our results bring about the estimates for critical exponents, governing the scaling behavior of the models. We found that the statistics of SAWs is governed by the same scaling exponent both on a percolation cluster and its backbone: since the walks, which visit the dead ends are eventually terminated after a certain number of steps, the walks that survive in the limit $N\to \infty$ on a percolation cluster are those confined within its backbone. For simple random walks, however, the picture is different: they cannot be trapped in “dead ends". However, visiting these parts of a cluster will lead to some “slowing down" of the diffusion process in comparison with the behavior on the backbone where all the dead ends are removed. We found that the inequality $\nu_{{\rm RW}}^{p_c} < \nu_{{\rm RW}}^{B}$ holds, and the quantitative difference between these two values grows with increasing space dimension $d$.
The presence of disorder leads to a stretching of the trajectory of self-avoiding walks: the value of $\nu_{{\rm SAW}}^{p_c}$ is larger than the corresponding exponent on the pure lattice. However, the exponent $\nu_{{\rm RW}}^{p_c}$, governing scaling of random walks on percolative lattices is smaller than that on a pure lattice: presence of disorder slows down the diffusion process. This can be explained as follows: due to the presence of defects, the randomly walking particle returns back to already visited sites more often, thus its walking distance is shorter than on the pure lattice.
Acknowledgments
===============
Work supported in part by the German Science Foundation (DFG) through the Research Group FOR877. V.B. is grateful for support through the “Marie Curie International Incoming Fellowship" EU Programme and to the Institut für Theoretische Physik, Universität Leipzig, for hospitality.
Appendix
========
To estimate the critical exponents $\nu_{{\rm SAW}}^{p_c}$, $\gamma_{{\rm SAW}}^{p_c}$, linear least-square fits with varying lower cutoff for the number of steps $N_{{\rm min}}$ are used in order to detect possible corrections to scaling. For estimating $\nu_{{\rm SAW}}^{p_c}$ we use linear fits for the averaged end-to-end distance $\ln({\overline{\langle r(N) \rangle}})=\ln(A)+\nu_{\rm SAW}^{p_c}\ln N$, and for $\gamma_{{\rm SAW}}^{p_c}$ we employ Eq. (\[ggg\]). Since this is an important aspect for assessing the quality of our final exponent estimates discussed in the main text, we have compiled in this Appendix these more detailed results in Tables \[2d\]-\[g4d\]. The $\chi^2$ value (sum of squares of normalized deviation from the regression line) divided by the number of degrees of freedom, DF, given in the last rows, serves as a test of the goodness of fit.
[@ rlll]{} $N_{\rm min}$ & $\nu_{{\rm SAW}}^{p_c}$ & $A$ & $\chi^2/DF$\
11 & 0.790 $\pm$ 0.005 & 0.829 $\pm$ 0.003 & 2.396\
16 & 0.786 $\pm$ 0.005 & 0.841 $\pm$ 0.005 & 1.910\
21 & 0.782 $\pm$ 0.004 & 0.847 $\pm$ 0.005 & 1.479\
26 & 0.783 $\pm$ 0.003 & 0.842 $\pm$ 0.006 & 1.262\
31 & 0.782 $\pm$ 0.003 & 0.840 $\pm$ 0.007 & 0.839\
\[2d\]
[@rlll]{} $N_{{\rm min}}$ & $\nu_{{\rm SAW}}^{p_c}$ & $A$ & $\chi^2/DF$\
11 & 0.668 $\pm$ 0.003 & 0.935 $\pm$ 0.004 & 2.269\
16 & 0.669 $\pm$ 0.003 & 0.930 $\pm$ 0.004 & 2.054\
21 & 0.669 $\pm$ 0.003 & 0.924 $\pm$ 0.004 & 1.345\
26 & 0.667 $\pm$ 0.002 & 0.930 $\pm$ 0.006 & 0.743\
31 & 0.668 $\pm$ 0.002 & 0.934 $\pm$ 0.008 & 0.844\
[@rlll]{} $N_{\rm min}$ & $\nu_{{\rm SAW}}^{p_c}$ & $A$ & $\chi^2/DF$\
8 & 0.588 $\pm$ 0.002 & 1.025 $\pm$ 0.004 & 2.615\
10 & 0.587 $\pm$ 0.002 & 1.023 $\pm$ 0.006 & 1.777\
12 & 0.586 $\pm$ 0.003 & 1.021 $\pm$ 0.01 & 0.978\
14 & 0.586 $\pm$ 0.003 & 1.031 $\pm$ 0.01 & 0.767\
\[4d\]
[@rlll]{} $N_{\rm min}$ & $\gamma_{{\rm SAW}}^{p_c}$ & $A$ & $\chi^2/DF$\
16 & $1.341 \pm 0.005$ & $1.219 \pm 0.003$ & 3.135\
21 & $ 1.349 \pm 0.005$ & $1.189 \pm 0.003$ & 2.682\
26 & $1.351 \pm 0.007$ & $1.168 \pm 0.002$ & 1.913\
31 & $1.352 \pm 0.008$ & $ 1.172 \pm 0.002$ & 1.621\
36 & $1.350 \pm 0.008 $ & $1.163 \pm 0.001$ & 0.704\
\[g2d\]
[@rlll]{} $N_{\rm min}$ & $\gamma_{{\rm SAW}}^{p_c}$ & $A$ & $\chi^2/DF$\
11 & 1.265 $\pm$ 0.004 & 1.82 $\pm$ 0.003& 2.767\
16 & 1.268 $\pm$ 0.005 & 1.192 $\pm$ 0.003 & 2.135\
21 & 1.270 $\pm$ 0.006 & 1.184 $\pm$ 0.003 & 1.968\
26 & 1.267 $\pm$0.008 & 1.176 $\pm$0.002 & 1.513\
31 & 1.269 $\pm$0.008 & 1.172 $\pm$ 0.002 $\pm$ & 0.762\
\[g3d\]
[@rlll]{} $N_{\rm min}$ & $\gamma_{{\rm SAW}}^{p_c}$ & $A$ & $\chi^2/DF$\
8 & 1.251 $\pm$ 0.005 & 1.77 $\pm$ 0.003& 1.767\
10 & 1.252 $\pm$ 0.007 & 1.182 $\pm$ 0.003 & 1.135\
12 & 1.250 $\pm$ 0.008 & 1.184 $\pm$ 0.003 & 0.968\
\[g4d\]
[90]{}
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| {
"pile_set_name": "ArXiv"
} |
**Orbit functions of ${\rm SU}(n)$ and Chebyshev polynomials**
*Maryna NESTERENKO $^\dag$, Jiří PATERA $^\ddag$ and Agnieszka TERESZKIEWICZ $^\S$*
$^\dag$ Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivs’ka Str., Kyiv-4, 01601 Ukraine
$^\ddag$ Centre de recherches mathématiques, Université de Montréal, C.P.6128-Centre ville, Montréal,\
H3C3J7, Québec, Canada
$^\S$ Institute of Mathematics, University of Bialystok, Akademicka 2, PL-15-267 Bialystok, Poland $\phantom{^\S}$ E-mail: [email protected], [email protected], [email protected]
Introduction {#Introduction}
============
The history of the Chebyshev polynomials dates back over a century. Their properties and applications have been considered in many papers. We refer to [@Shahat; @Rivlin1974] as a basic reference. Studies of polynomials in more than one variable were undertaken by several authors, namely [@Dunkl; @DunnLidl1980; @EierLidl; @Koornwinder1-4; @MasonHandscomb2003; @suetin; @suetin2]. Of these, none follow the path we have laid down here.
In this paper, we demonstrate that the classical Chebyshev polynomials in one variable are naturally associated with the action of the Weyl group of ${\rm SU}(2)$, or equivalently with the action of the Weyl group $W(A_1)$ of the simple Lie algebra of type $A_1$. The association is so simple that it has been ignored so far. However, by making $W(A_1)$ the cornerstone of our rederivation of Chebyshev polynomials, we have gained insight into the structure of the theory of polynomials. In particular, the generalization of Chebyshev polynomials to any number of variables was a straightforward task. It is based on the Weyl group $W(A_n)$, where $n<\infty$. This only recently became possible, after the orbit functions of simple Lie algebras were introduced as useful special functions [@Patera] and studied in great detail and generality [@KlimykPatera2006; @KlimykPatera2007-1; @KlimykPatera2008].
We proceed in three steps. In Section 2, we exploit the isomorphism of the group of permutations of $n+1$ elements ${\rm S}$ and the Weyl group of ${\rm SU}(n+1)$, or equivalently of $A_n$, and define the orbit functions of $A_n$. This opens the possibility to write the orbit functions in two rather different bases, the orthnormal basis, and the basis determined by the simple roots of $A_n$, which considerably alters the appearance of the orbit functions. In the paper, we use the non-orthogonal basis because of its direct generalization to simple Lie algebras of other types than $A_n$.
In Section 3 we consider classical Chebyshev polynomials of the first and second kind, and compare them with the $C$- and $S$-orbit functions of $A_1$. We show that polynomials of the first kind are in one-to-one correspondence with $C$-functions. Polynomials of the second kind coincide with the appropriate $S$-function divided by the unique lowest non-trivial $S$-function. We point out that polynomials of the second kind can be identified as irreducible characters of finite dimensional representations of ${\rm SU}(2)$. Useful properties of Chebyshev polynomials can undoubtedly be traced to that identification, because the fundamental object of representation theory of semisimple Lie groups/algebras is character. In principle, all one needs to know about an irreducible finite dimensional representation can be deduced from its character. An important aspect of this conclusion is that characters are known and uniformly described for all simple Lie groups/algebras.
In Section 4 we provide details of the recursive procedure from which the analog of the trigonometric form of Chebyshev polynomials in $n$ variables can be found. Thus there are $n$ generic recursion relations for $A_n$, having at least $n+2$ terms, and at most $\left(\begin{smallmatrix}n+1\\ [(n+1)/2]\\\end{smallmatrix}\right)+1$ terms. Irreducible polynomials are divided into $n+1$ exclusive classes with the property that monomials within one irreducible polynomial belong to the same class. This follows directly from the recognition of the presence and properties of the underlying Lie algebra.
In subsection 4.2, the simple substitution $z=e^{2\pi ix}$, $x\in\R^n$, is used in orbit functions to form analogs of Chebyshev polynomials in $n$ variables in their non-trigonometric form. It is shown that, in the case of 2 variables, our polynomials coincide with those of Koorwider [@Koornwinder1-4](III), although the approach and terminology could not be more different, ours being purely algebraic, having originated in Lie theory.
In Section 5, we present the orbit functions of $A_n$ disguised as polynomials built from multivariate orbit functions of the symmetric group. In Section 2, such a possibility is described in terms of related bases, one orthonormal (symmetric group), the other non-orthogonal (simple roots of $A_n$ and their dual $\omega$-basis). Both forms of the same polynomials appear rather different but may prove useful in different situations.
The last section contains a few comments and some questions related to the subject of this paper that we find intriguing.
Preliminaries
=============
This section is intended to fix notation and terminology. We also briefly recall some facts about ${\rm S}_{n+1}$ and $A_{n}$, dwelling particularly on various bases in $\R^{n+1}$ and $\R^n$. In Section \[ssec\_Weyl\_group\], we identify elementary reflections that generate the $A_n$ Weyl group $W$, with the permutation of two adjacent objects in an ordered set of $n+1$ objects. And, finally, we present some standard definitions and properties of orbit functions.
Permutation group ${\rm S}_{n+1}$
---------------------------------
The group ${\rm S}_{n+1}$ of order $(n+1)!$ transforms the ordered number set $[l_1,l_2,\dots,l_n,l_{n+1}]$ by permuting the numbers.
We introduce an orthonormal basis in the real Euclidean space $\R^{n+1}$, $${e_i}\in\R^{n+1}\,,\qquad
\l e_i , e_j\r=\delta_{ij}\,,\qquad 1\leq i,j\leq n+1\,,$$ and use the $l_k$’s as the coordinates of a point $\mu$ in the $e$-basis: $$\mu=\sum_{k=1}^{n+1}l_ke_k\,,\qquad l_k\in\R\,.$$
The group ${\rm S}_{n+1}$ permutes the coordinates $l_k$ of $\mu$, thus generating other points from it. The set of all distinct points, obtained by application of ${\rm S}_{n+1}$ to $\mu$, is called the orbit of ${\rm S}_{n+1}$. We denote an orbit by $W_\lambda$, where $\lambda$ is a unique point of the orbit, such that $$l_1\geq l_2\geq\cdots\geq l_n\geq l_{n+1}\,.$$ If there is no pair of equal $l_k$’s in $\lambda$, the orbit $W_\lambda$ consists of $(n+1)!$ points.
Further on, we will only consider points $\mu$ from the $n$-dimensional subspace ${\mathcal H}\subset\R^{n+1}$ defined by the equation $$\begin{gathered}
\label{plane H}
\sum_{k=1}^{n+1}l_k=0.\end{gathered}$$
Lie algebra $A_n$
-----------------
Let us recall basic properties of the simple Lie algebra $A_n$ of the compact Lie group ${\rm SU}(n+1)$. Consider the general value $(1\leq n<\infty)$ of the rank. The Coxeter-Dynkin diagram, Cartan matrix $\mathfrak{C}$, and inverse Cartan matrix $\mathfrak{C}^{-1}$ of $A_n$ are as follows: $$\begin{gathered}
\parbox{.6\linewidth}
{\setlength{\unitlength}{1pt}
\def\kr{\circle{10}}
\thicklines
\begin{picture}(20,30)
\put(10,14){\kr}
\put(6,0){$\alpha_1$}
\put(15,14){\line(1,0){10}}
\put(30,14){\kr}
\put(26,0){$\alpha_2$}
\put(35,14){\line(1,0){10}}
\put(50,14){\kr}
\put(46,0){$\alpha_3$}
\put(55,14){\line(1,0){10}}
\put(70,13.5){$\ \,\ldots$}
\put(95,14){\line(1,0){10}}
\put(110,14){\kr}
\put(104,0){$\alpha_{n\!-\!1}$}
\put(115,14){\line(1,0){10}}
\put(130,14){\kr}
\put(126,0){$\alpha_n$}
\end{picture}
}
\hspace{-110 pt}
\mathfrak{C}{=}\left(\begin{smallmatrix}
\phantom{-}2&{-}1&\phantom{-}0&\phantom{-}0&\phantom{-}0&\dots&\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0\\
{-}1&\phantom{-}2&{-}1&\phantom{-}0&\phantom{-}0&\dots&\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0\\
\phantom{-}0&{-}1&\phantom{-}2&{-}1&\phantom{-}0&\dots&\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0\\[-1ex]
\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\[0.5ex]
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0&\dots&\phantom{-}0&{-}1&\phantom{-}2&{-}1\\
\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0&\phantom{-}0&\dots&\phantom{-}0&\phantom{-}0&{-}1&\phantom{-}2
\end{smallmatrix}\right),
\\[2ex]
\mathfrak{C}^{-1}=\frac{1}{n+1}\left(\begin{smallmatrix}
1\cdot n\; &1\cdot(n-1)\;&1\cdot(n-2)\;&1\cdot(n-3)\;&\dots &1\cdot 3\; &1\cdot 2\; &1\cdot 1\\
1\cdot (n-1)\;&2\cdot(n-1)\;&2\cdot(n-2)\;&2\cdot(n-3)\;&\dots &2\cdot 3\; &2\cdot 2\; &2\cdot 1\\
1\cdot (n-2)\;&2\cdot(n-2)\;&3\cdot(n-2)\;&3\cdot(n-3)\;&\dots &3\cdot 3\; &3\cdot 2\; &3\cdot 1\\
1\cdot (n-3)\;&2\cdot(n-3)\;&3\cdot(n-3)\;&4\cdot(n-3)\;&\dots &4\cdot 3\; &4\cdot 2\; &4\cdot 1\\[-1 ex]
\vdots &\vdots &\vdots &\vdots &\ddots&\vdots &\vdots &\vdots\\[0.5ex]
1\cdot 3\; &2\cdot3\; &3\cdot3\; &4\cdot3\; &\dots &(n-2)\cdot3\;&(n-2)\cdot2\;&(n-2)\cdot 1\\
1\cdot 2\; &2\cdot2\; &3\cdot2\; &4\cdot2\; &\dots &(n-2)\cdot2\;&(n-1)\cdot2\;&(n-1)\cdot 1\\
1\cdot 1\; &2\cdot1\; &3\cdot1\; &4\cdot1\; &\dots &(n-2)\cdot1\;&(n-1)\cdot1\;&n\cdot 1\
\end{smallmatrix}\right).\end{gathered}$$
The simple roots $\alpha_i$, $1\le i\le n$ of $A_n$ form a basis ($\a$-basis) of a real Euclidean space $\R^n$. We choose them in ${\mathcal H}$: $$\begin{gathered}
\alpha_i = e_i - e_{i+1}, \quad i=1,\dots,n.\end{gathered}$$ This choice fixes the lengths and relative angles of the simple roots. Their length is equal to $\sqrt 2$ with relative angles between $\alpha_k$ and $\alpha_{k+1}$ $(1\leq k\leq n-1)$ equal to $\frac{2\pi}{3}$, and $\tfrac\pi2$ for any other pair.
In addition to $e$- and $\a$-bases, we introduce the $\w$-basis as the $\Z$-dual basis to the simple roots $\alpha_i$: $$\begin{gathered}
\l\alpha_i,\w_j\r=\delta_{ij},\quad 1\leq i,j\leq n.\end{gathered}$$ It is also a basis in the subspace ${\mathcal H}\subset\R^{n+1}$ (see (\[plane H\])). The bases $\alpha$ and $\w$ are related by the Cartan matrix: $$\begin{gathered}
\alpha=\mathfrak{C}\w,\quad \w=\mathfrak{C}^{-1}\alpha.\end{gathered}$$
Throughout the paper, we use $\lambda\in\mathcal H$. Here, we fix the notation for its coordinates relative to the $e$- and $\omega$-bases: $$\begin{gathered}
\label{notation_bases}
\lambda=\sum_{j=1}^{n+1}l_je_j=:(l_1,\ldots,l_{n+1})_e=\sum_{i=1}^n\lambda_i\w_i=:(\lambda_1,\ldots,\lambda_n)_\w,
\qquad \sum_{i=1}^{n+1}l_i=0.\end{gathered}$$
Consider a point $\lambda\in {\mathcal H}$ with coordinates $l_j$ and $\lambda_i$ in the $e$- and $\w$-bases, respectively. Using $\alpha=\mathfrak{C}\w$, i.e. $\w_i=\sum^n_{k=1}(\mathfrak{C}^{-1})_{ik}\alpha_k$, we obtain the relations between $\lambda_i$ and $l_j$: $$\begin{gathered}
l_1=\sum^n_{k=1}\lambda_k\mathfrak{C}^{{-}1}_{k1},\qquad
l_{n{+}1}=-\sum^n_{k=1}\lambda_k\mathfrak{C}^{{-}1}_{kn},
\\
l_j{=}\lambda_{1}(\mathfrak{C}^{{-}1}_{1\,j}{-}\mathfrak{C}^{{-}1}_{1\,j{-}1})
{+}\lambda_{2}(\mathfrak{C}^{{-}1}_{2\,j}{-}\mathfrak{C}^{{-}1}_{2\,j{-}1}){+}\!\cdots\!
{+}\lambda_n(\mathfrak{C}^{{-}1}_{n\,j}{-}\mathfrak{C}^{{-}1}_{n\,j{-}1}),
\quad j=2,\dots,n.
$$ or explicitly, $$\begin{gathered}
\label{l_to_lambda}
\lambda_i=l_i-l_{i+1},\qquad i=1,2,\ldots ,n.\end{gathered}$$ The inverse formulas are much more complicated $$\begin{gathered}
\label{lambda_to_l}
l=A\lambda,\end{gathered}$$ where $l=(l_1,\dots,1_{n+1})$, $\lambda=(\lambda_1,\dots,\lambda_n)$, and $A$ is the $(n{+}1)\times n$ matrix: $$\begin{gathered}
A=\tfrac{1}{n+1}\left(
\begin{smallmatrix}
n&n-1&n-2&\cdots&2&1\\
{-}1&n-1&n-2&\cdots&2&1\\
{-}1&{-}2&n-2&\cdots&2&1\\[-1 ex]
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\
{-}1&{-}2&{-}3&\cdots&{-}(n-1)&1\\
{-}1&{-}2&{-}3&\cdots&{-}(n-1)&{-}n
\end{smallmatrix}
\right).\end{gathered}$$
The Weyl group of $A_n$ {#ssec_Weyl_group}
-----------------------
The Weyl group $W(A_n)$ of order $(n+1)!$ acts in ${\mathcal H}$ by permuting coordinates in the $e$-basis, i.e. as the group ${\rm S}_{n+1}$. Indeed, let $r_i$, $1\le i\le n$ be the generating elements of $W(A_n)$, i.e, reflections with respect to the hyperplanes perpendicular to $\alpha_i$ and passing through the origin. Let $x=\sum\limits_{k=1}^{n+1}x_ke_k=(x_1,x_2,\dots,x_{n+1})_e$ and $\langle\cdot,\cdot\rangle$ denote the inner product. We then have the reflection by $r_i$: $$\begin{aligned}
r_i x&= x-\tfrac{2}{\langle \alpha_i,\alpha_i\rangle}\langle x,\alpha_i\rangle\alpha_i
=(x_1,x_2,\dots,x_{n+1})_e-(x_i-x_{i+1})(e_i-e_{i+1})\\
&=(x_1,\dots,x_{i-1},x_{i+1},x_i,x_{i+2},\dots,x_{n+1})_e.
\end{aligned}$$ Such transpositions generate the full permutation group ${\rm S}_{n+1}$. Thus, $W(A_n)$ is isomorphic to ${\rm S}_{n+1}$, and the points of the orbit $W_\lambda({\rm S}_{n+1})$ and $W_\lambda(A_n)$ coincide.
Definitions of orbit functions {#ssec_def_orb_func}
------------------------------
The notion of an orbit function in $n$ variables depends essentially on the underlying semisimple Lie group $G$ of rank $n$. In our case, $G={\rm SU}(n+1)$ (equivalently, Lie algebra $A_n$). Let the basis of the simple roots be denoted by $\alpha$, and the basis of fundamental weights by $\omega$.
The *weight lattice* $P$ is formed by all integer linear combinations of the $\omega$-basis, $$\begin{gathered}
P=\Z\omega_1+\Z\omega_2+\cdots+\Z\omega_n.\end{gathered}$$ In the weight lattice $P$, we define the *cone of dominant weights* $P^+$ and its subset of strictly dominant weights $P^{++}$ $$\begin{gathered}
P\;\supset\; P^+=\Z^{\ge 0}\omega_1+\cdots+\Z^{\ge 0}\omega_n
\;\supset\; P^{++}=\Z^{>0}\omega_1+\cdots+\Z^{>0}\omega_n.\end{gathered}$$
Hereafter, $W^e\subset W$ denotes the *even subgroup* of the Weyl group formed by an even number of reflections that generate $W$. $W_{\lambda}$ and $W^e_{\lambda}$ are the corresponding group orbits of a point $\lambda\in \R^n$.
We also introduce the notion of fundamental region $F(G)\subset \R^n$. For $A_n$ the *fundamental region* $F$ is the convex hull of the vertices $\{0,\omega_1,\omega_2,\ldots, \omega_n\}$.
The $C$ orbit function $C_{\lambda}(x)$, $\lambda\in P^+$ is defined as $$\begin{gathered}
\label{def_c-function1}
C_\lambda(x) := \sum_{\mu\in W_\lambda(G)} e^{2\pi i \l\mu, x\r},
\qquad
x\in\R^n.\end{gathered}$$
The $S$ orbit function $S_{\lambda}(x)$, $\lambda\in P^{++}$ is defined as $$\begin{gathered}
\label{def_s-function1}
S_\lambda(x) := \sum_{\mu\in W_\lambda(G)} (-1)^{p(\mu)}e^{2\pi i \l\mu , x\r},\qquad
x\in\R^n,\end{gathered}$$ where $p(\mu)$ is the number of reflections necessary to obtain $\mu$ from $\lambda$. Of course the same $\mu$ can be obtained by different successions of reflections, but all routes from $\lambda$ to $\mu$ will have a length of the same parity, and thus the salient detail given by $p(\mu)$, in the context of an $S$-function, is meaningful and unchanging.
We define $E$ orbit function $E_{\lambda}(x)$, $\lambda\in P^e$ as $$\begin{gathered}
\label{def_e-function1}
E_\lambda(x) := \sum_{\mu\in W^e_{\lambda}(G)} e^{2\pi i \l\mu, x\r},
\qquad
x\in\R^n,\end{gathered}$$ where $P^e:=P^+\cup r_i P^+$ and $r_i$ is a reflection from $W$.
If we always suppose that $\lambda, \mu\in P$ are given in the $\w$-basis, and $x\in\R^n$ is given in the $\alpha$ basis, namely $\lambda=\sum\limits^n_{j=1}\lambda_j\w_j$, $\mu=\sum\limits^n_{j=1}\mu_j\w_j$, $\lambda_j, \mu_j\in\Z$ and $x=\sum\limits^n_{j=1}x_j\alpha_j$, $x_j\in \R$, then the orbit functions of $A_n$ have the following forms $$\begin{gathered}
C_\lambda(x)
= \sum_{\mu\in W_\lambda} e^{2\pi i \sum\limits^n_{j=1}\mu_jx_j}
= \sum_{\mu\in W_\lambda} \prod\limits^n_{j=1} e^{2\pi i \mu_jx_j},\label{def_c-function2}
\\
S_\lambda(x)
= \sum_{\mu\in W_\lambda} (-1)^{p(\mu)}e^{2\pi i \sum\limits^n_{j=1}\mu_jx_j}
= \sum_{\mu\in W_\lambda} (-1)^{p(\mu)}\prod\limits^n_{j=1} e^{2\pi i \mu_jx_j},\label{def_s-function2}
\\
E_\lambda(x)
= \sum_{\mu\in W^e_{\lambda}} e^{2\pi i \sum\limits^n_{j=1}\mu_jx_j}
= \sum_{\mu\in W^e_{\lambda}} \prod\limits^n_{j=1} e^{2\pi i \mu_jx_j}.\label{def_e-function2}\end{gathered}$$
Some properties of orbit functions
----------------------------------
For $S$ functions, the number of summands is always equal to the size of the Weyl group. Note that in the 1-dimensional case, $C$-, $S$- and $E$-functions are respectively a cosine, a sine and an exponential functions up to the constant.
All three families of orbit functions are based on semisimple Lie algebras. The number of variables coincides with the rank of the Lie algebra. In general, $C$-, $S$- and $E$- functions are finite sums of exponential functions. Therefore they are continuous and have continuous derivatives of all orders in $\R^n$.
The $S$-functions are antisymmetric with respect to the $(n{-}1)$-dimensional boundary of $F$. Hence they are zero on the boundary of $F$. The $C$-functions are symmetric with respect to the $(n-1)$-dimensional boundary of $F$. Their normal derivative at the boundary is equal to zero (because the normal derivative of a $C$-function is an $S$-function).
For simple Lie algebras of any type, the functions $C_\lambda(x)$, $E_\lambda(x)$ and $S_\lambda(x)$ are eigenfunctions of the appropriate Laplace operator. The Laplace operator has the same eigenvalues on every exponential function summand of an orbit function with eigenvalue $-4\pi\langle \lambda,\lambda\rangle$.
### Orthogonality
For any two complex squared integrable functions $\phi(x)$ and $\psi(x)$ defined on the fundamental region $F$, we define a continuous scalar product $$\begin{gathered}
\label{def_cont_scalar_product}
\l\phi(x),\psi(x)\r:=\int\limits_{{F}}\phi(x)\overline{\psi(x)}{\rm d}x.\end{gathered}$$ Here, integration is carried out with respect to the Euclidean measure, the bar means complex conjugation and $x\in {F}$, where ${F}$ is the fundamental region of either $W$ or $W^e$ (note that the fundamental region of $W^e$ is $F^e=F\cup r_i F$, where $r_i\in W$).
Any pair of orbit functions from the same family is orthogonal on the corresponding fundamental region with respect to the scalar product (\[def\_cont\_scalar\_product\]), namely $$\begin{gathered}
\label{cont_orthog c funcs}
\l C_{\lambda}(x),C_{\lambda'}(x)\r=|W_\lambda|\cdot|F|\cdot\delta_{\lambda\lambda'},
\\\label{cont_orthog s funcs}
\l S_{\lambda}(x),S_{\lambda'}(x)\r=|W|\cdot|F|\cdot\delta_{\lambda\lambda'},
\\\label{cont_orthog e funcs}
\l E_{\lambda}(x),E_{\lambda'}(x)\r=|W^e_{\lambda}|\cdot|F^e|\cdot\delta_{\lambda\lambda'},\end{gathered}$$ where $\delta_{\lambda\lambda'}$ is the Kronecker delta, $|W|$ is the order of the Weyl group, $|W_{\lambda}|$ and $|W^e_{\lambda}|$ are the sizes of the Weyl group orbits (the number of distinct points in the orbit), and $|F|$ and $|F^e|$ are volumes of fundamental regions. The volume $|F|$ was calculated in [@HrivnakPatera2009].
Proof of the relations (\[cont\_orthog c funcs\],\[cont\_orthog s funcs\],\[cont\_orthog e funcs\]) follows from the orthogonality of the usual exponential functions and from the fact that a given weight $\mu\in P$ belongs to precisely one orbit function.
The families of $C$-, $S$- and $E$-functions are complete on the fundamental domain. The completeness of these systems follows from the completeness of the system of exponential functions; i.e., there does not exist a function $\phi(x)$, such that $\langle\phi(x),\phi(x)\rangle>0$, and at the same time $\langle\phi(x),\psi(x)\rangle=0$ for all functions $\psi(x)$ from the same system.
### Orbit functions of $A_n$ acting in $\R^{n+1}$ {#orb_func_n+1}
Relations (\[lambda\_to\_l\]) allow us to rewrite variables $\lambda$ and $x$ in an orbit function in the $e$-basis. Therefore we can obtain the $C$-, $S$- and $E$- functions acting in $\R^{n+1}$ $$\begin{gathered}
C_\lambda(x) = \sum_{s\in {\rm S}_{n+1}} e^{2\pi i (s(\lambda), x)},\\
C_\lambda(x) = \sum_{s\in {\rm S}_{n+1}} ({\rm sgn}\,s)e^{2\pi i (s(\lambda), x)},\\
E_\lambda(x) = \sum_{s\in {\rm Alt}_{n{+}1}}e^{2\pi i (s(\lambda), x)},\end{gathered}$$ where $(\cdot\,,\cdot)$ is a scalar product in $\R^{n+1}$, ${\rm sgn}\,s$ is the permutation sign, and ${\rm Alt}_{n{+}1}$ is the alternating group acting on an $(n+1)$-tuple of numbers. Note that variables $x$ and $\lambda$ are in the hyperplane $\mathcal{H}$.
Using the identity $\l\lambda,r_ix\r=\l r_i\lambda,x\r$ for the reflection $r_i$, $i=1,\ldots,n$, it can be verified that $$\begin{gathered}
\label{ri_symm_of_C_S}
C_\lambda(r_ix)=C_{r_i\lambda}(x)=C_\lambda(x),\quad
\text{and} \quad
{S}_{r_i\lambda}(x)={S}_\lambda(r_ix)=-{S}_\lambda(x).\end{gathered}$$
Note that it is easy to see for generic points that ${E}_\lambda(x)=\tfrac12\Big({C}_\lambda(x)+{S}_\lambda(x)\Big)$, and from the relations (\[ri\_symm\_of\_C\_S\]), we obtain $$\begin{gathered}
\label{ri_symm_of_E}
{E}_{r_i\lambda}(x)={E}_\lambda(r_ix)=\tfrac12\left( {C}_\lambda(x)-{S}_\lambda(x)\right)={E}_{\lambda}(x).\end{gathered}$$
A number of other properties of orbit functions are presented in [@KlimykPatera2006; @KlimykPatera2007-1; @KlimykPatera2008].
Orbit functions and Chebyshev polynomials {#sec_A1_and_Chebyshev}
=========================================
We recall known properties of Chebyshev polynomials [@Rivlin1974] in order to be subsequently able to make an unambiguous comparison between them and the appropriate orbit functions.
Classical Chebyshev polynomials {#ssec_Classical_Chebyshev}
-------------------------------
Chebyshev polynomials are orthogonal polynomials which are usually defined recursively. One distinguishes between Chebyshev polynomials of the first kind $T_n$: $$\begin{gathered}
T_0(x)=1,\quad
T_1(x)=x,\quad
T_{n+1}(x)=2xT_n-T_{n-1},\label{def_T0_T1_Tn}\\
\text{hence}\quad
T_2(x)=2x^2-1,\quad
T_3(x)=4x^3-3x, \dots\label{def_T2_T3}\end{gathered}$$ and Chebyshev polynomials of the second kind $U_n$: $$\begin{gathered}
U_0(x)=1,\quad
U_1(x)=2x,\quad
U_{n+1}(x)=2xU_n-U_{n-1},\label{def_U0_U1_Un}\\
\text{in particular}\quad
U_2(x)=4x^2-1,\quad
U_3(x)=8x^3-4x, \quad etc.\label{def_U2_U3}\end{gathered}$$
The polynomials $T_n$ and $U_n$ are of degree $n$ in the variable $x$. All terms in a polynomial have the parity of $n$. The coefficient of the leading term of $T_n$ is $2^{n-1}$ and $2^n$ for $U_n$, $n=1,2,3,\dots$.
The roots of the Chebyshev polynomials of the first kind are widely used as nodes for polynomial interpolation in approximation theory. The Chebyshev polynomials are a special case of Jacobi polynomials. They are orthogonal with the following weight functions: $$\begin{gathered}
\int\limits^1_{-1}\frac{1}{\sqrt{1-x^2}}T_n(x)T_m(x){\rm d}x=
\left\{
\begin{array}{l}
0,\quad n\ne m,\\
\pi,\quad n=m=0,\\
\frac{\pi}{2},\quad n=m\ne 0,
\end{array}
\right.
\\
\int\limits^1_{-1}\sqrt{1-x^2}U_n(x)U_m(x){\rm d}x=
\left\{
\begin{array}{l}
0,\quad n\ne m,\\
\frac{\pi}{2},\quad n=m.
\end{array}
\right.\end{gathered}$$
There are other useful relations between Chebyshev polynomials of the first and second kind. $$\begin{gathered}
\frac{\rm d}{{\rm d}x} T_n(x)=n U_{n-1}(x),\quad n=1,2,3,\dots \label{differentiation_Tn}
\\
T_n(x)=\frac 12 (U_{n}(x)-U_{n-2}(x)),\quad n=2,3,\dots \label{difference_Un_Un-2}
\\
T_{n+1}(x)=xT_n(x)-(1-x^2)U_{n-1},\quad n=1,2,3,\dots \label{difference_xTn_Un-1}
\\
T_{n}(x)=U_n(x)-xU_{n-1},\quad n=1,2,3,\dots \label{difference_Un_xUn-1}\end{gathered}$$
### Trigonometric form of Chebyshev polynomials
Using trigonometric variable $x=\cos y$, polynomials of the first kind become $$\begin{gathered}
\label{trigon_T}
T_n(x)=T_n(\cos y)=\cos(ny), \quad n=0,1,2,\dots\end{gathered}$$ and polynomials of the second kind are written as $$\begin{gathered}
\label{trigon_U}
U_n(x)=U_n(\cos y)=\frac{\sin((n+1)y)}{\sin y}, \quad n=0,1,2,\dots\end{gathered}$$
For example, the first few lowest polynomials are $$\begin{gathered}
T_0(x)=T_0(\cos y)=\cos(0y)=1, \quad
T_1(x)=T_1(\cos y)=\cos(y)=x, \\
T_2(x)=T_2(\cos y)=\cos(2y)=\cos^2y-\sin^2y=2cos^2y-1=2x^2-1; \quad \\
U_0(x)=U_0(\cos y)=\frac{\sin y}{\sin y}=1, \quad
U_1(x)=U_1(\cos y)=\frac{\sin(2y)}{\sin y}=2\cos y=2x, \\
U_2(x)=U_2(\cos y)=\frac{\sin(3y)}{\sin y}=\frac{\sin(2y)\cos y+\sin y \cos(2y)}{\sin y}
=4\cos^2y-1=4x^2-1.\end{gathered}$$
Orbit functions of $A_1$ and Chebyshev polynomials
--------------------------------------------------
Let us consider the orbit functions of one variable. There is only one simple Lie algebra of rank 1, namely $A_1$. Our aim is to build the recursion relations in a way that generalizes to higher rank groups, unlike the standard relations of the classical theory presented above.
### Orbit functions of $A_1$ and trigonometric form of $T_n$ and $U_n$
The orbit of $\lambda=m\omega_1$ has two points for $m\ne 0$, namely $W_\lambda=\{(m),(-m)\}$. The orbit of $\lambda=0$ has just one point, $W_0=\{0\}$.
One-dimensional orbit functions have the form (see (\[def\_c-function2\]), (\[def\_s-function2\]), (\[def\_e-function2\])) $$\begin{gathered}
C_\lambda(x)=e^{2\pi i m x}{+}e^{{-}2\pi i m x}{=}2\cos(2\pi m x){=}2\cos(m y),
\quad \text{where}\quad y=2\pi x,\; m\in\Z^{\geqslant 0};\label{c-func_A1}
\\
S_\lambda(x)=e^{2\pi i m x}{-}e^{-2\pi i m x}{=}2i\sin(2\pi m x){=}2i\sin(m y),
\quad \text{for}\quad m\in\Z^{> 0};\label{s-func_A1}
\\
E_\lambda(x)= e^{2\pi i m x}=y^m,
\quad \text{where}\quad y=e^{2\pi i x}, \; m\in\Z.\label{e-func_A1}\end{gathered}$$ From (\[c-func\_A1\]) and (\[trigon\_T\]) it directly follows that polynomials generated from $C_m$ functions of $A_1$ are doubled Chebyshev polynomials $T_m$ of the first kind for $m=0,1,2,\dots$.
Analogously, from (\[s-func\_A1\]) and (\[trigon\_U\]), it follows that polynomials $\frac{S_{m+1}}{S_1}$ are Chebyshev polynomials $U_m$ of the second kind for $m=0,1,2,\dots$.
The polynomials generated from $E_m$ functions of $A_1$, form a standard monomial sequence $y^m$, $m=0,1,2\dots$, which is the basis for the vector space of polynomials.
$C$- and $S$-orbit functions are orthogonal on the interval $F=[0,1]$ (see (\[cont\_orthog c funcs\]) and (\[cont\_orthog s funcs\])) what implies the orthogonality of the corresponding polynomials.
Comparing the properties of one-dimensional orbit functions with properties of Chebyshev polynomials, we conclude that there is a one-to-one correspondence between the Chebyshev polynomials and the orbit functions.
### Orbit functions of $A_1$ and their polynomial form {#recursive_polynomials}
In this subsection, we start a derivation of the $A_1$ polynomials in a way which emphasizes the role of the Lie algebra and, more importantly, in a way that directly generalizes to simple Lie algebras of any rank $n$ and any type, resulting in polynomials of $n$ variables and of a new type for each algebra. In the present case of $A_1$, this leads us to a different normalization of the polynomials and their trigonometric variables than is common for classical Chebyshev polynomials. No new polynomials emerge than those equivalent to Chebyshev polynomials of the first and second kind. Insight is nevertheless gained into the structure of the problem, which, to us, turned out to be of considerable importance. We are inclined to consider the Chebyshev polynomials, in the form derived here, as the canonical polynomials.
The underlying Lie algebra $A_1$ is often denoted $sl(2,\C)$ or $su(2)$. In fact, this case is so simple that the presence of the Lie algebras has never been acknowledged.
The orbit functions of $A_1$ are of two types (\[c-func\_A1\]) and (\[s-func\_A1\]); in particular, $C_0(x)=2$, and $S_0(x)=0$ for all $x$.
The simplest substitution of variables to transform the orbit functions into polynomials is $y=e^{2\pi i x}$, monomials in such a polynomial are $y^m$ and $y^{-m}$. Instead, we introduce new (‘trigonometric’) variables $X$ and $Y$ as follows: $$\begin{gathered}
X:=C_1(x){=}e^{2\pi i x}{+}e^{{-}2\pi i x}{=}2\cos(2\pi x),\label{def_X}\\
Y:=S_1(x){=}e^{2\pi i x}{-}e^{{-}2\pi i x}{=}2i\sin(2\pi x).\label{def_Y}\end{gathered}$$ We can now start to construct polynomials recursively in the degrees of $X$ and $Y$, by calculating the products of the appropriate orbit functions. Omitting the dependence on $x$ from the symbols, we have $$\begin{gathered}
\begin{array}{rlcrl}
X^2 \!\!\!&=C_2+2 &\Longrightarrow\quad& C_2\!\!\!&=X^2-2,\\
XC_2\!\!\!&=C_3+X &\Longrightarrow\quad& C_3\!\!\!&=X^3-3X,\\
XC_m\!\!\!&=C_{m+1}+C_{m-1} &\Longrightarrow\quad& C_{m+1}\!\!\!&=XC_m-C_{m-1},\quad m\geq3.
\end{array}\label{C_via_X}\end{gathered}$$ Therefore, we obtain the following recursive polynomial form of the $C$-functions $$\begin{gathered}
\label{C-form_of_Tn}
C_0=2,\quad
C_1=X,\quad
C_2=X^2{-}2,\quad
C_3=X^3{-}3X,\quad
C_4=X^4{-}4X^2+2,\dots .\end{gathered}$$ After the substitution $z=\tfrac 12 X$ we have $$\begin{gathered}
C_0{=}2\cdot1,\quad
C_1{=}2z,\quad
C_2{=}2(2z^2{-}1),\quad
C_3{=}2(4z^3{-}3z),\quad
C_4{=}2(8z^4{-}8x^2+1),\dots.\end{gathered}$$ Hence we conclude that $C_m\!=2T_m$, for $m=0,1,\dots$.
In our opinion, the normalization of orbit functions is also more ‘natural’ for the Chebyshev polynomials. For example, the equality $C_2^2=C_4+2$ does not hold for $T_2$ and $T_4$.
Each $C_m$ also can be written as a polynomial of degree $m$ in $X$,$Y$ and $S_{m-1}$. It suffices to consider the products $YS_m$, e.g., $C_2=Y^2+2$, $C_3=YS_2+X$, etc. Equating the polynomials obtained in such a way with the corresponding polynomials from (\[C\_via\_X\]), we obtain a trigonometric identity for each $m$. For example, we find two ways to write $C_2$, one from the product $X^2$ and one from $Y^2$. Equating the two, we get $$\begin{gathered}
X^2{-}Y^2=4\quad\Longleftrightarrow\quad \sin^2(2\pi x){+}\cos^2(2\pi x)=1\end{gathered}$$ because $Y$ is defined in (\[def\_Y\]) to be purely imaginary.
Just as the polynomials representing $C_m$ were obtained above, it is possible to to find polynomial expressions for $S_m$ for all $m$.
Fundamental relations between the $S$- and $C$- orbit functions follow from the properties of the character $\chi_m(x)$ of the irreducible representation of $A_1$ of dimension $m+1$.
The character can be written in two ways: as in the Weyl character formula and also as the sum of appropriate $C$-functions. Explicitly, we have the $A_1$ character: $$\begin{gathered}
\chi_m(x)=\frac{S_{m+1}(x)}{S_1(x)}=C_m(x)+C_{m-2}(x)+\cdots+
\begin{cases}
C_2(x)+1\quad &\text {for $m$ even},\\
C_3(x)+C_1(x)\quad &\text {for $m$ odd}.
\end{cases}\end{gathered}$$ Let us write down a few characters $$\begin{gathered}
\chi_0=\tfrac{S_{1}(x)}{S_1(x)}=1,\quad
\chi_1=\tfrac{S_{2}(x)}{S_1(x)}=C_1=X,\quad
\chi_2=\tfrac{S_{3}(x)}{S_1(x)}=C_2+C_0=X^2-1,\\
\chi_3=\tfrac{S_{4}(x)}{S_1(x)}=C_3+C_1=X^3-2X,\quad
\chi_4=\tfrac{S_{5}(x)}{S_1(x)}=C_4+C_2+C_0=X^4-3X^2+1,\dots\end{gathered}$$ Again, the substitution $z=\tfrac 12 X$ transforms these polynomials into the Chebyshev polynomials of the second kind $\frac{S_{m+1}}{S_1}=U_m$, $m=0,1,\dots$, indeed $$\begin{gathered}
\tfrac{S_{1}(x)}{S_1(x)}=1,\quad
\tfrac{S_{2}(x)}{S_1(x)}=2z,\quad
\tfrac{S_{3}(x)}{S_1(x)}=4z^2{-}1,\quad
\tfrac{S_{4}(x)}{S_1(x)}=8z^3{-}4z,\quad
\tfrac{S_{5}(x)}{S_1(x)}=16z^4{-}12z^2{+}1,\dots\end{gathered}$$
Note that in the character formula we used $C_0=1$, while above (see (\[def\_e-function2\]) and (\[C-form\_of\_Tn\])) we used $C_0=2$. It is just a question of normalization of orbit functions. For some applications/calculations it is convenient to scale orbit functions of non-generic points on the factor equal to the order of the stabilizer of that point in the Weyl group $W(A_1)$.
Orbit functions of $A_n$ and their polynomials
==============================================
This section proposes two approaches to constructing orthogonal polynomials of $n$ variables based on orbit functions. The first comes from the decomposition of Weyl orbit products into sums of orbits. Its result is the analog of the trigonometric form of the Chebyshev polynomials. The second approach is the exponential substitution in [@KlimykPatera2006].
Recursive construction
----------------------
Since the $C$- and $S$- functions are defined for $A_n$ of any rank $n=1,2,3,\dots$, it is natural to take $C$-functions and the ratio of $S$-functions as multidimensional generalizations of Chebyshev polynomials of the first and second kinds respectively $$\begin{aligned}
T_\lambda(x)&:=C_\lambda(x),\qquad \ x\in\R^n,\\
U_\lambda(x)&:=\tfrac{S_{\lambda+\rho}(x)}{S_{\rho}(x)},\qquad
\rho=\omega_1{+}\omega_2{+}\dots{+}\omega_n=(1,1,\dots,1)_\omega,\quad x\in\R^n,\end{aligned}$$ where $\lambda$ is one of the dominant weights of $A_n$.
The functions $T_\lambda$ and $U_\lambda$ can be constructed as polynomials using the recursive scheme proposed in Section \[recursive\_polynomials\]. In the $n$-dimensional case of orbit functions of $A_n$, we start from the $n$ orbit functions labeled by the fundamental weights, $$\begin{gathered}
X_1:=C_{\omega_1}(x),\quad X_2:=C_{\omega_2}(x),\quad\dots,\quad
X_n:=C_{\omega_n}(x)\,,
\qquad x\in\R^n\,.\end{gathered}$$ By multiplying them and decomposing the products into the sum of orbit functions, we build the polynomials for any $C$- and $S$-function.
The generic recursion relations are found as the decomposition of the products $X_{\w_j}C_{(a_1,a_2.\dots,a_n)}$ with ‘sufficiently large’ $a_1,a_2,\dots,a_n$. Such a recursion relation has $\left(\begin{smallmatrix}n+1\\ j\\\end{smallmatrix}\right)+1$ terms, where $\left(\begin{smallmatrix}n+1\\j\\\end{smallmatrix}\right)$ is the size of the orbit of $\w_j$.
An efficient way to find the decompositions is to work with products of Weyl group orbits, rather than with orbit functions. Their decomposition has been studied, and many examples have been described in [@HLP]. It is useful to be aware of the congruence class of each product, because all of the orbits in its decomposition necessarily belong to that class. The *congruence number* $\#$ of an orbit $\lambda$ of $A_n$, which is also the congruence number of the orbit functions $C_\lambda$ and $S_\lambda$, specifies the class. It is calculated as follows, $$\begin{gathered}
\#(C_{(a_1,a_2,\dots,a_n)}(x))=\#(S_{(a_1,a_2,\dots,a_n)}(x))=\sum_{k=1}^n ka_k\mod (n+1).\end{gathered}$$ In particular, each $X_j$, where $j=1,2,\dots,n$, is in its own congruence class. During the multiplication, congruence numbers add up $\mod n+1$.
Polynomials in two and three variables originating from orbit functions of the simple Lie algebras $A_2$, $C_2$, $G_2$, $A_3$, $B_3$, and $C_3$ are obtained in the forthcoming paper [@NesterenkoPatera2009].
Exponential substitution
------------------------
There is another approach to multivariate orthogonal polynomials, which is also based on orbit functions. Such polynomials can be constructed by the continuous and invertible change of variables $$\begin{gathered}
\label{subst_exp}
y_j=e^{2\pi i x_j}, \quad x_j\in\R,\quad j=1,2,\dots,n.\end{gathered}$$ Consider an $A_n$ orbit function $C_\lambda(x)$, $S_\lambda(x)$ or $E_\lambda(x)$, when $\lambda$ is given in the $\w$-basis and $x$ is given in the $\alpha$-basis. Each of these functions consists of summands $\prod\limits^n_{j=1}e^{2\pi i \mu_j x_j}$, where $\mu_j\in\Z$ are coordinates of an orbit point $\mu$. Then the summand is transformed by (\[subst\_exp\]) into a monomial of the form $\prod\limits^n_{j=1}y_j^{\mu_j}$. It is convenient to label these polynomials by non-negative integer coordinates $(m_1,m_2,\dots, m_n)$ of the point $\lambda=m_1\w_1+m_2\w_2+\dots m_n\w_n$ and to denote the polynomial obtained from the orbit function $C_\lambda$ as $P_{(m_1,\dots, m_n)}^{C}$ (analogously for $S$ and $E$ functions). Polynomials of two variables obtained from the orbit functions by the substitution (\[subst\_exp\]) are already described in the literature [@Koornwinder1-4], where they are derived from very different considerations. The detailed comparison is made in the following example.
Consider the $A_2$ Weyl orbits of the lower weights $(0,m)_\omega$, $(m,0)_\omega$ and the orbit of the generic point $(m_1,m_2)_\omega$, $m,m_1,m_2\in\Z^{>0}$ $$\begin{gathered}
W_{(0,m)}(A_2)=\{(0, m),\, ( {-}m, 0),\, (m, {-}m)\},
\quad W_{(m,0)}(A_2)=\{(m, 0),\, ({-}m, m),\, (0, {-}m)\},
\\
W_{(m_1,m_2)}(A_2)=\{
(m_1, m_2)^+,\ ({-}m_1, m_1{+}m_2)^-,\ (m_1{+}m_2, {-}m_2)^-,
\\
\phantom{W_{(m_1,m_2)}(A_2)=\{}
({-}m_2, {-}m_1)^-,\ ( {-}m_1{-}m_2, m_1)^+,\ (m_2, {-}m_1{-}m_2)^+\}.\end{gathered}$$ Suppose $x=(x_1,x_2)$ is given in the $\alpha$-basis, then the orbit functions assume the form $$\begin{gathered}
\begin{gathered}\label{C_S_for_A2}
C_{(0,0)}(x)=1,\quad
C_{(0,m)}(x)=\overline{C_{(m,0)}(x)}=e^{{-}2\pi i mx_1}{+}e^{2\pi i mx_1}e^{{-}2\pi i mx_2}{+}e^{2\pi i mx_2},
\\
C_{(m_1,m_2)}(x)
=e^{2\pi i m_1x_1}e^{2\pi i m_2x_2}{+}e^{{-}2\pi i m_1x_1}e^{2\pi i (m_1{+}m_2)x_2}
{+}e^{2\pi i (m_1{+}m_2)x_1}e^{{-}2\pi i m_2x_2}{+}
\\ \phantom{C_{(m_1,m_2)}(x)=}
e^{{-}2\pi i m_2x_1}e^{{-}2\pi i m_1x_2}{+}e^{{-}2\pi i (m_1{+}m_2)x_1}e^{2\pi i m_1x_2}
{+}e^{2\pi i m_2x_1}e^{{-}2\pi i (m_1{+}m_2)x_2},\\
S_{(m_1,m_2)}(x)
=e^{2\pi i m_1x_1}e^{2\pi i m_2x_2}{-}e^{{-}2\pi i m_1x_1}e^{2\pi i (m_1{+}m_2)x_2}
{-}e^{2\pi i (m_1{+}m_2)x_1}e^{{-}2\pi i m_2x_2}{-}
\\ \phantom{S_{(m_1,m_2)}(x)=}
e^{{-}2\pi i m_2x_1}e^{{-}2\pi i m_1x_2}{+}e^{{-}2\pi i (m_1{+}m_2)x_1}e^{2\pi i m_1x_2}
{+}e^{2\pi i m_2x_1}e^{{-}2\pi i (m_1{+}m_2)x_2}.\end{gathered}\end{gathered}$$ Using (\[subst\_exp\]) we have the following corresponding polynomials $$\begin{gathered}
P_{(0,0)}^C=1,\qquad
P_{0,m}^C=\overline{P_{0,m}^C}=y_1^{{-}m}{+}Y_1^{m}y_2^{{-}m}{+}y_2^{m},
\\
\begin{gathered}\label{Polyn_c-A2}
P_{(m_1,m_2)}^{C}
=y_1^{m_1}y_2^{m_2}{+}y_1^{-m_1}y_2^{(m_1{+}m_2)}{+}y_1^{(m_1{+}m_2)}y_2^{-m_2}{+}
\\ \phantom{P^{C(A_2)}=}
y_1^{-m_1}y_2^{-m_2}{+}y_1^{-(m_1{+}m_2)}y_2^{m_1}{+}y_1^{m_2}y_2^{-(m_1{+}m_2)},
\end{gathered}
\\
\begin{gathered}\label{Polyn_s-A2}
P_{(m_1,m_2)}^{S}
=y_1^{m_1}y_2^{m_2}{-}y_1^{-m_1}y_2^{(m_1{+}m_2)}{-}y_1^{(m_1{+}m_2)}y_2^{-m_2}{-}
\\ \phantom{P^{S(A_2)})=}
y_1^{-m_1}y_2^{-m_2}{+}y_1^{-(m_1{+}m_2)}y_2^{m_1}{+}y_1^{m_2}y_2^{-(m_1{+}m_2)}.
\end{gathered}\end{gathered}$$
The polynomials $e^+$ and $e^-$ given in (2.6) of [@Koornwinder1-4](III) coincide with those in (\[C\_S\_for\_A2\]) whenever the correspondence $\sigma=2\pi x_1$, $\tau=2\pi x_2$ is set up. So, both the orbit functions polynomials of $A_2$ and $e^{\pm}$ are orthogonal on the interior of Steiner’s hypocycloid.
It is noteworthy that the regular tessellation of the plane by equilateral triangles considered in [@Koornwinder1-4] is the standard tiling of the weight lattice of $A_2$. The fundamental region $R$ of [@Koornwinder1-4] coincides with the fundamental region $F(A_2)$ in our notations. The corresponding isometry group is the affine Weyl group of $A_2$.
Furthermore, continuing the comparison with the paper [@Koornwinder1-4], we want to point out that orbit functions are eigenfunctions not only of the Laplace operator written in the appropriate basis, e.g. in $\omega$-basis, the corresponding eigenvalues bring $-4\pi^2 \l\lambda ,\lambda\r$, where $\lambda$ is the representative from the dominant Weyl chamber, which labels the orbit function. This property holds not only for the Lie algebra $A_n$ and its Laplace operator, but also for the differential operators built from the elementary symmetric polynomials, see [@KlimykPatera2006; @KlimykPatera2007-1].
An independent approach to the polynomials in two variables is proposed in [@suetin2], and the generalization of classical Chebyshev polynomials to the case of several variables is also presented in [@EierLidl]. A detailed comparison would be a major task because the results are not explicit and contain no examples of polynomials.
Multivariate exponential functions
==================================
In this section, we consider one more class of special functions, which, as it will be shown, are closely related to orbit functions of $A_n$. Such a relation allows us to view orbit functions in the orthonormal basis, and to represent them in the form of determinants and permanents. At the same time, we obtain the straightforward procedure for constructing polynomials from multivariate exponential functions.
[@KlimykPatera2007-3] For a fixed point $\lambda=(l_1,l_2,\dots,l_{n+1})_e$, such that $l_1\geq l_2\geq\cdots\geq l_{n+1}$, $\sum\limits_{k=1}^{n+1}l_k=0$, the symmetric multivariate exponential function $D^+_{\lambda}$ of $x=(x_1,x_2,\dots,x_{n+1})_e$ is defined as follows $$\begin{gathered}
\label{E+def}
D^+_{\lambda}(x):= {\det}^+ \left(
\begin{array}{cccc}
e^{2\pi i l_1x_1}&e^{2\pi i l_1x_2}&\dots&e^{2\pi i l_1x_{n+1}}\\
e^{2\pi i l_2x_1}&e^{2\pi i l_2x_2}&\dots&e^{2\pi i l_2x_{n+1}}\\[-1ex]
\vdots&\vdots&\ddots&\vdots\\[1 ex]
e^{2\pi i l_{n+1}x_1}&e^{2\pi i l_{n+1}x_2}&\dots&e^{2\pi i l_{n+1}x_{n+1}}
\end{array}
\right).\end{gathered}$$ Here, ${\det}^+$ is calculated as a conventional determinant, except that all of its monomial terms are taken with positive sign. It is also called *permanent* [@Henryk] or *antideterminant*.
It was shown in [@KlimykPatera2007-3] that it suffices to consider $D_{\lambda}^+(x)$ on the hyperplane $x\in {\mathcal H}$ (see (\[plane H\])). Furthermore, due to the following property of the permanent $$\begin{gathered}
{\det}^+ (a_{ij})_{i,j=1}^m=\sum_{s\in {\rm S}_m}a_{1,s(1)}a_{2,s(2)}\cdots a_{m,s(m)}
=\sum_{s\in {\rm S}_m}a_{s(1),1}a_{s(2),2}\cdots a_{s(m),m}\end{gathered}$$ we have $$\begin{gathered}
D^+_{\lambda}(x)= \sum_{s\in {\rm S}_{n+1}}e^{2\pi i l_1 x_{s(1)}} \cdots
e^{2\pi i l_m x_{s(n+1)}} =\sum_{s\in {\rm S}_{n+1}}e^{2\pi i
(\lambda,s(x))} =\sum_{s\in {\rm S}_{n+1}}e^{2\pi i (s(\lambda),x)}.\end{gathered}$$
For all $\lambda, x\in \mathcal{H}\subset\R^{n+1}$, we have the following connection between the symmetric multivariate exponential functions in $n+1$ variables, and $C$ orbit functions of $A_n$ $D^+_{\lambda}(x)=kC_\lambda(x)$, where $k=\tfrac{|W|}{|W_\lambda|}$, $|W|$ and $|W_\lambda|$ are sizes of the Weyl group and Weyl orbit respectively. In particular, for generic points, $k=1$.
Proof follows from the definitions of the functions $C$ and $D^+$ (definitions 1 and 4 respectively) and properties of orbit functions formulated in Section \[orb\_func\_n+1\].
[@KlimykPatera2007-3] For a fixed point $\lambda=(l_1,l_2,\dots,l_{n+1})_e$, such that $l_1\geq l_2\geq\cdots\geq l_{n+1}$, ${\sum\limits_{k=1}^{n+1}l_k=0}$, the antisymmetric multivariate exponential function $D^-_{\lambda}$ of $x=(x_1,x_2,\dots,x_{n+1})_e\in \mathcal{H}$ is defined as follows $$\begin{gathered}
\label{E+def}
D^-_{\lambda}(x):=\det \left(
\begin{array}{cccc}
e^{2\pi i l_1x_1}&e^{2\pi i l_1x_2}&\dots&e^{2\pi i l_1x_{n+1}}\\
e^{2\pi i l_2x_1}&e^{2\pi i l_2x_2}&\dots&e^{2\pi i l_2x_{n+1}}\\[-1ex]
\vdots&\vdots&\ddots&\vdots\\[1 ex]
e^{2\pi i l_{n+1}x_1}&e^{2\pi i l_{n+1}x_2}&\dots&e^{2\pi i l_{n+1}x_{n+1}}
\end{array}
\right)=\sum_{s\in {\rm S}_{n+1}}({\rm sgn}\ s)e^{2\pi i(s(\lambda),x)},\end{gathered}$$ where ${\rm sgn}$ is the permutation sign.
For all generic points $\lambda\in \mathcal{H}\subset\R^{n+1}$, we have the following connection ${D^-_{\lambda}(x)=S_\lambda(x)}$.
The antisymmetric multivariate exponential functions $D^-$, and $S$ orbit functions, equal zero for non-generic points.
Proof directly follows from the definitions of functions $S$ and $D^-$ (definitions 2 and 5 respectively), and properties of $S$ functions formulated in Section \[orb\_func\_n+1\].
[@KlimykPateraAlternatingExp] The alternating multivariate exponential function $D^{{\rm Alt}}_\lambda(x)$, for $x=(x_1,\dots,x_{n+1})_e$, $\lambda=(l_1,\dots,l_{n+1})_e$, is defined as the function $$\begin{gathered}
\label{sdetD_Alt}
D^{{\rm Alt}}_\lambda(x):={\rm sdet}\left(
\begin{array}{cccc}
e^{2\pi i l_1x_1} & e^{2\pi i l_1x_2} & \cdots & e^{2\pi i l_1x_{n+1}}\\
e^{2\pi i l_2x_1} & e^{2\pi i l_2x_2} & \cdots & e^{2\pi i l_2x_{n+1}}\\[-1 ex]
\vdots & \vdots & \ddots & \vdots \\
e^{2\pi i l_{n+1}x_1}& e^{2\pi i l_{n+1}x_2}& \cdots & e^{2\pi i l_{n+1}x_{n+1}}
\end{array}
\right),\end{gathered}$$ where ${\rm Alt}_{n+1}$ is the alternating group (even subgroup of ${\rm S}_{n+1}$) and $$\begin{gathered}
{\rm sdet} \left(e^{2\pi i l_jx_k}\right)_{j,k=1}^{n+1}:=\!\!\!
\sum_{w\in {\rm Alt}_{n+1}}\!\!e^{2\pi il_1 x_{w(1)}}e^{2\pi il_2 x_{w(2)}}\cdots e^{2\pi il_{n+1} x_{w(n+1)}}=\!\!\!
\sum_{w\in {\rm Alt}_{n+1}}\!\!e^{2\pi i(\lambda,w(x))}.\end{gathered}$$ Here, $(\lambda,x)$ denotes the scalar product in the $(n+1)$-dimensional Euclidean space.
Note that ${\rm Alt}_{m}$ consists of even substitutions of ${\rm S}_m$, and is usually denoted as $A_{m}$; here we change the notation in order to avoid confusion with simple Lie algebra $A_n$ notations.
It was shown in [@KlimykPateraAlternatingExp] that it is sufficient to consider the function $D^{{\rm Alt}}_\lambda(x)$ on the hyperplane $\mathcal{H}\colon x_1+x_2+\cdots +x_{n+1}=0$ for $\lambda$, such that $l_1\ge l_2\ge l_3\ge \cdots \ge l_{n+1}$.
Alternating multivariate exponential functions are obviously connected with symmetric and antisymmetric multivariate exponential functions. This connection is the same as that of the cosine and sine, with the exponential function of one variable ${D^{{\rm Alt}}_\lambda(x)=\tfrac 12 (D^{+}_\lambda(x)+D^{-}_\lambda(x))}$.
For all generic points $\lambda\in \mathcal{H}\subset\R^{n+1}$, the following relation between the alternative multivariate exponential functions $D^{{\rm Alt}}$ and $E$-orbit functions of $A_n$ holds true: ${D^{{\rm Alt}}_\lambda(x)=E_\lambda(x)}$.
For non-generic points $\lambda$, we have $E_\lambda(x)=C_\lambda(x)$ and, therefore, $E_\lambda(x)=kD^+_{\lambda}(x)$, where $k=\tfrac{|W|}{|W_\lambda|}$.
Proof directly follows from definitions 3 and 6, from the relation $E=\tfrac 12(C+S)$, and from the properties of orbit functions formulated in Section \[orb\_func\_n+1\].
Concluding remarks
==================
1. Consequences of the identification of W-invariant orbit functions of compact simple Lie groups and multivariable Chebyshev polynomials merit further exploitation. It is conceivable that Lie theory may become a backbone of a segment of the theory of orthogonal polynomials of many variables.
Some of the properties of orbit functions translate readily into properties of Chebyshev polynomials of many variables. However there are other properties whose discovery from the theory of polynomials is difficult to imagine. As an example, consider the decomposition of the Chebyshev polynomial of the second kind into the sum of Chebyshev polynomials of the first kind. In one variable, it is a familiar problem that can be solved by elementary means. For two and more variables, the problem turns out to be equivalent to a more general question about representations of simple Lie groups. In general the coefficients of that sum are the dominant weight multiplocities. Again, simple specific cases can be worked out, but a sophisticated algorithm is required to deal with it in general [@MP1982]. In order to provide a solution for such a problem, extensive tables have been prepared [@BMP] (see also references therein).
2. Our approach to the derivation of multidimensional orthogonal polynomials hinges on the knowledge of appropriate recursion relations. The basic mathematical property underlying the existence of the recursion relation is the complete decomposability of products of the orbit functions. Numerous examples of the decompositions of products of orbit functions, involving also other Lie groups than ${\rm SU}(n)$, were shown elsewhere [@KlimykPatera2006; @KlimykPatera2007-1]. An equivalent problem is the decomposition of products of Weyl group orbits [@HLP].
3. Possibility to discretize the polynomials is a consequence of the known discretization of orbit functions. For orbit functions it is a simpler problem, in that it is carried out in the real Euclidean space $\R^n$. In principle, it carries over to the polynomials. But variables of the polynomials happen to be on the maximal torus of the underlying Lie group. Only in the case of $A_1$, the variables are real (the imaginary unit multiplying the $S$-functions can be normalized away). For $A_n$ with $n>1$ the functions are complex valued. Practical aspects of discretization deserve to be thoroughly investigated.
4. For simplicity of formulation, we insisted throughout this paper that the underlying Lie group be simple. The extension to compact semisimple Lie group and their Lie algebras is straightforward. Thus, orbit functions are products of orbit functions of simple constituents, and different types of orbit functions can be mixed.
5. Polynomials formed from $E$-functions by the same substitution of variables should be equally interesting once $n>1$. We know of no analogs of such polynomials in the standard theory of polynomials with more than one variable. Intuitively, they would be formed as ‘halves’ of Chebyshev polynomials although their domain of orthogonality is twice as large as that of Chebyshev polynomials [@KlimykPatera2008].
6. Orbit functions have many other properties [@KlimykPatera2006; @KlimykPatera2007-1; @KlimykPatera2008] that can now be rewritten as properties of Chebyshev polynomials. Let us point out just that they are eigenfunctions of appropriate Laplace operators with known eigenvalues.
7. Notions of multivariate trigonometric functions [@KP2007] lead us to the idea of new, yet to be defined classes of $W$-orbit functions based on trigonometric sine and cosine functions, hence also to new types of polynomials.
8. Analogs of orbit functions of Weyl groups can be introduced also for the finite Coxeter groups that are not Weyl groups of a simple Lie algebra. Many of the properties of orbit functions extend to these cases. Only their orthogonality, continuous or discrete, has not been shown so far.
9. Our choice of the $n$ dimensional subspace $\mathcal H$ in $\R^{n+1}$ by requirement , is not the only possibility. A reasonable alternative appears to be setting $l_{n+1}=0$ (orthogonal projection on $\R^n$).
Acknowledgements {#acknowledgements .unnumbered}
----------------
The work was partially supported by the Natural Science and Engineering Research Council of Canada. MITACS, and by the MIND Research Institute, Calif. Two of us, M.N. and A.T., are grateful for the hospitality extended to them at the Centre de Recherches mathématiques, Université de Montréal, where the work was carried out.
The authors are grateful to A. Kiselev and the anonymous Journal referee for critical remarks and comments on the previous version of this paper.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We prove that the energy dissipation property of gradient flows extends to the semigroup maximal operators in various settings. In particular, we show that the vertical maximal function relative to the $p$-parabolic extension does not increase the $\dot{W}^{1,p}$ norm of $\dot{W}^{1,p}({{\mathbb{R}^n}}) \cap L^{2}({{\mathbb{R}^n}})$ functions when $p > 2$. We also obtain analogous results in the setting of uniformly parabolic and elliptic equations with bounded, measurable, real and symmetric coefficients, where the solutions do not have a representation formula via a convolution.'
address:
- 'Moritz Egert, Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France'
- 'Simon Bortz, Department of Mathematics, University of Washington, Seattle, WA 98195, USA'
- 'Olli Saari, Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany'
author:
- Simon Bortz
- Moritz Egert
- Olli Saari
bibliography:
- 'Refs.bib'
date: 'October 29, 2019'
title: Sobolev Contractivity of Gradient Flow Maximal Functions
---
Introduction
============
Consider a positive continuously differentiable energy functional $\mathcal{F}$ on a Banach space $X$ embedded in a Hilbert space. We can define the gradient of $\mathcal{F}$ via the ambient inner product as $\mathcal{F'}(u)v = \langle \nabla \mathcal{F}u, v \rangle$ and study the related gradient flow obeying $$\begin{aligned}
\label{eq:gradient flow}
\dot{u} + \nabla \mathcal{F}(u) = 0.\end{aligned}$$ According to the fundamental Lyapunov principle, expressed by the formal calculation $$\begin{aligned}
\frac{d}{dt} \mathcal{F}(u(t)) = \mathcal{F}'(u(t)) \dot{u}(t) = - \langle \dot{u}(t), \dot{u}(t) \rangle \leq 0,\end{aligned}$$ solutions to such abstract diffusion equations dissipate energy as time passes. In other words, if $u$ is a solution to the Cauchy problem with initial data $f$, then the energy contraction property $\mathcal{F}(u(t)) \leq \mathcal{F}(f)$ holds for all $t \geq 0$. This setup can be made rigorous for countless examples, including the heat equation, the total variation flow and the mean curvature flow to mention a few. See for example [@Chill2010; @Lions; @Pazy; @Brezis] and references therein.
In the present paper, we propose a seemingly new paradigm. Suppose that $X$ is a space of real functions. Then not only does the energy decrease along the gradient flow, but also the related vertical maximal operator, mapping non-negative initial data $f$ to $$\begin{aligned}
u^*(x) = \sup_{t>0} u(t,x),\end{aligned}$$ is an energy contraction in the sense that $ \mathcal{F}(u^*) \leq \mathcal{F}(f)$.
The objective of this article is to implement this idea for two important energy quantities:
- The Sobolev $p$-energy $\mathcal{F}(u) = \frac{1}{p} \int_{{{\mathbb{R}^n}}} |\nabla u(x)|^p \, dx$ with $p>2$, whose gradient flow is the degenerate $p$-parabolic equation $$\dot{u} - \Delta_p u := \dot{u} - \div(|\nabla u|^{p-2}\nabla u) = 0 .$$
- The quadratic energy $\mathcal{F}(u) = \frac{1}{2} \int_{{{\mathbb{R}^n}}} A(x) \nabla u(x) \cdot \nabla u(x) \, dx$ with a bounded measurable, elliptic and symmetric conductivity matrix $A$, whose gradient flow is the linear uniformly parabolic equation $$\dot{u} - \div(A\nabla u) = 0.$$
Our main result for the $p$-energy flow relies on global well-posedness of the corresponding Cauchy problem in a natural class of continuous energy solutions. The preliminaries on that can be found in Section \[sec:p\_grad\_flow\] and the proof is given in Section \[sec:proof-p\].
\[thmintro:1\] Let $p > 2$, $n \geq 1$, $f \in L^{2}({{\mathbb{R}^n}}) \cap \dot{W}^{1,p}({{\mathbb{R}^n}})$ be non-negative and $S_t f$ the unique energy solution to the Cauchy problem $$\begin{aligned}
\begin{split}
\dot u(t,x) - \Delta_p u(t,x) &= 0& \quad &\text{for $(t,x) \in (0,\infty) \times {{\mathbb{R}^n}}$,} \\
u(0,x) &= f(x)& \quad &\text{for $x \in {{\mathbb{R}^n}}$.}
\end{split}
\end{aligned}$$ Define $S^*f(x) := \sup_{t > 0} S_t f (x)$. Then $S^*f$ is weakly differentiable and satisfies $$\int_{{{\mathbb{R}^n}}} | \nabla S^* f(x) |^{p} \, dx \leq \int_{{{\mathbb{R}^n}}} |\nabla f |^{p} \, dx .$$
In the case of linear divergence form equation with rough coefficients, we extend the initial data via the heat semigroup generated by $L := \div(A \nabla \cdot)$. The necessary background is given in Section \[sec:semigroups\] and the proof can be found in Section \[sec:heat and poisson\].
\[thm:A-para\] Let $L$ be a uniformly elliptic operator with bounded, measurable and symmetric coefficient matrix $A$. Let $f \in W^{1,2}({{\mathbb{R}^n}})$ be non-negative and define $H^*f(x) := \sup_{t > 0} e^{tL} f(x)$. Then $H^{*} f $ is weakly differentiable and $$\int_{{{\mathbb{R}^n}}} A \nabla H^* f (x)\cdot \nabla H^* f(x) \, dx \leq \int_{{{\mathbb{R}^n}}} A \nabla f(x) \cdot \nabla f(x) \, dx .$$
Our results were largely inspired by [@Carneiro2013] and [@Carneiro2018], where similar contractivity inequalities were established for several variants of heat and Poisson kernels relative to the Laplacian. Qualitative $\dot{W}^{1,2} \to \dot{W}^{1,2}$ bounds for maximal functions defined through convolution kernels easily follow from [@Kinnunen1997], but the main contribution of [@Carneiro2013; @Carneiro2018] was to show that some special maximal functions are indeed contractions on that space. This adds a co-equal perspective to the inequalities studied here. The first results on Sobolev contractivity appeared in [@Tanaka2002; @Aldaz2007], where the one-dimensional non-centred Hardy–Littlewood maximal function $M$ was shown to be a contraction in $\dot{W}^{1,1}(\mathbb{R})$. After the generalization to convolution kernels and $\dot{W}^{1,2}$ in [@Carneiro2013; @Carneiro2018], we take this program further to the nonlinear setting of $\dot{W}^{1,p}({{\mathbb{R}^n}})$ spaces with $p > 2$ (Theorem \[thmintro:1\]) and semigroups far beyond the convolution kernel case (Theorem \[thm:A-para\]).
General semigroup maximal functions appear naturally, for instance, in the context of Hardy spaces adapted to operators [@HofMay2009] and elliptic boundary value problems [@Yang-Yang; @Auscher-Russ]. Our third main result is about the Poisson semigroup, which has an equally important role in that setting [@Mayboroda].
\[thm:A-poisson\] Let $L$ be a uniformly elliptic operator with bounded, measurable and symmetric coefficients $A$. Let $f \in W^{1,2}({{\mathbb{R}^n}})$ be non-negative and define $P^*f(x) := \sup_{t > 0} e^{-t(-L)^{1/2}} f(x)$. Then $P^{*} f $ is weakly differentiable and $$\int_{{{\mathbb{R}^n}}} A \nabla P^* f (x)\cdot \nabla P^* f(x) \, dx \leq \int_{{{\mathbb{R}^n}}} A \nabla f(x) \cdot \nabla f(x) \, dx .$$
We conclude the introduction by sketching our main line of reasoning and how it can be adapted to different gradient flows. The key observation leading to the sharp bound for the one-dimensional Hardy–Littlewood maximal function in [@Aldaz2007] was to notice that $Mf$ cannot have local maxima in the *detachment set* $$\{M f > f \}.$$ This was understood as a generalized convexity property and reinterpreted in a clever way in [@Carneiro2013; @Carneiro2018], where it was shown that the heat maximal function of $L=\Delta$ is subharmonic in the detachment set. As our starting point, we reformulate this observation in an abstract context (Lemma \[lemma:abstract\_comparison\]): if solutions to a gradient system admit a suitable comparison principle, then the vertical maximal function is a comparison subsolution of the Euler–Lagrange equation of the relevant energy functional in the detachment set. General prequisites on this will be recalled in Section \[sec:energy\].
Once the maximal function is connected to the Euler–Lagrange equation, it follows from nonlinear potential theory that the comparison subsolutions in the correct energy class are weak subsolutions and hence energy subminimizers. The proof can then be concluded by the correct choice of competitor. However, making the passage from comparison principles to energy minimization rigorous, requires one to take the specific forms of the energy functionals into account. Without convolution structures as in [@Carneiro2013; @Carneiro2018], there is no ‘soft’ argument to guarantee the weak differentiability of $Mf$ even qualitatively. Instead, we have to take advantage of local regularity theory and various approximations to circumvent the fact that $Mf$ might not be an admissible competitor in the energy inequalities. For Theorem \[thmintro:1\], we rely on local Lipschitz continuity of solutions and finite speed of propagation, whereas the theory of analytic heat semigroups comes in handy for the proofs of Theorem \[thm:A-para\] and Theorem \[thm:A-poisson\].
To our knowledge, these are the first regularity results for vertical maximal functions without convolution structure. For the convolution case and the Hardy–Littlewood maximal function in particular, the literature on regularity is extensive. Sobolev bounds and continuity were first studied in [@Kinnunen1997; @Luiro2007], and further topics include endpoint Sobolev continuity [@Carneiro2017; @Madrid2017], fractional maximal functions [@Kinnunen2003; @Beltran2019], local maximal functions [@Kinnunen1998; @Heikkinen2015] and much more. We expect many of these themes to have their counterparts in the setting of gradient flow maximal functions.
**Acknowledgement.** This research was supported by the CNRS through a PEPS JCJC project. The third author was partially supported by DFG SFB 1060 and DFG EXC 2047. The authors would like to thank Katharina Egert for tolerating their persistent presence for the better part of two weeks (and longer in the case of the second named author).
Energy functionals {#sec:energy}
==================
We set up the definitions following Chapter 5 of [@Heinonen2018]. Let $p \in (1,\infty)$. A *variational kernel with $p$-growth* is a function $F: {{\mathbb{R}^n}}\times {{\mathbb{R}^n}}\to [0,\infty)$ such that
- the mapping $x \mapsto F(x,\xi)$ is measurable for all $\xi$,
- the mapping $\xi \mapsto F(x,\xi)$ is strictly convex and differentiable for all $x$,
- there is $\Lambda \in [1, \infty)$ such that for all $x, \xi \in {{\mathbb{R}^n}}$, $$\Lambda^{-1} |\xi|^{p} \leq F(x,\xi) \leq \Lambda |\xi|^{p},$$
- for all $\lambda \in \mathbb{R}$ it holds $F(x,\lambda \xi) = |\lambda|^{p} F(x,\xi)$.
Associated with a variational kernel $F$ and a measurable set $E \subset {{\mathbb{R}^n}}$, we define the *localized energies* $\mathcal{F}_{E} : W_{loc}^{1,p}({{\mathbb{R}^n}}) \to [0, \infty)$ as $$\mathcal{F}_E (u) = \int_{E } F(x, \nabla u(x)) \, dx$$ and we abbreviate the *global energy functional* by $\mathcal{F} := \mathcal{F}_{{{\mathbb{R}^n}}}$.
The Euler–Lagrange equation {#subsec:Euler--Lagrange}
---------------------------
Setting $\mathcal{A}(x,\xi) := (\nabla_\xi F)(x,\xi)$, we can write down the *Euler–Lagrange equation* $$\label{eq:aharmoniceq}
\div_x \mathcal{A}( x ,\nabla u (x) ) = 0$$ for the energy functional $\mathcal{F}$. This is an $\mathcal{A}$-harmonic equation in the sense of [@Heinonen2018], see Lemma 5.9 therein. In particular, the strict convexity of $F$ implies for a.e. $x \in {{\mathbb{R}^n}}$ the important inequality $$\begin{aligned}
\label{eq:convexity inequality}
F(x,\xi_1) - F(x, \xi_2) > \mathcal{A}(x,\xi_2) \cdot (\xi_1 - \xi_2)\end{aligned}$$ whenever $\xi_1, \xi_2 \in {{\mathbb{R}^n}}$, $\xi_1 \neq \xi_2$, see Lemma 5.6 in [@Heinonen2018]. A *(weak) solution* to in an open set $\Omega \subset {{\mathbb{R}^n}}$ is a function $u \in W_{loc}^{1,p}(\Omega)$ such that $$\begin{aligned}
\label{eq:aharmoniceq-weak}
\int_\Omega \mathcal{A}( x ,\nabla u (x) ) \cdot \nabla \varphi(x) \, dx = 0\end{aligned}$$ holds for all $\varphi \in C_c^\infty(\Omega)$. We speak of *supersolutions (subsolutions)* if the left-hand side above is non-negative (non-positive) for all non-negative $\varphi \in C_c^\infty(\Omega)$. More generally, the left-hand side in defines $-L u := -\div_x \mathcal{A}(x, \nabla u(x))$ as an operator $W^{1,p}_{loc}(\Omega) \to \mathcal{D}'(\Omega)$ and being a weak solution means that holds in the sense of distributions.
A-subharmonic functions {#subsec:A-subharmonic functions}
-----------------------
For the moment, fix an open set $\Omega$, relative to which the following definitions are given. An *$\mathcal{A}$-harmonic function* is a continuous weak solution to . An *$\mathcal{A}$-subharmonic function* is an upper semicontinuous function that is not identically $-\infty$ and satisfies the following *elliptic comparison principle* with respect to $u \mapsto \div_x \mathcal{A}( x ,\nabla u(x) )$.
\[def:comparison\] An upper semicontinuous function $u : \Omega \to [-\infty, \infty)$ is said to satisfy the comparison principle with respect to an operator $L : W_{loc}^{1,p}(\Omega) \to \mathcal{D}'(\Omega)$ if the following holds for all $G \Subset \Omega$ and all $h \in C(\overline{\Omega})$ with $Lh = 0$ in $\mathcal{D}'(G)$: $$\textrm{If $h(x) \geq u(x)$ for all $x \in \partial G$, then $h(x) \geq u(x)$ for all $x \in G$.}$$
\[rem:comparison\] If $Lu = \div_x \mathcal{A}( x ,\nabla u(x) )$, then every continuous subsolution to satisfies the comparison principle with respect to $L$. This is Theorem 7.1.6 in [@Heinonen2018].
$\mathcal{A}$-subharmonic functions are energy subminimizers in the following sense.
\[lemma:comparison\_submin\] If $u \in W_{loc}^{1,p}({{\mathbb{R}^n}})$ is $\mathcal{A}$-subharmonic in $\Omega$, then for all open $D \Subset \Omega$ and all non-negative $\varphi \in C_c^{\infty}(D) $, $$\mathcal{F}_D(u) \leq \mathcal{F}_D(u - \varphi).$$
By Corollary 7.21 of [@Heinonen2018] we know that $u$ is a subsolution to . Hence, for all non-negative $\varphi \in C_c^\infty(D)$ we have $$\begin{aligned}
\int_D \mathcal{A}(x, \nabla u(x)) \cdot (\nabla u(x) - \nabla (u-\varphi)(x)) \, dx
=\int_D \mathcal{A}(x, \nabla u(x)) \cdot \nabla \varphi(x) \, dx \leq 0\end{aligned}$$ and the claim follows from with $\xi_2 := \nabla u(x)$ and $\xi_1 := \nabla(u-\varphi)(x)$.
The A-parabolic equation {#subsec:Aparabolic equation}
------------------------
We turn to the corresponding parabolic problem $$\begin{aligned}
\label{eq:aparaboliceq}
\dot{u}(t,x) - \div_x \mathcal{A}( x ,\nabla u (t,x) ) = 0,\end{aligned}$$ where $\dot{u}$ denotes the derivative in $t$. Let $\Omega \subset {{\mathbb{R}^n}}$ be an open set and let $t_1<t_2$. We call $u$ a *weak solution* to in $(t_1,t_2) \times \Omega$, if $u\in~C(I; L^{2}(D)) \cap L^{p}(I;W^{1,p}(D))$ whenever $I \times D \Subset (t_1,t_2) \times \Omega$, and if $$\begin{aligned}
\label{eq:aparabolic-weak}
\int_{t_1}^{t_2} \int_\Omega - u \dot{\varphi} + \mathcal{A}( x ,\nabla u) \cdot \nabla \varphi \, dx \, dt = 0\end{aligned}$$ holds for all $\varphi \in C_c^\infty((t_1,t_2) \times \Omega)$. We speak again of *supersolutions (subsolutions)* if the left-hand side above is non-negative (non-positive) for all non-negative $\varphi$. Replacing $\varphi$ by $\eta \varphi$, where $\eta \in C_c^\infty(I)$, and passing to the limit as $\eta \to 1_I$, we obtain thanks to the continuity assumption on $u$ that $$\begin{aligned}
\int_\Omega u \varphi \, dx \bigg|_{\partial I} + \int_I \int_D - u \dot{\varphi} + \mathcal{A}( x ,\nabla u) \cdot \nabla \varphi \, dx \, dt = 0.\end{aligned}$$ Therefore, our notion of weak solutions coincides with the one in Chapter [II]{} of [@DiBenedetto1993a].
Equation has the fundamental DeGiorgi property that bounded weak solutions can be redefined on a set of measure zero to become Hölder continuous.
\[prop:DeGiorgi\] Let $u$ be a bounded weak solution to in an open cylinder $(t_1, t_2) \times \Omega$. Denote $ U^{p/(p-2)} = \|u\|_{L^{\infty} ((t_1,t_2) \times \Omega)}$. Then there exist $\alpha \in (0,1)$ and $C>0$ depending only on $n$ and $p$, so that whenever $t_1 < t_3<t_4< t_2$ and $D \Subset \Omega$, it holds for a.e. $(s,x), (t,y) \in (t_3,t_4) \times D$ that $$|u(s,x) - u(t,y)| \leq C U^{\frac{p}{p-2}} \left( \frac{|x-y| + U |s-t|^{1/p}}{ \min({\operatorname{dist}}(D, \Omega^{c}) , U |t_3 - t_1|^{1/p} ) } \right)^{\alpha} .$$
By a slight abuse of notation, we shall from now on identify bounded weak solutions to with their (Hölder) continuous representative. Then the bound in the proposition above holds for all $(s,x), (t,y) \in I \times D$.
Finally, we recall the parabolic comparison principle that holds in full generality for the $\mathcal{A}$-parabolic equations considered here.
\[lemma:p-parab\_comparison\] Suppose that $u$ is a weak supersolution and $v$ is a weak subsolution to in a cylinder $(t_1,t_2) \times \Omega$ , where $\Omega \subset {{\mathbb{R}^n}}$ is an open set. If $u$ and $-v$ are lower semicontinuous on $(t_1,t_2) \times \Omega$ and $v \leq u$ on the parabolic boundary $$(\{t_1\} \times \overline{\Omega} )\cup ([t_1,t_2] \times \partial \Omega),$$ then $v \leq u$ almost everywhere in $(t_1,t_2) \times \Omega$.
Two Concrete Energies {#subsec:concrete kernels}
---------------------
We apply the above results from nonlinear potential theory to two different energy quantities.
- The *$p$-energy* with variational kernel $$F(x,\xi) = \frac{1}{p} |\xi|^{p},$$ where $p > 2$. Then $\nabla_\xi F(x,\xi) = |\xi|^{p-2}\xi$ and the corresponding $\mathcal{A}$-harmonic equation is then the *$p$-Laplace equation* $\Delta_p u := \div(|\nabla u|^{p-2} \nabla u) = 0$.
- The *(quadratic) $A$-energy* defined as follows. Let $A: \mathbb{R}^{n} \to \mathbb{R}^{n \times n}$ be measurable in all entries and let $A(x)$ be symmetric for all $x \in {{\mathbb{R}^n}}$. Suppose uniform boundedness and ellipticity $$\Lambda^{-1} |\xi|^{2} \leq A(x) \xi \cdot \xi \quad \text{and} \quad |A(x) \xi| \leq \Lambda |\xi|$$ for all $x, \xi \in \mathbb{R}^{n}$ and set $$F(x,\xi) = \frac{1}{2} A(x) \xi \cdot \xi.$$ Then the $\mathcal{A}$-harmonic equation is linear and reads $Lu :=\div( A \nabla u) = 0$.
The gradient flow of $p$-energy {#sec:p_grad_flow}
===============================
In this section we extend a function $f \in L^{2}(\mathbb{R}^{n}) \cap \dot{W}^{1,p}(\mathbb{R}^{n})$ to the upper half space along the gradient flow of the $p$-energy functional. Our notation here means that $f \in L^2({{\mathbb{R}^n}})$ with $\nabla f \in L^p({{\mathbb{R}^n}})^n$. This is a non-linear analogue of the heat extension. It amounts to showing that the Cauchy problem $$\begin{aligned}
\label{eq:Cauchy-p}
\begin{split}
\dot u - \Delta_p u &= 0& \qquad &\text{in } (0,\infty) \times {{\mathbb{R}^n}}\\
u|_{t=0} &= f& \qquad &\text{in } {{\mathbb{R}^n}}\end{split}\end{aligned}$$ has a unique solution.
We begin by introducing a natural class of energy solutions. Throughout, we tacitly identify $W^{1,2}(0,T; L^2({{\mathbb{R}^n}}))$ with a subspace of $C([0,T]; L^2({{\mathbb{R}^n}}))$ via the one-dimensional Sobolev embedding whenever convenient.
\[def:energy-solution\] Let $f \in L^{2}(\mathbb{R}^{n}) \cap \dot{W}^{1,p}(\mathbb{R}^{n})$. A measurable function $u$ is an energy solution to the Cauchy problem if it
1. belongs to the function space $$W^{1,2}( 0,T ; L^{2}(\mathbb{R}^{n})) \cap L^{\infty}(0,T; \dot{W}^{1,p}(\mathbb{R}^{n}))$$ for every $T>0$;
2. satisfies for almost every $t>0$ the equation $$\int_{{{\mathbb{R}^n}}} \dot{u}(t,x) \varphi(x) + |\nabla u(t,x)|^{p-2} \nabla u(t,x) \cdot \nabla \varphi (x) \, dx = 0$$ for all $\varphi \in C_0^\infty({{\mathbb{R}^n}})$;
3. obtains the initial value in the $L^{2}$ sense $$\lim_{t \to 0} \int_{\mathbb{R}^{n}} | u(t,x) - f(x) |^2 \, dx = 0 .$$
Poincaré’s inequality implies $L^2({{\mathbb{R}^n}}) \cap \dot{W}^{1,p}({{\mathbb{R}^n}}) \subseteq W^{1,p}_{loc}({{\mathbb{R}^n}})$. Consequently, (ii) means $$\dot{u}(t) - \Delta_p u (t) = 0 \qquad (\text{a.e. } t>0),$$ where $\Delta_p: W^{1,p}_{loc}({{\mathbb{R}^n}}) \to \mathcal{D}'({{\mathbb{R}^n}})$ is the weak $p$-Laplace operator as in Section \[subsec:Euler–Lagrange\]. In particular, every energy solution is a weak solution to the $p$-parabolic equation in the sense of Section \[subsec:Aparabolic equation\].
\[prop:existence-energy-solutions\] Let $f \in L^{2}(\mathbb{R}^{n}) \cap \dot{W}^{1,p}(\mathbb{R}^{n})$. There exists a unique energy solution $u$ to the Cauchy problem . It satisfies for every $t> 0$ the energy estimates $$\begin{aligned}
\label{energyeq.eq}
\|u(t, \cdot) \|_{L^2({{\mathbb{R}^n}})}^{2} + 2 \int_0^t \|\nabla u(s,\cdot)\|_{L^p({{\mathbb{R}^n}})}^p \, ds
\le \|f\|_{L^2({{\mathbb{R}^n}})}^2
\end{aligned}$$ and $$\begin{aligned}
\label{energyeq.eq2}
\int_0^t\|\dot{u} ( s, \cdot) \|_{L^2({{\mathbb{R}^n}})}^{2} \, ds + \frac{1}{p} \|\nabla u(t,\cdot)\|_{L^p({{\mathbb{R}^n}})}^p
\le \frac{1}{p} \|\nabla f\|_{L^p({{\mathbb{R}^n}})}^p.
\end{aligned}$$ Moreover, $u\geq 0$ a.e. in $(0,\infty) \times {{\mathbb{R}^n}}$ provided $f \geq 0$ a.e. in ${{\mathbb{R}^n}}$.
Proposition \[prop:existence-energy-solutions\] is folklore but it cannot be read easily from the current literature. The nature of the time derivative is one of the main concerns here. Typically, the existence class is too large because of data unnecessarily general for our purposes, or the spatial domain is bounded (see e.g. [@Boegelein2014; @Chill2010; @DiBenedetto1989; @Fontes2009c; @Lions; @Wieser1987]). Therefore we give a self-contained proof at the end of the section.
With Proposition \[prop:existence-energy-solutions\] at hand, we can use *a priori* estimates for weak solutions to infer further regularity of $u$.
\[prop:Lipschitz\] Let $u$ be a weak solution to $\dot{u} - \Delta_p u = 0$ in an open cylinder $(t_1,t_2) \times \Omega$. Then $\nabla u \in L^\infty_{loc}(
(t_1,t_2) \times \Omega)$.
In fact, $\nabla u$ is even locally Hölder continuous [@Benedetto-Friedman85], but we do not need this more involved result.
\[prop:dibenedetto\_regularity\] Let $f \in L^{2}(\mathbb{R}^{n}) \cap \dot{W}^{1,p}(\mathbb{R}^{n})$ be non-negative and let $u$ be the energy solution to . Then $u \in L^\infty(({\epsilon},\infty) \times {{\mathbb{R}^n}})$ for every ${\epsilon}>0$.
Proposition \[prop:existence-energy-solutions\] guarantees that the solution $u$ is non-negative. Let ${\epsilon}> 0$ and fix $t_0 > {\epsilon}$, $x_0 \in {{\mathbb{R}^n}}$. By the local sup-estimate of Theorem V.4.2 in [@DiBenedetto1993a], there is a constant $C=C({\epsilon},n,p)$ such that $$\begin{aligned}
u(t,x) \leq C \max \bigg(1, \sup_{t_0 - {\epsilon}< s < t_0} \bigg(\int_{B(x_0,1)} u(s,y) \, dy \bigg)^{p/2} \bigg),\end{aligned}$$ for almost every $t \in (t_0-{\epsilon}/2,t_0)$ and $|x-x_0| < 1/2$. The right-hand side above is bounded independently of $(t_0,x_0)$ due to Hölder’s inequality and .
Proposition \[prop:dibenedetto\_regularity\] guarantees that the DeGiorgi property of Proposition \[prop:DeGiorgi\] applies to $u$ if $f$ is non-negative. From now on, we shall always use the Hölder continuous representative of $u$ in this case. There is no ambiguity with our earlier agreement since this representative also belongs to $C([0,\infty); L^2({{\mathbb{R}^n}}))$.
\[rem:regularity-representative\] By Proposition \[prop:Lipschitz\] the maps $x \mapsto u(t,x)$ satisfy a local Lipschitz condition on ${{\mathbb{R}^n}}$, locally uniformly in $t$. Moreover, since $u$ admits a weak derivative $\dot{u} \in L^2((0,\infty)\times{{\mathbb{R}^n}})$, we have the classical Beppo–Levi property that for a.e. $x \in {{\mathbb{R}^n}}$ the function $u(\cdot,x)$ is absolutely continuous on $[0,T]$ for every $T>0$. See Theorem 2.1.4 in [@Ziemer]. In particular, if $f$ is non-negative, then for such $x$ it follows from Proposition \[prop:dibenedetto\_regularity\] that $u(\cdot,x)$ is bounded on $[0,\infty)$.
We come to the central definition of our paper.
\[def:pgrad\_semi\] For $t \geq 0$ define the semigroup of operators $$S_t : L^{2}({{\mathbb{R}^n}}) \cap \dot{W}^{1,p}({{\mathbb{R}^n}}) \to L^{2}({{\mathbb{R}^n}}) \cap \dot{W}^{1,p}({{\mathbb{R}^n}})$$ by setting $$S_t f(x) := u(t,x),$$ where $u(t,x)$ is the energy solution of . If $f$ is non-negative, define the gradient flow maximal function as $$S^* f (x) := \sup_{t> 0} S_t f(x) .$$
The semigroup property $S_tS_s = S_{t+s}$ follows from Proposition \[prop:existence-energy-solutions\]. Remark \[rem:regularity-representative\] and the strong convergence $u(t,\cdot) \to f$ as $t\to 0$ imply $$\begin{aligned}
\label{eq:S*finite-ae}
0 \leq f(x) \leq S^*f(x) < \infty \qquad (\text{a.e. } x \in {{\mathbb{R}^n}}),\end{aligned}$$ so that $S^*$ is indeed a meaningful maximal function.
Finally, we recall that the gradient flow of $p$-energy for $p>2$ has *finite speed of propagation*.
\[prop:finitespeed\] If $f$ is bounded and compactly supported, then for every $T > 0$ there exists $R(T) < \infty$ such that if $u$ is the corresponding energy solution to , then $$\label{cmptstaycmpt.eq}
\operatorname{supp}u \cap ([0,T] \times {{\mathbb{R}^n}}) \subset [0,T] \times B(0, R(T) ) .$$
We come to the proof of Proposition \[prop:existence-energy-solutions\]. We adapt the Galerkin procedure in [@Chill2010; @Lions] to the unbounded spatial domain ${{\mathbb{R}^n}}$ by working in the anisotropic space $$V:=L^2({{\mathbb{R}^n}}) \cap \dot{W}^{1,p}({{\mathbb{R}^n}}).$$ To begin with, we need the following
\[lem:properties-V\] $V$ is separable, reflexive, and contains $C_c^\infty({{\mathbb{R}^n}})$ as a dense subspace.
The first two properties follow since $V$ is isomorphic to a closed subspace of $L^2({{\mathbb{R}^n}}) \times L^p({{\mathbb{R}^n}})^n$ via $u \mapsto (u,\nabla u)$. As for the third property, let $V_c$ be the space of compactly supported functions in $V$. It is enough to establish density of $V_c$. Indeed, let $f \in V$. Once we have approximants $f_k \in V_c$ of $f$ at hand, we obtain approximants in $C_c^\infty({{\mathbb{R}^n}})$ by smoothing $\min(\max(-k,f_k),k)$ via convolution. Furthermore, since $V_c$ is convex, it suffices to check density for the weak topology. To this end, let $\varphi$ be smooth with $1_{B(0,1)} \leq \varphi \leq 1_{B(0,2)}$, set $\varphi_k(x) := \varphi(x k^{-1})$ and define $f_k := f\varphi_k$. Then $f_k \in V_c$ satisfies $\|f_k\|_{L^2({{\mathbb{R}^n}})} \leq \|f\|_{L^2({{\mathbb{R}^n}})}$ and $$\begin{aligned}
\|\nabla f_k \|_{L^p({{\mathbb{R}^n}})}
&\le \|\varphi_k \nabla f\|_{L^p({{\mathbb{R}^n}})} + \|(f-f_{B(0,2k)})\nabla \varphi_k\|_{L^p({{\mathbb{R}^n}})} + |f_{B(0,2k)}|\|\nabla \varphi_k\|_{L^p({{\mathbb{R}^n}})} \\
&\lesssim \|\nabla f\|_{L^p({{\mathbb{R}^n}})} + k^{n/p-n/2-1} \|f\|_{L^2({{\mathbb{R}^n}})},\end{aligned}$$ where we have used Poincaré’s inequality. Due to $p>2$ the right-hand side stays bounded as $k \to \infty$. By reflexivity, we can extract a subsequence $f_{k_j}$ with weak limit $f_\infty$ in $V$. But $V$ embeds continuously into $L^2({{\mathbb{R}^n}})$ and $(f_k)$ converges strongly to $f$ in $L^2({{\mathbb{R}^n}})$. Hence, we must have $f_\infty = f$.
The argument is divided into 7 steps.
*Step 1: Finite dimensional approximation.* Due to Lemma \[lem:properties-V\] we can pick a countable dense subset $\{w_j : j \geq 1\}$ of $V$. For $k \geq 1$ we let $$\begin{aligned}
V_k := \mathrm{span} \{w_j: 1 \leq j \leq k\}\end{aligned}$$ and we pick $f_k \in V_k$ such that $f_k \to f$ in $V$ as $k \to \infty$.
For every $k$ we consider the variational problem of finding $u_k \in C^1([0,T_k); V_k)$ such that $$\begin{aligned}
\label{eq1:chill}
\begin{split}
\qquad \int_{{{\mathbb{R}^n}}} \dot{u}_k(t) v + |\nabla u_k(t)|^{p-2} \nabla u_k(t) \cdot \nabla v \, dx &=0 \qquad (v \in V_k, \, t \in (0,T_k)),\\
u_k(0) &=f_k.
\end{split}\end{aligned}$$ Since $V_k$ is finite dimensional, we can equivalently equip it with the Hilbert space norm of $L^2({{\mathbb{R}^n}})$ and identify its dual with $V_k$. In this way we obtain a continuous map $\iota: V_k \to V_k$ such that $$\int_{{{\mathbb{R}^n}}} |\nabla w|^{p-2} \nabla w \cdot \nabla v \, dx = \int_{{{\mathbb{R}^n}}} \iota(w) v \, dx \qquad (v,w \in V_k).$$ Consequently, is equivalent to solving the following initial value problem for an autonomous ODE in a finite dimensional space with continuous non-linearity: $$\begin{aligned}
\dot{u}_k(t) + \iota(u_k(t)) & = 0 \qquad (t \in (0,T_k)),\\
u_k(0) &=f_k.\end{aligned}$$ By Peano’s theorem there is a maximal solution $u_k$ such that either $T_k = \infty$ or $\|u_k(t)\|_{V_k} \to \infty$ as $t \to T_k$.
*Step 2: Uniform bounds for the $u_k$.* We have $$\begin{aligned}
\frac{d}{dt} \Big(\|u_k(t)\|_{L^2({{\mathbb{R}^n}})}^2 \Big)= 2 \int_{{{\mathbb{R}^n}}} \dot{u}_k(t) u_k(t) \, dx,\end{aligned}$$ and hence, taking $v= 2u_k(t)$ in and integrating over $(0,t)$ for any $t \in (0,T_k)$ gives $$\begin{aligned}
\label{eq2:chill}
\|u_k(t)\|_{L^2({{\mathbb{R}^n}})}^2 + 2 \int_0^t \|\nabla u_k(t)\|_{L^p({{\mathbb{R}^n}})}^p \, ds
= \|f_k\|_{L^2({{\mathbb{R}^n}})}^2.\end{aligned}$$ We conclude that $u_k$ is bounded on $(0,T_k)$ with values in $V_k$. Thus, we must have $T_k = \infty$. Likewise, $\nabla u_k \in C^1([0,T_k); L^p({{\mathbb{R}^n}}))$ along with Fréchet-differentiability of the $L^p({{\mathbb{R}^n}})$-norm yields $$\begin{aligned}
\frac{d}{dt} \Big(\frac{1}{p} \|\nabla u_k\|_{L^p({{\mathbb{R}^n}})}^p \Big)= \int_{{{\mathbb{R}^n}}} |\nabla u_k(t)|^{p-2} \nabla u_k(t) \cdot \nabla \dot{u}_k(t) \, dx,\end{aligned}$$ so that taking $v=u_k(t)$ in , we obtain for any $t \in (0,\infty)$ that $$\begin{aligned}
\label{eq3:chill}
\int_0^t \|\dot{u}_k(t)\|_{L^2({{\mathbb{R}^n}})}^2 \, ds + \frac{1}{p} \|\nabla u_k(t)\|_{L^p({{\mathbb{R}^n}})}^p
= \frac{1}{p} \|\nabla f_k\|_{L^p({{\mathbb{R}^n}})}^p.\end{aligned}$$ Since $(f_k)$ is a bounded sequence in $V$, these bounds imply that $(u_k)$ is bounded in $W^{1,2}(0,T; L^2({{\mathbb{R}^n}})) \cap L^\infty(0,T; V)$ for every $T>0$.
*Step 3: Extracting a convergent subsequence.* Step 2 and a diagonalization argument allow us to extract a subsequence, which we relabel so that $$\label{eq4:chill}
\begin{split}
& u_k \to u, \text{ weakly in } W^{1,2}(0,T; L^2({{\mathbb{R}^n}}))\\
&\nabla u_k \to v, \text{ weakly in } L^p(0,T; L^p({{\mathbb{R}^n}}))\\
&|\nabla u_k|^{p-2}\nabla u_k \to h \text{ weakly in } L^{p'}(0,T; L^{p'}({{\mathbb{R}^n}})),
\end{split}$$ for every $T>0$. In this manner $u$ is defined on $[0,\infty) \times {{\mathbb{R}^n}}$. By definition of weak convergence, we have $v = \nabla u$. The first weak limit also implies $$\label{eq5:chill}
\begin{split}
& \dot{u}_k \to \dot{u}, \text{ weakly in } L^{2}(0,T; L^2({{\mathbb{R}^n}})) \\
&u_k(t) \to u(t) \text{ weakly in } L^2({{\mathbb{R}^n}}) \text{ for every $t\geq 0$,}
\end{split}$$ and since we have $u_k(0) = f_k$, we conclude $u(0)=f$. This means that $u(t) \to f$ strongly in $L^2({{\mathbb{R}^n}})$ as $t \to 0$.
*Step 4: Energy inequalities.* For fixed $t > 0$ we use and the strong $L^2$-convergence of $(f_k)$ to pass to the limit inferior in . This results in $$\begin{aligned}
\|u(t)\|_{L^2({{\mathbb{R}^n}})}^2 + 2 \int_0^t \|\nabla u(t)\|_{L^p({{\mathbb{R}^n}})}^p \, ds \leq \|f\|_{L^2({{\mathbb{R}^n}})}^2.\end{aligned}$$ We also know from Step 2 that $(u_k(t))$ is bounded in $V$. Hence, we can extract a weakly convergent subsequence $(u_j(t))$ and identifies its weak limit as $u(t)$. Thus, we have $\nabla u_j(t) \to \nabla u(t)$ weakly in $L^p({{\mathbb{R}^n}})$ as $j \to \infty$, which in turn allows us to pass to the limit inferior in for $(u_j)$, so to obtain $$\begin{aligned}
\int_0^t \|\dot{u}(t)\|_{L^2({{\mathbb{R}^n}})}^2 \, ds + \frac{1}{p} \|\nabla u(t)\|_{L^p({{\mathbb{R}^n}})}^p \leq \frac{1}{p} \|\nabla f\|_{L^p({{\mathbb{R}^n}})}^p.\end{aligned}$$ In particular $u \in L^\infty(0,\infty; V)$, so that now $u$ has the required regularity.
*Step 5: Checking that $u$ is an energy solution.* It remains to verify that $u$ satisfies the variational formulation of the equation as in Definition \[def:energy-solution\]. To this end, it suffices to work on $(0,T) \times{{\mathbb{R}^n}}$ for an arbitrary finite $T$. Let $w \in V_j$ and $\phi \in L^p(0,T)$. For $k \geq j$ we take $v = \phi(t)w$ in and integrate in $t$ to give $$\begin{aligned}
\int_0^T \int_{{{\mathbb{R}^n}}} \dot{u}_k \phi w + |\nabla u_k|^{p-2} \nabla u_k \cdot \nabla (\phi w) \, dx \, dt =0.\end{aligned}$$ Due to , we can pass to the limit as $k \to \infty$ and obtain $$\begin{aligned}
\label{eq6:chill}
\int_0^T \int_{{{\mathbb{R}^n}}} \dot{u} \psi + h \cdot \nabla \psi \, dx \, dt =0,\end{aligned}$$ where $\psi(t,x) = \phi(t) w(x)$. Since the union of the $V_j$ is dense in $V$ and as simple functions (valued in $V$) are dense in $L^p(0,T; V)$, we conclude that test functions of that type are dense and that we can actually state for every $\psi$ in $L^p(0,T; V)$. The hard work is to identify $h$ with $|\nabla u|^{p-2} \nabla u$.
As in Section \[sec:energy\] we write $\mathcal{A}(\xi) = \nabla_\xi F(x,\xi) =|\xi|^{p-2} \xi$, where $F(x,\xi) = \frac{1}{p}|\xi|^p$ is the $p$-energy. From and the corresponding inequality with the roles of $\xi_1$ and $\xi_2$ reversed, we obtain the monotonicity inequality $$\begin{aligned}
\label{eq:monotonicity inequality}
(\mathcal{A}(\xi_1) - \mathcal{A}(\xi_2))\cdot (\xi_1- \xi_2) \geq 0.\end{aligned}$$ Thus, we have $$\begin{aligned}
I_k(v) := \int_0^T \int_{{{\mathbb{R}^n}}} (|\nabla u_k|^{p-2} \nabla u_k - |\nabla v|^{p-2} \nabla v)\cdot(\nabla u_k - \nabla v) \, dx \, dt \geq 0\end{aligned}$$ for every $v \in L^p(0,T; V)$. We use with $t=T$ in order to rewrite $I_k(v)$ as $$\begin{aligned}
I_k(v)
&= \int_0^T \int_{{{\mathbb{R}^n}}} -|\nabla u_k|^{p-2} \nabla u_k \cdot \nabla v - |\nabla v|^{p-2} \nabla v \cdot (\nabla u_k - \nabla v) \, dx \, dt \\
& \quad+ \frac{1}{2}\|f_k\|_{L^2({{\mathbb{R}^n}})}^2 - \frac{1}{2} \|u_k(T)\|_{L^2({{\mathbb{R}^n}})}^2.\end{aligned}$$ Strong convergence of $f_k$ and weak convergence from , yields $$\begin{aligned}
\limsup_{k \to \infty} I_k(v)
&\leq - \int_0^T \int_{{{\mathbb{R}^n}}} h \cdot \nabla v + |\nabla v|^{p-2} \nabla v \cdot (\nabla u - \nabla v) \, dx \, dt \\
&\quad +\frac{1}{2}\|f\|_{L^2({{\mathbb{R}^n}})}^2 - \frac{1}{2} \|u(T)\|_{L^2({{\mathbb{R}^n}})}^2.\end{aligned}$$ Here, the left-hand side is positive. To the right-hand side we can add with $\psi = u$, and integrate by parts in $t$, to finally arrive at $$\begin{aligned}
0 \leq \int_0^T \int_{{{\mathbb{R}^n}}} (h- |\nabla v|^{p-2} \nabla v) \cdot (\nabla u - \nabla v) \, dx \, dt.\end{aligned}$$ Applying this to $v= u -\lambda \psi$, where $\lambda>0$ and $\psi \in L^p(0,T; V)$ are arbitrary, and dividing out $\lambda$, yields $$\begin{aligned}
0 \leq \int_0^T \int_{{{\mathbb{R}^n}}} (h- |\nabla (u - \lambda \psi)|^{p-2} \nabla (u- \lambda \psi)) \cdot \nabla \psi \, dx \, dt.\end{aligned}$$ By Lebesgue’s dominated convergence we can pass to the limit as $\lambda \to 0$ to give $$\begin{aligned}
0 \leq \int_0^T \int_{{{\mathbb{R}^n}}} (h- |\nabla u|^{p-2} \nabla u) \cdot \nabla \psi \, dx \, dt.\end{aligned}$$ Since there is no sign restriction on $\psi$, we actually have equality for every $\psi$. Thus, yields $$\begin{aligned}
\label{eq6:chill}
\int_0^T \int_{{{\mathbb{R}^n}}} \dot{u} \psi + |\nabla u|^{p-2} \nabla u \cdot \nabla \psi \, dx \, dt =0\end{aligned}$$ and taking $\psi(t,x) = \phi(t) \varphi(x)$, where $\phi \in C_c^\infty(0,T)$ and $\varphi \in C_c^\infty({{\mathbb{R}^n}})$, confirms that $u$ satisfies the $p$-parabolic equation in the sense of Definition \[def:energy-solution\].
*Step 6: Uniqueness.* Let $u_1$ and $u_2$ be two energy solutions to . Thanks to Lemma \[lem:properties-V\] we can use $u_1(t) -u_2(t)$ as a testfunction for a.e. $t>0$. By the fundamental theorem of calculus, we find $$\begin{split}
\frac{1}{2} \|u_1(t)- u_2(t)\|_{L^2({{\mathbb{R}^n}})}^2\ &= \int_0^t\int_{{\mathbb{R}}} (\dot{u_1} - \dot{u_2})(u_1 - u_2) \, dx \,dt
\\ & = - \int_0^t\int_{{\mathbb{R}}} (|\nabla u_1|^{p-2} \nabla u_1 - |\nabla u_2|^{p-2} \nabla u_2 )\nabla (u_1 - u_2) \, dx \,dt,
\end{split}$$ which is non-positive due to . We conclude $u_1 = u_2$.
*Step 7: Non-negative initial data.* Suppose $f \geq 0$ a.e. on ${{\mathbb{R}^n}}$. Testing the equation for $u$ against $u_-= \min(u,0)$, we obtain as in the previous step $$\begin{split}
\frac{1}{2} \|u_-(t)\|_{L^2({{\mathbb{R}^n}})}^2 &= \int_0^t\int_{{\mathbb{R}}} \dot{u} u_- \, dx \, dt
= - \int_0^t\int_{{\mathbb{R}}} |\nabla u|^{p-2} \nabla u \cdot \nabla u_- \, dx \,dt
\\ & = - \int_0^t\int_{{\mathbb{R}}} |\nabla u_-|^p \, dx \,dt \le 0,
\end{split}$$ where we used $\|u_-(0)\|_{L^2({{\mathbb{R}^n}})}= 0$ and $\partial u_- = \mathbbm{1}_{u \le 0} \partial u$ for partial derivatives of $u$. This shows that $u \ge 0$ almost everywhere.
The method of proof reveals that $u$ depends on $f$ in a continuous fashion. Such a convergence result for weak solutions can also be found as Lemma 3.4 in [@Kilpelaeinen1996].
\[prop:convergence\_of\_data\] If let $(f_k)$ be a sequence of non-negative functions in $V$. If $f_k \to f$ strongly in $V$, then the corresponding energy solutions $u_k$ to admit a subsequence $u_{k_j}$ such that $$u_{k_j} \to u, \text{ locally uniformly in } (0,\infty) \times {{\mathbb{R}^n}},$$ where $u$ is the energy solution to with initial datum $f$.
Since the $u_k$ satisfy and , we have all the properties of Step 2 and Step 3 in the proof of Proposition \[prop:existence-energy-solutions\] for that new sequence. Following the same argument, we obtain weak convergence of a subsequence in the respective spaces to an energy solution of with data $f$. By uniqueness, this limit is $u$. To prove the locally uniform convergence, it suffices to note that given any compact set $K \subset (0,\infty) \times {{\mathbb{R}^n}}$, the family $u_k$ is equicontinuous and uniformly bounded in $K$ by Propositions \[prop:DeGiorgi\] and \[prop:dibenedetto\_regularity\], so that the claim follows from the Arzelà–Ascoli theorem, exhaustion of the upper half space by compact sets and a diagonalization argument.
$p$-energy of the maximal function {#sec:proof-p}
==================================
We begin with an abstract lemma that is crucial to proving Sobolev contractivity for maximal functions associated to any energy functional with $p$-growth as in Section \[sec:energy\].
\[lemma:abstract\_comparison\] Let $L : W_{loc}^{1,p}({{\mathbb{R}^n}}) \to \mathcal{D}'({{\mathbb{R}^n}})$ be an operator, $E \subset {{\mathbb{R}^n}}$ an open set and $T \in (0, \infty]$. Let $u \in C( [0, T) \times {{\mathbb{R}^n}}) $ satisfy the following comparison property:
Whenever $G \Subset E$ is open, $c \in \mathbb{R}$ is a constant and $h \in C(\overline{G})$ is such that $$\begin{aligned}
Lh &= 0 \quad \textrm{in} \quad \mathcal{D}'(G) \\
h(x) + c &\geq u(t,x) \quad \textrm{for all} \quad (t,x) \in \{0\} \times G \cup [0,T) \times \partial G ,\end{aligned}$$ then $u(t,x) \leq h(x) + c$ for all $(t,x) \in (0,T) \times G$.
Let $$u^{*}(x) := \sup_{0<t< T} u(t,x).$$ If $u^{*}$ is upper semicontinuous, finite a.e. and if $u^{*}(x) > u(0,x)$ for all $x \in E$, then $u^{*}$ satisfies the comparison principle (Definition \[def:comparison\]) relative to $L$ in $E$.
Let $G \Subset E$ be any open set. Let $h \in C(\overline{G})$ be a continuous solution to $$L h = 0$$ in $G$ with $h \geq u^{*}$ on $\partial G$.
Suppose, for contradiction, that $$c: = \max_{x \in G} ( u(0,x) - h(x) ) > 0.$$ Note that the maximum $c$ is finite and achieved in the interior since $h(x) \ge u^{*}(x) \geq u(0,x) $ for $x \in \partial G$ and $u$ and $h$ are continuous. Let $x_0 \in G$ be such that $$c = u(0,x_0) - h(x_0).$$ Consider now the function $h_c = h + c$. By choice of $c$, we have that $u(0,x) \le h_c(x)$ for $x \in G$, and by the counter-assumption $u \leq h_c$ also on $[0,T) \times \partial G$. It follows from the assumption on $u$ that $u \le h + c$ in $(0,T) \times G$. Thus, by the definition of $c$ and $x_0$, we have for all $t \in (0,T)$ that $$u(t,x_0) \le h(x_0) + c = u (0, x_0).$$ Taking the supremum in $t$, we obtain $u^{*}(x_0) \le u(0,x)$, which violates $x_0 \in G \subset E$. This contradiction implies $c \le 0$.
Consequently, we have $u(0,x) \le h(x)$ for $x \in \overline{G}$, and further $u(t,x) \leq h(x)$ for all $(t,x) \in [0,T) \times \partial G$ by the choice of $h$. The assumed comparison property of $u$ yields $u(t,x) \leq h(x)$ for all $(t,x) \in (0,T) \times G$. Taking the supremum in $t$ implies $u^{*}(x) \leq h(x)$ for all $x \in G$ as claimed.
We use this lemma to give the
Let $T > 1$ and define the truncated maximal function $$S^{*,T} f(x) := \sup_{0< t < T} S_t f(x) .$$
*Step 1.* We first assume that $f = S_{1/T} g$ for some non-negative, bounded and compactly supported function $g \in L^{2}({{\mathbb{R}^n}}) \cap \dot{W}^{1,p}({{\mathbb{R}^n}})$. In particular, $f$ is continuous. Our goal is to prove $$\label{eq:plaplace_goal}
\int_{{{\mathbb{R}^n}}} | \nabla S^{*,T} f(x)|^{p} \, dx \leq \int_{{{\mathbb{R}^n}}} | \nabla f|^{p} \, dx.$$ Let $E = \{ x \in {{\mathbb{R}^n}}: S^{*,T} f(x) > f(x) \}$. In view of Proposition \[prop:Lipschitz\], $S^{*,T} f$ is the pointwise supremum of functions that satisfy a local Lipschitz condition uniformly in $t$. Hence, $S^{*,T}f$ satisfies a local Lipschitz condition and we conclude that $E$ is open. By finite speed of propagation (Proposition \[prop:finitespeed\]), $S^{*,T} f = 0 $ outside of a bounded set, and hence $E$ is bounded. It also follows from the local Lipschitz condition that $S^{*,T}f \in W_{loc}^{1,\infty}({{\mathbb{R}^n}})$.
On ${}^c E$ we have $S^{*,T}f = f$ and hence $\nabla S^{*,T}f = \nabla f$ almost everywhere, see Corollary 1.21 in [@Heinonen2018]. Consequently, it suffices to show $$\int_{E} | \nabla S^{*,T} f(x)|^{p} \, dx \leq \int_{E} | \nabla f|^{p} \, dx .$$ To this end let $u(t,x) := S_t f(x)$ and $L := \Delta_p$. As $Lh = 0$ implies $(\partial_t - \Delta_p)(h+c) = 0$ for every $c \in {\mathbb{R}}$, the $p$-parabolic comparison principle from Lemma \[lemma:p-parab\_comparison\] guarantees that $u$ satisfies the assumptions of Lemma \[lemma:abstract\_comparison\]. Since $E$ is open and as $S^{*,T} f > f $ holds in $E$, it follows that $S^{*,T}$ satisfies the $p$-harmonic comparison principle in the sense of Definition \[def:comparison\] in $E$. By Lemma \[lemma:comparison\_submin\], it is a $p$-subminimizer in all bounded open subsets of $E$. Let $\{G_i\}$ be an exhaustion of $E$ by bounded open sets. For any non-negative $\varphi \in C_c^{\infty}(E)$, there is $G_i$ such that $\operatorname{supp}\varphi \subset G_i$. Hence, by the monotone convergence theorem $$\begin{aligned}
\label{eq:submin-exhaustion-argument}
\begin{split}
\int_{E} | \nabla S^{*,T}f |^{p} \, dx
&= \lim_{i \to \infty} \int_{G_i} | \nabla S^{*,T}f |^{p} \, dx \\
&\leq \lim_{i \to \infty} \int_{G_i} | \nabla (S^{*,T}f - \varphi ) |^{p} \, dx \\
&= \int_{E} | \nabla (S^{*,T} f - \varphi ) |^{p} \, dx.
\end{split}\end{aligned}$$ Plugging in a sequence of non-negative $\varphi$ with $\varphi \to S^{*,T} f - f$ in $W^{1,p}(E)$-norm, we conclude inequality . Such an approximation is possible because we have $S^{*,T} f - f \in W^{1,p}(E)$, since $E$ is bounded, and $S^{*,T}f-f$ is continuous up to the boundary with zero boundary values (Lemma 1.26 in [@Heinonen2018]).
*Step 2.* Take now an arbitrary non-negative $f \in L^2({{\mathbb{R}^n}}) \cap \dot{W}^{1,p}({{\mathbb{R}^n}})$ and let $(f_k)$ be an approximating sequence of bounded functions with compact support (Lemma \[lem:properties-V\]). Upon replacing $f_k$ with $\min(f_k,0)$, we can assume that the $f_k$ are non-negative. Inequality together with implies $$\begin{aligned}
\| \nabla S^{*,T} S_{1/T}f_k \|_{L^{p}({{\mathbb{R}^n}})}
\leq \| \nabla S_{1/T} f_k \|_{L^{p}({{\mathbb{R}^n}})} \leq \| \nabla f_k \|_{L^{p}({{\mathbb{R}^n}})} .\end{aligned}$$ The right-hand side converges to $\| \nabla f \|_{L^{p}({{\mathbb{R}^n}})}$ as $k \to \infty$, so $\{\nabla S^{*,T} S_{1/T}f_k : k \geq 0\}$ is bounded in $L^{p} ({{\mathbb{R}^n}})$. By reflexivity, we can extract a subsequence so that $$\begin{aligned}
\nabla S^{*,T} S_{1/T}f_k \to G, \text { weakly in } L^{p}({{\mathbb{R}^n}}). \end{aligned}$$ By Proposition \[prop:convergence\_of\_data\], we can extract a further subsequence, still indexed along $k$, such that $S_t f_k(x) \to S_t f(x)$ locally uniformly in $(0,\infty) \times {{\mathbb{R}^n}}$. We have for all $R > 0$, $$\sup_{x \in B(0,R)} |S^{*,T} S_{1/T} f_k(x) - S^{*,T} S_{1/T} f(x) | \leq \sup_{x \in B(0,R)} \sup_{ 1/T< t < T + 1/T} | S_t f_k(x) - S_t f(x) |,$$ so that $S^{*,T} S_{1/T} f_k \to S^{*,T} S_{1/T} f$ uniformly on compact sets. It follows from the dominated convergence theorem that $S^{*,T} S_{1/T} f_k \to S^{*,T} S_{1/T} f$ in the sense of distributions, and hence $G$ must be the weak gradient of $S^{*,T} S_{1/T} f$. By weak lower semicontinuity of the $L^{p}$-norm, we obtain $$\label{eq:p-laplace_intermediate1}
\| \nabla S^{*,T} S_{1/T}f \|_{L^{p}({{\mathbb{R}^n}})}
\leq \liminf_{k \to \infty} \| \nabla S^{*,T} S_{1/T} f_k \|_{L^{p}({{\mathbb{R}^n}})}
\leq \| \nabla f \|_{L^{p}({{\mathbb{R}^n}})} .$$
*Step 3.* It remains to take a limit in $T$. By reflexivity, allows us to extract a sequence $T_j \to \infty$ such that $$\begin{aligned}
\nabla S^{*,T_j} S_{1/T_j}f \to G', \text { weakly in } L^{p}({{\mathbb{R}^n}}).\end{aligned}$$ By definition, we have monotone convergence of $S^{*,T_j} S_{1/T_j} f(x) \to S^* f(x)$ for a.e. $x \in {{\mathbb{R}^n}}$. Suppose we already knew that $S^* f$ was locally integrable. Then the dominated convergence theorem implies $S^{*,T_j} S_{1/T_j}f \to S^* f$ in the sense of distributions and we conclude that $G'$ is the weak gradient of $S^*f$, whereupon weak lower semicontinuity of the $L^{p}$-norm and yield the claim $$\begin{aligned}
\|\nabla S^*f \|_{L^p({{\mathbb{R}^n}})}
\leq \liminf_{T_j \to \infty} \| \nabla S^{*,T_j} S_{1/T_j}f \|_{L^p({{\mathbb{R}^n}})}
\leq \| \nabla f \|_{L^{p}({{\mathbb{R}^n}})}.\end{aligned}$$
*Step 4.* This being said, we fix a ball $B(0,R)$ and show that the average $(S^*f)_{B(0,R)}$ is finite. We recall from that $S^* f$ is almost everywhere finite. Hence, we can find $M < \infty$ such that $$\begin{aligned}
\label{eq:p-laplace-Step3}
|\{ x \in B(0,R): S_* f(x) > M \} | \leq \frac{1}{2} |B(0,R)| .\end{aligned}$$ It follows from Poincaré’s and Chebyshev’s inequalities that for any $\lambda > 0$ and any $u \in \dot{W}^{1,p}(\mathbb{R}^{n})$ we have $$\begin{aligned}
| \{x \in B(0,R) : | u - u_{B(0,R)} | > \lambda \}|
\leq \frac{C_n^pR^p}{\lambda^p} \int_{{{\mathbb{R}^n}}} |\nabla u |^{p} \, dx\end{aligned}$$ for a dimensional constant $C_n>0$. Choosing $\lambda = 4^{1/p}C_n R |B(0,R)|^{1/p}\|\nabla f\|_{L^p({{\mathbb{R}^n}})}$, we obtain in combination with for all $j$ the bound $$\big|\big\{x \in B(0,R) : | S^{*,T_j} S_{1/T_j} f - (S^{*,T_j} S_{1/T_j} f)_{B(0,R)} | > \lambda \big \} \big| \leq \frac{1}{4} |B(0,R)|.$$ In particular, we can find points $x_j \in B(0,R)$ that neither satisfy this condition nor the one in . This means that $$(S^{*,T_j} S_{1/T_j} f)_{B(0,R)} \leq \lambda + S^{*,T_j} S_{1/T_j} f(x_j) \leq \lambda+ S^* f(x_j) \leq \lambda + M .$$ Monotone convergence yields $(S_* f)_{B(0,R)} \leq \lambda + M$ and the proof is complete.
Quadratic energies and related semigroups {#sec:semigroups}
=========================================
We consider quadratic energies with kernel $F(x,\xi) = \frac{1}{2}A(x) \xi \cdot \xi$, where $A: \mathbb{R}^{n} \to \mathbb{R}^{n \times n}$ is measurable, symmetric and satisfies $$\Lambda^{-1} |\xi|^{2} \leq A(x) \xi \cdot \xi \quad \text{and} \quad |A(x) \xi| \leq \Lambda |\xi|$$ for some constant $\Lambda \in (0,\infty)$ and for all $x, \xi \in \mathbb{R}^{n}$. The corresponding operator $L: W^{1,2}_{loc}({{\mathbb{R}^n}}) \to \mathcal{D}'({{\mathbb{R}^n}})$ as in Section \[sec:energy\] is the linear, uniformly elliptic divergence form operator $ L = \div_x( A(x) \nabla \cdot )$. For such operators the Cauchy problem $$\begin{aligned}
\label{eq:Cauchy-A}
\begin{split}
\dot u - L u &=0& \qquad &\text{in } (0,\infty) \times {{\mathbb{R}^n}}\\
u|_{t=0} &=f& \qquad &\text{in } {{\mathbb{R}^n}}\end{split}\end{aligned}$$ can be solved in the strong sense via the theory of $C_0$-semigroups. We assume basic familiarity with this topic and refer to [@Ouhabaz; @Haase] for background. To be precise, the maximal restriction of $L$ to an operator in $L^2({{\mathbb{R}^n}})$ with domain $D(L) \subset W^{1,2}({{\mathbb{R}^n}})$ is self-adjoint and generates the *heat semigroup* $(H_t)_{t \geq 0} := (e^{tL})_{t\geq 0}$.
This is a bounded analytic $C_0$-semigroup on $L^2({{\mathbb{R}^n}})$. For any semigroup $(T_t)_{t\geq 0}$ this terminology means that, given $f \in L^2({{\mathbb{R}^n}})$, the extension $u(t,x):= T_tf(x)$ to the upper half space has regularity $$\begin{aligned}
u \in C([0,\infty); L^2({{\mathbb{R}^n}})) \cap C^\infty((0,\infty); L^2({{\mathbb{R}^n}})),\end{aligned}$$ satisfies $u(0)=f$, and for every integer $k \geq 0$ there is a constant $C_k$ such that $$\begin{aligned}
\label{eq:analyticity}
\sup_{t \geq 0} t^k\|\partial_t^k u\|_{L^2({{\mathbb{R}^n}})} \leq C_k \|f\|_{L^2({{\mathbb{R}^n}})}.\end{aligned}$$ In the case of the heat semigroup, $C_k$ depends on $k$ and the *ellipticity parameter* $\Lambda$.
The heat extension $u(t,x) := H_t f(x)$ satisfies $\partial_t^k u = L^k u$ and in particular it is a weak solution to $\dot{u} - Lu = 0$ in the upper half space in the sense of Section \[subsec:Aparabolic equation\]. This extension is given by a heat kernel in the following sense.
\[prop:heatkernel\] Let $f \in L^2({{\mathbb{R}^n}})$. The operators $H_t$, $t>0$, are given by a kernel $K_{t,L}(x,y)$, measurable in $(t,x,y)$, via $$\begin{aligned}
\label{eq:heat-kernel-representation}
H_t f(x) = \int_{{{\mathbb{R}^n}}} K_{t,L}(x,y) f(y) \, d y \qquad (\text{a.e. } x \in {{\mathbb{R}^n}}).\end{aligned}$$ There are constants $c,C \in (0,\infty)$ depending only on $n$ and $\Lambda$ such that $$\begin{aligned}
0 \le K_{t,L}(x,y) \le C t^{-n/2} e^{-c|x-y|^2/t} \qquad (x,y \in {{\mathbb{R}^n}}).\end{aligned}$$
\[cor:boundedness heat extension\] Let $f \in L^2({{\mathbb{R}^n}})$ and $t>0$. There is a constant $C$ depending on only $n$ and $\Lambda$ such that the following hold for a.e. $x \in {{\mathbb{R}^n}}$:
1. $|H_tf(x)| \leq Ct^{-n/4} \|f\|_{L^2({{\mathbb{R}^n}})}$;
2. $|H_tf(x)| \leq C Mf(x)$, where $M$ is the Hardy-Littlewood maximal function.
The first item follows by applying the Cauchy–Schwarz inequality to . As for the second item, we split integration in into annuli $A_0:= B(x,2\sqrt{t})$ and $A_j := B(x, 2^{j+1}\sqrt{t}) \setminus B(x, 2^j \sqrt{t})$, $j \geq 1$, in order to obtain $$\begin{aligned}
H_tf(x) \leq \sum_{j=0}^\infty C |B(0,1)| 2^{(j+1)n} e^{-c (4^j-1)} f_{B(x,2^{j+1}\sqrt{t})}.\end{aligned}$$ Each average of $f$ is bounded by $Mf(x)$ and the remaining sum in $j$ converges.
Item (i) of the preceding corollary guarantees the DeGiorgi property of Proposition \[prop:DeGiorgi\] for $u(t,x) = H_tf(x)$. This function can be redefined on a set of measure zero to become bounded and Hölder continuous on $({\epsilon},\infty) \times {{\mathbb{R}^n}}$ for any ${\epsilon}> 0$. In fact, by dominated convergence, the continuous representative is the one given by and it is non-negative provided $f$ has this property.
In analogy with Section \[sec:p\_grad\_flow\], we define a vertical heat maximal function.
For non-negative $f \in L^2({{\mathbb{R}^n}})$ define the heat maximal function as $$\begin{aligned}
H^*f(x) := \sup_{t>0} H_t f(x).\end{aligned}$$
The strong continuity of the heat semigroup at $t=0$ and the Hardy–Littlewood maximal bound in Corollary \[cor:boundedness heat extension\] give $$0 \leq f(x) \leq H^*f(x) < \infty$$ for a.e. $x \in {{\mathbb{R}^n}}$ and in fact we have $H^* f \in L^2({{\mathbb{R}^n}})$. The latter property holds for more general diffusion semigroups (Chapter III in [@Stein]) but the Gaussian heat kernel bounds allowed us to give a particularly simple proof.
A self-adjoint generator of a bounded $C_0$-semigroup on $L^2({{\mathbb{R}^n}})$, such as $L$, admits self-adjoint fractional powers $(-L)^\alpha$ for $\alpha \in (0,1)$. In particular, the square root operator $-(-L)^{1/2}$ in $L^2({{\mathbb{R}^n}})$ generates an analytic $C_0$-semigroup given as an $L^2({{\mathbb{R}^n}})$-valued integral by the *subordination formula* $$\begin{aligned}
\label{eq:subordination}
e^{-t (-L)^{1/2}}f = \int_0^\infty \frac{t e^{-t^2/(4s)}}{2 \sqrt{\pi} s^{3/2}} e^{sL}f \, ds,\end{aligned}$$ whenever $f \in L^2({{\mathbb{R}^n}})$ and $t>0$. We refer to Section 3 and Example 3.4.6 in [@Haase]. We call $P_t := e^{-t (-L)^{1/2}}$ the *Poisson semigroup* for $L$.
The Poisson extension $u(t,x):= P_tf(x)$ of $f \in L^2({{\mathbb{R}^n}})$ has regularity as in and satisfies $$\begin{aligned}
\begin{split}
(\partial_t^2 + L) u &=& 0 \qquad &\text{in } (0,\infty) \times {{\mathbb{R}^n}}\\
u|_{t=0} &=& f \qquad &\text{in } {{\mathbb{R}^n}}\end{split}\end{aligned}$$ This is an $\mathcal{A}$-harmonic equation in the sense of Section \[subsec:Euler–Lagrange\] in dimension $n+1$. In particular, $u$ is a (stationary) solution of an $\mathcal{A}$-parabolic equation in dimension $n+2$. The DeGiorgi property from Proposition \[prop:DeGiorgi\] is guaranteed by the following
\[lem:boundedness of poisson extension\] Let $f \in L^2({{\mathbb{R}^n}})$ and $t>0$. There is a constant $C$ depending on $n$ and $\Lambda$ such that the following hold for a.e. $x \in {{\mathbb{R}^n}}$:
1. $|P_tf(x)| \leq Ct^{-n/2} \|f\|_{L^2({{\mathbb{R}^n}})}$;
2. $|P_tf(x)| \leq C Mf(x)$, where $M$ is the Hardy-Littlewood maximal function.
Simply use the bounds provided by Corollary \[cor:boundedness heat extension\] on the right-hand side of and calculate the integral in $s$ by a change of variable $r=t^2/s$.
As usual, the continuity of the Poisson extension in $(t,x)$ allows us to define a corresponding vertical maximal function.
For non-negative $f \in L^2({{\mathbb{R}^n}})$ define the Poisson maximal function as $$\begin{aligned}
P^*f(x) := \sup_{t>0} P_t f(x).\end{aligned}$$
As in the case of the heat maximal function, we obtain from Lemma \[lem:boundedness of poisson extension\] that $P^*f \in L^2({{\mathbb{R}^n}})$ and together with the strong continuity at $t=0$ we infer $$0 \leq f(x) \leq P^*f(x)<\infty.$$ for a.e. $x \in {{\mathbb{R}^n}}$.
We close with certain generic operator estimates for the heat and Poisson semigroups for $L$ that will turn out useful in the next section.
\[lem:operator bounds\] Let $f \in L^2({{\mathbb{R}^n}})$ and $t>0$. For every integer $k \geq 0$ there is a constant $C$ depending only on $k$, $n$ and $\Lambda$ such that $$\begin{aligned}
t^{1+2k} \| \partial_t^k \nabla H_t f\|_{L^2({{\mathbb{R}^n}})}^2 + t^{2+2k} \|\partial_t^k \nabla P_t f\|_{L^2({{\mathbb{R}^n}})}^2 \leq C \|f\|_{L^2({{\mathbb{R}^n}})}^2.\end{aligned}$$
By ellipticity, the definition of $L$ and the Cauchy–Schwarz inequality, we obtain for every $u \in D(L)$ that $$\begin{aligned}
\Lambda^{-1} \|\nabla u \|_{L^2({{\mathbb{R}^n}})}^2 \leq \int_{{{\mathbb{R}^n}}} A \nabla u \cdot \nabla u \, d x = \int_{{{\mathbb{R}^n}}} -Lu \cdot u \, dx \leq \|Lu\|_{L^2({{\mathbb{R}^n}})} \|u\|_{L^2({{\mathbb{R}^n}})}.\end{aligned}$$ The claim follows by taking $u= \partial_t^ke^{tL}f$ and $u=\partial_t^k e^{-t(-L)^{1/2}}f$ and using the generic analytic semigroup bounds from on the right-hand side.
For the next lemma we recall the notion of the energy associated with the kernel $F(x,\xi) = \frac{1}{2}A(x) \xi \cdot \xi$ as in Section \[sec:energy\]: $$\begin{aligned}
\mathcal{F}(u) = \frac{1}{2} \int_{{{\mathbb{R}^n}}} A \nabla u \cdot \nabla u \, dx.\end{aligned}$$
\[lem:semigroup energy dissipation\] Let $f \in W^{1,2}({{\mathbb{R}^n}})$. The energies $\mathcal{F}(H_tf)$ and $\mathcal{F}(P_tf)$ are decreasing for $t \in [0,\infty)$.
By the preceding lemma, the map $t \mapsto H_t f$ is smooth with values in $W^{1,2}({{\mathbb{R}^n}})$. Hence, we can differentiate $$\begin{aligned}
\frac{d}{dt} \mathcal{F}(H_t f)
= \int_{{{\mathbb{R}^n}}} A \nabla H_tf \cdot \nabla L H_tf \, dx
= \int_{{{\mathbb{R}^n}}} -L H_tf \cdot L H_tf \, dx
\leq 0,\end{aligned}$$ where we have used symmetry of $A$ in the first step. Likewise, we get $$\begin{aligned}
\frac{d}{dt} \mathcal{F}(P_t f)
= \int_{{{\mathbb{R}^n}}} L P_tf \cdot (-L)^{1/2} P_tf \, dx
= - \int_{{{\mathbb{R}^n}}} (-L)^{3/4} P_tf \cdot (-L)^{3/4} P_tf \, d x
\leq 0,\end{aligned}$$ where we have decomposed $-L = (-L)^{1/4} (-L)^{3/4}$ and used self-adjointness of the fractional powers.
$A$-energy of the heat and Poisson maximal functions {#sec:heat and poisson}
====================================================
We need the following property of Sobolev spaces from [@Hajlasz1996]. The formulation we use is not exactly the same as in the reference, but a brief inspection of the proof shows that the following lemma is valid.
\[lem:Hderv\] Let $p \in (1,\infty]$. If $f \in L^{p}({{\mathbb{R}^n}})$ and there exists a non-negative $g \in L^p({{\mathbb{R}^n}})$ such that $$|f(x) - f(y)| \le |x - y|(g(x) + g(y)) \qquad (\text{a.e. } x, y \in {{\mathbb{R}^n}}),$$ then $f \in W^{1,p}({{\mathbb{R}^n}})$ and $\|\nabla f\|_{L^p} \le C_{n,p} \|g\|_{L^p}$. Conversely, if $f \in W^{1,p}({{\mathbb{R}^n}})$, then the inequality above holds for all Lebesgue points and $g = M|\nabla f|$, where $M$ is the Hardy-Littlewood maximal function.
We use this lemma to prove qualitative Sobolev bounds for $H^*$ and $P^*$ away from the boundary. We recall from Section \[sec:semigroups\] that the maximal functions are bounded for the $L^2({{\mathbb{R}^n}})$-norm.
\[prop:qualitative\_sobolev\] Let $f \in W^{1,2}({{\mathbb{R}^n}})$ and $\epsilon >0$. Then $H^{*} H_\epsilon f$ and $P^{*} P_\epsilon f$ are weakly differentiable and there is a constant $C$ depending only on $n$ and $\Lambda$ such that $$\| \nabla H^{*} H_\epsilon f \|_{L^{2}({{\mathbb{R}^n}})} \leq \frac{C}{\epsilon^{1/2}} \|\nabla f\|_{L^{2}({{\mathbb{R}^n}})} , \quad \| \nabla P^{*} P_\epsilon f \|_{L^{2}({{\mathbb{R}^n}})} \leq \frac{C}{\epsilon} \|\nabla f\|_{L^{2}({{\mathbb{R}^n}})}.$$
Let $(S_t)_{t \geq 0}$ be either one of the two semigroups. For any $x,y \in {{\mathbb{R}^n}}$ we have $$\begin{aligned}
|S^* S_\epsilon f (x) -S^* S_\epsilon f(y)|
&\leq \sup_{t> 0} |S_t S_\epsilon f (x) - S_t S_\epsilon f(y)| \\
&= \operatorname*{ess\,sup}_{t > {\epsilon}} |S_t f (x) - S_t f(y)|,\end{aligned}$$ where the second step follows from the semigroup property and the continuity of $u(t,x) = S_tf(x)$ in the upper half-space. Since $S_t f \in C({{\mathbb{R}^n}}) \cap W^{1,2}({{\mathbb{R}^n}})$ for all $t > 0$, Lemma \[lem:Hderv\] yields $$\begin{aligned}
\label{eq0:qualitative_sobolev}
|S^* S_\epsilon f (x) - S^* S_\epsilon f(y)|
&\leq |x-y| (Mg(x) + Mg(y)),\end{aligned}$$ where $$\begin{aligned}
g(x) = \operatorname*{ess\,sup}_{t > \epsilon} | \nabla S_t f(x)|.\end{aligned}$$ Note carefully that the definition of $g$ is subject to having fixed a representative for $\nabla u$ but is the same for all $x,y$ and all representatives. By Lemma \[lem:operator bounds\] we have $\nabla u \in C^\infty(0,\infty; L^2({{\mathbb{R}^n}}))^n$ and in particular $\nabla \dot{u} \in L_{loc}^2((0,\infty) \times {{\mathbb{R}^n}})^n$. Hence, we can pick a representative $\overline{\nabla u}$ such that for a.e. $x \in {{\mathbb{R}^n}}$ the restriction $\overline{\nabla u}(\cdot,x)$ is absolutely continuous on all intervals $I \Subset (0,\infty)$, and such that $\partial_t \overline{\nabla u}$ is a representative of $\nabla \dot{u}$. See again Theorem 2.1.4 in [@Ziemer]. In the following we make no notational distinction between $\nabla u = \nabla S_t f$ and its special representative.
Since we already know $S^* S_{\epsilon}f \in L^2({{\mathbb{R}^n}})$, we can use Lemma \[lem:Hderv\] along with the $L^2$ boundedness of the Hardy–Littlewood maximal function, to conclude $$\begin{aligned}
\label{eq1:qualitative_sobolev}
\| \nabla S^*S_\epsilon f \|_{L^{2}({{\mathbb{R}^n}})}
\leq C_n \|g\|_{L^{2}({{\mathbb{R}^n}})}.\end{aligned}$$ It remains to estimate the right-hand side in . Fix $x \in {{\mathbb{R}^n}}$ such that $|\nabla S_t f (x)|$ is absolutely continuous in $t$ on compact intervals. Then we can apply the chain rule for absolutely continuous functions to $\partial_t |\nabla S_t f (x)|^2$ in order to obtain for all $t>{\epsilon}$ that $$\begin{aligned}
\label{eq2:qualitative_sobolev}
|\nabla S_t f (x)|^2
&= |\nabla S_\epsilon f (x)|^{2} + 2 \int_{\epsilon}^{t} \nabla S_s f(x) \cdot \partial _s \nabla S_s f (x) \, ds.\end{aligned}$$
In the case of the heat semigroup $S_t = H_t$ we distribute powers of $s$ and apply the elementary Young’s inequality to give $$\begin{aligned}
\sup_{t > \epsilon} |\nabla H_t f (x)|^2
&\leq |\nabla H_\epsilon f (x)|^{2} + \int_{\epsilon}^{\infty} |\nabla H_s f(x)|^2 \, \frac{ds}{s} + \int_{\epsilon}^{\infty} |\partial _s \nabla H_s f(x)|^2 \, s ds.\end{aligned}$$ Integration in $x$ and the operator bounds in Lemma \[lem:operator bounds\] lead us to $$\begin{aligned}
\| (\sup_{t > \epsilon} | \nabla H_t f|) \|_{L^{2}({{\mathbb{R}^n}})}^2 \leq \frac{C}{{\epsilon}}\end{aligned}$$ and in view of the proof is complete. The proof for the Poisson semigroup is exactly the same, with the difference in the powers of $\epsilon$ coming from the estimates for the Poisson semigroup in Lemma \[lem:operator bounds\].
We are in a position to prove our second main result.
We follow the pattern of the proof of Theorem \[thmintro:1\].
*Step 1.* We first assume that $f = H_\epsilon g$ for some $g \in W^{1,2}({{\mathbb{R}^n}})$ and some $\epsilon > 0$. Then $f$ is continuous. Our goal is to prove $$\begin{aligned}
\label{eq:A-para-goal}
\int_{{{\mathbb{R}^n}}} A \nabla H^*f \cdot H^*f \, dx \leq \int_{{{\mathbb{R}^n}}} A \nabla f \cdot \nabla f \, dx.\end{aligned}$$ By Proposition \[prop:qualitative\_sobolev\] we have $H^* f \in W^{1,2}({{\mathbb{R}^n}})$. Proposition \[prop:DeGiorgi\] and Corollary \[cor:boundedness heat extension\] yield that all $H_t f$ are locally Hölder continuous, uniformly in $t > 0$. Hence, their pointwise supremum $H^* f$ is (locally Hölder) continuous and therefore $E := \{x \in {{\mathbb{R}^n}}: H^* f(x) > f(x)\}$ is open. On the complement we have again $\nabla H^*f = \nabla f$, so that it suffices to prove the estimate for the localized energies $$\begin{aligned}
\mathcal{F}_E(H^*f) = \int_{E} A \nabla H^*f \cdot H^*f \, dx \leq \int_{E} A \nabla f \cdot \nabla f \, dx = \mathcal{F}(f).\end{aligned}$$ To this end let $u(t,x) := H_t f(x)$ and $L := \div(A\nabla \cdot)$. The comparison principle of Lemma \[lemma:p-parab\_comparison\] guarantees again that $u$ satisfies the assumptions of Lemma \[lemma:abstract\_comparison\]. Therefore $H^* f$ satisfies the comparison principle of Definition \[def:comparison\] with respect to $L$. By Lemma \[lemma:comparison\_submin\] it is a subminimizer for the $A$-energy in all bounded open subsets of $E$. This being said, the energy estimate on $E$ follows literally as in .
*Step 2.* Let now $f \in W^{1,2}({{\mathbb{R}^n}})$ be arbitrary. Ellipticity, and Lemma \[lem:semigroup energy dissipation\] imply $$\begin{aligned}
\label{eq:A-para-intermediate}
\begin{split}
\Lambda^{-1} \|\nabla H^* H_\epsilon f\|_{L^2({{\mathbb{R}^n}})}^2
&\leq \int_{{{\mathbb{R}^n}}} A \nabla H^* H_\epsilon f\cdot \nabla H^* H_\epsilon f \, d x \\
&\leq \int_{{{\mathbb{R}^n}}} A \nabla f\cdot \nabla f \, d x
\leq \Lambda \|\nabla f\|_{L^2({{\mathbb{R}^n}})}^2
\end{split}\end{aligned}$$ for all $\epsilon > 0$. Hence, we can extract a subsequence $\epsilon_j \to 0$ such that $$\begin{aligned}
\nabla H^* H_{\epsilon_j}f \to G', \text{ weakly in } L^2({{\mathbb{R}^n}}).\end{aligned}$$ By definition of the maximal function, we have monotone convergence $H^* H_{{\epsilon}_j} f \to H^* f$ a.e. on ${{\mathbb{R}^n}}$. Since $H^*f$ is finite almost everywhere, a literal repetition of Step 4 in the proof of Theorem \[thmintro:1\] reveals that $H^*f$ is locally integrable. By dominated convergence we obtain $H^* H_{\epsilon_j}f \to H^* f$ in the sense of distributions, whereupon $G' = \nabla H^*f$ follows.
Since $A(x)$ is symmetric, we can write $A(x) \xi \cdot \xi = |A(x)^{1/2}\xi|^{2}$ for all $\xi \in {{\mathbb{R}^n}}$, using the positive-definite square root of $A(x)$. From above we obtain weak convergence $A^{1/2} \nabla H^* H_{\epsilon_j} f \to A^{1/2} \nabla H^*f$ in $L^2({{\mathbb{R}^n}})$. Hence, using the square-root decomposition twice, we find $$\begin{aligned}
\int_{{{\mathbb{R}^n}}} A \nabla H^* f \cdot H^*f \, dx
\leq \liminf_{\epsilon_j \to 0} \int_{{{\mathbb{R}^n}}} A \nabla H^* H_{\epsilon_j} f \cdot H^* H_{\epsilon_j} f \, dx
\leq \int A \nabla f \cdot \nabla f\, dx,\end{aligned}$$ where the final step is due to .
Finally, we give the proof of the corresponding result for the Poisson maximal function.
In Section \[sec:semigroups\] and Proposition \[prop:qualitative\_sobolev\] we have seen that Poisson and heat semigroup share the exact same qualitative properties. Therefore we can follow the lines of the proof of Theorem \[thm:A-para\] upon replacing $H_t$ by $P_t$, with the one exception that we cannot appeal to Lemma \[lemma:p-parab\_comparison\] when verifying the assumptions of Lemma \[lemma:abstract\_comparison\] in Step 1. Indeed, $u(t,x) := P_t f(x)$, where $f = P_{\epsilon}g$ for some $\epsilon > 0$ and a non-negative $g \in W^{1,2}({{\mathbb{R}^n}})$, is a continuous weak solution of the elliptic equation $$\begin{aligned}
(\partial_t^2 + L)u = 0 \end{aligned}$$ in $[0,\infty) \times {{\mathbb{R}^n}}$, rather than a parabolic one.
In order to verify that the hypotheses of Lemma \[lemma:abstract\_comparison\] are still met, let $E \subset {{\mathbb{R}^n}}$ be an open set so that $P^*f > f$ in $E$. Let $G \Subset E$ and let $h \in C(\overline{G})$ with $Lh = 0$ in the sense of distributions and $c \in {\mathbb{R}}$ be such that $h(x) + c \geq u(t,x)$ for all $(t,x) \in \{0\} \times G \cup [0,T) \times \partial G$. In particular, we have $h(x)+c \geq 0$. Hence, the decay condition from Lemma \[lem:boundedness of poisson extension\] gives that for any $\eta > 0$ there exists $T_\eta >0$ such that $h(x) + c \geq u(T,x) - \eta$ for all $T \ge T_\eta$ and $x \in G$. By the comparison principle for $\partial_t^2 +L$ (see Remark \[rem:comparison\]) we conclude $h + c \geq u - \eta$ in $G \times [0,T]$ for all $T \ge T_\eta$ and hence in $G \times [0,\infty)$. Sending $\eta \to 0$, we have $h + c \ge u$ in $G \times [0, \infty)$, as desired.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'It is shown here how prior estimates on the local shape of the universe can be used to reduce, to a small region, the full parameter space for the search of circles in the sky. This is the first step towards the development of efficient estrategies to look for these matched circles in order to detect a possible nontrivial topology of our Universe. It is shown how to calculate the unique point, in the parameter space, representing a pair of matched circles corresponding to a given isometry $g$ (and its inverse). As a consequence, (i) given some fine estimates of the covering group $\Gamma$ of the spatial section of our universe, it is possible to confine, in a very effective way, the region of the parameter space in which to perform the searches for matched circles, and reciprocally (ii) once identified such pairs of matched circles, one could determine with greater precision the topology of our Universe and our location within it.'
author:
- |
G.I. Gomero[^1],\
\
Instituto de Física Teórica,\
Universidade Estadual Paulista,\
Rua Pamplona 145\
São Paulo, SP 01405–900, Brazil
title: '**‘Circles in the Sky’ in twisted cylinders**'
---
It has recently been suggested that the quadrupole and octopole moments of the CMB anisotropies are almost aligned, i.e. each multipole has a preferred axis along which power is suppressed and both axes almost coincide. In fact, the angle between the preferred directions of these lowest multipoles is $\sim \! 10^\circ$, while the probability of this occurrence for two randomly oriented axes is roughly 1/62. There is also at present almost no doubt that the extremely low value of the CMB quadrupole is a real effect, i.e. it is not an illusion created by foregrounds [@TOCH].
Traditionally, the low value of the quadrupole moment has been considered as indirect evidence for a non–trivial topology of the universe. Actually, it was the fitting to these low values of the quadrupole and octopole moments of the CMB anisotropy which motivated the recent proposal that our Universe would be a Poincare’s dodecahedron [@LWRLU]. On the other hand, the observed alignement of the quadrupole and the octopole moments has recently been used as a hint for determining the direction along which might occur the shortest closed geodesics characteristic of multiply connected spaces [@OCTZH].
However, in most of the studies reported, the model topology used for the comparison with data has been the $T^1$ topology, i.e. the torus topology with one scale of compactification of the order of the horizon radius, and the other two much larger. This is the simplest topology after the trivial one. Tests using $S$-statistics [@OCSS] and the *circles in the sky* method [@CSS] performed in [@OCTZH] yielded a null result for a non–trivial topology of our universe. However it should be reminded that multiply connected universe models cannot be ruled out on these grounds. In fact, $S$-statistics is a method sensitive only to translational isometries, while the search for the *circles in the sky*, which in principle is sensitive to detect any topology, was performed in a *three-parameter version* able to detect translations only.
If the topology of the Universe is detectable in the sense of [@Detect], then CMB anisotropy maps might present matched circles, i.e. pairs of circles along of which the anisotropy patterns match [@CSS]. These circles are actually the intersections (in the universal covering space of the spatial sections of spacetime) of the topological images of the sphere of last scattering, and hence are related by the isometries of the covering group $\Gamma$. Since matched circles will exist in CMB anisotropy maps of any universe with a detectable topology, i.e. regardless of its geometry and topology, it seems that the search for ‘circles in the sky’ might be performed without any *a priori* information of what the geometry and topology of the universe is. However, any pair of matched circles is described as a point in a six–dimensional parameter space, which makes a full–parameter search computationally expensive.[^2] Nevertheless, such a titanic search is currently being performed, and preliminary results have shown the lack of antipodal, and approximately antipodal, matched circles with radii larger than $25^\circ$ [@CSSK]. These results rule out the Poincare’s dodecahedron model [@LWRLU], and it has also been suggested that they rule out the possibility that we live in a small universe, since for the majority of detectable topologies we should expect antipodal or almost antipodal matched circles. In particular, it is argued that this claim is exact in all Euclidean manifolds with the only exception of the Hantzche–Wendt manifold ($\mathcal{G}_6$ in Wolf’s notation [@Wolf]).
The purpose of this letter is twofold. First, it is shown how to use prior estimates on the local shape of the universe to reduce the region of the full parameter space in a way that the search for matched circles might become practical. In fact, it is shown how to calculate the unique point in the parameter space representing a pair of matched circles corresponding to a given isometry $g$ (and its inverse). As a consequence, given some fine estimates of the covering group $\Gamma$ of the present spatial section of our Universe, we may be able to confine, in a very effective way, the region of the parameter space in which to perform the searches for circles in the sky. This is the first important step towards the development of efficient estrategies to look for these matched circles. Moreover, once such pairs of matched circles had been identified, it is a simple matter to use its location in the parameter space to determine with greater precision the topology of our Universe.
Second, it emerges from the calculations that we should not expect (nearly) antipodal matched circles from the majority of detectable topologies. In particular, any Euclidean topology, with the exception of the torus, might generate pairs of circles that are not even nearly antipodal, provided the observer lies out of the axis of rotation of the isometry that gives rise to the pair of circles. This result might be generalized to the spherical case, for which work is in progress.
The main motivation for this work is the suspicion that the alignement of the quadrupole and the octopole moments of CMB anisotropies observed by the satellite WMAP, together with the *anomalous* low value of the quadrupole moment, is the topological signature we should expect from a generic topology in a nearly flat universe, even if its size is slightly larger than the horizon radius. Moreover, as has been shown in [@LocSh], if topology is detectable in a very nearly flat universe, the observable isometries will behave nearly as translations. If we locally approximate a nearly flat constant curvature space $M$ with Euclidean space, the smallest isometries of the covering group of $M$ will behave as isometries in Euclidean space. Since these isometries are not translations, they must behave as screw motions, thus an appropriate model to get a feeling of what to expect observationally in a nearly flat universe with detectable topology is a *twisted* cylinder.
Thus, let us begin by briefly describing the geometry of twisted cylinders. An isometry in Euclidean 3-space can always be written as $(A,{\mathbf{a}})$, where ${\mathbf{a}}$ is a vector and $A$ is an orthogonal transformation, and its action on Euclidean space is given by $$\label{action}
(A,{\mathbf{a}}) : {\mathbf{x}} \mapsto A {\mathbf{x}} + {\mathbf{a}} \; ,$$ for any point ${\mathbf{x}}$. The generator of the covering group of a twisted cylinder is a screw motion, i.e. an isometry where its orthogonal part is a rotation and its translational part has a component parallel to the axis of rotation [@Wolf]. Thus we can always choose the origin and aligne the axis of rotation with the $z$–axis to write \[Rot\] A = (
[ccc]{} & - & 0\
& & 0\
0 & 0 & 1
) for the orthogonal part, and \[trans\] = (0,0,L) for the translational part of the generator $g = (A,{\mathbf{a}})$.
This is what is usually done when studying the mathematics of Euclidean manifolds, since it simplifies calculations. However, in cosmological applications this amounts to assume that the observer lies on the axis of rotation, which is a very unnatural assumption. In order to consider the arbitrariness of the position of the observer inside space, we parallel transport the axis of rotation, along the positive $x$–axis, a distance $\rho$ from the origin which remains to be the observer’s position. Thus the generator of the twisted cylinder is now $g = (A,{\mathbf{b}})$, with translational part given by $$\label{transGen}
{\mathbf{b}} = \rho (1-\cos \alpha) \, {\widehat{{\mathbf{e}}}_{x}} - \rho \sin \alpha \, {\widehat{{\mathbf{e}}}_{y}} +
L \, {\widehat{{\mathbf{e}}}_{z}} \; .$$
The pair of matched circles related by the generator $g=(A, {\mathbf{b}})$ are the intersections of the sphere of last scattering with its images under the isometries $g$ and $g^{-1}$ respectively, and the centers of these images are located at $g{\mathbf{0}} = {\mathbf{b}}$ and $g^{-1}{\mathbf{0}} = -A^{-1}{\mathbf{b}}$. Thus the angular positions of the centers of the matched circles are $$\label{centmatcirc}
{\mathbf{n}}_1 = \frac{{\mathbf{b}}}{|{\mathbf{b}}|} \qquad\mbox{and}\qquad
{\mathbf{n}}_2 = - \, \frac{A^{-1}{\mathbf{b}}}{|{\mathbf{b}}|} = - A^{-1} {\mathbf{n}}_1
\; .$$
There are four parameters we can determine using (\[Rot\]) and (\[transGen\]). The angle $\sigma$ between ${\mathbf{n}}_1$ and the axis of rotation, the angle $\mu$ between the centers of the pair of matched circles, the angular size $\nu$ of both matched circles, and the phase–shift $\phi$. It turns out that only three of them are independent, as should be expected since a screw motion has only three free parameters $(\rho,L,\alpha)$.
We easily obtain $$\label{angsep}
\cos \mu = {\mathbf{n}}_1 \cdot {\mathbf{n}}_2 = - \, \frac{1 + \tan^2 \sigma \cos
\alpha}{1 + \tan^2 \sigma}$$ for the angular separation between both directions ${\mathbf{n}}_1$ and ${\mathbf{n}}_2$, while $\sigma$ is given by $$\label{angb_rot}
\tan \sigma = \frac{\rho}{L} \, \sqrt{2(1- \cos \alpha)} \; .$$ One can see from (\[angsep\]) and (\[angb\_rot\]) that the matched circles will be antipodal only when the observer is on the axis of rotation ($\rho =
0$), or the isometry is a translation ($\alpha = 0$). In particular, this shows that in a universe with any topology $\mathcal{G}_2$–$\mathcal{G}_6$, if a screw motion of the covering group generates a pair of matched circles, they will not necessarily appear nearly antipodal to an observer located off the axis of rotation. As an example consider an observer in a $\mathcal{G}_4$ universe located at a distance $\rho = L/2$ from the axis of rotation of the generator screw motion ($\alpha = \pi/2$). From (\[angsep\]) and (\[angb\_rot\]) it follows that $\mu \approx 132^\circ$.
Next, to compute the angular size of these circles, let $R_{LSS}$ be the radius of the sphere of last scattering. Simple geometry shows that, since $|{\mathbf{b}}|$ is the distance between the two centers of the spheres whose intersections generate one of the matched circles, the angular size of this intersection is $$\label{angsize}
\cos \nu = \frac{|{\mathbf{b}}|}{2R_{LSS}} = \frac{L}{2R_{LSS} \cos \sigma} \; .$$
The computation of the phase–shift between the matched circles (the last parameter we wish to constrain) is more involved. First we need to have an operational definition of this quantity. This is simply accomplished if we realize that there is a great circle that passes through the centers, ${\mathbf{n}}_1$ and ${\mathbf{n}}_2$, of the matched circles. Orient this great circle such that it passes first through ${\mathbf{n}}_2$, and then through ${\mathbf{n}}_1$ along the shortest path, and let ${\mathbf{v}}_2$, ${\mathbf{u}}_2$, ${\mathbf{u}}_1$ and ${\mathbf{v}}_1$ be the intersections of the great circle with the matched ones following this orientation. If there were no phase–shift, then we would have $g(R_{LSS}{\mathbf{u}}_2) = R_{LSS}{\mathbf{u}}_1$ and $g(R_{LSS}{\mathbf{v}}_2) = R_{LSS}{\mathbf{v}}_1$. Hence we define the phase shift as the rotation angle, around the normal of the sphere at ${\mathbf{n}}_1$, that takes ${\mathbf{u}}_1$ to $\widehat{{\mathbf{u}}}_2 =
g(R_{LSS}{\mathbf{u}}_2)/R_{LSS}$, positive if the shift is counterclockwise, and negative otherwise.
In order to use this operational definition to compute the phase–shift, recall first that the great circle passing through ${\mathbf{n}}_2$ and ${\mathbf{n}}_1$, with the required orientation, is given by $$\label{greatcirc}
{\mathbf{n}}(t) = \frac{1}{\sin \mu} \, [ \, {\mathbf{n}}_2 \sin(\mu - t) +
{\mathbf{n}}_1 \sin t \,] \; ,$$ where $t$ is the angular distance between ${\mathbf{n}}(t)$ and ${\mathbf{n}}_2$. Thus we have $$\begin{aligned}
\label{prephase}
{\mathbf{u}}_1 & = & \frac{1}{\sin \mu} \, [ \, {\mathbf{n}}_2 \sin \nu +
{\mathbf{n}}_1 \sin (\mu - \nu) \,] \; , \nonumber \\
{\mathbf{u}}_2 & = & \frac{1}{\sin \mu} \, [ \, {\mathbf{n}}_2 \sin (\mu - \nu) +
{\mathbf{n}}_1 \sin \nu \,] \; , \qquad\mbox{and} \\
\widehat{{\mathbf{u}}}_2 & = & \frac{1}{\sin \mu} \, [ \, A{\mathbf{n}}_2
\sin (\mu - \nu) + A{\mathbf{n}}_1 \sin \nu \,] + \frac{{\mathbf{b}}}{R_{LSS}} \; ,
\nonumber\end{aligned}$$ Writing the positions of ${\mathbf{u}}_1$ and $\widehat{{\mathbf{u}}}_2$ with respect to their projections to the axis ${\mathbf{n}}_1$, as ${\mathbf{w}}_1 =
{\mathbf{u}}_1 - {\mathbf{n}}_1 \cos \nu$ and ${\mathbf{w}}_2 = \widehat{{\mathbf{u}}}_2
- {\mathbf{n}}_1 \cos \nu$, enables us to express easily the phase–shift as $$\label{phase}
\cos \phi = \frac{{\mathbf{w}}_1 \cdot {\mathbf{w}}_2}{\sin^2 \nu} \; ,$$ since $|{\mathbf{w}}_1| = |{\mathbf{w}}_2| = \sin \nu$. After a somewhat lengthy calculation one arrives at $$\label{postphase}
\cos \phi = \frac{2 (1 + \cos \alpha)}{1 - \cos \mu} - 1$$ for the phase–shift. It is easily seen that when the observer is on the axis of rotation ($\rho = 0$), the shift equals $\alpha$;[^3] while when the isometry is a translation ($\alpha = 0$), the shift vanishes. In general, however, the shift depends on the three parameters $(\rho, L,
\alpha)$, but only through the values of $\mu$ and $\alpha$, thus it is not an independent parameter. In fact, for a given pair $(\sigma, \mu)$, one can easily compute $\phi$, since $\alpha$ is readily obtained from (\[angsep\]).
Summarizing, given estimates of the parameters $(\rho, L, \alpha)$, and having determined an estimate of the axis of rotation of the screw motion, one can perform searches for pairs of circles both with centers at an angular distance $\sigma$ of this axis and separation $\mu$ between them, given by (\[angb\_rot\]) and (\[angsep\]) respectively. The phase–shift between the circles is fixed by these two parameters. Moreover, we can also limit the search of matched circles to only those with angular size $\nu$ given by (\[angsize\]). We have, thus, constrained three out of the six parameters needed to locate a pair of matched circles. The three missing parameters are the position ($\theta, \varphi$) of the axis of rotation of the screw motion and the azimuthal angle $\lambda$ of the center of one of the matched circles.
Under the hypothesis that the alignement of the quadrupole and the octopole moments is due to the topology of space, the position of the axis of rotation might be estimated from this alignement. Work is in progress in this direction. Finally, only the angle $\lambda$ remains totally unconstrained.
Interestingly, a consequent precise identification of a pair of matched circles will allow, reciprocally, to determine with greater precision the same topological parameters with which we started, together with our position and orientation in the Universe.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank FAPESP for the grant under which this work was carried out (contract 02/12328–6). I also thank B. Mota, A. Bernui and W. Hipólito–Ricaldi for useful conversations.
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G.I. Gomero, M.J. Rebouças & R. Tavakol, *Class. Quantum Grav.* [**18**]{}, 4461–4476 (2001); [**18**]{}, L145–L150 (2001); *Int. J. Mod. Phys. A* [**17**]{}, 4261–4272 (2002).\
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[^1]: [email protected]
[^2]: These parameters are the center of each circle as a point in the sphere of last scattering (four parameters), the angular radius of both circles (one parameter), and the relative phase between them (one parameter).
[^3]: We infere from this that, in the general case, both angles $\alpha$ and $\phi$ have the same sign.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We discuss in detail the pulsation properties of variable stars in globular clusters (GCs) and in Local Group (LG) dwarf galaxies. Data available in the literature strongly support the evidence that we still lack a complete census of variable stars in these stellar systems. This selection bias is even more severe for small-amplitude variables such as Oscillating Blue Stragglers (OBSs) and new exotic groups of variable stars located in crowded cluster regions. The same outcome applies to large-amplitude, long-period variables as well as to RR Lyrae and Anomalous Cepheids in dwarf galaxies.'
author:
- Giuseppe Bono
- Silvia Petroni
- Marcella Marconi
title: Variable stars in Stellar Systems
---
\#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} =
\#1 1.25in .125in .25in
Introduction
============
Variable stars in stellar systems such as GCs and dwarf galaxies have played a fundamental role in improving our knowledge on stellar populations (Baade 1958) as well as on the physical mechanism that drive the pulsation instability (Schwarzschild 1942). The main advantage of cluster variables when compared with field ones is that they are located at the same distance, and possibly the same reddening. Moreover, they formed from the same proto-globular cloud and therefore they have the same age, and chemical composition. Even though cluster variables present several undoubted advantages current knowledge concerning the pulsation properties of these objects is still limited. Recent estimates based on new data reduction procedures to perform differential photometry (ISIS, Alard 2000) suggest that the incompleteness factor in the detection of RR Lyrae stars is at least of the order of 30% (Kaluzny et al. 2001; Corwin & Carney 2001) in Galactic GCs characterized by high central densities. This limit is even more severe for OBSs, since the luminosity amplitude range from hundredths of a magnitude to a few tenths. Moreover, their radial distribution peaks toward the center of the cluster, and therefore ground based observations are strongly limited by crowding (Gilliland et al. 1998; Santolamazza et al. 2001). The same outcome applies to Miras and to Semi-Regular variables in GGCs, but for a different reason, quite often they are saturated in current CCD chips. This is a real limit for metal-rich clusters of the Galactic bulge, since they lack of RR Lyrae stars or host a few of them (Pritzl et al. 2002), and the detection of Miras could supply an independent distance estimate (Feast et al. 2002).
Variable stars in dwarf spheroidal (dSph) galaxies presents several pros and cons when compared with variables in GGCs. The star formation history as well as the dynamical evolution of dSph galaxies is much more complex than for GGCs. Typically the age of stellar populations in LG dSphs ranges from a few Gyr to 12-13 Gyr, i.e. as old as stars in GGCs (Da Costa 1999). Wide photometric surveys strongly support the evidence of extra-tidal stars near several dSphs (Irwin & Hatzidimitriou 1995; Martinez-Delgado et al. 2001). The observation of these stellar debris resembles the tidal tails detected in several GGCs (Leon et al. 2000). On the other hand, dSph galaxies apparently host large amounts of Dark Matter (DM), and indeed the mass-to-light ratios in these systems range from $(M/L)_V\sim 5$ (Fornax) to $\sim100$ (Ursa Minor). However, the scenario is still quite controversial and the evidence that dSphs present large DM central densities would suggest that they are not a large version of GGCs, since the latter present M/L ratios $\approx1-2$. Photometric and spectroscopic data on variable stars in dSphs might supply new insights on the impact that environmental effects have on their evolutionary and pulsation properties. Unfortunately, data available in the literature are limited, since these stellar systems cover wide sky regions. The use of wide field imagers and wide field, multifiber spectrographs might overcome these problems.
In the following we discuss the impact that variables in stellar systems might have on cosmic distances and on stellar populations.
Variables in globular clusters
==============================
RR Lyrae stars together with subdwarf main sequence fitting are the most popular standard candles to estimate the distance to GGCs (Carretta et al. 2000; Bono et al. 2001). Both of them require accurate evaluations of cluster metal abundance, but the latter ones are more sensitive to reddening corrections (Castellani 1999). RR Lyrae stars present the non trivial advantage that individual reddening can be estimated on the basis of mean colors. During the last few years have been suggested new methods that rely on observables that do not depend at all on color excess, namely the pulsation period and the luminosity amplitude (Kovacs & Walker 2001; Piersimoni et al. 2002). Even though these pulsation parameters can be easily estimated, the accuracy of individual reddenings might be affected by systematic uncertainties. Empirical evidence suggest that approximately the 30% of fundamental pulsators are affected by the Blazhko phenomenon (Kolenberg, this meeting), i.e. the light curve shows both amplitude and possibly phase modulation (Kurtz et al. 2000). The previous number fraction is supported by recent multiband investigation of RR Lyrae in NGC 3201 (Piersimoni et al. 2002) and in M3 (Corwin & Carney 2001).
= 10.0 cm
Fig. 1 shows the suspected Blazhko RR Lyrae detected in NGC 3201. Note that secondary Blazhko periods are only available for a few GGCs such as M3. Although, this pulsation feature was detected long time ago (Blazhko 1907) we still lack a firm knowledge of the physical mechanisms that drive the occurrence of such a phenomenon. Moreover, empirical data for cluster variables are poor, since they typically cover short-time intervals. This limits the use of the Bailey diagram (amplitude vs period) not only to estimate the intrinsic parameters of RR Lyrae (Bono et al. 1997) but also to estimate their individual color excesses.
This limit affects not only the detection of Semiregular (SR) and Long-Period-Variables (LPVs) but also variables along the RGB and long-period binary systems. On the other hand, the poor spatial resolution of ground based measurements and the limited accuracy hampered the detection of low-amplitude variables such as SX Phoenicis stars and BY Draconis in the innermost regions of GCs. The unprecedented amount of homogeneous and accurate time series data collected by Gilliland and collaborators to detect planets around G type stars in 47 Tuc demonstrated that current knowledge concerning cluster variables is still limited. In particular, they found a wealth of binary systems, as well as a new class of variable stars located at the base of the sub giant branch that they called “Red Stragglers” (see Table 1).
Table 1. Variable stars detected in 47 Tucanae
---------------- ------- --------------- ------------ ------------
Class N$^a$ $A_V^b$ Period$^c$ Source$^d$
mag days
SRs & LPVs 14 … … 1
RR Lyrae 1 $\approx 1$ 0.738 2
SX Phoenicis 6 0.01-0.09 0.03-0.1 3,4
Det. Ecl. Bin. 11 … 0.5-10 5
W UMa 15 … 0.2-0.53 5
Short-Period 10 … 0.1-1.5 5
BY Draconis 65 0.001-0.04 0.5-10 5
CVs 9 … … 5
Red Stragglers 6 0.003-0.12 1-9 5
Red Giants 27 … 3-10 5
LMXB 2 $\approx0.05$ 0.23-0.36 6
MSP 20 0.004 0.43 7
---------------- ------- --------------- ------------ ------------
$^a$ Number of variables. $^b$ Luminosity amplitude in the V band. $^c$ Pulsation period. $^d$ Sources: 1) Fox 1982; 2) Carney et al. 1993; 3) Gilliland et al. 1998; 4) Bruntt et al. 2001; 5) Albrow et al. 2001; 6) Edmonds et al. 2002; 7) Edmonds et al. 2001.
These facts further strengthen the evidence that the knowledge of periodic and aperiodic phenomena among cluster stars might be biased by selection effects (luminosity amplitudes and time resolution). Ground based observations can certainly help to overcome these limits for GGCs with low-central densities, but for high-central densities and post-core-collapse clusters the use of HST is mandatory.
Variables in dwarf galaxies
===========================
Photometric investigations of variable stars in nearby dwarf galaxies have been hampered by the reduced field of view of current CCDs. These stellar systems are characterized by low central densities and very large tidal radii (Mateo 1998). However, during the last few years wide field imagers (WFI) with fields of view of the order of 0.2-0.3 degree$^2$ become available[^1]. The absolute and the relative calibration of individual CCD chips is quite often challenging. Recent results concerning time series data collected with the [email protected] ESO/MPI telescope seem to suggest that these thorny problems can be overcome at the level of a few hundredths of a magnitude (Monelli et al. 2002).
The number of LG dwarf galaxies for which is available a detailed census of variable stars is limited (Mateo 1998; Cseresnjes 2001; Bersier & Wood 2002). This limit applies not only to long-period and aperiodic variables but also to classical ones such as RR Lyrae (Dall’Ora et al. 2002) and $\delta$ Scuti stars (Mateo, Hurley-Keller, & Nemec 1998). The reasons why dwarf galaxies might play a crucial role in improving current knowledge on stellar populations are manifold.
[**i)**]{} Dwarf galaxies harbor stellar populations whose age might range from less than 1 Gyr to more than 12 Gyr, i.e. the stellar masses range from $M/M_\odot\approx0.1$ to $M/M_\odot\approx2$. Therefore they are fundamental laboratories to investigate variable stars that are not present in GGCs such as Anomalous Cepheids (AC)[^2]. Recent findings based on evolutionary and pulsation properties support the evidence that these objects are intermediate-mass stars during their central He-burning phase (Dall’Ora et al. 2002). However, we cannot exclude that some ACs could be the result of mass transfer in old binary systems (Renzini, Mengel, & Sweigart 1977). Moreover, dwarf galaxies might supply fundamental constraints on the accuracy of the Period-Luminosity (PL) relation of $\delta$ Scuti. In fact, these stellar systems often host both RR Lyrae and $\delta$ Scuti, and therefore independent distances may be derived to reduce the systematic uncertainties. It is noteworthy that dwarf galaxies, in contrast with open clusters, host simultaneously $\delta$ Scuti variables, i.e. young intermediate-mass stars, and OBSs, i.e. intermediate-mass stars formed via binary collision or binary merging of two old, low-mass stars (Santolamazza et al. 2001).
[**ii)**]{} Even though GGCs are the template of low-mass population II stars, the HB morphology is affected by the second parameter problem. This means that two GGCs with the same metal abundance might have quite different stellar distributions on the ZAHB. Dwarf galaxies supply the unique opportunity to test whether the dynamical history somehow affects the HB morphology.
[**iii)**]{} The number of GGCs that host sizable samples of RR Lyrae is limited, while dwarf galaxies with old stellar populations present large samples of RR Lyrae. This means that they can be soundly adopted to constrain the accuracy of theoretical predictions concerning the topology of the instability strip.
= 10.0 cm
To investigate in more detail the last point, Fig. 2 shows the number fraction between first overtones and the total number of RR Lyrae as a function of the mean fundamental period. Data plotted in this figure show that a few dwarf galaxies such as Carina, Draco, and Leo II (see data listed in Table 2) can be hardly classified as Oosterhoff type I ($<P_{ab}>\approx0.55$) or Oosterhoff type II ($<P_{ab}>\approx0.64$) clusters. They present mean $<P_{ab}>$ values that are typical of Oo type II clusters, but the ratio between $RR_c$ and total number of RR Lyrae is more typical of Oo type I ($N_c/(N_{ab}+N_c)\approx0.2$) than of Oo type II ($N_c/(N_{ab}+N_c)\approx0.5$) clusters. On the other hand, Ursa Minor present a mean metallicity quite similar to Draco ($[Fe/H]=-2.2\pm0.1$ vs $[Fe/H]=-2.0\pm0.1$, Mateo 1998) but the mean $<P_{ab}>$ value and the number ratio of $RR_c$ variables is typical of Oo type II clusters. Unfortunately, the number of dwarf galaxies in which have been identified mixed-mode variables is still limited and no firm conclusion concerning their occurrence can be drawn.
Table 2. Catalogue of RR Lyrae variables in dwarf galaxies
------------- -------------- ----------------------- ---------- ------------------------------- ------- ------------
Name \[Fe/H\] $\sigma$ \[Fe/H\]$^a$ $N_{ab}$ $\log \langle P_{ab} \rangle$ $N_c$ Source$^b$
Carina -2.0$\pm$0.2 $<$0.1 52 -0.200 15 1
Draco -2.0$\pm$0.2 0.5$\pm$0.1 209 -0.211 54 2
LeoI -1.5$\pm$0.4 0.3$\pm$0.1 47 -0.220 7 3
LeoII -1.9$\pm$0.1 0.3$\pm$0.1 106 -0.210 34 4
Sculptor -1.8$\pm$0.1 0.3$\pm$0.05 134 -0.236 88 5
Sextans -1.7$\pm$0.2 0.2$\pm$0.05 26 -0.219 7 6
Ursa Minor -2.2$\pm$0.1 $\leq$0.2 47 -0.197 35 7
Sagittarius -1.0$\pm$0.2 0.5$\pm$0.1 1906 -0.241 464 8
Fornax -1.3$\pm$0.2 0.6$\pm$0.1 396 -0.233 119 9
Gal. Center … … 1496 -0.261 331 8
LMC -1.7$\pm$0.3 … 3499 -0.234 786 10
SMC -1.7 … 75 -0.231 22 11
------------- -------------- ----------------------- ---------- ------------------------------- ------- ------------
$^a$ Spread in metallicity; $^b$ Sources: 1) Dall’Ora et al. 2002; 2) Nemec 1985; 3) Held et al. 2001; 4) Siegel & Majewski 2000; 5) Kaluzny et al. 1995; 6) Mateo, Fischer & Krzeminski 1995; 7) Nemec, Wehlau & Oliveira 1988; 8) Cseresnjes 2001; 9) Bersier & Wood 2002; 10) Alcock et al. 1996; 11) Graham 1975; Smith et al. 1992.
Discussion
==========
The results presented in the previous sections bring forward the evidence that the empirical scenario for variable stars in stellar systems such as GCs and LG dwarf galaxies is far from being complete. The limit applies not only to aperiodic variables but also to long-period variables such as Miras and Semi-Regulars. The same outcome applies to RR Lyrae stars affected by the Blazhko effect.
Even though several LG dSphs present stellar populations with chemical compositions and stellar ages quite similar to stars in GCs, the RR Lyrae variables present pulsation properties that are a [*bridge*]{} between Oosterhoff type I and Oosterhoff type II clusters. This preliminary evidence seems to suggest that either the dynamical history and/or the chemical evolution in these stellar systems might play a role to explain this difference. In this context the use of the new wide field imagers will supply the unique opportunity to investigate on a star-by-star basis the stars and the variables in LG dwarf galaxies.
Although new theoretical frameworks have been developed to account for the occurrence of mixed-mode pulsators among RR Lyrae and OBSs we still lack a comprehensive explanation of the physical mechanisms that drive the occurrence of such a phenomena. It goes without saying that new sets of full amplitude nonlinear, convective models tightly connected with evolutionary models are highly requested to constrain the region of the instability strip where these pulsators present this intriguing behavior.
Alard, C. 2000, A&AS 144, 363 Albrow, M. D., Gilliland, R. L., Brown, T. M., Edmonds, P. D., Guhathakurta, P., Sarajedini, A. 2001, ApJ 559, 1060 Alcock, C., et al. 1996, AJ, 111, 1146 Bono, G., Caputo, F., Castellani, V., Marconi, M., & Storm, J. 2001, MNRAS, 326, 1183 Bono, G., Caputo, F., Santolamazza, P., Cassisi, S., & Piersimoni, A. 1997, AJ, 113, 2209 Baade, W. 1958, in Stellar Populations, (Amsterdam: North-Holland), ed. D.J.K. O’Connell., 3 Bersier, D., & Wood, P. R. 2002, AJ 123, 840 Bruntt, H. et al. 2001, A&A 371, 614 Cacciari, C. 1984, AJ 89, 1082 Carney, B. W., Storm, J., Williams, C. 1993, PASP 105, 294 Carretta, E., Gratton, R. G., Clementini, G., Fusi Pecci, F. 2000, ApJ, 533, 215 Castellani, V. in Globular Clusters, ed. C. Martinez Roger, I. Perez Fournon, & F. Sanchez (Cambridge, Cambridge Univ. Press), 109 Corwin, T. M., Carney, B. W., & Nifong, B. G. 1999, AJ 118, 2875 Corwin, T. M., Carney, B. W. 2001, AJ 122, 3183 Cseresnjes P. 2001, A&A 375, 909 Da Costa, G. S. 1999, in The Third Stromlo Symposium: The Galactic Halo, eds. Gibson, B.K., Axelrod, T.S. & Putman, M.E., (San Francisco: ASP), 153 Dall’Ora M. et al., submitted to AJ Edmonds, P. D., Grindlay, J. E., Cohn, H., Lugger, P. 2001, ApJ 547, 829 Feast, M., Whitelock, P., & Menzies, J. 2002, MNRAS, 329, L7 Fox, M. W. 1982, MNRAS 199, 715F Gilliland, R. L. et al. 1998, ApJ 507, 818 Graham, J. A. 1975, PASP, 87, 641 Held, E. V., Clementini, G., Rizzi, L., Momany, Y., Saviane, I., & Di Fabrizio, L. 2001, ApJ, 562, L39 Kaluzny, J., Kubiak, M., Szymanski, M., Udalski, A., Krzeminski, W., & Mateo, M. 1995, A&AS, 112, 407 Kaluzny, J., Olech, A., Stanek, K. Z. 2001, AJ, 121, 1533 Kovács, G., Walker, A. R. 2001, A&A 371, 579 Kurtz, D. W. 2000, in IAU Colloq. 176, The Impact of Large-Scale Surveys on Pulsating Star Research, ed. L. Szabados & D. Kurtz (San Francisco: ASP) 291 Irwin, M. & Hatzidimitriou, D. 1995, MNRAS 277, 1354 Leon, S., Meylan, G., & Combes, F. 2000, A&A 359, 907 Martínez-Delgado, D., Aparicio, A., Gómez-Flechoso, M. A., Carrera, R. 2001, ApJ 549, 199 Mateo, M. L. 1998, ARA&A 36, 435 Mateo, M., Fischer, P., & Krzeminski, W. 1995, AJ, 110, 2166 Mateo, M., Hurley-Keller, D., & Nemec, J. 1998, AJ 115, 1856 Monelli, M. et al., submitted to AJ Nemec, J. M. 1985, AJ, 90, 204 Nemec, J. M., Wehlau, A., & Mendes de Oliveira, C. 1988, AJ, 96, 528 Piersimoni, A. M., Bono, G., Ripepi, V. 2002, AJ 124, 1528 Pritzl, B. J., Armandroff, T. E., Jacoby, G. H., & Da Costa, G. S. 2002, AJ, 124, 1464 Renzini, A., Mengel, J. G., Sweigart, A. V. 1977, A&A 56, 369 Siegel, M. H., & Majewski, S. R. 2000, AJ, 120, 284 Santolamazza, P., Marconi, M., Bono, G., Caputo, F., Cassisi, S., Gilliland, R. L 2001, ApJ 554,1124 Schwarzschild, M. 1942, MNRAS 102, 152 Smith, H. A., Silbermann, N. A., & Baird, S. R., & Graham, J. A. 1992, AJ, 104, 1430
[^1]: In this site you can find more detailed information concerning present and future WFIs http://www.ls.eso.org/lasilla/sciops/2p2/E2p2M/WFI
[^2]: Only one AC is known among GGCs (NGC 5466), see Bono et al. (1997) and Corwin, Carney, & Nifong (1999).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The transverse Ising model (TIM), with pseudospins representing the lattice polarization, is often used as a simple description of ferroelectric materials. However, we demonstrate that the TIM, as it is usually formulated, provides an incorrect description of SrTiO$_{3}$ films and interfaces because of its inadequate treatment of spatial inhomogeneity. We correct this deficiency by adding a pseudospin anisotropy to the model. We demonstrate the physical need for this term by comparison of the TIM to a typical Landau-Ginzburg-Devonshire model. We then demonstrate the physical consequences of the modification for two model systems: a ferroelectric thin film, and a metallic LaAlO$_{3}$/SrTiO$_{3}$ interface. We show that, in both cases, the modified TIM has a substantially different polarization profile than the conventional TIM. In particular, at low temperatures the formation of quantized states at LaAlO$_{3}$/SrTiO$_{3}$ interfaces only occurs in the modified TIM.'
address: '$^1$Department of Physics and Astronomy, Trent University, Peterborough, Ontario, Canada, K9L 0G2'
author:
- Kelsey S Chapman$^1$ and W A Atkinson$^1$
bibliography:
- 'refs.bib'
title: 'Modified transverse Ising model for the dielectric properties of SrTiO$_3$ films and interfaces'
---
[*Keywords*]{}: strontium titanate, interface, two-dimensional electron gas, transverse Ising model, ferroelectric films
Introduction
============
The transverse Ising model (TIM) was developed by deGennes in 1963 to describe the ferroelectric transition in hydrogen-bonded materials like potassium dihydrogen phosphate (KDP) [@degennes63]. As suggested by its name, the model formally describes a system of magnetic Ising moments in a transverse magnetic field [@stinchcombe73], and since its discovery it has become significant because it is one of the simplest models to exhibit a quantum phase transition [@Sachdev:2011]. The focus of this work is more practical; we explore the use of the TIM to describe the dielectric properties of SrTiO$_3$. Indeed, the TIM has been used widely to model the low-energy physics of systems in which local degrees of freedom can be represented by pseudospins [@stinchcombe73]. In KDP, for example, the $S=\frac{1}{2}$ Ising spin states represent the two degenerate positions available to each hydrogen atom, while the transverse field represents the quantum mechanical tunneling between the states.
Because the TIM starts from a picture of fluctuating local dipole moments, it naturally describes materials, like KDP, with order-disorder transitions. However, the model has also been applied to materials like SrTiO$_3$, which are close to a displacive ferroelectric transition. While there are some clear discrepancies between the model and experiments [@Muller:1979wa], the mean-field TIM nonetheless gives a useful quantitative phenomenology for the dielectric properties of both pure [@hemberger95; @hemberger96] and doped[@kleemann00; @kleemann02; @kleemann98_di; @wu03; @guo12] SrTiO$_3$.
The local nature of the Ising pseudospins makes the TIM valuable as a model for inhomogeneous systems, including doped quantum paraelectrics [@kleemann00; @kleemann02; @kleemann98_di; @wu03; @guo12], ferroelectric thin films [@wangcl92; @sun08; @oubelkacem09; @wangCD10; @lu13; @li16], superlattices [@wangCL00; @yao02], and various low-dimensional structures [@xin99; @lang07; @lu14]. However, we show here that the TIM, as it is conventionally formulated, fails to correctly describe SrTiO$_3$ whenever nanoscale inhomogeneity is important. Most egregiously, the TIM fails to predict the formation of a quantized two-dimensional electron gas (2DEG) at LaAlO$_3$/SrTiO$_3$ interfaces, in contradiction with both theory and experiments [@gariglio15]. The goal of this paper is to propose a modification that we believe captures the essential physics of spatial inhomogeneity, and to compare it to the conventional TIM for model SrTiO$_3$ thin films and interfaces. In the TIM, the lattice polarization $P_{i}$ in unit cell $i$ is modelled by a pseudospin. This polarization is given by $$\label{P}
P_{i} = \mu \eta S^{(3)}_{i},$$ where $\mu$ sets the scale of the electric dipole moment, $\eta = a^{-3}$ is the volume density of dipoles, and $a$ is the lattice constant. The pseudospin is usually taken to be $S=\frac 12$, and $S^{(3)}_{i}$ is the third component of the corresponding three-dimensional pseudospin vector ${\mathbf{S}}_{i}$. The other two components, $S^{(1)}_{i}$ and $S^{(2)}_{i}$, are fictitious degrees of freedom, with only the projection of ${\bf S}_i$ onto the $(3)$-axis corresponding to the physical polarization. (The unpolarized state is therefore described by the pseudospin lying entirely in the $(1)$-$(2)$ plane.) In a quantum model, $S^{(3)}_{i}$ is the expectation value of the operator $\hat{S}^{(3)}_{i}$, which is identical to the spin matrix $\hat{S}^{z}$ but which acts within pseudospin space.
The simplest version of the $S = \frac 12$ TIM is [@hemberger96] $$\label{TIM_orig}
\hat{H} = - \Omega \sum_{i} \hat{S}^{(1)}_{i} - J_{1} \sum_{\langle i, i' \rangle} \hat{S}^{(3)}_{i} \hat{S}^{(3)}_{i'} - \mu \sum_{i} E_{i} \hat{S}^{(3)}_{i},$$ where $\Omega$ plays the role of a transverse magnetic field that flips the Ising spins, $J_1$ is a nearest-neighbour coupling constant with $\langle i,i' \rangle$ indicating nearest-neighbour sites, and $E_i$ is the electric field in unit cell $i$. For $J_1>0$, the model tends towards a ferroelectric state at low temperatures; however, this is limited by $\Omega$, which disorders the ferroelectric state. Under mean-field theory the model predicts a ferroelectric phase transition only if $\Omega < Z J_{1}$, where $Z$ is the coordination number of the lattice.
Although the TIM is only microscopically justified for order-disorder ferroelectrics, it is often used as a tool to characterize ferroelectrics of all types, and variations of this model have been applied to ferroelectricity in perovskites, including BaTiO$_{3}$ [@zhang00] and SrTiO$_3$ (STO) [@hemberger96]. As a phenomenological model, the TIM is more complex than simple Landau-Ginzburg-Devonshire theories; however, it is also more versatile. The TIM, for example, is particularly well-suited to doped quantum paraelectrics, namely Sr$_{1-x}$M$_x$TiO$_3$ with M typically representing Ca or Ba [@kleemann00; @kleemann02; @kleemann98_di; @wu03; @tao04; @guo12]. In these materials, small dopant concentrations are sufficient to induce a ferroelectric transition. Several groups have successfully modeled these materials as binary alloys of SrTiO$_3$ and MTiO$_3$ with doping-independent model parameters [@kleemann02; @kleemann98_di; @wu03; @tao04; @guo12].
The current work is motivated by the application of the TIM to metallic LaAlO$_{3}$/SrTiO$_{3}$ (LAO/STO) interfaces. These, and other related perovskite interfaces, have been widely studied since the discovery in 2004 that a 2DEG appears spontaneously at the interface when the LAO film is more than four unit cells thick [@ohtomo04]. This system is rich with interesting properties, including coexisting ferromagnetism and superconductivity [@Brinkman:2007fk; @Reyren:2007gv; @Dikin:2011gl], nontrivial spin-orbit effects [@BenShalom:2010kv; @Caviglia:2010jv], a metal-insulator transition [@thiel06; @Liao:2011bk], gate-controlled superconductivity [@Caviglia:2008uh], and a possible nematic transition at (111) interfaces [@Miao:2016hr; @Davis:2017; @Boudjada:2018; @Boudjada:2019]. Furthermore, STO’s proximity to the ferroelectric state has led to suggestions that quantum fluctuations shape its band structure [@atkinson17] and support superconductivity [@Edge:2015fj; @Dunnett:2018]. More generally, there has been a growing appreciation that lattice degrees of freedom play a key role in shaping the electronic structure near LAO/STO interfaces [@Behtash:2016dt; @Lee:2016dj; @Gazquez:2017bu; @raslan18]. With this in mind, the recent discovery that ferroelectric-like properties persist in some metallic perovskites [@Rischau:2017vj] naturally leads one to explore the effects of Ca or Ba doping on LAO/STO interfaces and, as described above, the TIM provides a natural framework in which to do this.
We found, however, that the TIM as it is usually formulated in equation cannot reproduce the interfacial 2DEG and therefore fails to describe even the simple LAO/STO interface. In this work, we explain the reason for this failure and propose a modification to the TIM. In , we introduce the modified model and by comparison with the standard Landau-Ginzburg-Devonshire (LGD) expansion, illustrate why the failure arises and how we fix it. As a simple example, we apply the modified model to ferroelectric thin films. In , we then apply the model to the LAO/STO interface, and show explicitly how the modification allows for the formation of the 2DEG.
Inhomogeneous Ferroelectrics {#sec:FE}
============================
We begin by describing a modified TIM () that contains an additional anisotropic interaction; depending on its sign, this interaction generates either a pseudospin easy axis or easy plane. We obtain mean-field equations for the pseudospin and susceptibility, and by comparison to the LGD theory () we show that the Landau parameters are under-determined by the conventional TIM. Essentially, the problem is that equation contains three adjustable parameters ($\Omega$, $J_1$, and $\mu$), while the simplest LGD model requires four parameters to describe an inhomogeneous system. The additional interaction in the modified TIM fixes this discrepancy. In sections \[sec:fit\] and \[sec:J1\] we obtain fits to the model parameters for the case of STO. As a simple application, in we explore how the new term modifies the polarization distribution of a ferroelectric thin film.
The Modified TIM {#sec:TIM}
----------------
The modified Hamiltonian for general pseudospin $S$ is $$\begin{aligned}
\label{TIM_full}
\fl \hat{H} = - \Omega \sum_{i} \hat{S}^{(1)}_{i} - \frac{J_{1}}{2S} \sum_{\langle i,i' \rangle} \hat{S}^{(3)}_{i} \hat{S}^{(3)}_{i'} - \frac{J_\mathrm{an}}{2S} \sum_{i} \hat{S}^{(3)}_{i} \hat{S}^{(3)}_{i} - \mu \sum_{i} E_{i} \hat{S}^{(3)}_{i}.\end{aligned}$$ This is equivalent to the Blume-Capel model in a transverse magnetic field [@Albayrak:2013]. The third term introduces an anisotropic pseudospin energy. If $J_\mathrm{an} > 0$, this term tends to align dipoles along the (3)-axis, making it an easy axis, which enhances the polarization; if $J_\mathrm{an} < 0$, the term tilts the dipole away from the (3)-axis, creating an easy plane and reducing the polarization.
The TIM is traditionally formulated with a spin-$\frac 12$ pseudospin. In that case, $\hat{S}^{(3)}_{i}$ is written in terms of a Pauli spin matrix, and $(\hat{S}^{(3)}_{i})^2$ is proportional to the identity operator. The new term therefore does not produce the desired anisotropy when $S = \frac{1}{2}$. This problem does not exist for higher spin models, and for this reason we formulate the TIM in terms of a general pseudospin $S$. However, we will show below that at the mean-field level, the model provides nearly the same results for any value of $S$, and for simplicity we revert to $S=1$ when we show results as a way of gaining insight into the general case.
Applying mean-field theory to equation gives the following self-consistent expression for $S^{(3)}_{i}$: $$\label{S3}
S^{(3)}_i = \frac{S h^{(3)}_i}{h_i} f_{S}(h_i),$$ where $$\label{f_S}
f_{S}(h_i) = \frac{1}{S} \frac{\sum\limits_{l=-S}^{S} l \rme^{\beta h_i l}}{\sum\limits_{n = -S}^{S} \rme^{\beta h_i n}} = B_S(\beta h_i S),$$ $B_S(x)$ is the Brillouin function, $\beta = (k_\mathrm{B} T)^{-1}$, $T$ is temperature, $h_{i} = | {\mathbf{h}}_{i} |$, and $h_i^{(3)}$ is the $(3)$-component of the Weiss mean field for lattice site $i$, $$\label{h_i}
\textbf{h}_i = \left( \Omega, 0, \frac{J_{1}}{S} \sum_{i'} S^{(3)}_{i'} + \frac{J_\mathrm{an}}{S} S^{(3)}_i + \mu E_{i} \right).$$ The summation $\sum_{i'}$ is a sum over the nearest neighbours of site $i$, and therefore depends on whether pseudospin $i$ is in a surface or bulk layer.
We linearize equation to obtain the condition that ensures ferroelectricity. In the uniform case, $$\label{h_uniform}
{\mathbf{h}} = \left( \Omega, 0, \frac{J_{0}}{S} S^{(3)} + \mu E \right),$$ where $$\label{J0}
J_0 = ZJ_1 + J_\mathrm{an},$$ for coordination number $Z$. At zero-temperature, $f_{S}(h_{i}) \rightarrow 1$, and from equation the model therefore predicts a paraelectric-ferroelectric phase transition when $$S^{(3)} = \frac{J_0S^{(3)}}{\Omega}.$$ From this one sees that, for any $S$, a ferroelectric transition occurs at nonzero temperature only when $J_0 > \Omega$. In the case of a paraelectric like STO, $J_0 < \Omega$.
![Inverse dielectric susceptibility versus temperature for SrTiO$_{3}$, modelled using three-, four- and five-component pseudospins. The fitting parameters were found separately for each pseudospin (). [*[Inset]{}*]{}: The SrTiO$_3$ unit cell is illustrated, showing that the polarization is primarily due to the soft phonon mode (black arrows), in which the oxygen cage moves opposite to the titanium ions. The inset is re-published from [@atkinson17].[]{data-label="fig:STOX_compspin"}](figure1){width="0.7\linewidth"}
To show that the choice of $S$ has a small effect at the mean-field level, the uniform inverse dielectric susceptibility of STO is plotted for different values of $S$ in . From equation , the susceptibility for a weak uniform electric field $E$ is $$\label{X_gen}
\chi (T) = \left . \frac{1}{\epsilon_0} \frac{dP}{dE} \right|_{E=0} =\left . \frac{\mu \eta}{\epsilon_{0}} \frac{d S^{(3)}}{dE} \right|_{E=0},$$ where $dS^{(3)}/dE$ is obtained from equation with ${\mathbf{h}}$ given by equation . shows results for $S=1$, $S=\frac 32$ and $S=2$. The fitting parameters $J_0$, $\mu$, and $\Omega$ depend on the value of $S$ and were determined by fitting to the experimental susceptibility, as described in below. (Note that $J_1$ is not explicitly used here because the calculations are for bulk STO.) The values of all these parameters are listed in .
Because the model was fitted to low- and high-temperature susceptibilities, the curves in are expected to be close in value at these limits. However, they also differ only slightly in between, indicating that STO is well-described by the simplest case shown, $S=1$, when using mean-field theory. In particular, the model accurately captures both Curie-Weiss behaviour at high temperature, and the saturation of the susceptibility at low temperature (where the ferroelectric transition is suppressed by quantum fluctuations).
[c c c c]{} & Spin-1 & Spin-3/2 & Spin-2\
$\Omega$ (meV) & 4.41 & 3.53 & 2.94\
$J_{0}$ (meV) & 3.88 & 3.10 & 2.58\
$J_{1}$ (meV) & 30-130 & 40-160 & 50-200\
$\mu$ ($e$Å) & 1.88 & 1.37 & 1.09\
\[tab:pars\]
Comparison to the Landau-Ginzburg-Devonshire Expansion {#sec:LGD}
------------------------------------------------------
While equation is the fundamental self-consistent equation for $S^{(3)}_{i}$, the role each parameter plays in determining the pseudospin is not transparent. For example, it is not immediately evident from this expression why the conventional TIM (with $J_\mathrm{an}=0$) is unable to describe inhomogeneous systems. To explore this point, we expand equation in powers of $h_{i}^{(3)}$ and compare the coefficients to those in a typical LGD expansion. We show that the transition temperature and correlation length cannot be set independently unless $J_\mathrm{an}$ is nonzero.
The typical LGD free energy with order parameter $S^{(3)}({\mathbf{r}})$ has the form $$\begin{aligned}
\label{F}
\mathcal{F} &=& \eta \int d^{3} r\, \Bigg[ \frac{A}{2} \left( S^{(3)}({\mathbf{r}}) \right)^{2} + \frac{B}{4} \left( S^{(3)}({\mathbf{r}}) \right)^{4} \nonumber \\ && + \frac{C}{2} \left( \nabla S^{(3)}({\mathbf{r}}) \right)^{2} - D E({\mathbf{r}}) S^{(3)}({\mathbf{r}}) \Bigg].\end{aligned}$$ $E({\mathbf{r}})$ is the electric field, $A$, $B$, $C$ and $D$ are the LGD coefficients that describe the material, and $\eta$ is the inverse volume of a unit cell. Minimizing equation with respect to $S^{(3)}({\mathbf{r}})$ gives the familiar equation $$\label{Fmin}
0 = A S^{(3)}({\mathbf{r}}) + B \left( S^{(3)}({\mathbf{r}})\right)^{3} -C \nabla^{2} S^{(3)}({\mathbf{r}}) - D E({\mathbf{r}}),$$ which can be solved for the pseudospin. The critical temperature is set by $A$, which changes sign at the ferroelectric transition, while $B$ determines the zero-temperature polarization. In the paraelectric phase, $D$ is determined by the dielectric susceptibility and $C$ and $A$ set the correlation length $\xi=\sqrt{C/A}$.
We expand equation in powers of $h_{i}^{(3)}$ to obtain $$\label{expand_2}
S^{(3)}_i = \frac{S f_S (\Omega)}{\Omega} h^{(3)}_i + \frac{1}{2 \Omega} \left( \frac{d}{d\Omega} \frac{Sf_{S}(\Omega)}{\Omega} \right) \left( h^{(3)}_{i} \right)^{3},$$ where ${h}_i$ and $h_i^{(3)}$ are defined by equation . To proceed further, we note that the discretized second derivative of a function $f_j=f(x_j)$ is $$\left . \frac{d^{2}f(x) }{dx^{2}} \right |_{x=x_j} \approx \frac{f_{j-1} - 2f_j + f_{j+1}}{a^{2}}.$$ Then, equation can be re-written as $$h^{(3)}_i = \frac{J_0}{S} S^{(3)}_i + \frac{J_1}{S} a^{2} \nabla^{2} S^{(3)}_i + \mu E_i,$$ with $J_0$ defined by equation . This can now be substituted into equation .
Keeping only terms that are directly comparable to those in equation , we obtain $$\begin{aligned}
A & = \frac{\Omega}{Sf_{S}(\Omega)} - \frac{J_{0}}{S}, \label{coeff_A} \\
B & = -\frac{1}{2 S f_{S}(\Omega)} \left( \frac{J_{0}}{S} \right)^{3} \frac{d}{d\Omega} \left( \frac{S f_S (\Omega)}{\Omega} \right), \label{coeff_B} \\
C & = \frac{J_{1} a^{2}}{S}, \label{coeff_C} \\
D & = \mu. \label{coeff_D}\end{aligned}$$ These equations show that $A$ and $B$ are determined by combinations of $J_0$ and $\Omega$, while $C$ and $D$ are determined by $J_1$ and $\mu$, respectively. The key point is that $J_0$ reduces to $ZJ_1$ for the conventional TIM, in which case $A$ and $C$ are not independent. Physically, this means that the correlation length, which sets the length scale over which the material responds to inhomogeneities, cannot be determined independently of the transition temperature and low-$T$ polarization. In other words, the four coefficients $A$, $B$, $C$ and $D$ are only described by three parameters, $\Omega$, $J_{1}$ and $\mu$.
In this case, the model predicts a significantly smaller correlation length at low temperatures than does the modified TIM. From equations (\[coeff\_A\]) and (\[coeff\_C\]), $$\xi = \sqrt{\frac{C}{A}} = \sqrt{\frac{J_1 a^2}{\frac{\Omega}{f_S(h)} - J_0}}.$$ At low temperatures, $f_S(\Omega) \rightarrow 1$. In this case, the conventional TIM ($J_1 = J_0/Z$) gives $\xi\approx4.3$ Å, independent of $S$. For $S=1$, the range of correlation lengths from the modified TIM, where the $J_1$ values are taken from , is 2.9-6.1 nm, which is an order of magnitude larger. The pseudospin anisotropy $J_\mathrm{an}$ is therefore an essential part of the TIM.
Fitting $\Omega$, $J_0$, and $\mu$ for SrTiO$_3$ {#sec:fit}
------------------------------------------------
Most of the TIM parameters can be fit to existing susceptibility data. We do this for STO, as it will form the basis of our discussion in .
Inserting equation into equation , we obtain the susceptibility $$\label{X}
\chi(T,0) = \frac{\mu^{2} \eta}{\epsilon_{0}} \frac{1}{L(h,T) - J_{0}/S} \Bigg\vert_{E=0},$$ where $h = |{\mathbf{h}}|$, ${\mathbf{h}}$ is given by equation , and $$\begin{aligned}
\fl L(h,T) = \Bigg[ S \left( \frac{1}{h} - \frac{\left( h^{(3)} \right)^{2}}{h^{3}} \right) f_S(h) + S \frac{\left( h^{(3)} \right)^{2}}{h^{2}} \frac{\partial f_S(h)}{\partial h} \Bigg]^{-1}.\end{aligned}$$
At high temperatures, this expression simplifies. Taking $L(h,T)|_{T \rightarrow \infty}=[\beta S(S+1)/3]^{-1}$, equation obtains a Curie-Weiss form, $$\label{X_0}
\chi(T,0) = \frac{\mu^{2} \eta S(S+1) }{3 \epsilon_{0} k_B} \frac{1}{T - T_\mathrm{CW}},$$ where $T_\mathrm{CW} = (S+1) J_{0}/3k_\mathrm{B} \approx 30$ K [@sakudo71] is the transition temperature implied by the high-temperature susceptibility. (In STO, this transition is suppressed by quantum fluctuations.) $J_0$ and $\mu$ are thus obtained by matching equation to high-$T$ experiments.
At low $T$, equation takes one of two forms depending on whether the system is ferroelectric or not. For a ferroelectric, $\Omega$ can be found from the behaviour of the susceptibility at $T \rightarrow T_\mathrm{c}^{+}$ for critical temperature $T_\mathrm{c}$. In this case, $h^{(3)}=0$, $h=\Omega$, and setting the denominator of equation to zero gives a self-consistent equation for $\Omega$, $$\Omega = J_0 f_S(\Omega).$$
For a paraelectric like STO, on the other hand, we obtain $\Omega$ from the zero-temperature susceptibility. In this limit $L(h,T)|_{T \rightarrow 0} = \Omega/S$ and equation may easily be inverted for $\Omega$. The values of $J_{0}$, $\mu$ and $\Omega$ for STO determined from equation are listed in .
The closeness in value between $J_0$ and $\Omega$ for STO can be understood from their physical meanings. $J_0$ sets the temperature at which a transition would occur in the absence of quantum fluctuations, while $\Omega$ sets the scale of the quantum fluctuations; that these two are close in value is because STO is close to a ferroelectric transition. Further, since the Curie-Weiss temperature is small, both of these parameters are small.
Estimating $J_{1}$ for SrTiO$_{3}$ {#sec:J1}
----------------------------------
As was shown in , $J_{1}$ sets the scale of the gradient term $C$ in the LGD expansion, and it can therefore be obtained from quantities related to spatial gradients of the polarization. In perovskites, the polarization is closely connected to an optical phonon mode [@cowley64; @atkinson17], pictured in . One can therefore obtain $J_1$ from the phonon dispersion.
Key to this analysis is that the optical phonon has a large dipole moment that is represented by the TIM pseudospins. The phonon spectrum can therefore be obtained from the dynamical pseudospin correlation function. In the paraelectric phase, the term proportional to $\Omega$ in equation ensures that the pseudospins lie primarily along the (1)-axis. Perturbations of this state can be viewed as the magnons of a fictitious ferromagnetic material in which the magnetic moments align along the (1)-axis. The phonons can then be described as spin-wave excitations.
The spin operators are difficult to work with, however, and it is useful to bosonize them. This is achieved with the Holstein-Primakoff transformation [@holstein; @holstein40]. This transformation maps the pseudospin operators on to the boson creation and annihilation operators, $\hat{a}^{\dag}_{i}$ and $\hat{a}_{i}$. Pseudospin projections on the (3)-axis are then modelled as boson excitations, with a pseudospin that is entirely polarized along the (1)-axis represented by the vacuum state.
In this representation, the raising and lowering operators for site $i$ differ from the typical set by a cyclic permutation of the pseudospin axes. We then define [@holstein] $$\begin{aligned}
\label{S+}
\hat{S}^{+}_{i} & = \hat{S}^{(2)}_{i} + i \hat{S}^{(3)}_{i} \\
& = \sqrt{2S} \left( 1 - \frac{1}{2S} \hat{a}^{\dag}_{i} \hat{a}_{i} \right)^{1/2} \ \hat{a}_{i},\end{aligned}$$ $$\begin{aligned}
\label{S-}
\hat{S}^{-}_{i} & = \hat{S}^{(2)}_{i} - i \hat{S}^{(3)}_{i} \\
& = \sqrt{2S} \hat{a}^{\dag}_{i} \left( 1 - \frac{1}{2S} \hat{a}^{\dag}_{i} \hat{a}_{i} \right)^{1/2}.\end{aligned}$$ Since the polarization lies close to the (1)-axis in the paraelectric state, only low bosonic excitation states are relevant. In this case, $\hat{S}^{+}_{i} \approx \sqrt{2S} \hat{a}_{i}$ and $\hat{S}^{-}_{i} \approx \sqrt{2S} \hat{a}^{\dag}_{i}$. Additionally, the (1)-component of the pseudospin is defined as [@holstein] $$\label{S1}
\hat{S}^{(1)}_{i} = S - \hat{a}^{\dag}_{i} \hat{a}_{i},$$ and the (3)-component is $$\begin{aligned}
\label{S3_a}
\hat{S}^{(3)}_{i} & = \frac{1}{2i} \left( \hat{S}^{+}_{i} - \hat{S}^{-}_{i} \right) \\
& = \frac{\sqrt{2S}}{2i} \left( \hat{a}_{i} - \hat{a}^{\dag}_{i} \right).\end{aligned}$$ Because $\hat{S}^{(3)}_{i}$ represents atomic displacements, $\hat{a}_i$ and $\hat{a}^{\dag}_i$ are therefore phonon operators.
Equations and can now be substituted into equation . We transform to reciprocal space using $\hat{a}_{i}$ = $\frac{1}{\sqrt{N}}\sum_{{\mathbf{k}}} e^{i{\mathbf{k}} \cdot {\mathbf{r}}_{i}} \hat{b}_{{\mathbf{k}}}$: $$\begin{aligned}
\fl \hat{H} = - N \left( \Omega S + \frac{J_\mathrm{an}}{4} \right) + \sum_{{\mathbf{k}}} \Gamma_{{\mathbf{k}}} \hat{b}^{\dag}_{{\mathbf{k}}} \hat{b}_{{\mathbf{k}}} + \sum_{{\mathbf{k}}} \frac{\Delta_{{\mathbf{k}}}}{2} \left( \hat{b}_{{\mathbf{k}}} \hat{b}_{-{\mathbf{k}}} + \hat{b}^{\dag}_{{\mathbf{k}}} \hat{b}^{\dag}_{-{\mathbf{k}}} \right),\end{aligned}$$ where $\gamma_{{\mathbf{k}}} = 2 \cos (k_{x}a) + 2 \cos (k_{y}a) + 2 \cos (k_{z} a)$, $N$ is the total number of lattice sites, and $$\begin{aligned}
\Delta_{{\mathbf{k}}} & = \frac{J_{1}}{2} \gamma_{{\mathbf{k}}} + \frac{J_\mathrm{an}}{2}, \\
\Gamma_{{\mathbf{k}}} & = \Omega - \Delta_{{\mathbf{k}}}.\end{aligned}$$ Note that we have set $E = 0$ here, since the phonon spectrum is measured at zero field.
It is convenient to formulate the dynamics of the pseudomagnons using Green’s functions. The Green’s functions are correlation functions between the pseudomagnon creation and annihilation operators, and the equations of motion of the Green’s functions therefore include the equations of motion of $\hat{b}_{{\mathbf{k}}}$ and $\hat{b}^{\dag}_{{\mathbf{k}}}$. The spin-wave excitation spectrum can then be obtained from the poles of the Green’s function.
The Green’s function and its equation of motion are, respectively, $$\label{G1}
D_{1}({\mathbf{k}}, t) = -i \left\langle \left[ \hat{b}_{{\mathbf{k}}}(t), \hat{b}^{\dag}_{{\mathbf{k}}}(0) \right] \right\rangle \theta (t),$$ $$\label{G1_eom}
\frac{dD_{1}({\mathbf{k}}, t)}{dt} = -i \delta (t) - i \Gamma_{{\mathbf{k}}} D_{1}({\mathbf{k}}, t) - i \Delta_{{\mathbf{k}}} D_{2}({\mathbf{k}}, t),$$ where $\theta(t)$ is the step function. The second Green’s function that appears in equation and its equation of motion are $$\label{G2}
D_{2}({\mathbf{k}}, t) = -i \left\langle \left[ \hat{b}^{\dag}_{-{\mathbf{k}}} (t), \hat{b}^{\dag}_{{\mathbf{k}}} (0) \right] \right\rangle \theta (t),$$ $$\label{G2_eom}
\frac{dD_{2}({\mathbf{k}}, t)}{dt} = i \Gamma_{-{\mathbf{k}}} D_{2}({\mathbf{k}}, t) + i \Delta_{{\mathbf{k}}} D_{1}({\mathbf{k}}, t).$$ Fourier transforming equations and in time and solving for $D_{1}({\mathbf{k}}, \omega_{{\mathbf{k}}})$ gives the following expression for the Green’s function: $$\label{G(om)}
D_{1}({\mathbf{k}}, \omega_{{\mathbf{k}}}) = \frac{\omega_{{\mathbf{k}}} + \Gamma_{{\mathbf{k}}}}{\omega^{2}_{{\mathbf{k}}} - \Gamma_{{\mathbf{k}}}^{2} + \Delta_{{\mathbf{k}}}^{2}}.$$ The phonon dispersion is therefore given by $$\begin{aligned}
\omega_{{\mathbf{k}}} & = \sqrt{\Gamma_{{\mathbf{k}}}^{2} - \Delta_{{\mathbf{k}}}^{2}} \\
& = \sqrt{\Omega \left( \Omega - 2\Delta_{{\mathbf{k}}} \right)}.\end{aligned}$$
We obtain an expression for $J_{1}$ by comparing the frequency at $k_{x} = \pi/2$ and the zone centre: $$\label{J1}
J_{1} = \frac{\hbar^{2} \left( \omega^{2}_{\pi/2} - \omega^{2}_{0} \right)}{\Omega \left( \gamma_{0} - \gamma_{\pi/2} \right)},$$ where the subscripts $\pi/2$ and 0 indicate ${\mathbf{k}}=(\pi/2,0,0)$ and ${\mathbf{k}}=(0,0,0)$, respectively. Since $\Omega$ is already known from bulk susceptibility data, $J_{1}$ can be estimated solely using the material’s phonon dispersion. Using neutron scattering data from [@cowley64], we obtained a range of $J_{1}$ values between 30 and 200 meV depending on $S$ and on how the fit was made. As will be shown in , these estimates are somewhat lower than the values required to produce a 2DEG at the LAO/STO interface, which is likely a limitation of the TIM. Nonetheless, this calculation shows that $J_{1}$ is orders of magnitude larger than the value $J_1 = J_0/Z$ that is implicit in the conventional TIM.
This large discrepancy between $J_1$ and $J_0$ is a key feature of STO, and that there is more than an order of magnitude difference between their values can be related to their different physical origins. Further, from equation it follows that $J_\mathrm{an}$ is not small; rather, it is negative and nearly cancels $ZJ_1$. $J_\mathrm{an}$ would however play less of a role in a material with a high transition temperature, where $J_1$ and $J_0$ would be closer in value.
Ferroelectric Thin Films {#sec:FEfilm}
------------------------
We first model the polarization in ferroelectric thin films as a simple application of the modified TIM. A ferroelectric’s properties can vary drastically between the bulk and thin-film forms, and the origins and applications of these differences have been increasingly studied in recent years [@setter06]. Ferroelectric thin films provide significant advantages in electronic devices such as increased efficiency in photovoltaic cells [@liu16; @zenkevich14; @kutes14] and decreased power usage in non-volatile memory storage [@muller11].
We focus on weakly ferroelectric materials, like those obtained by doping STO with $^{18}$O, Ca, or Ba. We take $S=1$, and we thus fix the parameters $J_{0}=3.88$ meV and $\mu=1.88$ $e$Å, which were determined in section \[sec:fit\] for STO. To obtain a ferroelectric transition, we take $\Omega=3.2$ meV, which yields a bulk transition temperature $T_\mathrm{c}\approx20$ K, similar to what is observed in Sr$_{1-x}$Ca$_x$TiO$_3$. We treat $J_1$ as an adjustable parameter.
Thin films have a layered geometry that simplifies calculations. Taking each layer to be one unit cell thick, and assuming translational invariance within the $xy$-plane, the pseudospin, electric field, and polarization depend only on the layer index $i_{z}$ (instead of site $i$). Equation becomes $$\label{S3_i}
S^{(3)}_{i_z} = \frac{S h^{(3)}_{i_z}}{h_{i_z}} f_{S}(h_{i_z}),$$ where $h_{i_z} = |{\mathbf{h}}_{i_z}|$ and the Weiss mean field is $$\label{eq:hz}
\textbf{h}_{i_z} = \left( \Omega, 0, \frac{J_{1}}{S} \sum_{i'} S^{(3)}_{i'} + \frac{J_\mathrm{an}}{S} S^{(3)}_{i_z} + \mu E_{i_z} \right),$$ where, for the cubic STO crystal structure, the sum over nearest neighbours of a pseudospin in layer $i_z$ is $\sum_{i'}S_{i'} = 4S^{(3)}_{i_z} + S^{(3)}_{i_z-1} + S^{(3)}_{i_z+1}$. The lattice polarization in layer $i_{z}$ is then $$\label{P_i}
P_{i_{z}} = \mu \eta S^{(3)}_{i_{z}}.$$ (Recall that $\mu S$ is the maximum dipole moment per unit cell and $\eta$ is the dipole moment density.)
We assume a short-circuit geometry, in which the top and bottom surfaces of the film are connected by a wire that maintains a zero voltage difference between them. This geometry is commonly adopted to minimize the effects of depolarizing electric fields. We thus have two kinds of charge: a bound charge $\rho^\mathrm{b}(z) = - \partial_z P_\mathrm{tot}(z)$ coming from a sum of atomic and lattice polarizations, and the external charges $\rho^\mathrm{ext}(z)$ on the top and bottom electrodes.
The electric field in equation is obtained from these charges via Gauss’ law, $$\epsilon_{0} \frac{d}{dz} E(z) = \rho^\mathrm{b}(z) + \rho^\mathrm{ext}(z).$$ We break the polarization into lattice and atomic pieces, $P(z)$ and $\epsilon_0 \alpha E(z)$ respectively, with $\alpha$ the atomic polarizability, and defining the optical dielectric constant $\epsilon_\infty=\epsilon_0(1+\alpha)\approx 5.5 \epsilon_0$ [@raslan17; @zollner00], we obtain the usual expression $$\label{Gauss}
\frac{d}{dz} \left[ \epsilon_{\infty} E(z) + P(z) \right] = \rho^\mathrm{ext}(z),$$ which can be integrated to find $E(z)$.
The charge density in the top and bottom electrodes is written as $$\rho^\mathrm{ext}(z) = \frac{en}{a^{2}} [ \delta(z) - \delta(z-L) ],$$ where $L$ is the film thickness, and $n$ is the positive charge per 2D unit cell on the top electrode. Integrating equation gives $$\label{E_Gauss}
\epsilon_{\infty} E(z) = - P(z) + \frac{en}{a^{2}}.$$ A second integration, of equation across the thickness of the film, gives $$\label{eq:ena2}
\frac{en}{a^{2}} = \frac{\int_{0}^{L}dz P(z) - \epsilon_{\infty} V}{L},$$ with $V$ the potential difference across the film. Using this to eliminate $en/a^2$ in equation , and setting $V=0$ for the short-circuit geometry, we obtain $$\label{E}
E(z) = \frac{P_\mathrm{ave} - P(z)}{\epsilon_{\infty}},$$ with $P_\mathrm{ave}$ the average polarization of the film. Equations and are evaluated at discrete positions $z = i_z a$, and together with equation form a closed set that can be solved self-consistently.
shows the results of simulations for a film that is $N_L = 50$ layers thick. The figure illustrates two main points: First, the results depend qualitatively on whether or not electric fields are included in the simulation, even in the short-circuit geometry (for which naive considerations suggest the field vanishes); second, for fixed $J_0$, the value of $J_1$ has a large impact on the polarization.
The effects of electric fields in thin films were discussed at length by Kretschmer and Binder [@kretschmer79], and the results in serve as a reminder of their importance. In (a), where electric fields are not included, the polarization is reduced at the surfaces and increases to its bulk value over a length scale set by the correlation length. In the ferroelectric phase, the correlation length is $\xi=\sqrt{-C/2A}$ (in terms of LGD parameters), which is proportional to $\sqrt{J_1}$. The conventional TIM with $J_\mathrm{an}=0$ has $J_1=0.65$ meV, which corresponds to a correlation length of $\xi=2.7$ Å. Consistent with this, (a) shows that for the conventional TIM, surface effects are confined to narrow regions near the edges of the film. Conversely, the modified TIM with a more realistic value of $J_1=100$ meV gives the correlation length $\xi=3.3$ nm, which is comparable to the film thickness. In this case, the polarization is inhomogeneous throughout the film. In contrast to both of these cases, the polarization is nearly constant across the film when electric fields are included \[(b)\]; the polarization decreases with increasing $J_1$, and is suppressed completely for $J_1=100$ meV.
The apparent uniformity of the polarization across the film in (b) is because the correlation length is replaced by a shorter length scale $\kappa^{-1}$ when electric fields are included, with [@kretschmer79] $$\label{kappa}
\kappa = \sqrt{\xi^{-2} + \frac{\mu^{2} \eta}{\epsilon_{0} C}}.$$ In STO, this length scale is less than a unit cell, and the polarization is therefore nearly constant, with only a small reduction in the surface layer. This slight reduction is, nonetheless, enough that the depolarizing fields are incompletely screened by the electrodes. There is thus a residual depolarizing field in the STO film that reduces the overall polarization of the film.
To make the dependence of $\kappa$ on the TIM parameters explicit, we substitute values for the LGD parameters from equations - into equation in the limit $T\rightarrow 0$. For spin-1 we find $$\kappa = \sqrt{-\frac{2(\Omega - J_{0})}{J_{1}a^{2}} + \frac{\mu^{2} \eta}{\epsilon_{0} J_{1} a^{2}}}.$$ For fixed $\Omega$ and $J_{0}$ (i.e. for a fixed value of the bulk $T_\mathrm{c}$), $\kappa^{-1}$ increases as $\sqrt{J_{1}}$. Because the difference between the polarizations at the film surface and interior depends on $\kappa^{-1}$, the depolarizing field also grows with $J_1$; it then follows immediately that $P_\mathrm{ave}$ decreases as $J_1$ increases. This suppression is illustrated in (a), which shows the dependence of both the average polarization and $\kappa^{-1}$ on $J_1$. The polarization equals its bulk value when $J_1=0$ and drops as $J_1$ increases. Notably, there is a critical value of $J_1$ (which depends on the number of layers, $N_L$, in the film) above which ferroelectricity is completely suppressed. For the 50-layer film modelled here, this value is approximately 17 meV.
Alternatively, one can fix $J_1$ and consider how $P_\mathrm{ave}$ depends on film thickness, as shown in (b). Here polarization increases and asymptotically approaches the bulk value with increasing $N_L$. Ferroelectricity is completely suppressed below a critical film thickness, with the value of this critical thickness depending on $J_1$. The results shown in (b) are for $J_1=10$ meV, and give a critical thickness of 30 layers. For $J_1=100$ meV, the critical thickness is closer to 300 layers.
Finally, the effect of increasing $J_{0}$ is shown in (c). Because the bulk value of polarization $P_\mathrm{bulk}$ depends on $J_0$, we show the ratio $P_\mathrm{ave}/P_\mathrm{bulk}$ as a function of $J_0/\Omega$. In bulk materials, the threshold for ferroelectricity is $J_0=$ $\Omega$, and this is increased by finite size effects in the 50-layer film as shown in (c). Size effects quickly become unimportant with increasing $J_0$, as $P_\mathrm{ave}$ rapidly increases towards its bulk value. Indeed, when $J_0$ is only twice $\Omega$, $P_\mathrm{ave}=0.93P_\mathrm{bulk}$.
These calculations show that doped quantum paraelectrics such as Sr$_{1-x}$Ca$_x$TiO$_3$, which have $J_0$ close to $\Omega$, should be highly sensitive to film thickness in the short-circuit geometry. While this might be naively anticipated based on the argument that the correlation length $\xi$ is comparable to the film thickness near a ferroelectric transition, this argument is wrong because the relevant length $\kappa^{-1}$ is actually rather short and does not diverge at the quantum critical point. Rather, the sensitivity is due to depolarizing fields, which can easily overwhelm the weak ferroelectricity.
(001) LAO/STO Interface {#sec:interface}
=======================
In the final section of this work, we apply the modified TIM to the (001) LAO/STO interface. For this calculation, the Hamiltonian must include an electronic term that describes the 2DEG that forms at the interface. The total Hamiltonian is thus $$\hat{H} = \hat{H}_\mathrm{e} + \hat{H}_\mathrm{TIM},$$ where $\hat{H}_\mathrm{TIM}$ is given by equation and $\hat{H}_\mathrm{e}$ is the electronic term discussed below. These two terms are linked through the electric field, which appears explicitly in $\hat{H}_\mathrm{TIM}$, and appears implicitly in $\hat{H}_\mathrm{e}$ through the electrostatic potential.
We outline the calculations in , and show results for the effect of $J_{1}$ on the interfacial 2DEG in . The main result from this section is that the conventional and modified TIM make very different predictions for the structure of the 2DEG.
Method {#sec:int_method}
------
We assume that the 2DEG arises due to a combination of top gating and the polar catastrophe. In this case a total charge density $-en_\mathrm{LAO}$ is donated from the LAO surface to the interface, where $n_\mathrm{LAO}$ is the surface hole density, in order to neutralize the polar discontinuity between the two materials. Top gating gives control over the number of free electrons doped into the system.
As shown in (a), we adopt a discretized model comprising alternating metallic TiO$_2$ layers with electron densities $n_{i_z}$ and dielectric layers with polarizations $P_{i_z}$. Translational invariance is assumed within the $xy$-plane, but not along the $z$-axis perpendicular to the interface. The system’s properties are therefore only dependent on layer.
The 2DEG is composed of electrons that occupy titanium $t_{2\mathrm{g}}$ orbitals in the STO substrate. Although the unit cell is tetragonally distorted both by unit cell rotations about the $c$-axis and by interfacial strains, to a good approximation we can assume STO has the cubic structure typical of a perovskite material, as shown in the inset of . We adopt a tight-binding model in which the conduction bands are made up of $t_{2\mathrm{g}}$ orbitals [@raslan17; @Stengel:2011hy; @Khalsa:2012fu], and assume that electrons only hop between orbitals of the same type (ie. from one $d_{xz}$ orbital to another $d_{xz}$ orbital; other hopping matrix elements vanish in the cubic phase by symmetry, and are generally small when lattice distortions are included).
### Electronic Hamiltonian
The electronic Hamiltonian is made up of a hopping kinetic energy $\hat{T}$ and an electrostatic potential energy $\hat{U}$: $$\hat{H}_\mathrm{e} = \hat{T} + \hat{U}.$$ The hopping energy is $$\begin{aligned}
\label{T_k}
\hat{T} = \sum_{i_{z} {\mathbf{k}} \alpha \sigma} \epsilon_{i_{z} \alpha} \hat{c}^{\dag}_{i_{z} {\mathbf{k}} \alpha \sigma} \hat{c}_{i_{z} {\mathbf{k}} \alpha \sigma} + \sum_{\langle i_{z},i_{z}' \rangle \alpha \sigma} \sum_{{\mathbf{k}} {\boldsymbol{\delta}}} t^{\alpha}_{{\boldsymbol{\delta}}} e^{-i {\mathbf{k}} \cdot {\boldsymbol{\delta}}} \hat{c}^{\dag}_{i_{z}' {\mathbf{k}} \alpha \sigma} \hat{c}_{i_{z} {\mathbf{k}} \alpha \sigma},\end{aligned}$$ where $\hat{c}^{\dag}_{i_{z} {\mathbf{k}} \alpha \sigma}$ creates an electron with spin $\sigma$ and orbital type $\alpha$ in the 2D plane-wave state ${\mathbf{k}}=(k_{x}, k_{y})$ in layer $i_{z}$. $\sum_{\langle i_{z}, i_{z}' \rangle}$ is a sum over nearest-neighbour layers $i_{z}$ and $i_{z}'$. $t^{\alpha}_{{\boldsymbol{\delta}}}$ is the hopping matrix element for an electron in orbital type $\alpha$ hopping along path ${\boldsymbol{\delta}}$ to a nearest-neighbour site. $\epsilon_{i_{z} \alpha}$ is the atomic energy of an orbital site in layer $i_{z}$, and can be set to zero in calculations.
In the tight-binding model, there are six possible hopping paths. Hopping along $\hat{x}$ corresponds to a displacement ${\boldsymbol{\delta}}_{x}=(\pm a, 0, 0)$ and hopping amplitude $t^{\alpha}_{x}$, and so on for hopping along $\hat{y}$ and $\hat{z}$. Then, equation simplifies to $$\begin{aligned}
\label{T_full}
\fl \hat{T} = \sum_{i_{z} {\mathbf{k}} \alpha \sigma} \Big( \epsilon_{{\mathbf{k}} \alpha} \hat{c}^{\dag}_{i_{z}{\mathbf{k}}\alpha \sigma} \hat{c}_{i_{z}{\mathbf{k}}\alpha \sigma} + t^{\alpha}_{z} \hat{c}^{\dag}_{i_{z}+1, {\mathbf{k}}\alpha \sigma} \hat{c}_{i_{z}{\mathbf{k}}\alpha \sigma} + t^{\alpha}_{z} \hat{c}^{\dag}_{i_{z}-1, {\mathbf{k}}\alpha \sigma} \hat{c}_{i_{z}{\mathbf{k}}\alpha \sigma} \Big),\end{aligned}$$ where $\epsilon_{{\mathbf{k}} \alpha} = 2 t^{\alpha}_{x} \cos (k_{x} a) + 2 t^{\alpha}_{y} \cos (k_{y} a)$. As illustrated in (b), the amplitudes $t^{\alpha}_{x}, t^{\alpha}_{y}$ and $t^{\alpha}_{z}$ are denoted by $t^\|$ for hopping paths that lie in the plane defined by $\alpha$, and $t^\perp$ for hopping paths that are perpendicular to this plane. We take $t^{\parallel}=-0.236$ eV and $t^{\perp}=-0.035$ eV as in [@raslan17].
The electrostatic potential energy is due to the charge on the LAO surface, the 2DEG, and the bound charge due to the polarization of the STO: $$\label{U_k}
\hat{U} = -e \sum_{i_{z} {\mathbf{k}} \alpha \sigma} V_{i_{z}} \hat{c}^{\dag}_{i_{z}{\mathbf{k}}\alpha \sigma} \hat{c}_{i_{z}{\mathbf{k}}\alpha \sigma},$$ where $e$ is electron charge and $V_{i_{z}}$ is the electrostatic potential in layer $i_{z}$.
Combining equations and gives the full electronic Hamiltonian: $$\begin{aligned}
\label{Hel_full}
\hat{H}_\mathrm{e} &=& \sum_{i_{z} {\mathbf{k}} \alpha \sigma} \Big\lbrace \Big( \epsilon_{{\mathbf{k}} \alpha} - e V_{i_{z}} \Big) \hat{c}^{\dag}_{i_{z}{\mathbf{k}}\alpha \sigma} \hat{c}_{i_{z}{\mathbf{k}}\alpha \sigma} \nonumber \\ && + t^{\alpha}_{z} \hat{c}^{\dag}_{i_{z}+1, {\mathbf{k}}\alpha \sigma} \hat{c}_{i_{z}{\mathbf{k}}\alpha \sigma} + t^{\alpha}_{z} \hat{c}^{\dag}_{i_{z}-1, {\mathbf{k}}\alpha \sigma} \hat{c}_{i_{z}{\mathbf{k}}\alpha \sigma} \Big\rbrace.\end{aligned}$$ The Hamiltonian can be written as an $N_L\times N_L$ matrix in the layer index, $\hat{H}_\mathrm{e} =$ $\sum_{{\mathbf{k}} \alpha \sigma} {\boldsymbol{\hat{c}}}^{\dag}_{{\mathbf{k}} \alpha \sigma} {\mathbf{H}}_{{\mathbf{k}} \alpha \sigma} {\boldsymbol{\hat{c}}}_{{\mathbf{k}} \alpha \sigma}$, with ${\boldsymbol{\hat{c}}}_{{\mathbf{k}} \alpha \sigma} =$ $(\hat c_{0{\mathbf{k}} \alpha \sigma}, \ldots, \hat c_{N_L-1, {\mathbf{k}}\alpha \sigma})$ and $${\mathbf{H}}_{{\mathbf{k}} \alpha \sigma} = {\mathbf{H}}_\alpha + \epsilon_{{\mathbf{k}} \alpha} {\mathbf{I}},$$ where ${\mathbf{H}}_\alpha$ is independent of ${\mathbf{k}}$ and ${\mathbf{I}}$ is the identity matrix. The eigenergies are particularly simple, with $$\epsilon_{n{\mathbf{k}}\alpha} = \lambda_{n\alpha} + \epsilon_{{\mathbf{k}} \alpha},$$ where $\lambda_{n\alpha}$ are the eigenvalues of ${\mathbf{H}}_\alpha$ and $n$ is the band index. The eigenvectors of ${\mathbf{H}}_{{\mathbf{k}} \alpha \sigma} $, which represent the layer-dependent wavefunctions, are ${\mathbf{k}}$-independent and satisfy $$\sum_{j_z} [{\mathbf{H}}_\alpha]_{i_z j_z} \psi_{j_z n \alpha} = \lambda_{n\alpha} \psi_{i_z n \alpha}.$$
From this, the free electron density (per unit cell) in layer $i_{z}$ is $$\label{n_i}
n_{i_{z}} = \frac{1}{ N} \sum_{n {\mathbf{k}} \alpha \sigma} f_\mathrm{FD} (\epsilon_{n {\mathbf{k}} \alpha}) | \psi_{i_{z} n \alpha} |^{2},$$ where $N$ is the total number of $k_{x}$- and $k_{y}$-points, and $f_\mathrm{FD} (\epsilon_{n {\mathbf{k}} \alpha})$ is the Fermi-Dirac distribution.
We note that the mean-field equations described in this section neglect thermal fluctuations of both the lattice and the charge density. Both of these broaden the electronic spectral functions, as in Fermi liquid theory, and can in principle mix the bands. These are perturbative effects, however, and band structure calculations like the one outlined here generally provide a good quantitative description of the electronic structure, even at room temperature.
### Electric Field
The electric potential in layer $i_{z}$ is obtained by integrating the electric field from layer 0 to layer $i_z$, which sets the interface to be the zero of potential. Then, $$\label{V}
V_{i_{z}} = -a \sum_{j_{z} < i_{z}} E_{j_{z}} + V_{0},$$ with $a=3.902$ Å the STO lattice constant.
Just as in , the electric field can be obtained using Gauss’ law, $$\label{Gauss_int}
\frac{d}{dz} (\epsilon_{\infty} E(z) + P(z)) = \rho^\mathrm{2DEG}(z) + \rho^\mathrm{ext}(z),$$ where $\rho^\mathrm{2DEG}(z)$ is the free charge density and $\rho^\mathrm{ext}$ is the external charge density along the LAO surface. The polarization $P(z)$ is obtained from the modified TIM.
Within the discretized model, the electrons are treated as if they are confined to two-dimensional TiO$_2$ layers, so $$\rho^\mathrm{2DEG}(z) = -\frac{e}{a^{2}} \sum_{i_{z}} n_{i_{z}} \delta(z - i_z a),$$ where $n_{i_{z}}$ is given by equation . Similarly, the external charge density is confined to the top LAO layer, $$\rho^\mathrm{ext} = \frac{en_\mathrm{LAO}}{a^{2}} \delta(z - z_\mathrm{LAO}),$$ where $z_\mathrm{LAO}$ is the distance from the interface to the LAO surface. Now, integrating equation over $z$ gives the electric field in layer $i_{z}$: $$\label{E_int}
\epsilon_{\infty} E_{i_{z}} = - P_{i_{z}} - \frac{e}{a^{2}} \sum_{j_{z} \leq i_{z}} n_{j_{z}} + \frac{en_\mathrm{LAO}}{a^{2}},$$ which is required for the TIM \[equation \] and the electric potential \[equation \].
Results {#sec:int_res}
-------
Here, we explore the effect that $J_{1}$ has on the electron distribution, eigenenergies, polarization and potential energy for the (001) LAO/STO interface. As a key point of comparison, these calculations include the case $J_1=0.65$ meV ($J_\mathrm{an}=0$), which corresponds to the conventional TIM, in order to clearly highlight why the modified TIM requires the term introduced in equation to correctly model interfaces.
![Electron density (per unit cell) for the first 20 layers of a 200-layer film. Results are for different $J_{1}$ values, at $T=10$ K and 300 K, and (a)-(b) $n_\mathrm{LAO}=0.01/a^{2}$, (c)-(d) $n_\mathrm{LAO}=0.05/a^{2}$, and (e)-(f) $n_\mathrm{LAO}=0.1/a^{2}$.[]{data-label="fig:int_nvsN"}](figure5a_5f){width="0.7\linewidth"}
Previous work has established that the 2DEG is composed of both interfacial and tail components. The interfacial component is tightly confined to the interface, and appears as a peak in the electron density extending over the first few layers of the substrate, while the tail component extends far into the STO substrate [@copie09; @dubroka10; @park13; @gariglio15; @raslan18]. Except at the very lowest dopings, the majority of the electrons are confined close to the interface, with as many as 70% of the electrons in the 2DEG found in approximately the first 10 nm [@raslan17; @copie09]. This interfacial peak in the electron density is strongly temperature- and electron doping-dependent, with the electrons spreading further out into the STO as temperature or doping decreases [@raslan17]. The first $d_{xy}$ band contributes the most electrons to the interface states, while the first $d_{yz}$ and $d_{xz}$ bands make up the majority of the tail states and are seen to have the most temperature-dependence [@raslan17].
The electron density is plotted in figures \[fig:int\_nvsN\] and \[fig:int\_nvsN\_one\]. explores the effect $J_{1}$ has on the electron density, focusing particularly on the interface region, while shows the full profile over the entire film for a typical set of model parameters.
We begin analyzing by focusing on the results of the conventional TIM. When $J_1=0.65$ meV, there is no evidence that electrons are confined to the interface region at 10 K at any doping, in disagreement with experiments. Weak confinement does appear at 300 K due to the reduced dielectric susceptibility at high $T$, and the 2DEG does move towards the interface with increasing doping; however, the density is expected to be strongly peaked at the interface, and this is not seen. The conventional TIM, therefore, does not capture the physics of STO interfaces.
The remaining curves in show how the charge profile changes with increasing $J_1$. These results are for fixed $J_0$ and $\Omega$ (which determine the uniform dielectric susceptibility), and the only difference between the curves is therefore the correlation length $\xi$. These curves show that increasing $J_{1}$ (or equivalently, increasing $\xi$) tends to increase electron density at the interface, except at the lowest doping.
At the lowest doping, $n_\mathrm{LAO}=0.01/a^{2}$, $J_{1}$ has little effect on the electron density at both high and low temperature. Indeed, interface states are absent for all $J_1$ values up to 400 meV. While this lack of interface states is consistent with previous calculations [@raslan18], it is not consistent with experiments [@yin19; @joshua12], and likely points to some additional missing physics in the model [@raslan18].
At intermediate doping, $n_\mathrm{LAO}=0.05/a^{2}$, the electron density does develop an interfacial component as $J_{1}$ increases. This interfacial state extends only a few unit cells from the interface, and is more tightly confined at large $J_1$. There is thus a clear qualitative distinction between the modified and conventional TIMs in this case. At high doping, $n_\mathrm{LAO} = 0.1/a^{2}$, the trends are similar. The electron density is confined closer to the interface and is less strongly temperature dependent than at lower doping, at least when $J_1\geq200$ meV. Both of these trends are consistent with results reported in [@raslan17].
![Profile of the electron density (solid) for $n_\mathrm{LAO}=0.05/a^{2}$ and $J_{1}=300$ meV, at both 10 K and 300 K. Probability distributions (dashed) for the two lowest-energy $d_{xy}$ bands ($n = 1,2$) are shown for 10 K.[]{data-label="fig:int_nvsN_one"}](figure6){width="0.7\linewidth"}
![Band structure for $n_\mathrm{LAO}=0.05/a^{2}$ at $J_1=0.65$ meV and 300 meV. Results are shown for a 200-layer substrate at both 10 K (top) and 300 K (bottom).[]{data-label="fig:int_bands05"}](figure7){width="0.7\linewidth"}
shows the electron density across the full thickness of the STO film for a typical $J_1$ value at intermediate doping for both 10 K and 300 K. We choose the value of $J_{1}=300$ meV as physically reasonable based on the results in . At 10 K, the charge profile shows a peak-dip-hump structure that has not been reported in previous calculations. To understand its origin, we plot also the wavefunctions $|\psi_{i_z n \alpha}|^2$ for the first ($n=1$) and second ($n=2$) $d_{xy}$ bands ($\alpha=xy$) at 10 K. These show that the dip comes from the extremely tight confinement of the first $d_{xy}$ band to the interface.
The band structure is shown in for intermediate doping for both the conventional TIM and the modified TIM ($J_1= 300$ meV). At 10 K, the band structures of the two models are quasi-continuous, which is indicative of deconfined tail states, except for a single $d_{xy}$ band that sits below the continuum in the modified TIM, and which corresponds to the interface state discussed above. At high $T$, the band structures are discrete, which is indicative of confinement to the interface region. At this temperature, the effects of $J_1$ are quantitative, rather than qualitative.
![(a) Polarization and (b) electron potential energy in the first 25 layers of a 200-layer SrTiO$_3$ substrate for different $J_{1}$ values at 300 K and $n_\mathrm{LAO}=0.05/a^{2}$.[]{data-label="fig:int_PvsN"}](figure8a "fig:"){width="0.5\linewidth"} ![(a) Polarization and (b) electron potential energy in the first 25 layers of a 200-layer SrTiO$_3$ substrate for different $J_{1}$ values at 300 K and $n_\mathrm{LAO}=0.05/a^{2}$.[]{data-label="fig:int_PvsN"}](figure8b "fig:"){width="0.5\linewidth"}
Finally, we plot in the polarization and potential energy at 300 K for intermediate doping. These plots show that there are clear distinctions between the conventional and modified TIMs in the interfacial region. In particular, the polarization near the interface is reduced, by up to 25%, as $J_1$ increases. This reduction is similar to that discussed in the case of the thin film, with one key difference: because electric fields are screened by the 2DEG, the relevant length scale over which differences between the curves decay in (a) is $\xi$, and not $\kappa^{-1}$ [@atkinson17].
Similar to the ferroelectric thin films discussed in , this reduced polarization incompletely screens the electric fields produced by the LAO surface charge and results in a large field at the interface. This is reflected in the potential energy profiles shown in (b). In particular, large values of $J_1$ generate a deep potential well that confines the lowest $d_{xy}$ band tightly to the interface. On the other hand, $J_{1}$ has little effect on the electric field away from the interface, and so each potential energy curve has roughly the same slope for $i_{z} > 2$. In summary, illustrates the mechanism by which the anisotropic pseudospin term in the modified TIM generates the interfacial component of the 2DEG that is observed at LAO/STO interfaces.
Conclusions
===========
We showed that the conventional transverse Ising model misses key features of spatially inhomogeneous STO-based nanostructures. To fix this we modified the TIM by adding an anisotropic pseudospin energy to the Hamiltonian. This corrects a deficiency of the TIM, namely that if one fits the model parameters to the bulk (homogeneous) susceptibility, the polarization correlation length is also fixed by the model and is at least an order of magnitude smaller than it should be.
To illustrate the effects of the new term, we considered two applications of the modified TIM: first, to thin films of an STO-like ferroelectric; and second, to a metallic LAO/STO interface. In both cases, the key point is that the conventional TIM underestimates the reduction of the polarization due to the surface; this reduced polarization leads to a reduced screening of electric fields in the interface region, which in turn has profound effects on the film or interface. In the case of the ferroelectric film, these fields depolarize the polarization in the film; in the case of the interface, they create a confining potential that generates tightly bound interface states.
This work has been supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.
References {#references .unnumbered}
==========
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Recently there has been sustained interest in modifying prediction algorithms to satisfy fairness constraints. These constraints are typically complex nonlinear functionals of the observed data distribution. Focusing on the causal constraints proposed by [@NabiShpitser18Fair], we introduce new theoretical results and optimization techniques to make model training easier and more accurate. Specifically, we show how to reparameterize the observed data likelihood such that fairness constraints correspond directly to parameters that appear in the likelihood, transforming a complex constrained optimization objective into a simple optimization problem with box constraints. We also exploit methods from empirical likelihood theory in statistics to improve predictive performance, without requiring parametric models for high-dimensional feature vectors.'
bibliography:
- 'references.bib'
---
INTRODUCTION {#sec:intro}
============
Predictive models trained on imperfect data are increasingly being used in socially-impactful settings. Predictions (such as risk scores) have been used to inform high-stakes decisions in criminal justice [@Perry2013policing], healthcare [@Kappen2018Clinical], and finance [@Khandani2010Finance]. While automation may bring many potential benefits – such as speed and accuracy – it is also fraught with risks. Predictive models introduce two dangers in particular: the illusion of objectivity and violation of fairness norms. Predictive models may appear to be “neutral,” since humans are less involved and because they are products of a seemingly impartial optimization process. However, predictive models are trained on data that reflects the structural inequities, historical disparities, and other imperfections of our society. Often data includes sensitive attributes (e.g., race, gender, age, disability status), or proxies for such attributes. A particular worry in the context of data-driven decision-making is “perpetuating injustice,” which occurs when unfair dependence between sensitive features and outcomes is maintained, introduced, or reinforced by automated tools.
We study how to construct fair predictive models by correcting for the unfair causal dependence of predicted outcomes on sensitive features. We work with the proposed fairness criteria in [@NabiShpitser18Fair], where the authors propose that fair prediction requires imposing hard constraints on the prediction problem in the form of restricting certain causal path-specific effects. Impermissible pathways are user-specified and context-specific, hence require input from policymakers, legal experts, or the general public. Some alternative but also causally-motivated constrained prediction methods are proposed in [@chiappa2018path; @Kusner17fair] and [@zhang18fairness]. For a survey and discussion of distinct fairness criteria (both causal and associative) see [@mitchell2018prediction].
We advance the state of the art in two ways. First, we give a novel reparameterization of the observed data likelihood in which unfair path-specific effects appear directly as parameters. This allows us to greatly simplify the constrained optimization problem, which has previously required complex or inefficient algorithms. Second, we demonstrate how tools from the empirical likelihood literature [@OwenEL] can be readily adapted to construct hybrid (semi-parametric) observed data likelihoods that satisfy given fairness criteria. With this approach, the entire likelihood is constrained, rather than only part of the likelihood as in past proposals [@NabiShpitser18Fair]. As a result, we use the data more efficiently and achieve better performance. Finally, we show how both innovations may be combined into a single procedure.
As a guiding example, we consider a setting such as automated hiring, in which we want to predict job success from applicant data. We have historical data on job success, resumes, and demographics, as well as new individuals for which we only see resumes and demographics for whom we would like to estimate a risk score with our predictive model. This may be considered a variant of semi-supervised learning or prediction with missing labels on a subset of the population. We aim to estimate those scores subject to path-specific fairness constraints. In order to describe the various components of this proposal, we must review some background on causal inference, path-specific effects, and constrained prediction.
CAUSAL INFERENCE AND A CAUSAL APPROACH TO FAIRNESS {#sec:prelim}
==================================================
Causal inference is concerned with quantities which describe the consequences of interventions. Causal models are often represented graphically, e.g. by directed acyclic graphs (DAGs). We will use capital letters ($V$) to denote sets of random variables as well as corresponding vertices in graphs and lowercase letters ($v$) to denote values or assignments to those random variables. A DAG consists of a set of vertices $V$ connected by directed edges ($V_i \to V_j$ for some $\{V_i,V_j\} \subseteq V$) such that there are no cycles. The set $\operatorname{pa}_{\cal G}(V_i) \equiv \{V_j \in V \mid V_j \to V_i \}$ denotes the parents of $V_i$ in DAG ${\cal G}$. ${\mathfrak X}_{A}$ denotes the statespace of $A \subseteq V$.
A causal model of a DAG ${\cal G}$ is a set of distributions defined on potential outcomes (a.k.a. counterfactuals). For example, we consider distributions $p(V(a))$ subject to some restrictions, where $V(a)$ represents the value of $V$ had all variables in $\operatorname{pa}_{\cal G}(V)$ been set, possibly contrary to fact, to value $a$. In this paper, we assume Pearl’s *functional model* [@pearl09causality] for a DAG $\mathcal{G}$ which stipulates that the sets of potential outcome variables $\big\{ \{V_i(a_i) \mid a_i \in {\mathfrak X}_{\operatorname{pa}_{\mathcal{G}}(V_i)} \} \mid V_i \in V \big\}$ are mutually independent. All other counterfactuals may be defined using *recursive substitution*. For any $A \subseteq V \setminus \{ V_i \}$, $$\begin{aligned}
V_i(a) \equiv V_i(a_{\operatorname{pa}_{\cal G}(V_i) \cap A},
\{ V_j(a) : V_j \in \operatorname{pa}_{\cal G}(V_i) \setminus A \}),
$$ where $\{ V_j(a) : V_j \in \operatorname{pa}_{\cal G}(V_i) \setminus A \}$ is taken to mean the (recursively defined) set of counterfactuals associated with variables in $\operatorname{pa}_{\cal G}(V_i) \setminus A$, had $A$ been set to $a$. Equivalently, Pearl’s model may be described by a system of nonparametric structural equations with independent errors.
A causal parameter is said to be *identified* in a causal model if it is a function of the observed data distribution $p(V)$. In the functional model of a DAG ${\cal G}$ (as well as some weaker causal models), all interventional distributions $p(V(a))$, for any $A \subseteq V$, are identified by the *extended g-formula*: $$p(V(a)) = \prod_{V_i \in V} \left. p(V_i | \operatorname{pa}_{\cal G}(V_i)) \right|_{A=a}.$$ For example, consider the DAG in Fig. \[fig:graphs\](a). $Y(a)$ is defined to be $Y(a,M(a,X),X)$ by recursive substitution and its distribution is identified as $\sum_{X,M} p(Y | a,M,X) p(M | a,X) p(X)$. The mean difference between $Y(a)$ and $Y(a')$ for some treatment value $a$ of interest and reference value $a'$ is ${\mathbb{E}}[Y(a)] - {\mathbb{E}}[Y(a')]$ and quantifies the *average causal effect* of treatment $A$ on the outcome $Y$.
Mediation Analysis and Path-Specific Effects
--------------------------------------------
An important goal in causal inference is to understand the mechanisms by which some treatment $A$ influences some outcome $Y$. A common framework for studying mechanisms is *mediation analysis* which seeks to decompose the effect of $A$ on $Y$ into the *direct effect* and the *indirect effect* mediated by a third variable, or more generally into components associated with particular causal pathways. As an example, the direct effect of $A$ on $Y$ in Fig. \[fig:graphs\](a) corresponds to the effect along the edge $A \rightarrow Y$ and the indirect effect corresponds to the effect along the path $A \rightarrow M \rightarrow Y$, mediated by $M$.
In the potential outcome notation, the direct and indirect effects can be defined using nested counterfactuals such as $Y(a, M(a'))$ for $a, a' \in {\mathfrak X}_A$, which denotes the value of $Y$ when $A$ is set to $a$ while $M$ is set to whatever value it would have attained had $A$ been set to $a'$. Given $p(a, M(a'))$, the *natural direct effect* (NDE) (on the expectation difference scale) is defined as ${\mathbb{E}}[Y(a, M(a'))] - {\mathbb{E}}[Y(a')]$, and the *natural indirect effect* (NID) is defined as ${\mathbb{E}}[Y(a)] - {\mathbb{E}}[Y(a, M(a'))]$. Under certain identification assumptions discussed by [@pearl01direct], the distribution of $Y(a, M(a'))$ (and thereby direct and indirect effects) can be nonparametrically identified from observed data by the following formula: [$$\begin{aligned}
p(Y(a, M(a')) = \sum_{X, M} \ p(Y \mid a, X, M) \ p(M \mid a', X) \ p(X).\end{aligned}$$ ]{}More generally, when there are multiple pathways from $A$ to $Y$ one may define various *path-specific effects* (PSEs). In this case, effect along a particular path will be obtained by comparing two potential outcomes, one where for the selected paths all nodes behave as if $A = a$, and along all other paths nodes behave as if $A = a'$.
PSEs are defined by means of nested, path-specific potential outcomes. Fix a set of treatment variables $A$, and a subset of *proper causal paths* $\pi$ from any element in $A$. A proper causal path only intersects $A$ at the source node. Next, pick a pair of value sets $a$ and $a'$ for elements in $A$. For any $V_i \in V$, define the potential outcome $V_i(\pi,a,a')$ by setting $A$ to $a$ for the purposes of paths in $\pi$, and to $a'$ for the purposes of proper causal paths from $A$ to $Y$ not in $\pi$. Formally, the definition is as follows, for any $V_i \in V$, $V_i(\pi, a, a') \equiv a \text{ if }V_i \in A$, otherwise [$$\begin{aligned}
\label{eqn:pse}
V_i(\pi, a, a') \equiv & V_i \Big( \big\{ V_j(\pi, a, a') \mid V_j \in \operatorname{pa}_{\cal G}^{\pi}(V_i) \big\}, \nonumber \\
& \hspace{2cm} \big\{ V_j(a') \mid V_j \in \operatorname{pa}_{\cal G}^{\overline{\pi}}(V_i) \big\} \Big)\end{aligned}$$ ]{}where $V_j(a') \equiv a'$ if $V_j \in A$, and given by recursive substitution otherwise, $\operatorname{pa}_{\cal G}^{\pi}(V_i)$ is the set of parents of $V_i$ along an edge which is a part of a path in $\pi$, and $\operatorname{pa}_{\cal G}^{\overline{\pi}}(V_i)$ is the set of all other parents of $V_i$.
A counterfactual $V_i(\pi, a, a')$ is said to be *edge inconsistent* if counterfactuals of the form $V_j(a_k, \ldots)$ and $V_j(a_k', \ldots)$ occur in $V_i(\pi, a, a')$, otherwise it is said to be *edge consistent*. It is well known that a joint distribution $p(V(\pi, a, a'))$ containing an edge-inconsistent counterfactual $V_i(\pi, a, a')$ is not identified in the functional causal model (nor weaker causal models) with a corresponding graphical criterion on $\pi$ and ${\cal G}(V)$ called the ‘recanting witness’ [@shpitser13cogsci; @shpitser15hierarchy]. Under some assumptions, PSEs are nonparametrically identified by means of the *edge g-formula* described in [@shpitser15hierarchy]. As an example, consider the DAG in Fig. \[fig:graphs\](b). The PSE of $A$ on $Y$ along the paths $\pi = \{A \rightarrow Y, A\rightarrow L \rightarrow Y \}$ is encoded by a counterfactual contrast of the form $Y(\pi, a, a') = Y(a, M(a'), L(a, M(a')))$. This counterfactual density is identified by the edge g-formula as follows: [ $$\begin{aligned}
&p(Y(a, M(a'), L(a, M(a'))) = \nonumber \\
& \sum_{X, M, L} \ p(Y \mid a, X, M) \ p(L \mid a, M, X) \ p(M \mid a', X) \ p(X).
\end{aligned}$$ ]{}For more details on PSEs, see [@shpitser13cogsci] and[@shpitser18medid], and [@nabi18pathpolicy].
Algorithmic Fairness via Constraining Path-Specific Effects {#sec:causal_fair}
-----------------------------------------------------------
There has been a growing interest in the issue of fairness in machine learning [@pedreshi08discrimination; @feldman15certifying; @hardt16equality; @kamiran13quantifying; @corbett2017algorithmic; @jabbari16fair; @kusner17counterfactual; @zhang18fairness; @Zhang17causal]. In this paper, we adopt the causal notion of fairness described in [@NabiShpitser18Fair] and [@nabi19fairpolicy], where unfairness corresponds to the presence of undesirable or impermissble path-specific effects of sensitive attributes on outcomes – a view which generalizes an example discussed in [@pearl09causality]. We provide a brief summary of their perspective on fairness in the following without defending it for lack of space; see @NabiShpitser18Fair for more details.
Consider an observed data distribution $p(Y,Z)$ induced by a causal model, where $Y$ is an outcome and $Z = \{X, A, M\}$ includes all baseline factors $X$, sensitive features $A$, and post-treatment pre-outcome mediators $M$. Context and background ethical considerations pick out some path-specific effect of the sensitive feature $A$ on the outcome $Y$ as unfair; we assume this effect is identified as a functional $g(p(Y,Z))$. Fix upper and lower bounds $\epsilon_l, \epsilon_u$ for the PSE, representing a tolerable range. The most relevant bounds in practice are $\epsilon_l = \epsilon_u = 0$ or approximately zero. @NabiShpitser18Fair propose to transform the inference problem on $p(Y,Z)$, the “unfair world,” into an inference problem on another distribution $p^*(Y,Z)$, called the “fair world,” which is close in the sense of minimal KL-divergence to $p(Y,Z)$ while also having the property that the PSE lies within $(\epsilon_l, \epsilon_u)$.
Given a dataset $\mathcal{D} = \{(Y_i,Z_i), i = 1, \ldots, n \}$ drawn from $p(Y,Z)$, a likelihood function $\cal{L}(\mathcal{D}; \alpha)$, an estimator $\widehat{g}(\mathcal{D})$ of the unfair PSE, and bounds $\epsilon_l, \epsilon_u$, @NabiShpitser18Fair suggest to approximate $p^*(Y,Z)$ by solving the following constrained maximum likelihood problem: [$$\begin{aligned}
&\widehat{\alpha} = \arg \max_{\alpha} \hspace{0.2cm} {\cal L}_{Y, Z}({\cal D}; {\alpha}), \hspace{0.3cm} \nonumber \\
&\text{subject to} \hspace{0.2cm} \epsilon_l \leq \widehat{g}({\cal D}) \leq \epsilon_u.
\label{eqn:c-mle}\end{aligned}$$ ]{}Having approximated the fair world $p^*(Y,Z; \widehat{\alpha})$ in this way, @NabiShpitser18Fair point out a key difficulty for using these estimated parameters to predict outcomes for new instances (e.g., new job applicants). A new set of observations $Z$ is not sampled from the “fair world" $p^*(Z)$ but from “unfair world" $p(Z)$. @NabiShpitser18Fair propose to map new instances from $p$ to $p^*$ and to use the result for predicting $Y$ with model parameters $\widehat{\alpha}$. They assume $Z$ can be partitioned into $Z_1$ and $Z_2$ such that $p^*(Y, Z) = p^*(Y, Z_1 | Z_2) p(Z_2)$. In other words, variables in $Z_2$ are shared between $p$ and $p^*$: $p^*(Z_2) = p(Z_2)$ but $p^*(Z_1 | Z_2) \neq p(Z_1 | Z_2)$. $Z_1$ typically corresponds to variables that appear in the estimator $\widehat{g}({\cal D})$. There is no obvious principled way of knowing exactly what values of $Z_1$ the “fair version" of the new instance would attain. Consequently, all such possible values are averaged out, weighted appropriately by how likely they are according to the estimated $p^*$. This entails predicting $Y$ as the expected value ${\mathbb{E}}^*[Y | Z_2]$ (with respect to the distribution $\sum_{{Z}_1} p^*(Y, Z_1 | Z_2))$.
Next, we explain some limitations of the inference procedure described here and present our main contributions to address these limitations.
FAIR PREDICTIVE MODELS IN A BATCH SETTING {#sec:fairpred}
=========================================
Prediction problems in machine learning are typically tackled from the perspective of nonparametric risk minimization and the “train-and-test” framework. Here, we instead take the perspective of maximum likelihood and missing data, i.e., we treat unknown outcomes as missing values which we hope to impute in a way that is consistent with our specified likelihood for the entire data set. Our motivation for doing so is the nature of our constrained prediction problem. Specifically, our causal constraints contain “nuisance” components (conditional expectations and conditional distributions derived from the observed data distribution) which must be modeled correctly to ensure the causal effects are reliably estimated. In the subsequent prediction step, we should predict in a way that is consistent with what has already been modeled – or else we fail to exploit all the information we have already committed to in the constraint estimation step. We chose the maximum likelihood framework as the most natural and simplest approach to accomplish this. Alternative methods for coherently combining nuisance estimation with nonparametric risk minimization are left to future work.
Unlike [@NabiShpitser18Fair], we consider a batch prediction setting – this allows us to avoid the inefficient averaging described in the previous section. In our case, historical data (of sample size $n_1$) consists of observations on $\{X,A,M,Y\}$ and new instances (of size $n_2$) comprise a set of observations with just $\{X,A,M\}$. The outcome labels for new instances are missing data which we aim to predict, subject to fairness constraints. Instead of training our constrained model on historical data alone, we train on the combination of historical data and new instances. This seems complicated since the observed data likelihood for the combined data set includes some complete rows and some partially incomplete rows. However, we can borrow ideas from the literature on missing data to accomplish this task. Specifically, we can impute missing outcomes (“labels”) using appropriate functions of observed data. In this paper we assume the labels are *missing at random* (MAR) [@little2002statistical]. However, our methods extend to any identifiable missing not at random (MNAR) model. Let the random variable $R$ denote the missingness status of the outcome variable $Y$ for each instance. That is, $R=1$ for all rows in the historical data (since $Y$ is observed) and $R=0$ for all rows in the new instances. Then the observed data likelihood is $\prod_{i = 1}^{n=n_1+n_2} p(X_i, A_i, M_i) \ p(Y_i | X_i, A_i, M_i)^{R_i}$.
This likelihood function describes the probability of the entire data set, though only uses $Y$ values from historical data. We can then maximize the likelihood subject to the specified path-specific constraints, and associate predicted values $\hat{Y}_{new}$ to the new instances. Note that the setting where new instances arrive sequentially one-at-a-time is a special case of this general setup, which would require retraining on the full combined data after the arrival of each instance. Though this is computationally more intensive than the proposal in [@NabiShpitser18Fair] (where they only train once), it will deliver significantly more accurate predictions because it uses all available information. We will elaborate on this point in Section \[sec:consMLE\].
The approach to fair prediction outlined in [@NabiShpitser18Fair] suffers from two problems: one general and one specific to our setting here. First, their approach requires solving a computationally challenging constrained optimization problem. Likelihood functions are not in general convex and the constraints on path-specific effects involve nonlinear and complicated functionals of the observed data distribution. This makes the proposed constrained optimization a daunting task that relies on complex optimization software (or computationally expensive methods such as rejection sampling), which does not always find high quality local optima. Second, @NabiShpitser18Fair propose to constrain only part of the likelihood. Specifically they do not constrain the density $p(X)$ over the baseline features (since this is high-dimensional and thus inplausible to model accurately in their parametric approach). The baseline density is instead estimated by placing $1/n$ mass at every observed data point. This is sub-optimal in the specific setting we consider, where we do not need to average over constrained variables. Constraining a larger part of the joint should lead to a fair world distribution KL-closer to the observed distribution, which leads to better predictive performance as long as the likelihood is correctly specified. This intuition is formalized in the following result:
Let $p(Z)$ denote the observed data distribution, $M_1 = \big\{p^*_1(Z) = \operatorname*{arg\,max}_{q(Z)} D_{KL}(p || q) \hspace{0.1cm} \text{ s.t. } \hspace{0.1cm} \epsilon_l \leq g(q(Z)) \leq \epsilon_u, \ q(Z_1) = p(Z_1) \big\}$, and $M_2 = \big\{p^*_2(Z) = \operatorname*{arg\,max}_{q(Z)} D_{KL}(p || q) \hspace{0.1cm} \text{ s.t. } \hspace{0.1cm} \epsilon_l \leq g(q(Z)) \leq \epsilon_u, \ q(Z_2) = p(Z_2) \big\}$. If $Z_2 \subseteq Z_1 \subseteq Z$, then $D_{KL}(p || p^*_2) \leq D_{KL}(p || p^*_1).$ \[lem:KLdis\]
In other words, if a larger part of the joint is being constrained in $M_2$ compared to $M_1$, then $p^*_2(Z)$ is at least as close to $p(Z)$ as $p^*_1(Z)$.
To address the first difficulty, we provide a novel reparameterization of the observed data likelihood such that the causal parameter corresponding to the unfair PSE appears directly in the likelihood. This approach generalizes previous work on reparameterizations implied by structural nested models [@robins99marginal; @tchetgen14semi] to apply to a wide class of PSEs. With such a reparameterization, the MLE with a constrained PSE simply corresponds to maximum likelihood inference in a submodel where a certain likelihood parameter is set to $0$. This type of inference can be implemented with standard software.
To address the second difficulty, we propose an approach to constraining the density $p(X)$. An alternative to fully parametric modeling is to consider nonparametric representations of $p(X)$. It is well known that the nonparametric maximum likelihood estimate of any $p(X)$ given a set of i.i.d draws is the empirical distribution which places mass $1/n$ at every observed point. Empirical likelihood methods have been developed for settings where the nonparametric and parametric (hybrid) likelihood must be maximized subject to moment constraints [@OwenEL]. We describe below how these methods may be adapted to our setting.
Finally, we show how both the reparameterization method and the empirical likelihood method can be combined to yield a constrained optimization method that maximizes a semi-parametric (hybrid reparameterized) likelihood using standard software.
EFFICIENT APPROXIMATION OF FAIR WORLDS {#sec:consMLE}
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Imposing Fairness Constraints With Reparameterized Likelihoods {#sec:likereparam}
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In this section, we describe how to reparameterize the observed data likelihood in terms of causal parameters that correspond to the effect of $A$ on $Y$ along certain causal pathways. The results presented in the following theorem greatly simplifies the constrained optimization problem shown in (\[eqn:c-mle\]) in settings where the PSE includes the direct influence of $A$ on $Y$. This is due to the fact that the constrained parameter, corresponding to the PSE of interest, now appears as a single coefficient in the outcome regression model.
Assume the observed data distribution $p(Y,Z)$ is induced by a *causal model*, where $Z = \{X, A, M\}$ includes pre-treatment measures $X$, treatment $A$, and post-treatment pre-outcome mediators $M$. Let $p(Y(\pi, a, a'))$ denote the potential outcome distribution that corresponds to the effect of $A$ on $Y$ along proper causal paths in $\pi$, where $\pi$ includes the direct influence of $A$ on $Y$, and let $p(Y_0(\pi, a, a'))$ denote the identifying functional for $p(Y(\pi, a, a'))$ obtained from the edge-formula in (\[eqn:pse\]), where the term $p(Y | Z)$ is evaluated at $\{Z \setminus A\} = 0$. Then ${\mathbb{E}}[Y | Z]$ can be written as follows: [ $$\begin{aligned}
{\mathbb{E}}[Y | Z] = f(Z) - \big( {\mathbb{E}}[Y(\pi, a, a')] - {\mathbb{E}}[Y_0(\pi, a, a')]\big) \ + \ \phi(A),
\end{aligned}$$ ]{} where $f(Z) := {\mathbb{E}}[Y | Z] - {\mathbb{E}}[Y | A, \{Z \setminus A\} = 0]$ and $\phi(A) = w_0 + w_aA$. Furthermore, $w_a$ corresponds to $\pi$-specific effect of $A$ on $Y$. \[theorem:reparam\]
To illustrate the above reparameterization, consider the graph in Fig. \[fig:graphs\](b), discussed in [@NabiShpitser18Fair] and [@chiappa2018path]. Assume the direct path and the paths through $M$ of $A$ on $Y$ are the impermissible pathways (depicted with green edges). The corresponding PSE is encoded by a counterfactual contrast with respect to [$Y(a, M(a), L(a', M(a)))$]{}. The reparameterization in Theorem \[theorem:reparam\] amounts to: [$$\begin{aligned}
&{\mathbb{E}}[Y | Z] = f(Z) \nonumber - \sum_{Z\setminus A} \big\{ f(Z) \times p(L | M, X, A=0) \times \nonumber \\
& \hspace{2cm} p(M | X, A = 1) \times p(X)\big\} + w_0 + {w_a} A,
\label{eq:exReparam}\end{aligned}$$ ]{}where $w_a$ represents the PSE of interest; see the appendix for more details. A special case of this reparameterization when $\pi$ includes only the direct edge $A \to Y$ is implicit in the work of [@tchetgen14semi].
Under linearity assumptions, the PSE of interest in Fig. \[fig:graphs\](b) has a simple form. Assume the data generating process in Fig. \[fig:graphs\](b) is the same as the one given in display (2) of [@chiappa2018path], where [${\text{PSE}} = \theta^y_a + \theta^y_m \theta^m_a + \theta^y_l \theta^l_m \theta^m_a$]{}. In this case, our reparameterization takes the following form: [ $$\begin{aligned}
& {\mathbb{E}}[Y | X, A, M, L] =
\underbrace{\Big(\theta^y_x X + \theta^y_m M + \theta^y_l L\Big)}_{f(Z)} - \\
& \underbrace{\Big( \big( \theta^m_0 \theta^y_m + (\theta^l_0 + \theta^l_m\theta^m_0)\theta^y_l \big) + \big(\theta^y_m \theta^m_a + \theta^y_l \theta^l_m \theta^m_a \big) A \Big)}_{\sum_{Z\setminus A} \big\{f(Z) p(L | M, X, A = 0) p(M | X, A = 1) p(X)\big\}} + \\
& \Big( \!\!\underbrace{\theta^y_0 \!+\! \big(\theta^m_0 \theta^y_m \!+\! (\theta^l_0 \!+\! \theta^l_m\theta^m_0)\theta^y_l \big)}_{w_0} + \underbrace{\big(\theta^y_a \!+\! \theta^y_m \theta^m_a \!+\! \theta^y_l \theta^l_m \theta^m_a \big)}_{w_a \equiv \text{PSE}} \!A\!\Big)
\end{aligned}$$ ]{}In order to move away from the linear setting and exploit more flexible techniques, [@chiappa2018path] makes assumptions on the latent variables. However, such assumptions are often hard to verify in practice. In contrast, our result is entirely nonparametric and does not rely on any assumptions beyond what is encoded in the causal DAG.
By Theorem \[theorem:reparam\], the constrained optimization problem in eq. (\[eqn:c-mle\]) simplifies significantly to the following optimization problem: [$$\begin{aligned}
\widehat{\alpha} = \arg \max_{\alpha} \hspace{0.2cm} {\cal L}_{Y, Z}({\cal D}; {\alpha}) \hspace{0.3cm} \text{subject to} \hspace{0.2cm} w_a = 0.\end{aligned}$$ ]{}In the prediction setting, i.e., finding optimal parameters for ${\mathbb{E}}[Y|Z; \alpha_y]$, this amounts to an unconstrained maximum likelihood problem with outcome regression taking the specific form: [$$\begin{aligned}
&{\mathbb{E}}[Y | Z; { \alpha_y}] = \label{eq:reparam_NDE} \\
&f(Z; \alpha_f) - \sum_{X, M} f(Z; \alpha_f) \ p(M | A = 0, X; {\alpha_m}) \ p(X) + w_0,
\nonumber \end{aligned}$$ ]{}where $f(Z) := {\mathbb{E}}[Y | Z] - {\mathbb{E}}[Y | A, X=0, M=0]$ and is parameterized by $\alpha_f$. In practice, for each $X_i$ in the data $p(X_i)$ is replaced with its empirical approximation $1/n$, since a parametric specification of $p(X)$ is not feasible. In next section, we explain how $p(X)$ can be incorporated into the constrained optimization problem using empirical likelihood methods.
Imposing Fairness Constraints With Hybrid Likelihoods {#sec:hybrid}
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In light of Theorem \[lem:KLdis\], we are interested in constraining the nonparameteric form of $p(X)$. Following work in [@OwenEL], we use hybrid/semi-parametric empirical likelihood methods to estimate $p(X)$ nonparametrically which is a novel idea in the fairness setting. First, according to Theorem \[lem:KLdis\], constraining $p(X)$ would bring our learned distribution closer to the observed (unfair) distribution, and hence results in improvement of model performance, as we demonstrate in our simulations. Second, $p(X)$ is often a high dimensional object that is difficult to estimate due to the curse of dimensionality. For simplicity of presentation, we focus on the DAG in Fig. \[fig:graphs\](a), and the constraint represented by the NDE, although the methods we describe generalize without difficulty to arbitrary causal models and constraints represented by arbitrary PSEs.
Let $(X_i, A_i, M_i, Y_i), i = 1, \ldots, n$ be independent random vectors with common distribution $p = p(X) \times p(A,M,Y | X)$. We assume a known parametric form for $p(A,M,Y | X)$, and leave $p(X)$ unrestricted. Assuming the unfair effect is NDE, the only constraint on the observed distribution is for NDE to be zero. Let $p(Y | M, A, X), p(M | A, X), p(A | X)$ be parameterized by $\alpha_y, \alpha_m, \alpha_a$, respectively. The direct effect can then be identified by ${\mathbb{E}}_x[ m(X; \alpha)]$, where [ $$\begin{aligned}
m(X; \alpha)& = \sum_{M} \Big\{{\mathbb{E}}[Y | A = 1, M, X; \alpha_y] - \label{eq:est_NDE} \\
&\qquad {\mathbb{E}}[Y | A = 0, M, X; \alpha_y]\Big\} p(M | A = 0, X; \alpha_m).
\nonumber
\end{aligned}$$ ]{}The *profile empirical likelihood ratio* parameters ($\{p_i, \widehat{ \alpha} \}^{opt}$) are then given by [$$\begin{aligned}
&\operatorname*{arg\,max}_{p_i, \alpha} \! \prod_{i = 1}^{n} p_i p(Y_i | M_i,\! A_i,\! X_i; \!\alpha_y\!) p(\!M_i | A_i,\! X_i; \!\alpha_m\!) p(\!A | X_i; \!\alpha_a\!) \nonumber \\
&\qquad \text{such that } \quad \sum_{i=1}^n p_i = 1, \quad \sum_{i=1}^n p_i \ m(X_i; \alpha) = 0
\label{eq:hybrid_fair}\end{aligned}$$ ]{}The above optimization problem involves a semi-parametric *hybrid* likelihood [@OwenEL], that contains both nonparametric and parametric terms. In order to solve the above optimization problem (formulated on both ${ \alpha}$ and $p_i$ parameters), we can apply the Lagrange multiplier method and solve its dual form (formulated on both ${ \alpha}$ and the Lagrange multipliers); see the appendix for more details. Empirical likelihood methods provide a natural extension to imposing constraints on arbitrary PSEs, since these can be written in the form of ${\mathbb{E}}_x[m(X; \alpha)]$ for some $m(\cdot)$.
If outcomes are missing at random, the NDE is identified by ${\mathbb{E}}_x[ m(X; \alpha)]$, where [ $$\begin{aligned}
&m(X; \alpha) = \sum_{M} \Big\{{\mathbb{E}}[Y | A = 1, M, X, R = 1; \alpha_y] - \nonumber \\
&\qquad {\mathbb{E}}[Y | A = 0, M, X, R = 1; \alpha_y]\Big\} p(M | A = 0, X; \alpha_m).
\end{aligned}$$ ]{}The resulting functional is then used in the profile empirical likelihood in (\[eq:hybrid\_fair\]).
Unlike the standard unconstrained prediction setting where it is common to use nonparametric methods to estimate an arbitrary regression function, our task requires a combination of prediction and estimation of the relevant causal parameter (constraint). Estimating the causal parameter requires estimating certain nuisance components (like $E[Y|Z]$ in eq. \[eq:est\_NDE\]) which we choose to do parametrically in part because we desire certain frequentist properties, namely fast rates of convergence. More fundamentally, the empirical likelihood optimization problem in (\[eq:hybrid\_fair\]) finds optimal parameter values $\alpha$, where $\alpha$ appears also in the constraint $\sum_i p_i m(X_i;p_i,\alpha)=0$. That is, the structure of the empirical likelihood optimization problem requires that $Y$ and $M$ models are specified parametrically. Though some combination of nonparametric risk minimization and empirical likelihood would be an interesting extension, how to accomplish this is an open question.
Imposing Fairness Constraints With Hybrid Reparameterized Likelihoods {#sec:unified}
---------------------------------------------------------------------
In Section \[sec:likereparam\], we reformulated the constrained optimization problem of interest by rewriting the likelihood in terms of the parameters we were interested in constraining, and directly setting those parameters to zero. However, we did not place any constraints on $p(X)$. In Section \[sec:hybrid\], we used hybrid likelihoods to constrain a nonparametric estimate of $p(X)$, but did not provide a convenient reparameterization of the likelihood in terms of relevant parameters. In this section we describe an approach to optimizing a *hybrid reparameterized likelihood* that combines the advantages of both proposals. This allows us to constrain the entire likelihood and do so with standard maximum likelihood software, since the constraint we must satisfy directly corresponds to a parameter in the hybrid likelihood.
For simplicity of presentation, we again focus on the constraining the NDE, although the methods we describe generalize without difficulty to arbitrary constraints represented by arbitrary PSEs. The direct effect can then be estimated by ${\mathbb{E}}_x[ m(X; \alpha)]$, where $m(X; \alpha)$ is given in (\[eq:est\_NDE\]), and ${\mathbb{E}}[Y | A, M, X; \alpha_y]$ is given in (\[eq:reparam\_NDE\]). Assuming $p(X_i = x_i) = p_i$ as in (\[eq:reparam\_NDE\]), $m(X; \alpha)$ will be a function of $p_i$s as well. The *profile empirical likelihood ratio* ($\{p_i, \widehat{ \alpha} \}^{opt} $) in this setting is then given by [$$\begin{aligned}
&\operatorname*{arg\,max}_{p_i, \alpha} \! \prod_{i = 1}^{n} p_i p(Y_i | M_i,\! A_i,\! X_i; \!\alpha_y\!) p(\!M_i | A_i,\! X_i; \!\alpha_m\!) p(\!A | X_i; \!\alpha_a\!)
\nonumber \\
& \quad \text{such that } \quad \sum_{i=1}^n p_i = 1, \quad \sum_{i=1}^n p_i \ m(X_i; p_i, \alpha) = 0
\label{eq:hybrid_reparam}\end{aligned}$$ ]{}Unlike the constrained optimization problem in (\[eq:hybrid\_fair\]), it is not straightforward to find the dual form of the optimization problem in (\[eq:hybrid\_reparam\]), which is a standard approach for solving such problems in the empirical likelihood literature. The reason is that $p_i$ appears in multiple places in the constraint corresponding to setting PSE to zero; that is $m(X; \alpha)$ is now a function of both $\alpha$ and $p_i$’s. As an alternative, we provide a heuristic approach for optimizing (\[eq:hybrid\_reparam\]) via an iterative procedure that starts with initialization of $\alpha$ and $p_i$s, and at the $k$th iteration updates the values for $\alpha^{k}$ and $p^{k}_i$s by treating $m(X_i; p_i, \ \alpha)$ as a function of $m(X_i; p^{k-1}_i, \alpha)$. The procedure terminates when the difference between the two updates is sufficiently small. In Algorithm \[alg:unified\], we provide a detailed description of our proposed iterative procedure to address this issue, which behaves well in experiments.
**Input:** ${\cal D} = \{X_i, A_i, M_i, Y_i\}, i = 1, \ldots, n$ and specification of a PSE of the form ${\mathbb{E}}_X[m(X; \alpha)]$. **Output:** $\widehat{\alpha}, \widehat{p}_i$ by solving
Pick starting values for $p^{(1)}_i$ and $\alpha^{(1)}$. At $k^{th}$ iteration, given fixed $p^{(k-1)}_i$ and $\alpha^{(k-1)}$, estimate the following (in order)
- $m\Big(X_i; \{p^{(k-1)}_i\}, \alpha^{(k-1)}\Big)$
- $\lambda$ by solving [$\quad \sum_{i = 1}^n \frac{m(X_i; \theta)}{1 + \lambda \ m(X_i; \theta)} = 0$]{}, which is a monotone function in $\lambda$.
- $p^{(k)}_i$ using [$p^{(k)}_i = \frac{1}{n} \frac{1}{1 + \lambda m(X_i; \theta)}, \forall i = 1, \ldots, n,$]{}
- $\alpha^{(k)}$ by maximizing the following [ $$\begin{aligned}
\widehat{\alpha}^{(k)} &= \arg \max_{\alpha} \hspace{0.2cm} {\cal L}_{Y, M, A | X}({\cal D}; {\alpha}) \hspace{0.3cm} \nonumber \\
& \text{subject to} \hspace{0.2cm} w_a = 0, \nonumber
\end{aligned}$$ ]{}
where $ {\mathbb{E}}[Y | X, A, M; \!{ \alpha_y}] = w_0 + f(Z; \! \alpha_f) - \sum_{i = 1}^n \!\big\{$ $
\!\sum_{m} f(Z_i; \alpha_f) p(M | A = 0, X_i; {\alpha_m})\big\} p^{(k)}_i$ and ${\small f(Z) := {\mathbb{E}}[Y | X, A, M] - {\mathbb{E}}[Y | X=0, A, M=0]}$. Repeat Step (2) until convergence.
EXPERIMENTS {#sec:exp}
===========
Given Theorem \[lem:KLdis\], the accuracy of the prediction procedure depends on what components of $p(Z, Y ; \alpha)$ are constrained, and following [@NabiShpitser18Fair] this depends on the chosen estimator $\widehat{g}(\mathcal{D})$. Here, we illustrate this dependence via experiments by considering four consistent estimators of the NDE presented in [@tchetgen12semi2] (assuming the model shown in Fig. \[fig:graphs\](a) is correct). We fit models $E[Y |A, M, X; \alpha_y]$, $p(M|A, X; \alpha_m)$, and $p(A|X; \alpha_a)$ by maximum likelihood. The first estimator (G-formula), is the MLE plug-in estimator and uses $Y$ and $M$ models to estimate NDE. The second one is the inverse probability weighted (IPW) estimator that uses $A$ and $M$ models. The “mixed” estimator uses the $A$ and $Y$ models, and the augmented IPW estimator (AIPW) uses all three models. See the appendix for details on these estimators.
We generated a sample of size $6,000$ using the data generating process described in the appendix. We approximate the fair world, $p^*$, by constraining MLE given in Section \[sec:prelim\]. We estimated the NDE using the four methods described above and evaluated the performance of the approximated $p^*$ for each case. In Table \[tab:NDE\_sims\] we show the estimated NDE with respect to $p^*$, the log likelihood, KL-divergence between $p^*$ and $p$, and the mean squared error between the observed outcomes and the predicted ones. We contrast these results with the unconstrained prediction model. Unconstrained MLE is KL-closest to the true distribution and yields the lowest MSE, as expected. However, it suffers from being unfair: $\text{NDE} = 2.235$. AIPW produces the second closest approximation to the true distribution while being fair. However, the MSE under AIPW is relatively large, since new instances are being mapped and more information are averaged out from the predictions. The approximated fair distributions under the other three estimators are KL-farther from the true distribution, and the accuracy of prediction varies, underscoring how the performance of the learned prediction model depends strongly on what part of the information is being averaged out and what estimator is being used.
Next, we illustrate that even in simple settings our proposed methods for solving constrained maximum likelihood problems considerably outperform the existing method described in [@NabiShpitser18Fair]. We will use continuous outcomes for simplicity, but our results are not substantially affected if outcomes are discrete. We generated synthetic data ($n=7000$ with $20\%$ missing outcomes) according to the causal model shown in Fig. \[fig:graphs\](a), where $A, M$ are binary and $X, Y$ are continuous variables. The model specification details are reported in Appendix D and the code is attached to the submission. For illustration purposes, we assume that the direct effect of sensitive feature $A$ on outcome $Y$ is unfair and estimate it via the g-formula. We approximate the fair world, $p^*$, by constrained MLE using the three methods described in Section \[sec:consMLE\] and contrast them with the constrained MLE described in Section \[sec:prelim\] as well as unconstrained MLE. We evaluated the performance of all five methods by computing the direct effect with respect to $p^*$, KL-divergence between $p^*$ and $p$, and the mean squared error between the observed and predicted outcomes.
Results are displayed in Table \[tab:sims\] (averaged over 20 repetitions). We see that all three proposed methods achieve an approximatation to the fair distribution $p^*$ KL-closer to the true unfair distribution $p$, compared to standard constrained MLE. Using the reparameterized MLE by itself requires averaging over the constrained covariates as in [@NabiShpitser18Fair], so there is only minimal improvement in prediction accuracy (measured by MSE). However, the last two methods involve prediction in batch mode as described above – that is, use all information in the data – and so can achieve substantial improvements in prediction accuracy.
CONCLUSION {#sec:conc}
==========
Imposing hard fairness constraints on predictive models involves a balance of parametric modeling, nonparametric methods, and constrained optimization. In this paper we have proposed two innovations to make the problem easier and make predictions more accurate: a reparameterization of the likelihood such that nonlinear constraints appear explictly as likelihood parameters constrained to be zero and an incorporation of techniques from empirical likelihood theory to make the constrained distribution closer to the unconstrained unfair distribution. Our simulations show that even in a relatively simple setting, we can improve significantly on prior proposals, achieving prediction performance comparable to unconstrained (unfair) maximum likelihood, particularly with the hybrid approach. Though we focus primarily on the path-specific fairness constraints proposed in [@NabiShpitser18Fair], the ideas presented here should be applicable more broadly to fair prediction proposals that require imposing constraints on predictive models. At this stage, our method which combines reparameterization with hybrid likelihood is somewhat heuristic; in future work, we hope to develop an approach for optimizing EL weights and likelihood parameters jointly without the need for iteration.
APPENDIX {#appendix .unnumbered}
========
In **Appendix A**, we provide additional details for the direct effect reparameterization example (illustrating Theorem 2) discussed in the main paper. In **Appendix B**, we provide a brief overview of empirical likelihood methods and some additional theoretical details useful for understanding our proposed hybrid likelihood approach. In **Appendix C**, we state the statistical modeling assumptions we made in our simulation experiments. In **Appendix D**, we give some relevant details for the first simulation reported in the main paper. **Appendix E** contains proofs of our theorems. For a clearer presentation of materials in this supplement, we use a one-column format.
A. Reparameterized Likelihood Example: Additional Details {#a.-reparameterized-likelihood-example-additional-details .unnumbered}
=========================================================
Consider the DAG in Fig. \[fig:graphs\](a), and assume the natural direct effect is the unfair PSE we wish to constrain to be $0$. Theorem \[theorem:reparam\] leads to the following reparameterization of the regression function: [ $$\begin{aligned}
{\mathbb{E}}[Y \mid X, A, M]
=& \ \underset{f(X, A, M)}{\underbrace{ {\mathbb{E}}[Y \mid X, A, M] - {\mathbb{E}}[Y \mid X = 0, A, M = 0]}} \\
& - \sum_{X, M} f(X, A, M) \ p(M \mid A = 0, X) \ p(X) \\
&+ \underset{\phi(A) = w_0 + w_a A}{\underbrace{ \sum_{X, M} {\mathbb{E}}[Y \mid X, A, M] \ p(M \mid A = 0, C) \ p(X)}}.
\end{aligned}$$ ]{}The coefficient $w_a$ corresponds to the direct effect, since [ $$\begin{aligned}
\text{NDE}
&= \sum_{X, M} \Big\{ {\mathbb{E}}[Y \mid X, A=1, M] - {\mathbb{E}}[Y \mid X, A=0, M] \Big\} \ p(M \mid A = 0, X) \ p(X) \\
&= \sum_{X, M} {\mathbb{E}}[Y \mid X, A=1, M] \ p(M \mid A = 0, X) \ p(X) - \sum_{X, M} {\mathbb{E}}[Y \mid X, A=0, M] \ p(M \mid A = 0, X) \ p(X) \\
&= \phi(A = 1) - \phi(A = 0) \\
&= w_a.
\end{aligned}$$ ]{} The observed data likelihood is given by [ $$\begin{aligned}
{\cal L}_{Y, M, A, X}({\cal D}; {\alpha})
= \prod_{i = 1}^n \ p(Y_i | M_i, A_i, X_i; {\alpha_y}) \ p(M_i | A_i, X_i; {\alpha_m}) \ p(A_i | X_i; {\alpha_a}) \ p(X_i),
\end{aligned}$$ ]{}where $p(Y | M, A, X; {\alpha_y})$ has mean [ $$\begin{aligned}
{\mathbb{E}}[Y | X, A, M; \alpha_y] = f(X, A, M; \alpha_f) - \sum_{x, m} f(X, A, M; \alpha_f) \ p(M | A = 0, X; { \alpha_m}) \ p(X) + w_0.
\end{aligned}$$ ]{}The constrained optimization problem in eq. (\[eqn:c-mle\]) then simplifies to the following optimization problem: [ $$\begin{aligned}
& \arg \max_{\alpha} \hspace{0.2cm} {\cal L}_{Y, M, A, X}({\cal D}; {\alpha}) \\
& \hspace{0.4cm} \text{subject to } \quad w_a = 0.
\end{aligned}$$ ]{}
B. Hybrid Likelihood: Overview and Details {#sec:EL .unnumbered}
==========================================
Empirical Likelihood {#empirical-likelihood .unnumbered}
--------------------
We briefly review empirical likelihood methods, described in detail in [@OwenEL]. Let $X_1, \ldots, X_n$ be independent random vectors with common distribution $F_0$. Let $F$ be any CDF, where $F(x) = p(X \leq x)$, and $F_n$ be the empirical distribution. Suppose that we are interested in $F$ through $\theta = T(F)$, where $T$ is a real-valued function of the distribution. The true unknown parameter is $\theta_0 = T(F_0)$. Proceeding by analogy to parametric MLE, the non-parametric MLE of $\theta$ is $\hat{\theta} = T(F_n)$. The nonparametric likelihood ratio, $R(F)= \frac{{\cal L}(F)}{{\cal L}(F_n)}$, is used as a basis for hypothesis testing and deriving confidence intervals. The ***profile likelihood ratio*** function is defined as $${\cal R}(\theta) = \sup \ \big\{ R(F) \ | \ T(F) = \theta, F \in {\cal F} \big\},$$ where $\cal F$ denotes the set of all distributions on $\mathbb{R}$.
Often, $\theta \equiv \theta(F )$ is the solution to an estimating equation of the form ${\mathbb{E}}[m(X, \theta)] = 0$. A natural estimator for $\theta$ is produced by solving the empirical estimating equation $\frac{1}{n} \sum_{i = 1}^{n} m(X_i, \widehat{\theta}) = 0$. Assuming $p_i = f(X = x_i) \text{ for }i = 1, \ldots, n$, the ***profile empirical likelihood ratio*** function of $\theta$ is defined as [ $$\begin{aligned}
{\cal R}(\theta) = \max \Big\{ \prod_{i = 1}^{n} n p_i \quad \text{such that} \quad \sum_{i = 1}^{n} p_i \ m(X_i, \theta) = 0, \ p_i \geq 0, \ \sum_{i = 1}^{n} p_i = 1 \Big\}.
\label{eq:ELest_eq0}
\end{aligned}$$ ]{}Since maximizing the likelihood is equivalent to maximizing the logarithm of the likelihood, the profile empirical likelihood ratio is rewritten in terms of log likelihood as follows. [ $$\begin{aligned}
{\cal R}(\theta) = \max \Big\{ \sum_{i = 1}^{n} \log p_i \quad \text{such that} \quad \sum_{i = 1}^{n} p_i \ m(X_i, \theta) = 0, \ p_i \geq 0, \ \sum_{i = 1}^{n} p_i = 1 \Big\}.
\label{eq:ELest_eq}
\end{aligned}$$ ]{}In order to solve the above optimization problem, we can apply the Lagrange multiplier method. [ $$\begin{aligned}
T(\{p_i\}, \lambda, \lambda_1) = \sum_{i = 1}^n \log p_i + \lambda_1 ( \sum_{i = 1}^n p_i - 1) - n \lambda \sum_{i = 1}^n p_i \ m(X_i; \theta),
\end{aligned}$$ ]{}where $\lambda, \lambda_1$ are the Lagrange multipliers. We take the derivative of $T(\{p_i\}, \lambda, \lambda_1)$, with respect to the $p_i$’s, and set them to zero. Solving the system of equations reveals that $\lambda_1 = -n$, and $$\begin{aligned}
p_i &= \frac{1}{n} \frac{1}{1 + \lambda m(X_i; \theta)}, \forall i = 1, \ldots, n,
\label{eq:pi_lagrange}\end{aligned}$$ where $\lambda$ is the solution to $$\begin{aligned}
\quad \sum_{i = 1}^n \frac{m(X_i; \theta)}{1 + \lambda \ m(X_i; \theta)} = 0,
\label{eq:get_lambda}\end{aligned}$$ which is a monotone function in $\lambda$. Maximizing the profile empirical log-likelihood ration in (\[eq:ELest\_eq\]) is equivalent to maximizing the following (substituting $p_i$ from (\[eq:pi\_lagrange\]) into (\[eq:ELest\_eq\])): [ $$\begin{aligned}
l(\theta) = - \sum_{i = 1}^n \log(1 + \lambda \ m(X_i ; \theta)) - n\log n.
\label{eq:logEL_est}
\end{aligned}$$ ]{}Maximizing $l(\theta)$ over a small set of parameters $\theta$, is a much simpler optimization problem than maximizing (\[eq:ELest\_eq\]) over $n$ unknowns. Equation \[eq:logEL\_est\] is known as the dual representation of \[eq:ELest\_eq\]. See [@OwenEL] for more details.
Hybrid Likelihood {#hybrid-likelihood .unnumbered}
-----------------
Now, consider independent pairs $(X_1, Y_1), \ldots, (X_n, Y_n)$. Suppose that all $n$ observations are independent, and that we have a correctly specified parametric model for $p(Y | X; \theta_y)$ but $p(X)$ is unspecified. Let $p_i = p(X = x_i)$. A natural approach for estimating $\theta_y$ and the $p_i$s is to form a ***hybrid*** likelihood that is nonparametric in the distribution of $X_i$ but is parametric in the conditional distribution of $Y_i | X_i$: $$\begin{aligned}
{\cal L} ({\cal D}; \{p_i\}, \theta) = \prod_{i = 1}^n p_i \ p(Y_i | X_i; \theta). \end{aligned}$$ Suppose we are interested in parameter $\theta$ through the estimating equation ${\mathbb{E}}[m(X, Y; \theta)] = 0$. Hence, the equivalent form of (\[eq:ELest\_eq\]) for the profile hybrid likelihood ratio function is as follows: [ $$\begin{aligned}
{\cal R}(\theta) = \max \Big\{ \sum_{i = 1}^{n} \Big( \log p_i + \log p(Y_i | X_i; \theta) \Big) \quad \text{such that} \quad \sum_{i = 1}^{n} p_i \ m(X_i, Y_i; \theta) = 0, \ p_i \geq 0, \ \sum_{i = 1}^{n} p_i = 1 \Big\}.
\label{eq:hybrid_el}
\end{aligned}$$ ]{}Similar to the empirical likelihood, we can apply the Lagrange multiplier method to solve the above optimization problem. For more details, see [@OwenEL] and [@Jing2017].
C. Simulation details {#app:sims .unnumbered}
=====================
Here we report the precise parameter settings used in our simulation studies. We trained our models on a batch size of $7,000$ using the following data generating process, where outcome $Y$ is treated as missing on $20\%$ of the data. Mean squared errors in Tables \[tab:NDE\_sims\] and \[tab:sims\] are computed only on the missing portion of the outcome $Y$.
$$\begin{aligned}
X & \sim \mathcal{N}(0, 1)
\nonumber \\
\text{logit}(p(A = 1 | X)) &\sim 0.5 + 0.5X
\nonumber \\
\text{logit}(p(M = 1 | A, X)) &\sim 0.5 + X + 0.5A - AX
\nonumber \\
Y& = 1 + X + 2A -2AX + M + 3XM + AM + XAM + \mathcal{N}(0, 1)
$$
D. Details on Estimation Strategies {#d.-details-on-estimation-strategies .unnumbered}
===================================
Given Theorem \[lem:KLdis\], the accuracy of the prediction procedure will depend on what parts of $p(Z, Y; \alpha)$ are constrained, and following [@NabiShpitser18Fair] this depends on the estimator $\widehat{g}(\mathcal{D})$. Here, we define several consistent estimators of the NDE (assuming the model shown in Fig. \[fig:graphs\](a) is correct) presented in [@tchetgen12semi2].
***G-formula***: The first estimator is the MLE plug in estimator, where we use the $Y$ and $M$ models to estimate NDE. We fit models ${\mathbb{E}}[Y |A, M, X; \alpha_y]$ and $p(M |A, X; \alpha_m)$ by maximum likelihood, and use the following formula: [ $$\begin{aligned}
\mathbb{P}_n\bigg( \sum_{m} \ \Big({\mathbb{E}}[Y_i \mid A = 1, X_i, M; \widehat{\alpha}_y] - {\mathbb{E}}[Y_i \mid A=0, X_i, M; \widehat{\alpha}_y] \Big) \ p(M \mid A=0, X_i; \widehat{\alpha}_m)\bigg).
\label{eq:nde_g}
\end{aligned}$$ ]{}Since solving (\[eqn:c-mle\]) using (\[eq:nde\_g\]) entails constraining $\mathbb{E}[Y | A, M, X]$ and $p(M | A, X)$, classifying a new instance entails using $\mathbb{E}[Y | A, X] = \sum_M \ \mathbb{E}[Y | A, M, X] \ p(M | A, X)$.
***Inverse probability weighting (IPW)***: The second estimator is the IPW estimator where we use the $A$ and $M$ models to estimate NDE. We can fit the models $p(A | X; \alpha_a)$ and $p(M | A, X; \alpha_m)$ by MLE, and use the following weighted empirical average as our estimate of the NDE: [ $$\begin{aligned}
\mathbb{P}_n
\left(
\frac{\mathbb I(A_i = 1) }{p(A_i = 1 | X_i; \widehat{\alpha}_a)} \cdot \frac{p(X_i | A = 0, X_i; \widehat{\alpha}_m)}{p(M_i | A = 1, X_i; \widehat{\alpha}_m)} \ Y_i
- \frac{ \mathbb I(A_i = 0)}{ p(A_i = 0 | X_i; \widehat{\alpha}_a)} \ Y_i
\right).
\label{eqn-IPW}
\end{aligned}$$ ]{}Since solving the constrained MLE problem using this estimator entails only restricting parameters of $A$ and $M$ models, predicting a new instance is done using $\mathbb{E}[Y | X] = \sum_{A, M} \mathbb{E}[Y | A, M, X] \ p(M | A, X) \ p(A | X).$
***Mixed approach***: The third way of computing the NDE is using $A$ and $Y$ models. In this estimator, we fit the models $p(A | X; \alpha_a)$ and $\mathbb{E}[Y | A, M, X; \alpha_y]$ by MLE, as usual, and combine the edge G-formula and IPW in the following way: [ $$\begin{aligned}
\mathbb{P}_n
\left(\frac{\mathbb I(A_i = 0)}{p(A_i = 0 | X_i ; \widehat{\alpha}_a)} \ \mathbb{E}[Y_i | A = 1, M_i, X_i; \widehat{\alpha}_y] - \mathbb{E}[Y_i | A = 0, M_i; \widehat{\alpha}_y] \right),
\label{eqn-mixed}
\end{aligned}$$ ]{} Since solving the constrained MLE problem using this estimator entails only restricting parameters of $A$ and $Y$ models, predicting a new instance is done using $\mathbb{E}[Y | M, X] = \sum_{A} \mathbb{E}[Y | A, M, X] \ \frac{p(M | A, X) \cdot p(A | X)}{\sum_A p(M | A, X) \cdot p(A | X)}.$
***Augmented inverse probability weighting (AIPW)***: The final estimator uses all three models, as follows: [ $$\begin{aligned}
\mathbb{P}_n \bigg(
& \frac{ \mathbb I(A_i=1) }{ p(A_i=1 | X_i; \widehat{\alpha}_a) }
\frac{ p(M_i \mid A=0, X_i; \widehat{\alpha}_m) }{ p(M_i | A = 1, X_i; \widehat{\alpha}_m)}
\Big\{ Y_i - \mathbb{E}[Y_i | A=1,M_i,X_i; \widehat{\alpha}_y] \Big\} \\
\notag
& + \frac{ \mathbb I(A_i = 0) }{p(A_i=0 | X_i )} \ \Big\{ \mathbb{E}[Y_i | A=1,M_i,X_i; \widehat{\alpha}_y] - \eta(1,0,X_i) \Big\} + \eta(1,0,X_i) \\
\notag
& - \frac{\mathbb I(A_i = 0) }{p(A_i=0 | X_i; \widehat{\alpha}_a)} \ \Big\{ Y_i - \eta(0,0,X_i) \Big\} + \eta(0,0,X_i)
\bigg),
\label{eqn:3-robust}
\end{aligned}$$ ]{}with $\eta(a,a',X) \equiv \sum_M \mathbb{E}[Y | a,M,X] p(M | a',X)$. Since the models of $A,M$, and $Y$ are all constrained with this estimator, predicting $Y$ for a new instance is via $\mathbb{E}[Y | X] = \sum_{A, M} \mathbb{E}[Y | A, M, X] \ p(M | A, X) \ p(A | X).$
E. Proofs {#app:proofs .unnumbered}
=========
[\[lem:KLdis\]]{} Let $p(Z)$ denote the observed data distribution, $M_1 = \big\{p^*_1(Z) = \operatorname*{arg\,max}_{q(Z)} D_{KL}(p || q) \hspace{0.1cm} \text{ s.t. } \hspace{0.1cm} \epsilon_l \leq g(q(Z)) \leq \epsilon_u, \ q(Z_1) = p(Z_1) \big\}$, and $M_2 = \big\{p^*_2(Z) = \operatorname*{arg\,max}_{q(Z)} D_{KL}(p || q) \hspace{0.1cm} \text{ s.t. } \hspace{0.1cm} \epsilon_l \leq g(q(Z)) \leq \epsilon_u, \ q(Z_2) = p(Z_2) \big\}$. If $Z_2 \subseteq Z_1 \subseteq Z$, then $D_{KL}(p || p^*_2) \leq D_{KL}(p || p^*_1).$ In other words, if more densities are being constrained in $M_2$ compared to $M_1$, then $p^*_2(Z)$ is closer to $p(Z)$ than $p^*_1(Z)$.
$M_1$ is a submodel of $M_2$, hence maximizing the likelihood under model $M_1$ yields a likelihood that is less than or equal to the one under model $M_2$: $\max {\cal L}_{M_1}({\cal D}) \leq \max {\cal L}_{M_2}(\cal D) $. Maximizing the likelihood of observed data with respect to the model parameters is equivalent to minimizing KL-divergence between the likelihood and the true distribution of the data [@wasserman2013all]. Consequently, KL-divergence between $p^*$ and $p$ is smaller in $M_2$ compared to $M_1$, i.e $D_{KL}(p || p^*_2) \leq D_{KL}(p || p^*_1).$
[\[theorem:reparam\]]{} Assume the observed data distribution $p(Y,Z)$ is induced by a *causal model*, where $Z = \{X, A, M\}$ includes pre-treatment measures $X$, treatment $A$, and post-treatment pre-outcome mediators $M$. Let $p(Y(\pi, a, a'))$ denote the potential outcome distribution that corresponds to the effect of $A$ on $Y$ along proper causal paths in $\pi$, where $\pi$ includes the direct influence of $A$ on $Y$, and let $p(Y_0(\pi, a, a'))$ denote the identifying functional for $p(Y(\pi, a, a'))$ obtained from the edge-formula in (\[eqn:pse\]), where the term $p(Y | Z)$ is evaluated at $\{Z \setminus A\} = 0$. Then ${\mathbb{E}}[Y | Z]$ can be written as follows: [ $$\begin{aligned}
{\mathbb{E}}[Y | Z] = f(Z) - \big( {\mathbb{E}}[Y(\pi, a, a')] - {\mathbb{E}}[Y_0(\pi, a, a')]\big) \ + \ \phi(A),
\end{aligned}$$ ]{} where $f(Z) := {\mathbb{E}}[Y | Z] - {\mathbb{E}}[Y | A, \{Z \setminus A\} = 0]$ and $\phi(A) = w_0 + w_aA$. Furthermore, $w_a$ corresponds to $\pi$-specific effect of $A$ on $Y$.
By letting $\phi(A=a) = {\mathbb{E}}[Y(\pi, a, a')]$, it suffices to show that ${\mathbb{E}}[Y_0(\pi, a, a')] = {\mathbb{E}}[Y | A, \{Z \setminus A\} = 0]$. Given the identification result for edge-consistent counterfactuals in [@shpitser15hierarchy], we can write the identification functional as follows. $$\begin{aligned}
{\mathbb{E}}[Y_0(\pi, a, a')] = \sum_{V \in {\mathfrak{X}_V} \setminus \{A, Y\}} {\mathbb{E}}[Y | A = a, \{Z \setminus A\} = 0] \times h(V \in {\mathfrak{X}_V} \setminus Y),
\end{aligned}$$ where $h(V \in {\mathfrak{X}_V} \setminus Y)$ is a function of all variables excluding $Y$. Note that $h$, does not include any density where $A$ appears on the LHS of the conditioning bar. Therefore, we have: $$\begin{aligned}
{\mathbb{E}}[Y_0(\pi, a, a')]
&= {\mathbb{E}}[Y | A = a, \{Z \setminus A\} = 0] \times \sum_{V \in {\mathfrak{X}_V} \setminus \{A, Y\}} h(V \in {\mathfrak{X}_V} \setminus Y) \\
&= {\mathbb{E}}[Y | A = a, \{Z \setminus A\} = 0].
\end{aligned}$$
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Lossy communication of correlated sources over a multiple access channel is studied. First, lossy communication is investigated in the presence of correlated decoder side information. An achievable joint source-channel coding scheme is presented, and the conditions under which separate source and channel coding is optimal are explored. It is shown that separation is optimal when the encoders and the decoder have access to a common observation conditioned on which the two sources are independent. Separation is shown to be optimal also when only the encoders have access to such a common observation whose lossless recovery is required at the decoder. Moreover, the optimality of separation is shown for sources with a common part, and sources with reconstruction constraints. Next, these results obtained for the system in presence of side information are utilized to provide a set of necessary conditions for the transmission of correlated sources over a multiple access channel without side information. The identified necessary conditions are specialized to the case of bivariate Gaussian sources over a Gaussian multiple access channel, and are shown to be tighter than known results in the literature in certain cases. Our results indicate that side information can have a significant impact on the optimality of source-channel separation in lossy transmission, in addition to being instrumental in identifying necessary conditions for the transmission of correlated sources when no side information is present.'
author:
- '[^1] [^2]'
bibliography:
- 'IEEEabrv.bib'
- 'ref.bib'
---
Introduction {#Sec:introduction}
============
We consider the transmission of two correlated memoryless sources over a multiple access channel with fidelity criteria. The encoders and/or the decoder may have access to side information correlated with the sources. We propose an achievable joint source-channel coding scheme in the presence of correlated decoder side information. We then focus on the case when the two sources are conditionally independent given the side information available at the encoders and/or the decoder. First, we identify the necessary and sufficient conditions under which separation is optimal when the side information is shared between the encoders and the decoder. Additionally, we show that separation is optimal for sources with reconstruction constraints, when only the decoder has access to the side information conditioned on which the two sources are independent. Next, we consider the case when the decoder is required to recover the common information shared by both encoders losslessly, but can tolerate some distortion for the parts known only at a single encoder. We show that separation is also optimal for this case. We then consider the transmission of sources with a common part in the sense of G[á]{}cs-K[ö]{}rner [@gacs1973common], and investigate the conditions under which separation is optimal in the absence of side information.
Next, we provide necessary conditions for the transmission of correlated sources over a multiple access channel. This is achieved by providing a particular side information to the receiver and the transmitters, and using our results for transmitting correlated sources in the presence of side information. In particular, when the two sources are independent conditioned on the side information, our initial results indicate that the necessary and sufficient conditions can be achieved by considering separate source and channel coding. For the special case of transmitting bivariate Gaussian sources over a Gaussian multiple access channel, we provide comparisons of the necessary conditions obtained from our approach with the conditions from [@lapidoth2010sending] and [@7541654]. Our results show that the proposed technique provides the tightest known bound in certain scenarios.
[*Related Work:*]{} Shannon proved the optimality of separate source and channel coding for transmitting a source through a noisy channel [@Shannon], known as the separation theorem. Separation was shown to be optimal for the lossy transmission of a source with decoder side information, in [@shamai1998systematic]. The point-to-point scenario was extended in [@gunduz2007correlated] to transmission of correlated sources through a multiple access channel, and separation was shown to be optimal when one of the sources is shared between the two encoders. For the lossless case, the optimality of separation was established in [@gunduz2009source] for transmitting correlated sources through a multiple access channel, whenever the decoder has access to some side information conditioned on which the two sources are independent. A joint source-channel coding scheme was proposed in [@minero2015unified] for the transmission of correlated sources over a multiple access channel based on hybrid coding. Transmission of correlated sources in the presence of common reconstruction constraints at the encoders is considered in [@steinberg2009coding]. Necessary conditions are derived in [@7541654] for the lossy transmission of correlated sources over a multiple access channel, and in [@lapidoth2010sending] for transmitting correlated sources over a Gaussian multiple access channel.
In the remainder of the paper, $X$ represents a random variable, and $x$ is its realization. $X^n=(X_1, \ldots, X_n)$ is a random vector of length $n$, and $x^n=(x_1, \ldots, x_n)$ denotes its realization. $\mathcal{X}$ is a set with cardinality $|\mathcal{X}|$. $\mathbb{E}[X]$ is the expected value and $\text{var}(X)$ is the variance of $X$.
![Communication of correlated sources over a multiple access channel.[]{data-label="Fig:Model1"}](SystemModel1v4.eps){width="0.7\linewidth"}
System Model {#Sec:SystemModel}
============
Consider the transmission of two discrete memoryless sources $S_1$ and $S_2$ in Fig. \[Fig:Model1\]. Encoder $1$ observes $S_1^n=(S_{11}, \ldots, S_{1n})$. Encoder $2$ observes $S_2^n=(S_{21}, \ldots, S_{2n})$. If switch $\text{SW}_2$ in Fig. \[Fig:Model1\] is closed, the two encoders also have access to a common observation $Z^n$ correlated with $S_1^n$ and $S_2^n$. Encoders $1$ and $2$ map their observations to the channel inputs $X_1^n$ and $X_2^n$, respectively. A discrete memoryless multiple access channel (DM-MAC) exists between the encoders and the decoder, characterized by the distribution $p(y|x_1, x_2)$. If switch $\text{SW}_1$ in Fig. \[Fig:Model1\] is closed, the decoder has access to side information $Z^n$. Upon observing the channel output $Y^n$ and side information $Z^n$ whenever it is available, the decoder constructs $\hat{S}_1^n$, $\hat{S}_2^n$, and $\hat{Z}^n$ such that $$\frac{1}{n}\sum_{i=1}^n\mathbb{E}[d_j(S_{ji}, \hat{S}_{ji})]\leq D_j \text{ for } j=1,2$$ where $D_j$ is the maximum average distortion allowed for $S_j$, given a distortion measure $d_j(s_{ji}, \hat{s}_{ji})$ for $j=1,2$, and $P(Z^n\neq \hat{Z}^n)\rightarrow 0$ as $n\rightarrow \infty$. Random variables $S_1$, $S_2$, $Z$, $X_1$, $X_2$, $Y$, $\hat{S}_1$, $\hat{S}_2$, $\hat{Z}$ are defined over the corresponding alphabets $\mathcal{S}_1$, $\mathcal{S}_2$, $\mathcal{Z}$, $\mathcal{X}_1$, $\mathcal{X}_2$, $\mathcal{Y}$, $\hat{\mathcal{S}}_1$, $\hat{\mathcal{S}}_2$, $\hat{\mathcal{Z}}$. Note that, when the switch $\text{SW}_1$ is closed, error probability in decoding $Z^n$ becomes irrelevant since it is readily available at the decoder, and serves as side information.
We use the following notation from [@shamai1998systematic], [@gunduz2007correlated]. Define the minimum average distortion for $S_j$ given $Q$ as $$\label{func1}
\mathcal{E}(S_j|Q)=\min_{f:Q\rightarrow \hat{S}_j} E[d_j(S_j, f(Q))], \quad j=1,2,$$ and the conditional rate distortion function [@gray1972conditional] for source $S_j$ when side information $Z$ is shared between the encoder and the decoder as $$\label{func2}
R_{S_j|Z} (D_j) = \min_{\substack{p(u_j| s_j, z)\\\mathcal{E}(S_j|U_j, Z)\leq D_j}} I(S_j; U_j|Z), \; \; j=1,2.$$
Joint Source-Channel Coding with Decoder Side Information {#section3}
=========================================================
We first assume that only $\text{SW}_1$ is closed, and present a general achievable scheme for the lossy communication of correlated sources in the presence of decoder side information.
\[lemma:hybrid\] The distortion pair $(D_1, D_2)$ is achievable for sending two discrete memoryless correlated sources $S_1$ and $S_2$ over a DM-MAC with $p(y|x_1, x_2)$ and decoder side information $Z$ if there exists a joint distribution $p(u_1, u_2, s_1, s_2, z)=p(u_1|s_1)p(u_2|s_2)p(s_1, s_2, z)$, and functions $x_j(u_j, s_j)$, $g_j(u_1, u_2, y,z)$ for $j=1,2$, such that $$\begin{aligned}
I(U_1; S_1|U_2, Z)&< I(U_1;Y|U_2, Z) \label{eq4n}\\
I(U_2; S_2|U_1, Z)&< I(U_2;Y|U_1, Z) \label{eq5n}\\
I(U_1, U_2; S_1, S_2| Z)&<I(U_1, U_2;Y|Z) \label{eq6n}\end{aligned}$$ and $\mathbb{E}[d_j(S_j, g_j(U_1, U_2, Y, Z))]\leq D_j$ for $j=1,2$.
Our achievable scheme builds upon the hybrid coding framework of [@minero2015unified], by generalizing it to the case with decoder side information. The detailed proof is available in Appendix \[appendix0\].
For the remainder of this section, we assume that the sources are independent when conditioned on the side information, i.e., the Markov condition $S_1-Z-S_2$ holds.
Optimality of Separation
------------------------
We now focus on the conditions under which separation is optimal when sources $S_1$ and $S_2$ are independent given the side information $Z$.
### Decoder Side Information Available at the Encoder
We first consider that both switches in Fig. \[Fig:Model1\] are closed. We show that whenever the two sources are independent given the side information that is shared between the encoders and the decoder, separation is optimal. The next theorem states the necessary and sufficient conditions.
\[lemma2\] Consider the communication of two correlated sources $S_1$ and $S_2$ with side information $Z$ shared between the encoders and the decoder. If $S_1-Z-S_2$ form a Markov chain, then separate source and channel coding is optimal, and a distortion pair $(D_1, D_2)$ is achievable if $$\begin{aligned}
R_{S_1|Z} (D_1)&< I(X_1; Y|X_2, Q) \label{lemm2cond1}\\
R_{S_2|Z} (D_2)&< I(X_2; Y|X_1, Q) \label{lemm2cond2} \\
R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2)&< I(X_1, X_2; Y|Q) \label{lemm2cond3}\end{aligned}$$ for some $p(x_1, x_2, y,q)=p(y|x_1, x_2) p(x_1|q) p(x_2|q)p(q)$.
Conversely, for any achievable $(D_1, D_2)$ pair, - must hold with $<$ replaced with $\leq$.
See Appendix \[appendixA\].
When side information $Z$ is available only at the decoder, i.e., when only switch $\text{SW}_1$ is closed, separation is known to be optimal for the lossless transmission of sources $S_1$ and $S_2$ whenever $S_1-Z-S_2$ [@gunduz2009source]. In light of Theorem \[lemma2\], we show that a similar result holds for the lossy case whenever the Wyner-Ziv rate distortion function of each source is equal to its conditional rate distortion function.
\[lemma2old\] Consider communication of two correlated sources $S_1$ and $S_2$ with decoder only side information $Z$. If $$R_{S_j|Z} (D_j)=R^{\text{WZ}}_{S_j|Z} (D_j), \label{secondcondition}$$ where $$R^{\text{WZ}}_{S_j|Z} (D_j) =\min_{\substack{p(u_j|s_j), g(u_j, z)\\ \mathbb{E}[d_j(S_j, g(U_j, Z))]\leq D_j\\U_j-S_j-Z}} I(S_j;U_j|Z) \text{ for } j=1,2 \notag$$ is the (Wyner-Ziv) rate distortion function of $S_j$ with decoder-only side information $Z$ [@wyner1976rate], and $S_1-Z-S_2$ form a Markov chain, then separation is optimal, with the necessary and sufficient conditions stated in -.
Corollary \[lemma2old\] follows from the fact that whenever holds, conditional rate distortion functions in Theorem \[lemma2\] are achievable by relying on decoder side information only.
\[remark1\] Gaussian sources are an example for . For instance, let $(S_1, S_2, Z)$ denote a zero-mean Gaussian random vector. Then, separation is optimal, from the necessary and sufficient conditions stated in Theorem \[lemma2\], if and only if $$\begin{aligned}
i) &\quad \frac{\mathbb{E}[S_1S_2]\mathbb{E}[Z^2]-\mathbb{E}[S_1Z] \mathbb{E}[S_2Z]}{\mathbb{E}[S^2_2] \mathbb{E}[Z^2]- \mathbb{E}^2[S_2Z]} =0 \label{finalcond1} \\
ii) &\quad \frac{\mathbb{E}[S_1Z]}{\mathbb{E}[Z^2]} = \frac{\mathbb{E}[S_1Z] \mathbb{E}[S_2^2]- \mathbb{E}[S_1S_2]\mathbb{E}[S_2Z]}{\mathbb{E}[S_2^2]\mathbb{E}[Z^2]-\mathbb{E}^2[S_2 Z]} \label{finalcond2} \\
iii) &\quad \frac{\mathbb{E}^2[S_1Z]}{\mathbb{E}[Z^2]}
=\frac{\mathbb{E}^2[S_1 S_2]\mathbb{E}[Z^2]+\mathbb{E}^2[S_1 S_3]\mathbb{E}[S_2^2]}{\mathbb{E}[S_2^2] \mathbb{E}[Z^2]-\mathbb{E}^2[S_2 Z]}
-\frac{2 \mathbb{E}[S_1 S_2]\mathbb{E}[S_1 S_3] \mathbb{E}[S_2 S_3]}{\mathbb{E}[S_2^2] \mathbb{E}[Z^2]-\mathbb{E}^2[S_2 Z]} \label{finalcond3}\end{aligned}$$ are satisfied.
Please see Appendix \[appendixExample\] for the details.
![Communication of correlated sources over a multiple access channel with reconstruction constraints.[]{data-label="Fig:Reconstruction"}](SystemModelReconstruction.eps){width="0.65\linewidth"}
### Sources with Reconstruction Constraints
We next consider transmission of correlated sources $(S_1, S_2)$ over a multiple access channel with reconstruction constraints in the presence of decoder side information $Z$ such that $S_1-Z-S_2$, and show that separation is optimal in this scenario. Accordingly, we consider in this section the system model illustrated in Fig. \[Fig:Reconstruction\] with $S_1-Z-S_2$. In addition to the encoding function $e_j: \mathcal{S}_j^n\rightarrow \mathcal{X}_j^n$ and decoding function $g_j: \mathcal{Y}^n \times \mathcal{Z}^n\rightarrow \hat{\mathcal{S}}^n_j$ such that $$\frac{1}{n}\sum_{i=1}^n \mathbb{E}[d_j(S_{ji}, \hat{S}_{ji})] \leq D_j \text{ for } j=1,2,$$ define the reconstruction constraints as in [@steinberg2009coding], by considering a reconstruction mapping $\psi_j: \mathcal{S}_j^n \rightarrow \bar{\mathcal{S}}_j^n$ for encoder $j=1,2$, such that, $$\label{eq:reconconst}
P(\bar{S}_j^n\neq \hat{S}_j^n)\rightarrow 0 \text{ as } n\rightarrow \infty.$$
\[Thm:Reconstruction\] For the lossy transmission of correlated sources $(S_1, S_2)$ over a MAC with reconstruction constraints from in the presence of decoder side information $Z$ such that $S_1-Z-S_2$, separation is optimal, and a distortion pair $(D_1, D_2)$ is achievable if $$\begin{aligned}
R^{RC}_{S_1| Z}(D_1)&< I(X_1; Y|X_2, Q) \label{eq:commonr1}\\
R^{RC}_{S_2| Z}(D_2)&< I(X_2; Y|X_1, Q) \label{eq:commonr2}\\
R^{RC}_{S_1| Z}(D_1)+R^{RC}_{S_2| Z}(D_2)&< I(X_1, X_2; Y|Q) \label{eq:commonr3}\end{aligned}$$ where $p(y, x_1, x_2, q)=p(y|x_1, x_2) p(x_1|q) p(x_2|q)p(q)$, and for $j=1,2$, $$\begin{aligned}
\label{eq:singlerecons}
R^{RC}_{S_j| Z}(D_j)\triangleq &\min_{\substack{p(\hat{s}_j|s_j)\\ \mathbb{E}[d(S_j, \hat{S}_j)]\leq D_j\\ \hat{S}_j-S_j-Z }} I(\hat{S}_j;S_j|Z) \end{aligned}$$ is the lossy source coding rate of a single source with decoder side information and reconstruction constraint [@steinberg2009coding]. Conversely, for any achievable distortion pair $(D_1, D_2)$ with reconstruction constraints in , the conditions - must hold with $<$ replaced with $\leq$.
We provide the proof in Appendix \[appendixD\].
Separation in the Presence of Common Observation
================================================
In this section, we assume that only switch $\text{SW}_2$ is closed in Fig. \[Fig:Model1\], and show the optimality of separation whenever lossless reconstruction of the common observation is required.
\[lemma3\] Consider the communication of correlated sources $S_1$, $S_2$, and $Z$, where $Z$ is observed by both encoders. If $S_1-Z-S_2$ form a Markov chain, and $Z$ is to be reconstructed at the receiver in a lossless fashion, then, separate source and channel coding is optimal, and the distortion pair $(D_1, D_2)$ is achievable if $$\begin{aligned}
R_{S_1|Z} (D_1) &< I(X_1;Y|X_2, W) \label{common1}\\
R_{S_2|Z} (D_2) &< I(X_2;Y|X_1, W)\label{common2}\\
R_{S_1|Z} (D_1)+R_{S_2|Z} (D_2) &< I(X_1, X_2;Y| W)\label{common3}\\
H(Z) +R_{S_1|Z} (D_1)+R_{S_2|Z} (D_2) &< I(X_1, X_2;Y) \label{common4}\end{aligned}$$ for some $p(x_1, x_2, y, w)=p(y|x_1, x_2) p(x_1|w) p(x_2|w)p(w)$.
Conversely, if a distortion pair $(D_1, D_2)$ is achievable, then - must hold with $<$ replaced with $\leq$.
We provide a detailed proof in Appendix \[appendixB\].
A special case of Theorem \[lemma3\] is the transmission of two sources over a DM-MAC with one distortion criterion, when one source is available at both encoders as considered in [@gunduz2007correlated], which corresponds to $S_2$ being a constant in Theorem \[lemma3\].
Separation for Sources with a Common Part {#Sec:CommonPart}
=========================================
In this section, we assume that both switches in Fig. \[Fig:Model1\] are open, $Z=\text{constant}$, and study the conditions under which separate source and channel coding is optimal for transmitting sources with a common part. We first review two notions that quantify common information between correlated sources.
[(G[á]{}cs-K[ö]{}rner common information)[@gacs1973common]]{}\[GKW\] Define the function $f_j: \mathcal{S}_j\rightarrow \{1, \ldots, k\}$ for $j=1,2$ with the largest integer $k$ such that $P(f_j(S_j)=u_0)>0$ for $u_0\in \{1, \ldots, k\}$, $j=1,2$ and $P(f_1(S_1)=f_2(S_2)) = 1$. Then, $U_0=f_1(S_1)=f_2(S_2)$ is defined as the common part between $S_1$ and $S_2$, and the G[á]{}cs-K[ö]{}rner common information is given by $$C_{GK}(S_1, S_2)=H(U_0) \label{GK}.$$
[(Wyner’s common information)[@wyner1975common]]{} Wyner’s common information between $S_1$ and $S_2$ is defined as, $$\label{commoninfo}
C_{W}(S_1, S_2) = \min_{\substack{p(v|s_1, s_2) \\ S_1-V-S_2}} I(S_1, S_2; V).$$
\[independence\] $C_{GK}(S_1, S_2)=C_{W}(S_1, S_2)$ if and only if there exists a random variable $U_0$ such that $U_0$ is the common part of $S_1$ and $S_2$ from Definition \[GKW\], and $S_1-U_0-S_2$ [@gacs1973common], [@wyner1975common].
We first state a separation result when common information is available as side information at the decoder.
\[corollary1\] Consider the transmission of two sources $S_1$ and $S_2$ with a common part $U_0=f_1(S_1)=f_2(S_2)$ from Definition \[GKW\]. Then, separation is optimal whenever $$\label{mycond}
C_{GK}(S_1, S_2)=C_W(S_1, S_2),$$ and the common part $U_0$ is available at the decoder.
We know from Remark \[independence\] that whenever holds, then $S_1-U_0-S_2$, where $U_0$ is the common part of $S_1$ and $S_2$ as in Definition \[GKW\]. Since the two encoders can extract $U_0$ individually, each encoder can achieve the corresponding conditional rate distortion function in which $U_0$ is shared between the encoder and the decoder. Corollary \[corollary1\] then follows from Theorem \[lemma2\] by letting $Z\leftarrow U_0$.
Our next result states that whenever G[á]{}cs-K[ö]{}rner common information between two sources is equal to Wyner’s common information, then separate source and channel coding is optimal if lossless reconstruction of the common part is required.
\[corollary1new\] Consider the transmission of two correlated sources $S_1$ and $S_2$ with a common part $U_0=f_1(S_1)=f_2(S_2)$ from Definition \[GKW\]. Let $C_{GK}(S_1, S_2)=C_W(S_1, S_2)$ and the common part $U_0$ of $S_1$ and $S_2$ is to be recovered losslessly. Then, separate source and channel coding is optimal.
We have from Definition \[GKW\] that the two encoders can separately reconstruct $U_0$, and from Remark \[independence\] that $S_1-U_0-S_2$. Then, the result follows from letting $Z\leftarrow U_0$ in Theorem \[lemma3\].
![Correlated sources over a MAC.[]{data-label="Fig:SystemModel100"}](SystemModel100.eps){width="0.65\linewidth"}
A necessary Condition for the Transmission of Correlated Sources over a MAC
===========================================================================
We consider in this section the lossy transmission of correlated sources over a multiple access channel without any side information at the receiver, i.e., both switches in Fig. \[Fig:Model1\] are assumed to be open; see Fig. \[Fig:SystemModel100\]. We provide necessary conditions for the achievability of a distortion pair $(D_1, D_2)$ using our results from Section \[section3\]. This will be achieved by providing a correlated side information to the encoders and the receiver, conditioned on which the two sources are independent. From Theorem \[lemma2\], separation is optimal in this setting, and the corresponding necessary and sufficient conditions for the achievability of a distortion pair serve as necessary conditions for the original problem. A set of necessary conditions for transmitting correlated sources over a MAC is presented in Theorem \[Thm:Necessary\] below.
We then specialize the necessary conditions to the transmission of correlated Gaussian sources over a Gaussian multiple access channel, and provide comparisons between the obtained necessary conditions with those presented in [@lapidoth2010sending] and [@7541654] for the same scenario.
\[Thm:Necessary\] Consider the communication of two correlated sources $S_1$ and $S_2$ over a MAC. If a distortion pair $(D_1, D_2)$ is achievable for sources $(S_1, S_2)$ and channel $p(y|x_1, x_2)$, then for every $Z$, for which $S_1-Z-S_2$ form a Markov chain, we have $$\begin{aligned}
R_{S_1|Z} (D_1)&\leq I(X_1; Y|X_2, Q), \label{lemm2cond1n}\\
R_{S_2|Z} (D_2)&\leq I(X_2; Y|X_1, Q), \label{lemm2cond2n} \\
R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2) &\leq I(X_1, X_2;Y|Q), \label{necesconst4} \\
R_{S_1 S_2} (D_1, D_2) &\leq I(X_1, X_2;Y), \label{necesconst5}\end{aligned}$$ for some $Q$ for which $X_1-Q-X_2$ form a Markov chain, where $$R_{S_1 S_2} (D_1, D_2) =\min_{\substack{p(\hat{s}_1, \hat{s}_2|s_1, s_2)\\ \mathbb{E}[d_1(S_1, \hat{S}_1)]\leq D_1 \\ \mathbb{E}[d_2(S_2, \hat{S}_2)]\leq D_2}} I(S_1, S_2;\hat{S}_1, \hat{S}_2) \notag$$ is the rate distortion function of the joint source $(S_1, S_2)$ with target distortions $D_1$ and $D_2$ for sources $S_1$ and $S_2$, respectively.
For any $Z$ that satisfies the Markov chain condition, we consider the genie-aided setting in which $Z^n$ is provided to both the encoders and the decoder. Then, we obtain the setting in Theorem \[lemma2\]. Conditions - follow from Theorem \[lemma2\], whereas condition follows from the cut-set bound.
By relaxing conditions and , we obtain the following necessary conditions.
\[corollary5\] If a distortion pair $(D_1, D_2)$ is achievable for sources $(S_1, S_2)$, then for every $Z$ that forms a Markov chain $S_1-Z-S_2$, we have $$\begin{aligned}
R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2), &\leq I(X_1, X_2;Y|Q) \label{newGGY1}\\
R_{S_1 S_2} (D_1, D_2) &\leq I(X_1, X_2;Y), \label{newGGY2}\end{aligned}$$ for some $Q$ such that $X_1-Q-X_2$.
In the following, we compare the necessary conditions in Corollary \[corollary5\] with the necessary conditions obtained by Lapidoth and Wigger in [@7541654], and the necessary conditions obtained by Lapidoth and Tinguely in [@lapidoth2010sending] for the transmission of Gaussian sources over a Gaussian MAC. To do so, we consider a bivariate Gaussian source $(S_1, S_2)$ such that $$\label{bivariate}
\left ( \begin{matrix} S_1 \\S_2\end{matrix} \right ) \sim
\mathcal{N}\left ( \left (\begin{matrix} 0 \\0\end{matrix} \right), \left ( \begin{matrix} 1 & \rho \\\rho & 1\end{matrix}\right) \right),$$ and a memoryless MAC with additive Gaussian noise: $$\label{GausMAC}
Y = X_1 + X_2 + N,$$ where $N$ is a standard Gaussian random variable. We impose input power constraints $\frac{1}{n} \sum_{i=1}^n\mathbb{E}[X_{ji}^2]$ $\leq P$, $j=1,2$, and consider squared error distortion functions $d_j(S_j, \hat{S}_j)= (S_j-\hat{S}_j)^2$ for $j=1,2$.
First, we consider the necessary conditions from Corollary \[corollary5\]. From [@bross2008gaussian], for the Gaussian MAC, we have $$\begin{aligned}
I(X_1, X_2; Y | Q) &\leq \frac{1}{2} \log ( 1 + \beta_1 P + \beta_2 P ) \label{twocond1} \\
I(X_1, X_2; Y) &\leq \frac{1}{2} \log ( 1 + 2P + 2P \sqrt{(1-\beta_1)(1-\beta_2)} ) \label{twocond2}\end{aligned}$$ for some $0\leq \beta_1, \beta_2 \leq 1$. From Corollary \[corollary5\] along with and , we obtain the following necessary conditions for the Gaussian example.
\[cor6\] If a distortion pair $(D_1, D_2)$ is achievable for sources $(S_1, S_2)$ over a Gaussian MAC given in , then for every $Z$ that forms a Markov chain $S_1-Z-S_2$, we have $$\begin{aligned}
R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2) &\leq \frac{1}{2} \log ( 1 + \beta_1 P + \beta_2 P ) \label{eq3GGY1}\\
R_{S_1 S_2} (D_1, D_2) &\leq \frac{1}{2} \log ( 1 + 2P + 2P \sqrt{(1-\beta_1)(1-\beta_2)} ) \label{eq3GGY2}\end{aligned}$$ for some $0\leq \beta_1, \beta_2 \leq 1$.
Next, observe that the Lapidoth-Tinguely necessary condition for this example is $$\label{LT}
R_{S_1 S_2}( D_1, D_2) \leq \frac{1}{2} \log (1 + 2P(1+\rho))$$ from [@lapidoth2010sending Theorem IV.1].
Lastly, we obtain the Lapidoth-Wigger necessary conditions from [@7541654 Corollary 1.1] and - for this example as follows, $$\begin{aligned}
R_{S_1S_2}(D_1, D_2) - \frac{1}{2} \log \frac{1+\rho}{1-\rho} &\leq \frac{1}{2} \log ( 1 + \beta_1 P + \beta_2 P ) \label{eq3LW1}\\
R_{S_1 S_2} (D_1, D_2) &\leq \frac{1}{2} \log ( 1 + 2P + 2P \sqrt{(1-\beta_1)(1-\beta_2)} ) \label{eq3LW2}\end{aligned}$$ for some $0\leq \beta_1, \beta_2 \leq 1$, since for the Gaussian source $C_{W}(S_1, S_2) = \frac{1}{2} \log \frac{1+\rho}{1-\rho}$ from [@wyner1975common].
In the following, we compare the necessary conditions from Corollary \[cor6\] with and - for Gaussian sources. To do so, we let $Z$ in Corollary \[cor6\] to be the common part of $(S_1, S_2)$ with respect to Wyner’s common information from . From [@xu2016lossy Proposition 1], the common part can be characterized as follows. Let $Z$, $N_1$, and $N_2$ be standard random variables. Then, $S_1$, and $S_2$ can be expressed as $$\begin{aligned}
S_1 &= \sqrt{\rho} Z + \sqrt{1-\rho} N_1 \\
S_2 &= \sqrt{\rho} Z + \sqrt{1-\rho} N_2 \end{aligned}$$ such that $I(S_1, S_2 ; Z) = \frac{1}{2} \log \frac{1+\rho}{1-\rho}$ and $I(S_1, S_2; Z') > \frac{1}{2} \log \frac{1+\rho}{1-\rho}$ for all $S_1-Z'-S_2$ for which $Z'\neq Z$.
The rate distortion function for source $S_1$ with encoder and decoder side information $Z$ is, $$R_{S_1|Z}(D_1) = \left \{ \begin{matrix}
\frac{1}{2} \log \frac{1-\rho}{D_1} & \text{ if } & \qquad 0 < D_1 < 1-\rho \\
0 & \text{ if } & D_1 \geq 1-\rho
\end{matrix}\right .$$ from [@wyner1978rate]. Similarly, $$R_{S_2|Z}(D_2) = \left \{ \begin{matrix}
\frac{1}{2} \log \frac{1-\rho}{D_2} & \text{ if } & \qquad 0 < D_2 < 1-\rho \\
0 & \text{ if } & D_2 \geq 1-\rho
\end{matrix}\right . .$$ Therefore, $$\begin{aligned}
\label{point1}
R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2) =
\left\{ \begin{matrix}
\frac{1}{2} \log \frac{(1-\rho)^2}{D_1 D_2} &\text{ if } & 0< D_1< 1-\rho, & 0< D_2< 1-\rho \\
\frac{1}{2} \log \frac{1-\rho}{D_1} &\text{ if } & 0< D_1< 1-\rho, & \quad \;\;\,D_2\geq 1-\rho \\
\frac{1}{2} \log \frac{1-\rho}{D_2} &\text{ if } & \quad \;\;\,D_1\geq 1-\rho, & 0< D_2< 1-\rho \\
0 &\text{ if } &\quad \;\;\, D_1\geq 1-\rho, & \quad \;\;\,D_2\geq 1-\rho
\end{matrix} \right . .\end{aligned}$$ We also have, from [@xiao2005compression; @lapidoth2010sending], that, $$\begin{aligned}
\label{RateGausLHS}
R_{S_1 S_2} (D_1, D_2) = \left\{ \begin{matrix}
\frac{1}{2} \log \left( \frac{1}{\min(D_1, D_2)}\right ) & \text{ if } (D_1, D_2) \in \mathcal{D}_1 \\
\frac{1}{2} \log^+ \left( \frac{1-\rho^2}{D_1 D_2}\right ) & \text{ if } (D_1, D_2) \in \mathcal{D}_2 \\
\frac{1}{2} \log^+ \left( \frac{1-\rho^2}{D_1 D_2 - \left(\rho - \sqrt{(1-D_1)(1-D_2)}\right)^2}\right ) & \text{ if } (D_1, D_2) \in \mathcal{D}_3 \\
\end{matrix}\right . , \end{aligned}$$ where $\log^+ (x) = \max\{0, \log(x)\}$ and $$\begin{aligned}
\mathcal{D}_1&=\bigg\{
(D_1, D_2): (0\leq D_1 \leq 1-\rho^2, D_2 \geq 1-\rho^2 + \rho^2D_1) \text{ or } \nonumber \\
&\qquad \qquad \qquad \qquad \Big(1-\rho^2 < D_1 \leq 1, D_2 \geq 1-\rho^2 + \rho^2 D_1, D_2 \leq \frac{D_1-(1-\rho^2)}{\rho^2}\Big)
\bigg\} \\
\mathcal{D}_2&=\bigg\{
(D_1, D_2): 0\leq D_1 \leq 1-\rho^2, 0\leq D_2 < (1-\rho^2-D_1) \frac{1}{1-D_1}
\bigg\} \\
\mathcal{D}_3&=\bigg\{
(D_1, D_2): \Big(0\leq D_1 \leq 1-\rho^2, (1-\rho^2-D_1) \frac{1}{1-D_1} \leq D_2 < 1-\rho^2 + \rho^2 D_1 \Big) \text{ or } \nonumber \\
&\qquad \qquad \qquad \qquad \qquad \Big(
1-\rho^2 < D_1 \leq 1, \frac{D_1-(1-\rho^2)}{\rho^2} < D_2 < 1-\rho^2 + \rho^2 D_1\Big)
\bigg\} \end{aligned}$$ Fig. \[Fig:1-regions\] illustrates the regions $\mathcal{D}_1$, $\mathcal{D}_2$, and $\mathcal{D}_3$ as in [@lapidoth2010sending].
![Regions $\mathcal{D}_1$, $\mathcal{D}_2$, and $\mathcal{D}_3$.[]{data-label="Fig:1-regions"}](regions.eps){width="0.4\linewidth"}
![Comparison of the necessary conditions from Corollary \[cor6\] with Lapidoth-Tinguely (LT) and Lapidoth-Wigger (LW) necessary conditions from and -, respectively, for $\rho=0.2$, $P=10$, and $0.05 \leq D_2\leq 1$.[]{data-label="FigRho02P10"}](FigRho02P10v2v15test4.eps){width="0.7\linewidth"}
![Comparison of the necessary conditions from Corollary \[cor6\] with Lapidoth-Tinguely (LT) and Lapidoth-Wigger (LW) necessary conditions from and -, for $\rho=0.5$, $P=10$, and $0.05 \leq D_2\leq 1$.[]{data-label="FigRho05P10"}](FigRho05P10v15testv3.eps){width="0.7\linewidth"}
In Fig. \[FigRho02P10\] and \[FigRho05P10\], we compare the necessary conditions from Corollary \[cor6\], , and - for the Gaussian example. To do so, we let the horizontal axis refer to $D_2$ and the vertical axis refer to $D_1$. Then, for each $D_2$ value, we evaluate and plot the minimum $D_1$ value that satisfies the corresponding necessary conditions, numerically. Accordingly, these values correspond to a lower bound on the achievable $D_1$ values for a given $D_2$. The higher the corresponding curve, the tighter the lower bound. We refer to the condition from as LT, and the condition from - as LW [^3].
Fig. \[FigRho02P10\] corresponds to a Gaussian example with $\rho=0.2$ and $P=10$. Fig. \[FigRho05P10\], on the other hand, corresponds to a Gaussian example for which $\rho=0.5$ and $P=10$. Fig. \[FigRho02P10\] and Fig. \[FigRho05P10\] suggest that the necessary condition from Corollary \[cor6\] provides the tightest bound in various regions, particularly, for large values of $D_2$.
In the following, we show that there exist $(D_1, D_2)$ values for which Corollary \[cor6\] gives the tightest bound, by analyzing the corresponding expressions from Corollary \[cor6\], , and -. To do so, we define $$\begin{aligned}
r_1 (\beta_1, \beta_2) &\triangleq \frac{1}{2} \log ( 1 + 2P + 2P \sqrt{(1-\beta_1)(1-\beta_2)} ),
\\
r_2 (\beta_1, \beta_2) &\triangleq \frac{1}{2} \log ( 1 + \beta_1 P + \beta_2 P ), \end{aligned}$$ and consider the region $$\label{region}
\mathcal{R} = \bigcup_{0\leq \beta_1, \beta_2\leq 1} \left \{(R_1, R_2): R_1\leq r_1(\beta_1, \beta_2), R_2\leq r_2(\beta_1, \beta_2) \right \}.$$ Then, the necessary conditions in Corollary \[cor6\] state that, if a distortion pair is achievable, then $$\label{checkGGY}
\left (R_{S_1 S_2} (D_1, D_2), R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2) \right ) \in \mathcal{R}.$$ The necessary conditions from -, on the other hand, state that, if a distortion pair $(D_1, D_2)$ is achievable, then $$\label{checkLW}
\left(R_{S_1 S_2} (D_1, D_2), R_{S_1 S_2} (D_1, D_2) - \frac{1}{2} \log \frac{1+\rho}{1-\rho} \right) \in \mathcal{R}.$$
![Partitioned distortion regions for $(D_1, D_2)$.[]{data-label="regions2"}](regions4.eps){width="0.4\linewidth"}
In the following, we let $\rho = 0.5$ and $P=2$. We partition the set of all distortion pairs $(D_1, D_2)$, $0\leq D_1, D_2\leq 1$, as in Fig. \[regions2\]. Consider first Region $\mathcal{B}$, for which $D_1 \leq 1-\rho$ and $1-\rho \leq D_2 \leq \frac{1-\rho^2-D_1}{1-D_1}$. We let $D_1 = 0.14 < 1-\rho$. For a $(D_1, D_2)$ pair in Region $\mathcal{B}$, i.e., $D_1=0.14$ and $1-\rho\leq D_2\leq \frac{1-\rho^2-D_1}{1-D_1}$, from and we have $$\label{pointDDYcheck}
\left ( R_{S_1 S_2} (D_1, D_2) , R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2)\right ) = \left (\frac{1}{2}\log \frac{1-\rho^2}{D_1D_2}, \frac{1}{2}\log \frac{1-\rho}{D_1} \right ).$$ The points obtained for increasing $D_2$ values within Region $\mathcal{B}$ are illustrated with a green “+” sign in Fig. \[fig:p06\]. On the other hand, the region $\mathcal{R}$ from is the region shaded in blue in the same figure. Whenever a green point from $\eqref{pointDDYcheck}$ falls outside the blue region $\mathcal{R}$, then we conclude that the corresponding distortion pair $(D_1, D_2)$ is not achievable, according to Corollary \[cor6\]. We also evaluate $$\label{pointLWheck}
\left(R_{S_1 S_2} (D_1, D_2), R_{S_1 S_2} (D_1, D_2) - \frac{1}{2} \log \frac{1+\rho}{1-\rho} \right) = \left (\frac{1}{2}\log \frac{1-\rho^2}{D_1D_2} , \frac{1}{2}\log \frac{(1-\rho)^2}{D_1D_2} \right )$$ for points $(0.14, D_2)$ in Region $\mathcal{B}$, using . The points for different $D_2$ values are illustrated with a dark blue “\*” marking in Fig. \[fig:p06\]. Accordingly, whenever such a point from $\eqref{pointLWheck}$ is not contained within $\mathcal{R}$ from , then the corresponding distortion pair $(D_1, D_2)$ is not achievable, according to the LW conditions from -.
![Comparison of the necessary conditions from Corollary \[cor6\] with LT and LW necessary conditions from and -, respectively, for $P=2$, $\rho=0.5$, and $D_1=0.14$.[]{data-label="fig:p06"}](D114v4lb.eps){width="0.62\linewidth"}
![Comparison of the necessary conditions from Corollary \[cor6\] with LT and LW necessary conditions from and -, respectively, for $P=2$, $\rho=0.5$, and $D_1=0.145$.[]{data-label="fig:p065"}](D1145v4lb.eps){width="0.62\linewidth"}
![Comparison of the necessary conditions from Corollary \[cor6\] with LT and LW necessary conditions from and -, respectively, for $P=2$, $\rho=0.5$, and $D_1=0.15$.[]{data-label="fig:p10"}](D115v4lb.eps){width="0.62\linewidth"}
![Comparison of the necessary conditions from Corollary \[cor6\] with LT and LW necessary conditions from and -, respectively, for $P=2$, $\rho=0.5$, and $D_1=0.16$.[]{data-label="fig:p16"}](D116v4lb.eps){width="0.62\linewidth"}
Next, we consider $(D_1, D_2)$ pairs from Region $\mathcal{D}$, for which $D_1 \leq 1-\rho$ and $\frac{1-\rho^2-D_1}{1-D_1}\leq D_2 \leq 1-\rho^2+\rho^2D_1$. We can evaluate $$\begin{aligned}
\label{pointDDYcheck1}
&\left (R_{S_1 S_2} (D_1, D_2), R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2) \right ) \notag \\
&\qquad \qquad = \left (\frac{1}{2} \log^+ \Bigg( \frac{1-\rho^2}{D_1 D_2 - \left(\rho - \sqrt{(1-D_1)(1-D_2)}\right)^2}\Bigg), \frac{1}{2}\log \frac{1-\rho}{D_1} \right )\end{aligned}$$ using and . The values obtained for $D_1=0.14$ and $D_2$ going from $\frac{1-\rho^2-D_1}{1-D_1}$ to $1-\rho^2+\rho^2D_1$ are illustrated with a purple “+” sign in Fig. \[fig:p06\]. Similarly, from , for $(D_1, D_2) \in \text{Region }\mathcal{D}$, $$\begin{aligned}
\label{pointLWheck1}
&\left(R_{S_1 S_2} (D_1, D_2), R_{S_1 S_2} (D_1, D_2) - \frac{1}{2} \log \frac{1+\rho}{1-\rho} \right) \notag \\
& = \!\left (\!
\frac{1}{2} \log^+\!\! \Bigg( \frac{1\!-\!\rho^2}{D_1 D_2 \!-\! \left(\!\rho \!-\!\! \sqrt{(1\!-\!D_1)(1\!-\!D_2)}\right)^2}\Bigg),
\frac{1}{2} \log^+\!\! \Bigg( \frac{(1\!-\!\rho)^2}{D_1 D_2 \!-\! \left(\!\rho \!-\! \sqrt{(1\!-\!D_1)(1\!-\!D_2)}\right)^2}\!\!\Bigg)\!\!\!
\right ) \!. \end{aligned}$$ Corresponding points for $D=0.14$ and increasing $D_2$ values in Region $\mathcal{D}$ are illustrated with a red “x” marking in Fig. \[fig:p06\].
Finally, we consider Region $\mathcal{G}$, where $D_1 \leq 1-\rho$ and $ 1-\rho^2+\rho^2D_1\leq D_2\leq 1$. For $(D_1, D_2)\in \text{Region } \mathcal{G}$, we have $$\label{pointDDYcheck12}
\left (R_{S_1 S_2} (D_1, D_2), R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2) \right )
= \left(\frac{1}{2} \log \frac{1}{D_1}, \frac{1}{2} \log \frac{1-\rho}{D_1}\right ).$$ Corresponding points are illustrated with the pink “+” sign in Fig. \[fig:p06\]. Note that since depends only on $D_1$, these points coincide and appear as a single point in Fig. \[fig:p06\]. We also evaluate $$\label{pointLWheck12}
\left(R_{S_1 S_2} (D_1, D_2), R_{S_1 S_2} (D_1, D_2) - \frac{1}{2} \log \frac{1+\rho}{1-\rho} \right) =
\left(\frac{1}{2} \log \frac{1}{D_1}, \frac{1}{2} \log \frac{1-\rho}{D_1(1+\rho)}\right)$$ for $1-\rho^2+\rho^2D_1\leq D_2\leq 1$ using . This is illustrated with a black “\*” marking in Fig. \[fig:p06\]. Since also depends only on $D_1$, these points coincide and appear as a single point.
One can observe from -, as well as from - and -, that the points that share the same value on the horizontal axis in Fig. \[fig:p06\] correspond to the same $(D_1, D_2)$ pairs, as the first terms of both - and - as well as - are equal.
Lastly, we illustrate the right-hand side (RHS) of with a straight line in Fig. \[fig:p06\]. It can be observed from that the points on the RHS of this line correspond to distortion pairs $(D_1, D_2)$ that are not achievable, based on the necessary condition in , since for these points one has $$R_{S_1 S_2}( D_1, D_2) > \frac{1}{2} \log (1 + 2P(1+\rho)).$$ In Figs. \[fig:p065\], \[fig:p10\], and \[fig:p16\], we illustrate the corresponding regions for $D_1=0.145$, $D_1=0.15$, and $D_1=0.16$, respectively, by keeping the remaining parameters fixed.
In order to compare the necessary conditions in Corollary \[cor6\] with LT and LW necessary conditions, we investigate the distortion pairs $(D_1, D_2)$ that cannot be achieved by Corollary \[cor6\], , and -, respectively, in Figs. \[fig:p06\], \[fig:p065\], \[fig:p10\], and \[fig:p16\]. From Fig. \[fig:p06\] we observe that when $P=0.14$, for all the $(D_1, D_2)$ pairs considered, Corollary \[cor6\] and the LT bound from both state that none of these distortion pairs is achievable, whereas some distortion pairs satisfy the necessary conditions for the LW bound from -, as several points labeled with a “x” marking fall into the blue region $\mathcal{R}$. From Fig. \[fig:p065\], we find that when $D_1=0.145$, some $(D_1, D_2)$ pairs from Regions $\mathcal{G}$ and $\mathcal{D}$ (from Fig. \[regions2\]) satisfy both LT and LW conditions from and -, but not Corollary \[cor6\], as can be observed from the pink and purple points marked with the “+” sign that are on the left-hand side (LHS) of the straight line, but outside the blue region $\mathcal{R}$. Fig. \[fig:p10\] demonstrates similar results for $D_1=0.15$. Therefore, we can conclude that there exist distortion pairs for which Corollary \[cor6\] provides tighter conditions than both the LT and LW bounds in Regions $\mathcal{G}$ and $\mathcal{D}$. Similarly, from Fig. \[fig:p10\], we find that for $D_1=0.16$, there exist $(D_1, D_2)$ pairs in Region $\mathcal{B}$ that satisfy the necessary conditions from LT and LW but not from Corollary \[cor6\], which can be observed from the green points marked with the “+” sign that are on the LHS of the straight line but are outside the blue region $\mathcal{R}$. Hence, for Region $\mathcal{B}$ we also observe that there exist distortion pairs for which Corollary \[cor6\] provides tighter conditions than both the LT bound from and the LW necessary conditions from -.
In the following, we compare Corollary \[cor6\] with the LW conditions from - by investigating the LHS of both conditions for various regions in Fig. \[regions2\], as the region defined by the RHS of both - and - are the same.
For $(D_1, D_2)\in \mathcal{A}$, we observe from and that, $$\begin{aligned}
R_{S_1 S_2} (D_1, D_2) - C_W(S_1, S_2)
&= \frac{1}{2} \log \left( \frac{1-\rho^2}{D_1 D_2}\right ) -
\frac{1}{2} \log \left( \frac{1+\rho}{1-\rho}\right ) \\
& = \frac{1}{2} \log \frac{(1-\rho)^2}{D_1 D_2} \nonumber \\
&= R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2), \end{aligned}$$ hence, in this region, Corollary \[cor6\] and the LW bound are equivalent.
For $(D_1, D_2)\in \mathcal{B}$, we find from and that, $$\begin{aligned}
R_{S_1 S_2} (D_1, D_2) - C_W(S_1, S_2)
&=\frac{1}{2} \log \left( \frac{1-\rho^2}{D_1 D_2}\right ) - \frac{1}{2} \log \left( \frac{1+\rho}{1-\rho}\right ) \\
& = \frac{1}{2} \log \frac{(1-\rho)^2}{D_1 D_2} \\
&\leq \frac{1}{2} \log \frac{1-\rho}{D_1} \\
&=R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2),\end{aligned}$$ since $D_1\leq 1-\rho$ and $D_2\geq 1-\rho$ for $(D_1, D_2)\in \mathcal{B}$. Hence, in this region, Corollary \[cor6\] is at least as tight as the LW bound. By swapping the roles of $D_1$ and $D_2$, we can extend the same argument to Region $\mathcal{C}$ as well.
For $(D_1, D_2)\in \mathcal{D}$, we have from and that, $$\begin{aligned}
\label{eqcompare2}
R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2) = \frac{1}{2} \log \frac{1-\rho}{D_1}, \end{aligned}$$ whereas $$\begin{aligned}
&R_{S_1 S_2} (D_1, D_2)- C_W(S_1, S_2)\notag \\
&\qquad=\frac{1}{2} \max \Bigg\{ \log \frac{1-\rho}{1+\rho}, \log \frac{(1-\rho)^2}{D_1 D_2 - \left(\rho - \sqrt{(1-D_1)(1-D_2)}\right)^2}\Bigg \} \label{obs}\\
&\qquad=\frac{1}{2} \max \left \{ \log \frac{1-\rho}{1+\rho}, \log \frac{(1-\rho)^2}{D_1 + D_2 - (1+\rho^2) + 2\rho \sqrt{(1-D_1)(1-D_2)}} \right \} \\
&\qquad=\frac{1}{2} \log \frac{(1-\rho)^2}{D_1 + D_2 - (1+\rho^2) + 2\rho \sqrt{(1-D_1)(1-D_2)}}
\label{general1}\end{aligned}$$ where the last equation follows from $$\begin{aligned}
{(2-D_1-D_2)}^2 - 4\rho^2 (1-D_1)(1-D_2) \label{2d1d2}
& = (1-\rho^2)(2-D_1-D_2)^2 + \rho^2(D_1-D_2)^2\\
&\geq 0 \label{3d1d2}\end{aligned}$$ and therefore, $$D_1 + D_2 - (1+\rho^2)+ 2\rho \sqrt{(1-D_1)(1-D_2)}\leq 1-\rho^2 .$$ Then, by comparing with , we find that, Corollary \[cor6\] provides necessary conditions at least as tight as the LW bound if $$\rho \in \left\{\rho: \tau - \sqrt{D_2-1+\tau^2} \leq \rho \leq \tau + \sqrt{D_2-1+\tau^2}, \quad D_2+\tau^2\geq 1\right\}$$ where $$\tau = \frac{D_1}{2} + \sqrt{(1-D_1)(1-D_2)}.$$ By symmetry, it then follows for region $(D_1, D_2)\in \mathcal{F}$ that, Corollary \[cor6\] is at least as tight as the LW bound if $$\rho \in \left\{\rho: \lambda - \sqrt{D_1-1+\lambda^2} \leq \rho \leq \lambda + \sqrt{D_1-1+\lambda^2}, \quad D_1+\tau^2\geq 1 \right\},$$ where $$\lambda = \frac{D_2}{2} + \sqrt{(1-D_1)(1-D_2)}.$$
For $(D_1, D_2)\in \mathcal{G}$, we observe from and that, $$\begin{aligned}
R_{S_1 S_2} (D_1, D_2)- C_W(S_1, S_2)&=
\frac{1}{2} \log \left ( \frac{1}{D_1}\right ) - \frac{1}{2} \log \left( \frac{1+\rho}{1-\rho}\right ) \\
&=\frac{1}{2} \log \frac{1-\rho}{D_1(1+\rho)} \\
&\leq \frac{1}{2} \log \frac{1-\rho}{D_1} \\
&=R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2).\end{aligned}$$ Therefore, Corollary \[cor6\] is again at least as tight as the LW bound. It follows by symmetry that Corollary \[cor6\] is at least as tight as the LW bound in Region $\mathcal{I}$ as well.
For $(D_1, D_2)\in \mathcal{H}$, we have from and that, $$\begin{aligned}
R_{S_1 S_2} (D_1, D_2)- C_W(S_1, S_2)&=
\frac{1}{2}
\log \left( \frac{1}{\min(D_1, D_2)}\right )
- \frac{1}{2} \log \left( \frac{1+\rho}{1-\rho}\right ) \\
&=\frac{1}{2}
\log \frac{1-\rho}{\min(D_1, D_2) (1+\rho)} \\
&\leq 0 \label{triv}\\
&=R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2) \end{aligned}$$ since $\min(D_1, D_2)\geq 1-\rho$ when $(D_1, D_2)\in H$. From , we find that both condition and condition are trivially satisfied in this region, and therefore Corollary \[cor6\] and the LW bound are equivalent. Same conclusion follows for Region $\mathcal{J}$.
For region $(D_1, D_2)\in \mathcal{E}$, we have from and that, $$\begin{aligned}
\label{eqcompare}
R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2) = 0,\end{aligned}$$ hence, condition is trivially satisfied, whereas $R_{S_1 S_2} (D_1, D_2)- C_W(S_1, S_2)$ is as given in and .
If $D_1=D_2$, we have from and $D_1\geq 1-\rho$ that, $$\begin{aligned}
R_{S_1 S_2} (D_1, D_2)- C_W(S_1, S_2)&=\frac{1}{2} \max \left \{ \log \frac{1-\rho}{1+\rho}, \log \frac{(1-\rho)^2}{D_1^2 - \left(\rho - (1-D_1)\right)^2} \right \} \\
&=\frac{1}{2} \max \left \{ \log \frac{1-\rho}{1+\rho}, \log \frac{(1-\rho)^2}{(1-\rho)(2D_1 - (1-\rho))}\right \} \\
& \leq 0\\
&=R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2),\end{aligned}$$ and is also trivially satisfied. Hence, for all $D_1=D_2$ in Region $\mathcal{E}$, Corollary \[cor6\] and the LW bound perform the same.
We next consider the case when $\rho\leq 0.5$ for $(D_1, D_2)\in E$. Without loss of generality, we assume that $D_1 \geq D_2$. Noting that $D_2\geq 1-\rho$, we have $$\begin{aligned}
D_1 + D_2 - (1+\rho^2) + 2\rho \sqrt{(1-D_1)(1-D_2)}
&\geq D_1 + D_2 - (1+\rho^2) + 2\rho (1-D_1) \\
& = D_1 (1-2\rho) + D_2 - (1-\rho)^2\\
& \geq D_2 (1-2\rho) + D_2 - (1-\rho)^2\\
& = (1-\rho) (2D_2 - (1-\rho))\\
& \geq (1-\rho)^2 \label{compfirst}\end{aligned}$$ from which, along with and , we find that $$\begin{aligned}
R_{S_1 S_2} (D_1, D_2)- C_W(S_1, S_2) \leq 0 = R_{S_1|Z} (D_1) + R_{S_2|Z} (D_2). \label{complast}\end{aligned}$$ Therefore, for all $\rho\leq 0.5$, Corollary \[cor6\] and the LW bound perform the same. By comparing with , in general we can show that, Corollary \[cor6\] and the LW bound perform the same if $$\rho \in \left\{\rho: \Delta - \sqrt{\frac{D_1+D_2}{2} -1+ \Delta^2} \leq \rho \leq \Delta + \sqrt{\frac{D_1+D_2}{2} -1+ \Delta^2}, \;\frac{D_1+D_2}{2} + \Delta^2 \geq 1 \right\}$$ where $$\Delta = \frac{1+\sqrt{(1-D_1)(1-D_2)}}{2}$$ We can therefore state that the necessary conditions from Corollary \[cor6\] are at least as tight as those suggested by the LW bound in all regions but Region $\mathcal{E}$, Region $\mathcal{D}$, and Region $\mathcal{F}$. We remark that this does not necessarily mean that Corollary \[cor6\] is strictly tighter in any of these regions, since the necessary conditions involve also the RHS of - and -, and they can be used to claim the impossibility of achieving certain distortion pairs based on the relative value of the rate distortion functions with respect to the rate region characterized by the RHS. It is possible that, even though the LHS of Corollary \[cor6\] is lower than the LHS of the LW bound, either both or none of the necessary conditions may be satisfied, leading exactly to the same conclusion regarding the achievability of the corresponding distortion pair. On the other hand, while we have shown numerically cases where Corollary \[cor6\] provides strictly tighter bounds than the LW conditions, we have not come across any case in which the opposite holds, that is, the LW conditions show that a certain distortion pair is not achievable, while the necessary conditions of Corollary \[cor6\] hold.
Conclusion {#Sec:Conc}
==========
We have considered lossy transmission of correlated sources over a multiple access channel. We have investigated the conditions under which separate source and channel coding is optimal when the encoder and/or decoder has access to side information. In addition, we have derived necessary conditions for the lossy transmission of correlated sources over a multiple access channel. We have shown that this technique provides in certain cases the tightest known necessary conditions for the Gaussian setting, i.e., Gaussian sources transmitted over a Gaussian MAC. Current and future directions include identifying other multiple access scenarios for which separation is (sub)optimal as well as the same for other multi-terminal scenarios with side information.
Proof of Theorem \[lemma:hybrid\] {#appendix0}
=================================
Our achievable scheme is along the lines of [@minero2015unified]. For completeness, we provide the details in the sequel.
*Generation of the codebook:* Choose $\epsilon > \epsilon' > 0$. Fix $p(u_1|s_1)$, $p(u_2|s_2)$, $x_1(u_1, s_1)$, $x_2(u_2, s_2)$, $\hat{s}_1(u_1, u_2, y, z)$ and $\hat{s}_2(u_1, u_2, y, z)$ with $\mathbb{E}[d_j(S_j, \hat{S}_j)]\leq \frac{D_j}{1+\epsilon}$ for $j=1,2$.
For each $j=1,2$, generate $2^{nR_j}$ sequences $u_j^n(m_j)$ for $m_j\in \{1,\ldots, 2^{nR_j}\}$ independently at random conditioned on the distribution $\prod_{i=1}^n p_{U_j}(u_{ji})$. The codebook is known by the two encoders and the decoder.
*Encoding:* Encoder $j=1,2$ observes a sequence $s_j^n$ and tries to find an index $m_j\in \{1, \ldots, 2^{nR_j}\}$ such that the corresponding $u_j^n(m_j)$ is jointly typical with $s_j^n$, i.e., $(s_j^n, u_j^n(m_j))\in \mathcal{T}_{\epsilon'}^{(n)}$. If more than one index exist, the encoder selects one of them uniformly at random. If no such index exists, it selects a random index uniformly. Upon selecting the index, encoder $j$ sends $x_{ji}=x_j (u_{ji}(m_j), s_{ji})$ for $i=1,\ldots, n$ to the decoder.
*Decoding:* The decoder observes the channel output $y^n$ and side information $z^n$, and tries to find a unique pair of indices $(\hat{m}_1, \hat{m}_2)$ such that $(u_1^n(\hat{m}_1), u_2^n(\hat{m}_2), y^n, z^n)\in \mathcal{T}_{\epsilon}^{(n)}$ and sets $\hat{s}_{ji}=\hat{s}_j(u_{1i}(m_1), u_{2i}(m_2), y_i, z_i)$ for $i=1, \ldots, n$ for $j=1, 2$.
*Expected Distortion Analysis:* Let $M_1$ and $M_2$ denote the indices selected by encoder $1$ and encoder $2$. Define $$\mathcal{E} \{(S_1^n, S_2^n, U_1^n(\hat{M}_1), U_2^n(\hat{M}_2), Y^n, Z^n)\notin \mathcal{T}_{\epsilon}^{(n)}\}$$ such that the distortion pair $(D_1, D_2)$ is satisfied if $P(\mathcal{E})\rightarrow 0$ as $n\rightarrow\infty$. Let $$\begin{aligned}
\mathcal{E}_j& = \{(S_j^n, U_j^n(m_j))\notin \mathcal{T}_{\epsilon'}^{(n)} \;\;\forall m_j\} , \quad j=1,2\\
\mathcal{E}_3& = \{(S_1^n, S_2^n, U_1^n(M_1), U_2^n(M_2), Y^n, Z^n)\notin \mathcal{T}_{\epsilon}^{(n)}\} \\
\mathcal{E}_4& = \{(U_1^n(m_1), U_2^n(m_2), Y^n, Z^n)\in \mathcal{T}_{\epsilon}^{(n)} \text{ for some } m_1\neq M_1, m_2\neq M_2\} \\
\mathcal{E}_5& = \{(U_1^n(m_1), U_2^n(M_2), Y^n , Z^n) \in \mathcal{T}_{\epsilon}^{(n)} \text{ for some } m_1 \neq M_1\} \label{error25}\\
\mathcal{E}_6& = \{(U_1^n(M_1), U_2^n(m_2), Y^n , Z^n) \in \mathcal{T}_{\epsilon}^{(n)} \text{ for some } m_2 \neq M_2\} \end{aligned}$$ Then, $$\begin{aligned}
&P(\mathcal{E})\leq P(\mathcal{E}_1) + P(\mathcal{E}_2) + P(\mathcal{E}_3 \cap \mathcal{E}_1^c \cap \mathcal{E}_2^c) + P(\mathcal{E}_4) + P(\mathcal{E}_5) + P(\mathcal{E}_6).\end{aligned}$$
We have for $j=1,2$ that $P(\mathcal{E}_j)\rightarrow 0$ as $n\rightarrow \infty$ if $$R_j> I(U_j; S_j) + \delta(\epsilon'), \label{Rate1}$$ from the covering lemma [@el2011network].
We have from the Markov lemma [@el2011network Section $12.1.1$] that $P(\mathcal{E}_3)\rightarrow 0$ as $n\rightarrow \infty$. In particular, the result follows from applying Markov lemma to: i) $U_2-S_2-S_1Z$, ii) $U_1-S_1-S_2U_2Z$, iii) $Y-U_1U_2S_1S_2-Z$ in the given order.
We next define the event $\mathcal{M}=\{M_1=1, M_2=1\}$. From the symmetry of the codebook and the encoding procedure, we have that $$\begin{aligned}
P(\mathcal{E}_4)& = P(\mathcal{E}_4|\mathcal{M})\nonumber \\
& \leq \sum_{m_1=2}^{2^{nR_1}}
\sum_{m_1=2}^{2^{nR_2}} \sum_{(u_1^n, u_1^n, y^n, z^n)\in \mathcal{T}_{\epsilon}^{(n)}} \hspace{-0.5cm}P\{U_1^n(m_1) = u_1^n, U_2^n(m_2) = u_2^n, Y^n = y^n, Z^n = z^n|\mathcal{M}\} \label{eqthen}\end{aligned}$$ where is from the union bound. By denoting $\tilde{U}^n=(U_1^n(1), U_2^n(1), S_1^n, S_2^n)$ and $\tilde{u}=(\tilde{u}_1^n, \tilde{u}_2^n, s_1^n, s_2^n)$, we observe the following. $$\begin{aligned}
&P\{U_1^n(m_1)=u_1^n, U_2^n(m_2)=u_2^n, Y^n=y^n, Z^n= z^n|\mathcal{M}\} \notag \\
&=\sum_{\tilde{u}^n} P\{U_1^n(m_1)=u_1^n, U_2^n(m_2)=u_2^n, Y^n=y^n, Z^n= z^n, \tilde{U}^n=\tilde{u}^n | \mathcal{M}\} \\
&=\sum_{\tilde{u}^n}
P\{\tilde{U}^n=\tilde{u}^n| Y^n=y^n, Z^n= z^n , \mathcal{M}\} \notag\\
&\quad P\{U_1^n(m_1) = u_1^n , U_2^n(m_2) = u_2^n| Y^n = y^n , Z^n = z^n , \tilde{U}^n = \tilde{u}^n , \mathcal{M}\} P\{Y^n=y^n, Z^n= z^n | \mathcal{M}\} \label{e4lasteq}\end{aligned}$$ Given $\mathcal{M}$, we have for all $m_1\neq 1$ and $m_2\neq 1$, $U_1^n(m_1) U_2^n(m_2)- U_1^n(1) U_2^n(1)S_1^n S_2^n-Y^nZ^n$. Then, is equal to $$\begin{aligned}
&\sum_{\tilde{u}^n} P\{U_1^n(m_1)=u_1^n | \mathcal{M}, \tilde{U}^n=\tilde{u}^n\}
P\{U_2^n(m_2)=u_2^n|\mathcal{M}, U_1^n(m_1)=u_1^n, \tilde{U}^n=\tilde{u}^n \}\notag\\
&\qquad \qquad P\{\tilde{U}^n = \tilde{u}^n| Y^n = y^n, Z^n= z^n , \mathcal{M}\}
P\{Y^n = y^n, Z^n = z^n | \mathcal{M}\} \\
&=\sum_{\tilde{u}^n} \!P\{U_1^n(m_1)\!=\!u_1^n | M_1\!=\!1, \!U_1^n(1)\!=\!\tilde{u}_1^n, \!S_1^n\!=\!s_1^n\}
P\{U_2^n(m_2)\!=\!u_2^n | M_2\!=\!1, \!U_2^n(1)\!=\!\tilde{u}_2^n, \!S_2^n\!=\!s_2^n\} \notag\\
&\qquad \qquad P\{\tilde{U}^n=\tilde{u}^n| Y^n = y^n, Z^n = z^n , \mathcal{M}\}
P\{Y^n = y^n, Z^n = z^n | \mathcal{M}\} \label{eqb}\\
& \leq (1\!+\!\epsilon) \sum_{\tilde{u}^n} \Big (\prod_{i=1}^{n} p_{U_1}(u_{1i}) p_{U_2}(u_{2i})\Big )
P\{\tilde{U}^n \!=\! \tilde{u}^n| Y^n \!=\! y^n, Z^n \!=\! z^n , \mathcal{M}\}
P\{Y^n\!=y^n, Z^n \!=\! z^n | \mathcal{M}\}
\label{eqc}\\
&= (1+\epsilon) \Big (\prod_{i=1}^{n} p_{U_1}(u_{1i})p_{U_2}(u_{2i})\Big ) P\{Y^n= y^n, Z^n = z^n|\mathcal{M}\} \label{eqfollow}\end{aligned}$$ where follows from the independent encoding protocol and follows from Lemma $1$ in [@minero2015unified] by setting $U\leftarrow U_1$, $S\leftarrow S_1$, and $M\leftarrow M_1$ and then setting $U\leftarrow U_2$, $S\leftarrow S_2$, and $M\leftarrow M_2$.
Using , we write as follows. $$\begin{aligned}
&P(\mathcal{E}_4) \notag \\
&\leq (1+\epsilon) \sum_{m_1=2}^{2^{nR_1}}
\sum_{m_1=2}^{2^{nR_2}} \sum_{(u_1^n, u_1^n, y^n, z^n)\in \mathcal{T}_{\epsilon}^{(n)}} p(u_1^n) p(u_2^n) P\{Y^n=y^n, Z^n=z^n|\mathcal{M}\} \\
&\leq (1+\epsilon) 2^{n(R_1+R_2)} \sum_{(y^n, z^n)\in \mathcal{T}_{\epsilon}^{(n)}}
P\{Y^n=y^n, Z^n=z^n|\mathcal{M}\}
\sum_{u_1\in \mathcal{T}_{\epsilon}^{(n)}(U_1|y^n, z^n)} \hspace{-0.7cm}p(u_1^n)
\sum_{u_2^n\in \mathcal{T}_{\epsilon}^{(n)}(U_2|u_1^n, y^n, z^n)} \hspace{-0.7cm}p(u_2^n) \label{eqTdefine}\\
&\leq (1+\epsilon) 2^{n(R_1+R_2)} \sum_{(y^n, z^n)\in \mathcal{T}_{\epsilon}^{(n)}}
P\{Y^n=y^n, Z^n=z^n|\mathcal{M}\} \notag \\
&\hspace{5cm} \sum_{u_1\in \mathcal{T}_{\epsilon}^{(n)}(U_1|y^n, z^n)} p(u_1^n)
| \mathcal{T}_{\epsilon}^{(n)}(U_2|u_1^n, y^n, z^n)| 2^{-n(H(U_2) -\delta(\epsilon))}\\
&\leq (1+\epsilon) 2^{n(R_1+R_2)} \sum_{(y^n, z^n)\in \mathcal{T}_{\epsilon}^{(n)}}
P\{Y^n=y^n, Z^n=z^n|\mathcal{M}\} \notag \\
&\hspace{5cm} \sum_{u_1\in \mathcal{T}_{\epsilon}^{(n)}(U_1|y^n, z^n)} p(u_1^n)
2^{n(H(U_2|U_1, Y, Z)+\delta(\epsilon))} 2^{-n(H(U_2) -\delta(\epsilon))}\\
&\leq (1+\epsilon) 2^{n(R_1+R_2)} \sum_{(y^n, z^n)\in \mathcal{T}_{\epsilon}^{(n)}}
P\{Y^n=y^n, Z^n=z^n|\mathcal{M}\} \notag \\
&\hspace{5cm}2^{n(H(U_1|Y, Z)+\delta(\epsilon))} 2^{-n(H(U_1)-\delta(\epsilon)\!)}
2^{n(H(U_2|U_1, Y, Z)-H(U_2)+2\delta(\epsilon))} \\
&\leq (1+\epsilon) 2^{n(R_1+R_2)} 2^{n(H(U_1|Y, Z)+H(U_2|U_1, Y, Z)-H(U_1)-H(U_2)+4\delta(\epsilon))} \label{eqp4end}\end{aligned}$$ where in we define $$\mathcal{T}_{\epsilon}^{(n)} (U_1|y^n, z^n) =
\{u_1^n: (u_1^n, y^n, z^n)\in \mathcal{T}_{\epsilon}^{(n)}\}$$ and $$\mathcal{T}_{\epsilon}^{(n)} (U_2|u_1^n, y^n, z^n) =
\{u_2^n: (u_2^n, u_1^n, y^n, z^n)\in \mathcal{T}_{\epsilon}^{(n)}\}$$ as in [@el2011network Section $2.5.1$]. Since $$\begin{aligned}
H(U_1|Y, Z) &+ H(U_2|U_1, Y, Z) -H(U_1) - H(U_2) \notag \\
&= H(U_1, U_2|Y, Z) -H(U_1) -H(U_2) \\
&= H(U_1, U_2|Y, Z) -H(U_1) -H(U_2) + H(U_1, U_2) - H(U_1, U_2) \\
&= -(H(U_1, U_2)-H(U_1, U_2|Y, Z)) -H(U_1)-H(U_2) +H(U_1)+H(U_2|U_1) \\
& =- I(U_1, U_2 ; Y, Z) -I(U_1 ; U_2) \label{eqp4endI}\end{aligned}$$ we observe from and that, if $$R_1+R_2 < I(U_1, U_2; Y, Z) + I(U_1; U_2) -4 \delta (\epsilon) \label{rate4}$$ then, $P(\mathcal{E}_4)\rightarrow 0$ as $n\rightarrow \infty$.
Similarly, we observe for that $$\begin{aligned}
\!\!P(\mathcal{E}_5) &=P(\mathcal{E}_5|\mathcal{M}) \notag\\
&= P\{(U_1^n(m_1), U_2^n(1), Y^n, Z^n)\in \mathcal{T}_{\epsilon}^{(n)} \text{ for some } m_1\neq 1|\mathcal{M} \} \\
&\leq\sum_{m_1=2}^{2^{nR_1}}
P\{(U_1^n(m_1), U_2^n(1), Y^n, Z^n)\!\in \!\mathcal{T}_{\epsilon}^{(n)}|\mathcal{M} \} \\
&=\sum_{m_1=2}^{2^{nR_1}}
\sum_{(u_1^n, u_2^n, y^n, z^n)\in \mathcal{T}_{\epsilon}^{(n)}}
P\{U_1^n(m_1)=u_1^n, U_2^n(1)=u_2^n, Y^n=y^n, Z^n=z^n|\mathcal{M} \} \label{stepA}\end{aligned}$$ by letting $\tilde{U}^n=(U_1^n(1), S_1^n, S_2^n)$ and $\tilde{u}^n=(\tilde{u}_1^n, s_1^n, s_2^n)$, we find that $$\begin{aligned}
&P\{ U_1^n(m_1) = u_1^n, U_2^n(1)=u_2^n, Y^n=y^n, Z^n=z^n|\mathcal{M}\} \notag \\
&=\sum_{\tilde{u}^n} \{ U_1^n(m_1) = u_1^n, U_2^n(1)=u_2^n, Y^n=y^n, Z^n=z^n, \tilde{U}^n=\tilde{u}^n|\mathcal{M}\} \\
&=\sum_{\tilde{u}^n} P\{ U_1^n(m_1) = u_1^n| U_2^n(1)=u_2^n, \tilde{U}^n=\tilde{u}^n, \mathcal{M}\} \notag \\
&\quad \; \; \times P\{ \tilde{U}^n=\tilde{u}^n| U_2^n(1)=u_2^n, Y^n=y^n, Z^n=z^n, \mathcal{M}\} P\{U_2^n(1)=u_2^n, Y^n=y^n, Z^n=z^n| \mathcal{M}\} \label{neweqb} \\
&=\sum_{\tilde{u}^n} P\{ U_1^n(m_1) = u_1^n| M_1=1, U_1^n(1)=u_1^n, S_1^n=s_1^n\} \notag \\
&\quad \;\; \times P\{\tilde{U}^n=\tilde{u}^n|U_2^n(1)=u_2^n, Y^n=y^n, Z^n=z^n, \mathcal{M}\} P\{ U_2^n(1)=u_2^n, Y^n=y^n, Z^n=z^n | \mathcal{M}\} \label{neweqc} \\
&=(1+\epsilon) \sum_{\tilde{u}^n} \bigg( \prod_{i=1}^n p_{U_1}(u_{1i}) \bigg) \notag \\
&\quad \;\;\times P\{\tilde{U}^n=\tilde{u}^n|U_2^n(1)=u_2^n, Y^n=y^n, Z^n=z^n, \mathcal{M}\}
P\{U_2^n(1) = u_2^n, Y^n=y^n, Z^n=z^n| \mathcal{M}\} \label{neweqd} \\
&=(1+\epsilon) \Big ( \prod_{i=1}^n p_{U_1}(u_{1i}) \Big ) P\{U_2^n(1)=u_2^n, Y^n=y^n, Z^n=z^n | \mathcal{M} \} \label{neweqdlast}\end{aligned}$$ where is from $U_1^n(m_1)-U_2^n(1) U_1^n(1) S_1^n S_2^n \mathcal{M}-Y^nZ^n$, is from the independent encoding protocol, and is again from Lemma $1$ in [@minero2015unified].
Using , we can write as $$\begin{aligned}
&\!\!\!P(\mathcal{E}_5)\notag \\
&\!\!\!\!\leq (1+\epsilon) \sum_{m_1=2}^{2^{nR_1}} \sum_{(u_1^n, u_2^n, y^n, z^n)\in \mathcal{T}_{\epsilon}^{(n)}} p(u_1^n) P\{Y^n=y^n, Z^n=z^n, U_2^n(1)=u_2^n |\mathcal{M}\} \\
&\!\!\!\!=(1+\epsilon) \sum_{m_1=2}^{2^{nR_1}} \sum_{(y^n, z^n, u_2^n)\in \mathcal{T}_{\epsilon}^{(n)}} P\{Y^n=y^n, Z^n=z^n, U_2^n(1)=u_2^n|\mathcal{M}\}
\sum_{u_1^n\in \mathcal{T}_{\epsilon}^{(n)}(U_1|y^n, z^n, u_2^n)} \hspace{-0.8cm}p(u_1^n) \\
&\!\!\!\!\leq (1\!+\!\epsilon) \! \sum_{m_1=2}^{2^{nR_1}} \sum_{(y^n, z^n, u_2^n)\in \mathcal{T}_{\epsilon}^{(n)}} \hspace{-0.8cm}P\{Y^n=y^n, Z^n=z^n, U_2^n(1)=u_2^n|\mathcal{M}\} 2^{n(H(U_1|Y, Z, U_2)+\delta(\epsilon))} 2^{-n(H(U_1)-\delta(\epsilon))}\\
&\!\!\!\!=(1+\epsilon) 2^{nR_1} 2^{-n(I(U_1; Y, Z, U_2)-2\delta(\epsilon))}\end{aligned}$$ hence, $P(\mathcal{E}_5)\rightarrow 0$ as $n\rightarrow \infty$ if $$R_1< I(U_1; Y, Z, U_2) - 2\delta(\epsilon). \label{rateR1}
\vspace{-0.2cm}$$
From similar steps, we find that $P(\mathcal{E}_6)\rightarrow 0$ as $n\rightarrow \infty$ if $$R_2< I(U_2; Y, Z, U_1) - 2\delta(\epsilon). \label{rateR2S2}$$
Therefore, we conclude that $P(\mathcal{E})\rightarrow 0$ as $n\rightarrow \infty$ as $P(\mathcal{E}_i)\rightarrow 0$ for $i=1, \ldots, 6$ as $n\rightarrow \infty$. Lastly, from the typical average lemma [@el2011network Section $2.4$], we can bound the expected distortions for $\mathcal{E}^c$ for the two sources $S_1$ and $S_2$.
Lastly, we combine the conditions , , and to obtain - as follows.
Combining and , we can show that $$I(U_1; S_1)< I(U_1; Y, Z, U_2)$$ leads to , since $$\begin{aligned}
I(U_1; Y| Z, U_2)
&> I(U_1; S_1)- I(U_1; Z, U_2) \\
&= H(U_1)-H(U_1| S_1)- H(U_1) + H(U_1| Z, U_2) \\
&= H(U_1)-H(U_1| S_1, Z, U_2) - H(U_1) + H(U_1| Z, U_2) \label{U1S1ZU2}\\
&= I(U_1; S_1| U_2, Z)\end{aligned}$$ where is from $U_1-S_1-ZU_2$.
Similarly, by combining and , we obtain , $$I(U_2; S_2| U_1, Z) < I(U_2; Y| Z, U_1).$$
Lastly, by comparing and we obtain $$I(S_1; U_1) + I(S_2; U_2) < I(U_1, U_2; Y, Z) + I(U_1; U_2) \label{eqstate}$$ which leads to $$I(S_1; \!U_1) \!+\! I(S_2; \!U_2) \!-\! I(U_1, \!U_2; \!Z) \!-\!I(U_1; \!U_2) \!<\! I(U_1, \!U_2; \!Y |Z) \label{eqapp48}$$ where $$\begin{aligned}
I(S_1; U_1) + I(S_2; U_2) &- I(U_1, U_2; Z) -I(U_1; U_2) \\
& = H(U_1)-H(U_1|S_1) +H(U_2)-H(U_2|S_2) -H(U_1, U_2) \notag \\
& \qquad \quad + H(U_1, U_2|Z) - H(U_2) + H(U_2|U_1) + H(U_1) - H(U_1) \\
& =H(U_1, U_2|Z)- H(U_1|S_1) -H(U_2|S_2) \\
& =H(U_1, U_2|Z)- H(U_1|S_1, S_2) -H(U_2|S_2, S_1, U_1) \label{midtrm}\\
& =H(U_1, U_2|Z) -H(U_1, U_2|S_2, S_1) \\
& =H(U_1, U_2|Z) -H(U_1, U_2|S_2, S_1, Z) \label{lasttrm} \\
& =I(U_1, U_2;S_2, S_1| Z) \label{verylasttrm}\end{aligned}$$ is from $U_1-S_1-S_2$ and $U_2-S_2-S_1U_1$, is from $U_1U_2-S_1S_2-Z$.
Combining and , we conclude that yields , i.e., $$I(U_1, U_2;S_2, S_1| Z) < I(U_1, U_2; Y|Z).$$
Proof of Theorem \[lemma2\] {#appendixA}
===========================
Achievability
-------------
The source coding part is based on lossy source coding at the two encoders conditioned on the side information $Z$ shared between the encoder and decoder [@gray1972conditional], after which the conditional rate distortion functions given in can be achieved for $S_1$ and $S_2$, respectively. The channel coding part is performed based on classical multiple access channel coding with independent channel inputs [@cover2012elements].
Converse
--------
Suppose there exist encoding functions $e_j: \mathcal{S}_j^n\times \mathcal{Z}^n\rightarrow \mathcal{X}_j^n$ for encoder $j=1,2$, and decoding functions $g_j: \mathcal{Y}^n\times \mathcal{Z}^n\rightarrow \hat{\mathcal{S}}_j^n$ such that $\frac{1}{n}\sum_{i=1}^n E[d_j(S_{ji}, \hat{\mathcal{S}}_{ji} )]\leq D_j+\epsilon$ for $j=1,2$, where $\epsilon \rightarrow 0$ as $n\rightarrow \infty$.
Define $U_{ji}=(Y^n, S_j^{i-1}, Z_i^c)$ for $j=1,2$, where $Z_i^c=(Z_1, \ldots, Z_{i-1}, Z_{i+1}, \ldots, Z_n)$. Then, $$\begin{aligned}
\frac{1}{n} I(X_1^n;Y^n|X_2^n, Z^n)
&\geq \frac{1}{n} I(S_1^n; Y^n|X_2^n, Z^n) \label{eq2}\\
&= \frac{1}{n} I(S_1^n; Y^n, X_2^n| Z^n) \label{eq3}\\
&\geq \frac{1}{n} I(S_1^n; Y^n| Z^n) \label{eq4}\\
&= \frac{1}{n} \sum_{i=1}^{n} I(S_{1i}; Y^n|S_1^{i-1}, Z^n) \label{eq5}\\
&= \frac{1}{n} \sum_{i=1}^{n} (I(S_{1i}; Y^n, S_1^{i-1}, Z_i^c| Z_i) - I(S_{1i};S_{1}^{i-1}, Z_i^c|Z_i )) \label{eq6}\\
&=\frac{1}{n} \sum_{i=1}^n I(S_{1i}; U_{1i}|Z_i) \label{eq7}\\
&\geq\frac{1}{n} \sum_{i=1}^n R_{S_1|Z} (\mathcal{E}(S_{1i}|U_{1i}, Z_i)) \label{eq7v2} \\
&\geq \frac{1}{n} \sum_{i=1}^n R_{S_1|Z} (\mathcal{E}(S_{1i}|Z^n, Y^n)) \label{eq8}\\
&\geq \frac{1}{n} \sum_{i=1}^n R_{S_1|Z} (E[d_1(S_{1i}, \hat{S}_{1i})]) \label{eq9}\\
&\geq R_{S_1|Z} (D_1+\epsilon) \label{eq10}\end{aligned}$$ is from $Y^n-X_1^nX_2^n-S_1^nZ^n$ and conditioning cannot increase entropy, and is from $X_2^n-Z^n-S_1^n$ which holds since $$\begin{aligned}
p(x_2^n, s_1^n|z^n)&=\sum_{s_2^n} p(x_2^n, s_1^n, s_2^n|z^n) \\
&=\sum_{s_2^n} p(x_2^n| s_1^n, s_2^n,z^n) p(s_1^n|z^n) p(s_2^n|z^n)\label{eqstar0} \\
&=\sum_{s_2^n} p(x_2^n| s_2^n,z^n) p(s_1^n|z^n) p(s_2^n|z^n)\label{eqstar}\\
&=\sum_{s_2^n} p(x_2^n, s_2^n|z^n) p(s_1^n|z^n) \\
&=p(x_2^n|z^n) p(s_1^n|z^n) \end{aligned}$$ where is from $S_1^n-Z^n-S_2^n$ and is from $X_2^n-S_2^nZ^n-S_1^n$. Equation is from the nonnegativity of mutual information; is from the chain rule; is from the definition of $U_{1i}$ and the memoryless property of the sources and the side information. Equation is from and ; is from the fact that further conditioning cannot increase ; follows since $\hat{S}_{1i}$ is a function of $(Y^n, Z^n)$; holds as $R_{S_1|Z}(D_1)$ is convex and monotone in $D_1$.
By defining a discrete random variable $\tilde{Q}$ uniformly distributed over $\{1,\ldots, n\}$ independent of everything else, we find that $$\begin{aligned}
\frac{1}{n} I(X_1^n; Y^n|X_2^n, Z^n)
&\leq\frac{1}{n} \sum_{i=1}^n (H(Y_i|X_{2i}, Z^n) - H(Y_i|X_{1i}, X_{2i}, Z^n) ) \label{lemma2Ix1yn1} \\
&=\frac{1}{n} \sum_{i=1}^n I(X_{1i};Y_i|X_{2i}, \tilde{Q}=i, Z^n) \\
&=I(X_{1\tilde{Q}};Y_{\tilde{Q}}|X_{2\tilde{Q}}, \tilde{Q}, Z^n) \\
&= I(X_1;Y|X_2, Q) \label{lemma2Ix1yn5}\end{aligned}$$ where we let $X_1=X_{1\tilde{Q}}$, $X_2=X_{2\tilde{Q}}$, $Y=Y_{\tilde{Q}}$ and $Q=(\tilde{Q}, Z^n)$. Combining , , and with yields .
Following similar steps we obtain , $$\label{eqsource2}
R_{S_2|Z} (D_2+\epsilon) \leq I(X_2;Y|X_1, Q).$$
Lastly, we have $$\begin{aligned}
\frac{1}{n} I(X_1^n, X_2^n;Y^n|Z^n)
&=\frac{1}{n} I (X_1^n;Y^n|X_2^n, Z^n)+ \frac{1}{n}I(X_2^n;Y^n|Z^n) \label{eqsumrate2}\\
&\geq R_{S_1|Z} (D_1+\epsilon) +\frac{1}{n} I(S_2^n;Y^n|Z^n)\label{eqsumrate3}\\
&\geq R_{S_1|Z} (D_1+\epsilon) + R_{S_2|Z} (D_2+\epsilon)\label{eqsumrate21}\end{aligned}$$ where the first term in is from -, and follows from the same steps in - with the role of $S_1$ changed with $S_2$. To obtain the second term in , we first show that $Y^n-Z^nX_2^n-S_2^n$: $$\begin{aligned}
p(y^n, s_2^n|z^n, x_2^n)
&= p(s_2^n|z^n, x_2^n) p(y^n|s_2^n, z^n, x_2^n) \label{Markov1}\\
&= p(s_2^n|z^n, x_2^n) \sum_{s_1^n, x_1^n} p(y^n, s_1^n, x_1^n|s_2^n, z^n, x_2^n) \label{Markov2}\\
&= p(s_2^n|z^n, x_2^n) \sum_{s_1^n, x_1^n} p(y^n|x_1^n, x_2^n, s_1^n, s_2^n, z^n) \notag \\
& \qquad \qquad \qquad p(x_1^n|s_1^n, s_2^n, z^n, x_2^n) p(s_1^n|s_2^n, z^n, x_2^n) \label{Markov3}\\
&=p(s_2^n|z^n, x_2^n) \sum_{s_1^n, x_1^n} p(y^n|x_1^n, x_2^n) p(x_1^n|s_1^n, z^n) p(s_1^n|z^n)\label{Markov4} \\
& =p(s_2^n|z^n, x_2^n) \sum_{s_1^n, x_1^n} p(y^n|x_1^n, x_2^n) p(x_1^n,s_1^n| z^n) \label{Markov5} \\
& =p(s_2^n|z^n, x_2^n) \sum_{ x_1^n} p(y^n|x_1^n, x_2^n) p(x_1^n| z^n) \label{Markov5} \end{aligned}$$ is from $Y^n-X_1^nX_2^n- S_1^n S_2^n Z^n$ and $X_1^n-S_1^nZ^n-S_2^nX_2^n$ as well as $S_1^n-Z^n-S_2^nX_2^n$ which holds since $$\begin{aligned}
p(s_1^n, s_2^n, x_2^n|z^n) &=p(x_2^n| s_1^n, s_2^n, z^n) p( s_2^n| z^n)p( s_1^n| z^n) \label{eqfirst}\\
&=p(x_2^n| s_2^n, z^n) p( s_2^n| z^n)p( s_1^n| z^n) \label{eqsecond}\\
&=p(x_2^n, s_2^n| z^n) p( s_1^n| z^n). \end{aligned}$$ is from $S_1^n-Z^n-S_2^n$ and from $X_2^n-S_2^nZ^n-S_1^n$. Note that $$\begin{aligned}
p(y^n|z^n, x_2^n)
&= \sum_{s_1^n, x_1^n} p(y^n, s_1^n, x_1^n|z^n, x_2^n) \\
&= \sum_{s_1^n, x_1^n} p(y^n|x_1^n, x_2^n) p(x_1^n| s_1^n,z^n, x_2^n)p(s_1^n|z^n, x_2^n)\\
&= \sum_{s_1^n, x_1^n} p(y^n|x_1^n, x_2^n) p(x_1^n| s_1^n,z^n)p(s_1^n|z^n)\label{eq:markovbeforelast}\\
&= \sum_{x_1^n} p(y^n|x_1^n, x_2^n) p(x_1^n|z^n). \label{eq:markovlast}\end{aligned}$$ is from $X_1^n-S_1^nZ^n-X_2^n$ and $S_1^n-Z^n-X_2^n$ holds since $S_1^n-Z^n-S_2^nX_2^n$. Combining and , we have $$p(y^n, s_2^n|z^n, x_2^n)=p(s_2^n|z^n, x_2^n)p(y^n|z^n, x_2^n)$$ hence $Y^n-Z^nX_2^n-S_2^n$. Then, we use the following in , $$\begin{aligned}
I(X_2^n;Y^n|Z^n)&=H(Y^n|Z^n)-H(Y^n|X_2^n, Z^n) \label{eqMarkovcont1}\\
&=H(Y^n|Z^n)-H(Y^n|X_2^n, Z^n, S_2^n) \label{eqMarkovcont2} \\
&\geq H(Y^n|Z^n)-H(Y^n|Z^n, S_2^n) \label{eqMarkovcont3} \\
&= I(S_2^n;Y^n|Z^n). \label{eqMarkovcont4} \end{aligned}$$ is from $Y^n-Z^nX_2^n-S_2^n$ and holds since conditioning cannot increase entropy, which leads to . Moreover, we have $$\begin{aligned}
\frac{1}{n}I(X_1^n, X_2^n; Y^n | Z^n)
&\leq \frac{1}{n} \sum_{i=1}^n(H(Y_i|Z^n)- H(Y_i|X_{1i}, X_{2i}, Z^n)) \label{eqlemma2sumrate1} \\
&= \frac{1}{n} \sum_{i=1}^nI(X_{1i}, X_{2i}; Y_i|\tilde{Q}=i, Z^n) \\
&\leq I(X_{1\tilde{Q}}, X_{2\tilde{Q}}; Y_{\tilde{Q}}| \tilde{Q}, Z^n)\\
&\leq I(X_{1}, X_{2}; Y| Q) \label{eqlemma2sumrate4}\end{aligned}$$ where $X_1=X_{1\tilde{Q}}$, $X_2=X_{2\tilde{Q}}$, $Y=Y_{\tilde{Q}}$ and $Q=(\tilde{Q}, Z^n)$.
Combining , and with recovers .
Lastly, we show that $p(x_1, x_2|q)=p(x_1|q) p(x_2|q)$ along the lines of [@gunduz2009source]. For $q=(i, z^n)$, $$\begin{aligned}
P(X_1=x_1, X_2=x_2| Q=q)
&=P(X_{1i}=x_1, X_{2i}=x_2| \tilde{Q}=i, Z^n=z^n) \label{distx1x2q0}\\
&=P(X_{1i}\!=\!x_1| \tilde{Q}\!=\!i, Z^n\!=\!z^n) P(X_{2i}\!=\!x_2| \tilde{Q}\!=\!i, Z^n\!=\!z^n) \label{distx1x2qn}\\
&=P(X_{1}=x_1| Q=q) P(X_{2}=x_2| Q=q) \label{distx1x2q1}\end{aligned}$$ where holds since $X_{1i}-Z^n-X_{2i}$ for $i=1,\ldots, n$ as follows. $$\begin{aligned}
p(x_1^n, x_2^n|z^n)
&=\sum_{s_1^n, s_2^n} p(x_1^n, x_2^n, s_1^n, s_2^n|z^n) \label{eq:indept00} \\
&=\sum_{s_1^n, s_2^n} p(x_1^n| s_1^n, z^n) p(x_2^n| s_2^n, z^n) p(s_1^n| z^n) p(s_2^n| z^n) \label{eqmarkovdist}\\
&=\sum_{s_1^n, s_2^n} p(x_1^n, s_1^n| z^n) p(x_2^n, s_2^n| z^n) \\
&= p(x_1^n| z^n) p(x_2^n| z^n) \label{eq:indept}\end{aligned}$$ where is from $X_1^n-S_1^n Z^n -S_2^nX_2^n$ and $X_2^n-S_2^n Z^n -S_1^n$ as well as $S_1^n - Z^n -S_2^n$. From , we observe that $X_{1}^n-Z^n-X_{2}^n$, which implies $X_{1i}-Z^n-X_{2i}$.
Proof of Remark \[remark1\] {#appendixExample}
===========================
$S_1-Z-S_2$ holds if and only if $p(s_1|s_2, z)=p(s_1|z)$. Note that for the Gaussian vector $(S_1, S_2, Z)$, we have that $(S_2, Z)\sim \mathcal{N}(\mu_{S_2Z}, K_{S_2Z})$ with mean $\mu_{S_2Z} = (0,0)$ and covariance $$K_{S_2 Z} = \begin{bmatrix}
\mathbb{E}[S_2^2] & \mathbb{E}[S_2 Z] \\
\mathbb{E}[Z S_2] & \mathbb{E}[Z^2]
\end{bmatrix}.$$ It follows that $S_1|S_2, Z\sim \mathcal{N}(\mathbb{E}[S_1|S_2, Z], \text{var}(S_1|S_2, Z))$ is Gaussian distributed with mean $$\begin{aligned}
\mathbb{E}[S_1|S_2, Z]
&=\begin{bmatrix}\mathbb{E} [S_1 S_2] & \mathbb{E} [S_1 Z] \end{bmatrix}
\begin{bmatrix}
\mathbb{E}[S_2^2] & \mathbb{E} [S_2 Z] \\
\mathbb{E}[Z S_2] & \mathbb{E} [Z^2]
\end{bmatrix}^{-1}
\begin{bmatrix}
S_2 \\ Z
\end{bmatrix} \\
&= \left(\frac{\mathbb{E}[S_1S_2]\mathbb{E}[Z^2]-\mathbb{E}[S_1Z] \mathbb{E}[S_2Z]}{\mathbb{E}[S^2_2] \mathbb{E}[Z^2]- \mathbb{E}^2[S_2Z]}\right) S_2
+ \left(\frac{\mathbb{E}[S_1Z] \mathbb{E}[S_2^2]- \mathbb{E}[S_1S_2]\mathbb{E}[S_2Z]}{\mathbb{E}[S_2^2]\mathbb{E}[Z^2]-\mathbb{E}^2[S_2 Z]}\right) Z \label{compare1}\end{aligned}$$ and variance $$\begin{aligned}
\text{var}(S_1|S_2, Z)
&=\mathbb{E}[S_1^2]
- \begin{bmatrix}
\mathbb{E}[S_1 S_2] & \mathbb{E}[S_1 Z]
\end{bmatrix}
\begin{bmatrix}
\mathbb{E}[S_2^2] & \mathbb{E}[S_2 Z] \nonumber \\
\mathbb{E}[Z S_2] & \mathbb{E}[Z^2]
\end{bmatrix}^{-1}
\begin{bmatrix}
\mathbb{E}[S_2 S_1] \\ \mathbb{E}[Z S_1]
\end{bmatrix} \\
&= \mathbb{E}[S_1^2]
-\frac{\mathbb{E}^2[S_1 S_2]\mathbb{E}[Z^2]+\mathbb{E}^2[S_1 S_3]\mathbb{E}[S_2^2]}{\mathbb{E}[S_2^2] \mathbb{E}[Z^2]-\mathbb{E}^2[S_2 Z]}
+\frac{ 2 \mathbb{E}[S_1 S_2]\mathbb{E}[S_1 S_3] \mathbb{E}[S_2 S_3]}{\mathbb{E}[S_2^2] \mathbb{E}[Z^2]-\mathbb{E}^2[S_2 Z]} \label{compare3}\end{aligned}$$
Similarly, since $(S_1, Z)\sim \mathcal{N}(\mu_{S_1Z}, K_{S_1Z})$ is jointly Gaussian with mean $\mu_{S_1Z} = (0,0)$ and covariance $$K_{S_1 Z} = \begin{bmatrix}
\mathbb{E}[S_1^2] & \mathbb{E}[S_1 Z] \\
\mathbb{E}[Z S_1] & \mathbb{E}[Z^2]
\end{bmatrix}$$ we have that $S_1|Z \sim \mathcal{N}(\mathbb{E}[S_1|Z], \text{var}(S_1|Z))$ such that $$\mathbb{E}[S_1|Z] = \frac{\mathbb{E}[S_1Z]Z}{\mathbb{E}[Z^2]} \label{compare2}$$ $$\text{var}(S_1|Z) = \mathbb{E}[S_1^2] -\frac{\mathbb{E}^2[S_1Z]}{\mathbb{E}[Z^2]}. \label{compare4}$$
Then, we know that $p(s_1|s_2, z)=p(s_1|z)$ if and only if $$\begin{aligned}
& i) \; \mathbb{E}[S_1| S_2, Z] = \mathbb{E}[S_1| Z] \label{compcond1} \\
& ii) \; \text{var}(S_1| S_2, Z) = \text{var}(S_1| Z) \label{compcond2}\end{aligned}$$ hold. By comparing with , condition leads to $$\begin{aligned}
&\frac{\mathbb{E}[S_1Z]}{\mathbb{E}[Z^2]} Z = \left(\frac{\mathbb{E}[S_1S_2]\mathbb{E}[Z^2]-\mathbb{E}[S_1Z] \mathbb{E}[S_2Z]}{\mathbb{E}[S^2_2] \mathbb{E}[Z^2]- \mathbb{E}^2[S_2Z]}\right) S_2
+ \left(\frac{\mathbb{E}[S_1Z] \mathbb{E}[S_2^2]- \mathbb{E}[S_1S_2]\mathbb{E}[S_2Z]}{\mathbb{E}[S_2^2]\mathbb{E}[Z^2]-\mathbb{E}^2[S_2 Z]}\right) Z \end{aligned}$$ which holds if and only if $$\frac{\mathbb{E}[S_1S_2]\mathbb{E}[Z^2]-\mathbb{E}[S_1Z] \mathbb{E}[S_2Z]}{\mathbb{E}[S^2_2] \mathbb{E}[Z^2]- \mathbb{E}^2[S_2Z]} =0$$ and $$\frac{\mathbb{E}[S_1Z]}{\mathbb{E}[Z^2]} = \frac{\mathbb{E}[S_1Z] \mathbb{E}[S_2^2]- \mathbb{E}[S_1S_2]\mathbb{E}[S_2Z]}{\mathbb{E}[S_2^2]\mathbb{E}[Z^2]-\mathbb{E}^2[S_2 Z]}$$ from which and follows, respectively.
By combining and , we find that leads to $$\begin{aligned}
&\mathbb{E}[S_1^2] -\frac{\mathbb{E}^2[S_1Z]}{\mathbb{E}[Z^2]}
= \mathbb{E}[S_1^2]
-\frac{\mathbb{E}^2[S_1 \!S_2]\mathbb{E}[Z^2]\!-\!\! 2 \mathbb{E}[S_1 \!S_2]\mathbb{E}[S_1 \!S_3] \mathbb{E}[S_2 S_3]\!+\!\mathbb{E}^2[S_1 \!S_3]\mathbb{E}[S_2^2]}{\mathbb{E}[S_2^2] \mathbb{E}[Z^2]\!-\!\mathbb{E}^2[S_2 Z]} \end{aligned}$$ from which we obtain .
Proof of Theorem \[Thm:Reconstruction\] {#appendixD}
=======================================
Our proof below heavily uses techniques from [@steinberg2009coding].
Achievability
-------------
Source coding is based on lossy compression of a source with decoder side information and common reconstruction constraint, [@steinberg2009coding], which is based on a special case of Wyner-Ziv source coding as $\hat{S}_j$ satisfies the Markov relation $\hat{S}_j-S_j-Z$ for $j=1,2$. Channel coding is based on multiple access channel coding.
Converse
--------
Suppose there exist encoding functions $e_j: \mathcal{S}_j^n\rightarrow \mathcal{X}_j^n$ for encoder $j=1,2$, decoding functions $g_j: \mathcal{Y}^n \times \mathcal{Z}^n\rightarrow \hat{\mathcal{S}}^n_j$ such that $\frac{1}{n}\sum_{i=1}^n \mathbb{E}[d_j(S_{ji}, \hat{S}_{ji})] \leq D_j+\epsilon$ for $j=1,2$ where $\epsilon\rightarrow 0$ as $n\rightarrow \infty$. Lastly, suppose that reconstruction functions $\psi_j: \mathcal{S}_j^n \rightarrow \bar{\mathcal{S}}_j^n$ exist for encoder $j=1,2$ such that $P(\bar{S}_j^n\neq \hat{S}_j^n)\leq P_{ej}$ where $P_{ej}\rightarrow 0$ as $n\rightarrow \infty$.
From Fano’s inequality, $$\begin{aligned}
H(\psi_j(S_j)|g_j(Y^n, Z^n)) &\leq h(P_{ej}) + P_{ej} \log (|\psi(S_j^n)|-1) \\
& \leq h(P_{ej}) + nP_{ej} \log (|\bar{\mathcal{S}}_j|) \\
&=n\delta (P_{ej}) \label{eqFanoCR} \end{aligned}$$ where $\delta (P_{ej})\rightarrow 0$ as $P_{ej}\rightarrow 0$ for $j=1,2$.
Next, define the event $$\label{eqCR8}
\mathcal{B} = \{\psi_1(S_1^n)\neq g_1(Y^n, Z^n)\}.$$ By definition, $P(\mathcal{B})\leq P_{e1}$. Let $\psi_{1i}(S_1^n)$ denote the $i$th entry of $\psi_{1}(S_1^n)$, and define $$\label{defnCR}
\hat{S}_{1i} = \psi_{1i}(S_1^n).$$ Lastly, define $g_{1i}(Y^n, Z^n)$ as the $i$th entry of $g_{1}(Y^n, Z^n)$. Then, $$\begin{aligned}
&\frac{1}{n} \sum_{i=1}^n \mathbb{E} [d_1(S_{1i}, \hat{S}_{1i})] \notag \\
&= \frac{1}{n}\sum_{i=1}^n \Big(\! \mathbb{E} [d_1(S_{1i}, \hat{S}_{1i})|\mathcal{B}^c] P(\mathcal{B}^c) \!+ \mathbb{E} [d_1(S_{1i}, \hat{S}_{1i})|\mathcal{B}] P(\mathcal{B}) \!\Big)\\
&\leq \frac{1}{n}\sum_{i=1}^n \mathbb{E} [d_1(S_{1i}, \hat{S}_{1i})|\mathcal{B}^c] P(\mathcal{B}^c) + d_{1, \text{max}} P_{e1} \\
&= \frac{1}{n}\sum_{i=1}^n \mathbb{E} [d_1(S_{1i}, g_{1i}(Y^n, Z^n))|\mathcal{B}^c] P(\mathcal{B}^c) + d_{1, \text{max}} P_{e1} \\
&\leq \frac{1}{n}\sum_{i=1}^n (\mathbb{E}[d_1(S_{1i}, g_{1i}(Y^n, Z^n)|\mathcal{B}] P(\mathcal{B})
+\mathbb{E}[d_1(S_{1i}, g_{1i}(Y^n, Z^n)|\mathcal{B}^c] P(\mathcal{B}^c) )+ d_{1, \text{max}} P_{e1} \\
&= \frac{1}{n}\sum_{i=1}^n \mathbb{E}[d_1(S_{1i}, g_{1i}(Y^n, Z^n)]+ d_{1, \text{max}} P_{e1} \\
&\leq D_1+ \epsilon + d_{1, \text{max}} P_{e1} \label{CRS11}\end{aligned}$$ where $d_{1, \text{max}}=\max_{s_1\in \mathcal{S}_1, \hat{s}_1\in \hat{\mathcal{S}}_1} d_1(s_1, \hat{s}_1)$.
Then, we can find that $$\begin{aligned}
\frac{1}{n}I(X_1^n;Y^n|X_2^n, Z^n)
&\geq \frac{1}{n} I(S_1^n;Y^n|X_2^n, Z^n) \label{eqfirstCR}\\
&=\frac{1}{n}I(S_1^n;Y^n, X_2^n|Z^n) \\
&\geq \frac{1}{n}I(S_1^n ; g_1(Y^n, Z^n)|Z^n) \label{eqCR13}\\
&= \frac{1}{n}\big[I(S_1^n; g_1(Y^n, Z^n), \psi_1(S_1^n)| Z^n) - I(S_1^n; \psi_1(S_1^n)|Z^n, g_1(Y^n, Z^n))\big] \\
&= \frac{1}{n}\big[I(S_1^n; g_1(Y^n, Z^n), \psi_1(S_1^n)| Z^n) - H(\psi_1(S_1^n)|Z^n, g_1(Y^n, Z^n))\big] \\
&\geq\frac{1}{n} \big[I(S_1^n; g_1(Y^n, Z^n), \psi_1(S_1^n)| Z^n) - H(\psi_1(S_1^n)|g_1(Y^n, Z^n)) \big]\\
&\geq \frac{1}{n}\big[I(S_1^n; g_1(Y^n, Z^n), \psi_1(S_1^n)| Z^n) - n\delta(P_{e1})\big] \label{CRFano}\\
&\geq \frac{1}{n}\big[I(S_1^n; \psi_1(S_1^n)|Z^n)-n\delta(P_{e1})\big] \\
&=\frac{1}{n}\big[I(\psi_1(S_1^n); S_1^n) - I(\psi_1(S_1^n); Z^n)-n\delta(P_{e1})\big] \label{eqlastCR}\end{aligned}$$ where follows from ; and follows since $\psi(S_1^n)-S_1^n-Z_1^n$ form a Markov chain. Then, we have $$\begin{aligned}
&\frac{1}{n}\big[I(\psi_1(S_1^n); S_1^n) - I(\psi_1(S_1^n); Z^n)-n\delta(P_{e1} )\big] \\
&=\!\frac{1}{n}\sum_{i=1}^n \!\big[I(\psi_1(S_1^n); S_{1i}|S_1^{i-1}) \!-\! I(\psi(S_1^n);Z_i|Z_{i+1}^n) ) \!-\!\delta(P_{e1} \big]\\
&=\frac{1}{n}\sum_{i=1}^n \big[I(\psi_1(S_1^n), Z_{i+1}^n; S_{1i}|S_1^{i-1}) \notag \\
&\quad -I(Z_{i+1}^n;S_{1i}|\psi_1(S_1^n), S_1^{i-1})
- I(\psi(S_1^n), S_1^{i-1};Z_i|Z_{i+1}^n)
+ I(S_1^{i-1};Z_i|Z_{i+1}^n, \psi(S_1^n)) \big]
-\delta(P_{e1} )\label{eqS63}\\
&\geq \frac{1}{n}\sum_{i=1}^n \big[I(\psi_1(S_1^n), Z_{i+1}^n; S_{1i}|S_1^{i-1})
- I(\psi_1(S_1^n), S_1^{i-1}; Z_i|Z_{i+1}^n)\big] -\delta(P_{e1})\label{eqS64} \\
& =\frac{1}{n}\sum_{i=1}^n \big[I(\psi_1(S_1^n), Z_{i+1}^n, S_1^{i-1}; S_{1i})
- I(\psi_1(S_1^n), S_1^{i-1}, Z_{i+1}^n; Z_i)\big] -\delta(P_{e1})\label{eqCRa} \\
&= \frac{1}{n}\sum_{i=1}^n I(\psi_1(S_1^n), S_1^{i-1}, Z_{i+1}^n;S_{1i}|Z_i) -\delta(P_{e1})\label{eqCRb} \\
&\geq \frac{1}{n} \sum_{i=1}^n I(\psi_{1i}(S_1^n);S_{1i}|Z_i) -\delta(P_{e1})\label{eqCRc} \\
&\geq \frac{1}{n} \sum_{i=1}^n I(\bar{S}_{1i};S_{1i}|Z_i) -\delta(P_{e1})\label{eqCRlast} \\
&\geq \frac{1}{n} \sum_{i=1}^n R^{RC}_{S_{1}|Z}(\mathbb{E}[d(S_{1i}, \hat{S}_{1i})])-\delta(P_{e1}) \label{eqCR37old}\\
&\geq R^{RC}_{S_{1}|Z}\Big(\frac{1}{n} \sum_{i=1}^n \mathbb{E}[d(S_{1i}, \hat{S}_{1i})]\Big)-\delta(P_{e1}) \label{eqCR37} \\
&\geq R^{RC}_{S_{1}|Z}(D_1 + \epsilon + d_{1,max} P_{e1})-\delta(P_{e1}) \label{eqCR38}\end{aligned}$$ where follows from Csiszar’s sum equality, follows from the fact that $\psi_1(S_1^n) S_1^{i-1}Z_{i+1}^n-S_{1i}-Z_i$ forms a Markov chain, follows from , follows from the definition of $R^{RC}_{S_{1}|Z}$ and the fact that $\bar{S}_{1i}-S_{1i}-Z$ form a Markov chain, and and follow from the convexity and monotonicity of .
Lastly, we define an independent discrete random variable $\tilde{Q}$ uniformly distributed over $\{1,\ldots, n\}$ such that $$\begin{aligned}
\frac{1}{n} I(X_1^n; Y^n|X_2^n, Z^n)
&\leq \frac{1}{n} \sum_{i=1}^n I(X_{1i};Y_i|X_{2i}, \tilde{Q}=i, Z^n) \notag \\ &=I(X_{1\tilde{Q}};Y_{\tilde{Q}}|X_{2\tilde{Q}}, \tilde{Q}, Z^n) \notag \\
&= I(X_1;Y|X_2, Q) \label{lemma2Ix1yn5}\end{aligned}$$ where $X_1=X_{1\tilde{Q}}$, $X_2=X_{2\tilde{Q}}$, $Y=Y_{\tilde{Q}}$ and $Q=(\tilde{Q}, Z^n)$, and $$p(y, x_1, x_2|q)=p(y| x_1, x_2) p(x_1|q)p(x_2|q).$$
By combining , , , , and , we have that $$R^{RC}_{S_{1}|Z}(D_1 + \epsilon + d_{1,max} P_{e1})-\delta(P_{e1})
\leq I(X_1;Y|X_2, Q)$$ where $P_{e1}\rightarrow 0$ and $\epsilon\rightarrow 0$ as $n\rightarrow \infty$.
By following similar steps for source $S_2$, we can obtain $$R^{RC}_{S_{2}|Z}(D_2 + \epsilon + d_{2,max} P_{e2})-\delta(P_{e2}) \leq I(X_2;Y|X_1, Q)$$ where $P_{e2}\rightarrow 0$ and $\epsilon\rightarrow 0$ as $n\rightarrow \infty$.
Lastly, we have $$\begin{aligned}
&\frac{1}{n}I(X_1^n, X_2^n;Y^n|Z^n) \notag \\
& \geq \frac{1}{n} I(S_1^n, S_2^n; Y^n|Z^n) \\
& = \frac{1}{n}(I(S_1^n; Y^n|Z^n) + I(S_2^n;Y^n|S_1^n, Z^n)) \\
& = \frac{1}{n}(I(S_1^n; Y^n|Z^n) + I(S_1^n; g_1(Y^n, Z^n)| Y^n, Z^n) \nonumber \\
&\qquad + I(S_2^n;Y^n|S_1^n, Z^n) +I(S_2^n; g_2(Y^n, Z^n)|S_1^n, Y^n, Z^n)) \\
& = \frac{1}{n}(I(S_1^n; Y^n, g_1(Y^n, Z^n)|Z^n) + I(S_2^n; Y^n, g_2(Y^n, Z^n)|S_1^n, Z^n)) \\
& \geq \frac{1}{n}(I(S_1^n; g_1(Y^n, Z^n)|Z^n) + I(S_2^n; g_2(Y^n, Z^n)|S_1^n, Z^n)) \\
& \geq \frac{1}{n}(I(S_1^n; g_1(Y^n, Z^n)|Z^n) + H(S_2^n|Z^n) - H(S_2^n|g_2(Y^n, Z^n), Z^n)) \\
& = \frac{1}{n}(I(S_1^n; g_1(Y^n, Z^n)|Z^n) + I(S_2^n; g_2(Y^n, Z^n)|Z^n)) \label{eqCRSumlast} \\
&\geq R^{RC}_{S_{1}|Z}(D_1 \!+\! \epsilon + d_{1,max} P_{e1})
\!+\! R^{RC}_{S_{2}|Z}(D_2 \!+\! \epsilon + d_{2,max} P_{e2}) -\delta(P_{e1}) -\delta(P_{e2}) \label{eqCR50}\end{aligned}$$ and the steps between and follow as in -. Lastly, by using the fact that $$\begin{aligned}
\frac{1}{n}I(X_1^n, X_2^n; Y^n| Z^n) &\leq \frac{1}{n} \sum_{i=1}^nI(X_{1i}, X_{2i}; Y_i|\tilde{Q}=i, Z^n) \notag \\
&=I(X_{1}, X_{2}; Y| Q) \label{eqlemma2sumrate4}\end{aligned}$$ we have $$\begin{aligned}
&R^{RC}_{S_{1}|Z}(D_1 + \epsilon + d_{1,max} P_{e1})
+ R^{RC}_{S_{2}|Z}(D_2 + \epsilon + d_{2,max} P_{e2}) -\delta(P_{e1}) -\delta(P_{e2}) \leq I(X_{1}, X_{2}; Y| Q). \end{aligned}$$
Then the converse follows from letting $n\rightarrow \infty$.
Proof of Theorem \[lemma3\] {#appendixB}
===========================
Achievability
-------------
Our source coding part is based on the distributed source coding scheme with a common part from [@wagner2011distributed Theorem $1$]. Initially, let $Y_0\leftarrow Z$, $Y_j\leftarrow (S_j, Z)$ for $j=1,2$, and $X\leftarrow Z$ in Fig. $3$ in [@wagner2011distributed] and observe that any achievable rate for this system is also achievable by our system. This follows from the fact that $Y_0$, which is to be reconstructed losslessly in [@wagner2011distributed], is known by both encoders in our system as $Y_0\leftarrow Z$ and that $Z$ is available to both encoders. Hence, the encoders can cooperate to send $Y_0$ to the decoder and realize any achievable scheme in [@wagner2011distributed].
Letting $U= X$ in [@wagner2011distributed Theorem $1$] and substituting $X\leftarrow Z$, $Y_0\leftarrow Z$, $\hat{Y}_0\leftarrow \hat{Z}$, $Y_j\leftarrow (S_j, Z)$, $V_j\leftarrow U_j$, $\hat{Y}_j\leftarrow \hat{S}_j$, and $d_j(Y_j, \hat{Y}_j)\leftarrow d_j(S_j, \hat{S}_j)$ for $j=1,2$, we find that a distortion pair $(D_1, D_2)$ is achievable for the rate triplet $(R_0, R_1, R_2)$ if $$\begin{aligned}
R_0&\geq H(Z|Z, U_1, U_2) \label{eqCommon1}\\
R_1&\geq I(S_1, Z; U_1|Z, U_2) \label{eqCommon2}\\
R_2&\geq I(S_2, Z; U_2|Z, U_1) \label{eqCommon3}\\
R_0+R_1&\geq H(Z|Z, U_2) + I(S_1, Z; U_1|Z, U_2)\label{eqCommon4}\\
R_0+R_2&\geq H(Z|Z, U_1) + I(S_2, Z; U_2|Z, U_1)\label{eqCommon5}\\
R_1+R_2&\geq I(S_1, S_2, Z; U_1, U_2, Z|Z)\label{eqCommon6}\\
R_0+R_1+R_2&\geq H(Z)+I(S_1, S_2, Z; U_1, U_2, Z|Z)\label{eqCommon7}\end{aligned}$$ and $\mathbb{E}[d_j(S_j, \hat{S}_j)]\leq D_j$ for $j=1,2$, for some distribution $$\begin{aligned}
\label{eq:distr}
&p(z, s_1, s_2, u_1, u_2, \hat{s}_1, \hat{s}_2) =p(z, s_1, s_2) p(u_1|s_1, z) p(u_2|s_2, z) p(\hat{s}_1, \hat{s}_2| z, u_1, u_2).\end{aligned}$$ Note that since $H(Z|Z, U_1, U_2)=0$, the condition can be removed without loss of generality. We next write as, $$\begin{aligned}
R_1&\geq I(S_1, Z; U_1|Z, U_2) \label{combine1} \\
&=I(S_1; U_1|Z, U_2) + I(Z; U_1|Z, U_2, S_1) \label{combineext} \\
&=H(U_1|Z, U_2)- H(U_1|S_1, Z, U_2) \label{combine12}\\
&=H(U_1|Z)- H(U_1|S_1, Z) \label{combine13}\\
&= I(S_1;U_1|Z) \label{combine14}\end{aligned}$$ where holds because $I(Z; U_1|Z, U_2, S_1)=0$ since $$\label{rec}
I(Z; A|Z, B)=H(Z|Z, B)-H(Z|Z, B, A)=0$$ for any random variable $A$ and $B$, is from $U_1-S_1Z-U_2$ and $U_1-Z-U_2$ since $$\begin{aligned}
p(u_1, u_2|z) &= \sum_{s_1, s_2} p(u_1, u_2, s_1, s_2|z) \label{markovcond1}\\
&= \sum_{s_1, s_2} \!p(u_1| s_1, z) p(u_2| s_2, z) p(s_1| z)p(s_2| z) \label{markovcond2}\\
&= \sum_{s_1, s_2} p(u_1, s_1|z) p(u_2, s_2|z) \label{markovcond3}\\
&= p(u_1|z) p(u_2|z) \label{markovpu1u2}\end{aligned}$$ where is from $U_1-S_1Z-S_2U_2$ and $U_2-S_2Z-S_1$ as well as $S_1-Z-S_2$.
Noting that $H(Z|Z, U_2)=0$ and following the steps in -, we can write as $$\label{combine1old}
R_0+R_1\geq I(S_1;U_1|Z)$$ which, by comparing it with , indicates that condition can be removed without loss of generality.
Following similar steps, we can write as $$\label{combine2old}
R_2\geq I(S_2; U_2|Z)$$ and as $$\label{combine2}
R_0+R_2\geq I(S_2; U_2|Z)$$ which, by comparing it with , shows that condition can also be removed. For and , we find that $$\begin{aligned}
I(S_1, S_2, Z; U_1, U_2, Z|Z)
&=I(S_1, S_2; U_1, U_2, Z|Z) + I(Z; U_1, U_2, Z|Z, S_1, S_2) \label{beginning}\\
&=I(S_1, S_2; U_1, U_2|Z)+I(S_1, S_2; Z|Z, U_1, U_2) \label{beginning2} \\
&=I(S_1, S_2; U_1, U_2|Z) \label{simplified}\\
&=H(U_1, U_2|Z)- H(U_1, U_2|Z, S_1, S_2) \\
&=H(U_1|Z)+H(U_2|Z, U_1)- H(U_1|Z, S_1, S_2)- H(U_2|Z, S_1, S_2, U_1)\\
&=H(U_1|Z)+H(U_2|Z, U_1)\!-\! H(U_1|Z, S_1)- H(U_2|Z, S_2) \label{eq:decompn} \\
&=H(U_1|Z)\!+\!H(U_2|Z)\!-\! H(U_1|Z, S_1)\!-\! H(U_2|Z, S_2) \label{eq:decompn2}\\
&=I(S_1;U_1|Z)+I(S_2;U_2|Z) \label{eq:decompn3}\end{aligned}$$ where holds as a result of $I(Z; U_1, U_2, Z|Z, S_1, S_2)=0$ from ; holds since $I(S_1, S_2; Z|Z, U_1, U_2)=0$ from ; holds as $U_1-ZS_1-S_2$ and $U_2-ZS_2-S_1U_1$; and follows from $U_1-Z-U_2$ shown in .
Combining , , , and with , we can now restate - as follows.
A distortion pair $(D_1, D_2)$ is achievable for the rate triplet $(R_0, R_1, R_2)$ if $$\begin{aligned}
R_1&\geq I(S_1;U_1|Z) \label{rr1}\\
R_2&\geq I(S_2;U_2|Z) \label{rr2}\\
R_1+R_2&\geq I(S_1;U_1|Z)+I(S_2;U_2|Z) \label{rr3}\\
R_0+R_1+R_2&\geq H(Z) \!+ \!I(S_1;U_1|Z)+I(S_2;U_2|Z) \label{rr4}\end{aligned}$$ and $\mathbb{E}[d_j(S_j, \hat{S}_j)]\leq D_j$ for $j=1,2$, for some distribution $$p(z, s_1, s_2) p(u_1|s_1, z) p(u_2|s_2, z) p(\hat{s}_1, \hat{s}_2| z, u_1, u_2).$$
We next show that one can set $\hat{S}_j=f_j(Z, U_1, U_2)$ for $j=$ $1,2$ without loss of optimality. To do so, we write $$\begin{aligned}
\mathbb{E}[d_1(S_1, \hat{S}_1)]&=\sum_{s_1, \hat{s}_1} p(s_1, \hat{s}_1) d_1(s_1, \hat{s}_1)\\
&=\sum_{s_1, \hat{s}_1, \hat{s}_2, z, u_1, u_2} \hspace{-0.4cm}p( \hat{s}_1, \hat{s}_2|z, u_1, u_2, s_1) p(z, u_1, u_2, s_1) d_1(s_1, \hat{s}_1)
\\
&=\sum_{s_1, \hat{s}_1, \hat{s}_2, z, u_1, u_2} p( \hat{s}_1, \hat{s}_2|z, u_1, u_2) p(z, u_1, u_2, s_1) d_1(s_1, \hat{s}_1)
\\
&=\sum_{z, u_1, u_2} \sum_{\hat{s}_1} \sum_{s_1} p( \hat{s}_1|z, u_1, u_2) p(z, u_1, u_2, s_1) d_1(s_1, \hat{s}_1) \\
&\geq \sum_{z, u_1, u_2, s_1} p(z, u_1, u_2, s_1) d_1(s_1, f_1(z, u_1, u_2)) \label{eq:f}\\
&=\mathbb{E}[d_1(S_1, f_1(Z, U_1, U_2))] \label{eqson93}\end{aligned}$$ where we define a function $f_1: \mathcal{Z}\times \mathcal{U}_1\times \mathcal{U}_2\rightarrow \hat{\mathcal{S}}_1$ in such that, $$f_1(z, u_1, u_2)=\arg \min_{\hat{s}_1} \sum_{s_1} p(z, u_1, u_2, s_1) d_1(s_1, \hat{s}_1)$$ and set $p( \hat{s}_1|z, u_1, u_2)=1$ for $\hat{s}_1=f_1(z, u_1, u_2)$ and $p( \hat{s}_1|z, u_1, u_2)=0$ otherwise.
A similar argument follows for $S_2$ by defining a function $f_2: \mathcal{Z}\times \mathcal{U}_1\times \mathcal{U}_2\rightarrow \hat{\mathcal{S}}_2$ leading to $$\label{functionfors2}
\mathbb{E}[d_2(S_2, \hat{S}_2)]\geq \mathbb{E}[d_2(S_2, f_2(Z, U_1, U_2))].$$ Therefore, we can set $\hat{S}_j=f_j(Z, U_1, U_2)$ for $j=1,2$.
We next show for $j=1,2$ that whenever there exists a function $f_j(Z, U_1, U_2)$ such that $$\mathbb{E}[d_j(S_j, f_j(Z, U_1, U_2))]\leq D_j,$$ then there exists a function $g_j(Z, U_j)$ such that $$\mathbb{E}[d_j(S_j, g_j(Z, U_j))]\leq \!\mathbb{E}[d_j(S_j, f_j(Z, U_1, U_2))]\!\leq\! D_j.$$
We show this result along the lines of [@gastpar2004wyner]. Consider a function $f_1(Z, U_1, U_2)$ such that $\mathbb{E}[d_1(S_1, f_1(Z, U_1, U_2))]\leq D_1$. From the law of iterated expectations, $$\begin{aligned}
\mathbb{E}[d_1(S_1, f_1(Z, U_1, U_2))]
&= \mathbb{E}_{S_2, U_2, Z} [\mathbb{E}_{S_1, U_1|S_2, U_2, Z} [d_1(S_1, f_1( Z, U_1, U_2))]] \\
&= \mathbb{E}_{S_2, U_2, Z} [\mathbb{E}_{S_1, U_1|Z} [d_1(S_1, f_1(Z, U_1, U_2))]] \label{eq:LOIE}\end{aligned}$$ holds due to $U_1S_1-Z-U_2S_2$, which can be observed from -. Define a function $
\phi: \mathcal{Z}\rightarrow \mathcal{U}_2$ such that $$\phi(z)=\arg \min_{u_2} \mathbb{E}_{S_1, U_1|Z=z} [d_1(S_1, f_1(z, U_1, u_2))],$$ then for each $Z=z$, $$\begin{aligned}
& \mathbb{E}_{S_2, U_2| Z=z} [\mathbb{E}_{S_1, U_1|Z=z} [d_1(S_1, f_1(z, U_1, U_2))]] \notag \\
&\qquad \qquad \qquad \geq \mathbb{E}_{S_1, U_1|Z=z} [d_1(S_1, f_1(z, U_1, \phi(z)))]\end{aligned}$$ hence $$\begin{aligned}
\mathbb{E}[d_1(S_1, f_1(Z, U_1, U_2))]
& = \mathbb{E}_{Z} [\mathbb{E}_{S_2, U_2| Z=z} [\mathbb{E}_{S_1, U_1|Z=z} [d_1(S_1, f_1(z, U_1, U_2))]]] \\
&\geq \mathbb{E}_{Z} [\mathbb{E}_{S_1, U_1|Z=z} [d_1(S_1, f_1(z, U_1, \phi(z)))]] \\
&= \mathbb{E}_{S_1, U_1,Z} [d_1(S_1, f_1(Z, U_1, \phi(Z)))] \\
&= \mathbb{E}[d_1(S_1, g_1(Z, U_1))] \label{eqf1}\end{aligned}$$ where $g_1(Z, U_1)=f_1(Z, U_1, \phi(Z))$.
Following similar steps, for any $f_2(Z, U_1, U_2)$ that achieves $\mathbb{E}[d_2(S_2, f_2(Z, U_1, U_2))]\leq D_2$ we can find a function $g_2(Z, U_2)$ such that $$\label{eqdist2}
\mathbb{E}[d_2(S_2, f_2(Z, U_1, U_2))] \geq \mathbb{E}[d_2(S_2, g_2(Z, U_2))]$$
Combining , , , and with and , we can state the rate region in - as follows.
A distortion pair $(D_1, D_2)$ is achievable for the rate triplet $(R_0, R_1, R_2)$ if $$\begin{aligned}
R_1&\geq R_{S_1|Z} (D_1) \label{rrn1}\\
R_2&\geq R_{S_2|Z} (D_2) \label{rrn2}\\
R_1+R_2&\geq R_{S_1|Z} (D_1)+R_{S_2|Z} (D_2) \label{rrn3}\\
R_0+R_1+R_2&\geq H(Z) + R_{S_1|Z} (D_1)+ R_{S_2|Z} (D_2) \label{rrn4}\end{aligned}$$ since for any $p(s_j, u_j, z)=p(u_j|s_j, z) p(s_j|z)p(z)$ and $g_j(z, u_j)$ with $\mathbb{E}[d_j(S_j, g_j(Z, U_j))]\leq D_j$, $$I(S_j;U_j|Z) \geq R_{S_j|Z} (D_j), \quad j=1,2$$ where $R_{S_j|Z} (D_j)$ is defined in . This completes the source coding part.
Our channel coding is based on multiple access channel coding with a common message [@slepian1973coding], for which any triplet of rates $(R_0, R_1, R_2)$ is achievable if $$\begin{aligned}
R_1 &\leq I(X_1;Y|X_2, W) \label{common1MAC}\\
R_2 &\leq I(X_2;Y|X_1, W)\label{common2MAC}\\
R_1+R_2&\leq I(X_1, X_2;Y| W)\label{common3MAC}\\
R_0+R_1+R_2 &\leq I(X_1, X_2;Y) \label{common4MAC}\end{aligned}$$ for some $p(x_1, x_2, y, w)=p(y|x_1, x_2) p(x_1|w) p(x_2|w)p(w)$.
Converse
--------
Our proof is along the lines of [@shamai1998systematic] and [@gunduz2007correlated]. Suppose there exist encoding functions $e_j: \mathcal{S}_j^n\times \mathcal{Z}^n\rightarrow \mathcal{X}_j^n$ for $j=1,2$, decoding functions $g_j: \mathcal{Y}^n\rightarrow \hat{\mathcal{S}}_j^n$ for $j=1,2$ and $g_0: \mathcal{Y}^n\rightarrow \hat{Z}^n$ such that $\frac{1}{n}\sum_{i=1}^n E[d_j(S_{ji}, \hat{\mathcal{S}}_{ji} )]\leq D_j+\epsilon$ for $j=1,2$ and $P(Z^n\neq \hat{Z}^n)\leq P_e$ where $\epsilon \rightarrow 0$, $P_e\rightarrow 0$ as $n\rightarrow \infty$.
Define $U_{ji}=(Y^n, S_j^{i-1}, Z_{i}^c)$ for $j=1,2$ where $Z_{i}^c=$ $(Z_{1}, \ldots, Z_{(i-1)}, Z_{(i+1)}, \ldots, Z_{n})$. Then, $$\begin{aligned}
\frac{1}{n} I(X_1^n; Y^n|X_2^n, Z^n)
&= \frac{1}{n} (H(Y^n|X_2^n, Z^n)- H(Y^n|X_1^n, X_2^n, Z^n, S_1^n)) \label{eq888n2}\\
&\geq \frac{1}{n} (H(Y^n|X_2^n, Z^n)- H(Y^n|X_2^n, Z^n, S_1^n)) \label{eq888n2n1}\\
&= \frac{1}{n} I(S_1^n; Y^n|X_2^n, Z^n) \\
&= \frac{1}{n} I(S_1^n; Y^n, X_2^n|Z^n) \label{eq888n5} \\
&\geq \frac{1}{n} I(S_1^n; Y^n|Z^n) \label{eq888n6} \\
&= \frac{1}{n} \sum_{i=1}^n I(S_{1i}; Y^n|S_{1}^{i-1}, Z^n) \label{eq888n62} \\
&= \frac{1}{n} \sum_{i=1}^n (I(S_{1i}; Y^n, S_1^{i-1}, Z_{i}^c|Z_{i})
-I(S_{1i}; S_1^{i-1}, Z_{i}^c|Z_{i}))\nonumber \\
&=\frac{1}{n} \sum_{i=1}^n I(S_{1i}; U_{1i}|Z_{i}) \label{eq888n7} \\
&\geq \frac{1}{n} \sum_{i=1}^n R_{S_1|Z}(\mathcal{E} (S_{1i}|U_{1i}, Z_{i})) \label{eq888n8} \\
&\geq \frac{1}{n} \sum_{i=1}^n R_{S_1|Z}(\mathcal{E} (S_{1i}|Y^n)) \label{eq888n9} \\
&\geq \frac{1}{n} \sum_{i=1}^n R_{S_1|Z}(\mathbb{E} [d_1(S_{1i}, \hat{S}_{1i} )] ) \label{eq888n10} \\
&\geq R_{S_1|Z}(D_1+\epsilon) \label{eq888n11} \end{aligned}$$ is from $Y^n-X_1^n X_2^n-Z^n S_1^n$, holds since conditioning cannot increase entropy, and is from $I(S_1^n;X_2^n|Z^n)=0$ since $S_1^n-Z^n-X_2^n$ as follows. $$\begin{aligned}
p(x_2^n, s_1^n|z^n)&=\sum_{s_2^n} p(x_2^n, s_2^n, s_1^n|z^n) \\
&=\sum_{s_2^n} p(x_2^n| s_2^n, s_1^n, z^n) p(s_2^n| s_1^n, z^n) p(s_1^n| z^n) \notag\\
&=\sum_{s_2^n} p(x_2^n| s_2^n, z^n) p(s_2^n| z^n) p(s_1^n| z^n) \label{neweq2}\\
&=\sum_{s_2^n} p(x_2^n, s_2^n| z^n) p(s_1^n| z^n) \\
&=p(x_2^n| z^n) p(s_1^n| z^n) \end{aligned}$$ where holds since $X_2^n-S_2^nZ^n -S_1^n$ and $S_1^n-Z^n-S_2^n$. Equation is from the chain rule; is from the definition of $U_{1i}$ and the memoryless property of the sources; is from and ; is from the fact that conditioning cannot increase ; follows as $\hat{S}_{1i}$ is a function of $Y^n$ and as $R_{S_1|Z}(D_1)$ is convex and monotone in $D_1$.
By defining a discrete random variable $\tilde{Q}$ uniformly distributed over $\{1,\ldots, n\}$ independent of everything else, and following steps - by $Q=(\tilde{Q}, Z^n)$ replaced with $W=(\tilde{Q}, Z^n)$, we find that $$\frac{1}{n} I(X_1^n; Y^n|X_2^n, Z^n)\leq I(X_1;Y|X_2, W) \label{eqbeforelast}$$ where $X_1=X_{1\tilde{Q}}$, $X_2=X_{2\tilde{Q}}$, $Y=Y_{\tilde{Q}}$.
Combining with and leads to .
We obtain by following similar steps.
Next, we show that $$\begin{aligned}
\frac{1}{n}I(X_1^n, X_2^n; Y^n|Z^n)
&= \frac{1}{n} (H(Y^n|Z^n)- H(Y^n|Z^n, X_1^n, X_2^n))\label{eq88802}\\
&= \frac{1}{n} (H(Y^n|Z^n)- H(Y^n|Z^n, X_1^n, X_2^n, S_1^n, S_2^n))\label{eq88802next}\\
&\geq \frac{1}{n} (H(Y^n|Z^n)- H(Y^n|Z^n, S_1^n, S_2^n))\label{eq88802next2}\\
&=\frac{1}{n} I(S_1^n, S_2^n; Y^n|Z^n) \\
&= \frac{1}{n} (I(S_1^n; Y^n|Z^n) + I(S_2^n; Y^n|S_1^n, Z^n))\notag\\
&= \frac{1}{n} ( I(S_1^n; Y^n|Z^n) + H(S_2^n|Z^n)- H(S_2^n| Y^n, S_1^n, Z^n))\label{eq8882} \\
&\geq \frac{1}{n} ( I(S_1^n; Y^n|Z^n) + H(S_2^n|Z^n) - H(S_2^n| Y^n, Z^n)) \label{eqson42}\\
&= \frac{1}{n} ( I(S_1^n; Y^n|Z^n) + I(S_2^n;Y^n | Z^n)) \label{88812}\\
&\geq R_{S_1|Z}(D_1+\epsilon) + R_{S_2|Z}(D_2+\epsilon) \label{88822}\end{aligned}$$ where is from $Y^n-X_1^n X_2^n-S_1^n S_2^n Z^n $, is from the fact that conditioning cannot increase entropy, is from $S_2^n-Z^n-S_1^n$, is from conditioning cannot increase entropy, is from following the steps - twice, where the role of $S_1^n$ and $S_2^n $ are changed for the second term.
Then, by replacing $Q=(\tilde{Q}, Z^n)$ with $W=(\tilde{Q}, Z^n)$ in -, we can show by following the same steps that, $$\frac{1}{n}I(X_1^n, X_2^n; Y^n | Z^n) \leq I(X_{1}, X_{2}; Y| W)$$ which, by combining with and leads to .
We lastly show that $$\begin{aligned}
\frac{1}{n}I(X_1^n, X_2^n; Y^n)
&\geq \frac{1}{n} I(S_1^n, S_2^n, Z^n; Y^n) \label{eq8880}\\
&=\frac{1}{n} I(S_1^n, Z^n; Y^n) + \frac{1}{n} I(S_2^n; Y^n|S_1^n, Z^n) \\
&= \frac{1}{n} (I(Z^n; Y^n) + I(S_1^n; Y^n|Z^n) + H(S_2^n|Z^n)- H(S_2^n| Y^n, S_1^n, Z^n))\label{eq888} \\
&\geq \frac{1}{n} (I(Z^n; Y^n) + I(S_1^n; Y^n|Z^n) + H(S_2^n|Z^n) - H(S_2^n| Y^n, Z^n)) \label{eqson4}\\
&= \frac{1}{n} (H(Z^n)\!-\!H(Z^n|Y^n) \!+\! I(S_1^n; Y^n|Z^n) \!+\! I(S_2^n;Y^n | Z^n))\notag\\
&\geq \frac{1}{n} (H(Z^n) + I(S_1^n; Y^n|Z^n) + I(S_2^n;Y^n | Z^n)-n\delta(P_e)) \label{8881}\\
&\geq H(Z) \!+\! R_{S_1|Z}(D_1\!+\!\epsilon) \!+\! R_{S_2|Z}(D_2\!+\!\epsilon) - \delta(P_e) \label{8882}\end{aligned}$$ where is from $Y^n-X_1^n X_2^n-S_1^n S_2^n Z^n $, is from $S_2^n-Z^n-S_1^n$, is from the fact that conditioning cannot increase entropy, is from Fano’s inequality combined with the data processing inequality, i.e., $$\begin{aligned}
H(Z^n|Y^n) \leq H(Z^n|\hat{Z}^n) \leq n\delta(P_e) \label{eq888last}\end{aligned}$$ where $\delta(P_e)\rightarrow 0$ as $P_e\rightarrow 0$ [@cover2012elements]. Equation is from the memoryless property of $Z^n$ and from following the steps - twice, the second one is when the role of $S_1^n$ is replaced with $S_2^n $.
Lastly, using random variable $\tilde{Q}$ that has been defined uniformly over $\{1, \ldots, n\}$ and independent of everything else, we derive the following. $$\begin{aligned}
\frac{1}{n}I(X_1^n, X_2^n; Y^n)
&\leq \frac{1}{n} \sum_{i=1}^n(H(Y_i)\!-\! H(Y_i|X_{1i}, X_{2i})) \label{eq888beginnew} \\
&= \frac{1}{n} \sum_{i=1}^nI(X_{1i}, X_{2i}; Y_i|\tilde{Q}=i) \\
&\leq I(X_{1\tilde{Q}}, X_{2\tilde{Q}}; Y_{\tilde{Q}}| \tilde{Q})\\
&= I(X_{1}, X_{2}; Y| \tilde{Q}) \label{eq8871}\\
&\leq H(Y) - H(Y|X_{1}, X_{2}) \\
&= I(X_{1}, X_{2}; Y) \label{eqcommon4last}\end{aligned}$$ where $X_1=X_{1\tilde{Q}}$, $X_2=X_{2\tilde{Q}}$, $Y=Y_{\tilde{Q}}$. Combining , , , and leads to .
In order to complete our proof, we demonstrate that $p(x_1, x_2|w)=p(x_1|w) p(x_2|w)$ for $w=(i, z^n)$ as in -. To this end, we show that $$\begin{aligned}
P(X_1=x_1, X_2=x_2| W=w)
&=P(X_{1i}=x_1, X_{2i}=x_2| \tilde{Q}=i, Z^n=z^n) \\
&=P(X_{1i}\!=\!x_1| \tilde{Q}\!=\!i, Z^n\!=\!z^n) P(X_{2i}\!=\!x_2| \tilde{Q}\!=\!i, Z^n\!=\!z^n) \label{distx1x2q2}\\
&=P(X_{1}=x_1| W=w) P(X_{2}=x_2| W=w) \end{aligned}$$ where holds since $X_{1i}-Z^n-X_{2i}$ for $i=1,\ldots, n$ due to $S_{1}^n-Z^n-S_{2}^n$, which can be observed from following the steps in -, which completes the proof.
[^1]: The material in Sections \[section3\]-\[Sec:CommonPart\] of this paper was presented in part at the 2016 IEEE International Symposium on Information Theory (ISIT’16).
[^2]: This research is sponsored in part by the U.S. Army Research Laboratory under the Network Science Collaborative Technology Alliance, Agreement Number W911NF-09-2-0053, and by the European Research Council Starting Grant project BEACON (project number 677854).
[^3]: We note that the bound obtained here from - slightly differs from the one reported in $(15)$ in [@7541654] due to a minor error in [@7541654].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Hardy’s proof is considered the simplest proof of nonlocality. Here we introduce an equally simple proof that (i) has Hardy’s as a particular case, (ii) shows that the probability of nonlocal events grows with the dimension of the local systems, and (iii) is always equivalent to the violation of a tight Bell inequality.'
author:
- 'Jing-Ling Chen'
- Adán Cabello
- 'Zhen-Peng Xu'
- 'Hong-Yi Su'
- Chunfeng Wu
- 'L. C. Kwek'
title: 'Hardy’s Paradox for High-Dimensional Systems: Beyond Hardy’s Limit'
---
[*Introduction.—*]{}Nonlocality, namely, the impossibility of describing correlations in terms of local hidden variables [@Bell64], is a fundamental property of nature. Hardy’s proof [@Hardy92; @Hardy93], in any of its forms [@Goldstein94; @Mermin94a; @Mermin94b; @KH05], provides a simple way to show that quantum correlations cannot be explained with local theories. Hardy’s proof is usually considered “the simplest form of Bell’s theorem” [@Mermin95].
On the other hand, if one wants to study nonlocality in a systematic way, one must define the local polytope [@Pitowsky89] corresponding to any possible scenario (i.e., for any given number of parties, settings, and outcomes) and check whether quantum correlations violate the inequalities defining the facets of the corresponding local polytope. These inequalities are the so-called [*tight*]{} Bell inequalities. In this sense, Hardy’s proof has another remarkable property: It is equivalent to a violation of a tight Bell inequality, the Clauser-Horne-Shimony-Holt (CHSH) inequality [@CHSH69]. This was observed in [@Mermin94a].
Hardy’s proof requires two observers, each with two measurements, each with two possible outcomes. The proof has been extended to the case of more than two measurements [@Hardy97; @BBDH97], and more than two outcomes [@KC05; @SG11; @RZS12]. However, none of these extensions is equivalent to the violation of a tight Bell inequality.
The aim of this Letter is to show that, if we remove the requirement that the measurements have two outcomes, then Hardy’s proof can be formulated in a much powerful way. The new formulation shows that the maximum probability of nonlocal events, which has a limit of $0.09$ in Hardy’s formulation and previously proposed extensions, actually grows with the number of possible outcomes, tending asymptotically to a limit that is more than four times higher than the original one. Moreover, for any given number of outcomes, the new formulation turns out to be equivalent to a violation of a tight Bell inequality, a feature that suggest that this formulation is more fundamental than any other one proposed previously. All this while preserving the simplicity of Hardy’s original proof.
[*A new formulation of Hardy’s paradox.—*]{}Let us consider two observers, Alice, who can measure either $A_1$ or $A_2$ on her subsystem, and Bob, who can measure $B_1$ or $B_2$ on his. Suppose that each of these measurements has $d$ outcomes that we will number as $0,1,2,\ldots,d-1$. Let us denote as $P(A_2 < B_1)$ the joint conditional probability that the result of $A_2$ is strictly smaller than the result of $B_1$, that is, $$P(A_2 < B_1)=\sum_{m<n}P(A_2=m, B_1=n),$$ with $m, n \in \{0, 1, \ldots, d-1\}$. Explicitly, for $d=2$, $P(A_2 < B_1)=P(A_2=0,B_1=1)$; for $d=3$, $P(A_2 <
B_1)=P(A_2=0,B_1=1)+P(A_2=0,B_1=2)+P(A_2=1,B_1=2)$, etc.
Then, the proof follows from the fact that, according to quantum theory, there are two-qudit entangled states and local measurements satisfying, simultaneously, the following conditions:
\[E1\] $$\begin{aligned}
&P(A_2 < B_1) = 0, \label{E1a}\\
&P(B_1 < A_1) = 0, \label{E1b}\\
&P(A_1 < B_2) = 0, \label{E1c}\\
&P(A_2 < B_2) > 0. \label{E1d}\end{aligned}$$
Therefore, if events $A_2<B_1$, $B_1<A_1$, and $A_1<B_2$ never happen, then, in any local theory, event $A_2
< B_2$ must never happen either. However, this is in contradiction with (\[E1d\]).
If $d=2$, the proof is exactly Hardy’s [@Hardy92; @Hardy93].
[*Beyond Hardy’s limit.—*]{}Let us define, $$P_{\rm Hardy}=\max P(A_2 < B_2)$$ satisfying conditions (\[E1a\])–(\[E1c\]). For $d=2$, $$\begin{aligned}
\label{Hlimit}
P^{(d=2)}_{\rm Hardy}=\frac{5\sqrt{5} - 11}{2}\approx 0.09,\end{aligned}$$ and is achieved with two-qubit systems [@Hardy92; @Hardy93].
In previous extensions of Hardy’s paradox to two-qudit systems [@KC05; @SG11; @RZS12], (\[Hlimit\]) is also the maximum probability of events that cannot be explained by local theories.
For example, the extension considered in Ref. [@KC05] is based on the following four probabilities: $P(A_1 = 0, B_1 = 0) = 0$, $P(A_1 \neq 0, B_2 = 0) = 0$, $P(A_2 = 0, B_1 \neq 0) = 0$, and $P(A_2 = 0, B_2 = 0) = P_{\rm KC} > 0$. Ref. [@SG11] proves that, for two-qutrit systems, $\max P_{\rm KC}$ equals (\[Hlimit\]), and conjectures that $\max P_{\rm KC}$ is always (\[Hlimit\]) for arbitrary dimension. Ref. [@RZS12] provides a proof of this conjecture.
Interestingly, in the proof presented in the previous section, $P_{\rm Hardy}$ equals Hardy’s limit (\[Hlimit\]) for $d=2$, but this is not longer true for higher dimensional systems.
To show this, we will consider pure states satisfying the three conditions (\[E1a\])–(\[E1c\]). An arbitrary two-qudit pure state can be written as $$\begin{aligned}
|\Psi\rangle = \sum\limits_{i = 0}^{d-1} \sum\limits_{j = 0}^{d-1}
h_{ij} |i\rangle_A|j\rangle_B,\end{aligned}$$ where the basis states $|i\rangle_A, |j\rangle_B \in \{|0\rangle, |1\rangle, \ldots, |d-1\rangle\}$, and $h_{ij}$ are coefficients satisfying the normalization condition $\sum_{ij} |h_{ij}|^2 = 1$.
The coefficients $h_{ij}$ completely determine the state $|\Psi\rangle$. We can associate any two-qudit state $|\Psi\rangle$ with a coefficient-matrix $H=(h_{ij})_{d \times d}$, where $i, j=0, 1, \ldots, d-1$, and $h_{ij}$ is the $i$-th row and the $j$-column element of the $d\times d$ matrix $H$. The connection between the coefficient-matrix $H$ and the two reduced density matrices of $|\Psi\rangle\langle\Psi|$ is
$$\begin{aligned}
\rho_A&={\mbox{tr}}_B(|\Psi\rangle\langle\Psi|)=HH^\dag, \\
\rho_B&={\mbox{tr}}_A(|\Psi\rangle\langle\Psi|)=H^T(H^T)^\dag,\end{aligned}$$
where $T$ for matrix transpose and $H^\dag$ is the hermitian conjugate matrix of $H$.
The probability $P(A_i=m, B_j=n)$ can be calculated as $$\begin{aligned}
\label{PRab}
P(A_i=m, B_j=n)={\mbox{tr}}[(\hat{\Pi}_{A_i}^m\otimes
\hat{\Pi}_{B_j}^n)\rho],\end{aligned}$$ where $\hat{\Pi}_{A_i}^m$ and $\hat{\Pi}_{B_j}^n$ are projectors, and $\rho=|\Psi\rangle\langle\Psi|$. Explicitly, the projectors are given by
\[PR1\] $$\begin{aligned}
&\hat{\Pi}_{A_1}^m=\mathcal {U}_1 \;|m\rangle\langle m| \;\mathcal {U}_1^\dag, \label{PR1a}\\
&\hat{\Pi}_{B_1}^n=\mathcal {V}_1 \;|n\rangle\langle n| \;\mathcal {V}_1^\dag, \label{PR1b}\\
&\hat{\Pi}_{A_2}^m=\mathcal {U}_2 \;|m\rangle\langle m| \;\mathcal {U}_2^\dag, \label{PR1c}\\
&\hat{\Pi}_{B_2}^n=\mathcal {V}_2 \;|n\rangle\langle n| \;\mathcal {V}_2^\dag, \label{PR1d}\end{aligned}$$
where $\mathcal {U}_1$, $\mathcal {V}_1$, $\mathcal {U}_2$, and $\mathcal {V}_2$ are, in general, $SU(d)$ unitary matrices. To calculate $P^{\rm opt}_{\rm Hardy}$, it is sufficient to choose the settings $A_1$ and $B_1$ as the standard bases, i.e., taking $\mathcal {U}_1=\mathcal {V}_1=\openone$, where $\openone$ is the identity matrix. Evidently, the condition (\[E1b\]) leads to $h_{ij}=0$, for $i>j$. This implies that the matrix $H$ is an upper-triangular matrix.
In Table \[Table1\], we list the optimal values of $P^{\rm
opt}_{\rm Hardy}$ for $d=2,\dots, 7$. The corresponding optimal Hardy states $H^{\rm opt}$ are explicitly given in the Appendix.
The calculations for $d>7$ are beyond our computers capability. However, we observe that $H^{\rm opt}$, written in the representation of $H$, have reflection symmetry with respect to the anti-diagonal line, that is, $h_{ij}=h_{d-1-j, d-1-i}$. We use this to calculate approximately the maximum probability for nonlocal events $P^{\rm app}_{\rm Hardy}$, by using a special class of states $H^{\rm app}$. The explicit form of states $H^{\rm app}$ is given in the Appendix. This allows us to go beyond $d=7$ and compute $P^{\rm app}_{\rm Hardy}$ from $d=2$ to $d=28000$. In Fig. \[fig1\], we have plotted $P^{\rm app}_{\rm Hardy}$ from $d=2$ to $d=1000$, showing that $P^{\rm app}_{\rm Hardy}$ increases with the dimension. Values for higher dimensions are given in the Appendix.
In Table \[Table1\], we also compare the $P_{\rm Hardy}$ for the optimal states and the approximate optimal states. This allows us to speculate that the asymptotic limit may be a little higher than the one showed in Fig. \[fig1\]. However, the limit $1/2$ can never be surpassed since $P(A_2>B_2)$ is always bigger than $P(A_2<B_2)$ as observed in the numerical computations. At this point, we do not know whether or not $1/2$ may be the asymptotic limit.
$d$ 2 3 4 5 6 7
--------------------------- ---------- ---------- ---------- ---------- ----------- ----------
$P^{\rm opt}_{\rm Hardy}$ 0.090170 0.141327 0.176512 0.203057 0.224221 0.241728
$P^{\rm app}_{\rm Hardy}$ 0.088889 0.138426 0.171533 0.195869 0.214825 0.230172
Error Rates 0.014207 0.020527 0.020288 0.035399 0.0419051 0.047807
![(Color online) $P^{\rm
app}_{\rm Hardy}$ from $d=2$ to $d=1000$.[]{data-label="fig1"}](phardyto1000.eps "fig:"){width="80mm"}\
[*Degree of entanglement.—*]{}Hardy’s proof does not work for maximally entangled states. The same is true for the proof introduced here. However, in out proof, as $d$ increases, the degree of entanglement tends to 1. To show this, we use the generalized concurrence or degree of entanglement [@AF01] for two-qudit systems given by $$\begin{aligned}
\mathcal
{C}=\sqrt{\frac{d}{d-1}\biggr[1-{\mbox{tr}}(\rho_A^2)\biggr]}=\sqrt{\frac{d}{d-1}\biggr[1-{\mbox{tr}}(\rho_B^2)\biggr]}.\end{aligned}$$
In Table \[Table2\], we have plotted $\mathcal{C}$ for the optimal Hardy’s states and the approximate Hardy’s states. From Table \[Table2\], we observe that, for $d=2$, the optimal Hardy’s state occurs at $\mathcal {C}^{\rm opt}\approx 0.763932$, and this value increases to $\mathcal {C}^{\rm opt}\approx 0.827702$ when $d=5$. For a fixed $d$, the corresponding $\mathcal {C}^{\rm app}$ is larger than that of $\mathcal {C}^{\rm app}$, and it also increases with the dimension $d$. For $d=800$, $\mathcal {C}^{\rm app}\approx
0.998062$, and tends to 1 as $d$ grows.
$d$ 2 3 4 5 6 7
-------------------- ---------- ---------- ---------- ---------- ---------- ----------
Optimal States 0.763932 0.793888 0.813483 0.827702 0.838679 0.847510
Approximate States 0.825885 0.845942 0.861735 0.874459 0.884926 0.893695
Finally, we can prove that the proof cannot work for two-qudit maximally entangled states, $$\begin{aligned}
|\Psi\rangle_{\rm MES}=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}|j\rangle_A |j\rangle_B.\end{aligned}$$
*Proof:* ${\mbox{tr}}[(\hat{\Pi}_{A_1}^m\otimes \hat{\Pi}_{B_1}^n)
|\Psi\rangle\langle\Psi|]$ can be expressed as $${\mbox{tr}}[ (|m\rangle\langle m| \otimes
|n\rangle\langle n| )(\mathcal {U}^\dag_1\otimes \mathcal
{V}^\dag_1) |\Psi\rangle\langle\Psi| (\mathcal {U}_1\otimes \mathcal
{V}_1)].$$ We will use $$\begin{aligned}
H_{\rm MES}\mapsto |\Psi\rangle_{\rm MES}, \;\; H_{1}\mapsto
(\mathcal {U}^\dag_1\otimes \mathcal {V}^\dag_1) |\Psi\rangle_{\rm
MES}.\end{aligned}$$ Taking into account that: (i) given a pure state $H\mapsto|\Psi\rangle_{AB}$ and a local action $U$ acting on Alice (the first part) and $V$ acting on Bob (the second part), then $$\begin{aligned}
H'\mapsto(U \otimes V) |\Psi\rangle_{AB} = U H V^T.\end{aligned}$$ (ii) Eq. (\[E1b\]) requires $H'$ to be an upper-triangular matrix, and (iii) $H_{\rm
MES}=\frac{1}{\sqrt{d}}\openone$. Then, we have the solution $$\mathcal {U}_1\mathcal {V}^T_1=\mathcal {D}_1,\label{D1}$$ where $\mathcal {D}_1={\rm
diag}(e^{i\chi_0},e^{i\chi_1},\ldots,e^{i\chi_{d-1}})$. Similarly, from (\[E1a\]) and (\[E1c\]), we have $$\mathcal {U}_1\mathcal {V}^T_2=\mathcal {D}_2,\;\mathcal
{U}_2\mathcal {V}^T_1=\mathcal {D}_3,\label{D23}$$ where $\mathcal {D}_2,\mathcal {D}_3$ are diagonal matrices similar to $\mathcal {D}_1$. From (\[D1\]) and (\[D23\]) we have $$\mathcal {U}_2\mathcal {V}^T_2=\mathcal {D}_3\mathcal
{D}^\dagger_1\mathcal {D}_2,$$ which directly leads to $P(A_2<B_2)=0$ for $|\Psi\rangle_{\rm MES}$. This ends the proof.
[*Connection to tight Bell inequalities.—*]{}As it can be easily seen, for any $d$, our proof can be transformed into the following Bell inequality: $$\label{ZGineq}
\begin{split}
&P(A_2 < B_1)+P(B_1 < A_1) \\
&+ P(A_1 < B_2)-P(A_2 < B_2) \stackrel{\mbox{\tiny{ LHV}}}{\geq} 0,
\end{split}$$ where LHV indicates that the bound is satisfied by local hidden variable theories. The interesting point is that, for any $d$, inequality (\[ZGineq\]) is equivalent to the Zohren and Gill’s version [@ZG08] of the Collins-Gisin-Linden-Massar-Popescu inequalities (the plural because there is a different inequality for each $d$) [@CGLMP02], which are tight Bell inequalities for any $d$ [@Masanes03]. This feature distinguishes our proof from any previously proposed nonlocality proof having Hardy’s as a particular case.
[*Conclusions.—*]{}Hardy’s proof is considered the simplest proof of nonlocality. Here we have introduced an equally simple proof that reveals much more about nonlocality in the case that the local systems are qudits. When $d=2$, the proof is exactly Hardy’s, but for $d>2$ the probability of nonlocal events grows with $d$, so, for high $d$, this probability is more than four times larger than in Hardy’s and in previous extensions to two-qudit systems. Interestingly, we have showed that, for any $d$, our proof is always equivalent to the violation of a tight Bell inequality. This suggests that ours is the most natural and powerful generalization of Hardy’s paradox when higher-dimensional systems are considered.
J.L.C. is supported by the National Basic Research Program (973 Program) of China under Grant No. 2012CB921900 and the NSF of China (Grant Nos. 10975075 and 11175089). A.C. is supported by Project No. FIS2011-29400 with FEDER funds (MINECO, Spain). This work is also partly supported by the National Research Foundation and the Ministry of Education, Singapore.
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Appendix A: Optimal Hardy states
================================
The optimal Hardy states $H_d$ for $d = 2,\ldots,7$ are
$$\begin{aligned}
H_2 = \left(
\begin{array}{cc}
0.618034 & 0.485868 \\
0 & 0.618034 \\
\end{array}
\right),\end{aligned}$$
$$\begin{aligned}
H_3 = \left(
\begin{array}{ccc}
0.498328 & 0.316483 & 0.329301 \\
0 & 0.441108 & 0.316483 \\
0 & 0 & 0.498328 \\
\end{array}
\right),\end{aligned}$$
$$\begin{aligned}
H_4 = \left(
\begin{array}{cccc}
0.429796 & 0.262169 & 0.224332 & 0.249934 \\
0 & 0.376021 & 0.217224 & 0.224332 \\
0 & 0 & 0.376021 & 0.262169 \\
0 & 0 & 0 & 0.429796 \\
\end{array}
\right),\end{aligned}$$
$$\begin{aligned}
H_5 = \left(
\begin{array}{ccccc}
0.383613 & 0.230044 & 0.189636 & 0.175427 & 0.201533 \\
0 & 0.334102 & 0.185035 & 0.157012 & 0.175427 \\
0 & 0 & 0.33072 & 0.185035 & 0.189636 \\
0 & 0 & 0 & 0.334102 & 0.230044 \\
0 & 0 & 0 & 0 & 0.383613 \\
\end{array}
\right),\end{aligned}$$
$$\begin{aligned}
H_6 = \left(
\begin{array}{cccccc}
0.349686 & 0.207877 & 0.16845 & 0.150559 & 0.144455 & 0.16883 \\
0 & 0.303795 & 0.165105 & 0.134967 & 0.125208 & 0.144455 \\
0 & 0 & 0.29972 & 0.160666 & 0.134967 & 0.150559 \\
0 & 0 & 0 & 0.29972 & 0.165105 & 0.16845 \\
0 & 0 & 0 & 0 & 0.303795 & 0.207877 \\
0 & 0 & 0 & 0 & 0 & 0.349686 \\
\end{array}
\right),\end{aligned}$$
$$\begin{aligned}
H_7 = \left(
\begin{array}{ccccccc}
0.323377 & 0.191279 & 0.153539 & 0.135037 & 0.12545 & 0.122887 & 0.145233 \\
0 & 0.280442 & 0.150851 & 0.121193 & 0.108665 & 0.104707 & 0.122887 \\
0 & 0 & 0.276282 & 0.145271 & 0.117498 & 0.108665 & 0.12545 \\
0 & 0 & 0 & 0.275414 & 0.145271 & 0.121193 & 0.135037 \\
0 & 0 & 0 & 0 & 0.276282 & 0.150851 & 0.153539 \\
0 & 0 & 0 & 0 & 0 & 0.280442 & 0.191279 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.323377 \\
\end{array}
\right).\end{aligned}$$
Appendix B: Approximate optimal Hardy states
============================================
The form of $H_d$ for $d = 2,\ldots,7$ suggests to define the approximate optimal Hardy states as follows: $$\begin{aligned}
H_d^{\rm app}=\left(
\begin{array}{cccccc}
\alpha_1 & \alpha_2 & \alpha_3 & \cdots & \alpha_{d-1} & \alpha_d\\
& \alpha_1 & \alpha_2 & \cdots & \alpha_{d-2} & \alpha_{d-1}\\
& & \ddots & \ddots & \vdots & \vdots\\
& & & \alpha_1& \alpha_2 &\alpha_3\\
& & & & \alpha_1 &\alpha_2\\
& & & & &\alpha_1\\
\end{array}
\right),\end{aligned}$$ where $$\begin{aligned}
\alpha_r=\frac{\beta_r}{\sqrt{d+1-r}}, \;\; r=1, 2, \ldots, d,\end{aligned}$$ with $\beta_r>0$ satisfying the following relations:
$$\begin{aligned}
&\beta_1:\beta_2:\beta_3:\cdots:\beta_d=1:\frac{1}{2}:\frac{1}{3}:\cdots:\frac{1}{d},\\
&\sum_{r=1}^d \beta_r^2=1.\end{aligned}$$
In Table \[to28000\] we have listed $P^{\rm app}_{\rm Hardy}$ up to $d=28000$.
$d$ $P^{\rm app}_{\rm Hardy}$ $d$ $P^{\rm app}_{\rm Hardy}$ $d$ $P^{\rm app}_{\rm Hardy}$ $d$ $P^{\rm app}_{\rm Hardy}$
----- --------------------------- ------ --------------------------- ------ --------------------------- ------- ---------------------------
2 0.088889 300 0.405106 2000 0.414711 10000 0.416300
10 0.263168 400 0.407749 2200 0.414885 11000 0.416339
20 0.316491 500 0.409394 2400 0.415031 12000 0.416371
30 0.340836 600 0.410520 2600 0.415156 13000 0.416398
40 0.355158 700 0.411341 2800 0.415263 14000 0.416421
50 0.364700 800 0.411966 3000 0.415357 16000 0.416459
60 0.371554 900 0.412459 4000 0.415687 18000 0.416489
70 0.376736 1000 0.412857 5000 0.415889 20000 0.416513
80 0.380803 1200 0.413464 6000 0.416024 22000 0.416533
90 0.384085 1400 0.413903 6000 0.416024 24000 0.416549
100 0.386793 1600 0.414230 8000 0.416196 26000 0.416563
200 0.400116 1800 0.414499 9000 0.416254 28000 0.416575
: $P^{\rm app}_{\rm Hardy}$ from $d=2$ to $d=28000$.[]{data-label="to28000"}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We examine the elliptic system given by $$\label{system_abstract}
-\Delta u = v^p, \qquad -\Delta v = u^\theta, \qquad \mbox{ in } \IR^N,$$ for $ 1 < p \le \theta$ and the fourth order scalar equation $$\label{fourth_abstract}
\Delta^2 u = u^\theta, \qquad \mbox{in $ \IR^N$,}$$ where $ 1 < \theta$. We prove various Liouville type theorems for positive stable solutions. For instance we show there are no positive stable solutions of (\[system\_abstract\]) (resp. (\[fourth\_abstract\])) provided $ N \le 10$ and $ 2 \le p \le \theta$ (resp. $ N \le 10$ and $1 < \theta$). Results for higher dimensions are also obtained.
These results regarding stable solutions on the full space imply various Liouville theorems for positive (possibly unstable) bounded solutions of $$\label{eq_half_abstract}
-\Delta u = v^p, \qquad -\Delta v = u^\theta, \qquad \mbox{ in } \IR^{N-1},$$ with $ u=v=0$ on $ \partial \IR^N_+$. In particular there is no positive bounded solution of (\[eq\_half\_abstract\]) for any $ 2 \le p \le \theta$ if $ N \le 11$. Higher dimensional results are also obtained.
author:
- |
Craig Cowan\
[*Department of Mathematical Sciences*]{}\
[*University of Alabama in Huntsville*]{}\
[*258A Shelby Center*]{}\
*Huntsville, AL 35899\
[*[email protected]*]{}*
title: 'Liouville theorems for stable Lane-Emden systems and biharmonic problems'
---
\[0\][ ]{}
\[1\][ \#1 \_H ]{}
[*2010 Mathematics Subject Classification: 35J61, 35J47.*]{}\
[*Key words: Biharmonic, entire solutions, Liouville theorems, Stability, Lane-Emden Systems, Half-space*]{}.
Introduction
============
In this article we examine the nonexistence of positive classical stable solutions of the system given by $$\label{eq}
-\Delta u = v^p, \qquad -\Delta v = u^\theta, \qquad \mbox{ in } \IR^N,$$ where $ 1 < p \le \theta$. We also examine the nonexistence of positive classical stable solutions of the fourth order equation given by $$\label{fourth}
\Delta^2 u = u^\theta \qquad \mbox{ in } \IR^N,$$ where $ \theta>1$.
We now define the notion of a stable solution and for this we prefer to examine a slight generalization of (\[eq\]) given by $$\label{eq_2}
-\Delta u = f(v), \qquad -\Delta v = g(u), \qquad \mbox{ in $\IR^N$,}$$ where $ f,g$ are positive and increasing on $(0,\infty)$.
We say a smooth positive solution $(u,v)$ of (\[eq\_2\]) is stable provided there exists $ 0 < \zeta, \chi$ smooth with $$\label{stand}
-\Delta \zeta = f'(v) \chi, \qquad -\Delta \chi = g'(u) \zeta \quad \mbox{in $ \IR^N$}.$$
This definition is motivated from [@Mont], also see (\[mont\_sta\]).
\[equi\] Note that the standard notion of a stable positive solution of $ \Delta^2 u = u^\theta$ in $ \IR^N$, is that $$\label{standard}
\int \theta u^{\theta-1} \gamma^2 \le \int (\Delta \gamma)^2,$$ for all $ \gamma \in C_c^\infty(\IR^N)$. For our approach we prefer to recast (\[fourth\]) into the framework of (\[eq\]). So towards this suppose $ 1 < \theta$ and $ 0 < u $ is a smooth solution of (\[fourth\]). Define $ v:=-\Delta u$. By [@Wei_10] $ v >0$ and hence $(u,v):=(u, -\Delta u)$ is a smooth positive solution of (\[eq\]) with $p=1$. One now has two options for the notion of the stability of (\[fourth\]). Either one views the equation as a scalar equation and uses the standard notion (\[standard\]), when we do this we will say $u$ is a stable solution of (\[fourth\]) or we view the solution as a solution of the system and we use the notion defined in (\[stand\]), when we do this we will say $(u,v)$ is a stable solution of (\[eq\]) with $p=1$. See Lemma \[equivalence\] for a relationship between these notions of stability.
We define some parameters before stating our main results. Given $ 1 \le p \le \theta$ we define $$t_0^-:= \sqrt{ \frac{p \theta (p+1)}{\theta+1}} - \sqrt{ \frac{p \theta (p+1)}{\theta+1} - \sqrt{ \frac{p \theta (p+1)}{\theta+1}}},$$ $$t_0^+:= \sqrt{ \frac{p \theta (p+1)}{\theta+1}} + \sqrt{ \frac{p \theta (p+1)}{\theta+1} - \sqrt{ \frac{p \theta (p+1)}{\theta+1}}}.$$ Properties of $ t_0^-,t_0^+$:\
(i) $ t_0^- \le 1 \le t_0^+$ and these inequalities are strict except when $ p=\theta=1$.\
(ii) $t_0^-$ is decreasing and $ t_0^+$ is increasing in $ z:= \frac{p \theta (p+1)}{\theta+1}$ and $ \lim_{z \rightarrow \infty} t_0^-=\frac{1}{2}$.\
We now state out main theorem.
(Lane-Emden System) \[MAIN\]
1. Suppose $ 2 \le p \le \theta$ and $$\label{cond_syst}
N <2 + \frac{4( \theta+1)}{p \theta-1} t_0^+.$$ Then there is no positive stable solution of (\[eq\]). In particular there is no positive stable solution of (\[eq\]) for any $ 2 \le p \le \theta$ if $ N \le 10$; see Remark \[computations\].
2. Suppose $ 1 < p \le \theta$, $ 2 t_0^- <p$ and (\[cond\_syst\]) holds. Then there is no positive stable solution $(u,v)$ of (\[eq\]).
\[MAIN\_four\] (Fourth Order Scalar Equation) Suppose that $ 1=p < \theta$ and $$\label{four_Extremal}
N < 2 + \frac{4(\theta+1)}{\theta-1} t_0^+.$$ Then there is no positive stable solution of (\[eq\]). In particular there is no positive stable solution of (\[eq\]), when $p=1$, for any $ 1 < \theta$ if $ N \le 10$.
We now turn our attention to the case of half space. Consider the Lane-Emden system given by $$\label{eq_half}
\left\{ \begin{array}{rll}
\hfill -\Delta u &=& v^p \qquad \; \mbox{ in } \IR^N_+ \\
\hfill -\Delta v &=& u^\theta \qquad \; \mbox{ in } \IR^N_+, \\
\hfill u &=& v =0 \quad \mbox{ on } \partial \IR^N_+,
\end{array}\right.$$ where $ 1 < p \le \theta$.
This is an updated version of the original work which contained results only regarding stable solutions on the full space. All results on the half space, in particular Theorem \[system\_thm\_half\], did not appear in the original work. Since the original work appeared there have been many very nice improvements, extensions and or related works. In [@new_3] the range of exponents in Theorem \[MAIN\_four\] is improved. In [@new_1] they examine (\[eq\]) but without any stability assumptions. They obtain optimal results regarding the existence versus nonexistence of positive radial solutions of (\[eq\]). In [@new_2] the problem $ \Delta^2 u = |u|^{p-1} u$ in $ \IR^N$ is examined. They give a complete classification of stable and finite Morse index solutions (no positivity assumptions).
We now state our main theorem.
\[system\_thm\_half\] (Lane-Emden System in $\IR_+^N$)
1. Suppose $ 2 \le p \le \theta$ and $$\label{cond_syst_Half}
N-1 <2 + \frac{4( \theta+1)}{p \theta-1} t_0^+.$$ Then there is no positive bounded solution of (\[eq\_half\]). In particular there is no positive bounded solution of (\[eq\_half\]) for any $ 2 \le p \le \theta$ if $ N \le 11$; see Remark \[computations\].
2. Suppose $ 1 < p \le \theta$, $ 2 t_0^- <p$ and (\[cond\_syst\]) holds. Then there is no positive bounded solution of (\[eq\_half\]).
\[computations\]
We are interested in obtaining lower bounds on the right hand side of (\[cond\_syst\]), in the case where $ 2 \le p \le \theta$, and so we set $ f(p,\theta):= \frac{4(\theta+1)}{p \theta-1} t_0^+$. We rewrite $f$ using the change of variables $ z= \frac{p \theta(p+1)}{\theta+1}$ to arrive at $$\tilde{f}(p,z)= \frac{4p}{z-p} \left( \sqrt{z}+ \sqrt{ z - \sqrt{z}} \right),$$ and the transformed domain is given by $$\mathcal{D}=\{ (p,z): p \ge 2, \; p^2 \le z \le p^2+p \}.$$ A computer algebra system easily shows that $ \tilde{f}>8$ on $\partial \mathcal{D}$. Note that $ \partial_p \tilde{f} > 0$ on $ \mathcal{D}$ and so we have $ \tilde{f} >8$ on $ \mathcal{D}$ which gives us the desired result.
There has been much work done on the existence and nonexistence of positive classical solutions of the Lane-Emden equation given by $$\label{lane_class}
-\Delta u = u^\theta, \qquad \mbox{in $ \IR^N$,}$$ for instance see [@Caf], [@chen], [@gidas],[@Gidas]. It is known that there are no positive classical solutions of (\[lane\_class\]) provided that $$1< \theta < \frac{N+2}{N-2},$$ and in the case of $ N =2$ there is no positive solution for any $ \theta>1$. We further remark that this is an optimal result. It is well known that a Liouville theorem related to (\[lane\_class\]) implies apriori estimates of solutions of the same equation on a bounded domain, see [@Caf], [@gidas]. We remark that this equation originally appeared in astrophysics, where it was a model for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid.
In [@Wang_solo] and [@Gui_Ni_Wang] parabolic versions of (\[lane\_class\]) were examined and in particular they were interested in the asymptotics. They also obtained various properties of solutions of (\[lane\_class\]) and in particular their results easily imply that for $ N \ge 11$ and $ \theta \ge \theta_{JL}$, where $\theta_{JL}$ is the so called Joseph-Lundgren exponent, there exists positive stable radial solutions of (\[lane\_class\]).
In [@farina] the equation $$\label{far}
-\Delta u = |u|^{\theta-1} u, \qquad \mbox{in $ \IR^N$,}$$ was examined. They completely classified the finite Morse index solutions of (\[far\]). It was shown there exists nontrivial finite Morse index solutions of (\[far\]) if and only if $ N \ge 11$ and $ \theta \ge \theta_{JL}$.
In [@Wei_dong] the nonexistence of stable solutions of (\[fourth\]) was examined. It was shown that there is no positive stable solutions of (\[fourth\]) provided either: $ N \le 8$ or $ N \ge 9$ and $ 1 < \theta < \frac{N}{N-8} + { \varepsilon}_N$ where $ { \varepsilon}_N $ is some positive, but unknown parameter. We also alert the reader to the work of [@Warn], where the nonexistence of stable solutions of $ \Delta^2 u = f(u)$ in $ \IR^N$ was examined for general nonlinearities $f$. Many interesting results were obtained.
As mentioned above Liouville theorems are extremely useful for the existence of apriori estimates of solutions on bounded domains. The nonexistence of nontrivial stable solutions of $ -\Delta u = g(u)$ in $ \IR^N$ is closely related to the regularity of the extremal solution associated with $$(Q)_\lambda \qquad \left\{
\begin{array}{ll}
-\Delta u =\lambda f(u) &\hbox{in }\Omega \subset \subset \IR^N, \\
u =0 &\hbox{on } \pOm
\end{array}
\right.$$ where $ \lambda $ is a positive parameter and $ f(u)$ is a nonlinearity which is related to $g$. Here the extremal solution $u^*$ is the minimal solution, ie. smallest in the pointwise sense, of $ (Q)_{\lambda^*}$ where $ \lambda^*$ is the largest parameter $\lambda$ such that $(Q)_\lambda$ has a weak solution. The critical fact in proving the regularity of $u^*$ in certain cases is the fact that $ u^*$ is a stable solution of $(Q)_{\lambda^*}$. See [@bcmr; @BV; @Cabre; @CC; @advection; @CR; @EGG; @GG; @Martel; @MP; @Nedev] for results concerning $(Q)_\lambda$.
We now examine some bounded domain analogs of (\[eq\]) and (\[fourth\]). We begin by examining $$\begin{aligned}
(P)_{\lambda,\gamma}\qquad \left\{ \begin{array}{lcl}
\hfill -\Delta u &=& \lambda f(v)\qquad \Omega \\
\hfill -\Delta v &=& \gamma g(u) \qquad \Omega, \\
\hfill u &=& v =0 \qquad \pOm,
\end{array}\right.
\end{aligned}$$ where $ \Omega$ is a bounded domain in $ \IR^N$ and where $f,g$ are smooth, positive increasing nonlinearities which are superlinear at $ \infty$. Set $ \mathcal{Q}=\{ (\lambda,\gamma): \lambda, \gamma >0 \}$, $ \mathcal{U}:= \left\{ (\lambda,\gamma) \in \mathcal{Q}: \mbox{ there exists a smooth solution $(u,v)$ of $(P)_{\lambda,\gamma}$} \right\},$ and set $ \Upsilon:= \partial \mathcal{U} \cap \mathcal{Q}$. Note $\Upsilon$ plays the role of the extremal parameter $ \lambda^*$ in the case of the system. Using monotonicity one can define an extremal solution $(u^*,v^*)$ for each $ (\lambda^*,\gamma^*) \in \Upsilon$.
To show the regularity of $(u^*,v^*)$ we will need to use the minimality of the minimal solutions to obtain added regularity. In [@Mont] a generalization of $(P)_{\lambda,\gamma}$ was examined and various properties we obtained. One important result was that the minimal solutions $(u,v)=(u_{\lambda,\gamma}, v_{\lambda,\gamma})$ were stable in the sense that there was some nonnegative constant $ \eta$ and $ 0 < \zeta,\chi \in H_0^1(\Omega)$ such that $$\label{mont_sta} -\Delta \zeta = \lambda f'(v) \chi + \eta \zeta, \quad -\Delta \chi = \gamma g'(u) \zeta + \eta \chi, \quad \mbox{in $ \Omega$.}$$ It is precisely this result that motivates our definition of a stable solution of (\[eq\]). Until recently it was not known hot to utilize the stability of solutions to obtain results regarding the regularity of the extremal solutions associated with the system $(P)_{\lambda,\gamma}$, except in very special cases. For instance in [@craig0] results were obtained in the case of $ f(v)=e^v$, $ g(u)=e^u$. Very recently, in [@craig_lane], the regularity of the extremal solutions associated with $(P)_{\lambda,\gamma}$ was examined in the case of $ f(v)=(v+1)^p$ and $ g(u)=(u+1)^\theta$ where $ 1 < p \le \theta$. It was shown that the associated extremal solutions were bounded provided condition (\[cond\_syst\]) holds. In [@craig_four] the fourth order problem $$\begin{aligned}
(N)_{\lambda}\qquad \left\{ \begin{array}{lcl}
\hfill \Delta^2 u &=& \lambda f(u)\qquad \quad \Omega, \\
\hfill u &=& \Delta u =0 \qquad \pOm,
\end{array}\right.
\end{aligned}$$ where $ \Omega $ is a bounded domain in $ \IR^N$ was examined in the case where $ f(u)=e^u$ and $ f(u)=(u+1)^\theta$. In the case of $ f(u)=e^u$, the previous best known result was from [@craig1] where it was shown that $ u^*$ was bounded provided $ N \le 8$. In [@craig_four] this was improved to $ N \le 10$ but this still falls far short of the expected optimal result of $ N \le 12$ after one considers the results of [@DDGM] on radial domains.
Two weeks after this work was made available online we received the manuscript [@NEW_GELF] where they also use this new idea of system stability for a scalar equation, see Lemma 2 in [@craig_four] or Lemma \[stabb\] in the current work. They examined $(N)_\lambda$ and (\[fourth\]) in the case where the nonlinearity is given by $f(u)=e^u$. In the case of the bounded domain they obtained an optimal result.
In [@craig_four] we showed the extremal solution associated with $ (N)_{\lambda}$ in the case of $ f(u)=(u+1)^\theta$ is bounded provided condition (\[four\_Extremal\]) holds. Again these were major improvement over the best known previous results, again from [@craig1], but fall short of the expected optimal results, see [@DFG] for the radial case. For more works related to $(N)_\lambda$ see [@BG; @CDG; @CEG; @GW].
Note that our results in the current work are expected after viewing the recent works on the regularity of the extremal solutions on bounded domains. It should be noted that even if this is to be expected, these results are not straightforward adaptions of the regularity results on bounded domains. We finish off by mentioning three related works on systems. The first two results [@fg; @fazly_system] deal with elliptic systems and stability. The third work, [@fazly], has many results. One of the results is the nonexistence of nontrivial stable solutions of $$-\Delta u = |x|^\alpha v^p, \qquad -\Delta v = |x|^\beta u^\theta \qquad \mbox{ in $\IR^N$,}$$ under certain restrictions of the involved parameters. For this result the methods developed in the current work are extended to handle the case of nonzero $ \alpha$ and $ \beta$.
The half space
--------------
We are also interested in Liouville theorems related to positive bounded solutions of (\[eq\_half\]). There has been much work done on these and related equations see, for instance, [@half_1; @half_2; @half_3; @quittner; @half_4]. The best known result to date regarding a Liouville theorem for positive bounded solutions of (\[eq\_half\]) is given by [@quittner]. Theorem \[system\_thm\_half\] improves this non existence result.
A brief outline of the approach
-------------------------------
Here we give a brief outline of the approach we take. Suppose $ (u,v)$ is a smooth positive stable solution of (\[eq\]) with $ 2 < p < \theta$. We begin by showing that stability implies $$\sqrt{p \theta} \int u^\frac{\theta-1}{2} v^\frac{p-1}{2} \phi^2 \le \int | \nabla \phi|^2,$$ for all $ \phi \in C_c^\infty(\IR^N)$. Using as a test function $ \phi=v^t \gamma$ where $ \gamma \in C_c^\infty(\IR^N)$ and using the pointwise comparison $ (\theta +1) v^{p+1} \ge (p+1) u^{\theta+1}$ in $ \IR^N$, which holds without stability; see [@souplet_4], we obtain an inequality roughly of the form $$\label{ten}
\int u^\theta v^{2t-1} \gamma^2 \le C_t \int v^{2t} | \nabla \gamma|^2,$$ for all $ t_0^- <t<t_0^+$. One also has the following integral estimates, see [@Mid] $$\label{twenty}
\int_{B_R} v^p \le C R^{N-2- \frac{2(p+1)}{p \theta-1}},$$ which also holds without the stability assumption. As a first attempt we assume that $ t_0^- < \frac{p}{2} < t_0^+$ and so we can take $ 2t =p$ in (\[ten\]) and we assume that $ 0 \le \gamma \le 1$ with $ \gamma=1 $ in $ B_R$ and is compactly supported in $ B_{2R}$. We then use (\[ten\]) and (\[twenty\]) to see that $$\int_{B_R} u^\theta v^{p-1} \le \frac{C}{R^2} \int_{B_{2R}} v^p \le C R^{ N-4- \frac{2(p+1)}{p \theta-1}},$$ for all $ R >0$. Provided this exponent is negative then we get a contradiction by sending $ R \rightarrow \infty$. Note this implies there is no positive stable solution of (\[eq\]) for $ N \le 4$ for any $ 2 \le p \le \theta$. Now since we expect $v$ to decay to zero at $\infty$ one would expect to obtain better results if we can choose $ t > \frac{p}{2}$. To do this we examine the elliptic equation which $ v^p$ satisfies and we use $L^1$ elliptic regularity theory along with (\[ten\]) and (\[twenty\]) to obtain integral estimates on $ v^{p \alpha}$ for $1 < \alpha < \frac{N}{N-2}$. We can iterate this procedure, as long as the range of $t$ in (\[ten\]) allows, looking at increasing powers of $ v$ to obtain integral estimates of $v$ for higher powers. These higher power integral estimates allow one to obtain better results.
We would like to thank Philippe Souplet for bringing to our attention the work [@PHAN]. In a previous version of this paper we had needed to assume that either: $ 2 \le \theta$ or $ u$ is bounded in Lemma \[pointwise\].
Proof of Theorem \[MAIN\] and \[MAIN\_four\].
=============================================
We begin with some integral estimates which are valid for any positive solution of (\[eq\]).
\[Mid\_lemma\] *[@Mid]* Suppose $(u,v)$ is a positive solution of (\[eq\]) with $ 1 \le p \le \theta$. Then $$\int_{B_R} v^p \le CR^{N-2- \frac{2(p+1)}{p \theta-1}},$$ $$\int_{B_R} u^\theta \le CR^{N-2- \frac{2(\theta+1)}{p \theta-1}}.$$
A crucial ingredient in our proof of Theorem \[MAIN\] is given by the following pointwise comparisons.
\[pointwise\]
1. [@PHAN], [@souplet_4] Suppose that $(u,v)$ is a smooth solution of (\[eq\]) and $ 1 < p \le \theta$. Then $$\label{point}
(\theta +1) v^{p+1} \ge (p+1) u^{\theta+1} \qquad \mbox{ in $ \IR^N$.}$$
2. [@Wei_dong] Suppose that $u$ is a smooth stable solution of (\[eq\]) with $ 1=p < \theta$. Then there exists a smooth positive stable bounded solution of (\[eq\]), which we denote by $ \tilde{u}$ and which satisfies $$\label{point_four}
(\theta +1) \tilde{v}^{2} \ge (p+1) \tilde{u}^{\theta+1} \qquad \mbox{ in $ \IR^N$,}$$ where $ \tilde{v}:= - \Delta \tilde{u}>0$.
When attempting to prove the nonexistence of positive stable solutions $(u,v)$ of (\[eq\]), in the case $p=1$, we can use the above lemma 2) to assume that $u$ is bounded.
The following lemma transforms our notion of a stable solution of (\[eq\]) into an inequality which allows the use of arbitrary test functions. A bounded domain version of this was proven in [@craig_lane] but we include the proof here for the readers sake. We remark that this result was motivated by a similar result in [@craig2].
[@craig_lane] \[stabb\]
Let $(u,v)$ denote a stable solution of (\[eq\_2\]). Then $$\label{second}
\int \sqrt{f'(v) g'(u)} \phi^2 \le \int | \nabla \phi|^2$$ for all $ \phi \in C_c^\infty(\IR^N)$.
Let $(u,v)$ denote a stable solution of (\[eq\_2\]) and so there is some $ 0 < \zeta, \chi $ smooth such that $$\frac{-\Delta \zeta}{\zeta} = f'(v) \frac{\chi}{\zeta}, \qquad \frac{-\Delta \chi}{\chi} = g'(u) \frac{\zeta}{\chi}, \qquad \mbox{ in $ \IR^N$.}$$ Let $ \phi,\psi \in C_c^\infty(\IR^N)$ and multiply the first equation by $ \phi^2$ and the second by $\psi^2$ and integrate over $ \IR^N$ to arrive at $$\int f'(v) \frac{\chi}{\zeta} \phi^2 \le \int | \nabla \phi|^2, \qquad \int g'(u) \frac{\zeta}{\chi} \psi^2 \le \int | \nabla \psi|^2,$$ where we have utilized the result that for any sufficiently smooth $ E>0$ we have $$\int \frac{-\Delta E}{E} \phi^2 \le \int | \nabla \phi|^2,$$ for all $ \phi \in C_c^\infty(\IR^N)$. We now add the inequalities to obtain $$\label{thing}
\int ( f'(v) \phi^2) \frac{\chi}{\zeta} + ( g'(u) \psi^2 )\frac{\zeta}{\chi} \le \int | \nabla \phi|^2 + | \nabla \psi|^2.$$ Now note that $$2 \sqrt{ f'(v) g'(u)} \phi \psi \le 2t f'(v) \phi^2 + \frac{1}{2t} g'(u) \psi^2,$$ for any $ t>0$. Taking $ 2t = \frac{\chi(x)}{\zeta(x)}$ gives $$2 \sqrt{ f'(v) g(u)} \phi \psi \le ( f'(v) \phi^2) \frac{\chi}{\zeta} + ( g'(u) \psi^2 )\frac{\zeta}{\chi},$$ and putting this back into (\[thing\]) gives the desired result after taking $ \phi=\psi$.
For integers $ k \ge -1$ define $ R_k:= 2^k R$ for $ R >0$.
\[one\] Suppose $ (u,v)$ is a smooth, positive stable solution of (\[eq\]) satisfying the hypothesis from Lemma \[pointwise\]. Then for all $ t_0^- < t <t^+_0$ there is some $C_t< \infty$ such that $$\int_{B_{R_k}} u^\theta v^{2t-1} \le \frac{C_t}{2^{2k}R^2} \int_{B_{R_{k+1}}} v^{2t} ,$$ for all $ 0<R <\infty$.
Let $ (u,v)$ denote a positive smooth stable solution of (\[eq\]). Let $ \gamma$ denote a smooth cut-off function which is compactly supported in $ B_{R_{k+1}}$ and which is equal to one in $ B_{R_k}$. Put $ \phi:= v^t \gamma$ into (\[second\]) to obtain $$\sqrt{p \theta} \int v^\frac{p-1}{2} u^\frac{\theta-1}{2} v^{2t} \gamma^2 \le
t^2 \int v^{2t-2} | \nabla v|^2 \gamma^2 + \int v^{2t} | \nabla \gamma|^2 + 2 t \int v^{2t-1} \gamma \nabla v \cdot \nabla \gamma.$$ We now re-write the left hand side as $$\sqrt{p \theta} \int u^\frac{\theta-1}{2} v^\frac{p+1}{2} v^{2t-1} \gamma^2,$$ and we now use the the pointwise bound (\[point\]) to see the left hand side is greater than or equal to $$\sqrt{ \frac{p \theta (p+1)}{\theta+1}} \int u^\theta v^{2t-1} \gamma^2,$$ and hence we obtain $$\sqrt{ \frac{p \theta (p+1)}{\theta+1}} \int u^\theta v^{2t-1} \gamma^2 \le t^2 \int v^{2t-2} | \nabla v|^2 \gamma^2 + \int v^{2t} | \nabla \gamma|^2 + 2 t \int v^{2t-1} \gamma \nabla v \cdot \nabla \gamma.$$ Multiply $ -\Delta v = u^\theta$ by $ v^{2t-1} \gamma^2$ and integrate by parts to obtain, after some rearrangement $$t^2 \int v^{2t-2} | \nabla v|^2 \gamma^2 \le \frac{t^2}{2t-1} \int u^\theta v^{2t-1} \gamma^2 - \frac{2 t^2}{2t-1} \int v^{2t-1} \gamma \nabla v \cdot \nabla \gamma.$$ We now use this to replace the first term on the right in the above inequality to obtain $$\left( \sqrt{ \frac{p \theta (p+1)}{\theta+1}} - \frac{t^2}{2t-1} \right) \int u^\theta v^{2t-1} \gamma^2 \le \int v^{2t} | \nabla \gamma|^2 - \frac{t-1}{2(2t-1)} \int v^{2t} \Delta (\gamma^2),$$ and from this we easily get the desired result after considering the support of $ \gamma$ and how $ | \nabla \gamma|^2 $ and $ \Delta \gamma$ scale.
In what follows we shall need the following result, which is just an $L^1$ elliptic regularity result with the natural scaling.
\[reg\] For any integer $k \ge 0$ and $ 1 \le \alpha < \frac{N}{N-2}$ there is some $ C=C(k,\alpha)< \infty$ such that for any smooth $ w \ge 0$ we have $$\left( \int_{B_{R_k}} w^\alpha dx \right)^\frac{1}{\alpha} \le C
R^{2 + N ( \frac{1}{\alpha}-1)} \int_{B_{R_{k+1}}} | \Delta w| + C R^{ N( \frac{1}{\alpha}-1)} \int_{B_{R_{k+1}}} w.$$
We give a brief sketch of the proof even though the result is well known.
After a scaling argument it is sufficient to show there is some $C>0$ such that $$\left( \int_{B_1} w^\alpha dx \right)^\frac{1}{\alpha} \le C \int_{B_2} |\Delta w| + C \int_{B_2} w,$$ for all smooth nonnegative $ w$. Let $ 0 \le \phi \le 1$ denote a smooth cut off with $ \phi=1$ in $B_1$ and compactly supported in $ B_\frac{3}{2}$ and set $ v=w \phi$. Then note that $L^1$ elliptic regularity theory gives $ \| v\|_{L^\alpha(B_2)} \le C \| \Delta v\|_{L^1(B_2)}$ and writing this out gives $$\| w\|_{L^\alpha(B_1)} \le C \int_{B_2} |\Delta w| + C \int_{B_2} w + C \int_{B_\frac{3}{2}} | \nabla w|,$$ where $C$ is a changing constant independent of $w$. To finish the proof we just need to control the first order term on the right. We decompose $w$ as $ w=w_1 +w_2$ where $ \Delta w_1 = \Delta w$ in $B_2$ with $ w_1=0$ on $B_2$ and where $ w_2$ is harmonic in $ B_2$ with $ w_2=w$ on $ \partial B_2$. Then by elliptic regularity theory $ \| \nabla w_1 \|_{L^1(B_2)} \le C \| \Delta w\|_{L^1(B_2)}$ and since $w_2$ is harmonic $ \| \nabla w_2 \|_{L^1(B_\frac{3}{2})} \le C \| w_2\|_{L^1(B_2)}$. Combining these results gives $$\int_{B_\frac{3}{2}} | \nabla w| \le C \int_{B_2} |\Delta w| + C \int_{B_2} w + C \int_{B_2} |w_1|,$$ and the last term on the right can be controlled by $ \| \Delta w\|_{L^1(B_2)}$. Recombining the results completes the proof.
We will bootstrap the following result which is the key to obtain higher integral powers of $v$ controlled by lower powers and, as mentioned in the brief outline, this is key in obtaining better nonexistence results.
\[initial\] Let $ (u,v)$ denote a positive stable solution of (\[eq\]) with $ 1 \le p < \theta$. Then for all $ 1 < \alpha < \frac{N}{N-2}$, $ t_0^- <t< t_0^+$ and nonnegative integers $k$ there is some $ C< \infty$ such that for all $ R \ge 1$ we have $$\label{final}
\left( \int_{B_{R_k}} v^{ 2 t \alpha} \right)^\frac{1}{2 t \alpha} \le C R^{ \frac{N}{2t} ( \frac{1}{\alpha}-1)} \left( \int_{B_{R_{k+3}}} v^{2t} \right)^\frac{1}{2 t}.$$
**Proof of Proposition \[initial\].** Let $t$ and $ \alpha$ be as in the hypothesis. Set $ w=v^{2t}$ and note that $$|\Delta w| \le 2t (2t-1) v^{2t-2} | \nabla v|^2 + 2 t v^{2t-1} u^\theta,$$ and also note that $ 2t-1 >0$ after considering the restrictions on $t$. From Lemma \[reg\] we have
$$\begin{aligned}
\label{eee}
\left( \int_{B_{R_k}} v^{2t \alpha} \right)^\frac{1}{\alpha} & \le & C_t R^{2+N ( \frac{1}{\alpha}-1)} \int_{B_{R_{k+1}}} v^{2t-2} | \nabla v|^2 \nonumber \\
&&+ C_t R^{2+N( \frac{1}{\alpha}-1)} \int_{B_{R_{k+1}}} v^{2t-1} u^\theta \nonumber \\
&& + C_t R^{N(\frac{1}{\alpha}-1)} \int_{B_{R_{k+1}}} v^{2t}.\end{aligned}$$
We begin by getting an upper bound on the gradient term. Let $ \phi$ denote a smooth cut off with $ \phi=1$ in $ B_{R_{k+1}}$ and compactly supported in $ B_{ R_{k+2}}$. Multiply $ -\Delta v = u^\theta$ by $ v^{2t-1} \phi^2$ and integrate by parts and apply Young’s inequality to arrive at an inequality of the form $$\int v^{2t-2} | \nabla v|^2 \phi^2 \le C \int u^\theta v^{2t-1} \phi^2 + C \int v^{2t} | \nabla \phi|^2,$$ and after considering the support of $ \phi$ and how $ | \nabla \phi|^2$ scales with respect to $R$ we obtain $$\label{part_1}
\int_{B_{R_{k+1}}} v^{2t-2} | \nabla v|^2 \le C \int_{B_{R_{k+2}}} u^\theta v^{2t-1} + \frac{C}{R^2} \int_{B_{R_{k+2}}} v^{2t}.$$ Putting (\[part\_1\]) into (\[eee\]) gives $$\label{fff}
\left( \int_{B_{R_k}} v^{2t \alpha} \right)^\frac{1}{\alpha} \le C R^{N( \frac{1}{\alpha}-1)+2} \int_{B_{R_{k+2}}} u^\theta v^{2t-1} + C R^{N( \frac{1}{\alpha}-1)} \int_{B_{R_{k+2}}} v^{2t}.$$ We now use Lemma \[one\] to eliminate the first term on the right hand side of (\[fff\]) to obtain $$\left( \int_{B_{R_k}} v^{2t \alpha} \right)^\frac{1}{\alpha} \le C R^{N ( \frac{1}{\alpha}-1)} \int_{B_{R_{k+3}}} v^{2t},$$ and the proof is complete after raising both sides to the power $ \frac{1}{2t}$.
$ \Box$
If one performs an iteration argument of the above result and pays some attention to the allowable range of the various parameters they obtain the following.
\[gone\]
1. Suppose $ (u,v)$ is a smooth stable positive solution of (\[eq\]) satisfying the hypothesis of Theorem \[MAIN\]. Suppose $ 1< p < \beta < \frac{2 N t_0^+}{N-2}$. Then there is some integer $ n \ge 1$ and $ C<\infty$ such that $$\label{high}
\left( \int_{B_{R}} v^{\beta} \right)^\frac{1}{\beta} \le C R^{N( \frac{1}{\beta}-\frac{1}{p})} \left( \int_{B_{R_{3n}}} v^p \right)^\frac{1}{p},$$ for all $ 1 \le R$.
2. Suppose that $ (u,v)$ is a stable smooth positive solution of (\[eq\]) with $ p=1$. For all $ 2 < \beta < \frac{2 N}{N-2} t_0^+$ there is some $ C<\infty$ and integer $n \ge 1$ such that $$\label{scalar_cor}
\left( \int_{B_R} v^\beta \right)^\frac{1}{\beta} \le C R^{ N( \frac{1}{\beta}-\frac{1}{2})} \left( \int_{B_{R_{3n}}} v^2 \right)^\frac{1}{2},$$ for all $ R \ge 1$.
**Proof of Corollary \[gone\].** Let $ t_0^- <t_0 < t_0^+$ and let $ 1 \le \alpha_k < \frac{N}{N-2}$ and define $t_{k+1}=\alpha_k t_k$. Iterating the result in Proposition \[initial\] one obtains $$\label{iter_gen}
\left( \int_{B_{R_{3^n}}} v^{2 t_n \alpha_n} \right)^\frac{1}{2t_n \alpha_n} \le C R^{ \frac{N}{2} ( \frac{1}{t_n \alpha_n}- \frac{1}{t_0} )} \left( \int_{B_R} v^{2t_0} \right)^\frac{1}{2t_0},$$ for all $ R \ge 1$ and all positive integers $n$ provided $ t_n < t_0^+$. By suitably picking the $ \alpha_k$ for $ k \le n-1$ we see that $ 2 t_n \alpha_n$ can be made arbitrarily close to $ \frac{2N t_0^+}{N-2}$. Note that when one performs the iterations that the powers of $R$ form a telescoping series and only the first and last terms don’t cancel.
We now separate the cases. We first deal with case 1) and recall we are assuming the hypothesis from Theorem \[MAIN\]. So we either take $ 2 \le p < \theta$ and one can then show by a computation that $ t_0^- < \frac{p}{2}$ or we don’t assume $ p \ge 2$ but we then, by hypothesis, assume $ t_0^- < \frac{p}{2}$. Also a computation shows that $ \frac{p}{2} <t_0^+$. This allows one to pick $ t_0= \frac{p}{2}$. With this choice of $ t_0$ and provided the above conditions hold on $ \alpha_k, t_k$ then (\[iter\_gen\]) gives $$\left( \int_{B_{R_{3^n}}} v^{2 t_n \alpha_n} \right)^\frac{1}{2t_n \alpha_n} \le C R^{ \frac{N}{2} ( \frac{1}{t_n \alpha_n}- \frac{2}{p} )} \left( \int_{B_R} v^{p} \right)^\frac{1}{p}.$$ This gives the desired result after considering the above comments on how big $2 t_n \alpha_n$ can be. One should note that we will only be interested in the case of $ \beta$ close to $\frac{2N}{N-2} t_0^+$.
We now examine 2). In this case we follow exactly the same argument as part 1) but we now take $ t_0=1$, which is allowed since $ t_0^- <1<t_0^+$. Putting $t_0=1$ into (\[iter\_gen\]) gives the desired result.
$ \Box$
**Completion of the proof of Theorem \[MAIN\].** Suppose $(u,v)$ is a smooth positive stable solution of (\[eq\]) and the hypothesis of Theorem \[MAIN\] are satisfied. Let $ p < \beta < \frac{2 Nt_0^+}{N-2}$. Combining Corollary \[gone\] 1) and Lemma \[Mid\_lemma\] there is some $ C< \infty $ such that $$\left( \int_{B_R} v^\beta \right)^\frac{1}{\beta} \le C R^{ N ( \frac{1}{\beta}- \frac{1}{p} ) + \frac{1}{p} ( N-2 - \frac{2 (p+1)}{p \theta-1})},$$ for all $ R \ge 1$. If this exponent is negative then after sending $ R \rightarrow \infty$ we obtain a contradiction. Note the exponent is negative if and only if we have $$N < \frac{2 (\theta+1) \beta}{p \theta-1},$$ and after considering the allowable range of $ \beta$ we obtain the desired result.
$ \Box$
We now examine the case of the scalar equation; $p=1$. Critical to our approach in the following result:
\[Wei\_dong\_init\] [@Wei_dong] Suppose that $ (u,v)$ is a stable smooth positive solution of (\[eq\]) with $ p=1$. Then by Lemma \[equivalence\], $u$ is a positive stable solution of (\[fourth\]) and then the results of [@Wei_dong] imply there is some $ C< \infty$ such that $$\int_{B_R} v^2 \le C R^{ N-4- \frac{8}{\theta-1}},$$ for all $ R>0$.
To complete the proof of Theorem \[MAIN\_four\] we combine Corollary \[gone\] 2) and Lemma \[Wei\_dong\_init\] and argue as in the proof of Theorem \[MAIN\]. Our final result relates the usual notion of stability for the scalar equation to the systems notion of stability.
\[equivalence\] Suppose $(u,v)$ is a positive stable solution of (\[eq\]) with $ 1=p<\theta$. Then $u$ is a stable solution of (\[fourth\]).
Let $ (u,v)$ be as in the hypothesis. By definition there are smooth positive functions $ \zeta,\chi$ such that $ -\Delta \zeta = \chi,$ $ -\Delta \chi = \theta u^{\theta-1} \zeta$ in $ \IR^N$. Let $ \gamma $ be smooth and compactly supported. First note that $ -\Delta \zeta > 0$ and $ \Delta^2 \zeta = \theta u^{\theta-1} \zeta$ and so we $$\begin{aligned}
\int \theta u^{\theta-1} \gamma^2 &=& \int \Delta^2 \zeta ( \gamma^2 \zeta^{-1}) \\
&=& \int \Delta \zeta \Delta ( \gamma^2 \zeta^{-1}) \\
&=& 2 \int \frac{\Delta \zeta}{\zeta} | \nabla \gamma|^2 + 2 \int \frac{\Delta \zeta}{\zeta} \gamma \Delta \gamma \\
&&+ 2 \int (\Delta \zeta) \frac{\gamma^2 | \nabla \zeta|^2}{\zeta^3} - \int \frac{ (\Delta \zeta)^2}{\zeta^2} \gamma^2 +I\end{aligned}$$ where $$I= -4 \int (\Delta \zeta) \gamma \frac{ \nabla \gamma \cdot \nabla \zeta}{\zeta^2}.$$ Using Young’s inequality and the fact that $ -\Delta \zeta \ge 0$ we see that $$| I | \le -2 \int \frac{\Delta \zeta | \nabla \gamma|^2}{\zeta} - 2 \int \frac{\gamma^2 | \nabla \zeta|^2 \Delta \zeta}{\zeta^3}.$$ Using this upper bound we see that $$\int \theta u^{\theta-1} \gamma^2 \le 2 \int \frac{ \gamma \Delta \zeta}{\zeta} \Delta \gamma - \int \frac{(\Delta \zeta)^2}{\zeta^2} \gamma^2,$$ and this is bounded above, after using Young’s inequality again, by $ \int (\Delta \gamma)^2$, which is the desired result.
Results on the half space
=========================
In this section we are interested in Liouville theorems on the half space. For notational convenience all integrals will be over $ \IR^N_+$ unless otherwise indicated.
Our first result shows that monotonic solutions on the half space satisfy the stability like inequality given by (\[second\]) but in fact note we prove slightly more. The test functions need not be zero on the boundary of $ \IR^N_+$.
\[toboun\] Suppose $f,g$ are sufficiently smooth, positive increasing nonlinearities on $ (0,\infty)$ with $ f(0)=g(0)=0$. Suppose $(u,v)$ is a positive solution of $ -\Delta u = f(v)$, $ -\Delta v=g(u)$ in $ \IR^N_+$ which satisfies $ u_{x_N},v_{x_N} >0$ in $ \IR_N^+$. Then $$\int_{\IR^N_+} \sqrt{f'(v) g'(u)} \phi^2 \le \int_{\IR^N_+} | \nabla \phi|^2 \qquad \forall \phi \in C^2_c(\IR^N).$$ Note the test functions need not be zero on $ \partial \IR_+^N$.
By taking a derivative in $ x_N$ of (\[system\_thm\_half\]) we see that $ -\Delta u_{x_N} = f'(v) v_{x_N}$ and $ -\Delta v_{x_N}=g'(u) u_{x_N}$ in $ \IR^N_+$. Let $ \phi \in C_c^\infty(\IR^N)$ and multiply the first equation by $ \frac{\phi^2}{ u_{x_N}}$ and the second equation by $ \frac{\phi^2}{v_{x_N}}$ (and note by Hopf’s Lemma that $u_{x_N},v_{x_N}>0$ on $ \partial \IR^N_+$) and integrate over the half space to obtain $$\int \frac{f'(v) v_{x_N} \phi^2}{u_{x_N}} \le \int \nabla (u_{x_N}) \cdot \nabla ( \phi^2 u_{x_N}^{-1}) - \int_{\partial \IR^N_+} \partial_\nu u_{x_N} ( \phi^2 u_{x_N}^{-1}),$$ $$\int \frac{g'(u) u_{x_N} \phi^2}{v_{x_N}} = \int \nabla (v_{x_N}) \cdot \nabla ( \phi^2 v_{x_N}^{-1}) - \int_{\partial \IR^N_+} \partial_\nu v_{x_N} ( \phi^2 v_{x_N}^{-1}),$$ where $ \partial \nu$ is the outward pointing normal. We first examine the boundary integrals. Note that $ \partial_\nu u_{x_N}= - u_{x_N x_N}$ and from the fact that $u,v$ are sufficiently regular to the boundary we see that $ -u_{x_N x_N}= f(v) + \sum_{k=1}^{N-1} u_{x_k x_k} $ on $ \IR_+^N$. Now note that by the assumption of $f$ and the boundary condition on $u$ we see that we must have $ u_{x_N x_N}=0$ on $ \partial \IR^N_+$ and hence the boundary integral is zero. Similarly one shows the other boundary integral is also zero. We then use Young’s inequality on the integrals involving the gradients and add the results to see that $$\int \left( \frac{f'(v) v_{x_N} }{u_{x_N}} + \frac{g'(u) u_{x_N} }{v_{x_N}} \right) \phi^2 \le 2 \int | \nabla \phi|^2.$$ We then use the argument from Lemma \[stabb\] to obtain the desired result.\
An alternate proof can be given by extending the solutions and the nonlinearities to $ \IR^N$, using odd extensions. One then has that $(u,v)$ is a monotonic solution of the extended problem on $\IR^N$ and hence is stable. Fix $ \phi$ to be a smooth and compactly supported function in $ \IR^N_+$ but we allow $ \phi$ to be non zero on the boundary and then extend $ \phi$ to all $ \IR^N$ using an even extension. The extension is a sufficiently regular test function which can be inserted into (\[second\]) and this gives the desired result after writing all the integrals over the half space.
**Proof of Theorem \[system\_thm\_half\].** Suppose $(u,v)$ is a bounded positive classical solution of (\[eq\_half\]). By a moving plane argument, see [@dancer_2; @sirak] one has $ u_{x_N}, v_{x_N}>0$ in $ \IR^N_+$. For $x \in \IR^N$ we write $ x=(x',x_N)$ and we now define, for each $ t>0$, $ u_t(x)=u(x', x_N+t)$ and $ v_t(x)=v(x', x_N+t)$ for $x \in \IR_N^+$. Note that $u_t,v_t$ are monotonic solutions of (\[eq\_half\]) in $ \IR^N_+$ but without any assumptions on the boundary values. Using the same argument as in the proof of Lemma \[toboun\] one can easily show that $$\label{shoo}
\sqrt{p \theta} \int v_t^\frac{p-1}{2} u_t^\frac{\theta-1}{2} \le \int | \nabla \phi|^2, \qquad \forall \phi \in C_c^\infty( \IR^N_+).$$ Now note that since $u$ and $v$ are bounded, monotonic and positive we see that $$w_1(x'):=\lim_{t \nearrow \infty} u_t(x), \qquad w_2(x'):= \lim_{t \nearrow \infty} v_t(x),$$ defined in $ \IR^{N-1}$ are positive bounded solutions of $$-\Delta w_1 = w_2^p, \qquad -\Delta w_2 = w_1^\theta, \qquad \mbox{ in } \IR^{N-1}.$$ To complete the proof we will show that $ w_1,w_2$ preserve some of the stability properties of the solutions on the half space. We won’t show $w_1,w_2$ are stable solutions but we will instead show that $w_1,w_2$ satisfy the stability like inequality given by (\[second\]) on $ \IR^{N-1}$. One can then apply Theorem \[MAIN\] (note the only place stability is used in the proof of Theorem \[MAIN\] is to obtain (\[second\])).
Let $ \phi_1 \in C_c^\infty(\IR^{N-1})$ and let $0 \le \phi_R \le 1$ be smooth and compactly supported in $ (R,4R) \subset \IR$ where $ R \ge 1$ with $ \phi_R =1$ in $ (2R,3R)$. Note there is some positive finite $C$ such that $ | \phi'(x_{n}) | \le \frac{C}{R}$ for all $ R \ge 1$ and $ x_N \in (R,4R)$. Define $ \phi(x)= \phi_1(x') \phi_R(x_N)$ and putting $ \phi $ into (\[shoo\]) and using the fact that $ u_t(x) \ge u(x',t)$ and similarly for $v$ shows (after writing the integrals as iterated integrals then using some algebra) that $$\begin{aligned}
\sqrt{ p \theta} \int_{\IR^{N-1}} v(x',t)^\frac{p-1}{2} u(x',t)^\frac{\theta-1}{2} \phi_1(x')^2 d x' & \le & \int_{\IR^{N-1}} | \nabla \phi_1(x')|^2 d x' \\
&& + T_R \int_{\IR^{N-1}} | \phi_1(x') |^2 d x'\end{aligned}$$
where $$T_R:= \frac{ \int_{\IR} | \nabla \phi_R(x_N)|^2 d x_N }{ \int_{\IR} \phi_R(x_N)^2 d x_N}.$$ One easily sees that $ T_R \rightarrow 0$ as $ R \rightarrow \infty$ and so sending $ R \rightarrow \infty$ and then sending $ t \rightarrow \infty$ gives the desired result. $\Box$
We now give some comments on the above proof. Firstly, this idea of relating a monotonic problem on the half space to a problem in one dimension lower on the full space has been used by many authors. A key step in the above argument is to show the problem in $ \IR^{N-1}$ is stable. We first learned of this idea of showing the limiting solution is stable in the recent work [@new_far] where the context was a quasilinear scalar problem. After examining the literature we realized this result is contained in [@Wei_dong] where they are examining a biharmonic problem.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We report results from an intensive multi-wavelength monitoring campaign on the TeV blazar Mrk 421 over the period of 2003–2004. The source was observed simultaneously at TeV energies with the Whipple 10 m telescope and at X-ray energies with [*Rossi X-ray Timing Explorer*]{} (RXTE) during each clear night within the [*Whipple*]{} observing windows. Supporting observations were also frequently carried out at optical and radio wavelengths to provide simultaneous or contemporaneous coverages. The large amount of simultaneous data has allowed us to examine the variability of Mrk 421 in detail, including cross-band correlation and broad-band spectral variability, over a wide range of flux. The variabilities are generally correlated between the X-ray and gamma-ray bands, although the correlation appears to be fairly loose. The light curves show the presence of flares with varying amplitudes on a wide range of timescales both at X-ray and TeV energies. Of particular interest is the presence of TeV flares that have no coincident counterparts at longer wavelengths, because the phenomenon seems difficult to understand in the context of the proposed emission models for TeV blazars. We have also found that the TeV flux reached its peak days [*before*]{} the X-ray flux did during a giant flare (or outburst) in 2004 (with the peak flux reaching $\sim$135 mCrab in X-rays, as seen by the ASM/RXTE, and $\sim$3 Crab in gamma rays). Such a difference in the development of the flare presents a further challenge to the leptonic and hadronic emission models alike. Mrk 421 varied much less at optical and radio wavelengths. Surprisingly, the normalized variability amplitude in optical seems to be comparable to that in radio, perhaps suggesting the presence of different populations of emitting electrons in the jet. The spectral energy distribution of Mrk 421 is seen to vary with flux, with the two characteristic peaks moving toward higher energies at higher fluxes. We have failed to fit the measured SEDs with a one-zone SSC model; introducing additional zones greatly improves the fits. We have derived constraints on the physical properties of the X-ray/gamma-ray flaring regions from the observed variability (and SED) of the source. The implications of the results are discussed.'
author:
- |
M. B[ł]{}ażejowski, G. Blaylock, I. H. Bond, S. M. Bradbury, J. H. Buckley, D. A. Carter-Lewis, O. Celik, P. Cogan, W. Cui, M. Daniel, C. Duke, A. Falcone, D. J. Fegan, S. J. Fegan, J. P. Finley, L. Fortson, S. Gammell, K. Gibbs, G. G. Gillanders, J. Grube, K. Gutierrez, J. Hall, D. Hanna, J. Holder, D. Horan, B. Humensky, G. Kenny, M. Kertzman, D. Kieda, J. Kildea, J. Knapp, K. Kosack, H. Krawczynski, F. Krennrich, M. Lang, S. LeBohec, E. Linton, J. Lloyd-Evans, G. Maier, D. Mendoza, A. Milovanovic, P. Moriarty, T. N. Nagai, R. A. Ong, B. Power-Mooney, J. Quinn, M. Quinn, K. Ragan, P. T. Reynolds, P. Rebillot, H. J. Rose, M. Schroedter, G. H. Sembroski, S. P. Swordy, A. Syson, L. Valcarel, V. V. Vassiliev, S. P. Wakely, G. Walker, T. C. Weekes, R. White, and J. Zweerink,\
and\
B. Mochejska, B. Smith, M. Aller, H. Aller, H. Teräsranta, P. Boltwood, A. Sadun, K. Stanek, E. Adams, J. Foster, J. Hartman, K. Lai, M. Böttcher, A. Reimer, and I. Jung
title: 'A Multi-wavelength View of the TeV Blazar Markarian 421: Correlated Variability, Flaring, and Spectral Evolution'
---
Introduction
============
Over the past decade or so, one of the most exciting advances in high energy astrophysics has been the detection of sources at TeV energies with ground-based gamma ray facilities (see Weekes 2003 for a recent review). Among the sources detected, blazars are arguably the most intriguing. They represent the only type of active galactic nuclei (AGN) that has been detected at TeV energies (although a 4-$\sigma$ detection of M87 has been reported; Aharonian et al. 2003). To date, there are a total of six firmly established TeV blazars.
The emission from a blazar is generally thought to be dominated by radiation from a relativistic jet that is directed roughly along the line of sight (review by Urry & Padovani 1995 and references therein). Relativistic beaming is necessary to keep gamma-ray photons from being significantly attenuated by the surrounding radiation field (via photon-photon pair production). The spectral energy distribution (SED) of TeV blazars invariably shows two characteristic peaks in the $\nu F_{\nu}$ representation, with one located at X-ray energies and the other at TeV energies (Fossati et al. 1998). There seems to be a general correlation between the two SED peaks as the source varies (e.g., Buckley et al. 1996; Catanese et al. 1997; Maraschi et al. 1999; Petry et al. 2000).
A popular class of models associates the X-ray emission from a TeV blazar with synchrotron radiation from highly relativistic electrons in the jet and the TeV emission with inverse-Compton scattering of the synchrotron photons by the electrons themselves (i.e., synchrotron self-Compton or SSC for short; Marscher & Gear 1985; Maraschi et al. 1992; Dermer et al. 1992; Sikora et al. 1994; see Böttcher 2002 for a recent review). The SSC models can, therefore, naturally account for the observed X-ray–TeV correlation. Moreover, they have also enjoyed some success in reproducing the measured SEDs. However, the models still face challenges in explaining some of the observed phenomena, such as the presence of “orphan” TeV flares (Krawczynski et al. 2004; Cui et al. 2004).
Alternatively, the jet might be energetically dominated by the magnetic field and it is the synchrotron radiation from highly relativistic protons that might be responsible for the observed TeV gamma rays (Aharonian 2000; Mücke et al. 2003). Other hadronic processes have also been considered, including photo-meson production, neutral pion decay, and synchrotron-pair cascading (e.g., Mannheim & Biermann 1992; Mücke et al. 2003), but they are thought to be less important in TeV blazars (Aharonian 2000; Mücke et al. 2003). Another class of hadronic models invokes $pp$ processes, for instance, in the collision between the jet and ambient “clouds” (e.g., Dar & Laor 1997; Beall & Bednarek 1999) or inside the (dense) jet (Pohl & Schlickeiser 2000). In this case, the gamma-ray emission is mainly attributed to the decay of neutral pions produced in the $pp$ interactions. In both classes of hadronic models, the emission at X-ray and longer wavelengths is still attributed to the synchrotron radiation from relativistic electrons (and positrons) in the jet, as in the SSC models. Although the hadronic models may also be able to describe the observed SED of TeV blazars and accommodate the X-ray–TeV correlation, they are generally challenged by the most rapid gamma-ray variabilities observed in TeV blazars (Gaidos et al. 1996).
TeV blazars are also known to undergo flaring episodes both at X-ray and TeV energies. The flares have been observed over a wide range of timescales, from months down to less than an hour. The observed X-ray flaring hierarchy seems to imply a scale-invariant physical origin of the flares (Cui 2004; Xue & Cui 2005). Blazar flares are thought to be related to internal shocks in the jet (Rees 1978; Spada et al. 2001), or to the ejection of relativistic plasma into the jet (e.g., Böttcher et al. 1997; Mastichiadis & Kirk 1997). Recently, it is suggested that the flares could also be associated with magnetic reconnection events in a magnetically dominated jet (Lyutikov 2003) and thus they could be similar to solar flares in this regard. Such a model might offer a natural explanation for the hierarchical flaring phenomenon, again in analogy to solar flares.
To make further progress on distinguishing the emission models proposed for TeV blazars, we believe that a large amount of simultaneous or contemporaneous data is critically needed over a wide range of flux, especially in the crucial X-ray and TeV bands, for quantifying the SED and spectral variability of a source and for allowing investigations of such important issues as variability timescales, cross-band correlation, spectral variability, spectral hysteresis, etc. Such data are severely lacking at present, despite intense observational efforts over the years. In this paper, we present results from an intensive multi-wavelength monitoring campaign on Mrk 421. This source is the first TeV blazar discovered (Punch et al. 1992) and remains one of the few blazars that can be detected at TeV energies nearly all the time with ground-based imaging atmospheric Cherenkov telescopes (IACTs). Some of the preliminary results have appeared elsewhere (Cui et al. 2004); they are superseded by those presented in this work.
We have assumed the following values for the various cosmological parameters: $H=71\mbox{ }km\mbox{ }s^{-1}\mbox{ }Mpc^{-1}$, $\Omega_m=0.27$, and $\Omega_{\Lambda}=0.73$. The corresponding luminosity distance of Mrk 421 ($z=0.031$) is about 129.8 Mpc.
Observations and Data Reduction
===============================
Gamma-ray Observations
----------------------
From 2003 February to 2004 June, Mrk 421 was observed at TeV energies with the Whipple 10 m Telescope (on Mt. Hopkins, AZ) during each clear night within the dark moon observing periods. The typical exposure time of a nightly observation was 28 minutes, corresponding to one observing run, but more runs were taken on occasion, especially near the end of the campaign (in 2004 April) when the source was seen to undergo an usually large X-ray outburst as seen by the All-Sky Monitor (ASM) on [*RXTE*]{}. To achieve simultaneous coverages of the source both in the TeV and X-ray bands, we communicated with the [*RXTE Science Operations Facility*]{} to ensure that the [*Whipple*]{} and [*RXTE*]{} observing schedules were matched as closely as possible. A total of 306 runs were collected in good (empirically designated as “A” or “B”) weather, and roughly 80% of them were taken at zenith angles $\lesssim$30.
The procedure for reducing and analyzing [*Whipple*]{} data has been standardized over the years (Hillas 1985; Reynolds et al. 1993; Mohanty et al. 1998). A detailed description of the current hardware can be found in Finley et al. (2001). For clarity, we briefly summarize a few key points that are relevant to this work. The success of IACTs lies in the fact that the Cherenkov images of an air shower produced by a gamma-ray primary have different shapes and orientations than those found in an air shower produced by cosmic-ray particles (mostly protons). The images in gamma-ray events are typically more compact and are more aligned to point towards the position of the source than those in cosmic-ray events. In practice, the image shape of an event is characterized by the major and minor axes of the best-fit ellipse to the image. The orientation of the ellipse is characterized by the parameter “$\alpha$”, defined as the angle between the major axis and the line connecting the center of the ellipse to the center of the field-of-view (FOV). We have developed standard selection criteria based primarily on the image shape and orientation parameters that remove over 99% of the cosmic ray events while keeping about half of the gamma-ray events (see, e.g., Falcone et al. 2004 for more details).
The [*Whipple*]{} observations are conducted in one of the two modes: tracking and ON/OFF. In the tracking mode, the telescope tracks the target across the sky so that the source stays at the center of the FOV throughout the observation. In the ON/OFF mode, on the other hand, the telescope tracks the target only during the ON run; it is then offset by 30 minutes in right ascension during the OFF run, and tracks the field as it covers the same range of zenith and azimuthal angles. The OFF run provides a direct measurement of the background, although the difference in the sky brightness of the fields between the ON and OFF runs must be taken into account by using a technique known as software padding (Reynolds et al 1993). For tracking observations, the background is derived from the $\alpha$ histogram of the events that have passed all but the orientation cuts. Since real gamma-ray events from a source should all be concentrated at small $\alpha$ values ($<$ 15) for on-axis observations, the background level can be estimated from events with larger $\alpha$ values (20–65), if the ratio of the number of background events in the two ranges of the $\alpha$ parameter is known. This ratio is derived from observations of fields in which there is no evidence for a gamma-ray source (see, e.g., Falcone et al. 2004).
Nearly all of the [*Whipple*]{} observations reported in this work were carried out in the tracking mode. We followed the standard procedure just described to obtain count rates from individual runs (taken in the good weather) and thus the long-term light curves. To examine variability on short timescales, we sometimes sub-divided runs into time intervals shorter than the nominal 28 minutes and constructed light curves with correspondingly smaller time bins. To correct for the effects due to changes in the zenith angle and overall throughput of the telescope, we applied the method developed by LeBohec & Holder (2003) to the data. We should note that for a given season the corrections were made with respect to a (somewhat arbitrarily chosen) reference run taken at 30 zenith angle during a clear night. To correct for changes in the telescope throughput across seasons, we used the measured rates for the Crab Nebula to further calibrate the light curves. For reference, the rate of the Crab Nebula is about 2.40 and 2.93 $\gamma\mbox{ }min^{-1}$ for the 2002/2003 and 2003/2004 seasons, respectively.
The spectral analysis was carried out by following Method 1 described in Mohanty et al. (1998). The technical aspects and difficulties involved in finding matching pairs for the spectral analysis of tracking observations are explained in detail by Petry et al. (2002) and Daniel et al. (2005). Combined with the fact that the observations were taken only in the tracking mode, poor statistics made it extremely challenging to derive a TeV spectrum from observations taken at low fluxes. For those cases, the tolerance for the parameter cuts was tightened to be just 1.5 standard deviations from the average value of the simulations, as opposed to the usual 2 standard deviations, in order to further reduce the cosmic-ray background (Daniel et al. 2005). The tighter cuts still retained $\sim 80\%$ of gamma rays. The downside is that the effective collection area of the telescope is not as independent of energy as can be ideally hoped for (Mohanty et al. 1998).
X-ray observations
------------------
In coordination with each [*Whipple*]{} observation, we took a snapshot of Mrk 421 at X-ray energies with [*RXTE*]{} (with a typical exposure time of 2–3 ks). We should note, however, that not every X-ray observation was accompanied by a simultaneous [*Whipple*]{} observation (due, e.g., to poor weather) and vice versa (especially near the end of the campaign, when many more [*Whipple*]{} observations were made to monitor the source in an exceptionally bright state; see Fig. 1). For this work, we only used data from the PCA instrument on [*RXTE*]{}, which covers a nominal energy range of 2–60 keV. The PCA consists of five nearly identical proportional counter units (PCUs). However, only two of the PCUs, PCU 0 and PCU 2, were in use throughout our campaign, due to operational constraints. PCU 0 has lost its front veto layer, so the data from it are more prone to contamination by events caused by low-energy electrons entering the detector. The problem is particularly relevant to variability studies of relatively weak sources, such as Mrk 421. For this work, therefore, we have chosen PCU 2 as our “standard” detector for flux normalization and spectral analysis.
We followed Cui (2004) closely in reducing and analyzing the PCA data. Briefly, the data were reduced with [*FTOOLS 5.2*]{}. For a given observation, we first filtered data by following the standard procedure for faint sources,[^1] which resulted in a list of good time intervals (GTIs). We then simulated background events for the observation by using the latest background model that is appropriate for faint sources. Using the GTIs, we proceeded to extract a light curve for each PCU separately. We repeated the steps to construct the corresponding background light curves from the simulated events. We then subtracted the background from the total to obtain the light curves of the source. Following a similar procedure, for each observation, we also constructed the X-ray spectrum for each PCU and its associated background spectrum. In this case, however, we only used data from the first xenon layer of each PCU (which is most accurately calibrated), which limits the spectral coverage to roughly 2.5–25 keV. Since few counts were detected at higher energies, the impact of the reduced spectral coverage is very minimal.
Optical observations
--------------------
The optical data were obtained with the Fred Lawrence Whipple Observatory (FLWO) 1.2 m telescope (located adjacent to the Whipple 10 m gamma-ray telescope on Mt. Hopkins) and with the 0.4 m telescope at the Boltwood Observatory in Stittsville, Ontario, Canada. We note that we had no optical coverage of the source during the 2002/2003 [*Whipple*]{} observing season.
The FLWO 1.2 m was equipped with 4Shooter CCD mosaic with four $2048 \times 2048$ chips. Each chip covers a $11\farcm 4\times 11\farcm 4$ FOV. The data were collected during 31 nights, from 2003 December 14 to 2004 February 17. A total of 77 images were obtained in the $B$ band, 69 in the $V$ band, 67 in the $R$ band, and 78 in the $I$ band, with an exposure time of 30 s for each image. The preliminary processing of the CCD frames was performed with the standard routines in the IRAF ccdproc package.[^2]
Photometry was extracted using the [DAOphot/Allstar]{} package (Stetson 1987). The fitting radius for profile photometry was varied with the seeing. The median seeing in our dataset was 4.5. A detailed description of the applied procedure is given in § 3.2 of Mochejska et al. (2002). The derivation of photometry was complicated by the proximity of two very bright stars, HD 95934 (V=6 mag) and HD 95976 (V=7.5 mag), at angles of 2.1 and 4 from Mrk 421, respectively. These angles correspond to distances of 187 and 356 pixels on the images. The presence of these stars introduces a gradient in the background, approximately in the north-south direction. To estimate the magnitude of this gradient, we examined two $10 \times 10$ pixel regions located at distances of 30–40 pixels north and south of Mrk 421. These regions were chosen to coincide with the annulus of 18-45 pixels used by [Allstar]{} for background determination. The difference in counts between the two regions is at a level of 1.4%, 0.7%, 0.3%, and 0.2% of the peak value of Mrk 421 with BVRI filters, respectively, on our best seeing images, and of 3.9%, 1.9%, 1.1%, and 0.6% on images with seeing close to the upper $85^{th}$ percentile. Thus, it varies with the seeing in BVRI by 2.5%, 1.2%, 0.8%, and 0.4%, respectively, all very small compared with the variability amplitudes of the source. The transformations of instrumental magnitudes to the standard system were derived from observations of 27 stars in 3 Landolt (1992) standard fields, collected on 19 January 2004. The systematic errors associated with the calibration are estimated to be 0.02 for BVI and 0.01 for R.
The Boltwood 0.4 m is equipped with an Apogee AP7p CCD camera that uses a back-illuminated SITE 502A chip. Mrk 421 was observed during the period from 2003 November 8 to 2004 June 11. The data were taken with an uncalibrated Cousins R filter (designed by Bessell and manufactured by Omega). The photometric measurements are differential (with respect to the comparison star 4 in Villata et al. 1998). Aperture photometry was performed with custom software. The aperture used is of 10 in diameter. Data points were obtained from averaging over between four and six 2 minute exposures. The typical statistical error on the relative photometry of each data point is 0.02 in magnitude. The seeing-induced errors or other systematic errors were not taken into account. To cross check results from the two sets of measurements, we scaled up the Boltwood values (by adding a constant to the measured differential magnitudes) so that they agree, on average, with the FLWO fluxes for the overlapping time period. We found that the observed variation patterns agree quite well between the two data sets. This also implies that any variability caused by systematic effects must be small compared with the intrinsic variability of the source.
Radio observations
------------------
We observed Mrk 421 frequently with the 26-meter telescope at the University of Michigan Radio Astronomy Observatory (UMRAO) and the 13.7-meter Metsähovi radio telescope at the Helsinki University of Technology. The UMRAO telescope is equipped with transistor-based radiometers operating at center frequencies of 4.8, 8.0, and 14.5 GHz; their bandwidths are 560, 760, and 1600 MHz, respectively. All three frequencies utilize rotating, dual-horn polarimeter feed systems, which permit both total flux and linear polarizations to be measured using an ON-OFF observing technique at 4.8 GHz and an ON-ON technique (switching the target source between the two feed horns closely spaced on the sky) at the other two frequencies. The latter technique reduces the contribution of tropospheric interference by an order of magnitude. Observations of Mrk 421 were intermixed with observations of a grid of calibrator sources: 3C 274 is the most frequently used calibrator, except during a period each fall when the sun is within 15$^\circ$ of the calibrator. The flux scale is set by observations of Cas A (e.g. see Baars et al. 1977). Details of the calibration and analysis techniques are described in Aller et al. (1985).
The Metsähovi observations were carried out as part of a long-time monitoring program. The observations were made with dual-horn receivers and the ON-ON technique, and they covered the 22 and 37 GHz bands. The flux measurements have been calibrated against DR 21 (with the adopted fluxes 19.0 and 17.9 Jy at 22 and 37 GHz, respectively). The full description of the receiving system can be found in Teräsranta et. al. (1998). The data were obtained during the period from 2003 January 1 to 2004 June 28.
Results
=======
Light Curves
------------
Figure 1 shows the light curves of Mrk 421 in the representative bands covered by the campaign. The source was relatively quiet during the 2002/2003 season, although it clearly varied at X-ray and TeV energies. The largest TeV flare occurred around MJD 52728 (= 30 March 2003), reaching a peak count rate of nearly 4 $\gamma\mbox{ }min^{-1}$ and lasting for about a week. The TeV flare was accompanied by a flare in X-rays, which reached a peak count rate of $\sim$65$\mbox{ }cts\mbox{ }s^{-1}\mbox{ }PCU^{-1}$ (or about 24 mCrab). There is no apparent time offset between the X-ray and gamma-ray flares.
Mrk 421 became much more active in the 2003/2004 season, with several major flares observed. In particular, an usually large flare (or outburst) took place near the end of the campaign, with the source reaching peak fluxes of $\sim$135 mCrab in X-rays, as seen by the ASM/RXTE,[^3] and $\sim$3 Crabs in the TeV band, respectively. An expanded view of this flaring episode is shown in Figure 2. The flare lasted for more than two weeks (from $<$ MJD 53104 to roughly MJD 53120), although its exact duration is difficult to quantify due to the presence of a large data gap between MJD 53093 and MJD 53104. Interestingly, during this giant flare, the TeV emission appears to reach the peak much sooner than the X-ray emission. Although the X-ray light curve is not as densely sampled as the gamma-ray light curve, the fact that the X-ray measurements were made at the [*same*]{} time as the corresponding gamma-ray measurements (see Fig. 2) makes it quite unlikely that the difference in the rise time between the two bands is caused by some sampling bias.
The light curves also show that Mrk 421 varies much less at optical and radio wavelengths. The variability amplitude in the R band is less than about 0.9 magnitude; the variability is even less obvious in the radio bands due to relatively large measurement errors. There is no apparent correlation between the optical and radio bands or between either radio or optical band and the high-energy (X-ray and TeV) bands. For completeness, Figure 3 shows (FLWO) optical and radio light curves for all of the bands covered. The figure shows highly correlated variability among the optical bands, while the situation is not nearly as clear among the radio bands, due both to large measurement errors and sparse coverages.
### Energy Dependence of the Source Variability
To quantify the energy dependence of variability amplitudes, we computed the so-called normalized variability amplitude (NVA) for each spectral band. The NVA is defined as (Edelson et al. 1996): $$NVA \equiv \frac{{({\sigma_{tot}}^2 -{\sigma_{err}}^2)}^{1/2}}{\bar{F}},$$ where $\bar{F}$ represents the mean count rate or flux, $\sigma_{tot}$ the standard deviation, and ${\sigma_{err}}$ the mean measurement error in a given spectral band.
To facilitate comparison of the NVAs for different bands, we only used light curves that are relatively well sampled and cover the entire 2003/2004 season (see Fig. 1). Consequently, the results are obtained only for the following spectral bands: 14.5 GHz, R, X-ray, and gamma-ray. We first rebinned the selected light curves with the same bin size. To examine variabilities on different timescales, two different bin sizes were used: 1 day and 7 days. We then computed the NVAs from the rebinned light curves. The results are shown in Figure 4. The variability of Mrk 421 shows a general increasing trend toward high energies. The source is highly variable in the X-ray and TeV bands, with the NVA reaching up to about 65% (with 1-day binning), but varies much less in the radio and optical bands, with the NVA equal to $\sim$11% and $\sim$16%, respectively, almost independent of the binning schemes used.
There are several caveats in the cross-band comparison. First of all, the density of data sampling is quite different for different bands. For instance, the sampling is more sparse in the 14.5 GHz band than in any of the other bands. However, the under-sampling of the radio data would probably only lead to an [*under-estimation*]{} of the radio NVA. Secondly, the measurement errors for the optical data are probably underestimated, due to possible systematic effects caused by, e.g., the presence of bright stars in the field. This would result in an [*over-estimation*]{} of the optical NVA. Finally, the measured optical flux includes contribution from the host galaxy, which is at a $\sim$15% level (Nilsson et al. 1999). Since it only affects the average flux, not the absolute variability amplitude, the optical NVA should be about 15% higher, which is a small correction. Therefore, the conclusion that the NVA is comparable in the radio and optical bands seems secure.
### X-ray–TeV Correlation
From the light curves (Fig. 1), we can see that the X-ray and TeV variabilities of Mrk 421 are roughly correlated, although they are clearly not always in step. Figure 5 shows the X-ray and gamma-ray count rates for all simultaneous measurements. Although a positive correlation between the rates seems apparent, it is only a loose one. We should note that the dynamical range of the data is quite large ($\sim$30 in both energy bands) which is important for studying correlative variability of the source.
To be more rigorous, we computed the Z-transformed discrete correlation function (ZDCF; Alexander 1997) from light curves in the two bands. The ZDCF makes use of the Fisher’s z-transform of the correlation coefficient (see Alexander 1997 for a detailed description). Its main advantage over the more commonly used DCF (Edelson & Krolik 1988) is that it is more efficient in detecting any correlation present. Figure 6 shows the ZDCF (in 1-day bins) derived from the 2002/2003 data set. The ZDCF seems to peak at a negative lag. Fitting the peak (in the narrow range of -7–7 days) with a Gaussian function, we found its centroid at $-1.8 \pm 0.4$ days, which is of marginal significance. If real, the result would imply that the X-ray variability [*leads*]{} the gamma-ray variability. Other ZDCF peaks are most likely caused by the [*Whipple*]{} observing pattern, such as the quasi-periodic occurrences of the dark moon periods. Similarly, we computed a ZDCF for the 2003/2004 data set. The results are shown in Figure 7. In this case, the main feature is very broad and significantly skewed toward [*positive*]{} lags. The feature appears to be a composite of multiple peaks, although large error bars preclude a definitive conclusion. A positive ZDCF peak means that the X-ray emission [*lags*]{} behind the gamma-ray emission. We checked the results with different binning schemes and found no significant changes.
The detection of X-ray [*lags*]{} in the 2003/2004 data set should not come as a total surprise, because we have already seen (from Fig. 2) that the X-ray flux rose more slowly than the gamma-ray flux during the 2004 giant flare. The difference in the rise times can be the cause of the broad ZDCF peak that is skewed toward positive lags. To check that, we excluded the flare from the X-ray and gamma-ray light curves (by removing all data points after MJD 53100; see Fig. 1) and computed a new ZDCF. The results are also shown in Fig. 7 (bottom panel) for a direct comparison. There is a narrow peak at around -1 day. Fitting it (in the range of -7–3 days) with a Gaussian function yields the centroid at $-1.2 \pm 0.5$ days, which is not inconsistent the measured value for the 2002/2003 data set though it is even less significant statistically. The features at around +7 and -15 days (which are also discernable in the overall ZDCF) are, again, most likely caused by the [*Whipple*]{} observing pattern for the season. Despite the complications, it is almost certain, by comparing the two ZDCFs, that the X-ray lags in the 2003/2004 data set are indeed associated with the difference between the X-ray and gamma-ray rise times of the giant flare.
To investigate the effects of data gaps (which are present both in the X-ray and gamma-ray data) on ZDCF, we did the following experiment. We created three light curves: the actual X-ray light curve of Mrk 421 in the 2003/2004 season([*lc1*]{}), an artificial light curve ([*lc2*]{}) made by shifting [*lc1*]{} by +8 days, and another artificial light curve ([*lc3*]{}) made by modulating [*lc2*]{} with the [*Whipple*]{} sampling pattern. Figure 8 shows the ZDCFs between [*lc1*]{} and [*lc2*]{} and between [*lc1*]{} and [*lc3*]{} separately. The artificially-introduced 8-day lag is easily recovered in both cases. We also shifted [*lc1*]{} by different amounts and the lags are always recovered correctly. Therefore, the coverage gaps associated with the [*RXTE*]{} and [*Whipple*]{} monitoring campaigns do not wash out intrinsic offsets between the X-ray and gamma-ray light curves, although the effects of the data gaps on the shape of the ZDCFs are measurable.
### X-ray and TeV Flares
Examining the X-ray and TeV light curves more closely, we noticed that some of the TeV flares have no [*simultaneous*]{} X-ray counterparts (or counterparts at long wavelengths). They are, therefore, similar to the reported “orphan” TeV flare in 1ES1959+650 (Krawczynski et al. 2004). Figure 9 shows an example of the phenomenon. The TeV flare peaks at almost 8 $\gamma\mbox{ }min^{-1}$ at around MJD 53033.4 when the X-ray flux is low. It is interesting to note, however, that the source is clearly variable in X-rays during this time period and that the X-ray flux seems to have peaked about 1.5 days before the TeV flux. Therefore, the TeV flare might not be a true orphan event; it might simply lag behind its X-ray counterpart. Alternatively, it is also possible that the TeV flare is a composite of two sub-flares, with one being the counterpart of the X-ray flare and the other a true, lagging orphan flare. The sparse sampling of the data prevents us from drawing a definitive conclusion in this regard. We note the remarkable similarities between the TeV flare shown in Fig. 9 and the reported “orphan” flare in 1ES1959+650 (see Fig. 4 in Krawczynski et al. 2004), including similar variation patterns in X-rays.
From the X-ray and TeV light curves, we also detected flares on much shorter timescales. Figure 10 shows examples of sub-hour X-ray flares. The most rapid X-ray flare detected lasted only for $\sim$20 minutes and shows substantial sub-structures, implying variability on even shorter timescales. Only on one occasion was a sub-hour X-ray flare detected during a (longer-duration) gamma-ray flare. No counterpart is apparent at TeV energies (see Fig. 10), although the error bars on gamma-ray measurements are quite large. We note that, if a strong rapid TeV flare, for example, like the one detected by Gaidos et al. (1995), had occurred, we should have easily detected it, given the improved instrumentation. The rapid X-ray flare shown in Fig. 10 is relatively weak, with a peak-to-peak amplitude of only about 5% of the average flux level. The data do not allow any direct comparison on long timescales, due to the short exposure of the X-ray observation.
Spectral Energy Distributions
-----------------------------
We first divided the [*RXTE*]{} observations into 8 groups based on the X-ray count rate of Mrk 421, with an increment of 20 $cts\mbox{ }s^{-1}\mbox{ }PCU^{-1}$. For this work, we focused on three of the groups: low (0–20 $cts\mbox{ }s^{-1}\mbox{ }PCU^{-1}$), medium (40–60 $cts\mbox{ }s^{-1}\mbox{ }PCU^{-1}$), and high (140–160 $cts\mbox{ }s^{-1}\mbox{ }PCU^{-1}$), with the average count rate increasing roughly by a factor of three from low to medium and from medium to high. Then, for each observation in a group we searched for an observation at TeV energies that was made within an hour. Only matched pairs were kept for further analysis. We ended up with a total of 16, 9, and 3 pairs in the low-flux, medium-flux, and high-flux group, respectively. It turns out that the low-flux group consists of observations taken between 2003 March 8 and May 3, the medium-flux group between 2004 January 27 and March 26, and the high-flux group between 2004 April 16–20.
For each group, we proceeded to construct a flux-averaged SED at X-ray and TeV energies. The flux-averaged X-ray spectrum can be fitted satisfactorily by a power law with an exponential roll-over (with reduced $\chi^2 \lesssim 1$). We should note that we added a 1% systematic uncertainty to the X-ray data for spectral analysis (which is a common practice), to take into account any residual calibration uncertainty. Also, we fixed the hydrogen column density at $1.38\times 10^{20}\mbox{ }cm^{-2}$ (Dickey & Lockman, 1990). The best-fit photon index is about $2.51^{+0.03}_{-0.05}$, $2.38^{+0.02}_{-0.02}$, and $1.99^{+0.02}_{-0.02}$ for the low-, medium-, and high-flux group, respectively, and the roll-over energy about $26^{+3}_{-5}$, $32^{+3}_{-3}$, and $32^{+2}_{-2}$ keV. Therefore, the X-ray spectrum of Mrk 421 hardens toward high fluxes. Using the best-fit model we then derived the X-ray SED for each data group.
For the gamma-ray spectral fits, a bin size of 0.1667 in $\log_{10}(E)$ was adopted for the medium- and high-flux groups and a wider bin size of 0.4 for the low-flux group. As shown in Fig. 12, the low-flux SED still has very large error bars. The first energy bin corresponds to an energy of $\sim$260GeV. The gamma-ray spectra can all be satisfactorily fit by a power law, with a photon index of $2.84\pm0.58$, $2.71\pm0.15$, and $2.60\pm0.11$ for the low-, medium-, and high-flux groups, respectively. The errors bars only include statistical contributions. For the purpose of comparison with some of the published results, we also fitted the spectra with a cut-off power law (see, e.g., Krennrich et al. 2002) but with the cut-off energy fixed at 4.96 TeV. The photon index is $2.73\pm0.56$, $2.40\pm0.18$, and $2.11\pm0.14$ for the low-, medium-, and high-flux groups, respectively. Like the X-ray spectrum, therefore, the TeV spectrum also seems to harden toward high fluxes, although the uncertainty here is quite large.
Finally, we searched for radio and optical observations that fall in one of the groups and computed the average fluxes to complete the SED for the group. Given the fact that Mrk 421 did not vary as significantly at these wavelengths, we believe that the derived SEDs are quite reliable. Figure 12 summarizes the results.
Spectral Modeling
-----------------
We experimented with using a one-zone SSC model (see Krawczynski et al. 2004 for a detailed description of the code, which we revised to use the adopted cosmological parameters) to fit the measured flux-averaged SEDs. Briefly, the model calculates the SED of a spherical blob of radius $R$. The blob moves down the conical jet with a Lorentz factor $\Gamma$. The emitting region is filled with relativistic electrons with a broken power-law spectral distribution: $S_e \propto E^{-p_1}$, for $E_{min}<E<E_{b}$, and $S_e
\propto E^{-p_2}$, for $E_{b} \le E<E_{max}$ (although the code does not evolve the electron spectrum self-consistently). The model accounts for the attenuation of the very-high-energy $\gamma$-rays by the diffuse infrared background (as modelled by MacMinn & Primack 1996) .
We first found an initial “best fit” to each SED by visual inspections. We then performed a systematic grid search around the “best fit” that involves the following parameters: magnetic field $B$ in the range of 0.045–0.45 G, Doppler factor $\delta$ in 10–20, $p_1$ in 1.8–2.2, $p_2$ in 2.9–3.7, $log{E_b}$ in 9.9–12.2, $log{E_{max}}$ in 9.9–12.2, and the normalization ($w_e$) of $S_e$ in 0.00675–0.44325 $ergs\mbox{ }cm^{-3}$. Variability constraints (see the next section) were taken into account in the choice of some of the parameter ranges. All other parameters in the model (e.g., $R$) were fixed at the nominal values determined by the visual inspections. We found roughly where the global $\chi^2$ minimum lies through a coarse-grid search, and then conducted a finer-grid search through much smaller parameter ranges around the minimum to find the best fit. Figure 11 shows the results for the high-flux group. It is apparent that the model severely underestimates the radio and optical fluxes. Large deviations are also apparent at TeV energies. We should stress that our grid searches are, by no means, exhaustive. However, the results should be adequate for revealing gross discrepancies between the model and the data.
To investigate whether the fit could be improved by introducing additional zones (which are assumed to be independent of each other), we introduced a new zone to account for the radio fluxes and another for the optical fluxes. To be consistent with the observed variability, the additional zones must be placed much further down the jets. Figure 12 shows fits to the SEDs with such a three-zone SSC model. This ad-hoc approach does seem to yield a reasonable fit to the data. In such a scenario, the observed variability at X-ray and TeV energies would be associated with zones close to the central black hole, while radio and optical emission are expected to vary on longer timescales further down the jet.
Discussion
==========
X-ray–TeV Correlation
---------------------
The success of the standard SSC model lies partly in the fact that it provides a natural explanation for the correlation between the X-ray and gamma-ray emission. However, we found that the correlation is not as tight as one might naively expect from the SSC model. It seems unlikely that the loose correlation is caused by the choice of spectral bands used in the cross-correlation analysis, given how broad the [*PCA*]{} and [*Whipple*]{} passing bands are. In fact, both the derived variability amplitudes (Fig. 4) and SEDs (Fig. 12) indicate that the X-ray and gamma-ray photons are likely to have originated from the same population of electrons in the context of the SSC model.
In the standard SSC model, X-ray photons are Compton upscattered to produce gamma-ray photons, so an X-ray lead is naturally expected. However, the characteristic timescale of the SSC process would be much too short to account for the X-ray lead of 1–2 days (see Fig. 6), although we cannot be sure about whether the lead is actually real, due to large uncertainties and possible systematic effects. What is certain is that the SSC model cannot explain the difference between the X-ray and gamma-ray rise times of the 2004 giant flare. It is conceivable that the flaring episode could have started with an orphan TeV flare and followed by a pair of simultaneous X-ray and TeV flares. If this is the case, it would be opposite to the theoretical scenario recently put forth by Böttcher (2004). In the hadronic models, both X-ray lag and lead could, in principle, occur, even at the same time, depending on the relative roles of the primary and secondary electrons. However, it would also seem challenging for these models to account for the different X-ray and gamma-ray rise times of the 2004 giant flare or to quantitatively explain the long X-ray lead (of 1–2 days) if it is real.
Variability Constraints
-----------------------
Combined with the measured SED, rapid X-ray flares pose severe constraints on some of the physical properties of the flaring region, in a relatively model-independent manner, because the X-ray emission from Mrk 421 is almost certainly of synchrotron origin. The most rapid X-ray flare detected during our campaign has a duration of about 20 minutes and reaches a peak amplitude of $\sim$15% of the (local) non-flaring flux level (see Fig. 10). Therefore, the size of the flaring region must satisfy: $l' \lesssim c t_{flare} \delta/(1+z) = 3.6\times 10^{14} \delta_1\mbox{ }cm$, where $t_{flare}$ is the duration of the flare ($=1200$ s), $\delta$ is the Doppler factor of the jet ($\delta = 10\delta_1$), and $z$ is the redshift of Mrk 421 ($z = 0.031$). Here and in the remainder of the paper, we use the primed symbols to denote quantities in the co-moving frame of the jet and unprimed ones corresponding quantities in the frame of the observer. It is interesting to note that the derived upper limit is already comparable to the radius of the last stable orbit around the $2\times 10^8 M_{\odot}$ black hole in Mrk 421 (Barth et al. 2003), for $\delta \sim 10$. Since the peak flux of the flare is a significant fraction of the steady-state flux, the size of the flaring region is probably comparable to the lateral extent of the jet (to within an order of magnitude). If the jet originates from accretion flows, as is often thought to be the case, the result would also represent an upper limit on the inner boundary of the flows.
The decaying time of the most rapid X-ray flare (about 600 s) sets a firm upper limit on the synchrotron cooling time of the emitting electrons. The cooling time is given by (Rybicki & Lightman 1979), $\tau'_{syn} \approx 6 \pi m_e c/\sigma_T \gamma'_b B'^2$, where $m_e$ is the electron rest mass, $\sigma_T$ is the Thomson cross section, $B'$ is the strength of the magnetic field in the region, and $\gamma'_b$ is the characteristic Lorentz factor of those electrons that contribute to the bulk of the observed X-ray emission. By requiring $\tau'_{syn} < t_d \delta$, where $t_d$ is the measured decaying time of the flare ($\approx 600 s$), we derived a lower limit on $B'$, $B' > 1.1 \delta_1^{-1/2} {\gamma'}_{b,5}^{-1/2} \mbox{ }G$, where $\gamma'_b = 10^5\gamma'_{b,5}$. From modeling the SEDs, we found $2 \times 10^5 \le \gamma'_b \le 4 \times 10^5$. It should be noted that the limits derived are only appropriate for the region that produced the 20-min X-ray flare and that not all flares are necessarily associated with the same region. In our attempt to model the SEDs (§ 3.2), we only require that the model parameters be consistent with variability timescales of hours. Results from more sophisticated modeling will be presented elsewhere.
Further constraints can be derived from the detected TeV flares. The fact that we detect TeV photons from Mrk 421 requires that the opacity due to $\gamma\gamma \rightarrow e^{+} e^{-}$ must be sufficiently small near the TeV emitting region(s). This requirement leads to (Dondi & Ghisellini 1995): $$\delta \geq \left[ \frac{\sigma_T d^2_L}{5hc^2} \frac{F(\nu_t)}{t_{var}} \right]^{1/(4+2\alpha)},$$ where $d_L$ is the luminosity distance, $t_{var}$ is the TeV flux doubling time, $\alpha$ and $F(\nu_t)$ are the local spectral index and the energy flux, respectively, at the target photon frequency, at which the pair production cross section peaks, $${\nu} {\nu_{t}} \sim \left( \frac{m_e c^2}{h} \frac{\delta}{1+z} \right)^2,$$ where $\nu$ is the frequency of the gamma-ray photon. For $h\nu \sim 1$ TeV, we have $\nu_t \sim 6 \times 10^{15}$ Hz. From the high-flux SED (Fig. 12), we found $\alpha \sim 0.5$ and $\nu_t F_{\nu_t}\sim 1\times 10^{-10}\mbox{ }erg\mbox{ }cm^2\mbox{ }s^{-1}$. Given $d_L = 129.8$ Mpc and $t_{var} \simeq 1$ hour (see Fig. 10), we derived a lower limit on the Doppler factor, $\delta \gtrsim 10$.
“Orphan” TeV Flares
-------------------
Since TeV emission is the consequence of inverse-Compton scattering of (synchrotron) X-ray photons by the electrons themselves in the standard SSC model, a flare at TeV energies must be [*preceded*]{} by an accompanying flare at X-ray energies. While the model might still be able to accommodate the presence of orphan X-ray flares, the presence of orphan TeV flares seems problematic. On the other hand, the “orphan” TeV flares in Mrk 421 (as shown Fig. 9) and in 1ES1959+650 (Krawczynski et al. 2004) might not be orphan events after all. In both cases, the gamma-ray flares seem to be preceded by X-ray flares, which could be attributed to the SSC process although the X-ray lead (of 1–2 days) would seem too long. Alternatively, it might be possible to attribute the X-ray lead to some sort of feedback between different emission regions in the jet (i.e., an orphan X-ray flare in one region triggers an orphan TeV in another).
Genuine orphan TeV flaring would not be easy to understand in the hadronic models either, despite looser coupling between the X-ray and TeV emission, unless the injected e/p ratio changes significantly from flare to flare. Any change in the proton population will likely be accompanied by a change in the electron (primary or secondary) population, so TeV and X-ray flares should both occur as a result. Interestingly, orphan TeV flares may be understood in a hybrid scenario in which protons are present in the jet in substantial quantities but are not necessarily dominant compared to the lepton component (Böttcher 2004). In this case, an orphan TeV flare is associated with (and [**]{}follows) a pair of simultaneous X-ray and TeV flares that originate in the standard SSC process. The synchrotron (X-ray) photons are then reflected off some external cloud and return to the jet (following a substantial delay with respect to the initial flares). The returned photons interact with the protons in the jet to produce pions and subsequent $\pi^0$ decay produces TeV photons in the flare. The model was shown to be able to account for the “orphan” TeV flare observed in 1ES1959+650 (Böttcher 2004). It might also explain some of the similar flares in Mrk 421, for example, the one shown in Fig. 9, if it is a composite of two gamma-ray flares (see discussion in § 3.1.3).
Spectral Energy Distribution
----------------------------
The SED of Mrk 421 varies greatly both at X-ray and TeV energies but only weakly at radio and optical wavelengths. The much reduced variability at long wavelengths is expected for short timescales, because of long synchrotron cooling times of the radio or optical emitting electrons. In other words, there is no reason to expect a tight correlation between the low-energy (radio or optical) emission and the high-energy (X-ray or TeV) emission. Of course, the lack of such a correlation could also be due to the fact that the low-energy emission originates in regions further down the jet. Between the radio and optical bands, we found that the NVAs of the source are comparable (see Fig. 4), which is somewhat surprising because the synchrotron cooling time of the electrons responsible for the optical emission is expected to be much shorter than that of those for the radio emission. On the other hand, this could be evidence for the presence of different populations of electrons that produce the emission in the radio and optical bands. The populations may be located in physically separated regions and may have different spectral energy distributions (e.g., $E_{max}$).
We found that both SED peaks move to higher energies as the luminosity of the source increases. Moreover, the X-ray spectrum becomes flatter, which seems to be common among high-frequency-peaked BL Lacs (e.g., Giommi et al. 1990; Pian et al. 1998; Xue & Cui 2004). Similar spectral hardening has also been see at TeV energies (Krennrich et al. 2002; Aharonian et al. 2002). In our data set, we have seen some indication of spectral hardening but we cannot be certain because of the low statistics of the data. The evolution of the SED could be driven by a hardening in the electron energy distribution and/or an increase in the strength of the magnetic field at high luminosities.
Summary
=======
In this paper, we have presented a substantial amount of multi-wavelength data that have been collected on Mrk 421 as the result of a long-term monitoring campaign. The large dynamical range of the data has allowed us to carry out some detailed investigations on variability timescales, cross-band correlation, and spectral variability. The main results from this work are summarize as follows:
- We have shown that the emission from Mrk 421 varied greatly at both X-ray and gamma-ray energies and that the variabilities in the two energy bands are generally correlated. This is consistent with some of the earlier results but the dynamical range of the data and statistics are much improved here. Equally important is the finding that the correlation is only a fairly loose one (see Figures 5–7). The emission clearly does not always vary in step between the two bands.
- We have discovered that the X-ray emission reached the peak days [*after*]{} the TeV emission during the giant flare in April 2004 (see Fig. 1). Such a difference in the rise times for the two energy bands poses a serious challenge to the standard SSC model, as well as the hadronic models. Whether it can be accommodated by some hybrid model remains to be seen. In addition, there is tentative evidence that the X-ray variability leads the gamma-ray variability by 1–2 days. If proven real, such a long X-rad lead would also be difficult for either the SSC or hadronic models to explain quantitatively.
- We have detected X-ray and TeV flares on relatively short timescales. For example, the most rapid X-ray flare detected has the duration of only $\sim$20 minutes and a peak amplitude of $\sim$15% of the local non-flaring flux level. Although no similarly rapid TeV flares were detected during our campaign, we did observe a TeV flare with duration of hours. Physical constraints on the flaring regions have been derived from the rapid X-ray and TeV variabilities.
- We have seen TeV flares in Mrk 421 that are similar to the reported “orphan” flare in 1ES1959+650. Genuinely orphan or not, this presents another serious challenge to the proposed emission models (see § 4.3). If they are true orphan events, they may be explained by a hybrid model in which both electrons and protons are present in the jet in substantial quantities. On the other hand, if the preceding X-ray flare turns out to be the counterpart, the 1–2 day X-ray lead would challenge the proposed emission models (see § 3.1.3).
- We have derived high-quality SEDs of Mrk 421 from simultaneous or nearly simultaneous observations at TeV, X-ray, optical, and radio energies. The X-ray spectrum clearly hardens toward high fluxes; the gamma-ray spectrum also appears to evolve similarly (although we cannot be sure due to large uncertainties of the data). A one-zone SSC model fails badly to fit the measured fluxes at the radio and optical wavelengths, and, to a lesser extent, also underestimates fluxes at the highest TeV energies. The introduction of additional zones improves the fit significantly.
- Further evidence for the presence of multiple populations of emitting electrons is provided by the comparable radio and optical NVAs (see Fig. 4 and discussion in § 4.4).
We acknowledge the technical assistance of E. Roache and J. Melnick. The VERITAS collaboration is supported by the U.S. Department of Energy, National Science Foundation (NSF), the Smithsonian Institution, NSERC (Canada), PPARC (UK), and Science Foundation Ireland. W. Cui and M. Błażejowski also gratefully acknowledge financial support from NASA and thank Peter Biermann for useful discussions on the subject and comments on the manuscript. UMRAO is supported in part by funds from NSF and from the Department of Astronomy at University of Michigan.
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[^1]: See the online [*RXTE*]{} Cook Book, available at http://heasarc.gsfc.nasa.gov/docs/xte/recipes/cook\_book.html.
[^2]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the NSF.
[^3]: based on data from the MIT archive at http://xte.mit.edu/asmlc/srcs/mkn421.html\#data
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Identifying groups of variables that may be large simultaneously amounts to finding out which joint tail dependence coefficients of a multivariate distribution are positive. The asymptotic distribution of a vector of nonparametric, rank-based estimators of these coefficients justifies a stopping criterion in an algorithm that searches the collection of all possible groups of variables in a systematic way, from smaller groups to larger ones. The issue that the tolerance level in the stopping criterion should depend on the size of the groups is circumvented by the use of a conditional tail dependence coefficient. Alternatively, such stopping criteria can be based on limit distributions of rank-based estimators of the coefficient of tail dependence, quantifying the speed of decay of joint survival functions. Numerical experiments indicate that the algorithm’s effectiveness for detecting tail-dependent groups of variables is highest when paired with a criterion based on a Hill-type estimator of the coefficient of tail dependence.'
author:
- Maël Chiapino
- Anne Sabourin
- Johan Segers
date: 'Received: date / Accepted: date'
title: Identifying groups of variables with the potential of being large simultaneously
---
Introduction
============
A question that often arises when monitoring several variables is which groups of variables are prone to be large simultaneously. In food risk management, for instance, the variables under consideration may be the concentrations of different contaminants in blood samples of consumers. In environmental applications, one may be interested in several physical variables such as wind speed and precipitation recorded at several locations, with the purpose of setting off a regional warning when several of these variables exceed a high threshold. In the context of semi-supervised anomaly detection, when the training sample is mostly made of normal instances, identifying the groups of variables which are likely to be large together allows to label certain new instances as abnormal.
The latter use case is the motivation behind the DAMEX algorithm [@goixsparse; @goix2017sparse]. In a regular variation framework, identifying those groups among $d$ variables that may be large simultaneously amounts to identifying the support of the exponent measure. The algorithm returns the list of groups of features $\alpha\subset\{1,\ldots, d\}$ such that the mass of the empirical exponent measure on certain cones exceeds a user-defined threshold. However, when the empirical version of the exponent measure is scattered over a large number of such cones, the DAMEX algorithm does not discover a clear-cut structure. @chiapinofeature encounter this difficulty for extreme streamflow data recorded at several locations of the French river system.
To overcome this issue, the same authors come up with the CLEF (CLustering Extreme Features) algorithm. Instead of partitioning the sample space, CLEF considers nested regions corresponding to increasing subsets of components. A group of variables is enlarged until there is no longer enough evidence that all features in it may be large together. In this respect, CLEF resembles the Apriori algorithm [@agrawal1994fast], which is a data-mining tool for discovering maximal sets of items among $d$ available items that are frequently bought together by consumers. Apriori considers increasing itemsets that are made to grow until their frequency falls below a user-defined threshold. In CLEF, the stopping criterion concerns the relative frequency of simultaneous occurrences of large values of all components in a considered subset compared to the frequency of simultaneous occurrences of larges values of all but one component in this subset. @chiapinofeature find the method to work well on real and simulated data but do not investigate the asymptotic properties of the statistic underlying the stopping criterion.
Our contributions are three-fold. First, we investigate the asymptotic behavior of the statistic underlying CLEF. In this way, the informal stopping criterion can be turned into a proper hypothesis test with controllable level. A second issue concerns the specification of the null hypothesis in the CLEF stopping criterion. Originally, a certain conditional tail dependence coefficient, $\kappa_\alpha$, related to a given group of variables $\alpha \subset \{1, \ldots, d\}$ is supposed to be above a strictly positive, user-defined and therefore somewhat arbitrary threshold. We propose instead to base the stopping criterion on the hypothesis that a multivariate version of the coefficient of @ledford1996statistics and @ramos2009new is equal to one. The test is based on the limit distributions of multivariate extensions of nonparametric estimators in @peng1999estimation and @draisma2001tail [@draisma2004bivariate]. Third, we conduct a numerical experiment to compare the finite-sample performance of the DAMEX algorithm and the CLEF algorithm with the various stopping criteria. We find that overall, the multivariate extension of the Hill-type estimator in [@draisma2004bivariate] yields the most reliable procedure to detect maximal groups of asymptotically dependent variables.
Section \[sec:taildep\_background\] casts the problem in the language of regular variation and introduces the tail dependence coefficients upon which the CLEF stopping criteria will be based. Necessary background on empirical tail dependence functions and processes is reviewed in Section \[sec:etdf\], including a new result for the empirical joint tail function. In Section \[sec:test-kappa\], we derive the asymptotic distribution of the statistic used in CLEF and turn the heuristic stopping criterion implemented in [@chiapinofeature] into a statistical test with asymptotically controllable level. Two alternative tests based on the asymptotic distributions of estimators of the Ledford–Tawn–Ramos coefficient of tail dependence are constructed in Sections \[sec:mult-extens-peng\] and \[sec:hill\]. We report the results of our simulation experiments in Section \[sec:simu-study\]. Section \[sec:conclusion\] concludes. Proofs are gathered in Appendix \[sec:appendix\] while the pseudo-code for the CLEF algorithm and variations is provided in Appendix \[sec:appendix-CLEF\].
Regular variation and tail dependence coefficients {#sec:taildep_background}
==================================================
Bold letters denote vectors and binary operations between vectors are understood componentwise. The indicator function of a set $A$ is denoted by ${{\mathbbm{1}}}_A$. For $t\in {\mathbb{R}}\cup\{\infty\}$, we let ${\bm}t_\alpha$ denote the constant vector of $({\mathbb{R}}\cup \{\infty\})^\alpha$ with all coordinates equal to $t$. In the special case $\alpha = \{1, \ldots, d\}$, the index $\alpha$ is usually omitted for brevity when clear from the context: for instance, ${\boldsymbol{0}}= {\boldsymbol{0}}_{\{1,\ldots,d\}} = (0, \ldots, 0) \in {{\mathbb{R}}}^d$.
Let ${{\bm}X} = (X_1, \ldots, X_d)$ be a random vector in ${{\mathbb{R}}}^d$ with cumulative distribution function $F$, whose margins $F_1, \ldots, F_d$ are continuous. We assume that the transformed vector ${{\bm}V} = (V_1, \ldots, V_d)$ with $V_j = 1/\{1 - F_j(X_j)\}$ for all $j \in \{1,\ldots,d\}$ is regularly varying on the cone $[0,\infty]^d \setminus \{{\boldsymbol{0}}\}$ with (nonzero) limit or exponent measure $\mu$. This means that $\mu$ is finite on Borel sets of $[0,\infty]^d \setminus \{{\boldsymbol{0}}\}$ bounded away from the origin and that $$\label{eq:reg-var}
\lim_{t \to \infty} t {\operatorname{\mathbb{P}}}[ {{\bm}V} \in t A] = \mu(A),$$ for all Borel sets $A \subset [0,\infty]^d \setminus \{{\boldsymbol{0}}\}$ such that ${\boldsymbol{0}}\notin\partial A$ and $\mu(\partial A)=0$. The measure $\mu$ is homogeneous, i.e., $\mu(s \,\cdot\,) = s^{-1} \mu(\,\cdot\,)$ for all $0 < s < \infty$, and therefore assigns no mass to hyperplanes parallel to the coordinate axes. As a consequence, applies to finite and infinite rectangles that are bounded away from the origin and whose sides are parallel to the coordinate axes. The measure $\mu$ characterizes the extremal dependence structure of ${{\bm}X}$. The reader is referred to @resnick:2007 [@resnick2013extreme] for an introduction to regular variation.
Let $\varnothing \ne \alpha \subset {\{1,\ldots,d\}}$. Particular instances of include the extremal coefficient $\lambda_\alpha$ [@schlather2003dependence] and the joint tail coefficient $\rho_\alpha$: $$\begin{aligned}
\label{eq:lambda_alpha}
\lambda_\alpha
&=
\lim_{t \to \infty} t {\operatorname{\mathbb{P}}}[ \exists j \in \alpha : V_j > t ]
=
\mu ( \{ {\bm}{u} \in [0, \infty)^d \mid \exists j \in \alpha : u_j > 1 \} ),
\\
\label{eq:rho_alpha}
\rho_\alpha
&=
\lim_{t \to \infty} t {\operatorname{\mathbb{P}}}[ \forall j \in \alpha : V_j > t ]
=
\mu ( \{ {\bm}{u} \in [0, \infty)^d \mid \forall j \in \alpha : u_j > 1 \} ).\end{aligned}$$ In the bivariate case, $|\alpha|=2$, and with our choice of Pareto margins, we have $\rho_\alpha = \lim_{t\to \infty} {\operatorname{\mathbb{P}}}(V_{\alpha_1}>t \mid V_{\alpha_2}>t)$, the upper tail dependence coefficient denoted by $\chi$ in [@coles1999dependence].
Our general objective is to propose statistically sound procedures to recover maximal subgroups $\alpha$ of components that are likely to be concomitantly large. Our aim can thus be phrased as recovering the maximal subsets $\alpha\subset{\{1,\ldots,d\}}$ such that $\rho_\alpha>0$.
Since $\rho_\alpha {\leqslant}\rho_\beta$ as soon as $\alpha \supset \beta$, any positive tolerance level with which we would like to compare an estimate of $\rho_\alpha$ should depend on $\alpha$ and in particular be decreasing as a function of the cardinality $|\alpha|$. To circumvent this issue, @chiapinofeature consider for $\alpha$ such that $|\alpha| {\geqslant}2$ the conditional tail dependence coefficient $$\label{eq:kappa}
\kappa_\alpha =
\lim_{t\to \infty} {\operatorname{\mathbb{P}}}\left[
\forall j \in \alpha : V_j > t
\; \Big|\;
\textstyle\sum_{j\in\alpha} {\mathbbm{1}}{\{V_j > t\}} {\geqslant}|\alpha| -1
\right],$$ which is the limiting conditional probability that all variables in $\alpha$ exceed a large threshold given that all but at most one already do. In contrast to $\rho_\alpha$, the coefficient $\kappa_\alpha$ has no particular reason to decrease as a function of $|\alpha|$. Note that $\rho_\alpha = \mu(\Gamma_\alpha)$ while $\kappa_\alpha = \mu(\Gamma_\alpha) / \mu(\Delta_\alpha) = \rho_\alpha / \mu(\Delta_\alpha)$ where $\Gamma_\alpha = \{ {\bm}{x} \in [0, \infty)^d \mid \forall j \in \alpha : x_j > 1 \}$ and $\Delta_\alpha = \{{\bm}x \in [0, \infty)^d \mid \textstyle\sum_{j \in \alpha} {\mathbbm{1}}_{\{x_j{\geqslant}1\}}{\geqslant}|\alpha|-1\}$, provided $\lvert \alpha \rvert {\geqslant}2$. If $\mu(\Delta_\alpha) = 0$, then $\mu(\Gamma_\beta) = 0$ for all $\beta \subset \alpha$ with $|\beta| = |\alpha| - 1$; in that case, we define $\kappa_\alpha = 0$. In the CLEF algorithm [@chiapinofeature], the criterion to decide whether $\rho_\alpha>0$ or not is that $\widehat\kappa_\alpha {\geqslant}C$, where $C$ is a user-defined tolerance level, $\widehat\kappa_\alpha = \widehat\mu(\Gamma_\alpha)/\widehat\mu(\Delta_\alpha)$, and $\widehat\mu$ is the empirical exponent measure in below. The level $C$ can be chosen independently of $\alpha$. Still, its choice is somewhat arbitrary, and in particular, the user has no control of false positives. In Section \[sec:test-kappa\], we will provide the asymptotic distribution of $\widehat\kappa_\alpha$ and propose a test statistic with a guaranteed asymptotic level.
If $\rho_\alpha=0$ (or $\kappa_\alpha=0$), the limiting distributions of the statistics $\sqrt{k} (\widehat{\rho}_\alpha - \rho_\alpha)$ and $\sqrt{k} (\widehat{\kappa}_\alpha - \kappa_\alpha)$ are degenerate at zero. We therefore have no control on the asymptotic levels of tests based on those statistics under $H_0 : \kappa_0 = 0$. This is why will have to define a CLEF stopping criterion in terms of a test of $H_0 : \kappa_\alpha {\geqslant}\kappa_{\min}$ versus $H_1 : \kappa_\alpha < \kappa_{\min}$ instead, in terms of a user-defined level $\kappa_{\min} > 0$. The choice of $\kappa_{\min}$ is somewhat arbitrary; in the simulation experiments (Section \[sec:simu-study\]), we choose $\kappa_{\min} = 0.08$. In Sections \[sec:mult-extens-peng\] and \[sec:hill\], we consider alternative CLEF stopping criteria based on estimators of the coefficient of tail dependence $\eta_\alpha\in (0,1]$. For bivariate distributions, the coefficient has been introduced by @ledford1996statistics and extended by @ramos2009new in order to model situations in between asymptotic dependence ($\rho_{\{1,2\}} > 0$) and full independence of $X_1$ and $X_2$. @HaanZhou11residual and @eastoe2012subasymptotic proposed and studied a multivariate extension of $\eta_\alpha$ for $|\alpha| {\geqslant}3$. The model assumption is that there exist $\eta_\alpha\in(0,1]$ and a slowly varying function $\mathcal{L}_\alpha$ such that $$\label{eq:taildep-multi}
{\operatorname{\mathbb{P}}}[ \forall j \in \alpha : V_j > t ] = t^{-1/\eta_\alpha} \mathcal{L}_\alpha(t).$$ Suppose that the limit $\rho_\alpha$ in exists and that holds. Then $\rho_\alpha > 0$ implies $\eta_\alpha = 1$. The converse is true as well, provided $\liminf_{t\to\infty}\mathcal{L}_\alpha(t) > 0$. Modulo this side condition, which we will take for granted, the null hypothesis $\rho_\alpha>0$ corresponds to the simple hypothesis $\eta_\alpha = 1$.
We will test the null hypothesis $\eta_\alpha = 1$ via multivariate extensions of nonparametric estimators of $\eta_\alpha$ in @peng1999estimation and @draisma2004bivariate. The null limit of the test statistic is non-degenerate, so that the asymptotic level of the test can be controlled, with no need to introduce an additional tolerance parameter $\kappa_{\min}$. The estimators that we will study are related to the Pickands estimator and the Hill estimator for the extreme value index of $T_\alpha = \min_{j\in\alpha} V_j$, respectively. The maximum likelihood estimator, also considered in [@draisma2004bivariate], is less suitable to our context due to its relative computational complexity, since the test is destined to be performed on a large number of subsets of $\{1,\ldots,d\}$. See also the review [@bacro2013measuring] and the references therein.
\[rm:relationship-goix\] The DAMEX algorithm [@goix2017sparse] is designed to recover the family $\mathcal{M}$ of non-empty subsets $\alpha$ of ${\{1,\ldots,d\}}$ with the property that $$\mu \Bigl(
\Bigl\{ {\bm}x \in [0, \infty)^d \;\Big|\;
\|{\bm}x\|_\infty {\geqslant}1 ;\;
\forall j \in \alpha\,, x_j > 0\; \text{ and } \forall j
\notin\alpha, \,x_j = 0 \Bigr\}
\Bigr)
> 0.$$ In contrast, our focus is on $\mathbb{M} = \{ \alpha \mid \rho_\alpha > 0\} = \{ \alpha \mid \kappa_\alpha > 0 \}$. Still, the maximal elements of $\mathbb{M}$ for the inclusion order are also the maximal elements of $\mathcal{M}$ [@chiapinofeature Lemma 1]. The two problems of finding the maximal elements of $\mathbb{M}$ or $\mathcal{M}$ are thus equivalent.
Empirical tail dependence functions and processes {#sec:etdf}
=================================================
To find the asymptotic distribution of nonparametric estimators of the various dependence coefficients, we rely on empirical tail processes. Let the random vector ${\bm}{X} \sim F$ be as in Section \[sec:taildep\_background\]; in particular, assume regular variation as in with exponent measure $\mu$. Let $\Lambda$ be the push-forward measure of $\mu$ on $[0, \infty]^d \setminus \{ {\bm}{\infty} \}$ induced by the transformation ${\bm}{x} \mapsto 1/{\bm}{x} = (1/x_1, \ldots, 1/x_d)$, i.e., $\Lambda(\,\cdot\,) = \mu( \{ {\bm}{x} \in [0, \infty]^d \setminus \{
\bm{0} \} \mid 1/{\bm}{x} \in \, \cdot \, \} )$.
For $\varnothing \ne \alpha \subset {\{1,\ldots,d\}}$, consider the stable tail dependence function $\ell_\alpha : [0, \infty)^\alpha \to [0, \infty)$ and the joint tail dependence function $r_\alpha : [0, \infty]^\alpha \setminus \{ {\bm}{\infty}_\alpha \} \to [0, \infty)$ given by $$\begin{aligned}
\nonumber
\ell_\alpha({\bm}x)
&= \lim_{t \to 0} t^{-1} {\operatorname{\mathbb{P}}}[ \exists j \in \alpha : F_j(X_j) > 1 - t x_j]
= \Lambda( \{ {\bm}{y} \mid \exists j \in \alpha : y_j < x_j \} ),
\\
\label{eq:r_alpha}
r_\alpha({\bm}x)
&= \lim_{t \to 0} t^{-1} {\operatorname{\mathbb{P}}}[ \forall j \in \alpha : F_j(X_j) > 1 - t x_j]
= \Lambda( \{ {\bm}{y} \mid \forall j \in \alpha : y_j < x_j \} ).\end{aligned}$$ From and , clearly $\lambda_\alpha = \ell_\alpha({\bm}{1}_\alpha)$ and $\rho_\alpha = r_\alpha({\bm}{1}_\alpha)$. For brevity, we write $\ell = \ell_{{\{1,\ldots,d\}}}$ and $r = r_{{\{1,\ldots,d\}}}$. Note that $\ell_\alpha( \bm{x} ) = \ell( \bm{x} \bm{e}_\alpha )$ for $\bm{x} \in [0, \infty)^\alpha$, where ${\bm}e_\alpha \in \{0, 1\}^d$ has components ${\bm}e_{\alpha, j} = {{\mathbbm{1}}}_{\alpha}(j)$. Similarly, $r_\alpha( \bm{x} ) = r( \bm{x} \bm{\iota}_\alpha )$ for $\bm{x} \in [0, \infty]^\alpha \setminus \{ \bm{\infty}_\alpha \}$, where ${\boldsymbol{\iota}}_\alpha \in \{1,\infty\}^d$ denotes the vector such that ${\boldsymbol{\iota}}_{\alpha,j} = 1 $ if $j \in \alpha$ and ${\boldsymbol{\iota}}_{\alpha,j} = +\infty $ otherwise. By the inclusion–exclusion formula, for $\bm{x} \in [0, \infty)^\alpha$, writing $\bm{x}_\beta = (x_j)_{j \in \beta}$, we have $$\begin{aligned}
\label{eq:ell_beta2r_alpha}
r_\alpha({\bm}x)
&=
\sum_{\varnothing \ne \beta \subset \alpha}
(-1)^{|\beta|+1} \ell_\beta ({\bm}x_\beta), &
\ell_\alpha({\bm}x)
&=
\sum_{{\varnothing}\neq \beta \subset \alpha}
(-1)^{|\beta|+1} r_\beta ({\bm}x_\beta).\end{aligned}$$ Let ${{\bm}X}_i = (X_{i,1}, \ldots, X_{i,d})$, for $i \in \{1,\ldots,n\}$, be an independent random sample from $F$, having continuous margins and satisfying . Let $k = k(n)\to \infty$ as $n\to\infty$, while $k(n) = {\mathrm{o}}(n)$. Following for instance [@einmahl2012m; @goix2017sparse; @qi1997almost], we rely on ranks to obtain an approximately Pareto-distributed sample $\widehat {{\bm}V}_i = (\widehat{V}_{i,1}, \ldots, \widehat{V}_{i,d})$. Let $\widehat F_j(x) = n^{-1} \sum_{i=1}^n {\mathbbm{1}}_{\{X_{i,j} < x\}}$ be the (left-continuous) empirical distribution function of component $j \in \{1,\ldots,d\}$ and put $\widehat V_{i,j} = 1/\{1 - \widehat F_j(X_{i,j})\} = n/(n + 1 - R_{i,j})$, where $R_{i,j}$ is the rank of $X_{i,j}$ among $X_{1,j}, \ldots, X_{n,j}$. The empirical counterparts to $\mu$ and $\Lambda$ are $$\begin{aligned}
\label{eq:mun}
\widehat{\mu}(\,\cdot\,)
&=
\frac{1}{k}\sum_{i=1}^n \delta_{ (k/n)\widehat{{\bm}V}_{i} }(\,\cdot\,), &
\widehat{\Lambda}(\,\cdot\,)
&=
\frac{1}{k}\sum_{i=1}^n \delta_{ (n/k)/\widehat{{\bm}V}_{i} }(\,\cdot\,),\end{aligned}$$ respectively, with $\delta_a$ the Dirac measure at the point $a$. Replacing $\Lambda$ by $\widehat{\Lambda}$ in the definition of $\ell_\alpha$ and $r_\alpha$ produces the empirical tail dependence function $$\begin{aligned}
\widehat{\ell}_\alpha( \bm{x} )
&= k^{-1} \textstyle\sum_{i=1}^n {{\mathbbm{1}}}\{\exists j \in \alpha : n + 1 - R_{i,j} {\leqslant}\lfloor k x_j \rfloor \} \\
&= k^{-1} \textstyle\sum_{i=1}^n {{\mathbbm{1}}}\{\exists j \in \alpha : X_{i,j} {\geqslant}X_{(n-\lfloor k x_j \rfloor +1),j} \} \end{aligned}$$ and the empirical joint tail function $$\begin{aligned}
\label{eq:r_alpha:estim}
\widehat{r}_\alpha( \bm{x} )
&= k^{-1} \textstyle\sum_{i=1}^n {{\mathbbm{1}}}\{\forall j \in \alpha : n + 1 - R_{i,j} {\leqslant}\lfloor k x_j \rfloor \} \\
\nonumber
&= k^{-1} \textstyle\sum_{i=1}^n {{\mathbbm{1}}}\{\forall j \in \alpha : X_{i,j} {\geqslant}X_{(n-\lfloor k x_j \rfloor +1),j} \}, \end{aligned}$$ where $X_{(1),j} {\leqslant}\ldots {\leqslant}X_{(n),j}$ are the ascending order statistics of $X_{1,j}, \ldots, X_{n,j}$ and $\lfloor \,\cdot\, \rfloor$ is the floor function. The identities hold for $\widehat{\ell}_\alpha$ and $\widehat{r}_\alpha$ as well.
@einmahl2012m [Theorem 4.6] find the weak limit of the empirical process $\sqrt{k} ( \widehat{\ell} - \ell )$ on $[0, T]^d$ for any $T > 0$. We leverage their theorem to show a similar result for $\sqrt{k}( \widehat{r}_\alpha - r_\alpha )$, jointly in $\alpha$. The following conditions stem from the cited article.
\[as:bias\] There exists $\gamma > 0$ such that, uniformly in ${\bm}x \in [0,1]^d$ with $\sum_{j=1}^d x_j = 1$, we have $$t^{-1}{\operatorname{\mathbb{P}}}[ \exists j = 1, \ldots, d : \, F_j(X_j) > tx_j ] - \ell({\bm}x)
= {\mathrm{O}}(t^{\gamma}), \qquad t \to \infty.$$
\[as:small-k\] The sequence $k = k(n)$ satisfies $k = {\mathrm{o}}(n^{2\gamma/(1+2\gamma)})$ as $n \to \infty$, with $\gamma > 0$ as in Condition \[as:bias\].
\[as:partialDeriv\] For all $j \in\{1,\ldots,d\}$, the partial derivative $\partial_j \ell = \partial \ell / \partial x_j$ exists and is continuous on the set $\{ {\bm}x \in [0,\infty)^d \mid x_j > 0\}$.
Since $\ell$ is convex, it is continuously differentiable Lebesgue almost everywhere [@rockafellar:1970 Theorem 25.5]. Condition \[as:partialDeriv\] is satisfied for many popular max-stable models (logistic, asymmetric logistic, Brown–Resnick) but fails for max-linear models. Under Condition \[as:partialDeriv\], the partial derivative $\partial_j r_\alpha = \partial r_\alpha / \partial x_j$ ($j \in \alpha$) exists and is continuous on $\{ {\bm}{x} \in [0, \infty)^\alpha \mid x_j > 0 \}$ and satisfies $\partial_j r_\alpha( {\bm}{x} ) = \sum_{\beta : j \in \beta \subset \alpha} (-1)^{|\beta|+1} \partial_j \ell_\beta({\bm}{x}_\beta)$, where ${\bm}{x}_\beta = (x_s)_{s \in \beta}$.
@einmahl1997poisson and @einmahl2012m consider a centered Gaussian process $W$ indexed by the Borel sets of $[0,\infty]^d\setminus\{{\bm}{\infty} \}$ bounded away from ${\bm}{\infty}$ with covariance function $$\label{eq:covW}
{\operatorname{\mathbb{E}}}[W(A) \, W(B)] = \Lambda(A \cap B ).$$ Note that $W({\varnothing}) = 0$ almost surely. For ${\varnothing}\neq \alpha\subset\{1,\ldots, d\}$ and $\bm{x} \in [0, \infty)^\alpha$, write $$W_\alpha( \bm{x} ) = W(\{ {\bm}{y} \in [0, \infty]^d \mid \forall j \in \alpha : y_j < x_j \}).$$ We consider weak convergence as in [@van2000asymptotic; @van1996weak]; notation ${\rightsquigarrow}$. We work in the metric space $\ell^\infty(S)$ of bounded, real functions $f$ on an arbitrary set $S$, the metric being the one induced by the supremum norm, $\| f \|_\infty = \sup_{x \in S} \lvert f(x) \rvert$; the double use of the symbol $\ell$ should not give rise to any confusion. The proof of the following proposition and of other results in the paper is deferred to Appendix \[sec:appendix\].
\[prop:rnx\] Let ${{\bm}X}_i = (X_{i,1}, \ldots, X_{i,d})$, for $i \in \{1,\ldots,n\}$, be an independent random sample from $F$, having continuous margins and satisfying . Let $k = k(n)\to \infty$ as $n\to\infty$, while $k(n) = {\mathrm{o}}(n)$. If Conditions \[as:bias\], \[as:small-k\] and \[as:partialDeriv\] hold, then, for $T > 0$, in the product space $\prod_{\varnothing \ne \alpha \subset {\{1,\ldots,d\}}} \ell^\infty( [0, T]^\alpha )$, we have, as $n \to \infty$, the weak convergence $$\label{eq:Z_alpha(x)}
\sqrt k\left\{\widehat r_\alpha({\bm}x) - r_\alpha({\bm}x)\right\}
{\rightsquigarrow}W_\alpha({\bm}x) - \sum_{j\in\alpha}
\partial_j r_{\alpha}({\bm}x) \, W_{\{j\}}(x_j)
=
Z_\alpha( {\bm}{x} ).$$
Estimating the conditional tail dependence coefficient {#sec:test-kappa}
======================================================
This section investigates the asymptotic distribution of the empirical conditional dependence coefficient $\widehat{\kappa}_\alpha$ based on the empirical exponent measure $\widehat{\mu}$. This is achieved by re-writing $\widehat\kappa_\alpha$ as a function of the empirical joint tail coefficients $\widehat\rho_\alpha$, the distribution of which follows from Proposition \[prop:rnx\]. We also propose consistent estimators of the asymptotic variance of $\widehat{\kappa}_\alpha$. Combining the two yields a test for the null hypothesis $\kappa_\alpha {\geqslant}\kappa_{\min}$ where $\kappa_{\min}\in(0,1)$ is a tolerance level fixed by the user, to be seen as the minimal limiting conditional probability that all components in a random vector exceed a threshold, given that all of them but at most one already do.
Let $\varnothing \ne \alpha \subset {\{1,\ldots,d\}}$ and recall the sets $\Gamma_\alpha = \{ {\bm}{x} \in [0, \infty)^d \mid \forall j \in \alpha : x_j > 1 \}$ and, provided $\alpha$ has at least two elements, $\Delta_\alpha = \{{\bm}x \in [0, \infty)^d \mid \textstyle\sum_{j \in \alpha} {\mathbbm{1}}_{\{x_j{\geqslant}1\}}{\geqslant}|\alpha|-1\}$. Write $\alpha \setminus j = \alpha \setminus \{j\}$ for $j \in \alpha$. Since $\Delta_\alpha$ is the disjoint union of the sets $\Gamma_{\alpha \setminus j} \setminus \Gamma_\alpha$ and $\Gamma_\alpha$, where $j \in \alpha$, we find, for every Borel measure $\nu$, the equality $$\label{eq:B-Gamma}
\nu( \Delta_\alpha )
= \sum_{j \in \alpha} \nu(\Gamma_{\alpha \setminus j}) - (|\alpha| -1) \, \nu(\Gamma_\alpha).$$ Recall $\rho_\alpha = \mu(\Gamma_\alpha)$ and $\kappa_\alpha = \mu(\Gamma_\alpha) / \mu(\Delta_\alpha)$ in . By applied to $\nu = \mu$, we have $$\label{eq:rewriteKappa}
\kappa_\alpha
=
\frac {\rho_\alpha} {\sum_{j \in \alpha} \rho_{\alpha \setminus j} - (|\alpha| - 1)\rho_{\alpha}}.$$
Recall the joint tail function $r_\alpha$ and its nonparametric estimator $\widehat{r}_\alpha$ in and , respectively. Since $\rho_\alpha = r_\alpha( {\bm}{1}_\alpha )$, we define the estimators $\widehat{\rho}_\alpha = \widehat{\mu}( \Gamma_\alpha ) = \widehat{r}_\alpha( \bm{1}_\alpha )$ and, provided $\lvert \alpha \rvert {\geqslant}2$, $$\widehat{\kappa}_\alpha
=
\frac{\widehat{\mu}( \Gamma_\alpha )}{\widehat{\mu}( \Delta_\alpha )}
=
\frac {\widehat{\rho}_\alpha} {\sum_{j \in \alpha} \widehat{\rho}_{\alpha \setminus j} - (|\alpha| - 1) \widehat{\rho}_{\alpha}}.$$ The asymptotic distribution of the vector of empirical joint tail coefficients follows immediately from Proposition \[prop:rnx\]. Write $\dot{\rho}_{\alpha, j} = \partial_j r_{\alpha}(\bm{1}_\alpha)$.
\[prop:asymptotic-hatRho\] In the setting of Proposition \[prop:rnx\], we have, jointly in $\varnothing \ne \alpha \subset {\{1,\ldots,d\}}$, the weak convergence $$\label{eq:G_alpha}
\sqrt{k_n} \left( \widehat{\rho}_{\alpha} - \rho_\alpha \right)
{\rightsquigarrow}Z_\alpha( {\bm}{1}_\alpha ) = G_\alpha,
\qquad n \to \infty.$$ The limit distribution is centered Gaussian with covariance matrix $$\label{eq:covG}
{\operatorname{\mathbb{E}}}[ G_{\alpha} G_{\alpha'} ]
= \rho_{\alpha\cup\alpha'} -
\sum_{j \in \alpha} \dot{\rho}_{j,\alpha}
\rho_{\alpha' \cup \{j\}} - \sum_{j' \in \alpha'}
\dot{\rho}_{j',\alpha'} \rho_{\alpha \cup \{j'\}}
+\sum_{j \in \alpha} \sum_{j' \in \alpha'}
\dot{\rho}_{j,\alpha} \, \dot{\rho}_{j',\alpha'} \, \rho_{\{j, j'\}}.$$
The asymptotic distribution of $\widehat{\kappa}_\alpha$ follows from the one of $(\widehat{\rho}_\beta)_\beta$ via the delta method. The asymptotic variance involves the partial derivative $\partial_j \kappa_\alpha = \partial \kappa_\alpha / \partial x_j$ of the function $$\label{eq:kappax}
\kappa_\alpha({\bm}x)
= \frac{r_\alpha({\bm}x)}
{ \sum_{j \in\alpha} r_{\alpha \setminus j}({\bm}x_{\alpha \setminus j}) - (\lvert\alpha\rvert -1) r_\alpha({\bm}x)}$$ for ${\bm}x \in [0,\infty)^\alpha$. Note that $\kappa_\alpha({\bm}1_\alpha) = \kappa_\alpha$. Write $\dot{\kappa}_{j,\alpha} = \partial_j \kappa_\alpha( {\bm}{1}_\alpha )$.
\[theo:asymptot-kappa\] In the setting of Corollary \[prop:asymptotic-hatRho\], we have, as $n \to \infty$ and jointly in $\alpha\subset\{1,\ldots,d\}$ such that $|\alpha| {\geqslant}2$ and $\mu(\Delta_\alpha) > 0$, the weak convergence $$\label{eq:kappalimit}
\sqrt{k}\left( \widehat\kappa_\alpha - \kappa_\alpha\right)
{\rightsquigarrow}\mu(\Delta_\alpha)^{-2}
\left\{
\left( \textstyle\sum_{j\in\alpha}\rho_{\alpha\setminus j} \right) G_\alpha
- \rho_\alpha \textstyle\sum_{j\in\alpha} G_{\alpha\setminus j}
\right\}.$$ For a fixed such $\alpha$, the limit distribution is $\mathcal{N}(0, \sigma_{\kappa,\alpha}^2)$ with $$\begin{gathered}
\label{eq:sigmakappa2}
\sigma^2_{\kappa, \alpha} = \big(1 - \kappa_\alpha)\kappa_\alpha
\left\{ \mu(\Delta_\alpha)^{-1} - \textstyle\sum_{j\in\alpha}\dot{\kappa}_{j,\alpha} \right\}
+ \sum_{i\in\alpha}\sum_{j\in\alpha} \dot{\kappa}_{i,\alpha}\dot{\kappa}_{j,\alpha} \rho_{\{i,j\}} \\
+ \kappa_{\alpha} \sum_{ j \in\alpha}\dot{\kappa}_{j,\alpha}
\left\{ 1 - \mu(\Delta_\alpha)^{-1} \rho_{\alpha\setminus j} \right\}.
\end{gathered}$$
Following [@peng1999estimation], the asymptotic variance $\sigma^2_{\kappa,\alpha}$ in can be estimated consistently by estimating the partial derivatives $\dot{\kappa}_{i,\alpha}$ via finite differencing applied to the empirical version of $\kappa_{\alpha}({\bm}{x})$ in obtained by replacing $r_\alpha$ and $r_{\alpha \setminus j}$ by $\widehat{r}_\alpha$ and $\widehat{r}_{\alpha \setminus j}$, respectively: $$\widehat \kappa_{\alpha}({\bm}x)
=
\frac {\sum_{i=1}^n {{\mathbbm{1}}}\{\forall j \in\alpha : X_{i,j } {\geqslant}X_{(n - \lfloor kx_j \rfloor +1 ), j} \} } {\sum_{i=1}^n {{\mathbbm{1}}}\{\exists m\in\alpha: \forall j \in\alpha \setminus m : X_{i,j } {\geqslant}X_{(n - \lfloor kx_j \rfloor +1 ), j} \}}$$ Define $$\label{eq:dotkappan}
\dot{\kappa}_{j, \alpha, n} = \frac{1}{2 k^{-1/4}}
\left\{ \widehat \kappa_{\alpha}({\bm}1_\alpha + k^{-1/4}{\bm}e_j) - \widehat \kappa_{\alpha}({\bm}1_\alpha - k^{-1/4}{\bm}e_j ) \right\},$$ with ${\bm}e_j$ the canonical unit vector of ${{\mathbb{R}}}^\alpha$ pointing in direction $j \in \alpha$, and put $$\begin{gathered}
\label{eq:Hatvar-kappa}
\widehat{\sigma}^2_{\kappa, \alpha} = \big(1 -
\widehat\kappa_\alpha)\widehat\kappa_\alpha
\left\{ \widehat\mu(\Delta_\alpha)^{-1} - \textstyle\sum_{j\in\alpha}\dot{\kappa}_{j,\alpha, n} \right\}
+
\sum_{i,j\in\alpha}\dot{\kappa}_{i,\alpha, n}\dot{\kappa}_{j,\alpha,n} \widehat\rho_{\{i,j\}} \\
+
\widehat\kappa_{\alpha} \sum_{ j \in\alpha} \dot{\kappa}_{j,\alpha,n}
\left\{ 1 - \widehat\mu(\Delta_\alpha)^{-1} \widehat\rho_{\alpha \setminus j} \right\}.\end{gathered}$$
\[prop:estim-sigma-kappa\] Under the conditions of Proposition \[theo:asymptot-kappa\], we have $\widehat{\sigma}^2_{\kappa,\alpha} = \sigma^2_{\kappa,\alpha} + {\mathrm{o}}_{{\operatorname{\mathbb{P}}}}(1)$ as $n \to \infty$, so that $\sqrt{k}(\widehat \kappa_\alpha - \kappa_\alpha) / \widehat{\sigma}_{\kappa,\alpha} {\rightsquigarrow}\mathcal{N}(0,1)$, provided $\sigma^2_{\kappa,\alpha} > 0$.
The proof relies on the weak convergence of the empirical process $\sqrt{k} \{ \widehat{\kappa}_{\alpha}(\,\cdot\,) - \kappa_\alpha(\,\cdot\,) \}$ on $[0,T]^\alpha$ for any $T>0$. This property follows in turn from Proposition \[prop:rnx\] and the functional delta method.
We consider a tolerance level $\kappa_{\min} \in (0, 1)$ under which the tail dependence between components $j\in\alpha$ is deemed negligible compared to the one between components $j\in\beta\subsetneq \alpha$. In other words, we aim at testing $H_0: \kappa_\alpha{\geqslant}\kappa_{\min}$. Since $\kappa_\alpha = \rho_\alpha / \mu(\Delta_\alpha)$, the null hypothesis is that $\rho_\alpha$ is greater than some level depending on $\alpha$. Let $0<\delta<1$ be a (small) probability, and consider the test $$\label{eq:testKappa-ge}
\tau_{\alpha,n}
=
{{\mathbbm{1}}}\left\{
\widehat\kappa_\alpha < \kappa_{\min} + q_\delta k^{-1/2} \widehat{\sigma}_{\kappa,\alpha}
\right\}$$ where $q_\delta$ is the $\delta$-quantile of the standard normal distribution. By Proposition \[prop:estim-sigma-kappa\], if $\sigma_{\kappa,\alpha} > 0$, the test in has asymptotic level $\delta$ for $H_0$ against $H_1: \kappa_{\alpha}< \kappa_{\min}$.
If $\rho_\alpha = 0$, then, in Proposition \[prop:asymptotic-hatRho\], we have $\sqrt{k}( \widehat{\rho}_\alpha - \rho_\alpha ) = {\mathrm{o}}_{{\operatorname{\mathbb{P}}}}(1)$ as $n \to \infty$: indeed, on the one hand, we have $\sqrt{k}( \widehat{\rho}_\alpha - \rho_\alpha ) = \sqrt{k} \widehat{\rho}_\alpha {\geqslant}0$, and on the other hand, its limit distribution is centered Gaussian. Likewise, we have $\sqrt{k}( \widehat{\kappa}_\alpha - \kappa_\alpha ) = {\mathrm{o}}_{{\operatorname{\mathbb{P}}}}(1)$ as $n \to \infty$ in Proposition \[theo:asymptot-kappa\] if $\kappa_\alpha = 0$. As a consequence, under the simple hypothesis $H_0 : \rho_\alpha = 0$, the asymptotic level of a test based on the asymptotic distribution of $\sqrt{k}(\widehat{\rho}_\alpha - \rho_\alpha)$ or $\sqrt{k}(\widehat{\kappa}_\alpha - \kappa_\alpha)$ cannot be controlled. This is why the test in concerns the null hypothesis $H_0 : \kappa_\alpha {\geqslant}\kappa_{\min}$ for some $\kappa_{\min} > 0$ instead. Alternatively, we propose tests based on estimators of the coefficient of tail dependence $\eta_\alpha$ in . In Sections \[sec:mult-extens-peng\] and \[sec:hill\], we consider two such estimators, extending the ones of @peng1999estimation and @draisma2004bivariate, respectively, to the multivariate setting.
Coefficient of tail dependence: Peng’s estimator {#sec:mult-extens-peng}
================================================
For bivariate distributions, Peng’s [@peng1999estimation] estimator of the coefficient of tail dependence $\eta = \eta_{\{1,2\}}$ is based on the property that the curve $t\mapsto (\log t, \log{\operatorname{\mathbb{P}}}[V_1>t, V_2>t])$ has an affine asymptote with slope $- 1/\eta$. A similar idea motivates Pickands’ [@pickands1975statistical] estimator for the extreme value index. Estimating the ordinate of the curve at $t = n/k$ and $t = n/(2k)$ allows to estimate that slope. Under a second-order regular variation condition, @peng1999estimation shows that his estimator is asymptotically normal, both if $\eta = 1$ and if $ \eta < 1$. In the former case, the asymptotic variance depends on the tail dependence function and its partial derivatives, which are unknown but may be estimated consistently, thus leading to tests whose asymptotic levels can be controlled.
Let $\alpha \subset {\{1,\ldots,d\}}$ have at least two elements. Recall the empirical joint tail function $\widehat{r}_\alpha$ in . We define the multivariate extension of Peng’s [@peng1999estimation] estimator of $\eta_\alpha$ in as $$\label{eq:multi-peng}
\widehat\eta_\alpha^P
= \log(2) /
\log \{ \widehat r_{\alpha}({\bm}2_\alpha) / \widehat r_{\alpha}({\bm}1_\alpha) \}.$$ The asymptotic normality of $\widehat\eta_\alpha^P$ follows from Proposition \[prop:rnx\] and the delta method.
\[thm:multi-peng\] In the setting of Proposition \[prop:rnx\], we have, as $n \to \infty$ and jointly in $\alpha \subset {\{1,\ldots,d\}}$ such that $\lvert \alpha \rvert {\geqslant}2$ and $\rho_\alpha > 0$, the weak convergence $$\sqrt{k} (\widehat\eta_{\alpha}^P - 1) {\rightsquigarrow}\frac{-1}{2\rho_\alpha \log 2} \left\{ Z_\alpha({\bm}2_\alpha) - 2 Z_\alpha({\bm}1_\alpha) \right\}.$$ The right-hand side is a $\mathcal{N}(0, \sigma_{\alpha,P}^2)$ random variable with variance $$\begin{gathered}
\label{eq:var-hateta}
\sigma_{\alpha,P}^2 = \frac{1}{2(\rho_\alpha\log 2)^2}
\biggl[
\rho_\alpha - 4 \rho_\alpha^2 + 2 \sum_{j\in\alpha} \dot{\rho}_{j,\alpha}
r_\alpha({\bm}2_\alpha \wedge {\boldsymbol{\iota}}_{j}) \\
+ \sum_{j\in\alpha} \sum_{j'\in\alpha}
\dot{\rho}_{j,\alpha}\dot{\rho}_{j',\alpha} \left\{ 3 \rho_{\{j, j'\}} - 2
r_{\{j, j' \}}(2,1) \right\}
\biggr],
\end{gathered}$$ where $\rho_{\{j, j'\}} = r_{\{j, j'\}}(2, 1) = 1$ if $j = j'$ and where ${\boldsymbol{\iota}}_j \in \{1,\infty\}^\alpha$ is the vector which all coordinates equal to $1$ except for the $j$-th one which equals $\infty$, so that $({\bm}2_\alpha \wedge {\boldsymbol{\iota}}_j)_m = 1$ if $m \in \alpha\setminus j$ and $({\bm}2_\alpha \wedge {\boldsymbol{\iota}}_j)_m = 2$ if $m = j$.
By extending the proof of [@peng1999estimation Theorem 2.1], it is also possible to obtain asymptotic normality of $\widehat{\eta}_\alpha^P$ in the case $\rho_\alpha=0$ and $\eta_\alpha < 1$ in . This would require a multivariate extension of the second-order regular variation condition in [@peng1999estimation] in the style of Condition \[as:draisma04\] below. For the application as a stopping criterion in the CLEF algorithm, we are only interested in the asymptotic distribution of $\widehat\eta_\alpha^P$ under the hypothesis $\rho_\alpha>0$, so we do not pursue this idea any further.
As in Proposition \[theo:asymptot-kappa\], the asymptotic variance $\sigma_{\alpha,P}^2$ in involves unknown quantities, all of which we can estimate consistently. For $\alpha\subset\{1,\ldots,d\}$ and $j\in\alpha$, define $$\label{eq:dotrho_j,alpha,n}
\dot{\rho}_{j,\alpha,n}
=
\frac{1}{2 k^{-1/4}}
\left\{
\widehat r_\alpha({\bm}1_\alpha + k^{-1/4}{\bm}e_j)
-
\widehat r_\alpha({\bm}1_\alpha - k^{-1/4}{\bm}e_j)
\right\},$$ where ${\bm}{e}_j$ is the canonical unit vector in ${{\mathbb{R}}}^\alpha$ pointing in dimension $j$. Define $$\begin{gathered}
\label{eq:Hatvar-hateta}
\widehat \sigma^2_{\alpha,P}
= \frac{1}{2(\widehat \rho_\alpha\log 2)^2}
\biggl[
\widehat \rho_\alpha
+
\sum_{j\in\alpha}
\dot{\rho}_{j,\alpha, n}
\{
- 4 \widehat \rho_\alpha
+ 2 \widehat r_\alpha({\bm}2_\alpha \wedge {\boldsymbol{\iota}}_{j})
\}
\\
+
\sum_{j\in\alpha} \sum_{j'\in\alpha}
\dot{\rho}_{j,\alpha,n} \dot{\rho}_{j',\alpha,n}
\left\{ 3 \widehat{\rho}_{\{j, j'\}} - 2 \widehat{r}_{\{j, j'\}}(2,1) \right\}
\biggr].
$$
\[prop:estimVarTerms\] In the setting of Proposition \[prop:rnx\], we have $\widehat{\sigma}_{\alpha, P}^2 = \sigma_{\alpha, P}^2 + {\mathrm{o}}_{{\operatorname{\mathbb{P}}}}(1)$ as $n \to \infty$, where $\alpha \subset {\{1,\ldots,d\}}$ is such that $\lvert \alpha \rvert {\geqslant}2$ and $\rho_\alpha > 0$. If $\sigma_{\alpha, P}^2 > 0$, then $\sqrt{k} (\widehat \eta_\alpha^P - 1) / \widehat{\sigma}_{\alpha, P} {\rightsquigarrow}\mathcal{N}(0,1)$ as $n \to \infty$.
The proof parallels the one of Proposition \[prop:estim-sigma-kappa\] and is omitted for brevity. The main step is to verify that $\dot{\rho}_{j,\alpha, n} = \dot{\rho}_{j,\alpha} + {\mathrm{o}}_{{\operatorname{\mathbb{P}}}}(1)$ as $n \to \infty$, which follows from Proposition \[prop:rnx\].
To test the hypothesis $H_0 : \rho_\alpha > 0$ at significance level $\delta \in (0, 1)$, we propose $$\label{eq:test-eta}
\tau_{\alpha,\eta^P,n}
=
{{\mathbbm{1}}}\left\{
\widehat\eta_\alpha^P
< 1 - q_{1-\delta} k^{-1/2} \widehat{\sigma}_{\alpha,P}
\right\},$$ where $q_{1-\delta}$ is the $(1-\delta)$-quantile of the standard normal distribution. In the setting of Proposition \[prop:estimVarTerms\], the test in has asymptotic level $\delta$ for $H_0$ against $H_1: \eta_{\alpha} < 1$.
Coefficient of tail dependence: Hill estimator {#sec:hill}
==============================================
The coefficient of tail dependence $\eta_\alpha$ in is the tail index of the random variable $T_\alpha = \min_{j \in \alpha} V_j$: the function $t \mapsto {\operatorname{\mathbb{P}}}[T_\alpha > t]$ is regularly varying at infinity with index $-1/\eta_\alpha$. A tractable alternative to Peng’s estimator for $\eta_\alpha$ is a Hill-type estimator as in @draisma2001tail [@draisma2004bivariate]. Replacing the unobservable Pareto variables $V_{i,j}$ by the rank-based versions $\widehat V_{i,j} = n / (n + 1 - R_{ij})$ in Section \[sec:etdf\] yields an approximate sample $$\widehat T_{i, \alpha} = \min_{j\in\alpha} \widehat V_{i,j},
\qquad i = 1, \ldots, n,$$ from the distribution of $T_\alpha$. Let $\widehat T_{(1), \alpha} {\leqslant}\ldots {\leqslant}\widehat T_{(n), \alpha}$ denote the order statistics of $\widehat T_{1,\alpha}, \ldots, \widehat{T}_{n,\alpha}$. The Hill estimator for $\eta_\alpha$ is defined as $$\label{eq:Hill}
\widehat \eta_\alpha^H
= \frac{1}{k} \sum_{i=1}^k
\log \frac {\widehat T_{(n-i+1), \alpha}} {\widehat T_{(n-k), \alpha}}.$$ Under the second-order regular variation conditions stated below, the asymptotic normality of $\widehat \eta_\alpha^H $ follows from [@draisma2004bivariate proof of Theorem 2.1]. The results in the cited reference cover the bivariate case only. In this section, we verify that they remain valid in any dimension $d{\geqslant}2$, and we provide the general expression for the asymptotic variance. Put $E_\alpha = [0, \infty]^\alpha \setminus \{ \bm{\infty}_\alpha \}$.
\[as:draisma04\] For each $\alpha\subset\{1,\ldots,d\}$ with $|\alpha|{\geqslant}2$, there exist functions $c_\alpha, c_{1,\alpha}: E_\alpha\to [0, \infty)$ such that $c_{1,\alpha}$ is neither constant nor a multiple of $c_\alpha$, and there exists $q_{1,\alpha}: (0, \infty) \to (0, \infty)$, with $q_{1,\alpha}(t) \to 0$ as $t\to 0$, such that, for all ${\bm}x \in E_\alpha$, we have $$ \lim_{t \to 0}
\left\{
\frac {{\operatorname{\mathbb{P}}}[ \forall j \in \alpha : 1 - F_j(X_j) {\leqslant}tx_j]} {{\operatorname{\mathbb{P}}}[ \forall j \in \alpha : 1 - F_j(X_j) {\leqslant}t ]}
- c_\alpha({\bm}x)
\right\}
\Big/
q_{1, \alpha}(t)
= c_{1,\alpha}({\bm}x).$$
Under Condition \[as:draisma04\], the function $q_\alpha(t) = {\operatorname{\mathbb{P}}}[ \forall j \in \alpha : 1 - F_j(X_j) {\leqslant}t ]$ is regularly varying at $0$ with some index $1/\eta_\alpha$. Condition \[as:draisma04\] implies that the first-order condition holds with the same index $1/\eta_\alpha$. In addition, $c_\alpha({\bm}1_\alpha) = 1$ and $c_\alpha$ is homogeneous of order $1/\eta_\alpha$, i.e., $c_\alpha(t {\bm}x ) =
t^{1/\eta_\alpha}c_\alpha({\bm}x)$ for $t>0$, see [@draisma2001tail; @draisma2004bivariate]. Under the regular variation assumption , we have $\rho_\alpha = \lim_{t \to 0} q_\alpha(t)/t$, so that, under Condition \[as:draisma04\], $\rho_\alpha > 0$ implies $\eta_\alpha = 1$, as in [@draisma2004bivariate] for the bivariate case. Finally, if $\rho_\alpha>0$, then $c_\alpha({\bm}x )= r_\alpha({\bm}x) /r_\alpha({\bm}1_\alpha) = r_\alpha({\bm}x) / \rho_\alpha$. Note that in [@draisma2004bivariate], our $\rho_\alpha$ is denoted by $l$ for $\alpha = \{1,2\}$.
The asymptotic variance of the Hill estimator involves a Gaussian process whose distribution depends on whether $\rho_\alpha=0$ or $\rho_\alpha > 0$. As in [@draisma2004bivariate], introduce a centered Gaussian process $W_1$ on $E_\alpha$ with covariance function ${\operatorname{\mathbb{E}}}[ W_1({\bm}x) \, W_1({\bm}y) ] = c_\alpha( {\bm}x\wedge {\bm}y)$ for ${\bm}x,{\bm}y \in E_\alpha$. Recall the stochastic process $Z_\alpha$ in and the random variable $G_\alpha = Z_\alpha({\bm}{1}_\alpha)$ in .
\[prop:normality-hill\] Let ${{\bm}X}_i = (X_{i,1}, \ldots, X_{i,d})$, for $i \in \{1,\ldots,n\}$, be an independent random sample from $F$, having continuous margins and satisfying . Let $k = k(n)\to \infty$ as $n\to\infty$, while $k(n) = {\mathrm{o}}(n)$. If Conditions \[as:bias\], \[as:small-k\], \[as:partialDeriv\], and \[as:draisma04\] hold, then, as $n \to \infty$, $$\sqrt{k} \left( \widehat \eta_\alpha^H - \eta_\alpha \right)
{\rightsquigarrow}\mathcal N(0,\sigma_{\alpha,H}^2),$$ with $\sigma_{\alpha,H}^2 = \eta_\alpha^2 {\operatorname{\mathbb{V}\mathrm{ar}}}\{\tilde W({\bm}1_\alpha) \}$, where $\tilde W({\bm}x) = W_1({\bm}x)$ if $\rho_\alpha = 0$ and $\tilde W({\bm}{x}) = \rho_\alpha^{-1/2} Z_\alpha({\bm}x)$ if $\rho_\alpha > 0$. In particular, if $\rho_\alpha > 0$, we have $$\label{eq:sigma_alpha,H}
\sigma_{\alpha,H}^2
=
\rho_\alpha^{-1} {\operatorname{\mathbb{V}\mathrm{ar}}}(G_\alpha)
= 1 - 2 \rho_\alpha +
\rho_\alpha^{-1}
\sum_{j \in \alpha} \sum_{j' \in \alpha}
\dot{\rho}_{j,\alpha}\dot{\rho}_{j',\alpha}
\rho_{\{j,j'\}}.$$
The proof of Proposition \[prop:normality-hill\] is based on the arguments developed in the proofs of [@draisma2004bivariate Theorem 2.1], [@drees1998smooth Theorem 3.2], and [@drees1998evindex Example 3.1], which we gather in Appendix \[sec:appendix\]. Again, the unknown terms in may be replaced by their empirical counterparts, leading to an asymptotically consistent test. Recall $\dot{\rho}_{j,\alpha,n}$ in and define $$\widehat\sigma^2_{\alpha, H}
= 1 - 2 \widehat\rho_\alpha +
\widehat\rho_\alpha^{-1}
\sum_{j\in\alpha} \sum_{j'\in\alpha}
\dot{\rho}_{j,\alpha,n}\dot{\rho}_{j',\alpha, n}
\widehat\rho_{\{j,j'\}}.$$
The proof of the consistency of the variance estimator follows the same lines as the proofs of Propositions \[prop:estim-sigma-kappa\] and \[prop:estimVarTerms\] and is omitted.
\[cor:testHill-estimVar\] Under the conditions of Proposition \[prop:normality-hill\], if $\rho_\alpha > 0$, we have $\widehat\sigma_{\alpha,H}^2 = \sigma_{\alpha,H}^2 + {\mathrm{o}}_{{\operatorname{\mathbb{P}}}}(1)$ as $n \to \infty$ and thus $\sqrt{k} (\widehat \eta_\alpha^H - 1) / \widehat{\sigma}_{\alpha, P} {\rightsquigarrow}\mathcal{N}(0,1)$, provided $\sigma_{\alpha,H}^2 > 0$.
We may exploit Corollary \[cor:testHill-estimVar\] to test $H_0 : \rho_\alpha > 0$ in the same way as we did by using Peng’s estimator in : at significance level $\delta \in (0, 1)$, the null hypothesis is rejected in favour of $H_1: \eta_\alpha < 1$ when $\widehat\eta_\alpha^H < 1 - q_{1-\delta} k^{-1/2} \widehat{\sigma}_{\alpha,H}$.
The condition $\sigma_{\alpha,H}^2 > 0$ in Corollary \[cor:testHill-estimVar\] is satisfied whenever $0<\rho_\alpha<1$. Indeed, in , we have $\rho_{\{j,j'\}} {\geqslant}\rho_\alpha$ and $\dot{\rho}_{j,\alpha} \dot{\rho}_{j',\alpha} {\geqslant}0$, whence $\sigma_{\alpha,H}^2$ ${\geqslant}1 - 2\rho_\alpha + \sum_{(j,j')\in\alpha^2} \dot{\rho}_{j,\alpha} \dot{\rho}_{j',\alpha}$ $= 1 - 2 \rho_\alpha + \rho_\alpha^2 = (1 - \rho_\alpha)^2$.
Simulation study {#sec:simu-study}
================
Our aim is to compare the finite sample performance of the various tests proposed in Sections \[sec:test-kappa\], \[sec:mult-extens-peng\] and \[sec:hill\] within the framework of the CLEF algorithm, the pseudo-code of which is given in Appendix \[sec:appendix-CLEF\]. Three variants of the CLEF algorithm are obtained by varying the criterion according to which a subset $\alpha$ is declared as tail-dependent: $\widehat \kappa_\alpha > \kappa_{\min} - q_{\delta} \widehat \sigma_{\kappa,\alpha} / \sqrt{k}$ for CLEF-asymptotic; $\widehat\eta_{\alpha,P} > 1 - q_{\delta} \widehat \sigma_{\alpha,P} / \sqrt{k}$ for CLEF-Peng; and $\widehat\eta_{\alpha,H} > 1 - q_{\delta} \widehat \sigma_{\alpha,H} / \sqrt{k}$ for CLEF-Hill. The original CLEF criterion was $\widehat{\kappa}_\alpha > C$ for some constant $C$ chosen by the user. For completeness, the output of the DAMEX algorithm [@goixsparse] is included in the comparison. In practice, the dependence tests based on the tail dependence coefficient should not be carried out to the letter when the test statistic is not defined or when its estimated variance is infinite. Thus, in our experiments, CLEF-Peng and CLEF-Hill are modified so as to take into account additional, common-sense stopping criteria. A subset $\alpha$ will *not* be part of the list returned by the algorithms under the following conditions:
1. Concerning CLEF-Hill, when $\widehat \rho_\alpha = 0$, that is, no extreme record impacts all coordinates in $\alpha$, the estimated variance of the Hill estimator of $\eta_\alpha$ is infinite. Therefore, $\widehat \rho_\alpha = 0$ is considered as a stopping criterion in CLEF-Hill.
2. Concerning CLEF-Peng, when $\widehat r_{\alpha}({\bm}2_\alpha) = \widehat r_{\alpha}({\bm}1_\alpha)$, the Peng estimator is ill-defined. Such a case arises when there are very few points in the joint tail within the subspace generated by $\alpha$. When the estimated derivatives $\dot{\rho}_{j,\alpha,n}$ are close to zero, and when $ \widehat\rho_\alpha\ll 1$, the estimated variance $\widehat \sigma_{\alpha,P}^2$ in becomes large, preventing rejection of the null hypothesis. To prevent these issues, each of the conditions $\widehat\rho_\alpha< 0.05$ and $\widehat r_{\alpha}({\bm}2_\alpha) = \widehat r_{\alpha}({\bm}1_\alpha)$ are declared as a stopping criterion in CLEF-Peng.
#### Experimental setting.
CLEF [@chiapinofeature] is designed to face situations where DAMEX [@goixsparse] fails to exhibit a clear-cut dependence structure. A major issue reported in [@chiapinofeature] for certain hydrological data is the high variability of the groups of features for which large values occur simultaneously. Because of this, the empirical exponent measure $\widehat\mu$ assigns low mass to any sub-region partitioning the sample space, see Remark \[rm:relationship-goix\]. The empirical finding motivating the latter work is that the various subsets $\alpha$ involved in simultaneous extreme records could nevertheless be clustered, meaning that many of them have a significant intersection, whereas many symmetric differences comprise just a single or at most a few features.
A natural assumption in this context is that a ‘true’ list of dependent subsets $\mathcal{M} = \{\alpha_1,\ldots, \alpha_K\}$ exists such that $\mu(\mathcal{C}_\alpha)>0$ for $\alpha\in\mathcal{M}$ and that noisy features are involved in each extreme event. Observed large records then concern groups of the kind $\alpha' = \alpha\cup\{j\}$, where $\alpha\in\mathcal{M}$ and $j\in{\{1,\ldots,d\}}\setminus\alpha$.
In our experiments, datasets are generated as follows: The dimension is fixed to $d=100$. A family of ‘true’ dependent subsets $\mathcal{M}= \{\alpha_1,\ldots,\alpha_K\}$ of cardinality $K = 80$ is randomly chosen: the subset sizes $|\alpha|$ follow a truncated geometric distribution, with a maximum subset size set to $8$. For simplicity, we forbid nested subsets, so $\alpha_j\not\subset\alpha_k$ whenever $j \ne k$. The maximal elements of $\mathbb{M} = \{\alpha\subset{\{1,\ldots,d\}}\mid \rho_\alpha>0\}$ are then precisely the elements of $\mathcal{M}$, as explained in Remark \[rm:relationship-goix\]. Finally, two different subsets may have at most two features in common.
Once the dependence structure $\mathcal{M}$ has been fixed, the data ${\bm}{X}_1, \ldots, {\bm}{X}_n$ are sampled independently from $d$-dimensional asymmetric logistic distributions [@tawn1990modelling], using Algorithm $2.2$ in [@stephenson2003simulating]. The underlying ‘true’ distribution function is $$\label{eq:asym-logistic}
G({\bm}x) = \exp\biggl[- \sum_{m=1}^K\Bigl\{ \sum_{j\in\alpha_{m}} (\lvert{\cal A}(j)\rvert x_j)^{-1/w_{\alpha_m}} \Bigr\}^{w_{\alpha_m}}\biggr],$$ where ${\cal A}(j) = \{\alpha \in \mathcal{M} \mid j \in \alpha\}$ and $w_{\alpha_m}$ is a dependence parameter which is set to $0.1$ in our simulations. Actually, to mimic the noisy situation described above, each point ${\bm}X_i$ is simulated according to a slightly different version, $G_i$, of $G$. For each $i = 1, \ldots, n$ and $k = 1, \ldots, K$, we randomly select an additional ‘noisy feature’ $j_{i,k} \in \{1, \ldots, d\} \setminus \alpha_k$ and set $\alpha_{i,k}' = \alpha_k \cup \{ j_{i,k} \}$. Then $\mathcal{M}_i' = \{ \alpha_{i,1}', \ldots, \alpha_{i,K}' \}$ is the collection of ‘noisy subsets’ for ${\bm}{X}_i$ and $G_i({\bm}{x})$ is as in with $\mathcal{A}(j)$ replaced by $\mathcal{A}'_i(j) = \{\alpha' \in \mathcal{M}'_i \mid j \in \alpha'\}$.
#### Results.
We generate datasets of size $n=5\mathrm{e}4$ and $n=1\mathrm{e}5$. For each sample size, $50$ independent datasets are simulated according to the procedure summarized in the preceding paragraph. We compare the average performance of the three proposed versions of CLEF, together with the original CLEF and DAMEX algorithms, for different choices of $k$ and confidence level $\delta$.
$n=5\mathrm{e}4$ $k/n$ recovered subset errors superset errors other errors
-------------------- ------- --------------------- -------------------- ----------------- -------------------
CLEF-asymptotic 0.003 71.1 (3.0) 7.4 (4.7) 5.1 (2.1) 28.0 (13.3)
0.005 73.0 (3.7) 8.0 (6.3) 2.4 (1.7) 14.6 (8.9)
\[.7ex\] CLEF-Peng 0.003 [**79.70 (0.7)**]{} [**1.00 (2.5)**]{} [**0. (0.)**]{} 3.9 (2.7)
0.005 [**79.98 (0.1)**]{} [**0.06 (0.4)**]{} [**0. (0.)**]{} 0.9 (0.9)
\[.7ex\] CLEF-Hill 0.003 79.0 (1.4) 2.4 (3.5) 0.04 (0.2) 17.9 (7.0)
0.005 75.7 (2.4) 9.2 (6.8) [**0. (0.)**]{} [**0. (0.)**]{}
\[.7ex\] CLEF 0.003 69.9 (4.4) 16.2 (8.1) 0.5 (0.6) [**2.3 (2.2)**]{}
0.005 75.0 (3.6) 8.1 (6.4) 0.2 (0.5) 0.9 (1.2)
\[.7ex\] DAMEX 0.003 0.6 (0.2) 1.7 (1.4) 32.9 (5.6) 45.4 (5.9)
0.005 0.1 (0.4) 2.4 (1.5) 18.3 (5.5) 59.1 (5.9)
$n=1\mathrm{e}5$
CLEF-asymptotic 0.003 73.2 (3.7) 9.5 (6.7) 0.9 (0.8) 4.7 (2.7)
0.005 72.6 (4.4) 11.7 (7.6) 0.1 (0.4) 0.5 (0.9)
\[.7ex\] CLEF-Peng 0.003 [**79.9 (0.2)**]{} [**0.2 (1.0)**]{} [**0. (0.)**]{} 0.1 (0.4)
0.005 [**80.0 (0.)**]{} [**0. (0.)**]{} [**0. (0.)**]{} [**0. (0.)**]{}
\[.7ex\] CLEF-Hill 0.003 77.0 (2.0) 6.1 (4.6) [**0. (0.)**]{} [**0. (0.)**]{}
0.005 67.2 (4.8) 22.8 (10.4) [**0. (0.)**]{} [**0. (0.)**]{}
\[.7ex\] CLEF 0.003 75.2 (3.2) 7.5 (5.9) 0.0 (0.2) 0.2 (0.5)
0.005 77.9 (2.3) 3.2 (3.9) 0.02 (0.1) 0.02 (0.1)
\[.7ex\] DAMEX 0.003 0.04 (0.2) 1.3 (1.0) 24.4 (6.7) 54.2 (7.0)
0.005 0.1 (0.3) 1.9 (1.6) 10.3 (3.7) 67.6 (4.7)
: Average number of recovered clusters and errors of CLEF-asymptotic ($\kappa_{\min}=0.08$), CLEF-Peng, CLEF-Hill, CLEF and DAMEX on $50$ datasets. Confidence level for the tests: $\delta=0.001$. Standard deviations over the 50 samples in brackets. Bold face indicates the best performing algorithm on average for a given $n$ and a given choice of $k/n$, the proportion of extreme data used.[]{data-label="hpc-noise-table-deltaBig"}
$n=5\mathrm{e}4$ $k/n$ recovered subset errors superset errors other errors
-------------------- ------- -------------------- ------------------- ----------------- -------------------
CLEF-asymptotic 0.003 71.8 (2.4) 2.3 (2.5) 7.8 (2.8) 41.9 (19.3)
0.005 73.5 (2.8) 3.7 (3.8) 4.8 (2.5) 25.8 (12.2)
\[.7ex\] CLEF-Peng 0.003 [**79.7 (0.7)**]{} [**1.0 (2.5)**]{} [**0. (0.)**]{} 3.9 (2.7)
0.005 [**80.0 (0.1)**]{} [**0.1 (0.4)**]{} [**0. (0.)**]{} 0.9 (0.9)
\[.7ex\] CLEF-Hill 0.003 79.5 (0.8) [**0.3 (1.1)**]{} 0.5 (0.8) 142.2 (33.2)
0.005 79.2 (1.0) 1.6 (2.3) [**0. (0.)**]{} [**0.2 (0.5)**]{}
\[.7ex\] CLEF 0.003 69.9 (4.4) 16.2 (8.1) 0.5 (0.6) 2.3 (2.2)
0.005 75.0 (3.6) 8.1 (6.4) 0.2 (0.5) 0.9 (1.2)
\[.7ex\] DAMEX 0.003 0.6 (0.2) 1.7 (1.4) 32.9 (5.6) 45.4 (5.9)
0.005 0.1 (0.4) 2.4 (1.5) 18.3 (5.5) 59.1 (5.9)
$n=1\mathrm{e}5$
CLEF-asymptotic 0.003 75.7 (2.8) 3.7 (3.8) 2.0 (1.4) 11.0 (5.5)
0.005 76.0 (2.9) 5.6 (4.5) 0.4 (0.7) 1.9 (1.9)
\[.7ex\] CLEF-Peng 0.003 [**79.9 (0.2)**]{} [**0.2 (1.0)**]{} [**0. (0.)**]{} 0.1 (0.4)
0.005 [**80. (0.)**]{} [**0. (0.)**]{} [**0. (0.)**]{} [**0. (0.)**]{}
\[.7ex\] CLEF-Hill 0.003 79.5 (1.0) 1.2 (2.3) [**0. (0.)**]{} [**0.1 (0.2)**]{}
0.005 75.4 (2.8) 8.7 (5.2) [**0. (0.)**]{} [**0. (0.)**]{}
\[.7ex\] CLEF 0.003 75.2 (3.2) 7.5 (5.9) 0.0 (0.2) 0.2 (0.5)
0.005 77.9 (2.3) 3.2 (3.9) 0.02 (0.1) 0.02 (0.1)
\[.7ex\] DAMEX 0.003 0.04 (0.2) 1.3 (1.0) 24.4 (6.7) 54.2 (7.0)
0.005 0.1 (0.3) 1.9 (1.6) 10.3 (3.7) 67.6 (4.7)
: Same setting as Table \[hpc-noise-table-deltaBig\] with $\delta=0.0001$[]{data-label="hpc-noise-table-deltaSmall"}
Tables \[hpc-noise-table-deltaBig\] and \[hpc-noise-table-deltaSmall\] gather the results for a confidence level $\delta$ equal to $0.001$ and $0.0001$, respectively. In both tables, the results obtained with the original version of CLEF and DAMEX are included in the comparison with an identical choice of tuning parameters, so that the last two lines of the two tables are the same. In CLEF, the threshold $C$ was chosen by trial and error in the interval $(0, \kappa_{\min})$, namely $C=0.05$. Imposing that $C < \kappa_{\min}$ is intended to reproduce the effect of the variance term upon the stopping criterion in CLEF-asymptotic. In DAMEX, the $80$ subsets with highest empirical mass are retained and the subspace thickening parameter $\epsilon$ is set to the default value of $0.1$, following the guidelines of the authors. Each algorithm produces a list, $\widehat{\mathbb{M}}$, of groups of features $\alpha \in \{1, \ldots, d\}$. This list is to be compared with the one of $K = 80$ ‘true’ subsets $\mathcal{M}$. The performance of each algorithm is measured in terms of two criteria: the number of ‘true’ subsets $\alpha\in\mathcal{M}$ that appear in $\widehat{\mathbb{M} }$ (third column of Tables \[hpc-noise-table-deltaBig\] and \[hpc-noise-table-deltaSmall\]); the number of ‘errors’, that is, the subsets $\alpha\in \widehat{\mathbb{M}}$ that do not belong to $\mathcal{M}$. These can be understood as ‘false positives’. Among these errors, we make the distinction between those which are respectively proper subsets (fourth column of Tables \[hpc-noise-table-deltaBig\] and \[hpc-noise-table-deltaSmall\]) or proper supersets (fifth column) of some true $\beta\in\mathcal{M}$, and the other errors (sixth column).
CLEF-Peng obtains the best overall scores for both values of $\delta$, but as explained above, a special treatment is reserved for the case $\widehat\rho_\alpha {\leqslant}0.05$, and this threshold constitutes an arbitrary tuning parameter, which can impact the performance significantly. On the other hand, CLEF-Hill does not require any other adjustment than for the special case $\widehat\rho_\alpha=0$ and performs nearly as well as CLEF-Peng with $\delta=0.0001$ and $k/n=0.005$. In addition, CLEF-Hill outperforms all the other methods. In particular, CLEF-asymptotic is globally less accurate than CLEF-Peng and CLEF-Hill. This reflects the fact that the null hypothesis in this algorithm involves an arbitrary $\kappa_{\min}>0$ fixed by the user. Our own choice $\kappa_{\min} = 0.08$ was fixed by trial and error, which is straightforward with synthetic data and could also be achieved by cross-validation in a real use case. Finally, as expected, DAMEX obtains very low scores, because it is not designed to handle the addition of noisy features, as explained earlier.
Conclusion {#sec:conclusion}
==========
In this work, we propose three variants of the CLEF algorithm [@chiapinofeature], replacing the heuristic criterion in the original version with a formal test for asymptotic dependence, and this for all possible subsets of features among $\{1,\ldots,d\}$. As in the original CLEF implementation, only a small proportion of all $2^d-1$ subsets has to be examined, while the computational complexity for each such subset is low. Experimental results indicate that the CLEF algorithm is most effective when based on a test constructed from an extension of the Hill estimator [@draisma2004bivariate] of the multivariate coefficient of tail dependence.
The procedure we propose is nonparametric and rank-based. Parametric approaches, based for instance on the nested asymmetric logistic distribution [@tawn1990modelling], could have a greater sensitivity, at the cost of increased model risk and greater computational complexity. We have also assumed that the observations are serially independent; in the contrary case, the asymptotic variances of the various estimator need to be estimated by some form of bootstrap, which, in high dimensions, poses important theoretical and computational challenges; see [@bucher+d:2013] for the bivariate and serially independent case.
Proofs {#sec:appendix}
======
For ${\varnothing}\ne \alpha \subset {\{1,\ldots,d\}}$ and $\bm{x} \in [0, \infty)^\alpha$, put $$\begin{aligned}
L_\alpha(\bm{x})
&=
\{ \bm{y} \in [0, \infty]^d \mid \exists j \in \alpha : y_j < x_j \}, \\
R_\alpha(\bm{x})
&=
\{ \bm{y} \in [0, \infty]^d \mid \forall j \in \alpha : y_j < x_j \}.\end{aligned}$$ If $\alpha = {\{1,\ldots,d\}}$, then just write $L$ rather than $L_{{\{1,\ldots,d\}}}$. Note that $L_\alpha(\bm{x}) = L(\bm{x} \bm{e}_\alpha)$ with $\bm{e}_\alpha = ({{\mathbbm{1}}}_\alpha(j))_{j=1}^d$ and that $L_{\{j\}}(x_j) = R_{\{j\}}(x_j)$ and thus $W(L_{\{j\}}(x_j)) = W_{\{j\}}(x_j)$. @einmahl2012m [Theorem 4.6] show that, in the space $\ell^\infty([0, T]^d)$ and under Conditions \[as:bias\], \[as:small-k\] and \[as:partialDeriv\], we have weak convergence $$\sqrt{k}\{ \widehat{\ell}(\bm{x}) - \ell(\bm{x}) \}
{\rightsquigarrow}W(L(\bm{x})) - \sum_{j=1}^d \partial \ell_j(\bm{x}) W_{\{j\}}(x_j)$$ as $n \to \infty$. Here, we have taken a version of the Gaussian process $W$ such that the trajectories ${\bm}{x} \mapsto W(L(\bm{x}))$ are continuous almost surely.
As in , we have, for ${\varnothing}\ne \alpha \subset {\{1,\ldots,d\}}$ and $\bm{x} \in [0, \infty)^\alpha$, the identity $$\widehat{r}_\alpha(\bm{x})
= \sum_{{\varnothing}\ne \beta \subset \alpha} (-1)^{|\beta|+1} \widehat\ell( \bm{x}_\beta \bm{e}_\beta )$$ where $\bm{x}_\beta = (x_j)_{j \in \beta}$. Hence, we can view the vector $(\sqrt{k}( \widehat{r}_\alpha - r_\alpha ))_{{\varnothing}\ne \alpha \subset {\{1,\ldots,d\}}}$ as the result of the application to $\sqrt{k}(\widehat{\ell}-\ell)$ of a bounded linear map from the space $\ell^\infty([0, T]^d)$ to the product space $\prod_{{\varnothing}\ne \alpha \in {\{1,\ldots,d\}}} \ell^\infty([0, T]^\alpha)$. By the continuous mapping theorem, we obtain, in the latter space, the weak convergence $$\sqrt k\left\{\widehat r_\alpha({\bm}x) - r_\alpha({\bm}x)\right\}
{\rightsquigarrow}\sum_{{\varnothing}\ne \beta \subset \alpha}
(-1)^{|\beta|+1}
\left\{
W(L_\beta(\bm{x}_\beta))
-
\textstyle\sum_{j=1}^d \partial_j \ell_\beta(\bm{x}_\beta) W_{\{j\}}(x_j {{\mathbbm{1}}}_\beta(j))
\right\}.$$ Here we used $\ell(\bm{x}_\beta \bm{e}_\beta) = \ell_\beta(\bm{x}_\beta)$.
The set-indexed process $W$ satisfies the remarkable property that $W(A \cup B) = W(A) + W(B)$ almost surely whenever $A$ and $B$ are disjoint Borel sets of $[0, \infty]^d \setminus \{ \bm{\infty} \}$ that are bounded away from $\bm{\infty}$: indeed, implies ${\operatorname{\mathbb{E}}}[\{W(A \cup B) - W(A) - W(B)\}^2] = 0$. It follows that the trajectories of $W$ obey the inclusion-exclusion formula, so that, for ${\varnothing}\ne \alpha \subset {\{1,\ldots,d\}}$ and $\bm{x} \in [0, \infty)^\alpha$, we have, almost surely, $$\begin{aligned}
\sum_{{\varnothing}\ne \beta \subset \alpha}
(-1)^{|\beta|+1}
W(L_\beta(\bm{x}_\beta))
&=
\sum_{{\varnothing}\ne \beta \subset \alpha}
(-1)^{|\beta|+1}
W\left( \textstyle\bigcup_{j\in\beta} R_{\{j\}}(x_j) \right) \\
&=
W\left( \textstyle\bigcap_{j\in\alpha} R_{\{j\}}(x_j) \right)
=
W(R_\alpha(\bm{x}))
=
W_\alpha(\bm{x}).\end{aligned}$$ We can make this hold true almost surely jointly for all such $\alpha$ and $\bm{x}$: first, consider points $\bm{x}$ with rational coordinates only and then consider a version of $W$ by extending $W_\alpha$ to points $\bm{x}$ with general coordinates via continuity. Similarly, since $W_{\{j\}}(0) = W({\varnothing}) = 0$ almost surely, we have $$\begin{aligned}
\sum_{{\varnothing}\ne \beta \subset \alpha} (-1)^{|\beta|+1}
\sum_{j=1}^d \partial_j \ell_\beta(\bm{x}_\beta) W_{\{j\}}(x_j {{\mathbbm{1}}}_\beta(j))
&=
\sum_{j \in \alpha} \sum_{\beta : j \in \beta \subset \alpha} \partial_j \ell_\beta(\bm{x}_\beta) W_{\{j\}}(x_j) \\
&=
\sum_{j \in \alpha} \partial r_j(\bm{x}) W_{\{j\}}(x_j).\end{aligned}$$ We have thus shown weak convergence as stated in .
The weak convergence statement is a special case of : set $\bm{x} = \bm{1}_\alpha$. The covariance formula follows from the fact that
$$\begin{aligned}
{\operatorname{\mathbb{E}}}[W_\alpha(\bm{1}_\alpha) W_{\alpha'}(\bm{1}_{\alpha'})]
&=
\Lambda( \{ \bm{y} \in [0, \infty]^d \mid \forall i \in \alpha \cup \alpha' : y_i < 1 \} ) \\
&=
\mu( \{ \bm{u} \in [0, \infty)^d \mid \forall i \in \alpha \cup \alpha' : u_i > 1 \} )
=
\rho_{\alpha\cup\alpha'};\end{aligned}$$
the first equality follows from and the last one from . We obtain by expanding $G_\alpha = Z_\alpha(\bm{1}_\alpha)$ using and working out ${\operatorname{\mathbb{E}}}[ G_\alpha G_{\alpha'} ]$ with the above identity.
Let $\alpha = \{\alpha_1, \ldots, \alpha_S \} \subset \{1, \ldots, d\}$ with $S = \lvert \alpha \rvert {\geqslant}2$ and such that $\mu(\Delta_\alpha) > 0$. In view of , we have $\kappa_\alpha = g_\alpha( \theta_\alpha )$ and $\widehat{\kappa}_\alpha = g_\alpha( \widehat{\theta}_\alpha )$ where $\theta_\alpha = ( \rho_\alpha, \rho_{\alpha\setminus\alpha_1}, \ldots, \rho_{\alpha\setminus\alpha_S} )$, $\widehat\theta_\alpha = (\widehat\rho_\alpha, \widehat\rho_{\alpha\setminus\alpha_1}, \ldots, \widehat\rho_{\alpha\setminus\alpha_S})$, and $$\label{eq:transfoKappa}
g_\alpha( x_0, x_1, \ldots, x_{S}) = \frac{x_0}{\sum_{j=1}^{S} x_j - (S-1) x_0}, \qquad x\in[0,\infty)^{1+S}.$$ Let $\nabla g_\alpha(x)$ denote the gradient vector of $g_\alpha$ evaluated $x$ and let $\langle \, \cdot \, , \, \cdot \, \rangle$ denote the scalar product in Euclidean space. Proposition \[prop:asymptotic-hatRho\] combined with the delta method as in [@van2000asymptotic Theorem 3.1] gives, as $n \to \infty$, $$\begin{aligned}
\sqrt{k} ( \widehat\kappa_\alpha - \kappa_\alpha )
=
\sqrt{k} \{ g_\alpha( \widehat{\theta}_\alpha ) - g_\alpha( \theta_\alpha ) \}
&=
\left\langle
\nabla g_\alpha(\theta_\alpha), \,
\sqrt{k} (\widehat{\theta}_\alpha - \theta_\alpha)
\right\rangle
+
{\mathrm{o}}_{{\operatorname{\mathbb{P}}}}(1)
\\
&{\rightsquigarrow}\left\langle
\nabla g_\alpha(\theta_\alpha), \,
(G_\alpha,G_{\alpha\setminus\alpha_1},\ldots,G_{\alpha\setminus\alpha_S})
\right\rangle,\end{aligned}$$ the weak convergence holding jointly in $\alpha$ by Slutsky’s lemma and Proposition \[prop:asymptotic-hatRho\]. The partial derivatives of $g_\alpha$ are $$\begin{aligned}
\frac{\partial g}{\partial x_0}( x )
&= \frac{\sum_{j=1}^S x_j }{\{ \sum_{j=1}^S x_j - (S-1) x_0 \}^2}, \\
\frac{\partial g}{\partial x_j}( x )
&= \frac{- x_0}{\{ \sum_{j=1}^S x_j - (S-1) x_0 \}^2},
\qquad j = 1, \ldots, S.\end{aligned}$$ Evaluating these at $x = \theta_\alpha$ and using $\sum_{j \in \alpha} \rho_{\alpha \setminus j} - (S-1) \rho_\alpha = \mu(\Delta_\alpha)$ as in and , we find that $$\left\langle
\nabla g_\alpha(\theta_\alpha), \,
(G_\alpha,G_{\alpha\setminus\alpha_1},\ldots,G_{\alpha\setminus\alpha_S})
\right\rangle
=
\mu(\Delta_\alpha)^{-2}
\left\{
\left( \textstyle\sum_{j\in\alpha}\rho_{\alpha\setminus j} \right) G_\alpha
- \rho_\alpha \textstyle\sum_{j\in\alpha} G_{\alpha\setminus j}
\right\},$$ in accordance to the right-hand side in .
To calculate the asymptotic variance $\sigma_{\kappa,\alpha}^2$, we introduce a few abbreviations: we write $R_\beta = R_\beta(\bm{1}_\beta)$ and $W_\beta^\cap = W_\beta(\bm{1}_\beta) = W(R_\beta)$ for ${\varnothing}\ne \beta \in {\{1,\ldots,d\}}$ and we put $W_j = W_{\{j\}}(1)$ for $j = 1, \ldots, d$, so that $G_\alpha = W_\alpha^\cap - \sum_{j\in\alpha}\dot{\rho}_{j,\alpha} W_j$. We find $$\begin{aligned}
H_\alpha
&=
\Big( \sum_{i\in\alpha}\rho_{\alpha\setminus i} \Big) G_\alpha
-
\rho_\alpha \sum_{i\in\alpha} G_{\alpha\setminus i} \\
&=
\Big(\sum_{i\in\alpha}\rho_{\alpha\setminus i}\Big)
\Big( W^\cap_{\alpha} - \sum_{j\in\alpha}\dot{\rho}_{j,\alpha}W_{j}\Big)
-
\rho_\alpha \sum_{i\in\alpha}
\Big( W^\cap_{\alpha\setminus i} - \sum_{j\in\alpha\setminus i} \dot{\rho}_{j,\alpha\setminus i} W_{j} \Big).\end{aligned}$$ From the proof of Proposition \[prop:rnx\], recall that $W(A \cup B) = W(A) + W(B)$ almost surely for disjoint Borel sets $A$ and $B$ of $[0, \infty]^d \setminus \{\bm\infty\}$ bounded away from $\bm{\infty}$; moreover, for such $A$ and $B$, the variables $W(A)$ and $W(B)$ are uncorrelated. Since $R_{\alpha \setminus i}$ is the disjoint union of $R_\alpha$ and $R_{\alpha \setminus i} \setminus R_\alpha$, we have therefore $W_{\alpha\setminus i}^\cap = W_\alpha^\cap + W(R_{\alpha\setminus i} \setminus R_\alpha)$ almost surely. In addition, $\sum_{i \in \alpha} \rho_{\alpha\setminus i} = \mu(\Delta_\alpha) + (S-1)\rho_\alpha$ by applied to $\nu = \mu$. As a consequence, $$H_\alpha
=
\{ \mu(\Delta_\alpha) - \rho_\alpha \}
W_{\alpha}^\cap
- \rho_\alpha \sum_{j \in\alpha} W(R_{\alpha\setminus j} \setminus R_\alpha)
+ \sum_{j\in\alpha} K_{\alpha, j} W_j$$ where $$K_{\alpha,j}
=
\rho_\alpha\Big(\sum_{i\in\alpha\setminus j}\dot{\rho}_{j,\alpha\setminus i}\Big)
-
\Big(\sum_{i\in\alpha}\rho_{\alpha\setminus i}\Big)\dot{\rho}_{j,\alpha},
\qquad j \in \alpha.$$ The $S+1$ variables $W_\alpha^\cap = W(R_\alpha)$ and $W(R_{\alpha\setminus j} \setminus R_\alpha)$, $j \in \alpha$, are all uncorrelated, since they involve evaluating $W$ at disjoint sets; $W_j = W(R_{\{j\}})$ is uncorrelated with $W(R_{\alpha \setminus j} \setminus R_\alpha)$, for the same reason. Moreover, ${\operatorname{\mathbb{E}}}[ W_\alpha^\cap W_j ] = \Lambda(R_\alpha \cap R_{\{j\}}) = \Lambda(R_\alpha) = \rho_\alpha$ and similarly ${\operatorname{\mathbb{E}}}[ W(R_{\alpha\setminus i} \setminus R_\alpha) W_j ] = \Lambda(R_{\alpha\setminus i} \setminus R_{\alpha}) = \rho_{\alpha \setminus i} - \rho_\alpha$ if $i,j\in\alpha$ and $i \ne j$. Hence
$$\begin{gathered}
{\operatorname{\mathbb{V}\mathrm{ar}}}(H_\alpha)
=
\{\mu(\Delta_\alpha) - \rho_\alpha\}^2 \rho_\alpha
+
\rho_\alpha^2 \sum_{j \in \alpha} (\rho_{\alpha \setminus j} - \rho_\alpha)
+
\sum_{i,j \in \alpha} K_{\alpha,i} K_{\alpha,j} \rho_{\{i,j\}} \\
+
\{\mu(\Delta_\alpha) - \rho_\alpha\} \rho_\alpha\sum_{j\in\alpha}K_{\alpha,j}
-
\rho_\alpha \sum_{j\in\alpha}K_{\alpha,j} \sum_{i\in\alpha\setminus j} (\rho_{\alpha\setminus i} - \rho_\alpha).\end{gathered}$$
As $\sum_{j \in \alpha} (\rho_{\alpha \setminus j} - \rho_\alpha) = \mu(\Delta_\alpha)- \rho_\alpha$ and $\sum_{i \in \alpha \setminus j}(\rho_{\alpha \setminus i} - \rho_\alpha) = \mu(\Delta_\alpha) - \rho_{\alpha,j}$, we get
$$\begin{gathered}
\label{eq:varmuZ}
{\operatorname{\mathbb{V}\mathrm{ar}}}(H_\alpha)
= \{\mu(\Delta_\alpha) - \rho_\alpha\}\rho_\alpha
\Big\{ \mu(\Delta_\alpha) + \sum_{j\in\alpha}K_{\alpha,j} \Big\}
+ \sum_{i,j\in\alpha}K_{\alpha,i}K_{\alpha,j} \rho_{\{i,j\}} \\
-\rho_{\alpha} \sum_{j \in\alpha}K_{\alpha,j}\{\mu(\Delta_\alpha) - \rho_{\alpha\setminus j} \}.
$$
Recall $\kappa_\alpha(\bm{x})$ in . We have $$\begin{aligned}
\frac{\partial}{\partial x_j}
\left(
\frac{1}{\kappa_\alpha(\bm{x})}
\right)_{\bm{x} = \bm{1}_\alpha}
&=
\frac{\partial}{\partial x_j}
\left(
\frac{\sum_{i \in \alpha} r_{\alpha \setminus i}(\bm{x}_{\alpha \setminus i})}{r_{\alpha}(\bm{x})}
\right)
\\
&=
\rho_\alpha^{-2}
\bigg(
\rho_\alpha \sum_{i \in \alpha \setminus j} \dot{\rho}_{j, \alpha \setminus i}
-
\dot{\rho}_{j, \alpha} \sum_{i \in \alpha} \rho_{\alpha \setminus i}
\bigg)
=
\rho_\alpha^{-2} K_{\alpha, j}.\end{aligned}$$ It follows that $\dot{\kappa}_{j,\alpha} = - \rho_{\alpha}^{-2} K_{\alpha, j} / (1/\kappa_{\alpha})^2 = - K_{\alpha,j} / \mu(\Delta_\alpha)^2$. By , we find that $\sigma_{\kappa,\alpha}^2 = \mu(\Delta_\alpha)^{-4} {\operatorname{\mathbb{V}\mathrm{ar}}}( H_{\alpha} )$ is equal to the right-hand side of .
We only need to prove that $\widehat{\sigma}^2_{\kappa,\alpha} = \sigma_{\kappa, \alpha}^2 + {\mathrm{o}}_{{\operatorname{\mathbb{P}}}}(1)$ as $n \to \infty$. In view of the expressions and for $\sigma_{\kappa, \alpha}^2$ and $\widehat{\sigma}_{\kappa,\alpha}$, it is enough to show that $\dot{\kappa}_{j,\alpha, n} = \dot{\kappa}_{\alpha,j} + {\mathrm{o}}_{{\operatorname{\mathbb{P}}}}(1)$, with $\dot{\kappa}_{j,\alpha, n}$ in ; indeed, Corollary \[prop:asymptotic-hatRho\] already gives consistency of $\widehat{\mu}(\Delta_\alpha)$ and $\widehat{\rho}_\beta$. Now since $2^{-1} k^{1/4} \{ \kappa_{\alpha}({\bm}1_\alpha + k^{-1/4}{\bm}e_j) - \kappa_{\alpha}({\bm}1_\alpha - k^{-1/4}{\bm}e_j ) \} \to \dot{\kappa}_{\alpha,j}$ as $n \to \infty$, a sufficient condition is that for some $\epsilon>0$, $$\label{eq:toshowKappa4}
\sup_{[1-\epsilon,2+\epsilon]^\alpha} k^{1/4}\big|\widehat \kappa_\alpha({\bm}x) - \kappa_\alpha({\bm}x)\big|
= {\mathrm{o}}_{{\operatorname{\mathbb{P}}}}(1), \qquad n \to \infty.$$ In turn, follows from weak convergence of $k^{1/2}( \widehat{\kappa}_\alpha - \kappa_\alpha )$ as $n \to \infty$ in the space $\ell^{\infty}([1-{\varepsilon}, 1+{\varepsilon}]^\alpha)$. In light of the expressions of $\widehat{\kappa}_\alpha$ and $\kappa_\alpha$ in terms of the (empirical) joint tail dependence functions $\widehat{r}_\beta$ and $r_\beta$, respectively, weak convergence of $k^{1/2}( \widehat{\kappa}_\alpha - \kappa_\alpha )$ follows from Proposition \[prop:rnx\] and the functional delta method [@van2000asymptotic Theorem 20.8]. The calculations are similar to the ones for the Euclidean case in the proof of Proposition \[theo:asymptot-kappa\]; an extra point to be noted is that if $\alpha$ is such that $\mu(\Delta_\alpha) > 0$, then the denominator in the definition of $\kappa_\alpha(\bm{x})$ in is positive for all $\bm{x}$ in a neighbourhood of $\bm{1}_\alpha$.
Proposition \[prop:rnx\] implies, as $n \to \infty$, the weak convergence
$$\bigl(
\sqrt k\{ \widehat r_\alpha({\bm}{2}_\alpha) - r_\alpha({\bm}{2}_\alpha \},
\sqrt k\{ \widehat r_\alpha({\bm}{1}_\alpha) - r_\alpha({\bm}{1}_\alpha \}
\bigr)
{\rightsquigarrow}\bigl(
Z_\alpha({\bm}2_\alpha), Z_\alpha({\bm}1_\alpha)
\bigr).$$ Now $\widehat \eta_\alpha^P = g(\widehat r_\alpha({\bm}2_\alpha), \widehat r_\alpha ({\bm}1_\alpha))$ and $\eta_\alpha = 1 = g(r_\alpha({\bm}2_\alpha), r_\alpha ({\bm}1_\alpha)) = g(2\rho_\alpha,\rho_\alpha)$, with $g(x,y) = \log(2)/ \log (x/y)$; note that the function $r_\alpha$ is homogeneous. Since the gradient of $g$ is $\nabla g(x,y) = \log(2) (\log(x/y))^{-2} ( - x^{-1}, y^{-1} )$, the delta method gives $$\begin{aligned}
\sqrt k (\widehat \eta^P - 1) \; {\rightsquigarrow}\;
&
\left\langle
\nabla g(2\rho_\alpha, \rho_\alpha) ,\, \big( Z_\alpha({\bm}2_\alpha),\, Z_\alpha({\bm}1_\alpha) \big)
\right\rangle \\
&=
\frac{1}{\rho_\alpha \log 2 }
\left\langle ( -1/2, 1 ) , \, \big( Z_\alpha({\bm}2_\alpha), Z_\alpha({\bm}1_\alpha)\big) \right\rangle \\
&=
\frac{-1}{2 \rho_\alpha \log 2 }
\{ Z_\alpha({\bm}2_\alpha) - 2 Z_\alpha({\bm}1_\alpha) \}.\end{aligned}$$ The first part of the assertion follows. As for the variance, $$\begin{aligned}
{\operatorname{\mathbb{V}\mathrm{ar}}}( Z_\alpha({\bm}2_\alpha) - 2Z_\alpha({\bm}1_\alpha) )
&=
{\operatorname{\mathbb{V}\mathrm{ar}}}( Z_\alpha({\bm}2_\alpha) )
+
4{\operatorname{\mathbb{V}\mathrm{ar}}}( Z_\alpha({\bm}1_\alpha) )
-
4 {\operatorname{\mathbb{C}\mathrm{ov}}}(Z_\alpha({\bm}2_\alpha), Z_\alpha({\bm}1_\alpha)), \end{aligned}$$ The function $r_\alpha$ is homogeneous of order $1$, so that $\partial_j{r}_{\alpha}$ is constant along rays, that is, the function $0 < t \mapsto \partial_j {r}_{\alpha}(t \,{\bm}x)$ is constant. Moreover, the measure $\Lambda$ is homogeneous of order $1$ too. In view of and , it follows that ${\operatorname{\mathbb{V}\mathrm{ar}}}( Z_\alpha(t \bm{x}) ) = t {\operatorname{\mathbb{V}\mathrm{ar}}}( Z_\alpha( \bm{x} ) )$ for $t > 0$; in particular ${\operatorname{\mathbb{V}\mathrm{ar}}}( Z_\alpha(\bm{2}_\alpha) = 2 {\operatorname{\mathbb{V}\mathrm{ar}}}( Z_\alpha(\bm{1}_\alpha )$. Further, $\rho_\alpha = (\mathrm{d} r_\alpha(t, \ldots, t) / \mathrm{d}t)_{t=1} = \sum_{j \in \alpha} \dot{\rho}_{j,\alpha}$ and thus $$\begin{aligned}
{\operatorname{\mathbb{V}\mathrm{ar}}}(Z_\alpha({\bm}1_\alpha))
& = \rho_\alpha - 2\sum_{j\in\alpha}\dot{\rho}_{j,\alpha} \rho_\alpha + \sum_{j\in\alpha} \sum_{j'\in\alpha}\dot{\rho}_{j,\alpha}\dot{\rho}_{j',\alpha} \rho_{\{j,j'\}} \\
& = \rho_\alpha - 2 \rho_\alpha^2 + \sum_{j\in\alpha} \sum_{j'\in\alpha}\dot{\rho}_{j,\alpha}\dot{\rho}_{j',\alpha} \rho_{\{j,j'\}}.
\end{aligned}$$ The covariance term is $$\begin{gathered}
{\operatorname{\mathbb{C}\mathrm{ov}}}(Z_\alpha({\bm}2_\alpha), Z_\alpha({\bm}1_\alpha))
=\rho_\alpha
- \sum_{j\in\alpha} \dot{\rho}_{j,\alpha} \rho_\alpha
- \sum_{j\in\alpha}\dot{\rho}_{j,\alpha}
r_\alpha({\bm}2_\alpha \wedge {\boldsymbol{\iota}}_{j}) \\
+ \sum_{j\in\alpha} \sum_{j'\in\alpha}
\dot{\rho}_{j,\alpha}\dot{\rho}_{j',\alpha}r_{\{j,j'\}}(2,1),
$$ with ${\bm}2_\alpha \wedge {\boldsymbol{\iota}}_{j}$ as explained in the statement of the proposition. Since $\sum_{j\in\alpha} \dot{\rho}_{j,\alpha} = \rho_\alpha$, we can simplify and find $$\begin{aligned}
{\operatorname{\mathbb{V}\mathrm{ar}}}( Z_\alpha({\bm}2_\alpha) - 2Z_\alpha({\bm}1_\alpha) )
&=
6 {\operatorname{\mathbb{V}\mathrm{ar}}}( Z_\alpha(\bm{1}_\alpha) ) - 4 {\operatorname{\mathbb{C}\mathrm{ov}}}(Z_\alpha({\bm}2_\alpha), Z_\alpha({\bm}1_\alpha)) \\
&=
2 \rho_\alpha - 8 \rho_\alpha^2 + 4\sum_{j\in\alpha} \dot{\rho}_{j,\alpha} r_\alpha({\bm}2_\alpha \wedge {\boldsymbol{\iota}}_{j})
\\
&\qquad \null
+
\sum_{j\in\alpha} \sum_{j'\in\alpha}
\dot{\rho}_{j,\alpha}\dot{\rho}_{j',\alpha} \big[ 6 \rho_{\{j, j'\}} - 4 r_{\{j, j' \}}(2,1)\big].\end{aligned}$$ Divide the right-hand side by $(2 \rho_\alpha \log 2)^2$ to obtain .
To alleviate notations, ${\varnothing}\ne \alpha \subset\{1,\ldots, d\}$ is fixed and the subscript $\alpha$ is omitted throughout the proof. Introduce the tail empirical process $Q_n(t) = \widehat T_{(n - \lfloor kt \rfloor)}$ for $0 < t < n/k$. The key is to represent the Hill estimator as a statistical tail functional [@drees1998evindex Example 3.1] of $Q_n$, i.e., $\widehat \eta^H = \Theta(Q_n)$, where $\Theta$ is the map defined for any measurable function $ z: (0,1]\to {\mathbb{R}}$ as $\Theta(z) = \int_{0}^1 \log^+ \{ z(t) / z(1) \} {\,\mathrm{d}}t$ when the integral is finite and $\Theta(z) = 0$ otherwise. Let $z_\eta: t\in(0,1]\mapsto t^{-\eta}$ denote the quantile function of a standard Pareto distribution with index $1/\eta$; it holds that $\Theta(z_\eta) = \eta$. The map $\Theta$ is scale invariant, i.e., $\Theta(t z) = \Theta(z), t>0$.
The proof consists of three steps:
1. Introduce a function space $D_{\eta,h}$ allowing to control $Q_n(t)$ and $z_\eta(t)$ as $t\to 0$. In this space and up to rescaling, $Q_n - z_\eta$ converges weakly to a Gaussian process.
2. Show that the map $\Theta$ is Hadamard differentiable at $z_\eta$ tangentially to some well chosen subspace of $D_{\eta,h}$.
3. Apply the functional delta method to show that $\eta^H = \Theta(Q_n) $ is asymptotically normal and compute its asymptotic variance via the Hadamard derivative of $\Theta$.
####
Let $\epsilon>0$ and $h(t) = t^{1/2+ \epsilon},\, t\in[0,1]$. Then $h\in\mathcal{H}$, where $$\mathcal{H} = \{z: [0,1]\to {\mathbb{R}}\mid z \text{ continuous, }
\lim_{t\to 0} z(t) t^{-1/2} (\log\log(1/t))^{1/2} = 0
\}.$$ Introduce the function space $$D_{\eta, h} = \{z: [0,1]\to {\mathbb{R}}\mid \lim_{t\to 0} t^{\eta} h(t)
z(t) = 0\,;\; t\mapsto t^{\eta}h(t) z(t) \in D[0,1] \},$$ where $D[0,1]$ is the space of càdlàg functions. Notice that $z_\eta \in D_{\eta,h}$. Equip $D_{\eta, h}$ with the seminorm $\|z\|_{\eta,h} = \sup_{t\in(0,1]}|t^\eta h(t) z(t)|$. Let $m = \lceil n q^\leftarrow (k/n)\rceil $, with $\lceil \, \cdot \, \rceil$ the ceil function, so that $k/m\to \rho$; for self-consistency of the present paper, the roles of $k$ and $m$ are reversed compared to the notation in [@draisma2004bivariate]. From [@draisma2004bivariate Lemma 6.2], we have, for all $t_0>0$, in the space $D_{\eta,h}$, the weak convergence $$\label{eq:cv-tailQuantile}
\sqrt{k} \left( \frac{m}{n}Q_n - z_\eta \right) {\rightsquigarrow}\left( \eta t^{-(\eta +1)} \bar W(t)\right)_{t \in [0,t_0]}$$ where $\bar W (t) = \tilde W({\bm}t_\alpha)$, and $\tilde W$ is defined as in the statement of Proposition \[prop:normality-hill\]. Indeed, the process $\bar W$ in the statement from [@draisma2004bivariate Lemmata 6.1 and 6.2] has same distribution as $W_1({\bm}t_\alpha)$ in the case $\rho=0$; recall that our $\rho$ is denoted by $l$ in [@draisma2004bivariate]. Put $U_{i,j} = 1 - F_j(X_{i,j})$, and let $U_{(1),j} {\leqslant}\ldots {\leqslant}U_{(d),j}$ be the order statistics of $U_{1,j}, \ldots, U_{n,j}$. In the case $\rho>0$, $\bar W$ equals in distribution $W_{\mathrm{dra}} ({\bm}t_\alpha)$ where $W_{\mathrm{dra}}$ appears in Lemma 6.1 in the cited reference as the limit in distribution (for $\alpha= \{1,2\}$), for ${\bm}x\in E_\alpha$, of $$\begin{aligned}
\Delta_{n,k,m} ({\bm}x)
&= \sqrt{k} \Bigg[ \frac{1}{k} \sum_{i=1}^n {{\mathbbm{1}}}\{ \forall j \in \alpha : U_{i,j} {\leqslant}U_{(\lfloor m x_j \rfloor), j} \} - c({\bm}x)\Bigg] \\
&=
\underbrace{ \sqrt{ \frac{m}{k}} }_{\to \rho^{-1/2}}
\sqrt{m}
\Biggl[
\underbrace{\frac{1}{m} \sum_{i=1}^n {{\mathbbm{1}}}\{ \forall j \in \alpha : U_{i,j} {\leqslant}U_{(\lfloor m x_j \rfloor),j} \} }_{\text{$r_n({\bm}x)$ with $k$ replaced by $m$}}
\null -
r({\bm}x)\underbrace{\frac{k}{m\rho}}_{\to 1}
\Biggr].
\end{aligned}$$ From Proposition \[prop:rnx\] and Slutsky’s Lemma, we have $\Delta_{n,k,m} {\rightsquigarrow}\rho^{-1/2} Z_\alpha$ in $\ell^\infty([0, 1]^\alpha)$. Therefore, $W_{\mathrm{dra}} = \rho^{-1/2} Z_\alpha$, as claimed.
####
The right-hand side of belongs to $\mathcal{C}_{h,\eta} = \{ z \in D_{\eta,h} \mid \text{$z$ is continuous}\}$. To apply the functional delta-method [@van2000asymptotic Theorem 20.8], we must verify that the restriction of $\Theta$ to $\bar D_{\eta, h}$ is Hadamard-differentiable tangentially to $\mathcal{C}_{\eta,h}$, with derivative $\Theta'$, where $\bar D_{\eta, h}$ is a subspace of $D_{\eta,h}$ such that ${\operatorname{\mathbb{P}}}(Q_n \in \bar D_{\eta,h})\to 1$ as $n\to \infty$; see the remark following Condition 3 in [@drees1998evindex]. Then it will follow from the scale invariance of $\Theta$, the identities $\Theta(Q_n) = \widehat \eta^H$ and $\Theta(z_\eta) = \eta$, and the weak convergence in that $$\label{eq:etaH_W}
\sqrt{k} \left( \widehat\eta^H- \eta \right)
=
\sqrt{k} \left( \Theta(\frac{m}{n}Q_n) - \Theta(z_\eta) \right)
{\rightsquigarrow}\Theta'\left[\left( \eta t^{-(\eta +1)} \bar W(t)\right)_{t \in [0,1]}\right]$$ as $n \to \infty$. From [@drees1998evindex Example 3.1], the restriction of $\Theta$ to $\bar D_{\eta,h}$, the subset of functions on $D_{\eta,h}$ which are positive and non increasing, is indeed Hadamard differentiable; letting $\nu$ denote the measure ${\,\mathrm{d}}\nu (t) = t^{\eta}{\,\mathrm{d}}t + {\,\mathrm{d}}\epsilon_1(t)$, with $\epsilon_1$ a point mass at $1$, the derivative is $$\Theta'(z) = \int_0^1 t^\eta z(t) {\,\mathrm{d}}t - y(1) = \int_{[0,1]} z(t)
{\,\mathrm{d}}\nu (t).$$
####
The weak limit in is thus equal to $\int_{[0,1]} \eta t^{-(\eta +1)} \bar W(t) {\,\mathrm{d}}\nu(t)$. From [@shorack2009empirical Proposition 2.2.1], the latter random variable is centered Gaussian with variance $$\sigma^2 = \iint_{[0,1]^2} \eta^2 (st)^{-(\eta +1)} {\operatorname{\mathbb{C}\mathrm{ov}}}(\bar
W(s), \bar W(t)){\,\mathrm{d}}\nu(s){\,\mathrm{d}}\nu(t).$$ By definition of $\nu$ and by symmetry of the covariance,
$$\begin{gathered}
\sigma^2 / \eta^2 = 2 \underbrace{\int_{s=0}^1\int_{t=0}^s (st)^{-1} {\operatorname{\mathbb{C}\mathrm{ov}}}(\bar W(s), \bar W(t)) {\,\mathrm{d}}t {\,\mathrm{d}}s}_{A} \\
- 2 \underbrace{\int_{s=0}^1 {\operatorname{\mathbb{C}\mathrm{ov}}}(\bar W(s), \bar W(1)) s^{-1}{\,\mathrm{d}}s}_B + {\operatorname{\mathbb{V}\mathrm{ar}}}(\bar W(1)).
\end{gathered}$$
For any $s\in(0,1)$, $$\begin{aligned}
\int_{t=0}^s {\operatorname{\mathbb{C}\mathrm{ov}}}(\bar W(s), \bar W(t))(st)^{-1} {\,\mathrm{d}}t
& =\int_{u=0}^1 {\operatorname{\mathbb{C}\mathrm{ov}}}(\bar W(s), \bar W(us))(su)^{-1} {\,\mathrm{d}}u \\
&= \int_{u=0}^1 {\operatorname{\mathbb{C}\mathrm{ov}}}(\bar W(1), \bar W(u))(u)^{-1} {\,\mathrm{d}}u
= B.
\end{aligned}$$ The penultimate equality follows from ${\operatorname{\mathbb{C}\mathrm{ov}}}(\bar W(\lambda s),\bar W( \lambda t)) = \lambda {\operatorname{\mathbb{C}\mathrm{ov}}}(\bar W(s),\bar W( t))$ for $\lambda>0$ and $s,t\in(0,1]$. Therefore $A=B$ and $\sigma^2 = \eta^2 {\operatorname{\mathbb{V}\mathrm{ar}}}(\bar W(1))$, as required.
CLEF algorithm and variants {#sec:appendix-CLEF}
===========================
The CLEF algorithm is described at length in [@chiapinofeature]. For completeness, its pseudo-code is provided below. The underlying idea is to iteratively construct pairs, triplets, quadruplets… of features that are declared ‘dependent’ whenever $\widehat\kappa_\alpha {\geqslant}C$ for some user-defined tolerance level $C > 0$. Varying this criterion produces three variants of the original algorithm, namely CLEF-Asymptotic, CLEF-Peng, and CLEF-Hill. The pruning stage of the algorithm is the same for all three variants.
**Input**: Tolerance parameter $\kappa_{\min}>0$.\
**Step 1:** Put $\hat{\mathcal{A}}_1 = \{ \{1\}, \ldots, \{d\} \}$ and $S = 1$. **Step $\bm{s = 2, \ldots, d}$**: If $\hat{\mathcal{A}}_{s-1}={\varnothing}$, end **STAGE 1**. Otherwise:
- Generate candidates of size $s$:\
$\mathcal{A}'_{s} = \{\alpha\subset\{1,\ldots,d\} : |\alpha|=s \text{ and }
\alpha\setminus j \in\hat{\mathcal{A}}_{s-1} \text{ for all } j\in\alpha\}$.
- Put $ \hat{\mathcal{A}}_{s}
= \big\{\alpha\in\mathcal{A}'_{s} : \hat{\kappa}_{\alpha}>\kappa_{\min} \big\}$.
- If $\hat{\mathcal{A}}_{s} \ne {\varnothing}$, put $S = s$.
***Output***: $\widehat{\mathbb{M}} = {\varnothing}$ if $S = 1$ and $\widehat{\mathbb{M}} =
\bigcup_{s=2}^{S} \hat{\mathcal{A}}_s$ if $S {\geqslant}2$.\
\
If $S = 1$, then $\widehat{\mathbb{M}}_{\max} = {\varnothing}$. Otherwise: *Initialization:* $\mathbb{\widehat M}_{\max} \leftarrow \hat{\mathcal{A}}_S$. for $s= (S-1) : 2$, for $\alpha\in\hat{\mathcal{A}}_s$, If there is no $\beta\in\widehat{\mathbb{M}}_{\max}$ such that $\alpha\subset\beta$, then $\widehat{\mathbb{M}}_{\max} \leftarrow
\widehat{\mathbb{M}}_{\max} \cup\{\alpha \}$. : $\widehat{\mathbb{M}}_{\max} $
This work was supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: '[ ]{}'
author:
- 'Alexander Marx[^1]'
- Jilles Vreeken
bibliography:
- 'abbrev.bib'
- 'bib-jilles.bib'
- 'bib-paper.bib'
- 'bib-alex.bib'
title: '[Causal Inference on Multivariate and Mixed-Type Data]{}'
---
Introduction {#sec:intro}
============
Telling cause from effect is one of the core problems in science. It is often difficult, expensive, or impossible to obtain data through randomized trials, and hence we often have to infer causality from, what is called, observational data [@pearl:09:book]. We consider the setting where, given data over the joint distribution of two random variables $X$ and $Y$, we have to infer the causal direction between $X$ and $Y$. In other words, our task is to identify whether it is more likely that $X$ causes $Y$, or vice versa, that $Y$ causes $X$, or that the two are merely correlated.
In practice, $X$ and $Y$ do not have to be of the same type. The altitude of a location (real-valued), for example, determines whether it is a good habitat (binary) for a mountain hare. In fact, neither $X$ nor $Y$ have to be univariate. Whether or not a location is a good habitat for an animal, is not just caused by a single aspect, but by a *combination* of conditions, which not necessarily are of the same type. We are therefore interested in the general case where $X$ and $Y$ may be of any cardinality, and may be single or mixed-type.
To the best of our knowledge there exists no method for this general setting. Causal inference based on conditional independence tests, for example, requires three variables, and cannot decide between $\XtoY$ and $\YtoX$ [@pearl:09:book]. All existing methods that consider two variables are only defined for single-type pairs. Additive Noise Models (ANMs), for example, have only been proposed for univariate pairs of real-valued [@peters:14:continuousanm] or discrete variables [@peters:11:dr], and similarly so for methods based on the independence of $P(X)$ and $P(Y\mid X)$ [@sgouritsa:15:cure; @liu:16:dc]. Trace-based methods require both $X$ and $Y$ to be strictly multivariate real-valued [@janzing:10:ltr; @chen:13:ktr], and whereas [@vreeken:15:ergo] also works for univariate pairs, these again have to be real-valued. We refer the reader to Sec. \[sec:rel\] for a more detailed overview of related work.
Our approach is based on algorithmic information theory. That is, we follow the postulate that if $\XtoY$, it will be easier—in terms of Kolmogorov complexity—to first describe $X$, and then describe $Y$ given $X$, than vice-versa [@janzing:10:algomarkov; @vreeken:15:ergo; @budhathoki:16:origo]. Kolmogorov complexity is not computable, but can be approximated through the Minimum Description Length (MDL) principle [@rissanen:78:mdl; @grunwald:07:book], which we use to instantiate this framework. In addition, we develop a causal indicator that is able to handle multivariate and mixed-type data.
To this end, we define an MDL score for coding forests, a model class where a model consists of classification and regression trees. By allowing dependencies from $X$ to $Y$, or vice versa, we can measure the difference in complexity between $\XtoY$ and $\YtoX$. Discovering a single optimal decision tree is already NP-hard [@murthy:97:decision-trees], and hence we cannot efficiently discover the coding forest that describes the data most succinctly. We therefore propose [<span style="font-variant:small-caps;">Crack</span>]{}, an efficient greedy algorithm for discovering good models directly from data.
Through extensive empirical evaluation on synthetic, benchmark, and real-world data, we show that [<span style="font-variant:small-caps;">Crack</span>]{}performs very well in practice. It performs on par with existing methods for univariate single-type pairs, is the first to handle pairs of mixed data type, and outperforms the state of the art on multivariate pairs with a large margin. It is also very fast, taking less than 4 seconds over any pair in our experiments.
Preliminaries {#sec:prelim}
=============
First, we introduce notation and give brief primers to Kolmogorov complexity and the MDL principle.
Notation
--------
In this work we consider data $D$ over the joint distribution of random variables $X$ and $Y$. Such data $D$ contains $n$ records over a set $A$ of $|A| = |X| + |Y| = m$ attributes, $a_1, \dots, a_m \in A$. An attribute $a$ has a type $\textit{type}(a)$ where $\textit{type}(a) \in \{ \text{\textit{binary}, \textit{categorical}, \textit{numeric}} \}$. We will refer to binary and categorical attributes as *nominal* attributes. The size of the domain of an attribute $a$ is defined as $$|\dom(a)| = \begin{cases}
\#\textit{values} &\text{if \textit{type}$(a)$ is nominal}\\
\frac{\max(a) - \min(a)}{\res(a)} + 1&\text{if \textit{type}$(a)$ is numeric} \; ,
\end{cases}$$ where $\res(a)$ is the resolution at which the data over attribute $a$ was recorded. For example, a resolution of 1 means that we consider integers, of $0.01$ means that $a$ was recorded with a precision of up to a hundredth.
We will consider decision and regression trees. In general, a tree $T$ consist of $|T|$ nodes. We identify internal nodes as $\node \in \internal(T)$, and leaf nodes as $\leaf \in \lvs(T)$. A leaf node $l$ contains $|l|$ data points.
All logarithms are to base 2, and we use $0 \log 0 = 0$.
Kolmogorov Complexity, a brief primer
-------------------------------------
The Kolmogorov complexity of a finite binary string $x$ is the length of the shortest binary program $p^*$ for a universal Turing machine $\mathcal{U}$ that generates $x$, and then halts [@kolmogorov:65:information; @vitanyi:93:book]. Formally, we have $$K(x) = \min \{ |p| \mid p \in \{0,1\}^*, \mathcal{U}(p) = x \} \; .$$ Simply put, $p^*$ is the most succinct *algorithmic* description of $x$, and the Kolmogorov complexity of $x$ is the length of its ultimate lossless compression. Conditional Kolmogorov complexity, $K(x \mid y) \leq K(x)$, is then the length of the shortest binary program $p^*$ that generates $x$, and halts, given $y$ as input. For more details see [@vitanyi:93:book].
MDL, a brief primer
-------------------
The Minimum Description Length (MDL) principle [@rissanen:78:mdl; @grunwald:07:book] is a practical variant of Kolmogorov Complexity. Intuitively, instead of all programs, it considers only those programs that we know that output $x$ and halt. Formally, given a model class $\models$, MDL identifies the best model $M \in \models$ for data $\data$ as the one minimizing $$L(\data, M) = L(M) + L(\data \mid M) \; ,$$ where $L(M)$ is the length in bits of the description of $M$, and $L(\data\mid\model)$ is the length in bits of the description of data $\data$ given $M$. This is known as two-part MDL. There also exists one-part, or *refined* MDL, where we encode data and model together. Refined MDL is superior in that it avoids arbitrary choices in the description language $L$, but in practice only computable for certain model classes. Note that in either case we are only concerned with code *lengths* — our goal is to measure the *complexity* of a dataset under a model class, not to actually compress it [@grunwald:07:book].
Causal Inference by Compression {#sec:causal}
===============================
We pursue the goal of causal inference by compression. Below we give a short introduction to the key concepts.
Causal Inference by Complexity
------------------------------
The problem we consider is to infer, given data over two correlated variables $X$ and $Y$, whether $X$ caused $Y$, whether $Y$ caused $X$, or whether $X$ and $Y$ are only correlated. As is common, we assume causal sufficiency. That is, we assume there exists no hidden confounding variable $Z$ that is the common cause of both $X$ and $Y$.
The Algorithmic Markov condition, as recently postulated by Janzing and Schölkopf [@janzing:10:algomarkov], states that factorizing the joint distribution over and into $P(\cause)$ and $P(\effect \mid \cause)$, will lead to simpler—in terms of Kolmogorov complexity—models than factorizing it into $P(\effect)$ and $P(\cause \mid \effect)$. Formally, they postulate that if $X$ causes $Y$, $$K(P(X)) + K(P(Y \mid X)) \le K(P(Y)) + K(P(X \mid Y)) \; . \label{eq:janzing}$$ While in general the symmetry of information, $K(x)+K(y\mid x) = K(y) + K(x \mid y)$, holds up to an additive constant [@vitanyi:93:book], Janzing and Schölkopf [@janzing:10:algomarkov] showed it does *not* hold when $X$ causes $Y$, or vice versa. Based on this, Budhathoki & Vreeken [@budhathoki:16:origo] proposed $$\DXY^{*} = \frac{K(P(\dataX)) + K(P(\dataY \mid \dataX))}{K(P(\dataX)) + K(P(\dataY))} \; , \label{eq:origo}$$ as a causal indicator that uses this asymmetry to infer that $X \rightarrow Y$ as the most likely causal direction if $\DXY^* < \DYX^*$, and vice versa. The normalisation has no function during inference, but does help to interpret the confidence of the indicator.
Both scores assume access to the true distribution $P(\cdot)$, whereas in practice we only have access to empirical data. Moreover, following from the halting problem, Kolmogorov complexity is not computable. We can approximate it, however, via MDL [@vitanyi:93:book; @grunwald:07:book], which also allows us to directly work with empirical distributions.
Causal Inference by MDL
-----------------------
For causal inference by MDL, we will need to approximate both $K(P(\dataX))$ and $K(P(\dataY \mid \dataX))$. For the former, we need to consider the model classes $\models_{X}$ and $\models_{Y}$, while for the latter we need to consider class $\models_{Y\mid X}$ of models $\model_{Y \mid X}$ that describe the data of $Y$ dependent the data of $X$.
That is, we are after the *causal* model $\model_{\XtoY}= (\model_{X}, \model_{Y\mid X})$ from the class $\models_\XtoY = \models_{X} \times \models_{Y \mid X}$ that best describes the data $Y$ by exploiting as much as possible structure of $X$ to save bits. By MDL, we identify the optimal model $\model_\XtoY \in \models_\XtoY$ for data $\data$ over $X$ and $Y$ as the one minimizing $$L(\data,\model_\XtoY) = L(\dataX, \model_{X}) + L(\dataY, M_{Y \mid X} \mid \dataX) \; ,$$ where the encoded length of the data of $X$ under a given model is encoded using two-part MDL, similarly so for $Y$, if we consider the inverse direction.
To identify the most likely causal direction between $X$ and $Y$ by MDL we can now simply rewrite Eq. $$\DXY = \frac{L(\dataX, \model_{X}) + L(\dataY, M_{Y\mid X} \mid X)}{L(\dataX, \model_{X}) + L(\dataY, \model_{Y})} \; .$$ Similar to the original score, we infer that $X$ is a likely cause of $Y$ if $\DXY < \DYX$, $Y$ is a likely cause of $X$ if $\DYX < \DXY$, and that $X$ and $Y$ are only correlated or might have a common cause if $\DXY = \DYX$.
Normalized Causal Indicator
---------------------------
Although $\Delta$ has nice theoretical properties, it has a mayor drawback. It assumes that a bit gain in the description of the data over one attribute has the same importance as one bit gain in the description of the data over another attribute. This does not hold true if these attributes have different intrinsic complexities, such as when their domain sizes strongly differ. For example, a continuous valued attribute is very likely to have a much higher intrinsic complexity than a binary attribute. This means that gaining $k$ bits from an attribute with a large domain is not comparable to gaining $k$ bits from an attribute with a small domain. Since the $\Delta$ indicator compares the absolute difference in bits, it does not account for differences w.r.t. the intrinsic complexity. Hence, $\Delta$ is highly likely to be a bad choice when $X$ and $Y$ are of different, or of mixed-type data.
We therefore propose an alternative indicator for causal inference on mixed-type data. Instead of taking the absolute difference between the conditioned and unconditioned score, we instead consider relative differences w.r.t. the marginal. We can derive the *Normalized Causal Indicator* () starting from the numerator of the $\Delta$ indicator. By subtracting the conditional costs on both sides, we have $$L(\dataX, \model_{X}) - L(\dataX, M_{X| Y} | Y) < L(\dataY, \model_{Y}) - L(\dataY, M_{Y| X} | X).$$ Since the aim of the is to measure the relative gain, we divide by the costs of the unconditioned data $$\frac{L(\dataX, \model_{X}) - L(\dataX, M_{X| Y} | Y)}{L(\dataX, \model_{X})} = 1 - \frac{L(\dataX, M_{X| Y} | Y)}{L(\dataX, \model_{X})} \; .$$ After this step, we can conclude that for the relative gain it holds, if $\XtoY$ $$\frac{L(\dataX, M_{X\mid Y} \mid Y)}{L(\dataX, \model_{X})} > \frac{L(\dataY, M_{Y\mid X} \mid X)}{L(\dataY, \model_{Y})} \; .$$ This score can be understood as an instantiation of the indicator proposed by Vreeken [@vreeken:15:ergo]. From the derivation, we can easily see that the difference between the score of both indicators depends only on the normalization factor and hence both are based on the Algorithmic Markov condition. It turns out, however, that the indicator is also biased. Although it balances the gain between $X$ and $Y$, we need a score that does not impose prior assumptions to the individual attributes of $X$ and $Y$. With the indicator, it could happen that a single $X_i \in X$ dominates the whole score for $X$. To account for this, we assume independence among the variables within $X$ and $Y$, meaning that the domain of two individual attributes within $X$ or $Y$ is allowed to differ. Hence, we formulate the , which we from now on denote by $\delta$, from $X$ to $Y$ as $$\delta_{\XtoY} = \frac{1}{|Y|} \sum_{Y_i \in Y} \frac{L(Y_i, M_{Y_i \mid X} \mid X)}{L(Y_i, \model_{Y_i})}\;$$ and analogously $\delta_{\YtoX}$. To avoid bias towards dimensionality, we normalize by the number of attributes. As above, we infer $\XtoY$ if $\delta_{\XtoY} < \delta_{\YtoX}$ and vice versa.
In practice, we expect that $\Delta$ performs well on data where $X$ and $Y$ are of the same type, especially when $|X|=|Y|$ and the domain sizes of their attributes are balanced. For unbalanced domains, dimensionality, and especially for mixed-type data, we expect $\delta$ to perform much better. The experiments indeed confirm this.
MDL for Tree Models {#sec:score}
===================
To use the above defined causal indicators in practice, we need to define a casual model class $\models_\XtoY$, how to encode a model $\model \in \models$ in bits, and how to encode a dataset $\data$ using a model $\model$. As models we consider tree models, or, *coding forests*. A coding forest $\model$ contains per attribute $a_i \in A$ one coding tree $T_i$. A coding tree $T_i$ encodes the values of $a_i$ in its leaves, splitting or regressing the data of $a_i$ on attribute $a_j$ ($i \neq j$) in its internal nodes to encode the data of $a_i$ more succinctly.
We encode the data over attribute $a_i$ with the corresponding coding tree $T_i$. The encoded length of data $D$ and $\model$ then is $L(D, \model) = \sum_{a_i \in \attributes} L(\tree_i)$, which corresponds to the sum of costs of the individual trees.
To ensure lossless decoding, there needs to exist an order on the trees $T \in \model$ such that we can transmit these one by one. In other words, in a *valid* tree model there are no cyclic dependencies between the trees $\tree \in \model$, and a valid model can hence be represented by a DAG. Let $\models(D)$ be the set of all valid tree models for $D$, that is, $\model \in \models(D)$ is a set of $|A|$ trees such that the data types of the leafs in $\tree_i$ corresponds to the data type of attribute $a_i$, and its dependency graph is acyclic.
![Toy data set with ground truth $\XtoY$. Shown is the dependency DAG (right). More dependencies go from $X$ to $Y$ than vice versa. Left: Example coding tree for $Y_2$. $X_1$ splits the values of $Y_2$ into two subsets. In addition, the subset belonging to the left child can be further compressed by regressing on $X_2$.[]{data-label="fig:toy:example"}](lepus_ytox_large.pdf "fig:") \[toy:dag\]
![Toy data set with ground truth $\XtoY$. Shown is the dependency DAG (right). More dependencies go from $X$ to $Y$ than vice versa. Left: Example coding tree for $Y_2$. $X_1$ splits the values of $Y_2$ into two subsets. In addition, the subset belonging to the left child can be further compressed by regressing on $X_2$.[]{data-label="fig:toy:example"}](lepus_xtoy_tree.pdf "fig:") \[toy:tree\]
We write $\models_{X}(X)$ and $\models_{Y}(Y)$ to denote the subset of valid coding forests for $X$ and $Y$, where we do not allow dependencies. To describe the possible set of models where we allow attributes of $X$ to only depend on attributes of $Y$ we write $\models_{X \mid Y}(X)$ and do so accordingly for $Y$ depending only on $X$. If an attribute does not have any incoming dependencies, its tree is a stump. Fig. \[fig:toy:example\] shows the DAG for a toy data set, and an example tree for $Y_2$. From the DAG, the set of purple edges would be a valid model in $\models_{Y \mid X}(Y)$, whereas the orange edges are a valid model from $\models_{X \mid Y}(X)$.
### Cost of a Tree {#cost-of-a-tree .unnumbered}
The encoded cost of a tree consists of two parts. First, we transmit the topology of the tree. From the root node on we indicate with one bit per node whether it is a leaf or an internal node, and if the latter, one further bit to identify whether it is a split or regression node. Formally we have that $$L(\tree) = |\tree| + \sum_{\node \in \internal(\tree)} (1+L(\node)) + \sum_{\leaf \in \lvs(\tree)} L(\leaf) \; .$$ Next, we explain how we encode internal nodes and then specify the encoding for leaf nodes.
### Cost of a Single Split {#cost-of-a-single-split .unnumbered}
The length of a split node $\node$ is $$L_{1\tsplit}(\node) = 1 + \log |\attributes| + \begin{cases}
\log |\dom(a_j)| \text{ if $a_i$ is categorical,}\\
\log |\dom(a_j) - 1| \text{ else.}
\end{cases}$$ whereas we first need one bit to indicate this is a single-split node, then identify in $\log |\attributes|$ bits on which attribute $a_j$ we split, and third the split condition.
The split condition can be any value in the domain for categorical, and can lie in between two consecutive values of a numeric attribute ($|\dom(a_j) - 1|$ choices). For binary we only have one option, resulting in zero cost.
### Costs of a Multiway split {#costs-of-a-multiway-split .unnumbered}
A multiway split is only possible for categorical and real valued data. As there are exponentially many multiway splits, we consider only a subset. The costs for a multiway split are $$L_{\text{k}\tsplit}(\node) = 1 + \log |\attributes| + \begin{cases}
0 \text{ if $a_i$ is categorical,}\\
L_{\mathbb{N}}(k) \text{ numeric,}
\end{cases}$$ where the first two terms are similar to above. For categorical data, we only consider splitting on all values, and hence have no further cost. For numeric data, we only split non-deterministic cases, i.e. if there exist duplicate values. To do so, we split on every such value that occurs at least $k$ times, and one residual split for all remaining data points. To encode such a split, we transmit $k$ using $L_{\mathbb{N}}(k)$ bits, where $L_{\mathbb{N}}$ is the MDL optimal encoding for integers $z \geq 1$ [@rissanen:83:integers].
### Cost of Regressing {#cost-of-regressing .unnumbered}
For a regression node we also first encode the target attribute, and then the parameters of the regression, i.e. $$L_{\tregress}(\node) = \log |\attributes| + \sum_{\phi \in \Phi(\node)} \left( \, 1 + \LN(s) + \LN(\lfloor \phi \cdot 10^{s}\rfloor) \, \right) ,$$ where $\Phi(\node)$ denotes the set of parameters for the regression. For linear regression, it consists of $\alpha$ and $\beta$, while for quadratic regression it further contains $\gamma$. To describe each parameter $\phi \in \Phi$ we transmit it up to a user defined precision, e.g. $0.001$, we first encode the corresponding number of significant digits $s$, e.g. $3$, and then the shifted parameter value.
Next, we describe how to encode the data in a leaf $l$. As we consider both nominal and numeric attributes, we need to define $L_\nom(l)$ for nominal and $L_\num(l)$ for numeric data.
### Cost of a Nominal Leaf {#cost-of-a-nominal-leaf .unnumbered}
To encode the data in a leaf of a nominal attribute, we can use refined MDL [@kontkanen:07:histo]. That is, we encode minimax optimally, without having to make design choices [@grunwald:07:book]. In particular, we encode the data of a nominal leaf using the normalized maximum likelihood (NML) distribution as $$\begin{aligned}
L_\nom(\leaf) =& \log \left( \sum_{\substack{h_1 + \cdots + h_{k} = |\leaf|}} \frac{|\leaf|!}{h_1! h_2! \cdots h_{k}!} \right)\\
& - |l| \sum_{c \in \dom(a_i)} \Pr(a_i = c \mid \leaf) \log \Pr(a_i = c \mid \leaf) \; .\end{aligned}$$ Kontkanen & Myllymäki [@kontkanen:07:histo] derived a recursive formula to calculate this in linear time.
### Cost of a Numerical Leaf {#cost-of-a-numerical-leaf .unnumbered}
For numeric data existing refined MDL encodings have high computational complexity [@kontkanen:07:histo]. Hence, we encode the data in numeric leaves using two-part MDL, using point models with Gaussian or uniform noise. A split or a regression on an attribute aims to reduce the variance or the domain in the leaf. We encode the costs of a numeric leaf as $$\begin{aligned}
L_\num(\leaf \mid \sigma, \mu) =& \frac{|l|}{2} \left( \frac{1}{\ln 2} + \log 2 \pi \sigma^2 \right) - |l| \log \res(a_i) ,\end{aligned}$$ given empirical mean $\mu$ and variance $\sigma$ or as uniform given $\min$ and $\max$ as $$\begin{aligned}
L_\num(\leaf \mid \min, \max) =& |l| \cdot \log \left( \frac{\max - \min}{\res(a_i)} + 1 \right) \; .\end{aligned}$$ We encode the data as Gaussian if this costs fewer bits than encoding it as uniform. To indicate this decision, we use one bit and encode the minimum of both plus the corresponding parameters. As we consider empirical data, we can safely assume that all parameters lie in the domain of the given attribute. Since we do not have any preference on the parameter values, the encoded costs of a numeric leaf $\leaf$ are $$\begin{aligned}
L_\num(l) &= 1 + 2 \log |\dom(a_j)| \\
&+ \min \{ L_\num(l \mid \sigma, \mu), L_\num(\leaf \mid \min, \max) \} \; .\end{aligned}$$
Putting it all together, we now know how to compute $L(D,\model)$, by which we can formally define the Minimal Coding Forest problem.
**Minimal Coding Forest Problem** *Given a data set $\data$ over a set of attributes $A = \{a_1, \ldots, a_m\}$, and $\models$ a valid model class for $A$. Find the smallest model $\model \in \models$ such that $L(\data,\model)$ is minimal.*
From the fact that both inferring optimal decision trees and structure learning of Bayesian networks—to which our tree-models reduce for nominal-only data and splitting on all values—are NP-hard [@murthy:97:decision-trees], it trivially follows that the Minimal Coding Forest problem is also NP-hard. Hence, we resort to heuristics.
The [<span style="font-variant:small-caps;">Crack</span>]{}Algorithm {#sec:algo}
====================================================================
Knowing the score $L(\data,\model)$ and the problem, we can now introduce the [<span style="font-variant:small-caps;">Crack</span>]{}algorithm, which stands for **c**lassification and **r**egression based p**ack**ing of data. [<span style="font-variant:small-caps;">Crack</span>]{}is an efficient greedy heuristic for discovering a coding forest $\model$ from given model class $\models$ with low $L(\data,\model)$. It builds upon the well-known ID3 algorithm [@quinlan:86:id3]. In the next section we explain the main aspects of the algorithm.
### Greedy algorithm {#greedy-algorithm .unnumbered}
We give the pseudocode of [<span style="font-variant:small-caps;">Crack</span>]{}as Algorithm \[alg:crack\]. Before running the algorithm, we set the resolution per attribute, which is $1$ for nominal data (line \[alg:crack:res\]). For numeric data, we calculate the differences between adjacent values, and to reduce sensitivity to outliers take the $k^{\mathit{th}}$ smallest difference as resolution. In general, setting $k$ to $0.1n$ works well in practice.
starts with an empty model consisting of only trivial trees, i.e. leaf nodes containing all records, per attribute (line \[alg:crack:trivialtrees\]). The given model class $\models$ implicitly defines a graph $\depgraph$ of dependencies between attributes that we are allowed to consider (line \[alg:crack:depgraph\]). To make sure the returned model is valid, we need to maintain a graph representing its dependencies (lines \[alg:crack:checkgraph1\]–\[alg:crack:checkgraph2\]). We iteratively discover that refinement of the current model that maximizes compression. To find the best refinement, we consider every attribute (line \[alg:crack:everyattrib\]), and every legal additional split or regression of its corresponding tree (line \[alg:crack:splitregress\]). A refinement is only legal when the dependency is allowed by the model family (line \[alg:crack:depcheck\]), the dependency graph remains acyclic, and we do not split or regress twice on the same attribute (line \[alg:crack:validcheck\]). We keep track of the best found refinement.
The key subroutine of [<span style="font-variant:small-caps;">Crack</span>]{}is , in which we discover the optimal refinement of a leaf $l$ in tree $T_i$. That is, it finds the optimal split of $l$ over all candidate attributes $a_j$ such that we minimize the encoded length. In case both $a_i$ and $a_j$ are numeric, also considers the best linear and quadratic regression and decides for the variant with the best compression—choosing to split in case of a tie. In the interest of efficiency, we do not allow splitting or regressing multiple times on the same candidate.
$\res(a_i) \leftarrow \RobustMinDiff(a_i)$ \[alg:crack:res\] $\tree_i \leftarrow \TrivialTree(a_i)$ for all $a_i \in A$ \[alg:crack:trivialtrees\] $\depgraph \leftarrow$ dependency graph for $\models$ \[alg:crack:depgraph\] $V \leftarrow \{ v_i \mid i \in A \}, \; E \leftarrow \emptyset$ \[alg:crack:checkgraph1\] $\depgraph \leftarrow (V,E)$ \[alg:crack:checkgraph2\] \[alg:crack:return\]
Since we use a greedy heuristic to construct the coding trees, we have a worst case runtime of $O(2^{m}n)$, where $m$ is the number of attributes and $n$ is the number of rows. Although the worst case runtime is exponential, in practice, [<span style="font-variant:small-caps;">Crack</span>]{}takes only a few seconds.
### Causal Inference with {#causal-inference-with .unnumbered}
To compute our causal indicators we have to run [<span style="font-variant:small-caps;">Crack</span>]{}twice on $D$. First with model class $\models_{X \mid Y}(X)$ to obtain $\model_{X \mid Y}(X)$ and second with $\models_{Y\mid X}(Y)$, to obtain $\model_{Y\mid X}(Y)$. To estimate $\models_{X}(X)$, we assume a uniform prior $L(X \mid \model_{X}) = -n\sum_{a_i \in X} \log res(a_i)$ and similarly for $\model_{Y}(Y)$. We can use these scores to calculate both the $\delta$ score and the $\Delta$ score. We will refer to [<span style="font-variant:small-caps;">Crack</span>]{}using the $\delta$ indicator as [$\textsc{Crack}_{\delta}$]{}, and [<span style="font-variant:small-caps;">Crack</span>]{}with the $\Delta$ indicator as [$\textsc{Crack}_{\Delta}$]{}.
Related Work {#sec:rel}
============
Causal inference on observational data is a challenging problem, and has recently attracted a lot of attention [@pearl:09:book; @janzing:10:algomarkov; @shimizu:06:anm; @budhathoki:16:origo]. Most existing proposals, however, are highly specific in the type of causal dependencies and type of variables they can consider.
Clasical constrained-based approaches, such as conditional independence tests, require three observed random variables [@spirtes:00:book; @pearl:09:book], cannot distinguish Markov equivalent causal DAGs [@verma:90:markov-equiv] and hence cannot decide between $\XtoY$ and $\YtoX$. Recent approaches use properties of the joint distribution to break the symmetry.
Additive Noise Models (ANMs) [@shimizu:06:anm], for example, assume that the effect is a function of the cause and cause-independent additive noise. ANMs exist for univariate real-valued [@shimizu:06:anm; @hoyer:09:nonlinear; @zhang:09:ipcm; @peters:14:continuousanm] and discrete data [@peters:10:discreteanm]. A related approach considers the asymmetry in the joint distribution of $\cause$ and $\effect$ for causal inference. The linear trace method () [@janzing:10:ltr] and the kernelized trace method () [@chen:13:ktr] aim to find a structure matrix $A$ and the covariance matrix $\Sigma_X$ to express $Y$ as $AX$. Both methods are restricted to multivariate continuous data. Sgouritsa et al. [@sgouritsa:15:cure] show that the marginal distribution $P(\cause)$ of the cause is independent of the conditional distribution $P(\effect \mid \cause)$ of the effect. They proposed , using unsupervised reverse regression on univariate continuous pairs. Liu et al [@liu:16:dc] use distance correlation to identify the weakest dependency between univariate pairs of discrete data.
The algorithmic information-theoretic approach views causality in terms of Kolmogorov complexity. The key idea is that if $X$ causes $Y$, the shortest description of the joint distribution $P(X, Y)$ is given by the separate descriptions of the distributions $P(X)$ and $P(Y \mid X)$ [@janzing:10:algomarkov], and justifies additive noise model based causal inference [@janzing:10:justifyanm]. However, as Kolmogorov complexity is not computable [@vitanyi:93:book], causal inference using algorithmic information theory requires practical implementations, or notions of independence. For instance, the information-geometric approach [@janzing:12:igci] defines independence via orthogonality in information space for univariate continuous pairs. Vreeken [@vreeken:15:ergo] instantiates it with the cumulative entropy to infer the causal direction in continuous univariate and multivariate data. Mooij instantiates the first practical compression-based approach [@mooij:10:mml] using the Minimum Message Length. Budhathoki and Vreeken approximate $K(X)$ and $K(Y \mid X)$ through MDL, and propose , a decision tree based approach for causal inference on multivariate binary data [@budhathoki:16:origo]. Marx and Vreeken[@marx:17:slope] proposed , an MDL based method employing local and global regression for univariate numeric data.
In contrast to all methods above, [<span style="font-variant:small-caps;">Crack</span>]{}can consider pairs of any cardinality, univariate or multivariate, and of same, different, or even mixed-type data.
Experiments {#sec:exps}
===========
In this section, we evaluate [<span style="font-variant:small-caps;">Crack</span>]{}empirically. We implemented [<span style="font-variant:small-caps;">Crack</span>]{}in C++, and provide the source code including the synthetic data generator along with the tested datasets for research purposes.[^2] The experiments concerning [<span style="font-variant:small-caps;">Crack</span>]{}were executed single-threaded. All tested data sets could be processed within seconds; over all pairs the longest runtime for [<span style="font-variant:small-caps;">Crack</span>]{}was $3.8$ seconds.
We compare [<span style="font-variant:small-caps;">Crack</span>]{}to [@sgouritsa:15:cure], [@janzing:12:igci], [@janzing:10:ltr], [@budhathoki:16:origo], [@vreeken:15:ergo] and [@marx:17:slope] using their publicly available implementations and recommended parameter settings.
Synthetic data
--------------
The aim of our experiments on synthetic data is to show the advantages of either score. In particular, we expect [$\textsc{Crack}_{\Delta}$]{}to perform well on nominal data and numeric data with balanced domain sizes and dimensions. On the other hand, [$\textsc{Crack}_{\delta}$]{}should have an advantage when it comes to numeric data with varying domain sizes and mixed-type data.
We generate synthetic data with assumed ground truth $\XtoY$ with $|X| = k$ and $|Y| = l$, each having $n=5 \, 000$ rows, in the following way. First, we randomly assign the type for each attribute in $X$. For nominal data, we randomly draw the number of classes between two (binary) and five and distribute the classes uniformly. Numeric data is generated following a normal distribution taken to the power of $q$ by keeping the sign, leading to a sub-Gaussian ($q < 1.0$) or super-Gaussian ($q > 1.0$) distribution.[^3]
To create data with the true causal direction $\XtoY$, we introduce dependencies from $X$ to $Y$, where we distinguish between splits and refinements. We call the probability threshold to create a dependency $\varphi \in [ 0, 1 ]$. For each $j \in \{ 1, \dots, l \}$, we throw a biased coin based on $\varphi$ for each $X_i \in X$ that determines if we model a dependency from $X_i$ to $Y_j$. A split means that we find a category (nominal) or a split-point (numeric) on $X_i$ to split $Y_j$ into two groups, for which we model its distribution independently. As refinement, we either do a multiway split or model $Y_j$ as a linear or quadratic function of $X_i$ plus independent Gaussian noise.
#### Accuracy
First, we compare the accuracies of [$\textsc{Crack}_{\delta}$]{}and [$\textsc{Crack}_{\Delta}$]{}with regard to single-type and mixed-type data. To do so, we generate $200$ synthetic data sets with $|X| = |Y| = 3$ for each dependency level where $\varphi \in \{0.0,0.1,\dots 1.0 \}$. Figure \[fig:dependency\] shows the results for numeric, nominal and mixed-type data. At $\varphi = 1.0$ both approaches reach nearly $100\%$ accuracy on single-type data. For single-type data, the accuracy of both methods increases with the dependency. At $\varphi = 0$, both approaches correctly do not decide instead of taking wrong decisions. As expected [$\textsc{Crack}_{\delta}$]{}strongly outperforms [$\textsc{Crack}_{\Delta}$]{}on mixed-type data, reaching near $100\%$ accuracy, whereas [$\textsc{Crack}_{\Delta}$]{}reaches only $72\%$. On nominal data, [$\textsc{Crack}_{\Delta}$]{}picks up the correct signal faster than [$\textsc{Crack}_{\delta}$]{}.
![Accuracy for $\Delta$ and $\delta$ on nominal, numeric and mixed-type data based on the dependency.[]{data-label="fig:dependency"}](dependency_nom.pdf "fig:") \[fig:dep:nom\]
![Accuracy for $\Delta$ and $\delta$ on nominal, numeric and mixed-type data based on the dependency.[]{data-label="fig:dependency"}](dependency_num.pdf "fig:") \[fig:dep:num\]
![Accuracy for $\Delta$ and $\delta$ on nominal, numeric and mixed-type data based on the dependency.[]{data-label="fig:dependency"}](dependency_mixed.pdf "fig:") \[fig:dep:mixed\]
#### Dimensionality
Next, we check how sensitive both scores are to dimensionality, whereas we discriminate between asymmetric $k \neq l$ and symmetric $k=l$. We evaluated $200$ data sets per dimensionality. For the symmetric case, both methods are near to $100\%$ on single-type data, whereas only [$\textsc{Crack}_{\delta}$]{}also reaches this target on mixed-type data, as can be seen in the appendix.We now discuss the more interesting case for asymmetric pairs in detail.
To test asymmetric pairs, we keep the dimension of one variable at three, $k = 3$, while we increase the dimension of the second variable $l$ from one to eleven. To avoid bias, we assigned the dimension $k$ to $X$ and $l$ to $Y$ and swap the dimensions in every other test. We show the results in Figure \[fig:asymmetric\]. As expected, we observe that [$\textsc{Crack}_{\delta}$]{}has much fewer difficulties with the asymmetric data sets than [$\textsc{Crack}_{\Delta}$]{}. From $l = 3$ onwards, [$\textsc{Crack}_{\delta}$]{}is close to $100\%$. On nominal data, [$\textsc{Crack}_{\Delta}$]{}performs near perfect and also has the clear advantage for $l=1$.
![Accuracy of $\Delta$ (left) and $\delta$ (right) on asymmetric dimensions $k\in \{1,3,5,7,11\}$ and $3$ for nominal, numeric and mixed-type data.[]{data-label="fig:asymmetric"}](asymmetric_origo.pdf "fig:") ![Accuracy of $\Delta$ (left) and $\delta$ (right) on asymmetric dimensions $k\in \{1,3,5,7,11\}$ and $3$ for nominal, numeric and mixed-type data.[]{data-label="fig:asymmetric"}](asymmetric_nci.pdf "fig:")
Real world data
---------------
Based on the evaluation on synthetic data, we test our approach on univariate benchmark data and multivariate data consisting of known test sets and new causal pairs with known ground truth that we present in the current paper.
#### Univariate benchmark
To evaluate [<span style="font-variant:small-caps;">Crack</span>]{}on univariate data, we apply it to the well-known Tuebingen benchmark data set that consists of $100$ univariate pairs.[^4] The pairs mainly consist of numeric data and a few categoric instances. Therefore, we apply [$\textsc{Crack}_{\Delta}$]{}. We compare to the state of the art methods that are applicable to multivariate and univariate data, [@budhathoki:16:origo] and [@vreeken:15:ergo], and methods specialized for univariate pairs, [@sgouritsa:15:cure], [@janzing:12:igci] and [@marx:17:slope]. For each approach, we sort the results by their confidence. According to this order, we calculate for each position $k$ the percentage of correct inferences up to this point, called the decision rate. We weigh the decisions as specified by the benchmark, plot the results in Fig. \[fig:decision\_rate\] and show the $95\%$ confidence interval of a fair coin flip as a grey area. Except to [<span style="font-variant:small-caps;">Crack</span>]{}and , all methods are insignificant w.r.t. the fair coin flip. In particular, [<span style="font-variant:small-caps;">Crack</span>]{}has an accuracy of over $90\%$ for the first $41\%$ of its decisions and reaches $77.2\%$ overall. Regarding the whole decision rate, [<span style="font-variant:small-caps;">Crack</span>]{}is nearly on par with , which is as far as we know, the current state of the art for univariate continuous data.
![\[Higher is better\] Decision rates of [<span style="font-variant:small-caps;">Crack</span>]{}, , , , and on univariate Tuebingen pairs (100) weighted as defined. Approaches that are only applicable to univariate data are drawn with dotted lines.[]{data-label="fig:decision_rate"}](decision_rate_weighted.pdf)
#### Multivariate data
To test [$\textsc{Crack}_{\delta}$]{}on multivariate mixed-type and single-type data, we collected 17 data sets. The information of the dimensionality for each data set is listed in Table \[tab:mv\_comparison\]. The first six data sets belong to the Tuebingen benchmark data set [@mooij:16:pairs] and the next four were published by Janzing et al. [@janzing:10:ltr]. Further, we extracted cause-effect pairs form the *Haberman* [@haberman:76:dataset], *Iris* [@fisher:36:dataset], *Mammals* [@heikinheimo:07:mammals] and *Octet* [@ghiringhelli:15:octet; @vechten:69:quantum] data sets. Those are described in more detail in the appendix.
We compare [$\textsc{Crack}_{\delta}$]{}with , and . and do not consider categoric data, and are hence not applicable on all data sets. In addition, is only applicable to strictly multivariate data sets. [$\textsc{Crack}_{\delta}$]{}is applicable to all data sets, infers $15/17$ causal directions correctly, by which it has an overall accuracy of $88.2\%$. Importantly, the two wrong decisions have low confidences compared to the correct inferences.
------------- ----------- ----- -------- -------- -------- -------- ---------------------------------------------------------
**Dataset** $m$ $k$ $l$ [<span style="font-variant:small-caps;">Crack</span>]{}
Climate $10\,226$ 4 4 – –
Ozone $989$ 1 3
Car $392$ 3 2 –
Radiation $72$ 16 16 – – –
Symptoms $120$ 6 2 –
Brightness $1\,000$ 9 1 –
Chemnitz $1\,440$ 3 7
Precip. $4\,748$ 3 12 – –
Stock 7 $2\,394$ 4 3 – –
Stock 9 $2\,394$ 4 5 – –
Haberman $306$ 3 1 – –
Iris flower $150$ 4 1 –
Canis $2\,183$ 4 2
Lepus $2\,183$ 4 3
Martes $2\,183$ 4 2
Mammals $2\,183$ 4 7
Octet $82$ 1 10
$0.56$ $0.82$ $0.47$ $0.88$
------------- ----------- ----- -------- -------- -------- -------- ---------------------------------------------------------
: Comparison of , , and [<span style="font-variant:small-caps;">Crack</span>]{}on eleven multivariate data sets. We write whenever a method is not applicable on the pair.[]{data-label="tab:mv_comparison"}
Conclusion {#sec:conc}
==========
We considered the problem of inferring the causal direction from the joint distribution of two univariate or multivariate random variables $X$ and $Y$ consisting of single-, or mixed-type data. We point out weaknesses of known causal indicators and propose the Normalized Causal Indicator for mixed-type data and data with highly unbalanced domains. Further, we propose a practical encoding based on classification and regression trees to instantiate these causal indicators and provide a fast greedy heuristic to compute good solutions.
In the experiments we evaluate the advantages of the NCI and the common indicator and give advice on when to use them. On real world benchmark data, we are on par with the state of the art for univariate continuous data and beat the state of the art on multivariate data with a wide margin.
For future work, we aim to investigate in the application of [<span style="font-variant:small-caps;">Crack</span>]{}for causal discovery, meaning that we would like to infer causal networks. In addition, we only selected a subset of possible refinements to exploit dependencies from candidates. This choice could be expanded by considering more complex functions, finding combinations of categories for splitting. However, unless specific care is taken many of such extensions will likely have repercussions on the runtime of our algorithm, which is why besides being out of scope here, we leave this for future work.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors wish to thank Kailash Budhathoki for insightful discussions. Alexander Marx is supported by the International Max Planck Research School for Computer Science (IMPRS-CS). Both authors are supported by the Cluster of Excellence “Multimodal Computing and Interaction” within the Excellence Initiative of the German Federal Government.
[^1]:
------------------------------------------------------------------------
Max Planck Institute for Informatics and Saarland University, Saarbrücken, Germany. `{amarx,jilles}@mpi-inf.mpg.de`
[^2]: [[<http://eda.mmci.uni-saarland.de/crack/>]{}]{}
[^3]: We use super- and sub-Gaussians to ensure identifiability.
[^4]: https://webdav.tuebingen.mpg.de/cause-effect/
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The out-of-equilibrium electron transport of carbon nanotube semiconducting quantum dot placed in a magnetic field is studied in the Kondo regime by means of the non-equilibrium Green functions. The equation of motion method is used. For parallel magnetic field the Kondo peak splits in four peaks, following the simultaneous splitting of the orbital and spin states. For perpendicular field orientation the triple peak structure of density of states is observed with the central peak corresponding to orbital Kondo effect and the satellites reflecting the spin and spin-orbital fluctuations.'
author:
- 'D. Krychowski'
- 'S. Lipinski'
title: 'Kondo effect in carbon nanotube quantum dot in a magnetic field\'
---
Introduction
============
Carbon nanotubes (CNTs) have emerged as a viable electronic material for molecular electronic devices because they display a large variety of behavior depending on their intrinsic properties and on the characteristics of their electrical contacts \[1\]. These systems also form the powerful tool for the study of fundamental many-body phenomena. An example is the observed Kondo effect in semiconducting carbon nanotube quantum dots (CNTQD) \[2,3\]. The long spin lifetimes, the relatively high Kondo temperature and the fact that this effect can be seen over a very wide range of gate voltage encompassing hundreds of Coulomb oscillations \[4\] make CNTQDs interesting candidates for spintronic applications.
The purpose of the present work is to discuss magnetic field dependence of the Kondo conductance of CNTQD. Perpendicular field couples only to spin and parallel field influences both spin and orbital magnetic moments. For vanishing magnetic field and orbitally degenerate states the Kondo effect appears simultaneously in spin and orbital sectors resulting in SU(4) Fermi liquid ground state with totally entangled spin and orbital degrees of freedom \[5\]. Magnetic field breaks the spin-orbital symmetry and in accordance to the experiment \[2,3\] our calculations show the occurrence of the multi-peak structure of the differential conductance reflecting the spin, orbital and spin-orbital fluctuations.
Model
=====
The low energy band structure of semiconducting carbon nanotubes is orbitally doubly degenerate at zero magnetic field. This degeneracy has been interpreted in a semiclassical fashion as the degeneracy between clockwise and counterclockwise propagating electrons along the nanotube circumference \[1\]. In the present considerations we restrict to the single shell and the dot is modeled by double orbital Anderson Hamiltonian with additional interorbital interaction:
$$\begin{aligned}
\lefteqn{{\mathcal{H}} = \sum_{k \alpha m \sigma}\epsilon_{k \alpha m \sigma} c^{+}_{k \alpha m \sigma}c_{k \alpha m \sigma}}
\nonumber\\
&&+\sum_{k \alpha m \sigma}t_{\alpha}(c^{+}_{k \alpha m \sigma}d_{m\sigma}+c.c.)\nonumber\\
&& +\sum_{k \alpha m
\sigma}\epsilon_{m \sigma}d^{+}_{m \sigma}d_{m \sigma}+\sum_{m}U n_{m+}n_{m-}
\nonumber\\
&&+\sum_{\sigma \sigma'}U_{12}
n_{1\sigma}n_{-1\sigma'}\end{aligned}$$
where $m = \pm1$ numbers the orbitals, the leads channels are labeled by $(m,\alpha)$, $\alpha =L,R$. $\epsilon_{m\sigma}=\epsilon_{0}+\mu_{orb}m h cos(\Theta)+g\sigma
\mu_{B} h$, $\Theta$ specifies the orientation of magnetic field $h$ relative to the nanotube axis, $\mu_{orb}$ is the orbital moment. The first term of (1) describes electrons in the electrodes, the second describes tunneling to the leads, the third represents the dot and the last two terms account for intra ($U$) and interorbital ($U_{12}$) Coulomb interactions. Current flowing through CNTQD can be expressed in terms of the Green functions \[6\]:
$$\begin{aligned}
I_{\alpha} = \frac{\imath e}{2 \hbar}\int_{-\infty}^{+\infty}\frac{d \omega}{2 \pi}
\sum_{m \sigma}\Gamma_{\alpha m \sigma}(\omega)\cdot G^{<}_{m
\sigma}(\omega)+\nonumber
\\+\Gamma_{\alpha m \sigma}(\omega)\cdot f_{\alpha}(\omega)\cdot\ [G^{+}_{m \sigma}(\omega)-G^{-}_{m \sigma}(\omega)]\end{aligned}$$
where $G^{<}$,$G^{+}$ and $G^{-}$ are lesser, retarder and advanced Green functions, respectively, $f_{\alpha}$ is the Fermi function of $\alpha$ lead and tunneling rate $\Gamma_{\alpha
m \sigma}=2\pi|t_{\alpha}|^{2}\varrho_{\alpha m \sigma}$, where $\varrho_{\alpha m \sigma}$ is the density of states of the leads. The total current is given by $I=(I_{L}-I_{R})/2$. The lesser Green function $G^{<}$ is found using Ng ansatz \[7\], according to which the lesser self-energy $\Sigma^{<}$ is proportional to the self-energy of the corresponding noninteracting system $\Sigma^{<}(\omega)=A\cdot \Sigma^{<}_{0}(\omega)$, and A can be found by the Keldysh requirement $\Sigma^{<}-\Sigma^{>}=\Sigma^{+}-\Sigma^{-}$. The Green functions are found by the equation of motion method using the self-consistent decoupling procedure proposed by Lacroix \[8\].\
Results and discussion
======================
The first point of our numerical analysis is addressed to the experiment of Jarillo-Herrero et al. \[2\], in which the conductance of CNTQD for the almost parallel field orientation was examined ($\Theta \simeq 21^{\circ}$). The calculations were performed with Coulomb interaction parameters $U = U_{12} = 40
meV$, inferred from the size of Coulomb diamonds. The addition energy spectrum indicates that the level spacing of examined CNTQDs $\Delta\epsilon \simeq 4.3 meV$ \[9\], what corresponds to the length of NCT $L \sim 400nm$. The estimated Kondo temperature is $T_{K} \sim7.7 K$ \[2\]. Our discussion is based on the single shell model (1) with the level placed in the centre of Coulomb valley ($\epsilon_{0} = -20meV$). Such an oversimplified approach, which gives only a first crude insight is justified since $\Delta
\epsilon/k_{B}T_{K} \sim6.5$ is large and the higher levels do not play an important role \[10\]. To get the experimental value of the Kondo temperature one has to assume a value of coupling to the leads $\Gamma = 3.2meV$, which is slightly higher than the observed broadening of atomic or Coulomb lines for NCTs examined by Jarillo-Herrero et al.\[2,9\]. The fact that the single level description of the multilevel systems underestimates Kondo temperature is well known in literature \[10,11\]. Orbital moment is estimated from the average slope between the two Coulomb peaks that correspond to the addition of the electrons to the same orbital state and reads $\mu_{orb} \sim 13 \mu_{B}$ \[2\]. We focus on the regime, where the quantum dot is occupied by a single electron. Fig.1a presents the calculated gray-scale plot of conductance versus magnetic field and bias voltage for $T = 0.34 K$ compared with the corresponding experimental plot (inset). The central bright spot of dimension determined by $T_{K}$ is the region of spin-orbital Kondo effect. For vanishing bias and magnetic field the Kondo effect appears simultaneously in spin and orbital sectors resulting in a SU(4) Fermi liquid ground state.
![Calculated differential conductance $dI/dV$ of CNTQD versus bias voltage $V$ and magnetic field $h$ in the centre of Coulomb valley for $T = 0.34 K$. The parameters used are: $U = U_{12} = 40 meV$, $ \Gamma= 3.2 meV$, $\epsilon_{0} = -20 meV$ and $\mu_{orb} = 13 \mu_{B}$. The angle between the nanotube axis and the field $\Theta = 21^{\circ}$. Colorscale: $0.1$ to $1.5 e^{2}/h$. Inset shows the corresponding (V,h) conductance map obtained from the data of Jarillo- Herrero et al.\[2\].[]{data-label="fig1"}](fig2.eps){width="0.7\columnwidth"}
The conductance reaches in the centre a value $G = 1.3 \times
e^{2}/h$. Magnetic field breaks the degeneracy and four high intensity lines appear. A pair of inner lines observed for small bias corresponds to orbital conserving fluctuations and the outer lines reflect the orbital fluctuations and simultaneous spin and orbital fluctuations. The latter two processes are not resolved for the assumed values of $\Gamma$ and temperature.
![Calculated linear conductance $G = dI/dV|_{V\rightarrow0}$ versus gate voltage at $T = 0.34 K$ and 8 K for the CNTQD specified by parameters as in Fig. 1a. Inset shows the corresponding curves obtained from the data reported in \[2\].[]{data-label="fig2"}](fig1.eps){width="0.7\columnwidth"}
Fig. 1b presents the linear conductance versus gate voltage at $T = 8K$ and $0.34 K$. $\Delta V_{G} =0$ corresponds to the centre of Coulomb valley. Our calculations reasonably well reproduce the shape of the dependence but underestimate all the values of conductance roughly by a constant value $0.5 \times e^{2}/h$ . The source of this discrepancy is not clear, but we suggest that a possible explanation is a neglect of higher orbital levels in our description. Apart from the earlier mentioned renormalization of Kondo temperature they also can cause a formation of shoulders in the density of states above the Fermi edge on the scale much larger than the Kondo temperature \[10\] and this might lead to additional weakly bias dependent contribution to the conductance. A detailed discussion of the mentioned point will be given in the following paper \[12\].
![Differential conductance of CNTQD in perpendicular magnetic field for $T = 1.2 K$ calculated for the centre of Coulomb valley. The parameters used are: $U = U_{12} =15meV$, $\Gamma = 1.25 meV$, $\epsilon_{0} = -8 meV$ with asymmetry of the leads $\Gamma_{L}/\Gamma_{R}=6$. Inset shows the corresponding curves for the same values of magnetic fields obtained from the data of Makarowski et al. \[3\]. []{data-label="fig3"}](fig3.eps){width="0.8\columnwidth"}
Now let us turn to the discussion of the influence of perpendicular field, which breaks only the spin degeneracy. Our numerical analysis describes the results of Makarovski et al. \[3\] for 600nm-long nanotube quantum dot ($\Delta \epsilon \sim 3 meV$). The Coulomb parameters estimated again from the size of Coulomb diamonds are taken as $U = U_{12}
= 15 meV$ and the orbital level is placed in the centre of Coulomb valley $\epsilon_{0} = -8 meV$. $\Gamma$ is taken as the width of orbital or Coulomb peaks. $\Gamma = 1.25 meV$ , together with the above parameters, well reproduces the experimental value of $T_{K}\sim 13 K$. The calculated differential conductance for several values of magnetic field is compared with experiment on Fig.2. A quasi SU(4) type behavior is still observed in the low field range, what reflects in a moderate change of conductance and a single peak structure. For higher magnetic fields the spin-orbital Kondo effect SU(4) is transformed to SU(2) orbital Kondo effects for each spin orientations seperately. This results in the occurrence of the central peak. For bias voltage $V =\pm2(\mu_{B}/e)h$ there occur also the satellites induced by tunneling processes which mix different spin channels.
Summarizing, the present paper provides a simple picture of the influence of magnetic field on the conductance of carbon nanotube QDs in the Kondo regime. Although the experiments under consideration concern multilevel dots our calculations show that the essence of the transport properties can be inferred from the effective single shell spin-orbital Kondo physics.
[12]{}
M.S. Dresselhaus, G. Dresselhaus, Ph. Avouris, Carbon Nanotubes, Springer-Verlag, Berlin (2000). P. Jarillo-Herrero, J. Kong, H.S.J van der Zant, C. Dekker, L.P. Kouwenhoven, S. De Franceschi, Nature 434, 484 (2005). A. Makarovski, A. Zhukov, J. Liu, G. Finkelstein, Phys. Rev. B 75, 241407 (2007).
J. Nygard, D.H. Cobden, P.E. Lindelof, Nature 408,342 (2000). M. Choi, R. Lopez, R. Aquado, Phys. Rev. Lett. 95, 0672041 ( 2005). H. Haug, A.P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, Springer-Verlag, Berlin, Heidelberg, New York (1998) T.K. Ng, Phys. Rev. Lett. 76, 487 (1996). C. Lacroix, J. Phys. F 11, 2389 (1998). P. Jarillo-Herrero, S. Sapmaz, C.Dekker, L.P. Kouvenhoven, H.S.J. van der Zant, Nature 429, 389 (2004).
D. Boese, W. Hofstetter, H. Schoeller, Phys. Rev. B 66, 125315 (2002). K. Yamada, K. Yosida, K. Hanzawa, Progr. Theor. Phys. 71, 450 (1984). S. Lipinski, D. Krychowski: (to be published)
| {
"pile_set_name": "ArXiv"
} |
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ł
Preprint Padova, DFPD 97/TH\
December 1997\
Introduction
============
Chiral bosons are described by $p$–form gauge potentials $B_p$ whose curvatures $H_{p+1}=dB_p$ satisfy, as equation of motion, a Hodge (anti)self–duality condition in a space–time with dimension $D=2(p+1)$. In space–times with Minkowskian signature $\eta_{ab}=(1,-1,\cdots,-1)$ the self–consistency of such an equation restricts $p$ to even values and hence the relevant dimensions are $D=2,6,10,\ldots$
Such fields populated superstring and supergravity theories, and more recently M theory, from their very beginning. Two dimensional chiral bosons (scalars) are basic ingredients in string theory, six–dimensional ones belong to the supergravity– and tensor–multiplets in $N=1$, $D=6$ supergravity theories and are necessary to complete the $N=2$, $D=6$ supermultiplet of the M-theory five–brane; finally a ten–dimensional chiral boson appears in $IIB$, $D=10$ supergravity.
A peculiar feature of the (manifestly Lorentz covariant) self–duality equation of motion of those fields is that a manifestly Lorentz invariant lagrangian formulation for them was missing for long time. The absence of a Lorentz invariant action from which one can derive the equations of motion leads in principle to rather problematic situations e.g. the conservation of the energy–momentum tensor is not guaranteed a priori and the coupling to gravity can not be performed via the usual minimal coupling.
For previous attempts in facing this problem and for a more detailed discussion of the problematic aspects involved, see in particular [@probl].
Recently a new manifestly Lorentz–invariant lagrangian approach for chiral bosons has been proposed in [@PST; @PST3]. The most appealing features of this approach are the introduction of [*only one*]{} single scalar auxiliary field, its natural compatibility with all relevant symmetries, in particular with diffeomorphisms and with $\kappa$–invariance [@M5], and its general validity in all dimensions $D=2(p+1)$ with $p$ even. Another characteristic feature of this approach is the appearance of two new local bosonic symmetries: one of them reduces the scalar auxiliary field to a non propagating “pure gauge” field and the other one reduces the second order equation of motion for the $p$–form to the first order (anti)self–duality equation of motion.
A variant of this approach allowed to write manifestly duality invariant actions for Maxwell fields in four dimensions [@PSTDUAL] and to construct a covariant effective action for the M theory five–brane [@M5]. On the other hand, the actions obtained through the non manifestly covariant approach developed in [@schw] can be regarded as gauge fixed versions of the actions in [@M5; @PSTDUAL] where the scalar auxiliary field has been eliminated.
The coupling of all these models with chiral bosons to gravity can be easily achieved since the new approach is manifestly covariant under Lorentz transformations; as a consequence it is obvious that the two above mentioned bosonic symmetries, which are a crucial ingredient of the new approach, are compatible with diffeomorphism invariance. To test the general validity of the approach, it remains to establish its compatibility with global and local supersymmetry. This is the aim of the present talk.
In the next section we review the covariant method, for definiteness, for chiral two–forms in six dimensions. In section three we test its compatibility with supersymmetry by writing a covariant action for the most simple cases, i.e. the rigid tensor supermultiplet and the free supergravity multiplet in six dimensions. Section four is devoted to some concluding remarks and to a brief discussion of the general case i.e. the supergravity multiplet coupled to an arbitrary number of tensor multiplets and super Yang–Mills multiplets.
The general strategy developed in this paper extends in a rather straightforward way to two and ten dimensions. Particularly interesting is the case of $IIB$, $D = 10$ supergravity whose covariant action we hope to present elsewhere. The bosonic part of this action has already been presented in [@IIB].
For more details on the results presented here and for more detailed references, see [@DLT].
Chiral bosons in six dimensions: the general method
===================================================
In this section we present the method for a chiral boson in interaction with an external or dynamical gravitational field in six dimensions. To this order we introduce sechsbein one–forms $e^a = d x^m {e_m}^a(x)$. With $m,n =0,\ldots,5$ we indicate curved indices and with $a,b=0,\ldots,5$ we indicate tangent space indices, which are raised and lowered with the flat metric $\eta_{ab}=(1,-1,\cdots,-1)$.
To consider a slightly more general self-duality condition for interacting chiral bosons we introduce the two-form potential $B$ and its generalized curvature three–form $H$ as $$H=dB+C\equiv {1\over 3!}e^a e^b e^c H_{cba},
\label{forms}$$ where $C$ is a three-form which depends on the fields to which $B$ is coupled, such as the graviton, the gravitino and so on, but not on $B$ itself. The free (anti)self–dual boson will be recovered for $C=0$ and $e_m{}^a=\delta_m{}^a$.
The Hodge–dual of the three–form $H$ is again a three–form $H^*$ with components $H^*_{abc} = \frac{1}{3!} \e_{abcdef} H^{def}.$ The self–dual and anti self–dual parts of $H$ are defined respectively as the three–forms $H^{\pm} \equiv \frac{1}{2} (H \pm H^*)$. The equations of motion for interacting chiral bosons in supersymmetric and supergravity theories, as we will see in the examples worked out in the next section, are in general of the form $H^{\pm}=0,$ for a suitable three–form $C$ whose explicit expression is determined by the model.
To write a covariant action which eventually gives rise to the equation $H^{\pm}=0$ we introduce as new ingredient the scalar auxiliary field $a(x)$ and the one–form v=[1]{} dae\^b v\_b. In particular we have $v_b={\partial_b a\over
\sqrt{-\partial_c a \partial^c a}}$ and $v_bv^b=-1$. Using the vector $v^b$, to the three–forms $H,H^*$ and $H^\pm$ we can then associate two-forms $h,h^*$ and $h^\pm$ according to $$h_{ab}=v^cH_{abc}, \qquad h={1\over 2} e^a e^b h_{ba},$$ and similarly for $h^*$ and $h^\pm$.
The action we search for can now be written equivalently in one of the following two ways \[S0\] S\_0\^= (v h\^ H + [12]{} dB C) = d\^6x([124]{}H\_[abc]{}H\^[abc]{} +[12]{}h\_[ab]{}\^h\^[ab]{}) dBC. $S_0^+$ will describe anti self–dual bosons ($H^+=0$) and $S_0^-$ self–dual bosons ($H^-=0$). The last term, $\int dBC$, is of the Wess–Zumino type and is absent for free chiral bosons.
What selects this form of the action are essentially the local symmetries it possesses. Under a general variation of the fields $B$ and $a$ it varies, in fact, as \[dS0\] S\_0\^= 2(vh\^dB + [v]{} h\^h\^ da). From this formula it is rather easy to see that $\delta S^\pm_0$ vanishes for the following three bosonic transformations, with transformation parameters $\Lambda$ and $\psi$, which are one–forms, and $\varphi$ which is a scalar: \[bos\] &I)&B=d,a =0\
&II)&B= -[2h\^]{} ,a =\
&III)&B=da ,a =0.
For what concerns $I)$ and $III)$ invariance of the action is actually achieved also for finite transformations. This fact will be of some importance below.
The transformation $I)$ represents just the ordinary gauge invariance for abelian two–form gauge potentials. The symmetry $II)$ implies that $a(x)$ is an auxiliary field which does, therefore, not correspond to a propagating degree o freedom[^1]. Finally, the symmetry $III)$ eliminates half of the propagating degrees of freedom carried by $B$ and allows to reduce the second order equation of motion for this field to the desired first order equation, i.e. $H^{\pm}=0$. To see this we note that the equations of motion for $B$ and $a$, which can be read from [(\[dS0\])]{}, are given respectively by d(vh\^)=0\[emb\],d([v]{}h\^h\^)=0. First of all it is straightforward to check that the $a$–equation is implied by the $B$-equation, as expected, while the general solution of the $B$–equation is given by $vh^\pm={1\over 2}d\tilde{\psi}da$, for some one–form $\tilde{\psi}$. On the other hand, under a (finite) transformation $III)$ we have $\delta\left(vh^\pm\right)={1\over 2}d\psi
da$ and therefore, choosing $\psi=-\tilde\psi$, we can use this symmetry to reduce the $B$-equation to $vh^\pm=0$. But $vh^\pm=0$ amounts to $h^\pm=0$, and this equation, in turn, can easily be seen to be equivalent to $H^\pm=0$, the desired chirality condition.
This concludes the proof that the actions $S_0^\pm$ describe indeed correctly the propagation of chiral bosons.
In a theory in which the $B$ field is coupled to other dynamical fields, for example in supergravity theories, we can now conclude that the complete action has to be of the form $$S=S_0^\pm+S_6,$$ where $S_6$ contains the kinetic and interaction terms for the fields to which $B$ is coupled. To maintain the symmetries $I)$–$III)$ one has to assume that those fields are invariant under these transformations and, moreover, that $S_6$ is independent of $B$ and $a$.
For more general chirality conditions describing self–interacting chiral bosons, like e.g. those of the Born–Infeld type, see ref. [@PST3].
To conclude this section we introduce two three–form fields, $K^\pm$, which will play a central role in the next section due to their remarkable properties. They are defined as \[k\] K\^H+2vh\^and are uniquely determined by the following peculiar properties: i) they are (anti) self–dual: $K^{\pm*} = \pm K^{\pm}$; ii) they reduce to $H^\pm$ respectively if $H^\mp= 0$; iii) they are invariant under the symmetries $I)$ and $III)$, and under $II)$ modulo the field equations [(\[emb\])]{}. These fields constitute therefore a kind of off–shell generalizations of $H^\pm$.
$N=1$, $D=6$ supersymmetric chiral bosons
=========================================
In this section we illustrate the compatibility of the general method for chiral bosons with supersymmetry in the six–dimensional case by means of two examples: the first concerns the free tensor supermultiplet in flat space–time and the second concerns pure supergravity. The strategy developed in these examples admits natural extensions to more general cases [@M5; @IIB; @DLT].
0.5truecm[*1) Free tensor multiplet.*]{} 0.5truecm
An $N=1,D=6$ tensor multiplet is made out of an antisymmetric tensor $B_{[ab]}$, a symplectic Majorana–Weyl spinor $\l_{\a i}$ ($\a = 1,\ldots,4; i = 1,2$) and a real scalar $\f$. The equations of motion for this multiplet and its on–shell susy transformation rules are well known. The scalar obeys the free Klein–Gordon equation, the spinor the free Dirac equation and the $B$–field the self–duality condition $H^-=0,$ where $H=dB$, which means that in this case we have $C=0$.
The on-shell supersymmetry transformations, with rigid transformation parameter $\xi^{\a i}$, are given by \[susy\] \_&=& \^i ł\_i,\
\_ł\_[ i]{} &=& ( \^a \_a + \^[abc]{}H\_[abc]{}\^+ )\_i,\
\_B\_[ab]{} &=& - \^i \_[ab]{} ł\_i. The $USp(1)$ indices $i,j$ are raised and lowered according to $ K_i = \e_{ij} K^j, K^i = - \e^{ij} K_j, $ where $\e_{12} =1$ and the $\Gamma^a$ are $4\times 4$ Weyl matrices.
Since the equations of motion are free our ansatz for the action, which depends now also on the auxiliary field $a$, is \[SH\] S=S\_0\^-+S\_6 =- v h\^- H +[12]{}d\^6x (ł\^i \^a \_a ł\_i + \_a \^a ). This action is invariant under the symmetries $I)$–$III)$ if we assume that $\f$ and $\l$ are invariant under these transformations.
For what concerns supersymmetry we choose first of all for $a$ the transformation $\delta_\xi a=0,$ which is motivated by the fact that $a$ is non propagating and does therefore not need a supersymmetric partner. Next we should find the off–shell generalizations of [(\[susy\])]{}. For dimensional reasons only $\delta_\xi\l$ allows for such an extension. To find it we compute the susy variation of $S_0^-$, which depends only on $B$ and $a$, as $$\delta_\xi S_0^-=-2\int vh^-d\delta_\xi B=-\int K^+d\delta_\xi B$$ in which the self-dual field $K^+$, defined in the previous section, appears automatically. Since $\delta_\xi S_0^-$ should be cancelled by $\delta S_6$ this suggests to define the off–shell susy transformation of $\l$ by making the simple replacement $H^+\ra K^+$, i.e. $$\delta_\xi \l_{ i}\ra \bar\delta_\xi \l_{ i} = \left( \G^a \partial_a \f +
\frac{1}{12} \G^{abc}K_{abc}^+ \right)\xi_i.$$ With this modification it is now a simple exercise to show that the action [(\[SH\])]{} is indeed invariant under supersymmetry. The relative coefficients of the terms in the action are actually fixed by supersymmetry.
The [*general rules*]{} for writing covariant actions for supersymmetric theories with chiral bosons, which emerge from this simple example, are the following. First one has to determine the on–shell susy transformations of the fields and their equations of motion, in particular one has to determine the form of the three-form $C$. For more complicated theories this can usually be done most conveniently using superspace techniques. The off–shell extensions of the susy transformation laws are obtained by substituting in the transformations of the fermions $H^\pm\ra K^\pm$. The action has then to be written as $S_0^\pm+S_6$ where the relative coefficients of the various terms in $S_6$ have to be determined by susy invariance. The field $a$, finally, is required to be supersymmetry invariant.
0.5truecm[*2) Pure supergravity.*]{} 0.5truecm
The supergravity multiplet in six dimensions contains the graviton, a gravitino and an antisymmetric tensor with anti–selfdual (generalized) field strength. The graviton is described by the vector–like vielbein $e^a = dx^m
{e_m}^a$, the gravitino by the spinor–like one–form $e^{\a i} =
dx^m {e_m}^{\a i}$ and the tensor by the two–form $B$.
The supersymmetry transformations of these fields and their equations of motion, obtained from the superspace approach [@DFR], are conveniently expressed in terms of a super–covariant differential, $D=d+\omega$, with respect to a super–covariant Lorentz connection one–form $\omega^{ab} = dx^m \omega_{m}{}^{ab}$. This connection is defined by $d e^a + e^{b}{\omega_b}^a = - e^i\G^a e_i,$ and results in the metric connection augmented by the standard gravitino bilinears.
Among the equations of motion we recall the generalized anti–selfduality condition for $B$. This reads $H^+=0,$ where now $$H=dB+ \left(e^i\G_a e_i\right) e^a,$$ meaning that in this case the three–form $C$ is non vanishing being given by $C=\left(e^i\G_a e_i\right) e^a.$
The on–shell supersymmetry transformations of $e^a$, $e^{\a i}$ and $B$ [@DFR], with local transformation parameter $\xi^{\a i} (x)$, are given by \[susyi\] \_e\^a &=& -2 \^i \^a e\_i, \_e\^[i]{} = D\^[i]{} - \^[i]{} e\^a (\^[bc]{})\_\^H\^-\_[abc]{}, \[28b\]\
\_B &=& -2 (\^i \_a e\_i) e\^a, \_a = 0.
According to our general rule, in the gravitino transformation we have to make the off–shell replacement $H^{-}\rightarrow K^{-}$ obtaining $$\delta_\xi e^{\a i}\rightarrow
\bar\delta_\xi e^{\a i}= D\xi^{\a i} - \frac{1}{8} \xi^{\b i} e^a
(\G^{bc})_\b{}^\a K^-_{abc}.$$
In the above relations we added the trivial transformation law for the auxiliary field $a$. As it stands, this trivial transformation law does not seem to preserve the susy algebra in that the commutator of two supersymmetries does not amount to a translation. On the other hand it is known that the supersymmetry algebra closes on the other symmetries of a theory; in the present case it is easily seen that the anticommutator of two susy transformations on the $a$ field closes on the bosonic transformations $II)$.
The covariant action for pure $N = 1$, $D = 6$ supergravity can now be written as $S=S_0^+ +\int L_6$, where \[azsu\] S\_0\^+&=& (v h\^+ H + [12]{} dB C)\
L\_6 &=& \_[a\_1 …a\_6 ]{} e\^[a\_1]{} e\^[a\_2]{} e\^[a\_3]{} e\^[a\_4]{} R\^[a\_5a\_6]{} - e\^[a\_1]{}e\^[a\_2]{}e\^[a\_3]{} (De\^i\_[a\_1 a\_2 a\_3]{}e\_i). For convenience we wrote the term $S_6$ as an integral of a six–form, $L_6$. This six–form contains just the Einstein term, relative to the super–covariantized spin connection $R^{ab}=d\omega^{ab} +\omega^a{}_c\omega^{cb}$, and the kinetic term for the gravitino. The relative coefficients are fixed by susy invariance. In this case $S_0^+$ contains also the couplings of $B$ to the gravitino and the graviton.
This action is invariant under the symmetries $I)$–$III)$ because $L_6$ is independent of the fields $B,a$ and we assume the graviton and the gravitino to be invariant under those transformations.
The evaluation of the supersymmetry variation of $S$ is now a merely technical point and can indeed be seen to vanish. In particular, as in the previous example, the susy variation of $S^+_0$ depends on the fields $B,a$ only through the combination $K^-$ and these contributions are cancelled by the gravitino variation, justifying again our rule for the modified susy transformation rules for the fermions.
Concluding remarks
==================
The covariant lagrangians presented in this talk for six–dimensional supersymmetric chiral bosons admit several extensions. The lagrangian for $n$ tensor multiplets coupled to the supergravity multiplet, which involves $n+1$ mixed (anti)self–duality conditions, has been worked out in [@DLT]. The introduction of hyper multiplets, on the other hand, does not lead to any new difficulty. The coupling to Yang–Mills fields in the presence of $n$ tensor multiplets requires some caution. In this case it turns out that an action, and therefore a consistent classical theory, can be constructed only if the $n+1$ two–forms can be arranged such that only one of them carries a Chern–Simons correction while the $n$ remaining ones have as invariant field strength $dB^{(n)}$.
In conclusion the covariant method illustrated in this talk appears compatible, at the classical level, with all relevant symmetries explored so far, in particular with supersymmetry.
Among the open problems at the quantum level one regards the existence of a covariant quantization procedure. A quantum consistency check of the covariant method for chiral bosons coupled to gravity, which has still to be performed, consists in a perturbative computation of the Lorentz anomaly and in a comparison with the result predicted by the index theorem.
#### Acknowledgements.
It is a pleasure for me to thank G. Dall’Agata and M. Tonin for their collaboration on the results presented in this talk. I am also grateful to I. Bandos, P. Pasti and D. Sorokin for their interest in this subject and for useful discussions. This work was supported by the European Commission TMR programme ERBFMPX-CT96-0045.
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[^1]: Notice however that, since the action becomes singular in the limit of a vanishing or constant $a(x)$, the gauge $d a(x) = 0$ is not allowed.
| {
"pile_set_name": "ArXiv"
} |
[**Some Comments on Wheeler De Witt Equation for Gravitational Collapse and the Problem of Time**]{}
by\
and [*F.C. Khanna$^{+}$[^1]*]{}
$^{\ddag}$Universidade Federal do Espírito Santo, UFES.\
Centro de Ciências Exatas\
Av. Fernando Ferrari s/n$^{\underline{0}}$\
Campus da Goiabeiras 29060-900 Vitória ES – Brazil.\
$^{+}$Theoretical Physics Institute, Dept. of Physics\
University of Alberta,\
Edmonton, AB T6G2J1, Canada\
and\
TRIUMF, 4004, Wesbrook Mall,\
V6T2A3, Vancouver, BC, Canada.
Abstract
We write the Hamiltonain for a gravitational spherically symmetric scalar field collapse with massive scalar field source, and we discuss the application of Wheeler De Witt equation as well as the appearence of time in this context. Using an Ansatz for Wheeler De Witt equation, solutions are discussed including the appearence of time evolution.
Introduction {#introduction .unnumbered}
============
#### {#section .unnumbered}
In this letter we discuss the problem of gravitational collapse of a star using the Wheeler-De Witt equation.
In accordance with [@dois] we assume a scalar field, $\phi$, with a mass term and we assume that the super hamiltonian has a constraint \[1-5\] such that $H\simeq 0$. Ordering of operators is assumed.
A particular ansatz for the functional is chosen to show qualitatively the appearance of the notion or concept of “time” after quantization.
As in the case of the hydrogen atom the discrete index is identified with an “internal time” just as in any relativistic field theory or general relativity but different from the usual quantum mechanics, where “time” appears as a Galilean time.
We apply the Wheeler-De Witt equation for a special collapse condition despite the fact that the question related to the Copenhagen interpretation for product of functional $\psi (\Lambda
,R,\phi )$ is not understood.
Let us begin by writing the super Hamiltonian for a gravitational spherically symmetric scalar field collapse with massive scalar field source such as [@dois]. $$H={\cal H}+\frac{1}{2}\ m^2R^2\Lambda \phi^2\ ,$$ where $$\begin{aligned}
{\cal H} &=& -R^{-1}P_RP_{\Lambda}+\frac{1}{2}\ R^{-2}\Lambda P^2_{\Lambda}+
\Lambda^{-1}RR''-\Lambda^{-2}RR'\Lambda '+\frac{1}{2}\ \Lambda^{-1}
R{'}^2+\nonumber \\
&-&\frac{1}{2}\ \Lambda +\frac{1}{2}\ R^{-2}\Lambda^{-1}P^2_{\phi}+
\frac{1}{2}\ R^2\Lambda^{-1}\phi{'}^2\ .\end{aligned}$$
In the expression above $P_R,P_{\Lambda},P_{\phi}$ imply respectively conjugate momenta associated with $R,\Lambda$ and $\phi$ variables.
Furthermore $R=R(r,t)$, $\Lambda =\Lambda (r,t)$, $\phi =\phi (r,t)$. We define conjugate momentum as $$\pi_x=-i\ \frac{\partial}{\partial x}$$ where $\underline{x}$ means $R,\Lambda$ or $\phi$ variable.
It is a known fact that using the Hamiltonain (2) some operator ordering problems appear [@um; @dois].
A simple form to represent the ambiguous order of factors $\left(x\ , \ \displaystyle{\frac{\partial}{\partial x}}\right)$ and $\left(y\ , \ \displaystyle{\frac{\partial}{\partial y}}\right)$ is given by [@um]. Applying such an ordering for operators in (2) we can find the following squared conjugate momenta $$\begin{aligned}
\pi^2_x &=&-\frac{\partial^2}{\partial x^2}-\frac{p}{x}\
\frac{\partial}{\partial x}\nonumber \\
\\
\pi^2_y &=&-\frac{\partial^2}{\partial y^2}-\frac{q}{y}\
\frac{\partial}{\partial y}\nonumber \end{aligned}$$ where $(p,q)$ are $c$-numbers.
It is assumed that the Hamiltonian (2) is a constraint for a classical Hamiltonain with the mass term present for the scalar field $\phi$. In other words, the canonical quantization needs the annihilation of the wave function $\psi$ by the corresponding quantum operator $$\hat{H}\psi =0$$ that results in the Wheeler-De Witt equation. Using eq. (2-5) we get $$\frac{\Lambda}{2R^2}\left(\frac{\partial^2\psi}{\partial \Lambda^2}+
\frac{p}{\Lambda}\ \frac{\partial \psi}{\partial \Lambda}\right)+
\frac{1}{2R^2\Lambda}\left(\frac{\partial^2\Lambda}{\partial \phi^2}+
\frac{q}{\phi}\frac{\partial \psi}{\partial \phi}\right)-
\frac{1}{R}\ \frac{\partial^2\psi}{\partial R\partial \Lambda}\equiv
V \psi$$ where $\psi$ is a functional of $\Lambda$, $\phi$ and $R$ functions, and $V$ is a potential term written as $$V=\frac{R}{\Lambda}\ R''-\frac{R}{\Lambda^2}\ R'\Lambda '+\frac{1}{2\Lambda}\
R{'}^2-\frac{1}{2}\ \Lambda +\frac{1}{2}\ \frac{R^2}{\Lambda}\ \phi{'}^2 +
\frac{1}{2}\ m^2R^2\Lambda \phi^2$$ The prime means derivative with respect to the coordinate $\underline{r}$. Observe that in equation (6) we don’t have any derivative with respect to time. This means that the equation (6) could be describing a spherically symmetric gravitational collapse but without any explicit time dependence for functional $\psi$. The concept of “time” in this case may appear only after quantization in accordance with [@tres].
This suggests that eq. (6) is like the usual Schrödinger equation of quantum mechanics applied to gravitational collapse but with a difference depending on the operator ordering \[1-5\].
The usual Schrödinger equation is written as $$H\psi =i\frac{\partial \psi}{\partial t}$$ where $H$ means the Hamiltonian of the system. It means that the wave function of the system has an important difference with equation (6) besides the fact that $\psi$ in (8) to be a function while $\psi$ in (6) being a functional $\psi (\Lambda ,\phi ,R)$. The parameter “time” $\underline{t}$ in (8) is a universal time-“external time” in the sence of Galili-Newton time, while in equation (6) “time” is an internal parameter. In some sense there is no “time” with which we could describe the evolution of gravitational collapse of the star for exemplo. Thus, in principle we might apply the equation for a static case such as Schwarszchild solution but not for a dynamic case where the functions $R,\Lambda
,\phi$ might be time dependent. In other words, one can apply Wheeler-De Witt equation (6) for static Schwarszchild case where $R=R(r)$, $\Lambda =\Lambda (r)$ and $\phi =\phi (r)$ but shall we apply the same equation for the general case, with $R=R(r,t)$, $\Lambda =\Lambda (r,t)$ and $\phi =\phi (r,t)$?
How does the conception of “time” appear in this case?
How can we get the notion of evolution in time of a collapsing star using equation (6) without explicit time dependence of the functional $\psi$?
The equation (8) can be applied for steady systems such as hydrogen atom where the right side is zero and we have $$\hat{H}\psi =E\psi =0$$ where $E$ is the energy. In the particular case of $E=0$ this equation has a strong resemblance to the Wheeler-De Witt equation.
It is a well known fact that stationary solution can be find from equation (9) in terms of $R(r),\ \Theta (\theta ),\ \phi (\varphi )$ with $R$, the radial solution and $\Theta (\theta )\phi (\varphi
)=Y(\theta ,\varphi )$ being the spherical harmonics. The obvious similarity of eq. (9) and eq. (5) leads us to think that eq. (6) can be solved in the general case, with an “internal time” and the idea of “evolution” being identified with some discrete index $i=1,2,3\cdots $. after solving eq. (6).
We know that there are many different $\psi_{k\ell m}(r,\theta
,\varphi )$ for different values of $k,\ell ,m$ for the hydrogen atom and in some sense “the evolution of the system” can be seen as a changing of wave function for a stationary situation. There is no “external time” in eq. (8) for the hydrogen atom.
In the same way we can think of applying in eq. (6) with an “internal time” or without an external time any way and to obtain the functional $\psi (\Lambda ,\phi ,R)$.
We may take an appropriate ansatz for the eq. (6) and to verify if it really does satisfy eq. (6). But immediately two questions can be raised.
First, which ansatz? There are an inifinite number of possibilities.
Second, the introduction of a mass term in (1) for scalar field $\phi$ can break the “constraint” character for $H$ and eq. (5) may not be valid anymore. We must remind that we are assuming the presence of mass of the scalar field and it does not break the constraint of super Hamiltonian as in [@dois].
In general the Wheeler-De Witt equation can be separated depending on the potential term (7). The role of $V\left(R,R',R'',\Lambda ,\Lambda ',\phi ,\phi ',m\right)$ is similar to the coordinates system for decoupling of the Schrödinger equation. It is a known fact that the Schrödinger Equation can be separated in several coordinates systems. In the same manner eq. (6) may decouple for $\psi (R,\Lambda ,\phi )$ depending on the potential term and the particular choice of the [*ansatz*]{} for the $\psi$ functional. But eq. (6-9) is too complicated and again there is no derivative in “time”.
Qualitatively the problem can be solved in the following way. Suppose that $\psi$ functional reads as $$\psi \left(\Lambda ,\phi ,R\right)=\Lambda (r+c)
\sqrt{\phi (r+c)}\ R(r+c)$$ where $\Lambda ,R,\phi$ are functions of $\ \underline{r}\ $ only since there is no“external time” as in eq. (8) or an “internal time” as in general Relativity theory or in the relativistic Klein Gordon equation.
In eq. (9) $\underline{c}$ is a constant that can be identified with “time” after quantization.
A class of solutions such as is shown below may be found
In reality we can find a sequence of $R_i(r),\ \Lambda_j(r)$ and $\phi_k(r)$ where $i,j,k=1,2,3\cdots$ the concept of “time” being identified with $i,j,k\sim t$ (Time).
In the Schrödinger equation for the hydrogen atom the wave function $\psi_{k\ell m}(r,t)$ can be written as a product of $R_{k\ell}(r),\Theta_{\ell m}(\theta )$ and $\phi_m(\varphi )$ for stationary states and one may see a notion of “evolution” through the different configurations is possible given by different values of $k,\ell ,m$.
In our case the same idea can be utilised by identifying with a discrete index $(i=1,2,3\cdots )$ as the “time” where $i,j,k$ are the different functions that contribute to our functional $\psi$.
Finally, we need to be clear that eq. (6) has a infinite number of solutions with the proposal given by eq. (10) being one of them.
The Wheeler-De Witt equation itself has many different possibilities depending of the operator ordering [@um; @dois; @tres; @quatro; @cinco]. Then, in principle one can write different mathematics (different Wheeler-De Witt equations) and each one of them with infinite number of ansatz. Each possibility is given us a notion of “Time” after quantization.
The natural question that we can put is:
Shall we find the same “physics” for different Wheeler-De Witt equations?
Can we find the same notion of “time” from different Wheeler-De Witt equation with infinite possibilities of the ansatz ?
The physical “time” is the same for each possibility or do we have many times in physics as in [@seis]?
Admitting that our equation (6) has some meaning and that the ansatz eq. (10) can provide us with a notion of “time” arises from the discretization of the index $i,j,k\sim t$. The next question we need to resolve is: if eq. (6) implies the Schrödinger equation for a global Universe in general and in our particular case it is a Schrödinger equation for a gravitational collpse of a body like a star how can we improve the Copenhagen interpretation for the functional $\psi (\Lambda ,\phi ,R)$?
Maybe the answer can be found as in eq. (6) and the ansatz given by eq. (10) describing the possibility of finding the star between $\underline{m}$ and $\underline{m+dm}$ mass states.
But if so, can it be supported by the condition $m\neq 0$ for the scalar field $\phi$ in (1)?
Should the superhamiltonian be a real constraint $H\sim 0$ on that condition?
In any case we need to understand the real meaning of operator ordering in quantum mechanics as well as the meaning of time in all of physics. While we don’t know the final answer for these open questions there have been uncertain consequences for a complete understanding of physics and our interpretation for the world.
Acknowledgements: {#acknowledgements .unnumbered}
-----------------
#### {#section-1 .unnumbered}
I would like to thank the Department of Physics, University of Alberta for their hospitality. This work was supported by CNPq (Governamental Brazilian Agencie for Research.
I would like to thank also Dr. Don N. Page for his kindness and attention with me at Univertsity of Alberta.
[20]{} Nuclear Phys. B.V. 264 (1986) 185-196.\
Hawking, S.W. and Page, D.N. Sperically Symmetric Scalar Field Collapse: An Example of Spacetime Problem of time.\
Joseph D. Romano. (Preprint).\
Department of Phys. – University of Utah. Canonical Quantum Gravity and the Problem of Time.\
C.J. Isham – Proceedings of NATO\
Edited by L.A. Ibort and M.A. Rodriguez\
Series C: Mathematical and Phys. Sciences – V. 409\
Kluwer Academic Publishers. (1992) – Salamanca – Spain. Two Component Formulation on the Wheeler-De Witt Equation for FRW Massive Scalar Field Minisuperspace.\
Ali Mostafazadeh.\
Internal Notes – Theoretical Physics Institute,\
University of Alberta,\
Edmonton, Alberta, Canadá T6G 2J1, The Appearence of Time in Quantum Gravity.\
M.A. Castagnino\
Institute de Astronomia y Física del Espacio.\
Casilla de Correos 67, SucursaL 28\
1428 Buenos Aires, Argentina and\
Boundary Conditions and the Wave Function o the Universe.\
L.P. Grishchuk, Yu.V. Sidorov\
Stenberg Astronomical Institute\
119899 Moscow-V-234, USSR. Two Times in Physics.\
Itzak Bars, 1999\
Southern California University/UCS.
[^1]: khanna@@phys.ualberta.ca
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Given an elliptic curve $E/\mathbb{Q}$, it is a conjecture of Goldfeld that asymptotically half of its quadratic twists will have rank zero and half will have rank one. Nevertheless, higher rank twists do occur: subject to the parity conjecture, Gouvêa and Mazur constructed $X^{1/2-\epsilon}$ twists by discriminants up to $X$ with rank at least two. For any $d\geq 3$, we build on their work to consider twists by degree $d$ $S_d$-extensions of $\mathbb{Q}$ with discriminant up to $X$. We prove that there are at least $X^{c_d-\epsilon}$ such twists with [positive rank]{}, where $c_d$ is a positive constant that tends to $1/4$ as $d\to\infty$. [Moreover, subject to a suitable parity conjecture, we obtain the same result for twists with rank at least two.]{}'
address:
- 'Department of Mathematics, Tufts University, 503 Boston Ave, Medford, MA 02155'
- 'Department of Mathematics, University of South Carolina, 1523 Greene St, Columbia, SC 29201'
author:
- 'Robert J. Lemke Oliver'
- Frank Thorne
bibliography:
- 'nonabeliantwists.bib'
title: Rank growth of elliptic curves in nonabelian extensions
---
Introduction and statement of results
=====================================
Let $E/\mathbb{Q}$ be an elliptic curve and let $E_D/\mathbb{Q}$ be its twist by the field $\mathbb{Q}(\sqrt{D})$. Goldfeld [@Goldfeld1979] has conjectured that as $D$ ranges over fundamental discriminants, asymptotically 50% of the twists $E_D/\mathbb{Q}$ will have rank zero and 50% will have rank one. Following the work of Gross and Zagier [@GrossZagier1986] and Kolyvagin [@Kolyvagin1988] on the Birch and Swinnerton-Dyer conjecture in the late 1980’s, it became of critical importance to demonstrate the existence of a twist, satisfying some additional splitting conditions, with *analytic* rank one. This was first achieved independently by Bump, Friedberg, and Hoffstein [@BumpFriedbergHoffstein1990] and Murty and Murty [@MurtyMurty1991]. Together, these results imply that if the analytic rank of an elliptic curve $E/\mathbb{Q}$ is at most one, then its algebraic rank is equal to its analytic rank.
In the wake of these results, it became natural to search for twists of rank two or greater. By employing an explicit construction, the squarefree sieve, and the then recently proven cases of the Birch and Swinnerton-Dyer conjecture, Gouvêa and Mazur [@GouveaMazur1991] were able to produce $\gg X^{1/2-\epsilon}$ discriminants $D$ with $|D|\leq X$ for which the analytic rank of $E_D/\mathbb{Q}$ is at least two; under the parity conjecture, these twists also have algebraic rank at least two. Unconditional results on twists with algebraic rank at least two were estbalished by Stewart and Top [@StewartTop1995], though with a worse exponent.
Motivated by the program of Mazur and Rubin on Diophantine stability (see, e.g., [@MazurRubin2015]), we may cast the above results as being about the growth of the rational points $E(K)$ relative to $E(\mathbb{Q})$ in quadratic extensions $K/\mathbb{Q}$. In this work, we are interested in the analogous problem when $K$ is a degree $d$ $S_d$-extension of $\mathbb{Q}$. Let $$\mathcal{F}_d(X) := \{ K/\mathbb{Q} : [K:\mathbb{Q}] = d, \mathrm{Gal}(\widetilde{K}/\mathbb{Q}) \simeq S_d, |\mathrm{Disc}(K)| \leq X\}$$ where $\mathrm{Disc}(K)$ denotes the absolute discriminant of the extension $K/\mathbb{Q}$ and $\widetilde{K}$ denotes its Galois closure. Our main theorem is the following analogue of Gouvêa and Mazur’s work:
\[thm:general\] Let $E/\mathbb{Q}$ be an elliptic curve and let $d \geq 2$. There is a constant $c_d>0$ such that for each $\varepsilon = \pm 1$, the number of fields $K \in \mathcal{F}_d(X)$ for which $\mathrm{rk}(E(K))> \mathrm{rk}(E(\mathbb{Q}))$ and the root number $w(E,\rho_K) = \varepsilon$ is $\gg X^{c_d-\epsilon}.$
We may take $c_d = 1/d$ for $d\leq 5$, $c_6 = 1/5$, $c_7=c_8=1/6$, and $$c_d = \frac{1}{4} - \frac{d^2+4d-2}{2d^2(d-1)}$$ in general. In particular, we may take $c_d > 0.16$ always, and $c_d > 1/4 - \epsilon$ as $d\to\infty$.
Here the [root number]{} $w(E,\rho_K) = \frac{w(E_K)}{w(E)}$ is related to the analytic ranks of $E/\mathbb{Q}$ and $E/K$ as follows. Let $L(s, E)$ and $L(s, E_K)$ be the Hasse-Weil $L$-functions associated to $E/\mathbb{Q}$ and its base change to $K$. Under the Birch and Swinnerton-Dyer Conjecture, the ranks $\mathrm{rk}(E(\mathbb{Q}))$ and $\mathrm{rk}(E(K))$ are equal to the analytic ranks of these $L$-functions. Therefore, $\mathrm{rk}(E(K) - \mathrm{rk}(E({\mathbb{Q}}))$ is conjecturally equal to the order of vanishing of $\frac{L(s, E_K)}{L(s, E)}$ at the central point $s = 1/2$.
This quotient is an $L$-function in its own right, the [*non-abelian twist*]{} $L(s, E, \rho_K)$ of $E$ by the standard representation $\rho_K$ of $\mathrm{Gal}(\widetilde{K}/\mathbb{Q}) \simeq S_d$. (See Section \[sec:twist-props\].) This $L$-function is conjectured, and is in some cases known, to be analytic and to satisfy a self-dual functional equation sending $s\mapsto 1-s$ with root number $w(E, \rho_K)$. (For example, this holds whenever $L(s,\rho_K)$ satisfies the strong Artin conjecture.) This root number thus controls the parity of ${{\text {\rm ord}}}_{s = 1/2} \frac{L(s, E_K)}{L(s, E)}$. Under either the Birch and Swinnerton-Dyer conjecture or the parity conjecture, this is the same as the parity of $\mathrm{rk}(E(K)) - \mathrm{rk}(E(\mathbb{Q}))$, and we obtain the following.
Assuming the parity conjecture, the number of $K \in \mathcal{F}_d(X)$ for which $\mathrm{rk}(E(K)) \geq 2 + \mathrm{rk}(E(\mathbb{Q}))$ is $\gg X^{c_d-\epsilon}$, with $c_d$ as in Theorem \[thm:general\].
Using known progress toward the Birch and Swinnerton-Dyer conjecture, we also obtain the following unconditional result on analytic ranks in the case $d=3$.
\[thm:analytic\] Assume that the elliptic curve $E/\mathbb{Q}$ has at least one odd prime of multiplicative reduction. Then the number of $K \in \mathcal{F}_3(X)$ for which the analytic rank of $L(s,E,\rho_K)$ is at least $2$, is $\gg X^{1/3-\epsilon}$.
A curious feature of Theorem \[thm:general\] is that the constant $c_d$ approaches $1/4$ from below. One might therefore hope that there is some easy improvement to Theorem \[thm:general\] that resolves this quirk. In fact, the value of $c_d$ presented is not always optimal: the proof of Theorem \[thm:general\] makes use of the Schmidt bound $\# \mathcal{F}_d(X) \ll X^{(d+2)/4}$, and this has been improved for large values of $d$. However, the net effect of this is minor, and the following result is not obviously improved by any stricter assumption on $\#\mathcal{F}_d(X)$.
\[thm:field-improvement\] Let $d \geq 7$. If $\#\mathcal{F}_d(X) \ll X^{\frac{d-3}{4}+\frac{1}{2d}+\epsilon}$, then we may take $$c_d = \frac{1}{4} - \frac{1}{2d}.$$ in Theorem \[thm:general\]. In particular, this is unconditional for $d\geq 16052$.
In fact, while our method in principle might have the ability to produce exponents $c_d$ slightly larger than $1/4$, we presently only see how to do so under rather heavy assumptions.
\[thm:overkill\] Assume either that the $L$-functions $L(s,E_K)$ for $K \in \mathcal{F}_d(X)$ are automorphic and satisfy the generalized Riemann hypothesis and the Birch and Swinnerton-Dyer conjecture, or that the bound $\#\mathrm{Cl}(K(E[2]))[2] \ll D_K^\epsilon$ holds for all $K \in \mathcal{F}_d(X)$ and all $\epsilon>0$. Then Theorem \[thm:general\] holds with $$c_d = \frac{1}{4} + \frac{1}{2(d^2-d)}.$$
We now comment on what we expect to be true. It is a folklore conjecture, strengthened by Bhargava [@Bhargava2007], that there is a positive constant $a_d$ such that $\#\mathcal{F}_d(X) \sim a_d X$. Based on the minimality philosophy, since fields $K \in \mathcal{F}_d(X)$ admit no nontrivial subfields and the root numbers $w(E,\rho_K)$ assume both signs, it is reasonable to expect that a version of Goldfeld’s conjecture should hold. That is, that the number of $K \in \mathcal{F}_d(X)$ for which $\mathrm{rk}(E(K))=\mathrm{rk}(E(\mathbb{Q}))$ and the number for which $\mathrm{rk}(E(K))=1+\mathrm{rk}(E(\mathbb{Q}))$ should each be asymptotic to $\frac{1}{2}a_d X$. Furthermore, a naïve heuristic based on quantization of Tate-Shafarevich groups and Tate’s version of the Birch and Swinnerton-Dyer conjecture over number fields suggests that perhaps the number of $K$ for which $\mathrm{rk}(E(K))=2+\mathrm{rk}(E(\mathbb{Q}))$ should be $X^{3/4+o(1)}$.
Thus, Theorem \[thm:general\] – which, to the best of our knowledge, provides the first general bounds as $d\to\infty$ for the number of $K \in \mathcal{F}_d(X)$ for which $\mathrm{rk}(E(K))>\mathrm{rk}(E(\mathbb{Q}))$ – is presumably very far from the truth. However, it is only known at present for $d\geq 6$ that $\#\mathcal{F}_d(X) \gg X^{1/2+1/d}$ due to recent work of Bhargava, Shankar, and Wang [@BhargavaShankarWang]. This result is the culmination of a natural line of thought (constructing fields via writing down polynomials), so producing a stronger lower bound for $\#\mathcal{F}_d(X)$ will require a substantial new idea. In particular, since we are conjecturally accessing in Theorem \[thm:general\] fields for which the rank increases by at least two, based on the above discussion, it is reasonable to expect that the best possible version of Theorem \[thm:general\] available with current methods can do no better than $c_d = 1/4 + 1/d$. We therefore view Theorem \[thm:general\] as nearly optimal, though it would surely be desirable to bridge the small gap between our results and this limit. It is not clear to us at this time how to do so.
Finally, we discuss briefly other results on the growth of the Mordell–Weil group in non-quadratic extensions $K/\mathbb{Q}$. Most notably for our purposes, V. Dokchitser [@Dokchitser2005] analyzed the root numbers of $L(s,E_K)$ for general $K$ and obtained many corollaries about analytic ranks. His work is a crucial ingredient in controlling the root numbers in Theorem \[thm:general\]. Quite recently, Fornea [@Fornea] has shown that for many curves $E/\mathbb{Q}$, the analytic rank of $E$ increases over a positive proportion of $K \in \mathcal{F}_5(X)$, though his work does not control the algebraic rank nor does it access twists for which the rank increases by two. In the large rank direction, in earlier work, by a consideration of root numbers, Howe [@Howe1997] showed that in Galois $\mathrm{PGL}_2(\mathbf{Z}/p^n\mathbf{Z})$-extensions, the rank increases dramatically if $-N_E$ is a quadratic nonresidue modulo $p$, where $N_E$ denotes the conductor of the curve $E/\mathbb{Q}$. However, this result is of a somewhat different flavor than Theorem \[thm:general\], as Howe is specifically exploiting the fact that such fields admit many nontrivial subfields. (We recall again that a field $K \in \mathcal{F}_d(X)$ admits no such subfields.) Lastly, in the complementary direction, Mazur and Rubin [@MazurRubin2015] show that for every prime power $\ell^n$, there are infinitely many cyclic degree $\ell^n$ extensions over which the Mordell–Weil group does not grow, and David, Fearnley, and Kisilevsky [@DavidFearnleyKisilevsky2007] have formulated conjectures about the frequency with which the rank increases over prime degree cyclic extensions.
Organization of the paper and the strategy of the proof
=======================================================
We begin by explaining the ideas that go into the proof of Theorem \[thm:general\].
To construct points on $E$ over degree $d$ number fields, we construct points in parametrized families over degree $d$ extensions of certain function fields $\mathbb{Q}(\mathbf{t})$ where $\mathbf{t}=(t_1,\dots,t_r)$ for some $r$. For example, for $d = 3$ we find a Weierstrass model $E\colon y^2 = f(x)$ for which $P_f(x, t) :=
f(x) - (x + t)^2$ defines an $S_3$-extension of ${\mathbb{Q}}(t)$. By Hilbert irreducibility, most specializations $t = t_0 \in {\mathbb{Q}}$ define $S_3$-extensions $K/{\mathbb{Q}}$, over which $E$ visibly gains a point. Lemma \[lem:fractional-points\] then establishes that these ‘new’ points usually increase the rank.
After proving some preliminary lemmas in Section \[sec:galois-background\], we devote Section \[sec:galois\] to constructing $S_d$-extensions of ${\mathbb{Q}}(t)$ along the lines discussed above for $d = 3$. The strategy is to prove that the Galois groups of specializations contain various cycle types. We first use Newton polygons to exhibit ‘long’ cycles. We then argue that, for a suitable Weierstrass model of $E$, there exists a prime $p$ and a specialization $P_f(x, t_0)$ such that $p$ divides the discriminant $P_f(x, t)$ and $p^2$ does not. This proves that the Galois group of $P_f$ contains a transposition, and (after a bit of group theory) that it is therefore $S_d$.
We thus obtain $S_d$-extensions $K$ over which $E$ gains a point of infinite order. We must then bound the multiplicity with which a given field arises. We present two ways of doing so. The first method is via an analysis of the squarefree part of the discriminant of $K$ and is carried out in Section \[sec:small-degree\]. This requires the transcendence degree of the function field $\mathbb{Q}(\mathbf{t})$ to be quite small, and so is the more efficient of the two methods only for $d\leq 8$.
The second method, presented in Section \[sec:large-degree\], is based on a slight improvement to a geometry-of-numbers argument due to Ellenberg and Venkatesh [@EV] that was originally used to bound $\#\mathcal{F}_d(X)$ from below. We adapt their construction to only count fields over which $E$ gains a point. This allows the transcendence degree of the field $\mathbb{Q}(\mathbf{t})$ to be large, but with some loss of control over the multiplicities. The added freedom gained by the number of parameters outweighs this small loss once $d\geq 9$.
Finally, it remains to control the root numbers $w(E,\rho_K)$. We do so using work of V. Dokchitser [@Dokchitser2005]. We review his work, along with other useful properties of the twist, in Section \[sec:twist-props\]. The net effect is that to show that both root numbers occur frequently it suffices to show that we construct many fields $K$ and $K^\prime$ that are “$p$-adically close” for each $p \mid N_E$ but for which the discriminants $D_K$ and $D_{K^\prime}$ have different signs. Assembling all of this, we obtain Theorem \[thm:general\]. The proof of Theorem \[thm:analytic\] relies on similar arguments from Section \[sec:small-degree\] for small degrees, but it requires a slightly different handling of the root number. This is provided to us by a different lemma of Dokchitser.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Michael Filaseta, Jan Nekovář, Jeremy Rouse, David Smyth, Stanley Yao Xiao, and David Zureick-Brown for useful insights on this problem.
This work was supported by NSF Grant DMS-1601398 (R.J.L.O.), by a NSA Young Investigator Grant (H98230-16-1-0051, F.T.), and by a grant from the Simons Foundation (563234, F.T.).
Properties of the twist {#sec:twist-props}
=======================
Let $E/\mathbb{Q}$ be an elliptic curve and let $K \in \mathcal{F}_d(X)$. Formally, the non-abelian twist $L(s, E, \rho_K)$ may be defined by the relation $$\label{eq:l_factor}
L(s,E_K) = L(s,E) L(s,E,\rho_K).$$ In Dokchitser [@Dokchitser2005], $L(s, E, \rho_K)$ is given a more intrinsic definition that we now briefly recall. Let $\rho_K$ be the standard $d - 1$ dimensional representation of ${{\text {\rm Gal}}}(\widetilde{K}/{\mathbb{Q}}) \simeq S_d$, which we also regard as a continuous representation of ${{\text {\rm Gal}}}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$. The usual Artin formalism provides a factorization $$\label{eq:artin}
\zeta_K(s) = \zeta(s) L(s, \rho_K)$$ of the Dedekind zeta function $\zeta_K(s)$, where $L(s, \rho_K)$ is the Artin $L$-function associated to $\rho_K$.
Now, let $T_\ell(E)$ be the $\ell$-adic Tate module associated to $E$, and write $$H_\ell(E) = \textnormal{Hom}(T_\ell(E) \otimes {\mathbb{Q}}_\ell, {\mathbb{Q}}_\ell) \otimes_{{\mathbb{Q}}_\ell} {\mathbb{C}},$$ which is a $2$-dimensional $G_{\mathbb{Q}}$-module. Then the $L$-function $L(s, E)$ is defined, as usual, in terms of the action of $G_{\mathbb{Q}}$ on $H_\ell(E)$; its twist $L$-function $L(s, E, \rho_K)$ is defined analogously in terms of the representation on $H_\ell(E) \otimes \rho_K$. The formula is then the exact analogue of , and is similarly proved. We may also regard $L(s,E,\rho_K)$ as the Rankin–Selberg $L$-function $L(s, E \times \rho_K)$. The analytic properties of Rankin–Selberg products are known when the two $L$-functions are attached to cuspidal automorphic forms; for example, see Cogdell [@Cogdell] for a wonderful summary. The modularity theorem establishes that $L(s,E)$ is attached to a classical modular form, and the strong Artin conjecture asserts that every $L(s,\rho_K)$ is attached to an automorphic form. Thus, we expect that $L(s,E,\rho_K)$ is always entire, but this is at present wide open in general.
In the special case that $K/\mathbb{Q}$ is an $S_3$ cubic, the strong Artin conjecture is known for $L(s,\rho_K)$, whereby the $L$-function $L(s, E, \rho_K)$ is known to be holomorphic. We may further connect this $L$-function to the Mordell–Weil group, as we now explain.
Given a field $K \in \mathcal{F}_3(X)$, there is a unique quadratic subfield $F$ of the Galois closure $\widetilde{K}$ known as the quadratic resolvent of $K$. If $\psi_K$ is the cubic ray class character of $F$ corresponding to the extension $\widetilde{K}/F$, then $L(s,\psi_K) = L(s,\rho_K)$. Correspondingly, there is an equality of $L$-functions $L(s,E,\rho_K) = L(s,E_F,\psi_K)$. As in , it follows that $$L(s,E_{\widetilde{K}}) = L(s,E_F) L(s,E_F,\psi_K) L(s,E_F,\overline{\psi}_K),$$ where $\overline{\psi}_K$ is the character conjugate to $\psi_K$. In fact, even though $\psi_K$ and $\overline{\psi}_K$ are distinct characters, their associated $L$-functions are the same. Similarly, $L(s,E,\psi_K) = L(s,E,\overline{\psi}_K)$ as analytic functions, so we conclude in particular that $$\begin{aligned}
\mathrm{ord}_{s=1/2} L(s,E_{\widetilde{K}}) - \mathrm{ord}_{s=1/2} L(s,E_F)
&= 2\cdot \mathrm{ord}_{s=1/2} L(s,E_F,\psi_K) \\
&= 2\cdot \mathrm{ord}_{s=1/2} L(s,E,\rho_K).\end{aligned}$$ In other words, the analytic rank of $L(s,E,\rho_K)$ controls the growth of the analytic rank of $E$ in the extension $\widetilde{K}/F$.
There is an arithmetic manifestation of this story as well. Viewing the Mordell–Weil group $E(\widetilde{K}) \otimes \mathbb{C}$ as a finite dimensional Galois representation and decomposing it into isotypic components, a bit of Galois theory shows that the $\rho_K$-isotypic component $E(\widetilde{K})^{\rho_K}$ satisfies $$\begin{aligned}
\dim_\mathbb{C} E(\widetilde{K})^{\rho_K}
&= \mathrm{rk}(E(\widetilde{K})) - \mathrm{rk}(E(F)) \\
&= 2\cdot (\mathrm{rk}(E(K)) - \mathrm{rk}(E(\mathbb{Q}))).\end{aligned}$$ The first line follows because $E(F) \otimes {\mathbb{C}}$ is the direct sum of the remaining isotypic components; the second because, for each element $\tau \in {{\text {\rm Gal}}}(\widetilde{K}/{\mathbb{Q}})$ of order two, $\rho(\tau)$ has eigenvalues $1$ and $-1$. In particular, we see that the growth of the rank of the Mordell–Weil group in the extension $\widetilde{K}/F$ is controlled by its growth in $K/\mathbb{Q}$.
Combining these two perspectives, the Birch and Swinnerton-Dyer conjecture predicts that the analytic rank of $L(s,E,\rho_K)$ controls the multiplicity of $\rho_K$ in the representation $E(\widetilde{K})\otimes \mathbb{C}$, and thereby the growth of the rank. While this conjecture is certainly still wide open, it is known in the case that the analytic rank is $0$ and the field $F$ is imaginary:
\[thm:cubic-bsd\] With notation as above, suppose that $F$ is an imaginary quadratic field and that $E$ does not have CM by an order in $F$. If $L(1/2,E_F,\psi_K) \neq 0$, then $\mathrm{rk}(E(\widetilde{K})) = \mathrm{rk}(E(F))$.
Here the $L$-function is again normalized so that $s = \frac{1}{2}$ is at the center of the critical strip.
From Theorem \[thm:cubic-bsd\] and the above discussion, we obtain the following corollary.
\[cor:cubic-ranks\] Let $E/\mathbb{Q}$ be an elliptic curve and let $K \in \mathcal{F}_3(X)$ have negative discriminant. Suppose that $E$ does not have CM by an order in the quadratic resolvent of $K$. If $w(E,\rho_K) = +1$ and $\mathrm{rk}(E(K)) \neq \mathrm{rk}(E(\mathbb{Q}))$, then the analytic rank of $L(s,E,\rho_K)$ is at least $2$.
Since $w(E,\rho_K) = +1$, the analytic rank of $L(s,E,\rho_K)$ must be even. On the other hand, the requirement that $K$ have negative discriminant guarantees that the quadratic resolvent $F$ is an imaginary quadratic field. Thus, since $\mathrm{rk}(E(K)) \neq \mathrm{rk}(E(\mathbb{Q}))$ and $L(s,E,\rho_K) = L(s,E_F,\psi_K)$, Theorem \[thm:cubic-bsd\] precludes the possibility that $L(s,E,\rho_K) \neq 0$. This implies that $L(s,E,\rho_K)$ must have rank at least $2$, as claimed.
We now recall the work of Dokchitser [@Dokchitser2005] on the root numbers $w(E,\rho_K)$. In many cases (see his Theorem 16, for example), he determined exactly the value of $w(E,\rho_K)$. We require only the following properties, obtained as a consequence of [@Dokchitser2005 Theorem 16] and its surrounding discussion.
If $K \in \mathcal{F}_d(X)$, then there is a factorization $$w(E,\rho_K) = w(E)^{d-1} w_\infty(E,\rho_K) \prod_p w_p(E,\rho_K)$$ such that:
1. $w_p(E,\rho_K) = 1$ if $E$ has good reduction at $p$;
2. $w_\infty(E,\rho_K) = \mathrm{sgn}(\mathrm{Disc}(K))$, the sign of the discriminant of $K$; and
3. if $p \mid N_E$, then $w_p(E,\rho_K)$ depends only on $\rho_E\!\mid_{G_{\mathbb{Q}_p}}$ and $\rho_K\!\mid_{G_{\mathbb{Q}_p}}$, where $\rho_E$ is the Galois representation attached to $E$ and $G_{\mathbb{Q}_p} = \mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p) \subseteq \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.
From this, we derive the following important corollary.
\[cor:sign\] Let $K$ and $K^\prime \in \mathcal{F}_d(X)$ be such that $K \otimes \mathbb{Q}_p \simeq K^\prime \otimes \mathbb{Q}_p$ for all $p \mid N_E$. Suppose that $\mathrm{sgn}(\mathrm{Disc}(K)) = - \mathrm{sgn}(\mathrm{Disc}(K^\prime))$. Then $w(E,\rho_K) = - w(E,\rho_{K^\prime})$.
In proving Theorem \[thm:analytic\], we will need a slightly different way to control the root number $w(E,\rho_K)$. In particular, we have [@Dokchitser2005 Corollary 2]:
\[lem:sign-jacobi\] Suppose that the conductor $N_E$ of $E$ is relatively prime to the discriminant $\mathrm{Disc}(K)$ of $K \in \mathcal{F}_d(X)$. Then $$w(E,\rho_K) = w(E)^{d-1} \mathrm{sgn}(\mathrm{Disc}(K)) \left(\frac{\mathrm{Disc}(K)}{N_E}\right),$$ where $(\frac{\cdot}{\cdot})$ denotes the Kronecker symbol.
We close this section by showing that for almost all $K \in \mathcal{F}_d(X)$, if $E(K) \neq E(\mathbb{Q})$, then $\mathrm{rk}(E(K)) > \mathrm{rk}(E(\mathbb{Q}))$.
\[lem:fractional-points\] Let $E/\mathbb{Q}$ be an elliptic curve. There is a constant $C_{E,d}$, depending only on $E$ and $d$, such that $$\#\{K\in\mathcal{F}_d(X): E(K) \neq E(\mathbb{Q}) \text{ but } \mathrm{rk}(E(K))=\mathrm{rk}(E(\mathbb{Q}))\} \leq C_{E,d}.$$
For each $K$ counted, there must exist some prime $\ell\geq 2$ and some point $P \in E(K) \setminus E({\mathbb{Q}})$ for which $\ell P\in E(\mathbb{Q})$ but $mP \not \in E(\mathbb{Q})$ for any $m < \ell$. Since any field in $\mathcal{F}_d(X)$ has no non-trivial subfields, we must have $\mathbb{Q}(P) = K$ and, recalling our notation for the Galois closure, $\widetilde{\mathbb{Q}(P)} = \widetilde{K}$. Now, any conjugate of $P$ differs from $P$ by some point of order $\ell$ in $E(\overline{\mathbb{Q}})$, so there must be at least one point of order $\ell$ defined over $\widetilde{\mathbb{Q}(P)} = \widetilde{K}$.
By work of Merel [@Merel], there is an absolute constant $T(d!)$ such that $|E(L)_\mathrm{tors}|\leq T(d!)$ for any field $L$ of degree $d!$. We therefore have $\ell \leq T(d!)$. For each such $\ell$ and point $P$ as above, the field $\mathbb{Q}(P)$ depends only on the class of $\ell P$ in $E(\mathbb{Q})/\ell E(\mathbb{Q})$ and possibly the choice of an $\ell$-torsion point in $E(\overline{\mathbb{Q}})$. Hence only finitely many such fields arise, and this yields the lemma.
Useful results from Galois theory {#sec:galois-background}
=================================
In this section, we recall several useful results from Galois theory and we prove a few preliminary lemmas that will be useful in what is to come.
We start off by recalling the Hilbert irreducibility theorem in the following context. Let $f(\mathbf{t},x) \in \mathbb{Q}(\mathbf{t})[x]$ be an irreducible polynomial of degree $d$ over $\mathbb{Q}(\mathbf{t})$ where $\mathbf{t} = (t_1,\dots,t_k)$. This defines an extension $K = \mathbb{Q}(t)[x]/f(\mathbf{t},x)$ which need not be Galois closed over $\mathbb{Q}(\mathbf{t})$. Let $L$ be its Galois closure, which we take to be generated by the polynomial $g(\mathbf{t},x)$, and we write $G = \mathrm{Gal}(L/\mathbb{Q}(\mathbf{t}))$. For any $\mathbf{t}_0 \in \mathbb{Q}^k$, we let $f_{\mathbf{t}_0}$, $g_{\mathbf{t}_0}$, $K_{\mathbf{t}_0}$, $L_{\mathbf{t}_0}$, and $G_{\mathbf{t}_0}$ denote the associated objects obtained under specialization.
\[thm:hit\] With notation as above, suppose $\mathbf{t}_0$ is such that $g_{\mathbf{t}_0}$ is irreducible over $\mathbb{Q}$. Then the permutation representations of $G$ and $G_{\mathbf{t}_0}$ acting on the roots of $f$ and $f_{\mathbf{t}_0}$ are isomorphic. Moreover, the above hypothesis holds for a proportion $1-o_H(1)$ of $\mathbf{t}$ inside any rectangular region in $\mathbb{Z}^k$ whose shortest side has length $H$.
This is classical, and we take the last claim (i.e., that $g_{\mathbf{t}_0}$ is irreducible for almost all $\mathbf{t}_0$) as ‘well known’. However, we will make frequent use of the isomorphism of permutation representations, and this feature is less commonly stated. Therefore, we provide a short proof of this fact.
Let $\alpha \in \overline{\mathbb{Q}(\mathbf{t})}$ be a root of $g(\mathbf{t},x)$, so that $L = \mathbb{Q}(\mathbf{t})(\alpha)$. Similarly, let $\beta \in \overline{\mathbb{Q}}$ be a root of $g_{\mathbf{t}_0}$ with $L_{\mathbf{t}_0} = \mathbb{Q}(\beta)$.
Since $L$ is Galois closed over $\mathbb{Q}(\mathbf{t})$, each automorphism $\sigma \in G$ is determined by the unique polynomial $P_\sigma(x) \in \mathbb{Q}(\mathbf{t})[x]$ for which $\deg(P_\sigma) < |G|$ and $\sigma(\alpha) = P_\sigma(\alpha)$. Writing $P_{\sigma, \mathbf{t}_0}(x) \in \mathbb{Q}[x]$ for the polynomial obtained by specializing $\mathbf{t}$ to $\mathbf{t_0}$, we see at once that the map $\widetilde{\sigma} \colon \beta \mapsto P_{\sigma, \mathbf{t}_0}(\beta)$ is an automorphism of $L_{\mathbf{t}_0}$.
The map $\sigma \mapsto \widetilde{\sigma}$ is thus a homomorphism from $G$ to $G_{\mathbf{t}_0}$. It is injective since $g_{\mathbf{t}_0}$ is irreducible, forcing each of the $P_{\sigma, \mathbf{t}_0}(\beta)$ to be distinct. Since $|G| = \mathrm{deg}(g_{\mathbf{t}_0})$, the set $\{P_{\sigma, \mathbf{t}_0}(\beta)\}_{\sigma \in G}$ forms a complete set of conjugates of $\beta$. Thus, the map $\sigma \mapsto \widetilde{\sigma}$ is surjective and hence an isomorphism.
The roots of $f$ can be written in the form $h_i(\alpha)$, where $h_i$ ranges over a set of $d$ polynomials in $\mathbb{Q}(\mathbf{t})[x]$, each of degree less than $|G|$. By construction, if $h$ and $h'$ are any two such polynomials with $\sigma(h(\alpha)) = h'(\alpha)$, we must have $\widetilde{\sigma}(h_{\mathbf{t}_0}(\beta)) = h'_{\mathbf{t}_0}(\beta)$. But the roots of $f_{\mathbf{t}_0}$ are exactly the $h_{i, \mathbf{t}_0}(\beta)$, so that the action of $\sigma$ on the $h_i(\alpha)$ corresponds exactly to the action of $\widetilde{\sigma}$ on the $h_{i, \mathbf{t}_0}(\beta)$. This is our desired isomorphism of permutation representations.
We derive the following important corollary to Theorem \[thm:hit\] that will enable us to populate the Galois groups $\mathrm{Gal}(f(\mathbf{t},x)/\mathbb{Q}(\mathbf{t}))$.
\[cor:cycles\] Suppose $f(\mathbf{t},x)$ is irreducible over $\mathbb{Q}(\mathbf{t})$. If the permutation representation $\mathrm{Gal}(f(\mathbf{t}_0,x) / \mathbb{Q})$ contains an element of a given cycle type for a positive proportion of $\mathbf{t}_0 \in \mathbb{Q}^k$ when ordered by height, then the permutation representation of $\mathrm{Gal}(f(\mathbf{t},x)/\mathbb{Q}(\mathbf{t}))$ must contain an element of the same cycle type.
Corollary \[cor:cycles\] gives a means to show that the Galois group $\mathrm{Gal}(f(\mathbf{t},x)/\mathbb{Q}(\mathbf{t}))$ contains elements with many different cycle types. The following lemma then enables us to show that in many cases, this suffices to guarantee that $\mathrm{Gal}(f(\mathbf{t},x)/\mathbb{Q}(\mathbf{t})) \simeq S_d$.
\[lem:S\_d-criterion\] Suppose that $G$ is a subgroup of $S_d$ such that:
- $G$ contains a $d$-cycle and a transposition; and,
- Either $G$ contains a $(d - 1)$-cycle, or $d \geq 5$ is odd and $G$ contains a $(d - 2)$-cycle.
Then $G = S_d$.
When $G$ contains a $(d - 1)$-cycle, we recall the proof from [@milne Lemma 8.26]. After renumbering, suppose that the $(d - 1)$-cycle is $(1 \ 2 \ 3 \cdots d - 1)$. Since $G$ is transitive, it will contain a conjugate of the transposition of the form $(i \ d)$, for some $i < d$. Conjugating by the $(d - 1)$-cycle and its powers, we see that $G$ will contain $(i \ d)$ for all $i < d$, and these elements generate $S_n$.
Now, suppose instead that $d \geq 5$ is odd and $G$ contains a $(d - 2)$-cycle. If $G$ contains a transposition $(i \ j)$, where the $(d - 2)$-cycle fixes $i$ but not $j$, then an argument similar to that above establishes that $G$ contains the full symmetric group on $i$ and the elements permuted by the $(d - 2)$-cycle. So $G$ contains a $(d - 1)$-cycle and we are reduced to the first case.
Finally, we prove that $G$ must contain such a transposition. By transitivity, $G$ will contain a transposition $(i \ j)$ where the $(d - 2)$-cycle fixes at least one of $i$ and $j$. If it fixes exactly one of them, we’re done. Otherwise, choose a suitable power $\sigma$ of the $d$-cycle so that $\sigma(i) = j$; since $d$ is odd, we have $\sigma(j) = k$ for some $k \neq i, j$. Then $G$ contains $(j \ k)$, which is the desired transposition.
We establish the existence of elements with various cycle types in a few different ways. To obtain transpositions, we make use of the following well-known lemma.
\[lem:trans-criterion\] Let $f(x) \in \mathbb{Z}[x]$ be irreducible and suppose that for some prime $p$ not dividing the leading coefficient of $f$, $p\mid\mid \mathrm{Disc}(f)$. Then the natural permutation representation of $\mathrm{Gal}(f(x)/\mathbb{Q})$ contains a transposition.
Let $L_p$ be the splitting field of $f(x)$ over $\mathbb{Q}_p$. The claim follows upon observing that $L_p$ is a ramified quadratic extension of an unramified extension of $\mathbb{Q}_p$, so that $\mathrm{Gal}(L_p/\mathbb{Q}_p)$ contains a transposition, and recalling the inclusion $\mathrm{Gal}(L_p/\mathbb{Q}_p) \hookrightarrow \mathrm{Gal}(f(x)/\mathbb{Q})$.
To find long cycle types, we use of the theory of Newton polygons. Given a rational polynomial $f(x) = a_d x^d + \dots + a_0$, its [*$p$-adic Newton polygon*]{} is defined to be the lower convex hull of the points $(i,v_p(a_i))$. It is a union of finitely many line segments whose slopes match the valuations of the roots of $f$ over $\overline{\mathbb{Q}_p}$, with multiplicities equal to their horizontal lengths. See [@Neukirch Ch. II.6] for a good reference.
The Newton polygon controls much of the behavior of $\mathrm{Gal}(f(x)/\mathbb{Q}_p)$. For our purposes, the following lemma suffices.
\[lem:newton\] Suppose that the Newton polygon of $f(x)$ as described above contains a line segment of slope $m/n$ with $\mathrm{gcd}(m,n)=1$. Assume that the length of this segment is $n$ and that the denominator of every other slope is coprime to $n$. Then $\mathrm{Gal}(f(x)/\mathbb{Q})$ contains an $n$-cycle.
The hypotheses ensure that the roots of valuation $m/n$ form a set of Galois conjugates over $\mathbb{Q}_p$. Thus, $f(x)$ admits a factorization $f(x) = f_0(x) f_1(x)$ over $\mathbb{Q}_p$, say, where the roots of $f_0(x)$ are the roots of valuation $m/n$. Since the degree of $f_0(x)$ is $n$ by assumption, it must cut out a totally ramified extension of $\mathbb{Q}_p$. The result now follows from the inclusions $\mathrm{Gal}(f_0(x)/\mathbb{Q}_p) \subseteq \mathrm{Gal}(f(x)/\mathbb{Q}_p) \subseteq \mathrm{Gal}(f(x)/\mathbb{Q})$.
Finally, we recall some basic facts concerning polynomial resultants. The [*resultant*]{} of two polynomials $f(x) = a_0 x^n + a_1x^{n-1} + \dots + a_n$ and $g(x) = b_0 x^m + \dots + b_m$ is given by $$\label{eq:result}
\mathrm{Res}(f,g) = a_0^m b_0^n \prod_{f(\alpha)=g(\beta)=0} (\alpha-\beta) \ = (-1)^{nm} b_0^n \prod_{g(\beta) = 0} f(\beta),$$ where the products run over roots of $f$ and $g$, counted with multiplicity. The key lemma here is the following.
\[lem:disc-res\] Let $F(x) = a_0 x^n + \dots + a_n$ be a polynomial. Then $$\mathrm{Disc}(F) = \frac{(-1)^{n(n-1)/2}}{a_0} \mathrm{Res}(F,F^\prime) = (-1)^{n(n-1)/2} n^n a_0^{n-1} \prod_{\beta: F^\prime(\beta)=0} F(\beta).$$
See, e.g., [@Lang Proposition IV.8.5] for the first equality, and the second follows from .
Analysis of Galois groups in a family {#sec:galois}
=====================================
We are finally ready to discuss the family of polynomials we will use to construct points on elliptic curves over number fields. Let $E$ be an elliptic curve given by a Weierstrass equation $y^2 = f(x)$. We define a polynomial $P(x, t) = P_f(x, t) \in \mathbb{Z}[x, t]$ by $$\label{eq:Pf_def}
P_f(x, t) =
\begin{cases}
t^2 x^d - f(x), & \text{ $d$ even}, \\
x^{d - 3} f(x) - t^2, & \text{ $d$ odd, $d\geq 5$}, \\
f(x)-(x +t)^2, & d=3.
\end{cases}$$ By construction, for each specialization $t = t_0 \in \mathbb{Q}$, each of $(x, t_0x^{d/2})$, $(x, t_0 x^{\frac{3 - d}{2}})$, and $(x,x+t_0)$ is respectively a point on $E(K)$, where $$K := \mathbb{Q}[x] / (P(x, t_0)).$$ This construction is exactly what we will use for small degrees, and it is a specialization of the construction we will use for larger $d$. In either case, we wish to argue that, for many choices of $t_0$, $K$ will indeed define an $S_d$-number field. In view of the Hilbert irreducibility theorem, Theorem \[thm:hit\], the key result in this section is thus the following.
\[prop:Sd\] Given $E$, there exists a Weierstrass model $y^2 = f(x)$ of $E$, integral except possibly at a single prime, for which $\mathbb{Q}(t)[x]/ (P(x, t))$ is a field extension of $\mathbb{Q}(t)$ of degree $d$ whose Galois closure has Galois group $S_d$ over ${\mathbb{Q}}(t)$.
The first step is to construct a Weierstrass model for $E$ with various properties to be exploited later.
\[lem:exists\_w\] Given an elliptic curve $E/{\mathbb{Q}}$, an integer $a$, a real number $\alpha$, and any positive $\epsilon > 0$, there exists a rational Weierstrass model $E \colon y^2 = f(x) = x^3 + Bx^2 + Cx + D$ and distinct primes $p_1, p_2, p_3 \nmid 6d(d - 3)N_E$ satisfying the following properties:
(i) The coefficients $B, C, D$ are all in ${\mathbb{Z}}[\frac{1}{p_1}]$.
(ii) We have $p_2 \mid \mid D$ and $p_2 \nmid C$.
(iii) We have $f(x) \equiv (x + a)^3 \pmod{p_3}$.
(iv) The polynomial $f(x)$ is ‘close to’ $(x + \alpha)^3$ in the Euclidean metric; namely, we have $$|B - 3\alpha| < \epsilon, \ \ |C - 3 \alpha^2| < \epsilon, \ \ |D - \alpha^3| < \epsilon.$$
We begin with (ii). Starting with an integral model $y^2 = g(x) := x^3 + ax + b$ for $E$, upon substituting $x + r$ for $x$ we obtain a model of the form $$\label{eq:nice_w_model}
y^2 = f_r(x) = x^3 + 3rx^2 + \big( 3r^2 + a \big) x + \big( r^3 + ar + b \big).$$ By Chebotarev density, we may choose a prime $p_2 \nmid {{\text {\rm Disc}}}(g)$ and some $r \in {\mathbb{Z}}/p_2{\mathbb{Z}}$ for which $g(r) \equiv 0 \pmod {p_2}$ and $g'(r) = 3r^2 + a \not \equiv 0 \pmod {p_2}$. Because $p_2 \nmid g'(r)$, distinct lifts of $r \pmod {p_2^2}$ will yield distinct values of $g(r) \pmod{p_2^2}$, so we may choose a lift of $r$ to ${\mathbb{Z}}$ such that $f_r(x)$ satisfies (ii).
To also obtain (iii), let $p_3$ be any prime not dividing $6d(d-3)\Delta_E p_2$ and replace $f_r(x)$ with $\widetilde{f_r}(x) := p_3^6 f_r\big(\frac{x + a p_2^2 \overline{p_2}^2}{p_3^2} \big)$, where $p_2 \overline{p_2} \equiv 1 \pmod{p_3^2}$.
Finally, let $p_1$ be any prime not dividing $6d(d-3)\Delta_E p_2p_3$. Let $u \in {\mathbb{Z}}[\frac{1}{p_1}]$ be such that $p_2^2 p_3 \mid u$ and such that $|u^i - \alpha^i| < \frac{\epsilon}{4}$ for $i = 1, 2, 3$. Then, for a sufficiently large positive integer $k$, $y^2 = p_1^{-6k} \widetilde{f_r}\big(p_1^{2k}(x + u)\big)$ is a Weierstrass model for $E$ satisfying all the stated properties.
\[lem:large-cycles\] Let $E/\mathbb{Q}$ be an elliptic curve with Weierstrass model in the form guaranteed by Lemma \[lem:exists\_w\]. Then $P(x,t)$ is irreducible over $\mathbb{Q}(t)$. Moreover, if $d$ is even, then the Galois group $\mathrm{Gal}(P(x,t))$ contains both a $d$-cycle and a $(d-1)$-cycle, while if $d$ is odd, it contains both a $d$-cycle and a $(d-2)$-cycle.
Arguing separately for $d$ even and odd, we make various substitutions $t = t_0$ in $P_f(x, t)$, and inspect the resulting Newton polygons over $\mathbb{Q}_p$ with $p = p_2$ as in Lemma \[lem:exists\_w\](ii). We will conclude that $P_f(x, t_0)$ is irreducible over ${\mathbb{Q}}_p$ (and hence over ${\mathbb{Q}}$), and we will exhibit various cycles in the Galois group of $\mathbb{Q}(t)[x]/ (P(x, t)))$ over ${\mathbb{Q}}(t)$ thereby using Corollary \[cor:cycles\].
[*$d \geq 4$ even*]{}: We consider two specializations, namely $t=p^{-d/2}$ and $t=p^{-1}$, from which we obtain a $d$-cycle and a $(d-1)$-cycle, respectively, using Lemma \[lem:newton\] and Corollary \[cor:cycles\]. We present these two $p$-adic Newton polygons in turn.
(0,-1.6) \[xstep = 1, ystep = 0.2\] grid (8.0,0.2); (0, 0) – (8, 0); (0, -1.6) – (8, 0.2); (0,-1.6) circle (0.05) node\[left\] [$(d,-d)$]{}; (8,0.2) circle (0.05) node\[right\] [$(0,1)$]{};
Newton polygon over $\mathbb{Q}_p$ with $t = p^{-d/2}$: a $d$-cycle.
(0,-.4) \[xstep = 1, ystep = 0.2\] grid (8.0,0.2); (0, 0) – (8, 0); (0, -.4) – (7, 0); (7, 0) – (8, 0.2); (0,-0.4) circle (0.05) node\[left\] [$(d,-2)$]{}; (7,0) circle (0.05) node\[below right\] [$(1,0)$]{}; (8,0.2) circle (0.05) node\[right\] [$(0,1)$]{};
Newton polygon over $\mathbb{Q}_p$ with $t = p^{-1}$: a $(d - 1)$-cycle.
[*$d = 3$*]{}: Immediate.
[*$d \geq 5$ odd*]{}: We take $t=p^{-1}$ and $t=p$, obtaining a $d$-cycle and a $(d-2)$-cycle, respectively, again using Lemma \[lem:newton\] and Corollary \[cor:cycles\].
(0,-0.4) \[xstep = 1, ystep = 0.2\] grid (7.0,0); (0, 0) – (7, 0); (0, 0) – (7, -0.4); (0,0) circle (0.05) node\[left\] [$(d,0)$]{}; (7,-0.4) circle (0.05) node\[right\] [$(0,-2)$]{};
Newton polygon over $\mathbb{Q}_p$, with $t = p^{-1}$: a $d$-cycle.
(0,0) \[xstep = 1, ystep = 0.2\] grid (7.0,0.4); (0, 0) – (7, 0); (2, 0) – (7, 0.4); (0,0) circle (0.05) node\[left\] [$(d,0)$]{}; (2,0) circle (0.05) node\[below\] [$(d-2,0)$]{}; (7,0.4) circle (0.05) node\[right\] [$(0,2)$]{};
Newton polygon over $\mathbb{Q}_p$, with $t = p$: a $(d - 2)$-cycle.
This completes the proof.
In view of Lemma \[lem:S\_d-criterion\], to show that $\mathrm{Gal}(P(x,t)/\mathbb{Q}(t)) \simeq S_d$, it remains to show that the Galois group contains a transposition. The key is the following computation. We also recall from Corollary \[cor:sign\] that to control the root numbers of these twists, we wish to control the sign of the discriminant of $P$. We subsume the proof that we may do so into the following lemma.
\[lem:disc-factor\] Given $E$, there exists a Weierstrass model of $E$ of the form given in Lemma \[lem:exists\_w\], such that with $P_f(x, t)$ defined as in , the discriminant of $P_f$ (taken in the variable $x$) is a non-squarefull polynomial in $t$ that assumes both positive and negative values in the interval $|t| \leq 1$. This discriminant is of degree $4$ when $d = 3$ and is otherwise of the form $${{\text {\rm Disc}}}(P_f) = t^{2d - 8} h(t)$$ for a non-squarefull polynomial $h(t)$ of degree $6$.
We consider first the case that $d \geq 5$ is odd. In this case, $P_f$ is monic and its discriminant is found via Lemma \[lem:disc-res\] by taking the resultant of $P_f$ with its derivative $P_f^\prime$; namely, we have $$\mathrm{Disc}(P_f)
= (-1)^{(d-1)/2} d^d \prod_{\beta: P_f^\prime(\beta)=0} P_f(\beta)$$ where the roots are taken with multiplicity. For any Weierstrass model $y^2 = f(x)$ of $E$, we have $P_f^\prime = x^{d-4}[(d-3)f(x) + x f^\prime(x)] =: x^{d-4}g(x)$ for some cubic polynomial $g \in \mathbb{Q}[x]$. Thus, $x=0$ is a root of $P_f^\prime$ with multiplicity $d-4$, and we conclude $$\mathrm{Disc}(P_f) = (-1)^{(d+1)/2}d^d t^{2d-8} \prod_{\beta: g(x) = 0} (\beta^{d-3}f(\beta) - t^2) = (-1)^{(d-1)/2} d^d t^{2d-8} h(t)$$ for some monic degree $6$ polynomial $h \in \mathbb{Q}[t]$. Choosing $f(x) \equiv (x + 1)^3 \pmod{p_3}$ in Lemma \[lem:exists\_w\](iii), we have $P_f \equiv x^{d-3}(x+1)^3 - t^2 \pmod{p_3}$ and ${{\text {\rm Disc}}}(P_f) \equiv {{\text {\rm Disc}}}(x^{d-3}(x+1)^3 - t^2) \pmod{p_3}$. By an argument with resultants similar to the above, we find $$\label{eqn:disc-1}
\mathrm{Disc}(x^{d-3}(x+1)^3 - t^2) = (-1)^{(d-1)/2} t^{2d-4} (d^d t^2 - 27 (d-3)^{d-3}),$$ which is not squarefull when reduced $\pmod{p_3}$. Thus, $\mathrm{Disc}(P_f)$ cannot be squarefull.
To ensure that $\mathrm{Disc}(P_f)$ assumes both positive and negative values in the interval $|t|\leq 1$, choose $f$ close to $(x + 1)^3$ in the Euclidean topology, by Lemma \[lem:exists\_w\](iv). As $\mathrm{Disc}(x^{d-3}(x+1)^3-t^2)$ visibly has the desired property thanks to , so does $P_f(x,t)$ by continuity.
In the case that $d\geq 4$ is even, we exploit the fact that the discriminant of a polynomial and its reciprocal polynomial are the same, i.e. $\mathrm{Disc}(P_f(x)) = \mathrm{Disc}(x^d P_f(1/x))$. The polynomial $x^d P_f(1/x)$ is of essentially the same form as the polynomials $P_f(x)$ for $d$ odd, and exactly the same argument shows that $\mathrm{Disc}(x^d P_f(1/x)) = t^{2d-8} h(t)$ for some sextic polynomial $h$.
To show that $\mathrm{Disc}(P_f)$ is not squarefull, choose $f(x) \equiv (x-1)^3 \pmod{p_3}$. As $P_f \equiv t^2 x^d - (x-1)^3 \pmod{p_3}$ and $$\mathrm{Disc}(t^2x^d - (x-1)^3) = \mathrm{Disc}(x^{d-3}(x-1)^3 + t^2) = (-1)^{d/2}t^{2d-4}(d^dt^2 -27 (d-3)^{d-3}),$$ it follows as in the odd case that $\mathrm{Disc}(P_f)$ is not squarefull. Similarly, by choosing $f$ close to $(x - 1)^3$ in the Euclidean topology, we ensure that $\mathrm{Disc}(P_f)$ assumes both positive and negative values in the interval $|t|\leq 1$.
Finally, if $d=3$, $P_f(x,t) = f(x) - (x+t)^2$ and $\mathrm{Disc}(P_f)=h(t)$ is a degree four polynomial in $t$. Choose a Weierstrass model for $f$ close, in $\mathbb{R}$, to $y^2 = x^3$; since $\mathrm{Disc}(x^3-(x+t)^2) = -t^3(27t+4)$, $h(t)$ will assume positive and negative values inside $|t|\leq 1$. Since a squarefull degree polynomial of degree four is either a square or a fourth power, this also proves that $h(t)$ is not squarefull.
We are now ready to argue that the Galois group of $K$ contains a transposition.
\[lem:exists-trans\] Let $E/\mathbb{Q}$ be an elliptic curve with Weierstrass model given by Lemma \[lem:disc-factor\]. Then $\mathrm{Gal}(P_f(x,t)/\mathbb{Q}(t))$ contains a transposition in its natural permutation representation.
As expected, we use Lemma \[lem:trans-criterion\]. If $E$ is given by a Weierstrass model of the form given by Lemma \[lem:disc-factor\], then $P_f(x,t)$ is irreducible and $\mathrm{Disc}(P_f) = t^{2d-8} h(t)$ for some non-squarefull polynomial $h(t) \in \mathbb{Z}[t]$ of degree $6$, or degree $4$ in the special case $d=3$. Since $h(t)$ is not squarefull, it admits an irreducible factor $h_0(t)$ of multiplicity one. Moreover, the proof of Lemma \[lem:disc-factor\] shows that we may take $h_0(t) \neq t$. If we write $h(t) = h_0(t) h_1(t)$, then only finitely many primes divide the resultant $\mathrm{Res}(h_0(t), th_1(t))$. By the Chebotarev density theorem, there are infinitely many primes $p$ for which $h_0(t)$ admits a root. Let $p$ be such a prime for which $p\nmid \mathrm{Disc}(h_0(t))$ and $p\nmid \mathrm{Res}(h_0(t),th_1(t))$. By the definition of the resultant, we may thus find an integer $t_0$ for which $p\mid\mid h_0(t_0)$ and $p\nmid t_0 h_1(t_0)$. Thus, $p\mid\mid \mathrm{Disc}(P_f(x,t_0))$ and $\mathrm{Gal}(P_f(x,t_0))$ contains a transposition by Lemma \[lem:trans-criterion\]. In particular, this construction shows that $\mathrm{Gal}(P_f(x,t_0)/\mathbb{Q})$ has a transposition for a positive proportion of $t_0 \in \mathbb{Q}$, which by Corollary \[cor:cycles\] implies that $\mathrm{Gal}(P_f(x,t)/\mathbb{Q}(t))$ must also contain a transposition.
Combining Lemmas \[lem:large-cycles\] and \[lem:exists-trans\] with Lemma \[lem:S\_d-criterion\], we conclude Proposition \[prop:Sd\].
Disambiguation via discriminants and small degree fields {#sec:small-degree}
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The main point of this section is to establish the following theorem, which forms part of our main theorem. At the end of this section, we then tweak the proof to obtain a proof of Theorem \[thm:analytic\].
\[thm:small-degree\] Let $E/\mathbb{Q}$ be an elliptic curve let $d\geq 3$ be an integer. There is a constant $c_d>0$ such that for each $\varepsilon =\pm 1$, there are $\gg X^{c_d - \epsilon}$ fields $K\in\mathcal{F}_d(X)$ with $w(E,\rho_K)=\varepsilon$ and $\mathrm{rk}(E(K)) > \mathrm{rk}(E(\mathbb{Q}))$. In particular, we may take $$c_d = \left\{ \begin{array}{ll} 1/3, & \text{if } d=3, \\ 1/4, & \text{if } d=4, \text{ and} \\ (\lceil \frac{d}{2}\rceil +2)^{-1}, & \text{if } d\geq 5. \end{array} \right.$$
Recall that Proposition \[prop:Sd\] yielded a Weierstrass model $y^2 = f(x)$ of $E$ and a polynomial $P_f(x, t)$ of defining an $S_d$-extension of ${\mathbb{Q}}(t)$, such that each specialization $t = t_0 \in \mathbb{Q}$ yields a point on $E(K)$ with $K := \mathbb{Q}[x]/(P_f(x, t_0))$.
We will choose specializations $t_0 = u/v$ where $u$ and $v$ range over integers in a suitably sized box. The next two lemmas, applied to a homogenization of the polynomial $h(t)$ from Lemma \[lem:disc-factor\], will be used to show that the discriminants of the $P_f(x, u/v)$, as polynomials in $x$, represent many different square classes in ${\mathbb{Q}}^{\times} / ({\mathbb{Q}}^{\times})^2$ – and hence that these polynomials generate many different field extensions.
\[lem:greaves\] Let $F(u,v)$ be an integral binary form with each irreducible factor of degree $\leq 6$. Let $M\geq 1$ be a fixed positive integer and let classes $a,b\pmod{M}$ be chosen so that $F(u,v)$ does not admit a constant square factor whenever $u\equiv a\pmod{M}$ and $v \equiv b \pmod{M}$. Let $\Omega \subset [-1,1]^2$ be a smooth domain with volume $\mathrm{vol}(\Omega)$ and for any $U>1$, let $U\cdot\Omega$ denote the dilation of $\Omega$ by $U$. Then there is a positive constant $c_F$, depending on $M$ but independent of $\Omega$, for which $$\label{eq:greaves}
\#\{u,v \in U\cdot \Omega : (u, v) \equiv (a, b) \ ({{\text {\rm mod}}}\ M), F(u,v) \text{ squarefree}\} = c_F \mathrm{vol}(\Omega) U^2 + O\left(\frac{U^2}{(\log U)^{1/3}}\right).$$
This is essentially the main theorem of [@Greaves1992], which is stated in the slightly simpler case $\Omega = (0 ,1]^2$. The result is easily extended to $\Omega = [-1, 1]^2$ by considering $F(\pm u, \pm v)$. Greaves’s proof is then easily modified as follows:
Writing $N(U)$ for the quantity in , Greaves writes $$N(U) = N'(U) + O(E(U)),$$ where the ‘principal term’ $N'(U)$ counts those $(u, v)$ such that $F(u, v)$ has no square factor $p^2$ with $p \leq \frac{1}{3} \log(x)$, and where the ‘tail estimate’ $E(U)$ is an error term.
The quantity $N'(U)$ is easily estimated using inclusion-exclusion and the geometry of numbers, and these methods extend immediately when $[-1, 1]^2$ is replaced with a more general $\Omega$. Meanwhile, the tail estimate for $\Omega$ is bounded by that for $[-1, 1]^2$, and thus the error term may be quoted from [@Greaves1992] without change.
With a further generalization of Lemma \[lem:greaves\] to skew boxes, we could improve our main result for small $d$. For example, when $d = 3$, we have ${{\text {\rm Disc}}}(K) \mid v^2 H(u, v)$ for a quartic form $H$, and we would improve our results if we could replace $U \cdot \Omega$ with a region approximating $[-X^{1/4}, X^{1/4}] \times [- X^{1/6}, X^{1/6}]$.
\[lem:multiplicity\] Let $F(u,v)$ be a homogeneous rational binary form of degree $m$, and let $U,V\geq 1$. For any integer $n$, there are $O_F(U^\epsilon V^\epsilon |n|^\epsilon)$ integral solutions to the equation $F(u,v)=n$ with $|u|\leq U$ and $|v|\leq V$.
We may choose a fixed finite extension $L/{\mathbb{Q}}$ and factorization $$F(u,v) = \frac{1}{k} \prod_{i=1}^m (\alpha_i u + \beta_i v),$$ for some integer $k$ and algebraic integers $\alpha_i, \beta_i \in \mathcal{O}_L$. Observe that if $u,v\in\mathbb{Z}$, then $|\alpha_i u + \beta_i v|_\nu \ll U+V$ for each infinite place $\nu$ of $L$.
Each solution to $F(u,v)=n$ determines a factorization $nk\mathcal{O}_L = \mathfrak{a}_1 \dots \mathfrak{a}_m$ into principal ideals $\mathfrak{a}_i$ of $\mathcal{O}_L$, and there are $O(n^\epsilon)$ such factorizations. Moreover, writing $r$ for the unit rank of $L$, there are at most $O(\log(U+V)^r)$ generators $\gamma_i = \alpha_i u + \beta_i v$ of each ideal $\mathfrak{a}_i$ for which $|\gamma_i|_\nu \ll U+V$ for each infinite place $\nu$. The result follows.
We are now ready to prove the main theorem of this section.
Let $E$ be given by the Weierstrass model produced in Proposition \[prop:Sd\], so that the polynomial $P_f(x,t)$ defined in cuts out an $S_d$ extension of $\mathbb{Q}(t)$. The polynomial $v^2 P_f(x,u/v)$ has coefficients integral away from a single fixed prime, and by Lemma \[lem:disc-factor\], it has discriminant of the form $u^{2d-8}v^{2d-2} H(u,v)$ for some binary sextic form $H(u,v)$ that is not squarefull. For $d = 3$, the discriminant is instead of the form $v^4 H(u, v)$ with $H$ quartic instead of sextic.
By Hilbert irreducibility (Theorem \[thm:hit\]), for asymptotically 100% of pairs $(u, v)$ with $|u|, |v| \leq U$, we will have that $K={\mathbb{Q}}[x]/(P_f(x,u/v))$ is an $S_d$-field extension of ${\mathbb{Q}}$. We have $v_p({{\text {\rm Disc}}}(K)) \leq p - 1$ for any tamely ramified prime $p$, and ${{\text {\rm Disc}}}(K)$ and ${{\text {\rm Disc}}}(v^d P_f(x,u/v))$ differ by a rational square. Therefore, ${{\text {\rm Disc}}}(K)$ divides a bounded factor times either $u^{d-2}v^{d-2} H(u,v)$ or $u^{d-1}v^{d-1} H(u,v)$, depending on whether $d$ is even or odd. Thus, there is some constant $q_{E,d} > 0$ such that taking $U = q_{E,d} X^{c_d/2}$ guarantees that $|D_K| \leq X$. Finally, Lemmas \[lem:greaves\] and \[lem:multiplicity\] guarantee that $H(u, v)$, and hence ${{\text {\rm Disc}}}(K)$, represents $\gg X^{c_d - \epsilon}$ distinct square classes, so that $\gg X^{c_d - \epsilon}$ distinct fields $K$ are produced.
By Lemma \[lem:fractional-points\], we have ${{\text {\rm rk}}}(E(K)) > {{\text {\rm rk}}}(E({\mathbb{Q}}))$ for all but a bounded number of these $K$. It remains to control the sign of the root number. Lemma \[lem:disc-factor\] shows that both regions $\Omega^\pm:=\{(u,v) \in [-1,1]^2 : \pm \mathrm{Disc}(P_f(x,u/v))>0\}$ have positive volume. By Corollary \[cor:sign\], there exists a residue class $(u_0, v_0) \pmod{M}$ (with $M$ a suitably large power of $N_E$), for which $w(E,\rho_{K_0})$ is determined by the sign of $\mathrm{Disc}(P_f(x,u/v))$ whenever $(u, v) \equiv (u_0, v_0) \pmod{M}$. We incorporate the conditions that $(u, v) \in \Omega^\pm$ and that $(u, v) \equiv (u_0, v_0) \pmod{M}$ into our application of Lemma \[lem:greaves\], and the remainder of our proof is unchanged.
Using very similar ideas, we prove Theorem \[thm:analytic\] on non-abelian cubic twists with analytic rank two.
The proof follows that of Theorem \[thm:small-degree\], except that to apply Corollary \[cor:cubic-ranks\] we must produce [complex]{} cubic fields $K$ for which $w(E, \rho_K) = +1$. Accordingly, we use Lemma \[lem:sign-jacobi\] instead of Corollary \[cor:sign\] to control the root number $w(E,\rho_K)$. In the event that $E$ has CM, there is one exceptional quadratic resolvent for which we may not apply Corollary \[cor:cubic-ranks\]. However, the quadratic resolvent of $K \in \mathcal{F}_3(X)$ is determined by the squarefree part of its discriminant. We distinguish fields in the above proof precisely by the squarefree part of their discriminant, so this one possible exceptional field has no impact on the result.
In Lemma \[lem:exists\_w\], after (ii) but before the remaining steps, we replace $f(x)$ with $N_E^6 f(x N_E^{-2})$, allowing us to demand that $f(x) \equiv x^3 \pmod{N_E}$, so that $$\mathrm{Disc}(P_f(x,t)) \equiv \mathrm{Disc}(x^3-(x+t)^2) \equiv -t^3(27t+4) \pmod{N_E}.$$ For each odd prime $p$ for which $p \mid \mid N_E$, an easy argument shows that the polynomial $27t^2+4t$ represents both squares and nonsquares $\pmod{p}$. Since by hypothesis there is at least one such prime, suitable congruence conditions on $t \pmod{N_E}$ may be chosen to guarantee that both $\mathrm{gcd}(\mathrm{Disc}(P_f(x,t)),N_E) = 1$ and $\left(\frac{\mathrm{Disc}(P_f(x,t))}{N_E}\right) = -1$. The result now follows as in the proof of Theorem \[thm:small-degree\].
Geometry of numbers and large degree fields {#sec:large-degree}
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In this section we prove the following complement to Theorem \[thm:small-degree\]:
\[thm:large-degree\] Let $E/\mathbb{Q}$ be an elliptic curve let $d\geq 5$ be an integer. Then, for each $\varepsilon =\pm 1$, there are $\gg X^{c_d - \epsilon}$ fields $K\in\mathcal{F}_d(X)$ with $w(E,\rho_K)=\varepsilon$ and $\mathrm{rk}(E(K)) > \mathrm{rk}(E(\mathbb{Q}))$, with $$c_d = \frac{1}{4} - \frac{d^2+4d-2}{2d^2(d-1)}.
$$ If $d \geq 16052$, then we may take $$c_d = \frac{1}{4} - \frac{1}{2d}.
$$
The result is identical to Theorem \[thm:small-degree\] except for the value of $c_d$. Here it is an increasing function of $d$, and this result improves upon Theorem \[thm:small-degree\] for $d \geq 9$.
Our strategy is to adapt Ellenberg and Venkatesh’s proof of a lower bound [@EV] for $\#\mathcal{F}_d(X)$. They produce many algebraic integers $\alpha$ for which $|{{\text {\rm Disc}}}({\mathbb{Z}}[\alpha])| < X$, and then, for each field $K$, bound from above the number of $\alpha$ so constructed with ${\mathbb{Q}}(\alpha) = K$. We adapt their construction so as to produce only those $\alpha$ for which there are polynomials $F(x),G(x) \in \mathbb{Z}[x]$ such that $\big(\alpha,\frac{F(\alpha)}{G(\alpha)}\big)$ is a point on $E(\overline{\mathbb{Q}})$. Equivalently, if $E$ is given by the Weierstrass model $E\colon y^2=f(x)$, we only count those $\alpha$ arising as solutions to $F(x)^2 - f(x)G(x)^2 = 0$ for some $F$ and $G$.
The construction
----------------
Let $d \geq 4$. Using Lemma \[lem:exists\_w\] to choose a Weierstrass model for $E/\mathbb{Q}$, we consider the following family of polynomials. Fix a parameter $Y$ to be chosen shortly. The construction is slightly different depending on whether $d$ is odd or even.
If $d$ is even, we take:
- $F(x) = x^{\frac{d}{2}} + a_1 x^{\frac{d}{2} - 1} + a_2 x^{\frac{d}{2} - 2} + \dots + a_{d/2}$, an integral monic polynomial of degree $\frac{d}{2}$ with and $|a_k| \leq Y^k$ for each $k$.
- $G(x) = b_2 x^{\frac{d}{2} - 2} + b_3 x^{\frac{d}{2} - 3} + \dots + b_{d/2}$, an integral polynomial of degree $\frac{d}{2} - 2$ with $|b_k| \leq Y^{k - \frac{3}{2}}$ for each $k$.
If $d$ is odd, we instead take:
- $G(x) = x^{\frac{d - 3}{2}} + b_1 x^{\frac{d - 3}{2} - 1} + b_2 x^{\frac{d - 3}{2} - 2}
+ \dots +
b_{\frac{d - 3}{2}}$, with $|b_k| \leq Y^k$ for each $k$.
- $F(x) = a_0 x^{\frac{d - 1}{2}} + a_1 x^{\frac{d - 1}{2} - 1}
+ \dots +
a_{\frac{d - 1}{2}}$, with $|a_k| \leq Y^{k + \frac{1}{2}}$ for each $k$.
In either case, the polynomial $$\label{eq:def_H}
H(x) := F^2 - f G^2 = x^d + c_1 x^{d - 1} + c_2 x^{d - 2} + \cdots + c_d$$ has $|c_k| \ll_{f, d} Y^k$ for each $k$, so that $|{{\text {\rm Disc}}}(H)| \ll_{f, d} Y^{d(d -1)}$. Thus, we will ultimately take $Y=q_{f,d} X^{1/d(d-1)}$ for a suitable constant $q_{f,d}$. In general $H$ is not required to have integral coefficients (because $f$ isn’t), but $H$ will have rational coefficients whose denominators are bounded above by a fixed constant (depending on $E$ and $d$).
Let $R$ be the polynomial ring obtained by adjoining all the $a_i$ and $b_j$ as indeterminates to ${\mathbb{Z}}[\frac{1}{p_1}]$.
Then, as a polynomial in $R[x]$, $H$ is irreducible with Galois group $S_d$.
It suffices to exhibit specializations of the $a_i$ and $b_j$ to the polynomials described in , proved to be irreducible over ${\mathbb{Q}}(t)$ with Galois group $S_d$.
When $d$ is odd, choose $F = t$ and $G = x^{\frac{d - 3}{2}}$. This yields $H = - (x^{d - 3} f(x) - t^2)$, which is the same as up to a sign.
When $d$ is even, choose $F = x^{d/2}$ and $G = t$, obtaining $H(x, t) = x^d - t^2 f(x)$. The polynomial $t^2 H(x, t^{-1}) = t^2 x^d - f(x)$ also appeared in and was previously proved irreducible over ${\mathbb{Q}}(t)$ with Galois group $S_d$. Since the map $t \rightarrow t^{-1}$ induces an automorphism of ${\mathbb{Q}}(t)$, the same is true of $H(x, t)$.
The following lemma establishes that we can control the discriminant of $H$, thereby allowing us to use Corollary \[cor:sign\] to control the root number $w(E, \rho_K)$.
\[lem:control\_sign\] Suppose we are given a fixed choice of $H_0(x)$ as in , a positive integer $M$ coprime to the denominators of the coefficients of $f$, and a choice of sign $\delta \in \pm 1$.
Then, as $Y \rightarrow \infty$, a positive proportion of the polynomials $H$ constructed above satisfy $H \equiv H_0 \pmod{M}$ and $\textnormal{sgn}({{\text {\rm Disc}}}(H)) = \delta$.
We will exhibit choices of $F$ and $G$ with the $a_i$ and $b_j$ real numbers in $(-1, 1)$ for which ${{\text {\rm Disc}}}(F^2 - x^3 G^2)$ is positive and for which it is negative.
Once this is done, the lemma quickly follows: for each $H$, define $H_Y(x) = Y^{-d} H(xY)$; equivalently, divide each $c_i$ in by $Y^i$. Then $\textnormal{sgn}({{\text {\rm Disc}}}(H)) = \textnormal{sgn}({{\text {\rm Disc}}}(H_Y))$. Since $Y^{-3} f(xY)$ tends to $x^3$ as $Y \rightarrow \infty$, and since the discriminant of a polynomial is a continuous function of the coefficients, a positive proportion of the $H$ constructed will satisfy $H \equiv H_0 \pmod{M}$ and will have $H_Y$ sufficiently close to $F^2 - x^3 G^2$ as to guarantee that their discriminants are of the same sign.
Our $F$ and $G$ are chosen in an ad hoc manner. When $d$ is even, choose $$F(x) = \Big( x^{d/2} + \frac{1}{100} \Big), \ \ G(x) = \lambda,$$ and set $T(x) := F(x)^2 - x^3G(x)^2$. We now recall Descartes’s *rule of signs*, that the number of positive roots of a real polynomial is bounded by the number of sign changes in its consecutive non-zero coefficients. When $\lambda = \frac{1}{100}$, $T(x)$ is always positive and has no real roots. When $\lambda = \frac{9}{10}$, $T(x)$ has exactly two real roots by Descartes’s rule of signs and because $T(\frac{1}{2})$ is negative. Therefore, these two choices of $\lambda$ lead to opposite signs for ${{\text {\rm Disc}}}(T)$.
Similarly, when $d$ is odd, choose $$F(x) = (x + \lambda)^2, \ \ G(x) = x^{\frac{d - 3}{2}},$$ and again set $T(x) = F(x)^2-x^3G(x)$. Then $T(x)$ has an odd number of real roots. When $\lambda = \frac{1}{10}$, $T$ has exactly one real root by Descartes’s rule. When $\lambda = - \frac{1}{10}$, $T$ may have either one or three roots. We have $T(0) > 0$, $T(\frac{1}{10}) < 0$, and $T(\frac{1}{5}) > 0$, so that $T(x)$ has three real roots in this case. We once again obtain opposite signs for ${{\text {\rm Disc}}}(T)$.
Bounding multiplicities
-----------------------
There are two sources of multiplicity with which a single field $K$ can arise from multiple choices of the $a_i$ and $b_j$. We first bound the number of times in which a given polynomial $H$ can occur in the construction .
\[lem:bound\_mult\] Let $H(x)$ and $f(x)$ be polynomials in ${\mathbb{Z}}[\frac{1}{p_1}][x]$ of degree $d$ and $3$ respectively. Then the number of polynomials $F(x), G(x) \in {\mathbb{Z}}[x]$ with $F^2 - fG^2 = H$ and with at least one of $F$ and $G$ monic, is $O_d(1)$.
To each way of writing $H = F^2 - fG^2$ we associate the factorization $H = (F - G \sqrt{f})(F + G \sqrt{f})$ in the coordinate ring ${\mathbb{C}}[x][\sqrt{f}] = {\mathbb{C}}[x, y]/(y^2 - f)$ of our elliptic curve. This ring is a Dedekind domain [@lorenzini Theorem II.5.10], so the ideal $(H)$ factors uniquely as a product of prime ideals, each of the form $(x - x_i, y - y_i)$ with $y_i^2 = f(x_i)$. Moreover, the curves $H = 0$ and $y^2 = f$ intersect in $2d$ points, counted with multiplicity, which implies that at most $2d$ prime ideals can occur in this factorization.
Since the ideal $(F - G \sqrt{f})$ is a product of some subset of these primes, there are at most $2^{2d}$ possibilities for it, and this ideal determines $F$ and $G$ up to a constant multiple. Since one of $F$ or $G$ is required to be monic, $F$ and $G$ are therefore determined in at most $2^{2d + 1}$ ways.
We now bound the number of different polynomials $H$ yielding the same field $K$. This is a variation of [@EV Lemma 3.1], incorporating an improvement that was suggested there.
The restriction won’t be used in this bound, so we consider the larger set of polynomials $$S(Y; S_d) := \{ f = x^d + c_1 x^{d - 1} + \cdots + c_d \in {\mathbb{Z}}\Big[\frac{1}{p_1}\Big][x] \ : \ |c_i| \leq (CY)^d \}$$ whose denominators are bounded by those of $f(x)$, subject to the condition that $K := \mathbb{Q}[x]/(f(x))$ is a field with Galois group $S_d$, and where $C$ is a constant depending only on $f$ and $d$. By construction, this set contains all polynomials constructed in . For each number field $K$ of degree $d$, we then define $$M_K(Y) := \#\{ f \in S(Y;S_d) : \mathbb{Q}[x]/(f(x)) \simeq K\}$$ to be the multiplicity with which $K$ is so constructed.
\[prop:shape\] We have $$\label{eq:MK_bound}
M_K(Y) \ll \max(Y^{d}\mathrm{Disc}(K)^{-1/2}, Y^{d/2}).$$
Embed $\mathcal{O}_K \hookrightarrow \mathbb{R}^n$ in the usual way, and let $\lambda_0, \lambda_1,\dots,\lambda_{d-1}$ denote the successive minima of $\mathcal{O}_K$, corresponding to vectors $\alpha_0 = 1, \alpha_1, \dots, \alpha_{d - 1} \in \mathcal{O}_K$. Note that all roots $\alpha$ of polynomials counted by $S(Y; S_d)$ are bounded rational multiples of algebraic integers with $|\alpha| \ll Y$.
If $\lambda_{d - 1} \ll Y$, then an integral basis for ${\mathcal{O}}_K$ fits inside a box of side length $O(Y)$, so that $M_K(Y) \ll Y^d \mathrm{Disc}(K)^{-1/2}$. Otherwise, let $k < d - 1$ be the largest integer for which $\lambda_k \leq Y$. Then $$\label{eq:gon}
M_K(Y) \ll \frac{Y^{k + 1}}{\lambda_1 \lambda_2 \cdots \lambda_k}
\ll \frac{Y^{k + 1}}{\mathrm{Disc}(K)^{1/2}} \cdot \lambda_{k+1}\dots\lambda_{d-1},$$ since $\lambda_1 \dots \lambda_{d-1} \asymp \mathrm{Disc}(K)^{1/2}$. If $k \leq \frac{d}{2} - 1 $, then $M_K(Y) \ll Y^{\frac{d}{2}}$ by the first bound above. Otherwise, by [@BSTTTZ Theorem 3.1], we have $Y < \lambda_{d-1} \ll \mathrm{Disc}(K)^{1/d}$, so that $$\begin{aligned}
M_K(Y)
\ll & \ \frac{Y^{k + 1}}{\mathrm{Disc}(K)^{1/2}} \mathrm{Disc}(K)^{\frac{d -k - 1}{d}} \\
= & \ \mathrm{Disc}(K)^{\frac{1}{2}} \big( Y /\mathrm{Disc}(K)^{\frac{1}{d}} \big)^{k + 1} \\
\ll & \ \mathrm{Disc}(K)^{\frac{1}{2}} \big( Y/ \mathrm{Disc}(K)^{\frac{1}{d}} \big)^{\frac{d}{2}} \\
= & \ Y^{\frac{d}{2}}.\end{aligned}$$
Finally, we require bounds on the number of $S_d$-fields of bounded discriminant.
\[prop:ev\][@schmidt; @EV] We have $$\#\mathcal{F}_d(X) \ll X^{\alpha(d)},$$ where we may take $$\label{eq:nf_bound}
\alpha(d) =
\begin{cases}
\frac{d + 2}{4} & \textnormal{for any $d \geq 3$, and} \\
\frac{d}{4} - \frac{3}{4} + \frac{1}{2d} & \textnormal{for any $d \geq 16052$}.
\end{cases}$$
The first bound is due to Schmidt [@schmidt]. In [@EV (2.6)], Ellenberg and Venkatesh prove for any $d$ that for any positive integers $r$ and $k$ satisfying $$\label{eqn:ev-constraint}
{r + k \choose r} > \frac{d}{2}$$ one may take $$\label{eq:EV_details}
\alpha(d) = \frac{4k}{d - 2} \cdot {r + 4k \choose r }.$$ One immediately checks that the choice $r = 2$, $k = \lceil \sqrt{d} - 1 \rceil$ satisfies for $d > 129^2 = 16641$ and that is stronger than . By computer one further checks that for $d \geq 16052$, there is some $k$ satisfying with $r=2$ for which yields .
The above bounds are far from sharp, but the second bound on $\alpha(d)$ in is enough in our proof. We expect that improvements to [@EV (2.6)], and hence to the range $d \geq 16052$, should be possible.
Assembling the ingredients
--------------------------
Write $N_{E,d}(X)$ for the number of degree $d$, $S_d$-number fields $K$ with $|\mathrm{Disc}(K)| < X$ that are cut out by a $\overline{\mathbb{Q}}$-point of $E$.
We put the preceding steps together as follows:
- The number of choices for the $a_i$ and $b_j$ is $\asymp Y^c$, where for $d$ even we compute that $$c = \sum_{i = 1}^{d/2} i + \sum_{j = 2}^{d/2} \left( j - \frac{3}{2} \right)
= \frac{d^2}{4} - \frac{d}{4} + \frac12,$$ and a similar computation with $d$ odd yields the same result.
- By Hilbert irreducibility (Theorem \[thm:hit\]) and Lemma \[lem:bound\_mult\], we therefore obtain $\asymp Y^c$ different $\alpha$ as roots of polynomials $H(x)$ which generate $S_d$ fields, and for which $(\alpha, \frac{F(\alpha)}{G(\alpha)})$ is a point on $E(\overline{\mathbb{Q}})$. Since these polynomials have bounded denominators, the discriminant of each of these polynomials, and thus of the fields themselves, is $\ll Y^{d^2 - d}$. Write $$X := C_1 Y^{d^2 - d}$$ for a bound on these discriminants, where $C_1$ is a constant depending only on $f$ and $d$.
- Following the strategy in (3.2) of [@EV], by Proposition \[prop:shape\] we therefore have $$\label{eq:EV_strategy}
\sum_{|{{\text {\rm Disc}}}(K)| \leq X} M_K(Y)
\gg Y^c,$$ where the sum is over the fields ${\mathbb{Q}}(\alpha)$ generated by the $\alpha$ as described above, which is a subset of the fields counted by $N_{E,d}(X)$.
We are now ready to finish. We first use Propositions \[prop:shape\] and \[prop:ev\] to bound the contribution to from fields of small discriminant. With $\alpha(d) = \frac{d+2}{4}$ in , we have for $T \leq Y^d$ that $$\label{eq:ld_schmidt}
\sum_{\mathrm{Disc}(K) \leq T} M_K(Y) \ll \sum_{\mathrm{Disc}(K)\leq T} \frac{Y^{d}}{\mathrm{Disc}(K)^{1/2}} \ll Y^d T^{\frac{d+2}{4}-\frac{1}{2}} = Y^d T^{d/4},$$ which is $o(Y^c)$ with the choice $T = Y^{d - 5 + \frac{2}{d} - \epsilon}$. We thus have from that $$\label{eq:EV_strategy_2}
\sum_{T < |{{\text {\rm Disc}}}(K)| \leq X} M_K(Y)
\gg Y^c.$$ By Proposition \[prop:shape\], $M_K(Y) \ll Y^d/T^{1/2}$ for each $K$ in the sum, and a bit of algebra shows that $$N_{E,d}(X)
\gg Y^c \big( Y^d/T^{1/2} \big)^{-1} \gg X^{\gamma - \epsilon}$$ with $$\label{eq:gamma_schmidt}
\gamma
= \frac{c - d + \frac{1}{2}(d-5+2/d)}{d^2-d}
= \frac{1}{4} - \frac{d^2+4d-2}{2d^2(d-1)}.$$ This yields the stated value of $c_d$ in Theorem \[thm:large-degree\] and Theorem \[thm:general\].
If we instead assume the slightly better bound $\alpha(d) = \frac{d}{4} - \frac{3}{4} + \frac{1}{2d}$ as in the hypotheses of Theorem \[thm:field-improvement\], then we find that the contribution from those fields $K$ with $\mathrm{Disc}(K) \leq T$ is $o(Y^c)$ for any $T \ll Y^{d-\epsilon}$. In we now have $M_K(Y) \ll Y^{d/2+\epsilon}$ for each $K$, yielding $N_d(X) \gg X^{\gamma - \epsilon}$ with $$\label{eq:gamma_linnik}
\gamma = \frac{c - \frac{d}{2}}{d^2 - d}
=\frac{1}{4} - \frac{1}{2d}.$$
Combined with Lemma \[lem:fractional-points\], this yields Theorem \[thm:large-degree\] apart from the claim about the root number $w(E,\rho_K)$. To control the root number, we use Lemma \[lem:control\_sign\]. By Corollary \[cor:sign\], there is some fixed power $M$ of the conductor $N_E$ such that if $F(x)$ and $G(x)$ lie in fixed congruence classes $\pmod{M}$, then the root number $w(E,\rho_K)$ depends only on the sign of the discriminant of $H(x)$. Therefore, Lemma \[lem:control\_sign\] implies, for each $\varepsilon = \pm 1$, that a positive proportion of the fields $K$ counted by $N_{E,d}(X)$ have $w(E,\rho_K) = \varepsilon$. This is Theorem \[thm:large-degree\].
Limitations and conditional improvements
----------------------------------------
Let $M_{E,K}(Y)$ be the multiplicity with which a given field $K$ arises from the construction . If we had the bound $M_{E, K}(Y) \ll Y^\epsilon$, then this would yield Theorem \[thm:general\] with $$\label{eqn:cd-limit}
c_d = \frac{c}{d^2-d} = \frac{1}{4}+ \frac{1}{2(d^2-d)},$$ the limitation of our method at present. We do not know how to establish this bound on $M_{E,K}(Y)$ unconditionally, even on average over $K$, but we can show that this follows from well known open conjectures.
One feature we have heretofore ignored is that the points $(\alpha, \frac{F(\alpha)}{H(\alpha)})$ are integral away from a single rational prime. We can bound the number of such points using a special case of a result of Helfgott and Venkatesh [@HelfgottVenkatesh Theorem 3.8]. Let $S$ be a finite set of places of $\mathbb{Q}$. Then, for each degree $d$ field $K$, the number of $K$-rational points on $E$ with canonical height at most $h$ and which are integral at all places not lying over $S$, is $$\label{eqn:helfvenk}
O_{S,f,d}\left((1+\log h)^2 e^{.28 \cdot \mathrm{rk}(E(K))}\right),$$ where the implied constant depends on $d$, $S$, and the Weierstrass equation $E \colon y^2 = f(x)$. In our case, the points $(\alpha, \frac{F(\alpha)}{G(\alpha)})$ have canonical height $\ll \log Y$. Thus, implies that $$M_{E,K}(Y) \ll Y^{\epsilon + 0.28 \frac{\mathrm{rk}(E(K))}{\log Y}}.$$ We expect the rank of $E(K)$ to be $o(\log D_K) = o(\log Y)$ for every $K$, from which we would obtain $M_{E,K}(Y) \ll Y^\epsilon$. This would yield Theorem \[thm:general\] with $c_d$ as in .
Unfortunately, this pointwise bound on the rank appears to be out of reach of algebraic methods. However, we may deduce it from the conjectural bound $\#\mathrm{Cl}(K(E[2]))[2] \ll \mathrm{Disc}(K(E[2]))^\epsilon$ for each $K\in \mathcal{F}_d(X)$. In particular, the rank of $E(K)$ is bounded by that of the $2$-Selmer group $\mathrm{Sel}_2(E_K)$. By a classical $2$-descent [@Silverman Proposition X.1.4] we have in turn that $|\mathrm{Sel}_2(E_K)| \ll |\mathrm{Cl}(K(E[2]))[2]|^2$. The field $K(E[2])$ is at most a degree $6$ extension of $K$ and is unramified away from $2\Delta_E$, so its discriminant satisfies $\mathrm{Disc}(K(E[2])) \ll \mathrm{Disc}(K)^6$. We therefore have the chain of inequalities $$\mathrm{rk}(E(K))
\leq \mathrm{rk}(\mathrm{Sel}_2(E_K))
\ll_{d,f} \log |\mathrm{Cl}(K(E[2]))[2]|
\ll_! \epsilon \log (\mathrm{Disc}(K))
\ll_d \epsilon \log Y,$$ where only the inequality marked $\ll_!$ is conjectural. Combined with , this would yield $M_{E,K}(Y) \ll Y^\epsilon$, and thereby that is admissible in Theorem \[thm:general\].
Alternatively, if we assume that the $L$-function $L(s,E,\rho_K)$ is entire, then the Birch and Swinnerton-Dyer conjecture provides an analytic way of accessing the rank of $E(K)$. Unfortunately, here too we run into an obstacle, with unconditional methods only being able to show that the analytic rank is $O(\log N_E^{d-1}D_K^2) = O(\log Y)$. However, if we are willing to assume that $L(s,E,\rho_K)$ satisfies the generalized Riemann hypothesis, then from [@IwaniecKowalski Proposition 5.21], we obtain the slight improvement $$\mathrm{ord}_{s=1/2}L(s,E,\rho_K) \ll \frac{\log Y}{\log\log Y},$$ which is sufficient to conclude that $M_{E,K}(Y) \ll Y^\epsilon$.
The outcome of this discussion is the following proposition.
Let $E/\mathbb{Q}$ be an elliptic curve and let $K \in \mathcal{F}_d(X)$. Suppose that either $L(s,E,\rho_K)$ is entire and satisfies both the Birch and Swinnerton-Dyer conjecture and the generalized Riemann hypothesis, or that $\#\mathrm{Cl}(K(E[2]))[2] \ll D_K^\epsilon$. Then $M_{E,K}(Y) \ll Y^\epsilon$.
In particular, if either holds for all $K \in \mathcal{F}_d(X)$, then Theorem \[thm:general\] holds with $$c_d = \frac{1}{4}+ \frac{1}{2(d^2-d)}.$$
| {
"pile_set_name": "ArXiv"
} |
DESY 05-127\
IPPP/05/45\
DCPT/05/90
[**Telltale Traces of U(1) Fields\
in Noncommutative Standard Model Extensions\
**]{}
[**Joerg Jaeckel$^1$, Valentin V. Khoze$^2$ and Andreas Ringwald$^1$**]{}
*$^1$Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, D-22607 Hamburg, Germany\
$^2$Department of Physics and IPPP, University of Durham, Durham, DH1 3LE, UK*
[[email protected], [email protected], [email protected]]{}
[Restrictions imposed by gauge invariance in noncommutative spaces together with the effects of ultraviolet/infrared mixing lead to strong constraints on possible candidates for a noncommutative extension of the Standard Model. In this paper, we study a general class of 4-dimensional noncommutative models consistent with these restrictions. Specifically we consider models based upon a gauge theory with the gauge group ${\textrm U}(N_1)\times {\textrm U}(N_2) \times \ldots \times {\textrm U}(N_m)$ coupled to matter fields transforming in the (anti)-fundamental, bi-fundamental and adjoint representations. Noncommutativity is introduced using the Weyl-Moyal star-product approach on a continuous space-time. We pay particular attention to overall trace-U(1) factors of the gauge group which are affected by the ultraviolet/infrared mixing. We show that, in general, these trace-U(1) gauge fields do not decouple sufficiently fast in the infrared, and lead to sizable Lorentz symmetry violating effects in the low-energy effective theory. Making these effects unobservable in the class of models we consider would require pushing the constraint on the noncommutativity mass scale far beyond the Planck mass ($M_{\textrm{NC}}\gtrsim 10^{100}\, M_{\textrm{P}}$) and severely limits the phenomenological prospects of such models.]{}
Introduction and discussion of results
======================================
Gauge theories on spaces with noncommuting coordinates $$[x^\mu,x^\nu]=i\,\theta^{\mu\nu} \ ,$$ provide a very interesting new class of quantum field theories with intriguing and sometimes unexpected features. These noncommutative models can arise naturally as low-energy effective theories from string theory and D-branes. As field theories they must satisfy a number of restrictive constraints detailed below, and this makes them particularly interesting and challenging for purposes of particle physics model building. For general reviews of noncommutative gauge theories the reader can consult e.g. Refs. [@Seiberg:1999vs; @Douglas:2001ba; @Szabo:2001kg].
There are two distinct approaches used in the recent literature for constructing quantum field theories on noncommutative spaces. The first approach uses the Weyl-Moyal star-products to introduce noncommutativity. In this case, noncommutative field theories are defined by replacing the ordinary products of all fields in the Lagrangians of their commutative counterparts by the star-products $$(\phi * \varphi) (x) \equiv \phi(x)\ e^{{i\over 2}\theta^{\mu\nu}
\stackrel{\leftarrow}{\partial_\mu}
\stackrel{\rightarrow}{\partial_\nu}} \ \varphi(x) \ . \label{stardef}$$ Noncommutative theories in the Weyl-Moyal formalism can be viewed as field theories on ordinary commutative spacetime. For example, the noncommutative pure gauge theory action is $$S = -{1\over 2g^2}\int d^{4} x \ \Tr ( F_{\mu \nu}* F^{\mu \nu}
) \ , \label{pureym}$$ where the commutator in the field strength also contains the star-product. The important feature of this approach is the fact that phase factors in the star-products are not expanded in powers of $\theta$ and the $\theta$ dependence in the Lagrangian is captured entirely. This ability to work to all orders in $\theta$ famously gives rise to the ultraviolet/infrared (UV/IR) mixing [@Minwalla:1999px; @Matusis:2000jf] in the noncommutative quantum field theory which we will review below.
The second approach to noncommutativity does not employ star-products. It instead relies [@Madore:2000en; @Calmet:2001na] on the Seiberg-Witten map which represents noncommutative fields as a function of $\theta$ and ordinary commutative fields. This approach essentially reduces noncommutativity to an introduction of an infinite set of higher-dimensional (irrelevant) operators, each suppressed by the corresponding power of $\theta$, into the action. There are two main differences compared to the Weyl-Moyal approach. First, in practice one always works with the first few terms in the power series in $\theta$ and in this setting the UV/IR mixing cannot be captured. Second, the Seiberg-Witten map is a non-linear field transformation. Therefore, one expects a non-trivial Jacobian and possibly a quantum theory different from the one obtained in the Weyl-Moyal approach. In the rest of this paper we will concentrate on the Weyl-Moyal approach.
In the context of Weyl-Moyal noncommutative Standard Model building, a number of features of noncommutative gauge theories have to be taken into account which are believed to be generic [@Khoze:2004zc]:
1. the mixing of ultraviolet and infrared effects [@Minwalla:1999px; @Matusis:2000jf] and the asymptotic decoupling of U(1) degrees of freedom [@Khoze:2000sy; @Hollowood:2001ng] in the infrared;
2. the gauge groups are restricted to U($N$) groups [@Matsubara:2000gr; @Armoni:2000xr] or products of thereof;
3. fields can transform only in (anti-)fundamental, bi-fundamental and adjoint representations [@Gracia-Bondia:2000pz; @Terashima:2000xq; @Chaichian:2001mu];
4. the charges of matter fields are restricted [@Hayakawa:1999zf] to $0$ and $\pm 1$, thus requiring extra care in order to give fractional electric charges to the quarks;
5. gauge anomalies cannot be cancelled in a chiral noncommutative theory [@Hayakawa:1999zf; @Ardalan:2000cy; @Gracia-Bondia:2000pz; @Bonora:2000he; @Martin:2000qf; @Intriligator:2001yu; @Armoni:2002fh], hence the anomaly-free gauge theory must be vector-like.
Building upon an earlier proposal by Chaichian [*[et al.]{}*]{} [@Chaichian:2001py], the authors of Ref. [@Khoze:2004zc] constructed an example of a noncommutative embedding of the Standard Model with the purpose to satisfy all the requirements listed above. The model of [@Khoze:2004zc] is based on the gauge group $\textrm{U}(4)\times \textrm{U}(3) \times \textrm{U}(2)$ with matter fields transforming in noncommutatively allowed representations. Higgs fields break the noncommutative gauge group down to a low-energy commutative gauge theory which includes the Standard Model group $\textrm{SU}(3)\times \textrm{SU}(2) \times \textrm{U}(1)_Y$. The $\textrm{U}(1)_Y$ group here corresponds to ordinary QED, or more precisely to the hypercharge $Y$ Abelian gauge theory. The generator of $\textrm{U}(1)_Y$ was constructed from a linear combination of [*traceless*]{} diagonal generators of the microscopic theory $\textrm{U}(4)\times \textrm{U}(3) \times \textrm{U}(2).$ Because of this, the UV/IR effects – which can affect only the overall trace-$\textrm{U}(1)$ subgroup of each $\textrm{U}(N)$ – were not contributing to the hypercharge $\textrm{U}(1)_Y.$ However some of the overall trace-$\textrm{U}(1)$ degrees of freedom can survive the Higgs mechanism and thus contribute to the low-energy effective theory, in addition to the Standard Model fields. These additional trace-$\textrm{U}(1)$ gauge fields logarithmically decouple from the low-energy effective theory and were neglected in the analysis of Ref. [@Khoze:2004zc]. The main goal of the present paper is to take these effects into account.
We will find that the noncommutative model building constraints, and, specifically, the UV/IR mixing effects in the trace-U(1) factors in the item 1 above, lead to an unacceptable defective behavior of the low-energy theory, when we try to construct a model having the photon as the only massless colourless U(1) gauge boson. Our findings rule out a class of noncommutative extensions of the Standard Model.
\(a) This class is based on a noncommutative quantum gauge theory defined on a four-dimensional continuous space-time (UV cutoff sent to infinity). Within the Weyl-Moyal approach there are two ways to avoid our conclusions. Either one can introduce extra dimensions [@AJKR] or one can give up the continuous space-time.
\(b) Noncommutative models we concentrate on are similar to the example in [@Khoze:2004zc] and should be distinguished from earlier ones studied in [@Chaichian:2001py] for two reasons.[^1] First, we include the effects of the UV/IR mixing in our analysis. Second, is that our models preserve full noncommutative gauge invariance including the Higgs and Yukawa sectors. As such, the difficulties related to unitarity violation discussed in [@Hewett:2001im] do not apply in our case.
\(c) Finally, as already mentioned earlier, we are not pursuing the Seiberg-Witten map approach and as such our conclusions cannot be directly applied to the class of noncommutative models which rely on Taylor expansion in powers of $\theta$ in [@Calmet:2001na; @Madore:2000en; @Carroll:2001ws; @Carlson:2001sw; @Behr:2002wx; @Schupp:2002up; @Calmet:2004dn; @Ohl:2004tn; @Melic:2005su].
The UV/IR mixing in noncommutative theories arises from the fact that certain classes of Feynman diagrams acquire factors of the form $e^{i k_\mu \theta^{\mu\nu} p_\nu}$ (where $k$ is an external momentum and $p$ is a loop momentum) compared to their commutative counter-parts. These factors directly follow from the use of the Weyl-Moyal star-product . At large values of the loop momentum $p$, the oscillations of $e^{i k_\mu \theta^{\mu\nu} p_\nu}$ improve the convergence of the loop integrals. However, as the external momentum vanishes, $k \to 0,$ the divergence reappears and what would have been a UV divergence is now reinterpreted as an IR divergence instead. This phenomenon of UV/IR mixing is specific to noncommutative theories and does not occur in the commutative settings where the physics of high energy degrees of freedom does not affect the physics at low energies.
There are two important points concerning the UV/IR mixing [@Matusis:2000jf; @Khoze:2000sy; @Hollowood:2001ng; @Armoni:2000xr] which we want to stress here. First, the UV/IR mixing occurs only in the trace-U(1) components of the noncommutative $\textrm{U}(N)$ theory, leaving the $\textrm{SU}(N)$ degrees of freedom unaffected. Second, there are two separate sources of the UV/IR mixing contributing to the dispersion relation of the trace-U(1) gauge fields: the $\Pi_1$ effects and the $\Pi_2$ effects, as will be explained momentarily.
A study of the Wilsonian effective action, obtained by integrating out the high-energy degrees of freedom using the background field method, and keeping track of the UV/IR mixing effects, has given strong hints in favour of a non-universality in the infrared [@Khoze:2000sy; @Hollowood:2001ng]. In particular, the polarisation tensor of the gauge bosons in a noncommutative $\textrm{U}(N)$ gauge theory takes form [@Matusis:2000jf; @Khoze:2000sy; @Hollowood:2001ng] $$\label{poltensor}
\Pi_{\mu\nu}^{AB} = \Pi_1^{AB}(k^2,\tilde k^2) \, \left( k^2 g_{\mu\nu} - k_\mu k_\nu \right)
+ \Pi_{2}^{AB} (k^2, \tilde k^2)\, \frac{\tilde{k}_{\mu}\tilde{k}_{\nu}}{\tilde{k}^2}
\,, \hspace{4ex} {\rm with\ } \tilde{k}_\mu = \theta_{\mu\nu} k^\nu \,.$$ Here $A,B=0,1,\ldots N^2-1$ are adjoint labels of $\textrm{U}(N)$ gauge fields, $A_\mu^A$, such that $A,B=0$ correspond to the overall $\textrm{U}(1)$ subgroup, i.e. to the trace-U(1) factor. The term in proportional to $\tilde{k}_\mu\tilde{k}_\nu /\tilde{k}^2 $ would not appear in ordinary commutative theories. It is transverse, but not Lorentz invariant, as it explicitly depends on $\theta_{\mu\nu}.$ Nevertheless it is perfectly allowed in noncommutative theories. It is known that $\Pi_2$ vanishes for supersymmetric noncommutative gauge theories with unbroken supersymmetry, as was first discussed in [@Matusis:2000jf].
In general, both $\Pi_1$ and $\Pi_2$ terms in are affected by the UV/IR mixing. More precisely, as already mentioned earlier, the UV/IR mixing affects specifically the $\Pi_1^{0\,0}$ components and generates the $\Pi_2^{0\,0}$ components in . The UV/IR mixing in $\Pi_1^{0\,0}$ affects the running of the trace-U(1) coupling constant in the infrared, $$\frac{1}{g(k,\tilde{k})_{\textrm{U}(1)}^2} = 4 \Pi_{1}^{0\,0}( k^2,\tilde k^2)
\, \rightarrow \,
-\,\frac{b_0}{(4\pi)^2} \, \log { k^2} \ , \qquad {\rm as} \
k^2\to 0
\,,$$ leading to a logarithmic decoupling of the trace-U(1) gauge fields from the $\textrm{SU}(N)$ low-energy theory, see Refs. [@Khoze:2004zc; @Khoze:2000sy; @Hollowood:2001ng] for more detail.
For nonsupersymmetric theories, $\Pi_2^{0\,0}$ can present more serious problems. In theories without supersymmetry, $\Pi_2^{0\,0} \sim 1/{\tilde{k}^2},$ at small momenta, and this leads to unacceptable quadratic IR singularities [@Matusis:2000jf]. In theories with softly broken supersymmetry (i.e. with matching number of bosonic and fermionic degrees of freedom) the quadratic singularities in $\Pi_2^{0\,0}$ cancel [@Matusis:2000jf; @Khoze:2000sy; @Hollowood:2001ng]. However, the subleading contribution $\Pi_2^{0\,0} \sim const,$ survives [@Alvarez-Gaume:2003] unless the supersymmetry is exact. For the rest of the paper we will concentrate on noncommutative Standard Model candidates with softly broken supersymmetry, in order to avoid quadratic IR divergencies. In this case, $\Pi_2^{0\,0} \sim \Delta M^2_{\rm susy},$[^2] as explained in [@Alvarez-Gaume:2003]. The presence of such $\Pi_2$ effects will lead to unacceptable pathologies such as Lorentz-noninvariant dispersion relations giving mass to only one of the polarisations of the trace-U(1) gauge field, leaving the other polarisation massless.
The presence of the UV/IR effects in the trace-U(1) factors makes it pretty clear that a simple noncommutative U(1) theory taken on its own has nothing to do with ordinary QED. The low-energy theory emerging from the noncommutative U(1) theory will become free at $k^2 \to 0$ (rather than just weakly coupled) and in addition will have other pathologies [@Khoze:2004zc; @Khoze:2000sy; @Hollowood:2001ng; @Alvarez-Gaume:2003]. However, one would expect that it is conceivable to embed a commutative $\textrm{SU}(N)$ theory, such as e.g. QCD or the weak sector of the Standard Model into a supersymmetric noncommutative theory in the UV, but some extra care should be taken with the QED U(1) sector [@Khoze:2004zc]. We will show that the only realistic way to embed QED into noncommutative settings is to recover the electromagnetic U(1) from a [*traceless*]{} diagonal generator of some higher $\textrm{U}(N)$ gauge theory. So it seems that in order to embed QED into a noncommutative theory one should learn how to embed the whole Standard Model [@Khoze:2004zc]. We will see, however, that the additional trace-U(1) factors remaining from the noncommutative $\textrm{U}(N)$ groups will make the resulting low-energy theories unviable (at least for the general class of models considered in this paper).
In order to proceed we would like to disentangle the mass-effects due to the Higgs mechanism from the mass-effects due to non-vanishing $\Pi_2.$ Hence we first set $\Pi_2= 0$ (this can be achieved by starting with an exactly supersymmetric theory). It is then straightforward to show (see Sec. \[prove\]) that the Higgs mechanism alone cannot remove all of the trace-U(1) factors from the massless theory. More precisely, the following statement is true: [*Consider a scenario where a set of fundamental, bifundamental and adjoint Higgs fields breaks $\textrm{U}(N_1)\times \textrm{U}(N_2)\times\cdots \times \textrm{U}(N_m) \rightarrow H,$ such that $H$ is non-trivial. Then there is at least one generator of the unbroken subgroup $H$ with [*non-vanishing trace*]{}. This generator can be chosen such that it generates a U(1) subgroup.*]{}
We can now count all the massless U(1) factors in a generic noncommutative theory with $\Pi_2= 0$ and after the Higgs symmetry breaking. In general we can have the following scenarios for massless U(1) degrees of freedom in $H$:
1. \[possa\]$\textrm{U}(1)_{Y}$ is traceless and in addition there is one or more factors of trace-U(1) in $H$.
2. \[possb\] $\textrm{U}(1)_{Y}$ arises from a mixture of traceless and trace-U(1) generators of the noncommutative product group $\textrm{U}(N_1)\times \textrm{U}(N_2)\times\cdots \times \textrm{U}(N_m).$
3. \[possc\] $\textrm{U}(1)_{Y}$ has an admixture of trace-U(1) generators as in (\[possb\]) plus there are additional massless trace-U(1) factors in $H$.
In the following sections we will see that none of these options lead to an acceptable low-energy theory once we have switched on $\Pi_2 \neq 0$, i.e. once we have introduced mass differences between superpartners. It is well-known [@Matusis:2000jf; @Alvarez-Gaume:2003] that $\Pi_2 \neq 0$ leads to strong Lorentz symmetry violating effects in the dispersion relation of the corresponding trace-U(1) vector bosons, and in particular, to mass-difference of their helicity components. If option (\[possa\]) was realised in nature, it would lead (in addition to the standard photon) to a new colourless vector field with one polarisation being massless, and one massive due to $\Pi_2.$
The options (b) and (c) are also not viable since an admixture of the trace-U(1) generators to the photon would also perversely affect photon polarisations and make some of them massive[^3].
In the rest of the paper we will explain these observations in more detail.
We end this section with some general comments on noncommutative Standard Modelling. This paper refines the earlier analysis of [@Khoze:2004zc]. In that work the trace-U(1) factors were assumed to be completely decoupled in the extreme infrared and, hence, were neglected. However, it is important to keep in mind that the decoupling of the trace-U(1)’s is logarithmic and hence slow. Even in presence of a huge hierarchy between the noncommutative mass scale $M_{\textrm{NC}}$, say of the order of the Planck scale $M_{\textrm{P}}\sim 10^{19}\ \textrm{GeV}$, and the scale $\Lambda \sim (10^{-14}-10^{9})\,\textrm{eV}$ (electroweak and QCD scale, respectively), where the SU($N$) subgroup becomes strong, the ratio $$\label{ration1}
\frac{g^2_{\rm U(1)}}{g^2_{\textrm{SU}(N)}}\sim \frac{\log\left(\frac{k^2}
{\Lambda^{2}}\right)}{\log\left(\frac{M^4_{\textrm{NC}}}{\Lambda^{2}k^2}\right)}
\gtrsim 10^{-3}\,$$ is not negligible. In particular, the above inequality holds for any $M_{\textrm{NC}}>k\gtrsim 2\Lambda$. Hence the complete decoupling of the trace-U(1) degrees of freedom at small non-zero momenta does not appear to be fully justified and the trace-U(1) would leave its traces in scattering experiments at accessible momentum scales $k\sim 1\,\textrm{eV}-10^{10}\,\textrm{eV}$ (see Sec. 2 for more detail).
UV/IR mixing and properties of the trace-U(1) {#example}
=============================================
UV/IR mixing manifests itself only in the trace-U(1) part of the full noncommutative U($N$). For this part it strongly affects $\Pi_{1}$ and is responsible for the generation of nonvanishing $\Pi_{2}$ (if SUSY is not exact). In this section we will briefly review how the UV/IR mixing arises in the trace-U(1) sector and how this leads us to rule out options (a) and (c) discussed in Sec. 1.
Running gauge coupling
----------------------
Following Refs. [@Khoze:2000sy; @Hollowood:2001ng], we will consider a U($N$) noncommutative theory with matter fields transforming in the adjoint and fundamental representations of the gauge group. We use the background field method, decomposing the gauge field $A_\mu = B_\mu + N_\mu$ into a background field $B_\mu$ and a fluctuating quantum field $N_\mu$, and the appropriate background version of Feynman gauge, to determine the effective action $S_{\rm eff}(B)$ by functionally integrating over the fluctuating fields.
To determine the effective gauge coupling in the background field method, it suffices to study the terms quadratic in the background field. In the effective action these take the following form (capital letters denote full U($N$) indices and run from $0$ to $N^{2}-1$) [^4], $$S_{\rm eff} \ni
2\int \frac{d^{4}k}{(2\pi)^4} B^{A}_{\mu}(k)B^{B}_{\nu}(-k)\Pi^{AB}_{\mu\nu}(k).$$ At tree level, $\Pi^{AB}_{\mu\nu}=(k^2 g_{\mu\nu}-k_{\mu}k_{\nu})\,\delta^{AB}/g^2_0$ is the standard transverse tensor originating from the gauge kinetic term. In a commutative theory, gauge and Lorentz invariance restrict the Lorentz structure to be identical to the one of the tree level term. In noncommutative theories, Lorentz invariance is violated by $\theta$. The most general allowed structure is then given by Eq. . The second term may lead to the strong Lorentz violation mentioned in the introduction. This term is absent in supersymmetric theories [@Matusis:2000jf; @Khoze:2000sy].
Let us start with a discussion of the effects noncommutativity has on $\Pi_{1}$ and the running of the gauge coupling. That is, for the moment, we postpone the study of $\Pi_2$-effects by considering a model with unbroken supersymmetry[^5]. As usual, we define the running gauge coupling as $$\label{defcoupling}
\left(\frac{1}{g^{2}}\right)^{AB}=\left(\frac{1}{g^{2}_{0}}\right)^{AB}+4\Pi^{AB}_{1\,\,\textrm{loop}}(k).$$ where $g^{2}_{0}$ is the microscopic coupling (i.e. the tree level contribution) and $\Pi_{\textrm{loop}}$ includes only the contributions from loop diagrams. Henceforth, we will drop the loop subscript.
To evaluate $\Pi$ at one loop order one has to evaluate the appropriate Feynman diagrams. The effects of noncommutativity appear via additional phase factors $\sim \exp(i \frac{p \tilde{k}}{2})$ in the loop-integrals. Using trigonometric relations one can group the integrals into terms where these factors combine to unity, the so called planar parts, and those where they yield $\sim \cos ({p\tilde{k}})$, the so called non-planar parts.
For fields in the fundamental representation, the phase factors cancel exactly[^6] and only the planar part is non-vanishing. Fundamental fields therefore contribute as in the commutative theory [@Khoze:2000sy]. In all loop integrals[^7] involving adjoint fields one finds the following factor [@Hollowood:2001ng], $$M^{AB}(k, p) =
(-d \sin {k \tilde{p}\over 2}+ f \cos {k \tilde{p}\over 2})^{ALM}
(d \sin {k \tilde{p}\over 2}+ f \cos {k \tilde{p}\over 2})^{BML}.$$ Using trigonometric and group theoretic relations this collapses to $$M^{AB}(k,p) = - N \ \delta^{AB} (1-\delta_{0A}\cos k\tilde{p}).$$ We can now easily see that all effects from UV/IR mixing, marked by the presence of the $\cos k\tilde{p}$, appear only in the trace-U(1) part of the gauge group. The planar parts, however, are equal for the U(1) and SU($N$) parts.
Summing everything up we find the planar contribution (the coefficients $\alpha_{j},C_j,d_j$ are given in Table \[coefficients\] and $C({\bf r})$ is the Casimir operator in the representation ${\bf r}$) $$\begin{aligned}
\label{planarsusy2}
&&\Pi_{1\,\textrm{planar}} (k^2) =
-{2 \over (4\pi )^2 }\bigg( \sum_{j, {\bf r}} \alpha_{j} C({\bf r})
\bigg[2C_j+\frac{8}{9}d_j
\\\nonumber
&&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+
\int_{0}^{1} dx \left(C_j-(1-2x)^2d_j\right)\ \log {A(k^2 , x,m^2_{j, {\bf r}}) \over \Lambda^2} \bigg]\bigg),\end{aligned}$$ where $m_{j, {\bf r}}$ is the mass of a spin $j$ particle belonging to the representation $\bf r$ of the gauge group, $$A(k^2,x,m^2_{j, {\bf r}})=k^2 x(1-x)+m^2_{j, {\bf r}},$$ and $\Lambda$ appears via dimensional transmutation similar to $\Lambda_{\overline{\textrm{MS}}}$ in QCD. We have chosen the renormalisation scheme, i.e. the finite constants, such that $\Pi_{1\,\textrm{planar}}$ vanishes at $k=\Lambda$.
j= scalar Weyl fermion gauge boson ghost
-------------- -------- --------------- ---------------- -------
$\alpha_{j}$ -1 $\frac{1}{2}$ $-\frac{1}{2}$ 1
$C_j$ 0 $\frac{1}{2}$ 2 0
$d_j$ 1 2 4 1
: Coefficients appearing in the evaluation of the loop diagrams.[]{data-label="coefficients"}
For the trace-U(1) part the nonplanar parts do not vanish and we find $$\Pi_{1\,{\rm nonplanar}} = { 1\over 2 k^2}\left(\hat{\Pi} - \tilde{\Pi}\right) ,$$ with $$\begin{aligned}
\hat{\Pi} &=& {C({\bf G}) \over (4\pi)^2}\left\{
{8 d_j \over \tilde{k}^2} - k^2\left[ 12C_j - d_j\right]
\int_{0}^{1}dx \
K_{0} (\sqrt{A} |\tilde{k}|)\right\}
\ \ ,
\\
\tilde{\Pi}& =& {4C({\bf G})\over (4\pi)^2}\left\{
{ d_j \over \tilde{k}^2}- \left(C_j k^2 -
d_j
{\partial^2 \over \partial^2 |\tilde{k}| } \right)
\int_{0}^{1}dx \
K_0 (\sqrt{A} |\tilde{k}|)\right\} ,\end{aligned}$$ where $C({\bf G})=N$ is the Casimir operator in the adjoint representation.
\[0.95\]
(190,180)(40,0) ![The running gauge couplings $g_{\textrm{U(1)}}$ (solid) and $g_{\textrm{SU(2)}}$ (dashed) for a U(2) theory with two matter multiplets and all particles of equal mass $m=0,10^4,10^8,10^{12},10^{16}\,\Lambda$, from top to bottom (left side, solid), as a function of the momentum $k$, for a choice of $|\tilde{k}|=\theta_\textrm{eff} |k|$, with $\theta_{\textrm{eff}}=10^{-20}\Lambda^{-2}$. []{data-label="u1gaugecoupling"}](coupling.eps "fig:"){width="9.5cm"} (-40,-15)\[c\][\[1.2\][$\log_{10}(k/\Lambda)$]{}]{} (-280,150)\[c\][\[1.7\][$\frac{1}{g^2}$]{}]{}
For illustration, we plot in Fig. \[u1gaugecoupling\] the coupling for a toy model which is a supersymmetric U(2) gauge theory with two matter multiplets and all masses (of all fields) taken to be equal. We observe that even for large masses the running of the U(1) part (solid lines) does not stop in the infrared. For masses smaller than the noncommutative mass scale $m^2\ll M_{\textrm{NC}}$ the trace-U(1) gauge coupling has a sharp bend at $M_{\textrm{NC}}$ where the nonplanar parts start to contribute. For larger masses the running stops at the mass scale $m^2$ only to resume running at a scale $\sim M^{4}_{\textrm{NC}}/m^2$ which is, of course, again due to the nonplanar parts. The dashed lines in Fig. \[u1gaugecoupling\] give the running of the SU(2) part which receives no nonplanar contributions and behaves like in an ordinary commutative theory. For $m^2=0$ the SU(2) gauge coupling reaches a Landau pole at $k=\Lambda$, for all non vanishing masses the running stops at the mass scale. We observe that the ratio between the SU(2) coupling and the trace-U(1) coupling is not incredibly small over a wide range of scales, in support of our assertion in Sec. 1.
Further support comes from looking at the following approximate form for the running of the gauge coupling. We assume the hierarchy $\Lambda^{2}\ll m^2\ll M^{2}_{\textrm{NC}}$, $$\begin{aligned}
\label{run1}
&&\!\!\!\!\!\frac{4\pi^2}{g^2_{\textrm{U(1)}}}=b^{\textrm{p}}_{0}\log\left(\frac{k^2}{\Lambda^2}\right),
\,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\textrm{for}\quad
k^2 \gg M^{2}_{\textrm{NC}},
\\\nonumber
&&\!\!\!\!\!\frac{4\pi^2}{g^2_{\textrm{U(1)}}}=b^{\textrm{p}}_{0}\log\left(\frac{k^2}{\Lambda^2}\right)
-b^{\textrm{np}}_{0} \log\left(\frac{k^2}{M^{2}_{\textrm{NC}}}\right),\,
\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\, \textrm{for}\quad m^{2}\ll k^2\ll
M^{2}_{\textrm{NC}},
\\\nonumber
&&\!\!\!\!\!\frac{4\pi^2}{g^2_{\textrm{U(1)}}}=b^{\textrm{p}}_{0}\log\left(\frac{m^2}{\Lambda^2}\right)
-b^{\textrm{np}}_{0} \left[\log\left(\frac{m^2}{M^{2}_{\textrm{NC}}}\right)
+\frac{1}{2}\log\left(\frac{k^2}{m^2}\right)\right],\,\quad\,\,\,\, \textrm{for}\quad k^2 \ll m^2.\end{aligned}$$ The gauge coupling for the SU($N$) subgroup $g^2_{\textrm{SU}(N)}$ is obtained by setting $b^{\textrm{np}}_{0}=0$. For simplicity let us now consider a situation where we have only fields in the adjoint representation. One finds [@Khoze:2004zc; @Hollowood:2001ng] that $b^{\textrm{np}}_{0}=2b^{\textrm{p}}_{0}$, and $$\begin{aligned}
\label{run2}
&&\frac{g^{2}_{\textrm{U}(1)}}{g^{2}_{\textrm{SU}(N)}}=1,
\quad\quad\quad\quad\quad\quad \textrm{for} \quad k^2\gg M^{2}_{\textrm{NC}},
\\\nonumber
&&\frac{g^{2}_{\textrm{U}(1)}}{g^{2}_{\textrm{SU}(N)}}
=\frac{\log\left(\frac{k^2}{\Lambda^{2}}\right)}{\log\left(\frac{M^{4}_{\textrm{NC}}}{\Lambda^{2}k^2}\right)},
\quad\,\,\,\,\, \textrm{for}\quad m^{2}\ll k^2\ll M^{2}_{\textrm{NC}},
\\\nonumber
&&\frac{g^{2}_{\textrm{U}(1)}}{g^{2}_{\textrm{SU}(N)}}
=\frac{\log\left(\frac{m^2}{\Lambda^{2}}\right)}{\log\left(\frac{M^{4}_{\textrm{NC}}}{\Lambda^{2}k^2}\right)},
\quad\,\,\,\,\, \textrm{for} \quad k^{2}\ll m^2.\end{aligned}$$ To reach $$\frac{g^{2}_{\textrm{U}(1)}}{g^{2}_{\textrm{SU}(N)}}<\epsilon = 10^{-3}$$ we need $\log\left(\frac{M^{4}_{\textrm{NC}}}{\Lambda^{2}k^2}\right)$ and in turn $M_{\textrm{NC}}$ to be large.
As a generic example let us use $\Lambda=\Lambda_{W}\sim 10^{-14}\,\textrm{eV}$ (the scale where the ordinary electroweak SU(2) would become strong, in absence of electroweak symmetry breaking) and $k=1\,\textrm{eV}$[^8]. We find $$\label{abschaetz}
M_{\textrm{NC}}>
\Lambda^{\frac{1}{2}}k^{\frac{1}{2}}\exp\left(\frac{1}{4\epsilon}\log\left(\frac{k^2}{\Lambda^2}\right)\right)
\sim 10^{6974}\,M_{\textrm{P}}.$$ Taking electroweak symmetry breaking into account we have to replace $\log\left(\frac{k^2}{\Lambda^2}\right)$ by $\log\left(\frac{M^2_{\textrm{EW}}}{\Lambda^2}\right)$ with $M_{\textrm{EW}}\sim 100\,\textrm{GeV}$ in . We find $$M_{\textrm{NC}}>10^{12474}\,M_{\textrm{P}}.$$ Let us increase the coupling strength of the SU($N$) by using $\Lambda=0.5\,\textrm{eV}$. $k=1\,\textrm{eV}$ is now quite close to the strong coupling scale of the SU($N$). Without symmetry breaking we find $$M_{\textrm{NC}}>10^{131}\,M_{\textrm{P}}.$$ We might be able to reduce this number by some orders of magnitude but without using an extreme field content it remains always incredibly large. Indeed, one can typically find a scale $k$ which is not too close to the strong coupling scale of the SU($N$) which strengthens the bounds dramatically. Therefore, as a conservative estimate we propose[^9] $$M_{\textrm{NC}}>10^{100}\,M_{\textrm{P}}.$$
To conclude this subsection, let us point out that, in a scattering experiment (as depicted in Fig. \[scattering\]), $k$ is really the scale of the internal momentum, and therefore, non-vanishing. $\tilde{k}$, too, is non-vanishing in appropriate (remember that we have Lorentz symmery violation) directions of $t$-channel scattering.
\[0.9\]
(190,140)(0,0) (3,10) (40,60)(0,100) (0,20)(40,60) (40,60)(100,60)[2]{} (140,100)(100,60) (100,60)(140,20) (40,60)[2]{} (20,60)\[c\][\[1.0\][$g(k)$]{}]{} (120,60)\[c\][\[1.0\][$g(k)$]{}]{} (100,60)[2]{} (70,75)\[c\][\[1.1\][$k$]{}]{} (70,65)\[c\][\[1.2\][$\longrightarrow$]{}]{}
The effects of a non vanishing $\Pi_{2}$ from SUSY breaking {#eom}
-----------------------------------------------------------
In the previous subsection we made $\Pi_{2}$ vanish by working in a supersymmetric theory. Let us now study, what happens, when supersymmetry is (softly) broken.
Looking only at the trace-U(1) degrees of freedom of a generic noncommutative theory we have $$\Pi_{2}=\sum_{j}\alpha_{j}\left[\frac{1}{2}(3\tilde{\Pi}_{j}-\hat{\Pi}_{j})\right].$$ One easily checks that $$\Pi_{2}\sim \sum_{j} \alpha_{j} d_j f(k^2,\tilde{k}^2,m_j).$$ If SUSY is unbroken, all masses are equal. Using supersymmetric matching between bosonic and fermionic degrees of freedom, $$\label{susycanc}
\sum_{j} \alpha_{j} d_j=0,$$ we reproduce the vanishing of $\Pi_{2}$. If SUSY is softly broken this cancellation is not complete anymore (in fact still holds and this removes the leading power-like IR divergence in $\Pi_{2}$, however, the subleading effects in $\Pi_{2}$ survive). $\Pi_{2}$ gets a contribution [@Alvarez-Gaume:2003] $$\begin{aligned}
\label{pi2}
\Pi_{2}\!\! &=& \!\!D\sum_{j}\alpha_{j}d_jm^{2}_{j}\left[K_{0}(m\tilde{k})+K_{2}(m\tilde{k})\right]+O(k^2)
\\\nonumber
\!\! &=& \!\! C \Delta M^2_{\textrm{SUSY}}+C'\sum_{j}\alpha_{j}d_j m^{2}_{j}\log(m^2_{j}\tilde{k}^2)+\cdots,\end{aligned}$$ with known constants $C$, $C'$ and $D$. This has dire consequences for the gauge boson. Let us look at the equations of motion resulting from this additional Lorentz symmetry violating contribution to the polarisation tensor (we briefly review the equations of motion for ordinary photons in Appendix \[polarisation\]).
In presence of a Higgs field which generates a mass term $m^2$ and using unitary gauge the field equations in presence of non vanishing $\Pi_2$ read $$\label{fieldnc} \left(\Pi_{1}(k^{2}g_{\mu\nu}-k_{\mu}k_{\nu})
+\Pi_{2}\frac{\tilde{k}_{\mu}\tilde{k}_{\nu}}{\tilde{k}^{2}}-m^{2}g_{\mu\nu}\right)A^{\nu}=0.$$ Using that unitary gauge implies Lorentz gauge, $k_{\mu}A^{\mu}=0$, we can simplify $$\label{fieldnc2}
(\Pi_{1}k^{2}-m^{2})A_{\mu}+\Pi_{2}\frac{\tilde{k}_{\mu}\tilde{k}_{\nu}}{\tilde{k}^{2}}A^{\nu}=0.$$ To proceed further it is useful to specify a direction for the momentum and the noncommutativity parameters. The photon flies in 3-direction and we have $$k^{\mu}=(k^{0},0,0,k^{3}).$$ What is the corresponding value of $\tilde k$? Since $\theta^{\mu\nu}$ breaks Lorentz invariance, we need to specify $\theta^{\mu\nu}$ in a particular frame. For the latter, a natural one is the system where the cosmic microwave background is at rest. In this frame, we assume that the only non-vanishing components of $\theta^{\mu\nu}$ are $$\theta^{13}=-\theta^{31}=\theta .$$
This yields, $$\tilde{k}_{\mu}=\theta_{\mu\nu}k^{\nu}=(0,\theta k^{3},0,0),\quad
\tilde{k}^{\, 2}=(\theta k^{3})^{2}.$$ We start with the ordinary transverse components of $A^{\nu}$, $$A^{\nu}_{1}=(0,1,0,0).$$ In this direction, yields $$\label{direction1}
(\Pi_{1}k^{2}-m^{2}-\Pi_{2})A_{1,\nu}=0.$$ In the other transverse direction, $$A^{\mu}_{2}=(0,0,1,0),$$ we find $$(\Pi_{1}k^{2}-m^{2})A_{2,\nu}.$$ Finally we have the third polarisation (which can be gauged away if and only if $m^2=0$), $$A^{\mu}=(a,0,0,b),\quad k^{0}a-k^{3}b=0$$ which results in $$(\Pi_{1}k^{2}-m^{2})A_{3,\nu}.$$ We note that the different polarisation states do not mix due to the presence of $\Pi_{2}$. The second and the third polarisation state behave more or less like in the ordinary commutative case. However, the first has a modified equation of motion, , in presence of a non-vanishing $\Pi_{2}$[^10].
This is another strong argument against a trace-U(1) being the photon [@Alvarez-Gaume:2003]. If the gauge symmetry is unbroken and $m^2=0$ we usually have two massless polarisations. However, a non vanishing $\Pi_{2}$ reduces this to one. The other one gets an additional mass $\Pi_{2}$. Since only one polarisation is affected this is a strong Lorentz symmetry violating effect. Moreover, a negative $\Pi_{2}$ would lead to tachyons while a positive mass is phenomenologically ruled out by the constraint [@Eidelman:2004wy] $$m_{\gamma}<6\times 10^{-17}\,\textrm{eV}$$ on the photon mass[^11].
If we take the trace-U(1) as an additional (to the photon) gauge boson from the unbroken subgroup $H$, we would still get strong Lorentz symmetry violation since the trace-U(1) is not completely decoupled.
In summary, we found in this section that additional trace-U(1) subgroups are not completely decoupled and should lead to observable effects. In particular, if SUSY is not exact we have non-vanishing $\Pi_{2}$ which gives rise to strong Lorentz symmetry violation which has not been observed. This rules out possibilities (\[possa\]) and (\[possc\]) of Sec. 1. Moreover, we confirmed that a trace-U(1) is not suitable as a photon candidate.
Mixing of trace and traceless parts {#u2example}
===================================
From the previous section we concluded that the trace-U(1) groups are unviable as candidates for the SM photon. Therefore, it has been suggested to construct the photon from traceless U(1) subgroups [@Khoze:2004zc]. It turns out, however, that typically trace and traceless parts mix and the trace parts contribute their Lorentz symmetry violating properties to the mixed particle.
For U(2) broken by a fundamental Higgs, the standard Higgs mechanism yields the symmetry breaking $U(2)\to U(1)$. However, the remaining U(1) is a mixture of trace and traceless parts. If SUSY is broken, the trace-U(1) has a $\Pi_{2}$ part in the polarisation tensor. Taking this into account we find the following matrix for the equations of motion $$\begin{aligned}
\label{final}
\begin{tiny}
\left(\begin{array}{cccccc}
\Pi^{\textrm{U(1)}}_{1}k^{2}-\Pi_{2}-m^{2} & m^2 & & & & \\
m^2 & \Pi^{\textrm{SU(2)}}_{1}k^2-m^{2} & & & & \\
& & \Pi^{\textrm{U(1)}}_{1}k^{2}-m^{2} & m^2 & & \\
& & m^2 &\Pi^{\textrm{SU(2)}}_{1}k^2-m^{2} & & \\
& & & & \Pi^{\textrm{U(1)}}_{1}k^2-m^{2} &m^{2} \\
& & & & m^{2} &\Pi^{\textrm{SU(2)}}_{1}k^2-m^{2}
\end{array} \right),
\end{tiny}\end{aligned}$$ where the adjoint U(2) and polarisation indices are $(0,1),(3,1),(0,2),(3,2),(0,3),(3,3)$. We omitted the values $1$ and $2$ for the adjoint U(2) indices which do not mix with the trace-U(1) and are not qualitatively different from the commutative case.
The matrix is block diagonal and the second and third polarisation (lower right corner) behave more or less like their commutative counterparts. We can concentrate on the upper left $2\times2$ matrix corresponding to the transverse polarisations affected by $\Pi_{2}$.
This $2\times2$ matrix admits two solutions for the equations of motion. Expanding for small $\Pi_{2}$ we find, $$\begin{aligned}
\label{u2solution}
\left(\Pi^{\textrm{U(1)}}_{1}+\Pi^{\textrm{SU}(N)}_{1}\right)k^2\!\!&=&\!\!\Pi_{2}+O(\Pi^{2}_{2}),
\\\nonumber
\left(\Pi^{\textrm{U(1)}}_{1}+\Pi^{\textrm{SU}(N)}_{1}\right)k^2\!\!&=&\!\!\
\frac{\left(\Pi^{\textrm{U(1)}}_{1}+\Pi^{\textrm{SU}(N)}_{1}\right)^2}{\Pi^{\textrm{U}(1)}_{1}\Pi^{\textrm{SU}(N)}_{1}} m^2 +\frac{\Pi^{\textrm{SU}(N)}_{1}}{\Pi^{\textrm{U}(1)}_{1}}\Pi_{2}+O(\Pi^{2}_{2}),\end{aligned}$$ in analogy to . In absence of $\Pi_{2}$ the first solution in Eq. is a massless one corresponding to the massless combination of gauge bosons (think of it as the photon). The second is a massive combination (similar to the $Z$ boson). The presence of non-vanishing $\Pi_{2}$ again leads to a mass $\frac{\Pi_{2}}{\Pi^{\textrm{U(1)}}}$ for the first solution and rules out the “massless” combination as a reasonable photon candidate.
This example demonstrates that the disastrous effects of $\Pi_{2}$ are also present in any combination which has an admixture of trace-U(1) degrees of freedom. Hence, this rules out possibilities (\[possb\]) and (\[possc\]) from the introduction.
Trace-U(1) factors in the unbroken subgroup {#prove}
===========================================
In the previous section, we learned in a specific example that even a small admixture of a trace part spoils the masslessness of the gauge boson corresponding to the unbroken gauge symmetry. This shows that a viable photon candidate must have a generator with vanishing (small is not enough) trace.
In our U(2) example with the gauge symmetry broken by a fundamental Higgs field the trace does not vanish. The generator corresponding to the unbroken U(1) is $$\sim \left(
\begin{array}{cc}
1 & 0 \\
0 & 0
\end{array}
\right),$$ which obviously has non-vanishing trace.
One can try to construct other symmetry breaking mechanisms with larger groups and products of groups as well as the other representations for the Higgs fields allowed by the condition 3 of the introduction. However, one always encounters one of the following situations. Either the remaining U(1) has a generator with non-vanishing trace or there is more than one unbroken U(1) subgroup. Both situations are in contradiction of observations, as our discussion of the previous sections shows.
This is generalised and more precisely formulated by the following proposition (already stated in the introduction): [*Consider a scenario where a set of fundamental, bifundamental and adjoint Higgs fields breaks $\textrm{U}(N_1)\times \textrm{U}(N_2)\times\cdots \times \textrm{U}(N_m) \rightarrow H,$ such that $H$ is non-trivial. Then there is at least one generator of the unbroken subgroup $H$ with [*non-vanishing trace*]{}. This generator can be chosen such that it generates a U(1) subgroup.*]{}
Let us now turn to a proof of the proposition. Let us start with the simple situation of one U($N$) group. Since we have only one group, we have only fundamental and adjoint Higgs fields at our disposal. We proceed by switching on one Higgs field (component) after the other. Let us start with the fundamental field. U($N$) symmetry allows us to chose this field as $$\label{fundamental}
\phi_{\textrm{f}}=(0,\ldots,0,a)^{\textrm{T}}.$$ [*[Case 1:]{}*]{} If $a=0$ we have no breaking with a fundamental Higgs. In this case we are finished, because the generator of the original trace-U(1) is proportional to the $N\times N$ unit matrix and therefore commutes with any adjoint Higgs field. Therefore this generator continues to generate an unbroken trace-U(1) subgroup, as stated in the proposition.
[*[Case 2:]{}*]{} If $a\neq0$ gauge symmetry is broken down to the U($N-1$) living in the upper $N-1$ components of any field. A set of generators for this group are the ordinary U($N-1$) in the upper left $(N-1)\times (N-1)$ submatrix and zero in the other components. In particular, there is a new trace-U(1) with generator $$T^{1}_{\textrm{trace}}=
\left(\begin{array}{cccc}
1 & & & \\
& \ddots & & \\
& & 1 & \\
& & & 0
\end{array}
\right).$$ Under this subgroup an adjoint field decomposes into $$\label{adjoint}
\phi_{\textrm{ad}}=
\left(\begin{array}{c|c}
\phi^{2}_{\textrm{ad}} & \phi^{2}_{\textrm{f}}\\
\hline (\phi^{2}_{\textrm{f}})^{\dagger} &\phi^{2}_{\textrm{s}} \\
\end{array}\right),$$ where $\phi^{2}_{\textrm{ad}}$, $\phi^{2}_{\textrm{f}}$ and $\phi^{2}_{\textrm{s}}$ are adjoint, fundamental and singlett fields under the remaining U($N-1$) symmetry. An additional fundamental field $\hat{\phi}_{\textrm{f}}$ decomposes as $$\label{fundamental}
\hat{\phi}_{\textrm{f}}=\left(
\begin{array}{c}
\hat{\phi}^{2}_{\textrm{f}} \\
\hat{\phi}^{2}_{\textrm{s}}
\end{array} \right)$$ into an additional fundamental $\hat{\phi}^{2}_{\textrm{f}}$ and another singlett $\hat{\phi}^{2}_{\textrm{s}}$. We can now repeat the argument for the remaining U($N-1$) group starting, again, with the fundamental fields.
This procedure has to stop at some point, i.e. at one point the fundamental $\phi^{n}_{\textrm{f}}$ has to be zero, or the symmetry is broken completely and $H$ would be the trivial group in violation of the assumptions.
For a product of more than one group the proof is analogous only that we have additional bifundamental fields. Let us briefly consider the situation with a product of two groups $\textrm{U}(M)\times \textrm{U}(N)$. Switching on fundamental fields we can end up with:
[*[Case 1:]{}*]{} If all fundamentals are zero the symmetry remains unbroken $\textrm{U}(M)\times \textrm{U}(N)$. One can easily see that bifundamental and adjoint fields cannot break the trace-U(1) generated by the $(N+M)\times (N+M)$ unit matrix[^12].
[*[Case 2:]{}*]{} Let us switch on one fundamental field. Without loss of generality we can take it to be an $N$ fundamental. The symmetry is broken down to $\textrm{U}(M)\times \textrm{U}(N-1)$ with a new trace-U(1) for the U($N-1$) in analogy to the simple U($N$) situation discussed above. All fields transforming under the U($M$) remain unaffected. The fundamental and adjoint fields for U($N$) are decomposed according to Eqs. ,. Finally the bifundamental decomposes as $$\phi_{b}=
\left(\begin{array}{c|c}
\phi^{2}_{\textrm{b}} & \phi^{2}_{\textrm{b, f}} \\
\end{array} \right)$$ into a bifundamental $\phi^{2}_{\textrm{b}}$ under $\textrm{U}(M)\times \textrm{U}(N-1)$ and a fundamental $\phi^{2}_{\textrm{b, f}}$ under U($N-1$).
The argument proceeds by induction. The case of more than two U($N$) factors is completely analogous.
Conclusions
===========
Noncommutative gauge symmetry in the Weyl-Moyal approach leads to two main features which have to be taken into account for sensible model building. First, there are strong constraints on the dynamics and the field content. The only allowed gauge groups are U($N$). In addition, the matter fields are restricted to transform as fundamental, bifundamental and adjoint representations of the gauge group. Finally, anomaly freedom for noncommutative theories requires the theory to be vector like[^13]. Second, there are the effects of ultraviolet/infrared mixing. Those lead to asymptotic infrared freedom of the trace-U(1) subgroup and, if the model does not have unbroken supersymmetry, to Lorentz symmetry violating terms in the polarisation tensor for this trace-U(1) subgroup.
We have demonstrated that, although the trace-U(1) decouples in the limit $k\to 0$, the coupling is not negligibly small at finite momentum scales $k$, as they appear, for example, in scattering experiments. Therefore, observations rule out additional unbroken (massless) trace-U(1) subgroups. An example is the model considered in Ref. [@Khoze:2004zc]. In Ref. [@Khoze:2004zc], the trace-U(1) groups were completely discarded before the symmetry breaking scheme was discussed. A more careful investigation which takes takes into account these subgroups yields the symmetry breaking $\textrm{U}(4)\times \textrm{U}(3)\times \textrm{U} (2) \to \textrm{SU}(3)\times\textrm{SU}(2)\times (\textrm{U}(1))^4$ instead of $\textrm{U}(4)\times \textrm{U}(3)\times \textrm{U} (2) \to \textrm{SU}(3)\times\textrm{SU}(2)\times \textrm{U}(1)$. Therefore we have superfluous U(1) subgroups. Following the above lines explicitely one easily finds that one of the $\textrm{U}(1)$’s has a generator which is proportional to the $9\times 9$ unity matrix. Noncommutativity explicitly breaks Lorentz invariance. Therefore an additional Lorentz symmetry violating structure is allowed in the polarisation tensor. This structure is absent only in supersymmetric models. If supersymmetry is (softly) broken, this additional structure is present in the polarisation tensor of the trace-U(1). It leads to an additional mass $\sim \Delta M^2_{\textrm{SUSY}}$ for one of the transverse polarisation states [@Alvarez-Gaume:2003]. The tight constraints on the photon mass therefore exclude trace-U(1)’s as a candidate for the photon. It turns out that even a small admixture of a trace part to a traceless part (unaffected by these problems) is fatal. The only way out seems to be the construction of the photon from a completely traceless generator. A group theoretic argument shows, that this is impossible whithout having additional unbroken U(1) subgroups. However, those are already excluded from the arguments given above.
This result severely restricts the possibilities to construct a noncommutative Standard model extension. If all of the constraints given at the beginning are fulfilled the noncommutativity scale is pushed to scales far beyond $M_{\textrm{P}}$. This is to be compared to the less restrictive constraints $M_{\textrm{NC}}\gtrsim 0.1-10\,\textrm{TeV}$ (conservative estimate) obtained from tree level amplitudes [@Hewett:2000zp] or from an approach where a Taylor expansion in the noncommutativity parameters is used before quantization, thereby ignoring effects of ultraviolet/infrared mixing and possibly constraints on the field content [@Calmet:2001na; @Madore:2000en; @Carroll:2001ws; @Carlson:2001sw; @Behr:2002wx; @Schupp:2002up; @Calmet:2004dn; @Ohl:2004tn; @Melic:2005su]. We stress, however, that the latter approach may lead to a completely different quantum theory and therefore our bounds may not be applicable.
We would like to conclude with a more optimistic prospect.
In general there is no reason to assume that the simple noncommutative model used here describes correctly the physics at energies ranging from a few eV up to the Planck mass. In fact, due to the ultraviolet/infrared mixing, a different ultraviolet embedding of the theory would modify the theory not only in the ultraviolet, but also in the infrared which can drastically alter our conclusions [@AJKR]. In particular, our conclusions are tied to a slow logarithmic decoupling of the trace-U(1), but if it is changed to a power-like decoupling, the U(1) factors would safely decouple and leave the Standard Model in peace. We expect that this can be achieved by embedding the noncommutative theory into a higher dimensional theory in the ultraviolet (which will have a power-like beta function) and then appeal to the ultraviolet/infrared mixing to transport this power-like behaviour to the infrared region for the trace-U(1) gauge coupling (see later work [@AJKR]).
**Acknowledgements**
We would like to thank Steve Abel and Jürgen Reuter for useful discussions. VVK acknowledges the support of PPARC through a Senior Fellowship.
Polarsation directions in gauge theories {#polarisation}
========================================
In this section we review some basics about the counting of degrees of freedom in gauge theories. In particular, we show how gauge invariance reduces the number of degrees of freedom from the naive 4 (4 components of the vector field) to 2 and 3 for the massless and massive case, respectively.
The massless case
-----------------
In ordinary QED, the field equations read $$\label{field} \Box
A^{\mu}-\partial^{\mu}(\partial_{\nu}A^{\nu})=0.$$ Using Lorentz gauge, $$\label{lorentz}
\partial_{\mu}A^{\mu}=0$$ Eq. simplifies to the wave equation $$\label{wave} \Box A^{\mu}=0.$$ Writing $$A^{\mu}=C \epsilon^{\mu} \exp(ikx),$$ any $\epsilon^{\mu}$ is a solution to as long as $$k^{2}=0.$$ So far we have all $4$ polarisations. However, implies 4-dimensional transversality, $$\label{4dtransverse} k_{\mu}\epsilon^{\mu}=0,$$ and reduces the allowed number of polarisations to three. This is still more than the two polarisation states a photon should have.
However, Lorentz gauge does not completely fix the gauge. We can still use a gauge transformation $\Omega$ with $\Box\Omega=0$. This allows us to choose $A^{0}=0$. Together with this leads us to the ordinary 3-dimensional transversality, $$\label{3dtransverse} \overrightarrow{k}\cdot \overrightarrow{\epsilon}=0.$$
The case with a Higgs field
---------------------------
The presence of a Higgs field modifies , $$\label{field2}
\Box A^{\mu}-\partial^{\mu}(\partial_{\nu}A^{\nu})+m^{2}A^{\mu}+m\partial^{\mu}\phi_{2}=0.$$ Moreover it supplies an additional equation for the Goldstone boson $\phi_{2}$, $$\label{goldstone} \Box \phi_{2}+m\partial_{\mu}A^{\mu}=0.$$ One convenient choice of gauge is unitary gauge where $$\label{unitary} \phi_{2}=0.$$ We stress from the beginning that unitary gauge implies , as can be seen from . In this gauge Eq. simplifies, $$\label{wave2} \Box A^{\mu} +m^{2} A^{\mu}=0.$$ Now everything runs in a similar fashion to the massless case, only that $$k^{2}-m^{2}=0.$$ The important difference is that unitary gauge fixes the gauge completely. We cannot make an additional gauge choice. Therefore it is impossible to get rid of the 3rd polarisation state which satisfies Lorentz gauge $k_{\mu}\epsilon^{\mu}=0$. Stated differently we cannot require 3-dimensional transversality for $\epsilon^{\mu}$ and we have therefore three allowed polarisation states with equal masses.
[99]{}
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[^1]: The construction in [@Khoze:2004zc] of correct values of hypercharges of the Standard Model from the product gauge group was influenced by [@Chaichian:2001py]. The authors of Ref. [@Chaichian:2001py] advocated a noncommutative model which satisfied criteria 2, 3 and 4 listed in the beginning of this section. Their model was based on the noncommutative gauge group $\textrm{U}(3)\times \textrm{U}(2) \times \textrm{U}(1)$ with matter fields transforming only in (bi-)fundamental representations, and remarkably, it predicted correctly the hypercharges of the Standard Model. In many respects their model is similar to the bottom-up approach of [@bottomup] to the string embedding of the Standard Model in purely commutative settings. Unfortunately, the noncommutative $\textrm{U}(3)\times \textrm{U}(2) \times \textrm{U}(1)$ model of [@Chaichian:2001py] ignores all the effects of the UV/IR mixing which alters the infrared behavior of the U(1) hypercharge sector.
[^2]: $\Delta M^2_{\rm SUSY}
=\frac{1}{2}\sum_s M_s^2-\sum_f M_f^2$ is a measure of SUSY breaking.
[^3]: One could hope that the trace-U(1) factors could be made massive at the string scale by working in a theory where these factors are anomalous. Then one could use the Green-Schwarz mechanism [@Green:1984sg] to cancel the anomaly and simultaneously give a large stringy mass to these U(1) factors. This scenario which is often appealed to in ordinary commutative theories to remove unwanted U(1) factors cannot be used in the noncommutative setting. The reason is that at scales above the noncommutative mass, the noncommutative gauge invariance requires the gauge group to be U($N$). It cannot become just an SU($N$) theory (above the noncommutative scale) and remain noncommutative, see e.g. [@Armoni:2002fh]. Therefore we require vector-like theories as stated in item 5.
[^4]: We use euclidean momenta when appropriate and the analytic continuation when considering the equations of motion in subsection \[eom\].
[^5]: Nevertheless, we will give general expressions for $\Pi_{1}$ valid also in the non-supersymmetric case.
[^6]: One may roughly imagine that for each fundamental field that appears in a Feynman diagram there is also the complex conjugate field which cancels the exponential factor.
[^7]: To keep the equations simple we consider in this section a situation where all particles of a given spin and representation have equal diagonal masses. Please note that the masses for fermions and bosons in the same representation may be different as required for SUSY breaking.
[^8]: It is obvious that $k^2\ll M^{2}_{\textrm{NC}}$. In this regime our formulas and approximate the full result to a very high precision since threshold effects are negligible.
[^9]: Of course, this constraint should not be taken overly serious. Above the string scale one should perform a string theory analysis. The main point is that the scale we find is way beyond the Planck scale.
[^10]: One might argue that instead of Eq. one has to use the rescaled equation (we set $m^2=0$ for simplicity) $k^2-\frac{\Pi_{2}(k^2,\tilde{k}^2)}{\Pi_{1}(k^2,\tilde{k^2})}=0$. For $k^2\to 0$, the second term vanishes since $\Pi_{1}$ diverges in this limit. Therefore, we find an additional solution. However, this solution is rather strange. It does not correspond to a pole in the propagator (it goes like a $\log$). Moreover, if one calculates the cross section $\Pi_{2}$ still upsets the angular dependence quite severely compared to the ordinary commutative case.
[^11]: Even fine-tuning of to zero is not an option. Since we have only a finite number of masses this is at best possible for a finite number of values of $|\tilde{k}|$ and we will surely find values of $|\tilde{k}|$ where $\Pi_{2}$ is nonzero.
[^12]: We can think of $\textrm{U}(M)\times \textrm{U}(N)$ embedded into $\textrm{U}(N+M)\times\textrm{U}(N+M)$
[^13]: In turn, this eliminates the Green-Schwarz mechanism [@Green:1984sg] as a possible source for a (large) mass term for the trace-U(1) part of the gauge group.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this article, we merge celebrated results of Kesten and Spitzer \[*Z. Wahrsch. Verw. Gebiete* **50** (1979) 5–25\] and Kawazu and Kesten \[*J. Stat. Phys.* **37** (1984) 561–575\]. A random walk performs a motion in an i.i.d. environment and observes an i.i.d. scenery along its path. We assume that the scenery is in the domain of attraction of a stable distribution and prove that the resulting observations satisfy a limit theorem. The resulting limit process is a self-similar stochastic process with non-trivial dependencies.'
address:
- 'Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstr. 150, 44780 Bochum, Germany. '
- |
Department of Mathematics, Keio University 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama-shi City,\
Kanagawa-ken Prefecture, 223-8522, Japan.
author:
-
-
title: 'A self-similar process arising from a random walk with random environment in random scenery'
---
Introduction {#sec1}
============
The following model for a random walk in a random environment can be found in the physics literature; see Anshelevic and Vologodskii ([-@AnsVol1981]), Alexander *et al.* ([-@Aleetal1981]), Kawazu and Kesten ([-@KawKes1984]). Let $ \{\lambda_j;j\in\mathbb{Z}\} $ be a family of positive i.i.d. random variables and $ \mathcal{A} $ the $ \sigma$-algebra generated by those random variables. Let $ \{X(t);t\geq0\} $ be a continuous-time random walk on $ \mathbb
{Z} $ having the following asymptotic transition rates for $ h\rightarrow0$: $$\begin{aligned}
\label{Formel1}
\mathbb{P} \bigl(X(t+h)=j+1|X(t)=j,\mathcal{A}\bigr) &=& \lambda_jh+\mathrm{o}(h),\\
\mathbb{P} \bigl(X(t+h)=j-1|X(t)=j,\mathcal{A}\bigr) &=& \lambda_{j-1}h+\mathrm{o}(h),\\
\mathbb{P} \bigl(X(t+h)=j|X(t)=j,\mathcal{A}\bigr) &=& 1-(\lambda_j+\lambda_{j-1})h+\mathrm{o}(h).\end{aligned}$$ In other words, the process $ \{X(t);t\geq0\} $ is a birth–death process with possibly negative population size, where, for a population with $ j $ individuals, birth occurs at rate $ \lambda_j $ and death at rate $ \lambda_{j-1} $. We will assume that the process $ \{X(t);t\geq0\} $ starts at zero at time zero. The resulting process is symmetric, in the sense that the permeability of the edge connecting the vertices $ j $ and $ j+1 $ does not depend on the direction of the motion. This physical background motivates the name ‘random environment’ for the sequence $ \{\lambda
_j;j\in\mathbb{Z}\} $. In what follows, we denote the distribution of the random environment on the sequence space by $ P_\lambda$. The following convergence results are described in Kawazu and Kesten ([-@KawKes1984]).
If $ c:=\mathbb{E} [\lambda_0^{-1}]<\infty$, then for $
P_\lambda$-almost all environments, the distributions (after conditioning on the environment) of the processes $$X_n(t):=\frac{1}{n}X(n^2t),\qquad t\geq0,$$ converge weakly with respect to the Skorohod topology toward the distribution of the process $ \{c^{-1/2}B(t);t\geq0\} $, where $ \{B(t);t\geq0\} $ is standard Brownian motion on $ \mathbb{R} $.
(See also Papanicolaou and Varadhan ([-@PapVar1981]) for some related results.)
If there exists a slowly varying function $ L_1 $ such that $$\frac{1}{nL_1(n)}\sum_{j=1}^n\frac{1}{\lambda_j}\longrightarrow1\qquad
\mbox{in probability},$$ then the distributions of the processes $$X_n(t):=\frac{1}{n}X(n^2L_1(n)t)$$ converge weakly with respect to the Skorohod topology toward the distribution of standard Brownian motion.
If there exists a slowly varying function $ L_2 $ such that the sequence of random variables $$R_n:=\frac{1}{n^{1/\alpha}L_2(n)}\sum_{j=1}^n\frac{1}{\lambda_j}$$ converges in distribution toward a one-sided stable distribution $
\vartheta_\alpha$ with index $ \alpha\in(0,1) $, then the distributions of the processes $$X_n(t):=\frac{1}{n}X\bigl(n^{(1+\alpha)/\alpha}L_2(n)t\bigr)$$ converge weakly with respect to the Skorohod topology toward the distribution of a continuous self-similar process $ \{X_\ast(t);t\geq0\} $ with scaling exponent $
\eta=\frac{\alpha}{\alpha+1} $.
\(1) In the next section, we will give a representation for the process $
X_\ast$ in terms of a standard Brownian motion and a stable subordinator associated with the measure $ \vartheta_\alpha$.
\(2) We note that the results from Kawazu and Kesten ([-@KawKes1984]) are generalized in Kawazu ([-@Kaw1989]).
He considered random walks in random environments defined by the following transition asymptotics: $$\begin{aligned}
\mathbb{P} \bigl(X(t+h)=j+1|X(t)=j,\mathcal{A}\bigr) &=& (\lambda_j/\eta
_j)h+\mathrm{o}(h),\\
\mathbb{P} \bigl(X(t+h)=j-1|X(t)=j,\mathcal{A}\bigr) &=& (\lambda_{j-1}/\eta
_j)h+\mathrm{o}(h),\\
\mathbb{P} \bigl(X(t+h)=j|X(t)=j,\mathcal{A}\bigr) &=& 1-\bigl((\lambda_j+\lambda
_{j-1})/\eta_j\bigr)h+\mathrm{o}(h),\end{aligned}$$ where $ \{\eta_j,j\in\mathbb{N}\} $ is an i.i.d. family of positive random variables satisfying suitable assumptions. Similarly to the situation studied in Kawazu and Kesten ([-@KawKes1984]), the resulting random walks converge toward appropriate continuous processes after scaling.
In Kesten and Spitzer ([-@KesSpi1979]), new classes of continuous self-similar processes are described. Moreover, it was proven therein that those processes are weak limits of random walks in random scenery. Those random walks are defined as follows.
Let $ \{\xi(x);x\in\mathbb{Z}\} $ and $ \{Z_i;i\in\mathbb{N}\} $ be two independent families of i.i.d. random variables, where the random variables $ Z_i $ are assumed to be $ \mathbb{Z} $-valued. One can think of the sequence $ \{Z_i;i\in\mathbb{N}\} $ as increments of a classical $ \mathbb{Z} $-valued random walk $ S_k:=\sum_{i=1}^kZ_i $. The stationary sequence $ \{\xi(S_k);k\in\mathbb{N}\} $ has some non-trivial long-range dependencies if the underlying random walk $ \{S_k;k\in\mathbb{N}\} $ is recurrent. This is the case, for example, if $ Z_1 $ is in the domain of attraction of an $ \alpha$-stable distribution with $ \alpha\in(1,2] $. The random sequence $ D(n):=\sum_{k=1}^n\xi
(S_k) $ is called a *random walk in random scenery*. In Kesten and Spitzer ([-@KesSpi1979]), the following convergence result was proven for those processes.
If $ \xi(0) $ is in the domain of attraction of a $
\beta$-stable distribution with $ \beta\in(0,2] $ and if $ Z_1 $ is in the domain of attraction of an $ \alpha$-stable distribution with $ \alpha\in(0,1) $, then the distributions of the processes $$D_n(t):=n^{-1/\beta}\sum_{k=1}^{\lfloor nt\rfloor}\xi(S_k)$$ *converge weakly with respect to the Skorohod topology toward $
\beta$-stable Lévy motion.*
(See also Spitzer ([-@Spi1976]) for a special case.)
If $ \xi(0) $ is in the domain of attraction of a $
\beta$-stable distribution with $ \beta\in(0,2] $ and if $ Z_1 $ is in the domain of attraction of an $ \alpha$-stable distribution with $ \alpha\in(1,2] $, then the distributions of the processes $$D_n(t):=n^{-\delta}\sum_{k=1}^{\lfloor nt\rfloor}\xi(S_k)$$ converge weakly with respect to the Skorohod topology toward a continuous self-similar process $ D_\ast$ with scaling exponent $ \delta=1-\frac{1}{\alpha}+\frac
{1}{\alpha\beta}$.
The statement in KS1 corresponds to the transient case and is not difficult to prove since, in that case, the sequence $ \{\xi(S_k);k\in\mathbb{N}\} $ has only weak dependencies. This is the reason why one obtains $ \beta$-stable Lévy noise in the limit. We also mention that the case $ \beta=1 $ is still open.
There exist various generalizations of the results of Kesten and Spitzer ([-@KesSpi1979]). We will only mention Shieh ([-@Shi1995]), where the limiting process is generalized to higher dimensions, Lang and Nguyen ([-@LanNgu1983]), which deals with multidimensional random walks and some special random scenery, Maejima ([-@Mae1996]), where the random scenery belongs to the domain of attraction of an operator-stable distribution, Arai ([-@Ara2001]), where the random scenery belongs to the domain of partial attraction of a semi-stable distribution, and Saigo and Takahashi ([-@SaiTak2005]), where the random scenery and the random walk belong to the domain of partial attraction of semi-stable and operator semi-stable distributions.
In this article, we investigate whether it is possible to substitute the classical random walk in the result of Kesten and Spitzer ([-@KesSpi1979]) by the random walk in random environment which was introduced in Kawazu and Kesten ([-@KawKes1984]). We will restrict our attention to the result KK3 since this is the case where a new type of self-similar process arises at the end. For simplicity and in order to avoid complicating notation, we will assume that the slowly varying function $ L_2 $ which appears in KK3 is constant and equal to one. The general case involving non-constant $ L_2 $ can be treated in a similar way.
We now fix a probability space $ (\Omega,\mathcal{F},\mathbb{P} ) $ which is sufficiently large to support a family of i.i.d. random variables $ \{\lambda_j;j\in\mathbb{Z}\} $, a birth–death process $ \{X(t);t\geq0\} $ with asymptotic transition rates given by equations (1)–(3) and a family of i.i.d. random variables $ \{\xi(k),k\in\mathbb{Z}\} $.
We assume that the families $ \{\xi(k),k\in\mathbb{Z}\} $ and $ \{
X(t);t\geq0\} $ are independent and that $ t\mapsto X(t) $ is cadlag $ \mathbb{P} $-almost surely.
Further, we assume that $ \lambda^{-1}_1 $ is in the domain of normal attraction of a one-sided $ \alpha$-stable distribution $ \vartheta_\alpha$ with $ \alpha\in(0,1) $.
Moreover, we assume that $ \xi(0) $ is in the domain of normal attraction of a $ \beta$-stable distribution $ \vartheta_\beta$ with $ \beta\in(0,2] $. Its characteristic function is given by $$\psi(\theta)=\exp\bigl(-|\theta|^\beta\bigl(A_1+\mathrm{i}A_2\operatorname{sgn}(\theta)\bigr)\bigr) ,$$ where $ 0<A_1<\infty$ and $ |A_1^{-1}A_2|\leq\tan(\uppi\beta/2) $. For $ \beta>1 $, it follows from those assumptions that $ \mathbb{E}
[\xi(0)]=0 $.
For $ \beta=1 $, we make the further assumption that there exists a $
K>0 $ such that $$\bigl|\mathbb{E} \bigl[\xi(0)\mathbh{1}_{[-\rho,\rho]}(\xi(0)) \bigr]
\bigr|\leq K\qquad
\mbox{for all } \rho>0 .$$ We can now define the following continuous-time version of the random walk in random scenery: $$\Xi(t):=\int_0^t\xi(X(s))\,\mathrm{d}s .$$ In the following, we will use the space $$D[0,\infty):= \{\gamma\dvtx[0,\infty)\rightarrow\mathbb{R}\dvtx\gamma
\mbox{ is cadlag} \}$$ with the Skorohod topology. We will prove the following theorem.
\[MT\] For $ \kappa:=\frac{1}{\alpha}+\frac{1}{\beta} $ and $
k_n:=n^{(1+\alpha)/\alpha}$, the distributions of the processes $$\Xi_n(t):=n^{-\kappa}\int_0^{k_nt}\xi(X(s))\,\mathrm{d}s$$ converge weakly with respect to the Skorohod topology toward the distribution of a self-similar stochastic process $ \{\Xi_\ast(t);t\geq0\} $ with scaling exponent $ \mu=1-\frac{\alpha}{\alpha+1}+\frac{\alpha}{(\alpha+1)\beta} $.
The stochastic process $ \{\Xi_\ast(t);t\geq0\} $ can be constructed as follows. Let $ Z_+ $ and $ Z_- $ be two independent copies of the $ \beta$-stable Lévy process which can be associated with the characteristic function $$\psi(\theta)=\exp\bigl(-|\theta|^\beta\bigl(A_1+\mathrm{i}A_2\operatorname{sgn}(\theta)\bigr) \bigr) .$$ Further, let $ \{L_\ast(\tau,x);\tau\geq0,x\in\mathbb{R}\} $ be the local time of the stochastic process $ \{X_\ast(\tau);\tau\geq0\} $; that is, the random variable $
L_\ast(\tau,x) $ is the derivative with respect to $ x $ of the occupation time $$\Gamma_\ast(\tau,(-\infty,x]):=\int_0^\tau\mathbh{1}_{(-\infty,x]}(X_\ast(\sigma))\,\mathrm{d}\sigma.$$ We will see in the next section that the local time exists for all but a countable number of points $ x\in\mathbb{R} $. Moreover, for all $ \tau\geq0 $, the processes $$\{L_\ast(\tau,x-);x\geq0\} \quad \mbox{and} \quad \{L_\ast(\tau
,-(x-));x\geq0\}$$ are predictable with respect to the natural filtrations of $ Z_+ $ (resp., $ Z_-$). The following integral representation of the process $ \Xi_\ast$ can be given: $$\Xi_\ast(\tau):=\int_0^\infty L_\ast(\tau,x-)\,\mathrm{d}Z_+(x)+\int
_0^\infty L_\ast(\tau,-(x-))\,\mathrm{d}Z_-(x) .$$
The convergence of the birth–death process
==========================================
The goal of this section is to prove Corollary \[PrinceKor\], which is the main ingredient needed to show that the finite-dimensional distributions of $ \Xi_n $ converge toward the finite-dimensional distributions of $ \Xi_\ast$. This corollary contains a statement on the weak convergence of certain functionals of the occupation times of the rescaled processes $ X_n $. A result corresponding to Corollary \[PrinceKor\] is also proved in Kesten and Spitzer ([-@KesSpi1979]); however, we have to adopt a totally different approach since we do not have such precise information on the potential theory related to the random walk $ X $. Instead, we will understand the occupation times of $ X_n $ and prove that they converge in an appropriate sense toward the local time of the limit process $ X_\ast$.
We describe some of the main arguments from the proof in Kawazu and Kesten ([-@KawKes1984]) for the convergence of the processes $$X_n(t):=\frac{1}{n}X\bigl(n^{(1+\alpha)/{\alpha}}t\bigr)$$ toward the self-similar process $ X_\ast$ defined in Kawazu and Kesten ([-@KawKes1984]). We can enlarge our underlying probability space $ (\Omega,\mathcal{F},\mathbb{P}
) $ in such a way that it contains a standard Brownian motion $ \{
B(t);t\geq0\} $ and a cadlag version of the stable Lévy subordinator $ \{W(x);x\in\mathbb{R}\} $ which can be associated with the one-sided $ \alpha$-stable distribution $ \vartheta_\alpha$.
Furthermore, we assume that $ \{B(t);t\geq0\} $, $ \{W(x);x\in\mathbb
{R}\} $, $ \{X(t);t\geq0\} $ and $ \{\xi(n);n\in\mathbb{Z}\} $ are independent. Moreover, we assume that $ W(0)=0 $ and $ B(0)=0 $ hold $ \mathbb{P} $-almost surely.
In the future, we will denote by $ \{L(t,x);t\geq0,x\in\mathbb{R}\}
$ the local time of the Brownian motion $ \{B(t);t\geq0\} $. The process $$V_\ast(t):=\int_\mathbb{R}L(t,W(x))\,\mathrm{d}x$$ is non-decreasing $ \mathbb{P} $-almost surely. Therefore, we can define the following pseudo-inverse: $$W^{-1}(y):=\inf\{x\in\mathbb{R};W(x)>y\} \quad \mbox{and} \quad
V_\ast^{-1}(\tau):=\inf\{t\geq0;V_\ast(t)>\tau\} .$$ In Kawazu and Kesten ([-@KawKes1984]), the following representation for the self-similar process $ X_\ast$ is given: $$X_\ast(\tau):=W^{-1}(B(V_\ast^{-1}(\tau))).$$ We now sketch the main arguments from the proof in Kawazu and Kesten ([-@KawKes1984]). We will need some of those ideas in our proof of the convergence of $
\Xi_n $ toward $ \Xi_\ast$. Their approach is based on the natural scale of the birth–death process. One defines $$S(j):= \cases{\displaystyle
\sum_{k=0}^{j-1}\lambda_k^{-1} &\quad for $ j>0$,\vspace*{2pt}\cr
0 & \quad for $j=0$,\cr
\displaystyle-\sum_{k=j}^{-1}\lambda_k^{-1} &\quad for $ j<0$.
}
%$$ This implies that conditioned on $ \mathcal{A}:=\{\lambda_j;j\in\mathbb
{Z}\}, $ the process $ S(X(t)) $ is on natural scale (see Kawazu and Kesten ([-@KawKes1984]), page 565). This means that for all $ a,b,x\in\mathbb{R} $ with $ a<x<b $, one has $$\mathbb{P} \bigl(S(X(t)) \mbox{ hits } \{a,b\} \mbox{ first at } a
\mid S(X(0))=x,\mathcal{A}\bigr)=\frac{b-x}{b-a}.$$ It is then possible to represent the process $ S(X(t)) $ as the time change of standard Brownian motion $ \{B(t);t\geq0\} $ as follows.
One defines $ m(\mathrm{d}x):=\sum_{i\in\mathbb{Z}}\delta_{S(i)}(\mathrm{d}x) $ and $$V(t):=\int_\mathbb{R} L(t,x)m(\mathrm{d}x)=\sum_{i\in\mathbb{Z}}L(t,S(i)) ,$$ where $ \{L(t,x);t\geq0,x\in\mathbb{R}\} $ is again the local time of the standard Brownian motion $ B $. One can see that $ \{B(V^{-1}(t));t\geq0\} $ and $ \{S(X(t));t\geq0\}
$ are both cadlag and have the same distribution (see Kawazu and Kesten ([-@KawKes1984]), page 566).
One then has to scale the above constructions. $$S_n(x):=n^{-1/\alpha}S(\lfloor nx\rfloor),\qquad n\in\mathbb{N},
x\in\mathbb{R} ,$$ where, for a positive real number $ x $, we denote by $ \lfloor
x\rfloor$ its integer part. It follows from the assumptions on the environment $ \{\lambda_j;j\in
\mathbb{Z}\} $ that for $ n\rightarrow\infty$, the processes $ \{S_n(x);x\in\mathbb{R}\} $ converge in distribution toward an $ \alpha$-stable Lévy process $ \{W(x);x\in\mathbb{R}\} $. Moreover, the process $ W $ is strictly increasing $ \mathbb{P}
$-almost surely since $ \vartheta_\alpha$ is a one-sided stable distribution and $ \alpha
\in(0,1) $. By a method given in Skorohod ([-@Sko1956]) and Dudley ([-@Dud1968]), it is possible to construct a suitable probability space $ (\tilde{\Omega},\tilde\mathcal{F},\tilde{\mathbb{P} }) $ with suitable $ D $-valued random variables $ \tilde{S}_n $ and $ \tilde{W} $ having the properties that $ \tilde
{S}_n $ converges toward $ \tilde{W} $ almost surely with respect to $ \tilde{\mathbb{P} } $ and that $
\tilde{S}_n
$ and $ \tilde{W} $ have the same distributions as $ S_n $ (resp., $ W $) (see Kawazu and Kesten ([-@KawKes1984]), page 567). One then defines $$\tilde{V}_n(t):=\int_\mathbb{R} L(t,x)\tilde{m}_n(\mathrm{d}x) \quad
\mbox{and}\quad
\tilde{V}_\ast(t):=\int_\mathbb{R} L(t,x)\tilde{m}_\ast(\mathrm{d}x)$$ with $$\int_\mathbb{R} f(x)\tilde{m}_n(\mathrm{d}x):=\int_{\mathbb{R}}f(\tilde
{S}_n(x))\,\mathrm{d}x\quad \mbox{and}\quad
\int_\mathbb{R} f(x)\tilde{m}_\ast(\mathrm{d}x):=\int_{\mathbb{R}}f(\tilde
{W}(x))\,\mathrm{d}x$$ for all measurable $ f\geq0 $. We then define $ \tilde{S}_n^{-1} $, $
\tilde{W}^{-1} $, $ \tilde{V}_n^{-1} $ and $ \tilde{V}_\ast^{-1} $ in the same way as $ W^{-1} $ (resp., $ V_\ast^{-1} $) above.
In Kawazu and Kesten ([-@KawKes1984]) (see page 568) they prove that $ \{B(\tilde{V}^{-1}_n(t));t\geq0\} $ converges $ \tilde{\mathbb{P} }
$-almost surely toward $ \{B(\tilde{V}^{-1}_\ast(t));t\geq0\} $ in the $ J_1 $-topology. For convenience, we define $$\tilde{X}_n(t):=\tilde{S}_n^{-1}(B(\tilde{V}^{-1}_n(t))), \qquad
\tilde{X}_\ast(t):=\tilde{W}^{-1}(B(\tilde{V}^{-1}_\ast(t))) .$$ We note that the process $ \{\tilde{X}_n(t);t\geq0\} $ is defined on $ (\Omega\times\tilde{\Omega},\mathcal{F}\times\tilde\mathcal{F},\mathbb{P}
\times\tilde{\mathbb{P} }) $. It is proved in Kawazu and Kesten ([-@KawKes1984]) that $ \{\tilde{X}_n(t);t\geq
0\} $ converges toward $ \{\tilde{X}_\ast(t);t\geq0\} $ with respect to the $ J_1 $-topology almost surely with respect to $ \mathbb{P}
\times
\tilde{\mathbb{P} } $ (see page 569).
Moreover, for $ B_n(t):=n^{-1/2}B(nt) $ one has that (see Kawazu and Kesten ([-@KawKes1984]), page 572) $$|X_n(t)-S_n^{-1}(B_n(V_n^{-1}(t)))|\leq1/n$$ and $$\{S_n^{-1}(B_n(V^{-1}_n(t)));t\geq0 \}\stackrel{\mathcal{D}}{=}
\{\tilde{S}^{-1}_n(B(\tilde{V}_n^{-1}(t)));t\geq0 \}
= \{\tilde{X}_n(t);t\geq0 \} .$$ If we define $ \hat{X}_n(t):=S_n^{-1}(B_n(V^{-1}_n(t))) $, then the previous observations imply that both processes $ \{X_n(t);t\geq0\} $ and $ \{\hat{X}_n(t);t\geq0\} $ converge in distribution toward $ \{\tilde{X}_\ast(t);t\geq0\} $, which has the same distribution as $ \{X_\ast(t);t\geq0\} $.
In the rest of this section, we analyze the distributional behavior of the occupation times for the process $ X_n $ (see Proposition \[PrinceProp\]). In order to obtain this result, we prove an analogous result for the process $ \tilde{X}_n $ (see Lemma \[PrinceLem\]), which can be reduced to Proposition \[CardTowardMeasureProp\]. The advantage of this detour is that we can prove almost sure convergence for the occupation times of the process $ \tilde{X}_n $ toward the local time of $ \tilde{X}_\ast$ (see Proposition \[OkTimeLokTimeConvProp\]). This result is based on the fact that we have explicit formulas for the occupation times of $ \tilde{X}_n $ and the local time of $ \tilde{X}_\ast$ (see Proposition \[OkTimePro\] and Corollary \[LokTimeKor1\]). The explicit expression of the occupation time of $ \tilde{X}_n $ and the local time of $ \tilde{X}_\ast$ reveals that in order to prove Proposition \[OkTimeLokTimeConvProp\], it is sufficient to prove the almost sure convergence of $ \tilde{S}_n $ and $ \tilde{V}_n^{-1} $ toward $ \tilde{W}_\ast$ (resp., $ \tilde{V}^{-1}_\ast$). The convergence of $ \tilde{S}_n $ toward $ \tilde{W}_\ast$ holds by construction. The convergence of $ \tilde{V}_n $ toward $ \tilde
{V}_\ast$ is obtained in Lemma \[TimeChangeConvLem\] and then used to obtain the convergence of $
\tilde{V}_n^{-1} $ toward $ \tilde{V}^{-1}_\ast$ in Lemma \[InversTimeChangeConvLem\].
The local times of $ X_\ast$ and $ \tilde{X}_\ast$
--------------------------------------------------
We define the time that the processes $ \tilde{X}_\ast$ and $ X_\ast
$ spend in the measurable set $ A $ until time $ \tau$ as $$\Gamma_\ast(\tau,A):=\int_0^\tau\mathbh{1}_{A}(X_\ast
(\sigma))\,\mathrm{d}\sigma\qquad \biggl(\mbox{resp.},\
\tilde{\Gamma}_\ast(\tau,A):=\int_0^\tau\mathbh{1}_{A}(\tilde{X}_\ast(\sigma))\,\mathrm{d}\sigma\biggr).$$ We denote by $ \{L_\ast(\tau,x);\tau\geq0,x\in\mathbb{R}\} $ and $ \{\tilde{L}_\ast(\tau,x);\tau\geq0,x\in\mathbb{R}\} $ the local times of $ X_\ast$ (resp., $ \tilde{X}_\ast$) if they exist. In this subsection, we prove that both local times exist almost surely and relate them to the local time $ \{L(t,x);t\geq0,x\in\mathbb{R}\} $ of the underlying Brownian motion $ \{B(t);t\geq0\} $.
\[LokTimePro\] One has $ \mathbb{P} $-almost surely that for $ \tau\geq0 $ and all $ x\in
\mathbb{R} $, $$\Gamma_\ast(\tau,(-\infty,x))=\int_{-\infty}^xL(V_\ast^{-1}(\tau
),W(y))\,\mathrm{d}y .$$ Further, $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely for all $ \tau\geq0
$ and all $ x\in\mathbb{R} $, $$\tilde{\Gamma}_\ast(\tau,(-\infty,x))=\int_{-\infty}^xL(\tilde
{V}_\ast^{-1}(\tau),\tilde{W}(y))\,\mathrm{d}y .$$
We have $ \mathbb{P} $-almost surely that $ x\mapsto W(x) $ is increasing. It follows that the set $ \mathcal{N}_1 $ of $ x\in\mathbb{R} $ where $
W $ is not continuous is countable. We define the set $$\mathcal{N}_2:= \bigl\{x\in\mathbb{R}\dvtx\ell\bigl(\sigma;B(V_\ast^{-1}(\sigma
))=W(x)\bigr)>0 \bigr\} ,$$ where $ \ell$ denotes the Lebesgue measure on $ \mathbb{R} $. The set $ \mathcal{N}_2 $ is countable since for $ x_1\neq x_2 $, one has that the sets $ \{\sigma;B(V_\ast^{-1}(\sigma))=W(x_1)\} $ and $\{\sigma
;B(V_\ast^{-1}(\sigma))=W(x_2)\} $ are disjoint. The statement then follows since there cannot be an uncountable number of disjoint subsets of $ \mathbb{R} $ with positive Lebesgue measure. Thus the set $ \mathcal{N}:=\mathcal{N}_1\cup\mathcal{N}_2 $ is countable. Since the function $ x\mapsto\Gamma_\ast(\tau,(-\infty,x)) $ is increasing and since $$x\mapsto\int_{-\infty}^xL(V_\ast^{-1}(\tau),W(y))\,\mathrm{d}y$$ is continuous, it is sufficient to prove the statement of the proposition for $ x\in\mathcal{N}^c $.
The fact that $ W $ is increasing and continuous in $ x $ implies the equivalence of the statement $ W(x)>y $ with the statement $ \exists z_0<x\dvtx W(z_0)>y $.
The latter statement is then equivalent to the statement $ W^{-1}(y):=\inf\{z\dvtx W(z)>y\}<x $.
This then implies that $ \mathbh{1}_{(-\infty,x)}(X_\ast(\sigma))
=\mathbh{1}_{(-\infty,W(x))}(B(V_\ast^{-1}(\sigma))) $.
We also note that $ t\mapsto V(t) $ is continuous and non-decreasing. This implies that $ V_\ast\circ V_\ast^{-1}=\operatorname{id}_\mathbb{R} $.
In the following, we want to compute the derivative of the non-decreasing function $$M\dvtx \sigma\mapsto\int_{-\infty}^xL(V_\ast^{-1}(\sigma),W(y))\,\mathrm{d}y .$$ Since $ W $ is increasing and continuous in $ x $, we have that $ B(V_\ast^{-1}(\sigma_0))<W(x) $ implies that $$\sigma\mapsto\int_x^\infty L(V_\ast^{-1}(\sigma),W(y))\,\mathrm{d}y$$ is locally constant, say equal to $c_0$, in a neighborhood of $ \sigma_0 $.
Thus $$\sigma\mapsto\int_{-\infty}^xL(V_\ast^{-1}(\sigma),W(y))\,\mathrm{d}y
= V_\ast(V_\ast^{-1}(\sigma))-c_0=\sigma-c_0$$ in a neighborhood of $ \sigma_0 $.
Moreover, since $ W $ is increasing and continuous in $ x $, we have that $ B(V_\ast^{-1}(\sigma_0))>W(x)$ implies $$\sigma\mapsto\int_{-\infty}^xL(V_\ast^{-1}(\sigma),W(y))\,\mathrm{d}y$$ is locally constant in a neighborhood of $ \sigma_0 $.
It therefore turns out that $$M'(\sigma)= \cases{
1, &\quad if $ B(V_\ast^{-1}(\sigma))<W(x)$,\cr
0, &\quad if $ B(V_\ast^{-1}(\sigma))>W(x)$.
}
%$$ Moreover, for all $ \sigma_1,\sigma_2\in\mathbb{R}^+ $ with $
\sigma_1\leq\sigma_2 $, we have that $$\int_{-\infty}^xL(V_\ast^{-1}(\sigma_1),W(y))\,\mathrm{d}y
\leq\int_{-\infty}^xL(V_\ast^{-1}(\sigma_2),W(y))\,\mathrm{d}y$$ and $$\int_x^\infty L(V_\ast^{-1}(\sigma_1),W(y))\,\mathrm{d}y
\leq\int_x^\infty L(V_\ast^{-1}(\sigma_2),W(y))\,\mathrm{d}y .$$ This implies that $$\begin{aligned}
&&\int_{-\infty}^xL(V_\ast^{-1}(\sigma_2),W(y))\,\mathrm{d}y
-\int_{-\infty}^xL(V_\ast^{-1}(\sigma_1),W(y))\,\mathrm{d}y\\
&&\quad
\leq V_\ast(V_\ast^{-1}(\sigma_2))-V_\ast(V_\ast^{-1}(\sigma
_1))=\sigma_2-\sigma_1.\end{aligned}$$ It follows that $$\sigma\mapsto\int_{-\infty}^xL(V_\ast^{-1}(\sigma),W(y))\,\mathrm{d}y$$ is Lipschitz continuous with Lipschitz constant smaller than one.
Since the set $ \{\sigma\dvtx B(V_\ast^{-1}(\sigma))=W(x)\} $ is a zero set with respect to the Lebesgue measure $ \ell$ for all $ x\in\mathcal{N}^c $, it follows that $$\int_0^\tau\mathbh{1}_{(-\infty,x)}(X_\ast(\sigma
))\,\mathrm{d}\sigma=
\int_0^\tau\mathbh{1}_{(-\infty,W(x))}(B(V_\ast
^{-1}(\sigma)))\,\mathrm{d}\sigma=
\int_0^\tau M'(\sigma)\,\mathrm{d}\sigma=M(\tau) .$$ The second statement is proved in the same way.
\[LokTimeKor1\] One has $ \mathbb{P} $-almost surely that the local time $ L_\ast
(\tau,x) $ is defined for all $ \tau\geq0 $ and all $ x $, where $ x\mapsto W(x) $ is continuous. Further, one has $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely that the local time $ \tilde{L}_\ast(\tau,x) $ is defined for all $ \tau\geq0 $ and all $ x $, where $ x\mapsto
\tilde{W}(x) $ is continuous. In those points, one has $$L_\ast(\tau,x)=L(V_\ast^{-1}(\tau),W(x)) \qquad \bigl(\mbox{resp.}, \
\tilde{L}_\ast(\tau,x)=L(\tilde{V}_\ast^{-1}(\tau),\tilde
{W}(x))\bigr) .$$
Differentiation in Proposition \[LokTimePro\] proves this corollary.
The occupation time of $ \tilde{X}_n $
--------------------------------------
For a measurable set $ A\subset\mathbb{R} $, we define $$\hat{\Gamma}_n(t,A):=\int_0^t \mathbh{1}_{A}(\hat
{X}_n(\sigma))\,\mathrm{d}\sigma, \qquad
\tilde{\Gamma}_n(t,A):=\int_0^t \mathbh{1}_{A}(\tilde
{X}_n(\sigma))\,\mathrm{d}\sigma$$ and $$\Gamma_n(t,A):=\int_0^t \mathbh{1}_{A}(X_n(\sigma
))\,\mathrm{d}\sigma.$$ These are the respective times that the processes $ \hat{X}_n $, $
\tilde{X}_n $ and $ X_n $ spend in the set $ A $ until time $ t $. In this section, we give an explicit expression for the occupation time of $ \tilde{X}_n $ in terms of the local time $ \{L(t,x);t\geq0,x\in\mathbb{R}\} $ of the underlying Brownian motion $ \{B(t);t\geq0\} $.
\[OkTimePro\] One has $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely for all $ \tau\geq0
$ and all $ x\in\mathbb{R} $ that $$\tilde{\Gamma}_n(\tau,\{x\})= \cases{
\displaystyle\frac{1}{n}L\biggl(\tilde{V}_n^{-1}(\tau),\tilde{S}_n\biggl(x-\frac{1}{n}\biggr)\biggr),
&\quad if $ nx\in\mathbb{Z},$\cr
0, &\quad if $ nx\notin\mathbb{Z} $.
}
%$$
First, we note that $$S_n^{-1}(S_n(x))=x+1/n \qquad \mbox{for all } x \mbox{ satisfying }
nx \in\mathbb{Z}.$$ If we use the fact that $ \{B_n(V_n^{-1}(t));t\geq0\}\}\stackrel{\mathcal{D}}{=}\{S_n(X_n(t));t\geq0\}, $ then we can see that $ \{\hat{X}_n(t);t\geq0\}\stackrel{\mathcal{D}}{=}\{
X_n(t)+1/n;t\geq0\} $. Therefore, we see that $ \hat{X}_n $ only takes values in the lattice $ \frac{1}{n}\mathbb{Z} $. Moreover, we have that $ \tilde{S}_n $ and $ \tilde{V_n} $ have the same joint distribution as $ S_n $ and $ V_n $. Therefore, $ \hat{X}_n=S_n^{-1}(B_n(V_n^{-1}(\cdot))) $ has the same distribution as $ \tilde{X}_n=\tilde{S}_n^{-1}(B(\tilde{V}_n^{-1}(\cdot))) $. From this, it also follows that $ \tilde{X}_n $ stays for all time in the countable state space $ \{x\in\mathbb{R};nx\in\mathbb{Z}\} $. This implies that $ \tilde{\Gamma}_n(\tau,\{x\})=0 $ for $ nx\notin
\mathbb{Z} $. This proves one part of the statement.
For the proof of the other part of the statement, we will need the derivative of the function $$\tilde{M}(\sigma):= \frac{1}{n}L\bigl(\tilde{V}_n^{-1}(\sigma),\tilde
{S}_n(x-1/n)\bigr) .$$ We first collect some useful facts which help to compute the derivative of $ \tilde{M} $.
Since $ \tilde{S}_n $ is constant on the intervals $ [\frac
{k}{n},\frac{k+1}{n}) $ for all $ k\in\mathbb{Z} $, we have $$\label{SumGleich}
\tilde{V}_n(t)=\int_\mathbb{R}L(t,\tilde{S}_n(x))\,\mathrm{d}x=\frac
{1}{n}\sum_{i\in\mathbb{Z}}L\bigl(t,\tilde{S}_n(i/n)\bigr) .$$ Since the $ (t,x)\mapsto L(t,x) $ is jointly continuous and non-decreasing $ \mathbb{P} $-almost surely (see Boylan ([-@Boy1964]) or Getoor and Kesten ([-@GetKes1972])), it follows that $
t\mapsto\tilde{V}_n(t) $ is continuous and non-decreasing $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely. This then gives rise to $$\label{VnId}
\tilde{V}_n\circ\tilde{V}_n^{-1}=\operatorname{id}_{\mathbb{R}^+}
\qquad \mathbb{P} \times\tilde{\mathbb{P} }\mbox{-almost surely}.$$ By construction, one has for all $ b\in\{\tilde{S}_n(x);x\in\mathbb
{R}\} $ that $ \tilde{S}_n^{-1}(b)=x $ is equivalent to $ b=\tilde{S}_n(x-\frac{1}{n}) $. Moreover, one has that $ B(\tilde{V}_n^{-1}(\sigma))\in\{\tilde{S}_n(x);x\in\mathbb{R}\}
$ for all $ \sigma\geq0 $ almost surely with respect to $ \mathbb{P} \times\tilde{\mathbb{P} } $. Hence, $$\label{equival}
\tilde{X}_n(\sigma)=\tilde{S}_n^{-1}(B(\tilde{V}_n^{-1}(\sigma
)))=x \mbox{ is equivalent to }
B(\tilde{V}_n^{-1}(\sigma))=\tilde{S}_n\biggl(x-\frac{1}{n}\biggr) .$$ Moreover, the random variables $ \{\lambda_i^{-1};i\in\mathbb{N}\} $ are positive $ \mathbb{P} $-almost surely and therefore $$\begin{aligned}
\label{inject}
\mbox{the restriction of } x\mapsto\tilde{S}_n(x) \mbox{ to the
set } \frac{1}{n}\mathbb{Z}
\mbox{ is injective almost surely with respect to } \tilde{\mathbb
{P} } .\end{aligned}$$ Since, conditioned on $ \mathcal{A}=\sigma\{\lambda_j;j\in\mathbb{N}\}
$, the process $ X $ is a Markov process, it follows that for $ nx\in\mathbb{Z} $, there exist non-negative random variables $
a_1<b_1<a_2<b_2<\cdots$ with the property $$\begin{aligned}
\{\sigma\geq0;\tilde{X}_n(\sigma)=x \}=\bigcup_{i\in\mathbb
{N}}[a_i,b_i) \qquad
\mathbb{P} \times\tilde{\mathbb{P} }\mbox{-a.s.}\end{aligned}$$ This implies that for all $ \sigma_0\notin\{a_i;i\in\mathbb{N}\} $, there exists a neighborhood $ \mathcal{U}(\sigma_0)$ containing $ \sigma_0 $ with the property that $ \sigma\mapsto\tilde{X}_n(\sigma)=\tilde{S}_n^{-1}(B(\tilde
{V}_n^{-1}(\sigma))) $ is constant on $ \mathcal{U}(\sigma_0) $. Equations (\[equival\]) and (\[inject\]) then imply that $ \sigma\mapsto
B(\tilde{V}_n^{-1}(\sigma)) $ must be constant on $ \mathcal{U}(\sigma_0) $.
Therefore, for $ \sigma_0\notin\{a_i;i\in\mathbb{N}\} $ and $ B(\tilde{V}_n^{-1}(\sigma_0))\neq\tilde{S}_n(x-\frac{1}{n}) $, we have $ B(\tilde{V}_n^{-1}(\sigma))\neq\tilde{S}_n(x-\frac{1}{n}) $ for all $ \sigma$ in a neighborhood of $ \sigma_0 $. Hence $$\sigma\mapsto L\bigl(\tilde{V}_n^{-1}(\sigma),\tilde{S}_n(x-1/n)\bigr)$$ is constant in a neighborhood of $ \sigma_0$. The previous argument and the fact that $ \tilde{X}_n $ only jumps to nearest neighbors in $ \frac{1}{n}\mathbb{Z} $ leads to the fact that $ \sigma_0\notin\{
a_i;i\in\mathbb{N}\} $ and $ B(\tilde{V}_n^{-1}(\sigma_0))=\tilde{S}_n(x-\frac{1}{n}) $ imply the existence of a suitable $ c_0>0 $ with the property $$\sigma\mapsto\frac{1}{n}\sum_{z\neq nx-1}L\bigl(\tilde{V}_n^{-1}(\sigma
),\tilde{S}_n(z/n)\bigr)=c_0$$ in a neighborhood of $ \sigma_0$. Therefore, we can use (\[VnId\]) to see that $ B(\tilde
{V}_n^{-1}(\sigma_0))=\tilde{S}_n(x-\frac{1}{n}) $ implies that $$\sigma\mapsto\frac{1}{n}L\bigl(\tilde{V}_n^{-1}(\sigma),\tilde
{S}_n(x-1/n)\bigr)=\tilde{V}_n(\tilde{V}_n^{-1}(\sigma))-c_0=\sigma-c_0$$ in a neighborhood of $ \sigma_0 $. Consequently, the function $$\tilde{M}(\sigma):= \frac{1}{n}L\bigl(\tilde{V}_n^{-1}(\sigma),\tilde
{S}_n(x-1/n)\bigr)$$ is differentiable for all $ \sigma\notin\{a_i;i\in\mathbb{N}\} $, and for $ nx\in\mathbb{Z} $, we have $$\tilde{M}'(\sigma)= \cases{
1, &\quad if $\displaystyle B(\tilde{V}_n^{-1}(\sigma))=\tilde{S}_n\biggl(x-\frac
{1}{n}\biggr)$,\cr
0, &\quad if $\displaystyle B(\tilde{V}_n^{-1}(\sigma))\neq\tilde{S}_n\biggl(x-\frac{1}{n}\biggr)$.
}
%$$ Moreover, it is possible to prove that the function $ \tilde{M} $ is Lipschitz continuous with Lipschitz constant one. From those properties, it follows that $$\int_0^\tau\mathbh{1}_{\{x\}}(\tilde{X}_n(\sigma
))\,\mathrm{d}\sigma=
\int_0^\tau\mathbh{1}_{\{\tilde{S}_n(x-1/n)\}
}(B(\tilde{V}_n^{-1}(\sigma)))\,\mathrm{d}\sigma=
\int_0^\tau\tilde{M}'(\sigma)\,\mathrm{d}\sigma=\tilde{M}(\tau) .$$
The convergence of the occupation times
---------------------------------------
In this section, we investigate whether the occupation times of $
\tilde{X}_n $ converge toward the local time of $ \tilde{X}_\ast$ in an appropriate way as $ n\rightarrow
\infty$. For this, we first need some auxiliary results.
\[TimeChangeConvLem\] One has $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely that $ \tilde{V}_n(t)
$ converges toward $ \tilde{V}_\ast(t) $ for all $ t\in\mathbb{R} $.
We fix a $ T>0 $ and define $ w_o:=\sup\{x\dvtx L(T,x)>0\} $ and $
w_u:=\inf\{x\dvtx L(T,x)>0\} $. Those two random variables are independent of $ \tilde{\mathbb{P} } $. We know that $ \{\tilde{S}_n(x);x\in\mathbb{R}\} $ converges toward $ \{\tilde{W}(x);x\in\mathbb{R}\} $ with respect to the $ J_1 $-topology $ \tilde{\mathcal{F}} $-almost surely. We note that the local time of Brownian motion $ (x,t)\mapsto L(t,x) $ is jointly continuous $ \mathbb{P} $-almost surely (see Boylan ([-@Boy1964]) or Getoor and Kesten ([-@GetKes1972])).
It follows that $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely $ \{L(t,\tilde
{S}_n(x));x\in\mathbb{R}\} $ converges toward $ \{L(t,\tilde{W}(x));x\in\mathbb{R}\} $ with respect to the $ J_1 $-topology for all $ t\in[0,T] $.
We fix a pair $ (\omega,\tilde{\omega})\in\Omega\times\tilde
{\Omega} $ with the property that $ \{L(t,\tilde{S}_n(x))(\omega,\tilde{\omega});x\in\mathbb{R}\} $ converges toward $ \{L(t,\tilde{W}(x))(\omega,\tilde{\omega});x\in\mathbb{R}\} $ with respect to the $ J_1 $-topology for all $ t\in[0,T] $.
There then exist suitable $ x_u,x_o\in\mathbb{R} $ with $ \tilde
{W}(x_u)\leq w_u $ and $ \tilde{W}(x_o)\geq w_o $, and there exists a sequence of increasing, absolutely continuous, surjective Lipschitz maps $ \lambda_n\dvtx [x_u,x_o]\rightarrow[x_u,x_o] $ with the properties $$\sup_{x\in[x_u,x_o]} |L(t,\tilde{W}(x))-L(t,\tilde{S}_n(\lambda
_n(x))) |\longrightarrow0
\qquad \mbox{as } n\rightarrow\infty$$ and $$\operatorname{esssup}\limits_{x\in[x_u,x_o]} |\lambda_n'(x)-1 |\longrightarrow0
\qquad \mbox{as } n\rightarrow\infty.$$ We should emphasize that the derivative of the function $ \lambda_n $ may not exist everywhere. However, those points where it does not exist form a zero set since $
\lambda_n $ is an absolutely continuous Lipschitz function.
By a change of variables for all $ t\in[0,T] $, one then has $$\begin{aligned}
&& \int_{x_u}^{x_o} L(t,\tilde{S}_n(x))\,\mathrm{d}x-\int_{x_u}^{x_o}
L(t,\tilde{S}_n(\lambda_n(x)))\,\mathrm{d}x \\
&&\quad= \int_{x_u}^{x_o} L(t,\tilde{S}_n(x)) \biggl(1-\frac{1}{\lambda
_n'(\lambda_n^{-1}(x))} \biggr)\,\mathrm{d}x
+\mathrm{O} \Bigl(\sup_{x\in[x_u,x_o]}|\lambda_n(x)-x| \Bigr).\end{aligned}$$ It follows from the assumptions on the sequence $ \lambda_n $ that the above difference converges toward zero. Further, for all $ t\in[0,T] $, we have that $$\int_{\mathbb{R}} L(t,\tilde{S}_n(\lambda_n(x)))\,\mathrm{d}x\longrightarrow
\int_{\mathbb{R}} L(t,\tilde{W}(x))\,\mathrm{d}x\qquad \mbox{as } n\rightarrow
\infty.$$ Hence, one has $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely that $ \tilde
{V}_n(t) $ converges toward $ \tilde{V}_\ast(t) $ for all $ t\in[0,T] $. Thus, for every $ T>0
$, we obtain an zero set $ N_T $ in $ \Omega\times\tilde{\Omega} $ where this convergence does not hold. The lemma now follows since the union $$N_\infty:=\bigcup_{T\in\mathbb{N}}U_T$$ is also a zero set with respect to $ \mathbb{P} \times\tilde{\mathbb
{P} } $.
Let $ f\dvtx\mathbb{R}\rightarrow\mathbb{R} $ be a function. We call $
\tau\in f(\mathbb{R}) $ a *critical value* for $ f $ if there exist at least two distinct points $ t_1,t_2\in\mathbb{R} $ such that $ f(t_1)=f(t_2)=\tau$. Further, we call a point $ \tau\in f(\mathbb{R}) $ a *regular value* for $ f $ if it is not a critical value. It is straightforward to see that the preimages of critical values contain an open interval if the function $ f $ is non-decreasing. This implies that the set of critical values of a non-decreasing function is at most countable.
\[InversTimeChangeConvLem\] One has $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely that $ \tilde
{V}_n^{-1}(\tau) $ converges toward $ \tilde{V}_\ast^{-1}(\tau) $ for all regular values $ \tau$ of $
\tilde{V}_\ast$.
We note that $ \mathbb{P} $-almost surely the local time $ L(t,x) $ of the Brownian motion $ B $ is continuous and non-decreasing in $ t $ for all $ x\in\mathbb{R} $ (see Boylan ([-@Boy1964]) or Getoor and Kesten ([-@GetKes1972])) for the continuity). It follows that $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely the function $$t\mapsto\tilde{V}_\ast(t):=\int_\mathbb{R} L(t,x)m_\ast(\mathrm{d}x)$$ is continuous and non-decreasing.
Therefore, $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely the function $ \tilde{V}_\ast^{-1}(\tau):=\inf\{t;\tilde{V}(t)>\tau\} $ is strictly increasing and right-continuous.
We use Lemma \[TimeChangeConvLem\] to fix a pair $ (\omega,\tilde
{\omega})\in\Omega\times\tilde{\Omega} $ with the properties that:
$ \tau\mapsto\tilde{V}_\ast^{-1}(\tau) $ is strictly increasing and right-continuous;
$ \tilde{V}_n(t) $ converges toward $ \tilde{V}_\ast(t) $ for all $ t\geq0 $.
Since the set where $ \tilde{V}_\ast$ is not continuous is countable, the set where $ \tilde{V}_\ast$ is continuous is dense in $ [0,\infty) $.
We denote by $ K $ the set of critical values of $ \tilde{V}_\ast$. As was pointed out before, $ K $ is at most countable. For an arbitrary point $ \tau\in[0,\infty)\cap K^c $ and for any $ \epsilon>0 $, one can find points $ t_{\epsilon,0}, t_{\epsilon,1}\in(\tilde{V}_\ast^{-1}(\tau
)-\epsilon,\tilde{V}_\ast^{-1}(\tau)) $ and $ t_{\epsilon,2}, t_{\epsilon,3}\in(\tilde{V}_\ast^{-1}(\tau
),\tilde{V}_\ast^{-1}(\tau)+\epsilon) $ with the property $$\tilde{V}_\ast(t_{\epsilon,0})<\tilde{V}_\ast(t_{\epsilon
,1})<\tau<\tilde{V}_\ast(t_{\epsilon,2})
<\tilde{V}_\ast(t_{\epsilon,3}) .$$ We can now choose a $ \delta>0$ such that $$\tilde{V}_\ast(t_{\epsilon,0})+\delta<\tilde{V}_\ast(t_{\epsilon
,1})-\delta
<\tilde{V}_\ast(t_{\epsilon,1})+\delta<\tau<\tilde{V}_\ast
(t_{\epsilon,2})-\delta
<\tilde{V}_\ast(t_{\epsilon,2})+\delta
<\tilde{V}_\ast(t_{\epsilon,3})-\delta.$$ Since $ \tilde{V}_n $ converges toward $ \tilde{V}_\ast$ in all points where $ \tilde{V}_\ast$ is continuous, there exists an $ n_0\in\mathbb{N} $ such that for all $ n\geq n_0 $, we have $$\tilde{V}_n(t_{\epsilon,0})<\tilde{V}_\ast(t_{\epsilon,0})+\delta
<\tilde{V}_\ast(t_{\epsilon,1})-\delta
<\tilde{V}_n(t_{\epsilon,1})<\tilde{V}_\ast(t_{\epsilon,1})+\delta
<\tau$$ and $$\tau<\tilde{V}_\ast(t_{\epsilon,2})-\delta<\tilde
{V}_n(t_{\epsilon,2})
<\tilde{V}_\ast(t_{\epsilon,2})+\delta
<\tilde{V}_\ast(t_{\epsilon,3})-\delta<\tilde{V}_n(t_{\epsilon
,3}) .$$ By definition of $ t_{\epsilon,0}$, we have that $ z\leq\tilde
{V}_\ast^{-1}(\tau)-\epsilon$ implies $ z\leq t_{\epsilon,0} $. From monotonicity and the first of both inequalities above, it follows that $$\tilde{V}_n(z)\leq\tilde{V}_n(t_{\epsilon,0})\leq\tilde{V}_\ast
(t_{\epsilon,0})
+\delta<\tilde{V}_\ast(t_{\epsilon,1}) .$$ We have thus seen that $ z\leq\tilde{V}_\ast^{-1}(\tau)-\epsilon$ implies $ \tilde{V}_n(z)<\tilde{V}_\ast(t_{\epsilon,1}) $. If we reverse the implication, then we obtain that $ \tilde
{V}_n(z)\geq\tilde{V}_\ast(t_{\epsilon,1}) $ implies $ z>\tilde{V}_\ast^{-1}(\tau)-\epsilon$. From this implication, it follows that $$\tilde{V}_n^{-1}(\tilde{V}_\ast(t_{\epsilon,1}))=\inf\{z\dvtx\tilde
{V}_n(z)>\tilde{V}_\ast(t_{\epsilon,1})\}
>\tilde{V}_\ast^{-1}(\tau)-\epsilon.$$ For $ z=t_{\epsilon,3} $, we have $ \tilde{V}_n(z)=\tilde
{V}_n(t_{\epsilon,3})>\tilde{V}_\ast(t_{\epsilon,2}) $. In other words, there exists a $ z<\tilde{V}_\ast^{-1}(\tau
)+\epsilon$ with $ \tilde{V}_n(z)>\tilde{V}_\ast(t_{\epsilon,2}) $. This proves that $$\tilde{V}_\ast^{-1}(\tau)+\epsilon>\tilde{V}_n^{-1}(\tilde
{V}_\ast(t_{\epsilon,2})) .$$ Altogether, we have proven that for all $ n\geq n_0 $, $$\tilde{V}_\ast^{-1}(\tau)-\epsilon<\tilde{V}_n^{-1}(\tilde
{V}_\ast(t_{\epsilon,1}))
<\tilde{V}_n^{-1}(\tilde{V}_\ast(t_{\epsilon,2}))<\tilde{V}_\ast
^{-1}(\tau)+\epsilon.$$ By monotonicity, for all $ n\geq n_0 $ and all $ \tau'\in[\tilde{V}_\ast(t_{\epsilon,1}),\tilde{V}_\ast
(t_{\epsilon,2})], $ one has $$\tilde{V}_\ast^{-1}(\tau)-\epsilon<\tilde{V}_n^{-1}(\tau')<\tilde
{V}_\ast^{-1}(\tau)+\epsilon.$$ Since $ \tau\in[\tilde{V}_\ast(t_{\epsilon,1}),\tilde{V}_\ast
(t_{\epsilon,2})] $, the proof is complete.
\[RegValLem1\] For all $ \tau\geq0 $, one has that $ \tau$ is a regular value of $
\tilde{V}_\ast$ almost surely with respect to $ \mathbb{P} \times\tilde{\mathbb{P} } $.
By the invariance properties of Brownian motion, we have that for all $
\gamma>0 $, $$\{L(t,w);w\in\mathbb{R},t\geq0\}\stackrel{\mathcal{D}}{=}
\{\gamma^{-1}L(\gamma^2t,\gamma w);w\in\mathbb{R},t\geq0\} .$$ By the invariance of the $ \alpha$-stable Lévy process, we have that $$\begin{aligned}
\{L(t,\tilde{W}(x));x\in\mathbb{R},t\geq0\}&\stackrel{\mathcal{D}}{=}&
\{\gamma^{-1}L(\gamma^2t,\gamma\tilde{W}(x));x\in\mathbb{R},t\geq
0\} \\
& \stackrel{\mathcal{D}}{=}& \{\gamma^{-1}L(\gamma^2t,\tilde{W}(\gamma
^\alpha x));x\in\mathbb{R},t\geq0\} .\end{aligned}$$ Substitution then yields $$\begin{aligned}
\biggl\{\int_{\mathbb{R}}L(t,\tilde{W}(x))\,\mathrm{d}x;t\geq0 \biggr\}
& \stackrel{\mathcal{D}}{=}& \biggl\{\gamma^{-1}\int_{\mathbb{R}}L(\gamma
^2t,\tilde{W}(\gamma^\alpha x))\,\mathrm{d}x;t\geq0 \biggr\} \\
& \stackrel{\mathcal{D}}{=}& \biggl\{\gamma^{-1-\alpha}\int_{\mathbb
{R}}L(\gamma^2t,\tilde{W}(x))\,\mathrm{d}x;t\geq0 \biggr\} .\end{aligned}$$ By definition, this means that $$\{\tilde{V}_\ast(t);t\geq0\} \stackrel{\mathcal{D}}{=}
\{\gamma^{-1-\alpha}\tilde{V}_\ast(\gamma^2t);t\geq0\} .$$ We define $ \ell_\ast$ to be the image measure of the Lebesgue measure $ \ell$ with respect $ \tilde{V}_\ast$. The previous considerations imply that $$\ell_\ast(\mathrm{d}t)\stackrel{\mathcal{D}}{=}\gamma^2\ell_\ast(\gamma^{-1-\alpha}\,\mathrm{d}t) .$$ This identity implies that no $ \tau>0 $ satisfies $ \ell_\ast(\{
\tau\})>0 $ with a positive probability with respect to $ \mathbb{P} \times\tilde{\mathbb{P} } $. To a critical value $ \tau$ corresponds an interval where $ t\mapsto
\tilde{V}_\ast$ is constant, which implies that $ \ell_\ast(\{\tau\})>0 $. For a particular point $
\tau>0 $, this cannot happen with positive probability. This finishes the proof of the statement.
\[OkTimeLokTimeConvProp\] For all $ \tau\geq0 $, the sequence of functions $ x\mapsto L(\tilde
{V}_n^{-1}(\tau),\tilde{S}_n(x+1/n)) $ converges toward the function $ x\mapsto L(\tilde{V}_\ast^{-1}(\tau
),\tilde{W}(x)) $ in the $ J_1 $-topology $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely.
It is known that $ \tilde{S}_n $ converges toward $ \tilde{W} $ in the $ J_1 $-topology almost surely with respect to $ \tilde{\mathbb{P} } $. Moreover, by Lemmas \[InversTimeChangeConvLem\] and \[RegValLem1\], for all $ \tau\geq0 $, the sequence $ \tilde{V}_n^{-1}(\tau) $ converges toward $ \tilde
{V}_\ast^{-1}(\tau) $ almost surely with respect to $ \mathbb{P} \times\tilde{\mathbb{P} } $. The proposition follows since it is well known that $ (t,x)\mapsto L(t,x) $ is jointly continuous $ \mathbb{P} $-almost surely; see Boylan ([-@Boy1964]) or Getoor and Kesten ([-@GetKes1972]).
\[LokalNullMenge\] For all $ k\in\mathbb{N} $, $ \theta_1,\ldots,\theta_k\in\mathbb{R}
$ and all $ \tau_1,\ldots,\tau_k\geq0 $, the set $$\mathcal{C}:=
\Biggl\{c>0\dvtx\ell\Biggl(x\in\mathbb{R}; \Biggl|\sum_{i=1}^k\theta_iL(\tilde{V}_\ast
^{-1}(\tau_i),\tilde{W}(x)) \Biggr|=c \Biggr)>0 \Biggr\}$$ is countable $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely, where $ \ell$ denotes the Lebesgue measure on $ \mathbb{R} $.
It is well known that $ x\mapsto\tilde{W}(x) $ is strictly increasing $ \tilde{\mathbb{P} } $-almost surely. For $ c>0 $, we define the level-sets $$\mathcal{N}_c:= \Biggl\{w\in\mathbb{R}; \Biggl|\sum_{i=1}^k\theta_i
L(\tilde{V}_\ast^{-1}(\tau_i),w) \Biggr|=c \Biggr\} .$$ Fix a strictly increasing path $ f\dvtx x\mapsto\tilde{W}(x) $ and assume that there exist an uncountable number of $ c>0 $ with the property that $ \ell(f^{-1}(\mathcal{N}_c))>0 $. For $ c\neq c' $, the sets $ f^{-1}(\mathcal{N}_c) $ and $ f^{-1}(\mathcal{N}_{c'}) $ are disjoint. We would obtain an uncountable number of disjoint sets with positive Lebesgue measure. This is, of course, not possible.
\[CardTowardMeasureProp\] For all $ k\in\mathbb{N} $, $ \theta_1,\ldots,\theta_k\in\mathbb
{R} $ and all $ \tau_1,\ldots,\tau_k\geq0 $, one has $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely that $$\begin{aligned}
&&\frac{1}{n}\operatorname{card}
\Biggl\{x\in\mathbb{Z}\dvtx n \Biggl|\sum_{i=1}^k\theta_i\tilde{\Gamma}_n(\tau
_i,\{x/n\}) \Biggr|>c \Biggr\}\\
&&\quad\longrightarrow
\ell\Biggl(x\in\mathbb{R}\dvtx \Biggl|\sum_{i=1}^k\theta_i\tilde{L}_\ast(\tau
_i,x) \Biggr|>c \Biggr) \qquad
\mbox{as } n\rightarrow\infty\end{aligned}$$ for all but a countable number of $ c>0 $.
We can find a $ K>0 $ such that $ \{y\in\mathbb{R}\dvtx L(\tau_i,y)\neq0
\ {\rm for\ all}\ i=1,\ldots,k \} $ is a subset of the interval $ (\tilde{W}(-K),\tilde{W}(K)) $. By Propositions \[OkTimePro\], \[OkTimeLokTimeConvProp\] and Corollary \[LokTimeKor1\], the sequence $$\begin{aligned}
\tilde{A}_n(x)&:=&n \Biggl|\sum_{i=1}^k\theta_i\tilde{\Gamma}_n(\tau_i,\{
x\}) \Biggr|\\
&\hspace*{3pt}=& \Biggl|\sum_{i=1}^k\theta_iL\bigl(\tilde{V}_n^{-1}(\tau_i),\tilde
{S}_n(x-1/n)\bigr) \Biggr|\end{aligned}$$ converges $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely in the $ J_1$-topology toward $$\tilde{A}_\ast(x):= \Biggl|\sum_{i=1}^k\theta_i\tilde{L}_\ast(\tau
_i,x) \Biggr|
= \Biggl|\sum_{i=1}^k\theta_iL(\tilde{V}_\ast^{-1}(\tau_i),\tilde
{W}(x)) \Biggr| .$$ There then exists a sequence of continuous increasing maps $ \lambda
_n\dvtx[-K,K]\rightarrow[-K,K] $ such that $$\sup_{x\in[-K,K]} |\tilde{A}_\ast(x)-\tilde{A}_n\circ\lambda
_n(x) |\longrightarrow0
\qquad \mbox{as } n\rightarrow\infty$$ and such that each $ \lambda_n $ is Lipschitz continuous and satisfies $$\operatorname{esssup}\limits_{x\in[-K,K]} |\lambda_n'(x)-1 |\longrightarrow0 .$$ We should emphasize that the derivative of the function $ \lambda_n $ may not exist everywhere. However, those points where the derivative does not exist form a zero set since $ \lambda_n $ is an absolutely continuous Lipschitz function. We note that for suitably large $ n\in\mathbb{N}$, one has $$\begin{aligned}
&&\frac{1}{n}\operatorname{card} \Biggl\{x\in\mathbb{R};
\Biggl|\sum_{i=1}^k\theta_iL\bigl(\tilde{V}_n^{-1}(\tau_i),\tilde
{S}_n(x-1/n)\bigr) \Biggr|>c \Biggr\}\\
&&\quad=\ell\bigl(x\in[-K,K];\tilde{A}_n(x)>c \bigr)=\int_{-K}^K\mathbh{1}_{(c,\infty)}(\tilde{A}_n(x))\,\mathrm{d}x .\end{aligned}$$ It then follows that $$\begin{aligned}
&& \frac{1}{n}\operatorname{card} \Biggl\{x\in[-K,K];
n \Biggl|\sum_{i=1}^k\theta_i\tilde{\Gamma}_n(\tau_i,\{x\}) \Biggr|>c \Biggr\}
-\int_{-K}^K\mathbh{1}_{(c,\infty)}(\tilde{A}_n(\lambda
_n(x)))\,\mathrm{d}x\\
&&\quad= \int_{-K}^K\mathbh{1}_{(c,-\infty)}(\tilde{A}_n(x))\,\mathrm{d}x
\biggl(1-\frac{1}{\lambda_n'(\lambda_n^{-1}(x))} \biggr)\,\mathrm{d}x
+\mathrm{O} \Bigl(\sup_{x\in[-K,K]}|\lambda_n(x)-x| \Bigr).\end{aligned}$$ By the assumptions on the sequence $ \{\lambda_n;n\in\mathbb{N}\} $, the previous difference converges toward zero. Furthermore, $$\int_{-K}^K\mathbh{1}_{(c,\infty)}(\tilde{A}_n(\lambda
_n(x)))\,\mathrm{d}x\longrightarrow
\int_{-K}^K\mathbh{1}_{(c,\infty)}(\tilde{A}_\ast
(x))\,\mathrm{d}x\qquad \mbox{as } n\rightarrow\infty$$ whenever the set $ \{x\in[-K,K];\tilde{A}_\ast(s)=c\} $ is a zero set with respect to the Lebesgue measure $ \ell$ on $ \mathbb{R} $. Since this was proven in Lemma \[LokalNullMenge\], the statement of the proposition follows.
Subsequently, we will make use of the following notation: $$A_n^+:= \Biggl\{x\in\mathbb{Z}\dvtx\sum_{i=1}^k\theta_i\tilde{\Gamma
}_n(\tau_i,\{x/n\})>0 \Biggr\} ,\qquad
A_n^-:= \Biggl\{x\in\mathbb{Z}\dvtx\sum_{i=1}^k\theta_i\tilde{\Gamma
}_n(\tau_i,\{x/n\})<0 \Biggr\}$$ and $$A^+:= \Biggl\{x\in\mathbb{R}\dvtx\sum_{i=1}^k\theta_i\tilde{L}_\ast(\tau
_i,x)>0 \Biggr\} ,\qquad
A^-:= \Biggl\{x\in\mathbb{R}\dvtx\sum_{i=1}^k\theta_i\tilde{L}_\ast(\tau
_i,x)<0 \Biggr\} .$$ Later, we will need the following version of Proposition \[CardTowardMeasureProp\].
\[SignedCardTowardMeasureProp\] For all $ k\in\mathbb{N} $, $ \theta_1,\ldots,\theta_k\in\mathbb
{R} $ and all $ \tau_1,\ldots,\tau_k\geq0 $, one has $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely that $$\frac{1}{n}\operatorname{card}
\Biggl\{x\in\mathbb{Z}\cap A_n^\pm\dvtx
n \Biggl|\sum_{i=1}^k\theta_i\tilde{\Gamma}_n(\tau_i,\{x/n\}) \Biggr|>c \Biggr\}
\longrightarrow
\ell\Biggl(x\in\mathbb{R}\cap A^\pm\dvtx \Biggl|\sum_{i=1}^k\theta_i\tilde
{L}_\ast(\tau_i,x) \Biggr|>c \Biggr)$$ for all but a countable number of $ c>0 $.
The proof uses essentially the same arguments as the proof of Proposition \[CardTowardMeasureProp\].
With the same proof as for Proposition \[CardTowardMeasureProp\], we can show that $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely $$\frac{1}{n}\operatorname{card} \bigl\{x\in\mathbb{Z}\dvtx n^2\tilde{\Gamma}_n^2(\tau
_i,\{x/n\})>c \bigr\}\longrightarrow
\ell\bigl(x\in\mathbb{R}\dvtx\tilde{L}_\ast^2(\tau_i,x)>c \bigr) \qquad
\mbox{as } n\rightarrow\infty$$ for all but a countable number of $ c>0 $.
A useful lemma on integrated powers of local time
-------------------------------------------------
\[PrinceLem\] For $ \tau_1,\ldots,\tau_k\geq0 $ and $ \theta_1,\ldots,\theta
_k\in\mathbb{R} $, the two sequences of random variables $$\begin{aligned}
&&n^{\beta-1}\sum_{x\in\mathbb{Z}} \Biggl|\sum_{i=1}^k\theta_i\tilde
{\Gamma}_n(\tau_i,\{x/n\}) \Biggr|^\beta\quad
\mbox{and }\\
&& n^{\beta-1}\sum_{x\in\mathbb{Z}}
\Biggl( \Biggl|\sum_{i=1}^k\theta_i\tilde{\Gamma}_n(\tau_i,\{x/n\}) \Biggr|^\beta
\operatorname{sgn} \Biggl(\sum_{i=1}^k\theta_i\tilde{\Gamma}_n(\tau_i,\{x/n\}) \Biggr)
\Biggr)\end{aligned}$$ converge $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely toward the respective random variables $$\int_{-\infty}^\infty\Biggl|\sum_{i=1}^k\theta_i\tilde{L}_\ast(\tau
_i,x) \Biggr|^\beta \,\mathrm{d}x \quad
\mbox{and}\quad
\int_{-\infty}^\infty\Biggl( \Biggl|\sum_{i=1}^k\theta_i\tilde{L}_\ast(\tau
_i,x) \Biggr|^\beta
\operatorname{sgn} \Biggl(\sum_{i=1}^k\theta_i\tilde{L}_\ast(\tau_i,x) \Biggr) \Biggr)\,\mathrm{d}x .$$
We use the layer cake representation of the integrals (see Lieb and Loss ([-@LieLos2001])) to write $$\sum_{x\in\mathbb{Z}} \Biggl|\sum_{i=1}^k\theta_in\tilde{\Gamma
}_n(\tau_i,\{x/n\}) \Biggr|^\beta=
\beta\int_0^\infty c^{\beta-1}\operatorname{card}
\Biggl\{x\in\mathbb{Z}\dvtx n \Biggl|\sum_{i=1}^k\theta_i\tilde{\Gamma}_n(\tau
_i,\{x/n\}) \Biggr|>c \Biggr\}\,\mathrm{d}c$$ and $$\int_{-\infty}^\infty\Biggl|\sum_{i=1}^k\theta_i\tilde{L}_\ast(\tau
_i,x) \Biggr|^\beta \,\mathrm{d}x
=\beta\int_0^\infty c^{\beta-1}\ell\Biggl(x\in\mathbb{R}\dvtx
\Biggl|\sum_{i=1}^k\theta_i\tilde{L}_\ast(\tau_i,x) \Biggr|>c \Biggr)\,\mathrm{d}c .$$ We note that the convergence of $ \tilde{V}_n^{-1}(\tau_i) $ toward $
\tilde{V}_\ast^{-1}(\tau_i) $ and the fact that $ t\mapsto L(t,y) $ is increasing for every $ y\in
\mathbb{R} $ imply that there exists an $ n_0\in\mathbb{N} $ with $$L(\tilde{V}_n^{-1}(\tau_i),y)\leq
L\bigl(\tilde{V}_\ast^{-1}(\tau_i)+1,y\bigr)\qquad
\mbox{for all } y\in\mathbb{R}, 1\leq i\leq k, n\geq n_0 .$$ Moreover, for all $ i\in\{1,\ldots,k\} $, the functions $ y\mapsto
L(\tilde{V}_\ast^{-1}(\tau_i)+1,y) $ are continuous and their supports are contained in $ [-K,K] $ for a suitable $ K>0 $. Hence, there exists a $ C>0 $ such that for $ n\geq n_0 $, one has $$\begin{aligned}
n \Biggl|\sum_{i=1}^k\theta_i\tilde{\Gamma}_n(\tau_i,\{x/n\}) \Biggr|
&\leq&\Biggl|\sum_{i=1}^k\theta_iL\bigl(\tilde{V}_n^{-1}(\tau_i),\tilde
{S}_n\bigl((x-1)/n\bigr)\bigr) \Biggr|\\
&\leq&\sum_{i=1}^k\theta_i\sup_{y\in\mathbb{R}}L\bigl(\tilde{V}_\ast
^{-1}(\tau_i)+1,y\bigr)\leq C.\end{aligned}$$ This implies that all of the functions $$c\mapsto\operatorname{card} \Biggl\{x\in\mathbb{Z}\dvtx
n \Biggl|\sum_{i=1}^k\theta_i\tilde{\Gamma}_n(\tau_i,\{x/n\}) \Biggr|>c \Biggr\}$$ have support contained in $ [0,C]$. Moreover, for all $ c>0 $, we have $$\begin{aligned}
\operatorname{card}
\Biggl\{x\in\mathbb{Z}\dvtx n \Biggl|\sum_{i=1}^k\theta_i\tilde{\Gamma}_n(\tau
_i,\{x/n\}) \Biggr|>c \Biggr\}
\leq\operatorname{card} \bigl\{x\in\mathbb{Z}\dvtx-K\leq\tilde{S}_n\bigl((x-1)/n\bigr)\leq K \bigr\}.\end{aligned}$$ Since $$\ell\bigl(x;\tilde{W}(x)\in\{-K,K\} \bigr)=0$$ and since $ \tilde{S}_n $ converges toward $ \tilde{W} $ with respect to the Skorohod metric, we have that $$\frac{1}{n}\operatorname{card} \bigl\{x\in\mathbb{Z}\dvtx-K\leq\tilde
{S}_n\bigl((x-1)/n\bigr)\leq K \bigr\}
\longrightarrow\ell\bigl(x\in\mathbb{R}\dvtx-K\leq\tilde{W}(x)\leq K \bigr).$$ This implies that there exists an $ R>0 $ such that for all $ n\in
\mathbb{N} $ and all $ c>0 $, we have $$\frac{1}{n}\operatorname{card} \Biggl\{x\in\mathbb{Z}\dvtx
n \Biggl|\sum_{i=1}^k\theta_i\tilde{\Gamma}_n(\tau_i,\{x/n\}) \Biggr|>c \Biggr\}
\leq R.$$ The first statement of the lemma then follows from dominated convergence and Proposition \[CardTowardMeasureProp\].\
The second statement is proved in the same way by separating the positive and the negative parts of the integrals and using the statements from Proposition \[SignedCardTowardMeasureProp\] instead of Proposition \[CardTowardMeasureProp\].
\[PrinceProp\] For $ \tau_1,\ldots,\tau_k\geq0 $ and $ \theta_1,\ldots,\theta
_k\in\mathbb{R} $, the two sequences of random variables $$\begin{aligned}
&&n^{\beta-1}\sum_{x\in\mathbb{Z}} \Biggl|\sum_{i=1}^k\theta_i\Gamma
_n(\tau_i,\{x/n\}) \Biggr|^\beta\quad
\mbox{and}\\
&&n^{\beta-1}\sum_{x\in\mathbb{Z}} \Biggl( \Biggl|\sum_{i=1}^k\theta_i\Gamma
_n(\tau_i,\{x/n\}) \Biggr|^\beta
\operatorname{sgn} \Biggl(\sum_{i=1}^k\theta_i\Gamma_n(\tau_i,\{x/n\}) \Biggr) \Biggr)\end{aligned}$$ converge jointly in distribution toward the respective random variables $$\int_{-\infty}^\infty\Biggl|\sum_{i=1}^k\theta_iL_\ast(\tau_i,x)
\Biggr|^\beta \,\mathrm{d}x \quad
\mbox{and}\quad
\int_{-\infty}^\infty\Biggl( \Biggl|\sum_{i=1}^k\theta_iL_\ast(\tau_i,x)
\Biggr|^\beta
\operatorname{sgn} \Biggl(\sum_{i=1}^k\theta_iL_\ast(\tau_i,x) \Biggr) \Biggr)\,\mathrm{d}x .$$
We know that $$\{L_\ast(t,x);t\geq0,x\in\mathbb{R} \} \stackrel{\mathcal{D}}{=}
\{\tilde{L}_\ast(t,x);t\geq0,x\in\mathbb{R} \}$$ and $$\{S_n^{-1}(B_n(V_n^{-1}(t)));t\geq0 \} \stackrel{\mathcal{D}}{=}
\{\tilde{S}_n^{-1}(B(\tilde{V}_n^{-1}(t)));t\geq0 \} .$$ Therefore, by Lemma \[PrinceLem\], the sequences of random variables $$\begin{aligned}
&&n^{\beta-1}\sum_{x\in\mathbb{Z}} \Biggl|\sum_{i=1}^k\theta_i\hat
{\Gamma}_n(\tau_i,\{x/n\}) \Biggr|^\beta\quad
\mbox{and}\\
&&n^{\beta-1}\sum_{x\in\mathbb{Z}} \Biggl( \Biggl|\sum_{i=1}^k\theta_i\hat
{\Gamma}_n(\tau_i,\{x/n\}) \Biggr|^\beta
\operatorname{sgn} \Biggl(\sum_{i=1}^k\theta_i\hat{\Gamma}_n(\tau_i,\{x/n\}) \Biggr) \Biggr)\end{aligned}$$ converge jointly in distribution toward the respective random variables $$\int_{-\infty}^\infty\Biggl|\sum_{i=1}^k\theta_iL_\ast(\tau_i,x)
\Biggr|^\beta \,\mathrm{d}x \quad
\mbox{and}\quad
\int_{-\infty}^\infty\Biggl( \Biggl|\sum_{i=1}^k\theta_iL_\ast(\tau_i,x)
\Biggr|^\beta
\operatorname{sgn} \Biggl(\sum_{i=1}^k\theta_iL_\ast(\tau_i,x) \Biggr) \Biggr)\,\mathrm{d}x .$$ Moreover, $ S_n^{-1}(S_n(x/n))=(x+1)/n $ for all $ x\in\mathbb{Z} $. This implies that $$\hat{X}_n(\tau)\stackrel{\mathcal{D}}{=}S_n^{-1}(S_n(X_n(\tau
)))=X_n(\tau)+1/n .$$ Hence, we have $ \hat{\Gamma}_n(\tau,\{x/n\})\stackrel{\mathcal{D}}{=}\Gamma_n(\tau,\{(x+1)/n\}) $ for all $ x\in\mathbb{Z} $. Therefore, $$n^{\beta-1}\sum_{x\in\mathbb{Z}} \Biggl|\sum_{i=1}^k\theta_i\hat
{\Gamma}_n(\tau_i,\{x/n\}) \Biggr|^\beta\stackrel{\mathcal{D}}{=}
n^{\beta-1}\sum_{x\in\mathbb{Z}} \Biggl|\sum_{i=1}^k\theta_i\Gamma
_n(\tau_i,\{x/n\}) \Biggr|^\beta$$ and $$\begin{aligned}
&& n^{\beta-1}\sum_{x\in\mathbb{Z}} \Biggl( \Biggl|\sum_{i=1}^k\theta_i\hat
{\Gamma}_n(\tau_i,\{x/n\}) \Biggr|^\beta
\operatorname{sgn} \Biggl(\sum_{i=1}^k\theta_i\hat{\Gamma}_n(\tau_i,\{x/n\}) \Biggr) \Biggr)
\\
&&\quad\stackrel{\mathcal{D}}{=} n^{\beta-1}\sum_{x\in\mathbb{Z}} \Biggl( \Biggl|\sum
_{i=1}^k\theta_i\Gamma_n(\tau_i,\{x/n\}) \Biggr|^\beta
\operatorname{sgn} \Biggl(\sum_{i=1}^k\theta_i\Gamma_n(\tau_i,\{x/n\}) \Biggr) \Biggr) .\end{aligned}$$ This proves the proposition.
For the sequel, we define the occupation time $$\Gamma(t,A):=\int_0^t\mathbh{1}_{A}(X(s))\,\mathrm{d}s$$ of the process $ X $ in the measurable set $ A\subset\mathbb{R} $. Consequently, we have $$\Xi(t)=\sum_x\Gamma(t,\{x\})\xi(x) .$$ We will use this fact and the following corollary in the proofs of the next section.
\[PrinceKor\] For $ \tau_1,\ldots,\tau_k\geq0 $ and $ \theta_1,\ldots,\theta
_k\in\mathbb{R} $, the two sequences of random variables $$\begin{aligned}
&&n^{-1-{\beta/\alpha}}\sum_{x\in\mathbb{Z}} \Biggl|\sum
_{i=1}^k\theta_i\Gamma(k_n\tau_i,\{x\}) \Biggr|^\beta\quad
\mbox{and}\\
&&n^{-1-{\beta/\alpha}}\sum_{x\in\mathbb{Z}} \Biggl( \Biggl|\sum
_{i=1}^k\theta_i\Gamma(k_n\tau_i,\{x\}) \Biggr|^\beta
\operatorname{sgn} \Biggl(\sum_{i=1}^k\theta_i\Gamma(k_n\tau_i,\{x\}) \Biggr) \Biggr)\end{aligned}$$ converge jointly in distribution toward the respective random variables $$\begin{aligned}
&&\int_{-\infty}^\infty\Biggl|\sum_{i=1}^k\theta_iL_\ast(\tau_i,x)
\Biggr|^\beta \,\mathrm{d}x \quad
\mbox{and}\\
&&\int_{-\infty}^\infty\Biggl( \Biggl|\sum_{i=1}^k\theta_iL_\ast(\tau_i,x)
\Biggr|^\beta
\operatorname{sgn} \Biggl(\sum_{i=1}^k\theta_iL_\ast(\tau_i,x) \Biggr) \Biggr)\,\mathrm{d}x .\end{aligned}$$
If we let $ k_n:=n^{(1+\alpha)/\alpha} $, then for all $ n\in
\mathbb{N} $ and $ x\in\mathbb{Z} $, we have that $$\Gamma_n(\tau,x/n)=\int_0^\tau\mathbh{1}_{\{x/n\}}(X_n(t))\,\mathrm{d}t
=k_n^{-1}\int_0^{k_n\tau} \mathbh{1}_{\{x\}}(X(t))\,\mathrm{d}t
= n^{-(\alpha+1)/\alpha}\Gamma(k_n\tau,\{x\}) .$$ The result then follows from Proposition \[PrinceProp\].
The finite-dimensional distributions
====================================
In this section, we prove the convergence of the finite-dimensional distributions of $ \Xi_n $ toward the finite-dimensional distributions of $ \Xi_\ast$. In order to do so, we first compute the exact expression of the finite-dimensional distributions of $ \Xi_\ast$. The proofs in this section follow the ideas given in Kesten and Spitzer ([-@KesSpi1979]).
In the , we defined $$\Xi_\ast(\tau):=\int_0^\infty L_\ast(\tau,x-)\,\mathrm{d}Z_+(x)+\int
_0^\infty L_\ast(\tau,-(x-))\,\mathrm{d}Z_-(x) ,$$ where $ \{Z_+(t);t\geq0\} $ and $ \{Z_-(t);t\geq0\} $ are independent copies of the $ \beta$-stable Lévy process, which can be associated with the stable distribution $ \vartheta_\beta$ with characteristic function given by $$\psi(\theta)=\exp\bigl(-|\theta|^\beta\bigl(A_1+\mathrm{i}A_2\operatorname{sgn}(\theta)\bigr) \bigr) .$$
\[FinitDistriLem\] For $ t_1,\ldots,t_k\geq0 $ and $ \theta_1,\ldots,\theta_k\in
\mathbb{R} $, we have that $$\begin{aligned}
&&\mathbb{E} \Biggl[\exp\Biggl(\mathrm{i}\sum_{j=1}^k\theta_j\Xi_\ast(t_j) \Biggr)
\Biggr]\\
&&\quad= \mathbb{E} \Biggl[\exp\Biggl(-A_1\int_{-\infty}^\infty\Biggl|\sum_{j=1}^k\theta_jL_\ast
(t_j,x) \Biggr|^\beta \,\mathrm{d}x \Biggr)\\
&&\qquad\hphantom{\mathbb{E} \Biggl[}{}\times\exp\Biggl(-\mathrm{i}A_2\int_{-\infty}^\infty\Biggl|\sum_{j=1}^k\theta_jL_\ast
(t_j,x) \Biggr|^\beta \,\mathrm{d}x
\operatorname{sgn} \Biggl(\sum_{j=1}^k\theta_jL_\ast(t_j,x) \Biggr) \Biggr) \Biggr] .\end{aligned}$$
The proof is similar to that given in Kesten and Spitzer ([-@KesSpi1979]) (see page 16ff). Let $ \nu$ be the Lévy measure of $ Z_+ $. One can truncate the Lévy measure as follows: $$\nu_1(B)=\nu(B\cap\{y\in\mathbb{R};|y|\leq1\}) \quad \mbox{and} \quad
\nu_2(B)=\nu(B\cap\{y\in\mathbb{R};|y|>1\}).$$ Let $ M(t) $ and $ A(t) $ be independent Lévy processes, with respective characteristic functions $$\mathbb{E} \bigl[\mathrm{e}^{\mathrm{i}\theta M(t)} \bigr]
=\exp\biggl(t\int_{|y|\leq1} (\mathrm{e}^{\mathrm{i}\theta y}-1-\mathrm{i}\theta y )\nu_1(\mathrm{d}y) \biggr)$$ and $$\mathbb{E} \bigl[\mathrm{e}^{\mathrm{i}\theta A(t)} \bigr]=\exp\biggl(t\int_{|y|\leq1} (\mathrm{e}^{\mathrm{i}\theta
y}-1 )\nu_2(\mathrm{d}y) \biggr),$$ such that $$Z^+(t)=M(t)+A(t)+Dt ,$$ where $ D $ is a suitable real constant. This decomposition exists and is called the Lévy–Itô representation of $ Z^+ $. The advantage of this representation is that $ M(t) $ is a martingale and has all moments and $ A(t) $ is a process with bounded variation. Since the process $ \{L_\ast
(t,x-);x\geq0\} $ is left-continuous and independent with respect to the filtration $ \mathcal{F}_t $ generated by $ Z^+(t) $, the process $ \{L_\ast(t,x-);x\geq0\} $ is $ \mathcal{F}_t $-predictable. Moreover, $ \{L_\ast(t,x-);x\geq0\} $ has bounded support $ \mathbb{P} $-almost surely. Therefore, we can find a suitable sequence of partitions $ \{x_l^{(n)};l\in\mathbb{N}\}$, $n\in\mathbb{N} $, with $
x^{(n)}_l<x^{(n)}_{l+1} $ for all $ l,n\in\mathbb{N} $ satisfying $$\lim_{l\rightarrow\infty} x_l^{(n)}=\infty\quad \mbox{and}\quad
\lim_{n\rightarrow\infty}\max_{l\in\mathbb{N}} \bigl(x_{l+1}^{(n)}-x_l^{(n)} \bigr)=0$$ such that $$\int_0^\infty L_\ast(t,x-)\,\mathrm{d}M(x)=\lim_{n\rightarrow\infty}
\sum_{l=1}^\infty L_\ast\bigl(t,x_l^{(n)}-\bigr) \bigl(M\bigl(x_{l+1}^{(n)}\bigr)-M\bigl(x_l^{(n)}\bigr)\bigr)$$ with probability 1 (see Meyer ([-@Mey1976]), Chapter II, Section 23). Moreover, we can also assume that $$\int_0^\infty L_\ast(t,x-)\,\mathrm{d}A(x)=\lim_{n\rightarrow\infty}
\sum_{l=1}^\infty L_\ast\bigl(t,x_l^{(n)}-\bigr) \bigl(A\bigl(x_{l+1}^{(n)}\bigr)-A\bigl(x_l^{(n)}\bigr)
\bigr)$$ with probability 1.
From those considerations, it follows that there exists a sequence of partitions $ (x_l^{(n)})_{l\in\mathbb{N}} $ such that $$\int_0^\infty L_\ast(t,x-)\,\mathrm{d}Z_+(x)=\lim_{n\rightarrow\infty}
\sum_{l=1}^\infty L_\ast\bigl(t,x_l^{(n)}-\bigr)
\bigl(Z_+\bigl(x_{l+1}^{(n)}\bigr)-Z_+\bigl(x_l^{(n)}\bigr) \bigr)$$ with probability 1. Since the increments $ D^{(n)}_l:=Z_+(x_{l+1}^{(n)})-Z_+(x_l^{(n)}),\
l\in\mathbb{N}, $ are independent and have characteristic function $$\mathbb{E} \bigl[\mathrm{e}^{\mathrm{i}\theta D^{(n)}_l} \bigr]
=\exp\bigl(-\bigl(x_{l+1}^{(n)}-x_l^{(n)}\bigr)|\theta|^\beta\bigl(A_1+\mathrm{i}A_2\cdot\operatorname{sgn}(\theta)\bigr) \bigr)$$ by dominated convergence, we have $$\begin{aligned}
&& \mathbb{E} \Biggl[\exp\Biggl(\mathrm{i}\sum_{j=1}^k\theta_j\int_0^\infty L_\ast
(t_j,x-)\,\mathrm{d}Z_+(x) \Biggr) \Biggr]\\
&&\quad=\lim_{n\rightarrow\infty}
\mathbb{E} \Biggl[\exp\Biggl(
\sum_{l=1}^\infty\sum_{j=1}^k\mathrm{i}\theta_jL_\ast\bigl(t_j,x_l^{(n)}-\bigr)
\bigl(Z_+\bigl(x_{l+1}^{(n)}\bigr)-Z_+\bigl(x_l^{(n)}\bigr) \bigr) \Biggr) \Biggr]\\
&&\quad=\lim_{n\rightarrow\infty}
\mathbb{E} \Biggl[\exp\Biggl(-\sum_{l=1}^\infty\bigl(x_{l+1}^{(n)}-x_l^{(n)} \bigr)
\Biggl|\sum_{j=1}^k\theta_jL_\ast\bigl(t_j,x_l^{(n)}-\bigr) \Biggr|^\beta\\
&&\qquad\hphantom{\lim_{n\rightarrow\infty}
\mathbb{E} \Biggl[\exp\Biggl(-\sum_{l=1}^\infty}
{}\times\Biggl(A_1+\mathrm{i}A_2\cdot\operatorname{sgn} \Biggl(\sum_{j=1}^k\theta_jL_\ast\bigl(t_j,x_l^{(n)}-\bigr)
\Biggr) \Biggr) \Biggr) \Biggr].\\
&&\quad= \mathbb{E} \Biggl[\exp\Biggl(-A_1\int_0^\infty\Biggl|
\sum_{j=1}^k\theta_jL_\ast\bigl(t_j,x_l^{(n)}\bigr) \Biggr|^\beta \,\mathrm{d}x\\
&&\qquad\hphantom{\mathbb{E} \Biggl[\exp\Biggl(}
{}-\mathrm{i}A_2\int_0^\infty\Biggl|\sum_{j=1}^k\theta_jL_\ast\bigl(t_j,x_l^{(n)}\bigr)
\Biggl|^\beta
\operatorname{sgn} \Biggl(\sum_{j=1}^k\theta_jL_\ast\bigl(t_j,x_l^{(n)}\bigr) \Biggr)\,\mathrm{d}x \Biggr) \Biggr].\end{aligned}$$ For $ Z_- $, one can proceed with similar arguments.
\[FinitDistriConvProp\] The finite-dimensional distributions of the processes $ \{\Xi
_n(t);t\geq0\} $ converge toward the finite-dimensional distributions of the process $ \{\Xi_\ast(t);t\geq0\} $.
As in the previous sections, we define $ k_n:=n^{(1+\alpha)/\alpha} $ and $ \kappa:=\frac{1}{\alpha}+\frac{1}{\beta} $. We already saw that we can use the occupation time $ \{\Gamma(t,\{x\}
);t\geq0,x\in\mathbb{R}\} $ of the process $ \{X(t);t\geq0\} $ to represent the process $ \{\Xi(t);t\geq
0\} $ as follows: $$\Xi(t)=\sum_{x\in\mathbb{Z}}\Gamma(t,\{x\})\xi(x) .$$ It follows that $$\Xi_n(t)=n^{-\kappa}\Xi(k_nt)=n^{-\kappa}\sum_{x\in\mathbb
{Z}}\Gamma(k_nt,\{x\})\xi(x) .$$ Let $ \varphi(\theta):=\mathbb{E} [\exp(\mathrm{i}\theta\xi(1)) ] $ be the characteristic function of the scenery random variable $ \xi(1) $. It then follows from the above representation that $$\sum_{j=1}^k\theta_j\Xi_n(t_j)
=n^{-\kappa}\sum_{x\in\mathbb{Z}}\sum_{j=1}^k\theta_j\Gamma
(k_nt_j,\{x\})\xi(x)$$ and $$\begin{aligned}
R_n:= \mathbb{E} \Biggl[\exp\Biggl(\mathrm{i}\sum_{j=1}^k\theta_j\Xi_n(t_j) \Biggr) \Biggr]
=\mathbb{E} \Biggl[\prod_{x\in\mathbb{Z}}\varphi\Biggl(n^{-\kappa}\sum
_{j=1}^k\theta
_j\Gamma(k_nt_j,\{x\}) \Biggr) \Biggr].\end{aligned}$$ The random scenery $ \{\xi(z);z\in\mathbb{Z}\} $ is in the domain of attraction of a $ \beta$-stable distribution with characteristic function given by $$\psi(\theta)=\exp\bigl(-|\theta|^\beta\bigl(A_1+\mathrm{i}A_2\cdot\operatorname{sgn}(\theta)
\bigr)\bigr) .$$ This implies that $$1-\varphi(\theta)\sim|\theta|^\beta
\bigl(A_1+\mathrm{i}A_2\cdot\operatorname{sgn}(\theta)\bigr)\qquad
\mbox{as } \theta\rightarrow0 .$$ Thus $$\begin{aligned}
\log(\varphi(\theta)) \sim\log(\psi(\theta)) \qquad\mbox{as }
\theta\rightarrow0 .\end{aligned}$$ Therefore, for $ |\theta|\leq1 $, we have that $$\begin{aligned}
\biggl|\frac{\log(\varphi(\theta))-\log(\psi(\theta))}{\log(\psi
(\theta))} \biggr|=\mathrm{o}(\theta) .\end{aligned}$$ If we define $$\varphi_{x,n}:=\varphi\Biggl( n^{-\kappa}\sum_{j=1}^k\theta_j\Gamma
(k_nt_j,\{x\}) \Biggr)$$ and $$\psi_{x,n}:=\exp\Biggl(-n^{-\kappa\beta} \Biggl|\sum_{j=1}^k\theta_j\Gamma
(k_nt_j,\{x\}) \Biggr|^\beta
\Biggl(A_1+\mathrm{i}A_2\cdot\operatorname{sgn} \Biggl(\sum_{j=1}^k\theta_j\Gamma(k_nt_j,\{x\}) \Biggr)
\Biggr) \Biggr)$$ for all $ x\in\mathbb{Z} $, one has $$\begin{aligned}
\biggl|\frac{\log(\varphi_{x,n})-\log(\psi_{x,n})}{\log(\psi_{x,n})} \biggr|
=\mathrm{o} \Biggl(n^{-\kappa}\sum_{j=1}^k\theta_j\Gamma(k_nt_j,\{x\}) \Biggr) .\end{aligned}$$ This implies that $$\begin{aligned}
\biggl|\log\biggl(\prod_{x\in\mathbb{Z}}\varphi_{x,n} \biggr)
-\log\biggl(\prod_{x\in\mathbb{Z}}\psi_{x,n} \biggr) \biggr|
&=& \biggl|\sum_{x\in\mathbb{Z}}\log(\varphi_{x,n})-\sum_{x\in\mathbb
{Z}}\log(\psi_{x,n}) \biggr| \\
&\leq&\sum_{x\in\mathbb{Z}}\log(\psi_{x,n})
\mathrm{o} \Biggl(n^{-\kappa}\sum_{j=1}^k\theta_j\Gamma(k_nt_j,\{x\}) \Biggr).\end{aligned}$$ By Corollary \[PrinceKor\], the right-hand side of the previous inequality converges toward zero in probability. The continuity of the logarithm then implies that $$\begin{aligned}
\biggl|\prod_{x\in\mathbb{Z}}\varphi_{x,n}-\prod_{x\in\mathbb{Z}}\psi
_{x,n} \biggr|\longrightarrow0
\qquad \mbox{in probability as } n\rightarrow\infty.\end{aligned}$$ We use this and dominated convergence to prove that the limit of the sequence $ \{R_n;n\in\mathbb{N}\} $ exists and is equal to the limit of the sequence $$\begin{aligned}
Q_n:=\mathbb{E} \Biggl[\exp\Biggl(-\sum_{x\in\mathbb{Z}}n^{-\kappa\beta} \Biggl|
\sum_{j=1}^k\theta_j\Gamma(k_nt_j,\{x\}) \Biggr|^\beta
\Biggl(A_1+\mathrm{i}A_2\cdot\operatorname{sgn} \Biggl(\sum_{j=1}^k\theta_j\Gamma(k_nt_j,\{x\}) \Biggr)
\Biggr) \Biggr) \Biggr].\end{aligned}$$ By Corollary \[PrinceKor\] and Lemma \[FinitDistriLem\], the sequence $ \{Q_n;n\in\mathbb{N}\} $ converges toward $$\begin{aligned}
Q_\ast&:=&\mathbb{E} \Biggl[\exp\Biggl(-\int_{-\infty}^\infty\Biggl|\sum
_{j=1}^k\theta
_jL_\ast(t_j,x) \Biggr|^\beta
\Biggl(A_1+\mathrm{i}A_2\cdot\operatorname{sgn} \Biggl(\sum_{j=1}^k\theta_jL_\ast(t_j,x) \Biggr) \Biggr)\,\mathrm{d}x \Biggr)
\Biggr]\\
&=&\mathbb{E} \Biggl[\exp\Biggl(\mathrm{i}\sum_{j=1}^k\theta_j\Xi_\ast(t_j) \Biggr) \Biggr].\end{aligned}$$ As we have seen in Lemma \[FinitDistriLem\], $ Q_\ast$ is the characteristic function for the finite-dimensional distributions of $ \{\Xi_\ast(t);t\geq0\} $. This completes the proof of the proposition.
The tightness
=============
In this section, we prove that the sequence $ \{\Xi_n(t);t\geq0\} $ is tight. The proof of Theorem \[MT\] then follows since we have already obtained the convergence of the finite-dimensional distributions in the previous section. The main proof of tightness also follows the ideas given in Kesten and Spitzer ([-@KesSpi1979]). We first need some suitable inequalities for the occupation times of $
X_\ast$. However, the proofs of those inequalities differ from those given in Kesten and Spitzer ([-@KesSpi1979]).
\[Lem2\] There exists a function $ \epsilon\dvtx\mathbb{R}^+\rightarrow\mathbb
{R}^+ $ with the properties $ \epsilon(A)\rightarrow0 $ as $ A\rightarrow\infty$ and $$\mathbb{P} \bigl(\Gamma(s,\{x\})>0\ \mbox{for\ some}\ x\ \mbox{with}\ |x|>As^{
{\alpha
}/({1+\alpha})} \bigr)\leq\epsilon(A) \qquad \mbox{for all } s\geq0.$$
For a positive real number $ x $, we denote by $ \lceil x\rceil$ the smallest integer which is greater or equal to $ x $. Obviously, for all $ s\geq0 $, we have $$\begin{aligned}
&& \mathbb{P} \bigl(\Gamma(s,\{x\})>0 \mbox{ for some } x\mbox{ with }
|x|>As^{\alpha/(1+\alpha)} \bigr) \\
&&\quad\leq \mathbb{P} \bigl(|X(r)|> As^{\alpha/(1+\alpha)}\mbox{ for
some }
r\leq s \bigr) \\
&&\quad\leq \mathbb{P} \bigl(|X(r)|> A \bigl( \bigl\lceil s^{\alpha/(1+\alpha)}
\bigr\rceil-1
\bigr)\mbox{ for some } r\leq\bigl\lceil s^{\alpha/(1+\alpha)} \bigr\rceil
^{(1+\alpha)/\alpha} \bigr) \\
&&\quad= \mathbb{P} \bigl( \bigl|X \bigl( \bigl\lceil s^{{\alpha}/({1+\alpha})} \bigr\rceil
^{(1+\alpha)/\alpha}u \bigr) \bigr|>
A \bigl\lceil s^{\alpha/(1+\alpha)} \bigr\rceil-A \mbox{ for some }
u\leq1 \bigr) \\
&&\quad\leq \mathbb{P} \Bigl(\sup_{r\leq1}\bigl|X_{n(s)}(r)\bigr|> A/2 \Bigr) \qquad
\mbox{for } s>1,\end{aligned}$$ with $ n(s):= \lceil s^{\alpha/(1+\alpha)} \rceil\rightarrow
\infty$ as $ s\rightarrow\infty$. Since $$\mathbb{P} \Bigl(\sup_{r\leq1}|X_n(r)|> A/2 \Bigr)\longrightarrow\mathbb{P}
\Bigl(\sup_{r\leq
1}|X_\ast(r)|> A/2 \Bigr)\qquad \mbox{as }
n\rightarrow\infty,$$ we can define $$\epsilon(A):=\sup_{s\geq0}\mathbb{P} \Bigl(\sup_{r\leq1}\bigl|X_{n(s)}(r)\bigr|>
A/2 \Bigr)\qquad
\mbox{for all } A>0 .$$ This proves the statement of the lemma.
\[Lem3\] There exists a $ C>0 $ such that for all $ s\geq0 $, one has $$\sum_{x\in\mathbb{Z}}\mathbb{E} [\Gamma^2(s,\{x\}) ]\sim
Cs^{2-{\alpha}/({1+\alpha})} .$$
For a positive real number $ x $, we denote by $ \lfloor x\rfloor$ its integer part. We know that for $ w(s):= \lfloor s^{\alpha/(\alpha+1)}
\rfloor$, one has $$\begin{aligned}
\frac{(w(s))^{2(\alpha+1)/\alpha}}{s^2}\sum_{x\in\mathbb
{Z}}\Gamma_{w(s)}^2\bigl(1,\{x/w(s)\}\bigr)
=s^{-2}\sum_{x\in\mathbb{Z}}\Gamma^2 \bigl((w(s))^{(\alpha+1)/{\alpha}},\{x\} \bigr)
\leq s^{-2}\sum_{x\in\mathbb{Z}}\Gamma^2(s,\{x\})\end{aligned}$$ and $$\begin{aligned}
s^{-2}\sum_{x\in\mathbb{Z}}\Gamma^2(s,\{x\})
& \leq& s^{-2}\sum_{x\in\mathbb{Z}}\Gamma^2 \bigl(\bigl(w(s)+1\bigr)^{
({\alpha+1})/{\alpha}},\{x\} \bigr)\\
&=& \frac{(w(s)+1)^{2(\alpha+1)/{\alpha}}}{s^2}\sum_{x\in
\mathbb{Z}}\Gamma_{w(s)+1}^2\bigl(1,\bigl\{x/\bigl(w(s)+1\bigr)\bigr\}\bigr) .\end{aligned}$$ Consequently, one has $$s^{-2}\sum_{x\in\mathbb{Z}}\mathbb{E} [\Gamma^2(s,\{x\}) ]
\sim\sum_{x\in\mathbb{Z}}\mathbb{E} \bigl[\Gamma_{w(s)}^2\bigl(1,\{x/w(s)\}
\bigr) \bigr]
= \sum_{x\in\mathbb{Z}}\mathbb{E} \bigl[\tilde{\Gamma}_{w(s)}^2\bigl(1,\{
x/w(s)\}\bigr) \bigr] .$$ It follows from the layer cake representation and the remark after the proof of Proposition \[SignedCardTowardMeasureProp\] that $$\begin{aligned}
w(s)\sum_{x\in\mathbb{Z}}\tilde{\Gamma}_{w(s)}^2\bigl(1,\{x/w(s)\}\bigr)
=\frac{1}{w(s)}\int_0^\infty\operatorname{card}
\bigl\{x\in\mathbb{Z}\dvtx
w^2(s)\tilde{\Gamma}^2_{w(s)}\bigl(1,\{x/w(s)\}\bigr)>c \bigr\}\,\mathrm{d}c\end{aligned}$$ converges $ \mathbb{P} \times\tilde{\mathbb{P} } $-almost surely toward $$\int_0^\infty\ell\bigl(x\in\mathbb{R}\dvtx\tilde{L}^2(1,x)>c \bigr)\,\mathrm{d}c
=\int_{\mathbb{R}}\tilde{L}_\ast^2(1,x)\,\mathrm{d}x.$$ Dominated convergence and Fubini’s theorem imply that $$w(s)\sum_{x\in\mathbb{Z}}\mathbb{E} \bigl[\tilde{\Gamma}_{w(s)}^2\bigl(1,\{
x/w(s)\}\bigr)
\bigr]\longrightarrow
\int_{\mathbb{R}}\mathbb{E} [\tilde{L}_\ast^2(1,x) ]\,\mathrm{d}x\qquad
\mbox{as }s\rightarrow\infty.$$ Therefore, $$w(s)s^{-2}\sum_{x\in\mathbb{Z}}\mathbb{E} [\Gamma^2(s,\{x\})
]\longrightarrow
\int_{\mathbb{R}}\mathbb{E} [\tilde{L}_\ast^2(1,x) ]\,\mathrm{d}x\qquad \mbox{as }
s\rightarrow\infty.$$ This proves the statement of the lemma.
\[Lem4\] For all $ \beta\in(0,2] $ and $ \rho>0 $, there exists a $ C_1>0
$ such that as $ n\rightarrow\infty$, we have $$\bigl|\mathbb{E} \bigl[\xi(0)\mathbh{1}_{[-\rho,\rho]}
(n^{-1/\beta}\xi(0)) \bigr] \bigr|\sim
C_1n^{(1-\beta)/\beta} .$$
For all $ \beta\in(0,2) $ and $ \rho>0 $, there exists a $ C_2>0
$ such that as $ n\rightarrow\infty$, we have $$\bigl|\mathbb{E} \bigl[\xi^2(0)\mathbh{1}_{[-\rho,\rho]}
\bigl(n^{-{1/\beta}}\xi(0)\bigr) \bigr] \bigr|\sim
C_2n^{(2-\beta)/{\beta}}.$$
The random variable $ \xi(0) $ is in the domain of attraction of a $
\beta$-stable random variable with characteristic function given by $$\psi(\theta)=\exp\bigl(-|\theta|^\beta\bigl(A_1+\mathrm{i}A_2\operatorname{sgn}(\theta)\bigr)\bigr) ,$$ with $ 0<A_1<\infty$ and $ |A_1^{-1}A_2|\leq\tan(\uppi\beta/2) $. A consequence of this setting is that for $\beta>1 $, we have $
\mathbb{E}
[\xi(0)]=0 $. Further, if $ \beta\in(0,2] $, then there exist $ B_1,B_2\geq0 $ such that $$\lim_{\rho\rightarrow\infty}\rho^\beta\mathbb{P} \bigl(\xi(0)\geq
\rho\bigr)= B_1\quad
\mbox{and}\quad
\lim_{\rho\rightarrow\infty}\rho^\beta\mathbb{P} \bigl(\xi(0)\leq
-\rho\bigr)= B_2 .$$ For $ \beta=2 $, we have $ B_1=B_2=0$ since the decay of the tail probabilities is exponential in that case. For $ \beta\neq1 $, we then have that $$\begin{aligned}
\bigl|\mathbb{E} \bigl[\xi(0)\mathbh{1}_{[-\rho,\rho]}(n^{-
{1}/{\beta
}}\xi(0)) \bigr] \bigr|
&=& \int_0^{\rho n^{{1}/{\beta}}}\mathbb{P} \bigl(|\xi(0)|\geq
c\bigr)\,\mathrm{d}c\\
&\sim& (B_1+B_2)\int_0^{\rho n^{{1}/{\beta}}}c^{-\beta}\,\mathrm{d}c\\
&=& (B_1+B_2)(1-\beta)^{-1}\rho^{1-\beta}n^{({1}/{\beta
})(1-\beta)}.\end{aligned}$$ This proves the first statement for $ \beta\neq1 $. For $ \beta=1 $, the statement is just our assumption from the .
Moreover, by similar arguments for $ \beta\neq2 $, we have that $$\begin{aligned}
\bigl|\mathbb{E} \bigl[\xi^2(0)\mathbh{1}_{[-\rho,\rho
]}(n^{-
{1}/{\beta}}\xi(0)) \bigr] \bigr|
&\sim& (B_1+B_2)\int_0^{\rho n^{{1}/{\beta}}}c^{1-\beta}\,\mathrm{d}c\\
&=& (B_1+B_2)(2-\beta)^{-1}\rho^{2-\beta}n^{(1/\beta)
(2-\beta)}.\end{aligned}$$ This completes the proof of the second statement.
The distributions of the sequence $ \{\Xi_{n};n\in\mathbb{N}\} $ are tight with respect to the Skorohod topology.
We follow the method given in Kesten and Spitzer ([-@KesSpi1979]). Let $ \epsilon
>0 $ be given. By Lemma \[Lem2\], there exists an $ A>0 $ such that $ \epsilon(AT^{-\alpha/(1+\alpha)} )\leq\epsilon/4 $. This implies that $$\begin{aligned}
&& \mathbb{P} \biggl(\Xi_n(t)\neq n^{-\kappa}\sum_{|x|\leq An}\Gamma
(k_nt,\{x\}
)\xi(x)\mbox{ for some } t\leq T \biggr) \\
&&\quad\leq \mathbb{P} \bigl(\Gamma(k_nT,\{x\})>0\mbox{ for some } x \mbox{ with }
|x|>Ak_n^{\alpha/(1+\alpha)} \bigr) \\
&&\quad\leq \epsilon\bigl(AT^{-\alpha/(1+\alpha)} \bigr)\\
&&\quad\leq \epsilon/4.\end{aligned}$$ There exists a $ \rho_0>0 $ with the property that for all $ \rho
>\rho_0 $ and all $ n\in\mathbb{N} $, we have $$3An\bigl(1-\mathbb{P} \bigl(-\rho n^{1/\beta}\leq\xi(0)\leq\rho
n^{1/\beta} \bigr)\bigr) \leq\epsilon/4 .$$ This is valid since for suitable $ B_1,B_2\geq0 $, we have $$\lim_{\rho\rightarrow\infty}\rho^\beta\mathbb{P} \bigl(\xi(0)\geq
\rho\bigr)= B_1
\quad \mbox{and}\quad
\lim_{\rho\rightarrow\infty}\rho^\beta\mathbb{P} \bigl(\xi(0)\leq
-\rho\bigr)= B_2 .$$ For all $ x\in\mathbb{Z} $, we have the random variables $$\begin{aligned}
\bar{\xi}_n(x)&:=&\xi(x)\mathbh{1}_{[-\rho,\rho
]}(n^{-1/\beta}\xi(x)) ,
\\
E_n&:=&n^{-\kappa}\frac{1}{T}\mathbb{E} \biggl[\sum_{x\in\mathbb
{Z}}\Gamma(k_nt,\{
x\})\bar{\xi}_n(x) \biggr]
=n^{-\kappa}\frac{1}{T}\mathbb{E} \biggl[\sum_{x\in\mathbb{Z}}
\Gamma(k_nt,\{x\})\mathbb{E} [\bar{\xi}_n(x) ] \biggr]\end{aligned}$$ and $$\bar{\Xi}_n(t):=n^{-\kappa}\sum_{x\in\mathbb{Z}}\Gamma(k_nt,\{x\})
\bigl(\bar{\xi}_n(x)-\mathbb{E} [\bar{\xi}_n(x) ] \bigr) .$$
*Claim* 1. The family of random variables $ \{E_n(t);n\in\mathbb
{N}\} $ is bounded. This is true since, by Lemma \[Lem4\], we have $$\begin{aligned}
\biggl|\sum_{x\in\mathbb{Z}}\Gamma(k_nt,\{x\})\mathbb{E} [\bar{\xi }_n(x) ] \biggr|
&=& |\mathbb{E} [\bar{\xi}_n(0) ] |\sum_{x\in\mathbb{Z}}\Gamma
(k_nt,\{x\})\\
&=&k_nt |\mathbb{E} [\bar{\xi}_n(0) ] |
\leq Ctn^{(\alpha+1)/\alpha}n^{(1/\beta)(1-\beta)}\end{aligned}$$ and $ \frac{\alpha+1}{\alpha}+\frac{1}{\beta}(1-\beta)-\kappa=0.$
*Claim* 2. For all $ \eta>0 $, there exists an $ n_0\in\mathbb
{N} $ such that for all $ n\geq n_0 $, we have $$\mathbb{P} \biggl(\sup_{t\leq T}|\Xi_n(t)-\bar{\Xi}_n(t)-E_nt|>\frac
{\eta}{2}
\biggr)\leq\frac{\epsilon}{2} .$$ To see this, we first note that $$\Xi_n(t)-\bar{\Xi}_n(t)-E_nt = n^{-\kappa}\sum_{x\in\mathbb
{Z}}\Gamma(k_nt,\{x\})
\bigl(\xi(x)-\bar{\xi}_n(x) \bigr)$$ since $$\begin{aligned}
&& \Xi_n(t)-\bar{\Xi}_n(t)-E_nt-n^{-\kappa}\sum_{x\in\mathbb
{Z}}\Gamma(k_nt,\{x\})
\bigl(\xi(x)-\bar{\xi}_n(x) \bigr) \\
&&\quad= n^{-\kappa} \biggl(\sum_{x\in\mathbb{Z}}\Gamma(k_nt,\{x\})\mathbb
{E} [\bar
{\xi}(x) ]
-\frac{t}{T}\mathbb{E} \biggl[\sum_{x\in\mathbb{Z}}\Gamma(k_nt,\{x\}
)\mathbb{E} [\bar
{\xi}(x) ] \biggr] \biggr)\\
&&\quad= n^{-\kappa}\mathbb{E} [\bar{\xi}(0) ] \biggl(\sum_{x\in\mathbb
{Z}}\Gamma
(k_nt,\{x\})
-\frac{t}{T}\mathbb{E} \biggl[\sum_{x\in\mathbb{Z}}\Gamma(k_nt,\{x\}) \biggr]
\biggr)\\
&&\quad= n^{-\kappa}\mathbb{E} [\bar{\xi}(0) ] \biggl(k_nt-\frac{t}{T}k_nT \biggr)\\
&&\quad=0.\end{aligned}$$ Lemma \[Lem4\] implies that $$\begin{aligned}
&& \mathbb{P} \biggl(n^{-\kappa}\sum_{x\in\mathbb{Z}}\Gamma(k_nt,\{x\})
\bigl(\xi
(x)-\bar{\xi}_n(x) \bigr)\neq0
\mbox{ for some } t\leq T \biggr) \\
&&\quad\leq \mathbb{P} \bigl(\Gamma(k_nT,\{x\})>0\mbox{ for some } x
\mbox{ with }|x|>Ak_n^{\alpha/(1+\alpha)} \bigr) \\
&&\qquad{}+ \mathbb{P} \bigl(\xi(x)\neq\bar{\xi}_n(x)\mbox{ for some }
|x|\leq Ak_n^{\alpha/(1+\alpha)} \bigr) \\
&&\quad\leq \epsilon\bigl(AT^{-\alpha/(1+\alpha)} \bigr)
+3Ak_n^{\alpha/(1+\alpha)}\mathbb{P} \bigl(\xi(0)\neq\bar{\xi
}_n(0) \bigr) \\
&&\quad\leq \frac{\epsilon}{4}+3An
\bigl(1-\mathbb{P} \bigl(-\rho n^{1/\beta}\leq\xi(0)\leq\rho n^
{1/\beta} \bigr) \bigr) \\
&&\quad\leq \frac{\epsilon}{2}.\end{aligned}$$
*Claim* 3. There exists a $ K_0>0 $ such that for all $ n\in
\mathbb{N} $, we have $$\mathbb{E} [ |\bar{\Xi}_n(t_2)-\bar{\Xi}_n(t_1) |^2 ]\leq
C_0(t_2-t_1)^{2-({1+\alpha})/{\alpha}} .$$ We define the $ \sigma$-field $ \mathcal{X}=\{X(t);t\geq0\} $. It then follows from the independence of $ \{X(t);t\geq0\} $ and $ \{\xi(x);x\in\mathbb
{Z}\} $ that $$\begin{aligned}
&& \mathbb{E} \biggl[ \biggl(\sum_{x\in\mathbb{Z}}\bigl(\Gamma(k_nt_2,\{x\})-\Gamma
(k_nt_1,\{x\})\bigr)\bar{\xi}_n(x) \biggr)^2 \biggr]\\
&&\quad=\mathbb{E} \biggl[\mathbb{E} \biggl[ \biggl(
\sum_{x\in\mathbb{Z}}\bigl(\Gamma(k_nt_2,\{x\})
-\Gamma(k_nt_1,\{x\})\bigr)\bar{\xi}_n(x) \biggr)^2 \bigg|\mathcal{X} \biggr] \biggr]\\
&&\quad= \mathbb{E} \biggl[\sum_{x\in\mathbb{Z}}\bigl(\Gamma(k_nt_2,\{x\})
-\Gamma(k_nt_1,\{x\})\bigr)^2\mathbb{E} [\bar{\xi}^2_n(x) |\mathcal{X} ] \biggr]\\
&&\quad= \sum_{x\in\mathbb{Z}} \mathbb{E} \bigl[\bigl(\Gamma(k_nt_2,\{x\})
-\Gamma(k_nt_1,\{x\})\bigr)^2 \bigr]\mathbb{E} [\bar{\xi}^2_n(x) ].\\\end{aligned}$$ This implies that $$\begin{aligned}
\mathbb{E} [ |\bar{\Xi}_n(t_2)-\bar{\Xi}_n(t_1) |^2 ]
&\leq& n^{-2\kappa}
\sum_{x\in\mathbb{Z}}\mathbb{E} \bigl[\bigl(\Gamma(k_nt_2,\{x\})-\Gamma
(k_nt_1,\{x\}
)\bigr)^2 \bigr]\mathbb{E} [\bar{\xi}_n^2(x) ]\\
&=& n^{-2\kappa}\mathbb{E} \biggl[\sum_{x\in\mathbb{Z}}\bigl(\Gamma(k_nt_2,\{
x\})
-\Gamma(k_nt_1,\{x\})\bigr)^2 \biggr]\mathbb{E} [\bar{\xi}_n^2(0) ] .\end{aligned}$$ Conditioned on $ \mathcal{A}:=\{\lambda_i;i\in\mathbb{Z}\} $, the process $ X $ has the strong Markov property. Using this, we can prove that for $ t_1\leq t_2 $, the conditional distribution of $ \sum_x(\Gamma(t_2,\{x\})-\Gamma(t_1,\{x\}))^2 $ with respect to $
\mathcal{A} $ equals the conditional distribution of $ \sum_x \Gamma^2(t_2-t_1,\{x\}) $ with respect to $ \mathcal{A} $. Hence, $$\begin{aligned}
\mathbb{E} \biggl[\sum_{x\in\mathbb{Z}}\bigl(\Gamma(t_2,\{x\})-\Gamma(t_1,\{
x\})\bigr)^2 \biggr]
&=&\mathbb{E} \biggl[\mathbb{E} \biggl[\sum_{x\in\mathbb{Z}}\bigl(\Gamma(t_2,\{x\}
)-\Gamma(t_1,\{x\}
)\bigr)^2 \big|\mathcal{A} \biggr] \biggr]\\
&=& \mathbb{E} \biggl[\mathbb{E} \biggl[\sum_{x\in\mathbb{Z}}\Gamma
^2(t_2-t_1,\{x\}) \big|\mathcal{A}
\biggr] \biggr]\\
&=& \mathbb{E} \biggl[\sum_{x\in\mathbb{Z}}\Gamma^2(t_2-t_1,\{x\}) \biggr].\end{aligned}$$ By Lemma \[Lem3\], it follows that $$\begin{aligned}
\mathbb{E} \biggl[\sum_{x\in\mathbb{Z}}\bigl(\Gamma(k_nt_2,\{x\})-\Gamma
(k_nt_1,\{x\}
)\bigr)^2 \biggr]
&\leq& Ck_n^{2-\alpha/(1+\alpha)}(t_2-t_1)^{2-\alpha
/(1+\alpha)} \\
&=& Cn^{2(1+\alpha)/\alpha-1}(t_2-t_1)^{2-\alpha
/(1+\alpha)}.\end{aligned}$$ Moreover, we know that $$\mathbb{E} [\bar{\xi}_n^2(0) ]\leq\tilde{C}n^{(2-\beta)
(1/\beta)} .$$ Putting this all together, we obtain $$\mathbb{E} [ |\bar{\Xi}_n(t_2)-\bar{\Xi}_n(t_1) |^2 ]
\leq C_0n^{(2-\beta)(1/\beta)}
n^{-2\kappa}n^{2(1+\alpha)/\alpha-1}(t_2-t_1)^{2-
{\alpha}/({1+\alpha})} .$$ Since $ (2-\beta)\frac{1}{\beta}-2\kappa+2\frac{1+\alpha}{\alpha
}-1=0 $, Claim 3 follows.
Since $ 2-\frac{\alpha}{1+\alpha}> 1 $, the tightness in the Skorohod topology of the family $ \{\Xi_n;n\in\mathbb{N}\} $ now follows from Claims 1–3 and a theorem of Billingsley ([-@Bil1968]) (see page 95).
Acknowledgements {#acknowledgements .unnumbered}
================
The authors wish to express their deepest gratitude toward the probability group and the staff of Academia Sinica and National Taiwan University for mathematical and administrational help during their visit to Taiwan. Special thanks go to Shieh Narn-Rueih, Hwang Chii-Ruey and Sheu Shuenn-Jyi for many interesting discussions on probability theory. Moreover, the authors would like to thank the referee for his very detailed report which helped to improve the manuscript.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this article we study the symmetry breaking phenomenon of solutions of non-cooperative elliptic systems. We apply the degree for $G$-invariant strongly indefinite functionals to obtain simultaneously a symmetry breaking and a global bifurcation phenomenon.'
address: |
Faculty of Mathematics and Computer Science\
Nicolaus Copernicus University\
PL-87-100 Toruń\
ul. Chopina $12 \slash 18$\
Poland
author:
- Piotr Stefaniak
title: |
Symmetry breaking of solutions\
of non-cooperative elliptic systems
---
[^1]
Introduction
============
In this paper, we consider a symmetry breaking of solutions of non-cooperative elliptic systems of the form: $$\label{problem1}
\left\{ \begin{array}{rcl}
-\Delta w_1 = \nabla_{w_1} F(w_1,w_2)+f_1& \text{in}&\Omega\\
\Delta w_2=\nabla_{w_2} F(w_1,w_2)+f_2& \text{in}&\Omega\\
\frac{\partial w_1}{\partial \nu}=\frac{\partial w_2}{\partial \nu}=0 & \text{on}& \partial \Omega,
\end{array}\right.$$ where ${\mathbb{R}}^n$ is an orthogonal representation of a compact Lie group $G$, $\Omega\subset{\mathbb{R}}^n$ is an open, bounded, $G$-invariant set with a smooth boundary and $F\in C^2({\mathbb{R}}^2,{\mathbb{R}})$. That is we discuss the existence of a $G$-symmetric function $(f_1,f_2)$ such that there is a $K$-symmetric solution $(w_1,w_2)$ of system , where $K$ is a closed subgroup of $G$. If such a solution exists, we say that occurs a symmetry breaking of solutions of problem .
The problem of symmetry breaking has been studied by many authors under various assumptions on $F$ and $\Omega$, see for instance [@Budd]-[@Dancer1], [@Srikanth1; @Jager; @Lauterbach], [@Srikanth]-[@Srikanth2], [@Srikanth4]. Of course this list is far from being complete. The authors have used different tools to obtain their results: Rybakowski’s homotopy index, the equivariant Conley index or the Leray-Schauder degree. We have applied the degree for $G$-invariant strongly indefinite functionals, see [@degree], to obtain our results. Using this degree we have formulated conditions on $F$ which enable us to decide whether there is a connected set of solutions of the main problem.
The idea of the proof of our main result is to reduce the problem to a bifurcation one. We follow the idea from [@Dancer], due to Dancer. The author has used a different tool, that is Rybakowski’s homotopy index, see [@Rybakwoski], which cannot be used to prove our results, because the functional corresponding to system is strongly indefinite. Moreover, using Rybakowski’s or Conley indices it is only possible to obtain a sequence of solutions of the symmetry breaking problem. Using the degree for $G$-invariant strongly indefinite functionals we have obtained a global bifurcation of solutions that problem. Moreover, our method can be used to handle a number of related problems.
After this introduction our article is organised as follows.
In section 2 we introduce our notation and reduce the symmetry breaking problem to a bifurcation problem.
In section 3 we consider a system of elliptic equations and recall basic properties of the operator induced by this system. We formulate the symmetry breaking and the corresponding bifurcation problem for this system. We calculate the degree for $N(K)$-invariant strongly indefinite functionals for an operator associated with a linear system of equations, where $N(K)$ is the normalizer of a subgroup $K$ of $G$. We use this results to proceed some computations in a nonlinear case.
In section 4 we formulate and prove the main results of this article. To do it we use the abstract results from the previous sections.
In section 5 we illustrate our method.
To make this article self-contained, we have included in section 6 the definition of the Euler ring $U(G)$ of a compact Lie group $G$ and the definition and basic properties of the degree for $G$-invariant strongly indefinite functionals, due to Go[ł]{}ȩbiewska and Rybicki, see [@degree].
Preliminaries
=============
Throughout this article $G$ stands for a compact Lie group and ${\overline{\operatorname{sub}}}(G)$ for the set of closed subgroups of $G$. Let $({\mathcal{H}},\langle \cdot,\cdot\rangle)$ be a separable Hilbert space, which is an orthogonal representation of $G$ and let ${\mathcal{H}}^K = \{x\in {\mathcal{H}}: \forall_{g\in K}\ gx=x\}$ be the set of all fixed points of the action of a subgroup $K\in{\overline{\operatorname{sub}}}(G)$. The set $N(K)$ is the normalizer of a subgroup $K\in{\overline{\operatorname{sub}}}(G)$, i.e. $N(K)=\{g\in G: gK=Kg\}$. Fix $k\in{\mathbb{N}}$. Let $C^k_G({\mathcal{H}},{\mathbb{R}})$ denote the set of all $G$-invariant functionals of class $C^k$, i. e. $\Psi (g x)=\Psi (x)$, where $\Psi \in C^k_G({\mathcal{H}},{\mathbb{R}}),\ g\in G$ $x\in{\mathcal{H}}$, and $C^{k-1}_G({\mathcal{H}},{\mathcal{H}})$ the set of all $G$-equivariant operators of class $C^{k-1}$, i. e. $T(gx)=gT(x)$, where $T \in C^{k-1}_G({\mathcal{H}},{\mathcal{H}}),\ g\in G$, $x\in {\mathcal{H}}$. It can be easily shown that for a fixed $K\in{\overline{\operatorname{sub}}}(G)$, ${\mathcal{H}}^G\subset{\mathcal{H}}^K$ and if $\Psi \in C^k_G({\mathcal{H}},{\mathbb{R}})$, then the gradient $\nabla\Psi \in C^{k-1}_G({\mathcal{H}},{\mathcal{H}})$, $k\in{\mathbb{N}}$. We denote by $B_{\gamma}({\mathcal{H}},p)$ the open unit ball in ${\mathcal{H}}$ centered at a point $p$ of radius $\gamma$. Moreover, we put $B({\mathcal{H}}, p)=B_{1}({\mathcal{H}},p)$, $B_{\gamma}({\mathcal{H}})=B_{\gamma}({\mathcal{H}},0)$ and $B({\mathcal{H}})=B_{1}({\mathcal{H}},0)$. Suppose that $\Lambda$ is a linear space of parameters, $\Psi \in C^k_G({\mathcal{H}}\times\Lambda,{\mathbb{R}})$ is such that $\nabla_u \Psi(0,\lambda)=0$ for every $\lambda\in \Lambda$. Consider the equation $$\label{bifogol}
\nabla_u \Psi(u,\lambda)=0.$$ Define a set of non-zero solutions of by $\mathcal{N}=\{(u,\lambda)\in ({\mathcal{H}}\setminus\{0\})\times\Lambda: \nabla_u \Psi(u,\lambda)=0\}$, fix $\lambda_0\in\Lambda$ and denote by $C(\lambda_0)$ a connected component of the closure $\operatorname{cl}(\mathcal{N})$ such that $(0,\lambda_0)\in C(\lambda_0)$.
A point $(0,\lambda_0)\in \{0\}\times\Lambda$ is said to be a local bifurcation point of solutions of equation , if $(0, ƒ\lambda_0)\in \operatorname{cl}(\mathcal{N})$. A point $(0, ƒ\lambda_0)\in \{0\}\times\Lambda$ is said to be a branching point of non-zero solutions of equation , if $C(\lambda_0) \neq \{(0, \lambda_0)\}$. A point $(0, ƒ\lambda_0)\in \{0\}\times\Lambda$ is said to be a global bifurcation point of non-zero solutions of equation , if either $C(\lambda_0) \cap ({0}\times(\Lambda \setminus \{\lambda_0\}) \neq \emptyset$ or $C(\lambda_0)$ is not bounded.
\[pr1\] Let $T\in C^0_G({\mathcal{H}},{\mathcal{H}})$. Does there exist $w\in {\mathcal{H}}^K\backslash {\mathcal{H}}^G$ such that $T(w)\in {\mathcal{H}}^G$?
For subspaces ${\mathcal{H}}_2\subset {\mathcal{H}}_1\subset{\mathcal{H}}$ set ${\mathcal{H}}_1\ominus{\mathcal{H}}_2=\{u\in{\mathcal{H}}_1:\langle u,v \rangle=0\ \forall_{v\in {\mathcal{H}}_2}\}$.
Consider a $G$-equivariant projection $\pi\colon {\mathcal{H}}\to {\mathcal{H}}$ such that ${\operatorname{im}}\pi = ({\mathcal{H}}^G)^{\bot}$. Then ${\operatorname{im}}(I-\pi)={\mathcal{H}}^G$ and note that $\pi({\mathcal{H}}^K)\subset {\mathcal{H}}^K$ for every $K\in{\overline{\operatorname{sub}}}(G)$. Define $\pi_1\colon {\mathcal{H}}^K\to {\mathcal{H}}$ to be the composition $\pi_1 = \pi\circ i$, where $i\colon {\mathcal{H}}^K\to ({\mathcal{H}}\ominus{\mathcal{H}}^K)\oplus{\mathcal{H}}^K$ is the embedding given by $i(x)=(0,x)$. The mapping $i$ is $N(K)$-equivariant (the space ${\mathcal{H}}^K$ is $N(K)$-invariant and does not have to be $G$-invariant), so $\pi_1$ is also $N(K)$-equivariant. It is easy to verify that ${\operatorname{im}}\pi_1={\mathcal{H}}^K\ominus{\mathcal{H}}^G$. Let ${\mathcal{H}}^K={\operatorname{im}}\pi_1\oplus\Lambda$, where $\Lambda={\mathcal{H}}^G$.
In [@Dancer] it has been shown that Problem \[pr1\] is equivalent to the following
\[pr2\] Let $T\in C^0_G({\mathcal{H}},{\mathcal{H}})$. Do there exist $\lambda \in \Lambda$ and $u\in {\operatorname{im}}\pi_1 \backslash\{0\}$ satisfying the equation $
(\pi_1\circ T\circ i)(u,\lambda)=0?
$
Define the operator ${\mathcal{A}}\in C^0_{N(K)}({\operatorname{im}}\pi_1 \oplus \Lambda, {\operatorname{im}}\pi_1)$ by ${\mathcal{A}}(u,\lambda)=\pi_1(T(i(u,\lambda)))$. It is easy to verify that the operator ${\mathcal{A}}$ is well defined.
The following remark follows from the definition of $\pi_1$ and the equality ${\mathcal{A}}(0,\lambda)=0$ for every $\lambda \in \Lambda$.
\[symbrAbif\] If there exists a bifurcation point of solutions of the equation ${\mathcal{A}}(u,\lambda)=0$, then the answer to Problem \[pr1\] is affirmative.
In view of remark \[symbrAbif\] our aim is to study the bifurcations of solutions of the equation ${\mathcal{A}}(u,\lambda)=0$.
Throughout the rest of this section we will need the following assumptions:
1. $\Phi\in C^2_G({\mathcal{H}},{\mathbb{R}})$,
2. $\Phi(w) =\frac{1}{2}\langle L w, w \rangle-\eta(w)$,
3. $L\colon {\mathcal{H}}\to {\mathcal{H}}$ is a linear, bounded, self-adjoint, G-equivariant Fredholm operator of index 0,
4. $\nabla \eta\in C^1_G({\mathcal{H}},{\mathcal{H}})$ is a completely continuous operator.
From now on we put $T=\nabla\Phi$. Because the operator $L$ is $G$-equivariant, $L({\mathcal{H}}^G)\subset {\mathcal{H}}^G$, $L({\mathcal{H}}^K)\subset {\mathcal{H}}^K$. Since the operator L is self-adjoint, we obtain the following $$\left.\begin{array}{cccc}
&{\mathcal{H}}\ominus{\mathcal{H}}^K& & {\mathcal{H}}\ominus{\mathcal{H}}^K \\
& \oplus& & \oplus\\
L\colon &{\mathcal{H}}^K\ominus{\mathcal{H}}^G&\to & {\mathcal{H}}^K\ominus{\mathcal{H}}^G, \\
& \oplus& & \oplus\\
&{\mathcal{H}}^G& & {\mathcal{H}}^G
\end{array}\right.
\ L=
\left[ \begin{array}{ccc}
L_1&0&0 \\
0&L_2&0 \\
0&0&L_3
\end{array} \right].$$ From the above we get $\pi_1(L(i(u,\lambda)))=\pi_1(L(0,u,\lambda))=\pi_1(0,L_2u,L_3\lambda)=L_2u.$ Therefore ${\mathcal{A}}(u,\lambda)=\pi_1(\nabla_u\Phi(u,\lambda))=L_2u-\pi_1(\nabla\eta(i(u,\lambda))).$
\[Lemma1\] For every $\lambda \in \Lambda$ the operator ${\mathcal{A}}(\cdot,\lambda)\in C^1_{N(K)}({\operatorname{im}}\pi_1, {\operatorname{im}}\pi_1)$ is gradient.
We refer the reader to [@Dancer] for the proof of the above lemma.
From the above lemma it follows that the equation ${\mathcal{A}}(u,\lambda)=0$ has a variational and symmetric structure. To study bifurcations of solutions of the equation can be used the degree for $N(K)$-invariant strongly indefinite functionals $\nabla_{N(K)}$-$\deg(\cdot,\cdot)$, which is an element of the Euler ring $U(N(K))$ of a compact Lie group $N(K)$, see Appendix for the definitions and basic properties.
Elliptic system
===============
In this section we study the strongly indefinite functional associated with a system of elliptic equations.
Consider the following system $$\label{rowNeuSN}
\left\{ \begin{array}{rcl}
-\Delta w_1 = \nabla_{w_1} F(w_1,w_2)& \text{in}&\Omega\\
\Delta w_2=\nabla_{w_2} F(w_1,w_2)& \text{in}&\Omega\\
\frac{\partial w_1}{\partial \nu}=\frac{\partial w_2}{\partial \nu}=0 & \text{on}& \partial \Omega,
\end{array}\right.$$ where
1. $\Omega$ is an open, bounded and $G$-invariant subset of an orthogonal $G$-representation ${\mathbb{R}}^n$, with a smooth boundary,
2. $F\in C^2({\mathbb{R}}^2,{\mathbb{R}})$,
3. $|\nabla^2 F(y)|\leq a +b|y|^q$, where $a,\ b\in{\mathbb{R}},\ q<\frac{4}{n-2}$ for $n\geq 3$ and $q<\infty$ for $n=2$.
Put in the previous section ${\mathcal{H}}=H^1(\Omega)\oplus H^1(\Omega)$. Since $H^1(\Omega)$ is an orthogonal $G$-representation with the action given by $(g,u)(x)\mapsto u(g^{-1}x)$ for $g\in G,\ u\in H^1(\Omega),\ x\in\Omega$, so is ${\mathcal{H}}$, where $G$ acts on this space by $(g,(u,v))(x)\mapsto (u(g^{-1}x),v(g^{-1}x))$ for $g\in G,\ u,\ v\in H^1(\Omega),\ x\in\Omega$. Put $L=
\left[\begin{array}{cc}
1 &0\\
0& -1
\end{array}\right]$. For brevity we use the same notation for a matrix and the operator $H^1(\Omega)\oplus H^1(\Omega)\to H^1(\Omega)\oplus H^1(\Omega)$ induced by the matrix.
Recall that a weak solution of the system is a function $w \in {\mathcal{H}}$ such that $$\forall_{v\in {\mathcal{H}}}~\int\limits_{\Omega} \langle L\nabla w(x), \nabla v(x)\rangle - \langle \nabla F(w(x)),v(x) \rangle dx= 0,$$ where $\langle\cdot,\cdot\rangle$ are the standard inner products in ${\mathbb{R}}^{2n}$ and ${\mathbb{R}}^2$.
Put in the previous section $$\label{Phi}
\Phi(w)= \frac{1}{2} \int\limits_{\Omega}|\nabla w_1(x)|^2 - |\nabla w_2(x)|^2dx -\int\limits_{\Omega} F(w(x))dx$$ $$= \frac{1}{2} \int\limits_{\Omega}|\nabla w_1(x)|^2 - |\nabla w_2(x)|^2 +|w_1(x)|^2 - |w_2(x)|^2dx+$$ $$- \int\limits_{\Omega}\frac{1}{2}|w_1(x)|^2 - \frac{1}{2}|w_2(x)|^2 + F(w(x))dx=$$ $$= \frac{1}{2} \int\limits_{\Omega}\langle \nabla (Lw(x)),\nabla w(x)\rangle +\langle L w(x), w(x)\rangle dx-\eta(w)=
\frac{1}{2}\langle Lw,w \rangle_{{\mathcal{H}}}-\eta(w),$$ where $$\eta(w)=- \int\limits_{\Omega}\frac{1}{2}|w_1(x)|^2 - \frac{1}{2}|w_2(x)|^2 + F(w(x))dx$$ and therefore $$\label{eta}
\langle \nabla\eta(w),v \rangle_{{\mathcal{H}}}= \int\limits_{\Omega}\langle L w(x),v(x)\rangle + \langle\nabla F(w(x)),v(x)\rangle dx.$$
Then $\Phi(w)=
\frac{1}{2}\langle Lw,w \rangle_{{\mathcal{H}}}-\eta(w)$, $\nabla \Phi (w) = L w -\nabla\eta(w)$ and $\nabla \eta$ is a completely continuous operator (and consequently compact). Moreover, a function $w\in {\mathcal{H}}$ is a weak solution of system if and only if $\nabla \Phi(w)=0$, that is $w$ is a critical point of $\Phi$.
We study breaking of symmetries of critical orbits for the functional $\Phi$. To do this, fix $K\in {\overline{\operatorname{sub}}}(G)$ and recall that we have defined an equivariant orthogonal projection $\pi_1 \colon {\operatorname{im}}\pi_1 \oplus \Lambda \to {\operatorname{im}}\pi_1$, where ${\operatorname{im}}\pi_1= {\mathcal{H}}^K \ominus {\mathcal{H}}^{G}$ and $\Lambda={\mathcal{H}}^{G}$. We have also defined the operator ${\mathcal{A}}\in C^1_{N(K)}({\operatorname{im}}\pi_1
\oplus \Lambda, {\operatorname{im}}\pi_1)$ by ${\mathcal{A}}(u,\lambda)=\pi_1(\nabla\Phi(i(u,\lambda)))$, that is $${\mathcal{A}}(u,\lambda)= \pi_1(\nabla
\Phi(i(u,\lambda)))= L_2 u - \pi_1(\nabla\eta(i(u,\lambda))),$$ where $i$ is an embedding ${\mathcal{H}}^K$ in $({\mathcal{H}}^K)^{\bot}\oplus{\mathcal{H}}^K$ defined by $i(x)=(0,x)$. From lemma \[Lemma1\] it follows that the operator ${\mathcal{A}}(\cdot,\lambda)\in C^1_{N(K)}({\operatorname{im}}\pi_1, {\operatorname{im}}\pi_1)$ is gradient for every $\lambda\in \Lambda$.
Denote by $\sigma(-\Delta,\Omega)=\{0=\mu_1<\mu_2<\ldots\}$ the set of eigenvalues of the elliptic equation on $\Omega$ with the Neumann boundary condition and ${\mathbb{V}}_{-\Delta}(\mu_k)$ the eigenspace associated with $\mu_k\in\sigma(-\Delta;\Omega)$. We also use the following notation
1. ${\mathcal{H}}^0=\{0\}$,
2. ${\mathcal{H}}_k={\mathbb{V}}_{-\Delta}(\mu_k) \oplus {\mathbb{V}}_{-\Delta}(\mu_k)$ for $k\in{\mathbb{N}}$,
3. ${\mathcal{H}}^n=\bigoplus\limits_{k=1}^n{\mathcal{H}}_k$ for $n\in{\mathbb{N}}$.
Fix $\lambda\in\Lambda={\mathcal{H}}^G$. We will calculate the degree $\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda), B({\operatorname{im}}\pi_1))$, which is an element of the Euler ring $U(N(K))$. To do this we need to define an approximation scheme for the mapping ${\mathcal{A}}(\cdot,\lambda)$, see Appendix. Consider the sequence of $N(K)$-equivariant orthogonal projections $\Gamma=\{\tau_n\colon{\mathcal{H}}\to{\mathcal{H}}:n\in{\mathbb{N}}_0\}$ defined as follows
1. ${\mathcal{H}}'^0=\{0\}$,
2. ${\mathcal{H}}'_k=\left({\mathbb{V}}_{-\Delta}(\mu_k)^K \ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G\right)\oplus \left({\mathbb{V}}_{-\Delta}(\mu_k)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G \right)$ for $k\in{\mathbb{N}}$,
3. ${\mathcal{H}}'^n=\bigoplus\limits_{k=1}^n {\mathcal{H}}'_k$ for $n\in{\mathbb{N}}$,
4. $\tau_n$ is a projection such that ${\operatorname{im}}\tau_n={\mathcal{H}}'^n$, for $n\in{\mathbb{N}}$.
Then $\Gamma$ is an $N(K)$-equivariant approximation scheme on ${\operatorname{im}}\pi_1={\mathcal{H}}^K\ominus{\mathcal{H}}^G$. Moreover, $\ker L= {\mathcal{H}}^0$ and for every $n\in{\mathbb{N}}\cup \{0\}$ it follows that $\tau_n\circ L=L\circ \tau_n$. Note that $\pi_1({\mathcal{H}}^n)={\mathcal{H}}'^n$.
Consider the system: $$\label{lin}
\left\{ \begin{array}{rcl}
-\Delta w_1 =aw_1+bw_2& \text{in}&\Omega\\
\Delta w_2 = bw_1+cw_2& \text{in}&\Omega\\
\frac{\partial w_1}{\partial \nu}=\frac{\partial w_2}{\partial \nu}=0 & \text{on}& \partial \Omega
\end{array}\right.$$ and put $A=
\left[\begin{array}{cc}
a &b\\
b& c
\end{array}\right]$. Then $$\Phi(w)= \frac{1}{2} \int\limits_{\Omega}|\nabla w_1(x)|^2 - |\nabla w_2(x)|^2 - \langle A(w(x)),w(x)\rangle dx.$$ Note that from and it follows that $\nabla\Phi(w)=Lw-C_Aw$, where $C_A$ is given by $\langle C_Aw,v\rangle_{{\mathcal{H}}}= \int\limits_{\Omega} \langle (L+A)w(x), v(x)\rangle dx$ for $w,v \in {\mathcal{H}}$.
For every $w\in{\mathcal{H}}_k,\ v\in{\mathcal{H}}$, $\langle C_Aw, v\rangle_{{\mathcal{H}}}=\langle\frac{1}{1+\mu_k}(L+A)w,v\rangle_{{\mathcal{H}}}$.
Note that $L+A=\left[\begin{array}{cc}
1+a &b\\
b& -1+c
\end{array}\right]$ and consider the formula: $$\langle(L+A)w,v\rangle_{{\mathcal{H}}}=\int\limits_{\Omega}\langle\nabla (L+A)w(x), \nabla v(x) \rangle dx +\int\limits_{\Omega}\langle(L+A)w(x),v(x)\rangle dx.$$ Then $$\left.\begin{array}{l}
\int\limits_{\Omega}\langle\nabla (L+A)w(x), \nabla v(x) \rangle dx \\
=\int\limits_{\Omega}\nabla ((1+a)w_1(x)) \nabla v_1(x) +\nabla (bw_1(x)) \nabla v_1(x)dx\\
+\int\limits_{\Omega}\nabla (bw_2(x)) \nabla v_2(x)+\nabla ((-1+c)w_2(x)) \nabla v_2(x) dx \\
=\int\limits_{\Omega}(-\Delta) ((1+a)w_1(x)) v_1(x) +(-\Delta)(bw_1(x)) v_1(x)dx\\
+\int\limits_{\Omega}(-\Delta) (bw_2(x)) v_2(x)+(-\Delta) ((-1+c)w_2(x)) v_2(x) dx
\\
=\mu_k\int\limits_{\Omega}((1+a)w_1(x)) v_1(x) +(bw_1(x)) v_1(x)+ (bw_2(x)) v_2(x)+ ((-1+c)w_2(x)) v_2(x) dx
\\
=\mu_k\int\limits_{\Omega}\langle(L+A)w(x),v(x)\rangle dx.
\end{array}\right.$$ Therefore $$\langle(L+A)w,v\rangle_{{\mathcal{H}}}=(1+\mu_k)\int\limits_{\Omega}\langle(L+A)w(x)v(x)\rangle dx=(1+\mu_k) \langle C_Aw,v\rangle_{{\mathcal{H}}}.$$ Hence $\langle C_Aw, v\rangle_{{\mathcal{H}}}=\langle\frac{1}{1+\mu_k}(L+A)w,v\rangle_{{\mathcal{H}}}.$
From the above lemma we obtain $ C_A ({\mathcal{H}}_k)\subset {\mathcal{H}}_k$ and therefore $ C_A \colon {\mathcal{H}}_k\to {\mathcal{H}}_k$. To describe the restriction of ${\mathcal{A}}$ to subrepresentations of ${\operatorname{im}}\pi_1$, we first describe the restriction of $\nabla \Phi$ to subrepresentations of ${\mathcal{H}}$. Let $
T_k(A)=
\left[\begin{array}{cc}
1-\frac{a}{1+\mu_k} &-\frac{b}{1+\mu_k}\\-\frac{b}{1+\mu_k}& -1-\frac{c}{1+\mu_k}
\end{array}\right]$ and $\alpha_{1,k},\ \alpha_{2,k}$ be the eigenvalues of the matrix $T_k(A)$, $f_{1,k},\ f_{2,k}$ the corresponding eigenvectors. Because the matrix $T_k(A)$ is symmetric, $\alpha_{1,k},\ \alpha_{2,k}\in{\mathbb{R}}$. Denote by $\epsilon_1,\ \epsilon_2$ the standard base of ${\mathbb{R}}^2$. Then ${\mathcal{H}}_k=\{\varphi_1(x)\cdot \epsilon_1+\varphi_2(x)\cdot \epsilon_2: \varphi_i\in{\mathbb{V}}_{-\Delta}(\mu_k)\}.$ It is easy to check that $$\{\varphi_1(x)\cdot \epsilon_1+\varphi_2(x)\cdot \epsilon_2: \varphi_i\in{\mathbb{V}}_{-\Delta}(\mu_k)\}=\{\varphi_1(x)\cdot f_{1,k}+\varphi_2(x)\cdot f_{2,k}: \varphi_i\in{\mathbb{V}}_{-\Delta}(\mu_k)\}.$$ Hence we obtain $
(\nabla\Phi)_{|{\mathcal{H}}_k}=
\left[\begin{array}{cc}
\alpha_{1,k} {\operatorname{Id}}&0\\
0& \alpha_{2,k}{\operatorname{Id}}\end{array}\right],$ where ${\operatorname{Id}}\colon {\mathbb{V}}_{-\Delta}(\mu_{k}) \to {\mathbb{V}}_{-\Delta}(\mu_{k})$ is the identity map.
Now we are able to describe the action of the restrictions of ${\mathcal{A}}(\cdot,\lambda)$ on the subrepresentations of ${\operatorname{im}}\pi_1$. Fix $\lambda\in{\mathcal{H}}^G$ and assume that $\dim {\mathbb{V}}_{-\Delta}(\mu_k)^K \ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G>0$. Since ${\mathcal{A}}(u,\lambda)= \pi_1(\nabla\Phi(i(u,\lambda)))$, $${\mathcal{A}}_{|{\mathcal{H}}'_k}(u,\lambda)=\pi_1(\nabla\Phi_{|{\mathcal{H}}_k}(i((u_1,\lambda_1),(u_2,\lambda_2))))=\pi_1(\nabla\Phi_{|{\mathcal{H}}_k}((0,u_1,\lambda_1),(0,u_2,\lambda_2)))$$ $$=\pi_1(\alpha_{1,k} {\operatorname{Id}}(0,u_1,\lambda_1), (\alpha_{2,k}{\operatorname{Id}}(0,u_2,\lambda_2)))=(\alpha_{1,k} u_1, \alpha_{2,k}u_2),$$ where $(u,\lambda)= ((u_1,\lambda_1),(u_2,\lambda_2))$ and $(u_i,\lambda_i)\in\left(({\mathbb{V}}_{-\Delta}(\mu_k)^K \ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G)\oplus{\mathbb{V}}_{-\Delta}(\mu_k)^G\right)$ for $i=1,2$. Therefore $
({\mathcal{A}}_{|{\mathcal{H}}'_k}(\cdot,\lambda))=
\left[\begin{array}{cc}
\alpha_{1,k} {\operatorname{Id}}&0\\
0& \alpha_{2,k}{\operatorname{Id}}\end{array}\right],$ where ${\operatorname{Id}}\colon ({\mathbb{V}}_{-\Delta}(\mu_k)^K \ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G) \to ({\mathbb{V}}_{-\Delta}(\mu_k)^K \ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G)$.
Define $m^0(T_k(A))= \dim\ker T_k(A)$ and
$$m^0({\mathcal{A}}_{|{\mathcal{H}}'_k}(\cdot,\lambda))=
\left\{\begin{array}{lcl}
\dim\ker\left(\left[\begin{array}{cc}
\alpha_{1,k} &0\\
0& \alpha_{2,k}
\end{array}\right]\right)
& \text{if} & \dim({\mathbb{V}}_{-\Delta}(\mu_k)^K \ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G)>0\\
0& \text{if} & \dim({\mathbb{V}}_{-\Delta}(\mu_k)^K \ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G)=0.
\end{array}\right.$$ Put
$i^0(A)= \sum\limits_{k=1}^{\infty}m^0(T_k(A))$ and $\widetilde{i^0}(A)=
\sum\limits_{k=1}^{\infty}m^0({\mathcal{A}}_{|{\mathcal{H}}'_k}(\cdot,\lambda))$. It is easy to see that:
$\nabla\Phi$ is an isomorphism if and only if $i^0(A)= 0$. Fix $\lambda\in\Lambda$. ${\mathcal{A}}(\cdot,\lambda)$ is an isomorphism if and only if $\widetilde{i^0}(A)= 0$.
Naturally, if $\nabla\Phi$ is an isomorphism, so is ${\mathcal{A}}(\cdot,\lambda)$ for every $\lambda\in\Lambda$.
Denote by $m^-(T_k(A))$ the Morse index of the matrix $T_k(A)$. Note that $m^-(T_k(A))\in\{0,1,2\}$ and for a sufficiently large $k$, $m^-(T_k(A))=1$. Define the subspaces: $${\mathbb{V}}_0(A)=\bigoplus\limits_{k\colon m^-(T_k(A))=0}
{\mathbb{V}}_{-\Delta}(\mu_k)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G,\
\ {\mathbb{V}}_2(A)=\bigoplus\limits_{k \colon m^-(T_k(A))=2}
{\mathbb{V}}_{-\Delta}(\mu_k)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G.$$
\[Theorem1\] Consider system satisfying $\widetilde{i^0}(A)= 0$ and fix $\lambda\in{\mathcal{H}}^G$. Then $$\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda), B({\operatorname{im}}\pi_1))=\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B(V_2(A)))
\star\left(\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B(V_0(A)))\right)^{-1}.$$
From the definition of the degree, see formula , for sufficiently large $n$ the following equality holds $$\left.\begin{array}{l}
\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda), B({\operatorname{im}}\pi_1))\\=
\left(\nabla_{N(K)}\text{-}\deg(L_2, B({\mathcal{H}}'^n\ominus{\mathcal{H}}'^0))\right)^{-1}\star
\nabla_{N(K)}\text{-}\deg({\mathcal{A}}_{|{\mathcal{H}}'^n}(\cdot,\lambda), B({\mathcal{H}}'^n)).
\end{array}\right.$$ Note that from the product formula, see Appendix, and from the definition of the function $L_2$ we obtain $$\nabla_{N(K)}\text{-}\deg(L_2, B({\mathcal{H}}^n\ominus{\mathcal{H}}^0)) =
\nabla_{N(K)}\text{-}\deg(L_2, B(\bigoplus\limits^n_{k=1}
{\mathbb{V}}_{-\Delta}(\mu_k)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G))$$ $$=
\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}},B({\mathbb{V}}_{-\Delta}(\mu_1)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_1)^G))\star \nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}},B({\mathbb{V}}_{-\Delta}(\mu_2)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_2)^G))$$ $$\star\ldots\star\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B({\mathbb{V}}_{-\Delta}(\mu_n)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_n)^G)).$$ If $m^-(T_k(A))=2$, then $$\left.\begin{array}{l}
\nabla_{N(K)}\text{-}\deg({\mathcal{A}}_{|{\mathcal{H}}'_k}(\cdot,\lambda), B({\mathcal{H}}'_k))\\
=\nabla_{N(K)}\text{-}\deg((-{\operatorname{Id}},-{\operatorname{Id}}), B(({\mathbb{V}}_{-\Delta}(\mu_k)^K \ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G)\oplus ({\mathbb{V}}_{-\Delta}(\mu_k)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G)))\\
=
\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B({\mathbb{V}}_{-\Delta}(\mu_k)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G))\star\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B({\mathbb{V}}_{-\Delta}(\mu_k)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G))
\end{array}\right.$$ When $m^-(T_k(A))=1$ $$\nabla_{N(K)}\text{-}\deg({\mathcal{A}}_{|{\mathcal{H}}'_k}(\cdot,\lambda), B({\mathcal{H}}'_k))=
\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B({\mathbb{V}}_{-\Delta}(\mu_k)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G)).$$ In the remaining case $m^-(T_k(A))=0$ we have $\nabla_{N(K)}\text{-}\deg({\mathcal{A}}_{|{\mathcal{H}}'_k}(\cdot,\lambda), B({\mathcal{H}}'_k))={\mathbb{I}},$ which completes the proof.
Consider the characteristic polynomial of $T_k(A)$ given by $$W_k(\alpha_k)=(1-\frac{a}{1+\mu_k}-\alpha_k)(-1-\frac{c}{1+\mu_k}-\alpha_k)-\frac{b^2}{(1+\mu_k)^2}.$$ It is easy to verify that $$\left.\begin{array}{ll}
W_k(\alpha_k)&=\alpha_k^2 + \frac{a+c}{(1+\mu_k)}\alpha_k + \frac{a-c}{(1+\mu_k)} -1 +\frac{ac-b^2}{(1+\mu_k)^2} \\ & =\alpha_k^2 + \frac{a+c}{(1+\mu_k)}\alpha_k +\frac{-(1+\mu_k)^2+(a-c)(1+\mu_k)+ac-b^2}{(1+\mu_k)^2}.
\end{array}\right.$$ Since the matrix $T_k(A)$ is symmetric, the polynomial has two real roots, denote them by $\alpha_{1,k},\ \alpha_{2,k}$. From Viete’s formulae we get $$\alpha_{1,k}\alpha_{2,k}=\frac{-(1+\mu_k)^2+(a-c)(1+\mu_k)+ac-b^2}{(1+\mu_k)^2}, \ \ \alpha_{1,k}+\alpha_{2,k}=-\frac{a+c}{(1+\mu_k)}.$$ Because the sign of the sum does not depend on $k$, the matrix $T_k(A)$ has the roots of the same sign if and only if $(1+\mu_k)^2-(a-c)(1+\mu_k)-(ac-b^2)<0$. Solving the inequality (with respect to $1+\mu_k$) we obtain the discriminant $\delta=(a-c)^2+4(ac-b^2)=(a+c)^2 - 4b^2$ and if $\delta\geq 0$, then $\beta_1=\frac{a-c-\sqrt{\delta}}{2}-1,\ \ \beta_2=\frac{a-c+\sqrt{\delta}}{2}-1$ are the roots of the polynomial $(1+\mu_k)^2-(a-c)(1+\mu_k)-(ac-b^2)$. If $\delta <0$, then we put $\beta_1=\beta_2=0$. Note that if $(1+\mu_k)^2-(a-c)(1+\mu_k)-(ac-b^2) = 0$ for $\mu_k\in\sigma(-\delta,\Omega)$, then $\alpha_{1,k}=0$ or $\alpha_{2,k}=0$, so $i^0(A)\neq 0$ and if also $\dim({\mathbb{V}}_{-\Delta}(\mu_k)^K \ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G)>0$, then $\widetilde{i}^0(A)\neq 0$.
Let $P=\sigma(-\Delta,\Omega)\cap (\beta_1,\beta_2)$ and note that for every $\mu_k\in P$ we have
1. If $a+c<0$, then $m^{-}(T_k(A))=2$.
2. If $a+c>0$, then $m^{-}(T_k(A))=0$.
Assume that $\widetilde{i^0}(A)=0$.
Under the above notations and assumptions:
1. if $a+c<0$, then $$\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda), B({\operatorname{im}}\pi_1))=\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B(\bigoplus\limits_{\mu\in P}{\mathbb{V}}_{-\Delta}(\mu)^K\ominus{\mathbb{V}}_{-\Delta}(\mu)^G)),$$
2. if $a+c>0$, then $$\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda), B({\operatorname{im}}\pi_1))=\left(\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B(\bigoplus\limits_{\mu\in P}{\mathbb{V}}_{-\Delta}(\mu)^K\ominus{\mathbb{V}}_{-\Delta}(\mu)^G))\right)^{-1},$$
3. if the set $P$ is empty, then $\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda), B({\operatorname{im}}\pi_1))={\mathbb{I}}.$
Note that the condition $i^0(A)=0$ is satisfied if and only if for every $k\in {\mathbb{N}}$ $$-(1+\mu_k)^2+(a-c)(1+\mu_k)+ac-b^2\neq 0.$$
From now on we consider the following nonlinear system $$\label{system}
\left\{ \begin{array}{rcl}
-\Delta w_1 = \nabla_{w_1} F(w_1,w_2)& \text{in}&\Omega\\
\Delta w_2=\nabla_{w_2} F(w_1,w_2)& \text{in}&\Omega\\
\frac{\partial w_1}{\partial \nu}=\frac{\partial w_2}{\partial \nu}=0 & \text{on}& \partial \Omega,
\end{array}\right.$$ Recall that with this system is associated the functional $\Phi \colon {\mathcal{H}}\to {\mathbb{R}}$ given by $$\Phi(w)= \int\limits_{\Omega}\frac{1}{2}(|\nabla w_1(x)|^2 - |\nabla w_2(x)|^2) - F(w(x))dx.$$ and $\nabla \Phi (w) = L w -\nabla\eta(w)$, where $\nabla \eta$ is an operator defined by $$\langle \nabla\eta(w),v \rangle_{{\mathcal{H}}}= \int\limits_{\Omega}\langle L w(x),v(x)\rangle + \langle\nabla F(w(x)),v(x)\rangle dx.$$ Moreover, $\nabla^2 \Phi (w) = L -C_{\nabla^2 F(w)}$ for $w\in{\mathcal{H}}$, where the operator $C_{\nabla^2 F(w)}\colon{\mathcal{H}}\to{\mathcal{H}}$ is given by the equality $$\langle C_{\nabla^2 F(w)}u,v\rangle_{{\mathcal{H}}}= \int\limits_{\Omega} \langle (L+\nabla^2 F(w(x)))u(x), v(x)\rangle dx \text{ for } u,v \in {\mathcal{H}}.$$ Denote ${\mathcal{Z}}=(\nabla F)^{-1}(0)$ and let $z\in {\mathcal{Z}}$ be a non-degenerate critical point of the functional $\Phi$. Define $$\langle C_{\nabla^2 F(z)}u,v\rangle_{{\mathcal{H}}}= \int\limits_{\Omega} \langle (L+\nabla^2 F(z))u(x), v(x)\rangle dx \text{ for } u,v \in {\mathcal{H}}.$$ Denote $\nabla^2 F(z)=
\left[\begin{array}{cc}
a(z) &b(z)\\
b(z)& c(z)
\end{array}\right]$. Then $T_k(\nabla^2 F(z))=
\left[\begin{array}{cc}
1-\frac{a(z)}{1+\mu_k} &-\frac{b(z)}{1+\mu_k}\\-\frac{b(z)}{1+\mu_k}& -1-\frac{c(z)}{1+\mu_k}
\end{array}\right]$ and let $\alpha_{1,k}(z),$ $\alpha_{2,k}(z)$ be the (real) eigenvalues of the matrix $T_k(\nabla^2 F(z))$. Then $$(\nabla^2\Phi(z))_{|H_k}=
\left[\begin{array}{cc}
\alpha_{1,k}(z) {\operatorname{Id}}&0\\
0& \alpha_{2,k}(z){\operatorname{Id}}\end{array}\right],$$ where ${\operatorname{Id}}\colon {\mathbb{V}}_{-\Delta}(\mu_{k}) \to {\mathbb{V}}_{-\Delta}(\mu_{k})$ is the identical function. Note that for a sufficiently large $k$, $m^-(T_k(\nabla^2 F(z)))=1$.
Define $\lambda\in{\mathcal{H}}^G$ by $\lambda(x)=z$ for every $x\in\Omega$. Since the derivative of ${\mathcal{A}}$ with respect to $u$ satisfies ${\mathcal{A}}_u'(0,\lambda)=\pi_1\circ\nabla^2\Phi(0,0,\lambda)\circ i$, it follows that if $\dim({\mathbb{V}}_{-\Delta}(\mu_k)^K \ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G)>0$, then $$({\mathcal{A}}_{|{\mathcal{H}}'_k}(\cdot,\lambda))=
\left[\begin{array}{cc}
\alpha_{1,k}(z) {\operatorname{Id}}&0\\
0& \alpha_{2,k}(z){\operatorname{Id}}\end{array}\right],$$ where ${\operatorname{Id}}\colon ({\mathbb{V}}_{-\Delta}(\mu_k)^K \ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G) \to ({\mathbb{V}}_{-\Delta}(\mu_k)^K \ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G)$.
Define the subspaces $${\mathbb{V}}_0(\nabla^2 F(z))=\bigoplus\limits_{k\colon m^-(T_k(\nabla^2 F(z)))=0}{\mathbb{V}}_{-\Delta}(\mu_k)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G,$$ $${\mathbb{V}}_2(\nabla^2 F(z))=\bigoplus\limits_{k\colon m^-(T_k(\nabla^2 F(z)))=2}{\mathbb{V}}_{-\Delta}(\mu_k)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G.$$
Let $z\in {\mathcal{Z}}$ be such that $\widetilde{i^0}(\nabla^2 F(z))= 0$. Then there exists $\gamma_0$ such that for every $0<\gamma<\gamma_0$ $$\left.\begin{array}{l}\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda), B_{\gamma}({\operatorname{im}}\pi_1))=\nabla_{N(K)}\text{-}\deg({\mathcal{A}}_u'(0,\lambda), B({\operatorname{im}}\pi_1))\\
=\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B(V_2(\nabla^2 F(z)))\star\left(\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B(V_0(\nabla^2 F(z)))\right)^{-1}.
\end{array}\right.$$
We refer the reader to [@degree] for the proof of the first equality. The second equality follows from theorem \[Theorem1\].
Put $\delta(z)=(a(z)-c(z))^2+4(a(z)c(z)-b(z)^2)=(a(z)+c(z))^2-4b(z)^2$ and if $\delta(z)>0$, then put $\beta_1(z)=\frac{a(z)-c(z)-\sqrt{\delta(z)}}{2}-1,\ \ \beta_2(z)=\frac{a(z)-c(z)+\sqrt{\delta(z)}}{2}-1$. If $\delta(z) <0$, then put $\beta_1(z)=\beta_2(z)=0$. Define $P(z)=\sigma(-\Delta,\Omega)\cap (\beta_1(z),\beta_2(z))$. Similarly as before it can be shown that the matrix $T_k(\nabla^2 F(z))$ has roots of the same sign if and only if $\mu_k\in P(z)$. Moreover, for any $\mu_k\in P(z)$:
1. If $a(z)+c(z)<0$, then $m^{-}(T_k(\nabla^2 F(z)))=2$.
2. If $a(z)+c(z)>0$, then $m^{-}(T_k(\nabla^2 F(z)))=0$.
Now we are able to give formulae of the degree of the functional associated with system . Assume that $\widetilde{i^0}(\nabla^2 F(z))=0$.
\[Theorem2\] Under the above notions and assumptions:
1. if $a(z)+c(z)<0$, then for a sufficiently small $\gamma>0$ $$\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda), B_{\gamma}({\operatorname{im}}\pi_1))=
\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B(\bigoplus\limits_{\mu\in P(z)}{\mathbb{V}}_{-\Delta}(\mu_k)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G)),$$
2. if $a(z)+c(z)>0$, then for a sufficiently small $\gamma>0$ $$\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda), B_{\gamma}({\operatorname{im}}\pi_1))=
\left(\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B(\bigoplus\limits_{\mu\in P(z)}{\mathbb{V}}_{-\Delta}(\mu_k)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_k)^G))\right)^{-1},$$
3. if the set $P(z)$ is empty, then for a sufficiently small $\gamma>0$ $$\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda), B_{\gamma}({\operatorname{im}}\pi_1))={\mathbb{I}}.$$
The following theorem will be used to formulate a bifurcation theorem. The proof is similar in spirit to that of [@Garza]. We use the notation from this article.
\[connected\] Let the above assumptions hold and assume that the group $G$ is connected. If ${\mathbb{W}}_1$ or ${\mathbb{W}}_2$ is a nontrivial $G$-representation, then $$\nabla_{G}\text{-}\deg(-{\operatorname{Id}}, B({\mathbb{W}}_1)) \neq \nabla_{G}\text{-}\deg(-{\operatorname{Id}}, B({\mathbb{W}}_2))^{-1}.$$
Recall that $\nabla_{G}\text{-}\deg(-{\operatorname{Id}}, B({\mathbb{W}}_i))=\chi_{G}(S^{{\mathbb{W}}_i})$ for $i=1,2$, see [@Geba], where $S^{{\mathbb{W}}_i} = D({\mathbb{W}}_i)/S({\mathbb{W}}_i)$ is a quotient of the closed unit ball and the unit sphere in ${\mathbb{W}}_i$ and $\chi_{G}$ denotes the $G$-invariant Euler characteristic of $S^{{\mathbb{W}}_i}$. Suppose the assertion of the lemma is false, that is $\chi_T(S^{{\mathbb{W}}_1})\star\chi_T(S^{{\mathbb{W}}_2})=\mathbb{I}\in U(G)$, the unit in $U(G)$. Denote by $T\subset G$ a maximal torus and notice that because ${\mathbb{W}}_i$ are $G$-representations, ${\mathbb{W}}_i$ can be also treated as $T$-representations. The natural homomorphism $\varphi\colon T \to G$ induces a ring homomorphism $\varphi^*\colon U(T) \to U(G)$ such that $\varphi^*(\chi_{G}(S^{{\mathbb{W}}_i}))=\chi_T(S^{{\mathbb{W}}_i})$. Hence $\chi_T(S^{{\mathbb{W}}_1})\star\chi_T(S^{{\mathbb{W}}_2})=\mathbb{I}\in U(T)$. From theorem 3.2 of [@Garza] it follows that there exist $k_0,\ k_1, \ldots, k_r,\ k'_0,\ k'_1, \ldots, k'_{r'}\in {\mathbb{N}}\cup\{0\}$, $K_{m_1},\ldots,K_{m_r},\ K_{m'_1},\ldots,K_{m'_{r'}}\in{\overline{\operatorname{sub}}}(T)$, $x,\ x',\ y,\ y'\in U(T)$ such that
1. $\dim K_{m_i}=\dim K_{m'_i}=\dim T - 1$ for every $i$,
2. $x=\sum\limits^r_{i=1} k_i \cdot \chi_T(T/K_{m_i})$,
3. $x'=\sum\limits^{r'}_{i=1} k'_i \cdot \chi_T(T/K_{m'_i})$,
4. $y=\sum\limits_{(K)\in\{(\mathcal{K})\in{\overline{\operatorname{sub}}}[T]:\dim \mathcal{K} <\dim T -1 \}} n(K)\cdot\chi_T(T/K^+)$, where $n(K)\in{\mathbb{Z}}$,
5. $y'=\sum\limits_{(K)\in\{(\mathcal{K})\in{\overline{\operatorname{sub}}}[T]:\dim \mathcal{K} <\dim T -1 \}} n'(K)\cdot\chi_T(T/K^+)$, where $n'(K)\in{\mathbb{Z}}$,
6. $\chi_T(S^{{\mathbb{W}}_1})=(-1)^{k_0}\mathbb{I}+(-1)^{k_0}x +y$,
7. $\chi_T(S^{{\mathbb{W}}_2})=(-1)^{k'_0}\mathbb{I}+(-1)^{k'_0}x' +y'$.
From the assumptions (since $G$ is connected) it follows that $x\neq\mathbb{I}$ or $x'\neq\mathbb{I}$. It is easy to see that $$\left.\begin{array}{l}
((-1)^{k_0}\mathbb{I}+(-1)^{k_0}x+y)((-1)^{k'_0}\mathbb{I}+(-1)^{k'_0}x'+y')\\
=(-1)^{k_0+k'_0}\mathbb{I}+(-1)^{k_0+k'_0}x+(-1)^{k_0+k'_0}x'\\
+(-1)^{k'_0}y+(-1)^{k_0}y'+(-1)^{k_0+k'_0}xx'+(-1)^{k_0}xy'+(-1)^{k'_0}x'y+yy'.\end{array}\right.$$ Note that if $k_0+k'_0$ is even, then $(-1)^{k_0+k'_0}=1$ and $$\left.\begin{array}{l}
(-1)^{k_0+k'_0}\mathbb{I}+(-1)^{k_0+k'_0}x+(-1)^{k_0+k'_0}x'= \mathbb{I}+x+x'\neq \mathbb{I}.
\end{array}\right.$$ If $k_0+k'_0$ is odd, then $(-1)^{k_0+k'_0}=-1$ and $$\left.\begin{array}{l}
(-1)^{k_0+k'_0}\mathbb{I}+(-1)^{k_0+k'_0}x+(-1)^{k_0+k'_0}x'= -\mathbb{I}-x-x'\neq \mathbb{I}.
\end{array}\right.$$ From the properties of the multiplication in $U(T)$, see [@Garza], it is easy to verify that $$(-1)^{k'_0}y+(-1)^{k_0}y'+xx'+xy'+x'y+yy'=\sum\limits_{(K)\in\{(\mathcal{K})\in{\overline{\operatorname{sub}}}[T]:\dim \mathcal{K} <\dim T -1 \}} n''(K)\cdot\chi_T(T/K^+),$$ where $n''(K)\in{\mathbb{Z}}$. Therefore $\chi_T(S^{{\mathbb{W}}_1})\star\chi_T(S^{{\mathbb{W}}_2})=((-1)^{k_0}\mathbb{I}+x+y)((-1)^{k'_0}\mathbb{I}+x'+y') \neq \mathbb{I}$. This contradiction completes the proof.
The main results
================
In this section we apply the bifurcation theory to formulate conditions implying breaking of symmetries. That, is we formulate theorems to answer the problem: does there exist a function $f=(f_1,f_2)\in{\mathcal{H}}^{G}$ such that the system $$\label{rowNeuSN1}
\left\{ \begin{array}{rcl}
-\Delta w_1 = \nabla_{w_1} F(w_1,w_2)+f_1& \text{in}&\Omega\\
\Delta w_2=\nabla_{w_2} F(w_1,w_2)+f_2& \text{in}&\Omega\\
\frac{\partial w_1}{\partial \nu}=\frac{\partial w_2}{\partial \nu}=0 & \text{on}& \partial \Omega,
\end{array}\right.$$ has a weak solution $w\in {\mathcal{H}}^K \backslash {\mathcal{H}}^{G}$?
Recall that $\Phi\colon{\mathcal{H}}^1(\Omega)\times{\mathcal{H}}^1(\Omega)\to{\mathbb{R}}$ is the functional associated with the above system and
1. $\Omega$ is an open, bounded and $G$-invariant subset of an orthogonal $G$-representation ${\mathbb{R}}^n$, with a smooth boundary,
2. $F\in C^2({\mathbb{R}}^2,{\mathbb{R}})$,
3. $|\nabla^2 F(y)|\leq a +b|y|^q$, where $a,\ b\in{\mathbb{R}},\ q<\frac{4}{n-2}$ for $n\geq 3$ and $q<\infty$ for $n=2$.
Assume that the set ${\mathcal{Z}}=(\nabla F)^{-1}(0)$ is finite and fix $z\in{\mathcal{Z}}$. In the previous sections we have introduced the following notation: $\nabla^2 F(z)=
\left[\begin{array}{cc}
a(z) &b(z)\\
b(z)& c(z)
\end{array}\right]$, $\delta(z)=(a(z)+c(z))^2-4b(z)^2$ and if $\delta(z)>0$, then $\beta_1(z)=\frac{a(z)-c(z)-\sqrt{\delta(z)}}{2}-1,\ \ \beta_2(z)=\frac{a(z)-c(z)+\sqrt{\delta(z)}}{2}-1$. If $\delta(z) <0$, then $\beta_1(z)=\beta_2(z)=0$. We have also put $P(z)=\sigma(-\Delta,\Omega)\cap (\beta_1(z),\beta_2(z))$.
Denote $$m_1=\min\{\beta_1(z):z\in{\mathcal{Z}}\},\ \ M_1=\max\{\beta_1(z):z\in{\mathcal{Z}}\},$$ $$m_2=\min\{\beta_2(z):z\in{\mathcal{Z}}\},\ \ M_2=\max\{\beta_2(z):z\in{\mathcal{Z}}\}.$$
Assume that $P(z)\neq\emptyset$ and $\gamma>0$ is a sufficiently small number. By theorem \[Theorem2\] we obtain $$\label{degree}
\left.\begin{array}{l}\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda), B_{\gamma}({\operatorname{im}}\pi_1))=\\
\\
=\left\{
\begin{array}{l}
\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B(\bigoplus\limits_{\mu\in P(z)}({\mathbb{V}}_{-\Delta}(\mu)^K\ominus{\mathbb{V}}_{-\Delta}(\mu)^G)))\ \text{, when } a(z)+c(z)<0\\
\left(\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B(\bigoplus\limits_{\mu\in P(z)}({\mathbb{V}}_{-\Delta}(\mu)^K\ominus{\mathbb{V}}_{-\Delta}(\mu)^G)))\right)^{-1}\ \text{, when } a(z)+c(z)>0.
\end{array}\right.
\end{array}\right.$$ If $P(z)= \emptyset$, then $\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda), B_{\gamma}({\operatorname{im}}\pi_1))=\mathbb{I}\in U(N(K)).$
Recall that for every $\lambda\in\Lambda$, ${\mathcal{A}}(0,\lambda)=0$. In the following theorems we formulate conditions implying a global bifurcation of solutions of the equation ${\mathcal{A}}(u,\lambda)=0$. To do this we use coefficients of the matrix $\nabla^2 F(z)$. We use the notation: if $z_1,\ z_2\in {\mathcal{Z}}$, then $\lambda_1,\ \lambda_2\in{\mathcal{H}}^G$ are defined by $\lambda_1(x)=z_1$ and $\lambda_2 (x)=z_2$ for every $x\in\Omega$.
\[lam1\] Let the above assumptions hold. Assume that there exist $z_1,\ z_2\in {\mathcal{Z}}$ such that
1. $P(z_1)\neq\emptyset$ and $\bigoplus\limits_{\mu\in P(z_1)}{\mathbb{V}}_{-\Delta}(\mu)$ is a nontrivial $N(K)$-representation or $P(z_2)\neq\emptyset$ and $\bigoplus\limits_{\mu\in P(z_2)}{\mathbb{V}}_{-\Delta}(\mu)$ is a nontrivial $N(K)$-representation,
2. $a(z_1)+c(z_1)<0<a(z_2)+c(z_2)$,
3. $\widetilde{i^0}(\nabla^2 F(z_1))=\widetilde{i^0}(\nabla^2 F(z_2))=0$,
4. $N(K)$ is a connect group.
Then there exists a global bifurcation point of solutions of the equation ${\mathcal{A}}(u,\lambda)=0.$
From formula it follows that for sufficiently small $\gamma_1,\ \gamma_2>0$ : $$\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda_1), B_{\gamma_1}({\operatorname{im}}\pi_1))=\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B(\bigoplus\limits_{\mu\in P(z_1)}({\mathbb{V}}_{-\Delta}(\mu)^K\ominus{\mathbb{V}}_{-\Delta}(\mu)^G)))$$ and $$\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda_2), B_{\gamma_2}({\operatorname{im}}\pi_1))=\nabla_{N(K)}\text{-}\deg(-{\operatorname{Id}}, B(\bigoplus\limits_{\mu\in P(z_2)}({\mathbb{V}}_{-\Delta}(\mu)^K\ominus{\mathbb{V}}_{-\Delta}(\mu)^G)))^{-1}$$ From lemma \[connected\] we obtain that $$\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda_1), B_{\gamma_1}({\operatorname{im}}\pi_1))\neq\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda_2),B_{\gamma_2}({\operatorname{im}}\pi_1)).$$ Therefore the theorem follows from theorem \[GLOB\].
Also, if between $m_1$ and $M_1$ exists $\mu\in\sigma(-\Delta,\Omega)$, then there can appear breaking of symmetries.
\[lam2\] Suppose that for every $z\in{\mathcal{Z}}$ either $a(z)+c(z)>0$ or $a(z)+c(z)<0$. Let $z_1,\ z_2\in {\mathcal{Z}}$ be such that $m_1=\beta_1(z_1),\ M_1=\beta_1(z_2)$ and assume that $\widetilde{i^0}(\nabla^2 F(z_1))=\widetilde{i^0}(\nabla^2 F(z_2))=0.$ Suppose that one of the following conditions is satisfied:
1. There exists $\mu_i\in\sigma(-\Delta,\Omega)$ such that $m_1<\mu_i<M_1$, where ${\mathbb{V}}_{-\Delta}(\mu_i)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_i)^G$ is a nontrivial $N(K)$-representation and for every $\mu_j\in\sigma(-\Delta,\Omega)$ if $m_2<\mu_j<M_2$, then ${\mathbb{V}}_{-\Delta}(\mu_j)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_j)^G$ is a trivial, even-dimensional $N(K)$-representation.
2. There exist $\mu_i,\ \mu_j\in\sigma(-\Delta,\Omega)$ such that $m_1<\mu_i<M_1$ and $\mu_j<m_2\leq M_2<\mu_{j+1}$, where ${\mathbb{V}}_{-\Delta}(\mu_i)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_i)^G$ is a nontrivial $N(K)$-representation.
Then there exists a global bifurcation point of solutions of the equation ${\mathcal{A}}(u,\lambda)=0.$
It easy to see that both the assumptions imply that for sufficiently small $\gamma_1,\ \gamma_2>0$ $$\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda_1), B_{\gamma_1}({\operatorname{im}}\pi_1))\neq\nabla_{N(K)}\text{-}\deg({\mathcal{A}}(\cdot,\lambda_2),B_{\gamma_2}({\operatorname{im}}\pi_1)).$$ Therefore the thesis follows from theorem \[GLOB\].
Analogously as the previous theorem, if between $m_2$ and $M_2$ exists $\mu\in\sigma(-\Delta,\Omega)$ we can formulate and prove conditions implying breaking of symmetries.
\[lam3\] Suppose that for every $z\in{\mathcal{Z}}$ either $a(z)+c(z)>0$ or $a(z)+c(z)<0$. Let $z_1,\ z_2\in {\mathcal{Z}}$ be such that $m_2=\beta_2(z_1),\ M_2=\beta_2(z_2)$ and assume that $\widetilde{i^0}(\nabla^2 F(z_1))=\widetilde{i^0}(\nabla^2 F(z_2))=0.$ Suppose that one of the following conditions is satisfied:
1. There exists $\mu_j\in\sigma(-\Delta,\Omega)$ such that $m_2<\mu_j<M_2$, where ${\mathbb{V}}_{-\Delta}(\mu_j)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_j)^G$ is a nontrivial $N(K)$-representation and for every $\mu_i\in\sigma(-\Delta,\Omega)$ if $m_1<\mu_i<M_1$, then ${\mathbb{V}}_{-\Delta}(\mu_i)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_i)^G$ is a trivial, even-dimensional $N(K)$-representation.
2. There exist $\mu_i,\ \mu_j\in\sigma(-\Delta,\Omega)$ such that $\mu_i<m_1\leq M_1<\mu_{i+1}$ and $m_2<\mu_j<M_2$, where ${\mathbb{V}}_{-\Delta}(\mu_j)^K\ominus{\mathbb{V}}_{-\Delta}(\mu_j)^G$ is a nontrivial $N(K)$-representation.
Then there exists a global bifurcation point of solutions of the equation ${\mathcal{A}}(u,\lambda)=0.$
Recall that if there exists a global bifurcation point of solutions of the equation ${\mathcal{A}}(u,\lambda)=0$, then there exists $\lambda_0\in\Lambda$ such that the connected component $C(\lambda_0)$ in the closure of the set $\mathcal{N}=\{(u,\lambda)\in ({\operatorname{im}}\pi_1\setminus\{0\})\times\Lambda: {\mathcal{A}}(u,\lambda)=0\}$ satisfies $C(\lambda_0) \neq \{(0, \lambda_0)\}$ and either $C(\lambda_0)$ is not bounded or $C(\lambda_0) \cap ({0}\times(\Lambda \setminus \{\lambda_0\}) \neq \emptyset$. In particular, we obtain a connected set of solutions of our main problem.
Moreover, for every $(u,\lambda)\in C(\lambda_0)\setminus(\{0\}\times\Lambda)$, the equality ${\mathcal{A}}(u,\lambda)=0$ implies that $\pi_1(\nabla\Phi(i(u,\lambda)))=0$, so $\nabla\Phi(i(u,\lambda))\in {\mathcal{H}}^G$ and therefore there exists $f=(f_1,f_2)\in{\mathcal{H}}^G$ such that the system $$\label{111}
\left\{ \begin{array}{rcl}
-\Delta w_1 = \nabla_{w_1} F(w_1,w_2)+f_1 & \text{in}&\Omega\\
\Delta w_2=\nabla_{w_2} F(w_1,w_2)+f_2 & \text{in}&\Omega\\
\frac{\partial w_1}{\partial \nu}=\frac{\partial w_2}{\partial \nu}=0 & \text{on}& \partial \Omega,
\end{array}\right.$$ has a solution in ${\mathcal{H}}^K\backslash {\mathcal{H}}^G$. That is, $i(C(\lambda_0)\setminus(\{0\}\times\Lambda))$ is a set of solutions of the main problem. Moreover, we have obtained that the whole connected set $i(C(\lambda_0))$ is mapped by $\nabla\Phi$ into a connected set in ${\mathcal{H}}^G$.
Examples
========
In this section we show an application of the bifurcation theorems. Consider the following system $$\label{rowNeukula}
\left\{ \begin{array}{rcl}
-\Delta w_1 = \nabla_{w_1} F(w_1,w_2)& \text{in}&B^n\\
\Delta w_2=\nabla_{w_2} F(w_1,w_2)& \text{in}&B^n\\
\frac{\partial w_1}{\partial \nu}=\frac{\partial w_2}{\partial \nu}=0 & \text{on}& \partial S^{n-1},
\end{array}\right.$$ where
1. $B^n$ is an open unit ball in an orthogonal ${\operatorname{SO}}(n)$-representation ${\mathbb{R}}^n$, where ${\operatorname{SO}}(n)$ is a special orthogonal group,
2. $F\in C^2({\mathbb{R}}^2,{\mathbb{R}})$,
3. $|\nabla^2 F(y)|\leq a +b|y|^q$, where $a,b\in{\mathbb{R}},\ q<\frac{4}{n-2}$ for $n\geq 3$ and $q<\infty$ for $n=2$.
Denote ${\mathcal{Z}}=(\nabla F)^{-1}(0)$ and define the function $\varphi\colon{\mathbb{R}}\times{\mathcal{Z}}\to{\mathbb{R}}$ by the formula $\varphi(x,z)=-x^2+(a(z)-c(z))x+a(z)c(z)-b(z)^2$. Put ${\mathcal{K}}=\{1+\mu:\mu\in\sigma(-\Delta,B^n)\}$. Recall that the set $\sigma(-\Delta,B^n)$ is discrete and therefore so is the set ${\mathcal{K}}$.
\[Example\] Suppose that there exist $z_1,\ z_2\in {\mathcal{Z}}$ such that
1. $\varphi(\cdot, z_i)^{-1}(0)\cap {\mathcal{K}}=\emptyset$ for $i=1, 2$,
2. $P(z_1)\neq\emptyset$ and $\bigoplus\limits_{\mu\in P(z_1)}{\mathbb{V}}_{-\Delta}(\mu)$ is a nontrivial $N(K)$-representation or $P(z_2)\neq\emptyset$ and $\bigoplus\limits_{\mu\in P(z_2)}{\mathbb{V}}_{-\Delta}(\mu)$ is a nontrivial $N(K)$-representation,
3. $a(z_1)+c(z_1)<0<a(z_2)+c(z_2)$.
Then there exists a connected set of points in ${\mathcal{H}}^K\backslash {\mathcal{H}}^G$ such that for each point from this set there exists $f\in{\mathcal{H}}^G$ such that these points are solutions of the system $$\label{1}
\left\{ \begin{array}{rcl}
-\Delta w_1 = \nabla_{w_1} F(w_1,w_2) +f_1& \text{in}&\Omega\\
\Delta w_2=\nabla_{w_2} F(w_1,w_2) +f_2& \text{in}&\Omega\\
\frac{\partial w_1}{\partial \nu}=\frac{\partial w_2}{\partial \nu}=0 & \text{on} &\partial \Omega.
\end{array}\right.$$
The first assumption implies that $i^0(\nabla^2 F(z_1))=i^0(\nabla^2 F(z_2))=0$, hence $\widetilde{i^0}(\nabla^2 F(z_1))=\widetilde{i^0}(\nabla^2 F(z_2))=0$. Theorem \[lam1\] completes the proof.
Elements of the set $\sigma(-\Delta, B^n)$ are well known, especially if $n=2,3$, see [@Michlin]. So are representations of ${\operatorname{SO}}(n)$, for $n=2,3$, and the degree for ${\operatorname{SO}}(2)$-invariant functionals. Therefore the assumptions of theorem \[Example\] are easy to verify.
1. If $G={\operatorname{SO}}(n)$ and $K=\{e\}$ (the trivial subgroup), then the normalizer of $K$ is equal to ${\operatorname{SO}}(n)$ and ${\mathcal{H}}^K={\mathcal{H}}$. Therefore using our method we can study existing of non-radial solutions such that $\nabla\Phi(w)\in {\mathcal{H}}^{{\operatorname{SO}}(n)}$.
2. If $G={\operatorname{SO}}(3)$ and $K={\mathbb{Z}}_m$ (a cyclic group), then the Weyl group of $K$ is equal to $\operatorname{O}(2)$ and the space ${\mathcal{H}}^K$ is an $\operatorname{O}(2)$-representation and therefore an ${\operatorname{SO}}(2)$-representation. Using the degree for ${\operatorname{SO}}(2)$-equivariant maps we can study the problem: does there exist $w\in {\mathcal{H}}^{{\mathbb{Z}}_m}\backslash {\mathcal{H}}^{{\operatorname{SO}}(3)}$ such that $\nabla\Phi(w)\in {\mathcal{H}}^{{\operatorname{SO}}(3)}$?
3. In [@Dancer] has been considered problem of breaking symmetries for elliptic equation using the homotopy index. The author has obtained a sequence of solution of the bifurcation problem. We emphasize that using the degree for gradient $G$-equivariant maps, we have obtained a connected set of solutions.
4. In this article we have considered a simple example of the method, but the same could be used for more complicated equations.
Appendix
========
To make this article self-contained we recall the definitions and properties of the Euler ring for a compact Lie group and the degree for $G$-invariant strongly indefinite functionals.
Denote by ${\mathcal{F}_{*}}(G)$ the class of pointed $G$-CW-complexes, see [@Dieck] for the definition, and by $[X]$ the $G$-homotopy class of a pointed $G$-CW-complex $X$. Let $\mathbf{F}$ be the free abelian group generated by the pointed $G$-homotopy types of finite $G$-CW-complexes and $\mathbf{N}$ the subgroup of $\mathbf{F}$ generated by all elements $[A]-[X]+[X/A]$ for pointed $G$-CW-subcomplexes $A$ of a pointed $G$-CW-complex $X$.
Put $U(G)=\mathbf{F}/\mathbf{N}$ and let $\chi_G(X)$ be the class of $[X]$ in $U(G)$. The element $\chi_G(X)$ is said to be the $G$-equivariant Euler characteristic of a pointed $G$-CW-complex $X$.
For $X,Y\in{\mathcal{F}_{*}}(G)$ let $[X\vee Y]$ denote a $G$-homotopy type of the wedge $X\vee Y\in{\mathcal{F}_{*}}(G)$. Since $[X]-[X\vee Y]+[(X\vee Y)/X]=[X]-[X\vee Y]+[Y] \in \mathbf{N}$ $$\label{plus}
\chi_G(X)+ \chi_G(Y)=\chi_G(X\vee Y).$$
For $X,Y\in{\mathcal{F}_{*}}(G)$ let $X\wedge Y=X\times/X\vee Y$, The assignment $(X,Y)\mapsto X\wedge Y$ induces a product $U(G)\times U(G)\to U(G)$ given by $$\label{razy}
\chi_G(X)\star \chi_G(Y)=\chi_G(X\wedge Y).$$
If $X$ is a $G$-CW-complex without a base point, then by $X^+$ we denote a pointed $G$-CW-complex $X\cup\{\star\}$ and consequently we put $\chi_G(X)=\chi_G(X^+)$.
$(U(G),+,\star)$ with an additive and multiplicative structures given by , , respectively, is a commutative ring with unit $\mathbb{I}=\chi_G(G/G^+)$.
We call $(U(G),+,\star)$ the Euler ring of $G$.
Denote by ${\overline{\operatorname{sub}}}[G]$ the set of conjugacy classes of subgroups of a group $G$.
$(U(G),+)$ is a free abelian group with basis $\chi_G(G/K^+),\ (K)\in{\overline{\operatorname{sub}}}[G]$.
See [@Dieck1; @Dieck] for the complete definition and more properties of the Euler ring.
An element of the Euler ring is the degree for $G$-invariant strongly indefinite functionals, we recall the definition and properties. Let $({\mathcal{H}},\langle\cdot,\cdot\rangle)$ be an infinite-dimensional, separable Hilbert space which is an orthogonal $G$-representation. Denote by $\Gamma=\{\tau_n\colon{\mathcal{H}}\to{\mathcal{H}}:n\in{\mathbb{N}}\cup \{0\}\}$ a sequence of $G$-equivariant orthogonal projections.
A set $\Gamma$ is said to be a $G$-equivariant approximation scheme on ${\mathcal{H}}$ if
1. for every $n\in{\mathbb{N}}\cup \{0\}$, ${\mathcal{H}}^n$ is a finite subrepresentation of the representation ${\mathcal{H}}$,
2. ${\mathcal{H}}^{n+1}={\mathcal{H}}^n\oplus{\mathcal{H}}_{n+1}$ and ${\mathcal{H}}^n\bot{\mathcal{H}}_{n+1}$,
3. for every $u\in{\mathcal{H}}\lim\limits_{n\rightarrow\infty}\tau_n(u)=u$.\
Assume that
(a1)
: $\Omega\subset{\mathcal{H}}$ is an open, bounded and $G$-invariant subset,
(a2)
: $L\colon{\mathcal{H}}\to{\mathcal{H}}$ is a linear, bounded, self-adjoint, $G$-equivariant Fredholm operator satisfying the following assumptions:
(a) $\ker L={\mathcal{H}}^0$,
(b) $\pi_n\circ L=L\circ\pi_n$, for all $n\in{\mathbb{N}}\cup \{0\}$,
(a3)
: $\nabla\eta\colon\Omega\to {\mathcal{H}}$ is a continuous, $G$-equivariant, compact operator,
(a4)
: $\Phi\in C^1_G(\Omega,{\mathbb{R}})$ satisfies the following assumptions:
(a) $\nabla\Phi(u)=Lu-\nabla\eta(u)$,
(b) $\operatorname{cl}((\nabla\Phi)^{-1}(0))\cap\partial\Omega=\emptyset$.
Under the above assumptions define the degree for $G$-invariant strongly indefinite functionals by $$\label{formulaofdegree}
\nabla_G\text{-}\deg(L-\nabla \eta, \Omega)=
(\nabla_G\text{-}\deg(L, B({\mathcal{H}}^n\ominus{\mathcal{H}}^0)))^{-1}\star
\nabla_G\text{-}\deg(L-\pi_n\nabla\eta, \Omega_{\epsilon}\cap{\mathcal{H}}^n),$$ where $\epsilon>0$ is sufficiently small and $n\in {\mathbb{N}}$ is sufficiently large, see [@degree] for details.
The degree has the following properties:
1. (a) if $\nabla_G\text{-}\deg(\nabla\Phi, \Omega)\neq\Theta\in U(G)$, then $(\nabla\Phi)^{-1}(0)\cap\Omega\neq\emptyset$,
(b) if $\Omega=\Omega_1\cup\Omega_2$ and $\Omega_1,\ \Omega_2$ are open, disjoint and $G$-invariant sets, then $$\nabla_G\text{-}\deg(\nabla\Phi, \Omega)=\nabla_G\text{-}\deg(\nabla\Phi, \Omega_1)+\nabla_G\text{-}\deg(\nabla\Phi, \Omega_2),$$
(c) if $\Omega_1\subset\Omega$ is an open and $G$-invariant set and $(\nabla\Phi)^{-1}(0)\cap\Omega\subset\Omega_1$, then $$\nabla_G\text{-}\deg(\nabla\Phi, \Omega)=\nabla_G\text{-}\deg(\nabla\Phi, \Omega_1),$$
(d) if $0\in\Omega$ and $\Phi\in C^2_G(\Omega,{\mathbb{R}})$ is such that $\nabla\Phi(0)=0$ and $\nabla^2\Phi(0)\colon{\mathcal{H}}\to{\mathcal{H}}$ is a $G$-equivariant self-adjoint isomorphism then there is $\gamma_0>0$ such that for every $\gamma<\gamma_0$ we have $$\nabla_G\text{-}\deg(\nabla\Phi, B_{\gamma}({\mathcal{H}}))=\nabla_G\text{-}\deg(\nabla\Phi^2(0), B({\mathcal{H}})).$$
2. Fix $\Phi\in C^1_G({\mathcal{H}}\times[0,1],{\mathbb{R}})$ such that $(\nabla_u\Phi)^{-1}(0)\cap(\partial\Omega\times[0,1])=\emptyset$ and $\nabla_u\Phi(u,t)=Lu-\nabla_u\eta(u,t)$, where $\nabla_u\eta\colon\Omega\times[0,1]\to{\mathcal{H}}$ is $G$-equivariant and compact. Then $$\nabla_G\text{-}\deg(\nabla_u\Phi(\cdot,0), \Omega)=\nabla_G\text{-}\deg(\nabla_u\Phi(\cdot,1), \Omega).$$
3. Let $\Omega_1\subset{\mathcal{H}}_1,\ \Omega_2\subset{\mathcal{H}}_2$ be open, bounded and $G$-invariant subsets of $G$-representations ${\mathcal{H}}_1,\ {\mathcal{H}}_2$. Assume that the functionals $\Phi_i\in C^1_G({\mathcal{H}}_i,{\mathbb{R}}),\ i=1,2$ are of the form $\Phi_i(u)=\frac{1}{2}\langle L_i u, u\rangle +\eta_i(u)$ and satisfy the assumptions (a1)-(a4). Define a functional $\Phi\in C^1_G({\mathcal{H}}_1\oplus{\mathcal{H}}_2,{\mathbb{R}})$ by $\Phi(u_1,u_2)=\Phi(u_1)+\Phi(u_2)$ and set $\Omega=\Omega_1\times\Omega_2$. Then $$\nabla_G\text{-}\deg(\nabla\Phi, \Omega)=\nabla_G\text{-}\deg(\nabla\Phi_1, \Omega_1)\star\nabla_G\text{-}\deg(\nabla\Phi_2, \Omega_2).$$
\[GLOB\] Fix $\Phi\in C^2_G({\mathcal{H}}\times\Lambda,{\mathbb{R}})$ such that $\nabla_u\Phi(u,\lambda)=Lu-\nabla_u\eta(u,\lambda)$, where the mapping $\nabla_u\eta\colon\Omega\times\Lambda\to{\mathcal{H}}$ is $G$-equivariant and compact. Suppose that $\nabla_u\Phi(0,\lambda)=0$ for every $\lambda\in\Lambda$. If there exist $\gamma_1,\ \gamma_2>0$ such that $$\nabla_G\text{-}\deg(\nabla_u\Phi(\cdot,\lambda_1), B_{\gamma_1}({\mathcal{H}}))\neq\nabla_G\text{-}\deg(\nabla_u\Phi(\cdot,\lambda_2), B_{\gamma}({\mathcal{H}})),$$ then at every path joining $(0,\lambda_1)$ and $(0,\lambda_2)$ exists a global bifurcation point of solutions of the equation $\nabla_u\Phi(u,\lambda)=0$.
See [@degree] for properties of the degree and [@Geba; @Rybicki] for the definition of the degree for gradient $G$-equivariant maps. For the general theory of the equivariant degree we refer the reader to [@BKS], [@BKR].
[99]{}
Z. Balanov, W. Krawcewicz and H. Steinlein, *Applied Equivariant Degree*, AIMS Series on Differential Equations and Dynamical Systems, American Institute of Mathematical Sciences (2006). Z. Balanov, W. Krawcewicz, H. Ruan *Applied equivariant degree. I. An axiomatic approach to primary degree.* Discrete Contin. Dyn. Syst. **15(3)** (2006), 983-1016. Ch. Budd *Symmetry breaking and semilinear elliptic equations* J. of Comp. and Appl. Math. **26** (1989), 79-96. G. Cerami, *Symmetry breaking for a class of semilinear elliptic problems*, Nonl. Anal. **10** (1986), 1-14. K-S Cheng, J. A. Smoller, *Symmetry- Breaking for Systems of Nonlinear Elliptic Equations* J. of Diff. Eq. **80** (1989), 315-342. E. N. Dancer, *Breaking of Symmetries for Forced Equations*, Math. Ann. **262** (1983), 473-486. E. N. Dancer, *On non-radialy symmetric bifurcation*, J. Math. Soc. **20(2)** (1979), 287-292. T. tom Dieck, *Transformation Groups and Representation Theory*, in: Lect. Not. in Math. **766** (1979). T. tom Dieck, *Transformation Groups*, Walter de Gruyter, Berlin-New York, 1987. G. L. Garza, S. Rybicki, *Equivariant bifurcations index*, Nonl. Anal. **73** (2010), 2779-2791. K. Gȩba, *Degree for gradient equivariant maps and equivariant Conley index*, Birkhäuser, TNA, Degree, Singularity and Variations, Eds. M. Matzeu and A. Vignoli, Progr. in Nonl. Diff. Equat. and Their Appl. **27**, Birkhäuser, (1997), 247-272. F. Gladiali, M. Grossi, F. Pacella, P. N. Srikanth, *Bifurcation and symmetry breaking for a class of semilinear eliptic equations in an annulus*, Calc. Var. **40** (2011), 295-317. A. Go[ł]{}ȩbiewska, S. Rybicki, *Global bifurcations of critical orbits of $G$-invariant strongly indefinite functionals*, Nonl. Anal. **74** (2011), 1823-1834. W. Jäger and K. Schmitt, *Symmetry breaking in semilinear elliptic problems*, Academic Press, Inc., Analysis, et cetera, Research Paper Published in Honour of Jürgen Moser’s 60th Birthday, Eds. P. H. Rabinowitz & E. Zehnder, (1990), 451-470. R. Lauterbach and S. Maier, *Symmetry breaking at non-positive solutions of semilinear elliptic equations*, Arch. Rational Mech. Anal. **126(4)** (1994), 299-331. C. W. Michlin, *Linear equations of mathematical of physics (in Russian)*, Science, Moscow, 1964. F. Pacella, P. N. Srikanth, *Nonradial solutions of a nonhomogeneous semilinear elliptic problem with linear growth*, J. Math Anal. Appl. **341** (2008), 131-139. J. Smoller, A. G. Wasserman, *Bifurcation and symmetry-breaking*, Invent. Math. **100** (1990), 63-95. J. Smoller, A. G. Wasserman, *Symmetry-Breaking for Positive Solutions of Semilinear Elliptic Equations*, Arch. Rational Mech, Anal. **95** (1986), 217-225. J. Smoller, A. G. Wasserman, *Symmetry-Breaking for Solutions of Semilinear Elliptic Equations with General Boundary Conditions*, Commun. Math. Phys. **105** (1986), 415-441. J. Smoller, A. G. Wasserman, *Symmetry, Degeneracy, and Universality in Semilinear Elliptic Equations. Infinitesimal Symmetry-Breaking*, J. of Func. Anal. **89** (1990), 364-409. M. Ramaswamy, P. N. Srikanth, *Symmetry breaking for a class of semilinear elliptic problems*, Trans. of AMS **304(2)** (1987), 839-845. K. Rybakowski. *On the homotopy index for infinite-dimensional semiflows*, Trans. Am. Math. Soc. **269**, 351-381 (1982). S. Rybicki, *Degree for equivariant gradient maps*, Milan J. Math. **73** (2005), 103-144. P. N. Srikanth, *Symmetry breaking for a class of semilinear elliptic problems.* Ann. Inst. H. Poincare **7** (1990), 107-112.
[^1]: Partially supported by the National Science Centre, Poland, under grant DEC-2012/05/B/ST1/02165
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Although latent factor models (e.g., matrix factorization) achieve good accuracy in rating prediction, they suffer from several problems including cold-start, non-transparency, and suboptimal recommendation for local users or items. In this paper, we employ textual review information with ratings to tackle these limitations. Firstly, we apply a proposed aspect-aware topic model (ATM) on the review text to model user preferences and item features from different *aspects*, and estimate the *aspect importance* of a user towards an item. The aspect importance is then integrated into a novel aspect-aware latent factor model (ALFM), which learns user’s and item’s latent factors based on ratings. In particular, ALFM introduces a weighted matrix to associate those latent factors with the same set of aspects discovered by ATM, such that the latent factors could be used to estimate aspect ratings. Finally, the overall rating is computed via a linear combination of the aspect ratings, which are weighted by the corresponding aspect importance. To this end, our model could alleviate the data sparsity problem and gain good interpretability for recommendation. Besides, an aspect rating is weighted by an aspect importance, which is dependent on the targeted user’s preferences and targeted item’s features. Therefore, it is expected that the proposed method can model a user’s preferences on an item more accurately for each user-item pair locally. Comprehensive experimental studies have been conducted on 19 datasets from Amazon and Yelp 2017 Challenge dataset. Results show that our method achieves significant improvement compared with strong baseline methods, especially for users with only few ratings. Moreover, our model could interpret the recommendation results in depth.'
author:
- Zhiyong Cheng
- Ying Ding
- Lei Zhu
- Mohan Kankanhalli
bibliography:
- 'www\_long.bib'
title: |
Aspect-Aware Latent Factor Model:\
Rating Prediction with Ratings and Reviews
---
<ccs2012> <concept> <concept\_id>10002951.10003260.10003261.10003270</concept\_id> <concept\_desc>Information systems Social recommendation</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003260.10003261.10003271</concept\_id> <concept\_desc>Information systems Personalization</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003317.10003347.10003350</concept\_id> <concept\_desc>Information systems Recommender systems</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003260.10003261.10003269</concept\_id> <concept\_desc>Information systems Collaborative filtering</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010258.10010260.10010268</concept\_id> <concept\_desc>Computing methodologies Topic modeling</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010293.10010309.10010311</concept\_id> <concept\_desc>Computing methodologies Factor analysis</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Introduction
============
When making comments on an item (e.g., *product*, *movie*, and *restaurant*) in the online review/business websites, such as Yelp and Amazon, reviewers also provide an overall rating, which indicates their overall preference or satisfaction towards the corresponding items. Hence, predicting users’ overall ratings to unrated items or *personalized rating prediction* is an important research problem in recommender systems. Latent factor models (e.g., matrix factorization [@koren2009matrix; @zhang2016discrete; @cheng2017exploiting]) are the most widely used and successful techniques for rating prediction, as demonstrated by the Netflix Prize contest [@bell2007lessons]. These methods characterize user’s interests and item’s features using *latent factors* inferred from rating patterns in user-item rating records. As a typical collaborative filtering technique, the performance of MF suffers when the ratings of items or users are insufficient ( also known as the cold-start problem) [@he2015trirank]. Besides, a rating only indicates the overall satisfaction of a user towards an item, it cannot explain the underlying rationale. For example, a user could give a restaurant a high rating because of its delicious food or due to its nice ambience. Most existing MF models cannot provide such fine-grained analysis. Therefore, relying solely on ratings makes these methods hard to explicitly and accurately model users’ preferences [@wang2018rec; @he2015trirank; @ling2014ratings; @mcauley2013hidden; @wu2015flame].
Moreover, MF cannot achieve optimal rating prediction locally for each user-item pair, because it learns the latent factors of users ($\bm{p_u}$) and items ($\bm{q_i}$) via a global optimization strategy [@christakopoulou2016local]. In other words, $\bm{p_u}$ and $\bm{q_i}$ are optimized to achieve a global optimization over all the user-item ratings in the training dataset.[^1] As a result, the performance could be severely compromised locally for individual users or items. MF predicts an unknown rating by the dot product of the targeted user $u$’s and item $i$’s latent factors (e.g., $\bm{p_u}^T\bm{q_i}$). The overall rating of a user towards an item ($\hat{r}_{u,i}$) is decided by the importance/contribution of all factors. Take the $k$-th factor as an example, its contribution is $p_{u,k}*q_{i,k}$. For accurate prediction, it is important to accurately capture the importance of each latent factor for a user towards an item. It is well-known that different users may care about different *aspects* of an item. For example, in the domain of restaurants, some users care more about the taste of *food* while others pay more attention to the *ambience*. Even for the same aspect, the preference of users could be different from each other. For example, in the *food* aspect, some users like *Chinese cuisines* while some others favor *Italian cuisines*. Similarly, the characteristics of items on an aspect could also be different from each other. Thus, it is possible that “a user $u$ prefers item $i$ but dislikes item $j$ on a specific aspect", while “another user $u'$ favors item $j$ more than item $i$ on this aspect". Therefore, in MF, the importance of a latent factor for users towards an item should be treated differently. At first glance, MF achieves the goal as the influence of a factor (e.g., $k$-th factor) is dependent on both $p_{u,k}$ and $q_{i,k}$ (i.e., $p_{u,k}*q_{i,k}$). However, it is suboptimal to model the importance of a factor by a fixed value of an item or a user. In fact, MF treats each factor of an item with the same importance to all users (i.e., $q_{i,k}$); and similarly, each factor of a user is equally important to all items (i.e., $p_{u,k}$) in rating prediction. Take the previous example, “*a user $u$ prefers item $i$ but dislikes item $j$ on an aspect*", i.e., a factor (e..g, $k$) in MF), which means $p_{u,k}*q_{i,k}$ should be larger than $p_{u,k}*q_{j,k}$ (i.e., $p_{u,k}*q_{i,k}>p_{u,k}*q_{j,k}$). On the other hand, “*user $u'$ favors item $j$ more than item $i$ on this aspect*", thus $p_{u',k}*q_{j,k}$ should be larger than $p_{u',k}*q_{i,k}$ (i.e., $p_{u',k}*q_{i,k} <p_{u',k}*q_{j,k}$). Because the values of $p_{u,k}$ and $p_{u',k}$ are kept the same when predicting ratings, it is impossible for MF to satisfy the local requirements $p_{u,k}*q_{i,k}>p_{u,k}*q_{j,k}$ and $p_{u',k}*q_{i,k} <p_{u',k}*q_{j,k}$ simultaneously for these user-item pairs. A straightforward solution is to assign different weights (e.g., $w_{u,i,k}$) to different user-item pairs (e.g., $p_{u,k}*q_{i,k}$). However, how to compute a proper weight for each user-item pair is challenging.
A large amount of research effort has been devoted to deal with these weaknesses of MF methods. For example, various types of side information have been incorporated into MF to alleviate the cold-start problem, such as tags [@shi2013mining; @zhang2014attribute], social relations [@ma2011recommender; @wang2017item], reviews [@ling2014ratings; @mcauley2013hidden; @zhang2016integrating], and visual features [@he2016vbpr]. Among them, the accompanied review of a rating contains important complementary information. It not only encodes the information of user preferences and item features but also explains the underlying reasons for the rating. Therefore, in recent years, many models have been developed to exploit reviews with ratings to tackle the cold-start problem and also enhance the explainability of MF, such as HFT [@mcauley2013hidden], CTR [@wang2011collaborative], RMR [@ling2014ratings], and RBLT [@tan2016rating]. However, a limitation of these models is that they all assume an *one-to-one correspondence relationship* between latent topics (learned from reviews) and latent factors (learned from ratings), which not only limits their flexibility on modeling reviews and ratings but also may not be optimal. In addition, they cannot deal with the suboptimal recommendation for local users or items in MF. In fact, very few studies in literature have considered this problem.
In this paper, we focus on the problem of *personalized rating prediction* and attempt to tackle the above limitations together by utilizing reviews with ratings. Specifically, an *A*spect-aware *T*opic *M*odel (ATM) is proposed to extract *latent topics* from reviews, which are used to model users’ preferences and items’ features in different *aspects*. In particular, each *aspect* of users/items is represented as a probability distribution of latent topics. Based on the results, the relative importance of an aspect (i.e., *aspect importance*) for a user towards an item can be computed. Subsequently, the aspect importance is integrated into a developed *A*spect-aware *L*atent *F*actor *M*odel (ALFM) to estimate *aspect ratings*. In particular, a weight matrix is introduced in ALFM to associate the latent factors to the same set of aspects discovered by ATM. In this way, our model avoids referring to external sentiment analysis tools for aspect rating prediction as in [@zhang2014explicit; @diao2014jointly]. The overall rating is obtained by a linear combination of the *aspect ratings*, which are weighted by the importance of corresponding aspects (i.e., *aspect importance*). Note that the latent topics and latent factors in our model are not linked directly; instead, they are correlated via the *aspects* indirectly. Therefore, the number of latent topics and latent factors could be different and separately optimized to model reviews and ratings respectively, which is fundamentally different from the *one-to-one* mapping in previous models [@mcauley2013hidden; @wang2011collaborative; @ling2014ratings; @tan2016rating; @bao2014topicmf; @zhang2016integrating]. Besides, our model could learn an aspect importance for each user-item pair, namely, assigning a different weight to each $p_{u,k}*q_{i,k}$, and thus could alleviate the suboptimal local recommendation problem and achieve better performance.
A set of experimental studies has been conducted on 19 real-world datasets from Yelp and Amazon to validate the effectiveness of our proposed model. Experimental results show that our model significantly outperforms the state-of-the-art methods which also use both reviews and ratings for rating prediction. Besides, our model also obtains better results for users with few ratings, demonstrating the advantages of our model on alleviating the cold-start problem. Furthermore, we illustrate the interpretability of our model on recommendation results with examples. In summary, the main contributions of this work include:
- We propose a novel aspect-aware latent factor model, which could effectively combine reviews and ratings for rating prediction. Particularly, our model relaxes the constraint of one-to-one mappings between the latent topics and latent factors in previous models and thus could achieve better performance.
- Our model could automatically extract explainable aspects, and learn the aspect importance/weights for different user-item pairs. By associating latent factors with aspects, the aspect weights are integrated with latent factors for rating prediction. Thus, the proposed model could alleviate the suboptimal problem of MF for individual user-item pairs.
- We conduct comprehensive experimental studies to evaluate the effectiveness of our model. Results show that our model is significantly better than previous approaches on tasks of rating prediction, recommendation for sparse data, and recommendation interpretability.
Related Work {#sec:relwork}
============
A comprehensive review on the recommender system is beyond the scope of this work. We mainly discuss the works which utilize both reviews and ratings for rating prediction. Some works assume that the review is available when predicting the rating score, such as SUIT [@li2014suit], LARAM [@wang2011latent], and recent DeepCoNN [@zheng2017joint]. However, in real world recommendation settings, the task should be predicting ratings for the uncommented and unrated items. Therefore, the review is unavailable when predicting ratings. We broadly classify the approaches for the targeted task in three categories: (1) sentiment-based, (2) topic-based, and (3) deep learning-based. Our approach falls into the second category.
**Sentiment-based.** These works analyze user’s sentiments on items in reviews to boost the rating prediction performance, such as [@pappas2013sentiment; @pero2013opinion; @diao2014jointly; @zhang2014explicit]. For example, [@pappas2013sentiment] estimated a sentiment score for each review to build a user-item sentiment matrix, then a traditional collaborative filtering method was applied. Zhang et al. [@zhang2014explicit] analyzed the sentiment polarities of reviews and then jointly factorize the user–item rating matrix. These methods rely on the performance of external NLP tools for sentiment analysis and thus are not self-contained.
**Topic-based.** These approaches extract latent topics or aspects from reviews. An early work [@ganu2009beyond] in this direction relied on domain knowledge to manually label reviews into different aspects, which requires expensive domain knowledge and high labor cost. Later on, most works attempt to extract latent topics or aspects from reviews automatically [@mcauley2013hidden; @bao2014topicmf; @diao2014jointly; @he2015trirank; @ling2014ratings; @mcauley2013hidden; @wu2015flame; @zhang2016integrating; @tan2016rating]. A general approach of these methods is to extract latent topics from reviews using topic models [@wang2011collaborative; @mcauley2013hidden; @ling2014ratings; @zhang2016integrating; @tan2016rating] or non-negative MF [@bao2014topicmf; @qiu2016aspect] and learn latent factors from ratings using MF methods. HFT [@mcauley2013hidden] and TopicMF [@bao2014topicmf] link the latent topics and latent factors by using a defined transform function. ITLFM [@zhang2016integrating] and RBLT [@tan2016rating] assume that the latent topics and latent factors are in the same space, and linearly combine them to form the latent representations for users and items to model the ratings in MF. CTR [@wang2011collaborative] assumes that the latent factors of items depend on the latent topic distributions of their text, and adds a latent variable to offset the topic distributions of items when modeling the ratings. RMR [@ling2014ratings] also learns item’s features using topic models on reviews, while it models ratings using a mixture of Gaussian rather than MF methods. Diao et al. [@diao2014jointly] propose an integrated graphical model called JMARS to jointly model aspects, ratings and sentiments for movie rating prediction. Those models all assume an one-to-one mapping between the learned latent topics from reviews and latent factors from ratings. Although we adopt the same strategy to extract latent topics and learn latent factors, our model does not have the constraint of one-to-one mapping. Besides, Zhang et al. [@zhang2014explicit] extracted aspects by decomposing the user–item rating matrix into item–aspect and user–aspect matrices. He et al. [@he2015trirank] extracted latent topics from reviews by modeling the user-item-aspect relation with a tripartite graph. **Deep learning-based**. Recently, there has been a trend of applying deep learning techniques in recommendation [@he2017neural; @covington2016deep]. For example, He et al. generalized matrix factorization and factorization machines to neural collaborative filtering and achieved promising performance [@he2017neural; @he2017fm]. Textual reviews have also been used in deep learning models for recommendation [@zhang2016collaborative; @zheng2017joint; @catherine2017transnets; @zhang2017joint]. The most related works in this direction are DeepCoNN [@zheng2017joint] and TransNet [@catherine2017transnets], which apply deep techniques to reviews for rating prediction. In DeepCoNN, reviews are first processed by two CNNs to learn user’s and item’s representations, which are then concatenated and passed into a regression layer for rating prediction. A limitation of DeepCoNN is that it uses reviews in the testing phase. [@catherine2017transnets] shows that the performance of DeepCoNN decreases greatly when reviews are unavailable in the testing phase. To deal with the problem, TransNet [@catherine2017transnets] extends DeepCoNN by introducing an additional layer to simulate the review corresponding to the target user-item pair. The generated review is then used for rating prediction.
[ll]{} Notation & Definition\
$\mathcal{D}$ & corpus with reviews and ratings\
$d_{u,i}$ & review document of user $u$ to item $i$\
$s$ & a sentence in a review $d_{u,i}$\
$\mathcal{U}$, $\mathcal{I}$, $\mathcal{A}$ & user set, item set, and aspect set, respectively\
$M,N,A$ & number of users, items, and aspects, respectively\
$N_{w,s}$ & number of words in a sentence $s$\
$K$ & number of latent topics in ATM\
$y$ & an indicator variable in ATM\
$a_s$ & assigned aspect $a$ to sentence $s$\
$\pi_u$ & the parameter of Bernoulli distribution $P(y=0)$\
$\eta$ & Beta priors ($\eta=\{\eta_0, \eta_1\}$)\
$\bm{\alpha_u}, \bm{\alpha_i}$ & Dirichlet priors for aspect-topic distributions\
$\bm{\gamma_u}, \bm{\gamma_i}$ & Dirichlet priors for aspect distributions\
$\bm{\beta_w} $ & Dirichlet priors for topic-word distributions\
$\bm{\theta_{u,a}}$ & user’s aspect-topic distribution: denoting user’s preference on $a$\
$\bm{\psi_{i,a}}$ & item’s aspect-topic distribution: denoting item’s features on $a$\
$\bm{\lambda_u}, \bm{\lambda_v}$ & aspect distributions of user and item, respectively\
$\bm{\phi_w}$ & topic-text word distribution\
$f$ & number of latent factors in ALFM\
$\mu_\cdot$ & regularization coefficients\
$b_\cdot$ & bias terms, e.g., $b_u, b_i, b_0$\
$w_a$ & weight vector for aspect $a$\
$p_u$, $q_i$ & latent factors of user $u$ and item $i$, respectively\
$r_{u,i}$ & rating of user $u$ to item $i$\
$r_{u,i, a}$ & aspect rating of user $u$ towards item $i$ on aspect $a$\
$\rho_{u,i,a}$ & aspect importance of $a$ for $u$ with respect to $i$\
$s_{u,i,a}$ & the degree of item $i$’s attributes matching user $u$’s preference\
& on aspect $a$\
\[tab:notation\]
Proposed Model {#sec:ourmodel}
==============
Problem Setting
---------------
Let $\mathcal{D}$ be a collection of reviews of item set $\mathcal{I}$ from a specific category (e.g., restaurant) written by a set of users $\mathcal{U}$, and each review comes with an overall rating $r_{u,i}$ to indicate the overall satisfaction of user $u$ to item $i$. The primary goal is to predict the unknown ratings of items that the users have not reviewed yet. A review $d_{u,i}$ is a piece of text which describes opinions of user $u$ on different aspects $a \in \mathcal{A}$ towards item $i$, such as *food* for *restaurants*. In this paper, we only consider the case that all the items are from the same category, i.e., they share the same set of aspects $\mathcal{A}$. Aspects that users care for items are latent and learned from reviews by our proposed topic model, in which each aspect is represented as a distribution of the same set (e.g., $K$) of latent topics. Table \[tab:notation\] lists the key notations. Before introducing our method, we would like to first clarify the concepts of *aspects*, *latent topics*, and *latent factors*.
- **Aspect** - it is a high-level semantic concept, which represents the attribute of items that users commented on in reviews, such as *“food”* for *restaurant* and *“battery"* for *mobile phones*.
- **Latent topic & latent factor** - in our context, both concepts represent a more fine-grained concept than *“aspect"*. A latent topic or factor can be regarded as a *sub-aspect* of an item. For instance, for the “food" aspect, a related latent topic could be “*breakfast*" or “*Italian cuisine*". We adopt the terminology of *latent topic* in topic models and *latent factor* in matrix factorization. Accordingly, “latent topics" are discovered by topic model on reviews, and “latent factors" are learned by matrix factorization on ratings.
Aspect-aware Latent Factor Model
--------------------------------
Based on the observations that (1) different users may care for different aspects of an item and (2) users’ preferences may differ from each other for the same aspect, we claim that the overall satisfaction of a user $u$ towards an item $i$ (i.e., the overall rating $r_{u,i}$) depends on $u$’s satisfaction on each aspect $a$ of $i$ (i.e., *aspect rating* $r_{u,i,a}$) and the importance of each aspect (of $i$) to $u$ (i.e., *aspect importance* $\rho_{u,i,a}$). Based on the assumptions, the overall rating $r_{u,i}$ can be predicted as: $$\vspace{-2pt}
\hat{r}_{u,i} = \sum_{a\in\mathcal{A}} \overbrace{\rho_{u,i,a}}^{{ \mbox{\scriptsize\begin{tabular}{@{}c@{}}aspect importance\end{tabular}}}} \underbrace{r_{u,i,a}}_{{ \mbox{\scriptsize\begin{tabular}{@{}c@{}}aspect rating\end{tabular}}}}
\vspace{-2pt}$$
### Aspect rating estimation.
Aspect rating (i.e., $r_{u,i,a}$) reflects the satisfaction of a user $u$ towards an item $i$ on the aspect $a$. To receive a high aspect rating $r_{u,i,a}$, an item should at least possess the characteristics/attributes in which the user is interested in this aspect. Moreover, the item should satisfy user’s expectations on these attributes in this aspect. In other words, the item’s attributes on this aspect should be of high quality such that the user likes it. Take the “food" aspect as an example, for a user who likes Chinese cuisines, to receive a high rating on the *“food"* aspect from the user, a restaurant should provide Chinese dishes and the dishes should suit the user’s taste. Based on user’s text reviews, we can learn users’ preferences and items’ characteristics on each aspect and measure *how the attributes of an item $i$ on aspect “$a$" suit a user $u$’s requirements on this aspect*, denoted by $s_{u,i,a}$. We compute $s_{u,i,a}$ based on results of the proposed Aspect-aware Topic Model (ATM) (described in Sect. \[sec:matm\]), in which user’s preferences and item’s characteristics on each aspect are modeled as multinomial distributions of latent topics, denoted by $\bm{\theta_{u,a}}$ and $\bm{\psi_{i,a}}$, respectively. $s_{u,i,a} \in [0,1]$ is then computed as : $$\label{eq:jsd}
\vspace{-2pt}
s_{u,i,a} = 1-JSD(\bm{\theta_{u,a}}, \bm{\psi_{i,a}})
\vspace{-1pt}$$ where $JSD(\bm{\theta_{u,a}}, \bm{\psi_{i,a}})$ denotes the Jensen–Shannon divergence [@endres2003new] between $\bm{\theta_{u,a}}$ and $\bm{\psi_{i,a}}$. Notice that a large value of $s_{u,i,a}$ does not mean a high rating $r_{u,i,a}$ - an item providing all the features that a user $u$ requires does not mean that it satisfies $u$’s expectations, since the provided ones could be of low quality. For instance, a restaurant provides all the Chinese dishes the user $u$ likes (i.e., high score $s_{u,i,a}$), but these dishes taste bad from $u$’s opinion (i.e., low rating $r_{u,i,a}$). Therefore, we can expect that for this restaurant: users discuss its Chinese dishes in reviews with negative opinions and thus give low ratings. Instead of analyzing the review sentiments for aspect rating estimation by using external NLP tools (such as [@zhang2014explicit]), we refer to the matrix factorization (MF) [@koren2009matrix] technique.
MF maps users and items into a latent factor space and represents users’ preferences and items’ features by $f$-dim latent factor vectors (i.e., $\bm{p_u} \in \mathbb{R}^{f \times 1}$ and $\bm{q_i} \in \mathbb{R}^{f \times 1}$). The dot product of the user’s and item’s vectors ($\bm{p_u}^T\bm{q_i}$) characterizes the user’s overall interests on the item’s characteristics, and is thus used to predict the rating $r_{u,i}$. To extend MF for aspect rating prediction, we introduce a binary matrix $\bm{W} \in \mathbb{R}^{f \times A}$ to associate the latent factors to different aspects, where $A$ is the number of aspects considered. We call this model aspect-aware latent factor model (ALFM), in which the weight vector $\bm{w_a}$ in the $a$-th column of $\bm{W}$ indicates which factors are related to the aspect $a$. Thus, $\bm{p_{u,a}} = \bm{w_a} \odot \bm{p_u}$ denotes user’s interests in the aspect $a$, where $\odot$ represents element-wise product between vectors. Therefore, $(\bm{p_{u,a}})^T(\bm{q_{i,a}})$ represents the aspect rating of user $u$ to item $i$ on aspect $a$. Finally, we integrate the matching results of aspects (i.e., $s_{u,i,a}$) into ALFM to estimate the aspect ratings: $$\small
\vspace{-2pt}
r_{u,i,a} = s_{u,i,a} \cdot (\bm{w_a} \odot \bm{p_u})^T(\bm{w_a} \odot \bm{q_i})
\vspace{-1pt}$$ As a high aspect rating $r_{u,i,a}$ requires large values of both $s_{u,i,a}$ and $(\bm{w_a} \odot \bm{p_u})^T(\bm{w_a} \odot \bm{q_i})$, it is expected that the results learned from reviews could guide the learning of latent factors.
### Aspect importance estimation.
We rely on user reviews to estimate $\rho_{u,i,a}$, as users often discuss their interest topics of aspects in reviews, such as different *cuisines* in the *food* aspect. In general, the more a user comments on an aspect in reviews, the more important this aspect is (to this user). Thus, we estimate the importance of an aspect according to the possibility of a user writing review comments on this aspect. When writing a review, some users tend to write comments from the aspects according to their own preferences, while others like commenting on the most notable features of the targeted item. Based on this consideration, we introduce (1) $\pi_u$ to denote the probability of user $u$ commenting an item based on his own preference and (2) $\lambda_{u,a}$ ($\sum_{a\in\mathcal{A}}\lambda_{u,a}=1$) to denote the probability of user $u$ commenting on the aspect $a$ based on his own preference. Accordingly, $(1-\pi_{u})$ denotes the probability of the user commenting from the item $i$’s characteristics ($\sum_{a\in\mathcal{A}}\lambda_{i,a}=1$), and $\lambda_{i,a}$ is the probability of user $u$ commenting item $i$ from the item’s characteristics on the aspect $a$. Thus, the probability of a user $u$ commenting an item $i$ on an aspect $a$ (i.e., $\rho_{u,i,a}$) is: $$\label{eq:rho}
\small
\vspace{-2pt}
\rho_{u,i,a} = \pi_{u}\lambda_{u,a} + (1-\pi_{u})\lambda_{i,a}
\vspace{-1pt}$$ $\lambda_{u,a}$, $\lambda_{i,a}$, and $\pi_u$ are estimated by ATM, which simulates the process of a user writing a review, as detailed in the next subsection.
Aspect-aware Topic Model {#sec:matm}
------------------------
Given a corpus $\mathcal{D}$, which contains reviews of users towards items $\{d_{u,i}|d_{u,i} \in \mathcal{D}, u \in \mathcal{U}, i \in \mathcal{I}\}$, we assume that a set of latent topics (i.e., $K$ topics) covers all the topics that users discuss in the reviews. $\bm{\lambda_u}$ is a probability distribution of aspects in user $u$’s preferences, in which each value $\lambda_{u,a}$ denotes the relative importance of an aspect $a$ to the user $u$. Similarly, $\bm{\lambda_i}$ is the probability distribution of aspects in item $i$’s characteristics, in which each value $\lambda_{i,a}$ denotes the importance of an aspect $a$ to the item $i$. As the $K$ latent topics cover all the topics discussed in reviews, an aspect will only relate to some of the latent topics closely. For example, topic “*breakfast*" is closely related to aspect “food", while it is not related to aspects like “*service*" or “*price*". The relation between aspects and topics is also represented by a probabilistic distribution, i.e., $\bm{\theta_{u,a}}$ for users and $\bm{\psi_{i,a}}$ for items. More detailedly, the interests of a user $u$ in a specific aspect $a$ is represented by $\bm{\theta_{u,a}}$, which is a multinomial distribution of the latent topics; the characteristics of an item $i$ in a specific aspect $a$ is represented by $\bm{\psi_{i,a}}$, which is also a multinomial distribution of the same set of latent topics. $\bm{\theta_{u,a}}$ is determined based on all the reviews $\{d_{u,i} | i \in \mathcal{I}\}$ of user $u$ writing for items. $\bm{\psi_{i,a}}$ is learned from all the reviews $\{d_{u,i} | u \in \mathcal{U}\}$ of $i$ written by users. A latent topic is a multinomial distribution of text words in reviews. Based on these assumptions, we propose an aspect-aware topic model ATM to estimate the parameters $\{\bm{\lambda_i}$, $\bm{\lambda_i}$, $\bm{\theta_{u,a}}$, $\bm{\psi_{i,a}}$, $\pi_u\}$ by simulating the generation of the corpus $\mathcal{D}$.
The graphical representation of ATM is shown in Fig. \[fig:matm\]. In the figure, the shaded circles indicate observed variables, while the unshaded ones represent the latent variables. ATM mimics the processing of writing a review sentence by sentence. A sentence usually discusses the same topic $z$, which could be from user’s preferences or from item’s characteristics. To decide the topic $z_s$ for a sentence $s$, our model introduces an indicator variable $y \in \{0,1\}$ based on a Bernoulli distribution, which is parameterized by $\pi_u$. Specifically, when $y=0$, the sentence is generated from user’s preference; otherwise, it is generated according to item $i$’s characteristics. $\pi_u$ is user-dependent, indicating the tendency to comment from $u$’s personal preferences or from the item $i$’s characteristics is determined by $u$’s personality. The generation process of ATM is shown in Algorithm \[alg:geneproc\]. Let $a_s$ denote the aspect assigned to a sentence $s$. If $y=0$, $a_s$ is drawn from $\lambda_u$ and $z_s$ is then generated from $u$’s preferences on aspect $a_s$: $\bm{\theta_{u,a_s}}$; otherwise, if $y=1$, $a_s$ is drawn from $\lambda_i$ and $z_s$ is then generated from $i$’s characteristics on aspect $a_s$: $\bm{\psi_{i,a_s}}$. Then all the words $w$ in sentence $s$ is generated from $z_s$ according to the word distribution: $\bm{\phi_{z_s, w}}$.
\[alg:geneproc\]
In ATM, $\bm{\alpha_u}$, $\bm{\alpha_i}$, $\bm{\gamma_u}$, $\bm{\gamma_i}$, $\bm{\beta}$, and $\eta$ are pre-defined hyper-parameters and set to be symmetric for simplicity. Parameters need to be estimated including $\bm{\lambda_i}$, $\bm{\lambda_i}$, $\bm{\theta_{u,a}}$, $\bm{\psi_{i,a}}$, and $\pi_u$. Different approximate inference methods have been developed for parameter estimation in topic models, such as variation inference [@blei2003latent] and collapsed Gibbs sampling [@griffiths2004]. We apply collapsed Gibbs sampling to infer the parameters, since it has been successfully applied in many large scale applications of topic models [@cheng2016effective; @cheng2017sigir]. Due to the space limitation, we omit the detailed inference steps in this paper.
Model Inference
---------------
With the results of ATM, $\rho_{u,i,a}$ and $s_{u,i,a}$ can be computed using Eq. \[eq:rho\] and \[eq:jsd\], respectively. With the consideration of bias terms (i.e., $b_u, b_i, b_0$) in ALFM, the overall rating can be estimated as[^2], $$\label{eq:re2}
\hat{r}_{u,i} = \sum_{a\in\mathcal{A}}(\rho_{u,i,a}\cdot s_{u,i,a} \cdot (\bm{w_a} \odot \bm{p_u})^T(\bm{w_a} \odot \bm{q_i}) ) + b_u + b_i + b_0$$ where $b_0$ is the average rating, $b_u$ and $b_i$ are user and item biases, respectively. The estimation of parameters is to minimize the rating prediction error in the training dataset. The optimization objective function is $$\label{eq:ojf}
\begin{split}
\vspace{-2pt}
\underset{p*,q*}{\text{min}} \frac{1}{2}\sum_{u,i} (r_{u,i} & -\hat{r}_{u,i})^2 + \frac{\mu_u}{2} ||\bm{p_u}||_2^2 + \frac{\mu_i}{2} ||\bm{q_i}||_2^2 \\
& + \mu_w \sum_a||\bm{w_a}||_1 + \frac{\mu_b}{2} (||b_u||_2^2 + ||b_i||_2^2);
\vspace{-2pt}
\end{split}$$ where $||\cdot||_2$ denotes the $\ell_2$ norm for preventing model overfitting, and $||\cdot||_1$ denotes the $\ell_1$ norm. $\mu_u, \mu_i, \mu_w$, and $\mu_b$ are regularization parameters, which are tunable hyper-parameters. In practice, we relax the binary requirement of $\bm{w_a}$ by using $\ell_l$ norm. It is well known that $\ell_l$ regularization yields sparse solution of the weights [@mairal2010online]. The $\ell_2$ regularization of $\bm{p_u}$ and $\bm{q_i}$ prevents them to have arbitrarily large values, which would lead to arbitrarily small values of $\bm{w_a}$.
**Optimization.** We use the stochastic gradient descent (SGD) algorithm to learn the parameters by optimizing the objective function in Eq. \[eq:ojf\]. In each step of SGD, the localized optimization is performed on a rating $r_{u,i}$. Let $L$ denote the loss, and the gradients of parameters are given as follows: $$\begin{aligned}
\label{eq:user}
\small
\vspace{-2pt}
\frac{\partial L}{\partial p_u}&=\sum_{i=1}^{N}(\sum_{a}\rho_{u,i,a}s_{u,i,a}w_a^2)(\hat{r}_{u,i}-r_{u,i})q_i + \mu_u p_u \\
\frac{\partial L}{\partial q_i}&=\sum_{u=1}^{M}(\sum_{a}\rho_{u,i,a}s_{u,i,a}w_a^2)(\hat{r}_{u,i}-r_{u,i})p_u + \mu_i q_i \\
\frac{\partial L}{\partial w_a}&=\sum_{u=1}^{M}\sum_{i=1}^{N}\rho_{u,i,a}s_{u,i,a}(\hat{r}_{u,i}-r_{u,i})p_uq_iw_a + \frac{\mu_w w_a} {\sqrt{(w_a^2+\epsilon)}}
\vspace{-2pt}\end{aligned}$$ Here, we omit the gradients of $b_u$ and $b_i$, as they are the same as in the standard biased MF [@koren2009matrix]. $M$ and $N$ are the total number of users and items in the dataset. Notice that in the deriving of the gradient for $w_a$, we use $\sqrt{w_a^2+\epsilon}$ in place of $||w_a||_1$, because $\ell_1$ norm is not differentiable at 0. $\epsilon$ can be regarded as a “smoothing parameter" and is set to $10^{-6}$ in our implementation.
Experimental Study {#sec:expconfig}
==================
To validate the assumptions when designing the model and evaluate our proposed model, we conducted comprehensive experimental studies to answer the following questions:
- **RQ1:** How do the important parameters (e.g., the number of latent topics and latent factors) affect the performance of our model? More importantly, is the setting $f=K$ optimal, which is a default assumption for many previous models? (Sect. \[sec:modelanalysis\])
- **RQ2:** Can our ALFM model outperform the state-of-the-art recommendation methods, which consider both ratings and reviews, on rating prediction? (Sect. \[sec:comp\])
- **RQ3:** Compared to other methods which also use textual reviews and ratings, how does our ALFM model perform on the cold-start setting when users have only few ratings? (Sect. \[sec:coldstart\])
- **RQ4:** Can our model explicitly interpret the reasons for a high or low rating? (Sect. \[sec:interpret\])
Datasets \#users \#items \#ratings Sparsity
--------------------- --------- --------- ----------- ----------
Instant Video 4,902 1,683 36,486 0.9956
Automotive 2,788 1,835 20,218 0.9960
Baby 17,177 7,047 158,311 0.9987
Beauty 19,766 12,100 196,325 0.9992
Cell Phones 24,650 10,420 189,255 0.9993
Clothing 34,447 23,026 277,324 0.9997
Digital Music 5,426 3,568 64,475 0.9967
Grocery 13,979 8,711 149,434 0.9988
Health 34,850 18,533 342,262 0.9995
Home & Kitchen 58,901 28,231 544,239 0.9997
Musical Instruments 1,397 900 10,216 0.9919
Office Products 4,798 2,419 52,673 0.9955
Patio 1,672 962 13,077 0.9919
Pet Supplies 18,070 8,508 155,692 0.9990
Sports & Outdoors 31,176 18,355 293,306 0.9995
Tools & Home 15,438 10,214 133,414 0.9992
Toys & Games 17,692 11,924 166,180 0.9992
Video Games 22,348 10,672 228,164 0.9990
Yelp 2017 169,257 63,300 1,659,678 0.9998
: Statistics of the evaluation datasets
\[tab:dataset\]
Dataset Description {#sec:dataset}
-------------------
We conducted experiments on two publicly accessible datasets that provide user review and rating information. The first dataset is Amazon Product Review dataset collected by [@mcauley2013hidden][^3], which contains product reviews and metadata from Amazon. This dataset has been widely used for rating prediction with reviews and ratings in previous studies [@mcauley2013hidden; @ling2014ratings; @tan2016rating; @catherine2017transnets]. The dataset is organized into 24 product categories. In this paper, we used 18 categories (See Table \[tab:dataset\]) and focus on the 5-core version, with at least 5 reviews for each user or item. The other dataset is from Yelp Dataset Challenge 2017[^4], which includes reviews of local business in 12 metropolitan areas across 4 countries. For the Yelp 2017 dataset, we also processed it to keep users and items with at least 5 reviews. From each review in these datasets, we extract the corresponding “userID", “itemID", a rating score (from 1 to 5 rating stars), and a textual review for experiments. Notice that for all the datasets, we checked and removed the duplicates, and then filtered again to keep them as 5-core. Besides, we removed the infrequent terms in the reviews for each dataset.[^5] Some statistics of the datasets are shown in Table \[tab:dataset\].
Experimental Settings
---------------------
For each dataset, we randomly split it into training, validation, and testing set with ratio 80:10:10 for each user as in [@mcauley2013hidden; @ling2014ratings; @catherine2017transnets]. Because we take the 5-core dataset where each user has at least 5 interactions, we have at least 3 interactions per user for training, and at least 1 interaction per user for validation and testing. Note that we only used the review information in the training set, because the reviews in the validation or testing set are unavailable during the prediction process in real scenarios. The number of aspect is set to 5 in experiments.[^6]
**Baselines:** We compare the proposed **ALFM** model with the following baselines. It is worth noting that these methods are tuned on the validation dataset to obtain their optimal hyper-parameter settings for fair comparisons.
- **BMF [@koren2009matrix].** It is a standard MF method with the consideration of bias terms (i.e., user biases and item biases). This method only leverages ratings when modeling users’ and items’ latent factors. It is typically a strong baseline model in collaborative filtering [@koren2009matrix; @ling2014ratings].
- **HFT [@mcauley2013hidden].** It models ratings with MF and review text with latent topic model (e.g., LDA [@blei2003latent]). We use it as a representative of the methods which use an exponential transformation function to link the latent topics with latent factors, such as TopicMF [@bao2014topicmf]. The topic distribution can be modeled on either users or items. We use the topic distribution based on items, since it achieves better results. Note that in experiments, we add bias terms into HFT, which can achieve better performance.
- **CTR [@wang2011collaborative].** This method also utilizes both review and rating information. It uses a topic model to learn the topic distribution of items, which is then used as the latent factors of items in MF with an addition of a latent variable.
- **RMR [@ling2014ratings].** This method also uses both ratings and reviews. Different from HFT and CTR, which use MF to model rating, it uses a mixture of Gaussian to model the ratings.
- **RBLT [@tan2016rating].** This method is the most recent method, which also uses MF to model ratings and LDA to model review texts. Instead of using an exponential transformation function to link the latent topics and latent factors (as in HFT [@mcauley2013hidden]), this method linearly combines the latent factors and latent topics to represent users and items, with the assumption that the dimensions of topics and latent factors are equal and in the same latent space. The same strategy is also adopted by ITLFM [@zhang2016integrating]. Here, we use RBLT as a representative method for this strategy.
- **TransNet [@catherine2017transnets].** This method adopts neural network frameworks for rating prediction. In this model, the reviews of users and items are used as input to learn the latent representations of users and items. More descriptions about this method could be found in Section \[sec:relwork\]. We used the codes published by the authors in our experiments and tuned the parameters as described in [@catherine2017transnets].
The standard root-mean-square error (**RMSE**) is adopted in evaluation. A smaller RMSE value indicates better performance.
[|c|C[0.6cm]{}C[0.6cm]{}C[0.6cm]{}C[0.6cm]{}C[0.6cm]{}C[0.9cm]{}C[0.75cm]{}|C[0.87cm]{}C[0.89cm]{}C[0.87cm]{}C[0.89cm]{}C[0.95cm]{}C[0.85cm]{}|]{} & BMF & HFT & CTR & RMR & RBLT & TransNet & ALFM &\
& (a) & (b) & (c) & (d) & (e) & (f) & (g) & g vs. a & g vs. b & g vs. c & g vs. d & g vs. e & g vs. f\
Instant Video & 1.162 & 0.999 & 1.014 & 1.039 & 0.978 & 0.996 & **0.967** & 16.79 & 3.19 & 4.63\* & 6.94\* & 1.12\*\* & 2.88\
Automotive & 1.032 & 0.968 & 1.016 & 0.997 & 0.924 & 0.918 & **0.885** & 14.26\* & 8.58\*\* & 12.86\* & 11.19\* & 4.24\*\* & 3.56\*\
Baby & 1.359 & 1.112 & 1.144 & 1.178 & 1.122 & 1.110 & **1.076** & 20.83\*\* & 3.24 & 5.98\* & 8.66\*\* & 4.11 & 3.05\*\
Beauty & 1.342 & 1.132 & 1.171 & 1.190 & 1.117 & 1.123 & **1.082** & 19.39\*\* & 4.47 & 7.65\*\* & 9.12\*\* & 3.18\*\* & 3.65\*\*\
Phones & 1.432 & 1.216 & 1.271 & 1.289 & 1.220 & 1.207 & **1.167** & 18.47\*\* & 3.98\* & 8.18\* & 9.4\*\* & 4.33 & 3.27\*\*\
Clothing & 1.073 & 1.103 & 1.142 & 1.145 & 1.073 & 1.064 & **1.032** & 3.8\*\* & 6.47\*\* & 9.65 & 9.9\*\* & 3.86\*\* & 2.96\*\
Digital Music & 1.093 & **0.918** & 0.921 & 0.960 & **0.918** & 1.061 & 0.920 & 15.82 & -0.15 & 0.13\* & 4.49\*\* & -0.15\*\* & 4.13\*\*\
Grocery & 1.192 & 1.016 & 1.045 & 1.061 & 1.012 & 1.022 & **0.982** & 17.66\*\* & 3.36\*\* & 6.07 & 7.46\*\* & 3.01\*\* & 3.94\*\
Health & 1.263 & 1.073 & 1.105 & 1.135 & 1.070 & 1.114 & **1.042** & 17.48\* & 2.83 & 5.65\* & 8.20 & 2.56\*\* & 6.46\*\*\
Home & Kitchen & 1.297 & 1.083 & 1.123 & 1.149 & 1.086 & 1.123 & **1.049** & 19.16\*\* & 3.15\*\* & 6.62 & 8.7\*\* & 3.41\*\* & 6.61\*\*\
Musical Instruments & 1.004 & 0.972 & 0.979 & 0.983 & 0.946 & 0.901 & **0.893** & 11.08 & 8.17\*\* & 8.83\*\* & 9.2\*\* & 5.61 & 0.95\
Office Products & 1.025 & 0.879 & 0.898 & 0.934 & 0.872 & 0.898 & **0.848** & 17.29\*\* & 3.55\*\* & 5.61\* & 9.26\*\* & 2.77\*\* & 5.67\*\*\
Patio & 1.180 & 1.041 & 1.062 & 1.077 & 1.032 & 1.046 & **1.001** & 15.19\*\* & 3.84\* & 5.7\* & 7.07\* & 2.96 & 4.33\*\*\
Pet Supplies & 1.367 & 1.137 & 1.177 & 1.200 & 1.139 & 1.149 & **1.099** & 19.64\*\* & 3.41\* & 6.67\* & 8.41 & 3.54\*\* & 4.38\*\*\
Sports & Outdoors & 1.130 & 0.970 & 0.998 & 1.019 & 0.964 & 0.990 & **0.933** & 17.42\*\* & 3.8\* & 6.47 & 8.4\* & 3.2\*\* & 5.77\*\*\
Tools & Home & 1.168 & 1.013 & 1.047 & 1.090 & 1.011 & 1.041 & **0.974** & 16.63\*\* & 3.90 & 6.98 & 10.68\*\* & 3.7\*\* & 6.51\*\*\
Toys & Games & 1.072 & 0.926 & 0.948 & 0.974 & 0.923 & 0.951 & **0.902** & 15.81\*\* & 2.59\* & 4.82\*\* & 7.39\*\* & 2.3\*\* & 5.11\*\
Video Games & 1.321 & 1.096 & 1.115 & 1.150 & 1.094 & 1.123 & **1.070** & 19.02\* & 2.43 & 4.03\*\* & 6.97\* & 2.24\*\* & 4.77\*\
Yelp 2017 & 1.415 & 1.174 & 1.233 & 1.266 & 1.202 & 1.190 & **1.155** & 18.35\* & 1.60\*\* & 6.33\*\* & 8.74\* & 3.88\*\* & 2.92\*\
Average & 1.207 & 1.044 & 1.074 & 1.097 & 1.037 & 1.049 & **1.004** & 14.56\*\* & 2.84\* & 7.16\*\* & 8.31\* & 3.37\*\* & 4.26\*\*\
The improvements with \* are significant with $p-value < 0.05$, and the improvements with \*\* are significant with $p-value < 0.01$ with a two-tailed paired t-test.
\[tab:comp\]
Effect of Important Parameters (RQ1) {#sec:modelanalysis}
------------------------------------
In this subsection, we analyze the influence of *the number of latent factors* and *the number of latent topics* on the final performance of ALFM. As we know, in MF, more latent factors will lead to better performance unless overfitting occurs [@he2016fast; @koren2009matrix]; while the optimal number of latent topics in topic models (e.g., LDA) is dependent on the datasets [@blei2012probabilistic; @arun2010finding]. Accordingly, the optimal number of latent topics in topic model and the optimal number of latent factors in MF should be tuned separately. However, in the previous latent factor models (e.g., HFT, TopicMF [@bao2014topicmf], RMR, CTR, and RBLT), the number of factors (i.e., \#factors) and the number of topics (i.e., \#topics) are assumed to be the same, and thus cannot be optimized separately. Since our model does not have such constraint, we studied the effects of \#factors and \#topics individually. Fig. \[fig:effects\] show the performance variations with the change of \#factors and \#topics by setting the other one to 5. We only visualize the performance variations of three datasets, due to the space limitation and the similar performance variation behaviors of other datasets. From the figure, we can see that with the increase of \#factors, RMSE consistently decreases although the degree of decline is small. Notice that in our model, the rating prediction still relies on MF technique (Eq. \[eq:re2\]). Therefore, the increase of \#factors could lead to better representation capability and thus more accurate prediction. In contrast, the optimal number of latent topics is different from dataset to dataset.
To better visualize the impact of \#factors and \#topics, we also present 3D figures by varying the number of factors and topics in $\{5, 10, 15, 20, 25\}$, as shown in Fig. \[fig:factorvstopic\]. In this figure, we use the performance of three datasets as illustration. From the figure, we can see that the optimal numbers of topics and latent factors are varied across different datasets. In general, more latent factors usually lead to better performance, while the optimal number of latent topics is dependent on the reviews of different datasets. This also reveals that setting \#factors and \#topics to be the same may not be optimal.
Model Comparison (RQ2) {#sec:comp}
----------------------
We show the performance comparisons of our ALFM with all the baseline methods in Table \[tab:comp\], where the best prediction result on each dataset is in bold. For fair comparison, we set the number of latent factors ($f$) and the number of latent topics ($K$) to be the same as $f=K=5$. Notice that our model could obtain better performance when setting $f$ and $K$ differently. Still, ALFM achieves the best results on 18 out of the 19 datasets. Compared with BMF, which only uses ratings, we achieve much better prediction performance (16.49% relative improvement on average). More importantly, our model outperforms CTR and RMR with large margins - 6.28% and 8.18% relative improvements on average, respectively. Compared to the recently proposed RBLT and TransNet, ALFM can still achieve 3.37% and 4.26% relative improvement on average respectively with significance testing. It is worth mentioning that HFT achieves better performance than RMR and comparable performance with recent RBLT, because we added bias terms to the original HFT in [@mcauley2013hidden]. TransNet applies neural networks, which has exhibited strong capabilities on representation learning, in reviews to learn users’ preferences and items’ characteristics for rating prediction. However, it may suffer from (1) noisy information in reviews, which would deteriorate the performance; and (2) errors introduced when generating fake reviews for rating prediction, which will also cause bias in the final performance. Compared to those baselines, the advantage of ALFM is that it models users’ preferences on different aspects; and more importantly, it captures a user’s specific attention on each aspect of a targeted item. The substantial improvement of ALFM over those baselines demonstrates the benefits of modeling users’ specific preferences on each aspect of different items.
Cold-Start Setting (RQ3) {#sec:coldstart}
------------------------
As shown in Table \[tab:dataset\], the datasets are usually very sparse in practical systems. It is inherently difficult to provide satisfactory recommendation based on limited ratings. In the matrix factorization model, given a few ratings, the penalty function tends to push $q_u$ and $p_i$ towards zero. As a result, such users and items are modeled only with the bias terms [@ling2014ratings]. Therefore, matrix factorization easily suffers from the cold-start problem. By integrating reviews in users’ and items’ latent factor learning, our model could alleviate the problem of cold-start to a great extent, since reviews contain rich information about user preferences and item features.
To demonstrate the capability of our model in dealing with users with very limited ratings, we randomly split the datasets into training, validation, and testing sets in ratio 80:10:10 based on the number of ratings in each set. In this setting, it is not guaranteed that a user has at least 3 ratings in the training set. It is possible that a user has no rating in the training set. For the users without any ratings in the training set, we also removed them in the testing set. Then we evaluate the performance of users who have the number of ratings from 1 to 10 in the training set. In Fig. \[fig:coldstart\], we show the **Gain in RMSE** ($y$-axis) grouped by the number of ratings ($x$-axis) of users in the training set. The value of **Gain in RMSE** is equal to the average RMSE of baselines *minus* that of our model (e.g., “BMF-TALFM"). A positive value indicates that our model achieves better prediction. As we can see, our ALFM model substantially improves the prediction accuracy compared with the BMF model. More importantly, our model also outperforms all the other baselines which also utilize reviews. This demonstrates that our model is more effective in exploiting reviews and ratings, because it learns user’s preferences and item’s features in different aspects and is capable of estimating the aspect weights based on the targeted user’s preferences and targeted item’s features.
**Value** **Comfort** **Accessories** **Shoes** **Clothing**
------------- ------------- ----------------- ----------- --------------
price size ring socks shirt
color fit pretty foot back
quality wear dress boots bra
worth comfortable time comfort top
cute bra beautiful sandals feel
comfortable small gift walk soft
fits color earrings toe black
ring fits compliments pairs jeans
dress perfect chain hold pants
shirt material jewelry strap tight
material long shoes pockets material
: Top ten words of each aspect for a user (index 1511) from *Clothing*. Each column is corresponding to an aspect attached with an “interpretation” label.
\[tab:aspects\]
Aspects Value Comfort Accessories Shoes Clothing
---------------- ------- --------- ------------- ------- ----------
Importance (1) 0.621 0.042 0.241 0.001 0.095
Matching (1) 0.982 0.596 0.660 0.759 0.638
Polarity (1) **+** **-** **+** **-** **+**
Importance (2) 0.621 0.042 0.241 0.001 0.094
Matching (2) 0.920 0.303 0.362 1.000 0.638
Polarity (2) **-** **-** **-** **-** **+**
: Interpretation for why the “user 1511" rated “item 1" and “item 2" with 5 and 2, respectively, from *Clothing*.
\[tab:explaination\]
Interpretability (RQ4) {#sec:interpret}
----------------------
In our ALFM model, a user’s preference on an item is decomposed into user’s preference on different aspects and the importance of those aspects. An aspect is represented as a distribution of latent topics discovered based on reviews. A user’s attitude/sentiment on an aspect of the targeted item is decided by the latent factors (learned from ratings) associating with the aspect. Based on the topic distribution of an aspect ($\bm{\theta_{u,a_s}}$) and the word distribution of topics ($\bm{\phi_{w}}$), we can semantically represent an aspect by the top words in this aspect. Specifically, the probability of a word $w$ in an aspect $a_s$ of a user $u$ can be computed as $\sum_{k=1}^K\theta_{u,a_s,k}\phi_{k,w}$. The top 10 aspect words (\#aspect $= 5$) of “user 1511" from *Clothing* dataset discovered by our model are shown in Table \[tab:aspects\]. Notice that in order to obtain a better visualization of each aspect, we removed the “background” words that belong to more than 3 aspects. As shown in Table \[tab:aspects\], the five aspects can be semantically interpreted to “value"[^7], “comfort", “accessories", “shoes", and “clothing". Next, we illustrate the interpretability of our ALFM model on high or low ratings by examples from the same dataset. Table \[tab:explaination\] shows the aspect importance (i.e., $\rho_{u,i,a}$ in Eq. \[eq:rho\]) of the “user 1511" , the aspect matching scores (i.e., $s_{u,i,a}$ in Eq. \[eq:jsd\]) as well as sentiment polarity (obtained by Eq. \[eq:re2\]) on the five aspects with respect to “item 1" and “item 2" in *Clothing* dataset. From the results, we can see that “user 1511" pays more attention to “Value" and “Accessories" aspects. On the “Value" aspect, both “item 1" and “item 2" highly match her preference, however, she has a positive sentiment on “item 1" while a negative sentiment on “item 2".[^8] For the “Accessories" aspect, “item 1" has a higher matching score than “item 2"; and more importantly, the sentiment is positive on “item 1" while negative on “item 2". As a result, “user 1511" rated “item 1" with 5 while rated “item 2" with 2. From the examples, we can see that our model could provide explanations for the recommendations in depth with *aspect semantics*, *aspect matching score*, as well as *aspect ratings* (which shows sentiment polarity).
Conclusions {#sec:concl}
===========
In this paper, we proposed an aspect-aware latent factor model for rating prediction by effectively combining reviews and ratings. Our model correlates the latent topics learned from review text and the latent factors learned from ratings based on the same set of aspects, which are discovered from textual reviews. Accordingly, our model does not have the constraint of one-to-one mapping between latent factors and latent topics as previous models (e.g., HFT, RMR, RBLT, etc.), and thus could achieve better user preference and item feature modeling. Besides, our model is able to estimate aspect ratings and assign weights to different aspects. The aspect weight is dependent on each user-item pair, since it is estimated based on user’s personal preferences on the corresponding aspect towards an item. Experimental results on 19 real-world datasets show that our model greatly improves the rating prediction accuracy compared to the state-of-the-art methods, especially for users who have few ratings. With the extracted aspects from textual reviews, estimated aspect weights, and aspect ratings, our model could provide interpretation for recommendation results in great detail.
This research is supported by the National Research Foundation, Prime Minister’s Office, Singapore under its International Research Centre in Singapore Funding Initiative. The authors would like to thank Rose Catherine Kanjirathinkal (from CMU)’s great help on fine-tuning the results of TransNet on all datasets.
[^1]: In the paper, unless otherwise specified, notations in bold style denote matrices or vectors, and the ones in normal style denote scalars.
[^2]: In our experiments, we tried to normalize $\rho_{u,i,a}$ or $\rho_{u,i,a} \cdot s_{u,i,a}$ in Eq. \[eq:re2\], but no improvement has been observed.
[^3]: http://jmcauley.ucsd.edu/data/amazon/
[^4]: http://www.yelp.com/dataset\_challenge/
[^5]: The thresholds of infrequent terms varied across different datasets. For example, for the “Yelp 2017" dataset, which is relatively large, a term that appears less than 10 times in reviews is defined as an infrequent term; and the thresholds are smaller for relatively small datasets (e.g., the threshold is 5 for the “Music Instruments" dataset).)
[^6]: We tuned the number of aspects from 1 to 8 for all the datasets, and found that the performance does not change much unless setting the aspect number to 1 or 2.
[^7]: “Value" means value for money
[^8]: As a reminder, the aspect matching is based on the reviews. It is possible that both item 1 and item 2 contains comments on aspect “value". However, “item 1" has a high value while “item 2" has a low value.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'M. Wernli'
- 'L. Wiesenfeld'
- 'A. Faure'
- 'P. Valiron'
bibliography:
- 'cyano\_v3.bib'
date: 'Received / Accepted '
title: 'Rotational Excitation of HC$_3$N by H$_2$ and He at low temperatures'
---
Introduction
============
Cyanopolyyne molecules, with general formula HC$_{2n+1}$N, $n\ge
1$, have been detected in a great variety of astronomical environments and belong to the most abundant species in cold and dense interstellar clouds [@bell97]. One of these, HC$_{11}$N, is currently the largest unambiguously detected interstellar molecule [@bell85]. The simplest one, [$\mathrm{HC_3N}$]{}(cyanoacetylene), is the most abundant of the family. In addition to interstellar clouds, [$\mathrm{HC_3N}$]{}has been observed in circumstellar envelopes [@pepe04], in Saturn satellite Titan [@kunde81], in comets [@bockelee00] and in extragalactic sources [@mauersberger90]. Furthermore, [$\mathrm{HC_3N}$]{}has been detected both in the ground level and in excited vibrational levels, thanks to the presence of low-lying bending modes [e.g. @wyrowski03]. Owing to a low rotational constant and a large dipole moment, cyanoacetylene lines are thus observable over a wide range of excitation energies and [$\mathrm{HC_3N}$]{}is therefore considered as a very good probe of physical conditions in many environments.
Radiative transfer models for the interpretation of observed [$\mathrm{HC_3N}$]{}spectra require the knowledge of collisional excitation rates participating to line formation. To the best of our knowledge, the only available collisional rates are those of @green78 for the rotational excitation of HC$_3$N by He below 100 K. In cold and dense clouds, however, the most abundant colliding partner is H$_2$. In such environments, para-[$\rm H_2$]{}is only populated in the $J=0$ level and may be treated as a spherical body. @green78 and @dickinson82 postulated that the collisional cross-sections with para-[$\rm H_2$]{}$(J=0)$ are similar to those with He (assuming thus an identical interaction and insensitivity of the scattering to the reduced mass). As a result, rates for excitation by para-[$\rm H_2$]{}were estimated by scaling the rates for excitation by He while rates involving ortho-[$\rm H_2$]{}were not considered.
In the present study, we have computed new rate coefficients for rotational excitation of [$\mathrm{HC_3N}$]{}by He, para-[$\rm H_2$]{}($J=0$) and ortho-[$\rm H_2$]{}($J=1$), in the temperature range 5$-$20 K for He and 5$-$100 K for H$_2$. A comparison between the different partners is presented and the collisional selection rules are investigated in detail. The next section describes details of the PES calculations. The cross-section and rate calculations are presented in Section \[sec:cross\]. A discussion and a first application of these rates is given in Section \[sec:disc\]. Conclusions are drawn in Section 5. The following units are used throughout except otherwise stated: bond lengths and distances in Bohr; angles in degrees; energies in cm$^{-1}$; and cross-sections in $\AA^2$.
Potential energy surfaces {#sec:pot}
=========================
Two accurate interatomic potential energy surfaces (PES) have recently been calculated in our group, for the interaction of [$\mathrm{HC_3N}$]{}with He and H$_2$. Both surfaces involved the same geometrical setup and similar *ab initio* accuracy. An outline of those PES is given below, while a detailed presentation will be published in a forthcoming article.
In the present work, we focus on low-temperature collision rates, well below the threshold for the excitation of the lower bending mode $\nu_7$ at 223 cm$^{-1}$. The collision partners may thus safely be approximated to be rigid, in order to keep the number of degrees of freedom as small as possible. For small van der Waals complexes, previous studies have suggested [@jeziorska00; @jankowski05] that properly averaged molecular geometries provide a better description of experimental data than equilibrium geometries ($r_e$ geometries). For the $\rm H_2O$ – [$\rm H_2$]{}system, geometries averaged over ground-state vibrational wave-functions ($r_0$ geometry) were shown to provide an optimal approximation of the effective interaction [@faure05; @wernlithese].
Accordingly, we used the [$\rm H_2$]{}bond separation $r_{\rm HH}= 1.44876$ Bohr obtained by averaging over the ground-state vibrational wave-function, similarly to previous calculations [@hodges04; @faure05; @wernli06]. For [$\mathrm{HC_3N}$]{}, as vibrational wave-functions are not readily available from the literature, we resorted to experimental geometries deduced from the rotational spectrum of [$\mathrm{HC_3N}$]{}and its isotopologues (@thor00; see also Table 5.8 in @gordy). The resulting bond separations are the following: $r_{\mathrm{HC_1}}= 1.998385$; $r_{\mathrm{C_1C_2}}=2.276364$;$r_{\mathrm{C_2C_3}}= 2.606688$; $r_{\mathrm{C_3N}}= 2.189625$, and should be close to vibrationally averaged values.
For the [$\mathrm{HC_3N}$]{}– He collision, only two coordinates are needed to fully determine the overall geometry. Let $\vec{R}$ be the vector between the center of mass of [$\mathrm{HC_3N}$]{}and He. The two coordinates are the distance $R=|\vec{R}|$ and the angle $\theta_1$ between the [$\mathrm{HC_3N}$]{}rod and the vector ***R***. In our conventions, $\theta_1 = 0$ corresponds to an approach towards the H end of the [$\mathrm{HC_3N}$]{}rod. For the collision with H$_2$, two more angles have to be added, $\theta_2$ and $\phi$, that respectively orient the [$\rm H_2$]{}molecule in the rod-***R*** plane and out of the plane. The [$\mathrm{HC_3N}$]{}– He PES has thus two degrees of freedom, the [$\mathrm{HC_3N}$]{}– [$\rm H_2$]{}four degrees of freedom.
As we aim to solve close coupling equations for the scattering, we need ultimately to expand the PES function $V$ over a suitable angular expansion for any intermolecular distance $R$. In the simpler case of the [$\mathrm{HC_3N}$]{}– He system, this expansion is in the form: $$\label{eq:pot}
V_{}(R,\theta_1) = \sum_{l_1}
v_{l_1}(R)\,P_{l_1}(\cos\theta_1)\quad ,$$ where $P_{l_1}(\cos\theta_1)$ is a Legendre polynomial and $v_{l_1}(R)$ are the radial coefficients.
For the [$\mathrm{HC_3N}$]{}– [$\rm H_2$]{}system, the expansion becomes: $$\label{eq:pot2}
V(R,\theta_1, \theta_2, \phi) = \sum_{l_1 l_2 l} v_{l_1 l_2 l}(R)
s_{l_1 l_2 l}(\theta_1, \theta_2, \phi),$$ where the basis functions $s_{l_1 l_2 l}$ are products of spherical harmonics and are expressed in Eq. (A9) of @green75. Two new indices $l_2$ and $l$ are thus needed, associated respectively with the rotational angular momentum of [$\rm H_2$]{}and the total orbital angular momentum, see also eq. (A2) and (A5) of @green75.
Because the Legendre polynomials form a complete set, such expansions should always be possible. However, @chapman77 failed to converge above expansion (\[eq:pot\]) due to the steric hindrance of He by the impenetrable [$\mathrm{HC_3N}$]{}rod, and @green78 abandoned quantum calculations, resorting to quasi classical trajectories (QCT) studies. Similar difficulties arise for the interaction with H$_2$. Actually, as can be seen on figure \[fig:PES\] for small $R$ values, the interaction is moderate or possibly weakly attractive for $\theta_1 \sim
90^{\circ}$ and is extremely repulsive or undefined for $\theta_1
\sim 0, 180^{\circ}$, leading to singularities in the angular expansion and severe Gibbs oscillations in the numerical fit of the PES over Legendre expansions.
Accordingly, we resorted to a cautious sampling strategy for the PES, building a spline interpolation in a first step, and postponing the troublesome angular Legendre expansion to a second step. All details will be published elsewhere. Let us summarize this first step for He, then for H$_2$.
For the [$\mathrm{HC_3N}$]{}– He PES, we selected an irregular grid in the $\left\{R,\theta_1\right\}$ coordinates. The first order derivatives of the angular spline were forced to zero for $\theta_1=0,180^{\circ}$ in order to comply with the PES symmetries. For each distance, angles were added until a smooth convergence of the angular spline fit was achieved, resulting to typical angular steps between 2 and 15$^{\circ}$. Then, distances were added until a smooth bicubic spline fit was obtained, amounting to 38 distances in the range 2.75 – 25 Bohr and a total of 644 geometries. The resulting PES is perfectly suited to run quasi classical trajectories.
We used a similar strategy to describe the interaction with H$_2$, while minimizing the number of calculations. We selected a few $\left\{\theta_2,\phi\right\}$ orientation sets, bearing in mind that the dependence of the final PES with the orientation of [$\rm H_2$]{}is weak. In terms of spherical harmonics, the PES depends only on $Y_{l_2m_2}(\theta_2,\phi)$, with $l_2=0,2,4,\ldots$ and $m_2=0,1,2,\ldots$, $|m_2|\leq l_2$. Terms in $Y_{l_2m_2}$ and $Y_{l_2 -m_2}$ are equal by symmetry. Previous studies [@faure05a; @wernli06] have shown that terms with $l_2> 2$ are small, and we consequently truncated the $Y_{l_2m_2}$ series to $l_2\leq 2$. Hence, only four basis functions remain for the orientation of [$\rm H_2$]{}: $Y_{00},Y_{20},Y_{21}$ and $Y_{22}$.
Under this assumption, the whole [$\mathrm{HC_3N}$]{}– [$\rm H_2$]{}surface can be obtained knowing its value for four sets of $\left\{\theta_2, \phi\right\}$ angles at each value of $R$. We selected actually five sets, having thus an over-determined system allowing for the monitoring of the accuracy of the $l_2$ truncation. Consequently, we determined five independent PES, each being constructed similarly to the [$\mathrm{HC_3N}$]{}– He one as a bicubic spline fit over an irregular grid in $\left\{R,
\theta_1\right\}$ coordinates. The angular mesh is slightly denser than for the [$\mathrm{HC_3N}$]{}– He PES for small $R$ distances to account for more severe steric hindrance effects involving [$\rm H_2$]{}. In total, we computed 3420 $\left\{R, \theta_1, \theta_2, \phi\right\}$ geometries. Finally, the [$\mathrm{HC_3N}$]{}– H$_2$ interaction can be readily reconstructed from these five PES by expressing its analytical dependence over $\left\{\theta_2, \phi\right\}$ [@wernlithese].
For each value of the intermolecular geometry $\left\{R,\theta_1\right\}$ or $\left\{R, \theta_1, \theta_2,
\phi\right\}$, the intermolecular potential energy is calculated at the conventional CCSD(T) level of theory, including the usual counterpoise correction of the Basis Set Superposition Error [@jansen69; @boys70]. We used augmented correlation-consistent atomic sets of triple zeta quality (Dunning’s aug-cc-pVTZ) to describe the [$\mathrm{HC_3N}$]{}rod. In order to avoid any possible steric hindrance problems at the basis set level, we did not use bond functions and instead chose larger Dunning’s aug-cc-pV5Z and aug-cc-pVQZ basis set to better describe the polarizable (He, H$_2$) targets, respectively. All calculations employed the direct parallel code <span style="font-variant:small-caps;">Dirccr12</span> [@dirccr12].
Comparison of the [$\mathrm{HC_3N}$]{}– He PES with existing surfaces [@akinojo03; @topic05] showed an excellent agreement. The [$\mathrm{HC_3N}$]{}– para-[$\rm H_2$]{}($J=0$) interaction (obtained by averaging the [$\mathrm{HC_3N}$]{}– H$_2$ PES over $\theta_2$ and $\phi$) is qualitatively similar to the [$\mathrm{HC_3N}$]{}– He PES with a deeper minimum (see values at the end of present Section). As illustrated in Figure \[fig:PES\], these PES are largely dominated by the rod-like shape of [$\mathrm{HC_3N}$]{}, implying a prolate ellipsoid symmetry of the equipotentials.
In a second step, let us consider how to circumvent the difficulty of the angular expansion of the above PES, in order to obtain reliable expansions for He and H$_2$ (eqs \[eq:pot\] and \[eq:pot2\]).
Using the angular spline representation, we first expressed each PES over a fine $\theta_1$ mesh suitable for a subsequent high $l_1$ expansion. As expected from the work of @chapman77, high $l_1$ expansions (\[eq:pot\]) resulted in severe Gibbs oscillations for $R$ in the range 5–7 Bohr, spoiling completely the description of the low energy features of the PES. Then, having in mind low energy scattering applications, we regularized the PES by introducing a scaling function $S_f$. We replaced $V(R,\theta_1,...)$ by $S_f(V(R,\theta_1,...))$, where $S_f(V)$ returns $V$ when $V$ is lower than a prescribed threshold, and then smoothly saturates to a limiting value when $V$ grows up into the repulsive walls. Consequently, the regularized PES retains only the low energy content of the original PES, unmodified up to the range of the threshold energy; it should not be used for higher collisional energies. However, in contrast to the original PES, it can be easily expanded over Legendre functions to an excellent accuracy and is thus suitable for quantum close coupling studies. We selected a threshold value of 300 cm$^{-1}$, and improved the quality of the expansion by applying a weighted fitting strategy [e.g. @hodges04] to focus the fit on the details of the attractive and weakly repulsive regions of the PES. Using $l_1\leq 35$, both the He and H$_2$ PES fits were converged to within 1 cm$^{-1}$ for $V \le 300$ cm$^{-1}$. These expansions still describe the range $300<V<1000$ cm$^{-1}$ to within an accuracy of a few $\rm cm^{-1}$.
The corresponding absolute minima are the following (in cm$^{-1}$ and Bohr): for [$\mathrm{HC_3N}$]{}– He, $V=-40.25$ for $R=6.32$ and $\theta_1=95.2^{\circ}$; for [$\mathrm{HC_3N}$]{}– para-[$\rm H_2$]{}($J=0$), $V=-111.24$ for $R=6.41$ and $\theta_1=94.0^{\circ}$; and for [$\mathrm{HC_3N}$]{}– [$\rm H_2$]{}, $V=-192.49$ for $R=9.59$, $\theta_1=180^{\circ}$, and $\theta_2=0^{\circ}$.
![The [$\mathrm{HC_3N}$]{}– para-[$\rm H_2$]{}PES. The [$\mathrm{HC_3N}$]{}molecule is shown at scale. Equipotentials (in $\rm cm^{-1}$) : in dashed red, -100, -30 -10, -3; in solid black, 0; in blue, 10, 30, 100, 300, 1000, 3000. The dotted circle centered at the [$\mathrm{HC_3N}$]{}center of mass with radius $R=6.41$ Bohr illustrates the angular steric hindrance problem occurring when the collider rotates from the vicinity of the minimum towards the [$\mathrm{HC_3N}$]{}rod. []{data-label="fig:PES"}](PES.eps){height="0.55\textheight"}
Inelastic cross section and rates {#sec:cross}
=================================
In the following $J_1, J^\prime_1$, denote the initial and final angular momentum of the [$\mathrm{HC_3N}$]{}molecule, respectively, and $J_2$ denote the angular momentum of H$_2$. We also denote the largest value of [$J_1, J^\prime_1$]{} as $J_{1\rm up}$.
The most reliable approach to compute inelastic cross sections $\sigma_{J_1J^{\,\prime}_1}(E)$ is to perform quantum close coupling calculations. In the case of molecules with a small rotational constant, like [$\mathrm{HC_3N}$]{}[$B=4549.059\mbox{~MHz}$, see e.g. @thor00], quantum calculations become soon intractable, because of the large number of open channels involved. While observations at cm-mm wavelengths culminates with $J_{1\rm up}
\lesssim 24$ [@kahane94], sub-mm observations can probe transition as high as $J_{1\rm up} = 40$, at a frequency of 363.785 GHz and a rotational energy of $202.08 \:\rm
cm^{-1}$ [@pepe04; @charnley04; @cauxpc]. It is thus necessary to compute rates with transitions up to $J_1=50$ ($E=
386.8$ cm$^{-1}$), in order to properly converge radiative transfer models. Also, we aim at computing rates up to a temperature of 100 K for H$_2$. We resorted to two methods in order to perform this task. For $J_{1\rm up}\leq 15$, we performed quantum inelastic scattering calculations, as presented in next subsection \[par:molscat\]. For $J_{1\rm up} > 15$, we used the QCT method, as presented in subsection \[par:rates\].
For He, of less astrophysical importance (\[He\]/\[H\]$\sim 0.1$), only quantum calculations were performed and were limited to the low temperature regime ($T$=5$-$20 K and $J_1<10$).
Rotational inelastic cross sections with <span style="font-variant:small-caps;">Molscat</span> {#par:molscat}
----------------------------------------------------------------------------------------------
All calculations were made using the rigid rotor approximation, with rotational constants $B_{\rm HC_3N}=0.151739$ cm$^{-1}$ and $B_{\rm
H_2}=60.853$ cm$^{-1}$, using the <span style="font-variant:small-caps;">Molscat</span> code [@molscat]. All quantum calculations for [$\mathrm{HC_3N}$]{}– ortho-[$\rm H_2$]{}were performed with $J_{\rm H_2}\equiv J_2=1$. Calculations for [$\mathrm{HC_3N}$]{}– para-[$\rm H_2$]{}were performed with $J_2=0$. We checked at $E_{\rm
tot}=E_{\rm coll}+E_{\rm rot} = 30 \,\rm cm^{-1}$ that the inclusion of the closed $J_2=2$ channel led to negligible effects.
The energy grid was adjusted to reproduce all the details of the resonances, as they are essential to calculate the rates with high confidence [@dubernet02; @dubernet03; @wernli06]. The energy grid and the quantum methods used are detailed in table \[tab:param\]. Using this grid, we calculated the whole resonance structure of all the transitions up to $J_1=15$ for the [$\mathrm{HC_3N}$]{}– para-[$\rm H_2$]{}collisions. At least 10 closed channels were included at each energy to fully converge the [$\mathrm{HC_3N}$]{}rotational basis. We used the hybrid log-derivative/Airy propagator [@alexander87]. We increased the parameter <span style="font-variant:small-caps;">STEPS</span> at the lowest energies to constrain the step length of the integrator below 0.1 to 0.2 Bohr, in order to properly follow the details of the radial coefficients. Other propagation parameters were taken as the <span style="font-variant:small-caps;">molscat</span> default values.
[ccc]{}\
$E_{\rm tot}\:\rm (cm^{-1})$ & Energy step ($\rm cm^{-1}$) & Method\
$0.3 \rightarrow 60$ & $0.1$ & CC\
$60 \rightarrow 110$ & $10$ & CC\
$40 \rightarrow 200$ & $10$ & CS\
$50 \rightarrow 800$ & $10-100 $ & IOS\
\
$0\rightarrow 30 $& $1$ & CC\
\
$0\rightarrow 25 $& $0.1$ & CC\
$25\rightarrow 100 $& $5$ & CC\
$100\rightarrow 150 $& $10$ & CC\
Two examples of deexcitation cross-sections are shown in figure \[fig:sectionop\]. We see that for energies between threshold and about 20 cm$^{-1}$ above threshold, the cross-section displays many shape resonances, justifying *a posteriori* our very fine energy grid. This behaviour is by no means unexpected and is very similar to most earlier calculations is many different systems, see e.g. @dubernet02 [@wernli06] for a discussion. In a semi-classical point of view, those shape resonances manifest the trapping of the wave-packet between the inner repulsive wall and the outer centrifugal barrier, see @wie03 [@abrol01]. At energies higher than about 20 cm$^{-1}$ above threshold, all cross-sections become smooth functions of the energy.
Figure \[fig:sectionop\] also shows that ortho-[$\rm H_2$]{}inelastic cross-sections follow very closely the para-[$\rm H_2$]{}ones, including the position of resonances. Examination of all cross-sections reveals that the relative difference between $\sigma_{J_1J'_1}(E, \mbox{para})$ and $\sigma_{J_1J'_1}(E, \mbox{ortho})$ is less than $5\%$. This justifies *a posteriori* the much smaller amount of computational effort devoted to ortho-[$\rm H_2$]{}collisions as well as the neglect of $J_2 = 2$ closed para-[$\rm H_2$]{}channels. A detailed discussion of this behaviour is put forward in section \[sec:paraortho\].
Quantum rates and classical rates {#par:rates}
---------------------------------
The quantum collisional rates are calculated for $J_{1\rm up}\leq
15$, at astrophysically relevant temperatures, from 5 K to 100 K. We average the cross-sections described in the preceding section over the Maxwell distribution of velocities, up to a kinetic energy at least 10 times $kT$. The quantum calculations at the higher end of the energy range are approximated at the IOS level (see table \[tab:param\]), which is justified at these energies by the smallness of the rotational constant $B_{\rm HC_3N}$. Also we used a coarse energy grid for the IOS calculations because the energy dependence of the cross-sections becomes very smooth.
For values of $J_1 > 15$, the close coupling approach enters a complexity barrier due to the rapid increase of the number of channels involved in calculations, while memory and CPU requirements scale as the square and the cube of this number, respectively. Resorting to quantum CS or IOS approximations is inaccurate, because the energy is close to threshold for high-$J_1$ channels. In the meanwhile, the accuracy of classical approximations improves for higher collisional energies. For the energy range where $J_1 > 15$ channels are open *and* for deexcitation processes involving those channels, we employ a Quasi-Classical Trajectory (QCT) method, which has been shown at several instances to be a valid approximation for higher collisional energies and large rates [@chapman77; @lepp95; @mandy04; @faure06].
For Monte-Carlo QCT methods, we must devise a way of defining an ensemble of initial conditions for classical trajectories, on the one hand, and of analyzing the final state of each trajectory, on the other hand. Contrary to the asymmetric rotor case [like water, see @faure04], the analysis of final conditions for a linear molecule is straightforward. Using the simplest quantization approximation, we bin the final classical angular momentum $J'_1$ of [$\mathrm{HC_3N}$]{}to the nearest integer. While the quantum formalism goes through a microcanonical calculation —calculating $\sigma_{J_1J_1'}(E)$ for fixed energies, then averaging over velocity distributions— it is possible for QCT calculations to directly resort to a canonical formalism, i.e. to select the initial velocities of the Monte-Carlo ensemble according to the relevant Maxwell-Boltzmann distribution and find the rates as: $$\label{eq:rate}
k_{J_1J'_1} = \left(\frac{8kT}{\pi\mu}\right)^{1/2}\,\pi
b_{max}^2\,\frac{N}{N_{\rm tot}}$$ where $b_{max}$ is the maximum impact parameter used (with the impact parameter $b$ distributed with the relevant $b\,\textmd{d}b$ probability density) and $N$ is the number of trajectories with the right final $J'_1$ value among all $N_{\rm tot}$ trajectories. The Monte-Carlo standard deviation is: $$\label{eq:error}
\frac{\delta k_{J_1J_1'}}{k_{J_1J_1'}} = \left(\frac{N_{\rm
tot}-N}{N_{\rm tot}N}\right)^{1/2} \quad ,$$ showing that the accuracy of the method improves for larger rates. The $b_{max}$ parameter was determined by sending small batches of $500$ to $1,000$ trajectories for fixed $b$ values; values of $20 \leq
b_{max} \leq 26$ Bohr were found. We then sent batches of $10,000$ trajectories for each temperature in the range $5-100$K, with a step of 5K. Trajectories are integrated by means of a Bürlich-Stoer algorithm [@numrec92], with a code similar to that of @faure05a. Precision is checked by conservation of total energy and total angular momentum.
Some illustrative results are shown in tables \[tab:rates\] and \[tab:rates2\] and are illustrated in figures \[fig:ratecompare\] and \[fig:rate12\].
As an alternative to QCT calculations, we tested J-extrapolation techniques, using the form of @depristo79 generally used by astrophysicists (see for example @lamda, section 6). We found that even if it reproduces the interference pattern, the extrapolation systematically underestimates the rates, for $J_1\ge
20$. Hence, QCT rates are more precise in the average.
--------- ------------------ ------------------ ------------- -------------------
$J_{1}$ $T = 10 \rm K\;$ $T = 20 \rm K\;$ $T = 50 \rm $T = 100 \rm K\;$
K\;$
1 2.03(-11) 1.59(-11) 1.32(-11) 1.24(-11)
2 4.94(-11) 4.83(-11) 6.23(-11) 8.04(-11)
3 1.20(-11) 1.04(-11) 8.23(-12) 7.43(-12)
4 2.25(-11) 2.57(-11) 2.85(-11) 2.87(-11)
5 7.01(-12) 6.80(-12) 5.62(-12) 4.77(-12)
6 9.15(-12) 1.18(-11) 1.42(-11) 1.38(-11)
7 3.14(-12) 3.40(-12) 3.46(-12) 3.26(-12)
8 2.45(-12) 3.71(-12) 5.92(-12) 6.61(-12)
9 1.63(-12) 1.63(-12) 1.95(-12) 2.18(-12)
10 5.35(-13) 8.13(-13) 2.00(-12) 2.96(-12)
11 7.81(-13) 7.01(-13) 9.42(-13) 1.36(-12)
12 1.37(-13) 1.58(-13) 6.17(-13) 1.32(-12)
13 2.74(-13) 2.51(-13) 4.17(-13) 8.26(-13)
14 4.14(-14) 4.65(-14) 2.24(-13) 6.28(-13)
15 7.63(-14) 8.26(-14) 1.76(-13) 4.85(-13)
--------- ------------------ ------------------ ------------- -------------------
: [$\mathrm{HC_3N}$]{}– para-[$\rm H_2$]{} s($J=0$)collisions. Quantum deexcitation rates in $\mathrm{cm^3\,s^{-1}}$, for $J_1'=0$, for successive initial $J_1$ and for various temperatures. Powers of ten are denoted in parenthesis.[]{data-label="tab:rates"}
[llcccc]{} &\
$J_{1}'$ &$J_{1}$& $T = 10 \rm K\;$ & $T = 20 \rm K\;$ & $T = 50 \rm K\;$ & $T = 100 \rm K\;$\
&\
&\
0 & 1 & 2.03(-11) & 1.59(-11) & 1.32(-11) & 1.24(-11)\
0 & 2 & 4.94(-11) & 4.83(-11) & 6.23(-11) & 8.04(-11)\
0 & 3 & 1.20(-11) & 1.04(-11) & 8.23(-12) & 7.43(-12)\
0 & 4 & 2.25(-11) & 2.57(-11) & 2.85(-11) & 2.87(-11)\
\
5 & 6 & 6.34(-11) & 5.48(-11) & 4.80(-11) & 4.64(-11)\
5 & 7 & 1.30(-10) & 1.38(-10) & 1.72(-10) & 2.04(-10)\
5 & 8 & 3.93(-11) & 3.66(-11) & 3.27(-11) & 3.21(-11)\
5 & 9 & 6.83(-11) & 7.61(-11) & 8.63(-11) & 8.93(-11)\
\
10 & 11 & 5.77(-11) & 5.35(-11) & 4.75(-11) & 4.61(-11)\
10 & 12 & 1.50(-10) & 1.53(-10) & 1.81(-10) & 2.11(-10)\
10 & 13 & 3.91(-11) & 3.80(-11) & 3.50(-11) & 3.40(-11)\
10 & 14 & 8.51(-11) & 8.84(-11) & 9.42(-11) & 9.47(-11)\
\
\
15 & 16 $^\dag$ & 1.45(-10) & 1.49(-10) & 1.80(-10) & 2.28(-10)\
15 & 17 $^\dag$ & 1.06(-10) & 1.03(-10) & 9.60(-11) & 1.18(-10)\
15 & 18 $^\dag$ & 8.71(-11) & 9.38(-11) & 7.93(-11) & 8.24(-11)\
15 & 19 $^\dag$ & 7.59(-11) & 6.81(-11) & 6.23(-11) & 7.10(-11)\
\
25 & 26 $^\dag$ & 1.14(-10) & 1.50(-10) & 1.83(-10) & 2.30(-10)\
25 & 27 $^\dag$ & 1.13(-10) & 1.05(-10) & 1.18(-10) & 1.31(-10)\
25 & 28 $^\dag$ & 8.55(-11) & 8.38(-11) & 8.34(-11) & 7.76(-11)\
25 & 29 $^\dag$ & 7.67(-11) & 7.45(-11) & 8.31(-11) & 7.02(-11)\
\
35 & 36 $^\dag$ & 1.16(-10) & 1.34(-10) & 1.73(-10) & 2.32(-10)\
35 & 37 $^\dag$ & 9.63(-11) & 1.12(-10) & 1.21(-10) & 1.11(-10)\
35 & 38 $^\dag$ & 8.33(-11) & 9.20(-11) & 8.56(-11) & 9.19(-11)\
35 & 39 $^\dag$ & 8.77(-11) & 8.51(-11) & 7.65(-11) & 7.51(-11)\
\
For H$_2$, all deexcitation rates $k_{J_1J_1'}(T)$, $J_1\neq J_1'\leq
50$, are fitted with the following formula [@wernli06]: $$\label{eq:fit}
\log_{10}\left(k_{J_1J_1'}(T)\right)=\sum_{n=0}^{
4}a^{(n)}_{J_1J_1'} x^n$$ where $x=T^{-1/6}$. As some transitions have zero probability within the QCT approach, the above formula was employed when rates were bigger than 10$^{-12}$ cm$^3$s$^{-1}$ for at least one grid temperature. For these rates, null grid values were replaced by a very small value, namely 10$^{-14}$ cm$^3$s$^{-1}$, to avoid fitting irregularities. All rates not fulfilling this condition are set to zero. Note that below 20 K, QCT rates for low-probability transitions may show a non physical behaviour. All $a^{(n)}_{J_1J_1'}$ coefficients are provided as online material, for a temperature range $5{\rm\; K}\leq T \leq 100{\rm\; K}$. We advise to use the same rates for collisions with ortho-[$\rm H_2$]{}as for para-H$_2$, since their difference is smaller than the uncertainty on the rates themselves. Rates with He were not fitted, but can be obtained upon request to the authors.
---------------------------------- ----------------------------------
{width="8cm"} {width="8cm"}
---------------------------------- ----------------------------------
Discussion {#sec:disc}
==========
Para and ortho [$\rm H_2$]{}cross-sections {#sec:paraortho}
------------------------------------------
A comparison of the $\sigma_{J_1J_1'}(E)$ cross sections for [$\mathrm{HC_3N}$]{}with ortho-[$\rm H_2$]{}and para-[$\rm H_2$]{}is given in figure \[fig:sectionop\]. It can be seen that the difference between the two spin species of [$\rm H_2$]{}may be considered as very small, in any case smaller than other PES and cross-section uncertainties. This is an unexpected result, as sizeable differences between para-[$\rm H_2$]{}and ortho-[$\rm H_2$]{}inelastic cross-sections exist for other molecules. These differences were expected to increase for a molecule possessing a large dipolar moment, in view of the results obtained for the C$_2$ molecule [@phillips94], the CO molecule [@wernli06], the OH radical [@offer94], the NH$_3$ molecule [@offer89; @flower94] and the $\rm H_2O$ molecule [@phillips96; @dubernet02; @dubernet03; @dubernet06], due to the interaction between the dipole of the molecule and the quadrupole of [$\rm H_2$]{}(J$_2$ $>$ 0).
This apparently null result deserves an explanation. We focus on equation (9) of @green75. This equation describes the different matrix elements that couple the various channels in the close-coupling equations. Some triangle rules apply which restrict the number of terms in the sum of equation (9); the relevant angular coupling algebra is represented there as a sum of terms of the type $$\label{eq:green75}
\left(\begin{array}[c]{ccc} l &L'& L \\ 0 & 0 & 0
\end{array}\right)
\;
\left(\begin{array}[c]{ccc} l_1 &J'_1& J_1 \\ 0 & 0 & 0
\end{array}\right)
\;
\left(\begin{array}[c]{ccc} l_2 &J'_2& J_2 \\ 0 & 0 & 0
\end{array}\right)
\;
\left\{\begin{array}[c]{ccc} L'& L & l \\ J_{12} & J'_{12} &
J
\end{array}\right\}\quad ,
$$ where we have the potential function expanded in terms of Eqs. (4) and (A2) in @green75, by means of the coefficients $v_{l_1l_2
l}$. The symbol $(\ldots)$ are 3-$j$ symbols, the $\{\ldots\}$ is a 6-$j$ symbol, see @messiah69. We also define $\vec{J}_{12}=\vec{J}_1+\vec{J}_2$. We have the following rules:
- The para-[$\rm H_2$]{}inelastic collisions are dominated by the $J_2 = 0$ channel (the $J_2=2$ channel is closed till $E_{coll} \gtrsim 365.12
\;\rm cm^{-1}$). Then, only the $l_2=0$ may be retained ($J_2=J'_2=0$), due to the third 3-$j$ symbol in eq.(\[eq:green75\]).
- The ortho-[$\rm H_2$]{}remains in $J_2=1$, implying $l_2=0,2$.
- For inelastic collisions, $J_1\neq J'_1$ implies potential terms with $l\neq 0$, because of the 6-$j$ term in Eq.(\[eq:green75\]). Indeed, $J_2=J'_2$ and $J_1\neq J'_1$ entail $J_{12}\neq J'_{12}$.
The key point is thus to compare the $v_{l_1l_2 l}(R)$ coefficients (eq. \[eq:pot2\]) with $l \neq 0$ in the two cases:
- [$l_2=0$]{} para and ortho contributions;
- [$l_2=2$]{} ortho contribution only.
Figure \[fig:comp\] displays such a comparison. We notice that the coupling is largely dominated by the $l_2=0$ contribution, terms which are common to collisions with para and ortho conformations. This is particularly true for $R<10$ Bohr, the relevant part of the interaction for collisions at temperatures higher than a few Kelvin. At a higher intermolecular separation, terms implied only in collisions with ortho-[$\rm H_2$]{}become dominant, but in this regime the potential is also less than a few cm$^{-1}$. Sizeable differences in rates between ortho and para forms are thus expected only either at very low temperatures, or possibly at much higher temperatures, with the opening of [$\rm H_2$]{}(J$_2=2,3$) channels.
Propensity rules
----------------
In figure \[fig:ratecompare\], we compare the various rates that we obtain here with the ones previously published by @green78. These authors used a coarse electron-gas approximation for the PES, and computed rates by a QCT classical approach. Despite these approximations, we see that the rates obtained by @green78 are qualitatively comparable with the quantum rates obtained here, in an average way. However, as table \[tab:rates2\] and figures \[fig:rate12\] and \[fig:ratecompare\] show clearly, only quantum calculations manifest the strong $\Delta J = 2$ propensity rule. This rule originates in the shape of the PES, being nearly a prolate ellipsoid, dominated by the rod shape of [$\mathrm{HC_3N}$]{}and *not dominated* by the large dipole of HC$_3$N molecule (3.724 Debye). Because of the very good approximate symmetry $\theta_1\leftrightarrow \pi - \theta_1$, the $l_1$ even terms (equation (\[eq:green75\]) and @green75) are the most important ones, directing the inelastic transition toward even $\Delta J_1$. This propensity has also been explained semi-classically by @miller77 in terms of an interference effect related to the even anisotropy of the PES. These authors show in particular that the reverse propensity can also occur if the odd anisotropy of the PES is sufficiently large. This reverse effect is indeed observed in Fig. \[fig:rate12\] for transitions with $\Delta J>10$. A similar propensity rule has been experimentally observed for CO–He collisions [@sims04].
Besides this strong $\Delta J = 2$ propensity rule, one can see from table \[tab:rates2\] and figures \[fig:ratecompare\], \[fig:rate12\] that the rod-like interaction drives large $\Delta J$ transfers. For instance, for T $> 20$ K, rates for $\Delta J > 6$ are generally larger than rates for $\Delta J = 1$, and rates for $\Delta J > 8$ are only one order of magnitude below those for $\Delta J = 2$. This behaviour is likely to emphasize the role of collisional effects versus radiative ones. This effect, of purely geometric origin, has been predicted previously [@bosanac80] and is of even greater importance for longer rods like $\rm HC_5N$, $\rm HC_7N$, $\rm HC_9N$, see @snell81 [@dickinson82].
We also observe that the ratio $k_{J_1J'_1}(\textrm{He})/k_{J_1J'_1}(\mbox{para-H$_2$}) $ is in average close to $1/1.4 \sim 1/\sqrt{2}$, thus confirming the similarity of He and para-[$\rm H_2$]{}as projectiles, as generally assumed. But it is also far from being a constant, as already observed for H$_2$O [@phillips96] or CO [@wernli06]. Our data shows that the $1/\sqrt{2}$ scaling rule results in errors up to 50%.
Population inversion and critical densities
-------------------------------------------
Because of the strong $\Delta J_1=0,2,4$ propensity rule, population inversion could be strengthened if LTE conditions are not met, even neglecting hyperfine effects[^1] [@hunt99]. In order to see the density conditions giving rise to population inversion, we solved the steady-state equations for the population of the $J=0,1,\dots,15$ levels of [$\mathrm{HC_3N}$]{}, including collisions with [$\rm H_2$]{}(densities ranging from $10^2$ to $10^6 \rm\; cm^{-3}$), a black-body photon bath at 2.7 K, in the optically thin approximation, [@goldsmith72] : $$\begin{aligned}
\label{eq:ss}
\frac{\textmd{d}n_i}{\textmd{d}t}=0&=&
+\sum_{j\neq i}n_j\,\left[ A_{ji} +
B_{ji} \;n_\gamma\left(\nu_{ji}\right)+k_{ji}\;
n_{\rm H_2}\right] \nonumber \\
& & - n_i\,\sum_{j\neq i}
\left[A_{ij}+B_{ij}\,n_\gamma\left(\nu_{ij}\right)+k_{ij}\; n_{\rm
H_2}\right]\end{aligned}$$ where $i,j$ are the levels, $n_\gamma$ is the photon density at temperature $T_\gamma$ and $n_{\rm H_2} $ is the hydrogen density at kinetic temperature $T_{\rm H_2}$. Figure \[fig:invers\] shows the results at $T_{\rm H_2} = 40$ K. The lines show the population per sub-levels $\left|J_1, m_{J_1}\right>$. For a consequent range of [$\rm H_2$]{}densities, $10^4\lesssim n_{\rm H_2}
\lesssim 10^6$, population inversion does occur, for $0\leq
J_1\leq 2, 3, 4$. Our new rates are expected to improve the interpretation of the lowest-lying lines of [$\mathrm{HC_3N}$]{}, especially so in the 9 - 20 GHz regions (cm-mm waves), see for example @walms86 [@takano98; @hunt99], and @kal04 for a recent study. Moreover, from the knowledge of both collision coefficients $k_{ij}$ and Einstein coefficients $A_{ij}$, it is possible to derive a critical density of [$\rm H_2$]{}, defined as: $$\label{eq:nstar}
n^{\star}_i(T)=\frac{\sum_{j<i}A_{ij}}{\sum_{j<i}k_{ij}}$$ The $n^\star$ density is the [$\rm H_2$]{}density at which photon deexcitation and collisional deexcitation are equal. The evolution of $n^\star$ with $J_1$ at $T= 40\;\rm K$ is given in figure \[fig:critical\]. It can be seen that for many common interstellar media, the LTE conditions are not fully met.
It must be underlined that similar effects should appear for the whole cyanopolyyne ($\rm HC_{5,7,9}N$) family, where cross-sections should scale approximately with the rod length [@dickinson82]. It is expected that the propensity rule $\Delta J_1=2,4,\dots$ should remain valid. Also, the critical density should decrease for the higher members of the cyanopolyyne family, as the Einstein A$_{ij}$ coefficients, hence facilitating the LTE conditions.
Conclusion
==========
We have computed two [*ab initio*]{} surfaces, for the [$\mathrm{HC_3N}$]{}– He and [$\mathrm{HC_3N}$]{}– [$\rm H_2$]{}systems. The latter was built using a carefully selected set of [$\rm H_2$]{}orientations, limiting the computational effort to approximately five times the [$\mathrm{HC_3N}$]{}– He one. Both surfaces were successfully expanded on a rotational basis suitable for quantum calculations using a smooth regularization of the potentials. This approach circumvented the severe convergence problems already noticed by @chapman77 for such large molecules. The final accuracy of both PES is a few cm$^{-1}$ for potential energy below 1000 cm$^{-1}$.
Rates for rotational excitation of [$\mathrm{HC_3N}$]{}by collisions with He atoms and [$\rm H_2$]{}molecules were computed for kinetic temperatures in the range 5 to 20 K and 5 to 100 K, respectively, combining quantum close coupling and quasi-classical calculations. The rod-like symmetry of the PES strongly favours even $\Delta J_1$ transfers and efficiently drives large $\Delta J_1$ transfers. Quasi classical calculations are in excellent agreement with close coupling quantum calculations but do not account for the even $\Delta J_1$ interferences. For He, results compare fairly with @green78 QCT rates, indicating a weak dependance to the details of the PES. For para-H$_2$, rates are compatible in average with the generally assumed $\sqrt{2}$ scaling rule, with a spread of about 50 %. Despite the large dipole moment of $\mathrm{HC_3N}$, rates involving ortho-H$_2$ are very similar to those involving para-H$_2$, due to the predominance of the rod interactions.
A simple steady-state population model shows population inversions for the lowest [$\mathrm{HC_3N}$]{}levels at [$\rm H_2$]{}densities in the range 10$^4-$10$^6$ cm$^{-3}$. This inversion pattern manifests the importance of large angular momentum transfer, and is enhanced by the even $\Delta J_1$ quantum propensity rule.
The [$\mathrm{HC_3N}$]{}molecule is large enough to present an original collisional behaviour, where steric hindrance effects hide the details of the interaction, and where quasi classical rate calculations achieve a fair accuracy even at low temperatures. With these findings, approximate studies for large and heavy molecules should become feasible including possibly the modelling of large $\Delta J$ transfer collisions and ro-vibrational excitation of low energy bending or floppy modes.
This research was supported by the CNRS national program “Physique et Chimie du Milieu Interstellaire” and the “Centre National d’Etudes Spatiales”. LW was partly supported by a CNRS/NSF contract. MW was supported by the Ministère de l’Enseignement Supérieur et de la Recherche. CCSD(T) calculations were performed on the IDRIS and CINES French national computing centers (projects no. 051141 and x2005 04 20820). <span style="font-variant:small-caps;">Molscat</span> and QCT calculations were performed on local workstations and on the “Service Commun de Calcul Intensif de l’Observatoire de Grenoble” (SCCI) with the valuable help from F. Roch.
[c c c c c c c]{}
\
[^1]: Hyperfine effects in [$\mathrm{HC_3N}$]{}inelastic collisions will be dealt with in a forthcoming paper, @wie06
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
In this paper we introduce the *persistent magnitude*, a new numerical invariant of (sufficiently nice) graded persistence modules. It is a weighted and signed count of the bars of the persistence module, in which a bar of the form $[a,b)$ in degree $d$ is counted with weight $(e^{-a}-e^{-b})$ and sign $(-1)^d$. Persistent magnitude has good formal properties, such as additivity with respect to exact sequences and compatibility with tensor products, and has interpretations in terms of both the associated graded functor, and the Laplace transform.
Our definition is inspired by Otter’s notion of blurred magnitude homology: we show that the magnitude of a finite metric space is precisely the persistent magnitude of its blurred magnitude homology. Turning this result on its head, we obtain a strategy for turning existing persistent homology theories into new numerical invariants by applying the persistent magnitude. We explore this strategy in detail in the case of persistent homology of Morse functions, and in the case of Rips homology.
address:
- 'Institute of Mathematics, University of Aberdeen, Aberdeen, United Kingdom AB24 3UE'
- 'Institute of Mathematics, University of Aberdeen, Aberdeen, United Kingdom AB24 3UE'
author:
- Dejan Govc
- Richard Hepworth
bibliography:
- 'rips-magnitude.bib'
title: Persistent Magnitude
---
Introduction
============
Magnitude is a numerical invariant of metric spaces arising from category theory and with nontrivial geometric content. In this paper we apply the theory of magnitude and its categorifications to the study of persistence modules and persistent homology theories.
Background
----------
Persistent homology, a fundamental notion in topological data analysis (or TDA), is a tool for measuring the shape of data sets and other objects. The general idea is to take a data set and produce an increasing sequence of topological spaces $X_s$, one for each value of a parameter $s$, where $X_s$ describes the shape of the data set ‘at scale $s$’. Taking the homology of the $X_s$ produces the homology groups $H_\ast(X_s)$ together with structure maps $H_\ast(X_s)\to H_\ast(X_{s'})$ whenever $s\leqslant s'$. This structure is called the *persistent homology* of the data set, and it is an example of an algebraic structure called a graded *persistence module*. Any (sufficiently nice) persistence module has a *barcode decomposition* describing its isomorphism class in terms of a collection of intervals called *bars*. Each bar is interpreted as a feature of the data set: the start point of the interval is the scale at which the feature first comes into being, and the end point is the scale at which the feature evaporates. Longer bars are interpreted as significant features, while shorter bars are interpreted as noise.
*Magnitude* is a numerical invariant of metric spaces introduced by Leinster [@LeinsterMetricSpace] (see also the survey [@survey]), as an instance of a general category theoretical construction. Despite its abstract origins, magnitude is a rich geometric invariant: Meckes [@MeckesMagnitudeDimensions] showed that magnitude can detect the Minkowski dimension of compact subsets of Euclidean space, Barceló-Carbery [@BarceloCarbery] showed that it can detect the volume of compact subsets of Euclidean space, and Gimperlein-Goffeng [@GimperleinGoffeng] showed that it can in addition detect surface area and the second intrinsic volume $V_2$ of appropriate subsets of odd-dimensional Euclidean space.
Magnitude of metric spaces has a categorification, called *magnitude homology*, which was introduced by Hepworth-Willerton [@richard] and Leinster-Shulman [@shulman]. The magnitude homology of a metric space is a bigraded abelian group, whose graded Euler characteristic recovers the magnitude of the metric space, at least when the space is finite. Thus the relationship between magnitude and magnitude homology is analogous to the relationship between Euler characteristic and singular homology. More recently, Otter [@Otter] has introduced a *blurred* or persistent version of magnitude homology, which relates magnitude homology to the Rips complex and, importantly, to ordinary homology.
Results
-------
Blurred magnitude homology assigns to each metric space $X$ a graded persistence module $\operatorname{BMH}_\ast(X)$. When $X$ is finite, we show that there is an attractive relationship between the barcode decomposition of $\operatorname{BMH}_\ast(X)$ and the magnitude $|X|$ of $X$: $$|X|
=
\sum_{k=0}^{\infty}
\sum_{i=1}^{m_k}
(-1)^k (e^{-a_{k,i}} - e^{-b_{k,i}})$$ where $\operatorname{BMH}_\ast(X)$ has bars $[a_{k,1},b_{k,1}),\ldots,[a_{k,m_k},b_{k,m_k})$ in degree $k\geq 0$.
Observe that the right hand side of the equation above makes sense for any graded persistence module, so long as it is subject to a finiteness condition such as being finitely presented. We turn this observation into a definition: The *persistent magnitude* or simply *magnitude* $|M_\ast|$ of a finitely presented graded persistence module $M_\ast$ is defined by $$|M_\ast|
=
\sum_{k=0}^{\infty}
\sum_{i=1}^{m_k}
(-1)^k (e^{-a_{k,i}} - e^{-b_{k,i}})$$ where $M_\ast$ has bars $[a_{k,1},b_{k,1}),\ldots,[a_{k,m_k},b_{k,m_k})$ in degree $k\geq 0$. Note that a bar $[a,b)$ makes a contribution of $\pm(e^{-a} - e^{-b})$ to the magnitude, so that longer bars make a greater contribution, in line with the general philosophy of persistent homology.
Persistent magnitude has good formal properties: we show that it is additive with respect to exact sequences, and that the magnitude of a tensor product of persistence modules is the product of the magnitudes of the factors, so long as the tensor product is understood in an appropriate derived sense.
Now suppose that we have a persistent homology theory defined for some class of mathematical objects, for example the Rips homology of metric spaces. By applying persistent magnitude to the persistent homology, we obtain a new numerical invariant of the mathematical objects in question. Our first example of this process is the case of the sublevel set persistent homology of Morse functions, where the resulting magnitude invariant is a (signed and weighted) count of the critical points of the original function.
Our most detailed example of persistent magnitude in action is the *Rips magnitude*. This is the numerical invariant of finite metric spaces obtained by taking the persistent magnitude of the Rips homology, and is given by the weighted simplex-count $$|X|_\mathrm{Rips}
=
\sum_{\emptyset\neq A\subseteq X}(-1)^{\#A-1}e^{-\operatorname{diam}(A)}.$$ We compute the Rips magnitude of cycle graphs with their path, Euclidean and geodesic metrics. In each case they are determined by a number-theoretical formula reminiscent of the sum of divisors function.
In the original setting of magnitude, defining the magnitude of infinite metric spaces is not straightforward: the simplest method is to take the supremum of the magnitude of all finite subspaces of the given infinite metric space, but there are alternatives, and currently the theory only works well in the case of *positive definite* spaces. We conclude the paper by investigating the question of whether Rips magnitude can be extended to infinite metric spaces. In the case of closed intervals in ${\mathbb{R}}$ the approach via a supremum works well and we find that $|[a,b]|_\mathrm{Rips}=1+(b-a)$. In the case of the circle with its Euclidean and geodesic metrics, which we study in detail, the results are attractive but inconclusive.
Organisation
------------
We begin the paper with a series of generous background sections: persistence modules and persistent homology in section \[section-background-persistence\], magnitude in section \[section-background-magnitude\], and magnitude homology in section \[section-background-mh\]. Section \[section-persistent-magnitude\] introduces the persistent magnitude of persistence modules, and studies its basic properties. Section \[section-sublevel\] applies persistent magnitude to the persistent homology of sublevel sets. The final part of the paper studies Rips magnitude: section \[section-rips\] introduces Rips magnitude and discusses its properties and some basic examples, section \[section-cycles\] computes it in the case of cycle graphs (with various metrics), and section \[section-infinite\] explores the possibility of defining Rips magnitude for infinite metric spaces.
Open Questions
--------------
The results obtained in this work raise several natural questions, that we have not yet been able to answer conclusively:
- What is the most general notion of tameness sufficient to develop the theory of persistent magnitude? (Our characterisation using the Laplace transform suggests that one might want to consider a notion of persistence modules of “exponential type”, meaning that the rank function is of exponential type.)
- Is there a general definition of Rips magnitude for (a suitable class of) infinite metric spaces? Can we establish asymptotic results similar to the case of the circle for higher-dimensional spheres or other manifolds? (Note that not much seems to be known about Rips filtrations of manifolds beyond the circle [@adamaszek2017vietoris].)
- The formulas for the Rips magnitudes of cycle graphs and Euclidean cycles seem reminiscent of the sum of divisors functions from number theory. Are there interesting connections between Rips magnitude of cycles and analytic number theory?
Background on persistence modules and persistent homology {#section-background-persistence}
=========================================================
Persistence modules
-------------------
Here we review some standard material on persistence modules, mostly following [@chazal2016structure; @bubenik2014categorification]. For a survey explaining the basic ideas and historical origin of persistence, see [@edelsbrunner2008persistent]. A modern exposition of the main ideas including the structure and stability theorems for persistence modules can be found in [@chazal2016structure]. For further background on persistence modules from the category theoretical perspective, see [@bubenik2014categorification]. A slightly more algebraic perspective, with a view towards multi-dimensional persistence, can be found in [@lesnick2015theory]. An account of some aspects of homological algebra for persistence modules can be found in [@BubenikMilicevic].
Throughout the paper, we will work with vector spaces over a fixed field ${\mathbbm{k}}$. The category of vector spaces over ${\mathbbm{k}}$ will be denoted by ${\mathbf{Vect}}$. In the most general setting, persistence modules can be considered over an arbitrary small category, see e.g. [@botnan2018decomposition; @bubenik2014categorification]; however, we will restrict attention to the case of $({\mathbb{R}},\leq)$-indexed persistence modules, as this is entirely sufficient for our purposes. Here $({\mathbb{R}},\leq)$ denotes either the poset ${\mathbb{R}}$ equipped with the partial order $\leq$, or the associated category with objects ${\mathbb{R}}$ and a unique morphism $x\to y$ whenever $x\leq y$.
A [*persistence module*]{} is a functor $M\colon({\mathbb{R}},\leq)\to{\mathbf{Vect}}$. A [*morphism*]{} of persistence modules is a natural transformation of such functors.
The category ${\mathbf{PersMod}}={\mathbf{Vect}}^{({\mathbb{R}},\leq)}$ of persistence modules has the structure of an abelian category. In particular, morphisms of persistence modules have well-defined kernels, and cokernels. These are again persistence modules and can be computed object-wise. The zero object of this abelian category is the persistence module $0\colon ({\mathbb{R}},\leq)\to{\mathbf{Vect}}$ all of whose components are $0$.
In some cases, we will also consider [*graded persistence modules*]{}, which are functors $M\colon ({\mathbb{R}},\leq)\to{\mathbf{GrVect}}$, where ${\mathbf{GrVect}}$ is the category of ${\mathbb{N}}_0$-graded vector spaces over ${\mathbbm{k}}$. Most of the content of this section generalises to the graded case in a completely straightforward way, so to avoid too much duplication, we only state it for the ungraded case.
To be able to extract any sort of useful information from persistence modules, we need to understand their structure. One way of doing this is by decomposing them into indecomposable summands. The indecomposables relevant in our case are known as interval modules.
A persistence module $M\colon({\mathbb{R}},\leq)\to{\mathbf{Vect}}$ is [*indecomposable*]{} if $M\cong M_1\oplus M_2$ implies that either $M_1\cong 0$ or $M_2\cong 0$.
Let $J\subseteq{\mathbb{R}}$ be an interval. The [*interval module*]{} ${\mathbbm{k}}J\colon ({\mathbb{R}},\leq)\to{\mathbf{Vect}}$ is defined as $${\mathbbm{k}}J(x)=\begin{cases}
{\mathbbm{k}};&\text{if $x\in J$,}\\
0;&\text{otherwise,}
\end{cases}$$ and $${\mathbbm{k}}J(x\leq y)=\begin{cases}
{\operatorname{id}}_{{\mathbbm{k}}};&\text{if $x,y\in J$,}\\
0;&\text{otherwise.}
\end{cases}$$
One of the main features of persistence modules over $({\mathbb{R}},\leq)$ that makes them useful in TDA is that they can frequently be decomposed as direct sums of interval modules. When such a decomposition exists, it is unique [@azumaya1950corrections]. The following version of the decomposition theorem is originally due to Crawley-Boevey [@crawley2015decomposition]. In the case of persistence modules over $({\mathbb{R}},\leq)$, it can be stated as follows.
Suppose $M\colon({\mathbb{R}},\leq)\to{\mathbf{Vect}}$ is a persistence module such that $M(x)$ is finite dimensional for every $x\in{\mathbb{R}}$. Then $M$ has a decomposition into interval modules.
Whenever a persistence module $M\colon({\mathbb{R}},\leq)\to{\mathbf{Vect}}$ decomposes as a sum of interval modules, we can represent it using a [*persistence barcode*]{}. This is defined as the multiset of all intervals that occur in the decomposition. Sometimes we represent these intervals as pairs $(a,b)$ where $a$ is the startpoint and $b$ is the endpoint of an interval in the decomposition. (These points are sometimes decorated to preserve information regarding which types of intervals the points correspond to, see [@chazal2016structure] for details.) The multiset of such pairs is called the [*persistence diagram*]{} corresponding to $M$. The notion of persistence diagram can be generalised to some cases where the interval decomposition does not exist [@chazal2016structure].
We will often concentrate on the case of *finitely presented* persistence modules. Note that a persistence module is finitely presented if and only if it is isomorphic to a finite direct sum of half-open interval modules ${\mathbbm{k}}[a,b)$, where $-\infty<a<b\leq\infty$.
The *tensor product* of two persistence modules $M$ and $N$ is given by $$(M\otimes N)(s) = \operatorname{colim}_{s_1+s_2\leq s}M(s_1)\otimes N(s_2).$$ Thus $(M\otimes N)(s)$ is the quotient of $\bigoplus_{s_1+s_2=s}M(s_1)\otimes M(s_2)$ obtained as follows. Suppose given $u_1,u_2$ with $u_1+u_2\leq s$. Then for any pair $v_1,v_2$ with $v_1+v_2=s$ and $u_1\leq v_1$, $u_2\leq v_2$, we have a composite $$\label{equation-tensor}
M(u_1)\otimes N(u_2)
\to
M(v_1)\otimes N(v_2)
\hookrightarrow
\bigoplus_{s_1+s_2=s}M(s_1)\otimes M(s_2).$$ Then $(M\otimes N)(s)$ is the largest quotient of $\bigoplus_{s_1+s_2=s}M(s_1)\otimes M(s_2)$ with the property that for all $u_1,u_2$ all such composites coincide, regardless of the choice of $v_1,v_2$. See Section 3.2 of [@BubenikMilicevic] or Section 2.2 of [@PSS].
The operation of tensoring with a fixed persistence module is right exact but not exact, and therefore induces *derived functors* denoted by $M,N\mapsto\operatorname{Tor}_i(M,N)$ for $i\geq 0$, with $\operatorname{Tor}_0(M,N)=M\otimes N$.
For finitely presented persistence modules the tensor products and $\operatorname{Tor}$-functors can be described explicitly. In order to do this, it suffices to explain what happens for interval modules. Given interval modules ${\mathbbm{k}}[a,b)$ and ${\mathbbm{k}}[c,d)$, we have $$\begin{aligned}
{\mathbbm{k}}[a,b)\otimes {\mathbbm{k}}[c,d) &= {\mathbbm{k}}[a+c,\min(a+d,b+c)),
\\
\operatorname{Tor}_1({\mathbbm{k}}[a,b), {\mathbbm{k}}[c,d)) &= {\mathbbm{k}}[\max(a+d,b+c),b+d),
\\
\operatorname{Tor}_i({\mathbbm{k}}[a,b), {\mathbbm{k}}[c,d)) &= 0
\text{ for }i\geq 2.\end{aligned}$$ See Example 7.1 of [@BubenikMilicevic].
Persistent homology
-------------------
Persistence modules have an important role in TDA, where they are used in order to study data sets in the form of finite metric spaces, also known as *point clouds*. The idea is to take a finite metric space $X$ and convert it into a simplicial complex (or topological space or other topological object) $Y$ equipped with an $({\mathbb{R}},\leq)$-filtration, i.e. a system of subsets $Y_r\subseteq Y$ for $r\in{\mathbb{R}}$, such that $\bigcup_{r\in{\mathbb{R}}}Y_r=Y$ and $Y_r\subseteq Y_{r'}$ for $r<r'$. There are many such constructions, and they are often based on the principle that $Y_r$ should capture the behaviour of $X$ ‘at length scale $r$’. Given such an $({\mathbb{R}},\leq)$-filtered complex $Y$, the assignment $r\mapsto Y_r$ defines a functor from $({\mathbb{R}},\leq)$ into simplicial complexes (or topological spaces, or other appropriate codomain). So taking the homology of the $Y_r$ then produces a graded persistence module $$r\longmapsto H_\ast(Y_r).$$ These persistence modules are called the *persistent homology* of the original object $X$. Once the persistent homology of $X$ has been obtained, the resulting barcode is then analysed. The bars are regarded as features of the metric space $X$. Longer or *persistent* bars are usually regarded as genuine features, while shorter bars are regarded as noise.
Here we will describe some important examples of this general construction, starting with the Vietoris-Rips filtration and the Čech filtration.
Suppose $(X,d)$ is a finite metric space. We define the [*Vietoris-Rips complex ${\mathcal{R}}(X)$*]{} of $(X,d)$ to be the $({\mathbb{R}},\leq)$-filtered simplicial complex with vertex set $X$, in which the simplices of the $r$-th filtration step ${\mathcal{R}}_r(X)$ are defined by the rule $$\sigma\in{\mathcal{R}}_r(X)\Leftrightarrow\operatorname{diam}\sigma\leq r.$$
In some cases, we consider $X$ as a subspace of some larger metric space $Y$, e.g. $Y={\mathbb{R}}^n$. In this case we can define the corresponding Čech complex as follows:
Suppose $(Y,d)$ is a metric space and $X\subseteq Y$ is a finite subset. We define the [*Čech complex ${\mathcal{\check{C}}}(X)$*]{} associated to $X$ to be the $({\mathbb{R}},\leq)$-filtered simplicial complex with vertex set $X$, in which the simplices of the $r$-th filtration step ${\mathcal{\check{C}}}_r(X)$ are defined by the rule $$\sigma\in{\mathcal{\check{C}}}_r(X)\iff\bigcap_{x\in\sigma}B(x,r)\neq\emptyset,$$ where $B(x,r)$ denotes the open ball in $Y$ with centre $x$ and radius $r$.
Note that both the Vietoris-Rips and the Čech complex are filtrations of the simplex spanned by the vertices of $X$.
A related source of persistence modules are sublevel set filtrations. These are associated to a function $f\colon X\to{\mathbb{R}}$, where $X$ is a topological space. They are motivated by ideas of Morse theory, where $X=M$ is assumed to be a smooth manifold and $f$ is a Morse function (smooth function whose critical points are nondegenerate).
Let $f\colon X\to{\mathbb{R}}$ be a (continuous) function on a topological space $X$. The [*sublevel set filtration*]{} associated to $(X,f)$ is the family $(X^a)_{a\in{\mathbb{R}}}$ where $X^a=f^{-1}(-\infty,a]$, which can also be viewed as a functor $S\colon ({\mathbb{R}},\leq)\to{\mathbf{Top}}$. Composing this functor with $k$-th singular homology yields a persistence module $H_k\circ S$ which is called the [*$k$-th sublevel set persistent homology of $(X,f)$*]{}.
Other examples of persistence modules that have been used are lower star filtrations of simplicial complexes, alpha (or Delaunay) complexes, wrap complexes, witness complexes, and many more besides [@edelsbrunner2010computational; @de2004topological; @bauer2017morse].
In order to ensure stability of persistence modules arising in applications despite the noise arising from imprecise measurements, it is important to be able to use approximation techniques. This is done using the notion of $\epsilon$-interleavings. These provide a way to formalise the intuitive notion of approximate isomorphism of persistence modules and can be used to define a notion of distance on the category of persistence modules. For details, see [@chazal2016structure; @bubenik2014categorification; @lesnick2015theory].
Background on magnitude of metric spaces {#section-background-magnitude}
========================================
In this section we will introduce the magnitude of metric spaces. This is a numerical invariant of metric spaces developed by Tom Leinster in [@LeinsterMetricSpace], building on earlier work defining numerical invariants of categories [@LeinsterEulerCharCategory]. Despite these abstract origins, magnitude turns out to be an interesting invariant containing meaningful geometric information. Here we will introduce the basics and attempt to give readers an impression of magnitude’s interest and reach. Readers who wish to know more are strongly recommended to take a look at Leinster’s original paper [@LeinsterMetricSpace] and Leinster and Meckes’s survey [@survey]. We note here that magnitude of metric spaces is just one instance of a more general invariant of enriched categories. The latter is developed in section 1 of [@LeinsterMetricSpace], and we will not say anything about it here.
Here, and in the rest of the paper, we will use the symbol $|X|$ to denote the magnitude of an object $X$. To avoid notational clashes, we will use the symbol $\#X$ to denote the cardinality of a finite set $X$.
Magnitude of finite metric spaces
---------------------------------
We begin with the magnitude of finite metric spaces. This is based almost entirely on section 2 of [@LeinsterMetricSpace].
Let $(X,d)$ be finite metric space. A *weighting* on $X$ is a function $w\colon X\to{\mathbb{R}}$ such that the equality $$\sum_{y\in X}e^{-d(x,y)}w(y)=1$$ is satisfied for every $x\in X$. If $X$ admits a weighting, then we define the *magnitude* of $X$ to be $$|X|=\sum_{x\in X}w(x).$$ This is independent of the choice of weighting. If no weighting exists, then the magnitude of $X$ is not defined.
Suppose that $X$ is a finite metric space with elements $x_1,\ldots,x_n$, and let $Z_X$ denote the $n\times n$ matrix with $(Z_X)_{ij}=e^{-d(x_i,x_j)}$. If it happens that $Z_X$ is invertible, then the magnitude of $X$ is defined and is given by the formula $$\label{equation-inverse}
|X| = \sum_{i,j=1}^n (Z_X^{-1})_{ij}.$$ It can happen that $|X|$ is defined (using weightings) in cases where $Z_X$ is not invertible. (See Lemma 1.1.4 of [@LeinsterMetricSpace].)
For $t>0$, we let $tX$ be the metric space $X$ rescaled by $t$, so that $d_{tX}(x,y)=td_X(x,y)$. There is no simple relationship between $|tX|$ and $|X|$, and as a consequence we gain information by considering all rescalings at once, as in the following definition.
\[definition-magnitude-function\] Let $X$ be a finite metric space. Its *magnitude function* is the (partially defined) function from $(0,\infty)$ to ${\mathbb{R}}$ given by $$t\mapsto |tX|.$$
Let $X$ denote the space consisting of a single point $x$. Then $Z_X$ is the $1\times 1$ matrix $(1)$, so that $Z_X^{-1}=(1)$ and formula gives us $|X| = 1$.
Let $X=\{x_1,x_2\}$ be the two-point space in which $d_X(x_1,x_2)=d$ for some $d>0$. Then $$Z_X = \begin{pmatrix} 1 & e^{-d} \\ e^{-d} & 1\end{pmatrix}$$ so that $$Z_X^{-1} = \frac{1}{1-e^{-2d}}
\begin{pmatrix} 1 & -e^{-d} \\ -e^{-d} & 1 \end{pmatrix}$$ and consequently $$|X|=\frac{2-2e^{-d}}{1-e^{-2d}}=\frac{2}{1+e^{-d}}.$$ The same computation shows that the magnitude function of $X$ is given by $$|tX|=\frac{2}{1+e^{-dt}}$$ with graph:
{width="200pt"}
We see in this case that $|tX|$ varies between $1$ and $2$, tending to $1$ as $t\to 0$ and to $2$ as $t\to \infty$. This suggests that magnitude is an ‘effective number of points’, regarding two points as essentially the same if they are very close, and essentially different if they are very far apart. The latter property generalises.
Let $X$ be a finite metric space. Then $|tX|\to \#X$ as $t\to\infty$, where $\#X$ denotes the cardinality of $X$.
- It is known that all metric spaces with four points or less have magnitude, but there exist spaces with five or more points that do not have magnitude (See pages 870-871 of [@LeinsterMetricSpace]).
- There is a simple formula due to Speyer for the magnitude of *homogeneous* metric spaces, i.e. those that admit a transitive group action (see Proposition 2.1.5 of [@LeinsterMetricSpace]). This allows one to compute magnitude of many simple spaces, for example complete graphs and cyclic graphs. (Graphs are always regarded as metric spaces by equipping them with the shortest path metric.)
- The magnitude function of a finite metric space $X$ can take negative values, it can take values greater than $\#X$, and it can have intervals on which it is increasing or decreasing. Example 2.2.7 of [@LeinsterMetricSpace] gives a demonstration of this on a space $X$ with 5 points.
- It is not always true that $|tX|\to 1$ as $t\to 0$. An example due to Willerton describes a metric space with $6$ points for which $|tX|\to 6/5$ as $t\to 0$. (See Example 2.2.8 of [@LeinsterMetricSpace].)
One may take a data set in the form of a finite subspace of Euclidean space, and take its magnitude or magnitude function, which in this case is always defined. The result is a potentially interesting invariant of such data sets. But for this to be useful, one would like to know that the invariant is stable under perturbations of the data set. In mathematical terms, one would like to know that magnitude is continuous with respect to the Hausdorff metric on subsets of Euclidean space. This is currently unknown, although Meckes has shown that in this situation the function $X\mapsto |X|$ is *lower semicontinuous*, meaning roughly that magnitude may jump upwards but not downwards. (See Theorem 2.6 of [@MeckesPositiveDefinite] and the paragraph that follows it.)
Magnitude of compact metric spaces
----------------------------------
Magnitude also makes sense for certain classes of compact, infinite metric spaces. Here we will recall the relevant definition and some of the main results. Good references for this section are section 3 of [@LeinsterMetricSpace] and the survey [@survey].
In the following we will consider *positive definite* metric spaces, which are metric spaces $X$ with the property that for every finite subspace $F$ the matrix $Z_F$ is positive definite. Any subset of Euclidean space, with its induced metric, is positive definite.
Let $(X,d)$ be a compact positive definite metric space. *The magnitude* of $X$ is defined by the formula: $$|X|=\sup\{|W| : W\subseteq X,\ W\text{ finite.}\}$$ The *magnitude function* of $X$ is defined by $t\mapsto |tX|$ for $t\in [0,\infty)$.
We let $S^1_\mathrm{eucl}$ denote the Euclidean circle, i.e. the unit circle in the plane with its induced metric. And we let $S^1_\mathrm{geo}$ denote the same circle with its geodesic metric of total arclength $2\pi$. Both are positive definite. Then the magnitude function of the Euclidean circle is given by $$|t\cdot S^1_\mathrm{eucl}|=\pi t+O(t^{-1})\qquad\text{as}\qquad t\to\infty$$ and the magnitude function of the geodesic circle is given by $$|t\cdot S^1_{\mathrm{geo}}|=\frac{\pi t}{1-e^{-\pi t}}.$$ See Theorems 13 and 14 of [@asymptotic]. It has been argued [@asymptotic] that the linear term $\pi t$ in these expressions corresponds to half the length of the circle, whereas the absence of the constant term corresponds to the fact that the Euler characteristic of the circle is zero.
In general, computing the magnitude of infinite spaces is difficult, and existing computations tend to require a significant amount of analysis. A useful survey on this subject is given in [@survey]. Important recent progress by Gimperlein and Goffeng [@GimperleinGoffeng] shows that for appropriate $X\subseteq{\mathbb{R}}^{2n+1}$, the asymptotics of the magnitude function as $t\to\infty$ encode geometric properties including volume, surface area and mean curvature.
Background on magnitude homology {#section-background-mh}
================================
Singular homology can be regarded as a *categorification* of the Euler characteristic: The Euler characteristic is a function taking values in the set of integers, whereas homology is a functor taking values in the category of graded abelian groups, and the function can be obtained from the functor by taking the alternating sum of the ranks: $$\chi(X) = \sum_{i=0}^\infty (-1)^i \operatorname{\mathrm{rank}}H_i(X)$$ This is a classical story, but there are more recent examples of such categorifications, notably Khovanov homology, which categorifies the Jones polynomial, and Knot Floer homology, which categorifies the Alexander polynomial.
Hepworth and Willerton [@richard] together with Leinster and Shulman [@shulman] introduced *magnitude homology*, a categorification of magnitude. (Precisely, Hepworth and Willerton first introduced magnitude homology in the case of graphs, and Leinster and Shulman later extended this to arbitrary metric spaces and very general enriched categories.) More recently, Nina Otter [@Otter] introduced a persistent version of magnitude homology called *blurred magnitude homology*.
In this section we will introduce magnitude homology and its blurred variant, and we will conclude by giving an explicit formula to extract the magnitude of a space from the barcode of its blurred magnitude homology. It is this story that we will *reverse* in the rest of the paper, using its conclusion as the *definition* of the magnitude of persistence modules. Applying this to persistent homology theories other than blurred magnitude homology, we will then obtain new notions of magnitude of metric spaces.
Magnitude homology
------------------
Given a metric space $X$ and elements $x_0,\ldots,x_k\in X$, we define $$\ell(x_0,\ldots,x_k)
=
d(x_0,x_1)+d(x_1,x_2)+\cdots+d(x_{k-1},x_k).$$ We think of this as the *length* of the tuple $(x_0,\ldots,x_k)$.
\[definition-mh\] The *magnitude chain complex* of a metric space $X$ consists of the abelian groups $$\operatorname{MC}_{k,l}(X)
=
\left\langle
(x_0,\ldots,x_k)\in X^{k+1}
\ \middle|\
\begin{array}{l}
x_{0}\neq x_1\neq\cdots\neq x_k,\\
\textstyle \ell(x_0,\ldots,x_k)=l
\end{array}
\right\rangle$$ with $l\in[0,\infty)$ and $k$ a non-negative integer. Here, and in what follows, angled brackets $\langle\quad\rangle$ denote free ${\mathbb{Z}}$-modules. The boundary operators $$\partial_{k,l}\colon \operatorname{MC}_{k,l}(X)\to \operatorname{MC}_{k-1,l}(X)$$ are defined by the rule $$\partial_{k,l}(x_0,\ldots,x_k)
=
\sum_{i=0}^k (-1)^i(x_0,\ldots,\widehat{x_i},\ldots,x_k),$$ where the term $(x_0,\ldots,\widehat{x_i},\ldots,x_k)$ is omitted if $\ell(x_0,\ldots,\widehat{x_i},\ldots,x_k)<l$. The *magnitude homology* $\operatorname{MH}_{k,l}(X)$ of $X$ is defined to be the homology of the magnitude chains $$\operatorname{MH}_{k,l}(X)=H_k(\operatorname{MC}_{\ast,l}(X))$$ where again $k$ is a non-negative integer and $l\in[0,\infty)$.
Magnitude homology is a categorification of the magnitude, in the sense that the graded Euler characteristic of magnitude homology coincides with the magnitude itself, as shown in the next proposition. This is categorification in the same sense that Khovanov homology categorifies the Jones polynomial, and that knot Floer homology categorifies the Alexander polynomial.
\[proposition-alternating\] Let $X$ be a finite metric space. Then $$|tX|=\sum_{l\in[0,\infty)}
\sum_{k=0}^\infty
(-1)^k\operatorname{\mathrm{rank}}(\operatorname{MH}_{k,l}(X))e^{-lt}$$ for $t$ sufficiently large.
\[remark-l-values\] The formula above requires some elaboration. Consider the set of real numbers occuring as $\ell(x_0,\ldots,x_k)$ for $x_0,\ldots,x_k\in X$, $k\geq 0$, with consecutive $x_i$’s distinct. Let us call these *$\ell$-values*. Since $X$ is finite, there is a positive minimum nonzero distance between elements of $X$, call it $\delta>0$, and then all $\ell$-values satisfy the following inequality: $$\label{equation-ell-bound}
\ell(x_0,\ldots,x_k)\geq \delta k$$
A first consequence of equation is that, for a fixed choice of $l\in[0,\infty)$, the set of $k$ for which $\operatorname{MH}_{k,l}(X)\neq 0$ is bounded above by $l/\delta$. That is because if $\operatorname{MH}_{k,l}(X)\neq 0$ then $l$ must be an $\ell$-value $\ell(x_0,\ldots,x_k)$. It follows that in Proposition \[proposition-alternating\] the inner sum is finite for each $l$.
The second consequence of equation is that for any positive real $N$, the collection of $\ell$-values satisfying $\ell(x_0,\ldots,x_k)\leq N$ is finite (because then $k\leq N/\delta$, and $X$ is finite). It follows that the set of all $\ell$-values can be totally ordered $0=l_0<l_1<l_2<\cdots$. Thus the outer series in Proposition \[proposition-alternating\] can be rewritten as the (infinite) sum over the $l_i$.
A graph can be regarded as a metric space by taking the set of vertices and equipping them with the shortest path metric. This is the original setting of magnitude homology in [@richard], where a number of explicit examples (done using computer algebra) are described. We include two of these here as an illustration. Figure \[TableFiveCycle\] shows the ranks of the magnitude homology $\operatorname{MH}_{k,l}(C_5)$ of the cyclic graph with $5$ vertices, and Figure \[TablePetersen\] shows the ranks of the magnitude homology $\operatorname{MH}_{k,l}(\mathit{P\!etersen})$ of the Petersen graph. (Note that the images and tables in Figures \[TableFiveCycle\] and \[TablePetersen\] are taken directly from [@richard].) Observe that in each case, the rank of $\operatorname{MH}_{0,0}(G)$ is the number of vertices, and the rank of $\operatorname{MH}_{1,1}(G)$ is the number of oriented edges. These are general features, but the question of what data is encoded in $\operatorname{MH}_{k,l}(G)$ for other choices of $k$ and $l$ remains mysterious. Another general feature visible here is that the nonzero magnitude homology groups lie in a range of pairs $(k,j)$ bounded by two diagonals, one of them the diagonal $k=j$, and the other determined by the diameter of the graph.
in [0,72,...,288]{} (+90:2cm) – (+72+90:2cm); in [0,72,...,288]{} (+90:2cm) circle (0.1cm);
[rrrrrrrrrrrrrr]{} &&&&&&&$k$\
&&0&1&2&3&4&5&6&7&8&9&10&11\
&0 & 5\
& 1 & & 10\
&2 & && 10\
& 3 &&& 10 & 10\
& 4 &&&& 30 & 10\
& 5 &&&&& 50 & 10\
$l$& 6 &&&&& 20 & 70 & 10\
& 7 &&&&&& 80 & 90 & 10\
& 8 &&&&&&& 180 & 110 & 10\
& 9 &&&&&&& 40 & 320 & 130 & 10\
&10 &&&&&&&& 200 & 500 & 150 & 10\
&11 &&&&&&&&& 560 & 720 & 170 & 10\
[rrrrrrrrrrr]{} &&&&&$k$\
&0&1&2&3&4&5&6&7&8\
& 0 & 10\
& 1 & & 30\
& 2 & && 30\
& 3 &&& 120 & 30\
$l$ & 4 &&&& 480 & 30\
& 5 &&&&& 840 & 30\
& 6 &&&&& 1440 & 1200 & 30\
& 7 &&&&&& 7200 & 1560 & 30\
& 8 &&&&&&& 17280 & 1920 & 30\
Magnitude homology has many good characteristics of homology theories and categorification:
- Magnitude homology refines magnitude: there are finite metric spaces with the same magnitude but non-isomorphic magnitude homologies [@Gu].
- Magnitude homology can contain torsion [@KanetaYoshinaga]. Thus the magnitude homology contains more data than just the ranks $\operatorname{\mathrm{rank}}(\operatorname{MH}_{k,l}(X))$.
- Magnitude homology has properties that categorify known properties of the magnitude. In the setting of graphs, a Künneth theorem categorifies the known product formula for magnitude, and a Mayer-Vietoris sequence categorifies the known inclusion-exclusion formula. See [@richard].
- Magnitude homology contains information about geometric features of a metric space, for example it can precisely detect the property of being Menger convex, and it contains obstructions to the existence of upper bounds on curvature, and to the existence of closed geodesics. See [@shulman], [@Gomi], [@Asao].
- Magnitude homology has been computed fully in several interesting examples, including trees, complete graphs, cycle graphs, and the icosahedral graph. See [@richard] and [@Gu].
Blurred magnitude homology
--------------------------
We now describe some recent work of Nina Otter [@Otter] that connects magnitude homology with persistent homology, specifically the Vietoris-Rips complex. We also give a new result that relates magnitude with barcodes for the first time.
\[definition-bmh\] The *blurred magnitude chain complex* of a metric space $X$ is the chain complex of persistence modules $\operatorname{BMC}_\ast(X)$ defined by the rule $$\operatorname{BMC}_k(X)(l)
=
\left\langle
(x_0,\ldots,x_k)\in V^{k+1}
\ \middle|\
\begin{array}{l}
x_{0}\neq x_1\neq\cdots\neq x_k,\\
\textstyle \ell(x_0,\ldots,x_k)\leq l
\end{array}
\right\rangle$$ where $l$ is the persistence parameter and $k$ is a non-negative integer. The boundary operators $$\partial_{k}\colon \operatorname{BMC}_k(X)\to \operatorname{BMC}_{k-1}(X)$$ are defined by the rule $$\partial_{k,l}(x_0,\ldots,x_k)
=
\sum_{i=0}^k (-1)^i(x_0,\ldots,\widehat{x_i},\ldots,x_k).$$ The *blurred magnitude homology* $\operatorname{BMH}_\ast(X)$ of $X$ is defined to be the homology of the blurred magnitude chains: $$\operatorname{BMH}_k(X)=H_k(\operatorname{BMC}_\ast(X))$$ for $k$ a non-negative integer.
One of the main results of Otter’s paper [@Otter] is that it compares the blurred magnitude homology of a metric space $X$ with the homology of its Rips complex. The main idea of this comparison is that there are maps $$\operatorname{BMC}_k(X)(s)\to C_k({\mathcal{R}}^\mathrm{sim}(X)(s))
\text{\ \ and\ \ }
C_k({\mathcal{R}}^\mathrm{sim}(X)(s))\to \operatorname{BMC}_k(X)(ks),$$ where ${\mathcal{R}}^\mathrm{sim}(X)$ denotes a variant of the Rips chain complex, having the same persistent homology. These comparison maps are a multiplicative version of an interleaving, and although the constant appearing here is the degree $k$ in the chain complex, and in particular is not constant, it is nevertheless sufficient for Otter to prove the following theorem.
\[theorem-nina\] $$\lim_{0\leftarrow\epsilon}\operatorname{BMH}_\ast(X)(\epsilon)
\cong
\lim_{0\leftarrow\epsilon}H_\ast({\mathcal{R}}(X)(\epsilon))$$
The quantity $\lim_{0\leftarrow\epsilon}H_\ast({\mathcal{R}}_\ast(X)(\epsilon))$ is the *Vietoris homology* of $X$, a version of homology developed for metric spaces. In good cases, e.g. when $X$ is a compact Riemannian manifold, it coincides with the singular homology of $X$. This theorem therefore demonstrates for the first time a concrete connection between magnitude homology and ordinary homology of spaces.
We now state a new result that gives the relation between magnitude and the barcode decomposition of the blurred magnitude homology.
\[theorem-magBMH\] Let $X$ be a finite metric space and let $\operatorname{BMH}_\ast(X)$ denote its blurred magnitude homology. Suppose that $\operatorname{BMH}_\ast(X)$ has barcode whose bars in degree $k\geq 0$ are $[a_{k,0},b_{k,0}),[a_{k,1},b_{k,1}),\ldots$. Then the magnitude of $X$ is given by the formula $$|tX|
=
\sum_{k=0}^\infty
\sum_{i=1}^{m_k}
(-1)^k (e^{-a_{k,i}t} - e^{-b_{k,i}t})$$ for $t$ sufficiently large.
The proof of Theorem \[theorem-magBMH\] itself is rather long and technical, thanks to the convergence issues, and so we have relegated it to Appendix \[section-long-proof\]. Instead, at this point we offer a sketch.
$$\begin{aligned}
|tX|
&\stackrel{1}{=}
\sum_{j=0}^\infty
\sum_{k=0}^\infty
(-1)^k\operatorname{\mathrm{rank}}(\operatorname{MH}_{k,l_j}(X))e^{-l_jt}
\\
&\stackrel{2}{=}
\sum_{j=0}^\infty
\sum_{k=0}^\infty
(-1)^k
\left[
\operatorname{\mathrm{rank}}(\operatorname{BMH}_{k}(X)(l_j))
-
\operatorname{\mathrm{rank}}(\operatorname{BMH}_{k}(X)(l_{j-1}))
\right]e^{-l_jt}
\\
&\stackrel{3}{=}
\sum_{k=0}^\infty
(-1)^k
\sum_{j=0}^\infty
\operatorname{\mathrm{rank}}(\operatorname{BMH}_{k}(X)(l_j))(e^{-l_jt}-e^{-l_{j+1}t})
\\
&\stackrel{4}{=}
\sum_{k=0}^\infty
(-1)^k
\sum_{i=0}^\infty
(e^{-a_it}-e^{-b_it})\end{aligned}$$
Step 1 is precisely the formula of Proposition \[proposition-alternating\]. Step 2 follows as a consequence of the long exact sequences $$\cdots
\to
\operatorname{BMH}_k(X)(l_{j-1})
\to
\operatorname{BMH}_k(X)(l_j)
\to
\operatorname{MH}_{k,l_j}(X)
\to
\cdots$$ which relate the blurred and original versions of magnitude homology. (See Proposition \[proposition-absolute-blurred\].) Step 3 amounts to exchanging the order of the sums and then rearranging the inner sum. And step 4 follows by rearranging the series in line 3 as a sum over bars rather than over the lengths $l_j$.
By standard homological algebra, these definitions have the following immediate consequence, which relates ordinary and blurred magnitude homology.
\[proposition-absolute-blurred\] Let $X$ be a finite metric space and let $0=l_0<l_1<l_2<\cdots$ be the distinct real numbers occuring as $\ell(x_0,x_1,\ldots,x_k)$ for $x_0,\ldots,x_k\in X$, $k\geq 0$. Then for each $k\geq 0$ and $j>0$ we have a short exact sequence: $$0\to\operatorname{BMC}_k(X)(l_{j-1})\to\operatorname{BMC}_{k}(X)(l_j)\to\operatorname{MC}_{k,l_j}(X)\to0$$ Consequently in homology there is a long exact sequence: $$\cdots\to\operatorname{BMH}_k(X)(l_{j-1})\to\operatorname{BMH}_k(X)(l_j)\to\operatorname{MH}_{k,l_j}(X)\to\cdots$$
Magnitude of persistence modules {#section-persistent-magnitude}
================================
In Theorem \[theorem-magBMH\] in the previous section, we saw a formula expressing the magnitude function of a finite metric space $X$ in terms of the barcode decomposition of its blurred magnitude homology. In this section we will turn that result on its head, and use the formula to *define* a numerical invariant of persistence modules and graded persistence modules, and explore its formal properties.
In subsequent sections we will apply our new invariant to persistent homology groups, in order to obtain new invariants of finite metric spaces (or of whatever input the persistent homology theory accepts).
In this section we will usually work with finitely presented persistence modules, and finitely presented graded persistence modules. In the latter case, we mean that the graded persistence module has finitely many generators and relations in total, so that it is nonzero in only finitely many degrees. Thus, our persistence modules will always be direct sums of finitely many interval modules of the form ${\mathbbm{k}}[a,b)$ where possibly $b=\infty$. This restriction allows us to work with the most relevant examples such as Rips and Čech complexes while keeping technicalities to a minimum. At the end of the section we will offer two different perspectives on the persistent magnitude, via the derived associated graded module and the Laplace transform. These offer potential for extending the scope of persistent magnitude beyond the present setting.
Persistent magnitude
--------------------
\[definition-mag\] Let $M$ be a finitely presented persistence module with barcode decomposition $$M
\cong
\bigoplus_{i=1}^n
{\mathbbm{k}}[a_i,b_i).$$ The *persistent magnitude* or simply *magnitude* of $M$ is the real number $$|M|
=
\sum_{i=1}^n (e^{-a_i}- e^{-b_i})$$ where by convention $e^{-\infty}=0$.
For interval modules we have $|{\mathbbm{k}}[a,b)|=e^{-a}-e^{-b}$ and $|{\mathbbm{k}}[a,\infty)| = e^{-a}$. Thus longer intervals have greater magnitude, in line with the general philosophy of persistent homology that longer bars — the features that persist longer — are the more significant. But note also that interval modules of fixed length have greater magnitude the closer they are to $0$, i.e. the sooner they begin.
\[proposition-ses\] If $$0\to M\to N\to P\to 0$$ is a short exact sequence of finitely presented persistence modules, then $|N| = |M| + |P|$.
We will give a proof of this proposition in section \[subsection-associated-graded\] below, and another proof in section \[subsection-laplace\].
Let $M_\ast$ be a finitely presented graded persistence module. The *persistent magnitude* of $M_\ast$ is defined as follows: $$\begin{aligned}
|M_\ast|
&=
\sum_{i} (-1)^i|M_i|
\end{aligned}$$
If $C_\ast$ is a chain complex of persistence modules, then we obtain two graded persistence modules, namely $C_\ast$ itself, and the homology $H_\ast(C)$. The persistent magnitude of these is related by the following result, whose proof is a standard consequence of additivity with respect to short exact sequences. (Compare with the proof of Theorem 2.44 of [@Hatcher].)
\[proposition-maghom\] Let $C_\ast$ be a finitely presented chain complex of persistence modules. Then $$|H_\ast(C)| = |C_\ast|.$$
Rescaling and the magnitude function
------------------------------------
In Definition \[definition-magnitude-function\] the magnitude of a finite metric space was extended from a number to a function by means of rescaling the metric space. We now do the same with persistent magnitude.
Given a persistence module $M$ and a real number $t\in(0,\infty)$, we can define the *rescaled module* $tM$ to be the new persistence module defined by $$tM (s) = M(s/t)$$ for $s\in[0,\infty)$. More precisely, $tM$ is obtained from $M$ by precomposing with the functor from $[0,\infty)$ to itself that sends $s$ to $s/t$. This operation extends to graded persistence modules and chain complexes of persistence modules in the evident way.
One can think of the definition of this rescaling operation as saying that features of $M$ that occur at $s$ become features of $tM$ that occur at $ts$.
One can check that $t{\mathbbm{k}}[a,b) = {\mathbbm{k}}[ta,tb)$. Thus the effect of the rescaling operation on the barcode of a finitely presented persistence module $M$ is to simply rescale it by $t$: the barcode of $tM$ is obtained from that of $M$ by applying a scale factor of $t$ in the horizontal direction.
Rescaling of persistence modules is designed to interact nicely with rescaling of metric spaces. Recall that if $X$ is a metric space and $t\in (0,\infty)$, then the rescaling $tX$ is the metric space with the same underlying set and with metric defined by $d_{tX}(a,b) = t d_X(a,b)$. Then one can check directly that the Rips and Čech complexes satisfy $$tC_\ast({\mathcal{R}}(X)) = C_\ast{\mathcal{R}}(tX)
\quad\text{and}\quad
tC_\ast({\mathcal{\check{C}}}(X)) = C_\ast({\mathcal{\check{C}}}(tX))$$ and similarly for the persistent homology.
\[definition-persistent-magnitude-function\] The *persistent magnitude function* or simply *magnitude function* of a finitely presented persistence module $M$ is the function $(0,\infty)\to{\mathbb{R}}$ defined by $$t\longmapsto |tM|.$$ If $M$ has direct sum decomposition $M \cong \bigoplus_{i=1}^n {\mathbbm{k}}[a_i,b_i)$, then the magnitude function is given by the formula $$|tM|
=
\sum_{i=1}^n (e^{-a_it}- e^{-b_it})$$ where again by convention $e^{-\infty}=0$.
The extremal behaviour of the magnitude function singles out two special classes of bars, as we see in the next proposition. Its proof is an immediate consequence of the definitions.
\[proposition-limiting\] Let $M$ be a finitely presented persistence module. Then:
- $\lim_{t\to 0}|tM|$ is the number of bars in $M$ of the form ${\mathbbm{k}}[a,\infty)$.
- if all bars of $M$ are contained in $[0,\infty)$ (or equivalently if $M(s)=0$ for $s<0$), then $\lim_{t\to\infty}|tM|$ is the number of bars in $M$ of the form ${\mathbbm{k}}[0,b)$, including the case $b=\infty$.
One can think of this as follows: As $t\to 0$, we are scaling down the barcode of $M$, so that any finite bars eventually disappear at $0$, while any infinite bars remain, but all become indistinguishable; in this limit the magnitude function simply counts the latter. As $t\to \infty$, we are scaling up the barcode of $M$, so that any bars which begin *after* $0$ eventually disappear at infinity, while all bars that begin at $0$ remain but become indistinguishable; in this limit the magnitude function again just counts the latter.
\[remark-homeo\] Occasionally, it is useful to reparameterise a persistence module $M$ by an orientation preserving homeomorphism $h\colon{\mathbb{R}}\to{\mathbb{R}}$. In this case, we can define the [*reparameterised module*]{} $hM$ by: $$hM(s)=M(h^{-1}(s))$$ The properties of this definition generalise the ones for rescaling by a positive real number in a natural way. For instance, $$h{\mathbbm{k}}[a,b)={\mathbbm{k}}[h(a),h(b))$$ and if $M\cong\bigoplus_{i=1}^n{\mathbbm{k}}[a_i,b_i)$ we have $$|hM|=\sum_{i=1}^n (e^{-h(a_i)}- e^{-h(b_i)}).$$
This definition has the following immediate but useful consequence:
\[lemma-homeo\] Suppose $M$ is a finitely presented persistence module with magnitude function $$|tM|=\sum_{i=1}^n \lambda_i e^{-r_i t}$$ and $h\colon{\mathbb{R}}\to{\mathbb{R}}$ is an orientation preserving homeomorphism. Then the reparameterised module $hM$ has magnitude function $$|thM|=\sum_{i=1}^n \lambda_i e^{-h(r_i)t}.$$
In section 6 of [@bobrowski], Bobrowski and Borman define the *Euler characteristic* of a barcode with no bars of length $\infty$, or in other words of a finitely presented graded persistence module $M$ with the property that $M(t)=0$ for $t$ sufficiently large. The definition is given by $$\chi(M_*)=\sum_{i=1}^b(-1)^{|\beta_i|}(b_i-a_i)$$ where $\beta_1,\ldots,\beta_b$ are the bars of $M_\ast$, and $\beta_i=[a_i,b_i)$. They then describe a connection between the Euler integral of a function and the Euler characteristic of the barcode of the persistent homology of that function. Observe that the Euler characteristic of a finitely presented persistence module is related to its magnitude as follows: $$\chi(M_*)=|M_\ast|'(0)$$ Thus the magnitude of a persistence module encodes its Euler characteristic. (In order to form $|M_\ast|'(0)$ above we have extended the magnitude function $t\mapsto |tM_\ast|$ to $0$ in the evident way (see the formula in Definition \[definition-persistent-magnitude-function\]) and then taken a one-sided derivative.)
Persistent magnitude and products
---------------------------------
Now we will explore how the persistent magnitude interacts with the tensor product of persistence modules, which was described in Section \[section-background-persistence\].
\[proposition-tensor\] Let $M$ and $N$ be finitely presented persistence modules. Then $$\label{equation-products}
|M|\cdot|N|
=
|M\otimes N|
-|\operatorname{Tor}_1(M,N)|.$$
It suffices to prove this when $M$ and $N$ are interval modules. In this case both sides of the equation can be computed directly using the results stated in Section \[section-background-persistence\].
Let us explain why equation states that ‘persistent magnitude respects tensor products’, since it may look a little odd from that point of view. Homological algebra tells us that to fully understand the tensor product of $M$ and $N$, we must consider the graded object $\operatorname{Tor}_\ast(M,N)$. (Serre’s intersection formula in algebraic geometry is a good example of this principle in action.) Thus, the ‘true’ statement that persistent magnitude respects products would be $$|M|\cdot |N|
=
|\operatorname{Tor}_\ast(M,N)|
=
\sum_{i=0}^\infty(-1)^i |\operatorname{Tor}_i(M,N)|.$$ But this reduces to exactly the equation appearing in the proposition.
The usefulness of Proposition \[proposition-tensor\] is that it leads to product preserving properties of the magnitude of persistent homology theories. Indeed, suppose given a theory that assigns to objects $X$ a graded persistence module $A_\ast(X)$, and suppose that objects $X$ and $Y$ can be equipped with a product $X\times Y$ in such a way that we have a Künneth theorem: $$0\to A_\ast(X)\otimes A_\ast(Y)
\to
A_\ast(X\times Y)
\to
\operatorname{Tor}_1(A_\ast(X),A_{\ast-1}(Y))
\to
0$$ Examples (and non-examples) of such Künneth theorems are discussed in [@BubenikMilicevic], [@PSS] and [@CarlssonFilippenko]. In such a setting, Proposition \[proposition-tensor\] can be used to prove the identity $$|A_\ast(X\times Y)|
=
|A_\ast(X)|\cdot|A_\ast(Y)|.$$ Indeed, we have $$\begin{aligned}
|A_\ast(X\times Y)|
&=
|A_\ast(X)\otimes A_\ast(Y)|
+
|\operatorname{Tor}_1(A_\ast(X),A_{\ast-1}(Y))|
\\
&=
|A_\ast(X)\otimes A_\ast(Y)|
-
|\operatorname{Tor}_1(A_\ast(X),A_{\ast}(Y))|
\\
&=
|A_\ast(X)|\cdot|A_\ast(Y)|+|\operatorname{Tor}_1(A_\ast(X),A_\ast(Y))|
- |\operatorname{Tor}_1(A_\ast(X),A_\ast(Y))|
\\
&=
|A_\ast(X)|\cdot |A_\ast(Y)|\end{aligned}$$ where the first equality comes from the short exact sequence (Proposition \[proposition-ses\]) and the third comes from .
Persistent magnitude via derived associated graded modules {#subsection-associated-graded}
----------------------------------------------------------
In this subsection we will give a perspective on the definition of persistent magnitude using the homological algebra of the ‘associated graded’ or ‘causal onset’ functor. This will allow us to give a proof of Proposition \[proposition-ses\]. It also gives a potential avenue for extending the definition of persistent magnitude beyond the case of finitely presented modules.
As described in section \[section-background-persistence\] we denote by ${\mathbf{PersMod}}$ the category of persistence modules and by ${\mathbf{GrMod}}$ the category of ${\mathbb{R}}$-graded modules. These are abelian categories, and ${\mathbf{PersMod}}$ has enough projectives — they are the interval modules ${\mathbbm{k}}[a,\infty)$ for $a\in{\mathbb{R}}$. For $a\in{\mathbb{R}}$ we let ${\mathbbm{k}}_a$ denote the object of ${\mathbf{GrMod}}$ consisting of ${\mathbbm{k}}$ in grading $a$ and $0$ in all other gradings.
The *associated graded functor* ${\mathrm{Gr}}\colon{\mathbf{PersMod}}\to{\mathbf{GrMod}}$ is defined by $${\mathrm{Gr}}(M)(s) = \frac{M(s)}{\sum_{s'<s}\mathrm{im}(M(s')\to M(s))}$$ for $s\in{\mathbb{R}}$. The functor ${\mathrm{Gr}}$ is right exact, and we denote its derived functors by ${\mathrm{Gr}}_i(M)$, $i\geq 0$, with ${\mathrm{Gr}}_0(M)={\mathrm{Gr}}(M)$.
The terminology above was chosen because if the persistence module $M$ is obtained from an ${\mathbb{R}}$-filtered vector space in the evident way, then ${\mathrm{Gr}}(M)$ is nothing other than the ${\mathbb{R}}$-graded vector space associated to $M$. The functor $M\mapsto {\mathrm{Gr}}(M)_s$ appears in [@vongmasacarlsson], where it is denoted by $\mathcal{O}_s$, and called the *causal onset* functor.
\[example-gr\] For a free module ${\mathbbm{k}}[a,\infty)$ we have ${\mathrm{Gr}}_0({\mathbbm{k}}[a,\infty))={\mathrm{Gr}}({\mathbbm{k}}[a,\infty))={\mathbbm{k}}_a$ and ${\mathrm{Gr}}_1({\mathbbm{k}}[a,\infty))=0$. And for an interval module ${\mathbbm{k}}[a,b)$ with $a<b$ we have a free resolution $$\cdots\to 0 \to {\mathbbm{k}}[b,\infty)\to {\mathbbm{k}}[a,\infty)\to {\mathbbm{k}}[a,b)\to 0$$ so that ${\mathrm{Gr}}_0({\mathbbm{k}}[a,b))={\mathbbm{k}}_a$ and ${\mathrm{Gr}}_1({\mathbbm{k}}[a,b))={\mathbbm{k}}_b$. It follows that for finitely presented modules, ${\mathrm{Gr}}_i(M)=0$ for $i>1$.
The *graded magnitude* of a finitely presented object of ${\mathbf{GrMod}}$, i.e. a module of the form $\bigoplus_{i=1}^n {\mathbbm{k}}_{a_i}$ is $$\left|\bigoplus_{i=1}^n {\mathbbm{k}}_{a_i}\right|
=
\sum_{i=1}^n e^{-a_i}.$$
The graded magnitude function is clearly additive with respect to short exact sequences of finitely presented graded modules. The computations in Example \[example-gr\] give the following.
\[lemma-graded-persistent\] The persistent and graded magnitude are related as follows. Let $M$ be a finitely presented persistence module. Then $$|M| = |{\mathrm{Gr}}_0(M)| - |{\mathrm{Gr}}_1(M)|.$$
We are now in a position to prove that magnitude is additive with respect to short exact sequences.
Like any derived functors, the ${\mathrm{Gr}}_i$ convert a short exact sequence $$0\to M\to N\to P\to 0$$ into a long exact sequence $$0\to {\mathrm{Gr}}_1(M)\to {\mathrm{Gr}}_1(N)\to {\mathrm{Gr}}_1(P)
\to {\mathrm{Gr}}_0(M)\to {\mathrm{Gr}}_0(N)\to {\mathrm{Gr}}_0(P)\to 0,$$ and the statement of the proposition amounts to the claim that the alternating sum of the magnitudes of the modules in this sequence is zero. But this is a standard consequence of additivity with respect to short exact sequences, which in the case of graded magnitude is immediate from the definitions. (Compare with the proof of Theorem 2.44 of [@Hatcher].)
The proof of Proposition \[proposition-ses\] shows that the magnitude of a persistence module $M$ depends only on its ‘derived associated graded’ modules ${\mathrm{Gr}}_0(M)$ and ${\mathrm{Gr}}_1(M)$. In simpler terms, the magnitude depends not on the lengths of the bars in the barcode, but only on the collection of start and end points of bars in the barcode. One may then argue that magnitude does not contain any ‘persistent’ information, only ‘graded’ information. However, the same comment can apply to *any* invariant of persistence modules that is additive with respect to short exact sequences, thanks to the short exact sequences $$0\to {\mathbbm{k}}[b,\infty)\to {\mathbbm{k}}[a,\infty)\to {\mathbbm{k}}[a,b)\to 0.$$ From the point of view of graded modules, persistence modules and persistent homology theories should perhaps then be regarded as an excellent source of interesting examples.
Persistent magnitude and the Laplace transform {#subsection-laplace}
----------------------------------------------
Here is an alternative approach to the persistent magnitude using the Laplace transform, that we believe is the ‘correct’ way to understand persistent magnitude.
Let $M$ be a finitely presented persistence module and let $$\operatorname{\mathrm{rank}}(M)\colon{\mathbb{R}}\to{\mathbb{R}}$$ be its associated rank function, i.e. $\operatorname{\mathrm{rank}}(M)(s)=\operatorname{\mathrm{rank}}(M(s))$ for $s\in{\mathbb{R}}$. This is a step function, given by the sum of the indicator functions of the bars of $M$. It has a derivative in the distributional sense, given by $$\operatorname{\mathrm{rank}}(M)' = \sum_{i=1}^n(\delta_{a_i}-\delta_{b_i})$$ where $M\cong\bigoplus_{i=1}^n {\mathbbm{k}}[a_i,b_i)$, and where $\delta_x$ denotes the Dirac delta distribution supported at $x$. Recall that the *bidirectional Laplace transform* $\mathcal{L}\{f\}$ of a function or distribution $f$ is given by $$\mathcal{L}\{f\}(t)=\int_{-\infty}^\infty f(s)e^{-st}ds$$ for $t\in[0,\infty)$. Then one can check directly that $$|tM| = \mathcal{L}\{\operatorname{\mathrm{rank}}'(M)\}(t).$$ In particular, the right hand side can be used as an alternative definition of magnitude function, whereas magnitude is recovered by evaluating at $1$. For example, if $M={\mathbbm{k}}[a,b)$ then $\operatorname{\mathrm{rank}}(M)$ is the step function $1_{[a,b)}$ and $\operatorname{\mathrm{rank}}'(M) = \delta_a - \delta_b$ so that $$\mathcal{L}\{\operatorname{\mathrm{rank}}'(M)\}(t)
=
\int_{-\infty}^\infty (\delta_a(s)-\delta_b(s))e^{-st}ds
=
e^{-at}-e^{-bt}
=
|tM|.$$
From this point of view, the additivity of persistent magnitude with respect to short exact sequences (Proposition \[proposition-ses\]) is an immediate consequence of the fact that if $$0\to M\to N\to P\to 0$$ is a short exact sequence of persistence modules, then $\operatorname{\mathrm{rank}}(N) = \operatorname{\mathrm{rank}}(M) + \operatorname{\mathrm{rank}}(P)$.
In the case of a finitely presented graded persistence module $M_\ast$, we can associate to it its Euler characteristic curve (see [@turner2014persistent Section 3.2] and [@heiss2017streaming; @bobrowski2014topology; @fasy2018challenges] for some related work), i.e. $\chi(M_*)(s)=\sum_{i=0}^{\infty}(-1)^i\operatorname{\mathrm{rank}}(M_i(s))$ for $s\in{\mathbb{R}}$. Then we have: $$|tM_\ast|=\sum_{i=0}^{\infty}(-1)^i|tM_i|
=\sum_{i=0}^{\infty}(-1)^i\mathcal{L}\{\operatorname{\mathrm{rank}}'(M_i)\}(t)
=\mathcal{L}\{\chi'(M_i)\}(t).$$ In other words, the magnitude of a finitely presented graded persistence module is precisely the Laplace transform of the derivative of its Euler characteristic curve. This provides yet another connection between magnitude and TDA.
More explicitly, whenever $M_*$ is a finitely presented graded persistence module and $r_1<\ldots<r_n<r_{n+1}=\infty$ is the sequence of all the startpoints and endpoints in its interval decomposition, we have $$\label{magnitudeviaeuler}
|tM_*|=\sum_{j=1}^n \chi(M_*)(r_j)(e^{-r_j t}-e^{-r_{j+1} t}).$$ One interpretation of this formula is that persistent magnitude of a graded persistence module can be regarded as the ‘filtered Euler characteristic’ associated to it. We include this perspective as a useful alternative point of view, as well as a potential avenue for generalising the magnitude from finitely presented modules to more general modules that, despite not being finitely presented, may nevertheless have a ‘rank function’ or ‘rank distribution’ that we can then differentiate and subject to the Laplace transform. (Here we recall [@chazal2016structure], where persistence modules that do not admit a barcode decomposition are nevertheless equipped with a persistence diagram.)
Magnitude and persistent homology of sublevel sets {#section-sublevel}
==================================================
An important class of filtrations that can be studied by methods of persistent homology are sublevel set filtrations; the study of these is to a large extent inspired by Morse theory. Recall that if $f\colon X\to{\mathbb{R}}$ is a continuous function, then the *sublevel set persistent homology* of $(X,f)$ is the graded persistence module defined by $s\mapsto H_\ast(f^{-1}(-\infty,s])$.
Consider the case of a Morse function $f\colon M\to{\mathbb{R}}$ on a closed smooth manifold. Being Morse, it only has finitely many nondegenerate critical points. The magnitude of the sublevel set persistence module associated to $(M,f)$, which we refer to as the [*Morse magnitude of $(M,f)$*]{} and write $|t(M,f)|_{\mathrm{Morse}}$, has an explicit formula in terms of the critical points.
Let $f\colon M\to{\mathbb{R}}$ be a Morse function on a closed smooth manifold $M$, Let $S\colon ({\mathbb{R}},\leq)\to{\mathbf{Top}}$ be the sublevel set filtration given by $S(s) = f^{-1}(-\infty,s]$. Then the magnitude function of the sublevel set persistent homology $H_\ast S\colon({\mathbb{R}},\leq)\to{\mathbf{GrVect}}$ is expressed as follows: $$|t(M,f)|_{\mathrm{Morse}}=|t(H_\ast S)|=\sum_{p} (-1)^{{\operatorname{ind}}(p)}e^{-f(p)t}$$ where the sum is over all critical points of $f$.
A basic result of Morse theory [@milnor2016morse Theorem 3.1] states that if $a<b$ are real numbers such that $f^{-1}(a,b]$ contains no critical points of $f$, then $M^b=f^{-1}(-\infty,b]$ deformation retracts onto $M^a=f^{-1}(-\infty,a]$. It follows that the critical values (i.e. the values $f(p)$ of $f$ at the critical points $p$) are the startpoints and endpoints of the interval decomposition of $H_\ast S$. List the critical values as $v_1<v_2<\cdots<v_k$. We may now use the description of magnitude as the filtered Euler characteristic : $$|t(H_\ast S)|=\sum_{i=1}^k\chi(M^{v_i})(e^{-v_it}-e^{-v_{i+1}t}),$$ where $v_{k+1}$ is interpreted as $\infty$. Another basic result of Morse theory [@milnor2016morse Theorem 3.2, Remark 3.3 & Remark 3.4] states the following. Suppose that $b$ is a critical value of $f$, and $a<b$ is such that there are no critical values of $f$ in $(a,b)$, and let $p_1,\ldots,p_r$ be the critical points of $f$ with critical value $b$. Then $M^b$ has a subspace of the form $M^a\cup e^{{\operatorname{ind}}(p_1)}\cup\cdots\cup e^{{\operatorname{ind}}(p_r)}$, and $M^b$ deformation retracts onto $M^a\cup e^{{\operatorname{ind}}(p_1)}\cup\cdots\cup e^{{\operatorname{ind}}(p_r)}$. Using this result, we then have $\chi(M^b) = \chi(M^a) + \sum_{j=1}^r (-1)^{{\operatorname{ind}}(p_j)}
= \chi(M^a) + \sum_{p\colon{\operatorname{ind}}(p)=b} (-1)^{{\operatorname{ind}}(p)}$. It follows that if $v$ is a critical value of $f$, then $$\chi(M^v) = \sum_{p\colon f(p)\leq v} (-1)^{{\operatorname{ind}}(p)}$$ where the sum is over critical points with critical value at most $v$. We now have $$\begin{aligned}
|t(H_\ast S)|
&=
\sum_{i=1}^k\chi(M^{v_i})(e^{-v_it}-e^{-v_{i+1}t})
\\
&=
\sum_{i=1}^k \sum_{p\colon f(p)\leq v_i}(-1)^{{\operatorname{ind}}(p)}
(e^{-v_i t}-e^{-v_{i+1}t})
\\
&=
\sum_p
(-1)^{{\operatorname{ind}}(p)}
\sum_{v_j\colon f(p)\leq v_j}
(e^{-v_j t}-e^{-v_{j+1}t})
\\
&=
\sum_p (-1)^{{\operatorname{ind}}(p)} e^{-f(p) t}\end{aligned}$$
This could be generalised in a straightforward way to the case of the sublevel set filtration associated to any tame function $f\colon X\to{\mathbb{R}}$ on a topological space $X$ using the concept of [*homological critical value [@bubenik2014categorification; @cohen2007stability; @govc2016definition]*]{}.
Consider a subset $A\subseteq {\mathbb{R}}^n$ and filter ${\mathbb{R}}^n$ by $B(A,r)=\bigcup_{a\in A}B(x,r)$. This is the sublevel set filtration associated to the distance function $x\mapsto d(x,A)$. Applying singular homology $H_*$ to this filtration yields a graded persistence module.
For example, for the standard embedding $i\colon S^{n-1}\hookrightarrow{\mathbb{R}}^n$ we obtain the persistence module consisting of a ${\mathbbm{k}}[0,\infty)$ bar in degree $0$ and a ${\mathbbm{k}}[0,1)$ bar in degree $n-1$. In particular, the associated magnitude function, which could also be called ‘the distance magnitude function’ is $|t S^{n-1}|_{\mathrm{dist}}=1+(-1)^{(n-1)}(1-e^{-t})$.
In the case $A$ is finite, the graded persistence module obtained is isomorphic to the Čech persistent homology module associated to $A$. The corresponding magnitude function could therefore reasonably be called the ‘Čech magnitude function’ of $A$ and denoted $|t A|_{\mathrm{\check{C}ech}}$.
Rips magnitude {#section-rips}
==============
In this section we will apply the persistent magnitude developed earlier to the persistent homology of the Rips complex in order to obtain a new, variant form of magnitude of a finite metric space. Here we explore the basic properties of this new invariant, before going into further detail in later sections.
Let $X$ be a finite metric space. Then the chains $C_\ast({\mathcal{R}}(X))$ of its Rips complex are a chain complex of finitely presented persistence modules, concentrated in degrees less than the cardinality of $X$. The same therefore holds for the homology $H_\ast({\mathcal{R}}(X))$. The *Rips magnitude* of $X$ is defined to be the magnitude of the chains of the Rips complex or equivalently the magnitude of its homology: $$|X|_{\mathrm{Rips}}
=|C_\ast({\mathcal{R}}(X))|
= |H_\ast({\mathcal{R}}(X))|$$ The *Rips magnitude function* of $X$ is defined as $t\mapsto|tX|_{\mathrm{Rips}}$, which is equal to $$|tX|_{\mathrm{Rips}}
=|tC_\ast({\mathcal{R}}(X))|
= |tH_\ast({\mathcal{R}}(X))|$$
The Rips magnitude (function) of a finite metric space $X$ has the following properties.
1. The Rips magnitude is computed by the formula: $$\label{subsets-formula}
|tX|_{\mathrm{Rips}} =
\sum_{A\subseteq X,\, A\neq\emptyset}
(-1)^{\#A-1}e^{-\operatorname{diam}(A)t}$$
2. If $H_\ast({\mathcal{R}}(X))$ has barcode with bars $[a_{k,0},b_{k,0}),\ldots,[a_{k,m_k},b_{k,m_k})$ in degree $k$, then: $$\label{barcode-formula}
|tX|_{\mathrm{Rips}}
=
\sum_{k=0}^{\#X-1}\sum_{j=0}^{m_k}
(-1)^k(e^{-a_{k,j}t}-e^{-b_{k,j}t})$$
3. If $0=d_0<d_1<\ldots<d_n$ is the set of all pairwise distances between elements of $X$ arranged in a sequence and $d_{n+1}=\infty$, then: $$\label{euler-formula}
|tX|_{\mathrm{Rips}}=\sum_{j=0}^n \chi({\mathcal{R}}_{d_j}(X))(e^{-d_j t}-e^{-d_{j+1} t}).$$
4. $\lim_{t\to 0}|tX|_{\mathrm{Rips}} = 1$ and $\lim_{t\to\infty}|tX|_{\mathrm{Rips}}=|X|$.
The fourth part of the proposition suggests that the Rips magnitude is an ‘effective number of points’, in the same spirit as the magnitude.
For the first part, we use the description $|tX|_{\mathrm{Rips}} = |t C_\ast({\mathcal{R}}(X))|$. Now $C_\ast({\mathcal{R}}(X))$ has barcode with one bar for each nonempty subset $A$ of $X$, and this bar lies in degree $\#A-1$, and has type $[\mathrm{diam}(A),\infty)$. The definition of persistent magnitude gives the result immediately. The second part follows from the description $|tX|_{\mathrm{Rips}}=|t H_\ast({\mathcal{R}}(X))|$. The third part follows from either the first or the second part using the formula . The fourth part follows directly from the first, and it can also be deduced from the barcode description given there using Proposition \[proposition-limiting\].
Let $X$ denote the space consisting of a single point $x$. Then it has precisely one nonempty subset, namely $X$ itself, and $\#X=1$ while $\operatorname{diam}(X)=0$. Using formula then gives us $|tX|_{\mathrm{Rips}}=1$.
Let $X=\{x_1,x_2\}$ be the two-point space in which $d_X(x_1,x_2)=d$ for some $d>0$.
Let us compute $|tX|_{\mathrm{Rips}}$ using . The nonempty subsets of $X$ are $A_1=\{x_1\}$, $A_2=\{x_2\}$ and $A_3=X$, with $\#A_1=1$, $\#A_2=1$, $\#A_3=2$, $\operatorname{diam}(A_1)=0$, $\operatorname{diam}(A_2)=0$ and $\operatorname{diam}(A_3)=d$. Thus we have $$\begin{aligned}
|tX|_{\mathrm{Rips}}
&=
(-1)^{1-1}\cdot e^{-0t}
+
(-1)^{1-1}\cdot e^{-0t}
+
(-1)^{2-1}\cdot e^{-dt}
\\
&=
1 + 1 -e^{-dt}
\\
&=
2-e^{-dt}.
\end{aligned}$$
Let us also compute $|tX|_{\mathrm{Rips}}$ using . The Rips-homology $H_\ast({\mathcal{R}}(X))$ has barcode with bars $[0,\infty)$ and $[0,d)$ in degree $0$, and no other bars. Thus gives us $$\begin{aligned}
|tX|_{\mathrm{Rips}}
&=
(e^{-0t}-e^{-\infty t})
+
(e^{-0t} - e^{-dt})
\\
&=
(1-0)
+
(1 - e^{-dt})
\\
&=
2-e^{-dt}.
\end{aligned}$$ (Recall our convention that $e^{-\infty}=0$.)
The Rips magnitude is not necessarily increasing or convex, it can attain negative values, and it can attain values greater than the cardinality of the space. For instance, for the complete bipartite graph $K_{5,6}$ (with the graph metric) the Rips magnitude is given by $$|t K_{5,6}|_{\mathrm{Rips}}=11-30e^{-t}+20e^{-2t}.$$ with graph:
{width="200pt"}
And for the complete tripartite graph $X=K_{4,4,4}$, Rips magnitude is given by $$|t K_{4,4,4}|_{\mathrm{Rips}}=12+16e^{-t}-27e^{-2t}$$ with graph:
{width="200pt"}
Note that in general, if the metric only assumes integer values, as in the case of a graph metric, the associated Rips magnitude function is a polynomial in $q=e^{-t}$.
Now we consider the case of subsets of the real line, where the computation is a little less trivial but accessible nonetheless.
\[proposition-subsets-of-R\] Let $A$ be a finite subset of the real line ${\mathbb{R}}$, with its induced metric. Order the elements of $A$ by size, $a_1<\ldots<a_n$. Then $$|t A|_{\mathrm{Rips}}=n-\sum_{j=1}^{n-1}e^{-(a_{j+1}-a_{j})t}.$$
We can think of this result as follows. Take $A_1=\{a_1\}$, $A_2=\{a_1,a_2\}$, $A_3=\{a_1,a_2,a_3\}$ and so on, so that $A=A_n$. Then the proposition tells us that $|t A_1|_{\mathrm{Rips}}=1$, $|t A_2|_{\mathrm{Rips}}
= |t A_1|_{\mathrm{Rips}}+ (1-e^{-(a_2-a_1)t})$, $|t A_3|_{\mathrm{Rips}} = |t A_2|_{\mathrm{Rips}} + (1-e^{-(a_3-a_2)t})$, and so on. In other words, adding a point at the end increases the Rips magnitude by $1-e^{-dt}$, where $d$ is the distance of the new end point from the old one. So if $d$ is very large, we increase the Rips magnitude by almost $1$, whereas if $d$ is very small, then we increase the Rips magnitude only a tiny amount.
Given $B\subseteq A$, let $B_\mathrm{max}$ and $B_\mathrm{min}$ denote the maximum and minimum elements of $B$ respectively. Given $a\leq a'$ in $A$, let $\mathcal{B}_{a,a'}$ denote the set of $B\subseteq A$ for which $B_\mathrm{min}=a$ and $B_\mathrm{max}=a'$, and note that $\operatorname{diam}(B)=a'-a$ for all $B\in\mathcal{B}_{a,a'}$. Thus equation gives us $$|t A|_{\mathrm{Rips}} =
\sum_{a\leq a'}
e^{-(a'-a)t}
\sum_{B\in\mathcal{B}_{a,a'}}
(-1)^{\#B-1}.$$ Now note the following:
- If $a=a'$, then $\mathcal{B}_{a,a'}$ consists of $\{a\}$ alone and $\sum_{B\in\mathcal{B}_{a,a'}}(-1)^{\#B-1}=1$.
- If $a$ and $a'$ are adjacent elements of $A$, then $\mathcal{B}_{a,a'}$ consists of $\{a,a'\}$ alone and $\sum_{B\in\mathcal{B}_{a,a'}}(-1)^{\#B-1}=-1$.
- If $a$ and $a'$ are non-adjacent elements of $A$, then let $A_{a,a'}$ denote the set of those elements of $A$ that lie strictly between $a$ and $a'$. Then any $B\in\mathcal{B}_{a,a'}$ is the disjoint union of $\{a,a'\}$ with a subset $C\subseteq A_{a,a'}$. Thus $\sum_{B\in\mathcal{B}_{a,a'}}
(-1)^{\#B-1}
=
\sum_{C\subseteq A_{a,a'}}(-1)^{\#C+1}
=
-\sum_{C\subseteq A_{a,a'}}(-1)^{\#C}
=0$.
We therefore have $$\begin{aligned}
|t A|_{\mathrm{Rips}}
&=
\sum_{a}
e^{-(a-a)t}\cdot 1
+
\sum_{\substack{a<a'\\ \text{adjacent}}}
e^{-(a'-a)t}\cdot (-1)
+
\sum_{\substack{a<a'\\ \text{non-adjacent}}}
e^{-(a'-a)t}\cdot 0
\\
&=
n
-
\sum_{j=1}^{n-1}
e^{-(a_{j+1}-a_j)t}
\end{aligned}$$ as required.
Rips magnitude of cycle graphs and Euclidean cycles {#section-cycles}
===================================================
Other than the cases treated in the previous section, the topology of Rips complexes seems to be understood at all scales only in the case of finite subsets of the circle. See [@adamaszek2017vietoris] and [@AAFPPJ]. In the case of Riemannian manifolds, Rips complexes are well understood at small scales thanks to a result of Hausmann [@Hausmann Theorem 3.5]. Recently, it has also been shown that Rips complexes can be understood as nerves of certain covers via Dowker duality [@Virk].
In this section we focus on the circle $S^1$, which is equipped either with the Euclidean metric obtained from the standard embedding into ${\mathbb{R}}^2$, and denoted $S^1_\mathrm{eucl}$, or the geodesic (arclength) metric of total length $2\pi$, and denoted $S^1_\mathrm{geo}$.
We will examine the subsets of equally spaced points in these spaces, whose corresponding Rips filtrations are well understood [@adamaszek]. Let $C_n^\mathrm{eucl}$ be the subset of $n$ equidistant points in $S^1_\mathrm{eucl}$ and let $C_n^\mathrm{geo}$ be the subset of $n$ equidistant points in $S^1_{geo}$.
Both of these can be related to cycle graphs. Let $C_n$ be the set of vertices of the $n$-cycle graph, equipped with the graph metric, where two adjacent vertices are considered to be at a distance of $1$. Note that this can be described as the subset of $n$ equidistant points in a geodesic circle of total arclength $n$. The Rips filtration of $C_n$ was studied in [@adamaszek] and the Rips filtrations of $C_n^\mathrm{geo}$ and $C_n^\mathrm{eucl}$ are just reparameterised versions of it.
More precisely, extend the function $[0,2]\to[0,\pi]$ given by $r\mapsto2\arcsin\frac{r}{2}$ to a homeomorphism $\phi\colon {\mathbb{R}}\to{\mathbb{R}}$. Then, by elementary trigonometry, we have the following relations: $$\label{reparameterised}
{\mathcal{R}}_r(C_n^\mathrm{geo})={\mathcal{R}}_{\frac{n}{2\pi} r}(C_n),\qquad{\mathcal{R}}_r(C_n^\mathrm{eucl})={\mathcal{R}}_{\frac{n}{2\pi}\phi(r)}(C_n)$$ and $$\label{euclvsgeo}
\quad{\mathcal{R}}_r(C_n^\mathrm{eucl})={\mathcal{R}}_{\phi(r)}(C_n^\mathrm{geo}).$$
We now state the main results of this section and then proceed to prove them.
\[ripsmag\_cycle\] Writing $q=e^{-t}$, we have: $$|t C_n|_{\mathrm{Rips}}=\sum_{\substack{\text{odd }r|n\\r\neq n}}\frac nr q^{\frac nr \frac{r-1}2}(1-q)+q^{\lfloor\frac n2\rfloor}.$$
This is reminiscent of certain functions that appear in analytic number theory, the simplest of which is probably the sum of divisors function $\sigma_k(n)=\sum_{d|n}d^k$. The appearance of sums over divisors of integers is quite surprising to us and seems to suggest that the Rips magnitudes of $n$-cycle graphs might be intimately connected to number theory in some way. We feel this connection could be worthy of further study:
What is the connection between Rips magnitude of cycles and various functions studied in analytic number theory?
Using Lemma \[lemma-homeo\] and equations and , Proposition \[ripsmag\_cycle\] immediately allows us to infer the following two corollaries:
\[ripsmag\_eucl\] The Rips magnitude of Euclidean cycles is given by[^1]: $$|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}=\sum_{\substack{\text{odd }r|n\\r\neq n}}\frac nr (e^{-\delta_r t}-e^{-\delta_{r,n}t})+e^{-\delta_n t},$$ where $$\delta_r=\operatorname{diam}(C^{\mathrm{eucl}}_r)=2\sin\left(\pi\frac{\lfloor\frac r2\rfloor}r\right)\quad\text{and}\quad\delta_{r,n}=2\sin\left(\pi\left(\frac1n+\frac{\lfloor\frac r2\rfloor}r\right)\right).$$
\[ripsmag\_geo\] The Rips magnitude of geodesic cycles is given by: $$|t C^{\mathrm{geo}}_n|_{\mathrm{Rips}}=\sum_{\substack{\text{odd }r|n\\r\neq n}}\frac nr (e^{-\eta_r t}-e^{-\eta_{r,n}t})+e^{-\eta_n t},$$ where $$\eta_r=2\pi\frac{\lfloor\frac r2\rfloor}r\qquad\text{and}\qquad\eta_{r,n}=2\pi\left(\frac1n+\frac{\lfloor\frac r2\rfloor}r\right).$$
The graph of $|t C^{\mathrm{eucl}}_{60}|_{\mathrm{Rips}}$
{width="200pt"}
looks deceptively similar to the one for $|t C^{\mathrm{geo}}_{60}|_{\mathrm{Rips}}$
{width="200pt"}
so it is perhaps more instructive to look at the difference $|t C^{\mathrm{geo}}_{60}|_{\mathrm{Rips}}-|t C^{\mathrm{eucl}}_{60}|_{\mathrm{Rips}}$:
{width="200pt"}
Note that the Rips filtration of $C_n$ only has “jumps” at the integers. More precisely: $${\mathcal{R}}_r(C_n)={\mathcal{R}}_{\lfloor r\rfloor}(C_n).$$ So it is sufficient to understand the Rips filtration at integer values of the filtration parameter $r$. For integer $r$ such that $0\leq r<\frac{n}2$, Adamaszek [@adamaszek Corollary 6.7] gives the following description of the homotopy types of the various stages of the Rips filtration: $${\mathcal{R}}_r(C_n)\simeq\begin{cases}\bigvee_{n-2r-1}S^{2l};&r=\frac l{2l+1}n,\\
S^{2l+1};&\frac l{2l+1}n<r<\frac{l+1}{2l+3}n.
\end{cases}$$ From this we can immediately infer the Euler characteristics: $$\chi({\mathcal{R}}_r(C_n))=\begin{cases}n-2r;&\text{if $\frac{n}{n-2r}$ is an odd integer,}\\
1;&\text{if $n=2r$,}\\
0;&\text{otherwise.}
\end{cases}$$ Using , this implies $$\begin{aligned}
|t C_n|_{\mathrm{Rips}}
&=\sum_{r=0}^{\lfloor\frac{n}2\rfloor-1}\chi({\mathcal{R}}_r(X))(e^{-r t}-e^{-(r+1) t})+\chi({\mathcal{R}}_{\lfloor\frac{n}2\rfloor}(X))e^{-(\lfloor\frac{n}2\rfloor) t})
\\
&=\sum_{\substack{\text{odd }d|n\\d\neq n}}\frac nd q^{\frac nd \frac{d-1}2}(1-q)+q^{\lfloor\frac n2\rfloor}\end{aligned}$$ where the indices in the first and second summations are related by $\frac{n}{n-2r}=d$. The claim follows.
Another way to calculate the Euler characteristic of ${\mathcal{R}}_r(C_n)$ would be from the simplex counts. We have computational evidence that the number of $i$-simplices in ${\mathcal{R}}_r(C_n)$ for $r<\operatorname{diam}C_n$ is given by $$N_{n,r,i}=\sum_{k=0}^{\lfloor\frac n2\rfloor}\frac n{2k+1}\binom{(2k+1)r-kn+2k}{2k}\binom{(2k+1)r-kn}{i-2k}.$$
Computational evidence seems to suggest that the Rips magnitude $|t C_n|_{\mathrm{Rips}}$ of a cycle graph is convex if and only if $$n\in\{1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 48, 51, 57\}.$$
Rips magnitude of infinite metric spaces {#section-infinite}
========================================
We will now try to understand the Rips magnitude of infinite metric spaces. In the original setting of magnitude, positive definite spaces $X$, such as subspaces of Euclidean space, have the property that if $A\subseteq B\subseteq X$ with $A,B$ finite, then $|A|\leq |B|$. Thus one definition of the magnitude of an infinite metric space $X$ is $$|X| = \sup_{\substack{A\subseteq X\\ A\text{ finite}}}|A|.$$ Proceeding by analogy, we could attempt to make the definition $$|X|_{\mathrm{Rips}} = \sup_{\substack{A\subseteq X\\ A\text{ finite}}}|A|_{\mathrm{Rips}}$$ or the stronger definition $$|X|_{\mathrm{Rips}} = \lim_{\substack{A\to X\\ A\text{ finite}}}|A|_{\mathrm{Rips}},$$ where $A\to X$ in the Hausdorff metric. The issues here are whether the supremum above exists, how one computes its actual values, and the corresponding questions for the limit.
In this section we will explore the above situation in the case of the unit interval, the circle with its Euclidean metric, and the circle with its geodesic metric. In the case of the interval the situation is as good as one can hope for, with the conclusion that the Rips magnitude of the interval ‘is’ the function $t\mapsto 1+t$. In the circle cases the situation is more ambiguous, and while the supremum above may well exist (the supremum taken over all *equally spaced* subsets certainly does), the limit does not.
To improve readability, we state the results for each of the cases treated in a separate subsection, while deferring all the proofs to one final subsection.
The unit interval
-----------------
First let us consider the unit interval. Here we can give the following definitive description of the situation.
\[theorem-interval\] Let $I=[0,1]$ denote the unit interval.
1. If $A,B\subseteq I$ are finite subsets with $A\subseteq B$, then $$|A|_{\mathrm{Rips}}\leq |B|_{\mathrm{Rips}}.$$
2. For any $t\in (0,\infty)$ we have $$\sup_{\substack{A\subseteq I\\\text{finite}}}|t A|_{\mathrm{Rips}}
= 1+t$$ and indeed $$\lim_{\substack{A\to I\\A\;\text{finite}}}|t A|_{\mathrm{Rips}}
=
1+t$$ with uniform convergence on compact subsets of $(0,\infty)$.
Thus it seems reasonable in this situation to declare the Rips magnitude of $I=[0,1]$ to be the function $|t[0,1]|_{\mathrm{Rips}}=1+t$.
Euclidean Circle
----------------
One possible approach to try and make sense of the notion of “Rips magnitude of the Euclidean circle” is by studying the behaviour of $|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}$ as $n\to\infty$. As it turns out, however, this behaviour is not as straightforward as one might hope.
For instance, we show that $|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}$ does not converge as $n\to\infty$, despite the fact that the Hausdorff distance $d_H(C^{\mathrm{eucl}}_n,S^1_{\mathrm{eucl}})=2\sin\left(\frac{\pi}{2n}\right)$ converges to $0$ as $n\to\infty$. We show this by first studying the behaviour of sequences of the form $|t C^{\mathrm{eucl}}_{mp}|_{\mathrm{Rips}}$, for fixed $m\in{\mathbb{N}}$ and where $p$ runs through all primes, and showing that the limit along each such subsequence exists. However, these limits are different for different values of $m$.
We then show that despite this inconsistency, the $|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}$ has finite upper and lower limits ($\liminf$ and $\limsup$) as $n\to\infty$ which can be expressed explicitly. These could be considered to be the ‘upper’ and ‘lower Rips magnitude’. We also show that the ‘upper Rips magnitude’ is equal to $\sup_{n\in{\mathbb{N}}}|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}$. (Compare this with the first proposed definition of Rips magnitude at the beginning of this section.)
\[theorem-EC-lim\] For any $m\in{\mathbb{N}}$: $$\lim_{\substack{p\to\infty\\p\text{ prime}}}|t C^{\mathrm{eucl}}_{mp}|_{\mathrm{Rips}}=e^{-2t}+2\pi t\sum_{\text{odd }r|m}\frac1r e^{-2t\cos\left(\frac{\pi}{2r}\right)}\sin\left(\frac{\pi}{2r}\right)$$
This result means in particular that $$\lim_{n\to\infty}|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}$$ cannot exist. For instance, we have $$\lim_{\substack{p\to\infty\\p\text{ prime}}}|t C^{\mathrm{eucl}}_p|_{\mathrm{Rips}}
=e^{-2t}+2\pi t$$ but $$\lim_{\substack{p\to\infty\\p\text{ prime}}}|t C^{\mathrm{eucl}}_{3p}|_{\mathrm{Rips}}
=
e^{-2t}+2\pi t+\frac{\pi t}3 e^{-\sqrt{3}t}$$ and the two limits do not coincide.
\[theorem-lim-sup-inf\] The Rips magnitudes of Euclidean cycles $C^{\mathrm{eucl}}_n$ satisfy $$\liminf_{n\to\infty}|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}=e^{-2t}+2\pi t$$ and $$\limsup_{n\to\infty}|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}=e^{-2t}+2\pi t\sum_{r\text{ odd}}\frac1r e^{-2t\cos\left(\frac\pi{2r}\right)}\sin\left(\frac\pi{2r}\right).$$ This series converges absolutely since its $r$-th term is bounded above by $\frac{\pi}{2r^2}$.
In fact, the upper limit is also the supremum of the sequence:
\[theorem-sup\] The Rips magnitudes of Euclidean cycles $C^{\mathrm{eucl}}_n$ satisfy $$\sup_{n\in{\mathbb{N}}}|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}=e^{-2t}+2\pi t\sum_{r\text{ odd}}\frac1r e^{-2t\cos\left(\frac\pi{2r}\right)}\sin\left(\frac\pi{2r}\right).$$
We include a plot of the $\liminf$ and $\limsup$ for small $t$ for comparison:
{width="300pt"}
It is visible from this graph that the difference is the most pronounced for small values of $t$. Now, fix $t=\frac12$. The behaviour of $|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}$ evaluated at $t=\frac12$ as $n$ grows larger can be pictured as follows, with $n$ on the horizontal axis and $|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}$ on the vertical. The values of the $\liminf$ and $\limsup$ at $t=\frac12$ are plotted as the two red lines.
{width="300pt"}
Note that the chaotic behaviour of the graph is reminiscent of various functions from analytic number theory such as for instance the sum of divisors function $\sigma(n)=\sum_{d|n}d$, and is the graphical expression of the behaviour discussed in the paragraph following Proposition \[ripsmag\_cycle\]. It does appear as though the points accumulate more along specific lines, which we suspect correspond to subsequences with certain divisibility properties.
Finally, note that restricting to equally spaced subsets of $S^1_{eucl}$ is somewhat unnatural; we leave open the following question, which would define the ‘upper and lower Rips magnitude’ of the circle intrinsically:
Does this asymptotic behaviour extend to arbitrary finite subsets of $S^1_{eucl}$? For instance, given any $\epsilon>0$, is there a $\delta>0$ such that for all finite $A\subseteq S^1_{eucl}$ with $d_H(A,S^1_{eucl})<\delta$ we have $$e^{-2t}+2\pi t-\epsilon<|tA|_\mathrm{Rips}<e^{-2t}+2\pi t\sum_{r\text{ odd}}\frac1r e^{-2t\cos\left(\frac\pi{2r}\right)}\sin\left(\frac\pi{2r}\right)+\epsilon$$ for all $t$ in a given interval?
Geodesic circle
---------------
Finally, we note that the case of the geodesic circle $S^1_{geo}$ of total arclength $2\pi$ can be treated using the same methods as we used for $S^1_{eucl}$. Namely, we restrict attention to equally spaced subsets $C_n^{geo}$ described in Section \[section-cycles\]. (Note that $C_n^{geo}$ is just the $n$-cycle graph $C_n$ rescaled by $\frac{2\pi}{n}$.)
We could calculate the limits along the same subsequences we examined in the case $S^1_{eucl}$ and find that they again exist, but instead we just state the final result regarding the lower and upper limit. In this case it turns out that the lower limit is still finite and can be expressed explicitly, whereas the upper limit becomes infinite. Thus the Rips magnitude of $S^1_{eucl}$ and $S^1_{geo}$ behave quite differently.
\[theorem-GC\] The Rips magnitudes of geodesic cycles $C^{\mathrm{geo}}_n$ satisfy $$\liminf_{n\to\infty}|t C^{\mathrm{geo}}_n|_{\mathrm{Rips}}=e^{-\pi t}+2\pi t$$ and $$\limsup_{n\to\infty}|t C^{\mathrm{geo}}_n|_{\mathrm{Rips}}=\infty.$$
The behaviour of $|t C^{\mathrm{geo}}_n|_{\mathrm{Rips}}$ evaluated at $t=\frac12$ as $n$ grows larger can be pictured as follows; the $\liminf$ evaluated at $t=\frac12$ corresponds to the red line and the $\limsup$ is infinite.
{width="300pt"}
We again note that it would be interesting to study the asymptotics over all finite subsets.
Proofs of the Results
---------------------
Here we prove the results stated in the preceding subsections. We only give a sketch of the proof of Theorem \[theorem-GC\] as the ideas are analogous to the ones used in the proof of Theorem \[theorem-lim-sup-inf\].
Proposition \[proposition-subsets-of-R\] shows that if $A\subseteq[0,1]$ is finite, with elements $a_1<\cdots<a_n$, then $$|A|_{\mathrm{Rips}} = n - \sum_{j=1}^{n-1} e^{-(a_{j+1}-a_j)}.$$ Suppose that $a_0<a_1$. Then $$|\{a_0\}\cup A|_{\mathrm{Rips}}
=
|A|_{\mathrm{Rips}} + (1-e^{-(a_1-a_0)})
\geq
|A|_{\mathrm{Rips}},$$ and similarly for $|A\cup\{a_{n+1}\}|_{\mathrm{Rips}}$ if $a_{n+1}>a_n$. Now suppose that $a_i<b<a_{i+1}$. Then $$\begin{aligned}
|A\cup\{b\}|_{\mathrm{Rips}}
&=
|A|_{\mathrm{Rips}} + 1 + e^{-(a_{i+1}-a_i)}- e^{-(a_{i+1}-b)}-e^{-(b-a_i)}
\geq |A|_{\mathrm{Rips}},\end{aligned}$$ the latter because one can easily see that $1+e^{-(x+y)}-e^{-x}-e^{-y}\geq 0$ for $x,y\geq 0$.
Suppose $A$ consisting of $a_1<\ldots<a_n$ is a finite subset of the interval such that $d_H(A,I)<\delta<1$. Further define $a_0=0$ and $a_{n+1}=1$. The assumption on the Hausdorff distance implies that $a_{j+1}-a_j<\delta$ for $j=0,\ldots,n$ and $\delta<1$ implies $\delta^k\leq\delta$ for $k\in{\mathbb{N}}$. Consider the function $f\colon(0,\infty)\to{\mathbb{R}}$ defined by $$\begin{aligned}
f(t)
&=
(1+t)-|t A|_{\mathrm{Rips}}
\\
&=
(1+t)-n+\sum_{j=1}^{n-1} e^{-(a_{j+1}-a_j)t}
\\
&=t+\sum_{j=1}^{n-1}(e^{-(a_{j+1}-a_{j})t}-1).\end{aligned}$$ We extend the domain of $f$ to $[0,\infty)$ in the evident way. We will show that:
1. $f(0)=0$
2. $f'(0)$ lies in the range $0\leq f'(0)\leq 2\delta$.
3. $f''(t)$ lies in the range $0\leq f''(t)\leq\delta$ for all $t\in[0,\infty)$.
It follows quickly that $$0\leq f(t)\leq \delta\cdot (2t+t^2/2)$$ for $t\in[0,\infty)$. Thus, as $\delta\to 0$ we have $f\to 0$ uniformly on any bounded subset of $[0,\infty)$, and the result follows.
It remains to check the three properties. The first and second derivatives of $f$ are as follows. $$\begin{aligned}
f'(t)
&=
1-\sum_{j=1}^{n-1}(a_{j+1}-a_j)e^{-(a_{j+1}-a_{j})t}
\\
f''(t)
&=
\sum_{j=1}^{n-1}(a_{j+1}-a_j)^2e^{-(a_{j+1}-a_{j})t}\end{aligned}$$ Then (1) is immediate, while $f'(0)=1-\sum_{j=1}^{n-1}(a_{j+1}-a_j)=(1-a_n)+a_1$ and (2) follows, and $0\leq f''(t)
\leq \sum_{j=1}^{n-1}\delta\cdot(a_{j+1}-a_j)\cdot 1$ and (3) follows.
Recall that by Corollary \[ripsmag\_eucl\], the Rips magnitudes of Euclidean cycles are given by: $$|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}=\sum_{\text{odd }r|n}\frac nr (e^{-\delta_r t}-e^{-\delta_{r,n}t})+e^{-\delta_n t},$$ where $$\delta_r=\operatorname{diam}(C^{\mathrm{eucl}}_r)=2\sin\left(\pi\frac{\lfloor\frac r2\rfloor}r\right)\quad\text{and}\quad\delta_{r,n}=2\sin\left(\pi\left(\frac1n+\frac{\lfloor\frac r2\rfloor}r\right)\right).$$ For an odd prime $p$, each odd divisor of $mp$ is of the form $r$ or $rp$ (or both), where $r$ is an odd divisor of $m$. Therefore, assuming $p$ is large enough, so that $p\nmid m$, we can split the Rips magnitude into three summands: $$|t C^{\mathrm{eucl}}_{m p}|_{\mathrm{Rips}}=\sum_{\text{odd }r|m}\frac{mp}r (e^{-\delta_r t}-e^{-\delta_{r,mp}t})+\sum_{\text{odd }r|m}\frac mr (e^{-\delta_{rp} t}-e^{-\delta_{rp,mp}t})+e^{-\delta_{mp} t}.$$ We can calculate the limit as $p\to\infty$ of each summand individually. To treat the first summand, define a function $\phi\colon {\mathbb{R}}\to{\mathbb{R}}$ by the formula $$\phi(x)=2\sin\left(\pi\left(x+\frac{r-1}{2r}\right)\right),$$ with derivative $$\phi'(x)=2\pi\cos\left(\pi\left(x+\frac{r-1}{2r}\right)\right).$$ Observe that $\delta_r=\phi(0)$ and $\delta_{r,mp}=\phi(\frac1{mp})$. Therefore, $$\begin{aligned}
\lim_{p\to\infty}\frac{mp}r (e^{-\delta_r t}-e^{-\delta_{r,mp}t})&=-\frac1r\lim_{p\to\infty}\frac{e^{-\phi(\frac1{mp})t}-e^{-\phi(0)t}}{\frac1{mp}}\\&=-\frac1r\frac{\mathrm d}{\mathrm dx}{\Big\vert}_{x=0}e^{-\phi(x)t}\\&=\frac1re^{-\phi(0)t}\phi'(0)t\\&=\frac{2\pi t}re^{-2t\sin\left(\pi\frac{r-1}{2r}\right)}\cos\left(\pi\frac{r-1}{2r}\right)\\&=\frac{2\pi t}re^{-2t\cos\left(\frac{\pi}{2r}\right)}\sin\left(\frac{\pi}{2r}\right).\end{aligned}$$ This takes care of the first summand. The second summand vanishes in the limit, because $$\lim_{p\to\infty}e^{-\delta_{rp} t}=\lim_{p\to\infty}e^{-\delta_{rp,mp} t}=e^{-2t}.$$ Finally, the limit of the third summand is $$\lim_{p\to\infty}e^{-\delta_{mp} t}=e^{-2t}.$$
Define $\phi_{n,r}(t)=\frac nr(e^{-\delta_r t}-e^{-\delta_{r,n}t})$ for $n\geq r$ and $0$ otherwise. Here $r$ will always be assumed to be odd. We can show that for $t>0$, $$\phi_{n,r}(t)\geq 0\qquad\text{and}\qquad\phi_{n+1,r}(t)\geq\phi_{n,r}(t)$$ always hold. The first of these inequalities reduces to the fact that $\delta_r\leq\delta_{r,n}$. For the second inequality, define $\psi(t)=1+ne^{(\delta_r-\delta_{r,n})t}-(n+1)e^{(\delta_r-\delta_{r,n+1})t}$ and observe that the inequality reduces to $\psi(t)\geq 0$ for $t>0$. To show that this is in fact true, one can then verify that:
- $\psi(0)=0$,
- $\lim_{t\to\infty}\psi(t)>0$,
- $\psi'(0)=(n+1)\delta_{r,n+1}-n\delta_{r,n}-\delta_r>0$ and
- $\psi'(t)=0$ for at most one $t\in(0,\infty)$.
To prove the third bullet point, we can write out the expression explicitly and use the addition theorem for the sine function: $$\begin{gathered}
2(n+1)\sin\left(\pi\left(\frac1{n+1}+\frac{r-1}{2r}\right)\right)-2n\sin\left(\pi\left(\frac1n+\frac{r-1}{2r}\right)\right)-2\sin\left(\pi\frac{r-1}{2r}\right)\\
=2\left[(n+1)\sin\left(\frac{\pi}{n+1}\right)-n\sin\left(\frac{\pi}{n}\right)\right]\cos\left(\pi\frac{r-1}{2r}\right)\\+2\left[(n+1)\cos\left(\frac{\pi}{n+1}\right)-n\cos\left(\frac{\pi}{n}\right)-1\right]\sin\left(\pi\frac{r-1}{2r}\right).\end{gathered}$$ One can now show that the expressions in square brackets are positive and conclude that the whole expression is positive. This proves the second inequality.
Using the explicit formula for the Rips magnitude, we now have: $$|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}=\sum_{\text{odd }r|n}\frac nr (e^{-\delta_r t}-e^{-\delta_{r,n}t})+e^{-\delta_n t}\leq\sum_{\text{odd }r\leq n}\frac nr (e^{-\delta_r t}-e^{-\delta_{r,n}t})+e^{-\delta_n t}.$$ Therefore, $$\begin{aligned}
\limsup_{n\to\infty}|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}&\leq\lim_{n\to\infty}\left(\sum_{\text{odd }r}\phi_{n,r}(t)+e^{-\delta_nt}\right)\\&=\sum_{\text{odd }r}\lim_{n\to\infty}\phi_{n,r}(t)+\lim_{n\to\infty}e^{-\delta_nt}\\&=\sum_{\text{odd }r}\lim_{n\to\infty}\frac nr (e^{-\delta_r t}-e^{-\delta_{r,n}t})+e^{-2t}\\&=\sum_{\text{odd }r}\frac {2\pi t}r e^{-2t\cos\left(\frac\pi{2r}\right)}\sin\left(\frac\pi{2r}\right)+e^{-2t},\end{aligned}$$ where the limit is calculated in the same way as in the proof of the previous theorem. The interchange of sum and limit is justified by the Lebesgue monotone convergence theorem. To prove the reverse inequality, let $N$ be an arbitrary positive integer. Let $m=N!$ and note that $m$ is divisible by every odd $r\leq N$. Therefore, by the previous theorem, $$\begin{aligned}
e^{-2t}+2\pi t\sum_{\text{odd }r\leq N}\frac1r e^{-2t\cos\left(\frac\pi{2r}\right)}\sin\left(\frac\pi{2r}\right)&\leq e^{-2t}+2\pi t\sum_{\text{odd }r|m}\frac1r e^{-2t\cos\left(\frac\pi{2r}\right)}\sin\left(\frac\pi{2r}\right)\\&=\lim_{\substack{p\to\infty\\p\text{ prime}}}|t C^{\mathrm{eucl}}_{mp}|_{\mathrm{Rips}}\\&\leq \limsup_{n\to\infty}|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}.\end{aligned}$$ Taking the limit as $N\to\infty$ establishes the lower bound.
To prove the statement about the lower limit, again start from the explicit formula and note that $r=1$ is a proper odd divisor of any integer $n>1$: $$|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}=\sum_{\text{odd }r|n}\frac nr (e^{-\delta_r t}-e^{-\delta_{r,n}t})+e^{-\delta_n t}\geq n (e^{-\delta_1 t}-e^{-\delta_{1,n}t})+e^{-\delta_n t}.$$ Therefore $$\liminf_{n\to\infty}|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}\geq\lim_{n\to\infty}\left(n (e^{-\delta_1 t}-e^{-\delta_{1,n}t})+e^{-\delta_n t}\right)=2\pi t+e^{-2t}.$$ To establish the other inequality, note that by the previous theorem, the prime indices yield a subsequence converging to the lower bound.
The upper limit of any sequence is a lower bound for its supremum, so $$\limsup_{n\to\infty}|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}\leq\sup_{n\in{\mathbb{N}}}|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}.$$ To show the converse inequality, we first observe that $$|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}\leq|t C^{\mathrm{eucl}}_{2n}|_{\mathrm{Rips}}$$ holds for any odd $n\in{\mathbb{N}}$. To see this, we look at their difference. Using the facts that $n$ and $2n$ have the same odd divisors, that $\delta_n=\delta_{n,n}$ and $\delta_{n,2n}=\delta_{2n}=2$, this can be simplified to: $$|t C^{\mathrm{eucl}}_{2n}|_{\mathrm{Rips}}-|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}=\sum_{\substack{\text{odd }r|n\\r\neq n}}\left(\phi_{2n,r}(t)-\phi_{n,r}(t)\right)+e^{-\delta_n t}-e^{-2 t},$$ where $\phi_{n,r}$ is as defined in the proof of Theorem \[theorem-lim-sup-inf\], where we also showed that $\phi_{n+1,r}\geq\phi_{n,r}$ for all $n$ and $r$. Using this latter fact and the fact that $\delta_n\leq2$, we now have $$\phi_{2n,r}(t)-\phi_{n,r}(t)\geq0\qquad\text{and}\qquad e^{-\delta_n t}-e^{-2 t}\geq 0,$$ so the difference is indeed nonnegative.
Therefore, the supremum may be calculated over the even numbers: $$\sup_{n\in{\mathbb{N}}}|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}=\sup_{n\text{ even}}|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}.$$ Now recall that for even $n$ we have: $$|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}\leq\sum_{\text{odd }r\leq n}\frac nr (e^{-\delta_r t}-e^{-\delta_{r,n}t})+e^{-2t}.$$ Furthermore, we saw in the proof of Theorem \[theorem-lim-sup-inf\] that the expression on the right hand side is increasing in $n$ (because $\phi_{n+1,r}\geq\phi_{n,r}$). Taking the supremum of both sides over all even integers $n$, we therefore have: $$\sup_{n\text{ even}}|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}}\leq\lim_{\substack{n\to\infty\\n\text{ even}}}\sum_{\text{odd }r\leq n}\frac nr (e^{-\delta_r t}-e^{-\delta_{r,n}t})+e^{-2t}\leq\limsup_{n\to\infty}|t C^{\mathrm{eucl}}_n|_{\mathrm{Rips}},$$ where the second inequality is immediate from the proof of Theorem \[theorem-lim-sup-inf\].
By Corollary \[ripsmag\_geo\], the explicit formulas for Rips magnitudes of geodesic cycles are $$|t C^{\mathrm{geo}}_n|_{\mathrm{Rips}}=\sum_{\substack{\text{odd }r|n\\r\neq n}}\frac nr (e^{-\eta_r t}-e^{-\eta_{r,n}t})+e^{-\eta_n t},$$ where $$\eta_r=2\pi\frac{\lfloor\frac r2\rfloor}r\qquad\text{and}\qquad\eta_{r,n}=2\pi\left(\frac1n+\frac{\lfloor\frac r2\rfloor}r\right).$$ A similar procedure as in the Euclidean case now allows for the calculation of limits of various subsequences, as well as the upper and the lower limit. The main difference is that in this case, the series obtained as the upper limit does not converge anymore: $$\limsup_{n\to\infty}|t C^{\mathrm{geo}}_n|_{\mathrm{Rips}}=\sum_{\text{odd }r}\frac{2\pi t}re^{-\pi\frac{r-1}{r}t}+e^{-\pi t}=\infty.
\qedhere$$
Proof of Theorem \[theorem-magBMH\] {#section-long-proof}
===================================
Throughout this proof we let $\delta$ denote the minimum nonzero distance between elements of $X$, and we let $n$ denote the cardinality of $X$. We let $l_0<l_1<l_2<\cdots$ be the distinct values of $l$ for which the inner sum appearing in Proposition \[proposition-alternating\] is nonzero, as in Remark \[remark-l-values\]. And we define $D(i,j,k)$ to be $1$ if $l_j$ is in $[a_{k,i},b_{k,i})$, and to be $0$ otherwise. We make a standing assumption that $t$ is large enough that $ne^{-\delta t}<1$; this is the assumption under which $t$ is large enough that the conclusions of Proposition \[proposition-alternating\] hold.
We will use the following fact several times. Let $(x_0,\ldots,x_k)$ be a tuple of elements of $X$ in which consecutive elements are distinct, and suppose that this tuple is a generator of $\operatorname{MC}_{k,l}(X)$ or $\operatorname{BMC}_k(X)(l)$. Then $\ell(x_0,\ldots,x_k)\leq l$, while $\ell(x_0,\ldots,x_k)\geq k\delta$, so that $k\delta \leq l$. It follows that, if $k$ and $l$ do not satisfy this relation, then the homology groups $\operatorname{MH}_{k,l}(X)$ and $\operatorname{BMH}_k(X)(l)$ vanish.
We now have the following computation, whose steps will be justified below. $$\begin{aligned}
|tX|
&\stackrel{1}{=}
\sum_{j=0}^\infty
\sum_{k=0}^\infty
(-1)^k\operatorname{\mathrm{rank}}(\operatorname{MH}_{k,l_j}(X))e^{-l_jt}
\\
&\stackrel{2}{=}
\sum_{j=0}^\infty
\sum_{k=0}^\infty
(-1)^k
\left[
\operatorname{\mathrm{rank}}(\operatorname{BMH}_{k}(X)(l_j))
-
\operatorname{\mathrm{rank}}(\operatorname{BMH}_{k}(X)(l_{j-1}))
\right]e^{-l_jt}
\\
&\stackrel{3}{=}
\sum_{k=0}^\infty
(-1)^k
\sum_{j=0}^\infty
\left[
\operatorname{\mathrm{rank}}(\operatorname{BMH}_{k}(X)(l_j))
-
\operatorname{\mathrm{rank}}(\operatorname{BMH}_{k}(X)(l_{j-1}))
\right]e^{-l_jt}
\\
&\stackrel{4}{=}
\sum_{k=0}^\infty
(-1)^k
\sum_{j=0}^\infty
\operatorname{\mathrm{rank}}(\operatorname{BMH}_{k}(X)(l_j))(e^{-l_jt}-e^{-l_{j+1}t})
\\
&\stackrel{5}{=}
\sum_{k=0}^\infty
(-1)^k
\sum_{j=0}^\infty
\sum_{i=0}^\infty
D(i,j,k)(e^{-l_jt}-e^{-l_{j+1}t})
\\
&\stackrel{6}{=}
\sum_{k=0}^\infty
(-1)^k
\sum_{i=0}^\infty
\sum_{j=0}^\infty
D(i,j,k)(e^{-l_jt}-e^{-l_{j+1}t})
\\
&\stackrel{7}{=}
\sum_{k=0}^\infty
(-1)^k
\sum_{i=0}^\infty
(e^{-a_{k,i}t}-e^{-b_{k,i}t})\end{aligned}$$
Step 1 is precisely the formula of Proposition \[proposition-alternating\]. The series here is absolutely convergent. That is because $$\begin{aligned}
\sum_{j=0}^J
\sum_{k=0}^\infty
\operatorname{\mathrm{rank}}(\operatorname{MH}_{k,l_j}(X))
e^{-l_jt}
&=
\sum_{k=0}^\infty
\sum_{j=0}^J
\operatorname{\mathrm{rank}}(\operatorname{MH}_{k,l_j}(X))
e^{-l_jt}
\\
&\leq
\sum_{k=0}^\infty
\sum_{x_0,\ldots,x_k}
e^{-\ell(x_0,\ldots,x_k)t}
\\
&\leq
\sum_{k=0}^\infty
n^{k+1}e^{-k\delta t}
\\
&=
n\cdot\sum_{k=1}^\infty
(ne^{-\delta t})^k.\end{aligned}$$ Here, in the second line the inner sum is over all tuples $(x_0,\ldots,x_k)$ with consecutive elements distinct, and there are at most $n^{k+1}$ of these, where $n$ denotes the cardinality of $X$. Now we have $ne^{-\delta t}<1$ by our standing assumption, so that the latter sum converges and is bounded above independent of $J$. This shows absolute convergence.
To explain step 2, recall that by Proposition \[proposition-absolute-blurred\] there is a long exact sequence $$\cdots
\to
\operatorname{BMH}_k(X)(l_{j-1})
\to
\operatorname{BMH}_k(X)(l_j)
\to
\operatorname{MH}_{k,l_j}(X)
\to
\cdots$$ This sequence terminates in both directions, because the relation described in the second paragraph above fails for all three groups when $k$ is large enough. A standard fact from homological algebra then guarantees that $$\begin{gathered}
\sum_{k=0}^\infty
(-1)^k\operatorname{\mathrm{rank}}(\operatorname{MH}_{k,l_j}(X))
=
\\
\sum_{k=0}^\infty
(-1)^k
\left[\operatorname{\mathrm{rank}}(\operatorname{BMH}_{k}(X)(l_j))
-
\operatorname{\mathrm{rank}}(\operatorname{BMH}_{k}(X))(l_{j-1})
\right].\end{gathered}$$
For step 3, we have exchanged the order of summation. This is valid because the series is absolutely convergent (indeed, it is the same series as the one appearing in step 1).
For step 4, we have ‘telescoped’ the sum, using the fact that $$\operatorname{\mathrm{rank}}(\operatorname{BMH}_k(X)(l_j))e^{-l_jt}\to 0
\text{ as }
j\to\infty.$$ The latter holds because $\operatorname{\mathrm{rank}}(\operatorname{BMH}_k(X)(l_j))$ is at most the number of tuples $(x_0,\ldots,x_k)$ with consecutive entries distinct and $\ell(x_0,\ldots,x_k)\le l_j$. But then $l_j\ge k\delta$ so that $\operatorname{\mathrm{rank}}(\operatorname{BMH}_{k}(X)(l_j))e^{-l_jt}\leq n^{k+1}e^{-l_jt}
\le n\cdot n^{l_j/\delta}e^{-l_j t}=n\cdot(n^{1/\delta}e^{-t})^{l_j}$. But $(n^{1/\delta}e^{-t})<1$ by our standing assumption. Since $l_j\to\infty$ as $j\to\infty$, the claim follows.
Step 5 follows by simply describing $\operatorname{\mathrm{rank}}(\operatorname{BMH}_k(X)(l_j))$ as the number of bars in the barcode decomposition for $\operatorname{BMH}_k(X)$ that contain $l_j$. For step 6 we have again exchanged the order of summation, which is valid because the series consists of non-negative numbers and is convergent. Step 7 is then a direct computation of the series $\sum_{j=0}^\infty D(i,j,k)(e^{-l_jt}-e^{-l_{j+1}t})$.
[^1]: In fact, $\delta_n=\delta_{n,n}$ for odd $n$, so the condition $r\neq n$ can be omitted for Euclidean cycles.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We utilize a high quality calcium fluoride whispering-gallery-mode resonator to stabilize a simple erbium doped fiber ring laser with an emission frequency of 196[$\,\mathrm{THz}$]{} (wavelenght 1530[$\,\mathrm{nm}$]{}) to a linewidth below 650[$\,\mathrm{Hz}$]{}. This corresponds to a relative stability of $3.3\times10^{-12}$ over 16[$\,\mathrm{\mu s}$]{}. In order to characterize the linewidth we use two identical self-built lasers and a commercial laser to determine the individual lasing linewidth via the three-cornered hat method.'
address: |
$^1$Max Planck Institute for the Science of Light, G.-Scharowsky-Str. 1/Bau 24, 91058 Erlangen, Germany\
$^2$Institute of Optics, Information and Photonics, University of Erlangen-Nuremberg, G.-Scharowsky-Str. 1/Bau 24, 91058 Erlangen, Germany\
$^3$Humboldt-Universität zu Berlin, Institut für Physik, AG Nanooptik, Newtonstraße 15, 12489 Berlin, Germany\
$^4$Physics Department and Joint Inst. of Measurement Science (JMI), Tsinghua University, Beijing, 100084 China\
$^5$SAOT, School in Advanced Optical Technologies, Paul-Gordan-Stra[ß]{}e 6, 91052 Erlangen, Germany\
$^*$Corresponding author: +49 9131 6877-134, [email protected]
author:
- 'M. C. Collodo$^{1,2}$, F. Sedlmeir$^{1,2,5}$, B. Sprenger$^3$, S. Svitlov$^2$, L. J. Wang$^4$, and H. G. L. Schwefel$^{1,2,*}$'
title: |
Sub-kHz lasing of a CaF$_2$ Whispering Gallery Mode Resonator\
Stabilized Fiber Ring Laser
---
Compact and stable light sources are in high demand in metrology[@lea_limits_2007] and biochemical sensing[@baaske_optical_2012] to mention just two predominant fields. Optical resonators are at the heart of both applications. Whispering gallery mode (WGM) resonators are dielectric cavities that confine light due to total internal reflection at their dielectric interface[@vahala_optical_2003]. Their quality factor ($Q$) is mainly limited by surface scattering and by material absorption. For highly transparent materials such as calcium fluoride (CaF$_2$) quality factors up to $10^{11}$[@grudinin_ultra_2006; @savchenkov_optical_2007] have been shown. Operability exists throughout the whole transparency window of the host material, in case of calcium fluoride from 150[$\,\mathrm{nm}$]{} to 10[$\,\mathrm{\mu m}$]{}. With their resulting very narrow linewidth, these resonators serve as excellent optical frequency filters[@matsko_whispering-gallery-mode_2007] and are suitable to enhance the lasing modes of a conventional primary lasing module[@liang_whispering-gallery-mode-resonator-based_2010].
Before reaching the fundamental thermal noise floor limit[@alnis_thermal-noise-limited_2011; @numata_thermal-noise_2004; @chijioke_thermal_2011; @matsko_whispering-gallery-mode_2007], the stability of a reference cavity is mainly determined by its deformation due to mechanical vibration or thermal effects, whose influence scales with the cavity’s dimensions [@sprenger_caf2_2010]. Therefore, due to their compact sizes, WGM resonators are eminently suitable as a frequency reference.
In this Letter we report the setup and characterization of a free running fiber ring laser providing lasing linewidths below 1[$\,\mathrm{kHz}$]{}. This is achieved by resonantly filtering the broad emission spectrum of an erbium doped fiber via the narrow-linewidth modes of a WGM resonator. Only these narrow modes can pass the resonator and achieve gain in the following round trip. This establishes a narrow linewidth lasing behavior.
In our experiment we observed a suppression of the lasing linewidth to sub-kilohertz in comparison with the cold cavity linewidth of the resonator (sub-megahertz). This can be equivalently described by an improvement of the resonator’s $Q$ by a factor of $10^3$ due to active lasing. In order to verify these results, we modeled our ring laser setup analytically following and extending the approach by Wang [@wang_causal_2002]. We modeled the filtering mechanism due to the WGM resonator and implemented this model in an iterative numerical simulation, taking into account gain saturation. The experimentally observed characteristics could be reproduced by appropriate choice of parameters, the most crucial being the fiber cavity’s and the WGM resonator’s $Q$ factors and the saturated intracavity power. The laser’s emission spectrum narrows with increasing intracavity power. This outcome agrees well with a fully analytic approach by Eichhorn et al. [@Eichhorn12]. Our simulation does not cover further aspects regarding the WGM resonator’s instability due to an increased lasing power, and is therefore not able to determine the optimal intracavity power.
![(Color online) Sketch of one of the whispering gallery mode resonator filtered lasers. A WGM resonator is used to filter the emission spectrum of a conventional telecom fiber ring laser. The active medium is an erbium doped fiber with an emission spectrum in the telecommunication C-band. Passive filtering of the fiber loop lasing modes is provided by a WGM resonator. Single mode lasing is obtained without further active stabilization techniques.[]{data-label="fig:setup"}](wglaser2.pdf){width="8.4cm"}
*Experimental setup.* We set up a conventional fiber ring laser using an erbium doped fiber with a broad band emission spectrum in the telecommunication C-band wavelength regime ($\sim$1530[$\,\mathrm{nm}$]{}). It is pumped by a 980[$\,\mathrm{nm}$]{} laser diode with 200[$\,\mathrm{mW}$]{} output power through a wavelength-division-multiplexer. Light is coupled out via a 99/1 fiber coupler and prevented from clockwise circulation by an optical Faraday isolator. By inserting the WGM resonator into the fiber loop the final whispering gallery laser (WGL) is set up (figure \[fig:setup\]). We fabricate the WGM resonators on a home built diamond lathe. Mono-crystalline CaF$_2$ is cut into disks with an optimized surface curvature via diamond turning. The disks are 4[$\,\mathrm{mm}$]{} in diameter. An optimal surface quality is achieved via polishing with grain sizes down to 50[$\,\mathrm{nm}$]{}. Loaded cold cavity $Q$ factors of our millimeter sized resonators measure a few $10^8$. Evanescent coupling through the polarization dependent WGMs can be achieved via a pair of piezo controlled coupling prisms (SF11 glass). Prism coupling allows for a more rigid construction, which is less influenceable by mechanical vibrations. In comparison, previous approaches utilized tapered or angle polished fibers for the resonator coupling [@sprenger_whispering-gallery-mode-resonator-stabilized_2009]. For the coupling into and out of the fiber loop a pair of gradient index (GRIN) lenses is used, where the numerical aperture and the focal point are adjustable. The coupling efficiency from the fiber loop transmitted through the WGM resonator was approximately 20%. For optimal coupling to distinct whispering gallery modes a fiber polarisation controller is necessary. Laser output power is in the range of tens of microwatts. This solely passively stabilized lasing systems provides a straightforward setup design, featuring easy assembly and tight packaging.
As our main task will be the linewidth measurement, the tunability of the lasing frequency is paramount. With a broadband ($1{\ensuremath{\,\mathrm{nm}}}$ full width at half maximum) optical bandpass filter (not depicted) a coarse tuning of the lasing mode’s wavelength over the whole emission spectrum of the erbium doped fiber in steps of the WGM resonator’s free spectral range ($\sim$20[$\,\mathrm{GHz}$]{}) is possible. Further fine tuning can be achieved via temperature control of the resonator.
![(Color online) Allan Deviation values (corresponding to lasing linewidth in [$\,\mathrm{Hz}$]{}) and relative stabilities (corresponding to lasing $Q$ factor) for the whispering gallery lasers (WGL). (a) Direct evaluation of the beat note signal between WGL1 and WGL2 reports the combination of both noise sources. (b) Individual noise components were obtained via the three-cornered hat method, a possible correlation was taken into account.[]{data-label="fig:allan_3ch"}](allan_w1w2.pdf "fig:"){width="8.4cm"} ![(Color online) Allan Deviation values (corresponding to lasing linewidth in [$\,\mathrm{Hz}$]{}) and relative stabilities (corresponding to lasing $Q$ factor) for the whispering gallery lasers (WGL). (a) Direct evaluation of the beat note signal between WGL1 and WGL2 reports the combination of both noise sources. (b) Individual noise components were obtained via the three-cornered hat method, a possible correlation was taken into account.[]{data-label="fig:allan_3ch"}](correlation_w1.pdf "fig:"){width="8.4cm"}
*Measurement procedure.* As the frequency stability of the individual lasers cannot be measured directly, two identical systems were built. The beat note generated by mixing the emission specta of the two lasers allows us to reconstruct the frequency stability of the combined system. We recorded 50[$\,\mathrm{ms}$]{} (sampling rate is 200megasamples/second, with 8 bit vertical resolution) of the beat note traces of the two approximately 10[$\,\mathrm{MHz}$]{} detuned WGLs. For the computation of the beat note’s frequency contiguous, non overlapping basic time intervals of 1${\ensuremath{\,\mathrm{\mu s}}}$ length are chosen from the trace in order to obtain a frequency stream.
This basic time interval is chosen such that it covers at least ten periods of the beat note signal. We perform a nonlinear least-square fit of a sine function over these separate time domain traces. The only fitting parameters in the used model are frequency, amplitude, offset and phase, which are all assumed to be constant over the basic 1${\ensuremath{\,\mathrm{\mu s}}}$ fit interval. We prefer this method in comparison to a conventional frequency counter because of the higher flexibility and robustness in case of a low signal to noise ratio. Furthermore, we avoid ambiguity in the interpretation of the resulting values and we are free to compute different types of variances[@rubiola_measurement_2005].
In order to determine the frequency stability of our lasers with respect to the averaging time the Allan Deviation[@allan_standard_1988; @barnes_characterization_1971] $$\sigma^2 = \frac{1}{2(M-1)} \sum_{i=1}^{M-1} \left(f_i - f_{i+1} \right)^2$$ is used, where $\sigma$ is the Allan Deviation of the lasing frequency, $f_i$ denotes the value of the lasing frequency referring to the $i$-th basic time interval and $M$ is the total amount of basic time intervals. Pre-averaging of the frequency stream over multiple basic time intervals allows us to compute Allan Deviations for different averaging time scales.
*Results.* The resulting Allan Deviation values are shown in figure \[fig:allan\_3ch\] (a). They directly correspond to the lasing linewidth. The relative stabilities comparing the laser’s linewidth to its emission frequency of 196[$\,\mathrm{THz}$]{} are also presented, they correspond to the inverse lasing $Q$ factor. For the deviation values of the beat note trace evaluation a minimum of 1056[$\,\mathrm{Hz}$]{} is reached for an averaging time of 18[$\,\mathrm{\mu s}$]{}. Since the individual WGM laser setups are slightly different we are reporting an upper limit for the more stable WGL. This can be estimated by 1056/$\sqrt{2}$[$\,\mathrm{Hz}$]{} = 750[$\,\mathrm{Hz}$]{}. Thereby we assume no negative correlation in the lasing behavior. The graph (figure \[fig:allan\_3ch\] (a)) reveals a $\tau^{-1/2}$-slope in the short timescale regime (less than 18[$\,\mathrm{\mu s}$]{}), before the optimal averaging time is attained. This slope can be associated with a white frequency noise behavior [@allan_standard_1988]. After the optimal averaging time a drift to larger Allan Deviation values is predominant. The timescale suggests that temperature fluctuations alone cannot be responsible. Also the curve’s slope does not fit to the therewith related random frequency walk ($\tau^{1/2}$). A directed frequency shift due to heating of the modal volume seems more plausible here. The measurement noise curve in figure \[fig:allan\_3ch\] (a) is a measure for the quality of the signal and of the measurement procedure as a whole. It combines the frequency errors computed from the least square residuals and thus reflects the signal to noise ratio of the acquired beat note, time jitter and quantization errors of the oscilloscope, errors due to the method of frequency calculation and the short term (less than 1[$\,\mathrm{\mu s}$]{}) instability of the lasers, which is ignored by the fit model.
In order to extract the individual frequency stabilities we add a third lasing system, namely a commercial Toptica DLpro design laser and perform a three-cornered hat measurement [@gray_method_1974]. This is done by recording the beat notes from the three possible combinations of laser pairs simultaneously and solving for the single laser variances $$2\sigma^2_{\text{\tiny{WGL1}}} = \sigma^2_{\text{\tiny{WGL1+WGL2}}} + \sigma^2_{\text{\tiny{WGL1+DLpro}}} - \sigma^2_{\text{\tiny{WGL2+DLpro}}}$$ (and permutations of this equation). This holds only if no correlation in the lasing frequency characteristics is predominant. For a more incontestable approach taking into account possible correlations, a correlation removal algorithm by Premoli et al. [@premoli_revisited_1993] has been used with WGL1 as a reference (choice of reference is not significant). The individual frequency stabilities, revised in this manner, are shown in figure \[fig:allan\_3ch\] (b).
The more stable WGM resonator laser reaches a relative stability of $\left(1.67 \pm 1.60 \right) \times 10^{-12}$ for an averaging time of 16[$\,\mathrm{\mu s}$]{}. This corresponds to a lasing linewidth of $\left(328 \pm 314 \right) {\ensuremath{\,\mathrm{Hz}}}$ at the laser’s emission frequency of 196[$\,\mathrm{THz}$]{}. In a conservative estimate we report a relative stability of $3.3 \times 10^{-12}$ and a linewidth of 650[$\,\mathrm{Hz}$]{}, respectively. These values agree well with the results of the prior estimation using the combined frequency stability directly.
*Conclusions.* To summarize, we demonstrated a sub-kilohertz linewidth lasing behavior in a solely passively stabilized erbium doped fiber ring laser. The stabilization arises through filtering via high-$Q$ modes of a crystalline calcium fluoride whispering gallery mode resonator. Our evaluation method for the lasing stability is based on the analysis of the digitized time domain beat note traces and avoids the standard frequency counter approach. The experimentally observed linewidth enhancement during the lasing process to finite lasing $Q$ factors is confirmed theoretically. Thus, the final lasing linewidth can be influenced either by the cold cavity $Q$ factor of the filtering resonator or by an increased circulating intracavity power.
The authors would like to thank Dmitry V. Strekalov and Josef U. Fürst for stimulating discussions and Gerd Leuchs for his support.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We report growth of superconducting [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} films by oxide molecular beam epitaxy (MBE). Careful tuning of the Ru flux with an electron beam evaporator enables us to optimize growth conditions including the Ru/Sr flux ratio and also to investigate stoichiometry effects on the structural and transport properties. The highest onset transition temperature of about 1.1 K is observed for films grown in a slightly Ru-rich flux condition in order to suppress Ru deficiency. The realization of superconducting [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} films via oxide MBE opens up a new route to study the unconventional superconductivity of this material.'
author:
- 'M. Uchida'
- 'M. Ide'
- 'H. Watanabe'
- 'K. S. Takahashi'
- 'Y. Tokura'
- 'M. Kawasaki'
title: 'Molecular beam epitaxy growth of superconducting [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} films'
---
The layered perovskite [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} has attracted enduring interest since Y. Maeno [*et al.*]{} found superconductivity in its single-crystalline bulk [@SRO]. Its fascinating properties as a possible two-dimensional chiral $p$-wave superconductor, classified into a topological superconductor, have been intensively studied from both the experimental and theoretical sides [@SROsymmetry; @SROreview1; @reviewdesiringfilm1; @reviewdesiringfilm2]. In spite of the lasting experimental progress as represented by strain effects [@strain1; @strain2], its underlying physics has not been entirely understood. In this context, reproducible growth of superconducting thin films has long been desired in order to enable junction and microfabricated device experiments for determining pairing symmetry and topological aspects of the superconductivity [@reviewdesiringfilm1; @reviewdesiringfilm2].
Growth of superconducting [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} thin films is known to be extremely difficult, because the low transition temperature ($T_{\mathrm{c}}\sim1.5$ K) in [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} bulks is highly sensitive to impurities [@sensitivity] and sample nonstoichiometry [@SROsymmetry]. Among the many [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} films grown by the pulsed laser deposition (PLD) method [@YoshiharuPLD; @RobinsonPLD; @antiphaseboundary1PLD; @antiphaseboundary2PLD; @otherSROfilm1PLD; @otherSROfilm2PLD; @otherSROfilm3PLD; @otherSROfilm4PLDlaser; @otherSROfilm5PLD; @otherSROfilm6PLD; @otherSROfilm8PLD], successful growth of superconducting films have been very limited [@YoshiharuPLD; @RobinsonPLD]. An alternative growth method is molecular beam epitaxy (MBE), which has traditionally delivered high-quality and high-reproducibility thin films in the field of semiconductors, but has now been adapted for the growth of oxides [@SchlomMBEreview]. In particular, it has found success in the growth of clean systems, such as those which display high mobility or unconventional superconductivity [@oxideMBEpower1; @oxideMBEpower2; @oxideMBEpower3; @oxideMBEpower4]. Nonetheless, MBE growth of superconducting [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} films has been highly challenging in spite of recent developments [@otherSROfilm7MBE; @SchlomSROMBE; @StemmerSROMBE; @WOE]. The primary challenge in the MBE growth is to evaporate high-purity Ru while maintaining its stable flux through film deposition.
Here we demonstrate the growth of superconducting [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} films using MBE with an electron beam evaporator. Careful tuning of the Ru flux enables us to perform systematic optimization of growth conditions to realize superconducting [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} films.
![ (a)–(c) XRD $\theta$–2$\theta$ scans, (d)–(f) rocking curves of the [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} (006) peak, and (g)–(i) AFM images, for the samples A–C grown with different ratios between the Ru and Sr fluxes. LSAT substrate peaks in the XRD scans are marked with an asterisk. Tiny peaks denoted by a triangle or a diamond are respectively ascribed to RuO$_2$ or other Ruddlesden-Popper phase. []{data-label="fig1"}](fig1.eps){width="13.5cm"}
The $c$-axis oriented [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} films were grown with a Veeco GEN10 oxide MBE system on as-received single crystalline (001) (LaAlO$_{3}$)$_{0.3}$(SrAl$_{0.5}$Ta$_{0.5}$O$_{3}$)$_{0.7}$ (LSAT) substrates supplied by Furuuchi Chemical Co. 4N Sr and 3N Ru elemental fluxes were simultaneously provided from a conventional Knudsen cell and a Telemark TT-6 electron beam evaporator, respectively. While the Sr flux $I_{\mathrm{Sr}}$, measured by an INFICON quartz crystal microbalance system, was set to $6.9\times10^{13}$ $\mathrm{atoms}/\mathrm{cm}^2 \mathrm{s}$, the Ru flux $I_{\mathrm{Ru}}$ was tuned to $3.3$, $3.4$, and $3.6\times10^{13}$ $\mathrm{atoms}/\mathrm{cm}^2 \mathrm{s}$ for samples A, B, and C, which correspond to $I_{\mathrm{Ru}}/I_{\mathrm{Sr}}=$ 0.48 (Ru-deficient), 0.50 (stoichiometric), and 0.53 (Ru-rich). Superconducting [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} films were not grown out of this flux ratio range ($I_{\mathrm{Ru}}/I_{\mathrm{Sr}}=$ 0.48–0.53). Other conditions were the same for these three samples. The deposition was performed in 100% $\mathrm{O}_{3}$ with a pressure of $1\times 10^{-6}$ Torr, supplied from a Meidensha Co. MPOG-104A1-R pure ozone generator, and at a substrate temperature of 900 $^{\circ}$C, achieved with a semiconductor-laser heating system [@otherSROfilm4PLDlaser]. The film thickness was about 58 nm and the growth rate was about 1.4 nm/min.
Figure 1 summarizes structural characterization of the three samples A–C grown with the different Ru/Sr flux ratios. As seen in x-ray diffraction (XRD) $\theta$–2$\theta$ scans (Figs. 1(a)–(c)), sharp (00$l$) [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} peaks ($l$: even integer) are commonly observed up to the (0014) peak, indicating $c$-axis oriented epitaxial film growth. Tiny peaks assigned to Ru-rich phases such as RuO$_2$ or other Ruddlesden-Popper phases [@RP1; @RP2] appear for samples B and C, while no impurity peaks are confirmed for sample A. From a thermodynamical standpoint, this result can be conversely interpreted as sample A having a non-negligible amount of Ru deficiency. In fact, sample B shows the sharpest film rocking curve among them in spite of the impurity peaks (Figs. 1(d)–(f)), although all the three values of the full width at half maximum (FWHM) are small enough to demonstrate high-quality oxide MBE growth. While the $a$-axis lattice constant is fixed to 3.87 [Å]{} ($-0.07$% compared to the bulk value [@latticeconstant]) on the LSAT substrate as confirmed in the reciprocal space mapping (not shown), the $c$-axis lattice constant estimated from the $\theta$–2$\theta$ scans is slightly elongated to 12.76 [Å]{} ($+0.17$%) for the three samples.
Surface topography taken by atomic force microscopy (AFM) (Figs. 1(g)–(i)) also consistently indicates changes reflecting the used Ru/Sr flux ratio. An extremely flat surface is confirmed for sample A. With increasing the Ru flux, on the other hand, ridge structures begin to be seen in sample B, and then some segregations presumably ascribed to RuO$_2$ appear on the surface of sample C. Accordingly, root mean square roughness $R_{\mathrm{RMS}}$ becomes much larger.
![ (a) Cross-sectional TEM image of the sample C, showing no secondary phase segregations, stacking faults, nor extended defects including out-of-phase boundary in a wide film region, which is quite a contrast to previously reported [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} films grown by PLD [@YoshiharuPLD; @antiphaseboundary1PLD; @antiphaseboundary2PLD]. Higher-resolution HAADF-STEM images showing (b) epitaxially connected lattice structures and (c) an out-of-phase boundary at the interface between the film and substrate. []{data-label="fig2"}](fig2.eps){width="13.5cm"}
Figure 2 shows cross-sectional transmission electron microscope (TEM) images of sample C. As shown in a low magnification image (Fig. 2(a)), there can be seen almost no segregations, stacking faults, nor extended defects inside the wide film region, even while the secondary phases due to excess supply of Ru are detected in XRD and AFM as noted above. In particular, extended defects characteristic to layered perovskite oxides, out-of-phase boundaries [@antiphaseboundary1PLD; @antiphaseboundary2PLD; @OPBs], are almost entirely eliminated, in stark contrast to the superconducting [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} film grown by PLD [@YoshiharuPLD]. Lattice structures of the [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} film and the LSAT substrate and their epitaxial relation can be clearly confirmed in the magnified image taken by high angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) (Fig. 2(b)). By searching in wider regions, an out-of-phase boundary is detected as indicated by an arrow (Fig. 2(c)), which may intrinsically originate from the 3.87 [Å]{}-high unit-cell step of the LSAT substrate. While the out-of-phase boundaries have been considered to strongly suppress the superconductivity in [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} films [@antiphaseboundary1PLD; @YoshiharuPLD], the defect-free regions in the grown films exist over about 500 nm, which is an order of magnitude longer than the in-plane superconducting coherence length of 66 nm [@coherencelength].
![ (a) Resistivity of the samples A–C as a function of temperature up to 300 K. (b)–(d) Low-temperature resistivity of the respective samples, measured with applying a magnetic field parallel to the $c$-axis at intervals of 200 Oe. []{data-label="fig3"}](fig3.eps){width="13.5cm"}
Figure 3 summarizes the transport characteristics of the three samples A–C. Longitudinal resistivity was measured with a standard four-probe method in a Quantum Design PPMS cryostat equipped with a 9 T superconducting magnet and a 3He refrigerator. The samples show residual resistivity ratio (RRR = $\rho_{\mathrm{300 K}}/\rho_{\mathrm{2 K}}$) as high as 22, 37, and 30 (Fig. 3(a)). A clear superconducting transition is observed for all the samples, which is systematically suppressed with applying a magnetic field parallel to the $c$-axis. The highest transition temperature for sample C is $T_{\mathrm{c,zero}}\sim0.8$ K (zero resistivity) and $T_{\mathrm{c,onset}}\sim1.1$ K (onset), which exceeds $T_{\mathrm{c,zero}}\sim0.5$ K and $T_{\mathrm{c,onset}}\sim0.9$ K of previously reported superconducting films grown by PLD [@YoshiharuPLD]. Here it is notable that sample A shows a lower $T_{\mathrm{c}}$ than the other two samples having the surface segregations, indicating that the suppression of Ru deficiency in films is crucially important for realizing better superconductivity as also pointed out in the bulk experiments [@SROsymmetry].
In conclusion, we have found a set of conditions to grow superconducting [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} films in MBE. By carefully tuning the Ru flux supplied from the electron beam evaporator, we have systematically investigated the relationship between film stoichiometry and structure and transport properties. The highest transition temperature $T_{\mathrm{c,zero}}\sim0.8$ K and $T_{\mathrm{c,onset}}\sim1.1$ K is observed for the film grown in the slightly Ru-rich flux condition. Although segregations due to the excess supply of Ru are confirmed on the surface, it is concluded that the suppression of the Ru deficiency in the film is crucially important. Further precise control of the flux ratio as well as optimization of other growth conditions will be necessary to increase $T_{\mathrm{c}}$ to the bulk level ($T_{\mathrm{c}}\sim1.5$ K). The ability to grow high-quality superconducting [[Sr$_{\mathrm{2}}$RuO$_{\mathrm{4}}$]{}]{} films using oxide MBE opens a new avenue for verifying pairing symmetry and topological aspects of its superconductivity, for example through phase-sensitive transport measurements of mesoscopic systems [@mesoscopic1] and Josephson junctions [@josephson1; @josephson2].
This work was supported by and by a Grant-in-Aid for Scientific Research on Innovative Areas “Topological Materials Science" No. JP16H00980 from MEXT, Japan and JST CREST Grant No. JPMJCR16F1, Japan. We thank J. Falson for proofreading of the manuscript.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the problem of visualizing large-scale and high-dimensional data in a low-dimensional (typically 2D or 3D) space. Much success has been reported recently by techniques that first compute a similarity structure of the data points and then project them into a low-dimensional space with the structure preserved. These two steps suffer from considerable computational costs, preventing the state-of-the-art methods such as the t-SNE from scaling to large-scale and high-dimensional data (e.g., millions of data points and hundreds of dimensions). We propose the LargeVis, a technique that first constructs an accurately approximated K-nearest neighbor graph from the data and then layouts the graph in the low-dimensional space. Comparing to t-SNE, LargeVis significantly reduces the computational cost of the graph construction step and employs a principled probabilistic model for the visualization step, the objective of which can be effectively optimized through asynchronous stochastic gradient descent with a linear time complexity. The whole procedure thus easily scales to millions of high-dimensional data points. Experimental results on real-world data sets demonstrate that the LargeVis outperforms the state-of-the-art methods in both efficiency and effectiveness. The hyper-parameters of LargeVis are also much more stable over different data sets.'
author:
- |
Jian Tang$^1$, Jingzhou Liu$^2$[^1], Ming Zhang$^{2}$, Qiaozhu Mei$^3$\
\
\
\
bibliography:
- 'sigproc.bib'
title: 'Visualizing Large-scale and High-dimensional Data'
---
[^1]: This work was done when the second author was an intern at Microsoft Research Asia.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Near a bifurcation point, the response time of a system is expected to diverge due to the phenomenon of critical slowing down. We investigate critical slowing down in well-mixed stochastic models of biochemical feedback by exploiting a mapping to the mean-field Ising universality class. This mapping allows us to quantify critical slowing down in experiments where we measure the response of T cells to drugs. Specifically, the addition of a drug is equivalent to a sudden quench in parameter space, and we find that quenches that take the cell closer to its critical point result in slower responses. We further demonstrate that our class of biochemical feedback models exhibits the Kibble-Zurek collapse for continuously driven systems, which predicts the scaling of hysteresis in cellular responses to more gradual perturbations. We discuss the implications of our results in terms of the tradeoff between a precise and a fast response.'
author:
- 'Tommy A. Byrd'
- Amir Erez
- 'Robert M. Vogel'
- Curtis Peterson
- Michael Vennettilli
- 'Grégoire Altan-Bonnet'
- Andrew Mugler
title: 'Biochemical feedback and its application to immune cells II: dynamics and critical slowing down'
---
[^1]
[^2]
Introduction
============
Critical slowing down is the phenomenon in which the relaxation time of a dynamical system diverges at a bifurcation point [@strogatz2018nonlinear]. Biological systems are inherently dynamic, and therefore one generally expects critical slowing down to accompany transitions between their dynamic regimes. Indeed, signatures of critical slowing down, including increased autocorrelation time and increased fluctuations, have been shown to precede an extinction transition in many biological populations [@scheffer2009early; @scheffer2012anticipating], including bacteria [@veraart2012recovery], yeast [@dai2012generic], and entire ecosystems [@wang2012flickering]. Similar signatures are also found in other biological time series, including dynamics of protein activity [@sha2003hysteresis] and neural spike dynamics [@meisel2015critical].
Canonically, critical slowing down depends on scaling exponents that define divergences along particular parameter directions in the vicinity of a critical point [@hohenberg1977theory]. Therefore, connecting the theory of critical slowing down to biological data requires identification of thermodynamic state variables, their scaling exponents, and a principled definition of distance from the critical point. However, in most biological systems it is not obvious how to define the thermodynamic state variables, let alone scaling exponents and distance from criticality. In a previous study [@erez2018universality] we showed how near its bifurcation point, a class of biochemical systems can be mapped to the mean-field Ising model, thus defining the state variables and their associated scaling exponents. This provides a starting point for the investigation of critical slowing down in such systems, as well as how to apply such a theory to experimental data.
Additionally, most studies of critical slowing down in biological systems investigate the response to a sudden experimental perturbation (a “quench”), such as a dilution or the addition of a nutrient or drug. This leaves unexplored the response to gradual environmental changes, a common natural scenario. When a gradual change drives a system near its critical point, critical slowing down delays the system’s response such that no matter how gradual the change, the response lags behind the driving. In physical systems this effect is known as the Kibble-Zurek mechanism [@kibble1976topology; @zurek1985cosmological], which predicts these nonequilibrium lagging dynamics in terms of the exponents of the critical point. It remains unclear whether and how the Kibble-Zurek mechanism applies to biological systems.
Here we investigate critical slowing down for well-mixed biochemical networks with positive feedback, and we use our theory to interpret the response of immune cells to an inhibitory drug. Using our previously derived mapping [@erez2018universality], we show theoretically that critical slowing down in our class of models proceeds according to the static and dynamic exponents of the mean-field Ising universality class. The mapping identifies an effective temperature and magnetic field in terms of the biochemical parameters, which defines a distance from the critical point that can be extracted from experimental fluorescence data. We find that drug-induced quenches that take an immune cell closer to its critical point result in longer response times, in qualitative agreement with our theory. We then show theoretically that our system, when driven across its bifurcation point, falls out of steady state in the manner predicted by the Kibble-Zurek mechanism, thereby extending Kibble-Zurek theory to a biologically relevant nonequilibrium setting. Our work elucidates the effects of critical slowing down in biological systems with feedback, and provides insights for interpreting cell responses near a dynamical transition point.
Results
=======
We consider a well-mixed reaction network in a cell where $X$ is the molecular species of interest, and the other species $A$, $B$, $C$, etc. form a chemical bath for $X$ \[Fig. \[fig:setup\](a)\]. Whereas previously we considered only the steady state distribution of $X$ [@erez2018universality], here we focus on dynamics in and out of steady state. Specifically, as shown in Fig. \[fig:setup\](b), we consider (i) steady state, where the bath is constant in time; (ii) a quench, where the bath changes its parameters suddenly; and (iii) driving, where the bath changes its parameters slowly and continuously. In each case we are interested in a corresponding timescale: (i) the autocorrelation time $\tau_c$ of $X$, (ii) the response time $\tau_r$ of $X$, and (iii) the driving time $\tau_d$ of the bath.
![(a) Inside a cell, a chemical species $X$ with molecule number $n$ exists in a bath of other species. (b) We consider steady-state, quench, and driven dynamics for the bath, and focus on the autocorrelation time $\tau_c$, response time $\tau_r$, and driving time $\tau_d$, respectively.[]{data-label="fig:setup"}](fig1){width="\linewidth"}
First we review the key features of our stochastic framework for biochemical feedback and its mapping to the mean-field Ising model [@erez2018universality]. We consider an arbitrary number of reactions $r$ in which $X$ is produced from bath species $Y_r^\pm$ and/or $X$ itself (feedback), $$\label{eq:rxns}
j_rX + Y_r^+ \rightleftharpoons (j_r+1)X + Y_r^-,$$ where $j_r$ are stoichiometric integers. The probability of observing $n$ molecules of species $X$ in steady state according to Eq. \[eq:rxns\] is $$\label{eq:pn}
p_n = \frac{p_0}{n!} \prod_{j=1}^n f_j,$$ where $p_0^{-1} = \sum_{n=0}^\infty(1/n!)\prod_{j=1}^n f_j$ is set by normalization, and $f_n$ is a nonlinear feedback function governed by the reaction network. The inverse of Eq. \[eq:pn\], $$\label{eq:fn}
f_n = \frac{np_n}{p_{n-1}},$$ allows calculation of the feedback function from the distribution. The function $f_n$ determines an effective order parameter, reduced temperature, and magnetic field, $$\label{eq:cparam}
m \equiv \frac{n_*-n_c}{n_c}, \quad
h \equiv \frac{2(f_{n_c} - n_c)}{-f'''_{n_c}n_c^3}, \quad
\theta \equiv \frac{2(1-f'_{n_c})}{-f'''_{n_c}n_c^2},$$ respectively, where $n_c$ is defined by $f''_{n_c} = 0$, and $n_*$ are the maxima of $p_n$. Qualitatively, $n_c$ sets the typical molecule number, $\theta$ drives the system to a unimodal ($\theta > 0$) or bimodal ($\theta < 0$) state, and $h$ biases the system to high ($h > 0$) or low ($h < 0$) molecule numbers. The critical point occurs at $\theta = h = 0$. The state variables $m$, $\theta$, and $h$ scale according to the exponents $\alpha=0$, $\beta=1/2$, $\gamma=1$, and $\delta=3$ of the mean-field Ising universality class. Detailed analysis of this mapping in steady state is found in our previous work [@erez2018universality].
Near the critical point, all specific realizations of a class of systems scale in the same way, and therefore it suffices to consider a particular realization of Eq. \[eq:rxns\] from here on. We choose Schlögl’s second model [@erez2018universality], a simple and well-studied case [@schlogl1972chemical; @dewel1977renormalization; @nicolis1980systematic; @brachet1981critical; @grassberger1982phase; @prakash1997dynamics; @liu2007quadratic; @vellela2009stochastic] in which $X$ is either produced spontaneously from bath species $A$, or in a trimolecular reaction from two existing $X$ molecules and bath species $B$, $$\label{eq:schlogl_rxns}
A \xrightleftharpoons[k_1^-]{k_1^+} X, \quad 2X+B \xrightleftharpoons[k_2^-]{k_2^+} 3X.$$ In this case the feedback function is $$\label{eq:schlogl_fn}
f_n = \frac{aK^2 + s(n-1)(n-2)}{(n-1)(n-2)+K^2},$$ where we have introduced the dimensionless quantities $a \equiv k_1^+n_A/k_1^-$, $s \equiv k_2^+ n_B/k_2^-$, and $K^2 \equiv k_1^-/k_2^-$ in terms of the reaction rates and the numbers of $A$ and $B$ molecules. Given Eqs. \[eq:cparam\] and \[eq:schlogl\_fn\], the effective thermodynamic variables $n_c$, $\theta$, and $h$ can be written in terms of $a$, $s$, and $K$ or vice versa [@erez2018universality], with $1/k_1^-$ setting the units of time.
Critical slowing down in steady state
-------------------------------------
In steady state, critical slowing down causes correlations to become long-lived near a dynamical transition point. Qualitatively, the fixed point is transitioning from stable to unstable, and therefore the basin of attraction is becoming increasingly wide. As a result, a dynamic trajectory takes increasingly long excursions from the mean, making it heavily autocorrelated. The autocorrelation time $\tau_c$ diverges at the critical point according to [@pathria2011statistical] $$\begin{aligned}
\label{eq:tauc1}
\tau_c|_{h=0} &\sim |\theta|^{-\nu z}, \\
\label{eq:tauc2}
\tau_c|_{\theta=0} &\sim |h|^{-\nu z/\beta\delta},\end{aligned}$$ where we expect $\nu z = 1$ for mean-field dynamics [@hohenberg1977theory; @kopietz2010introduction]. Here the autocorrelation time $\tau_c$ is defined as $$\label{eq:tauc_def}
\tau_c = \frac{1}{\kappa(0)}\int_0^\infty dt\ \kappa(t),$$ where $\kappa(t) = \langle n(0)n(t)\rangle - \bar{n}^2$ is the steady-state autocorrelation function, $\kappa(0) = \sigma^2$ is the variance, and we have taken the start time to be $t=0$ without loss of generality because the system is in steady state.
To confirm the value of $\nu z$, we plot $\tau_c$ vs. $h$ at $\theta = 0$ (Eq. \[eq:tauc2\]). We calculate $\tau_c$ either directly from the master equation or from stochastic simulations [@gillespie1977exact] using the method of batch means [@thompson2010comparison] (see Appendix \[app:time\]). The results are shown in Fig. \[fig:ss\]. We see in Fig. \[fig:ss\](a) that $\tau_c$ indeed diverges with $h$, and that the location of the divergence approaches the expected value $h = 0$ as the molecule number $n_c$ increases. We also see that the height of the peak increases with $n_c$ due to the rounding of the divergence [@stephens2013statistical]. The inset of Fig. \[fig:ss\](b) plots this dependence: we see that $\tau_c$ at the critical point $\theta = h = 0$ scales like $n_c^{1/2}$ for large $n_c$ (the application of this dependence to dynamic driving will be discussed in Section \[sec:KZ\]). Finally, we see in the main panel of Fig. \[fig:ss\](b) that when $n_c$ is sufficiently large, $\tau_c$ falls off with $|h|$ with the expected scaling exponent of $\nu z/\beta\delta = 2/3$. Taken together, these results confirm that the divergence of the autocorrelation time in the Schlögl model obeys the static exponents of the mean-field Ising universality class ($\beta\delta = 3/2$) and the dynamic expectation for mean-field systems ($\nu z = 1$).
![Critical slowing down in steady state. (a) Autocorrelation time $\tau_c$ in Schlögl model (Eq. \[eq:tauc\_def\]) peaks with field $h$ when reduced temperature $\theta = 0$. Height increases and location moves to $h=0$ as molecule number $n_c$ increases. Time is in units of $1/k_1^-$. (b) At large $n_c$, $\tau_c$ scales with $|h|$ with expected exponent of $\nu z/\beta\delta = 2/3$. Inset: $\tau_c$ at $\theta=h=0$ scales as $n_c^{1/2}$. In a and inset of b, $\tau_c$ is calculated using eigenfunctions with cutoff $N = \max(100,3n_c)$; in main panel of b, $\tau_c$ is calculated using batch means with 250 trajectories, duration $T = 10^5$, and batch time $\tau_b =$ 2,222 (see Appendix \[app:time\]).[]{data-label="fig:ss"}](fig2){width="\linewidth"}
Quench response and application to immune cells
-----------------------------------------------
When subjected to a sudden environmental change (a quench), the system will take some finite amount of time to respond \[Fig. \[fig:setup\](b), middle\]. We expect that if a quench takes the system closer to its critical point, the response time should be longer due to critical slowing down. To make this expectation quantitative, we define the response time $\tau_r$ in terms of the dynamics of the mean molecule number $\bar{n}$ as $$\label{eq:taur1}
\tau_r = \frac{1}{\Delta\bar{n}(0)}\int_0^{t_{\max}} dt\ \Delta\bar{n}(t),$$ where the quench occurs at $t=0$, we define $\Delta \bar{n}(t) = \bar{n}(t) - \bar{n}(t_{\max})$, and we ensure that $t_{\max} \gg \tau_r$. We compute $\bar{n}(t)$ from the time-dependent distribution $p_n(t)$ using the stochastic simulations. Examples of $p_n(t)$ for a small and a large quench are shown in Fig. \[fig:quench\](a).
We define the distance from the critical point in terms of the state variables $\theta$ and $h$. Specifically, $\tau_c$ scales identically with $\theta^{\beta\delta}$ as it does with $h$ (Eqs. \[eq:tauc1\] and \[eq:tauc2\]), which defines the Euclidean distance $d_c$ from the critical point as $$\label{eq:dc}
d_c = \left[(\theta^{\beta\delta})^2 + h^2\right]^{1/2}.$$ This measure will be important when comparing with the experiments because, as opposed to in most condensed matter experiments, it is difficult in the biological experiments we describe to manipulate only one parameter ($\theta$ or $h$) independently of the other.
![Quench response in theory (left) and in immune cell experimental data (right). (a) Stochastic simulations of Shlögl model show effect of small and large parameter quenches on distribution. Time is in units of $1/k_1^-$. (b) Initial (black square) and quenched (colored circles) parameter values in $\theta$ and $h$ space in model; $n_c = 500$. Dotted lines show contours of equal $d_c$ (Eq. \[eq:dc\]), distance from critical point ($\theta = h = 0$). Response time $\tau_r$ in model (c) decreases with $d_c$ and (g) increases with entropy $S$. (d) Experimental distributions of T cell ppERK fluorescence intensity measured at times after addition of SRC inhibitor (see Fig. \[fig:doses\] for all doses). (e) $\theta$ and $h$ extracted from initial distribution (black square) and final distributions (colored circles) for all \[SRCi\] doses (color bar). Experimental response time $\tau_r$ (f) decreases with $d_c$ and (h) increases with $S$. Error bars: for $\theta$ and $h$, standard error from Savitzky-Golay [@savitzky1964smoothing] filter windows $25 \le W \le 35$ [@erez2018universality]; for $d_c$, propagated in quadrature from e; for $\tau_r$, standard deviation of Riemann sums spanning left- to right-endpoint methods to approximate integral in Eq. \[eq:taur2\]. In h, fluorescence of one molecule set to $I_1 = 10$.[]{data-label="fig:quench"}](fig3){width="\linewidth"}
To test whether the response time increases with proximity to the critical point, we must define initial values $\theta_0$ and $h_0$ for the environment before the quench, and a series of values $\theta$ and $h$ for the environment after the quench that are varying distances from the critical point $\theta = h = 0$. There are many such choices for these values, but anticipating the experimental results that we will describe shortly, we choose the initial point (black square) and final points (colored circles) shown in Fig. \[fig:quench\](b). Dotted curves of equal $d_c$ are also shown, which make clear that larger quenches (yellow circles) take the system farther from the critical point than smaller quenches (blue circles). The dependence of $\tau_r$ on $d_c$ is shown in Fig. \[fig:quench\](c), and we see that indeed $\tau_r$ decreases as $d_c$ increases, or equivalently the response time increases with proximity to the critical point.
We now compare our theory with data from immune cells. We focus on the abundance in T cells of doubly phosphorylated ERK (ppERK), a protein that initiates cell proliferation and is implicated in the self/non-self decision between mounting an immune response or not [@vogel2016dichotomy; @altan2005modeling]. Specifically, we use flow cytometry to measure the ppERK distribution at various times after the addition of a drug that inhibits SRC, a key enzyme in the cascade that leads to ERK phosphorlyation (see Appendix \[app:expt\] for experimental methods). When the dose of the drug is small, the distribution hardly changes \[Fig. \[fig:quench\](d), top\]; whereas when the dose is large, the distribution changes significantly \[Fig. \[fig:quench\](d), bottom\]. The responses to all doses are shown in Appendix \[app:expt\].
After the addition of the drug, the cells reach a new steady-state ppERK distribution \[green curves in Fig. \[fig:quench\](d)\]. The distribution corresponds to an effective feedback function via Eq. \[eq:fn\], from which the effective temperature $\theta$ and field $h$ can be calculated via Eq. \[eq:cparam\] [@erez2018universality]. The values of $\theta$ and $h$ calculated from the experimental distributions are shown in Fig. \[fig:quench\](e). We see that larger doses take the cells farther from their initial distribution (black square), as expected. We also see that larger doses take the system farther from the critical point $\theta = h = 0$. The general shape of the $\theta$ and $h$ values motivated our choice of theoretical values in Fig. \[fig:quench\](b).
We define the response time to the drug as in Eq. \[eq:taur1\], here in terms of the mean fluorescence intensity of ppERK, $$\label{eq:taur2}
\tau_r = \frac{1}{\Delta\bar{I}(0)}\int_0^{t_{\max}} dt\ \Delta\bar{I}(t),$$ where $\Delta \bar{I}(t) = \bar{I}(t) - \bar{I}(t_{\max})$ and $t_{\max} = 30$ min. We calculate the distance from criticality using Eq. \[eq:dc\] as before, here using the experimental values of $\theta$ and $h$. We see in Fig. \[fig:quench\](f) that the response time $\tau_r$ decreases with the distance from criticality $d_c$, consistent with the prediction from the theory \[Fig. \[fig:quench\](c)\]. This suggests that critical slowing down occurs in the response of the T cells to the drug.
Although in Fig. \[fig:quench\](f) the response time $\tau_r$ comes directly from the experimental data, the distance from criticality $d_c$ is calculated from the experimental data using expressions from the theory (Eqs. \[eq:fn\] and \[eq:cparam\]). This makes the results in Figs. \[fig:quench\](c) and \[fig:quench\](f) not entirely independent. To confirm that the agreement between Figs. \[fig:quench\](c) and \[fig:quench\](f) is not a result of an implicit co-dependence, we seek a measure that is related to distance from criticality but that is not dependent on the theory. We choose the entropy of the distribution $S = -\sum_n p_n\log p_n$ because near criticality, the distribution is broad and flat, and therefore we expect the entropy to be large; whereas far from criticality, the distribution has either one or two narrow peaks, and therefore we expect the entropy to be small [@erez2018universality]. Indeed, we see in Fig. \[fig:quench\](g) that in the theory, the response time $\tau_r$ increases with the entropy $S$, consistent with the fact that it decreases with the distance from criticality \[Fig. \[fig:quench\](c)\]. The same is evident in the experiments: we see in Fig. \[fig:quench\](h) that low drug doses correspond to long response times and high entropies, whereas high drug doses correspond to short response times and low entropies, resulting in an increase of response time $\tau_r$ with entropy $S$. Calculating the entropy in Fig. \[fig:quench\](h) requires a conversion between intensity $I$ and molecule number $n$, and we have checked that the results in Fig. \[fig:quench\](h) are qualitatively unchanged for different choices of this conversion factor over several orders of magnitude. The agreement between Figs. \[fig:quench\](g) and \[fig:quench\](h) offers further evidence that the T cells experience critical slowing down, with the data analysis completely independent from our theory.
Dynamic driving and Kibble-Zurek collapse {#sec:KZ}
-----------------------------------------
While some environmental changes are sudden, many changes in a biological context are gradual \[Fig. \[fig:setup\](b), right\]. When a gradual change drives a system through its critical point, critical slowing down delays the system’s response such that no matter how gradual the change, the response lags behind the driving. Although in a biological setting the driving protocol could take many forms, terms beyond the leading-order linear term do not change the critical dynamics [@chandran2012kibble]. This is a major theoretical advantage because it allows us to specialize to linear driving without loss of biological realism. Specifically, we focus on linear driving across the critical point with driving time $\tau_d$, setting either $\theta(t) = \theta_i-(\theta_f - \theta_i)t/\tau_d$ and $h=0$, or $h(t) = h_i-(h_f-h_i)t/\tau_d$ and $\theta = 0$, where $i$ and $f$ denote the initial and final parameter values, respectively.
In a traditional equilibrium setting, the dynamics of lagging trajectories are described in terms of the critical exponents by the Kibble-Zurek mechanism [@kibble1976topology; @zurek1985cosmological]. The idea of the Kibble-Zurek mechanism is that far from the critical point, the change in the system’s correlation time due to the driving, over a correlation time, is small compared to the correlation time itself, $(d\tau_c/dt)\tau_c \ll \tau_c$, and therefore the system responds adiabatically. However, as the system is driven closer to the critical point, these two quantities are on the same order, or $d\tau_c/dt \sim 1$, and the system begins to lag. Applying this condition to Eqs. \[eq:tauc1\] and \[eq:tauc2\], and using the above expressions for $\theta(t)$ and $h(t)$, one obtains $$\begin{aligned}
\label{eq:kz1}
\theta &\sim \tau_d^{-1/(\nu z+1)}, \\
\label{eq:kz2}
h &\sim \tau_d^{-\beta\delta/(\nu z + \beta\delta)},\end{aligned}$$ respectively. Because $m \sim (-\theta)^\beta$ or $m \sim h^{1/\delta}$ near criticality in the mean-field Ising class, we have $$\begin{aligned}
\label{eq:kzm1}
m &\sim \tau_d^{-\beta/(\nu z+1)}, \\
\label{eq:kzm2}
m &\sim \tau_d^{-\beta/(\nu z + \beta\delta)},\end{aligned}$$ respectively. Therefore, if the system is driven at different timescales $\tau_d$, the Kibble-Zurek mechanism predicts that plots of the rescaled variables $m\tau_d^{\beta/(\nu z+1)}$ vs. $\theta\tau_d^{1/(\nu z+1)}$ or $m\tau_d^{\beta/(\nu z + \beta\delta)}$ vs. $h\tau_d^{\beta\delta/(\nu z + \beta\delta)}$ will collapse onto universal curves.
![Dynamic driving and Kibble-Zurek collapse. (a) As reduced temperature $\theta$ is driven over time $\tau_d$ in Schlögl model, order parameter $m$ lags behind due to critical slowing down. Decreasing $\theta$ causes supercooling (left curves), while increasing $\theta$ causes superheating (right curves), resulting in hysteresis. (b) Same, for driving $h$. (c, d) Rescaled curves collapse as predicted. Each point is computed via Eq. \[eq:cparam\] from the mode $n_*$ in b, or the modes $n_*^{(1)}<n_c$ and $n_*^{(2)}>n_c$ in a, of $10^5$ simulation trajectories. For finite-size correction we use $n_c = 10\tau_d$ in a and $n_c = 22\tau_d^{4/5}$ in b. Time is in units of $1/k_1^-$.[]{data-label="fig:kz"}](fig4){width="\linewidth"}
When testing these predictions using simulations of a spatially extended physical system, the finite size of the system causes a truncation of the autocorrelation time. This truncation is usually accounted for using a finite-size correction [@chandran2012kibble]. In our system, a similar truncation of the autocorrelation time is caused by the finite number of molecules. Specifically, the inset of Fig. \[fig:ss\](b) shows that at criticality we have $\tau_c \sim n_c^{1/2}$ for large $n_c$, where $n_c$ sets the typical number of molecules in the system. Therefore, we interpret $n_c$ as a “system size,” and we correct for finite-size effects in the following way. Combining the relation $\tau_c \sim n_c^{1/2}$ with Eqs. \[eq:tauc1\] and \[eq:tauc2\], and Eqs. \[eq:kz1\] and \[eq:kz2\], we obtain $$\begin{aligned}
\label{eq:finite1}
n_c &\sim \tau_d^{2\nu z/(\nu z+1)},\\
\label{eq:finite2}
n_c &\sim \tau_d^{2\nu z/(\nu z + \beta\delta)},\end{aligned}$$ for the driving of $\theta$ or $h$, respectively. We choose $n_c$ arbitrarily for a particular driving time $\tau_d$, and when we choose a new $\tau_d$, we scale $n_c$ appropriately according to Eqs. \[eq:finite1\] and \[eq:finite2\].
This procedure allows us to test the predictions of the Kibble-Zurek mechanism using simulations of the Schlögl model. The results are shown in Fig. \[fig:kz\]. We see in Fig. \[fig:kz\](a) that as $\theta$ is driven from a positive to a negative value, the bifurcation response is lagging, occurring at a value less than the critical value $\theta = 0$ (supercooling). Conversely, when $\theta$ is driven from a negative to a positive value, the convergence occurs at a value greater than $\theta = 0$ (superheating). In both directions, the lag is larger when the driving is faster, corresponding to smaller values of $\tau_d$ (from yellow to dark brown). We see in Fig. \[fig:kz\](b) that similar effects occur for the driving of $h$. Yet, we see in Figs. \[fig:kz\](c) and (d) that the rescaled variables collapse onto single, direction-dependent curves within large regions near criticality. Note that the direction dependence (i.e., hysteresis) is preserved as part of these universal curves, but the lags vanish in the collapse. This result demonstrates that our nonequilibrium birth-death model exhibits the Kibble-Zurek collapse predicted for critical systems. Together with our previous findings, this result suggests that such a collapse should emerge in biological experiments where environmental parameters (e.g., drug dose) are dynamically controlled in a gradual manner. More broadly, by phenomenologically collapsing such experimental curves, it should be possible to deduce the critical exponents of such biological systems without fine-tuning them to criticality, but instead by gradual parameter sweeps.
Discussion
==========
We have investigated critical slowing down in a minimal stochastic model of biochemical feedback. By exploiting a mapping to Ising-like thermodynamic variables, we have made quantitative predictions for the response of a system with feedback to both sudden and gradual environmental changes. In response to a sudden change (a quench), we have shown that the system will respond more slowly if the quench takes it closer to its critical point, in qualitative agreement with multiple-time-point flow cytometry experiments in immune cells. In response to more gradual driving, we have shown that the lagging dynamics of the system proceed according to the Kibble-Zurek mechanism for driven critical phenomena. Together, our results elucidate the consequences of critical slowing down for biochemical systems with feedback, and demonstrate those consequences on an example system from immunology.
For the immune cells, critical slowing down may present a tradeoff in terms of the speed vs. the precision of an immune response. Specifically, ppERK is implicated in the decision of whether or not to mount the immune response [@vogel2016dichotomy; @altan2005modeling], suggesting that ppERK dynamics near the bifurcation point are of key biological importance. Yet, the bifurcation point is the point where critical slowing down is most pronounced. In fact, the inset of Fig. \[fig:ss\](b) demonstrates that the system slows down as the number of molecules in the system increases. On the other hand, large molecule number is known to decrease intrinsic noise and thereby increase the precision of a response [@elowitz2002stochastic]. This suggests that cells may face a tradeoff in terms of speed vs. precision when responding to changes that occur near criticality, as suggested for other biological systems [@skoge2011dynamics; @mora2011biological].
Our work extends the Kibble-Zurek mechanism to a nonequilibrium biological context. Traditionally, the mechanism has been applied to physical systems from cosmology [@kibble1976topology] and from hard [@zurek1985cosmological; @del2014universality] or soft [@deutschlander2015kibble] condensed matter. Here, we extend the mechanism to the context of biochemical networks with feedback, where the system already exists in a nonequilibrium steady state, and the external protocol takes the system further out of equilibrium into a driven state. It will be interesting to see to what other nonequilibrium contexts the Kibble-Zurek mechanism can be successfully applied [@deffner2017kibble].
The theory we present here assumes only intrinsic birth-death reactions and neglects more complex mechanisms such as bursting [@friedman2006linking; @mugler2009spectral], parameter fluctuations [@shahrezaei2008colored; @horsthemke1984noise], or cell-to-cell variability [@cotari2013cell; @erez2018modeling] that may play an important role in the immune cells. Nonetheless, similar models that also focus only on intrinsic noise have successfully described ppERK in T cells in the past [@das2009digital; @prill2015noise]. Moreover, we expect that intrinsic fluctuations should play their largest role near the bifurcation point. Finally, we expect that near the bifurcation point, the essential behavior of the system should be captured by any model that falls within the appropriate universality class.
In this and previous work [@erez2018universality] we have explored the dynamic and static scaling properties of single cells subject to biochemical feedback. Natural extensions include generalizing the theory to cell populations or other systems that are not well-mixed such as intracellular compartments. This would allow one to investigate the spatial consequences of proximity to a bifurcation point, such as long-range correlations in molecule numbers and the associated implications for sensing, information transmission, patterning, or other biological functions.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Anushya Chandran for helpful communications. This work was supported by Simons Foundation grant 376198 (T.A.B. and A.M.), Human Frontier Science Program grant LT000123/2014 (Amir Erez), National Science Foundation Research Experiences for Undergraduates grant PHY-1460899 (C.P.), National Institutes of Health (NIH) grants R01 GM082938 (A.E.) and R01 AI083408 (A.E., R.V., and G.A.-B) and the NIH National Cancer Institute Intramural Research programs of the Center for Cancer Research (A.E. and G.A.-B.).
Autocorrelation time {#app:time}
====================
We calculate the autocorrelation time $\tau_c$ (Eq. \[eq:tauc\_def\]) for the Schlögl model in steady state using one of two methods, the first more efficient for small molecule numbers, and the second more efficient for large molecule numbers. The first method is to calculate $\tau_c$ numerically from the master equation for $p_n$ by eigenfunction expansion. The master equation follows from the reactions in Eq. \[eq:schlogl\_rxns\] [@erez2018universality] and can be written in vector notation as $$\label{eq:meL}
\dot{\vec{p}} = {\bf L}\vec{p}.$$ where ${\bf L}$ is a tridiagonal matrix containing the birth and death propensities for $X$. The eigenvectors of [**L**]{} satisfy $$\begin{aligned}
{\bf L}\vec{v}_j &= \lambda_j \vec{v}_j, \\
\vec{u}_j{\bf L} &= \lambda_j \vec{u}_j,\end{aligned}$$ where the eigenvalues obey $\lambda_j \le 0$ with only $\lambda_0$ vanishing for the steady state, and $\vec{v}_j^T \neq \vec{u}_j$ because [**L**]{} is not Hermitian [@walczak2009stochastic]. Because Eq. \[eq:meL\] is linear in $\vec{p}$, the solution is $$\label{eq:pt}
p_n(t) = \sum_{jn'} u_{jn'} p_{n'}(0) e^{\lambda_jt} v_{nj}$$ for initial condition $p_n(0)$. Calling $n(0) \equiv m$ and $n(t) \equiv n$, we write the autocorrelation function (see Eq. \[eq:tauc\_def\]) as $$\label{eq:kappa2}
\kappa(t) = -\bar{n}^2 + \sum_{mn} p_{mn} mn = -\bar{n}^2 + \sum_{mn} p_{n|m}p_m mn,$$ where $p_m = v_{m0}$ is the steady-state distribution, and $p_{n|m}$ is the dynamic solution at time $t$ assuming the system starts with $m$ molecules. That is, $p_{n|m}$ is given by Eq. \[eq:pt\] with initial condition $p_n(0) = \delta_{nm}$. Eq. \[eq:kappa2\] becomes $$\begin{aligned}
\kappa(t) &= -\bar{n}^2 + \sum_{mn} mn v_{m0} \sum_j u_{jm} e^{\lambda_jt} v_{nj} \\
\label{eq:kappa3}
&= \sum_{mn} mn v_{m0} \sum_{j=1}^\infty u_{jm} e^{\lambda_jt} v_{nj},\end{aligned}$$ where the second step uses orthonormality, $\sum_j v_{nj} u_{jn'} = \delta_{nn'}$, and probability conservation, $u_{0n} = 1$, to recognize that the $j=0$ term evaluates to $\bar{n}^2$. Inserting Eq. \[eq:kappa3\] into Eq. \[eq:tauc\_def\] and performing the integral (recalling that $\lambda_j < 0$ for $j > 0$), we obtain $$\tau_c = \frac{1}{\sigma^2} \sum_{mn} mn v_{m0}
\sum_{j=1}^\infty u_{jm} \left(\frac{1}{-\lambda_j}\right) v_{nj}.$$
![Autocorrelation time computed (a) numerically using eigenfunction expansion or (b) by simulation using method of batch means. For sufficient cutoff $N$ or trajectory duration $T$, respectively, both methods converge to same value (dashed line). Parameters: $\theta = h = 0$ and $n_c = 100$. Time is in units of $1/k_1^-$. In (b), $\tau_b = 1000$, and error bars are standard error from $50$ trajectories.[]{data-label="fig:tauc"}](fig5){width="\linewidth"}
In matrix notation, $$\label{eq:tauc_mat}
\tau_c = \sigma^{-2} \vec{n} {\bf V} {\bf F} {\bf U} \vec{w},$$ where $\vec{n}$ is a row vector, $\vec{w} = mv_{m0}$ is a column vector, and neither the eigenvector matrices ${\bf V}$ and ${\bf U}$ nor the diagonal matrix $F_{jj'} = -\delta_{jj'}/\lambda_j$ contain the $j=0$ term. Numerically, we compute $\tau_c$ via Eq. \[eq:tauc\_mat\] using a cutoff $N > n_c$ for the vectors and matrices.
The second method is to calculate $\tau_c$ from stochastic simulations [@gillespie1977exact] and the method of batch means [@thompson2010comparison]. The idea is to divide a simulation trajectory of length $T$ into batches of length $\tau_b$. In the limit $T\gg\tau_b\gg\tau_c$, the correlation time can be estimated by [@thompson2010comparison] $$\label{eq:batch}
\tau_c = \frac{\tau_b\sigma_b^2}{2\sigma^2},$$ where $\sigma_b^2$ is the variance of the means of the batches.
In Fig. \[fig:tauc\] we verify that the two methods converge to the same limit for sufficiently large $N$ or $T$, respectively. We find that the first method is more efficient until $n_c \sim 1000$, when numerically computing the eigenvectors for large $N > n_c$ becomes intractable.
Experimental methods {#app:expt}
====================
{width=".7\linewidth"}
The data in Fig. \[fig:doses\] \[of which the smallest and largest doses are reproduced in Fig. \[fig:quench\](d)\] were acquired at the same time and in a similar way as the data published in [@vogel2016dichotomy] and summarized in [@erez2018universality]. The difference is that, instead of only recording the data after steady state was reached, the time series was sampled by applying a chemical fixative to stop chemical reactions and preserve all biomolecular states. Specifically, we administered ice cold formaldehyde in PBS to each experimental well of a 96 well-v-bottom plate such that the final working dilution is 2%, and then transferred the cell-fixative solution to a new 96 well-v-bottom plate on ice. Cells were kept on ice for 10 minutes and then precipitated by centrifugation, resuspended in ice-cold 90% methanol, and placed in a $-20$ $^{\rm o}$C freezer until measurements were taken.
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[^1]: These authors contributed equally.
[^2]: These authors contributed equally.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
It is well-known that the Shannon entropies of some parameterized probability distributions are concave functions with respect to the parameter. In this paper we consider a family of such distributions (including the binomial, Poisson, and negative binomial distributions) and investigate the Shannon, Rényi, and Tsallis entropies of them with respect to the complete monotonicity.\
**keywords**: entropies; concavity; complete monotonicity; inequalities\
**subject class**: 94A17; 60E15; 26A51
author:
- |
Ioan Raşa\
Department of Mathematics, Technical University of Cluj-Napoca,\
Memorandumului Street 28,\
400114 Cluj-Napoca,\
Romania, ioan.rasa$@$math.utcluj.ro
title: Complete monotonicity of some entropies
---
Introduction {#intro}
============
Let $c\in \mathbb{R}$, $I_c := \left [ 0, -\frac{1}{c}\right ]$ if $c<0$, and $I_c:= [0,+\infty )$ if $c \geq 0$.
For $\alpha \in \mathbb{R}$ and $k \in \mathbb{N}_0$ the binomial coefficients are defined as usual by $${\alpha \choose k}:=\frac{\alpha (\alpha -1)\dots (\alpha-k+1)}{k!}\quad \text{if } k \in \mathbb{N}, \text{ and } {\alpha \choose 0}:=1.$$
Let $n> 0$ be a real number such that $n>c$ if $c\geq 0$, or $n=-cl$ with some $l\in \mathbb{N}$ if $c<0$.
For $k\in \mathbb{N}_0$ and $x\in I_c$ define $$p_{n,k}^{[c]}(x):=(-1)^k {-\frac{n}{c} \choose k}(cx)^k (1+cx)^{-\frac{n}{c}-k}, \quad \text{if } c\neq 0,$$ $$p_{n,k}^{[0]}(x):=\lim _{c\to 0} p_{n,k}^{[c]}(x)= \frac{(nx)^k}{k!}e^{-nx}.$$
Details and historical notes concerning these functions can be found in [@3], [@7], [@21] and the references therein. In particular, $$\frac{d}{dx}p_{n,k}^{[c]}(x) = n \left ( p_{n+c,k-1}^{[c]}(x) - p_{n+c,k}^{[c]}(x)\right ).\label{eq:1}$$
Moreover, $$\sum _{k=0}^\infty p_{n,k}^{[c]}(x) = 1;\label{eq:2}$$
$$\sum _{k=0}^\infty k p ^{[c]}_{n,k}(x)=nx,\label{eq:3}$$
so that $\left (p_{n,k}^{[c]}(x)\right )_{k\geq 0}$ is a parameterized probability distribution. Its associated Shannon entropy is $$H_{n,c}(x):=-\sum_{k=0}^\infty p_{n,k}^{[c]}(x) \log p_{n,k}^{[c]}(x),$$ while the Rényi entropy of order $2$ and the Tsallis entropy of order $2$ are given, respectively, by (see [@18], [@20]) $$R_{n,c}(x):= -\log S_{n,c}(x); \quad T_{n,c}(x):=1-S_{n,c}(x),$$ where $$S_{n,c}(x) := \sum _{k=0}^\infty \left (p_{n,k}^{[c]}(x)\right )^2, \quad x\in I_c.$$
The cases $c=-1$, $c=0$, $c=1$ correspond, respectively, to the binomial, Poisson, and negative binomial distributions. For other details see also [@15], [@16].
In this paper we investigate the above entropies with respect to the complete monotonicity.
Shannon entropy
===============
A. Let’s start with the case $c<0$. {#a.-lets-start-with-the-case-c0. .unnumbered}
-----------------------------------
$H_{n,-1}$ is a concave function; this is a special case of the results of [@19]; see also [@6], [@8], [@9] and the references therein.
Here we shall determine the signs of all the derivatives of $H_{n,c}$.
Let $c<0$. Then, for all $k\geq 0$, $$H_{n,c}^{(2k+2)}(x)\leq 0, \quad x \in \left ( 0,-\frac{1}{c} \right ),\label{eq:4}$$
$$H_{n,c}^{(2k+1)}(x) = \label{eq:5}
\begin{cases}
\geq 0 & x \in ( 0,-\frac{1}{2c} ],\\
\leq 0 & x \in [ -\frac{1}{2c}, - \frac{1}{c} ).\\
\end{cases}$$
**Proof** We have $n=-cl$ with $l \in \mathbb{N}$. As in [@10], let us represent $\log{(l!)}$ by integrals: $$\log{(l!)} = \int _0 ^\infty \left ( l - \frac{1-e^{-ls}}{1-e^{-s}} \right )\frac{e^{-s}}{s} ds = \int _0 ^1 \left ( \frac{1-(1-t)^l}{t} -l \right ) \frac{dt}{\log{(1-t)}}.\label{eq:6}$$
Now using , and we get
$$H_{n,c}(x) = H_{l,-1}(-cx) = - l \left [(-cx)\log{(-cx)}+(1+cx)\log{(1+cx)}\right ]+$$
$$\int _0 ^1 \frac{-t}{\log{(1-t)}} \frac{(1+cxt)^l+(1-t-cxt)^l-1-(1-t)^l}{t^2}dt.$$
It is a matter of calculus to prove that
$$\begin{aligned}
H''_{n,c}(x) &=& cl \left ( \frac{1}{x} - \frac{c}{1+cx}\right ) \\&+& c^2l(l-1)\int _0 ^1 \frac{-t}{\log{(1-t)}} \left [ (1+cxt)^{l-2} + (1-t-cxt)^{l-2}\right ] dt,\end{aligned}$$
and for $k\geq 0$
$$\begin{aligned}
&&H_{n,c}^{(2k+2)}(x)=cl(2k)! \left ( \frac{1}{x^{2k+1}} - \left ( \frac{c}{1+cx}\right )^{2k+1} \right )\\
&+& l(l-1)\dots (l-2k-1)c^{2k+2}\\&& \int _0 ^1 \frac{-t}{\log{(1-t)}} \left [ (1+cxt)^{l-2k-2} + (1-t-cxt)^{l-2k-2}\right ]t^{2k} dt.\end{aligned}$$
For $0<t<1$ we have $$0<\frac{-t}{\log{(1-t)}}<1, \label{eq:new7}$$ so that $$H_{n,c}^{(2k+2)}(x) \leq cl(2k)! \left ( \frac{1}{x^{2k+1}} - \left ( \frac{c}{1+cx}\right )^{2k+1} \right )+\label{eq:7}$$ $$+ l(l-1)\dots (l-2k-1)c^{2k+2} \int _0 ^1 \left [ (1+cxt)^{l-2k-2} + (1-t-cxt)^{l-2k-2}\right ]t^{2k} dt.$$
Repeated integration by parts yields $$\int _0 ^1 (1+cxt)^{l-2k-2}t^{2k}dt \leq \frac{(2k)!}{(l-2)(l-3)\dots (l-2k-1)(cx)^{2k}}\int _0 ^1 (1+cxt)^{l-2}dt,$$ and so $$\int _0 ^1 (1+cxt)^{l-2k-2}t^{2k}dt \leq \frac{(2k)!\left [ (1+cx)^{l-1}-1 \right ]}{(l-1)(l-2)\dots (l-2k-1)(cx)^{2k+1}}.\label{eq:8}$$
Replacing $x$ by $-\frac{1}{c}-x$ we obtain $$\int _0 ^1 (1-t-cxt)^{l-2k-2}t^{2k}dt \leq \frac{(2k)! \left [ 1-(-cx)^{l-1} \right ]}{(l-1)(l-2)\dots (l-2k-1)(1+cx)^{2k+1}}.\label{eq:9}$$
From , and it follows that $$H_{n,c}^{(2k+2)}(x)\leq cl(2k)! \left [ \frac{(1+cx)^{l-1}}{x^{2k+1}} - \frac{c^{2k+1}(-cx)^{l-1}}{(1+cx)^{2k+1}}\right ] \leq 0,$$ and this proves .
It is easy to verify that $H_{n,c}^{(2k+1)}\left ( -\frac{1}{2c} \right ) = 0$. Since $H_{n,c}^{(2k+2)}\leq 0$, it follows that $H_{n,c}^{(2k+1)}$ is decreasing, and this implies .
B. Consider the case $c=0$. {#b.-consider-the-case-c0. .unnumbered}
---------------------------
$H_{n,0}$ is the Shannon entropy of the Poisson distribution. The derivative of this function is completely monotonic: see, e.g., [@2 p. 2305]. For the sake of completeness we insert here a short proof.
$H'_{n,0}$ is completely monotonic, i.e., $$(-1)^k H_{n,0}^{(k+1)}(x)\geq 0, \quad k \geq 0,\quad x>0. \label{eq:10}$$
**Proof** Let us remark that $H_{n,0}(y) = H_{1,0}(ny)$; so it suffices to investigate the derivatives of $H_{1,0}(x)$.
According to [@10 (2.5)], $$\begin{aligned}
H_{1,0}(x) &=& x-x\log {x} + \int _0 ^\infty \frac{e^{-t}}{t} \left (x - \frac{1-\exp {(x(e^{-t}-1))}}{1-e^{-t}} \right )dt\\
&=& x-x\log{x} - \int _0 ^1 \left ( x - \frac{1-e^{-sx}}{s} \right ) \frac{ds}{\log{(1-s)}}.\end{aligned}$$
It follows that $$H'_{1,0}(x) = -\log{x} - \int _0^1 \left ( 1-e^{-sx}\right )\frac{ds}{\log{(1-s)}}$$ and for $k\geq 1$, $$H_{1,0}^{(k+1)}(x) = (-1)^k \left ( \frac{(k-1)!}{x^k} + \int _0^1 s^k e^{-sx} \frac{ds}{\log{(1-s)}}\right ). \label{eq:11}$$
By using we get $$\int _0 ^1 \frac{s^k e^{-sx}}{\log{(1-s)}}ds \geq - \int _0 ^1 s^{k-1}e^{-sx} ds =$$
$$=-\int _0 ^x \frac{t^{k-1}}{x^k} e^{-t}dt \geq - \int _0 ^\infty \frac{1}{x^k}t^{k-1}e^{-t}dt = -\frac{(k-1)!}{x^k}.$$
Combined with , this proves for $k\geq 1$. In particular, we see that $H_{n,0}$ is concave and non-negative on $[0,+\infty )$; it follows that $H'_{n,0}\geq 0$ and so is completely proved.
C. Let now $c>0$. {#c.-let-now-c0. .unnumbered}
-----------------
For $c>0$, $H'_{n,c}$ is completely monotonic.
**Proof** Since $H_{m,c}(y) = H_{\frac{m}{c},1}(cy)$, it suffices to study the derivatives of $H_{n,1}(x)$.
By using , and $$\log{A} = \int _0 ^\infty \frac{e^{-x}-e^{-Ax}}{x}dx, \quad A>0,$$ we get $$H_{n,1}(x) = n \left ( (1+x)\log{(1+x)} - x\log{x} \right ) +\int _0 ^\infty \frac{e^{-ns}-e^{-s}}{s(1-e^{-s})}\left ( 1-(1+x-xe^{-s})^{-n} \right )ds$$ $$= n \left ( (1+x)\log{(1+x)} - x\log{x} \right ) + \int _0 ^1 \frac{1-(1-t)^{n-1}}{t\log{(1-t)}} \left (1-(1+tx)^{-n} \right ) dt.$$
It follows that, for $j\geq 1$, $$\frac{1}{n}H_{n,1}^{(j+1)}(x) = (-1)^{j-1}(j-1)! \left ( (x+1)^{-j} - x^{-j} \right )+$$ $$+(-1)^{j-1}(n+1)(n+2)\dots (n+j) \int _0 ^1 \frac{-t}{\log{(1-t)}} \left [ 1-(1-t)^{n-1} \right ] (1+xt)^{-n-j-1}t^{j-1}dt.$$
Using again , we get $$\begin{aligned}
&&(-1)^{j-1}\frac{1}{n}H_{n,1}^{(j+1)}(x) \leq (j-1)! \left ( (x+1)^{-j}-x^{-j}\right )+\\
&&+(n+1)(n+2)\dots (n+j) \int _0 ^1 \left [1-(1-t)^{n-1}\right ] (1+xt)^{-n-j-1}t^{j-1}dt\\
&&=u(x) + v(x),\end{aligned}$$ where $$u(x):=\frac{(j-1)!}{(x+1)^j} - (n+1)(n+2)\dots (n+j) \int _0 ^1 t^{j-1}(1-t)^{n-1} (1+xt)^{-n-j-1}dt,$$ $$v(x):= (n+1)(n+2)\dots (n+j) \int _0 ^1 t^{j-1} (1+xt)^{-n-j-1}dt - \frac{(j-1)!}{x^j}.$$
We shall prove that $u(x)\leq 0$ and $v(x)\leq 0$, $x>0$. Let us remark that $$\int _0 ^1 t^{j-1}(1-t)^{n-1}(1+xt)^{-n-j-1}dt \geq \int _0 ^1 t^{j-1}(1-t)^n(1+xt)^{-n-j-1}dt, \label{eq:12}$$ and integration by parts yields $$\int _0 ^1 \frac{t^{j-1} (1-t)^n}{(1+xt)^{n+j+1}}dt = \frac{j-1}{(n+1)(x+1)} \int _0 ^1 \frac{t^{j-2}(1-t)^{n+1}}{(1+xt)^{n+j+1}}dt.$$
Applying repeatedly this formula we obtain $$\int _0 ^1 \frac{t^{j-1}(1-t)^n}{(1+xt)^{n+j+1}}dt = \frac{(j-1)!}{(n+1)(n+2)\dots (n+j)}\frac{1}{(x+1)^j}.\label{eq:13}$$
Now and imply $u(x) \leq 0$.
Using again integration by parts we get $$\begin{aligned}
\int _0 ^1 t^{j-1}(1+xt)^{-n-j-1}dt \leq \frac{j-1}{(n+j)x} \int _0 ^1 t^{j-2}(1+xt)^{-n-j}dt \\ \leq \dots \leq \frac{(j-1)!}{(n+1)(n+2)\dots (n+j)}\frac{1}{x^j},\end{aligned}$$ which shows that $v(x) \leq 0$.
We conclude that $$(-1)^{j-1}H_{n,1}^{(j+1)}(x)\leq 0, \quad j \geq 1, x>0. \label{eq:14}$$
In particular, shows that $H_{n,1}$ is concave on $[0,+\infty )$; it is also non-negative, which means that $H'_{n,1}\geq 0$. Combined with , this shows that $H'_{n,1}$ is completely monotonic, and the proof is finished.
can be obtained alternatively by using the change of variables $y=(1-t)/(1+xt)$ and the properties of the Beta function. An alternative proof of the inequality $v(x)\leq 0$ follows from $$\int _0^1 t^{j-1} (1+xt)^{-n-j-1} dt \leq \frac{1}{x^{j-1}}\int _0 ^\infty \frac{(xt)^{j-1}}{(1+xt)^{n+j+1}}dt=$$ $$=\frac{1}{x^j} \int _0 ^\infty \frac{s^{j-1}}{(1+s)^{j+n+1}}ds=\frac{1}{x^j}B(j,n+1)=\frac{1}{x^j}\frac{(j-1)!n!}{(n+j)!}.$$
The following inequalities are valid for $x>0$ and $c\geq 0$: $$\log{\frac{x}{cx+1}} \leq \sum _{k=0}^\infty p_{n+c,k}^{[c]}(x)\log{\frac{k+1}{ck+n}}\leq \log{\frac{nx+1}{ncx+n}}.\label{eq:15}$$
In particular, for $c=0$ and $n=1$, $$\log{x} \leq \sum _{k=0}^\infty e^{-x}\frac{x^k}{k!}\log{(k+1)}\leq \log{(x+1)}.$$
**Proof** We have seen that $H'_{n,c}(x)\geq 0$. An application of yields $$H'_{n,c}(x) = n \left ( \log{\frac{1+cx}{x}} + \sum _{k=0}^\infty p_{n+c,k}^{[c]}(x)\log{\frac{k+1}{n+ck}} \right ).$$
This proves the first inequality in ; the second is a consequence of Jensen’s inequality applied to the concave function $\log{t}$.
Rényi entropy and Tsallis entropy
=================================
The following conjecture was formulated in [@13]:
\[conj:3.1\] $S_{n,-1}$ is convex on $[0,1]$.
Th. Neuschel [@11] proved that $S_{n,-1}$ is decreasing on $\left [ 0, \frac{1}{2}\right ]$ and increasing on $\left [ \frac{1}{2}, 1\right ]$. The conjecture and the result of Neuschel can be found also in [@5].
A proof of the conjecture was given by G. Nikolov [@12], who related it with some new inequalities involving Legendre polynomials. Another proof can be found in [@4].
Using the important results of Elena Berdysheva [@3], the following extension was obtained in [@17]:
\[th:3.2\] ([@17 Theorem 9]). For $c<0$, $S_{n,c}$ is convex on $\left [ 0, -\frac{1}{c}\right ]$.
A stronger conjecture was formulated in [@14] and [@17]:
\[conj:3.3\] For $c\in \mathbb{R}$, $S_{n,c}$ is logarithmically convex, i.e., $\log S_{n,c}$ is convex.
It was validated for $c\geq 0$ by U. Abel, W. Gawronski and Th. Neuschel [@1], who proved a stronger result:
\[th:3.4\] ([@1]). For $c\geq 0$, the function $S_{n,c}$ is completely monotonic, i.e., $$(-1)^m \left ( \frac{d}{dx} \right )^m S_{n,c}(x)>0, \quad x\geq 0, m\geq 0.$$
Consequently, for $c\geq 0$, $S_{n,c}$ is logarithmically convex, and hence convex.
Summing up, for the Rényi entropy $R_{n,c} = -\log S_{n,c}$ and Tsallis entropy $T_{n,c}=1-S_{n,c}$, we can state
i) Let $c\geq 0$. Then $R_{n,c}$ is increasing and concave, while $T'_{n,c}$ is completely monotonic on $[0,+\infty )$.
ii) $T_{n,c}$ is concave for all $c\in \mathbb{R}$.
**Proof**
i) Apply Theorem \[th:3.4\].
ii) For $c<0$, apply Theorem \[th:3.2\]. For $c\geq 0$, Theorem \[th:3.4\] shows that $S_{n,c}$ is convex, so that $T_{n,c}$ is concave.
As far as we know, Conjecture \[conj:3.3\] is still open for $c<0$, so that the concavity of $R_{n,c}$, $c<0$, remains to be investigated.
Acknowledgement {#acknowledgement .unnumbered}
---------------
The author is grateful to the referee for valuable comments and very constructive suggestions. In particular, the elegant alternative proofs presented in Remark 1 were kindly suggested by the referee.
U. Abel, W. Gawronski, Th. Neuschel, Complete monotonicity and zeros of sums of squared Baskakov functions, Appl. Math. Comput., 258, 130-137 (2015) J.A. Adell, A. Lekuona and Y. Yu, Sharp bounds on the entropy of the Poisson Law and related quantities, IEEE Trans. Information Theory, 56, 2299-2306 (2010) E. Berdysheva, Studying Baskakov-Durrmeyer operators and quasi-interpolants via special functions, J. Approx. Theory, 149, 131-150 (2007) I. Gavrea, M. Ivan, On a conjecture concerning the sum of the squared Bernstein polynomials, Appl. Math. Comput., 241, 70-74 (2014) H. Gonska, I. Raşa, M.-D. Rusu, Chebyshev-Grüss-type inequalities via discrete oscillations, Bul. Acad. Stiinte Repub. Mold. Mat., 1, (74), 63-89; arxiv 1401.7908 \[math.CA\] (2014) P. Harremoës, Binomial and Poisson distributions as maximum entropy distributions, IEEE Trans. Information Theory, 47, 2039 - 2041 (2001) M. Heilmann, Erhöhung der Konvergenzgeschwindigkeit bei der Approximation von Funktionen mit Hilfe von Linearkombinationen spezieller positiver linearer Operatoren, Habilitationsschrift, Universität Dortmund (1992) E. Hillion, Concavity of entropy along binomial convolutions, Electron. Commun. Probab., 17, 1-9 (2012) E. Hillion, O. Johnson, A proof of the Shepp-Olkin entropy concavity conjecture, arXiv: 1503.01570v1, (2015) C. Knessl, Integral representations and asymptotic expansions for Shannon and Rényi entropies, Appl. Math. Lett., 11, (1998), 69-74. Th. Neuschel, Unpublished manuscript (2012) G. Nikolov, Inequalities for ultraspherical polynomials. Proof of a conjecture of I. Raşa, J. Math. Anal. Appl., 418, (2014), 852-860. I. Raşa, Unpublished manuscripts (2012) I. Raşa, Special functions associated with positive linear operators, arxiv: 1409.1015v2, (2014) I. Raşa, Rényi entropy and Tsallis entropy associated with positive linear operators, arxiv: 1412.4971v1, (2014) I. Raşa, Entropies and the derivatives of some Heun functions, arxiv: 1502.05570v1, (2015) I. Raşa, Entropies and Heun functions associated with positive linear operators, Appl. Math. Comput., 268, 422-431 (2015) A. Rényi, On measures of entropy and information, Proc. Fourth Berkeley Symp. Math. Statist. Prob., Vol. 1, Univ. of California Press, pp. 547-561 (1961) L.A. Shepp, I. Olkin, Entropy of the sum of independent Bernoulli random variables and of the multinomial distribution, Proc. Contributions to Probability, New York, pp. 201-206 (1981) C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys., 52, 479-487 (1988) M. Wagner, Quasi-Interpolaten zu genuinen Baskakov-Durrmeyer-Typ Operatoren, Shaker Verlag, Aachen (2013)
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We develop a method for the determination of thecdynamics of dissipative quantum systems in the limit of large number of quanta $N$, based on the $1/N$-expansion of Heidmann [*et al.*]{} \[ Opt. Commun. [**54**]{}, 189 (1985) \] and the quantum-classical correspondence. Using this method, we find analytically the dynamics of nonclassical states generation in the higher-order anharmonic dissipative oscillators for an arbitrary temperature of a reservoir. We show that the quantum correction to the classical motion increases with time quadratically up to some maximal value, which is dependent on the degree of nonlinearity and a damping constant, and then it decreases. Similarities and differences with the corresponding behavior of the quantum corrections to the classical motion in the Hamiltonian chaotic systems are discussed. We also compare our results obtained for some limiting cases with the results obtained by using other semiclassical tools and discuss the conditions for validity of our approach.'
address: |
$^a$Department of Optics and Joint Laboratory of Optics of\
Palacký University and Academy of Sciences of Czech Republic,\
17. listopadu 50, 772 07 Olomouc, Czech Republic\
$^b$Theory of Nonlinear Processes Laboratory, Kirensky Institute of Physics,\
Russian Academy of Sciences, Krasnoyarsk 660036, Russia
author:
- 'Kirill N. Alekseev$^{a,b}$[@email1], and Jan Peřina$^{a}$[@email2]'
title: 'The $1/N$-expansion, quantum-classical correspondence and nonclassical states generation in dissipative higher-order anharmonic oscillators'
---
Introduction {#sec:introduc}
============
The quantum anharmonic oscillator with the Hamiltonian in the interaction picture ($\hbar\equiv 1$) $$\label{1}
H=\Delta b^{\dag}b + \frac{\lambda_l}{l+1} \left( b^{\dag} b \right)^{l+1},
\quad [ b, b^{\dag} ]=1$$ is one of the simplest and the most popular models describing the quantum statistical properties of light interacting with a nonlinear medium [@1; @2]. In Eq. (\[1\]), the operators $b$ and $b^{\dag}$ describe a single mode of quantum field and the constant $\lambda_l$ is proportional to the $(2 l+1)$-order nonlinear susceptibility of a nonlinear medium ($l$ is an integer), $\Delta$ is the detuning of the light frequency from the characteristic frequency of quantum transition. We adopt the normal ordering of operators. For the case of a cubic nonlinearity ($l=1$), this model first was introduced by Tanaś [@3] for the investigation of self-squeezing of light propagating through a nonlinear Kerr medium without loss. Because of such a model is exactly integrable, the explicit time dependence of the quadrature variances necessary for the determination of squeezing condition has been found for any moments of time and for any number of photons [@3; @4]. The problem of a dissipative anharmonic oscillator is much more difficult. Nevertheless, Milburn and Holms obtained the exact solution for the damped Kerr oscillator ($l=1$) interacting with a reservoir of zero temperature [@5]. This result has been further generalized to the case of a reservoir of non zero temperature in [@6].
In the conditions of an experiment, as a rule, a large number of photons are involved in a nonlinear interaction between light and a nonlinear medium modelled by the anharmonic oscillator [@1; @3; @4]. The determination of squeezing conditions from the exact solution in this limiting case is straightforward for the model of Kerr oscillator without loss [@4]. In contrast, due to the complex form of the exact solution for the damped Kerr oscillator, the determination of photon statistics for the large number of photons in this model demands an application of numerical methods or special approximate analytical methods (for a review see [@2]). Moreover, there are no exact solutions for the model of the quantum dissipative oscillator with higher-order nonlinearity and a very little amount of the information on its dynamics is documented in the literature [@2].
In general, the situation when a large number of photons $N$ are involved in nonlinear interactions is a quite typical for many problems of quantum optics [@reynaud; @fabre]. Heidmann [*et al.*]{} suggested [@7] to use the method of $1/N$-expansion for the determination of nonclassical states generation dynamics. They originally applied the $1/N$-expansion technique to the problem of squeezing and antibunching of an electromagnetic field interacting with a collection of the Rydberg atoms inside a high-$Q$ cavity [@heidmann-prl], where a large number of atoms is of the same order as number of the photons $N$. The general scheme of the $1/N$-expansion method states that an exact or an approximate solution of the problem can be found in the classical limit $N\rightarrow\infty$ and then the quantum corrections could be added [@yaffe]. Because this method allows to find the motion equations for the mean values and the lower-order cumulants, it could also be considered as a variant of the cumulant expansion [@7; @schack]. Recently we further developed and applied the $1/N$-expansion technique [@7] to the investigation of an enhanced squeezing at the transition to quantum chaos[^1] [@8; @9; @9']. It should be noticed that only nondissipative quantum systems have been considered in papers [@7; @8; @9; @9'].
In this paper, using the method of $1/N$-expansion, we consider a dynamics of squeezing and a deviation from the Poissonian statistics for the damped anharmonic oscillators with arbitrary degree of nonlinearity $l$ in the limit of a large number of photons $N\gg 1$. We find the explicit time dependencies for the squeezing and the Fano factor for an arbitrary degree of nonlinearity and for an arbitrary temperature of a reservoir. Our consideration is based on the quantum-classical correspondence and the fact that the solution of classical equations of motion, obtained within the zero-order approximation of $1/N$, could be found analytically for the case of any linear damping. We show that for a weak damping the degree of squeezing is mainly determined by the nonlinear polarization of a nonlinear medium, modelled by the nonlinear oscillator. For the case of no damping, our time dependencies for squeezing are transformed to the corresponding formulas of work [@4], which have been found from an exact solution of the Hamiltonian problem. A finite damping decreases the degree of squeezing. The consideration of the Fano factor demonstrates that the quantum statistics is always a super-Poissonian for dissipative oscillators. Another restrictive factor having influence on the time dependencies of squeezing and the Fano factor are the thermal fluctuations of the reservoir.
Note that in spite of the fact that we find our main results for the the model of the higher-order oscillator, we present our self-consistent system of motion equations for the first- and the second-order cumulants in a form which is valid for the description of any single-mode quantum system in the semiclassical limit. One of the main finding of this general consideration consists in the influence of a specific quantum diffusive term on the dynamics of the expectation values and dispersions. We interpret this diffusion, which is proportional to the damping constant, as an influence of the zero-point energy of reservoir on the quantum system. Although the influence of quantum diffusion around the classical solution is insufficient for the description of time dependencies of squeezing and mean values for the particular system under study, and especially in the most interesting case of a short time of interactions, we think that the account of this quantum diffusion is important for the correct description of other dissipative quantum systems in the semiclassical limit.
We compare our basic equations of motion for dissipative systems with the equations arising within the so-called generalized Gaussian approximation [@1; @schack; @perina1] and find a one-to-one correspondence up to terms of $1/N^2$ for several popular models of quantum optics [@perina1; @szlachetka1; @perina2; @szlachetka2].
We also discuss the conditions for validity of the cumulant expansion in the form of the $1/N$-expansion for the description of the dissipative dynamics of nonlinear oscillators. This problem is related to the problem of finding the time interval for the quantum-classical correspondence, which attracts large attention nowadays, and especially in connection with the studies of quantum chaotic systems (see [@berman-book] and the references cited therein). For the Hamiltonian systems with regular dynamics, the quantum corrections to the corresponding classical equations grow in the course of time power-wise [@berman-zaslavsky; @berman-book; @sundaram; @8; @9; @9']. As a result, the time interval for the classical description has a power-wise dependence on the semiclassical parameter $N$ [@berman-zaslavsky; @berman-book; @sundaram]. In contrast, for the case of nondissipative quantum systems which are chaotic in the classical limit, the quantum corrections grow exponentially in time due to underlying local instability in the classical system [@berman-zaslavsky; @berman-book; @sundaram; @8; @9; @9']. Therefore, the time interval for the validity of the $1/N$-expansion method and the classical description is logarithmic in the semiclassical parameter $N$ [@berman-zaslavsky; @berman-book; @sundaram].
Our finding for the dissipative nonlinear oscillators is that the quantum correction first increases, then reaches some maximum and finally decreases in the course of time. As a consequence, the time interval for validity of the $1/N$-expansion could be divided into two subintervals with completely different dependencies on the semiclassical parameter $N$. The first subinterval scales up power-wise with $N$ and the second one has a logarithmic dependence on $N$. While a power dependence on $N$ originates from the same time behavior as in the Hamiltonian systems with regular dynamics, the $\log N$ scale has a different nature. It appears because of an exponential damping of the underlying classical dynamics.
Our work is organized as follows. We consider a single mode dissipative quantum system and present the derivation of motion equations for the first and second-order cumulants in section \[sec:Basic\_eq\]. In the same section we also compare our approach based on the $1/N$-expansion method with other semiclassical methods and find analytically the time dependencies for the mean value and the second-order cumulants for the model of the higher-order dissipative oscillator. Using these results, we focus on the time dependencies of the squeezing and the Fano factor in section \[sec:Squeez\_and\_Fano\]. The section \[sec:validity\] is devoted to the discussion of conditions for the validity of the $1/N$-expansion in the description of nonlinear oscillators dynamics. In the concluding section, we briefly summarize our results and outlook the main directions for the future developments. Some details related to the solution of motion equations for the cumulants and the determination of time interval for the validity of our approach are presented in two Appendices.
Basic equations {#sec:Basic_eq}
===============
The $1/N$-expansion method and comparison with other approaches {#subsec:1/N-expan}
---------------------------------------------------------------
First of all, we need to generalize the approach of [@7; @9; @9'] to the case of a single-mode quantum system with dissipation. Consider an oscillator with the Hamiltonian (\[1\]) which interacts with an infinite linear reservoir of finite temperature. The Hamiltonians of a reservoir and its interaction with oscillator are the following $$\label{2}
H_r=\sum_{j} \psi_j (d_j^{\dag}d_j+1/2), \quad
H_{int}=\sum_{j}\left(\kappa_j d_j b^{\dag} + {\rm H. c.}\right),$$ where the Bose operator $d_j$ ($[d_j, d_k^{\dag}]=\delta_{jk}$) describes an infinite reservoir with characteristic frequencies $\psi_j$, and $\kappa_j$ are coupling constants between the reservoir modes and the oscillator. Introduce new scaled operators $a=b/N^{1/2}$, $c_j=d_j/N^{1/2}$, and Hermitian conjugats to them with commutation relations $$\label{4}
[ a, a^{\dag} ]=1/N, \quad [c_j, c_k^{\dag}]=\delta_{jk}/N.$$ In the classical limit $N\rightarrow\infty$, we have commuting classical $c$-numbers instead of operators. Now the full Hamiltonian $H=H_0+H_r+H_{int}$ may be rewritten as $H=N {\cal H}$, where ${\cal H}$ has the same form as in the formulas (\[1\]) and (\[2\]) with an account of the following replacements $$\label{5}
b\rightarrow a,\quad b^{\dag}\rightarrow a^{\dag},\quad
d_j\rightarrow c_j,\quad d_j^{\dag}\rightarrow c_j^{\dag},\quad
\mbox{and}\quad \lambda_l\rightarrow g_l(N)\equiv \lambda N^l.$$ It could be shown that dependent on photon number constant $g_l(N)$ correctly gives the time scale of energy oscillations for the nonlinear oscillator (\[1\]) in the classical limit (for the case of the Kerr nonlinearity, see, [*e.g.*]{} [@10]).
Within the standard Heisenberg-Langevin approach the equation of motion has the form ([@1], chap. 7; [@mandel_wolf]) $$\label{5'}
i\dot{a}=\left(\Delta-i\frac{\gamma}{2}\right) a + V +L(t),$$ where $V=\partial {\cal H}_0/\partial a^{\dag}$, $\gamma=2\pi|\kappa(\omega)|^2\rho(\omega)$ is a damping constant, $\rho(\omega)$ being the density function of reservoir oscillators, which spectrum is considered to be flat. The Langevin force operator $L(t)=\sum_{j}\kappa_j d_j(0)\exp(-i\psi_j t)$ has properties [@1; @mandel_wolf], which in our notations (\[5\]) may be rewritten as $$\label{6}
\langle L(t)\rangle_R=\langle L^{\dag} (t)\rangle_R=0,\quad
\langle L^{\dag}a\rangle_R+\langle a^{\dag} L\rangle_R=
\gamma\frac{\langle n_d\rangle}{N},\quad
\langle L a\rangle_R+\langle a L\rangle_R=0,$$ where the average is performed over the reservoir variables and $\langle n_d\rangle$ is a mean number of reservoir quanta (phonons), related to the temperature $T$ as $\langle n_d\rangle=\langle c^{\dag}(0) c(0)\rangle=
\left[ 1-\exp\left(\frac{\omega}{k T}\right)
\right]^{-1}$, where $k$ is the Boltzmann’s constant. From the Heisenberg-Langevin equations for $a$, $a^2$ and Hermitian conjugated equations, we get using Eqs. (\[5’\]) and (\[6\]) $$\begin{aligned}
\label{7}
i \frac{d}{d\tau}\langle\alpha\rangle & =&
\langle V\rangle-i\frac{\Gamma}{2}\langle\alpha\rangle, \nonumber\\
%
i \frac{d}{d\tau}\langle\left(\delta\alpha\right)^2\rangle &= &
2\langle V\delta\alpha\rangle+
\langle W\rangle
-i\Gamma\langle\left(\delta\alpha\right)^2\rangle, \\
%
i \frac{d}{d\tau}\langle\delta\alpha^*\delta\alpha\rangle & = &
-\langle V^*\delta\alpha\rangle+
\langle\delta\alpha^* V\rangle
-i\Gamma\langle\delta\alpha^*\delta\alpha\rangle
+i\Gamma\frac{\langle n_d\rangle}{N}, \nonumber\end{aligned}$$ where $$\label{W}
W(\alpha, \alpha^*)=\frac{1}{N} \frac{\partial V}{\partial a^{\dag}},\quad
z\equiv\langle a\rangle,\quad
\langle\left(\delta\alpha\right)^2\rangle=\langle a^2\rangle-z^2,\quad
\langle\delta\alpha^*\delta\alpha\rangle =
\langle a^{\dag} a\rangle-|z|^2,$$ and we have introduced the scaled variables $\tau=g_l t$, $\Gamma=\gamma/g_l$, $\bar{\Delta}=\Delta/g_l$. Averaging in Eqs. (\[7\]) is performed over both the reservoir variables and the coherent state $|\alpha\rangle=\exp(N\alpha a^{\dag} -N\alpha^* a) |0\rangle$ corresponding to the mean photon number $\simeq N$. Such kind of minimum-uncertainty states are most suitable for a consideration of the semiclassical limit $N\gg 1$ [@yaffe]. In derivation of Eqs. (\[7\]) we neglect an insufficient additional detuning introduced to $\Delta$ by the interaction with the reservoir [@1; @mandel_wolf].
The set of equations (\[7\]) is not closed and actually is equivalent to the infinite hierarchic dynamical system for the cumulants of different order. To truncate it up to the cumulants of the second order, we make the substitution $a\rightarrow z +\delta\alpha$, where at least initially the mean $z\simeq1$ and the quantum correction $|\delta\alpha(0)|\simeq N^{-1/2}\ll 1$. Using the Taylor expansion of the functions $V(\alpha,\alpha^*)$ and $W(\alpha,\alpha^*)$ around the mean values $z$ and $z^*$, $$\label{expansion}
V=V_z + \left(\frac{\partial V}{\partial\alpha}\right)_z
\delta\alpha +
\left(\frac{\partial V}{\partial\alpha^*}\right)_z
\delta\alpha^* +\cdots, \quad
%
W=W_z + \left(\frac{\partial W}{\partial\alpha}\right)_z
\delta\alpha +
\left(\frac{\partial W}{\partial\alpha^*}\right)_z
\delta\alpha^* +\cdots,$$ and after some algebra analogous to that used in [@9; @9'], we get from (\[7\]) in the first order of $1/N$ the following self-consistent system of equations for the mean value and the second-order cumulants
\[details\] $$\label{details_a}
i \frac{d}{d\tau} z = -i\frac{\Gamma}{2} z+\langle V\rangle_z +
\frac{1}{N} Q\left( z, z^*,\langle\left(\Delta\alpha\right)^2\rangle,
\langle\left(\Delta\alpha^*\right)^2\rangle,
\langle|\Delta\alpha|^2\right),$$ $$\label{details_b}
i \frac{d}{d\tau}\langle\left(\Delta\alpha\right)^2\rangle =
2\left(\frac{\partial V}{\partial\alpha}\right)_z
\langle\left(\Delta\alpha\right)^2\rangle +
2\left(\frac{\partial V}{\partial\alpha^*}\right)_z
\langle|\Delta\alpha|^2\rangle + \langle w \rangle_z -
i\Gamma\langle\left(\Delta\alpha\right)^2\rangle,$$ $$\label{details_c}
i \frac{d}{d\tau}\langle|\Delta\alpha|^2\rangle =
-\left(\frac{\partial V^*}{\partial\alpha}\right)_z
\langle\left(\Delta\alpha\right)^2\rangle +
\left(\frac{\partial V}{\partial\alpha^*}\right)_z
\langle\left(\Delta\alpha^*\right)^2\rangle-
i\Gamma\langle|\Delta\alpha|^2\rangle+
i\Gamma\langle n_d\rangle,$$
where we have introduced the scaled second-order cumulants as $\langle\left(\Delta\alpha\right)^2\rangle=
N\langle\left(\delta\alpha\right)^2\rangle$ and $\langle|\Delta\alpha|^2\rangle=
N\langle\delta\alpha^*\delta\alpha\rangle$, as well as the function $w(z,z^*)=N W(z,z^*)$ (for definition of $W$ see Eq. (\[W\])). Now both the first-order cumulant $z$ and the second-order cumulants $\langle\left(\Delta\alpha\right)^2\rangle$, $\langle|\Delta\alpha|^2\rangle$ are of the order of unity and small parameter $1/N$ arises only as a prefactor for the quantum correction $Q$ to the classical motion equation in (\[details\_a\]). The expression for the quantum correction has the form of second order differential $Q=\frac{1}{2} d^2 V\mid_z$. In this formula and in Eqs. (\[details\]), the subscript $z$ means that the values of $V$ and its derivatives are calculated at the mean value $z$.
The equations (\[details\_b\]), (\[details\_c\]) are nonlinear due to the presence of the nonlinear term $\langle w \rangle_z$ in Eq. (\[details\_b\]). However, introducing new variables for the second-order cumulants $$\label{10}
B=\langle|\Delta\alpha|^2\rangle+\frac{1}{2},\quad
C=\langle\left(\Delta\alpha\right)^2\rangle,$$ the term $\langle w \rangle_z$ could be removed and the self-consistent equations (\[details\]) may be rewritten as
\[9\] $$i\dot{z}=-i\frac{\Gamma}{2} z+
\langle V\rangle_z + \frac{1}{N} Q(z, z^*, C, C^*, B),
\label{9a}$$ $$i\dot{C}=2\left(\frac{\partial V}{\partial\alpha}\right)_z C +
2\left(\frac{\partial V}{\partial\alpha^*}\right)_z B
-i \Gamma C,
\label{9b}$$ $$i\dot{B}=-\left(\frac{\partial V^*}{\partial\alpha}\right)_z C +
\left(\frac{\partial V}{\partial\alpha^*}\right)_z C^*
-i\Gamma\left( B-B^{(0)}\right),
\label{9c}$$
$$\label{10'}
Q(z, z^*, C, C^*, B)=
\frac{1}{2} \left(\frac{\partial^2 V}{\partial\alpha^2}\right)_z C +
\frac{1}{2} \left(\frac{\partial^2 V}{\partial\alpha^{*2}}\right)_z C^* +
\left(\frac{\partial^2 V}{\partial\alpha^{*}\partial\alpha}\right)_z
\left(B-\frac{1}{2}\right).$$
If initially the oscillator is in the coherent state, then the initial conditions for system (\[9\]) are $$\label{initial_cond}
B(0)=1/2,\quad C(0)=0,$$ and some arbitrary $z(0)\equiv z_0$ which is of the order of unity. Involved into equation (\[9c\]) an equilibrium value of cumulant $B$ is determined by the mean number of reservoir’s quanta and its zero-point energy as $$\label{B_0}
B^{(0)}=\langle n_d\rangle+1/2.$$ Note that the zero-point energy of a reservoir appears in the equations of motion for the cumulants though it was not presented in the Heisenberg equations of motion and even may be dropped from the Hamiltonian redefining a zero of the energy. Such “reappearence” of a zero-point field energy is rather often in other problems of quantum theory where a vacuum is responsible for the physical effects [@milonni_book].
It is sufficient that now our basic equations for the second-order cumulants (\[9b\]), (\[9c\]) are linear and this fact allows us to find analytically the solution of whole set (\[9\]) using the quantum-classical correspondence in the limit $N\rightarrow\infty$. The substitution (\[10\]) describes, of course, nothing but the transformation from a normal to a symmetric ordering. The finding that equations of motion for the lower-order cumulants in the semiclassical limit are solvable just within the symmetric ordering is one of the illustration of the fact that symmetric ordering is most suitable for the consideration of the quantum-classical correspondence [@fabre].
Before we will find a solution of equations (\[9\]), let us make several comments and make a comparison with other approaches.\
(i) It should be noticed that we derive the equations (\[7\]), (\[9\]) within the Heisenberg-Langevin approach, but alternatively, the same equations could be obtained [@kna_98] starting from the corresponding generalized Fokker-Planck equation [@1; @mandel_wolf].\
(ii) In the case of no damping $\Gamma=0$, our equations for the mean values and the second order cumulants (\[9\]) are reduced to the corresponding equations of ref. [@9; @9']. On other hand, for the Hamiltonians which can be presented as sum of a kinetic and a potential energy, our approach gives the same motion equations as the cumulant expansion method introduced in [@sundaram]. Moreover, Sundaram and Milonni showed [@sundaram] that the first-order cumulant approximation to the Heisenberg equations of motion gives the expectation values identical to those obtained by the methods of Gaussian wavepacket dynamics [@heller1; @heller2], semiquantum [@pattanayak; @ashkenazy], and based on the time-dependent variational principle [@zhang]. In this respect, our present work could be considered as some variant for the generalization of the semiquantum methods to the case of a dissipative dynamics.\
(iii) Another approach to the description of the dynamics of quantum fluctuations based on almost Gaussian wavepackets is the so-called generalized Gaussian approximation (GGA) [@perina1; @1; @schack]. This approximation is valid for both the Hamiltonian and the dissipative systems and consists in an assumption that the Fourier transform of quantum distribution function, [*i.e.*]{} quantum characteristic function, is Gaussian for any moment of time [@schack]. Within GGA only the first- and the second-order cumulants are non zero, and therefore it presents the higher-order cumulants in terms of only the first and the second-order cumulants and to truncate an infinite dynamic system for the cumulants. We compare our system (\[9\]) with the corresponding dynamic equations for the mean values and the cumulants obtained within GGA for a several popular dissipative models of quantum optics: a second harmonic generation [@szlachetka1], a nondegenerate optical three-wave mixing[^2] and a forced nonlinear oscillator with cubic nonlinearity [@szlachetka2]. For the forced nonlinear oscillator, we found that our self-consistent set of equations (\[9\]) coincides with the corresponding basic equations of [@szlachetka2] up to the terms of the order of $1/N^2$, and for the problems of a nondegenerate and a degenerate optical 3-wave mixing, our approach gives equations which are identical to the corresponding motion equations obtained within GGA.
There are yet the principal differences between our approach and GGA: First, our approach works good in the semiclassical limit $N\gg 1$, in contrast, GGA is suitable also for the quantum limit $N\simeq 1$. Second, in the general case, the equations of motion obtained within GGA are nonlinear and cannot be solved analytically. As we shall see in the next subsection, the solution of our system (\[9\]) could be obtained analytically and has simple physical meaning.
The quantum-classical correspondence and the dynamics of cumulants {#subsec:Dynamics_of_cumulants}
------------------------------------------------------------------
We need to find the time dependencies of $z$, $C$, and $B$ from the equations (\[9\]). First, suppose that we know the solution $z(\tau)$ of the motion equation (\[9a\]) for the mean value. Then, the equations of motion for the second-order cumulants (\[9b\]) and (\[9c\]) form the linear inhomogeneous equation for the vector variable ${\bf X}(\tau)=\left[C(\tau), C^*(\tau), B(\tau)\right]$ as $$\label{matrix}
i\dot{\bf X}=-i(\Gamma/2){\bf X}+\hat{A}{\bf X}+\epsilon{\bf X}_0,$$ where ${\bf X}_0=(0, 0, i\Gamma B^{(0)})$ and $B^{(0)}$ is given by Eq. (\[B\_0\]). The matrix $\hat{A}$ is formed by the partial derivatives of $V$ calculated at the mean value $z$, its form can be easily obtained from Eqs. (\[9b\]) and (\[9c\]) (to save space we do not present an explicit form of $\hat{A}$ here). We also introduced a parameter $\epsilon$ for the discussion of solutions of Eq. (\[matrix\]) (actual value $\epsilon\equiv 1$ in (\[matrix\])). The solution ${\bf X}(\tau)$ of linear Eq. (\[matrix\]) consists of two parts $$\label{X_full}
{\bf X}(\tau)=\overline{\bf X}(\tau)+\tilde{\bf X}(\tau),$$ where $\overline{\bf X}(\tau)$ is a general solution of the homogeneous equation (i.e., Eq. (\[matrix\]) with $\epsilon=0)$ and $\tilde{\bf X}(\tau)$ is a particular solution of the inhomogeneous equation. It easy to see that $$\label{X_tilde}
\tilde{\bf X}(\tau)=\left( 0, \Gamma B^{(0)}\tau\right).$$ To find $\overline{\bf X}(\tau)$ we need to solve the self-consistent set of equations (\[matrix\]) for $\epsilon=0$ together with (\[9a\]) and (\[10’\]). We seek for a solution by the perturbation theory using the smallness of the parameter $1/N$. Substituting the expression $$\label{16}
z(\tau)=z_{cl}(\tau)+\frac{1}{N} z^{(1)}(\tau),\quad
\frac{1}{N} |z^{(1)}(\tau)|\ll |z_{cl}(\tau)|$$ into the motion equation (\[9a\]) for the mean value, we get in the zero-order of $1/N$ the classical motion equation $$\label{17}
i\dot{z}_{cl}=-i\frac{\Gamma}{2} z_{cl}+ V(z_{cl},z^*_{cl}).$$ Now, it could be shown that solution $\overline{\bf X}(\tau)$ of equation of motion for the cumulants (\[matrix\]) with $\epsilon=0$ can be obtained directly from the solution of the classical equation (\[17\]) by the linearization near $z_{cl}$ (the substitution $z_{cl}\rightarrow z_{cl}+\delta z$, $|\delta z|\ll |z|$), if one writes the dynamical equations for the variables $(\delta z)^2$ and $|\delta z|^2$ (for the details of derivation, see Appendix \[appendix1\]). Thus, the solution for $\overline{\bf X}(\tau)$ is $$\label{X_overline}
\overline{\bf X}(\tau)=\left[ ({\rm d}z_{cl})^2,({\rm d}z^*_{cl})^2,
|{\rm d}z_{cl}|^2\right],$$ where ${\rm d}z_{cl}(\tau)$ is the differential of the classical variable $z_{cl}(\tau)$ governed by Eq. (\[17\]), and an initial conditions for (\[X\_overline\]) are (\[initial\_cond\]), [*i.e.*]{} one should take $$\label{initial_cond1}
({\rm d}z_{cl})^2(0)=0, \quad |{\rm d}z_{cl}|^2(0)=1/2.$$ Now we can demonstrate how to find the dynamics of cumulants $C$ and $B$, if the classical dynamics $z_{cl}(\tau)$ is known. Combining formulas (\[X\_full\]), (\[X\_tilde\]), (\[X\_overline\]) and (\[B\_0\]), we get $$\label{B_and_C_general}
C=({\rm d}z_{cl})^2, \quad B=|{\rm d}z_{cl}|^2+
(\langle n_d\rangle+1/2)\Gamma\tau.$$ Turn to the determination of the influence of the quantum correction (\[10’\]) on the dynamics of the mean value $z$. Substituting the expansion (\[16\]) into equation (\[9a\]), we have in the first order of $1/N$ the motion equation for $z^{(1)}$ as $$\label{z1_equation}
i\dot{z}^{(1)}=-i\frac{\Gamma}{2} z^{(1)}+ Q(\tau),\quad
Q(\tau)\equiv Q[z_{cl}(\tau), z^*_{cl}(\tau), C(\tau), C^*(\tau),
B(\tau)],$$ where the cumulants $C(\tau)$, $C^*(\tau)$, and $B(\tau)$ are determined by the formulas (\[B\_and\_C\_general\]) and $z_{cl}(\tau)$ is a solution of the classical equation (\[17\]). A formal solution of Eq (\[z1\_equation\]) is $$\label{z1_solution}
z^{(1)}(\tau)=z^{(1)}(0) \exp\left(-\frac{\Gamma}{2}\tau\right)
-i\int_0^\tau d\tau' Q(\tau').$$ For the initial conditions (\[initial\_cond\]), the initial value of quantum correction (\[10’\]) is zero. Therefore, $z(0)=z_{cl}(0)=z_0$ and $z^{(1)}(0)=0$. The condition of smallness of $z^{(1)}(\tau)$ in comparison with $z_{cl}(\tau)$ \[Eq. (\[16\])\] takes the form $$\label{25}
R(\tau)\equiv
\left|\frac{z(\tau)-z_{cl}(\tau)}{z_{cl}(\tau)}\right|=
\frac{1}{N} \left| \frac{z^{(1)}(\tau)}{z_{cl}(\tau)}\right|=
\frac{1}{N} \left| \frac{\int_0^\tau d\tau' Q(\tau')}
{z_{cl}(\tau)}\right|\ll 1.$$ We shall see when the condition (\[25\]) is valid for the model of anharmonic oscillator (\[1\]) in the next section.
We turn to the determination of the dynamics of the cumulants and the mean value in the case of anharmonic oscillator with the Hamiltonian (\[1\]). Here, the expression for $V(z,z^*)$ takes the form $$\label{26}
V(z,z^*)=\overline{\Delta} z +|z|^{2l}z.$$ The exact solution of the classical motion equation (\[17\]) with $V$ of the form (\[26\]) is $$\label{z_cl}
z(\tau)=z_0\exp\left[(-i\overline{\Delta} -\Gamma/2)\tau\right]
\exp\left[ -i |z_0|^{2 l}\mu_l(\tau)\right],\quad
\mu_l(\tau)=\left[ 1-\exp( -\Gamma l \tau ) \right]/\Gamma l.$$ Using the expression (\[z\_cl\]), we have from Eqs. (\[B\_and\_C\_general\]) the following time dependencies for the cumulants $$\begin{aligned}
\label{B_and_C}
C(\tau) &=&
-\mu_l(\tau) l z_0^2 |z_0|^{2(l-1)}\left( \mu_l(\tau) l |z_0|^{2 l}+i\right)
\exp\left[ (-\Gamma -i 2\overline{\Delta}) \tau -i 2 |z_0|^{2 l}
\mu_l(\tau) \right],
\nonumber\\
%
B(\tau) &=& \exp(-\Gamma\tau)\left[ 1/2+l^2 |z_0|^{4l}\mu_l^2(\tau) \right] +
\left( \langle n_d\rangle + 1/2\right) \Gamma\tau,\end{aligned}$$ where we took into an account the initial conditions (\[initial\_cond1\]). We shall use these time dependencies for the cumulants in the consideration of the squeezing and the Fano factor in the next section.
The squeezing and the Fano factor {#sec:Squeez_and_Fano}
=================================
The dynamics of squeezing {#subsec:squeez}
-------------------------
Define the general field quadrature as $X_\theta=a\exp(-i\theta)+
a^{\dag}\exp(i\theta)$, where $\theta$ is a local oscillator phase. A state is said to be squeezed if there exists some $\theta$ such that the variance of $X_\theta$ is smaller than the variance for a coherent state or the vacuum [@1; @2]. Minimizing the variance of $X_\theta$ over $\theta$, we get the condition for so-called principal squeezing [@1; @2; @4] $$\label{sq_def}
S\equiv 1+2 N (\langle|\delta\alpha|^2\rangle-
|\langle(\delta\alpha)^2\rangle|)=2 (B-|C|) < 1.$$ The determination of the principal squeezing $S$ is very useful because it gives the maximal squeezing measurable by the homodyne detection [@1; @2]. Substituting the expressions (\[B\_and\_C\]) for the cumulants $B$ and $C$ into the definition (\[sq\_def\]), we find $$\label{30}
S(\tau)=\exp(-\Gamma\tau) \left[1+\phi_l(x_0,\tau)\right]
+(\langle n_d\rangle +1/2) 2\Gamma\tau,$$ $$\label{31}
\phi_l(x_0,\tau)=2a\left[ a-(1+a^2)^{1/2}\right],\quad
a(\tau)=l x_0^{2l}\mu_l(\tau),$$ where $\mu_l(\tau)$ is defined in (\[z\_cl\]) and we assumed for the sake of simplicity that the initial condition $z_0$ is real $x_0={\rm Re} z_0$. First, consider some interesting particular cases.
### The weak dissipation
In the case of weak dissipation $\Gamma\tau\ll 1$, we have $\exp(-\Gamma\tau)\approx 1 -\Gamma\tau$ and $\mu_l(\tau)\approx\tau$. Thus, in this limit, from Eqs. (\[30\]) and (\[31\]), we get $$\label{34}
S(\tau)=1+(1-\Gamma\tau)\phi_l(x_0,\tau)+2\langle n_d\rangle\Gamma\tau,
\quad a(\tau)\approx l x_0^{2l}\tau\equiv l {\cal P}^{(2l+1)} t.$$ In the case of no loss ($\Gamma=0$), the formula (\[34\]) shows that the rate of squeezing is determined by the factor $2 l x_0^{2 l}\lambda_l N^l\equiv 2 l {\cal P}^{(2 l+1)}$. Because of $\lambda_l$ is proportional to the $(2 l+1)$-order nonlinear susceptibility, the factor ${\cal P}^{(2 l+1)}$ has a physical meaning of the nonlinear polarization. Therefore, the stronger is nonlinear polarization induced by a light in the medium, the more effective squeezing of light is possible. For a finite dissipation $\Gamma\not=0$, the squeezing is determined by an interplay between dissipation, the polarization of the nonlinear medium modelled by the anharmonic oscillator and the thermal fluctuations of a reservoir. Both dissipation and a finite temperature of reservoir are the destructive factors for the squeezing. It is interesting to note that the vacuum term does not appear in Eq. (\[34\]), what is characteristic of the weak dissipation limit.
### The short-time approximation $\tau\ll 1$
The short-time approximation $\tau\ll 1$, as well as the limit of large photon number $N\gg 1$, are quite realistic for a nonlinear medium modelled by the anharmonic oscillators (for numerical estimates, see [@1], chap. 10, and [@4]). In the limits of $\tau\ll 1$ and $\Gamma\tau\ll 1$, we have from Eq. (\[31\]): $\phi_l(x_0,\tau)\approx -2l x_0^{2l}\tau=
-2l {\cal P}^{(2 l+1)} t$. Thus, substituting this expression into (\[34\]), we get a very simple dependence of $S$ on time as $$\label{35}
S(t)=1-\left[ l x_0^{2l}-\Gamma\langle n_d\rangle\right]
2\tau+O(\tau^2).$$ As follows from Eq. (\[35\]), there exists the critical number of phonons $n_d^{(cr)}=(l/\gamma) {\cal P}^{(2 l+1)}$ such that at $n_d\ge n_d^{(cr)}$ any degree of squeezing is impossible.
### The lossless case $\Gamma=0$
In the case of no loss, the time dependence of $S$ is $$\label{32}
S(\tau)=1+\phi_l(x_0,\tau),$$ where $\phi_l$ is defined in (\[31\]) and we should take into account that now $\mu_l(\tau)=\tau$ and thus $a(\tau)=l x_0^{2l}\tau=
l {\cal P}^{(2l+1)} t$. For the case of Kerr nonlinearity ($l=1$), the formula (\[32\]) coincides with the corresponding formula from [@4] obtained in the limit $N\gg 1$ from the exact solution for the Hamiltonian case.
Now we consider the time dependence of $S$ \[Eq. (\[32\])\] in the limit of large time $\tau\gg 1$, which corresponds to $a\gg 1$. Rewriting $\phi_l$ in the form $\phi_l(x_0,\tau)=2 a^2 \left[ 1- (1+a^{-2})^{1/2}\right]$ and expanding $(1+a^{-2})^{1/2}$ up to the term of order of $a^{-3}$, we get $\phi_l\approx -1+(6 a)^{-1}$ for $a\gg 1$. Therefore $$\label{33}
S(\tau)\approx \left( l x_{0}^{2l}\tau\right)^{-1}
=\left( l {\cal P}^{(2l+1)} t \right)^{-1}$$ for $\tau\gg 1$. This is in good agreement with the statement that the principal squeezing $S$ is a power-wise decreasing function of time for the general class of integrable systems in the semiclassical limit [@9'].
Return to the case of an arbitrary dissipation. The dependence of $S$ on $\tau$ \[Eqs. (\[30\]), (\[31\])\] is shown in Fig. 1, where the solid curve corresponds to the lossless case ($\Gamma=0$), the dashed curve corresponds to the finite damping constant $\Gamma=5\times 10^{-2}$ with a reservoir of zero temperature $\langle n_d\rangle=0$, and, finally, the dotted curve corresponds to the reservoir with finite temperature ($\langle n_d\rangle=1$). As is evident from this figure, the squeezing is stronger for higher nonlinearity (compare Fig. 1a and Fig. 1b), a dissipation slows down the rate of squeezing (compare the solid and the dashed curves in Fig. 1). Moreover, a finite temperature of a reservoir fastly destroys the squeezing.
The Fano factor {#subsec:Fano}
---------------
Another important characteristic of the nonclassical properties of a light is the Fano factor $$\label{Fano_1}
F=\frac{\langle n^2\rangle-\langle n\rangle^2}{\langle n\rangle},$$ which determines the deviation of a probability distribution from the Poissonian distribution with $F=1$ [@1; @2]. In Eq. (\[Fano\_1\]) the mean foton number $\langle n\rangle$ and the mean of square of foton number $\langle n^2\rangle$ are $$\label{Fano_2}
\langle n\rangle\equiv \langle b^{\dag} b\rangle=
N\langle a^{\dag} a\rangle,\quad
\langle n^2\rangle=N^2\langle a^{\dag} a a^{\dag} a\rangle=
N^2\langle a^{\dag 2} a^2\rangle+\langle n\rangle.$$ Substituting expression $a\rightarrow z +\delta\alpha$ $(|\delta\alpha|\ll |z|\simeq 1)$ into formulas (\[Fano\_2\]), we have after Taylor expansions in the first order of $1/N$ $$\label{Fano_3}
\langle a^{\dag} a\rangle=|z|^2 + N^{-1} (B-1/2),\quad
\langle a^{\dag 2} a^2\rangle =|z|^4 + N^{-1} (z^* C + {\rm c.c.})
+N^{-1} 4 |z|^2 (B-1/2),$$ and, therefore, the dependence of the Fano factor on the cumulants and the mean values in the first order of $1/N$ is[^3] $$\label{Fano_4}
F=2 B+\left(\frac{z^*}{z} C+ {\rm c.c.}\right).$$ Substituting expressions (\[B\_and\_C\]) for $B$ and $C$ into Eq. (\[Fano\_4\]), we find the following time dependence for the Fano factor $$\label{Fano_5}
F(\tau)=\exp(-\Gamma \tau)+\left( \langle n_d\rangle+1/2
\right) 2\Gamma\tau.$$ As it is evident from Eq. (\[Fano\_5\]), the statistics is the super-Poissonian $F(\tau)>1$ at any time. The most simple form the time dependence for the Fano factor takes in the case of weak dissipation $$\label{Fano_6}
F(\tau)=1+2\langle n_d\rangle\Gamma\tau, \quad \Gamma\tau\ll 1.$$ Thus, the statistics is super-Poissonian for any $\Gamma\not=0$ and is independent on the degree of nonlinearity $l$. This is in good agreement with the previous result of [@2; @6] for the case of the dissipative Kerr oscillator ($l=1$), where from the exact solution the impossibility of sub-Poissonian statistics and the antibunching has been predicted.
The conditions of validity of the $1/N$-expansion and the quantum-classical correspondence {#sec:validity}
==========================================================================================
The procedure of the $1/N$-expansion may be considered self-consistently, if the influence of the quantum correction to classical motion on the dynamics of the mean values is small, [*i. e.*]{} the condition (\[25\]) is satisfied. From Eqs. (\[10’\]) and (\[26\]), we have the following expression for the quantum correction $Q$ in the case of an arbitrary nonlinearity $l$ $$\label{Q_osc}
Q(z,z^*)=\frac{1}{2}l(l+1) z^{*l} z^{l-1} C+
\frac{1}{2}l(l-1) z^{*(l-2)} z^{l+1} C^* +
l(l+1) z^{*(l-1)} z^l (B-1/2).$$ We start the consideration of the condition of validity (\[25\]) with some particular cases.
### The short-time approximation
First, we consider the short-time approximation. In this case, the integral over $Q(\tau)$ in (\[25\]) could be replaced by the product and we have $$\label{short_appl}
\left|\frac{z(\tau)-z_{cl}(\tau)}{z_{cl}(\tau)}\right|=
\frac{\tau}{N} \left| \frac{Q(\tau)}{z_{cl}(\tau)}\right|\ll 1.$$ It is easy to see that for $\tau\ll 1$, $Q(\tau)\simeq Q(0)$ and thus $|(z-z_{cl})/z_{cl}|$ is of the order of $1/N$. Therefore, the condition of validity within the short-time approximation (\[short\_appl\]) is fulfilled for any number of photons $N\gg 1$ large enough.
### The lossless case $\Gamma=0$
In Appendix \[appendix2\] we show that the time interval of validity of our approach for the lossless case $\Gamma=0$ is $$\label{lossless_appl}
\tau\ll\tau^*_{ham}=N^{1/2}\frac{|\bar{\Delta}+|z_0|^{2l}|^{1/2}}
{l^{3/2}|z_0|^{3l-1}}.$$ This result is in good agreement with our previous finding [@9'] that for the Hamiltonian integrable models the time scale for the validity of the $1/N$-expansion has a power-wise dependence in $N$ in the semiclassical limit $N\gg 1$. Now we turn to the general case $\Gamma\not=0$.
### The case of an arbitrary dissipation
The condition of validity (\[25\]) cannot be obtained analytically for an arbitrary time and for an arbitrary damping $\Gamma$, but combining the computations and simple analytic estimates for some limiting case, we can qualitatively understand the behavior of $R(\tau)$ \[Eq. (\[25\])\]. We start with the time dependence of the quantum correction $Q(\tau)$ \[Eqs. (\[z1\_equation\]), (\[Q\_osc\]), (\[z\_cl\]), (\[B\_and\_C\])\]. This function is shown in Fig. 2 for the different degrees of nonlinearity (the solid line for $l=1$ and the dashed line for $l=3$), as well as for a rather week (Fig. 2a) and for a relatively strong (Fig. 2b) damping constants. The quantum correction $Q(\tau)$ first increases, then reaches some maximum and finally decreases. The maximum of $Q$ increases with the increase of nonlinearity degree (compare the solid and the dashed curves in Fig. 2) and shifts to a shorter time with the increase of dissipation (compare Fig. 2a and Fig. 2b).
Such behavior is in a sharp contrast to the lossless (Hamiltonian) nonlinear systems, where the quantum correction always increases with time either power-wise for regular dynamics [@berman-book] or exponentially for chaotic dynamics [@berman-book]. Before $Q(\tau)$ gets to a maximum, during some time interval $[0,\tau_1]$ it could be well approximated by the time dependence for the lossless case $Q(\tau<\tau_1)\simeq l^3\tau^2$ (see formula (\[B8\])). If we estimate the time scale $\tau_1$ from the behavior of the most fastly decreasing function $\mu_l(\tau)$ \[Eq. (\[z\_cl\])\], we have $$\label{tau_1}
\tau_1=\left(\Gamma l\right)^{-1},\quad
Q(\tau_1)\simeq l^3\tau_1^2=l/\Gamma^2$$ for the position and the value of maximum of $Q(\tau)$. As is easy to see from Fig. 2, this naive estimate, however, well represents all main features of the time behavior of $Q(\tau)$.
Consider now the time dependence of the difference between the classical and the quantum mean values $z^{(1)}(\tau)=N|z(\tau)-z_{cl}(\tau)|$ caused by the existence of a quantum correction to the classical motion. Computing the integral over $Q(\tau)$ \[Eq. (\[z1\_solution\])\], we plot the time dependence $z^{(1)}(\tau)$ in Fig. 3 for weak (Fig. 3a) and for strong (Fig. 3b) damping, as well as for the different degrees of nonlinearity. This plot shows that the difference $z^{(1)}(\tau)$ first increases and then saturates at some level determined by the degree of nonlinearity and dissipation.
We start our analysis with the case of weak dissipation (Fig. 3a). For weak dissipation, the level of saturation of $z^{(1)}(\tau)$ increases with consequent consideration of higher nonlinearities. This result may be qualitatively understood as follows. For weak dissipation, the time interval of almost lossless behavior $\tau\lesssim\tau_1$ (\[tau\_1\]) is rather long and therefore the asymptotic behavior of $z^{(1)}(\tau)$ sufficiently depends on its behavior during the time $0\leq\tau\lesssim\tau_1$. For $\tau\lesssim\tau_1$, we can use the approximation (\[B9\]), i.e. $|z^{(1)}(\tau_1)|\simeq\tau_1^2
l^3=l/\Gamma^2$. This estimate shows that the level of saturation for $z^{(1)}(\tau)$ should increase with the growth of $l$.
Turn to the case of strong dissipation (Fig. 3b). For strong dissipation, the level of saturation of $z^{(1)}(\tau)$ is sufficiently lower in comparison with the case of weak dissipation (compare Fig. 3a and Fig. 3b). Moreover, the level of saturation for $l=1$ is slightly higher than for nonlinearities with $l>1$ and the difference of the saturation levels for the different nonlinearities with $l>1$ is practically invisible. To understand such behavior we note that for strong enough dissipation the time interval for almost lossless dynamics $\tau\lesssim\tau_1$ is short and the behavior of $Q(\tau)$ for another time interval $[\tau_1, \infty]$ is more sufficient (Fig. 2b). For the nonlinearity with $l=1$, $Q(\tau)$ has slowly decreasing tail, while it decays rapidly for $l>1$ (Fig. 2b). As a result, the area under the curve $Q(\tau)$ for $l=1$ is greater than the areas under the curves corresponding to different $l$ with $l>1$. Yet, because all $Q(\tau)$ are fastly decreasing functions at $\tau>\tau_1$ for all $l>1$, the levels of saturation of $z^{(1)}(\tau)$ corresponding to different $l$ are indistinguishable in the scale of Fig. 3.
Till now we studied the time dependence of the difference between quantum mean value and classical solution $|z(\tau)-z_{cl}(\tau)|$. However, the criterion of validity of the $1/N$-expansion in the form (\[25\]) includes also the $z_{cl}(\tau)$ as $R(\tau)=N^{-1}|z^{(1)}/z_{cl}|$. During the time interval of order of $\tau_1$, when $z^{(1)}(\tau)$ is growing, $z_{cl}$ is slowly decreasing and oscillating function. But when $z^{(1)}(\tau)$ saturates, simultaneously the classical solution should be considered as a rapidly decreasing function $z_{cl}\simeq\exp(-\Gamma\tau/2)$. Therefore, for the plateaus in Fig. 3, $R(\tau)\simeq N^{-1}\exp(\Gamma\tau/2)$ resulting in the corresponding time scale for validity of our approach on a plateau as $$\label{log}
\tau<\tau^*=\Gamma^{-1}\ln N.$$ Formally, this result looks like corresponding estimate for the time scale for validity of the semiclassical description (“breaking time”) in the quantum chaotic systems $\tau^*=\lambda^{-1}\ln N$, where $\lambda$ is the maximal Lyapunov exponent [@berman-book; @sundaram; @8; @9; @9']. However, the physical reason for the appearance of a very short, logarithmic breaking time is quite different in our case. When a classical oscillator spiraled around an equilibrium state, the amplitude of the oscillations becomes very small and the quantum description starts to differ significantly from classical one due to the uncertainty principle. That is the reason for the logarithmically short time scale of the semiclassical description of strongly damped oscillations.
However, the behavior of an oscillator near an equilibrium point is a physically not very interesting process. On the opposite, the physically interesting case when the oscillations are not overdamped, we can again apply the estimate (\[B9\]) during the time interval $\tau\simeq\tau_1=(\Gamma l)^{-1}$ and get the criterion of validity in the form $$\label{loss_appl}
R(\tau)=\frac{1}{N} \left|\frac{z^{(1)}(\tau_1)}{z_{cl}(\tau_1)}\right|
\simeq\frac{l}{N\Gamma^2}\ll 1.$$ For large enough $N$, the criterion (\[loss\_appl\]) could be always satisfied and our approach works well.
We can summarize our findings concerning the quantum-classical correspondence and the conditions for validity of the $1/N$-expansion as follows. In a lossless case, the quantum correction to the classical motion equation grows quadratically with time. For finite dissipation, the correction $Q(\tau)$ first increases and then decreases with time. The difference between the solution for the quantum expectation value and the classical solution $|z(\tau)-z_{cl}(\tau)|$ first increases a power-wise with time and then saturates at some level, which is dependent on the degree of nonlinearity and on the dissipation strength. The condition for validity has different scaling dependence on the semiclassical parameter $N$ for two different time intervals. During the time interval of growth of $|z(\tau)-z_{cl}(\tau)|$, the criterion of validity has the form (\[loss\_appl\]), i.e. $R\simeq 1/N\ll 1$, and it is certainly satisfied for large $N$. During the time interval when $|z(\tau)-z_{cl}(\tau)|$ saturates, our approach is valid only for a short time $\tau^*\simeq\ln N$. We believe that described above conditions for the semiclassical description of the quantum system are common for all dissipative systems with simple attractor.
Conclusion {#sec:Conclusion}
==========
In summary, using the quantum-classical correspondence, we find analytically the time dependencies of squeezing and the Fano factor for the dissipative anharmonic oscillators of an arbitrary degree of nonlinearity in the limit of large number of fotons. It should be noticed that our basic equations are rather general and could be considered as some generalization of the Gaussian wavepacket dynamics methods to the dissipative systems. Moreover, some effects mentioned in the present work are also rather general. For instance, quantum diffusion around classical trajectory in the dissipative system, which we interpret as the influence of the reservoir’s vacuum on the quantum system, should be observable not only in the dissipative nonlinear oscillators. In this respect, we would like to mention the work of Savage published ten years ago [@savage]. The author of [@savage] studied numerically, within the Gaussian approximation, the quantized version of the second harmonic generation problem for the self-oscillating regime [@mcneil]. He had found numerically a diffusive growth of the quantum variances near a classical limit cycle. However, neither an analytic treatment of the problem, nor a physical explanation of this “diffusion” were presented in [@savage]. We plan to devoted our future publication to the investigation of this problem together with more accurate study of the validity conditions for the application of the $1/N$-expansion to the dissipative quantum systems.
Acknowledgments {#acknowledgments .unnumbered}
===============
Discussions with Evgeny Bulgakov, Antoine Heidmann, Vlasta Peřinová are gratefully acknowledged. We also thank Serge Reynaud for stimulating interest to the activity. KNA thanks Department of Optics, Palacký University and Joint Laboratory for Optics, Olomouc, as well as Department of Physics, University of Illinois at Urbana-Champaign for hospitality during the work on this project. This work was partially supported by Czech Grant Agency, grant No 202/96/0421, Czech Ministry of Education, grant No VS96028 and Russian Fund for Basic Research, grant No 96-02-16564.
{#appendix1}
In this Appendix we show that the equations of motion for the cumulants (\[matrix\]) at $\epsilon=0$ could be obtained from the classical motion equation (\[17\]). This proof is valid for any dissipative system with one degree of freedom.
Linearization of classical equations (\[17\]) near $z_{cl}$ by means of the substitution $z_{cl}\rightarrow z_{cl}+\delta z$ gives $$\label{A1}
i \frac{d}{d\tau}\delta z =
-i\frac{\Gamma}{2} \delta z +
\frac{\partial V}{\partial z} \delta z +
\frac{\partial V}{\partial z^*}\delta z^*,\quad
%
i \frac{d}{d\tau}\delta z^* =
-i\frac{\Gamma}{2} \delta z^*
-\frac{\partial V^*}{\partial z}\delta z -
\frac{\partial V^*}{\partial z^*}\delta z^*,$$ where all derivatives are taken on the classical trajectory $z_{cl}(\tau)$. Using the equality $\left( \partial V/\partial z\right)=
\left( \partial V^* /\partial z^* \right)$, we have from Eq. (\[A1\]) for the quadratic variables $(\delta z)^2$ and $|\delta z|^2$ the following equations of motion $$\begin{aligned}
\label{A2}
i \frac{d}{d\tau}\left(\delta z\right)^2 &=&
-i\Gamma \left(\delta z\right)^2 +
2\frac{\partial V}{\partial z} \left(\delta z\right)^2 +
2\frac{\partial V}{\partial z^*} |\delta z|^2,\nonumber \\
%
i \frac{d}{d\tau} |\delta z|^2 &=&
-i\Gamma |\delta z|^2
-\frac{\partial V^*}{\partial z} \left(\delta z\right)^2 +
\frac{\partial V}{\partial z^*} \left(\delta z^*\right)^2.\end{aligned}$$ It is easy to see that Eqs. (\[A2\]) are equivalent to Eqs. (\[matrix\]) with $\epsilon=0$, if one makes substitutions $|\delta z|^2\rightarrow B$ and $(\delta z)^2\rightarrow C$.
It should be noticed that yet exists one difference between the linearization of the classical motion equations and the equations for quantum cumulants (\[9b\]), (\[9c\]): It is impossible to get the initial conditions (\[initial\_cond\]) for $C$ and $B$ from only initial conditions for the linearized classical equations of motion (see also discussion of this problem in [@9; @9']).
{#appendix2}
In this Appendix, we present the derivation of the time scale for the validity of the $1/N$-method for lossless ($\Gamma=0$) oscillators. Start with the case of Kerr nonlinearity $l=1$ and then generalize obtained results for arbitrary $l$. From Eqs. (\[z\_cl\]) and (\[B\_and\_C\]) for $\Gamma=0$ and $l=1$, we get $$\label{B1}
z_{cl}(\tau)=z_0\exp(-i\Omega\tau), \quad
\Omega\equiv\bar{\Delta}+|z_0|^2,$$ $$\label{B2}
C(\tau)=-\tau z_0^2 \left[ |z_0|^2\tau+i\right]
\exp(-i 2\Omega\tau), \quad
B(\tau)=1/2+|z_0|^4 \tau^2.$$ The expression (\[Q\_osc\]) for quantum correction in this case is $$\label{B3}
Q(z, z^*, C, C^*, B)=z^* C+2 z B -z.$$ Substituting (\[B1\]) and (\[B2\]) into (\[B3\]), we have for defined in Eq. (\[z1\_equation\]) function $Q(\tau)$: $$\label{B4}
Q(\tau)=z_0 |z_0|^4 \tau^2 \exp(-i\Omega\tau)
-i z_0 |z_0|^2 \tau \exp(-i\Omega\tau),$$ and involved in (\[25\]) integral is $$\begin{aligned}
\label{B5}
\int_0^\tau d\tau' Q(\tau') &=& z_0 |z_0|^4\Omega^{-3}\left[
2\Omega\tau\exp(-i\Omega\tau)+i (\Omega^2\tau^2-2)\exp(-i\Omega\tau)
+2 i \right] - \nonumber\\
& & i z_0 |z_0|^2 \Omega^{-2}\left[
\exp(-i\Omega\tau)+i \Omega\tau\exp(-i\Omega\tau)-1\right].\end{aligned}$$ The most rapidly increasing term in (\[B5\]) for $\tau\gg 1$ is $\Omega^{-1}z_0 |z_0|^4\tau^2\exp(-i\Omega\tau)$. Substituting this expression into Eq. (\[25\]), we have condition of validity in the form $$\label{B6}
\frac{|z_0|^4}{|\bar{\Delta}+|z_0|^2|}\frac{\tau^2}{N}\ll 1,$$ and therefore the time scale $\tau^*$ of validity is $$\label{B7}
\tau\ll\tau^*=N^{1/2}\frac{|\bar{\Delta}+|z_0|^2|^{1/2}}{|z_0|^2}.$$ Turn to the lossless case for an arbitrary $l$. Analogously to the case $l=1$, it could be shown that the expression for quantum correction (\[z1\_equation\]) in the limit of large time $\tau\gg 1$ is $$\label{B8}
Q(\tau)\approx z_0 |z_0|^{2(3l-1)} l^3 \tau^2\exp[-i\Omega_l\tau], \quad
\Omega_l\equiv\bar{\Delta}+|z_0|^{2l},$$ and the influence of quantum correction on mean value (\[z1\_solution\]) is $$\label{B9}
z^{(1)}(\tau)=\int_0^\tau d\tau' Q(\tau')\approx
\frac{l^3 z_0 |z_0|^{2(3l-1)} \exp(-i\Omega_l\tau)}{\Omega_l} \tau^2,$$ where we again presented only the term which is leading in time for $\tau\gg 1$. Finally, substituting (\[B9\]) to (\[25\]) with account of (\[z\_cl\]), we obtain (\[lossless\_appl\]).
E-mail: [email protected]\
E-mail: [email protected]
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[^1]: Recently a good agreement between the predictions for dynamics of squeezing obtained using the method of $1/N$-expansion and the results of numerical simulation for the model of kicked quantum rotator with $N\simeq 10^5$ levels were found in the works [@jetp; @glasgow].
[^2]: The equations of motion within GGA for the model of nondegenerate three-wave mixing could be easily obtained in an analogy with the case of second harmonic generation [@szlachetka1]. The study of lossless three-wave mixing within GGA has been presented in [@perina2].
[^3]: Note that this expression coincides with the corresponding result obtained within GGA (see formulas (3.153) and (10.42) in [@1]) up to the terms of order of $1/N^2$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers $\kappa_1, \ldots, \kappa_N$, quaternions\
$p_1, \ldots, p_N$ all of modulus $1$, so that the $2$-spheres determined by each point do not intersect and $p_u \neq 1$ for $u = 1,\ldots, N$, and quaternions $s_1, \ldots, s_N$, we wish to find a slice hyperholomorphic Schur function $s$ so that $$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}} s(r p_u) = s_u\quad {\rm for} \quad u=1,\ldots, N,$$ and $$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}}\frac{1-s(rp_u)\overline{s_u}}{1-r}\le\kappa_u,\quad
{\rm for} \quad u=1,\ldots, N.$$ Our arguments relies on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces.
address:
- |
(KA) Department of Mathematics\
Ben-Gurion University of the Negev\
Beer-Sheva 84105 Israel
- |
(DA) Department of Mathematics\
Ben-Gurion University of the Negev\
Beer-Sheva 84105 Israel
- |
(FC) Politecnico di Milano\
Dipartimento di Matematica\
Via E. Bonardi, 9\
20133 Milano, Italy
- |
(DPK) Department of Mathematics\
Ben-Gurion University of the Negev\
Beer-Sheva 84105 Israel
- |
(IS) Politecnico di Milano\
Dipartimento di Matematica\
Via E. Bonardi, 9\
20133 Milano, Italy
author:
- 'Khaled Abu-Ghanem'
- Daniel Alpay
- Fabrizio Colombo
- 'David P. Kimsey'
- Irene Sabadini
title: Boundary interpolation for slice hyperholomorphic Schur functions
---
Introduction
============
In the paper [@2013arXiv1308.2658A] the Nevanlinna-Pick interpolation problem for slice hyperholomorphic Schur functions has been solved using the FMI (fundamental matrix inequality) method (see [@kky] for details). By a Schur function we mean a function $f$ which is slice hyperholomorphic on the open unit ball $\mathbb B_1$ of the quaternions and is bounded in modulus by $1$, i.e. $\sup_{p\in\mathbb B_1}|f(p)|\leq 1$. In the present paper we solve a boundary interpolation problem for slice hyperholomorphic functions using the reproducing kernel Hilbert space method based on de Branges-Rovnyak spaces. We refer the reader to [@abds2; @abds3; @Dym_CBMS] for more information on the reproducing kernel Hilbert space approach to interpolation problems.\
We state the problem we will solve in this paper and introduce some notation and definitions. Let us denote by $\mathbb B_1$ and $\mathbb H_1$, the open unit ball and the unit sphere of $\mathbb H$, respectively. For a given element $p\in\mathbb H$ we denote by $[p]$ the associated 2-sphere: $$[p]=\left\{qpq^{-1}: q\in\mathbb H\setminus\left\{0\right\}\right\}.$$ Recall that two quaternions belong to the same sphere if and only if they have the same modulus and the same real part.
\[pb1\] Given $p_1,\ldots,p_N\in\mathbb H_1\setminus \left\{1\right\}$ such that $[p_u]\cap[p_v]=\emptyset$ for $u\not = v$ (the interpolation nodes), $s_1,\ldots,
s_N\in\mathbb H_1$, and $\kappa_1,\ldots, \kappa_N\in [0,\infty)$, find a necessary and sufficient condition for a slice hyperholomorphic Schur function $s$ to exist such that the conditions $$\begin{aligned}
\label{inter1}
\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}} s(rp_u)&=&s_u,\\
\lim_{\substack{r\rightarrow 1\\
r\in(0,1)}}\frac{1-s(rp_u)\overline{s_u}}{1-r}&\le& \kappa_u
\label{inter2}\end{aligned}$$ hold for $u=1, \ldots N$, and describe the set of all Schur functions satisfying - when this condition is in force.
We note that - imply that $$\lim_{\substack{r\rightarrow 1\\
r\in(0,1)}}\frac{1-|s(rp_u)|^2}{1-r^2}\le \kappa_u,\quad
u=1,\ldots, N, \label{wertyu}$$ since $$\label{richelieu-drouot1}
\frac{1-|s(rp_u)|^2}{1-r^2}=\frac{1-s(rp_u)\overline{s_u}}{(1-r)(1+r)}+
(s(rp_u)\overline{s_u})\frac{1-s_u\overline{s(rp_u)}}{(1-r)(1+r)}.$$
We also note that the fact that the limits is part of the requirement in the interpolation problem (in the complex case, the corresponding limit is well-known to be non-negative).\
As it appears from the statement of Problem \[pb1\], there is a major difference with the complex case. Here we have to require that not only the interpolation points are distinct, but also the spheres they determine. The fact that this hypothesis is necessary, and cannot be avoided, can be intuitively justified by the fact that the $S$-spectrum of a matrix, or in general of an operator (see Definition \[defspscandres\]), consists of spheres (which may reduce to real points). It is important to note that the notion of $S$-spectrum of a matrix $T$ coincides with the set of right eigenvalues of $T$, i.e. the set of $\lambda \in \mathbb{H}$ so that $Tx = x \lambda$ for a nonzero vector $x$.\
Another major difference is the lack of a Carathéodory theorem (see e.g. [@sarason94 p. 48]) in the quaternionic setting.\
Part of the arguments follow the classical case, taking into account the noncommutativity of the quaternions. As we shall see, even though the structure of the proof follows the the arguments from [@adubi1], it is necessary to suitably adapt the arugment to the quaternionic setting and often the needed modifications are not immediate.\
The paper consists of five sections, besides the introduction. In Section 2, we recall some basic material on slice hyperholomorphic functions which will be needed in the sequel. Section 3 illustrates the strategy and the various steps we will follow to solve Problem \[pb1\]. Section 4 contains detailed proofs of these steps and Section 5 deals with the degenerate case. Section 6 deals with an analogue of Carathéodory’s theorem in the quaternionic setting.
Some preliminaries
==================
In this section we collect some basic results, which will be used in the sequel. Let $\hh$ be the real associative algebra of quaternions with respect to the basis $\{1, i,j,k \}$ satisfying the relations $
i^2=j^2=k^2=-1,\
ij =-ji =k,\
jk =-kj =i ,
\ ki =-ik =j .
$ A quaternion $p$ is denoted by $p=x_0+ix_1+jx_2+kx_3$, $x_\ell\in \mathbb{R}$, $\ell=0,\ldots, 3$, its conjugate is $\bar p=x_0-ix_1-jx_2-kx_3$, and the norm of a quaternion is such that $|p|^2=p\overline{p}$. A quaternion $p$ can be written as $p={\rm Re}(p)+\underline{p}$ where the real part ${\rm Re}(p)$ is $x_0$ and $\underline{p} = i x_1 + j x_2 + k x_3$. The symbol $\mathbb{S}$ denotes the 2-sphere of purely imaginary unit quaternions, i.e. $$\mathbb{S}=\{ \underline{p}=ix_1+jx_2+kx_3\ |\ x_1^2+x_2^2+x_3^2=1\}.$$ Note that if $I\in\mathbb S$ then $I^2=-1$. Any nonreal quaternion $p=x_0+ix_1+jx_2+kx_3$ uniquely determines an element $I_p=(ix_1+jx_2+kx_3)/|ix_1+jx_2+kx_3|\in\mathbb S$. If $p=x_0\in\mathbb R$ then $p=x_0+I0$ for all $I\in\mathbb S$. Given $p\in\mathbb H$ we can write $p=p_0+I_pp_1$ and the 2-sphere $[p]$ coincides with the set of all elements of the form $p_0+Jp_1$ when $J$ varies in $\mathbb{S}$. The set $[p]$ is reduces to the point $p$ if and only if $p\in\mathbb{R}$.\
We now recall the definition of a slice hyperholomorphic function, for more details see [@MR2752913].
[Let $\Omega\subseteq\hh$ be an open set and let $f:\ \Omega\to\hh$ be a real differentiable function. Let $I\in\mathbb{S}$ and let $f_I$ be the restriction of $f$ to the complex plane $\mathbb{C}_I := \mathbb{R}+I\mathbb{R}$ passing through $1$ and $I$ and denote by $x+Iy$ an element on $\mathbb{C}_I$. We say that $f$ is a left slice hyperholomorphic (or slice hyperholomorphic, for short) function in $\Omega$ if, for every $I\in\mathbb{S}$, we have $$\frac{1}{2}\left(\frac{\partial }{\partial x}+I\frac{\partial
}{\partial y}\right)f_I(x+Iy)=0.$$ We say that $f$ is a right slice hyperholomorphic function in $\Omega$ if, for every $I\in\mathbb{S}$, we have $$\frac{1}{2}\left(\frac{\partial }{\partial x}f_I(x+Iy)+\frac{\partial
}{\partial y}f_I(x+Iy) I\right)=0.$$ ]{}
Slice hyperholomorphic functions have nice properties on some particular open sets which are defined below.
[Let $\Omega$ be a domain in $\mathbb{H}$. We say that $\Omega$ is a slice domain (s-domain for short) if $\Omega \cap \mathbb{R}$ is non empty and if $\Omega\cap \mathbb{C}_I$ is a domain in $\mathbb{C}_I$ for all $I \in \mathbb{S}$. We say that $\Omega$ is axially symmetric if, for all $p \in \Omega$, the sphere $[p]$ is contained in $\Omega$. ]{}
On an axially symmetric s-domain $\Omega$, a slice hyperholomorphic function satisfies the following formula, which is called the Structure formula or the Representation formula (see [@MR2752913 Theorem 4.3.2]): $$\label{repr}
f(x+J y)=\frac 12 \left[ f(x+Iy) +f(x-Iy) + J I
(f(x-Iy)- f(x+Iy))\right].$$ Formula is useful as it allows one to extend a holomorphic map $h:\ \Omega\subseteq \mathbb{C}\cong \mathbb{C}_I\to \mathbb H$ to a slice hyperholomorphic function. Let $U_\Omega$ be the axially symmetric completion of $\Omega$, i.e. $$U_\Omega=\bigcup_{J\in\mathbb S, \, x+Iy\in\Omega} \{x+Jy\}.$$ The left slice hyperholomorphic extension ${\rm ext}(h): \ U_\Omega\subseteq \mathbb H \to\mathbb H$ of $h$ is the function defined as (see [@MR2752913]): $$\label{ext}
{\rm ext}(h)(x+J y)=\frac 12 \left[ h(x+Iy) +h(x-Iy) + J I
(h(x-Iy)-h(x+Iy))\right].$$ It is immediate that ${\rm ext}(h+g)={\rm ext}(h)+{\rm ext}(g)$ and that if $h(z)=\sum_{n=0}^\infty h_n(z)$ then ${\rm
ext}(h)(z)=\sum_{n=0}^\infty {\rm ext}(h_n)(z)$.
Two left (resp. right) slice hyperholomorphic functions can be multiplied, on an axially symmetric s-domain, using the so called $\star$-product (resp. $\star_r$-product) in order to obtain another left (resp. right) slice hyperholomorphic function.\
Let $f,g:\ \Omega \subseteq\mathbb{H}$ be slice hyperholomorphic functions. Their restrictions to the complex plane $\mathbb{C}_I$ can be written as $f_I(z)=F(z)+G(z)J$, $g_I(z)=H(z)+L(z)J$ where $J\in\mathbb{S}$, $J\perp I$, i.e. $IJ = -JI$. The functions $F$, $G$, $H$, $L$ are holomorphic functions of the variable $z\in
\Omega \cap \mathbb{C}_I$, see [@MR2752913 p. 117]. We have the following:
[Let $f$ and $g$ be slice hyperholomorphic functions defined on an axially symmetric s-domain $\Omega\subseteq\mathbb{H}$. The $\star$-product of $f$ and $g$ is defined as the unique left slice hyperholomorphic function on $\Omega$ whose restriction to the complex plane $\mathbb{C}_I$ is given by $$\label{starproduct}
\begin{split}
(f\star g)_I(z)&=(F(z)+G(z)J)\star(H(z)+L(z)J)\\
&=
(F(z)H(z)-G(z)\overline{L(\bar z)})+(G(z)\overline{H(\bar z)}+F(z)L(z))J.
\end{split}$$ ]{}
If $f$ and $g$ are slice hyperholomorphic on a ball with center at the origin, they can be expressed in a power series, i.e. $f(p)=\sum_{n=0}^\infty p^n a_n$ and $g(p)=\sum_{n=0}^\infty p^n b_n$. Thus $(f\star g)(p)=\sum_{n=0}^\infty p^n c_n$, where $c_n=\sum_{r=0}^na_rb_{n-r}$ is obtained by convolution on the coefficients. For the construction of the $\star$-product of right slice hyperholomorphic functions and for more information on the $\star$-product, we refer the reader to [@MR3127378; @MR2752913].\
Given a slice hyperholomorphic function, it is possible to define its slice hyperholomorphic reciprocal, see [@MR2752913]. Here we limit ourselves to the case in which $f$ admits the power series expansion $f(p)=\sum_{n=0}^\infty p^n a_n$. In this case we set $$f^c(p)=\sum_{n=0}^\infty p^n \bar a_n,\qquad f^s(p)=(f^c\star f)(p
)=\sum_{n=0}^\infty p^nc_n,\quad
c_n=\sum_{r=0}^n a_r\bar a_{n-r},$$ so that the left slice hyperholomorphic reciprocal of $f$ is defined as $$f^{-\star}:=(f^s)^{-1}f^c.$$ In the general case, this formula is still valid with $f^s$, $f^c$ suitably defined.
[Let $k(p,q)$ be a function left slice hyperholomorphic in $p$ and right slice hyperholomorphic in $\bar q$. When taking the $\star$-product of a function $f(p)$ slice hyperholomorphic in the variable $p$ with a function $k(p,q)$, we will write $f(p)\star k(p,q)$ meaning that the $\star$-product is taken with respect to the variable $p$; similarly, the $\star_r$-product of $k(p,q)$ with functions right slice hyperholomorphic in the variable $\bar q$ is always taken with respect to $\bar q$. ]{}
The following proposition is taken from [@MR3127378 Proposition 4.3], where a proof can be found.
Let $\mathcal H(K_1)$ and $\mathcal H(K_2)$ be two reproducing kernel Hilbert spaces of $\mathbb H^m$ and $\mathbb H^n$-valued slice hyperholomorphic functions in $\Omega$, with reproducing kernels $K_1$ and $K_2$, respectively. Let $R$ be a $\mathbb H^{n\times m}$-valued function slice-hyperholomorphic in $\Omega$. Then the operator of left $\star$-multiplication $$M_R\,:\, \,\, f\,\,\mapsto\,\, R\star f$$ is continuous from $\mathcal H(K_1)$ into $\mathcal H(K_2)$ if and only if the kernel $$K_2(p,q)-R(p)\star K_1(q, p)\star_rR(q)^*$$ is positive definite in $\Omega$. Furthermore $$\label{grenelle}
M_R^*(K_2(\cdot, q)d)=K_1(\cdot, q)\star_r R(q)^*d,\quad d\in\mathbb H^n.$$ \[la-seine\]
Let us recall a few facts on the $S$-spectrum and on the $S$-resolvent operator.
\[defspscandres\] [Let $A$ be a bounded quaternionic linear operator acting on a quaternionic, two sided, Banach space $V$. We define the $S$-spectrum $\sigma_S(A)$ of $A$ as: $$\sigma_S(A)=\{ \text{$s\in \mathbb{H}$ : $A^2-2 {\rm Re}\,(s) A+|s|^2\mathcal{I}$ is not invertible} \},$$ where $\mathcal I$ denotes the identity operator on $V$. The $S$-resolvent set $\rho_S(A)$ is defined as $\rho_S(A)=\mathbb{H}\setminus\sigma_S(A)$. ]{}
From Definition \[defspscandres\] it follows that the $S$-spectrum consists of spheres (which may reduce to real points).\
The definition of $S$-spectrum arises from the following:
\[Ssinistro\] Let $A$ be a bounded quaternionic linear operator acting on a quaternionic, two sided, Banach space $V$. Then, for $\|A\|< |p|$, we have $$\label{SresolvR}
\sum_{n= 0}^\infty s^{-1-n}A^n=-
(A- \overline{s}\mathcal{I})(A^2-2{\rm Re}(s) A+|s|^2 \mathcal{I} )^{-1}.$$
[The operator $$\label{SresolvoperatorRdd}
S_R^{-1}(s,A):=-(A- \overline{s}\mathcal{I})(A^2-2{\rm Re}(s) A+|s|^2 \mathcal{I} )^{-1},$$ is called the right $S$-resolvent operator. ]{}
The right $S$-resolvent operator is obviously defined for $s\in\rho_S(A)$.\
In the sequel we will be in need of the result below:
Let $V$ be a two sided quaternionic Banach space and let $A$ be a bounded right linear operator from $V$ into itself. Then, for $|p| \,\|A\|< 1$ we have $$\label{eq:oberkampf_ligne_5}
\sum_{n=0}^\infty p^n A^n =( \mathcal{I} -\bar p A)(|p|^2A^2-2 {\rm Re}(p) A+ \mathcal{I} )^{-1}.$$
Another way to write the operator on the right hand side of (\[eq:oberkampf\_ligne\_5\]) is to observe that it corresponds to the function one obtains by constructing the right $\star$-reciprocal of the function $f(q)=(1-pq)$. Upon computing $f^{-\star}(A)$ using the quaternionic functional calculus, see [@MR2752913], one can write: $$\label{eq:oberkampf_ligne_55}
(\mathcal{I}-pA)^{-\star}=\sum_{n=0}^\infty p^n A^n.$$
Finally, we mention a result which is a restatement of [@2013arXiv1310.1035A Proposition 2.22] and which contains an identity that will be crucial in the sequel.
Let $p\in\mathbb H$, $1/p\in\rho_S(A)$ and $(G,A)\in\mathbb H^{n\times m}\times{\mathbb H}^{m\times m}$. Then $$\label{magic}
\sum_{t=0}^\infty p^tGA^t=(G-\overline{p}GA)(\mathcal{I}_m-2{\rm Re}(p)A+|p|^2A^2)^{-1},$$ where $\mathcal{I}_m$ denotes the $m \times m$ identity matrix.
[We note that if $m=1$ then $A$ is a quaternion $a$ and the condition $1/p\in\rho_S(A)$ translates to the condition $1/p\not\in[a]$. ]{}
The main result and the strategy
================================
For the convenience of the reader we recall the main steps of the reproducing kernel method. We first introduce some notation. We set $$\label{Dn:ACJ}
A={\rm diag}\,(\overline{p_1},\ldots,\overline{p_N})\in\mathbb H^{N\times N},\quad C=\begin{pmatrix}1&\cdots &1\\
\overline{s_1}&\cdots &\overline{s_N}\end{pmatrix}\in\mathbb H^{2\times N},$$ and $$\mathcal{J}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}\in \mathbb R^{2\times 2}.$$ Consider the matrix equation $$\label{steineq}
P-A^*PA=C^*\mathcal{J}C$$ where the unknown is $P\in\mathbb H^{N\times N}$. The off diagonal entries of the matrix equation are uniquely determined by the equation $$\label{puv}
P_{uv}-p_uP_{uv}\overline{p_v}=1-s_u\overline{s_v}$$ but, in view of the following lemma the diagonal entries can be arbitrary:
\[La:ph-hq\] Let $p$ and $q$ be quaternions of modulus $1$. Then, the equation $$ph-hq=0,
\label{trivial}$$ where $h\in\mathbb H$, has the only solution $h=0$ if and only if ${\rm Re}(p)\not={\rm Re}(q)$, that is, if and only if $[p]\cap[q]=\emptyset$.
If has a solution $h\not=0$, then $p=hqh^{-1}$ and so $p$ and $q$ are in the same sphere. So a necessary condition for to have only $h=0$ as solution is that $[p]\cap[q]=\emptyset$. We now show that this condition is also sufficient. Let $p=z_1+z_2j$ and $q=w_1+w_2j$, where $z_1,z_2,w_1,w_2\in\mathbb C$. Since ${\rm Re}(p)\not={\rm Re}(q)$ we have $$\label{complicated}
{\rm Re}( z_1)\pm i\sqrt{1-({\rm Re}( z_1))^2}\not={\rm Re}( w_1)\pm i\sqrt{1-({\rm Re}( w_1))^2}.$$ We now introduce the injective ring homomorphism $\chi : \mathbb H \to \mathbb C^{2 \times 2}$ given by $$\label{eq:Oct27jkl1}
\chi(p) = \begin{pmatrix} z_1 & z_2 \\ - \overline{z}_2 & \overline{z}_1 \end{pmatrix}.$$ Using the map $\chi$, equation becomes $$\label{detour}
\chi(p)\chi(h)-\chi(h)\chi(q)=0.$$ The eigenvalues of $\chi(p)$ are the solutions of $$\lambda^2-2({\rm Re}( z_1))\lambda+1=0,$$ that is $\lambda={\rm Re}( z_1)\pm i\sqrt{1-({\rm Re}( z_1))^2}$, and similarly for $\chi(q)$. By a well known result on matrix equations (see e.g., Corollary 4.4.7 in [@HornJohnson]), equation (\[detour\]) has only the solution $\chi(h)=0$ if and only if $\lambda-\mu\not=0$ for all possible choices of eigenvalues of $\chi(p)$ and $\chi(q)$, and this condition holds in view of . So the only solution of is $h=0$.
We denote by $P$ the $N\times N$ Hermitian matrix with entries $P_{uv}$ given by for $u\not= v$ and with diagonal entries equal to $P_{uu}=\kappa_u$, $u,v=1,\ldots, N$. When $P$ is invertible we define $$\label{parislelouvre}
\Theta(p)=\mathcal{I}_2-(1-p)\star C\star(\mathcal{I}_N-pA)^{-\star}P^{-1}(\mathcal{I}_N-A)^{-*}C^*\mathcal{J}=\begin{pmatrix}a(p)&b(p)\\ c(p)&d(p)\end{pmatrix}.$$ Note that $\Theta$ is well defined in $\mathbb B_1$ since we assumed that the interpolation nodes $p_u$ are all different from $1$. Finally we denote by $\mathcal M$ the span of the columns of the function $$\label{Fp}
F(p)=C\star (\mathcal{I}_N-pA)^{-\star}
=\sum_{t=0}^\infty p^tCA^t,$$ and endow $\mathcal M$ with the Hermitian form $$[F(p)c,F(p)d]_{\mathcal M}=d^*Pc,\quad c,d\in\mathbb H^N.$$ We prove the following theorem.
\
$(1)$ There always exists a Schur function so that holds.\
$(2)$ Fix $\kappa_1,\ldots, \kappa_N \geq 0$ and assume $P> 0$. Any solution of Problem \[pb1\] is of the form $$\label{lft1}
s(p)=(a(p)\star e(p)+b(p))\star(c(p)\star
e(p)+d(p))^{-\star},$$ where $a,b,c,d$ are as in (\[parislelouvre\]) and $e$ is a slice hyperholomorphic Schur function.\
$(3)$ Conversely, any function of the form satisfies . If $$\label{limits}
\lim_{\substack{r\rightarrow 1\\
r\in(0,1)}}\frac{1-s(rp_u)\overline{s_u}}{1-r}$$ exists and is real, then $s$ satisfies .\
$(4)$ If $e$ is a unitary constant, then the limit exists (but are not necessarily real) and satisfies $$\label{bastille}
\frac{|\beta_u-\overline{p_u}\beta_u\overline{p_u}|^2}{|1-\overline{p_u}^2|}\le
({\rm Re}\,\beta_u)\kappa_u.$$ \[tm1\]
The strategy of the proof is as follows:\
STEP 1: [*The condition $P\ge 0$ is necessary for Problem \[pb1\] to have a solution.*]{}\
STEP 2: [*Assume that $s$ is a solution of Problem \[pb1\]. Then the map $M_{\begin{pmatrix}1&-s\end{pmatrix}}$ of left $\star$-multiplication by $\begin{pmatrix}1&-s(p)\end{pmatrix}$ is a contraction from $\mathcal M$ into $\mathcal H(s)$, where $\mathcal H(s)$ denotes the reproducing kernel Hilbert space of quaternionic valued functions which are hyperholomorphic in the ball $\mathbb B_1$ and with reproducing kernel $$K_s(p,q)=\sum_{t=0}^{\infty} p^t(1-s(p)\overline{s(q)})\bar q^t.$$*]{}\
STEP 3: [*Assume that $s$ is a solution of Problem \[pb1\] and that $P>0$. Then, $s$ is of the form .*]{}\
STEP 4: [*Assume that $P>0$. Then any function of the form satisfies the interpolation condition and if, in addition, is in force, then $s$ satisfies* ]{}.\
The proofs of Steps 1-4 are given in Section \[sec123\]. The degenerate case is considered in Section \[sec345\].
Proofs of Steps 1-4
===================
\[sec123\]
[**Proof of Step 1:**]{} Assume a solution $s$ exists. Since $s$ is a Schur function the kernel $K_s(p,q)$ is positive definite and so for every $r\in(0,1)$ the $N\times N$ matrix $P(r)$ with $(u,v)$ entry equal to $$P_{uv}(r)=K_s(rp_u,rp_v)=\sum_{t=0}^\infty r^{2t}p_u^t(1-s(rp_u)\overline{s(rp_v)})p_v^t,\quad u,v=1,\ldots N$$ is positive. Setting $$G=(1-s(rp_u)\overline{s(rp_v)}),\quad p=r^2p_u,\quad\text{and}\quad A=\overline{p_v}$$ in formula we have $$P_{uv}(r)=\left((1-s(rp_u)\overline{s(rp_v)})-r^2\overline{p_u}(1-s(rp_u)\overline{s(rp_v)})\overline{p_v}\right)
(1-2r^2{\rm Re}(p_u)\overline{p_v}+r^4\overline{p_v}^2)^{-1}.$$ Furthermore, we note that $P(r)$ is a solution of the matrix equation $$P(r)-r^2A^*P(r)A=C(r)^*\mathcal{J}C(r)$$ where $$C(r)=\begin{pmatrix}1&\cdots &1\\
& &\\
\overline{s(rp_1)}&\cdots &\overline{s(rp_N)}\end{pmatrix},$$ and $A$ is as in (\[Dn:ACJ\]). In fact, with the above notation, the $(u,v)$ element of the matrix $P(r)-r^2A^*P(r)A$ can be computed as follows: $$\begin{split}
&P_{uv}(r)-r^2p_uP_{uv}(r)\overline{p_v}\\
&=\left(
\left(G-r^2\overline{p_u}G\overline{p_v}\right) -r^2p_u\left(G-r^2\overline{p_u}G\overline{p_v}\right)\overline{p_v}
\right)
(1-2r^2{\rm Re}(p_u)\overline{p_v}+r^4\overline{p_v}^2)^{-1}\\
&=\left(
G-r^2\overline{p_u}G\overline{p_v} -r^2p_uG \overline{p_v}+r^4G\overline{p_v}^2
\right)
(1-2r^2{\rm Re}(p_u)\overline{p_v}+r^4\overline{p_v}^2)^{-1}\\
&=G\left(
1-2r^2{\rm Re}({p_u})\overline{p_v} +r^4\overline{p_v}^2
\right)
(1-2r^2{\rm Re}(p_u)\overline{p_v}+r^4\overline{p_v}^2)^{-1}=(1-s(rp_u)\overline{s(rp_v)})\\
\end{split}$$ and so the $(u,v)$ element in the matrix $P(r)-r^2A^*P(r)A$ equals the $(u,v)$ element in $C(r)^* \mathcal{J}C(r)$ as stated. We now let $r$ tend to $1$. Since $s$ is assumed to be a solution of Problem \[pb1\], we have $$\lim_{\substack{r\rightarrow 1\\
r\in(0,1)}}K_s(rp_u,rp_u)=\lim_{\substack{r\rightarrow 1\\
r\in(0,1)}}\frac{1-|s(rp_u)|^2}{1-r^2} \le \kappa_u,\quad
u=1,\ldots N$$ and $$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}}C(r)=C,$$ where $C$ is as in (\[Dn:ACJ\]). Furthermore we note that $1-2{\rm Re}(p_u)\overline{p_v}+\overline{p_v}^2\not=0$ since $1-2{\rm Re}(p_u)x+x^2$ is the so-called minimal (or companion) polynomial associated with the sphere $[p_u]$ which vanishes exactly at points on the sphere $[p_u]$ and $p_v\not\in [p_u]$. This fact can also be obtained directly using Lemma \[La:ph-hq\]. Indeed, for indices $u\not= v$, we have $$\label{aveiro:2014}
1-2{\rm Re} ( p_u)\overline{p_v}+\overline{p_v}^2=p_u(\overline{p_u}-\overline{p_v})-(\overline{p_u}-\overline{p_v})
\overline{p_v}\not=0,$$ since $p_u$ and $p_v$ (and hence $p_u$ and $\overline{p_v}$) are assumed on different spheres for $u\not=v$. It follows that $\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}}
P_{uv}(r)$ exists and is in fact equal to $P_{uv}$ for $u\not =v$ by uniqueness of the solution of the equation $$\label{secretequation}
x-p_ux\overline{p_v}=0.$$ Hence $P\ge 0$ since $P(r)\ge 0$ for all $r\in(0,1)$.\
[**Proof of Step 2:**]{} Let $s$ be a solution (if any) of Problem \[pb1\], let $u\in\left\{1,\ldots, N\right\}$, and let $r\in(0,1)$. The functions $$g_{u,r}(p)=K_s(p, rp_u)=\sum_{t=0}^\infty
p^t(1-s(p)\overline{s(rp_u)})\overline{p_u}^t$$ belong to $\mathcal H(s)$ and have uniformly bounded norms since $$\lim_{\substack{r\rightarrow 1\\
r\in(0,1)}}\|g_{u,r}(rp_u)\|^2_{\mathcal H(s)}=
\lim_{\substack{r\rightarrow 1\\
r\in(0,1)}}K_s(rp_u,rp_u)\le\kappa_u.$$ Thus there is a sequence of numbers $r_0,r_1,\ldots \in(0,1)$ which tends to $1$ (without loss of generality we may assume that the sequence is the same for $p_1, \cdots, p_N$) and an element $g_u\in\mathcal H(s)$ such that the functions $g_{u,r_n}$ tend weakly to $g_u$. In a reproducing kernel Hilbert space weak convergence implies pointwise convergence, and so $$\begin{split}
g_u(p)&=\lim_{n\longrightarrow\infty}g_{u,r_n}(p)\\
&=\lim_{n\longrightarrow\infty}
\sum_{t=0}^\infty r_n^tp^t(1-s(p)\overline{s(r_np_u)})\overline{p_u}^t\\
&=\sum_{t=0}^\infty p^t(1-s(p)\overline{s_u})\overline{p_u}^t\\
&=\begin{pmatrix}1&-s(p)\end{pmatrix}\star f_u(p),\quad\forall p\in\mathbb B_1,
\end{split}$$ where $$\label{fu}
f_u(p)=\sum_{t=0}^\infty p^t\begin{pmatrix}1\\ \overline{s_u}\end{pmatrix}\overline{p_u}^t$$ denotes the $u$-th column of the matrix-function $F(p)$ and where the interchange of summation and limit is justified since $|p|<1$. Hence $M_{\begin{pmatrix}1&-s\end{pmatrix}}$ sends $\mathcal M$ into $\mathcal H(s)$. Note that for $Y = (y_{u,v})_{u,v=1}^N$ and $Z = (z_{u,v} )_{u,v=1}^N$ we define $Y \star Z$ to be the $N \times N$ matrix whose $(u,v)$ entry is given by $\sum_{t=1}^N y_{u,t} \star z_{t,v}$. To show that this operator is a contraction we first compute the inner product $\langle g_v,g_u
\rangle_{\mathcal H(s)}$ for $u\not= v$. By the definition of the weak limit and of the reproducing kernel, we can write $$\begin{split}
\langle g_v,g_u
\rangle_{\mathcal H(s)}&=\lim_{n\longrightarrow\infty}\langle g_v,g_{u,r_n}
\rangle_{\mathcal H(s)}\\
&=\lim_{n\longrightarrow\infty}g_v(r_np_u)\\
&=\lim_{n\longrightarrow\infty}\sum_{t=0}^\infty r_n^tp_u^t(1-s(r_np_u)\overline{s_v})\overline{p_v}^t\\
&=\lim_{n\longrightarrow\infty}\left((1-s(r_np_u)\overline{s_v})-r_n\overline{p_u}(1-s(r_np_u)\overline{s_v})\overline{p_v}\right)
(1-2r_n{\rm Re}( p_u)\overline{p_v}+r_n^2\overline{p_v}^2)^{-1}\\
&=\left((1-s_u\overline{s_v})-\overline{p_u}(1-s_u\overline{s_v})\overline{p_v}\right)
(1-2{\rm Re}( p_u)\overline{p_v}+\overline{p_v}^2)^{-1},
\end{split}$$ where we have used formula and, as in the proof of Step 1 (see ), the fact that $[p_u]\cap[p_v]=\emptyset$ (recall that we assume here $u\not=v$). We claim that $$\label{parisbastille}
P_{uv}=\left((1-s_u\overline{s_v})-\overline{p_u}(1-s_u\overline{s_v})\overline{p_v}\right)
(1-2{\rm Re} (p_u)\overline{p_v}+\overline{p_v}^2)^{-1}.$$ The proof is similar to the argument in the proof of step 1, and is as follows. Set $h_n=\langle g_v,g_{u,r_n}
\rangle_{\mathcal H(s)}$. Then $$h_n-r_np_uh_n\overline{p_v}=1-s(r_np_u)\overline{s_v}.$$ Letting $n\rightarrow\infty$ we see that $h=\lim_{n\rightarrow\infty}h_n$ satisfies equation . By the uniqueness of the solution of this equation we have $h=P_{uv}$. Furthemore, by the property of the weak limit versus the norm, $$\label{parisstmichel}
\|g_u\|^2_{\mathcal H(s)}\le\lim_{n\rightarrow\infty}\|g_{u,r_n}\|^2_{\mathcal H(s)}\le \kappa_u.$$ We can now show that $\|M_{\begin{pmatrix}1&-s\end{pmatrix}}\|\le 1$. Let $c\in \mathbb H^N$. Then, $$\left(M_{\begin{pmatrix}1&-s\end{pmatrix}}Fc\right)(p)=\sum_{u=1}^N g_u(p)c_u$$ and we have $$\begin{split}
\|(M_{\begin{pmatrix}1&-s\end{pmatrix}}Fc\|^2_{\mathcal H(s)}&=\sum_{u,v=1}^N
\overline{c_u}\left(\langle g_v,g_u\rangle_{\mathcal H(s)}\right)c_v\\
&=\sum_{u=1}^N
|c_u|^2\|g_u\|^2_{\mathcal H(s)}+\sum_{\substack{u,v=1\\ u\not=v}}^N
\overline{c_u}\left(\langle g_v,g_u\rangle_{\mathcal H(s)}\right)c_v\\
&=\sum_{u=1}^N
|c_u|^2\|g_u\|^2_{\mathcal H(s)}+\sum_{\substack{u,v=1\\ u\not=v}}^N
\overline{c_u}P_{uv}c_v\\
&\le \sum_{u=1}^N|c_u|^2\kappa_u+\sum_{\substack{u,v=1\\ u\not=v}}^N
\overline{c_u}P_{uv}c_v\\
&=c^*Pc\\
&=\|Fc\|^2_{\mathcal M},
\end{split}$$ where we have used and . Thus the $\star$-multiplication by $(1\ -s(p))$ is a contraction from $\mathcal M$ into $\mathcal{H}(s)$.\
[**Proof of Step 3:**]{} Let $\Theta$ be defined by , and $$\label{musee-d-orsay}
K_\Theta(p,q)=\sum_{t=0}^\infty
p^t\left(\mathcal{J}-\Theta(p)\mathcal{J}\Theta(q)^*\right)\overline{q}^t.$$ The formula $$\label{fpf}
F(p)P^{-1}F(q)^*=K_\Theta(p,q)$$ is proved as in the complex case when $p$ and $q$ are real, and is then extended to $p,q\in\mathbb B_1$ by a slice hyperholomorphic extension. Using we have $$\left(M_{\begin{pmatrix}1&-s\end{pmatrix}}^*K_s(\cdot, q)\right)(p)=\sum_{t=0}^\infty
p^n\left(\begin{pmatrix}1\\ -\overline{s(q)}\end{pmatrix}-\Theta(p)\mathcal{J}\Theta(q)^*\star_r\begin{pmatrix}1\\
-\overline{s(q)}\end{pmatrix}\right)\overline{q}^t,$$ and so $$\begin{split}
(M_{\begin{pmatrix}1&-s\end{pmatrix}}M_{\begin{pmatrix}1&-s\end{pmatrix}}^*K_s(\cdot, q))(p)&\\
&\hspace{-2cm}=K_s(p,q)-
\sum_{t=0}^\infty p^t\left(\begin{pmatrix}1&-s(p)\end{pmatrix}\star\Theta(p)
\mathcal{J}\Theta(q)^*\star_r\begin{pmatrix}1\\
-\overline{s(q)}\end{pmatrix}\right)\overline{q}^t\\
&\hspace{-2cm}\le K_s(p,q),
\end{split}$$ and therefore the kernel $$\sum_{t=0}^\infty p^t\left(\begin{pmatrix}1&-s(p)\end{pmatrix}\star\Theta(p)
\mathcal{J}\Theta(q)^*\star_r\begin{pmatrix}1\\
-\overline{s(q)}\end{pmatrix}\right)\overline{q}^t
\sum_{t=0}^\infty p^t\left(A(p)\overline{A(q)}-B(p)\overline{B(q)}\right)\overline{q}^t$$ is positive definite in $\mathbb B_1$, where $$A(p)=(a-s\star c)(p)\quad\text{and}\quad B(p)=(b-s\star d)(p).$$ The point $p=1$ is not an interpolation node, and so $\Theta$ is well defined at $p=1$. From we have $$\label{operabastille}
\Theta(1) = \mathcal{I}_2$$ and so $(a^{-1}c)(1) = 0$. Since $s$ is bounded by $1$ in modulus in $\mathbb B_1$ it follows that $(a-s\star c)(p)\not\equiv 0$. Thus $e=-(a-s\star c)^{-\star}\star(b-s\star d)$ is defined in $\mathbb B_1$, with the possible exception of spheres of poles. Since $$\sum_{t=0}^\infty p^t\left(A(p)\overline{A(q)}-B(p)\overline{B(q)}\right)\overline{q}^t=A(p)\star\left\{
\sum_{t=0}^\infty p^t(
1-e(p)\overline{e(q)})\overline{q}^t\right\}\star_r
\overline{A(q)},$$ we have from [@acs3 Proposition 5.3] that the kernel $$K_e(p,q)=\sum_{t=0}^\infty p^t(
1-e(p)\overline{e(q)})\overline{q}^t$$ is positive definite in its domain of definition, and thus $e$ extends to a Schur function (see [@acs1] for the latter assertion). From $$e=-(a-s\star c)^{-\star}\star(b-s\star d)$$ we get $s\star(c\star e +d)=a\star e+b$. To conclude we remark that implies that $$(d^{-1}c)(1)=0.$$ Thus, as just above $c\star e+d\not\equiv 0$ and we get that $s$ is of the form .\
[**Proof of Step 4:**]{} Assume that $s$ is of the form . Then the formula $$K_s(p,q)=\begin{pmatrix}1&-s(p)\end{pmatrix}\star K_\Theta(p,q)\star_r \begin{pmatrix}1\\-\overline{s(q)}\end{pmatrix}+
(a-s\star c)(p)\star K_e(p,q)\star_r\overline{(a-s\star c)(q)}$$ implies that $M_{\begin{pmatrix}1&-s\end{pmatrix}}$ is a contraction from $\mathcal H(\Theta)$ into $\mathcal H(s)$. In particular $$\label{gu}
g_u(p)=\begin{pmatrix}1&-s(p)\end{pmatrix}\star f_u(p)=\sum_{t=0}^\infty p^t(1-s(p)\overline{s_u})\overline{p_u}^t\in\mathcal H(s)$$ and $$\|g_u\|^2_{\mathcal H(s)}\le \kappa_u.$$ We want to infer from these facts that $s$ satisfies the interpolation conditions . We have $$\label{richelieu-drouot}
\begin{split}
|g_u(rp_u)|^2&=|\langle g_u(\cdot), K_s(\cdot, rp_u)\rangle_{\mathcal H(s)}|^2\\
&\le \left(\|g_u\|^2_{\mathcal H(s)}\right)\cdot K_s(rp_u,rp_u)\\
&\le \kappa_u\cdot\frac{1-|s(rp_u)|^2}{1-r^2}\\
&\le \frac{2\kappa_u}{1-r}.
\end{split}$$ In view of , we get $$\label{republique}
\begin{split}
g_u(rp_u)&=\sum_{t=0}^\infty r^tp_u^t(1-s(rp_u)\overline{s_u})\overline{p_u}^t\\
&=\left((1-s(rp_u)\overline{s_u})-r\overline{p_u}(1-s(rp_u)\overline{s_u})\overline{p_u}\right)(1-2r{\rm Re}( p_u)
\overline{p_u}+r^2\overline{p_u}^2)^{-1}\\
&=\left((1-s(rp_u)\overline{s_u})-r\overline{p_u}(1-s(rp_u)\overline{s_u})\overline{p_u}\right)((1-r)(1-r\overline{p_u}^2))^{-1},
\end{split}$$ and so we have $$\frac{|(1-s(rp_u)\overline{s_u})-r\overline{p_u}(1-s(rp_u)\overline{s_u})\overline{p_u}|}{|1-r\overline{p_u}^2|}\le \sqrt{2\kappa_u}\cdot
\sqrt{1-r}.$$ Let $\sigma_u$ be a limit, via a subsequence, of $s(rp_u)$ as $r\rightarrow 1$, and set $X_u=1-\sigma_u\overline{s_u}$. The above inequality implies that $X_u=\overline{p_u}X_u\overline{p_u}$, and so $$\label{richardlenoir}
X_up_u=\overline{p_u}X_u.$$ The conjugate of is $$\label{richardlenoir1}
\overline{X_u}p_u=\overline{p_u}\overline{X_u}.$$ Adding and we obtain $${\rm Re}( X_u)p_u=\overline{p_u}{\rm Re}( X_u).$$ Since $p_u$ is not real we get that ${\rm Re}( X_u)=0$. Let $X_u=\alpha i+\beta j+\gamma k$, where $\alpha, \beta,\gamma\in\mathbb R$. From $\sigma_u\overline{s_u}=1-X_u$ we have $$|\sigma_u\overline{s_u}|^2=1+\alpha^2+\beta^2+\gamma^2.$$ Since $\sigma_u\in\mathbb B_1$ we have $|\sigma_u\overline{s_u}|\le 1$ and so $\alpha=\beta=\gamma=0$. Thus, $X_u=0$ and $\sigma_u\overline{s_u}=1$. Hence $\sigma_u=s_u$ and the limit $\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}} s(rp_u)$ exists and is equal to $s_u$, and hence is satisfied.\
To prove that is met we proceed as follows. From we have in particular $$|g_u(rp_u)|^2
\le \kappa_u\cdot\frac{1-|s(rp_u)|^2}{1-r^2},$$ and using we obtain: $$\label{jussieu}
\frac{|X(r)-r\overline{p_u}X(r)\overline{p_u}|^2}{(1-r)^2|1-r\overline{p_u}^2|^2}\le
\kappa_u\cdot\frac{1-|s(rp_u)|^2}{1-r^2},$$ where we have set $X(r)=1-s(rp_u)\overline{s_u}$. Assume now that is in force and let $$\lim_{\substack{r\rightarrow 1\\
r\in(0,1)}}\frac{1-s(rp_u)\overline{s_u}}{1-r}=\beta_u\in\mathbb
R.$$ Then together with imply that $$\beta_u^2\le \beta_u\kappa_u,$$ from which we get that $\beta_u\ge0$ and $$\lim_{\substack{r\rightarrow 1\\
r\in(0,1)}} \frac{1-s(rp_u)\overline{s_u}}{1-r}\le \kappa_u.$$
The degenerate case
===================
\[sec345\]
We now consider the case where $P$ is singular. We need first a definition. A finite Blaschke product is a finite $\star$-product of terms of the form which are given by $$\label{eqBlaschke} b_a(p)=(1-p\bar
a)^{-\star}\star(a-p)\frac{\bar a}{|a|},$$ where $a\in\mathbb{H}$, $|a|<1$ (see [@MR3127378]).\
The purpose of this section is to prove the following theorem. First a remark. We denote by $r$ the rank of $P$ and assume that the main $r\times r$ minor of $P$ is invertible. This can be done by rearranging the interpolation points.
\[tm2\] Assume that $P$ is singular. Then Problem \[pb1\] has at most one solution, and the latter is then a finite Blaschke product. It has a unique solution satisfying for $u=1,\ldots, r$.
We begin with some preliminary results and definitions.
Let $f$ be a slice hyperholomorphic in a neighborhood $\Omega$ of $p=1$, and let $f(p)=\sum_{t=0}^\infty (p-1)^tf_t$ be its power series expansion at $p=1$. We define $$R_1f(p)=\sum_{t=1}^\infty (p-1)^t f_t.$$
Denoting by ${\rm ext}$ the slice hyperholomorphic extension we have $$R_1f(p)={\rm ext}\left(R_1f|_{p=x}\right).
\label{lisbonne}$$
\[lemma:R1\] Let $f(p)=F(p)\xi$ where $F(p)=C\star (\mathcal{I}_N-pA)^{-\star}$, then $$\label{place-des-vosges}
R_1f(p)=F(p)A(\mathcal{I}_N-A)^{-1}\xi.$$
First of all, recall that $$F(p)=C\star (\mathcal{I}_N-pA)^{-\star}=(C-\bar p CA)(I_n-2{\rm Re}(p) A +|p|^2 A^2)^{-1}$$ so $$F(1)=(C-CA)(\mathcal{I}_N-2 A + A^2)^{-1}=C(\mathcal{I}_N-A)^{-1}.$$ Let us compute $$\begin{split}
R_1 f(p)&=(p-1)^{-1}(f(p)-f(1))=(p-1)^{-1}(C\star (\mathcal{I}_N-pA)^{-\star}\xi- C (\mathcal{I}_N-A)^{-1}\xi)\\
&= C\star (p-1)^{-1}( (\mathcal{I}_N-pA)^{-\star}- (\mathcal{I}_N-A)^{-1})\xi\\
&= C\star (p-1)^{-1} \star (\mathcal{I}_N-pA)^{-\star}\star ((\mathcal{I}_N-A) - (\mathcal{I}_N-pA))(\mathcal{I}_N-A)^{-1}\xi\\
&= C\star (p-1)^{-1} \star (\mathcal{I}_N-pA)^{-\star}\star (p-1) A(\mathcal{I}_N-A)^{-1}\xi\\
&= C\star (\mathcal{I}_N-pA)^{-\star} A(\mathcal{I}_N-A)^{-1}\xi \\
&= F(p)A(\mathcal{I}_N-A)^{-1}\xi.
\end{split}$$
Let $f,g\in\mathcal M$. Then $$\label{dbspecial-case} [f,g]+[R_1f,g]+[f,R_1g]=g(1)^*\mathcal{J} f(1).$$
Let $f(p)=F(p)\xi$ and $g(p)=F(p)\eta$ with $\xi,\eta\in\mathbb H^N$. We have $$f(1)=C(\mathcal{I}_N-A)^{-1}\xi \quad\text{and }\quad g(1)=C(\mathcal{I}_N-A)^{-1}\eta.$$ These equations together with show that is equivalent to $$P+P(\mathcal{I}_N-A)^{-1}A+A^*(\mathcal{I}_N-A)^{-*}P=(\mathcal{I}_N-A)^{-*}C^*\mathcal{J}C(\mathcal{I}_N-A).$$ Multiplying this equation by $\mathcal{I}_N-A^*$ on the left and by $\mathcal{I}_N-A$ on the right we get the equivalent equation .
[Equation corresponds to a special case of a structural identity which characterizes $\mathcal H(\Theta)$ spaces in the complex setting. A corresponding identity in the half place case was first introduced by de Branges, see [@dbhsaf1], and improved by Rovnyak [@HM]. Ball introduced the corresponding identity in the setting of the open unit disk and proved the corresponding structure theorem. See [@ball-contrac]. See e.g. [@ad-jfa p. 17] for further discussions on this topic. ]{}
Let $a$ and $b$ be slice hyperholomorphic functions defined in an axially symmetric s-domain containing $p=1$. Then, $$\label{opera-garnier}
R_1(a\star b)(p)=\left(R_1a(p)\right)b(1)+(a\star R_1b)(p) .$$
By the Identity Principle, see [@MR2752913 Theorem 4.2.4] the equality holds if and only if it holds for the restrictions to a complex plane $\mathbb C_I$ i.e., using the notations in Section 2, if and only if $$\label{R1equality}
(R_1(a\star b))_{I}(z)=
\left(R_1a(z)\right)_{I}b(1)+(a\star R_1b)_{I}(z), \quad z\in\mathbb C_I.$$ Let $J\in\mathbb S$ be such that $J$ is orthogonal to $I$ and assume that $$a_{I}(z)=F(z)+G(z)J,\qquad b_{I}(z)=H(z)+L(z)J .$$ Let us compute the left-hand side of (\[R1equality\]), using the fact that $(R_1(a\star b))_{I}(z)= R_1((a\star b)_{I})$ and formula (\[starproduct\]): $$\begin{split}
R_1((a\star b)_{I})&=R_1\left(F(z)H(z)-G(z)\overline{L(\bar z)}+(G(z)\overline{H(\bar z)}+F(z)L(z))J\right) \\
&=(z-1)^{-1}\left(F(z)H(z)-G(z)\overline{L(\bar z)}+(G(z)\overline{H(\bar z)}+F(z)L(z))J\right.\\
&\left.-F(1)H(1)+G(1)\overline{L(1)}-(G(1)\overline{H(1)}+F(1)L(1))J)\right).
\end{split}$$ At the right hand side of (\[R1equality\]) we have $\left(R_1a(z)\right)_{I}b(1)=\left(R_1a_{I}(z)\right)b(1)$ which can be written as $$\begin{split}
\left(R_1a_{I}(z)\right)&b(1)=\left((z-1)^{-1}(F(z)+G(z)J-F(1)-G(1)J)\right)(H(1)+L(1)J)\\
&=(z-1)^{-1}\left(F(z)H(1)+F(z)L(1)J+G(z)\overline{H(1)}J-G(z)\overline{L(1)}-F(1)H(1)\right.\\
&\left.-F(1)L(1)J-G(1)\overline{H(1)}J+G(1)\overline{L(1)}\right),
\end{split}$$ moreover $$\begin{split}(a\star R_1b)_{I}(z)&=(F(z)+G(z)J)\star\left((z-1)^{-1}(H(z)+L(z)J-H(1)-L(1)J)\right)\\
&= (z-1)^{-1} (F(z)+G(z)J) \star (H(z)+L(z)J-H(1)-L(1)J)\\
&=(z-1)^{-1}(F(z)H(z)-G(z)\overline{L(\bar z)}+(G(z)\overline{H(\bar z)}+F(z)L(z))J )\\
& -F(z)H(1)+G(z)\overline{L(1)}-(G(z)\overline{H(1)}+F(z)L(1))J
\end{split}$$ from which the equality follows.
We will also need the following result, well known in the complex case. We refer to [@aron; @schwartz] for more information and to [@fw] for connections with operator ranges.
\[new\] Let $K_1(p,q)$ and $K_2(p,q)$ be two $\mathbb H$-valued functions positive definite in a set $\Omega$ and assume that the corresponding reproducing kernel Hilbert spaces have a zero intersection. Then the sum $$\mathcal H(K_1+K_2)=\mathcal H(K_1)+\mathcal H(K_2)$$ is orthogonal.
Let $K=K_1+K_2$. The linear relation in $\mathcal H(K)\times (\mathcal H(K_1)\times \mathcal H(K_2))$ spanned by the pairs $$(K(p,q), (K_1(p,q),K_2(p,q))),\quad q\in\Omega,$$ is densely defined and isometric. It therefore extends to the graph of an everywhere defined isometry, which we will call $T$. See [@MR3127378 Theorem 7.2]. From $$\begin{split}
(T^*(f_1,f_2))(q)&=\langle T^*(f_1,f_2),K(p,q)\rangle_{\mathcal H(K)}\\
&=\langle (f_1,f_2), TK(p,q)\rangle_{\mathcal H(K_1)\times \mathcal H(K_2)}\\
&=\langle f_1, K_1(p,q)\rangle_{\mathcal H(K_1)}+\langle f_2, K_2(p,q)\rangle_{\mathcal H(K_2)}\\
&=f_1(q )+f_2(q),\quad q\in\Omega,
\end{split}$$ we see that $\ker T^*=\left\{0\right\}$ since $\mathcal H(K_1)\cap \mathcal H(K_2)=\left\{0\right\}$. Thus $T$ is unitary and the result follows then easily.
We proceed in a number of steps. Recall that $r={\rm rank}\, P$.\
STEP 1: [*Assume $r=0$. Then, $s_1=\cdots=s_N$ and Problem \[pb1\] is solvable with the unique solution the constant unitary function $s(p)\equiv s_1$ .*]{}\
The matrix $P=0$, and equation imply that $C^*\mathcal{J}C=0$, and so $1-s_u\overline{s_v}=0$ for $u\not= v\in\left\{
1,\ldots, N\right\}$. Thus $s_1=\cdots=s_N$ and the function $s(p)\equiv s_1$ is clearly a solution. Assume that $s$ is a (possibly different) solution of Problem \[pb1\]. The map $M_{\begin{pmatrix}1&-s\end{pmatrix}}$ of slice multiplication by $\begin{pmatrix}1&-s(p)\end{pmatrix}$ is a contraction from $\mathcal M$ into $\mathcal H(s)$ (see the second step in the proof of Theorem \[tm1\]). Thus $$\begin{pmatrix}1&-s(p)\end{pmatrix}\star f_u(p)\equiv 0,\quad u=1,\ldots, N,$$ that is $g_u(p)\equiv 0$, where $f_u$ and $g_u$ have been defined in and respectively. From we have (for $|p|<1$) $$g_u(p)=\left((1-s(p)\overline{s_u})-\overline{p}(1-s(p)\overline{s_u})\overline{p_u}\right)(1-2{\rm Re}(p)\overline{p_u}+
|p|^2p_u^2)^{-1}$$ since $$1-2{\rm Re}(p)\overline{p_u}+|p|^2p_u^2\not=0$$ for $|p|<1$. Hence $$(1-s(p)\overline{s_u})=\overline{p}(1-s(p)\overline{s_u})\overline{p_u},\quad\forall p\in\mathbb H_1.$$ Taking absolute values of both sides of this equality we get $1-s(p)\overline{s_u}\equiv 0$, and so $s(p)\equiv s_u$. This ends the proof of Step 1.\
In the rest of the proof we assume $r > 0$. By reindexing the interpolating nodes we can assume that the principal minor of order $r$ is invertible. Thus the corresponding space is a $\mathcal H(\Theta_r)$ space, and we can write $$\mathcal M=\mathcal H(\Theta_r)\oplus\Theta_r\star \mathcal N.$$
STEP 2: [*The elements of $\mathcal N$ are slice hyperholomorphic in a neighborhood of $p=1$ and $R_1\mathcal N\subset\mathcal N$.*]{}\
We follow the argument in Step 1 in the proof of Theorem 3.1 in [@ad-laa-herm] (see p. 153). From we have $$(R_1(\Theta_r\star n))(p)=(R_1\Theta_r)(p)n(1)+(\Theta_r\star
R_1n)(p). \label{newlabel}$$ To prove that $R_1n\in\mathcal N$ we show that $$\label{au-clair-de-la-lune}
[(R_1(\Theta_r\star
n))(p)-(R_1\Theta_r)(p)n(1),g]_{\mathcal M}=0,\quad \forall
g\in\mathcal H(\Theta_r).$$ Using we have $$\begin{split}
[(R_1(\Theta_r\star n))(p),g]_{\mathcal
M}&=g(1)^*\mathcal{J}(R_1(\Theta_r\star n))(1)-[\Theta_r\star
n,g]_{\mathcal M}-[\Theta_r\star n,
R_1g]_{\mathcal M}\\
&=g(1)^*\mathcal{J}(R_1(\Theta_r\star n))(1)
\end{split}$$ since $$[\Theta_r\star n,g]_{\mathcal M}=0\quad\text{and}\quad
[\Theta_r\star n, R_1g]_{\mathcal M}=0,$$ where the second equality follows from $R_1g\in\mathcal M$. Moreover, for real $p=x$ we have the equality of real analytic functions $$(R_1\Theta_r)(x)=-K_{\Theta_r}(x,1)\mathcal{J}\Theta_r(1)^*,$$ and so, by slice hyperholomorphic extension, see [@2013arXiv1310.1035A Remark 2.18], in a suitable neighborhood of $p=1$ we have $$(R_1\Theta_r)(p)=-K_{\Theta_r}(p,1)\mathcal{J}\Theta_r(1)^*.$$ Note that $\Theta_r(1)$ is the identity. Thus $$\begin{split}
[(R_1\Theta_r)(p)n(1),g]_{\mathcal
M}&=-[K_{\Theta_r}(p,1)\mathcal{J}\Theta_r(1)^*n(1),g]_{\mathcal M}\\
&=-(n(1)^*\Theta_r(1)^*g(1)^*)\\
&=-g(1)^*\Theta_r(1)\mathcal{J}n(1),
\end{split}$$ and so is in force. This ends the proof of the second step.
Endow now $\mathcal N$ with the Hermitian form $$[n_1, n_2]_{\mathcal N}= [\Theta_r\star n_1,\Theta_r\star
n_2]_{\mathcal M}.$$
STEP 3: [*There exist matrices $(G,T)\in\mathbb H^{2\times
(N-r)}\times\mathbb H^{(N-r)\times (N-r)}$ such that $\mathcal N$ is spanned by the columns of the function $F_{\mathcal
N}(p)=G\star(\mathcal{I}_{N-r}-pT)^{-\star}$ and moreover for $\xi\in\mathbb H^{N-r}$. $$F_{\mathcal N}(p)\xi\equiv 0\quad\Longrightarrow\quad \xi=0.$$* ]{}
Indeed, we first note that the elements of $\mathcal N$ are well defined at $p=1$ since $\Theta$ is invertible at $p=1$ (see also the formulas in [@ad-laa-herm Theorem 3.3 (2)] ). Let $F_{\mathcal N}(p)$ be built from the columns of a basis of $\mathcal N$ and note that there exists $B\in\mathbb H^{(N-r)\times (N-r)}$ such that $$R_1F_{\mathcal N}=F_{\mathcal N}B.$$ Restricting to $p =x$, where $x$ is real, we have $$\frac{F(x)-F(1)}{x-1}=F(x)B,$$ and so $$\label{le-louvre}
F(x)(\mathcal{I}_{N-r}+B-xB)=F(1).$$ We claim that $\mathcal{I}_{N-r}+B$ is invertible. Let $\xi\in\mathbb H^{N-r}$ be such that $B\xi=-\xi$. Then, implies that $$xF(x)\xi=F(1)\xi,\quad x\in(-1,1).$$ Thus $F(1)\xi=0$ (by setting $x=0$) and so $F(x)\xi=0$ and so $\xi=0$. Hence $$F(x)=F(1)(\mathcal{I}_{N-r}+B)^{-1}(\mathcal{I}_{N-r}-xB(\mathcal{I}_{N-r}+B)^{-1})^{-1},$$ and the result follows.\
The following step is [@ad-laa-herm Step 2 of proof of Theorem 3.1, p. 154]. The proof uses and is similar to the above arguments.\
STEP 3: [*The space $\mathcal N$ is neutral and $G^*\mathcal{J}G=0$.*]{}\
$\mathcal N$ is neutral by construction since $r= {\rm rank} \, P$. We first show that the inner product in $\mathcal N$ satisfies . We may proceed as in [@ad-laa-herm p. 154] and using in $\mathcal
M$ we have for $n_1,n_2\in\mathcal M$: $$\begin{split}
[R_1n_1,n_2]_{\mathcal N}&=[\Theta\star R_1n_1,\Theta\star
n_2]_{\mathcal M}\\
&=[R_1(\Theta\star n_1),\Theta\star n_2]_{\mathcal M}-
[(R_1\Theta)(n_1(1)),\Theta\star n_2]_{\mathcal M}\quad
\text{(where we used \eqref{newlabel})}\\
&=[R_1(\Theta\star n_1),\Theta\star n_2]_{\mathcal M}
\end{split}$$ since $(R_1\Theta)(n_1(1))\in\mathcal H(\Theta)$, and so $[(R_1\Theta)(n_1(1)),\Theta\star n_2]_{\mathcal M}=0$.
Similarly, $$\begin{split}
[n_1,R_1n_2]_{\mathcal N}&= [\Theta\star n_1,\Theta\star
R_1n_2]_{\mathcal M}\\
&=[\Theta\star n_1,(R_1\Theta)(n_2(1))]_{\mathcal M}-
[\Theta\star n_1,(R_1\Theta)(n_2(1))]_{\mathcal M}\\
&=[\Theta\star n_1,(R_1\Theta)(n_2(1))]_{\mathcal M}.
\end{split}$$ Thus, with $m_1=\Theta\star n_1$ and $m_2=\Theta\star n_2$,
$$\begin{split}
[n_1,n_2]_{\mathcal N}+[R_1n_1,n_2]_{\mathcal N}+
[n_1,R_1n_2]_{\mathcal N}&=[m_1,m_2]_{\mathcal
M}+[R_1m_1,m_2]_{\mathcal M}+ [m_1,R_1m_2]_{\mathcal M}\\
&=m_2(1)^*\mathcal{J}m_1(1)\\
&=n_2(1)\mathcal{J}n_1(1)
\end{split}$$ since $m_v(1)=(\Theta\star n_v)(1)=\Theta(1)n_v(1)$ for $v=1,2$ and $\Theta(1)^*\mathcal{J}\Theta(1)=\mathcal{J}$.\
Proceeding as in Step 1 it follows that $$P_{\mathcal N}-T^*P_{\mathcal N}T=G^*\mathcal{J}G,$$ and so $G^*\mathcal{J}G=0$.\
STEP 4: [*Problem \[pb1\] has at most one solution.*]{}\
Let $$\Theta_r(p)=\begin{pmatrix}a_r(p)&b_r(p)\\c_r(p)&d_r(p)\end{pmatrix}.$$ From the study of the nondegenerate case, we know that, under the assumptions that ensure the existence of a solution, any solution is of the form $$\label{ttheta} s(p)=(a_r(p)\star e(p)+b_r(p))\star(c_r(p)\star
e(p)+d_r(p))^{-\star},$$ for some Schur function $e$. Furthermore as in step 1, for every $n\in\mathcal N$ we have $$\begin{pmatrix}1&-s\end{pmatrix}\star\Theta_r\star n\equiv 0.$$ Thus $$(a-sc)\star \begin{pmatrix}1&-e\end{pmatrix}\star n\equiv 0,$$ and so $$\begin{pmatrix}1&-e\end{pmatrix}\star n\equiv 0.$$ Since $G^*\mathcal{J}G=0$ we conclude in the way as in step 1. Indeed, let $$G=\begin{pmatrix}h_1 & \ldots &h_{N-r}\\k_1 & \ldots
&k_{N-r}\end{pmatrix}.$$ At least one of the $h_u$ or $k_u$ is different from $0$ and $G^*\mathcal{J}G=0$ implies that $$\overline{h_u}h_v=\overline{k_u}k_v,\quad\forall u,v=1,\ldots,
N-r,$$ and so $e$ is a unitary constant.\
We now show that the solution, when it exists, is a finite Blaschke product.\
STEP 5: [*Let $s$ be given by . Then the associated space $\mathcal H(s)$ is finite dimensional.*]{}
This follows from $$K_s(p,q)=\begin{pmatrix}1 &-s(p)\end{pmatrix}\star
K_{\Theta_r}(p,q)\star_r
\begin{pmatrix}1 \\ \overline{s(q)}\end{pmatrix}+
\underbrace{\begin{pmatrix}1 &-s(p)\end{pmatrix}\star
\Theta_r(p)\mathcal J\Theta_r(q)^* \star_r
\begin{pmatrix}1 \\ \overline{s(q)}\end{pmatrix}}_{\text{is equal to $0$ since
$|e|=1$}},$$ where $K_{\Theta_r}$ is defined as in (with $\Theta_r$ in place of $\Theta$).\
STEP 6: [*The space $\mathcal H(s)$ contains an element of the form $$\label{sevres-babylone} f(p)=x\star(1-p\overline{a})^{-\star},$$ where $x\in\mathbb H$ and $a\in\mathbb B_1$.*]{}
We first recall that (see [@acs3 Theorem 7.1]) $$\|R_0f\|_{\mathcal H(s)}^2\le\|f\|_{\mathcal
H(s)}^2-|f(0)|^2,\quad\forall f\in\mathcal H(s). \label{R0R0}$$ Here, the space $\mathcal H(s)$ is finite dimensional and $R_0$ invariant. Thus $R_0$ has a right eigenvector $f$ with eigenvalue $\overline{a}$; see [@MR97h:15020 p. 36]. Any eigenvector of $R_0$ is of the form , and equation implies that $$\label{contrac}
\|f\|^2\le \frac{|f(0)|^2}{1-|a|^2}.$$ We will see at the end of the proof of Step 8 that equality in fact holds in .\
STEP 7: [*It holds that $s(a)=0$.*]{}\
From [@acs1 p. 282-283] it follows that the span of $f$ endowed with the norm $\|f\|^2=\frac{|f(0)|^2}{1-|a|^2}$ is equal to $\mathcal
H(b_a)$, where $b_a$ is a Blaschke factor, see . From we get that $\mathcal
H(b_a)$ is contractively included in $\mathcal H(s)$ and from [@acs1 Lemme 5.1] we then have that the kernel $$\label{place-voltaire} K_s(p,q)-K_{b_a}(p,q)=\sum_{t=0}^\infty
p^t(b_a(p)\overline{b_a(q)}-s(p)\overline{s(q)})\overline{q}^t$$ is positive definite in $\mathbb B_1$. But $b_a(a)=0$. Thus, setting $p=q=a$ in leads to $s(a)=0$.\
STEP 8: [*We can write $s=b_a\star \sigma_1$, where $\sigma_1$ is a Schur function.*]{}\
In the argument we make use of the Hardy space $\mathbf H_2(\mathbb
B_1)$ which is the reproducing kernel Hilbert space with reproducing kernel $$(1-p\overline{q})^{-\star}=\sum_{t=0}^\infty p^t\overline{q}^t.$$ Note that this is the kernel $k_s$ with $s(p)\equiv 0$. For more information on this space we refer to [@2013arXiv1308.2658A; @acs1].\
Since a Schur function is bounded in modulus and thus belongs to the space $\mathbf H_2(\mathbb B_1)$ (see [@2013arXiv1308.2658A]), the representation $s=b_a\star \sigma_1$ with $\sigma_1\in\mathbf H_2(\mathbb B_1)$, follows from [@MR3127378 Proof of Theorem 6.2, p. 109]. To see that $\sigma_1$ is a Schur multiplier we note that $$\label{decom}
K_s(p,q)-K_{b_a}(p,q)=b_a(p)\star K_{\sigma_1}(p,q)\star_r
\overline{b_a(q)}$$ implies that $b_a(p)\star K_{\sigma_1}(p,q)\star_r \overline{b_a(q)}$ is positive definite in $\mathbb{B}_1$ and hence $K_{\sigma_1}(p,q)$ is as well by [@acs3 Proposition 5.3].\
STEP 9: [*It holds that ${\rm dim}\,(\mathcal H(\sigma_1))={\rm
dim}\,(\mathcal H(s))-1$.*]{}\
The decomposition gives the decomposition $$K_s(p,q)=K_{b_a}(p,q)+b_a(p)\star K_{\sigma_1}(p,q)\star_r \overline{b_a(q)}.$$ The corresponding reproducing kernel spaces do not intersect. Indeed, all elements in the reproducing kernel Hilbert space with reproducing kernel $b_a(p)\star K_{\sigma_1}(p,q)\star_r \overline{b_a(q)}$ vanish at the point $a$ while non zero elements in $\mathcal H(b_a)$ do not vanish. So the decomposition is orthogonal in $\mathcal H(s)$ by Theorem \[new\], and equality holds in . The claim on the dimensions follow.\
After a finite number of iterations, this procedure leads to a constant $\sigma_{\ell}$, for some positive integer $\ell$. This constant has to be unitary since the corresponding space $\mathcal H(\sigma_\ell)$ reduces to $\left\{0\right\}$, thus proving the theorem.
We conclude with two remarks and a corollary.
[Given a Blaschke factor the operator of multiplication by $b_a$ is an isometry from ${\bf H}_2(\mathbb{B}_1)$ into itself (see [@MR3127378 Theorem 5.17, p. 106]), and so is the operator of multiplication by a finite Blaschke product $B$. The degree of the Blaschke product is the dimension of the space $\mathbf H_2(\mathbb B_1)\ominus B \mathbf H_2(\mathbb B_1)$. Thus the previous argument shows in fact that $\mathcal H(s)$ is isometrically included inside $\mathbf
H_2(\mathbb B_1)$ and that $\mathcal H(s)=\mathbf H_2(\mathbb
B_1)\ominus M_s \mathbf H_2(\mathbb B_1)$.]{}
One can plug a unitary constant $e$ also in the linear fractional transformation and the same arguments lead to:
If Problem \[pb1\] has a solution, it is a Blaschke product of degree ${\rm rank}\, P$.
[The arguments in Steps 5-7 take only into account the fact that the space $\mathcal H(\Theta)$ is finite dimensional and that $e$ is a unitary constant. In particular, they also apply in the setting of [@2013arXiv1308.2658A], and in that paper too, the solution of the interpolation problem is a Blaschke product of degree ${\rm rank}\, P$ when the Pick matrix is degenerate. ]{}
An analogue of Carathéodory’s theorem in the quaternionic setting {#remarks}
=================================================================
Recall first that Carathéodory’s theorem states the following (see for instance [@burckel pp. 203-205], [@sarason94 p. 48]). We write the result for a radial limit, but the result holds in fact for a non tangential limit.
Let $s(z)$ be a Schur function and let $e^{it_0}$ be a point on the unit circle such that $$\liminf_{\substack{r\rightarrow
1\\r\in(0,1)}}\frac{1-|s(re^{it_0})|}{1-r}<\infty.$$ Then, the limits $$c=\lim_{\substack{r\rightarrow 1\\r\in(0,1)}}s(re^{it_0})\quad
and\quad \lim_{\substack{r\rightarrow
1\\r\in(0,1)}}\frac{1-s(re^{it_0})\overline{c}}{1-r}$$ exist, and the second one is positive.
This result plays an important role in the classical boundary interpolation problem for Schur functions. See for instance [@ADLW], [@MR98m:47017].\
We prove a related result in the setting of slice-hyperholomorphic functions. The condition will hold particular for rational functions $s$, as is proved using a realization of $s$ (see [@acs1] for the latter).
Let $s$ be a slice hyperholomorphic Schur function, and assume that at some point $p_u$ of modulus $1$ we have $$\label{place-de-la-republique}
\sup_{r\in(0,1)}\frac{1-|s(rp_u)|^2}{1-r^2}<\infty .$$ Assume moreover that the function $r\mapsto s(rp_u)$ has a development in series with respect to the [*real*]{} variable $r$ at $r=1$: $$s(rp_u)=s_u+(r-1)a_u+O(r-1)^2. \label{place-de-l-opera}$$ Then $$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}}
\sum_{t=0}^\infty
r^tp_u^t(1-s(r_np_u)\overline{s_u})\overline{p_u}^t=
(a_u\overline{s_u}-\overline{p_u}a_u\overline{s_u}\, \overline{p_u})(1-\overline{p_u}^2)^{-1}.$$
In view of , the family of functions $K_s(\cdot, rp_u)$ has a weakly convergent subsequence. Since weak convergence implies pointwise convergence the weak limit is readily seen to be the function $g_u$. Thus $$0\le \langle g_u,g_u \rangle_{\mathcal H(s)}=\lim_{n\rightarrow\infty}
\langle g_u,K_s(\cdot,r_np_u) \rangle_{\mathcal
H(s)}=\lim_{n\rightarrow\infty}g_u(r_np_u),$$ where $(r_n)_{n\in\mathbb N}$ is a sequence of numbers in $(0,1)$ with limit equal to $1$. Hence we have that $$\lim_{n\rightarrow\infty}\sum_{t=0}^\infty r_n^tp_u^t(1-s(r_np_u)\overline{s_u})\overline{p_u}^t\ge 0.$$ Using we have: $$\begin{split}
\sum_{t=0}^\infty
r^tp_u^t(1-s(r_np_u)\overline{s_u})\overline{p_u}^t&=
\sum_{t=0}^\infty r^tp_u^t((r-1)a_u\overline{s_u}+O(r-1)^2)\overline{p_u}^t\\
&=((r-1)a_u\overline{s_u}-r\overline{p_u}(r-1)a_u\overline{s_u}\,
\overline{p_u})(1-r)^{-1}(1-r\overline{p_u}^2)^{-1}+\\
&\hspace{5mm}+
\sum_{t=0}^\infty r^tp_u^nO(r-1)^2\overline{p_u}^t\\
&=(a_u\overline{s_u}-r\overline{p_u}a_u\overline{s_u}\, \overline{p_u})
(1-r\overline{p_u}^2)^{-1}+\\
&\hspace{5mm}+ \sum_{t=0}^\infty r^tp_u^nO(r-1)^2\overline{p_u}^t.
\end{split}$$ This expression tends to $$\label{barbes}
(a_u\overline{s_u}-\overline{p_u}a_u\overline{s_u}\, \overline{p_u})(1-\overline{p_u}^2)^{-1},$$ as $r\rightarrow 1$.
[The example $s(p)=\frac{1+pa}{2}$, where $a\in\mathbb B_1$ is such that $ap_u\not=p_ua$, shows that is different, in general, from $a_u\overline{s_u}$.]{}
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$ ($S=\frac{1}{2}$) is a promising new candidate for an ideal Kagomé Heisenberg antiferromagnet, because there is no magnetic phase transition down to $\sim$50 mK. We investigated its local magnetic and lattice environments with NMR techniques. We demonstrate that the intrinsic local spin susceptibility [*decreases*]{} toward $T=0$, but that slow freezing of the lattice near $\sim$50 K, presumably associated with OH bonds, contributes to a large increase of local spin susceptibility and its distribution. Spin dynamics near $T=0$ obey a power-law behavior in high magnetic fields.'
author:
- 'T. Imai$^{1,2}$, E. A. Nytko$^{3}$, B.M. Bartlett$^{3}$, M.P. Shores$^{3}$, and D. G. Nocera$^{3}$'
title: '$^{63}$Cu, $^{35}$Cl, and $^{1}$H NMR in the $S=\frac{1}{2}$ Kagomé Lattice ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$'
---
A major challenge in condensed matter physics today is identifying a model material for investigating [*spin liquid*]{} [@Anderson; @PhysicsToday]. Searching for exotic electronic states without magnetic long range order, such as Kagomé Heisenberg antiferromagnets, constitutes a common thread in a wide range of research fields, from high temperature superconductivity to low dimensional quantum magnetism. Over the last decade, many candidate materials have been investigated as model systems for a Kagomé lattice [@SrCr; @Jarosite; @CuV; @Review]. However, they mostly exhibit a magnetically ordered or spin-glass-like state at low temperatures. A recent breakthrough in the hunt for a spin liquid state[@PhysicsToday] is the successful synthesis [@Shores] and characterization [@Helton] of ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$ (herbertsmithite), a chemically pure spin $S=\frac{1}{2}$ Kagomé lattice. As shown in Fig. 1, three Cu$^{2+}$ ions form a triangle, and a network of corner-shared triangles form a Kagomé lattice. The $S=\frac{1}{2}$ spins on Cu sites are mutually frustrated by antiferromagnetic super-exchange interaction $J\sim 170$ K [@Helton; @Singh1], hence the possibility of a spin liquid ground state.
Recent measurements of ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$ with bulk magnetic susceptibility, $\chi_{bulk}$[@Helton], specific heat[@Helton], neutron scattering on powders[@Helton], $\mu$SR[@Ofer; @Mendels], and $^{35}$Cl NMR[@Ofer] have established that ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$ remains paramagnetic down to at least $\sim50~mK$ with no evidence of magnetic long range order. These findings indeed point towards the possible realization of a frustrated spin liquid state with the Kagomé symmetry. However, very little is known beyond the paramagnetic nature of the ground state. For example, the bulk averaged susceptibility, $\chi_{bulk}$, reveals a mysterious sharp [*increase*]{} below $\sim 50$ K [@Helton]. This clearly contradicts the predictions of various theoretical calculations: series expansions predict a [*decrease*]{} of $\chi_{bulk}$ below $T\sim J/6$ with a gap[@Elstner; @Mila], while the recent Dirac Fermion model predicts linear behavior in $T$ towards $T=0$ [@Lee]. Does this apparent contradiction mean that ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$ is not a good Kagomé model system after all, or that extrinsic effects other than Kagomé Heisenberg interaction, such as mixing of Zn ($S=0$) into Cu ($S=\frac{1}{2}$) sites[@Vries; @Bert] and Dzyaloshinsky-Moriya (DM) interactions[@Singh1; @Singh2], simply mask the intrinsic Kagomé behavior below $\sim 50~K$? What about spin dynamics? Do spin fluctuations slow down toward a critical point, or are they gapped [@Elstner; @Mila; @Lee]?
In this [*Letter*]{}, we report a $^{63}$Cu, $^{35}$Cl, and $^{1}$H NMR investigation of ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$ for a broad range of magnetic fields and frequencies. Taking full advantage of the local nature of NMR techniques, we uncover hitherto unknown properties of ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$. First, from the observation of the broadening of $^{35}$Cl NMR lineshapes, we will demonstrate that local spin susceptibility, $\chi_{loc}$, has a large distribution throughout the sample. Moreover, the smallest components of $\chi_{loc}$ actually [*saturate*]{} and even [*decrease*]{} with $T$ below $T\sim 0.2J$, even though the bulk averaged $\chi_{bulk}$ increases as $\sim 1/T$. The observed decrease of $\chi_{loc}$ is precisely what the intrinsic spin susceptibility of a Kagomé Heisenberg antiferromagnet is expected to show. Second, from the comparison of $^{1}$H and $^{35}$Cl nuclear spin-lattice relaxation rates, $^{1,35}(1/T_{1})$, we present unambiguous evidence for slow freezing of the lattice near $\sim50~K$, most likely due to orientational disorder of OH bonds. We suggest that this subtle freezing of lattice distortion enhances the DM interactions, and is key to understanding the aforementioned upturn of bulk-averaged spin susceptibility, $\chi_{bulk}$, below $\sim50~K$ that masks the intrinsic Kagomé behavior of $\chi_{loc}$. Third, we demonstrate from the measurements of the $^{63}$Cu nuclear spin-lattice relaxation rate, $^{63}(1/T_{1})$, that in the presence of a high magnetic field the low frequency Cu spin fluctuations [*grow*]{} without a gap below $\sim 30$ K satisfying a simple power-law.
In Fig. 2, we show representative $^{35}$Cl NMR lineshapes. For these measurements, we cured a powder sample in glue in a magnetic field of 9 Tesla. From powder x-ray diffraction measurements, we confirmed that approximately 20% of the powder is uniaxially aligned along the c-axis. In fact, we observe a sharp c-axis central peak near 35.02 MHz (marked as [*B//c*]{} in Fig. 2) arising from particles oriented along the c-axis. The “double horns” marked as \#1 and \#2 are split by the nuclear quadrupole interaction, and arise from the randomly oriented portion of the powder (i.e. 80% of the sample) [@Ofer]. Notice that the whole $^{35}$Cl NMR lineshape begins to tail-off toward lower frequencies below $\sim$50 K. The resonance frequency of the sharp c-axis central peak and its distribution depends on the NMR Knight shift, $^{35}K$, induced by $\chi_{loc}$. Hence the observed line broadening implies that [*$\chi_{loc}$ varies depending on the location within the sample*]{} below $\sim 50~K$.
In Fig. 3, we summarize the $^{35}$Cl NMR Knight shifts $^{35}K$ and $^{35}K_{1/2}$ deduced from the lineshapes, together with $\chi_{bulk}$ as observed by SQUID. $^{35}K$ corresponds to the central peak above $\sim$ 45 K as determined by FFT techniques. Below $\sim$ 45 K, where the central peak is smeared out by line broadening, we determined $^{35}K$ as the higher frequency edge of the central peak from point-by-point measurements, [*i.e.*]{} $^{35}K$ represents the smallest component of the distributed $\chi_{loc}$. $^{35}K_{1/2}$ corresponds to the half-intensity position of the central peak on the lower frequency side of the spectrum. Quite generally, $^{35}K = A_{hf}\chi_{local} + ^{35}K_{chem}$, where $A_{hf}$ is the magnetic hyperfine interaction between $^{35}$Cl nuclear spin and nearby Cu electron spins, and $^{35}K_{chem}$ is a very small, temperature independent chemical shift. In the present case, from the comparison with $\chi_{bulk}$, we can estimate $A_{hf}\sim -3.7 \pm 0.7$ kOe/$\mu_{B}$. The negative sign of $A_{hf}$ makes the overall sign of $^{35}K$ negative. Accordingly, we have inverted the vertical scale of Fig. 3.
We wish to comment on two important aspects of Fig. 3. First, $^{35}K$ follows Curie-Weiss behavior all the way from 295 K down to $\sim$25 K. This clearly differs from $\chi_{bulk}$ which begins to deviate from Curie-Weiss behavior below temperatures as high as $\sim$150 K [@Helton]. On the other hand, series-expansion calculations indicate that the Kagomé lattice follows Curie-Weiss behavior down to $T\sim J/6\sim 25$ K [@Elstner]. Our Knight shift data demonstrate that $\chi_{local}$ of some ideal segments of the Cu$^{2+}$ Kagomé lattice in this material indeed show such behavior. Theoretical models also predict that below $T\sim J/6\sim 25$ K, $\chi_{bulk}$ begins to decrease exponentially with a small gap[@Elstner], or linearly[@Lee]. As shown in Fig. 2, the $^{35}$Cl NMR lineshape begins to transfer some spectral weight to higher frequencies below 25 K. Recalling that $A_{hf}$ is negative, [*this is consistent with vanishing spin susceptibility, $\chi_{local}$, near $T=0$ for some parts of the Kagomé lattice*]{}. After the initial submission of this [*Letter*]{}, Olariu [*et al.*]{} also observed a similar decrease of $^{17}$O NMR Knight shift and arrived at the same conclusion [@Olariu].
Another important aspect of Fig. 3 is that $^{35}K_{1/2}$ begins to deviate from the aforementioned Curie-Weiss fit in a manner similar to $\chi_{bulk}$, as the $^{35}$Cl NMR line gradually broadens to lower frequencies. It is important to realize that $^{35}(1/T_{1})$ also begins to increase in the same temperature range below $\sim$150 K (see Fig. 4). Below 50 K, where $^{35}(1/T_{1})$ shows a peak, the $^{35}$Cl NMR line shows a dramatic broadening to lower frequencies. Fig. 2 and Fig. 3 establish that $^{35}K_{1/2}$ follows the same trend as $\chi_{bulk}$, i.e. [*some segments of the Kagomé lattice have large and distributed local spin susceptibility $\chi_{local}$*]{}, and [their temperature dependence is different from the smaller $\chi_{local}$ as represented by $^{35}K$.]{} $\chi_{bulk}$ simply represents a bulk average of $\chi_{local}$.
In passing, we recall that earlier $\mu$SR Knight shift $K_{\mu SR}$ measurements by Ofer [*et al.*]{} [@Ofer] showed identical behavior between $K_{\mu SR}$ and $\chi_{bulk}$. They concluded that the upturn of $\chi_{bulk}$ below 50 K is not caused by impurity spins but is a bulk phenomenon. Our new results in Fig. 3 do not contradict these $\mu$SR data. $K_{\mu SR}$ was deduced by assuming a Gaussian distribution of $\chi_{local}$, hence by default $K_{\mu SR}$ represents the central value of the presumed Gaussian distribution. That explains why $K_{\mu SR}$ shows behavior similar to $\chi_{bulk}$ and $^{35}K_{1/2}$.
Next, we turn our attention to the dynamics of lattice and spin degrees of freedom. Fig. 4 shows the temperature dependence of the $^{35}$Cl nuclear spin-lattice relaxation rate, $^{35}(1/T_{1})$, measured at the central peak frequency in various magnetic fields, $B$. We also plot $^{1}(1/T_{1})$ for $^1$H NMR in 0.9 Tesla, and $^{63}(1/T_{1})$ for $^{63}$Cu NMR in 8 Tesla. We have overlaid $^{1,63}(1/T_{1})$ on $^{35}(1/T_{1})$ measured in comparable magnetic fields by scaling the vertical axis.
We can draw a number of conclusions from Fig. 4. First, let us focus on $T$ and $B$ independent results of $^{35}(1/T_{1})$ above $\sim$150 K. This high temperature regime is easily understandable within Moriya’s theory for the exchange narrowing limit of Heisenberg antiferromagnets, where we should expect $^{35}(1/T_{1})_{exc}\sim A_{hf}^{2}/J = constant$ for $T>J\sim 170$ [@MoriyaExc]. If we assume $J\sim170$K and $A_{hf}\sim -4$ kOe/$\mu_{B}$, we can estimate $^{35}(1/T_{1})_{exc}\sim 4$ sec$^{-1}$. This is in excellent agreement with our result.
Another important feature is that $^{35}(1/T_{1})$ begins to increase below $\sim$150 K and peaks near $\sim 50~K$. Since $^{35}$Cl is a quadrupolar nucleus with nuclear spin $I=\frac{3}{2}$, the observed enhancement may be caused by slow fluctuations of the lattice via nuclear quadrupole interactions, as well as by Cu spin fluctuations. To discern the two possibilities, we show $^{1}(1/T_{1})$ of $^1$H measured at 0.9 Tesla for comparison. The $^{1}(1/T_{1})$ data nicely interpolate $^{35}(1/T_{1})$ in the field-independent regime above $\sim$150 K and $^{35}(1/T_{1})$ measured at a comparable magnetic field (1 Tesla) at low-temperatures, without a peak near $\sim 50~K$. Since $^1$H has $I=\frac{1}{2}$, $^{1}(1/T_{1})$ has no contributions from lattice fluctuations. Therefore we conclude that [*the peak of $^{35}(1/T_{1})$ near $\sim 50~K$ arises from enhancement of lattice fluctuations at the NMR frequency*]{}. The peak of $^{35}(1/T_{1})$ shifts to progressively lower temperatures as we lower the $^{35}$Cl NMR frequency, from $50~K$ (35 MHz at 8.3 Tesla), $46~K$ (18 MHz at 4.4 Tesla) to $40~K$ (10 MHz at 2.4 Tesla). This means that the typical frequency scale of lattice fluctuations is 35 MHz at 50 K, 18 MHz at 46 K, and 10 MHz at 40 K, and the Kagomé lattice becomes static below 40 K. Given that no structural phase transition has been detected by x-ray and neutron scattering techniques, the observed freezing of the lattice near $\sim 50~K$ must be a very subtle effect. In fact, our careful measurements of the quadrupole split $\pm \frac{1}{2}$ to $\pm \frac{3}{2}$ satellite transitions of $^{35}$ Cl NMR didn’t detect any noticeable changes, either. All pieces put together, we suggest that the lightest elements in the lattice, [*i.e.*]{} OH bonds, must be freezing with random orientations, with only subtle effects on heavier atoms.
Regardless of the exact nature of the freezing of the lattice, our observation provides a major clue to understanding the mysterious behaviors of $\chi_{bulk}$ and $\chi_{local}$. Recent structural studies revealed that up to 6 % of Cu sites may be occupied by Zn to create unpaired defect spins [@Vries]. There is no doubt that such antisite disorder would enhance $\chi_{bulk}$ and contribute to the large distribution of $\chi_{local}$ at low temperatures. However, it is important to realize that $\chi_{bulk}$ begins to deviate from high temperature Curie-Weiss behavior below $\sim$150 K [@Helton], exactly where $^{35}(1/T_{1})$ begins to grow due to slowing of lattice fluctuations toward $\sim 50~K$. Furthermore, the crossover between the two different Curie-Weiss behaviors of $\chi_{bulk}$ and $^{35}K_{1/2}$ takes place precisely when the lattice fluctuations die out below $\sim$50 K [@Helton]. Evidently, the anomaly of the lattice correlates well with that of the spins, suggesting it must be playing a major role in the deviation of spin susceptibility from the theoretically expected behavior of ideal Kagomé Heisenberg antiferromagnets. In fact, recent numerical simulations by Rigol and Singh showed that DM interactions can enhance spin susceptibility below $\sim$50 K [@Singh1; @Singh2]. Our experimental finding naturally fits with their theoretical picture: when OH bonds freeze with random orientation, the hexagonal symmetry of the Kagomé lattice slightly breaks down [*locally*]{}; this would progressively enhance the DM interaction from $\sim$150 K to $\sim$50 K, hence leading to an enhancement of $\chi_{bulk}$ and $^{35}K_{1/2}$, as well as a large distribution of $\chi_{local}$.
Finally, let us focus on the low temperature regime below 20 K, where only spin fluctuations contribute to $(1/T_{1})$. $^{35}(1/T_{1})$ at $\sim$4 K is independent of magnetic field from $B\sim$0.9 Tesla up to $B\sim$2 Tesla, but larger magnetic fields suppress $^{35}(1/T_{1})$. This suggests that unpaired paramagnetic spins are fluctuating in low magnetic fields, but applied magnetic field decouples the weak coupling between them. The exact origin of these paramagnetic spins is not clear, but one obvious possibility is unpaired free spins induced in the near neighbor sites of Zn ($S=0$) ions occupying the Cu sites, i.e. [*antisite disorder*]{} [@Vries; @Bert]. In fact, after the initial submission of the present work, Olariu et al. reported that approximately 30 % of the integrated intensity of $^{17}$O NMR signals is split off from the main line, which they attributed to the contribution of the 4 nearest neighbor O sites of Zn ions occupying the Cu sites [@Olariu]. Our observation of field dependence is also consistent with a recent report that the application of $B\sim8$ Tesla suppresses defect contributions to dc spin susceptibility in the low $T$ regime [@Bert].
All the $^{63,35,1}(1/T_{1})$ data measured at various magnetic fields and frequencies decrease towards the zero temperature limit. That is, we observe no hint of critical slowing down of spin fluctuations towards $T\sim0$, hence any magnetic critical point, if it exists, is still very far from our temperature region. This is in agreement with an earlier report based on a limited set of $^{35}$Cl NMR data[@Ofer], but in remarkable contrast with earlier NMR works on frustrated spin systems with Kagomé or triangular lattice geometry, where $(1/T_{1})$ data always show evidence for a magnetic long range order or freezing [@Review]. At $B\sim8$ Tesla, both $^{35}(1/T_{1})$ and $^{63}(1/T_{1})$ decrease toward $T=0$, obeying a simple power law, $^{63}(1/T_{1}) = T^{\eta}$ with $\eta \sim 0.5$. Since the slope of the log-log plot of $(1/T_{1})$ increases somewhat at higher fields, $\eta \sim 0.5$ may be somewhat underestimated. We note that spin fluctuation susceptibility is proportional to $^{63}(1/T_{1}T) = T^{\eta-1}$, hence spin fluctuations [*grow*]{} toward $T=0$. This is inconsistent with the exponential decrease expected for a gapped Kagomé lattice in some theoretical scenarios. On the other hand, recent field theoretical calculations based on the Dirac Fermion model predicted a power law, $^{63}(1/T_{1}) = T^{\eta}$, with unknown critical exponent $\eta$ [@Lee]. We note that the issue of power law behavior of spin dynamics is at the focus of the field theoretical approach toward spin liquid systems. It remains to be seen if the Dirac Fermion model would be consistent with $\eta \gtrsim 0.5$.
To summarize, we have presented a site-by-site picture of the new Kagomé material ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$ using NMR techniques. Our NMR data revealed that both local spin susceptibility and spin dynamics (in high magnetic fields) show aspects that are consistent with theoretical conjectures for ideal Kagome antiferromagnets, despite perturbations from lattice freezing and paramagnetic defects.
We thank Y.S. Lee, P.A. Lee, S. De’Soto, M. Rigol, R. Singh, and P. Mendels for helpful communications. The work at McMaster was supported by NSERC and CIFAR. We thank NSF for providing EAN and BMB with predoctoral fellowships, and DuPont for providing BMB with a Graduate Fellowship Award.
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Fig.1\
Left : A Kagomé lattice. Right : Cu$^{2+}$ Kagomé layer in ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$. Cl$^{-}$ site is above or below the middle of the Cu$^{2+}$ triangles.\
Fig.2\
$^{35}$Cl NMR lineshapes of the $I_{z}=+\frac{1}{2}$ to $-\frac{1}{2}$ central transition in 8.4 Tesla in a partially ($\sim$20 %) uniaxially aligned powder sample. The sharp peak near 35.02 MHz marked as “[*B//c*]{}” originates from the particles whose c-axis is aligned along the external magnetic field. Vertical lines specify the c-axis peak and edge corresponding to $^{35}K$. Vertical dashed lines specify the frequency corresponding to $^{35}K_{1/2}$.\
Fig.3\
$^{35}$Cl NMR Knight shift (left scale). $^{35}K$ (blue circles). $^{35}K_{1/2}$ (red triangles). The blue solid line is a fit to Curie-Weiss behavior, $^{35}K =
(22\pm7)/(T-\theta_{CW}) + ^{35}K_{chem}$, where $\theta_{CW} =
-240\pm 80$K. The constant background $^{35}K_{chem}=0.018\%$ is shown by a dashed line. $\chi_{bulk}$ measured by SQUID (dotted line) is also overlaid (right scale).\
Fig.4\
Temperature dependence of $^{35}$Cl NMR spin-lattice relaxation rate $^{35}(1/T_{1})$ at various magnetic fields (filled symbols). Solid line represents a fit to a power law, $^{35}(1/T_{1}) = T^{\eta }$ with $\eta = 0.47$ (8.3 T), 0.44 (4.4 T), 0.2 (2.4 T and 1.0 T). $^{1}$H relaxation rate in low field (0.9 T), $^{1}(1/T_{1})$, and $^{63}$Cu relaxation rate in high field (8 T), $^{63}(1/T_{1})$, are also superposed on $^{35}(1/T_{1})$ measured in comparable magnetic field.\
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We use *Hubble Space Telescope* (*HST*) to reach the end of the white dwarf (WD) cooling sequence (CS) in the solar-metallicity open cluster NGC6819. Our photometry and completeness tests show a sharp drop in the number of WDs along the CS at magnitudes fainter than $m_{\rm F606W} = 26.050 \pm 0.075$. This implies an age of $2.25\pm0.20$ Gyr, consistent with the age of $2.25\pm0.30$ Gyr obtained from fits to the main-sequence turn-off. The use of different WD cooling models and initial-final-mass relations have a minor impact the WD age estimate, at the level of $\sim$0.1 Gyr.\
As an important by-product of this investigation we also release, in electronic format, both the catalogue of all the detected sources and the atlases of the region (in two filters). Indeed, this patch of sky studied by *HST* (of size $\sim$70 arcmin$^2$) is entirely within the main $Kepler$-mission field, so the high-resolution images and deep catalogues will be particularly useful.
author:
- |
L. R. Bedin$^{1}$[^1], M. Salaris$^{2}$, J. Anderson$^{3}$, S. Cassisi$^{4}$, A. P. Milone$^{5}$, G. Piotto$^{6,1}$, I. R. King$^{7}$, and P. Bergeron$^{8}$.\
$^{1}$INAF-Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy\
$^{2}$Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK\
$^{3}$Space Telescope Science Institute, 3800 San Martin Drive, Baltimore, MD 21218\
$^{4}$INAF-Osservatorio Astronomico di Collurania, via M. Maggini, 64100 Teramo, Italy\
$^{5}$Research School of Astronomy and Astrophysics, The Australian National University, Cotter Road, Weston, ACT, 2611, Australia\
$^{6}$Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova, Vicolo dell’Osservatorio 3, I-35122 Padova, Italy\
$^{7}$Department of Astronomy, University of Washington, Box 351580, Seattle, WA 98195-1580\
$^{8}$Département de Physique, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, Québec H3C 3J7, Canada\
date: 'Accepted 2015 January 12. Received 2015 January 9; in original form 2014 November 27'
title: ' *Hubble Space Telescope* observations of the *Kepler*-field cluster NGC6819. I. The bottom of the white dwarf cooling sequence[^2] '
---
\[firstpage\]
open clusters and associations: individual (NGC6819) — Hertzsprung-Russell diagram — white dwarfs
Introduction
============
During the last few decades both observations and theory have improved to a level that has made it possible to employ white dwarf (WD) stars for estimating ages of stellar populations in the solar neighbourhood (i.e., Winget et al. 1987; García-Berro et al. 1988; Oswalt et al. 1996), open (i.e., Richer et al. 1998; von Hippel 2005; Bedin et al. 2008a, 2010) and globular (i.e., Hansen et al. 2004, 2007; Bedin et al. 2009) clusters.
Methods to determine stellar population ages from their WD cooling sequences are usually based on the comparison of the observed WD luminosity function (LF – star counts as a function of magnitude) with theoretical ones calculated from WD isochrones. When considering star clusters, owing to the single (and finite) age of their stars, the more massive WDs formed from higher-mass short-lived progenitors pile up at the bottom of the cooling sequence (CS), producing a turn to the blue (a turn towards lower radii) in the isochrones. At old ages, when the WD ${\rm T_{eff}}$ decreases below $\approx$5000 K, the contribution by collision-induced absorption of molecular hydrogen (Hansen 1998) to the opacity in the atmospheres reduces the infrared flux and increase the flux at shorter wavelengths. This produces a turn to the blue of the colours of individual cooling tracks, that enhances the blue excursion at the bottom of old WD isochrones. The existence of a minimum WD luminosity due to the cluster finite age, together with the accumulation of WDs of different masses and a general increase of WD cooling times with decreasing luminosity (at least before the onset of Debye cooling) translates into a peak and cut-off in the LF. Comparisons of observed and predicted absolute magnitudes of the WD LF cut-off provides the population age.
The discovery of a second, brighter peak in the WD LF of the metal-rich open cluster NGC6791 (see Bedin et al. 2005a, 2008a, 2008b for the discovery and possible interpretations) has raised questions about our understanding of the CS in simple systems like open clusters, and their use for age dating of stellar populations. In particular, this bright peak has been interpreted by Kalirai et al. (2007) as due to a population of massive He-core WDs, whilst Bedin et al. (2008b) have explained it in terms of a sizable population of WD+WD binaries. As for the fainter peak –expected to be the [*real*]{} age indicator– the age obtained from standard WD models is in conflict ($\sim$ 2Gyr younger) with that derived from the cluster main-sequence (MS) turn-off (TO), and the age later obtained from the cluster eclipsing binaries studied by Brogaard et al. (2012). This discrepancy has led to a detailed reevaluation of the effect of diffusion of ${\rm ^{22}Ne}$ in the CO core before crystallization sets in (e.g., Bravo et al. 1992, Deloye & Bildsten 2002). As shown by García-Berro et al. (2010, 2011) with full evolutionary calculations, at the old age and high metallicity of this cluster (about twice solar), the extra-energy contributed by Ne-diffusion in the liquid phase slows down substantially the cooling of the models and can bring into agreement WD, TO and eclipsing binary ages.
This result highlights very clearly the need for further observations, and the importance of studying WD ages in comparison with TO estimates in individual clusters. As WDs lie in one of the least-explored regions of the colour-magnitude diagram (CMD), we are carrying out a campaign to find out whether the case of NGC6791 is unique or whether other clusters with similar WD CSs might exist. Our purpose is to extend our knowledge of the dependence of WD LFs on cluster age and metallicity. So far we have investigated two other open clusters: NGC2158 from space (Bedin et al. 2010), and M67 from the ground (Bellini et al. 2010); both of them show canonical WD CSs (hence LFs).
The aim of the present work is to investigate the WD CS of another open cluster, NGC6819, that is within the $Kepler$-mission field. NGC6819 has solar metallicity (Bragaglia et al. 2001), is about a fourth as old as NGC6791 (Anthony-Twarog et al. 2013), and somewhat less massive (as can be inferred from their images in the Digital Sky Survey).
Section 2 will describe our observations and WD selection, whilst Section 3 presents the theoretical analysis of the WD LF. Sections 4 and 5 discuss our proper motion analysis and present the electronic material we make publicly available. Conclusions close the paper.
Observations, Measurements and Selections
=========================================
All data presented here were collected with two different instruments at the focus of the [*Hubble Space Telescope*]{} ([*HST*]{}) under two programs taken at different epochs, GO-11688 and GO-12669 (on both PI: Bedin).
For the first epoch (GO-11688) 8 orbits were allocated in two filters during October 2009, while the second epoch (4 orbits) was in October 2012 and used only the redder of the two filters. As primary instrument the Ultraviolet-Visible (UVIS) channel of the Wide Field Camera 3 (WFC3) gathered images in four contiguous fields (each 162$^{\prime\prime}$$\times$162$^{\prime\prime}$) organized in a 2$\times$2 array centered on the core of NGC6819. The same number of fields were also observed in parallel, with the Wide Field Channel (WFC) of the Advanced Camera for Surveys (ACS) (each 202$^{\prime\prime}$$\times$202$^{\prime\prime}$), which is located in the *HST* focal plane at about $6^\prime$ from UVIS, and with the detector axes oriented at $\sim45^\circ$ from the WFC3 axes. Thus the primary plus parallel exposures covered a total of about 70 arcmin square in NGC6819 (see top-left panel of Fig. \[f1\]).
{width="178mm"} {width="178mm"}
{width="178mm"}
All collected images were taken in the filters F606W & F814W, which are the optimal choice for our scientific goals, i.e., the study of the faintest WDs in a relatively-old open cluster, and are available for both instruments (although with different zero points and slightly different colour responses). Data were organized in 1-orbit visits per filter and per field. Each orbit consists of one 10s short and 4$\times$$\sim$600s long exposures for the primary instrument, and 3$\times$$\sim$470s long exposures for ACS. Within each orbit both instruments use analogous filters. The bottom panels of Fig.\[f1\] show a stacked image of the regions, after removal of cosmic rays and most of the artifacts.
All images were treated with the procedures described in detail by Anderson & Bedin (2010), to correct positions and fluxes for imperfections in the charge-transfer efficiency (CTE) of both ACS/WFC and WFC3/UVIS. Finally, in all images, we masked-out by hand the ghosts of the brightest stars in the fields.[^3]
Photometry and relative positions were obtained with the software tools described by Anderson et al. (2008). In addition to solving for positions and fluxes, we also computed two important diagnostic parameters: *quality-fit* and *rmsSKY*. The *quality-fit* essentially tells how well the flux distribution resembles the shape of the PSF (defined in Anderson et al. 2008); this parameter can be very useful for eliminating the faint blue galaxies that plague studies of WDs (see Bedin et al. 2009 for detailed discussions). The sky-smoothness parameter *rmsSKY* is the rms deviation from the mean, for the pixels in an annulus from 3.5 to 8 pixels from each source. As discussed in Bedin et al. (2008a), *rmsSKY* is invaluable in measuring a more effective completeness than has been used in most previous studies.
The photometry was calibrated to the Vega-mag system following the procedures given in Bedin et al. (2005b), and using the most updated encircled energy curves and zero points[^4]. In the following we will use for these calibrated magnitudes the symbols $m_{\rm F606W}$ and $m_{\rm F814W}$. The absolute accuracy of the calibration should be good to about 0.02 magnitude per filter.
The ACS/WFC parallel fields, although useful to derive the present-day mass function in the outskirts of the clusters from the MS stars, are not very useful to study the WD CS, as the handful of WDs are outnumbered by the field stars and background galaxies. There are also fewer ACS/WFC images than WFC3/UVIS images, and their larger pixels size complicates the identification of true point-sources, and the measurement of proper motions. Also, there are no short exposures for the WFC images, making these data useless for studying the evolved members of NGC6819. Nevertheless, for completeness and given the interest in this patch of sky, which falls inside the *Kepler* field, we release the catalogue and atlases for the ACS/WFC fields as well (see Sect.\[EM\]).
Hereafter, we will use only the WFC3/UVIS central field. In particular, the long exposures of the primary field will be used to study the WD CS of NGC6819, while the short exposures will allow us to study the brightest and evolved cluster members. The photometry obtained from the short exposures was corrected for differential reddening as in Milone et al. (2012, see their Sect. 3), and carefully registered to the zero points of the long exposures using unsaturated common stars. A precision of $\sim$0.01 magnitude per filter was estimated for this operation; this means that the photometry from short and long exposures should be consistent to this level.
Artificial-star (AS) tests were performed using the procedures described by Anderson et al. (2008). We chose to cover the magnitude range $\sim$$23 < m_{\rm F606W} < \, \sim$$28.5$, with colours that placed the artificial stars on the WD sequence. These tests played an important role in showing us what selection criteria should be used for the real stars. The quality of our results depends very much on making a good selection, the details for which are shown in Fig.\[f2\]. (See Bedin et al. 2009, 2010 for a detailed description of the selection procedures.) Panels with unprimed labels refer to real stars, while those with primed labels refer to the artificial stars.
We used the AS tests to show what combinations of magnitude and *quality-fit* are acceptable for valid star images \[panel (a$^\prime$)\], and we drew the blue lines to isolate the acceptable region. We then drew identical lines in panel (a), to separate the real stars from blends and probable galaxies. We went through similar steps for the *rmsSKY* parameter (panels b$^\prime$ and b). Finally we plotted CMDs and drew dividing lines in a similar way to isolate the white dwarfs (panels c$^\prime$ and c) from field objects. Thankfully, WDs occupy a relatively uncontaminated region of the CMD (Bedin et al. 2009).
As one can see from panels (a) and (a$^\prime$), the shape (i.e., the *quality-fit*) of the objects becomes increasingly confused with the noise at magnitudes fainter than $m_{\rm F606W}$$\sim$26, and therefore a few blue compact background galaxies might have fallen in the list of selected objects. However, these cannot affect the location of the drop of WD star counts, which is the relevant feature studied in our work. Furthermore, the proper motions described in Sect.4, confirm the WD LF cut-off magnitude. Our final step was to deal with completeness, a concept that appears in two different contexts: (1) The observed number of WDs must be corrected for a magnitude-dependent incompleteness. (2) It is customary to choose the limiting reliable magnitude at the level where the sample is 50% complete. In a star cluster these two aims need to be treated in quite different ways. (1$^\prime$) To correct for incompleteness the WD LF, we use the traditional ratio of AS recovered to the number inserted. (2$^\prime$) For the limit to which measures of faint stars are reliable, however, the 50%-completeness level needs to be chosen in a quite different way, because in a crowded star cluster more faint stars are lost in the brightness fluctuations around the bright stars, than are lost to the fluctuations of the sky background. The completeness measure that we should use to set the 50% limit is therefore the recovered fraction of ASs [*only*]{} among those ASs whose value of *rmsSKY*, as a function of magnitude, indicates that it [*was possible to recover the star*]{}. \[For a more detailed discussion, see Sect. 4 of Bedin et al. (2008a), and for other two similar applications and further discussions see Bedin et al. 2009, 2010\].
The distinction between the two completeness measures is illustrated in panel (d) of Fig.\[f2\], where the black line shows the overall completeness, while the blue line shows the completeness statistics that take *rmsSKY* into account; its 50% completeness level is at $m_{\rm F606W}=27.11$, rather than 26.87mag with the traditional statistics. To emphasize the contrast, the red line shows the fraction of the area (i.e., with low-background) in which faint stars could have been found.
Comparison with Theory
======================
As in our previous similar works on other clusters, the comparison with theory comprises the determination of a MS TO age, and the age implied by the WD LF.
BaSTI[^5] scaled-solar isochrones for both WD (Salaris et al. 2010) and pre-WD (Pietrinferni et al. 2004) evolutionary phases, including MS convective-core overshooting, were used to determine both MSTO and WD ages. We considered the available BaSTI (pre-WD and WD) isochrones for \[Fe/H\]=+0.06, consistent with \[Fe/H\]=$+$0.09$\pm$0.03 determined spectroscopically for a sample of cluster red clump stars by Bragaglia et al. (2001 – but see Anthony-Twarog et al. 2014 for a slightly different estimates, \[Fe/H\]=$-$0.06$\pm$0.04, obtained from $uvby\,Ca\,H_\beta$ observations).
First we determined the cluster distance modulus from isochrone fitting to the MS. Given that stars above the MSTO are saturated, we could not use the magnitude of the red clump and the colour of red giant branch stars as constraints. We have considered the unevolved cluster MS between $m_{\rm F606W}$=17 (about two magnitudes below the MSTO) and 19, and derived a fiducial line by taking the mode of the $(m_{\rm F606W}-m_{\rm F814W})$ colour distribution of MS stars in 0.25mag wide $m_{\rm F606W}$ bins. Before performing a least square fitting of the isochrone MSs to the fiducial line, we fixed the cluster $E(B-V)$ to values between 0.10 and 0.18 mag, in steps of 0.01 mag. We explored this range following the estimate by Bragaglia et al. (2001), who used temperatures of three red clump stars derived from line excitation (a reddening-free parameter) and the appropriate theoretical spectra, to determine their intrinsic $(B-V)$ colours. Comparison with the observed colours provided $E(B-V)$=0.14$\pm$0.04 (Anthony-Twarog et al. 2014 determined $E(B-V)$=0.160$\pm$0.007 from $uvby\,Ca\,H_\beta$ observations, within the range spanned by Bragaglia et al. estimate). These $E(B-V)$ values between 0.10 and 0.18 magnitudes were then transformed into $E(m_{\rm F606W}-m_{\rm F814W})$ and $A_{m_{\rm
F606W}}$ as described by Bedin et al. (2009), and assuming $R_V$=3.1. For each $E(m_{\rm F606W}-m_{\rm F814W})$ we finally fitted isochrones of various ages to the fiducial line. The procedure worked as follows. For a given choice of $E(B-V)$ the vertical shift required to fit the isochrones to the observed MS fiducial line provided the apparent distance modulus. This is unaffected by the isochrone age, given that we were fitting the unevolved MS. Once derived the apparent distance modulus for a given $E(B-V)$, we compared magnitude and shape of the isochrones’ TO with the observed CMD, for ages varying between 1.0 and 4.0 Gyr. This comparison allowed us to both estimate the TO age and narrow down the range of reddenings consistent with the isochrones. It turned out that for some $E(B-V)$ values (and the associated distance moduli), around the TO region the shape of the isochrone that matched the observed TO magnitude was very different from the shape of the CMD. This implied that the isochrone was too old or too young compared to the real cluster age, so that the TO magnitude had matched only on account of an inconsistent distance modulus, and hence an inconsistent reddening. Our isochrone fitting provided $E(B-V)=0.17\pm0.01$, $(m-M)_0=11.88\pm0.11$ \[corresponding to a distance of 2377pc, and to $(m-M)_V=12.41\pm0.12$\] and a MSTO age $t_{\rm MS\,TO}=2.25\pm0.30$Gyr. The total age range was fixed by the youngest and oldest isochrones that bracketed the TO region in the observed CMD for, respectively, the shortest and largest distance moduli obtained from the MS fitting. The error on the distance modulus is an internal error of our isochrone MS-fitting method that combines in quadrature the error of the least square fit at fixed reddening, and the effect of the error on the reddening on the least square fit value.
The reliability of our derived distance can be assessed by comparisons with completely independent methods. Sandquist et al. (2013) obtained $(m-M)_V=12.39\pm0.08$ \[employing $E(B-V)=0.12\pm0.01$\] from the cluster eclipsing binary WOCS 23009, whilst Jeffries et al. (2013) determined $(m-M)_V=12.44\pm0.07$ for the eclipsing binary WOCS 40007. Wu, Li & Hekker (2014) determined $(m-M)_0=11.88\pm0.14$ \[assuming $E(B-V)=0.14\pm0.04$\] through the global oscillation parameters ${\rm \Delta \nu}$ and ${\rm \nu_{max}}$ and $V$-band photometry of the cluster red giants.
Our isochrone-based distance estimate is in good agreement with these independent results based on completely different methods.
Regarding the age, our estimate from the MSTO agrees with $t=2.2\pm0.2$Gyr obtained by Jeffries et al. (2013) by comparing the MS mass-radius relationship of the same BaSTI isochrones used here, with the masses and radii determined for the components of the cluster eclipsing binary WOCS 23009.
![ *(Upper panel:)* Fit of theoretical isochrones \[1.95 Gyr, $(m-M)_0=11.99$, $E(B-V)$=0.18 and 2.55 Gyr, $(m-M)_0=11.77$, $E(B-V)$=0.16\] to the cluster MS in the $m_{\rm F606W}$-$(m_{\rm F606W} - m_{\rm F814W})$ CMD (see text for details). *(Lower panel:)* Fit of WD isochrones to the observed CS, for the same age-distance-reddening combinations. Solid lines denote DA WDs, dotted lines DB objects. \[thCMDs\] ](f3.ps){width="89mm"}
The upper panel of Fig. \[thCMDs\] shows the result of the isochrone fits, with the two isochrones that best bracket the MSTO region, i.e., 1.95Gyr \[for $(m-M)_0=11.99$ and $E(B-V)$=0.18\] and 2.55Gyr \[for $(m-M)_0=11.77$ and $E(B-V)$=0.16\], respectively. WD isochrones (for both DA and DB objects) with the same age-distance-reddening combinations are displayed in the lower panel of the same figure. Notice how the termination of the DB sequence is brighter than that for the DA WDs, contrary to the result for typical globular cluster ages, because in this regime He-envelope WDs cool down more slowly that the H-envelope counterparts (see, e.g., Salaris et al. 2010). To compare WD and MSTO ages we have used the completeness corrected WD LF, and matched with theoretical WD LFs of varying age the observed cut-off of the star counts. The observed LF –see points with error bars in Fig. \[thLFs\]– exhibits just one peak and a cut-off (at $m_{\rm F606W}$=26.050$\pm$0.0.075 at its faint end, as expected for a standard cluster CS. Theoretical LFs have been calculated with the Monte Carlo (MC) technique described in Bedin et al. (2008b), employing the BaSTI WD models for both DA and DB objects, and considering a Salpeter mass function (MF) for the progenitors. Our MC calculations account for the photometric error as found from the data-reduction procedure and arbitrary fractions of unresolved WD+WD binaries and DA and DB WDs. The effect of random fluctuations of the synthetic LF are minimized by using $\sim$100 times as many WDs as the observed sequence (which comprises about 200 objects after the completeness correction).
To give some more details, the MC code calculates the synthetic CMD of a WD population of fixed age (and initial chemical composition) by selecting first a value of the progenitor mass for a generic WD along the corresponding WD isochrone (we assume a burst of star formation with negligible age and metallicity dispersion, appropriate for open clusters), according to an IMF that is set to Salpeter as default choice. The corresponding mass and magnitudes of the [*synthetic*]{} WD are then determined by quadratic interpolation along the isochrone, that has been calculated assuming a WD initial-final mass relation (see Salaris et al. 2010 for details about the WD isochrones)[^6]. For the fraction of objects assumed to be in unresolved binaries, we extract randomly also a value for the ratio $q$ between the initial mass of the companion and the mass of the WD progenitor, according to a specified statistical distribution. If the initial mass of the companion has produced a WD, we determine its mass and magnitudes as for single WDs. In case the companion is in a different evolutionary stage, we determine its magnitudes by interpolating along the pre-WD isochrone. The fluxes of the two components are then added, and the total magnitudes and colours of the composite system are computed. We finally add the distance modulus and extinction to the magnitudes of both single and unresolved binary stars, and perturbe them randomly by using a Gaussian photometric error with the $\sigma$ derived from the artificial-star tests.
A theoretical LF is then computed for the synthetic population to be compared with the observed one.
![ Comparison of the completeness-corrected observed $m_{\rm F606W}$ WD LF (filled circles with error bars) with theoretical 2.25Gyr LFs, calculated with a Salpeter MF and different choices of parameters, for $E(B-V)=0.17$ and $(m-M)_0=11.88$. Panel a) displays DA LFs with no unresolved WD+WD binaries (dotted line), with a 22% fraction of progenitor binaries (solid line – our reference case), and with the same fraction of progenitor binaries but a different distribution of mass ratios of the two components (dashed line – see text for details). Panel b) displays our reference theoretical LF (solid line), and a LF with 85% DA and 15% DB objects, leaving everything else unchanged (dashed line). Panel c) displays the reference LF and one calculated with a flat MF (dashed line – see text for details). Panel d) displays our reference LF (solid line) and a 2.5Gyr old LF, everything else being the same (dashed line). \[thLFs\] ](f4.ps){width="89mm"}
In general, the exact shape of the WD LF (and of the WD sequence in the CMD) depends on a number of parameters in addition to the cluster age, e.g., the initial-final-mass relationship, the mass function of the progenitors, the relative fraction of the (dominant) DA and DB objects, the fraction of WD+WD binaries (and their mass distribution) and, very importantly, the dynamical evolution of the cluster, which selectively depletes the WDs according to their mass and their time of formation. Also, in open clusters, if a WD receives a sufficient velocity kick from asymmetric mass loss during its pre-WD evolution, it may become unbound (see, e.g., the introduction in Williams 2004).
On one hand, this is why we are not attempting to match precisely the shape of the whole WD LF, by trying to determine simultaneously the cluster age, the precise fraction of WD+WD binaries, DB objects, and the progenitor MF. These effects are difficult to disentangle using just the observed LF, and complicated by the limited statistic (total number of observed WDs, compared for example with the cases of M4 or NGC6791) and by a potential (although mild) residual field contamination.
On the other hand, the observed position of the WD LF cut-off, which is the primary WD age indicator, should have been affected very little by any of these parameters. In the following we quantitatively demonstrate how all the above uncertainties, although potentially affecting the shape of the WD LF, do not alter the magnitude location of the LF cut-off.
First of all, the effect of the dynamical evolution (see Yang et al. 2013 for indications about the dynamical evolution of this cluster) should be negligible, for the following reason. The WD isochrones predict that the brighter WDs should have nearly a single mass, whereas the region of the WD LF cut-off — i.e., at the blue-turn at the bottom of the CS — is expected to be populated by objects spanning almost the whole predicted WD mass spectrum (hence the presence of a blue turn towards lower stellar radii). A mass-dependent loss mechanism due to dynamical evolution would therefore alter the number distribution of WDs around the blue-turn, but it should have no major effect on the observed location of the cut-off in the WD LF. Figure \[thLFs\] compares the completeness-corrected observed $m_{\rm F606W}$ WD LF (filled circles with error bars) with theoretical LFs calculated to explore some of the other effects listed above.
Panel a) displays the theoretical LF for DA WDs (with the same number of objects as the observed candidate WDs), an age of 2.25 Gyr \[$(m-M)_0$=11.88 and $E(B-V)$=0.17\], and two different fractions of progenitor binaries – no binaries, and 22% respectively (dotted and solid lines in the figure)– corresponding to zero, and 14% WD+WD binary (supposed to be unresolved) systems on the final cooling sequence. The mass ratio $q$ for the binary companions (the primary components are assumed to follow a Salpeter MF) was chosen with a flat distribution between 0.5 and 1.0. As discussed, e.g., in Podsiadlowski (2014), the distribution of $q$ in binary systems is not well determined and appears to depend on the mass range. Massive binaries favour stars of comparable mass ($q$$\approx$1) whilst the situation is less clear for low-mass stars, and most studies show that $q$ is possibly consistent with a flat distribution. The 22% fraction of progenitor binaries was chosen after the recent results by Milliman et al. (2014), who determined a present fraction of $22\pm3$% MS binaries with period less than $10^4$ days.
The unresolved systems show up in the LF at ${m_{\rm F606W}}$ between 25 and 25.2 (the variation of the star counts at fainter magnitudes is a consequence of keeping the total number of objects fixed at the observed value) about 0.75 mag brighter than the LF cut-off, as for the case of NGC6791 (Bedin et al. 2008b). The inclusion of unresolved WD+WD binaries improves the fit of the LF at those magnitudes and leaves completely unchanged the magnitude of the LF cut-off.
We have also tested the effect of changing the distribution of $q$ by fixing the progenitor binary fraction to 22%, but this time employing a flat distribution between 0.1 and 1.0. The resulting LF (dashed line) has a slightly changed shape, the local maximum between 25 and 25.2 is less pronounced, but the magnitude of the cut-off is not affected. Note how the few contaminant sources in the sample that might alter our observed WD LF below $m_{\rm F606W}$$\sim$26.5, do not affect the location of the WD LF drop-off.
{width="178mm"}
Panel b) shows the effect of including a 15% fraction of DB objects (dashed line – DB isochrones are also from Salaris et al. 2010) for an age of 2.25 Gyr, and a reference 22% binary progenitor fraction. Also in this case the magnitude of the LF cut off is unchanged, and the signature of DB WDs appears at $m_{\rm F606W}\sim$25.5 (the variation of the star counts at fainter magnitudes is again a consequence of keeping the total number of objects fixed). According to the observed LF there is no room for a much higher fraction of DB objects, and actually a value close to zero would probably be a better match to the observations.
Panel c) investigates the combined effect of varying the progenitor initial MF plus the cluster dynamical evolution that progressively depletes the lower mass (both progenitors and WDs) objects. To this purpose we display the LF for the standard Salpeter MF (solid line – no DBs, 22% of WD+WD unresolved binaries) and a LF calculated with a close to flat MF for the WD progenitors (dashed line), i.e., a power law with exponent $-$0.15. Progenitor masses are between $\sim$1.7 and $\sim$6.5 ${\rm M_{\odot}}$ at our reference age of 2.25 Gyr, whilst the final WD masses range between $\sim$0.61 and 1.0 ${\rm
M_{\odot}}$ according to the initial-final mass relation adopted by the BaSTI WD isochrones (see Salaris et al. 2010 for details).
The comparison of the two LFs shows that the magnitude of the cut-off is unchanged, as expected. Only the overall shape is affected and the width of the peak of the theoretical LF is increased, but not the position of the drop of the star counts beyond the peak, that is the age indicator. Finally, panel d) displays the effect of age (2.25 and 2.5 Gyr, respectively), keeping the DB fraction to zero, and the reference 22% of progenitor binaries. The older LF has a fainter termination, and can be excluded by means of the comparison with the position of the observed cut-off.
After this reassessment of the solidity of the LF cut-off as age indicator, we have varied the cluster apparent distance modulus within the range established with the MS fitting and obtained a range of WD ages ${\rm t_{WD}}$=2.25$\pm$0.20 Gyr. This age range has been determined by considering all theoretical LFs that display after the peak a sharp drop in the star counts at the bin centred around $m_{\rm F606W}$=26.05. Notice that the range is narrower than the MSTO age range, but completely consistent with ${\rm t_{MS\,TO}}$. This narrower age range is due to the different sensitivity of TO and WD magnitude cut-off to the population age (Salaris 2009a).
We close our theoretical analysis presenting two additional tests about our derived WD ages. First, we have recalculated the BaSTI WD isochrones considering the WD initial-final mass relation (IFMR) by Kalirai (2009) instead of the default IFMR employed by the BaSTI isochrones (Salaris et al. 2009b). In the WD progenitor mass range typical of NGC6819 (progenitor masses above $\sim$1.6 $M_{\odot}$), this different IFMR predicts generally slightly lower progenitor masses at fixed WD mass, compared to the BaSTI default IFMR. This implies that with the Kalirai et al. (2009) IFMR WDs of a given mass start to cool along the CS at a later time, hence they reach somewhat brighter magnitudes for a given WD isochrone age (we recall that the WD isochrone age is equal to the sum of the progenitor age plus the WD cooling age), compared to the case of the Salaris et al. (2009b) IFMR. Comparisons with the observed LF show that the use of the Kalirai et al. (2009) IFMR increases the estimate of the WD age by just 40 Myr.
As a second test, we have considered the DA WD cooling tracks by Renedo et al. (2010 – their calculations for metallicity approximately half-solar, the closest to NGC6819 metallicity in their calculations), that are completely independent from the BaSTI WD models employed here. Renedo et al. (2010) calculations follow the complete evolution of the WD progenitors through the thermal pulse phases and include the effect of CO phase separation upon crystallization, like BaSTI WD models. The CO stratification and envelope thickness of Renedo et al. models (as well as the model input physics) are slightly different from BaSTI models, but the cooling times down to the luminosities of the bottom of the observed CS of NGC6819 are similar. From their calculations, the match of the cut-off of the observed LF provide an age older by $\sim$100Myr compared to the age determined with our reference BaSTI WD models.
Proper Motions
==============
In principle a mild contamination from field objects, might also affect the shape of the WD LF, but again, it cannot change the magnitude location of the sharp drop off in the WD LF. In this section we use proper motions to strengthen the cluster membership of the observed stars in the expected CMD location of the WD CS.
Proper motions were determined by means of the techniques described in Bedin et al. (2003, 2006, 2009, 2014). We used the deep exposures in F814W collected in October 2009 as first epoch, while we used the F814W exposures taken in October 2012 as second epoch. The proper motions are calculated as the displacement between the average positions between the two epochs, divided by the time base-line (about 3 years), and finally multiplied by the pixel scale. We assumed a WFC3/UVIS pixel scale of 39.773 maspixel$^{-1}$ (Bellini, Anderson & Bedin 2011). As the proper motions signals are a function of the F814W magnitude only, we plot proper motions properties as a function of this magnitude, rather than F606W.
Unfortunately, the cluster has a proper motion that is not significantly different from the bulk of the field objects, and our precision is too low for an accurate discrimination between field objects and cluster members, especially at the magnitudes of the faintest WDs. Furthermore, proper motions are not available for all sources, with an incompleteness that is hard to assess quantitatively (and reliably); the precision of the derived proper motions is also heterogeneous across the sample. Even though we cannot do quantitative work with the proper motions, we can still use them in a qualitative way to confirm: *(i)* the truncation of the WD LF, *(ii)* the observed shape of the end of the WD CS (as the stars having proper motions are also the ones with the best photometry), and *(iii)* the goodness of the employed isochrones (how well they reproduce the blue-turn).
The left panel of Fig. \[pms\] shows as small dots all stars in panel (c) of Fig. \[f2\], but this time plotting the colour [ *vs.*]{}$m_{\rm F814W}$ instead of [*vs.*]{}$m_{\rm F606W}$. Stars for which it was possible to estimate a proper motion are highlighted with open circles. Only members along the MS were used to define the transformations between the two epochs, and they are marked in blue. In the second panel, we show for this sub-sample of stars the magnitudes of the proper motions vs. $m_{\rm F814W}$. Cluster members should have a dispersion of the order of $\sim$1 km s$^{-1}$, which at a distance of $\sim$2.4kpc would correspond to a proper motion internal dispersion of less than 0.1masyr$^{-1}$, making them an ideal almost-fixed reference frame. For this reason our zeroes of the motion will coincide with the cluster bulk motion.
It can be seen that at high-signal-to-noise ratio, cluster stars have a sharp proper motion distribution which is consistent with a few times 0.1masyr$^{-1}$, but embedded in the middle of a cloud of field objects, mainly thin disk and thick disk stars \[NGC6819 is at (*l,b*)$\simeq$(74$^\circ$,8$^\circ$)\]. Note the increasing size of random errors with decreasing signal-to-noise ratio.
In order to separate members from non-members we used the results of the artificial star test illustrated in (f$_3$ and f$_4$) and derive the 1-D $\sigma$ as the 68.27th percentile of the added$-$recovered positions for each coordinate at different magnitude steps. The red line marks the 2.5$\sigma$ level.
Non-members, to the right of the red line, are marked in red. In the third panels from left, using the same symbols, we show the vector-point diagrams (in mas yr$^{-1}$) for four different magnitude bins ($m_{\rm F606W}$ = 18-20.5, 20.5-23, 23-25.5, and 25.5-28). The CMDs that results from the cleaning criterion of the second panel are shown in the rightmost panels of the same figure, using both filters. No matter how tight the circle is, even at the brightest magnitudes, there will always be field objects that accidentally have the same motions of NGC6819 members. Thankfully, the WDs of NGC6819 occupy a low-contamination region of the CMD, were few blue faint stars exist, field WDs and blue compact galaxies both being generally significantly fainter (Bedin et al. 2009).
Although affected by low completeness, it seems even more clear that the sharp drop at $m_{\rm F606W}\simeq26.05$ (i.e., $m_{\rm
F814W}\simeq25.6$) in Fig. \[f2\](c) is real and that the few objects along the continuation of the WD CS at $m_{\rm F606W}\sim27$ (i.e., $m_{\rm F814W}\simeq26$) are most likely field WDs or background unresolved galaxies.
Finally, in Fig. \[WDcln\] we show the CMDs of the decontaminated WD CS. Although very incomplete, these contain the best measured WDs in our sample, offering a higher-photometric-precision version of the complete sample used to derive the WD LF. Indeed, the shape of the observed blue-turn here appears to be significant, and most importantly consistent with the shape of the WD DA isochrones (indicated in red) and with the synthetic CMDs for the reference LF of panel b) in Fig. \[thLFs\]. Very interestingly, the synthetic CMDs show around $m_{\rm
F606W}\sim25$ and $(m_{\rm F606W}-m_{\rm F814W})\sim$0.5 a feature caused by the unresolved WD+WD binaries included in the simulation (see Sect. 3 for details), that corresponds to the end of the DA cooling sequence shifted 0.75mag brighter (see upper panels of the same figure). The observed CMDs exhibit a very similar feature located at the same magnitude and colour, hinting at the presence of a small fraction of unresolved WD+WD systems, as suggested also by the fit of the LF (see Section 3).
A future paper (involving other team-members) will deal with the absolute motions of NGC6819 with respect to background galaxies in order to compute the cluster Galactic orbit using as input the sources in the catalogue described in the next section. \[Similar to the study of the orbit of NGC6791 done in Bedin et al. 2006.\]
![ A zoom-in of the decontaminated CMDs in Fig. \[pms\] centred on the WD CS, with colour *vs.* $m_{\rm F814W}$ in left panels and *vs.* $m_{\rm F606W}$ in the right panel. The top panels display the observed CMDs, with over-imposed the WD isochrones employed in Fig. \[thCMDs\] (solid lines: DA in red, DB in blue). We also display, as dotted lines, the same isochrones but 0.75 mag brighter, to mark approximately the magnitude and colour range covered by a population of unresolved WD+WD binaries. The bottom panels display the synthetic CMDs corresponding to the reference LF shown as a solid line in panel b) of Fig. \[thLFs\] (see Sect. 3 for details) with approximately the same number of stars of the decontaminated CMDs (see Sect. 3 for details). \[WDcln\] ](f6.ps){width="89mm"}
Electronic Material: WFC3/UVIS and ACS/WFC {#EM}
==========================================
The catalogue is available electronically in this Journal (and also under request to the first author). In the catalogue, Col.(1) contains the running number; Cols.(2) and (3) give the J2000 equatorial coordinates in decimal degrees for the epoch J2000.0, while Cols.(4) and (5) provide the pixel coordinates x and y in a distortion-corrected reference frame. Columns (6) to (11) give photometric data, i.e., $m_{\rm F606W}$ and $m_{\rm F814W}$ magnitudes and their errors. If photometry in a specific band is not available, a flag equal to 99.999 is set for the magnitude and 0.999 - for the error. In Fig. \[acs\] we show the CMDs obtained from the unselected catalogue of the ACS/WFC parallel fields, which we have not analysed and used in this article. For ACS/WFC we provide the entire list of all detections, these still include blends and other artifacts. For the case of WFC3/UVIS main catalogue we also provide the subset of the selected stars as shown in Fig.\[f2\] and a WD flag. We also provide the photometry from short exposures.
The stack images shown in the bottom panels of Fig.\[f1\] provide a high-resolution representation of the astronomical scene that enables us to examine the region around each source. Indeed, the pixel-based rapresentation of the composite data set provides a nice objective cross-check on the catalog-based representations, which is a product of our subjective finding and selection criteria. The stacked images are 20000$\times$20000 for UVIS (and 24000$\times$24000 for WFC) super-sampled pixels (by a factor of 2, i.e., 20maspixel$^{-1}$ for UVIS, 25maspixel$^{-1}$ for WFC). We have included in the header of the image, as World Coordinate System keywords, our absolute astrometric solution based on the 2MASS point source catalogue. As part of the material of this article, we also electronically release these astrometrized stacked images.
![ Colour-Magnitude diagrams obtained from the ACS/WFC parallel fields shown in the bottom-right panel of Fig.\[f1\]. The bottom and left axes display instrumental magnitudes, while the top and right the calibrated ones. Dashed lines indicate the level where saturation sets-in. Note that it is possible to recover photometry for saturated stars up to about 3.5 magnitude above these limits (see Gilliland 2004 for details), but not enough to observe the MS TO stars reliably. \[acs\] ](f7.ps){width="89mm"}
CONCLUSIONS
===========
We have used *Hubble Space Telescope* to observe the WD sequence in NGC6819, a near solar-metallicity old open cluster within the $Kepler$ mission field of view. By means of our photometry and completeness tests we have determined the LF of the cluster WD CS, which exhibits a sharp drop at $m_{\rm F606W} = 26.050 \pm 0.075$, and a shape consistent with theoretical predictions for a canonical cluster CS.
Isochrone fits to the cluster MS have provided $E(B-V)=0.17\pm0.01$, $(m-M)_0=11.88\pm0.11$, and a MSTO age $t_{\rm MS\,TO}=2.25\pm0.30$Gyr. These distances and ages are in agreement with independent constraints from eclipsing binary and asteroseismological analyses.
We have reassessed the robustness of the WD LF cutoff magnitude as age indicator and determined an age of $2.25\pm0.20$Gyr from the CS, completely consistent with the age of $2.25\pm0.30$Gyr obtained from the MSTO. We have tested the effect of a different IFMR on our WD ages, obtaining an age larger by just 40 Myr when using the Kalirai et al. (2009) IFMR instead of our reference Salaris et al. (2009) IFMR. We have also employed the completely independent set of WD calculations by Renedo et al. (2010) and determined an age older by just 100Myr compared to the result with our reference WD models.
This agreement between WD and MSTO ages is in line with the results we obtained in our previous studies of NGC2158 and M67, two open clusters spanning the age range between $\sim$2 and $\sim$4 Gyr.
As a by-product of this work we release, in electronic format, both the catalogue of all the detected sources and the atlases of the region (in two filters), contained in the $Kepler$ field.
Acknowledgments
===============
J.A. and I.R.K. acknowledge support from STScI grant GO-11688 and GO-12669. P.B. was supported in part by the NSERC Canada and by the Fund FRQ-NT (Québec).
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\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: Based on observations with the NASA/ESA [*Hubble Space Telescope*]{}, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555, under GO-11688 & GO-12669.
[^3]: (from WFC3 instrument handbook, Sect. 6.5.3): *The WFC3 UVIS channel exhibits three different types of optical ghosts: a) those due to reflections between the CCD front surface and the two detector package windows; b) those due to reflections between the window surfaces; and c) those due to reflections within the particular filter in use.*
[^4]: ` http://www.stsci.edu/hst/wfc3/phot_zp_lbn; http://www.stsci.edu/hst/acs/analysis/zeropoints. `
[^5]: `http://www.oa-teramo.inaf.it/BASTI`
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present novel algorithms to estimate outcomes for qubit quantum circuits. Notably, these methods can simulate a Clifford circuit in linear time without ever writing down stabilizer states explicitly. These algorithms outperform previous noisy near-Clifford techniques for most circuits. We identify a large class of input states that can be efficiently simulated despite not being stabilizer states. The algorithms leverage probability distributions constructed from Bloch vectors, paralleling previously known algorithms that use the discrete Wigner function for qutrits.'
author:
- Patrick Rall
- Daniel Liang
- Jeremy Cook
- William Kretschmer
title: Simulation of Qubit Quantum Circuits via Pauli Propagation
---
Introduction
============
Simulating quantum circuits on classical hardware requires large computational resources. Near-Clifford simulation techniques extend the Gottesmann-Knill theorem to arbitrary quantum circuits while maintaining polynomial time simulation of stabilizer circuits. Their runtime analysis gives rise to measures of non-Cliffordness, such as the robustness of magic [@resource], magic capacity [@seddon], sum-negativity [@vmge13]. These algorithms evaluate circuits by estimating the mean of some probability distribution via the average of many samples, a process with favorable memory requirements and high parallelizability.
Previous work [@bennink; @resource] gives an algorithm based on quasiprobability distributions over stabilizer states; we refer to this algorithm as ‘stabilizer propagation’. In contrast to techniques based on stabilizer rank [@gosset; @extent], stabilizer propagation is appealing for simulation of NISQ-era hardware [@nisq] because it can simulate noisy channels. Moreover, depolarizing noise decreases the number of samples required, measured by robustness of magic and the magic capacity. However, bounding the number of required samples can be expensive: For example, the magic capacity of a 3-qubit channel is defined as a convex optimization problem over 315,057,600 variables [@enums; @seddon].
Pashayan et al. [@pash] showed that in qutrit systems, the discrete Wigner function provides a simpler simulation strategy. This strategy takes linear time to sample, and the number of samples required (measured by the sum-negativity) is tractable to compute for small systems. However, discrete Wigner functions do not yield efficient simulation of qubit Clifford circuits [@rbdobv15].
Our main result is that Bloch vectors yield simulation strategies for qubit circuits, similar to those in Pashayan et al. We present two algorithms, which we individually call **Schrödinger propagation** and **Heisenberg propagation**, and collectively call **Pauli propagation techniques**. They have several surprising properties:
1. They yield linear time simulation for qubit Clifford circuits without writing down stabilizer states.
2. Schrödinger propagation can efficiently simulate a new family of quantum states called ‘hyper-octahedral states’ which is significantly larger than the set of stabilizer mixtures in terms of the Hilbert-Schmidt measure.
3. The runtime of Heisenberg propagation does not depend on the input state at all.
4. Non-Cliffordness in both algorithms is measured via the stabilizer norm, which is a lower bound to the robustness of magic. This gives Pauli propagation techniques a strictly lower runtime than stabilizer propagation for all input states and most channels.
**Table: Circuit components that can be simulated efficiently**\
---------------------------- ---------------------------------------------- ------------------------------ -----------------------------
**Stabilizer propagation** **Heisenberg propagation** **Schrödinger propagation**
What input states Any separable state, Hyper-octahedral states,
are efficient to simulate? Stabilizer mixtures Noisy states reduce runtime
Depolarized $T$ gate Efficient when fidelity $ \lessapprox 0.551$
Reset channels Pauli reset channels efficient All reset channels efficient
Adaptive gates Adaptive Cliffords efficient Generally inefficient
Marginal observables
Pauli observables
---------------------------- ---------------------------------------------- ------------------------------ -----------------------------
: Summary of the results of Section III. All algorithms take polynomial time to sample, but the number of samples scales exponentially in the number of *inefficient* circuit components. *Efficient* components do not increase runtime.
\
We describe these algorithms in Section II. In Section III we perform a detailed comparison of Schrödinger, Heisenberg and stabilizer propagation which we summarize in the table below. In Section IV we briefly discuss the implications of the algorithms for resource theories of Cliffordness. This work is intended to supersede [quant-ph/1804.05404](https://arxiv.org/abs/1804.05404).
\[sec:intro\]Algorithms
=======================
In this section we describe two algorithms for estimating the expectation value of observables at the end of a quantum circuit. Schrödinger propagation involves propagating states forward though the circuit and taking inner products with the final observables. Heisenberg propagation involves propagating observables backward though the circuit and taking inner products with the initial states. At every step, both procedures sample from an unbiased estimator for the propagated state/observable that is distribution over Pauli matrices.
Sampling Pauli Matrices
-----------------------
The workhorse of both protocols is a subroutine that samples a random scaled tensor product of Pauli matrices as a proxy for an arbitrary $n$-qubit Hermitian matrix $A$. Let $\mathcal{P}_n = \{\sigma_1 \otimes \cdots \otimes \sigma_n : \sigma_i \in \{I, \sigma_X, \sigma_Y, \sigma_Z\}\}$ denote the set of $n$-qubit Pauli matrices. We define a pair of completely dependent random variables $\hat \sigma \in \mathcal{P}_n$ and $\hat c \in \mathbb{R}$ that satisfy $\mathbb{E}\left[\hat c \cdot \hat \sigma\right] = A$: $$\begin{aligned}
\label{eq:sigma_hat}\hat \sigma(A) &= \sigma \text{ with prob. } \frac{\left|\text{Tr}(\sigma A)\right|}{ 2^n \cdot \mathcal{D}(A)} \text{ for each }\sigma \in \mathcal{P}_n,\\
\label{eq:c_hat}\hat c(A) &= \mathrm{sign}\left( \mathrm{Tr}(\hat \sigma(A) A)\right) \cdot \mathcal{D}(A).\end{aligned}$$
The quantity $\mathcal{D}(A)$ is a normalization constant that makes $\frac{\left|\text{Tr}(\sigma A)\right| }{ 2^n \cdot \mathcal{D}(A)}$ for $\sigma \in \mathcal{P}_n$ a probability distribution.
The **stabilizer norm** $\mathcal{D}(A)$ is: $$\label{eq:stabnorm}\mathcal{D}(A) = \frac{1}{2^n} \sum_{\sigma \in \mathcal{P}_n} \left|\mathrm{Tr}(\sigma A) \right|.$$
The product of the random variables $\hat c(A) \cdot \hat \sigma(A)$ is an unbiased estimator for $A$ because the Pauli matrices form an operator basis for Hermitian matrices: $$\begin{aligned}
\mathbb{E}[\hat c(A) \cdot \hat \sigma(A)] &= \sum_{\sigma \in \mathcal{P}_n} \frac{\left|\text{Tr}(\sigma A)\right|}{ 2^n \cdot \mathcal{D}(A)} \cdot \text{sign}\left( \text{Tr}(\sigma A) \right) \cdot \mathcal{D}(A)\cdot \sigma\nonumber\\
&= \sum_{\sigma \in \mathcal{P}_n} \frac{\text{Tr}(\sigma A)}{2^n} \cdot \sigma = A.\end{aligned}$$
The time to compute the probabilities and sample from the distributions scales exponentially with the number of qubits of $A$. We say $A$ has **tensor product structure** if it can be written as a tensor product of several operators, each of which acts on a constant number of qubits: $$A = A_1\otimes A_2 \otimes \cdots$$
Then one can observe that: $$\hat\sigma(A) = \hat\sigma(A_1)\otimes \hat\sigma(A_2)\cdots \text{ and } \hat c(A) = \hat c(A_1) \cdot \hat c(A_2)\cdots$$ Since each $A_i$ acts on a constant number of qubits, each of the probability distributions for $\hat \sigma(A_i), \hat c(A_i)$ can be computed and sampled from in constant time. So $\hat\sigma(A)$ and $\hat c(A)$ can be sampled from in linear time if $A$ has tensor product structure, even if $A$ acts on many qubits.
Schrödinger Propagation
-----------------------
Suppose we want to apply a sequence of channels $\Lambda_1,\ldots,\Lambda_k$ to an $n$-qubit state $\rho_0$. These operations are given as a quantum circuit, so $\rho_0$ has tensor product structure and each of the $\Lambda_i$ non-trivially act on a constant-size subset of the qubits. Let $\rho_i$ be the state after applying the first $i$ channels: $$\rho_i = \Lambda_i(\Lambda_{i-1}(\cdots\Lambda_1(\rho_0)))$$
We are given an observable $E$ which also has tensor product structure. We want to estimate the expectation of $E$ on the final state: $$\langle E\rangle = \text{Tr}\left( E \rho_k \right) = \text{Tr}\left( E \Lambda_k(\Lambda_{k-1}(\cdots\Lambda_1(\rho_0))) \right)\label{eq:schrgoal}$$
We apply the sampling procedure defined by and to $\rho_0$. We define $\hat \sigma(\rho_0) = \hat\sigma_0$ and $\hat c(\rho_0) = \hat c_0$. Their product $\hat c_0 \cdot \hat \sigma_0$ is an unbiased estimator for $\rho_0$.
Given an unbiased estimator $\hat c_i \cdot \hat \sigma_i$ for $\rho_i$, we will obtain an unbiased estimator $\hat c_{i+1} \cdot \hat \sigma_{i+1}$ for $\rho_{i+1}$. Apply $\Lambda_{i+1}$ to $\hat c_i \cdot \hat \sigma_i$ and use linearity of $\Lambda_{i+1}$: $$\mathbb{E} \left[ \Lambda_{i+1}(\hat c_i \cdot \hat \sigma_i) \right] =\Lambda_{i+1}(\mathbb{E} \left[ \hat c_i \cdot \hat \sigma_i \right]) = \rho_{i+1}$$
We have $\Lambda_{i+1}(\hat c_i \cdot \hat \sigma_i) = \hat c_i \cdot \Lambda_{i+1}( \hat \sigma_i) $. Since $\Lambda_{i+1}$ acts non-trivially on a constant-size subset of the qubits, $\Lambda_{i+1}( \hat \sigma_i)$ has tensor product structure and we can sample using and again. Let: $$\hat \sigma_{i+1} = \hat \sigma\left(\Lambda_{i+1}( \hat \sigma_i)\right) \text{ and } \hat c_{i+1} = \hat c_{i}\cdot \hat c\left(\Lambda_{i+1}( \hat \sigma_i)\right)$$
Now we have $\hat c_{i+1} \cdot \hat \sigma_{i+1}$, an estimator for $\rho_{i+1}$, and can recursively obtain $\hat c_{k} \cdot \hat \sigma_{k}$ for $\rho_k$. Since $E$ and $\hat\sigma_k$ have tensor product structure, we can efficiently obtain their trace inner product. The protocol yields a sample from the distribution in time linear in $k+n$: $$\text{Output: sample from } \hat c_{k} \cdot \text{Tr}( \hat \sigma_k E ) \label{eq:shrout}$$ This distribution estimates the target quantity: $$\mathbb{E}\left[\hat c_{k} \cdot \text{Tr}( \hat \sigma_k E ) \right] = \text{Tr}( \mathbb{E}\left[\hat c_{k} \cdot \hat \sigma_k\right] E ) = \text{Tr}\left(\rho_k E \right) = \langle E\rangle$$
We estimate the mean of $ \hat c_{k} \cdot \text{Tr}( \hat \sigma_k E ) $ by taking the average of $N$ samples. The Hoeffding inequality [@hoeffding] provides a sufficient condition on $N$ for an additive error $\varepsilon$ with probability $1-\delta$ in terms of the range of the distribution: $$N \geq \frac{1}{2\varepsilon^2} \cdot \ln \frac{2}{\delta} \cdot (\text{range})^2\label{eq:hoeff}$$ The range of the output distribution is bounded by twice the maximum magnitude of the output distribution . $$\text{range} \leq 2 \cdot \left|\hat c_k \cdot \mathrm{Tr}(\hat \sigma_k E) \right| \leq 2\cdot |\hat c_k| \cdot \max_{\sigma \in \mathcal{P}_n}\left|\mathrm{Tr}(\sigma E)\right|$$
Observe that $\hat c(A) = \pm \mathcal{D}(A)$, so: $$\begin{aligned}
|\hat c_{i+1}| &= |\hat c_{i}|\cdot| \hat c\left(\Lambda_{i+1}( \hat \sigma_i)\right)| \nonumber\\
&= |\hat c_{i}|\cdot \mathcal{D}(\Lambda_{i+1}(
\hat \sigma_i)) \nonumber \\
&\le |\hat c_{i}| \cdot \max_{\sigma \in \mathcal{P}_n}\mathcal{D}(\Lambda_i(\sigma))\label{eq:ccost}\end{aligned}$$ Intuitively, $\mathcal{D}$ measures the “cost” of a Hermitian matrix in this algorithm. The above motivates a corresponding notion of the “cost” of a channel:
The **channel stabilizer norm** $\mathcal{D}(\Lambda)$ is defined by: $$\mathcal{D}(\Lambda) = \max_{\sigma \in \mathcal{P}_n} \mathcal{D}(\Lambda(\sigma))$$
Expanding the recursion in (\[eq:ccost\]) we obtain the bound: $$\left|\hat c_k \cdot \mathrm{Tr}(\hat \sigma_k E) \right| \le \underbrace{\vphantom{\prod_{i=1}^k}\mathcal{D}(\rho_0)}_{(1)} \cdot \underbrace{\prod_{i=1}^k \mathcal{D}(\Lambda_i)}_{(2)} \cdot \underbrace{\vphantom{\prod_{i=1}^k}\left|\max_{\sigma \in \mathcal{P}_n}\mathrm{Tr}(\sigma E)\right|}_{(3)}$$ The number of samples $N$ scales with the square of the above quantity. Thus, the cost of Schrödinger propagation on a circuit breaks into three parts: (1) the cost of the initial state, (2) the cost of each channel, and (3) the cost of the final observable.
Here are two observations:
- Say $\rho_0 = \rho^{\otimes m}$, so $\mathcal{D}(\rho_0) = \mathcal{D}(\rho)^m$. For many $\rho$ with short Bloch vectors, the cost $\mathcal{D}(\rho)$ can be strictly less than 1, meaning more copies of $\rho$ result in an exponential runtime improvement from cost term (1).
- Often we are interested in observables $E_\text{local}$ that act only on a small subset of the output qubits. Then $E$ is a tensor product of linearly many identity matrices and $E_\text{local}$, resulting in an exponential runtime blowup from cost term (3).
Loosely speaking, Schrödinger propagation works well when the input qubits are noisy and all output qubits are measured, like some supremacy circuits [@suprem].
Heisenberg Propagation
----------------------
Heisenberg propagation involves propagating the observable $E$ backwards through the circuit and taking the inner product with the initial state $\rho_0$. To do so we utilize the **channel adjoint** $\Lambda^\dagger$ which satisfies: $$\text{Tr}(E \Lambda(\rho)) = \text{Tr}(\Lambda^\dagger(E)\rho)$$
Applying this to , our goal is to estimate: $$\begin{aligned}
\langle E\rangle = \text{Tr}\left( \rho_0 \Lambda^{\dagger}_1(\cdots\Lambda^{\dagger}_{k-1}(\Lambda^{\dagger}_k(E))) \right) = \text{Tr}\left( \rho_0 E_1 \right) \nonumber\\
\text{where }E_i = \Lambda^{\dagger}_{i}(\Lambda^{\dagger}_{i+1}(\cdots\Lambda^{\dagger}_{k-1}(\Lambda^{\dagger}_k(E)))) \end{aligned}$$
For Heisenberg propagation we will define $\hat c_i, \hat \sigma_i$ differently from Schrödinger propagation. We use the sampling procedure defined by and and obtain $\hat \sigma(E) = \hat \sigma_{k+1}$ and $\hat c(E) = \hat c_{k+1}$. Then $\hat c_{k+1} \cdot \hat \sigma_{k+1}$ is an unbiased estimator for $E$.
With an unbiased estimator $\hat c_{i+1} \cdot \hat \sigma_{i+1}$ for $E_{i+1}$ we can obtain an unbiased estimator $\hat c_{i} \cdot \hat \sigma_{i}$ for $E_{i}$ from $\Lambda^{\dagger}_i(\hat c_{i+1} \cdot \hat \sigma_{i+1}) = \hat c_{i+1} \cdot \Lambda^{\dagger}_i( \hat \sigma_{i+1})$. Since $\Lambda^{\dagger}_i( \hat \sigma_{i+1})$ has tensor product structure we can sample using and , and obtain: $$\hat \sigma_{i} = \hat \sigma(\Lambda_i^\dagger(\hat\sigma_{i+1}))\text{ and } \hat c_{i} = \hat c_{i+1} \cdot \hat c(\Lambda_i^\dagger(\hat\sigma_{i+1}))$$
This operation is iterated until we obtain $\hat c_{1}\cdot \hat \sigma_1$, an unbiased estimator for $E_1$. Since $\rho_0$ has tensor product structure we can compute the trace inner product and produce a sample, again in time linear in $k+n$: $$\text{Output: sample from } \hat c_{1} \cdot \text{Tr}( \hat \sigma_1 \rho_0 ) \label{eq:heisout}$$ This estimates the target quantity: $$\mathbb{E}\left[\hat c_{1} \cdot \text{Tr}( \hat \sigma_1 \rho_0 ) \right] = \text{Tr}( \mathbb{E}\left[\hat c_{1} \cdot \hat \sigma_1\right] \rho_0 ) = \text{Tr}\left(E_1 \rho_0 \right) = \langle E\rangle$$
To bound the number of samples $N$ we bound the maximum magnitude of and utilize Hoeffding’s inequality . Since $\rho_0$ is a quantum state, we always have $\max_{\sigma \in \mathcal{P}_n}\left|\mathrm{Tr}(\sigma \rho_0)\right| = 1$ since the eigenvalues of $\sigma$ are $\pm 1$. This leaves the recursion relation:
$$\begin{aligned}
\left|\hat c_1 \cdot \mathrm{Tr}(\hat \sigma_1 \rho_0) \right| \leq |\hat c_1| &= |\hat c_{i+1}|\cdot \left| \hat c\left(\Lambda^\dagger_{i}( \hat \sigma_{i+1})\right)\right| \nonumber\\
&= |\hat c_{i+1}|\cdot \mathcal{D}(\Lambda^\dagger_{i}( \hat \sigma_{i+1})) \nonumber \\
&\le |\hat c_{i+1}| \cdot \max_{\sigma \in \mathcal{P}_n}\mathcal{D}(\Lambda^\dagger_i(\sigma))\nonumber\\
&= |\hat c_{i+1}| \cdot \mathcal{D}(\Lambda_i^\dagger) \label{eq:hccost}\end{aligned}$$
Expanding the recursion we obtain the bound: $$\left|\hat c_1 \cdot \mathrm{Tr}(\hat \sigma_1 \rho_0) \right| \le \underbrace{\vphantom{\prod_{i=1}^k}\mathcal{D}(E)}_{(1)} \cdot \underbrace{\prod_{i=1}^k \mathcal{D}(\Lambda^\dagger_i)}_{(2)}$$
The number of samples $N$ scales with the square of the cost of the observable (1) and the cost of channel adjoints (2), and is independent of the initial state.
Loosely speaking, Heisenberg propagation is efficient for *any* separable input state or stabilizer mixture and supports a wider range of observables than Schrödinger propagation. However, it cannot capitalize on particularly noisy input states for a runtime improvement.
A version of Heisenberg propagation appears in [@sampling], where they restrict operations to Clifford unitaries. Our work generalizes the technique to arbitrary quantum channels.
\[sec:states\]Efficient Circuit Components
==========================================
In this section we study which input states, channels and observables (collectively ‘circuit components’) can be simulated by Schrödinger, Heisenberg and stabilizer propagation without increasing runtime. This viewpoint helps address the practical question: “Given a particular quantum circuit, which near-Clifford algorithm is best?”
Straightaway, if the quantum circuit is unitary then stabilizer rank techniques [@extent] are the best choice due to their superior accuracy and runtime. The primary advantage of propagation algorithms is their ability to support arbitrary circuit components with noise, measurement, and adaptivity. Despite their flexibility, the propagation algorithms vary significantly in their performance.
Since the number of samples scales as the product of the square of the cost of the components, a component occurring linearly many times with cost $>1$ demands exponential runtime. In the following, when we say an algorithm **supports** or **can handle** a component, we mean that the cost of the component is $\leq 1$, although the protocols can be applied to any component possibly inefficiently.
Efficiency of Stabilizer Propagation
------------------------------------
For a self-contained description of stabilizer propagation see [@bennink; @resource; @seddon]. Just as the algorithms in section II decompose input states into a weighted sum of Pauli matrices, stabilizer propagation decomposes input states into a weighted sum of stabilizer states. A sampling process identical to equations and results in the number of samples required to be proportional to the square of the following normalization constant:
The **robustness of magic** $\mathcal{R}(\rho)$ of an $n$-qubit state $\rho$ is the outcome of a convex optimization program over real vectors $\vec q$: $$\mathcal{R}(\rho) = \min_{\vec q} \sum_i \lvert q_i \rvert \text{ s.t. } \rho = \sum_i q_i \ket{\phi_i} \bra{\phi_i} \text{ and } \sum_i q_i = 1,$$
where $\{\ket{\phi_i}\}$ are the $n$-qubit stabilizer states.
When $\mathcal{R}(\rho) = 1$ (the minimum value) then $\rho$ is a **stabilizer mixture**, since then the vector $\vec q$ is a probability distribution.
Due to the sheer number of stabilizer states, evaluating $\mathcal{R}(\rho)$ for even small $n$ is very expensive. As stated in [@bennink], evaluating the cost function for 3-qubit unitaries is impractical, although the performance can be improved for diagonal gates [@seddon].
The performance of stabilizer propagation gives a lens for the non-Cliffordness of channels, studied extensively in [@seddon]. In the appendix, we expand on this work by modifying the protocol to support all **postselective channels** which include all trace preserving channels and all ‘reasonable’ non-trace-preserving channels. There, we prove the following theorem:\
\[thm:stabprop\] Let $\Lambda$ be a postselective channel and let $\bar\phi_\Lambda$ be the channel’s normalized Choi state. $\Lambda$ does not increase the number of samples required for stabilizer propagation if and only if $\mathcal{R}(\bar\phi_\Lambda) = 1$.
This establishes simple and flexible criteria for when a circuit component does not increase the runtime of stabilizer propagation: states $\rho$ are cheap when $\mathcal{R}(\rho) = 1$ and channels $\Lambda$ are cheap if $\mathcal{R}(\bar\phi_\Lambda) = 1$.
Observables
-----------
Observables encountered in practice are usually computational basis measurements, or operators with bounded norm that can be expressed as sums of not too many Pauli matrices. Sometimes these observables are marginal: many of the qubits are not measured and traced out. Tracing out corresponds to measuring the identity observable, a kind of Pauli observable.
Stabilizer propagation outputs the inner product of the final observable with a stabilizer state. For all of the observables above, calculating inner products with stabilizer states is efficient: inner products with Pauli matrices can be obtained in $n^2$ time and marginal inner products with other stabilizer states in $n^3$ time [@tableau]. Crucially, these inner products remain bounded by the eigenvalues of the observable and thereby do not exponentially increase the range of the distribution.
Schrödinger propagation, which outputs the inner product with a Pauli matrix, does not have this property: although inner products between Pauli matrices are trivial to compute, the maximum inner product grows like $2^n$. Therefore, Schrödinger propagation is only viable when we are interested in the probability of measuring a particular state and only a constant number of discarded qubits. On the other hand, there exist contrived observables that only Schrödinger propagation can handle. If the observable is the tensor product of many non-stabilizer states, then neither Heisenberg propagation nor stabilizer propagation runs efficiently. (Indeed, calculating inner products of stabilizer states with tensor products of many non-stabilizer states is a key slow step in stabilizer rank techniques [@gosset; @extent].)
Heisenberg propagation applies the sampling method (1) (2) to the observable $E$, so cost is measured by $\mathcal{D}(E)$. The following facts, proven in [@resource], show that Heisenberg propagation can handle the observables most common in quantum circuits.
$\mathcal{D}(\sigma) = 1$ for $\sigma \in \mathcal{P}_n$.
If $\ket{\phi}$ is a stabilizer state, then $\mathcal{D}(\ket{\phi}\bra{\phi}) = 1$.
$\mathcal{D}$ is multiplicative: $\mathcal{D}(A \otimes B) = \mathcal{D}(A) \cdot \mathcal{D}(B)$.
Hyper-Octahedral States
-----------------------
A central observation of this work is that Pauli matrix decompositions can produce similar simulational power as decompositions over stabilizer states. Here we show that despite their simplicity, Pauli matrix decompositions are *more* powerful with regards to the input state of the circuit. The number of samples required for Heisenberg propagation does not depend at all on the input state (19). For Schrödinger propagation we observe:
1. there exist states supported by Schrödinger propagation unsupported by stabilizer propagation, and
2. sufficiently depolarized states can actively decrease the number of samples required.
From the definition of the stabilizer norm, $\mathcal{D}$ can be viewed as the L1 norm of the Bloch vector $\vec x$ of $\rho$. The equation $||\vec x||_1 \leq 1$ defines the surface and interior of a hyper-octahedron, motivating the following definition.
**Hyper-octahedral states** $\rho$ satisfy $\mathcal{D}(\rho) \leq 1$. These states do not increase the number of samples for Schrödinger propagation.
To see (B), we simply observe that the interior of the octahedron satisfies $\mathcal{D}(\rho) = ||\vec x||_1 < 1$. $\mathcal{D}$ is minimized at the $n$-qubit maximally mixed state where $\mathcal{D}(I/2^n) = 1/2^n$. The following result, proved in [@resource], shows that all stabilizer mixtures are hyper-octahedral.
For states $\rho$, $\mathcal{D}(\rho) \leq \mathcal{R}(\rho)$.
This fact classifies mixed states into three categories: stabilizer mixtures, non-stabilizer hyper-octahedral states, and magic states. For the single qubit, the first two categories coincide (the qubit stabilizer polytope is an octahedron). We plot a cross-section of the two-qubit Bloch sphere in FIG. \[fig:ZIIZ\], showing that all of these categories are non-empty. FIG. \[fig:states-pie\] shows the relative quantity of these states according to the Hilbert-Schmidt measure. Stabilizer mixtures occupy a tiny fraction of all mixed states, whereas more than half are hyper-octahedral.
From the standpoint of quantum resource theories, hyper-octahedral states are interesting because they are similar to the ‘bound’ states discussed in [@vcge12; @hwve14; @dh15; @acb12]: they contain non-stabilizer mixed states that can be efficiently simulated. But unlike $\mathcal{R}$, tracing out qubits can increase $\mathcal{D}$. Hadamard eigenstates $\ket{H}$ are magic states that let Clifford circuits attain universal quantum computation, but $\ket{H} \otimes (I/2)$ is hyper-octahedral. Hyper-octahedral states are not bound for magic state distillation in the same sense as those in [@vcge12]: there are operations that can be simulated efficiently by stabilizer propagation that increase $\mathcal{D}$. Schrödinger propagation cannot simulate operations that increase $\mathcal{D}$.
![\[fig:ZIIZ\] Visualization of a cross section of the two-qubit Bloch sphere, given by:\
$\rho(x,y) = \frac{\sigma_{II}}{4} + x(\sigma_{XX} + \sigma_{ZZ} - \sigma_{YY}) + y(\sigma_{ZI} + \sigma_{IZ})$ ](ZIIZ.pdf){width="35.00000%"}
![\[fig:states-pie\] Relative quantity of two-qubit mixed states, based on one million samples via the Hilbert-Schmidt measure. Hyper-octahederal states are plentiful for two-qubits, despite not existing for the single qubit.](states-pie.pdf){width="35.00000%"}
Channel Classification
----------------------
While the classification of states gave rise to only three categories, the classification of channels is not so simple. FIG. \[fig:Venn\] shows eight categories, all of which are non-empty. Here are examples of each:
M
: Non-Clifford unitaries, such as the $T$ gate.
CSH
: Clifford unitaries, measuring a qubit in a Pauli basis (without discarding it), and very depolarized non-Clifford unitaries.
SH
: Mildly depolarized non-Clifford unitaries, e.g. the $T$ gate with fidelity $0.551 \lessapprox f \leq 2^{-1/2}$ (FIG. \[fig:noisy\_z\_theta\]).
C
: Most adaptive Clifford gates: gates performed based on the outcome of a measurement (Proposition \[thm:adapt\]).
H
: Any non-Pauli reset channel (Proposition \[thm:reset\]).
CH
: Pauli reset channels [@bennink].
S, CS
: Channels adjoints for H, HC, respectively.
To obtain the relative proportions of these categories akin to FIG. \[fig:states-pie\] we leverage channel-state duality. Our definition of postselective channels in the appendix is specifically chosen to make the correspondence between two-qubit mixed states and qubit-to-qubit channels a bijection. We sample states according to the Hilbert-Schmidt measure and classify their corresponding channels. Most channels in practice are either unital, trace preserving or both. It is not obvious how to restrict sampling to these measure-zero subspaces. Instead, we sample from the full Hilbert Schmidt measure, and then project onto the Bloch-subspaces corresponding to unital and/or trace preserving channels.
FIG. \[fig:channel-pies\] shows the resulting proportions. For qubit-to-qubit channels, Pauli propagation techniques permit simulation of a significant fraction of the circuit components which are a superset of those simulable by stabilizer propagation. As before, it is not clear that this demonstrates that Pauli propagation is significantly more useful in practice, since most quantum circuits are dominated by a few specific types channels.
(3.5, 1.5) circle \[radius = 1.5\];; (2.5, 0) circle \[radius = 1.5\];; (4.5, 0) circle \[radius = 1.5\];; at (6, 2.5) [$\mathcal{R}(\bar \phi_\Lambda) = 1$]{}; at (6, 2) [Stabilizer]{}; at (6, 1.7) [Propagation]{}; at (0, -0.5) [$\mathcal{D}(\Lambda) \leq 1$]{}; at (0, -1) [Schrödinger]{}; at (0, -1.3) [Propagation]{}; at (7, -0.5) [$\mathcal{D}(\Lambda^\dag) \leq 1$]{}; at (7, -1) [Heisenberg]{}; at (7, -1.3) [Propagation]{}; at (2.25, -0.35) [S]{}; at (2.75, 0.85) [CS]{}; at (3.5, 2) [C]{}; at (3.5, 0.5) [CSH]{}; at (3.5, -0.35) [SH]{}; at (4.25, 0.85) [CH]{}; at (4.75, -0.35) [H]{}; at (1.2, 2) [M]{};
In the following we give evidence for the above examples. To do so, we phrase $\mathcal{D}(\Lambda)$ in terms of the Pauli transfer matrix of $\Lambda$.
The Pauli Transfer Matrix (PTM) of a quantum channel $\Lambda$ taking $n$ qubits to $m$ qubits has elements $(R_\Lambda)_{ij} = 2^{-m}\operatorname{Tr}(\sigma_i\Lambda(\sigma_j))$ such that $\Lambda(\rho) = 2^{-n}\sum_{i, j} (R_\Lambda)_{ij} \sigma_i \operatorname{Tr}(\rho \sigma_j)$. We take $\sigma_{1} = I$.
Intuitively, the columns of $R_\Lambda$ are the Bloch vectors of $\Lambda(\sigma_i)$. The following observations are useful and trivial to prove.
$D(\Lambda) = \lVert R_\Lambda \rVert_1$, where $\lVert \cdot \rVert_1$ is the induced L1-norm, i.e. the largest column L1-norm.
$R^T_\Lambda = R_{\Lambda^\dagger}$
[\[thm:D\_dag\] ]{} $D(\Lambda^\dagger) = \lVert R_\Lambda \rVert_\infty$, where $\lVert \cdot \rVert_\infty$ is the induced L$\infty$-norm, i.e. the largest row L1-norm.
The PTM of a Clifford gate is a signed permutation matrix and the PTMs of Pauli basis measurements are signed permutations of $\text{diag}(1,1,0,0)$. Their Choi states are also readily shown the be stabilizer mixtures, so these channels are CSH as claimed.\
**Relative Quantity of Qubit-To-Qubit Channels**\
{width="\textwidth"}
Depolarized Rotations
---------------------
Many useful unitaries take the form $e^{-i\theta \sigma/2}$ with $\sigma \in \mathcal{P}_n$. Via some Clifford transformations these can be obtained from the qubit unitary $e^{-i\theta \sigma_Z/2}$. In this section we consider composing this unitary with depolarizing noise, obtaining a family of channels $\Lambda_{\theta,f}$ where $f$ is the fidelity.
The PTMs of the unitary $e^{-\theta\sigma_Z/2}$ and depolarizing noise are respectively: $$R_{\theta} =
\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & \cos \theta & -\sin \theta & 0\\
0 & \sin \theta & \cos \theta & 0\\
0 & 0 & 0 & 1
\end{bmatrix}\\
\hspace{5mm} R_f =
\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & f & 0 & 0\\
0 & 0 & f & 0\\
0 & 0 & 0 & f
\end{bmatrix}$$
Composing these two channels simply involves multiplying the two PTMs, resulting in: $$\begin{aligned}
R_{\Lambda_{f, \theta}} &=&
\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & f\cos \theta & -f\sin \theta & 0\\
0 & f\sin \theta & f \cos \theta & 0\\
0 & 0 & 0 & f
\end{bmatrix}\\
\mathcal{D}(\Lambda_{f, \theta}) = \mathcal{D}(\Lambda_{r, \theta}^\dag) &=& \max\big(1, f\lvert \cos \theta \rvert + f\lvert \sin \theta \rvert\big)\end{aligned}$$
![\[fig:noisy\_z\_theta\] Qubit quantum channels $\Lambda_{f,\theta}$ obtained by an application of the unitary $e^{-i\theta \sigma_Z/2}$ followed by depolarizing noise with fidelity $f$. The region simulable by Pauli propagation (SH) is larger than that simulable by stabilizer propagation (CSH).](rtheta-family.pdf){width="40.00000%"}
We plot the family in FIG. \[fig:noisy\_z\_theta\], showing that there are channels simulable by Pauli propagation methods that are not simulable by stabilizer propagation. The boundary of $\mathcal{D} \leq 1$ given by $\lvert \cos \theta \rvert + \lvert \sin \theta \rvert = 1$ forms a diamond. The depolarized $T$ gate becomes SH when $f \leq 2^{-1/2} \approx 0.707$, and becomes CSH when $f \lessapprox 0.551$.
Reset Channels
--------------
Pauli reset channels can be described as projecting into the $+1$ eigenspace of some $\sigma \in \mathcal{P}_n $ as in [@bennink]. Alternatively we can use Clifford transformations to convert $\sigma$ to $\sigma_Z$, converting the channel to tracing out a single qubit and replacing it with $\ket{0}$. We generalize the notion of a reset channel $\Lambda_\rho$ to tracing out $n$ qubits and replacing them with an $n$-qubit state $\rho$. To make the channel trace preserving we write $\Lambda_\rho(\sigma) = \text{Tr}(\sigma) \cdot \rho$.
[\[thm:reset\]]{} If $\Lambda_\rho$ is a reset channel, $\mathcal{D}(\Lambda^\dagger) = 1$.
The entries of the PTM of $\Lambda_\rho$ are the following: $$(R_{\Lambda_\rho})_{ij} = 2^{-n}\operatorname{Tr}(\sigma_i\Lambda_\rho(\sigma_j)) =
\begin{cases}
2^{-n}\operatorname{Tr}(\sigma_i \rho) & \sigma_j = I\\
0 & \sigma_j \neq I
\end{cases}$$ All rows except for the first are zero. The entries are bounded $-1 \leq 2^{-n}\operatorname{Tr}(\sigma_i \rho) \leq 1$ and the top left entry is 1. Thus the maximum column L1 norm is 1, and Proposition \[thm:D\_dag\] tells us that $\mathcal{D}(\Lambda^\dag) = 1$.
Observe that the first row is actually the Bloch vector of $\rho$ (including the identity component) scaled by $2^n$. So unless $\rho$ is the maximally mixed state the first row’s L1 norm is $> 1$, so the channel is not simulable by Schrödinger propogation, and its adjoint is not simulable by Heisenberg propogation.
The Choi state of $\Lambda_\rho$ is $\frac{I}{2^n} \otimes \rho$, so $\Lambda_\rho$ is simulable by stabilizer propagation when $\rho$ is a stabilizer mixture.
Adaptive Channels
-----------------
Adaptive channels consist of making a $\sigma_Z$ measurement, and then conditionally applying a channel based on the measurement outcome. While Pauli propagation techniques are stronger than stabilizer propagation in many respects, adaptive channels are their key weak point. This remains true even if the measured qubit is *not* discarded, so we are not conflating the cost of tracing out qubits with the cost of adaptivity.
[\[thm:adapt\]]{} Let $\Lambda$ be a quantum channel with PTM $R_\Lambda$. Let $A(\Lambda)$ be the adaptive channel that conditionally applies $\Lambda$ based on a $\sigma_Z$ measurement on some qubit that is not discarded post-measurement. Then: $$\begin{aligned}
\mathcal{D}(A(\Lambda)) &=& 1 + \max_{i} \sum_{i\neq j} |R_{ij}| \leq 1 + \mathcal{D}(\Lambda^\dag)\\
\mathcal{D}(A(\Lambda)^\dagger) &=& 1 + \max_{j} \sum_{i\neq j} |R_{ij}| \leq 1 + \mathcal{D}(\Lambda)\end{aligned}$$
$A(\Lambda)$ is supported by Pauli propagation methods if and only if the PTM of $\Lambda$ is diagonal.
So Pauli propagation methods are not ‘closed under adaptivity’: $A(\Lambda)$ can be non-simulable even if $\Lambda$ is simulable. Stabilizer propagation on the other hand *is* closed under adaptivity.
Let $\Lambda$ take $n$ qubits to $m$ qubits. The measurement of the first qubit projects into the space spanned by $I,\sigma_Z$ on the first qubit.
$$\begin{aligned}
A(\Lambda)(I \otimes \sigma_j) &=
\begin{pmatrix}
\sigma_j & 0\\
0 & \Lambda(\sigma_j)
\end{pmatrix} &=
\begin{pmatrix}
\sigma_j & 0\\
0 & \sum_k R_{kj} \sigma_k
\end{pmatrix}\\
A(\Lambda)(\sigma_Z \otimes \sigma_j) &=
\begin{pmatrix}
\sigma_j & 0\\
0 & -\Lambda(\sigma_j)
\end{pmatrix} &=
\begin{pmatrix}
\sigma_j & 0\\
0 & -\sum_k R_{kj} \sigma_k
\end{pmatrix}\end{aligned}$$
The output remains in the space spanned by $I,\sigma_Z$ on the first qubit, so the only nonzero entries of the PTM are: $$\begin{aligned}
\frac{1}{2^{m+1}}\text{Tr}\left( (I \otimes \sigma_i) \cdot A(\Lambda)(I \otimes \sigma_j) \right) = \frac{1}{2}(\delta_{ij} + R_{ij}) \\
\frac{1}{2^{m+1}}\text{Tr}\left( (\sigma_Z \otimes \sigma_i) \cdot A(\Lambda)(I \otimes \sigma_j) \right) = \frac{1}{2}(\delta_{ij} - R_{ij}) \\
\frac{1}{2^{m+1}}\text{Tr}\left( (I \otimes \sigma_i) \cdot A(\Lambda)(\sigma_Z \otimes \sigma_j) \right) = \frac{1}{2}(\delta_{ij} - R_{ij}) \\
\frac{1}{2^{m+1}}\text{Tr}\left( (\sigma_Z \otimes \sigma_i) \cdot A(\Lambda)(\sigma_Z \otimes \sigma_j) \right) = \frac{1}{2}(\delta_{ij} + R_{ij}) \end{aligned}$$ Applying the definition of channel stabilizer norm: $$\begin{aligned}
\mathcal{D}(A(\Lambda)) = \frac{1}{2} \max_i \sum_j \left(|\delta_{ij} + R_{ij}| + |\delta_{ij} - R_{ij}|\right)\\ = 1 + \max_i \sum_{i\neq j} |R_{ij}| \hspace{4mm}\square\end{aligned}$$
Numerical Results
=================
Algorithms based on Monte Carlo averages have favorable memory requirements and admit massive parallelization. We demonstrate these practical advantages via the performance of a GPU implementation written in CUDA [@cuda].
Following previous tests of near-Clifford algorithms [@extent] we simulate the Quantum Approximate Optimization Algorithm (QAOA) on E3LIN2 [@QAOAE3LIN2]. We generate $m$ random independent linear equations acting on three qubits $a,b,c \in [n]$ of the form $x_a \oplus x_b \oplus x_c = d_j$ for $j \in [m]$. Each qubit appears in at most $m/10$ equations. Let $\sigma^{(j)}_Z = \sigma_{Z,a} \otimes \sigma_{Z,b} \otimes\sigma_{Z,c}$ be $\sigma_Z$ acting on the qubits corresponding to equation $j$. Our goal is to estimate the observable $$C = \frac{1}{2} \sum_{j \in [m]} (-1)^{d_j} \sigma^{(j)}_{Z}$$ since $C+m/2$ is the number of satisfied equations. We estimate the expectation of this observable with the state $$\ket{\gamma,\beta} = e^{-i \beta B} e^{-i\gamma C}\ket{+^{\otimes n}}$$ where $B = \sum_{i \in [n]} \sigma_{X,i}$ and $\beta = \pi/4$.
Heisenberg propagation is most appropriate for this problem, with performance $\mathcal{D}(C) = m/2$ and $\mathcal{D}(e^{\pm i\gamma \sigma^{(j)}_Z}) =|\sin \gamma| + |\cos \gamma| $. Although the unitary $e^{\pm i\gamma \sigma^{(j)}_Z}$ appears $m$ times in the circuit, at most $3(m/10 -1) + 1$ can act non-trivially on any term in $C$. Thus the accuracy of the simulation is given by: $$\varepsilon_\text{Heis} = \frac{m}{\sqrt{2N}} \cdot \sqrt{\ln \frac{2}{\delta}} \cdot (|\sin \gamma| + |\cos \gamma|)^{3(m/10 -1) + 1}$$
As pointed out by [@extent], a protocol by van den Nest [@nest] gives an *efficient* Monte Carlo protocol for estimating $\langle C \rangle$ with error $\varepsilon_\text{Nest} =\frac{m}{\sqrt{N}} \cdot \sqrt{\ln \frac{2}{\delta}} $. We utilize the van den Nest estimate $\langle C\rangle_\text{Nest}$ to verify the Heisenberg propagation estimate $\langle C\rangle_\text{Heis}$.
Writing effective CUDA applications demands careful memory management. Implementing stabilizer propagation via the Aaronson-Gottesman tableau algorithm would be a serious computer engineering task. In contrast, the increased simplicity of Pauli propagation algorithms permits a very simple implementation. We furthermore utilize bitwise operations to express the logic in a compact and efficient manner. Despite the better scaling it was ultimately necessary to also implement the van den Nest protocol in CUDA due to the sheer performance improvement over a Python implementation.
![\[fig:QAOA\] Comparison of Hoeffding error bound $\varepsilon_\text{Heis}$ to error as estimated by the van den Nest protocol $|\langle C\rangle_\text{Nest} - \langle C\rangle_\text{Heis}|$ for 32 qubits. Top: $m = 40$ and varying $\gamma$. Bottom: $\gamma = \pi/8$ and varying $m$.](QAOA.pdf){width="50.00000%"}
For every data point we collected $2^{30} \approx 1$ billion samples in 25 minutes using a laptop GPU (GeForce GTX 1050 Ti). We fix $n = 32$ qubits and $\delta = 0.01$ throughout, and vary $\gamma$ for a single instance with $m = 40$ equations (Figure \[fig:QAOA\], top). Then we set $\gamma = \pi/8$, maximizing $\mathcal{D}(e^{\pm i\gamma \sigma^{(j)}_Z}) $ at $\sqrt{2}$, and perform a scaling analysis with instances up to $m = 80$ (Figure \[fig:QAOA\], bottom).\
Hoeffding’s inequality gives a worst-case upper bound for the accuracy of the estimate, potentially very far from the actual error. This is the case here: for $m \gtrapprox 60$ we have $\varepsilon_\text{Heis} \geq 1$ predicting that $\langle C\rangle_\text{Heis}$ is useless, but we observe that the actual error is $\leq 0.01$. Furthermore the actual accuracy does not seem to scale proportionally with $\varepsilon_\text{Heis}$ as we vary $\gamma$ and $m$.
Conclusion {#conclusion .unnumbered}
==========
Recent interest in near-Clifford simulation [@bennink; @pash; @gosset; @extent] and the (non-)contextuality of Clifford circuits [@hwve14; @rbdobv15; @rbdobv16; @dovbr16] demonstrates that there is still much to be learned about embedding symmetry into Hilbert space. The qubit Clifford group appears different from the Clifford group in odd dimensions, where the discrete Wigner function [@gross] has led to well-behaved resource theories [@resource; @vcge12; @vmge13] and associated simulation algorithms [@sampling]. We observe that the qubit analogue of the Wigner function is just a Bloch vector, and our analysis of the resulting algorithms sheds further light into the differences between the even and odd-dimensional cases. Furthermore, the simplicity of Pauli propagation algorithms along with their improved performance for many quantum channels make them a compelling addition to near-Clifford simulation techniques.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by Dr. Scott Aaronson (UT Austin CS), who gave us invaluable advice throughout the project. We thank Dr. Antia Lamas-Linares (TACC) for giving us access to a plethora of supercomputing resources. We thank Dr. David Gross (Univ. of Colgone) as well as Dr. Earl Campbell, Dr. Mark Howard and James Seddon (Univ. of Sheffield) for useful suggestions and for coming up with creative names for some of the concepts introduced in this paper. We thank David McKnight, DeVon Ingram and Adrian Trejo Nuñez for incisive editing feedback.
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\[sec:Postselective\]Postselective Quantum Channels
===================================================
In this section we define postselective quantum channels. These include all trace preserving channels, and all ‘sensible’ non-trace-preserving channels. Furthermore, there is a bijection between postselective channels taking $\mathcal{H}^A$ to $\mathcal{H}^B$ and density operators on $\mathcal{H}^B \otimes \mathcal{H}^A$, which is essential for FIG. 5 and Theorem \[thm:stabprop\].
Completely positive maps $\Lambda$ with $0 \leq \text{Tr}(\Lambda(\rho)) \leq \text{Tr}(\rho)$ have an operational interpretation: the associated channels can ‘fail’ or ‘abort’ the computation by yielding 0. For example, let $\Lambda$ be the channel that measures in the $\sigma_Z$ basis and postselects on obtaining $\ket{0}$. Then $\Lambda(\ket{1}\bra{1}) = 0$, and $\Lambda(\ket{+}\bra{+}) = \frac{1}{2} \ket{0}\bra{0}$.
Let $\Lambda$ be a completely positive map from $\mathcal{H}^A$ to $\mathcal{H}^B$. Let $\ket{\text{Bell}_A} \in \mathcal{H}^A \otimes \mathcal{H}^A$ be a Bell state for $\mathcal{H}^A$, i.e. if $\{\ket{i}\}$ are a basis for $\mathcal{H}^A$ then:
$$\ket{\text{Bell}_A} = \frac{1}{\sqrt{\text{dim}(\mathcal{H}^A)}} \sum_i \ket{i} \otimes \ket{i}$$
The **un-normalized Choi state** $\phi_\Lambda$ of $\Lambda$ is the resulting state when $\Lambda$ is applied to one half of $\ket{\text{Bell}_A}$.
$$\phi_\Lambda = (\Lambda \otimes I)(\ket{\text{Bell}_A}\bra{\text{Bell}_A} ) \in \mathcal{H}^B \otimes \mathcal{H}^A$$
$\text{Tr}(\phi_\Lambda$) of $\Lambda$ can be less than 1 if $\Lambda$ is not trace preserving. Let $\bar \phi_\Lambda = \phi_\Lambda/\text{Tr}(\phi_\Lambda)$ be the **normalized Choi state** with trace 1. This distinction is crucial.
To calculate the output of a channel $\Lambda(\rho)$ given its Choi state $\phi_\Lambda$ we compute: $$\Lambda(\rho) = \text{dim}(\mathcal{H}^A)\cdot \text{Tr}_A\left(\phi_\Lambda ( I \otimes \rho^T)\right)\label{eq:choiapply}$$
Crucially we use $ \phi_\Lambda$, not $\bar \phi_\Lambda$. To explain why, consider a Choi state $\bar \phi_\Lambda = \ket{00}\bra{00}$. If we apply the equation above to $\bar \phi_\Lambda$ we obtain $\Lambda(\rho) = 2 \cdot \ket{0}\bra{0} \cdot \bra{0}\rho^T\ket{0}$, so $\Lambda(\ket{0}\bra{0}) = 2\ket{0}\bra{0}$ which makes no sense. The fact that $\phi_\Lambda$ is under-normalized takes care of this constant.
Given a normalized Choi state $\bar \phi_\Lambda$, e.g. $\ket{00}\bra{00}$, how do we determine $ \phi_\Lambda$? In general, $\phi_\Lambda$ is not unique. Consider channels $\Lambda(\rho)$ and $\Lambda'(\rho) = p\cdot0 + (1-p)\Lambda(\rho)$, i.e. $\Lambda'$ aborts with probability $p$ and otherwise applies $\Lambda$. Both channels have the same $\bar \phi_\Lambda$, but $\phi_{\Lambda'} = p \phi_\Lambda$.
However, $\Lambda'$ is somewhat silly: aborting the computation should be a tool for postselection and should not happen regardless of the input state. For all sensible channels there should exist an input state where the postselection succeeds with probability 1. To associate all $\bar \phi_\Lambda$ to a unique $\phi_\Lambda$ we restrict our attention to the following quantum channels.
A completely positive map $\Lambda$ represents a **postselective quantum channel** if:
1. $\Lambda$ is trace-non-increasing: for all positive-semidefinite $\rho$, $\Lambda$ satisfies $0 \leq \text{Tr}(\Lambda(\rho)) \leq \text{Tr}(\rho)$,
2. the postselection can be satisfied: there exists a normalized pure state $\ket{\psi}$ such that $\text{Tr}(\Lambda(\ket{\psi}\bra{\psi})) = 1$.
Among these channels we can uniquely obtain $\phi_{\Lambda}$ from $\bar \phi_{\Lambda}$, so there is a bijection between normalized mixed states and postselective quantum channels. Let $ \phi_{\Lambda} = p_\Lambda \bar \phi_{\Lambda}$. Then: $$\frac{1}{p_\Lambda} = \text{dim}(\mathcal{H}^A)\cdot \max_{\ket{\psi}} \text{Tr}\left( \bar\phi_{\Lambda} (I \otimes (\ket{\psi}\bra{\psi})^T) \right) \label{eq:plambda}$$
For example, if $\bar \phi_{\Lambda} = \ket{00}\bra{00}$ then $\ket{\psi} = \ket{0}$ maximizes $1/p_\Lambda$ at 2, so $\phi_{\Lambda} = \frac{1}{2}\ket{00}\bra{00}$ and $\Lambda(\rho) = \ket{0}\bra{0} \cdot \bra{0}\rho^T\ket{0}$. Incidentally, $p_\Lambda$ is the probability of postselection succeeding when $\Lambda$ is applied to the Bell state.
\[sec:R\_channel\_proof\]Simulating Channels whose Choi States are Stabilizer Mixtures
======================================================================================
In this appendix we prove Theorem \[thm:stabprop\]: Stabilizer propagation can efficiently simulate a quantum channel $\Lambda$ if and only if the robustness of its Choi state $\mathcal{R}(\phi_\Lambda)$ is 1. This criterion also captures postselective quantum channels, and thereby all sensible non-trace-preserving channels.
All results of [@seddon] generalize neatly to postselective channels. Assuming familiarity with the work, the definition of magic capacity $\mathcal{C}(\Lambda)$ remains identical and the channel robustness $\mathcal{R}_*(\Lambda)$ can be obtained via convex optimization over linear combinations of *un-normalized* Choi states of stabilizer channels. It is easy to see that Theorem 2, $\mathcal{R}(\phi_\Lambda) \leq \mathcal{C}(\Lambda) \leq \mathcal{R}_*(\Lambda)$, still holds. Their Lemma 2, $\mathcal{R}(\bar \phi_\Lambda) = 1$ implies $\mathcal{C}(\Lambda) = 1$, is our Theorem 3.2.
Consider a postselective channel $\Lambda: \mathcal{H}^A \to \mathcal{H}^B$. The following statements are equivalent.
1. The channel’s normalized Choi state $\bar \phi_\Lambda$ is a probabilistic mixture of stabilizer states, so $\mathcal{R}(\bar\phi_\Lambda) = 1$.
2. If $\Lambda$ is applied to any subset of the qubits of any large stabilizer state $\ket{\psi}$, one can efficiently sample from a probability distribution over stabilizer states and ‘abort’ whose mean is the resulting state.
Say a channel $\Lambda$ is simulable. Apply $\Lambda$ to one half of the state $\ket{\text{Bell}_A}$, a stabilizer state. The resulting Choi state is probabilistic mixture of stabilizer states and ‘abort’:
$$\begin{aligned}
\phi_\Lambda = p_0 \cdot 0 + \sum_i p_i \ket{\phi_i}\bra{\phi_i}\\
\bar\phi_\Lambda = \frac{\phi_\Lambda}{\text{Tr}(\phi_\Lambda)} = \frac{1}{1-p_0} \sum_i p_i \ket{\phi_i}\bra{\phi_i}\end{aligned}$$
Since $p_i / (1-p_0)$ is a probability distribution, $\bar\phi_\Lambda$ is also a probabilistic mixture of stabilizer states.
Say we are given $$\bar\phi_{\Lambda} = \sum_i p_i \bar\phi_{\Gamma_i}$$ where $\bar \phi_{\Gamma_i}$ are *pure* stabilizer states with corresponding pure operations $\Gamma_i$. Our goal is to obtain an efficiently computable probability distribution over stabilizer states and ‘abort’ of $\Lambda$ applied to some subset of the qubits of a stabilizer state $\ket{\psi}$. The channel acts on a constant number of qubits, so we can compute anything we want about it. The stabilizer state, however, may live in a Hilbert space of exponential dimension. Using $ \phi_{\Lambda} = p_\Lambda \bar \phi_{\Lambda}$: $$\phi_{\Lambda} = p_\Lambda \sum_i \frac{p_i}{p_{\Gamma_i}} \phi_{\Gamma_i}$$ All of the quantities $p_\Lambda, p_i$ and $p_{\Gamma_i}$ can be obtained quickly. Now we apply (\[eq:choiapply\]), but we extend $\Lambda$ and $\Gamma_i$ from the constant size Hilbert space to $\tilde\Lambda$ and $\tilde \Gamma_i$ which act on the large Hilbert space containing $\ket{\psi}$. $$\tilde\Lambda(\ket{\psi}\bra{\psi}) = p_\Lambda \sum_i \frac{p_i}{p_{\Gamma_i}} \tilde \Gamma_i(\ket{\psi}\bra{\psi})$$
Crucially, $\tilde \Gamma_i(\ket{\psi}\bra{\psi})$, the right-hand side of (\[eq:choiapply\]), is an inner product between pure stabilizer states $\phi_\Gamma$ and $\ket{\psi}$ and is therefore a pure stabilizer state that can be computed in polynomial time. Since $\tilde \Gamma_i$ may be non-trace-preserving, $\tilde \Gamma_i(\ket{\psi}\bra{\psi})$ may not be normalized. Let $\ket{\gamma_i}$ be the normalized pure stabilizer state: $$\ket{\gamma_i}\bra{\gamma_i} = \tilde \Gamma_i(\ket{\psi}\bra{\psi})\Big/ \text{Tr}(\tilde \Gamma_i(\ket{\psi}\bra{\psi}))\hspace{5mm}$$ We write $\tilde\Lambda(\ket{\psi}\bra{\psi})$ as a weighted sum over normalized pure stabilizer states $\ket{\gamma_i}$. $$\tilde\Lambda(\ket{\psi}\bra{\psi}) = \sum_i p_\Lambda \frac{p_i}{p_{\Gamma_i}} \text{Tr}(\tilde \Gamma_i(\ket{\psi}\bra{\psi})) \cdot \ket{\gamma_i}\bra{\gamma_i}\hspace{5mm}$$
The weights are positive and one can see that they sum to less than 1 by taking the trace of both sides. Furthermore since $\bar\phi_{\Gamma_i}$ are pure stabilizer states, the number $ \text{Tr}(\tilde \Gamma_i(\ket{\psi}\bra{\psi})) $ and stabilizer state $\ket{\gamma_i}\bra{\gamma_i}$ are efficiently computable.
Thus, to simulate $\Lambda$ acting on $\ket{\psi}$ we sample: $$\begin{aligned}
\ket{\gamma_i}\bra{\gamma_i} &\text{ w.p. }& p_\Lambda \frac{p_i}{p_{\Gamma_i}} \text{Tr}(\tilde \Gamma_i(\ket{\psi}\bra{\psi})) \\
0 &\text{ w.p. }& 1- \sum_i p_\Lambda \frac{p_i}{p_{\Gamma_i}} \text{Tr}(\tilde \Gamma_i(\ket{\psi}\bra{\psi})).\hspace{1mm}\square\end{aligned}$$
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Robin J. Evans\
University of Oxford\
`[email protected]`
title: 'Model selection and local geometry.'
---
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Component-based software development has posed a serious challenge to system verification since externally-obtained components could be a new source of system failures. This issue can not be completely solved by either model-checking or traditional software testing techniques alone due to several reasons: (1) externally obtained components are usually unspecified/partially specified; (2) it is generally difficult to establish adequacy criteria for testing a component; (3) components may be used to dynamically upgrade a system. This paper introduces a new approach (called [*model-checking driven black-box testing*]{}) that combines model-checking with traditional black-box software testing to tackle the problem in a complete, sound, and automatic way. The idea is to, with respect to some requirement (expressed in CTL or LTL) about the system, use model-checking techniques to derive a condition (expressed in a communication/witness graph) for an unspecified component such that the system satisfies the requirement iff the condition is satisfied by the component. The condition’s satisfiability can be established by testing the component with test-cases generated from the condition on-the-fly. In this paper, we present algorithms for model-checking driven black-box testing, which handle both CTL and LTL requirements for systems with unspecified components. We also illustrate the ideas through some examples.'
author:
- |
Gaoyan Xie [ and ]{} Zhe Dang\
\
\
bibliography:
- 'fse04.bib'
subtitle: '\[Extended Abstract\]'
title: 'Model-checking Driven Black-box Testing Algorithms for Systems with Unspecified Components'
---
\[Formal methods, Model-checking\] \[Black-box testing\] \[Temporal Logic\]
Introduction
============
Component-based software development [@KB98; @BW98] is a systematic engineering method to build software systems from prefabricated software components that are previously developed by the same organization, provided by third-party software vendors, or even purchased as commercial-off-the-shelf (COTS) products. Though this development method has gained great popularity in recent years, it has also posed serious challenges to the quality assurance issue of component-based software since externally obtained components could be a new source of system failures. The issue is of vital importance to safety-critical and mission-critical systems. For instance, in June 1996, during the maiden voyage of the Ariane 5 launch vehicle, the launcher veered off course and exploded less than one minute after taking off. The report [@Ariane] of the Inquiry Board indicates that the disaster resulted from insufficiently tested software reused from the Ariane 4. The developers had reused certain Ariane 4 software component in the Ariane 5 without substantially testing it in the new system, having assumed that there were no significant differences in these portions of the two systems.
Most of the current work addresses the issue from the viewpoint of component developers: how to ensure the quality of components before they are released. However, this view is obviously insufficient: an extensively tested component (by the vendor) may still not perform as expected in a specific deployment environment, since the systems where a component could be deployed may be quite different and diverse and they may not be tried out by its vendor. So, we look at this issue from system developers’ point of view:
> (\*) [*how to ensure that a component functions correctly in the host system where the component is deployed.*]{}
In practice, testing is almost the most natural resort to resolve this issue. When integrating a component into a system, system developers may have three options for testing: (1) trust the component provider’s claim that the component has undergone thorough testing and then go ahead to use it; (2) extensively retest the component alone; (3) hook the component with the system and conduct integration testing. Unfortunately, all of the three options have some serious limitations. Obviously, for systems requiring high reliability, the first option is totally out of the question. The second option may suffer from the following fact. Software components are generally built with multiple sets of functionality [@GL02], and indiscriminately testing all the functionality of a software component is not only expensive but sometimes also infeasible, considering the potentially huge state space of the component interface. Additionally, it is usually difficult to know when the testing over the component is adequate. The third option is not always applicable. This is because, in many applications, software components could be applied for dynamic upgrading or extending a running system [@SZY03] that is costly or not supposed to shut down for retesting at all. Even without all the above limitations, purely testing-based strategies are still not sufficient to establish the solid confidence for a reliable component required by mission-critical or safety-critical systems, where formal methods like model-checking are highly desirable. However, one fundamental obstacle for using a formal method to address the issue of (\*) is that design details or source code of an externally obtained software component is generally not fully available to the developers of its host system. Thus, existing formal verification techniques (like model-checking) are not directly applicable.
Clearly, this problem plagues both component-based software systems and some hardware systems with a modularized design. Generally, we call such systems as [*systems with unspecified components*]{} (in fact, in most cases, the components are partially specified to which our approach still applies.).
In this paper, we present a new approach, called [*model-checking driven black-box testing*]{}, which combines model-checking techniques and black-box testing techniques to deal with this problem. The idea is simple yet novel: with respect to some temporal requirement about a system with an unspecified component, a model-checking based technique is used to derive automatically a condition about the unspecified component from the rest of the system. This condition guarantees that the system satisfies the requirement iff the condition is satisfied by the unspecified component, which can be checked by adequate black-box testing over the unspecified component with test-cases generated automatically from the condition.
We provide algorithms for both LTL and CTL model-checking driven black-box testing. In the algorithms, the condition mentioned earlier is represented as communication graphs and witness graphs, on which a bounded and nested depth-first search procedure is employed to run black-box testing over the unspecified component. Our algorithms are both sound and complete.
Though we do not have an exact complexity analysis result, our preliminary studies show that, in the liveness testing algorithm for LTL, the maximal length of test-cases run on the component is bounded by $O(n\cdot m^2)$. For CTL, the length is bounded by $O(k\cdot n\cdot m^2)$. In here, $k$ is the number of CTL operators in the formula to be verified, $n$ is the state number in the host system, and $m$ is the state number in the component.
The advantages of our approach are obvious: a stronger confidence about the reliability of the system can be established through both formal verification and adequate functional testing; system developers can customize the testing with respect to some specific system properties; intermediate model-checking results (the communication and witness graphs) can be reused to avoid (repetitive) integration testing when the component is updated, if only the new component’s interface remains the same; our algorithms are both sound and complete; most of all, the whole process can be carried our in an automatic way.
The rest of this paper is organized as follows. Section \[prel\] provides some background on temporal logics LTL and CTL along with our model of systems containing unspecified components. The main body of the paper consists of Section \[ltltesting\] and Section \[ctltesting\], which propose algorithms for LTL and CTL model-checking driven black-box testing, respectively, over the system model. Section \[examples\] illustrates the algorithms through an example. Section \[relatedwork\] lists some of the related work. Section \[discussions\] concludes the paper with some further issues to be resolved in the future.
Details on some algorithms are omitted in this extended abstract. At http://www.eecs.wsu.edu/$\sim$gxie, a full version of this paper is available.
Preliminaries {#prel}
=============
The System Model
----------------
In this paper, we consider systems with only one unspecified component (the algorithms generalize to systems with multiple unspecified components). Such a system is denoted by $$Sys=\langle M, X \rangle,$$ where $M$ is the host system and $X$ is the unspecified component. Both $M$ and $X$ are finite-state transition systems communicating synchronously with each other via a finite set of input and output symbols.
Formally, the unspecified component $X$ is defined as a deterministic Mealy machine whose internal structure is unknown (but an implementation of $X$ is available for testing). We write $X$ as a triple $\langle\Sigma,
\nabla, m\rangle$, where $\Sigma$ is the set of $X$’s input symbols, $\nabla$ is the set of $X$’s output symbols, and $m$ is an upper bound for the number of states in $X$ (as a convention in black-box testing, the $m$ is given). Assume that $X$ has an initial state $s_{init}$. A [*run*]{} of $X$ is a sequence of symbols alternately in $\Sigma$ and $\nabla$: $\alpha_0\beta_0\alpha_1\beta_1...$, such that, starting from the initial state $s_{init}$, $X$ outputs exactly the sequence $\beta_0\beta_1...$ when it is given the sequence $\alpha_0\alpha_1...$ as input. In this case, we say that the input sequence is accepted by $X$.
The host system $M$ is defined as a $5$-tuple $$\langle S, \Gamma, R_{env}, R_{comm}, I\rangle$$ where
- $S$ is a finite set of states;
- $\Gamma$ is a finite set of events;
- $R_{env}\subseteq S\times\Gamma\times S$ defines a set of [*environment transitions*]{} where $(s, a, s^\prime)\in R_{env}$ means that $M$ moves from state $s$ to state $s^\prime$ upon receiving an event (symbol) $a\in\Gamma$ from the outside environment;
- $R_{comm}\subseteq S\times\Sigma\times\nabla\times S$ defines a set of [*communication transitions*]{} where $(s,\alpha,\beta,s^\prime)\in R_{comm}$ means that $M$ moves from state $s$ to state $s^\prime$ when $X$ outputs a symbol $\beta\in\nabla$ after $M$ sends $X$ an input symbol $\alpha\in\Sigma$; and,
- $I\subseteq S$ is $M$’s initial states.
Without loss of generality, we further assume that, there is only one transition between any two states in $M$ (but $M$, in general, could still be nondeterministic).
An [*execution path*]{} of the system $Sys=\langle M, X \rangle$ can be represented as a (potentially infinite) sequence $\tau$ of states and symbols, $s_0c_0s_1c_1...$, where each $s_i\in S$, each $c_i$ is either a symbol in $\Gamma$ or a pair $\alpha_i\beta_i$ (called a [*communication*]{}) with $\alpha_i\in\Sigma$ and $\beta_i\in\nabla$. Additionally, $\tau$ satisfies the following requirements:
- $s_0$ is an initial state of $M$, i.e., $s_0\in I$;
- for each $c_i\in\Gamma$, $(s_i,c_i,s_{i+1})$ is an environment transition of $M$;
- for each $c_i=\alpha_i\beta_i$, $(s_i,\alpha_i,\beta_i,s_{i+1})$ is a communication transition of $M$.
The [*communication trace*]{} of $\tau$, denoted by $\tau_X$, is the sequence obtained from $\tau$ by retaining only symbols in $\Sigma$ and $\nabla$ (i.e., the result of projecting $\tau$ onto $\Sigma$ and $\nabla$). For any given state $s\in S$, we say that the system $Sys$ can [*reach*]{} $s$ iff $Sys$ has an execution path $\tau$ on which $s$ appears and $\tau_X$ (if not empty) is also a run of $X$.
In the case when $X$ is fully specified, the system can be regarded as an I/O automaton [@lynch87hierarchical].
Model-checking
--------------
Model-checking[@CE81; @SC85; @CES86; @VW86; @M93] is an automatic technique for verifying a finite-state system against some temporal specification. The system is usually represented by a Kripke structure $K=\langle S,R,L\rangle$ over a set of atomic propositions $AP$, where
- $S$ is a finite set of states;
- $R\subseteq S\times S$ is the (total) transition relation;
- $L:S\rightarrow 2^{AP}$ is a function that labels each state with the set of atomic propositions that are true in the state.
The temporal specification can be expressed in, among others, a branching-time temporal logic (CTL) or a linear-time temporal logic (LTL). Both CTL and LTL formulas are composed of [*path quantifiers*]{} $A$ and $E$, which denote “for all paths” and “there exists a path”, respectively, and [*temporal operators*]{} $X$, $F$, $U$ and $G$, which stands for “next state”, “eventually”, “until”, and “always”, respectively.
More specifically, CTL formulas are defined as follows:
- Constants $true$ and $false$, and every atomic proposition in $AP$ are CTL formulas;
- If $f_1$ and $f_2$ are CTL formulas, then so are $\neg f_1$, $f_1\wedge f_2$, $f_1\vee f_2$, $f_1\rightarrow f_2$, $EX~f_1$, $AX~f_1$, $EF~f_1$, $AF~f_1$, $E[f_1~U~f_2]$, $A[f_1~U~f_2]$, $EG~f_1$, $AG~f_1$.
Due to duality, any CTL formula can be expressed in terms of $\neg,\vee, EX, EU$ and $EG$. A CTL model-checking problem, formulated as $$K,s\models f$$, is to check whether the CTL formula $f$ is true at a state $s$. For example, $AF~f$ is true at state $s$ if $f$ will be eventually true on all paths from $s$; $E[f~U~g]$ is true at state $s$ if there exists a path from $s$ on which $f$ is true at each step until $g$ becomes true.
LTL formulas, on the other hand, are all in the form of $A~f$ where $f$ is a [*path formula*]{} defined as follows:
- Constants $true$ and $false$, and every atomic proposition in $AP$ are path formulas;
- If $f_1$ and $f_2$ are path formulas, then so are $\neg f_1$, $f_1\wedge f_2$, $f_1\vee f_2$, $X~f_1$, $F~f_1$, $[f_1~U~f_2]$, $G~f_1$.
An LTL model-checking problem, formulated as $$K,s\models A~f$$, is to check whether the path formula $f$ is true on all paths from a state $s$. For example, $AFG~f$ is true at $s$ if on all paths from $s$, after a future point $f$ will be always true; $AGF~f$ is true at $s$ if on all paths from $s$, $f$ will be true infinitely often.
More detailed background in model-checking and temporal logics can be found in the textbook [@CGP99]. The system $Sys=\langle M, X\rangle$ defined earlier can be understood as a Kripke structure (with a given labeling function and atomic propositions over states in $M$). Since $X$ is an unspecified component, in the rest of the paper, we mainly focus on how to solve the LTL/CTL model-checking problems on the $Sys$ through black-box testing on $X$.
Black-box Testing
-----------------
Black-box testing (also called functional testing) is a technique to test a system without knowing its internal structure. The system is regarded as a “black-box” in the sense that its behaviour can only be determined by observing (i.e., testing) its input/output sequences. As a common assumption in black-box testing, the unspecified component $X$ (treated as a black-box) has a special input symbol $reset$ which always makes $X$ return to its initial state regardless of its current state. We use $Experiment(X,reset\pi)$ to denote the output sequence obtained from the input sequence $\pi$, when $X$ runs from the initial state (caused by the $reset$). After running this $Experiment$, suppose that we continue to run $X$ by providing an input symbol $\alpha$ following the sequence $\pi$. Corresponding to this $\alpha$, we may obtain an output symbol $\beta$ from $X$. We use $Experiment(X, \alpha)$ to denote the $\beta$. Notice that this latter $Experiment$ is a shorthand for “the last output symbol in\
$Experiment(X, reset\pi\alpha)$".
Studies have shown that if only an upper bound for the number of states in the system and the system’s inputs set are known, then its (equivalent) internal structure can be fully recovered through black-box testing. Clearly, a naive algorithm to solve the LTL/CTL model-checking problem over the $Sys$ is to first recover the full structure of the component $X$ through testing, and then to solve the classic model-checking problem over the fully specified system composed from $M$ and the recovered $X$. Notice that, in the naive algorithm, when we perform black-box testing over $X$, the selected test-cases have nothing to do with the host system $M$. Therefore, it is desirable to find more sophisticated algorithms such as the ones discussed in this paper, that only select “useful" test-cases wrt the $M$ as well as the temporal specification of $M$ that needs to be checked.
LTL Model-Checking Driven\
Black-Box Testing {#ltltesting}
==========================
In this section, we introduce algorithms for LTL model-checking driven black-box testing for the system $Sys=\langle M,X\rangle$ defined earlier. We first show how to solve a liveness analysis problem. Then, we discuss the general LTL model-checking problem.
Liveness Analysis
-----------------
The liveness analysis problem (also called the [*infinite-often*]{} problem) is to check: starting from some initial state $s_0\in I$, whether the system $Sys$ can reach a given state $s_f$ for infinitely many times.
When $M$ has no communications with the unspecified component $X$, solving the problem is equivalent to finding a path $p$ that runs from $s_0$ to $s_f$ and a loop $C$ that passes $s_f$. However, as far as communications are involved, the problem gets more complicated. The existence of the path $p$ does not ensure that the system can indeed reach $s_f$ from $s_0$ (e.g., communications with $X$ may never allow the system to take the necessary transitions to reach $s_f$). Moreover, the existence of the loop $C$ does not guarantee that the system can run along $C$ forever either (e.g., after running along $C$ for three rounds, the system may be forced to leave $C$ by the communications with $X$).
We approach this infinite-often problem in three steps. First, we look at whether a definite answer to the problem is possible. If we can find a path from $s_0$ to $s_f$ and a loop from $s_f$ to $s_f$ that involve only environment transitions, then the original problem (i.e., the infinite-often problem) is definitely true. If such a path and a loop, no matter what transitions they may involve, do not exist at all, then the original problem is definitely false. If no definite answer is possible, we construct a directed graph $G$ and use it to generate test-cases for the unspecified component $X$. The graph $G$, called a [*communication graph*]{}, is a subgraph of $M$, represents all paths and loops in $M$ that could witness the truth of the problem (i.e., paths that run from $s_0$ to $s_f$ and loops that pass $s_f$). The graph $G$ is defined as a pair $\langle
N,E\rangle$, where $N$ is a set of nodes and $E$ is a set of edges. Each edge of $G$ is annotated either by a pair $\alpha\beta$ that denotes a communication of $M$ with $X$, or has no annotation. We construct $G$ as follows.
- Add one node to $G$ for each state in $M$ that is involved in some path between $s_0$ and $s_f$ or in a loop that passes $s_f$;
- Add one edge between two nodes in $N$ if $M$ has a transition between two states corresponding to the two nodes respectively. If the transition involves a communication with $X$, then annotate the edge with the communication symbols.
It is easy to see that the liveness analysis problem is true if and only if the truth is witnessed by a path in $G$. Therefore, the last step is to check whether $G$ has a path along which the system can reach $s_f$ from $s_0$ first and then reach $s_f$ for infinitely many times. More details of this step are addressed in the next subsection.
See appendix \[algcheckio\] for details on the above operations.
Liveness Testing {#TestLTL}
----------------
To check whether the constructed communication graph $G$ has a path that witnesses the truth of the original problem, the straightforward way is to try out all paths in $G$ and then check, whether along some path, the system can reach $s_f$ from $s_0$ first and then reach $s_f$ for infinitely many times. The check is done by testing $X$ with the communication trace of the path to see whether it is a run of $X$. However, one difficulty is that $G$ may contain loops, and certainly we can only test $X$ with a finite communication trace. Fortunately, the following observations are straightforward:
- To check whether the system can reach $s_f$ from $s_0$, we only need to consider paths with length less than $mn_1$ where $n_1$ is the maximal number of communications on all [*simple paths*]{} (i.e., no loops on the path) between $s_0$ and $s_f$ in $G$, and $m$ is an upper bound for the number of states in the unspecified component $X$;
- To check whether the system can reach from $s_f$ to $s_f$ for infinitely many times, we only need to make sure that the system can reach $s_f$ for $m-1$ times, and between $s_f$ and $s_f$, the system goes through a path no longer than $n_2$ that is the maximal number of communications on all [*simple loops*]{} (i.e., no nested loops along the loop) in $G$ that pass $s_f$.
Let $n=max(n_1,n_2)$. The following procedure $TestLiveness$ uses a bounded and nested depth-first search to traverse the graph $G$ while testing $X$. It first tests whether the system can reach $s_f$ from $s_0$ along a path with length less than $mn$, then it tests whether the system can further reach $s_f$ to $s_f$ for $m-1$ more times. The algorithm maintains a sequence of input symbols that has been successfully accepted by $X$, an integer variable $level$ that records how many communications have been gone through without reaching $s_f$, and an integer variable $count$ that indicates how many times $s_f$ has been reached. At each step, it chooses one candidate from the set of all possible input symbols at a node, and feeds the input sequence concatenated with the candidate input symbol to $X$. If the candidate input symbol and the output symbol (corresponding to the candidate input symbol) of $X$ match the annotation of an edge originating from the node, the procedure moves forward to try the destination node of the edge with $level$ increased by 1. If there is no match, then the procedure tries other candidates. But before trying any other candidate, we need to bring $X$ to its initial state by sending it the special input symbol $reset$. The procedure returns $false$ when all candidates are tried without a match, or when more than $mn$ communications have been gone through without reaching $s_f$. After $s_f$ is reached, the procedure increases $count$ by $1$ and resets $level$ to $0$. The procedure returns $true$ when it has already encountered $s_f$ for $m$ times.
[**Procedure**]{} $TestLiveness(X,\pi,s_0,s_f,level,count)$
$level>mn$ [**Then**]{}
$false$;
$s_0=s_f$ [**Then**]{}
$count >= m$ [**Then**]{}
$true$;
$count := count+1$; $level := 0$;
$(s_0,s^\prime)\in E$ [**Do**]{}
$Experiment(X, reset\pi)$;
$TestLiveness(X,\pi, s^\prime,s_f,level,count)$ [**Then**]{}
$true$;
$Inputs := \{\alpha|(s_0,\alpha\beta, s^\prime)\in E\}$;
$\alpha\in Inputs$ [**Do**]{}
$Experiment(X, reset\pi)$;
$\beta := Experiment(X,\alpha)$;
$\exists s^\prime:(s_0,\alpha\beta, s^\prime)\in E$ [**Then**]{}
$TestLiveness(X,\pi\alpha,s^\prime,s_f,level+1,count)$
$true$;
$false$.
In summary, our liveness testing algorithm to solve the liveness analysis problem has two steps: (1) build the communication graph $G$; (2) return the truth of $$TestLiveness(X,reset,s_0,s_f,level=0,count=0).$$
LTL Model-Checking Driven Testing
---------------------------------
Recall that the LTL model-checking problem is, for a Kripke structure $K=\langle S,R,L\rangle$ with a state $s\in S$ and a path formula $f$, to determine if $K, s\models A~f$. Notice that $K, s\models A~f$ if and only if $K,s\models\neg
E~\neg f$. Therefore it is sufficient to only consider formulas in the form $E~f$. The standard LTL model-checking algorithm [@CGP99] first constructs a [*tableau*]{} $T$ for the path formula $f$. $T$ is also a Kripke structure and includes [*every*]{} path that satisfies $f$. Then the algorithm composes $T$ with $K$ and obtains another Kripke structure $P$ which includes exactly the set of paths that are in both $T$ and $K$. Thus, a state in $K$ satisfies $E~f$ if and only if it is the start of a path (in the composition $P$) that satisfies $f$.
Define $sat(f)$ to be the set of states in $T$ that satisfy $f$ and use the convention that $(s,s^\prime)\in sat(f)$ if and only if $s^\prime\in
sat(f)$. The LTL model-checking problem can be summarized by the following theorem [@CGP99]:
$K,s\models E~f$ if and only if there is a state $s^\prime$ in $T$ such that $(s,s^\prime)\in sat(f)$ and $P, (s, s^\prime)\models
EG~true$ under fairness constraints $\{sat(\neg(g~U~h)\vee
h)~|~g~U~h~$ occurs in $f \}$.
Note that the standard LTL model-checking algorithm still applies to the system $Sys=\langle M, X\rangle$, although it contains an unspecified component X. To see this, the construction of the tableau $T$ from $f$ and the definition of [*sat*]{} are not affected by the unspecified component $X$. The composition of $Sys$ and $T$ is a new system $Sys^\prime=\langle P, X\rangle$ where $P$ is the composition of $M$ and $T$. Then one can show
$\langle M,X\rangle,s\models E~f$ if and only if there is a state $s^\prime$ in $T$ such that $(s,s^\prime)\in sat(f)$ and $\langle P, X\rangle, (s, s^\prime)\models EG~true$ under fairness constraints $\{sat(\neg(g~U~h)\vee h)~|~g~U~h$ occurs in $f \}$.
Obviously, checking whether there is a state $s^\prime$ in $T$ such that $(s, s^\prime)\in sat(f)$ is trivial. To check whether $\langle P, X\rangle, (s, s^\prime)\models EG~true$ under the fairness constraints is equivalent to checking whether there is computation in $\langle P, X\rangle$ that starts from $(s,s^\prime)$ and on which the fairness constraints are true infinitely often. One can show that this is equivalent to the liveness analysis problem we studied in the previous subsection, and thus, the LTL model-checking problem can be solved by extending our algorithms for the liveness analysis problem. Moreover, the algorithms are both complete and sound.
CTL Model-Checking Driven\
Black-Box Testing {#ctltesting}
==========================
In this section, we introduce algorithms for CTL model-checking driven black-box testing for the system $Sys=\langle M,X\rangle$.
Ideas {#Ideas}
-----
Recall that the CTL model-checking problem is, for a Kripke structure $K=(S, R, L)$, a state $s_0\in S$, and a CTL formula $f$, to check whether $K,s_0\models f$ holds. The standard algorithm [@CGP99] for this problem operates by searching the structure and, during the search, labeling each state $s$ with the set of subformulas of $f$ that are true at $s$. Initially, labels of $s$ are just $L(s)$. Then, the algorithm goes through a series of stages—during the $i$-th stage, subformulas with the $(i-1)$-nested CTL operators are processed. When a subformula is processed, it is added to the labels for each state where the subformula is true. When all the stages are completed, the algorithm returns $true$ when $s_0$ is labeled with $f$, or $false$ otherwise.
However, if a system is not completely specified, the standard algorithm does not work. This is because, in the system $Sys=\langle M,X\rangle$, transitions of $M$ may depend on communications with the unspecified component $X$. In this section, we adapt the standard CTL model-checking algorithm [@CGP99] to handle the system $Sys$ (i.e., to check whether $$\langle M,X\rangle, s_0\models f$$ holds where $s_0$ is an initial state in $M$ and $f$ is a CTL formula over $M$).
The new algorithm follows a structure similar to the standard one. It also goes through a series of stages to search $M$’s state space and label each state during the search. However, during a stage, processing the subformulas is rather involved, since the truth of a subformula $h$ at a state $s$ can not be simply decided (it may depend on communications). Similar to the algorithm for the liveness analysis problem, our ideas here are to construct a graph representing all the paths that witness the truth of $h$ at $s$. But, the new algorithm is far more complicated than the liveness testing algorithm for LTL, since the truth of a CTL formula is usually witnessed by a tree instead of a single path. In the new algorithm, processing each subformula $h$ is sketched as follows.
When $h$ takes the form of $EX~g$, $E[g_1~U~g_2]$, or $EG~g$, we construct a graph that represents exactly all the paths that witness the truth of $h$ at some state. We call such a graph the subformula’s [*witness graph*]{} (WG), written as $\llbracket h\rrbracket$. We also call $\llbracket h\rrbracket$ an [*EX graph*]{}, an [*EU graph*]{}, or an [*EG graph*]{} if $h$ takes the form of $EX~g$, $E[g_1~U~g_2]$, or $EG~g$, respectively.
Let $k$ be the total number of CTL operators in $f$. In the algorithm, we construct $k$ WGs, and for each WG, we assign it with a unique ID number that ranges between $2$ and $k+1$. (The ID number 1 is reserved for constant $true$.) Let ${\cal I}$ be the mapping from the WGs to their IDs; i.e., ${\cal I}(\llbracket h\rrbracket)$ denotes the ID number of $h$’s witness graph, and ${\cal I}^{-1}(i)$ denotes the witness graph with $i$ as its ID number, $1<i\le k+1$. We label a state $s$ with ID number $1$ if $h$ is true at $s$ and the truth does not depend on communications between $M$ and $X$. Otherwise, we label $s$ with $2\le i\le k+1$ if $h$ could be true at $s$ and the truth would be witnessed only by some paths which start from $s$ in ${\cal I}^{-1}(i)$ and, on which, communications are involved.
When $h$ takes the form of a Boolean combination of subformulas using $\neg$ and $\vee$, the truth of $h$ at state $s$ is also a logic combination of the truths of the component subformulas at the same state. To this end, we label the state with an [*ID expression*]{} $\psi$ defined as follows:
- $ID := 1~|~2~|~\ldots~|~k+1$;
- $\psi := ID~|~\neg \psi~|~\psi\vee \psi$.
Let $\Psi$ denote the set of all ID expressions. For each subformula $h$, we construct a labeling (partial) function $L_h:S\rightarrow \Psi$ to record the ID expression labeled to each state during the processing of the subformula $h$, and the labeling function is returned when the subformula is processed.
The detailed procedure, called $ProcessCTL$, for processing subformulas will be given in Section \[ProcessCTL\]. After all subformulas are processed, a labeling function $L_f$ for the outer-most subformula (i.e., $f$ itself) is returned. The algorithm returns $true$ when $s$ is labeled with $1$ by $L_f$. It returns $false$ when $s$ is not labeled at all. In other cases, a testing procedure over $X$ is applied to check whether the ID expression labeled in $L_f(s)$ could be evaluated true. The procedure, called $TestWG$, will be given in Section \[TestWG\]. In summary, the algorithm (to solve the CTL model-checking problem $\langle M,X\rangle, s_0\models f$) is sketched as follows:
[**Procedure**]{} $CheckCTL(M,X,s_0,f)$
$L_f := ProcessCTL(M,f)$
$s_0$ is labeled by $L_f$ [**Then**]{}
$L_f(s_0)=1$ [**Then**]{}
$true$;
$TestWG(X, reset, s_0, L_f(s_0))$;
(i.e., $s_0$ is not labeled at all)
$false$.
Processing a CTL formula {#ProcessCTL}
------------------------
Processing a CTL formula $h$ is implemented through a recursive procedure $ProcessCTL$. Recall that any CTL formula can be expressed in terms of $\vee$, $\neg$, $EX$, $EU$, and $EG$. Thus, at each intermediate step of the procedure, depending on whether the formula $h$ is atomic or takes one of the following forms: $g_1\vee g_2$, $\neg g$, $EX~g$, $E[g_1~U~g_2]$, or $EG~g$, the procedure has only six cases to consider and when it finishes, a labeling function $L_h$ is returned for formula $h$.
[**Procedure**]{} $ProcessCTL(M,h)$
$h$ is atomic: Let $L_h$ label every state with 1
whenever $h$ is true on the state;
$h=g_1 \vee g_2$:
$L_{g_1} := ProcessCTL(M,g_1)$;
$L_{g_2} := ProcessCTL(M,g_2)$;
$L_h := HandleUnion(L_{g_1}, L_{g_2})$;
$h=\neg g$:
$L_g := ProcessCTL(M,g)$;
$L_h := HandleNegation(M, L_g)$;
$h=EX~g$:
$L_g := ProcessCTL(M,g)$;
$L_h := HandleEX(M, L_g)$;
$h=E~[g_1 ~U~ g_2]$:
$L_{g_1} := ProcessCTL(M,g_1)$;
$L_{g_2} := ProcessCTL(M,g_2)$;
$L_h := HandleEU(M, L_{g_1}, L_{g_2})$;
$h=EG~g$:
$L_g := ProcessCTL(M,g)$;
$L_h := HandleEG(M,L_g)$;
$L_h$.
In the above procedure, when $h=g_1 \vee g_2$, we first process $g_1$ and $g_2$ respectively by calling $ProcessCTL$, then construct a labeling function $L_h$ for $h$ by merging (i.e., $HandleUnion$, see Appendix \[alghandleunion\] for details)) $g_1$ and $g_2$’s labeling functions $L_{g_1}$ and $L_{g_2}$ as follows:
- For each state $s$ that is in both $L_{g_1}$’s domain and $L_{g_2}$’s domain, let $L_h$ label $s$ with $1$ if either $L_{g_1}$ or $L_{g_2}$ labels $s$ with $1$ and label $s$ with ID expression $L_{g_1}(s)\vee L_{g_2}(s)$ otherwise;
- For each state $s$ that is in $L_{g_1}$’s domain (resp. $L_{g_2}$’s domain) but not in $L_{g_2}$’s domain (resp. $L_{g_1}$’s domain), let $L$ label $s$ with $L_{g_1}(s)$ (resp. $L_{g_2}(s)$).
When $h=\neg g$, we first process $g$ by calling $ProcessCTL$, then construct a labeling function $L_h$ for $h$ by “negating" (i.e., $HandleNegation$, see Appendix \[alghandlenegation\] for details)) $g$’s labeling function $L_g$ as follows:
- For every state $s$ that is not in the domain of $L_g$, let $L_h$ label $s$ with $1$;
- For each state $s$ that is in the domain of $L_g$ but not labeled with $1$ by $L_g$, let $L_h$ label $s$ with ID expression $\neg L_g(s)$.
The remaining three cases (i.e., for $EX$, $EU$, and $EG$) in the above procedure are more complicated and are handled in the following three subsections respectively.
### Handling EX
When $h=EX g$, $g$ is processed first by $ProcessCTL$. Then, the procedure $HandleEX$ is called with $g$’s labeling function $L_g$ to construct a labeling function $L_h$ and create a witness graph for $h$ (we assume that, whenever a witness graph is created, the current value of a global variable $id$, which initially is 2, is assigned as the ID number of the graph, and $id$ is incremented by $1$ after it is assigned to the graph).
The labeling function $L_h$ is constructed as follows. For each state $s$ that has a successor $s^\prime$ in the domain of $L_g$, if $s$ can reach $s^\prime$ through an environment transition and $s^\prime$ is labeled with $1$ by $L_g$ then let $L_h$ also label $s$ with $1$, otherwise let $L_h$ label $s$ with the current value of the global variable $id$.
The witness graph for $h=EX g$, called an $EX$ graph, is created as a triple: $$\llbracket h\rrbracket = \langle N, E, L_g\rangle,$$ where $N$ is a set of nodes and $E$ is a set of annotated edges. It is created as follows:
- Add one node to $N$ for each state that is in the domain of $L_g$.
- Add one node to $N$ for each state that has a successor in the domain of $L_g$.
- Add one edge between two nodes in $N$ to $E$ when $M$ has a transition between two states corresponding to the two nodes respectively; if the transition involves a communication with $X$ then annotate the edge with the communication symbols.
When $HandleEX$ finishes, it increases the global variable $id$ by $1$ (since one new witness graph has been created).
See Appendix \[alghandleex\] for details.
### Handling EU
The case when $h=E~[g_1~U~g_2]$ is more complicated. We first process $g_1$ and $g_2$ respectively by calling $ProcessCTL$, then call procedure $HandleEU$ with $g_1$ and $g_2$’s labeling functions $L_{g_1}$ and $L_{g_2}$ to construct a labeling function $L_h$ and create a witness graph for $h$.
We construct the labeling function $L_h$ recursively. First, let $L_h$ label each state $s$ in the domain of $L_{g_2}$ with $L_{g_2}(s)$. Then, for state $s$ that has a successor $s^\prime$ in the domain of $L_{h}$, if both $s$ and $s^\prime$ is labeled with $1$ by $L_{g_1}$ and $L_{h}$ respectively and $s$ can reach $s^\prime$ through an environment transition then let $L_h$ also label $s$ with $1$, otherwise let $L_h$ label $s$ with the current value of the global variable $id$. Notice that, in the second step, if a state $s$ can be labeled with both $1$ and the current value of $id$, let $L_h$ label $s$ with $1$. Thus, we can ensure that the constructed $L_h$ is indeed a function.
The witness graph for $h$, called an $EU$ graph, is created as a $4$-tuple: $$\llbracket h \rrbracket := \langle N, E, L_{g_1}, L_{g_2}\rangle,$$ where $N$ is a set of nodes and $E$ is a set of edges. $N$ is constructed by adding one node for each state that is in the domain of $L_h$, while $E$ is constructed in the same way as that of $HandleEX$. When $HandleEU$ finishes, it increases the global variable $id$ by $1$.
See Appendix \[alghandleeu\] for details.
### Handling EG
To handle formula $h=EG g$, we first process $g$ by calling $ProcessCTL$, then call procedure $HandleEG$ with $g$’s labeling function $L_g$ to construct a labeling function $L_h$ and create a witness graph for $h$.
The labeling function $L_h$ is constructed as follows. For each state $s$ that can reach a loop $C$ through a path $p$ such that every state (including $s$) on $p$ and $C$ is in the domain of $L_g$, if every state (including $s$) on $p$ and $C$ is labeled with $1$ by $L_g$ and no communications are involved on the path and the loop, then let $L_h$ also label $s$ with $1$, otherwise let $L_h$ label $s$ with the current value of the global variable $id$.
The witness graph for $h$, called an $EG$ graph, is created as a triple: $$\llbracket h\rrbracket := \langle N, E, L_g\rangle,$$ where $N$ is a set of nodes and $E$ is a set of annotated edges. The graph is constructed in a same way as that of $HandleEU$. When $HandleEG$ finishes, it also increases the global variable $id$ by $1$.
See Appendix \[alghandleeg\] for details.
Testing a Witness Graph {#TestWG}
-----------------------
As mentioned in Section \[Ideas\], the procedure for CTL model-checking driven black-box testing, $CheckCTL$, consists of two parts. The first part, which was discussed in Section \[ProcessCTL\], includes $ProcessCTL$ that processes CTL formulas and creates witness graphs. The second part is to evaluate the created witness graphs through testing $X$. We will elaborate on this second part in this section.
In processing the CTL formula $f$, a witness graph is constructed for each CTL operator in $f$ and a labeling function is constructed for each subformula of $f$. As seen from the algorithm $CheckCTL$ (at the end of Section \[Ideas\]), the algorithm either gives a definite “yes” or “no” answer to the CTL model-checking problem, i.e., $\langle M,X\rangle, s_0\models f$, or it reduces the problem to checking whether the ID expression $\psi$ labeled to $s_0$ can be evaluated true at the state. The evaluation procedure is carried out by the following recursive procedure $TestWG$, after an input sequence $\pi$ has been accepted by the unspecified component $X$.
[**Procedure**]{} $TestWG(X,\pi,s_0,\psi)$
$\psi=\psi_1\vee \psi_2$:
$TestWG(X,\pi,s_0,\psi_1)$ [**Then**]{}
$true$;
$TestWG(X,\pi,s_0,\psi_2)$
$\psi=\neg \psi_1$:
$\neg TestWG(X,\pi,s_0, \psi_1)$
$\psi=1$:
$true$;
$\psi=i$ with $2\le i\le k+1$:
${\cal I}^{-1}(i)$ is an $EX$ graph
$TestEX(X,\pi,s_0,{\cal I}^{-1}(i))$;
${\cal I}^{-1}(i)$ is an $EU$ graph
$TestEU(X,\pi,s_0,{\cal I}^{-1}(i), level=0)$;
${\cal I}^{-1}(i)$ is an $EG$ graph
$TestEG(X,\pi,s_0,{\cal I}^{-1}(i)).$
In $TestWG$, the first three cases are straightforward, which are consistent with the intended meaning of ID expressions. The cases $TestEX, TestEU, TestEG$ for evaluating $EX, EU$, $EG$ graphs are discussed in the following three subsections.
### $TestEX$
The case for checking whether an $EX$ graph $G=\langle N,E, L_g\rangle$ can be evaluated true at a state $s_0$ is simple. We just test whether the system $M$ can reach from $s_0$ to another state $s^\prime\in {\bf dom}(L_g)$ through a transition in $G$ such that the ID expression $L_g(s^\prime)$ can be evaluated true at $s^\prime$.
See Appendix \[algtestex\] for details.
### $TestEU$
To check whether an $EU$ graph $G=\langle N,E, L_{g_1},
L_{g_2}\rangle$ can be evaluated true at a state $s_0$, we need to traverse all paths $p$ in $G$ with length less than $mn$ and test the unspecified component $X$ to see whether the system can reach some state $s^\prime
\in {\bf dom}(L_{g_2})$ through one of those paths. In here, $m$ is an upper bound for the number of states in the unspecified component $X$ and $n$ is the maximal number of communications on all simple paths between $s_0$ and $s^\prime$. In the meantime, we should also check whether $L_{g_2}(s^\prime)$ can be evaluated true at $s^\prime$ and whether $L_{g_1}(s_i)$ can be evaluated true at $s_i$ for each $s_i$ on $p$ (excluding $s^\prime$) by calling $TestWG$.
See Appendix \[algtesteu\] for details.
### $TestEG$
For the case to check whether an $EG$ graph $G=\langle N,E, L_g\rangle$ can be evaluated true at a state $s_0$, we need to find an infinite path in $G$ along which the system can run forever.
The following procedure $TestEG$ first decomposes $G$ into a set of SCCs. Then, for each state $s_f$ in the SCCs, it calls another procedure $SubTestEG$ to test whether the system can reach $s_f$ from $s_0$ along a path not longer than $mn$, as well as whether the system can further reach $s_f$ from $s_f$ for $m-1$ times. The basic idea of $SubTestEG$ (see Appendix \[algsubtesteg\] for details) is similar to that of the $TestLiveness$ algorithm in Section \[TestLTL\], except that we need also check whether $L_g(s_i)$ can be evaluated true at $s_i$ for each state $s_i$ that has been reached so far by calling $TestWG$. Here, $m$ is the same as before while $n$ is the maximal number of communications on all simple paths between $s_0$ and $s_f$.
[**Procedure**]{} $TestEG(X,\pi,s_0,G=\langle N,E,L_g\rangle)$
$SCC := \{C|C$ is a nontrivial SCC of $G\}$;
$T := \bigcup_{C\in SCC}\{s|s\in C\}$;
$s\in T$ [**Do**]{}
$Experiment(X,reset\pi)$;
$SubTestEG(X,\pi,s_0,s,G,level=0,count=0)$;
$true$;
$false$.
In summary, to solve the CTL model-checking problem $$(M, X), s_0\models f,$$ our algorithm $CheckCTL$ in Section \[Ideas\] either gives a definite yes/no answer or gives a sufficient and necessary condition in the form of ID expressions and witness graphs. The condition is evaluated through black-box testing over the unspecified component $X$. The evaluation process will terminate with a yes/no answer to the model-checking problem. One can show that our algorithm is both complete and sound.
Examples
========
In this section, to better understand our algorithms, we look at some examples[^1].
Consider a system $Sys=\langle M, X \rangle$ where $M$ keeps receiving messages from the outside environment and then transmits the message through the unspecified component $X$. The only event symbol in $M$ is $msg$, while $X$ has two input symbols $send$ and $ack$, and two output symbols $yes$ and $no$. The transition graph of $M$ is depicted in Figure 1 where we use a suffix $?$ to denote events from the outside environment (e.g., msg?), and use a infix $/$ to denote communications of $M$ with $X$ (e.g., $send/yes$).
Assume that we want to solve the following LTL model-checking problem $$(M,X),s_0\models EGF s_2$$ i.e., starting from the initial state $s_0$, the system can reach state $s_1$ infinitely often. Applying our liveness analysis algorithms, we can obtain the (minimized) communication graph in Figure 2.
From this graph and our liveness testing algorithms, the system satisfies the liveness property iff the communication trace $$send~yes(send~yes~ack~yes)^{m-1}$$ is a run of $X$, where $m$ is an upper bound for number of states in $X$. Now, we slightly modified the transition graph of $M$ into Figure 3 such that when a send fails, the system shall return to the initial state.
For this modified system, its (minimized) communication graph with respect to the liveness property would be as shown in Figure 4.
From Figure 4 and the liveness testing algorithms, the system satisfies the liveness property iff there exist $0\le k_1, k_2\le 2m$ such that the communication trace $$(send~no)^{k_1} send~yes ((send~yes~ack~yes)(send~no)^{k_2})^{m-1}$$ is a run of $X$.
Still consider the system in Figure 3, but we want to solve a CTL model-checking problem $(M,X),s_0\models AF s_2$; i.e., along all paths from $s_0$, the system can reach state $s_1$ eventually. The problem is equivalent to $$(M,X),s_0\models \neg EG \neg s_2.$$ Applying our CTL algorithms to formula $h=EG \neg s_2$, we construct an $EG$ witness graph $G=\langle N, E, L_{true}\rangle$ whose ID number is $2$ and a labeling function $L_h$, where $L_{true}$ labels all three states $s_0$,$s_1$, and $s_3$ with ID expression $1$ (as defined in Section \[Ideas\], which stands for $true$), and $L_{h}$ labels all three states $s_0$, $s_1$, and $s_3$ with $2$. The graph $G$ is depicted in Figure 5. From this graph as well as $L_h$, the algorithms conclude that the model-checking problem is true iff the communication trace $(send~no)^{m-1}$ is not a run of $X$.
Now we modify the system in Figure 1 into a more complicated one shown in Figure 6. For this system, we want to check $$(M,X),s_0\models \neg E [\neg s_2 U s_3]$$ i.e., starting from the initial state $s_0$, the system should never reach state $s_3$ earlier than it reaches $s_2$. Applying our CTL algorithms to formula $$h=E[\neg s_2 U s_3],$$ we obtain an $EU$ witness graph $G=\langle N, E, L_1, L_2\rangle$ whose ID number is $2$ and a labeling function $L_h$, where $L_1$ labels all four states $s_0$, $s_1$, $s_3$ and $s_4$ with $1$, $L_2$ just labels $s_3$ with $1$, and $L_{h}$ labels states $s_0$, $s_1$, and $s_4$ with $2$, and labels $s_3$ with $1$. The graph $G$ is depicted in Figure 7. From this graph as well as $L_h$, the algorithms conclude that the model-checking problem is true iff none of communication traces in the form of $send~no(ack~yes~send~no)^*$ and with length less than $3m$ is a run of $X$.
For the same system, we could consider more complicated temporal properties as follows:
- $(M,X)\models AG (s_2\rightarrow AF s_3)$; i.e., starting from the initial state $s_0$, whenever the system reaches $s_2$, it would eventually reach $s_3$.
- $(M,X),s_0\models AG (s_2\rightarrow AXA[\neg s_2 U s_3])$; i.e., starting from the initial state $s_0$, whenever it reaches state $s_2$, the system should never reach $s_2$ again until it reaches $s_3$.
We do not include the witness graphs and labeling functions for these two cases in this extended abstract. Nevertheless, it can be concluded that the two problems are true iff no communication traces with two consecutive symbol pairs $(send~yes)$ can be runs of $X$.
See Appendix \[example1\] and Appendix \[example2\] for details about the above two examples.
Related Work {#relatedwork}
============
The quality assurance problem for component-based software has attracted lots of attention in the software engineering community, as witnessed by recent publications in conferences like ICSE and FSE. However, most of the work is based on the traditional testing techniques and considers the problem from the viewpoint of component developers; i.e., how to ensure the quality of components before they are released.
Voas [@Voas98; @Voas00] proposed a component certification strategy with the establishment of independent certification laboratories performing extensive testing of components and then publishing the results. Technically, this approach would not provide much improvement for solving the problem, since independent certification laboratories can not ensure the sufficiency of their testing either, and a testing-based technique alone is not enough to a reliable software component. Some researchers [@SW00; @OHR01] suggested an approach to augment a component with additional information to increase the customer’s understanding and analyzing capability of the component behavior. A related approach [@WML02] is to automatically extract a finite-state machine model from the interface of a software component, which is delivered along with the component. This approach can provide some convenience for customers to test the component, but again, how much a customer should test is still a big problem. To address the issue of testing adequacy, Rosenblum defined in [@Rosenblum97] a conceptual basis for testing component-based software, by introducing two notions of $C$-[*adequate*]{}-[*for*]{}-${\cal P}$ and $C$-[*adequate*]{}-[*for*]{}-$M$ (with respect to certain adequacy criteria) for adequate unit testing of a component and adequate integration testing for a component-based system, respectively. But this is still a purely testing-based strategy. In practice, how to establish the adequacy criteria is an unclear issue.
Recently, Bertolino et. al. [@BP03] recognized the importance of testing a software component in its deployment environment. They developed a framework that supports functional testing of a software component with respect to customer’s specification, which also provides a simple way to enclose with a component the developer’s test suites which can be re-executed by the customer. Yet their approach requires the customer to have a complete specification about the component to be incorporated into a system, which is not always possible. McCamant and Ernst [@ME03] considered the issue of predicting the safety of dynamic component upgrade, which is part of the problem we consider. But their approach is completely different since they try to generate some abstract operational expectation about the new component through observing a system’s run-time behavior with the old component.
In the formal verification area, there has been a long history of research on verification of systems with modular structure (called modular verification [@Pnu85]). A key idea [@Lam83; @henzinger98you] in modular verification is the [*assume-guarantee*]{} paradigm: A module should guarantee to have the desired behavior once the environment with which the module is interacting has the assumed behavior. There have been a variety of implementations for this idea (see, e.g., [@AH+98]). However, the assume-guarantee idea does not immediately fit with our problem setup since it requires that users must have clear assumptions about a module’s environment.
In the past decade, there has also been some research on combining model-checking and testing techniques for system verification, which can be classified into a broader class of techniques called specification-based testing. But most of the work only utilizes model-checkers’ ability of generating counter-examples from a system’s specification to produce test cases against an implementation [@CSE96; @H97; @EFM97; @GH99; @ABM98; @BOY00; @AB02].
Peled et. al. [@PVY99; @GPY02; @Peled03CAV] studied the issue of checking a black-box against a temporal property (called black-box checking). But their focus is on how to efficiently establish an abstract model of the black-box through black-box testing , and their approach requires a clearly-defined property (LTL formula) about the black-box, which is not always possible in component-based systems. Kupferman and Vardi [@KV97] investigated module checking by considering the problem of checking an open finite-state system under [*all*]{} possible environments. Module checking is different from the problem in (\*) mentioned at the beginning of the paper in the sense that a component understood as an environment in [@KV97] is a specific one. Fisler et. al. [@FK01; @LKF02] proposed an idea of deducing a model-checking condition for extension features from the base feature, which is adopted to study model-checking feature-oriented software designs. Their approach relies totally on model-checking techniques; their algorithms have false negatives and do not handle LTL formulas.
Discussions
===========
In this paper, we present algorithms for LTL and CTL model-checking driven black-box testing. The algorithms create communication graphs and witness graphs, on which a bounded and nested depth-first search procedure is employed to run black-box testing over the unspecified component. Our algorithms are both sound and complete. Though we do not have an exact complexity analysis result, our preliminary studies show that, in the liveness testing algorithm for LTL, the maximal length of test-cases fed into the unspecified component $X$ is bounded by $O(n\cdot m^2)$. For CTL, the length is bounded by $O(k\cdot n\cdot m^2)$. In here, $k$ is the number of CTL operators in the formula to be verified, $n$ is the state number in the host system, and $m$ is the state number in the component.
The next natural step is to implement the algorithms and see how well they work in practice. In the implementation, there are further issues to be addressed.
Practical Efficiency
--------------------
Similar to the traditional black-box testing algorithms to check conformance between Mealy machines, the theoretical (worst-case) complexities are high in order to achieve complete coverage. However, worst-cases do not always occur in a practical system. In particular, we need to identify scenarios that our algorithms can be made more efficient. For instance, using existing ideas of abstraction [@cousotcousot-77], we might obtain a smaller but equivalent model of the host system before running the algorithms. We might also, using additional partial information about the component, to derive a smaller state number for the component and to find ways to expedite the model-checking process. Notice that the number is actually the state number for a minimal automaton that has the same set input/output sequences as the component. Additionally, in the implementation, we also need a database to record the test results that have been performed so far (so repeated testing can be avoided). Algorithms are needed to make use of the test results to aggressively trim the communication/witness graphs such that less test-cases are performed but the complete coverage is still achieved. Also, we will study algorithms to minimize communication/witness graphs such that duplicate test-cases are avoided. Lastly, it is also desirable to modify our algorithms such that the communication/witness graphs are generated with the process of generating test-cases and performing black-box testing over the unspecified component $X$. In this way, a dynamic algorithm could be designed to trim the graphs on-the-fly.
Coverage Metrics
----------------
Sometimes, a complete coverage will not be achieved when running the algorithms on a specific application system. In this case, a coverage metric is needed to tell how much the test-cases that have run so far cover. The metric will give a user some confidence on the partial model-checking results. Furthermore, such a metric would be useful in designing conservative algorithms to debug/verify the temporal specifications that sacrifice the complete coverage but still bring the user reasonable confidence.
More Complex System Models
--------------------------
The algorithms can be generalized to systems containing multiple unspecified components. Additionally, we will also consider cases when these components interacts between each other, as well as cases when the host system communicates with the components asynchronously. Obviously, when the unspecified component (as well as the host system) has an infinite-state space, both the traditional model-checking techniques and black-box techniques are not applicable. One issue with infinite-state systems is that, the internal structure of a general infinite-state system can not be learned through the testing method. Another issue is that model-checking a general infinite-state system is an undecidable problem. It is desirable to consider some restricted classes of infinite-state systems (such as real-time systems modeled as timed automata [@AD94]) where our algorithms generalize. This is interesting, since through the study we may provide an algorithm for model-checking driven black-box testing for a real-time system that contains an (untimed) unspecified component. Since the algorithm will generate test-cases for the component, real-time integration testing over the composed system is avoided.
Definitions
===========
$R_{env}^s := \{(s,s^\prime)|\exists~a\in\Gamma:(s,a,s^\prime)\in R\}$;
$R_{comm}^s := \{(s,s^\prime)|\exists~\alpha\in\Sigma,\beta\in\nabla:(s,\alpha,\nabla,s^\prime)\in R_{comm}\}$;
$R^s := R_{env}^s\cup R_{comm}^s$;
$R_{env}^T := TranstiveClosure(R_{env}^s)$;
$R^T := TranstiveClosure(R^s)$;
[**Integer**]{} $id := 1$;
Algorithms
==========
Liveness Analysis {#algcheckio}
-----------------
[**Procedure**]{} $CheckIO(\langle M, X\rangle, s_0, s_f)$
$N := \emptyset$; $E:= \emptyset$;
$(s_0,s_f)\in R_{env}^T\wedge(s_f,s_f)\in R_{env}^T$ [**Then**]{}
“Yes”;
$(s_0,s_f)\not\in R^T\wedge(s_f,s_f)\not\in R^T$ [**Then**]{}
“No”;
$N:=\{s|(s_0, s)\in R^T\wedge(s,s_f)\in R^T)\}$;
$E:=\{(s, s^\prime)|s,s^\prime\in N:(s,a,s^\prime)\in R_{env}\}\\\ttt
\cup\{(s,\alpha\beta,s^\prime)|s,s^\prime\in N:(s,\alpha,\beta,s^\prime)\in R_{comm}\}$;
$TestIO(X,reset,s_0,s_f,level=0,count=0)$;
[**End procedure**]{}
Union of Labeling Functions {#alghandleunion}
---------------------------
[**Procedure**]{} $Union(L_1,L_2)$
$L:=\emptyset$;
$s\in {\bf dom}(L_1)\cup {\bf dom}(L_2)$ [**Do**]{}
$s\in {\bf dom}(L_1)\cap {\bf dom}(L_2)$ [**Then**]{}
$L_1(s)=1\vee L_2(s)=1$ [**Then**]{}
$L := L\cup\{(s,1)\}$;
$L := L\cup\{(s,L_1(s)\vee L_2(s))\}$;
$s\in {\bf dom}(L_1)$ [**Then**]{}
$L := L\cup\{(s,L_1(s))\}$;
$L := L\cup\{(s,L_2(s))\}$;
$L$;
[**End procedure**]{}
Negation of a Labeling Function {#alghandlenegation}
-------------------------------
[**Procedure**]{} $Negation(M,L_1)$
$L:=\emptyset$;
$s\in S$ [**Do**]{}
$s\not\in {\bf dom}(L_1)$ [**Then**]{}
$L := L \cup\{(s,1)\}$;
$f(s)\not=1$ [**Then**]{}
$L := L \cup\{(s,\neg L_1(s))\}$;
$L$;
[**End procedure**]{}
Checking an EX Subformula {#alghandleex}
-------------------------
[**Procedure**]{} $HandleEX(M,L_1)$
$N:={\bf dom}(L_1)$; $L:=\emptyset$;
$t\in {\bf dom}(L_1)$ [**Do**]{}
$s:R^s(s,t)$ [**Do**]{}
$N := N\cup\{s\}$
$L_1(t)=1\wedge R_{env}^s(s,t)$ [**Then**]{}
$s\not\in {\bf dom}(L)$ [**Then**]{}
$L := L \cup \{(s, 1)\}$;
$L(s)\not=1$ [**Then**]{}
$L := L |_{s\leftarrow 1}$;
$s\not\in{\bf dom}(L)$ [**Then**]{}
$L := L\cup\{(s,id)\}$;
$E := \{(s,s^\prime)|s^\prime\in dom(f)\wedge\exists a:(s,a,s^\prime)\in R_{env}\}\\\ttt
\cup\{(s,\alpha\beta,s^\prime)|s^\prime\in dom(f)\wedge(s,\alpha,\beta,s^\prime)\in R_{comm}\}$;
$id$ [**with**]{} $G=\langle N,E,L_1\rangle$; $id := id+1$;
$L$;
[**End procedure**]{}
Checking an EU Subformula {#alghandleeu}
-------------------------
[**Procedure**]{} $HandleEU(M,L_1,L_2)$
$L := L_2$;
$T_1 := {\bf dom}(L_1)$; $T_2 := {\bf dom}(L)$;
$T_2\not = \emptyset$ [**Do**]{}
$t\in T_2$; $T_2 := T_2\setminus\{t\}$;
$s\in T_1 \wedge R^s(s,t)$ [**Do**]{}
$L_1(s)=1\wedge L(t)=1\wedge R_{env}^s(s,t)$ [**Then**]{}
$s\not\in {\bf dom}(L)$ [**Then**]{}
$T_2 := T_2 \cup \{s\}$; $L := L \cup \{(s, 1)\}$;
$L(s)\not=1$ [**Then**]{}
$T_2 := T_2 \cup \{s\}$; $L := L |_{s\leftarrow 1}$;
$s\not\in {\bf dom}(L)$ [**Then**]{}
$T_2 := T_2 \cup \{s\}$; $L := L \cup \{(s, id)\}$;
$N := {\bf dom}(L)$;
$E := \{(s,s^\prime)|s,s^\prime\in N\wedge\exists a:(s,a,s^\prime)\in R_{env}\}\\\ttt
\cup\{(s,\alpha\beta,s^\prime)|s,s^\prime\in N\wedge(s,\alpha,\beta,s^\prime)\in R_{comm}\}$;
$id$ [**with**]{} $G=\langle N,E,L_1,L_2\rangle$; $id := id+1$;
$L$;
[**End procedure**]{}
Checking an EG Subformula {#alghandleeg}
-------------------------
[**Procedure**]{} $HandleEG(X,\pi,s_0,G=\langle N,E,L_g\rangle)$
$SCC_{env} := \{C|C$ is a nontrivial SCC of $M$ and $C$ contains no communication transitions $\}$;
$SCC_{comm} := \{C|C$ is a nontrivial SCC of $M$ and $C$ contains some communication transitions $\}$;
$L := \{(s,1)|\exists C\in SCC_{env}:s\in C\}$
$\cup\{(s,id)|\exists C\in SCC_{comm}:s\in C\}$
$T := dom(L)$;
$T\not=\emptyset$ [**Do**]{}
$t\in T$; $T := T\setminus\{t\}$;
$s\in {\bf dom}(L_1) \wedge R^s(s,t)$ [**Do**]{}
$L(t)=1\wedge L_1(s)=1\wedge R_{env}^s(s,t)$ [**Then**]{}
$s\not\in {\bf dom}(L)$ [**Then**]{}
$T := T \cup \{s\}$; $L := L \cup \{(s, 1)\}$;
$L(s)\not=1$ [**Then**]{}
$T := T \cup \{s\}$; $L := L |_{s\leftarrow 1}$;
$s\not\in {\bf dom}(L)$ [**Then**]{}
$T := T \cup \{s\}$; $L := L \cup \{(s, id)\}$;
$N := {\bf dom}(L)$;
$E := \{(s,s^\prime)|s,s^\prime\in N\wedge\exists a:(s,a,s^\prime)\in R_{env}\}\\\ttt
\cup\{(s,\alpha\beta,s^\prime)|s,s^\prime\in N\wedge(s,\alpha,\beta,s^\prime)\in R_{comm}\}$;
$id$ [**with**]{} $G=\langle N,E,L_1\rangle$; $id := id+1$;
$L$;
[**End procedure**]{}
Testing an EX Graph {#algtestex}
-------------------
The algorithm for testing an $EX graph$ is simple. It first checks whether $L_1(s^\prime)$ can be evaluated true at any state $s^\prime$ such that the system can reach $s^\prime$ from $s_0$ through an environment transition. It returns true if it is the case. Otherwise,it chooses one candidate from the set of all possible input symbols from $s_0$, and feeds the sequence $\pi$ concatenated with the input symbol to $X$. If the output symbol of $X$ and the input symbol matches the annotation of an edge originating from the node, it moves forward to try the destination node of the edge. If there is no match, then it tries other candidates. But before trying any other candidate, it brings $X$ to its initial state by sending it the special input symbol, $reset$. The algorithm returns $false$ when all candidates are tried without a match.
[**Procedure**]{} $TestEX(X,\pi,s_0,G=\langle N,E,L_1\rangle)$
$(s_0,s^\prime)\in E:s^\prime\in dom(L_1)$ [**Do**]{}
$Experiment(X, reset\pi)$;
$TestWG(X,\pi, s^\prime, L_1(s^\prime))$ [**Then**]{}
$true$;
$Inputs := \{\alpha|\exists \beta:(s_0,\alpha\beta,s^\prime)\in E\}$;
$\alpha\in Inputs$ [**Do**]{}
$Experiment(X, reset\pi)$;
$\beta := Experiment(X, \alpha)$;
$\exists s^\prime:(s_0,\alpha\beta, s^\prime)\in E$ [**Then**]{}
$TestWG(X, \pi\alpha, s^\prime, L_1(s^\prime))$ [**Then**]{}
$true$;
;
$false$;
[**End procedure**]{}
Testing an EU Graph {#algtesteu}
-------------------
The procedure $TestEU$ keeps a sequence of input symbols $\pi$ that has been successfully accepted by $X$ and an integer $level$ that records how many communications have been gone through without reaching a destination state. And the algorithm works as follows. At first, it checks whether it has gone through more than $mn$ communications without success, it returns false if it is the case. Then, it checks whether it has reached a destination state (i.e., $s_0\in dom(L_2)$). If it is the case, it returns $true$ when $L_2(s_0)$ can be evaluated true $s_0$. Next, it checks whether $L_1(s_0)$ can be evaluated true at $s_0$, it returns false if it is not the case. After that, it checks whether $L_1(s^\prime)$ can be evaluated true at any state $s^\prime$ such that the system can reach $s^\prime$ from $s_0$ through an environment transition. It returns true if it is the case. Otherwise,it chooses one candidate from the set of all possible input symbols from $s_0$, and feeds the sequence $\pi$ concatenated with the input symbol to $X$. If the output symbol of $X$ and the input symbol matches the annotation of an edge originating from the node, it moves forward to try the destination node of the edge with $level$ increased by 1. If there is no match, then it tries other candidates. But before trying any other candidate, it brings $X$ to its initial state by sending it the special input symbol, $reset$. The algorithm returns $false$ when all candidates are tried without a match.
[**Procedure**]{} $TestEU(X, \pi, s_0, G=\langle
N,E,L_1,L_2\rangle,level)$
$level>mn$ [**Then**]{}[^2]
$false$;
$s_0\in dom(L_2)$ [**Then**]{}
$TestWG(X,\pi, s_0, L_2(s_0))$ [**Then**]{}
$true$;
$TestWG(X,\pi, s_0, L_1(s_0))$ [**Then**]{}
$false$;
$\exists s^\prime:(s_0,s^\prime)\in E$ [**Do**]{}
$Experiment(X, reset\pi)$;
$TestEU(X,\pi,s^\prime,G,level)$ [**Then**]{}
$true$;
$Inputs := \{\alpha|(s_0,\alpha\beta, s^\prime)\in E\}$;
$\alpha\in Inputs$ [**Do**]{}
$Experiment(X, reset\pi)$;
$\beta := Experiment(X, \alpha)$;
$\exists s^\prime:(s_0,\alpha\beta, s^\prime)\in E$ [**Then**]{}
$TestEU(X,\pi\alpha,s^\prime,G,level+1)$ [**Then**]{}
$true$;
;
$false$;
[**End procedure**]{}
Subroutine for Testing an EG Graph {#algsubtesteg}
----------------------------------
The procedure $SubTestEG$ keeps a sequence of input symbols that has been successfully accepted by $X$, an integer $level$ that records how many communications have been gone through without reaching $s_f$, and an integer $count$ that indicates how many times $s_f$ has been reached. It first checks whether it has gone through more than $mn$ communications without reaching $s_f$, it returns false if it is the case. Then, it checks whether it has reached the given state $s_f$. If it is the case, it returns $true$ when it has already reached $s_f$ for $m$ times, it increases $count$ by $1$ and resets $level$ to $0$ when otherwise. The next, it tests whether $L_1(s_0)$ can be evaluated true at $s_0$, and it returns false if it is not the case. After that it checks whether $L_1(s^\prime)$ can be evaluated true at any state $s^\prime$ such that the system can reach $s^\prime$ from $s_0$ through an environment transition. It returns true if it is the case. Otherwise, it chooses one candidate from the set of all possible input symbols from $s_0$, and feeds the sequence $\pi$ concatenated with the input symbol to $X$. If the output symbol of $X$ and the input symbol matches the annotation of an edge originating from the node, it moves forward to try the destination node of the edge with level increased by 1. If there is no match, it tries other candidates. But before trying any other candidate, it brings $X$ to its initial state by sending it the special input symbol $reset$. The algorithm returns $false$ when all candidates are tried without a match.
[**Procedure**]{}
$SubTestEG(X,\pi,s_0,s_f,G=\langle N,E,L_1\rangle,level,count)$
$level>mn$ [**Then**]{}
$false$;
$s_0=s_f$ [**Then**]{}
$count >= m$ [**Then**]{}
$true$;
$count := count+1$; $level := 0$;
$TestWG(X,\pi, s_0, L_1(s_0))$ [**Then**]{}
$false$;
$\exists s^\prime:(s_0,s^\prime)\in E$ [**Do**]{}
$Experiment(X, reset\pi)$;
$SubTestEG(X,\pi,s^\prime,s_f,G,level,count)$ [**Then**]{}
$true$;
$Inputs := \{\alpha|(s_0,\alpha\beta, s^\prime)\in E\}$;
$\alpha\in Inputs$ [**Do**]{}
$Experiment(X, reset\pi)$;
$\beta := Experiment(X,\alpha)$;
$\exists s^\prime:(s_0,\alpha\beta, s^\prime)\in E$ [**Then**]{}
$SubTestEG(X,\pi\alpha,s^\prime,s_f,G,level+1,count)$ [**Then**]{}
$true$;
;
$false$;
[**End procedure**]{}
Examples {#examples}
========
Check $(M,X)\models AG(s_2\rightarrow AF s_3)$ {#example1}
----------------------------------------------
To check whether $(M,X)\models AG(s_2\rightarrow AF s_3)$, is equivalent to checking whether $$(M,X)\models \neg E[true~U(s_2\wedge
EG\neg s_3)].$$ We describe how the formula $$f=E[true~U(s_2\wedge EG\neg s_3)]$$ is processed by $HandleCTL$ from bottom to up as follows.
1. the atomic subformula $s_2$ is processed by $HandleCTL$, and a labeling function $L_1=\{(s_2,1)\}$ is returned;
2. the atomic subformula $s_3$ is processed, and a labeling function $L_2=\{(s_3,1)\}$ is returned;
3. to process $\neg s_3$, $HandleNegation$ is called with $L_2$ to return a labeling function $L_3=\{(s_0,1),(s_1,1),(s_2,1),(s_4,1)\}$;
4. to process $EG\neg s_3$, $HandleEG$ is called with $L_3$ to construct an $EG~
graph$ $G_1=\langle N,E,L_3\rangle$ with id $2$ (see Figure 8) and return a labeling function $L_4=\{(s_0,2),(s_1,2),(s_2,2)\}$;
5. to process $s_2\wedge EG\neg s_3$, $HandleNegation$ and $HandleUnion$ are called with $L_1$ and $L_4$ to return a labeling function $L_5=\{(s_2,2)\}$;
6. to process $E[true~U(s_2\wedge EG\neg s_3)]$, $HandleEU$ is called with $L_5$ to construct an $EU graph$ $G_2=\langle N,E,L_5\rangle$ with id $3$ (see Figure 9) and return a labeling function $$L_f=\{(s_0,3),(s_1,3),(s_2,3),(s_3,3),(s_4,3)\}.$$
Since $s_0$ is labeled by $L_f$ with an ID expression $3$ instead of $1$ (i.e., $true$), we need to test whether the ID expression $3$ can be evaluated true at $s_0$ by calling $TestWG$ with $s_0$ and $G_2$. It’s easy to see that, essentially $TestWG$ would be testing whether some communication trace (with bounded length) with two consecutive symbol pairs $(send~yes)$ is a run of $X$. It returns $false$ if such trace exists, or vice versa.
Check $(M,X),s_0\models AG(s_2\rightarrow AXA[\neg s_2 U
s_3])$ {#example2}
--------------------------------------------------------
To check whether $(M,X),s_0\models AG (s_2\rightarrow AXA[\neg s_2 U s_3])$, is equivalent to checking whether $$(M,X)\models \neg E[true~U (s_2\wedge EX(E[\neg s_3 U (s_2\wedge\neg
s_3)]\vee EG\neg s_3))].$$ We describe how the formula $$f=E[true~U (s_2\wedge EX(E[\neg s_3 U (s_2\wedge\neg s_3)]\vee EG\neg s_3))]$$ is processed by $HandleCTL$ from bottom to up as follows.
1. the atomic subformula $s_2$ is processed by $HandleCTL$, and a labeling function $L_1=\{(s_2,1)\}$ is returned;
2. the atomic subformula $s_3$ is processed, and a labeling function $L_2=\{(s_3,1)\}$ is returned;
3. to process $\neg s_3$, $HandleNegation$ is called with $L_2$ to return a labeling function $L_3=\{(s_0,1),(s_1,1),(s_2,1),(s_4,1)\}$;
4. to process $s_2\wedge\neg s_3$, $HandleNegation$ and $HandleUnion$ are called with $L_1$ and $L_3$ to return a labeling function $L_4=\{(s_2,1)\}$;
5. to process $E[\neg s_3 U(s_2\wedge\neg s_3)]$, $HandleEU$ is called with $L_3$ and $L_4$ to construct an $EU~graph$ $G_1=\langle N,
E, L_3, L_4\rangle$ with id $2$ (see Figure 10) and return a labeling function $L_5=\{(s_0,2),(s_1,2),(s_2,1)\}$;
6. to process $EG\neg s_3$, $HandleEG$ is called with $L_3$ to construct an $EG~graph$ $G_2=\langle N,
E, L_3\rangle$ with id $3$ (see Figure 11) and return a labeling function $L_6=\{(s_0,3),(s_1,3),(s_2,3)\}$;
7. to process $E[\neg s_3 U(s_2\wedge\neg s_3)]\vee EG\neg s_3$, $HandleUnion$ is called with $L_5$ and $L_6$ to return a labeling function $L_7=\{(s_0,2\vee 3),(s_1,2\vee 3),(s_2,1)\}$;
8. to process $EX(E[\neg s_3 U(s_2\wedge\neg s_3)]\vee EG\neg s_3)$, $HandleEX$ is called with $L_7$ to construct an $EX~graph$ $G_3=\langle N,
E, L_7\rangle$ with id $4$ (see Figure 12) and return a labeling function $L_8=\{(s_0,4),(s_1,1),(s_2,4),(s_3,4)\}$;
9. to process $s_2\wedge EX(E[\neg s_3 U(s_2\wedge\neg s_3)]\vee
EG\neg s_3)$, $HandleNegation$ and $HandleUnion$ are called with $L_1$ and $L_8$ to return a labeling function $L_9=\{(s_2,4)\}$;
10. to process $E[true~U(s_2\wedge EX(E[\neg s_3 U(s_2\wedge\neg s_3)]\vee
EG\neg s_3))]$, $HandleEU$ is called with $L_9$ to construct an $EU graph$ $G_4=\langle N,E,L_5\rangle$ with id $5$ (see Figure 13) and return a labeling function $$L_f=\{(s_0,5),(s_1,5),(s_2,5),(s_3,5),(s_4,5)\}.$$
Since $s_0$ is labeled by $L_f$ with an ID expression $5$ instead of $1$ (i.e., $true$), we need to test whether the ID expression $5$ can be evaluated true at $s_0$ by calling $TestWG$ with $s_0$ and $G_4$. It’s easy to see that, essentially $TestWG$ would be testing whether some communication trace (with bounded lengtg) with two consecutive symbol pairs $(send~yes)$ is a run of $X$. It returns $false$ if such trace exists, or vice versa.
[^1]: The transition graphs in the figures in this section are not made total for the sake of readability.
[^2]: Here, $n$ always denotes the maximal number of communications on any simple paths in $G$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the Néel to four-fold columnar valence bond solid quantum phase transition in a sign free $S=1$ square lattice model. From quantum Monte Carlo simulations, we find evidence for a new kind of direct transition between these ordered phases. Even though both competing order parameters are [ *finite*]{} at the transition, it does not fit into the standard first order picture with its concomitant hysteresis and double peaked histograms. Instead the transition features diverging length scales and an emergent O(5) rotational symmetry between Néel and VBS order parameters. We argue that this striking behavior results crucially from a topological term that must be included in a field theoretic description of this system and is hence beyond a Landau order parameter analysis.'
author:
- Julia Wildeboer
- 'Jonathan D’Emidio'
- 'Ribhu K. Kaul'
title: Emergent symmetry at a transition between intertwined orders in a $S=1$ magnet
---
The study of the destruction of Néel order in $S=1/2$ magnets is a major field of theoretical condensed matter research inspired originally by the parent compounds of cuprate high temperature superconductors. Various theoretical arguments and extensive unbiased numerical calculations support the existence of a four fold degenerate columnar valence bond solid (VBS) phase on the destruction of Néel order, separated by the novel deconfined critical point [@read1989:vbs; @senthil2004:science; @senthil2004:deconf_long; @sandvik2007:deconf; @melko2008:jq; @nahum2015:so5]. Inspired by the iron pnictide superconductors, a number of studies of the destruction of Néel order in $S=1$ square lattice systems have appeared [@wang2015:s1; @yu2015:s1; @hu2017:nem], building on previous studies of the phase diagram of square lattice $S=1$ systems, (see [@toth2012:s1; @jiang2009:s1; @chen2018:s1; @harada2007:deconf; @michaud2013:s1] and references therein). It is thus interesting to extend the success of unbiased quantum Monte Carlo (QMC) studies of the destruction of Néel order in square lattice $S=1/2$ systems [@kaul2013:qmc] to the $S=1$ case, which we initiate here for the first time (phase transitions in coupled $S=1$ chains were considered previously in [@harada2007:deconf]). Since the subtle quantum effects that arise from topological terms depend crucially the microscopic value of the spin [@haldane1988:berry], one can expect striking differences between $S=1/2$ and $S=1$ even for phase transitions that appear identical with respect to the Landau-Ginzburg-Wilson criteria of dimensionality, symmetry and order parameters.
In this work we present unbiased numerical simulations of the Néel to four-fold columnar VBS transition in a $S=1$ square lattice model – the analogue for $S=1/2$ magnets is the well known deconfined critical point at which both order parameters simultaneously vanish [@senthil2004:science]. In contrast, for the $S=1$ system studied here we present extensive numerical evidence for a new kind of phase transition at which both order parameters are [*finite*]{}. Nonetheless, the transition does not fit into the conventional first order picture in which two minima in the free energy cross, giving rise to hysteresis. Instead the transition has diverging length scales and an emergent symmetry between the Néel and VBS vectors. We emphasize here that the symmetry is emergent at the transition and is absent in the microscopic model. The symmetry allows the system to rotate from Néel to VBS without encountering the free energy barriers that give rise to the characteristic hysteric behavior in a conventional first order transition. We argue that the unconventional phenomena cannot be explained by a naive LGW theory and trace this failure to the presence of a topological Wess-Zumino-Witten (WZW) term that captures the “intertwinement” of the order parameters and will appear in a field theory properly derived starting from $S=1$ spins.
[*Designer Model & Simulations:*]{} Our goal is to design a $S=1$ sign free model in which the Néel-VBS transition can be studied using Monte-Carlo simulations. We start with the square lattice $S=1$ Heisenberg model, $$\label{eq:j}
H_J = J \sum_{\langle ij \rangle}\vec{S_i} \cdot \vec{S_j}$$ This model is well known to be Néel ordered. Because we are working with $S=1$, it is possible to square the bilinear operator and obtain an independent “biquadratic operator,” $\left ( \vec S_i \cdot \vec S_j\right )^2$, also amenable to QMC [@harada2002:biq; @kaul2012:biq]. Using this term we can construct a Sandvik-like four spin interaction [@sandvik2007:deconf], $$\label{eq:qk}
H_{Q_K} = -Q_K \sum_{ijkl \in \square}\left (\left (\vec{S_i} \cdot
\vec{S_j}\right )^2-1\right ) \left (\left (\vec{S_k} \cdot
\vec{S_l}\right )^2-1\right )$$ We note that $H_{Q_K}$ has a higher staggered SU(3) symmetry because it is constructed from the biquadratic interaction, of which the physical SU(2) is a subgroup. However the model we study here $H_{JQ_K}=H_J+H_{Q_K}$ has only the generic SU(2) symmetry obtained by rotating the $\vec S$ vector in the usual way. Previous numerical studied have established that $H_{Q_K}$ on the square lattice has four-fold columnar VBS order [@lou2009:sun; @kaul2011:su34; @banerjee2010:su3]. Thus the single tuning parameter in $H_{JQ_K}$ gives us unbiased numerical access to the Néel-VBS transition in a $S=1$ system, as desired.
![\[fig:RNV\] Néel and VBS order parameters ratios, ${\cal
R}_N$ and ${\cal R}_V$ close to the quantum phase transition showing clear evidence for a direct transition. (inset) shows the value of $g_c$ obtained by analyzing crossings of $L$ and $2L$ values for both ratios. solid lines are a fit to the data giving $g_c=0.588(2)$.](ratios.pdf){width="1.0\columnwidth"}
Since our model is constructed to be Marshall sign positive, it can be simulated without a sign problem using the stochastic series expansion method [@sandvik2010:vietri]. To update the Monte Carlo configurations, we use an efficient directed loop algorithm [@syljuasen2002:dirloop]. Our simulations are carried out an $L\times L$ square lattices at an inverse temperature $\beta$ – all the data presented here with $\beta=L/4$ has been checked to be in the $T=0$ limit [@nbdrct2015:supmat]. We work in units in which $J=1$, and define the tuning parameter $g\equiv Q_K/J$ to access the phase transition. We study the Fourier transform of the Néel and VBS correlation functions, $S^N_{\bf k}=\frac{1}{L^2}\sum_r e^{i {\bf k \cdot r}}\langle S^z({\bf r}) S^z({\bf 0})\rangle$ and $S^V_{\bf k}=\frac{1}{L^2}\sum_r e^{i {\bf k \cdot r}}\langle S(\bf r) \cdot S(\bf r + \hat{\bf x}) S(\bf 0) \cdot S(\bf 0 + \hat{\bf x})\rangle$. We define the order parameters as ${\cal O}^2_N = S^N_{\bf (\pi,\pi) }$ and ${\cal O}^2_V = S^V_{\bf
(\pi,0) }$. For each of the order parameters we define ratios $R = 1-\frac{S_{{\bf K}+\frac{2\pi}{L}{\bf
y}} }{S_{\bf K}}$ (with ${\bf K}$ the ordering momentum); $R$ goes to 1 in a phase with long range order and 0 in a disordered phase.
Fig. \[fig:RNV\] shows the ratios $R$ for the Néel and VBS order parameters as a function of $g$ for different $L$. The data (see inset for finite size scaling) provides strong evidence that the Néel-VBS transition is direct – we can safely rule out co-existence or an intermediate phase. The crossing of the ratio at a finite value indicates a diverging length scale. However, the possibility of a direct continuous transition is contradicted by a study of the order parameters themselves, shown in Fig. \[fig:orderNV\]. In this finite size scaling plot of both order parameters, we have clear evidence that at the transition both order parameters are [*finite*]{}. This is in direct contradiction with the expectation of a continuous deconfined critical point for $S=1/2$ systems where they must both vanish. A guess then is that this could be a regular first order transition at which there is a free energy crossing of two independent minima (one for Néel and the other for VBS) at the transition. This scenario leads to the well known hysteric behavior and double peaked histograms in MC studies. We note that the finite values the order parameter extrapolate to are not small and hence this may not be thought of at all as a “weak” first order transition. In our case, as we show below a study of the histograms of the order parameters close to the transition do not show any evidence for double peaked or hysteric behavior.
![\[fig:orderNV\] Finite size scaling of the order parameters ${\cal O}^2_N$ and ${\cal O}^2_{V}$ close to the phase transition. The extrapolation to a finite value for both Néel and VBS order parameters at a common coupling $g=0.59$ establish that both order parameters are finite at the transition.](orderNV.pdf){width="1.0\columnwidth"}
We present evidence that the reason Néel and VBS order parameters are able to both be finite and still tunnel among each other without hysteresis is because of an enlarged O(5) symmetry joining three components of the Néel and two components of VBS at the phase transition. This symmetry is emergent (not enforced microscopically) since the VBS and Néel order parameters are expressed entirely differently in the $S=1$ spins. We now turn to a test of this symmetry in our $S=1$ model system. We are able to measure only one component of the Néel $N_z$ but both components $V_x$ and $V_y$ of the VBS order parameter for each configuration. These measurements suffice since $N_x$ and $N_y$ are microscopically symmetric with $N_z$ because of the global SU(2) of $H_{JQ_K}$. To proceed we define a vector $\vec \Psi=(aV_x,aV_y,bN_z)$ with $a$ and $b$ non-universal scale factors that we choose appropriately.
If the 5-component vector $\hat \Phi = (V_x,V_y,N_x,N_y,N_z)$ has O(5) symmetry it is clear that the 3-component $\vec \Psi$ vector must have O(3) at the transition, since the O(3) rotations are a sub-group of O(5). To check this we study the direction of the unit vector $\vec \Psi/ |\vec \Psi |$. Fig. \[fig:anghist\] shows the angular distribution of the unit vector on the unit sphere using standard polar co-ordinates, cos$\theta$ and $\phi$. The numerical data provides evidence that right at the transition there is an emergent symmetry which allows rotations among Néel and VBS. We also note that slightly away from the transition this symmetry is lost and we recover what one would expect for the histograms in either the Néel or VBS phases.
To test the emergence of O(5) symmetry in the thermodynamic limit more quantitatively we study the distribution of the magnitude $|\vec \Psi|$. We expect in the thermodynamic limit that at the phase transition point $\hat \Phi$ since it is ordered will be distributed uniformly on $S^4$ (unit sphere surface in 5-dimensions), if normalized properly. This distribution has direct consequences on the components of $\vec \Psi$ and its magnitude as can easily be worked out: $P(\Psi_\alpha)=
\frac{3}{4}(1-\Psi^2_\alpha)$ for $|\Psi_\alpha|<1$ and 0 otherwise. $P(|\vec \Psi|)=3 |\vec \Psi|^2$ for $|\vec \Psi|<1$ and 0 otherwise. Fig. \[fig:radius\] shows how these distributions are approached as $L$ is increased in our numerical simulations. This is our strongest evidence for the emergence of the O(5) symmetry in our model at the phase transition in the thermodynamic limit.
[*Landau Theory & Topological Term:*]{} We now argue that the enhanced symmetry cannot appear at a generic phase transition (by tuning one parameter) in a Landau analysis. Thus the phase transition found in our $S=1$ model simulation is beyond Landau theory.
We begin our discussion by noting that our transition is phenomenologically similar to one in an anisotropic five component classical spin model with energy, $$\label{eq:o5model}
E= - J_N\sum_{\langle ij \rangle} \vec {N_i}\cdot \vec {N_j}- J_V\sum_{\langle ij \rangle} \vec {V_i}\cdot \vec {V_j},$$ where $\vec N_i$ is a three component vector representing the Néel order and $\vec V_i$ is a two component vector representing the VBS with $\vec N^2+\vec V^2 =1$ and defined on a three-dimensional cubic lattice where as usual, $Z=\sum_c e^{-E/T}$. We assume for simplicity that the VBS order parameter has O(2) rather than $Z_4$ symmetry, but our arguments can easily be extended to include the lower symmetry. The model has $O(3) \times O(2)$ for $J_N\neq J_V$ and $O(5)$ symmetry when $J_N=J_V$. Working at low-$T$, we now consider the nature of broken symmetry phases. When $J_N/J_V=1$ both $\langle \vec N\rangle \neq 0$ and $\langle \vec V\rangle \neq 0$ and there is an O(5) symmetry that allows rotations between $\vec N$ and $\vec V$. When $J_N/J_V$ is increased above 1, $\langle \vec N\rangle \neq 0$ and $\langle \vec V\rangle =0$ (the Néel phase in this model) and $\langle \vec V\rangle \neq 0$ and $\langle \vec N\rangle =0$ (analogue of the VBS phase) when the ratio is lowered below 1. Now consider the transition from Néel ($\langle \vec N\rangle \neq 0$) to VBS ($\langle \vec V\rangle \neq 0$) as the parameter $J_N/J_V$ is tuned. This transition shares all of the features with what we have found in for the Néel-VBS transition in $H_{JQ_K}$: diverging length scales, higher symmetry at the phase transition, finite order parameters and an absence of hysteresis (since $\vec N$ can clearly rotate into $\vec V$ without encountering an energy barrier when $J_N=J_V$). There is one crucial difference however between this classical Landau theory and what we have found for $H_{JQ_K}$. In Eq. \[eq:o5model\] the O(5) symmetry is microscopically enforced at the transition when $J_N=J_V$, in $H_{JQ_K}$ no such microscopic symmetry is present. Indeed the Néel and VBS order parameters are implemented so distinctly in the $S=1$ spins, it is hard to imagine how such a symmetry would be enforced at the microscopic level. It is then natural to ask: could the O(5) symmetry emerge at a generic phase transition (tuning one parameter) between the phases where $\vec
N$ and $\vec V$ individually condense in a generic (with only $O(3)\times O(2)$ symmetry microscopically) classical model of the type Eq. \[eq:o5model\]?
We now provide a simple argument that proves this is not possible. We consider the stable fixed point that describes the ordered phase of an O(5) vector sigma model for $\hat \Phi= (\vec V,\vec N)$ (if we enforce the O(5) symmetry there are of course no relevant operators). This gaussian fixed point can be accessed by the usual Goldstone expansion $\hat
\Phi = ({\sqrt{1-\vec \pi^2}},\vec \pi)$ and expanding to quadratic order in $\vec \pi$. We now ask how many relevant operators we can add if we only enforce $O(3)\times O(2)$ symmetry? If there is only one such operator one could reach the symmetric order fixed point as a generic phase transition. However in the simple LGW theory one can add two strongly relevant anisotropy operators in 2+1 dimensions $\vec N^2 - \vec V^2 $ and $(\vec N^2 -
\vec V^2)^2 $. Thus the symmetric fixed point cannot emerge by tuning just one parameter in a classical model, one must generically have a regular first order transition with hysteresis between the two phases- thus a simple Landau theory cannot explain the emergence of O(5) symmetry like we have found in $H_{JQ_K}$.
![\[fig:radius\] Normalized histograms for individual components $\Psi_\alpha$ and its radius $|\vec \Psi|$ of the vector $\vec
\Psi=(aV_x,aV_y,bN_z)$, close to the phase transition and comparison with predictions of O(5) distribution with unit radius. (left) Comparison of individual components for Néel and VBS. (right) Showing how $P(|\vec \Psi |)$ approaches the O(5) prediction as the thermodynamic limit is approached (see text). We have fixed the constants $a$ and $b$ by choosing the convention $\langle
\Psi^2_\alpha\rangle=1/5$ to match the O(5) prediction. We used $g=0.588$ and $L=32$.](radius.pdf){width="1.0\columnwidth"}
Given the above argument, it is interesting to ask what goes wrong with the application of naive LGW theory to our $S=1$ model? As it turns out a microscopic derivation of the Landau theory for the $S=1$ model requires a $k=2$ Wess-Zumino-Witten (WZW) term to be included. Here we present only a heuristic argument for the topological term by building on the well-known result that the Néel-VBS transition in $S=1/2$ magnets can be described by the $k=1$ WZW theory in the order parameter supervector which combining Néel and VBS is a five component vector living in $S^4$ [@tanaka2005:nvbs; @senthil2006:topnlsm]. The approach is to construct the $S=1$ field theory by coupling two layers of $S=1/2$ field theories [@wang2015:s1]. The critical theory for each $S=1/2$ layer close to its own Néel- VBS deconfined critical point is described by the 2+1 dimensional vector O(5) sigma model with a WZW term at level one, $S_{S=1/2} = \int d^3x \frac{1}{g}\left (\partial_\mu \hat \Phi \right )^2 + i 2\pi \Gamma$, where the topological WZW term is written as $\Gamma = \frac{1}{{\rm vol}(S^4)}\int d^3x du \epsilon^{abcde} \hat
\Phi_a \partial_x \hat \Phi_b \partial_y \hat \Phi_c \partial_z \hat
\Phi_d \partial_u \hat \Phi_e $ with ${\rm vol}(S^4)= 8\pi^2/3$. Now it is clear that to combine the two $S=1/2$ layers to obtain the field theory for the $S=1$ transition between Néel and four fold VBS, we must couple both Néel and VBS vectors [*ferromagnetically*]{}, this guarantees that the correct states of matter are obtained for the $S=1$ system, Néel and four-fold columnar VBS. The ferromagnetic coupling will force the $\hat \phi$ in both the layers to track each other perfectly coupling causing the two $k=1$ WZW terms to simply add (see supplementary materials of [@wang2015:s1]). We thus obtain the same critical theory as $S_{S=1/2}$ except the level of the WZW term is doubled, $S_{S=1} = \int d^3x \frac{1}{g}\left (\partial_\mu \hat \Phi \right )^2 + i 4
\pi \Gamma$. In addition we have to add anisotropies to this field theory that reduce the symmetry down to the physical symmetry of $O(3)\times Z_4$. It is clearly the presence of the WZW term that invalidates a naive application of Landau theory, and which leads to the interesting behavior found here. Our work can hence shed valuable light on the poorly understood physics of sigma models with WZW terms in 2+1 dimensions. Interestingly, very recent work has found a similar phenomenology in an entirely different microscopic setting: $S=1/2$ systems and their transition to a two-fold VBS [@zhao2018:ssl] and certain classical loop models [@serna2018:first], suggesting this kind of emergent symmetry enhanced “first order” transition could appear in various settings. Our findings provides strong motivation for further studies of this new kind of phase transition and also more generally unbiased numerical studies of two-dimensional spins systems beyond $S=1/2$.
We thank A. Dymarsky, M. Levin, G. Murthy, A. Nahum and A. Sandvik for inspiring conversation. Partial financial support from NSF-DMR 1611161 is acknowledged. Computing resources were provided by the XSEDE consortium and the DLX computer at UK.
SUPPLEMENTAL MATERIALS
=======================
In these supplementary materials we elaborate on our measurement definitions, provide comparisons with exact diagonalization and present data confirming our convergence to zero temperature. Finally we provide data on a related $J$-$Q_J$ model which exhibits Néel order throughout the entire phase diagram. \
[**Algorithm:**]{} The numerical results presented in this work have been obtained using the stochastic series expansion (SSE) method [@sandvik_aip]. Since the method is standard and well documented we will not review it here. We work in the $S_z =-1,0,1$ basis for our $S=1$ problem. To update the SSE configurations we use both local diagonal updates and the non-local directed loop alorithm [@sandvik2] that allows us to switch between the allowed vertices while respecting the $S_z$ conservation.
[**QMC-ED comparison:**]{} We have tested our code by performing comparisons against exact diagonalization. For future reference, Tables \[table0\] and \[table1\] provide test comparisons between measurements obtained from a SSE study and exact diagonalization (ED) on a lattice of size $(L_{x},L_{y}) = (4,4)$, for various combinations of the bond and plaquette interactions $J$ and $Q_{K}$ for the $J-Q_{K}$ model under investigation in this work and for various combinations of the bond and plaquette interactions $J$ and $Q_{J}$ for the spin$-1$ version of Sandvik’s $J-Q_{J}$ model (see Appendix E). Due to the very large Hilbert space for this spin-1 model on a 4x4 lattice, we project out the ground state from a random state in the $S^z=0$ subspace, thus avoiding the need to diagonalize the sparse Hamiltonian matrix. We list values for the extensive ground state energy, the Néel order parameter $\mathcal{O}^{2}_{N}$ as well as the VBS order parameter $\mathcal{O}^{2}_{V}$. Also shown are the so-called ratios $\mathcal{R}_{N}$ and $\mathcal{R}_{V}$. All observables are defined below.
![The order parameters $\mathcal{O}^{2}_{V}$ and $\mathcal{O}^{2}_{N}$ are shown as a function of the inverse temperature $\beta$ for the $J-Q_{K}$ model for a system of size $(L_{x}, L_{y}) = (16,16)$. Again we checked their $\beta$-dependence in the Néel phase $(Q_K = 0.5)$, close to the transition $(Q_K = 0.6)$, and in the VBS phase $(Q_K = 0.7)$. $J$ is always set to $1$. We observe that the values of $\mathcal{O}^{2}_{V}$ and $\mathcal{O}^{2}_{N}$ are independent of $\beta$ as long as $\beta \geq 2$. []{data-label="Fig_1b"}](Bxx_SzSz_color_nogrid_L16.pdf){width="1.0\columnwidth"}
$L_{x}$ $L_{y}$ $J$ $Q_{K}$ $E$ (ED) $E$ (MC) $\mathcal{O}^{2}_{N}$ (ED) $\mathcal{O}^{2}_{N}$ (MC) $\mathcal{O}^{2}_{V}$ (ED) $\mathcal{O}^{2}_{V}$ (MC) $\mathcal{R}_{N}$ (ED) $\mathcal{R}_{N}$ (MC) $\mathcal{R}_{V}$ (ED) $\mathcal{R}_{V}$ (MC)
--------- --------- ----- --------- ------------ ------------- ---------------------------- ---------------------------- ---------------------------- ---------------------------- ------------------------ ------------------------ ------------------------ ------------------------
4 4 0.2 0.9 -96.15381 -96.147(8) 0.13590 0.13592(2) 0.50414 0.5044(5) 0.49940 0.4993(1) 0.75713 0.7570(7)
4 4 0.5 0.2 -49.02200 -49.024(4) 0.25596 0.25594(9) 0.28370 0.2838(2) 0.78679 0.7868(1) 0.59012 0.5907(7)
4 4 0.7 0.3 -70.29052 -70.288(5) 0.24879 0.24867(8) 0.29726 0.2971(2) 0.77611 0.7760(1) 0.60493 0.6054(6)
4 4 0.8 0.4 -85.17819 -85.180(6) 0.23283 0.23291(6) 0.32728 0.3269(2) 0.75040 0.7503(1) 0.63436 0.6346(5)
4 4 0.9 0.6 -109.00470 -109.001(7) 0.20556 0.20562(3) 0.37805 0.3777(2) 0.69897 0.6989(1) 0.67619 0.6761(4)
$L_{x}$ $L_{y}$ $J$ $Q_{J}$ $E$ (ED) $E$ (MC) $\mathcal{O}^{2}_{N}$ (ED) $\mathcal{O}^{2}_{N}$ (MC) $\mathcal{O}^{2}_{V}$ (ED) $\mathcal{O}^{2}_{V}$ (MC) $\mathcal{R}_{N}$ (ED) $\mathcal{R}_{N}$ (MC) $\mathcal{R}_{V}$ (ED) $\mathcal{R}_{V}$ (MC)
--------- --------- ----- --------- ------------ ------------- ---------------------------- ---------------------------- ---------------------------- ---------------------------- ------------------------ ------------------------ ------------------------ ------------------------
4 4 0.2 0.9 -157.24324 -157.251(8) 0.33323 0.3330(1) 0.12077 0.121(1) 0.87616 0.87610(8) 0.29722 0.295(9)
4 4 0.5 0.2 -66.86936 -66.861(3) 0.34103 0.3409(2) 0.10760 0.1073(3) 0.88539 0.8854(1) 0.21940 0.215(3)
4 4 0.7 0.3 -96.79576 -96.790(5) 0.34071 0.3406(2) 0.10814 0.1079(3) 0.88501 0.8850(1) 0.22295 0.225(3)
4 4 0.8 0.4 -119.70732 -119.707(4) 0.34001 0.3402(1) 0.10935 0.1090(2) 0.88417 0.8843(1) 0.23071 0.226(3)
4 4 0.9 0.6 -158.52300 -158.520(6) 0.33873 0.3388(1) 0.11153 0.1113(3) 0.88264 0.88268(9) 0.24436 0.241(3)
![Shown is the ratio $\mathcal{R}_{N}$ of the Néel order parameter of various values of the plaquette interaction coupling $Q_{J}$ with $J^{2} + Q_{J}^{2} = 1$ for systems of size $(L,L)$ with $L$ up to 32 lattice sites. $\mathcal{R}_{N}$ appears to be independent from $Q_{J}$ and approaches $1$ for increasingly large system sizes indicating a phase diagram consisting entirely of Néel order. The inset shows the Néel order parameter $\mathcal{O}_{N}^{2}$.[]{data-label="Fig_3"}](SzSzRatio_color_allL_Sandvik_II6a.pdf){width="1.0\columnwidth"}
![Shown is the ratio $\mathcal{R}_{V}$ of the VBS order parameter of various values of the plaquette interaction coupling $Q_{J}$ with $J^{2} + Q_{J}^{2} = 1$ for systems of size $(L,L)$ with $L$ up to 32 lattice sites. Confirming the findings from Fig. \[Fig\_3\], we see that $\mathcal{R}_{V}$ approaches zero for sufficiently large lattice sizes independent from the coupling $Q_{J}$ providing evidence for the absence of VBS order in the $J-Q_{J}$ model for spin$-1$.[]{data-label="Fig_4"}](BxxRatio_color_allL_Sandvik_II6a.pdf){width="1.0\columnwidth"}
[**Measurements:**]{} In order to simplify the QMC loop algorithm, we have shifted our $J$ bond operators by the identity, $J(S_i \cdot S_j - 1)$. The extensive energy quoted in the tables includes this shift. In order to characterize the Néel and the VBS phases, we measure the equal time bond-bond correlation function $\langle S_{\vec{r}}\cdot S_{\vec{r}+\hat{\alpha}} S_{\vec{r}^{'}}\cdot S_{\vec{r}^{'}+\hat{\alpha}} \rangle$. Here a bond is identified by its location on the lattice $\vec{r}$ and its orientation $\alpha$ with $\alpha = x,y$ in two-dimensions. In the VBS phase, lattice translational symmetry is broken. This gives rise to a Bragg peak in the Fourier transform of the bond-bond correlator defined as $$\tilde{C}^{\alpha}(\vec{q})=\frac{1}{N_{\mathrm{site}}^2}\sum_{\vec{r},\vec{r}'}e^{i(\vec{r}-\vec{r}')\cdot\vec{q}}\langle
S_{\vec{r}}\cdot S_{\vec{r}+\hat{\alpha}} S_{\vec{r}^{'}}\cdot S_{\vec{r}^{'}+\hat{\alpha}}\rangle \;.
\label{FT}$$ For a columnar VBS patterns, peaks appear at the momenta $(\pi,0)$ and $(0,\pi)$ for $x$ and $y$-oriented bonds, respectively. Thus, the VBS order parameter is given by $$\mathcal{O}_{VBS}=\frac{\tilde{C}^{x}(\pi,0)+\tilde{C}^{y}(0,\pi)}{2} \;.
\label{Ovbs}$$ Another useful quantity to locate a possible phase transitions is the above mentioned VBS ratio $\mathcal{R}_{V}$. We first distinguish between $x-$ and $y-$oriented bonds: $$\begin{aligned}
\mathcal{R}^x_{V}&=&1- \tilde{C}^{x}(\pi,2\pi/L)/\tilde{C}^{x}(\pi,0) \nonumber \\
\mathcal{R}^y_{V}&=&1- \tilde{C}^{y}(2\pi/L,\pi)/\tilde{C}^{y}(0,\pi)\;.
\label{RXvbs}\end{aligned}$$ Subsequently, we average over $x$ and $y$- orientations: $$\mathcal{R}_{V}=\frac{\mathcal{R}^x_{V}+\mathcal{R}^y_{V}}{2} \;.
\label{RXvbs1}$$ This quantity goes to $1$ in a phase with long-range VBS order and it approaches $0$ in a phase without VBS order present.
The Néel structure factor is, $$m^{2}_{z}(\vec{q})=\frac{1}{N_{\mathrm{site}}^2}\sum_{\vec{r},\vec{r}'}e^{i(\vec{r}-\vec{r}')\cdot\vec{q}}\langle
S^{z}_{\vec{r}}S^{z}_{\vec{r}'\alpha}\rangle \;.
\label{FT_m}$$ The Bragg peak appears at momentum $(\pi,\pi)$ and thus the Néel order parameter is given by $$\mathcal{O}_{N}=m^{2}_{z}(\pi,\pi)\;.
\label{ONeel}$$ To additionally provide a quantity that goes to $1$ in a Néel ordered phase and vanishes in a phase without, we study the The Néel ratio: $$\begin{aligned}
\mathcal{R}^{x}_{N}&=&1- m^{2}_{z}(\pi+2\pi/L,\pi)/m^{2}_{z}(\pi,\pi) \nonumber \\
\mathcal{R}^{y}_{N}&=&1- m^{2}_{z}(\pi,\pi+2\pi/L)/m^{2}_{z}(\pi,\pi)\;.
\label{RN}\end{aligned}$$ We can now average over both quantities: $$\mathcal{R}_{N}=\frac{\mathcal{R}^x_{N}+\mathcal{R}^y_{N}}{2}\;.
\label{RXYNeel}$$
[**Ground state convergence:**]{} Here we investigate the behavior of our observables when the SSE is carried out at different inverse temperatures $\beta$. We perform scaling as a function of inverse temperature $\beta$ for a system of size $(L_{x}, L_{y}) = (16,16)$. We checked the behavior of this observable in the Néel phase $(Q_K =
0.5)$, close to the transition $(Q_K = 0.6)$, and in the VBS phase $(Q_K = 0.7)$. We find that the measurements are essentially saturated already for $\beta \geq 2$. We observe analogous behavior for systems of size $(L,L)$ with $L = 8, 10, 12$. Therefore we fixed the inverse temperature to $\beta = L/4$ in our SSE simulation in the main manuscript.
[**$J-Q_{J}$ Model for spin 1**]{}: We now briefly discuss another designer model Hamiltonian and compare the phase diagram for the two cases of a spin$-1/2$ system and a spin$-1$ system.
The so-called “$J-Q$” model was introduced by Sandvik in 2007 [@sandvik3]. The model consists of a Heisenberg interaction between nearest neighbor sites (see equation $(1)$ in the main manuscript) on the square lattice and an additional plaquette term:
$$\begin{aligned}
\label{HQJ12}
H_{Q} = -Q\sum_{ijkl \in \square} \big(\vec{S}_{i}\cdot \vec{S}_{j} - \frac{1}{4}\big)\big(\vec{S}_{k}\cdot \vec{S}_{l} - \frac{1}{4}\big)\;.\end{aligned}$$
The spin$-1/2$ case of this model $H = H_{J} + H_{Q}$ was shown to have a phase transition from Néel order to VBS order at a critical point $ J/Q \approx 0.04$ [@sandvik3].
We now subject the same term structure to a SSE-MC simulation in order to determine the phase diagram. We note that for the spin$-1$ case the constant $\frac{1}{4}$ is replaced by $1$ in order by make the plaquette term amenable to the SSE-MC study: $$\begin{aligned}
H_{Q_{J}} = -Q_{J}\sum_{ijkl \in \square} \big(\vec{S}_{i}\cdot \vec{S}_{j} - 1\big)\big(\vec{S}_{k}\cdot \vec{S}_{l} - 1\big)\;.
\label{HQJ}\end{aligned}$$ The $J-Q_{J}$ model spin$-1$ Hamiltonian is then $H_{JQ_{J}} = H_{J} + H_{Q_{J}}$. We analyzed the phase diagram for various couplings $J$ and $Q_{J}$ with the condition $J^{2} + Q_{J}^{2} = 1$ and found that the phase diagram consists entirely of Néel order independent from the ratio of the two coupling strengths $J$ and $Q_{J}$. Fig. \[Fig\_3\] shows the ratio of the Néel order parameter. The ratio appears to be independent from $Q_{J}$ (with $J$ fixed by $J^{2} + Q_{J}^{2} = 1$). Further the ratio $\mathcal{R}_{N}$ approaches $1$ for increasingly large system sizes. This is a clear indicator that the entire phase diagram consists of Néel order. For completeness we also give the ratio $\mathcal{R}_{V}$ of the VBS order parameter $\mathcal{O}_{V}^{2}$. In compliance with our findings from Fig. \[Fig\_3\], we see the ratio $\mathcal{R}_{V}$ approaches zero for sufficiently large lattice sizes independent from the coupling $Q_{J}$ (again with $J$ fixed by $J^{2} + Q_{J}^{2} = 1$). This provides evidence for the absence of VBS order that was present in the spin$-1/2$ flavor of the model.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- Gurtina Besla
bibliography:
- 'BeslaSeychelles.bib'
title: The Orbits and Total Mass of the Magellanic Clouds
---
Introduction {#sec:1}
============
Owing to their proximity to our Galaxy, the Magellanic Clouds (MCs) have been observed in wavebands spanning almost the entire electromagnetic spectrum, allowing us to study the interstellar medium (ISM) of two entire galaxies in unprecedented detail. Such observations have facilitated studies of how turbulence, stellar feedback and star formation are interrelated and how these internal processes affect galactic structure on small to large scales [e.g., @Elm01; @Block10].
However, the MCs are also subject to environmental processes that can dramatically alter their internal structure. For example, the MCs are surrounded by a massive complex of HI gas in the form of a 150 degree long stream trailing behind them (the Magellanic Stream), a gaseous bridge connecting them (the Magellanic Bridge) and an HI complex that leads them (the Leading Arm) [@Nidever10]. This material once resided within the MCs and was likely stripped out by some combination of external tides and/or hydrodynamic processes.
Recently, @Fox14 revealed that these HI structures harbor a significant amount of ionized gas, increasing the total gas mass budget [*outside*]{} the MCs from $4.87 \times 10^{8}$ M$_\odot$ to $\sim2\times 10^{9}$ M$_\odot$. This extended gas complex thus represents a non-negligible fraction of the MW’s circumgalactic medium (CGM).
Identifying the formation mechanism of these structures depends sensitively on the amount of time the MCs have spent in close proximity to the MW. Constraining the dynamics of the MCs is thus critical to our understanding of the morphologies, star formation histories and ISM properties of these important galactic laboratories.
Our understanding of the orbital history of the MCs has evolved considerably over the past 10 years. The canonical view, wherein the MCs have completed multiple orbits about the MW over a Hubble time [@MF80], has changed to one where they are recent interlopers, just completing their first passage about our Galaxy [@B07].
This dramatic change has been driven by two factors. Firstly, high precision proper motions measured using the Hubble Space Telescope (HST) have enabled accurate 3D velocities of both the Large and Small Magellanic Clouds (LMC and SMC). We now know the MCs are moving faster than previously believed, relative to not only the MW, but also to [*each other*]{} [@K06a; @K06b; @K13].
Secondly, our understanding of the mass and structure of galactic dark matter halos has evolved. In the $\Lambda$ Cold Dark Matter paradigm, low mass galaxies reside within massive dark matter halos, having much larger mass-to-light ratios than expected for galaxies like the MW. This means that the MCs are likely significantly more massive than traditionally modeled. Furthermore, the dark matter halos of massive galaxies are now understood to be poorly represented by isothermal sphere profiles at large distances. Instead, the dark matter density profile falls off more sharply, making it easier for satellites to travel to larger Galactocentric distances.
However, debate still ensues concerning the orbital history of the MCs. While the canonical picture, where the MCs have completed $\sim$6 orbits about the MW with an orbital period of $\sim$2 Gyr, has been largely dismissed, there are new proposed models where the MCs have completed one or two orbits about the MW within a Hubble time [@Sha09; @Zha12; @Diaz11; @Diaz12]. The goal of this review is to explain why the controversy arises and why various lines of evidence support a first infall scenario.
Determining the Orbit of the MCs {#sec:3}
================================
Reconstructing the past orbital history of the MCs depends on 3 important factors. 1) An accurate measurement of the current 3D velocity vector and distance of the MCs with respect to the MW. 2) The mass of the MW and its evolution over time. 3) The masses of the MCs, which ultimately determines the importance of dynamical friction as the MCs orbit about of the MW and each other.\
[**The 3D Velocity of the MCs:**]{}
Recently, @K13 [ hereafter K13] used HST to measure the proper motions of stars in the LMC with respect to background quasars, obtaining 3 epochs of data spanning a baseline of $\sim$7 years and proper motion random errors of only 1-2% per field. This astonishing accuracy is sufficient to explore the internal stellar dynamics of the LMC, allowing for the first constraints on the large-scale rotation of [*any*]{} galaxy based on full 3D velocity measurements [@van14]. This analysis has resulted in the most accurate measurement of the 3D Galactocentric velocity of the LMC and SMC to date. The LMC is currently moving at 321 $\pm$ 23 km/s with respect to the MW. The SMC is moving at 217 $\pm$ 26 km/s with respect to the MW and 128 $\pm$ 32 km/s with respect to the LMC; the SMC cannot be on a circular orbit about the LMC. Errors on the velocity measurement are now limited by the errors on the distance measurement to the Clouds rather than the proper motions.\
[**The Mass of the MW:**]{}
The mass of the MW is uncertain within a factor of $\sim$2. Values for the virial mass range from M$_{\rm vir}=$(0.75-2.25) $\times 10^{12}$ M$_\odot$. Here, M$_{\rm vir}$ is defined as the mass enclosed within the radius where the dark matter density is $\Delta_{\rm vir} =$360 times the average matter density, $\Omega_m \rho_{\rm crit}$.
HST proper motions over a six year baseline revealed that the Leo I satelite is moving with a Galactocentric velocity of 196.0 $\pm$ 19.4 km/s [@Sohn13]. At 260 kpc away, this is faster than the local escape speed of $\sim$180 km/s for a M$_{\rm vir}$=$10^{12}$ M$_\odot$ MW model. Since unbound satellite orbits are statistically improbable within $\Lambda$CDM cosmology [@BK13], we do not explore MW models lower than $10^{12}$ M$_\odot$ .
Few upper limits on M$_{\rm vir}$ exist apart from the timing argument, which limits the combined total mass of the MW and M31. Using the HST proper motions of M31 and other mass arguments in the literature, @van12 estimate the Local Group mass to be 3.17 $\pm$ 0.57 $\times 10^{12}$ M$_\odot$. It is thus unlikely that the MW individually contributes much more than $2 \times 10^{12}$ M$_\odot$.
In the orbital analysis that follows, we explore 3 different mass models: $10^{12}$, $1.5 \times 10^{12}$ and $2 \times 10^{12}$ M$_\odot$. Using WMAP7 cosmology, the corresponding virial radii are R$_{\rm vir}$ = 250, 300 and 330 kpc. The MW is modeled as a static, axisymmetric, three-component model consisting of dark matter halo, exponential disk, and spheroidal bulge. Model parameters are listed in Table 2 of K13.
Note that the MW mass is expected to have grown by roughly a factor of 2 over the past 6 Gyr [@Fak10]. K13 found that this mass evolution causes the orbital period of the LMC to increase substantially relative to static models. The orbital periods discussed in the following sections are thus underestimated.\
[**The Mass of the LMC:**]{}
K13 found that the LMC’s mass is the dominant uncertainty in its orbital history, since dynamical friction, which is proportional to the satellite mass squared, changes the LMC’s orbit on timescales shorter than, e.g., the MW’s mass evolution. The mass of the LMC also controls the orbit of the SMC, ultimately determining how long the two galaxies have interacted with each other as a binary pair (see $\S$\[sec:5\]).
The LMC has a well defined rotation curve that peaks at Vc = 91.7 $\pm$ 18.8 km/s and remains flat out to at least 8.7 kpc [@van14], consistent with the baryonic Tully-Fisher relation. This implies a minimum enclosed total mass of M(8.7 kpc) = 1.7 $\pm 10^{10}$ M$_\odot$; the LMC is dark matter dominated.
There is strong evidence that the stellar disk of the LMC extends to 15 kpc [@Saha10]. If the rotation curve stays flat to at least this distance then the total mass enclosed is M(15 kpc) = $GVc^2/r \sim 3 \times 10^{10}$ M$_\odot$. This minimum value is consistent with LMC masses adopted by earlier models [e.g., @GN96].
The total dynamical mass of the LMC can also be estimated using its baryon fraction. Currently, the LMC has a stellar mass of $2.7\times10^{9}$ M$_\odot$ and a gas mass of $5.0 \times 10^{8}$ M$_\odot$. The baryonic mass of the LMC is thus M$_{\rm bar} = 3.2 \times 10^{9}$ M$_\odot$. Using the minimum total mass of M$_{\rm tot} = 3\times10^{10}$ M$_\odot$, the baryon fraction of the LMC becomes M$_{\rm bar}$/M$_{\rm tot}$ = 11%. This is much higher than the baryon fraction of disks in galaxies like the MW, which is on the order of 3-5%. In the shallower halo potentials of dwarf galaxies, stellar winds should be more efficient, making baryon fractions even lower, not higher.
This analysis is further complicated if material has been removed from the LMC. As mentioned earlier, @Fox14 have recently estimated the total gas mass (HI and ionized gas) outside the MCs at $2\times10^{9} (d/55 {\rm kpc})^2 $ M$_\odot$. If half of this material came from the LMC, as suggested by @Nidever08, its initial baryon fraction would be 14%, approaching the cosmic value. Note that the bulk of the Magellanic Stream likely resides at distances of order 100 kpc, rather than 55 kpc, in which case the baryon fraction would increase to $\sim$20%.
To get a baryon fraction matching observational expectations of $f_{\rm bar} \sim$3-5%, the total mass of the LMC (at least at infall) needs to have been $20-6 \times 10^{10}$ M$_\odot$. This higher total mass is consistent with cosmological expectations from halo occupation models that relate a galaxy’s observed stellar mass to its halo mass. Using relations from @Mos13, the mean halo mass for a galaxy with a stellar mass of $2.7 \times 10^{9}$ M$_\odot$ is $1.7 \times 10^{11}$ M$_\odot$, implying a baryon fraction of $f_{\rm bar}\sim$ 2-4% (see Table \[tab:1\]). Because there is large scatter in halo occupation models, we consider a maximal halo mass for the LMC of $2.5 \times 10^{11}$ M$_\odot$ in the analysis that follows.\
[**The Mass of the SMC**]{}
The current dynamical mass of the SMC within 3 kpc is constrained between 2.7-5.1 $\times 10^{9}$ M$_\odot$, using the velocity dispersion of old stars [@Harris06]. This is larger than the current gas mass of the SMC is $4.2 \times 10^{8}$ M$_\odot$ and stellar mass of $3.1 \times 10^{8}$ M$_\odot$; the SMC is dark matter dominated. In most orbital models, the total mass of the SMC is estimated at $M_{\rm DM} = 1.4 - 3 \times 10^{9}$ M$_\odot$ [@B07; @K13; @Diaz11; @GN96]. This yields $f_{\rm bar}\sim$50%-20%, well above cosmological expectations.
This issue gets a lot worse when we account for the substantial amount of gas the SMC must have lost to form the Magellanic Stream, Bridge and Leading Arm. If we estimate again that half the total gas mass outside the MCs comes from the SMC, then the initial baryon mass of the SMC must have been $M_{\rm bar} = 1.73 \times10^{9}$ M$_\odot$. Taking the traditional dark matter mass of $3\times 10^{9}$ M$_\odot$ yields $f_{\rm bar} \sim$ 60%. It would take a total mass order $M_{\rm SMC} = 3\times 10^{10}$ M$_\odot$ to obtain a baryon fraction of 5%. This value is consistent with the choice of $M_{\rm SMC} = 2\times 10^{10}$ M$_\odot$ in B12 and B10 (see Table \[tab:1\]). Using cosmological halo occupation models the SMC dark matter mass is expected to be even higher. The mean expectation from relations in [@Mos13] is $M_{\rm SMC} = 4.2\times10^{10}$ M$_\odot$.
[p[2.5cm]{}p[2.5cm]{}p[2cm]{}p[4cm]{}]{} Baryonic Mass ($10^9$ M$_\odot$) & Dark Matter Mass ($10^{10}$ M$_\odot$)& Baryon Fraction$^a$ & Motivation for Dark Matter Mass\
& & &\
3.2 & 3 & 0.11 & Traditional Models\
& 6 & 0.05 & Minimum mass to get f$_{\rm bar}<$ %5\
& 17 & 0.02 & B12, Mean $\Lambda$CDM$^c$\
& 25 & 0.01 & Maximal Model\
4.2-6.5$^b$ & 3 & 0.14-0.22\
& 6 & 0.07 - 0.11\
& 17 & 0.03-0.04\
& 25 & 0.02 - 0.03\
& & &\
7.3 & 0.3 & 0.24 & Traditional Models\
& 0.5 & 0.15 & Max Dynamical Mass $< $3 kpc\
& 2 & 0.04 & B12\
& 4.2 & 0.02 & Mean $\Lambda$CDM$^c$\
17.3-40.3$^b$ & 0.3 & 0.58-1.34 &\
& 0.5 & 0.35-0.81 &\
& 3 & 0.06-0.13 &\
& 4.2 & 0.04-0.1 &\
$^a$ Gas mass/ (Stellar Mass + Gas Mass) $^b$ Including half the mass in the Magellanic Stream (total $2\times10^{9}$ M$\odot$) [@Fox14] at a distance of 55 kpc, or where half the stream is at 100 kpc (total $6.6\times10^{9}$ M$_\odot$). $^c$ Mean value from relations in @Mos13 for LMC/SMC stellar masses.
Plausible Orbital Histories for the LMC {#sec:4}
=======================================
Following the methodology outlined in K13 and considerations for the mass of the MW and the mass and velocity of the LMC described in \[sec:3\], the orbit of the LMC can be integrated backwards in time. For various combinations of MW and LMC mass, Monte-Carlo drawings from the LMC’s 4$\sigma$ velocity error distribution are used to explore plausible orbital histories. Figure \[fig:1\] shows the resulting mean orbital solutions for each MW/LMC mass combination. The LMC is considered to be on its first infall if it has not experienced more than one pericentric approach within the past 10 Gyr. In all cases the LMC has made at least one pericentric passage, since it is just at pericenter now.
The illustrated dependence on LMC mass explains the discrepancy in the literature concerning different orbital solutions. Most studies have adopted low mass LMC models, which allows for orbital solutions with lower eccentricity. For example, [@Zha12] explore a variety of MW models to constrain the orbital history of the LMC, but they consider only one LMC mass model of $2 \times 10^{10}$ M$_\odot$. Numerical models of the Magellanic System by [@Diaz12; @Diaz11] consider a total mass of only $10^{10}$ M$_\odot$, which is discrepant with the dynamical mass determined from the LMC’s rotation curve. Other recent orbital studies adopt LMC masses of $3 \times 10^{10}$ M$_\odot$ [e.g., @Ruz10; @Sha09]. These low masses are at odds with mass estimates from arguments about the baryon fraction of the LMC, which require a minimum dark matter mass of at least $6 \times 10^{10}$ M$_\odot$ ($\S$\[sec:3\]).
The numerical models presented in [@B10; @B12 hereafter B10 and B12] were designed to account for cosmological expectations, and thus adopt an LMC mass of $1.8 \times 10^{12}$ M$_\odot$.
Note that while lower halo masses do allow for solutions where the LMC has completed one orbit about the MW, the mean orbital period is 5 Gyr. This timescale is much longer than the age of the Magellanic Stream. Furthermore, in this study, the MW mass is assumed to be static in time. If the mass evolution of the MW were included, the orbital period would be even longer.
![Typical orbital histories for the LMC are indicated as a function of LMC and MW mass. Orbits are determined by searching the 4$\sigma$ proper motion error space in Monte Carlo fashion and computing the mean number of pericentric passages completed within 10 Gyr. High mass LMC models experience greater dynamical friction and are consequently on more eccentric orbits, yielding first infall solutions (dark blue regions). Lower mass models allow for orbits where the LMC has made one pericentric passage (light purple regions); such solutions typically have orbital periods of order 5 Gyr. High MW mass and low LMC mass combinations are required for the LMC to have completed more than one orbit. The mass of the LMC needs to be larger than $3\times 10^{10}$ M$_\odot$ in order to account for LMC stars located at distances of $\sim$15 kpc from the LMC center and greater than $6\times 10^{10}$ M$_\odot$ for the baryon fraction of the LMC to approach $\sim$5%. Most studies have not considered such high mass models, apart from B12. []{data-label="fig:1"}](Besla_1.ps)
The LMC-SMC Binary {#sec:5}
==================
At a distance of $\sim$50 kpc from the Galactic center, the LMC and SMC are our closest example of an interacting pair of dwarf galaxies. Evidence of their ongoing interaction is clearly illustrated by the existence of the Magellanic Bridge that connects the two galaxies. This structure likely formed during their last close approach $\sim 100-300$ Myr ago [B12, @GN96].
The tidal field of the MW makes it statistically improbable that the LMC could have randomly captured the SMC some 300 Myr ago while in orbit about the MW. It is more likely that the two galaxies were accreted as a binary; but the longevity of their binary status is unclear.
All models for the Magellanic Stream and Bridge invoke tidal interactions between the MCs to some degree. The MCs must therefore have interacted for at least the lifetime of the Stream. Based on the current high rate of ionization of the Stream [@Weiner96] and large extended ionized component [@Fox14], it is unlikely that the Stream could have survived as a neutral HI structure for more than 1-2 Gyr [@Bland07].
The star formation histories (SFHs) of the MCs also suggest a common evolutionary history. @Weisz13 illustrate that $\sim$4 Gyr ago, the SFHs of both the LMC and SMC appear to increase in concert. It is thus reasonable to assume the MCs have maintained a binary status for at least the past 4 Gyr.\
[**Plausible Orbital Histories for the LMC-SMC Binary**]{}
The orbital analysis presented in Figure \[fig:1\] is revisited, exploring the same LMC/MW mass range, but this time also searching the 4$\sigma$ proper motion error space of the SMC in addition to that of the LMC. The goal is to identify the combinations of LMC and MW mass that allow for the relative velocity of the MCs to be lower than the local escape speed of the LMC for some time in the past (not necessarily including today). Figure \[fig:2\] illustrates the resulting mean longevity of the LMC-SMC binary as a function of LMC and MW mass.
The action of MW tides are detrimental to the longevity of the Magellanic binary. As the mass of the MW increases, its tidal field is stronger and thus long-lived binary configurations are rare. In particular, no long-lived solutions are found if the mass of the MW is greater than $2\times 10^{12}$ M$_\odot$. This places an interesting upper bound on the virial mass of the MW. If the MW is $> 2\times 10^{12}$ M$_\odot$, the MCs could not have interacted for an appreciable amount of time in the past and their current proximity would be a random happenstance. Few upper bounds exist on the mass of the MW, making this a novel constraint.
The SMC has historically been modeled in a circular orbit about the LMC, with a relative velocity of order 60 km/s [@GN96]. However, the new HST measurements reveal that the relative velocity between the Clouds is significantly larger. At a relative velocity of 128 $\pm$ 32 km/s (K13), the SMC is moving well above the escape speed of the LMC if its total mass is $3 \times 10^{10}$ M$_\odot$ (V$_{esc} \sim$110 km/s). This relative velocity measurement is a robust result that has been confirmed by other teams [@Vieira10; @Piatek08] and has not changed substantially from the earlier HST results [@K06b]. This high speed makes it very difficult to maintain a long-lived binary, unless the LMC is substantially more massive than traditionally modeled.
The preferred configuration for a long-lived LMC-SMC binary is a low/intermediate mass MW + a high mass LMC. This is exactly the opposite requirement for orbital solutions where the LMC makes at least one orbit about the MW. In fact, the colors in Figure \[fig:2\] correspond to the same as those in Figure \[fig:1\]; all binary solutions that last longer than 4 Gyr are first infall solutions.
The high relative velocity between the Clouds implies that the SMC is on an eccentric orbit about the LMC. Such binary configurations are easily disrupted by MW tides, meaning that even one previous pericentric passage is sufficient to have destroyed the binary. This result is consistent with the fact that only 3.5% of MW type galaxies host both an LMC and SMC stellar mass analog [@Liu11]. Similarly, statistics from cosmological simulations find only 2.5% MW type dark matter halos host both LMC and SMC mass analogs [@BK11]. Our MW galaxy is thus an oddity in that it hosts two massive satellites in close proximity to each other. However, this rare configuration can be understood if the MCs have only recently passed pericentric approach for the first time; only now are MW tides operating to disrupt this configuration.
This study implies that all existing models in the literature that invoke the new HST proper motions in combination with low mass LMCs [*do not allow for long-lived LMC-SMC binary solutions*]{}. In particular, because of the high speeds, no binary LMC-SMC solutions can exist in a MOND framework [@Zhao13].
![Similar to Figure \[fig:1\], except now the SMC is also included in the orbital analysis; its 4$\sigma$ proper motion error space is also searched in Monte Carlo fashion. Colored regions represent the same LMC orbital solutions as in Figure\[fig:1\], but here the longevity of the LMC-SMC binary is indicated. The SMC is assumed to be bound to the LMC if the relative velocity between the Clouds is less than the local escape speed of the LMC at any point in the past. Binary solutions are only viable in low/intermediate mass MW models + high LMC models; these are necessarily first infall orbits. Dynamical friction between the Clouds is not included in this analysis, but is expected to decrease the longevity of the binary orbit. []{data-label="fig:2"}](Besla_2.ps)
Further Support for a First Infall of a Binary LMC/SMC
=======================================================
The simplest argument in favor of a recent infall is the unusually high gas fractions of both the LMC and SMC, given their proximity to the MW. @vdBergh06 conducted a morphological comparison of the satellites of the MW and M31, finding that the L/SMC are the only two gas-rich dwarf Irregulars at close Galactocentric distance to their host. There are numerous environmental factors that work to quench star formation and morphologically change galaxies after they become satellite galaxies of massive hosts (see chapter by Carraro). It is thus remarkable that a satellite such as the SMC could have retained such a high gas content if it were accreted over 5 Gyr ago.
In this volume, Burkert discusses the gas consumption timescale for galaxies; upon accretion satellites are cut off from their gas supply and thus their star formation rates should decline over time. However, Gallart (also in this volume) has illustrated that the star formation rates of the LMC have been [*increasing*]{}, with no signs of quenching over the past 4 Gyr until very recently.
Taken together these arguments strongly support a scenario where the Clouds are on their first infall to our system, having only been within the virial radius of the MW for the past 1-2 Gyr.
Lessons from the Magellanic Clouds
====================================
The Magellanic Clouds are recent interlopers in our neighborhood. This statement is a consequence of the dramatically improved proper motions of the MCs made using the HST by K13 and our evolving understanding of the structure of dark matter halos. Specifically, these factors have forced us to reconsider the total dark matter masses of the MCs. The baryon fractions of the MCs and imply that their total masses must be at least a factor of 10 larger than traditionally modeled. Dynamical friction then requires their orbits to be highly eccentric, preventing short period orbits. Finally, the existence of a high relative velocity, LMC-SMC binary today strongly argues against their having completed a previous pericentric approach about our Galaxy, as MW tides can efficiently disrupt such tenuous configurations.
A first infall solution implies that the MCs have interacted as a binary pair prior to accretion; in B10 and B12 we argued that tidal interactions between the pair gave rise to the formation of the Magellanic Stream, Bridge and Leading Arm. As such, interactions between dwarfs galaxies are important drivers of their evolution and may explain the existence of Magellanic Irregular type galaxies (i.e. LMC analogs) in the field.
However, this scenario creates a large challenge for the theory that the satellite galaxies of the MW occupy a unique orbital plane and/or were formed in a common event (see the chapter in this volume by Pavel Kroupa). To be more concise, the biggest challenge posed to this plane of satellites and MOND orbital histories constructed for the MCs is the current gas fraction of the SMC and the consequent requirement that the [**SMC must be on a first infall**]{}. Orbits for the SMC in MOND have small periods and apocenters [@Zhao13]. Such orbital solutions cannot explain the absence of quenching in the star formation history of the SMC and the fact that there is currently just as much gas in the SMC as there is in its much larger companion the LMC.
| {
"pile_set_name": "ArXiv"
} |
hep-th/0106146
[**Elias Gravanis and Nick E. Mavromatos** ]{}\
Department of Physics, Theoretical Physics, King’s College London,\
Strand, London WC2R 2LS, United Kingdom.
[We demonstrate that an impulse action (‘recoil’) on a D-particle embedded in a (four-dimensional) cosmological Robertson-Walker (RW) spacetime is described, in a $\sigma$-model framework, by a suitably extended higher-order logarithmic world-sheet algebra of relevant deformations. We study in some detail the algebra of the appropriate two-point correlators, and give a careful discussion as to how one can approach the world-sheet renormalization group infrared fixed point, in the neighborhood of which the logarithmic algebra is valid. It is found that, if the initial RW spacetime does not have cosmological horizons, then there is no problem in approaching the fixed point. However, in the presence of horizons, there are world-sheet divergences which imply the need for Liouville dressing in order to approach the fixed point in the correct way. A detailed analysis on the subtle subtraction of these divergences in the latter case is given. In both cases, at the fixed point, the recoil-induced spacetime is nothing other than a coordinate transformation of the initial spacetime into the rest frame of the recoiling D-particle. However, in the horizon case, if one identifies the Liouville mode with the target time, which expresses physically the back reaction of the recoiling D-particle onto the spacetime structure, it is found that the induced spacetime distortion results in the removal of the initial cosmological horizon and the eventual stopping of the acceleration of the Universe. In this latter sense, our model may be thought of as a conformal field theory description of a (toy) Universe characterized by a sort of ‘phase transition’ at the moment of impulse, implying a time-varying speed of light.]{}
June 2001
Introduction and Summary
========================
Placing D-branes in curved space times is not understood well at present. The main problem originates from the lack of knowledge of the complete dynamics of such solitonic objects. One would hope that such a knowledge would allow a proper study of the back reaction of such objects onto the surrounding space time geometry (distortion), and eventually a consistent discussion of their dynamics in curved spacetimes. Some modest steps towards an incorporation of curved space time effects in D-brane dynamics have been taken in the recent literature from a number of authors [@curved]. These works are dealing directly with world volume effects of D-branes and in some cases string dualities are used in order to discuss the effects of space time curvature.
A different approach has been adopted in [@kogan; @kmw; @szabo; @recoil], in which we have attempted to approach some aspects of the problem from a world sheet view point, which is probably suitable for a study of the effects of the (string) excitations of the heavy brane. We have concentrated mainly on heavy $D$-particles, embedded in a [*flat*]{} target background space time. We have discussed the instantaneous action (impulse) of a ‘force’ on a heavy $D$-particle. The impulse may be viewed either as a consequence of ‘trapping’ of a [*macrosopic number*]{} of closed string states on the defect, and their eventual splitting into pairs of open strings, or, in a different context, as the result of a more general phenomenon associated with the [*sudden*]{} appearance of such defects. Our world sheet approach is a valid approximation only if one looks at times [*long after*]{} the event. Such impulse approximations usually characterize classical phenomena. In our picture we view the whole process as a [*semi-classical*]{} phenomenon, due to the fact that the process involves open string [*recoil*]{} excitations of the heavy $D$-particle, which are [*quantum*]{} in nature. It is this point of view that we shall adopt in the present article.
Such an approach should be distinguished from the problem of studying single-string scattering of a $D$-particle with closed string states in flat space times [@paban]. We have shown in [@kogan; @kmw; @szabo; @recoil] that for a $D$-particle embedded in a $d$-dimensional [*flat Minkowski*]{} space time such an impulse action is described by a world-sheet $\sigma$-model deformed by appropriate ‘recoil’ operators, which obey a logarithmic conformal algebra [@lcft]. The appearance of such algebras, which lie on the border line between conformal field theories and general renormalizable field theories in the two-dimensional world sheet, but can still be classified by conformal data, is associated with the fact that an impulse action (recoil) describes a [*change*]{} of the string/D-particle background, and as such it cannot be described by conformal symmetry all along. The [*transition*]{} between the two asymptotic states of the system before and (long) after the event is precisely described by deforming the associated $\sigma$-model by operators which [*spoil*]{} the conformal symmetry.
Indeed, the recoil operators are [*relevant*]{} from a world-sheet renormalization-group view point [@kmw], and thus the induced string theory becomes non-critical, in need of Liouville dressing [@ddk] in order to restore the conformal symmetry. The dressing results in the appearance of target-space metric distortion [@recoil], which - under the identification of the Liouville mode with the time [@emn] - is interpreted as a backreaction of the recoiling $D$-particle defect onto the surrounding (initially flat) space time. Under such an impulse/recoil, there is in general an induced vacuum energy, which can even become time dependent [@emncosmo]. Such time dependent vacuum energies in Cosmology have recently attracted a lot of attention as a challenge for string theory [@challenge], given that in certain cases the corresponding Universes are characterized by cosmological horizons, and hence a field-theoretic $S$ matrix cannot be defined for asymptotic states. From the point of view of Liouville string such a situation is expected [@emnsmatrix], due to the fact that in Liouville strings, with the time identified with the Liouville mode [@emn], a scattering matrix cannot be defined.
In this work we shall attempt to extend the flat space time results of [@kogan; @kmw; @recoil] to the physically relevant case of a Robertson-Walker (RW) cosmological background space time. Although, our results do not depend on the target space dimension, however, for definiteness we shall concentrate on the case of a $D$-particle embedded in a four-dimensional RW spacetime. It must be stressed that we shall not attempt here to present a complete discussion of the associated space time curvature effects, which - as mentioned earlier - is a very difficult task, still unresolved. Nevertheless, by concentrating on times much larger than the moment of impulse on the $D$-particle defect, one may ignore such effects to a satisfactory approximation. As we shall see, our analysis produces results which look reasonable and are of sufficient interest to initiate further research.
The vertex operators which describe the impulse in curved RW backgrounds obey a suitably extended (higher-order) logarithmic algebra. The algebra is valid at, and in the neighborhood of, a non-trivial infrared fixed point of the world-sheet Renormalization Group. For a RW spacetime of scale factor of the form $t^p$, where $t$ is the target time, and $p > 1$ in the horizon case, the algebra is actually a set of logarithmic algebras up to order $[2p]$, which are classified by the appropriate higher-order Jordan blocks [@lcft].
As in the flat case, which is obtained as a special limit of this more general case, the recoil deformations are relevant operators from a world-sheet Renormalization-Group viewpoint. One distinguishes two cases. In the first, the initial RW spacetime does not possess cosmological horizons. In this case it is shown that the limit to the conformal world-sheet non-trivial (infrared) fixed point can be taken smoothly without problems. On the other hand, in the case where the initial spacetime has cosmological horizons, such a limit is plagued by world-sheet divergences. These should be carefully subtracted in order to allow for a smooth approach to the fixed point. A detailed discussion of how this can be done is presented. In general, the divergences spoil the conformal invariance of the $\sigma$-model, thus implying the need for Liouville dressing [@ddk] in order to properly restore the conformal symmetry.
Moreover, a careful discussion of the matching between the results of the Liouville dressing and those implied by the logarithmic algebra is given, which supports the possibility of identifying the world-sheet zero mode of the Liouville field (viewed as a local renormalization-group scale on the world sheet) with the target time. One distinguishes various cases which depend on whether the underlying theory lives in its critical dimension, and thus the only source of not criticality is the impulse action, or not. Such an identification induces target-space metric deformations, which are responsible for the [*removal*]{} of the cosmological horizon of the initial spacetime background, and the stopping of the acceleration of the Universe. Essentially the situation implies an effective time-dependent light velocity after the moment of impulse, which is responsible for the removal of the cosmological horizon. From this point of view our work may thus seem to provide a conformal-field-theory framework for a proper treatment of such time-varying speed of light scenaria [@moffat] in the context of non-critical string theory [@emnsmatrix].
Recoiling D-particles in Robertson-Walker Backgrounds
=====================================================
Geodesic Paths and Recoil
-------------------------
Let us consider a $D$-particle, located (for convenience) at the origin of the spatial coordinates of a four-dimensional space time, which at a time $t_0$ experiences an impulse. In a $\sigma$-model framework, the trajectory of the $D$-particle $y^i(t)$, $i$ a spatial index, is described by inserting the following vertex operator $$\label{path}
V = \int _{\partial \Sigma} G_{ij}y^j(t)\partial_n X^i$$ where $G_{ij}$ denotes the spatial components of the metric, $\partial \Sigma$ denotes the world-sheet boundary, $\partial _n$ is a normal world-sheet derivative, $X^i$ are $\sigma$-model fields obeying Dirichlet boundary conditions on the world sheet, and $t$ is a $\sigma$-model field obeying Neumann boundary conditions on the world sheet, whose zero mode is the target time.
This is the basic vertex deformation which we assume to describe the motion of a $D$-particle in a curved geometry to leading order at least, where spacetime back reaction and curvature effects are assumed weak. Such vertex deformations may be viewed as a generalization of the flat-target-space case [@dparticle].
Perhaps a formally more desirable approach towards the construction of the complete vertex operator would be to start from a T-dual (Neumann) picture, where the deformation (\[path\]) should correspond to a proper Wilson loop operator of an appropriate gauge vector field. Such loop operators are by construction independent of the background geometry. One can then pass onto the Dirichlet picture by a T-duality transformation viewed as a canonical transformation from a $\sigma$-model viewpoint [@otto]. In principle, such a procedure would yield a complete form of the vertex operator in the Dirichlet picture, describing the path of a $D$-particle in a curved geometry. Unfortunately, such a procedure is not free from ambiguities at a quantum level [@otto], which are still unresolved for general curved backgrounds. Therefore, for our purposes here, we shall consider the problem of writing a complete form for the operator (\[path\]) in a RW spacetime background in the Dirichlet picture as an open issue. Nevertheless, for RW backgrounds at large times, ignoring curvature effects proves to be a satisfactory approximation, and in such a case one may consider the vertex operator (\[path\]) as a sufficient description for the physical vertex operator of a $D$-particle. As we shall show below, the results of such analyses appear reasonable and interesting enough to encourage further studies along this direction.
For times long after the event, the trajectory $y^i(t)$ will be that of free motion in the curved space time under consideration. In the flat space time case, this trajectory was a straight line [@dparticle; @kmw; @szabo], and in the more general case here it will be simply the associated [*geodesic*]{}. Let us now determine its form, which will be essential in what follows.
The space time assumes the form: $$\label{rwmetric}
ds^2 = -dt^2 + a(t)^2 (dX^i)^2$$ where $a(t)$ is the RW scale factor. We shall work with expanding RW space times with scale factors $$a(t) =a_0 t^p, \qquad p \in R^+$$ The geodesic equations in this case read: $$\begin{aligned}
{\ddot t} + pt^{2p-1}({\dot y}^i)^2 &=& 0 \nonumber \\
{\ddot y} + 2\frac{p}{t}({\dot y}^i) {\dot t}&=& 0\end{aligned}$$ where the dot denotes differentiation with respect to the proper time $\tau$ of the $D$-particle.
With initial conditions $y^i(t_0)=0$, and $dy^i/dt (t_0) \equiv v^i$, one easily finds that, for long times $t \gg t_0$ after the event, the solution acquires the form: $$\label{pathexpre}
y^i(t) =\frac{v^i}{1-2p}\left(t^{1-2p}t_0^{2p} - t_0 \right) +
{\cal O}(t^{1-4p}), \qquad t \gg t_0$$
To leading order in $t$, therefore, the appropriate vertex operator (\[path\]), describing the recoil of the $D$-particle, is: $$\label{path2}
V=\int _{\partial \Sigma} a_0^2 \frac{v^i}{1-2p}\Theta (t-t_0)\left(tt_0^{2p}-t_0t^{2p}\right)
\partial_n X^i$$ where $\Theta (t-t_0)$ is the Heaviside step function, expressing an instantaneous action ([*impulse*]{}) on the $D$-particle at $t=t_0$ [@kmw; @recoil]. As we shall see later on, such deformed $\sigma$-models may be viewed as providing rather generic mathematical prototypes for models involving phase transitions at early stages of the Universe, leading effectively to time-varying speed of light. In the context of the present work, therefore, we shall be rather vague as far as the precise physical significance of the operator (\[path2\]) is concerned, and merely exploit the consequences of such deformations for the expansion of the RW spacetime after time $t_0$, from both a mathematical and physical viewpoint.
\[cut\]
In [@kmw], we have studied the case $p=0$, $a_0=1$, where the operators assumed the form $t\Theta_\epsilon (t)$ to leading order in $t$, where $\Theta_\epsilon (t)$ is the regulated form of the step function, given by [@kmw]: $$\label{rep}
\Theta_\epsilon =-i\int_{-\infty}^{+\infty}
\frac{d\omega}{2\pi} \frac{1}{\omega -i\epsilon}
e^{i\omega\,t}, \qquad \epsilon \rightarrow 0^+$$ As discussed in that reference, this operator forms a logarithmic pair [@lcft] with $\epsilon \Theta_\epsilon (t)$, expressing physically fluctuations in the initial position of the $D$-particle.
In the current case, one may expand the integrand of (\[path2\]) in a Taylor series in powers of $(t-t_0)$, which implies the presence of a series of operators, of the form $(t-t_0)^q\Theta_\epsilon (t-t_0)$, where $q $ takes on the values $2p, 2p-1, \dots $, i.e. it is not an integer in general. In a direct generalization of the Fourier integral representation (\[rep\]), we write in this case: $$\begin{aligned}
\label{opd}
&~& {\cal D}^{(q)}
\equiv v_i (t-t_0)^q \Theta_\epsilon (t-t_0)\partial_n X^i
=v_i~N_q~\int _{-\infty}^{+\infty}d\omega
\frac{1}{(\omega -i\epsilon)^{q+1}}~e^{i\omega(t-t_0)}\partial_nX^i~,
\nonumber \\
&~& N_q \equiv \frac{i^q}{\Gamma (-q)(1-e^{-i2\pi q})}=
\frac{(-i)^{q+1}\Gamma (q+1)}{2\pi}~,\end{aligned}$$ where we have incorporated the velocity coupling $v_i$ in the definition of the $\sigma$-model deformation, and we have defined the integral along the contour of figure 1, having chosen the cut to be from $+i\epsilon$ to $+i\infty$.
Extended Logarithmic world-sheet Algebra of recoil in RW backgrounds
--------------------------------------------------------------------
Following the flat space time analysis of [@kmw], we now proceed to discuss the conformal structure of the recoil operators in RW backgrounds. We shall do so by acting on the operator ${\cal }D^{(q)}$ (\[opd\]) with the world-sheet energy momentum tensor operator $T_{zz} \equiv T$ (in a standard notation). Due to the form of the background space time (\[rwmetric\]), the stress tensor $T$ assumes the form $$\label{stress}
2T =-(\partial t)^2 + a^2(t) (\partial X^i)^2$$ where, from now on, $\partial \equiv \partial _z$, unless otherwise stated. One can then obtain the relevant operator-product expansions (OPE) of $T$ with the operators ${\cal D}^{(q)}$. For convenience in what follows we shall consider the action of each of the two terms in (\[stress\]) on the operators ${\cal D}^{(q)}$ separately. For the first (time $t$-dependent part), one has, as $z \to w$: $$\begin{aligned}
&~& -\frac{1}{2}(\partial t(z))^2\cdot{\cal D}^{(q)}(w)
= \frac{v_i}{(z-w)^2}\left[
N_q\int _{-\infty}^{+\infty}d\omega
\frac{\omega^2/2}{(\omega -i\epsilon)^{q+1}}~e^{i\omega~t(w)}\right]
\partial_nX^i
= \nonumber \\
&~& \frac{1}{(z-w)^2}\left[-\frac{\epsilon^2}{2}{\cal D}^{(q)} + q\epsilon {\cal D}^{(q-1)} + \frac{q(q-1)}{2}{\cal D}^{(q-2)} \right]\end{aligned}$$ The above formulæ were derived for asymptotically large time $t$, assuming the two-point correlators $$\label{correl}
\langle X^\mu(z) X^\nu(w) \rangle
= 2G^{\mu\nu}{\rm ln}|z-w|^2 + \dots~,$$ where the $\dots $ denote terms with negative powers of $t$, related to space-time curvature, which are subleading in the limit $t \to \infty$.
For the spatial part of (\[stress\]) we consider the OPE $a(t(z))^2 (\partial X^i(z))^2 {\cal D}^{(q)}(w)$ as $z \to w$. Again, for convenience we shall do the time and space contractions separately: $$\begin{aligned}
&~& t^{2p}(z)\cdot {\cal D}^{(q)}(w) = \int d\omega {\cal {\tilde D}}^{(q)}(\omega)
t^{2p}(z)\cdot
e^{i\omega (t(w)-t_0)} = \nonumber \\
&~& \int _0^{\infty} \frac{d\nu}{\Gamma (-2p)} \nu^{-1-2p}
\int d\omega {\cal {\tilde D}}^{(q)}(\omega) e^{-\nu t(z)}\cdot
e^{i\omega (t(w)-t_0)} \end{aligned}$$ Using the OPE $e^{-\nu t(z)} \cdot e^{-i\omega (t(w)-t_0)} \sim
|z-w|^{i\nu\omega}e^{-\nu t(z) - i\omega (t(z)-t_0) + {\cal O}(z-w)}$ one obtains (as $z \sim w$): $$\begin{aligned}
&~& t(z)^{2p}\cdot{\cal D}^{(q)}(w)=
\int _0^\infty \frac{d\nu}{\Gamma (-2p)}\nu^{-1-2p}e^{-\nu t(z)}
{\cal D}^{(q)} (t-t_0 -\nu {\rm ln}|z-w|) = \nonumber \\
&~& t^{2p}~\int _0^\infty \frac{d\nu}{\Gamma (-2p)}\nu^{-1-2p}e^{-\nu}
{\cal D}^{(q)} (t-t_0 -\frac{\nu}{t} {\rm ln}|z-w|) = \nonumber \\
&~& t^{2p}\left[{\cal D}^{(q)}(t-t_0) -\frac{1}{t}{\rm ln}|z-w|
\frac{\Gamma (1-2p)}{\Gamma (-2p)}\frac{d}{dt}{\cal D}^{(q)}(t-t_0)
+ {\cal O}(t-t_0)^{q-2}\right]\end{aligned}$$ We now observe that $\tfrac{d{\cal D}^{(q)}}{dt}=q{\cal D}^{(q-1)} -
\epsilon~{\cal D}^{(q)} $, where both terms have vacuum expectation values of the same order in $\epsilon$, as we shall see below, and hence both should be kept in our perturbative expansion.
Expanding the various terms around $t_0$, $t^{s}=(t-t_0)^{s} + s~t_0(t-t_0)^{s-1}+
\frac{t_0^{2}}{2}(s)(s-1)(t-t_0)^{s-2}+{\cal O}([t-t_0]^{s-3})$, one has: $$\begin{aligned}
&~& t^{2p}(z)\cdot{\cal D}^{(q)}(w) = {\cal D}^{(2p+q)}(t-t_0)+
\left(2p~t_0 - 2p~\epsilon~{\rm ln}|z-w|\right)~{\cal D}^{(2p+q-1)}+
\nonumber \\
&~& + \left(\frac{t_0^{2}}{2}2p(2p-1)+[2pq +
(2p-4p^2)\epsilon~t_0]~{\rm ln}|z-w|\right)
{\cal D}^{(2p+q-2)}(t-t_0)+ \nonumber \\
&~& + {\cal O}([t-t_0]^{2p+q-3})\end{aligned}$$ where it is worthy of mentioning that inside the subleading terms there are higher logarithms of the form ${\rm ln}^n|z-w|$, where $n =2,3,4, \dots$.
We now come to the OPE between the spatial parts. In view of (\[correl\]), upon expressing $\partial _z$ in normal $\partial_n$ and tangential parts, and imposing Dirichlet boundary conditions on the world-sheet boundary where the operators live on, we observe that such operator products take the form: $$(\partial X^j(z))^2 \cdot \partial_n X^i(w) \sim
G^{ii}\frac{1}{(z-w)^2}\partial_n X^i \sim \frac{t^{-2p}}{(z-w)^2}
\partial_n X^i, \qquad ({\rm no}~~{\rm sum}~~{\rm over}~~i)$$ Performing the last contraction with the $t^{-2p}$, following the previous general formulæ and collecting appropriate terms, one obtains: $$\begin{aligned}
\label{elalg}
&~& T(z)\cdot{\cal D}^{(q)}[(t-t_0)(w)]
=\frac{1-\frac{\epsilon^2}{2}}{(z-w)^2}{\cal D}^{(q)} [(t-t_0)(w)] +
\frac{q\epsilon}{(z-w)^2}{\cal D}^{q-1}[(t-t_0)(w)] + \nonumber \\
&~& + \frac{\frac{q(q-1)}{2} -2p^{2}~{\rm ln}|z-w|
-2p^{2}~\epsilon^2~{\rm ln}^2|z-w|}{(z-w)^2}{\cal D}^{(q-2)}
[(t-t_0)(w)]
+ {\cal O}([t-t_0]^{q-3})\end{aligned}$$ where again inside the subleading terms there are higher logarithms.
We next notice that, as a consistency check of the formalism, one can calculate the OPE (\[elalg\]) in case one considers matrix elements between [*on-shell*]{} physical states. In the context of $\sigma$-models, we are working with, the physical state condition implies the constraint of the vanishing of the world-sheet stress-energy tensor $2T=-(\partial t)^2 + a(t)^2(\partial X^i)^2 =0$. This condition allows $(\partial X^i)^2 $ to be expressed in terms of $(\partial t)^2$, which is consistent even at a correlation function level in the case of very target times $t \gg t_0$, since in that case, the correlator $\langle X^i t \rangle$ is subleading, as mentioned previously. Implementing this, it can be then seen that the OPE between the spatial parts of $T$ and ${\cal D}^{(q)}$ is: $$\begin{aligned}
&~& a^2(t)(\partial X^i)^2 \cdot {\cal D}^{(q)} =
t^{-2p}(\partial t)^2 \cdot
\{ {\cal D}^{(2p+q)}(t-t_0)+
\left(2p~t_0 - 2p~\epsilon~{\rm ln}|z-w|\right)~{\cal D}^{(2p+q-1)}+
\nonumber \\
&~& + \left(\frac{t_0^{2}}{2}2p(2p-1)+[2pq +
(2p-4p^2)\epsilon~t_0]~{\rm ln}|z-w|\right)
{\cal D}^{(2p+q-2)}(t-t_0)+ \nonumber \\
&~& +{\cal O}([t-t_0]^{2p+q-3})\}.\end{aligned}$$ Performing the appropriate contractions, and adding to this result the OPE of the temporal part of $T$ with ${\cal D}^{(q)}$, i.e. the quantity $-\frac{\epsilon^2}{2}{\cal D}^{(q)} + q\epsilon {\cal D}^{(q-1)}
+ \frac{1}{2}q(q-1){\cal D}^{(q-2)}$, we obtain: $$\begin{aligned}
&~& T\cdot {\cal D}^{(q)}|_{\rm on-shell}=\left(-2p\epsilon
-pt_0~\epsilon^2 + p\epsilon^2~{\rm ln}(a/L)\right)~{\cal D}^{(q-1)}
+ \nonumber \\
&~& + \{t_0^2\epsilon^2~2p(2p+1) - 3\epsilon^2p(2p+q){\rm ln}(a/L)
- 2\epsilon^3(p+p^2)t_0{\rm ln}(a/L) - 2p^2\epsilon^4{\rm ln}^2(a/L)
+ \nonumber \\
&~& + \epsilon (2p+q)2p~t_0 - (4p^2 + 4pq - 2p)\}
{\cal D}^{(q-2)} + {\cal O}([t-t_0]^{q-3})
\label{onshell})\end{aligned}$$ From the above we observe that the on-shell operators become marginal as they should, given that an on-shell theory ought to be conformal. Moreover, and more important, the world-sheet divergences [*disappear*]{} upon imposing the condition $$\label{xi0}
\epsilon^{2}{\rm ln}(L/a)^2 = \xi_0 =
{\rm constant~independent~of}~\epsilon,~a,~L$$ where $L$ ($a$) is the world-sheet (ultraviolet) infrared cut-off on the world sheet. As we shall discuss later on, this condition will be of importance for the closure of the logarithmic algebra, which characterizes the fixed point [@kmw]. Hence, the conformal invariance is preserved by the on-shell states, any dependence from it being associated with [*off-shell*]{} states.
We next notice that, in the context of the RW metric (\[rwmetric\]), there are two cases of expanding universes, one corresponding to $0< p \le 1$, and the other to $p > 1$. Whenever $p \le 1$ (which notably incorporates the cases of both radiation and matter dominated Universes) there is [*no horizon*]{}, given that the latter is given by: $$\label{horizon}
\delta (t) = a(t)\int_{t_0}^\infty \frac{dt'}{a(t')}$$ In this case the relevant value for $q$ is $q=2p \le 2$. On the other hand, for the case $p > 1$, i.e. $q > 2$ there is a non-trivial cosmological [*horizon*]{}, which as we shall see requires special treatment from a conformal symmetry viewpoint.
We commence with the no-horizon case, $1 < q \le 2$. We first notice that the linear in $t$ term in (\[path2\]) leads to the conventional logarithmic algebra, discussed in [@kmw], corresponding to a pair of impulse (‘recoil’) operators $C,D$. The main point of our discussion below is a study of the $t^{2p}$ terms in (\[path2\]), and their connection to other logarithmic algebras. Indeed, we observe that a logarithmic algebra [@lcft; @kogan; @kmw] can be obtained for these terms of the operators, if we define ${\cal D} \equiv {\cal D}^{(q)}$ and ${\cal C} \equiv q\epsilon {\cal D}^{(q-1)}$. In this case we have the following OPE with $T$: $$\begin{aligned}
\label{tdope}
&~& (z-w)^2~T\cdot {\cal D}=(1-\frac{\epsilon^2}{2}){\cal D} + {\cal C}, \nonumber \\
&~& (z-w)^2~T\cdot {\cal C}=(1-\frac{\epsilon^2}{2}){\cal C} + {\cal O}([t-t_0]^{q-2}),\end{aligned}$$ where throughout this work we ignore terms with negative powers in $t-t_0$ (e.g. of order $q-2$ and higher), for large $t\gg t_0$. Notice that in the case $q < 1$ (i.e. $p < 1/2$) the ${\cal C}$ operator defined above is absent.
In the second case $p > 1$ one faces the problem of having cosmological horizons (cf. (\[horizon\])), which recently has attracted considerable attention in view of the impossibility of defining a consistent scattering $S$-matrix for asymptotic states [@challenge; @emnsmatrix]. In this case the operator ${\cal D}^{(q-2)}$ is [*not subleading*]{} and one has an [*extended (higher-order) logarithmic algebra*]{} defined by (\[elalg\]). It is interesting to remark that now the logarithmic world-sheet terms in the coefficient of the ${\cal D}^{(q-2)}$ operator imply that the limit $z \to w$ is plagued by ultraviolet world-sheet divergences, and hence the world-sheet conformal invariance is spoiled. This necessitates Liouville dressing, in order to restore the conformal symmetry [@ddk]. As we shall show later, in such cases with horizons the recoil of the D-particle may induce a non-trivial backreaction on the spacetime geometry, which results in an effective spacetime in which the horizons [*disappear*]{}. This happens, as we shall discuss later, in the context of Liouville strings with the identification of the Liouville mode with time.
We now turn to a study of the correlators of the various ${\cal D}^{(q)}$ operators, which will complete the study of the associated logarithmic algebras, in analogy with the flat target-space case of [@kmw]. From the algebra (\[elalg\]) we observe that we need to evaluate correlators between ${\cal D}^{(q)}, {\cal D}^{(q-n)},~n=0,1,2, \dots$. We shall evaluate correlators $\langle \dots \rangle $ with respect to the free world-sheet action, since we work to leading order in the (weak) coupling $v_i$. For convenience below we shall restrict ourselves only to the time-dependent part of the operators ${\cal D}$. The incorporation of the $\partial_n X^i$ is trivial, and will be implied in what follows. With these in mind one has: $$\begin{aligned}
\label{qqn}
\langle {\cal D}^{(q)}(z) {\cal D}^{(q-n)}(w) \rangle =
N_q N_{q-n} \int \int_{-\infty}^{+\infty} \frac{d \omega d\omega '}{
(\omega - i\epsilon)^{q+1}(\omega' - i\epsilon)^{q-n+1}}
\langle e^{-i\omega t(z)}~e^{-i\omega' t(w)} \rangle\end{aligned}$$ where $\epsilon \to 0^+$. As already mentioned, we work to leading order in time $t \gg \infty$, and hence we can we apply the formula (\[correl\]) for two-point correlators of the $X^\mu$ fields to write [^1] $$\begin{aligned}
&~& \langle e^{-i\omega t(z)}~e^{-i\omega' t(w)}\rangle
=e^{-\frac{\omega^2}{2}\langle t(z) t(z)\rangle -
\frac{\omega^{'2}}{2}\langle t(w) t(w)\rangle -
\omega \omega'\langle t(z) t(w)\rangle} = \nonumber \\
&~& = e^{-(\omega+\omega')^2{\rm ln}(L/a)^2 + 2\omega\omega'{\rm ln}(|z-w|/a)^2}~,\end{aligned}$$ where we took into account that ${\rm Lim}_{z\to w}~\langle t(z)t(w)\rangle
= -2{\rm ln}(a/L)^2$. Given that ${\rm ln}(L/a)$ is very large, one can approximate $e^{-(\omega+\omega')^2{\rm ln}(L/a)^2} \simeq
\frac{\sqrt{\pi}}{\sqrt{{\rm ln}(L/a)^2}}\delta(\omega+\omega')$. Thus we obtain: $$\begin{aligned}
\langle {\cal D}^{(q)}(z) {\cal D}^{(q-n)}(w) \rangle
= (-1)^{-q+n-1}~N_q~N_{q-n}{\cal J}_n^{(q)}~,
\qquad {\cal J}_n^{(q)} \equiv \sqrt{\frac{\pi}{\alpha}}\int_{-\infty}^{+\infty}
\frac{d\omega~e^{-\omega^2
\lambda}~(\omega+i\epsilon)^{n}}{(\omega^2 + \epsilon^2)^{q+1}}\end{aligned}$$ where $\lambda \equiv 2{\rm ln}(|z-w|/a)^2$, and $\alpha \equiv {\rm ln}(L/a)^2$.
Below, for definiteness, we shall be interested in the case $2<q<3$, in which the relevant correlators are given by $n=0,1,2$. One has: $$\begin{aligned}
\label{defj}
&~&{\cal J}_{0}^{(q)}~=\sqrt{\frac{\pi}{\alpha}}~\epsilon^{-2q-1}~f_q (\epsilon^2 \lambda)~;
\nonumber \\
&~& f_q(\xi) =\sqrt{\pi}\frac{\Gamma (\frac{1}{2}+q)}{\Gamma (1 + q)}~F(\frac{1}{2},~\frac{1}{2}-q~;\xi)
+ ~\xi^{\frac{1}{2}+q}~\Gamma(-\frac{1}{2}-q)~F(1+q,~\frac{3}{2}+q~;\xi)
\nonumber \\
&~&{\cal J}_{1}^{(q)}~=~i\epsilon{\cal J}_{0}^{(q)}~, \nonumber \\
&~& {\cal J}_{2}^{(q)}=-2\epsilon^2~{\cal J}_0^{(q)} + {\cal J}_0^{(q-1)}=-\frac{\partial}{\partial \lambda}{\cal J}_0^{(q)}-\epsilon^2{\cal J}_0^{(q)}\end{aligned}$$ where $F(a,b;z)=1 + \frac{a}{b}\frac{z}{1!}+ \frac{a(a+1)}{b(b+1)}\frac{z^2}{2!} + \dots $ is the degenerate (confluent) hypergeometric function.\
Thus, the form of the algebra away from the fixed point (‘[ *off-shell form*]{}’), i.e. for $\epsilon^2 \ne 0$, is: $$\begin{aligned}
\label{offshellalg}
&~& \langle {\cal D}^{(q)}(z) {\cal D}^{(q)}(0) \rangle
= {\tilde N}_q^2~\sqrt{\frac{\pi}{\xi_0}}~\left(f_q(2\xi_0)
(\frac{\alpha}{\xi_0})^q + 2~f'_q(2\xi_0)(\frac{\alpha}{\xi_0})^{q-1}~
{\rm ln}(|z/L|^2) + \right.\nonumber \\
&~& \left.+ \frac{1}{2}~f''_q(2\xi_0)(\frac{\alpha}{\xi_0})^{q-2}~
4{\rm ln}^2(|z/L|^2)
+ {\cal O}(\alpha^{q-3})\right)~, \nonumber \\
&~& \epsilon~q~\langle {\cal D}^{(q)}(z) {\cal D}^{(q-1)}(0) \rangle =
{\tilde N}_q^2\sqrt{\frac{\pi}{\xi_0}}~\left(f_q(2\xi_0)
(\frac{\alpha}{\xi_0})^{q-1} + \right. \nonumber \\
&~& \left. + 2~f'_q(2\xi_0)(\frac{\alpha}{\xi_0})^{q-2}~
{\rm ln}(|z/L|^2) + {\cal O}(\alpha^{q-3})\right)~, \nonumber \\
&~& \epsilon^2~q^2~\langle {\cal D}^{(q-1)}(z) {\cal D}^{(q-1)}(0)\rangle
= {\tilde N}_q^2~\sqrt{\frac{\pi}{\xi_0}}~f_{q-1}(2\xi_0)
(\frac{\alpha}{\xi_0})^{q-2} + {\cal O}(\alpha^{q-3})
, \nonumber \\
&~& \epsilon^2~q~(q-1)~\langle {\cal D}^{(q)}(z) {\cal D}^{(q-2)}(0)\rangle
= -{\tilde N}_q^2~\sqrt{\frac{\pi}{\xi_0}}
\left(f_q(2\xi_0) + f'_q(2\xi_0)\right)~(\frac{\alpha}{\xi_0})^{q-2}
+ {\cal O}(\alpha^{q-3})~, \nonumber \\
&~& \epsilon^3~q^2(q-1)~\langle {\cal D}^{(q-1)}(z) {\cal D}^{(q-2)}(0)\rangle
= {\cal O}(\alpha^{q-3})~,
\nonumber \\
&~& \epsilon^4~q^2(q-1)^2~\langle {\cal D}^{(q-2)}(z) {\cal D}^{(q-2)}(0)\rangle
= {\cal O}(\alpha^{q-4})~\end{aligned}$$ where ${\tilde N}_q = \frac{\Gamma (1+q)}{2\pi}$, and $\xi_0$ has been defined in (\[xi0\]).
Notice that the above algebra is plagued by world-sheet ultraviolet divergences as $\epsilon^2 \to 0^+$, thereby making the approach to the fixed (conformal) point subtle. As becomes obvious from (\[xi0\]), the non-trivial fixed point $\epsilon \to 0^+$ corresponds to $L/a \to +\infty$, i.e. it is an infrared world-sheet fixed point. In order to understand the approach to the infrared fixed point, it is important to make a few remarks first, motivated by physical considerations.
From the integral expression of the regularized Heaviside function [@kmw] (\[rep\]) it becomes obvious that a scale $1/\epsilon$ for the target time is introduced. This, together with the fact that the scale $\epsilon$ is connected (\[xi0\]) to the world renormalization-group scales $L/a$, implies naturally the introduction of a ‘renormalized’ $\sigma$-model coupling/velocity $v_{R,i}(\tfrac{1}{\epsilon}) $ at the scale $\tfrac{1}{\epsilon}$: $$v_{R,i}(\frac{1}{\epsilon}) \sim \left(\frac{1}{\epsilon}\right)^{q-1}
\label{velrenorm}$$ for a trajectory $y_i (t) \sim t^q$. This normalization would imply the following rescaling of the operators $${\cal D}^{(q-n)} \rightarrow \epsilon^{q-1}{\cal D}^{(q-n)}$$ As a consequence, the factors $\epsilon^{2(1-q)}$ in (\[defj\]), (\[offshellalg\]) are removed. In the context of the world-sheet field theory this renormalization can be interpreted as a subtraction of the ultraviolet divergences by the addition of appropriate counterterms in the $\sigma$ model.
The approach to the infrared fixed point $\epsilon \to 0^+$ can now be made by looking at the [*connected*]{} two point correlators between the operators ${\cal D}^{(q)}$ defined by $$\label{connected}
\langle {\cal A}{\cal B} \rangle_c = \langle {\cal A}{\cal B} \rangle -
\langle {\cal A} \rangle~\langle {\cal B} \rangle~,$$ where the one-point functions are given by: $$\begin{aligned}
&~&\langle {\cal D}^{(s)}\rangle = N_s\int
\frac{d\omega}{(\omega -i\epsilon)^{s+1}}\langle e^{i\omega t} \rangle
=N_s\int \frac{d\omega}{(\omega -i\epsilon)^{s+1}} e^{-\omega^2\alpha} = {\tilde N}_s~\epsilon^{-s}~h_s(\epsilon^2\alpha)~, \nonumber \\
&~&h_s(x)
= -\frac{x^{s/2}}{2}\left(\frac{4\pi}{\Gamma (\frac{1+s}{2})}~
\sqrt{\pi}~F(1+\frac{s}{2}~, \frac{3}{2}~, x) - \frac{2\pi}{\Gamma (1 +
\frac{s}{2})}~F(\frac{1 + s}{2}~, \frac{1}{2}~, x)\right).\end{aligned}$$
For the two-point function of the ${\cal D}^{(q)}$ operator the result is: $$\begin{aligned}
&~& \langle {\cal D}^{(q)}(z){\cal D}^{(q)}(0) \rangle_c =
{\tilde N}_q \epsilon^{-2}\left(\frac{\sqrt{\pi}}{\xi_0}
f_q(2\xi_0+2\epsilon^2{\rm ln}|z/L|^2) - h^2_q(\xi_0)\right)~.\end{aligned}$$
Expanding in powers of $\epsilon$, we obtain $$\begin{aligned}
\label{dd}
&~& \langle {\cal D}^{(q)}(z) {\cal D}^{(q)}(0) \rangle_c =
{\tilde N}_q \epsilon^{-2}\left(\frac{\sqrt{\pi}}{\sqrt{\xi_0}}
f_q(2\xi_0) - h^2_q(\xi_0)\right) +
{\tilde N}^2_q~\frac{\sqrt{\pi}}{\sqrt{\xi_0}}~f'_q(2\xi_0)2
{\rm ln}|z/L|^2 + \nonumber \\
&~& +~ \epsilon^2 {\tilde N}_q^2 \sqrt{\frac{\pi}{\xi_0}}
\frac{1}{2}f''_q(2\xi_0)4 {\rm ln}^2|z/L|^2
+ \dots \end{aligned}$$ where $\dots $ denote terms that vanish as $\epsilon \to 0^+$.
To avoid the divergences coming from the $\epsilon^{-2}$ factors, the following condition must be satisfied: there must be a solution $\xi_0=\xi_0(q)$ of the equation: ${\cal H}(\xi_0) \equiv \frac{\sqrt{\pi}}{\sqrt{\xi_0}}
f_q(2\xi_0) - h^2_q(\xi_0)=0$. The existence of such a solution can be verified numerically (see figure 2). Analytically this can be confirmed by looking at the asymptotic behaviour of the function ${\cal H}(x)$ as $x \to \infty$, which yields a negative value: ${\cal H}(x\to \infty) \sim
-\frac{\pi^3x^{2q}e^{2x}}{\Gamma^2 (\frac{1+q}{2})
\Gamma^2(1 + \frac{q}{2})} < 0$. This behaviour comes entirely from the term $h_q^2(x)$, given that $f_q(x\to \infty) \to 0^+$.
\[sol\]
As we shall show below, for various values of $q$, near the fixed point $\epsilon \to 0^+$, one can construct higher order logarithmic algebras, whose highest power is determined by the dominant terms in the operator algebra of correlators (\[offshellalg\]), (\[tdope\]). To this end, we first remark that in the above analysis we have dealt with a small but otherwise arbitrary parameter $\epsilon$, which allows us to keep as many powers as required by (\[offshellalg\]) in conjunction with the value of $q$. The value of $\epsilon$ determines the distance from the fixed point.
For $1< q <2$, there are only two dominant operators as the time $t \to \infty$, ${\cal D}$,${\cal C}$. In this case one obtains a conventional logarithmic conformal algebra of two-point functions near the fixed point: $$\begin{aligned}
\label{logalg}
&~& \langle {\cal D}^{(q)}(z)~{\cal D}(0)^{(q)}
\rangle_c = \langle {\cal D}(z)~{\cal D}(0) \rangle_c
\sim {\tilde N}^2_q~\frac{\sqrt{\pi}}{\sqrt{\xi_0}}~f'_q(2\xi_0)2
{\rm ln}|z/L|^2~, \nonumber \\
&~& \epsilon~q~\langle {\cal D}^{(q-1)}(z)~D^{(q)}(0) \rangle_c =
\langle {\cal C}(z)~{\cal D}(0) \rangle_c \sim {\tilde N}^2_q~
\left(h^2_q(\xi_0) - h_{q-1}~h_q (\xi_0)\right)~, \end{aligned}$$ and all the other correlators are subleading as $t \to \infty$.
Therefore, the [*on shell algebra*]{} is of the conventional [*logarithmic form*]{} [@lcft], between a pair of operators, and hence, ${\cal D}^{(q-2)}$ and subsequent operators, which owe their existence to the non-trivial RW metric, do not modify the two-point correlators of the standard logarithmic algebra of ‘recoil’ (impulse) [@kmw] [^2].
Next, we consider the case where $2< q < 3$. In this case, from (\[offshellalg\]) we observe that there are now three operators which dominate in the limit $t \to \infty$, ${\cal D}$, ${\cal C}$ and ${\cal B} = \epsilon^2~q~(q-1)
{\cal D}^{(q-2)}$, whose form is implied from (\[tdope\]), in analogy with ${\cal C}$. The corresponding algebra of correlators consists of parts forming a conventional logarithmic algebra, and parts forming a second-order logarithmic algebra, the latter being obtained from terms of order $\epsilon^2$ in the appropriate two-point connected correlators ( cf. (\[dd\]) [*etc.*]{}), which are denoted by a superscript $\langle \dots \rangle_c^{(2)}$: $$\begin{aligned}
\label{higherord}
&~& \langle {\cal D}(z)~{\cal D}(0) \rangle_c^{(2)} =
{\tilde N}_q^2 \sqrt{\frac{\pi}{\xi_0}}
\frac{1}{2}f''_q(2\xi_0)4 {\rm ln}^2|z/L|^2~,
\nonumber \\
&~& \langle {\cal C}(z)~{\cal D}(0) \rangle_c^{(2)}=
{\tilde N}_q^2 \sqrt{\frac{\pi}{\xi_0}}
2f'_q(2\xi_0) {\rm ln}|z/L|^2~,
\nonumber \\
&~& \langle {\cal C}(z)~{\cal C}(0) \rangle_c^{(2)} =
{\tilde N}_q^2 \sqrt{\frac{\pi}{\xi_0}}
f_{q-1}(2\xi_0)~, \nonumber \\
&~& \langle {\cal B}(z)~{\cal D}(0) \rangle_c^{(2)} =
-{\tilde N}_q^2 \sqrt{\frac{\pi}{\xi_0}}
\left(f_{q}(2\xi_0) + f'_q(2\xi_0)\right)~, \nonumber \\
&~& \langle {\cal C}(z)~{\cal B}(0) \rangle_c^{(2)} =
\langle {\cal B}(z)~{\cal B}(0) \rangle_c^{(2)} = 0\end{aligned}$$ where the last two correlators are of order $\epsilon^4$ and $\epsilon^6$ respectively, that is of higher order than the $\epsilon^2$ terms, and hence they are viewed as zero to the order we are working here.
In general, if one considers $q > 3$ one arrives at higher order logarithmic algebras [@lcft], with the highest power given by the integer value of $q$, $[q]$. This is an interesting feature of the recoil-induced motion of $D$-particles in RW backgrounds with scale factors $\sim t^{p}$, $p >1$, corresponding to cosmological horizons and accelerating Universes. In such a case the order of the logarithmic algebra is given by $[2p]$. It is interesting to remark that radiation and matter (dust) dominated RW Universes would imply simple logarithmic algebras.
We now notice that, under a world-sheet finite-size scaling, $$\begin{aligned}
L \to L' = L~e^{{\cal T}\,{\cal K}(q)}~, \qquad \epsilon^{-2}
\to (\epsilon')^{-2} = \epsilon^{-2} + {\cal T} \end{aligned}$$ with ${\cal K}(q)$ a function of $q$ determined by (\[logalg\]), the operators ${\cal C},~{\cal D}, \dots $, and consequently the target-time $t$, transform in a non trivial way. In particular, for $t$ one has: $$\begin{aligned}
\label{timeshift2}
\left(\frac{\epsilon'}{\epsilon}\right)^{q-1}{\cal Z}~({\cal T})^q
t({\cal T})^q = t^q + q~\epsilon~{\cal T}~t^{q-1} + {\cal O}(\epsilon^2)\end{aligned}$$ where ${\cal Z}({\cal T})$ is a wave function renormalization of the world-sheet field $t(z)$, which can be chosen in a natural way so that $\left(\frac{\epsilon'}{\epsilon}\right)^{q-1}{\cal Z}\,({\cal T})^q =1$. This implies $$\begin{aligned}
\label{timeshift}
&~& t({\cal T})^q = (t + \epsilon {\cal T})^q + {\cal O}(\epsilon^2)~,
\nonumber \\
&~& t({\cal T}) = t + \epsilon {\cal T} + {\cal O}(\epsilon^2)~,\end{aligned}$$ i.e. that a shift in the target time is represented as $\epsilon\,{\cal T}$. Of course, at the fixed point, $\epsilon =0$, the field $t(z)$ does not run, as expected. As we shall see in the next section, the shift (\[timeshift\]) is consistent with the identification of the Liouville mode with the target time, in case one wishes to discuss certain aspects of slightly off-shell string physics.
Space Time Metrics
==================
Vertex Operator for the Path and associated SpaceTime Geometry
--------------------------------------------------------------
In this section we shall discuss the implications of the world-sheet deformation (\[path\]) for the spacetime geometry. In particular, we shall show that its rôle is to preserve the Dirichlet boundary conditions on the $X^i$ by changing coordinate system, which is encoded in an induced change in the space time geometry $G_{ij}$. The final coordinates, then, are coordinates in the rest frame of the recoiling particle, which naturally explains the preservation of the Dirichlet boundary condition.
To this end, we first rewrite the world-sheet boundary vertex operator (\[path\]) as a bulk operator: $$\begin{aligned}
\label{bulkop}
&~& V = \int _{\partial \Sigma} G_{ij}y^j(t)\partial_n X^i =
\int_{\Sigma} \partial_\alpha \left(y_i(t)\partial^\alpha X^i\right) =
\nonumber \\
&~& = \int _{\Sigma} \left({\dot y}_i(t)\partial_\alpha t \partial^\alpha X^i
+ y_i \partial^2 X^i \right)\end{aligned}$$ where the dot denotes derivative with respect to the target time $t$, and $\alpha$ is a world-sheet index. Notice that it is the covariant vector $y_i$ which appears in the formula, which incorporates the metric $G_{ij}$, $y_i=G_{ij}y^j$.
To determine the background geometry, which the string is moving in, it is sufficient to use the classical motion of the string, described by the world-sheet equations: $$\begin{aligned}
\label{sem}
\partial^2 X^i + {\Gamma^i}_{\mu\nu}\partial_\alpha X^\mu
\partial^\alpha X^\nu =0,\end{aligned}$$ where $\mu, \nu$ are space time indices, $\alpha=1,2$ is a world-sheet index, $\partial^2$ is the Laplacian on the world sheet, and $i$ is a target spatial index.
The relevant Christoffel symbol in our RW background case, is ${\Gamma^i}_{ti}$, and thus the operator (\[bulkop\]) becomes: $$\begin{aligned}
\int _{\Sigma} \left({\dot y}_i - 2y_i(t){\Gamma^i}_{ti}\right)\partial_\alpha t \partial^\alpha X^i \end{aligned}$$ from which we read an induced non-diagonal component for the space time metric $$\begin{aligned}
\label{indmetr}
2G_{0i}= {\dot y}_i - 2y_i(t){\Gamma^i}_{ti}\end{aligned}$$ In the RW background (\[rwmetric\]) the path $y_i(t)$ is described (\[pathexpre\]) by (notice again we work with covariant vector $y_i$): $$\begin{aligned}
y_i(t) = \frac{v_ia_0^2}{1-2p}\left(t t_0^{2p}-t_0 t^{2p}\right)\end{aligned}$$ which gives $2G_{0i}=a^2(t_0) v_i$, yielding for the metric line element: $$\label{fixedpoint}
ds^2=-dt^2 + v_ia^2(t_0)dtdX^i + a^2(t)(dX^i)^2, \qquad {\rm for}
\qquad t > t_0$$ As expected, this spacetime has precisely the form corresponding to a Galilean-boosted frame (the D-particle’s rest frame), with the boost occurring suddenly at time $t=t_0$.
This can be understood in a general fashion by first noting that (\[indmetr\]) can be written in a general covariant form as: $$\begin{aligned}
2G_{0i}=\nabla_t y_i~~(= \nabla_t y_i + \nabla_i t)\end{aligned}$$ which is the general coordinate transformation associated with $y_i$ from a passive (Lie derivative) point of view.
In general, given the boundary condition $\partial_n t=0$, one can write the operator (\[path\]), in a covariant form by expressing it as a world-sheet bulk operator: $$\begin{aligned}
V= \int _{\partial \Sigma} y_\mu \partial_n X^\mu =
\int _{\Sigma} \partial_\alpha \left(y_\mu \partial^\alpha X^\mu \right) = \int_{\Sigma} \nabla_{\mu} y_\nu
\partial_\alpha X^\mu \partial^\alpha X^\nu \end{aligned}$$ where in the last step, we have used again the string equations of motion (\[sem\]). From this expression, one then derives the induced change in the metric $$\begin{aligned}
\label{lie}
2 \delta G_{\mu\nu} = \nabla_\mu y_\nu + \nabla_\nu y_\mu \end{aligned}$$ which is the familiar expression of the Lie derivative under the coordinate transformation associated with $y_\mu$.
In all the above expressions we have taken the limit $\epsilon \to 0$, which corresponds to considering the ratio of world-sheet cut-offs $a/L \to 0$, implying that one approaches the infrared fixed point in a Wilsonian sense. As noted previously, in the context of the logarithmic conformal analysis of the path $y^i(t)$, we have seen that this limit can be reached without problems only in the case $p \le 1$, which corresponds to the absence of cosmological horizons. On the other hand, the case of non-trivial horizons, $p > 1$, implies ultraviolet divergences, which prevent one from taking this limit in a way consistent with conformal invariance of the underlying $\sigma$ model. In such a case, the operators are relevant, with finite anomalous dimensions $-\epsilon^2/2$, and thus Liouville dressing is required [@ddk; @recoil]. This is the topic of the next subsection.
Cosmological Horizons and Liouville Dressing
--------------------------------------------
In this subsection we shall discuss Liouville dressing of the relevant recoil deformations [@recoil]. There are two ways one can proceed in this matter. The first, concerns dressing of the boundary operators (\[path\]) $$\begin{aligned}
\label{boundary}
V_{L,{\rm boundary}} = \int _{\partial \Sigma} e^{\alpha_i \varphi} y_i(t)\partial_nX^i,
\qquad \alpha_i = -\frac{Q}{2} + \sqrt{\frac{Q^2}{2} + (1-h_i)} \end{aligned}$$ where $h_i$ is the boundary conformal dimension, and $Q^2$ is the induced central charge deficit on the boundary of the world-sheet.
In a similar spirit to the flat target-space case [@kmw], the rate of change of $Q^2$ with respect to world-sheet scale ${\cal T} \sim \epsilon^{-2}$ is given by means of Zamolodchikov C-theorem [@zam], and it is found to be of order [@recoil] $v_i^2~\epsilon^4 $, as being proportional to the square of the renormalization-group $\beta^i$ functions ($i = v_i$): $\tfrac{\partial Q^2}{\partial {\cal T}} \propto
-\beta^i {\cal G}_{ij} \beta^j $, where ${\cal G}_{ij} = \delta _{ij} + \dots$, is the Zamolodchikov metric in coupling constant space. This implies that $Q^2(t) = Q_0^2 + {\cal O}(\epsilon^2)$, where $Q_0^2$ is constant.
We shall distinguish two cases for $Q_0$. The first concerns the case where $Q_0 \ne 0$ (and by appropriate normalization may be assumed to be of order ${\cal O}(1)$). This is the case of strings living in a non-critical space time dimension. The other pertains to the case where the only source of non-criticality is the impulse deformation, i.e. $Q_0 =0$. In the former case, one has a Liouville dimension $\alpha_i \sim \epsilon^2 $, while in the latter $\alpha_i \sim \epsilon$. In [*both cases*]{} however, $\epsilon \sim \tfrac{1}{t}$, where $t$ is the target time.
In the second method [@recoil], one dresses by the Liouville field the bulk operator (\[bulkop\]), i.e. $$\begin{aligned}
\label{bulk}
V_{L, {\rm bulk}} = \int _{\Sigma} e^{\alpha_i \varphi}
\partial_\alpha \left(y_i(t)\partial^\alpha X^i \right), \qquad
\alpha_i = -\frac{Q}{2} + \sqrt{\frac{Q^2}{2} + (2-\Delta_i)}\end{aligned}$$ where $\Delta_i$ is the conformal dimension of the bulk operator. The central charge deficit $Q$ is of the same order $Q^2 = Q_0^2 + {\cal O}(\epsilon^2)$ as in the boundary case, which implies again that $\alpha_i \sim \epsilon^2$ if $Q_0 \ne 0$, and $\alpha_i \sim \epsilon$ if $Q_0 = 0$. An interesting question, which we shall answer in the affirmative below, concerns the equivalence between these two approaches either at the fixed point ($\epsilon \to 0$), or close to it ($\epsilon \ne 0$ but small).
We commence our analysis by first looking at the boundary operator (\[boundary\]). We may rewrite it as a bulk operator and then manipulate it as follows: $$\begin{aligned}
\label{boundary2}
&~& V_{L, {\rm boundary}} = \int _{\Sigma} \partial_\alpha
\left(e^{\alpha_i\varphi}y_i(t)\partial^\alpha X^i\right)=
\nonumber \\
&~& = \int_\Sigma \alpha_i e^{\alpha_i\varphi}~y_i(t)\partial_\alpha \varphi
\partial^\alpha X^i +
\int_\Sigma e^{\alpha_i\varphi} {\dot y}_i(t) \partial_\alpha
t \partial^\alpha X^i + \int _\Sigma e^{\alpha_i\varphi}y_i(t)\partial^2 X^i \end{aligned}$$ For the bulk operator (\[bulk\]) one has: $$\begin{aligned}
\label{bulk2}
&~& V_{L, {\rm bulk}} = \int _{\Sigma}
\partial_\alpha
\left(e^{\alpha_i\varphi}y_i(t)\partial^\alpha X^i\right) -
\int_\Sigma \alpha_i e^{\alpha_i\varphi}~y_i(t)\partial_\alpha \varphi
\partial^\alpha X^i = \nonumber \\
&~& =\int_{\partial \Sigma} e^{\alpha_i\varphi} y_i(t) \partial_n X^i
- \int_\Sigma \alpha_i e^{\alpha_i\varphi}~y_i(t)\partial_\alpha \varphi
\partial^\alpha X^i\end{aligned}$$
The logarithmic algebra, as discussed in [@kmw] and above, implies a non-trivial infrared fixed point, which in the case $Q_0 \ne 0$ is determined by $\varphi_0 = \epsilon^{-2} \sim {\rm ln}(L/a)^2 \to \infty$, where $\varphi_0$ is the Liouville field world-sheet zero mode. Thus, $\alpha_i \varphi_0 $ is finite as $\epsilon \to 0^+$. Therefore, as expected from the restoration of the conformal invariance by means of the Liouville dressing, one can now take safely the infra-red limit $\epsilon \to 0^+$ in the above expressions. It is then easy to see that one is left [*in both cases*]{} with the metric (\[fixedpoint\]), thereby proving the equivalence of both approaches at the infrared fixed point.
In the case $Q_0 =0$, the running central charge deficit $Q^2 ={\cal O}(\epsilon^2)$. Recalling [@ddk] that the above formulæ imply a rescaling of the Liouville mode by $Q \sim \epsilon$, so as to have a canonical kinetic $\sigma$-model term [^3], and that in this case it is the $\varphi_0/Q$ which is identified with ${\rm ln}(L/a)^2 \sim \epsilon^{-2}$ as pertaining to the covariant world-sheet cutoff, one observes that again $\alpha_i~\varphi$ is finite as $\epsilon \to 0^+$, and hence similar conclusions are reached concerning the equivalence of the two methods of Liouville dressing of the impulse operator.
This equivalence is also valid [*close to*]{}, but not exactly at, the infrared fixed point, as we demonstrate now. To this end, we discuss the two cases $Q_0 =0$ and $Q_0 \ne 0$ separately.
Consider first the case $Q_0 \ne 0$. In this case $\alpha_i \sim \epsilon^2$, $\varphi_0 \sim \epsilon^{-2}$ and hence $\alpha_i \varphi_0 \sim \epsilon t = {\rm const}$. We identify now the Liouville direction $\varphi$ with that of the target time [@emn; @recoil]. Given that $t \sim
\tfrac{1}{\epsilon}$ this implies that $\varphi \sim t^2$. Under this identification we observe [@recoil] in both cases (\[boundary2\]), (\[bulk2\]) that the terms $e^{\alpha\varphi_0}$, and the exponential factors $e^{-\epsilon~(t-t_0)}$ appearing in the regulated $\Theta$ functions [@kmw] are all of order one.
From these considerations one obtains an induced non-diagonal metric element $G_{0i}$, which in the case (\[boundary2\]) is $$\begin{aligned}
\label{ndmetric}
G_{0i}d\varphi dX^i =
\left(v_i~\alpha_i t^{2p} + v_ia^2(t_0)\right)~d(t^2)dX^i
\simeq v_i~\alpha_i t^{2p}
d(t^2)dX^i~,
\qquad t \gg t_0\end{aligned}$$ and in the case (\[bulk2\]): $$\begin{aligned}
G_{0i}d\varphi dX^i \simeq -v_i~\alpha_i t^{2p}d(t^2)dX^i~,
\qquad t \gg t_0\end{aligned}$$ We then observe that, up to an irrelevant sign, the two results are equivalent in the regime of large $t$, where our perturbative string (world-sheet) analysis is valid, thereby proving the equivalence of the two ways of Liouville dressing even away from the fixed point.
Under the fact that one identifies $\epsilon^{-1} =t-t_0 \sim t \gg t_0$, the non-diagonal element of the spacetime metric becomes: $$\begin{aligned}
\label{ndmetric2}
G_{0i} \sim v_i t^{2p-1} \end{aligned}$$ We remind the reader that we analyze here the case with horizon, which implies $p > 1$.
It is convenient now to diagonalize the metric, which implies the following line element $$\begin{aligned}
\label{nontrans}
ds^2 = -\frac{v_i^2}{a_0^2}t^{2p-2}dt^2 +
a_0^2t^{2p}(dX^i)^2 \end{aligned}$$ By redefining the time coordinate to $t'=\frac{{v}_i}{a_0 p}t^p$ one obtains the induced line element: $$\begin{aligned}
\label{rwfinal}
ds^2 = -(dt')^2 + \frac{a_0^4~p^2}{{v}_i^2}(t')^2~(dX^i)^2, \qquad t \gg t_0 \end{aligned}$$ From (\[horizon\]), we thus observe that the induced metric has [*no horizon*]{}, and no cosmic acceleration. In other words a recoiling $D$-particle, embedded in a space time which initially appeared to have an horizon, back reacted in such a way so as to remove it! Equivalently, we may say that recoiling $D$-particles are consistent only in spacetimes without cosmological horizons.
Similar conclusions are reached in the case $Q_0 =0$. In that case, $\varphi/Q \sim \epsilon^{-2}$, as explained above, and since $Q \sim \epsilon \sim \tfrac{1}{t}$, one now has that $\varphi \sim t$. Again, the exponential terms $e^{\alpha_i \varphi}$, $\alpha_i \sim \epsilon$, and those coming from the regulated $\Theta_\epsilon (t)$ are of order one. Evidently, the induced non-diagonal metric has the same form (\[ndmetric2\]) as in the case with $Q_0 \ne 0$, and one can thus repeat the previous analysis, implying removal of the cosmological horizon and stopping of cosmic acceleration.
The reader must have noticed that the same conclusion is reached already at the level of the metric (\[nontrans\]), before the time transformation, once one interprets the coefficient of the $(dt)^2$ as a time-dependent light velocity. The fact that such situations arise ‘suddenly’, after a time moment $t_0$, might prompt the reader to draw some analogy with the scenaria of time-dependent light velocity, involving some sort of phase transitions at a certain moment in the (past) history of our Universe [@moffat]. In our case, as we have seen, one can perform (at late times) a change in the time coordinate in order to arrive at a RW metric (\[rwfinal\]) [^4].
The removal of the horizon would seem to imply from a field-theoretic point of view that one can define asymptotic states and thus a proper $S$-matrix. However, in the context of Liouville strings, with the Liouville mode identified with the time [@emn], there is no proper $S$-matrix, independently of the existence of horizons [@emnsmatrix]. This has to do with the structure of the correlation functions of vertex operators in this construction, which are defined over steepest-descent closed time-like paths in a path-integral formalism, resembling closed-time paths of non-equilibrium field theory [@emn; @emnsmatrix]. In such constructions one can define properly only a (non factorizable) superscattering matrix \$$\ne S\,S^\dagger$.
Conclusions
===========
In this work we have analyzed in some detail the problem of impulse(recoil)-induced motion of a heavy $D$-particle in a Robertson-Walker spacetime, at large times $t$ after the moment of impact. We have shown, that for RW spacetimes with scale factors $\sim t^p$, there is an order $[2p]$-logarithmic algebra, involving a group of impulse operators, which are relevant from a world-sheet renormalization group point of view.
A detailed study of how one can approach the non-trivial infrared fixed point is given. In the case where $p > 1$, which is physically characterized by the presence of cosmological horizons, one encounters world-sheet divergences. A proper subtraction of such divergences is subtle, and a detailed discussion of how this can be done has been presented. The fact that away from the fixed point the deformed theory is plagued by relevant deformations, of anomalous dimension which itself depends on the world-sheet renormalization-group scale, implies the need for Liouville dressing.
Such a dressing results in the interesting possibility of identifying the Liouville mode with the target time, in which case one has a formal description -in terms of conformal field theory methods on the world sheet - of back reaction effects of the recoiling $D$-particle on the surrounding space time. It is interesting to notice that the effect is equivalent to a ‘phase transition’ at the moment of impact, in which there is induced a time-varying speed of light, effectively leading to the removal of the initial cosmological horizon, and the eventual stopping of the acceleration of the Universe.
From a field-theoretic view point, this would imply that in such models proper asymptotic states, and thus an $S$-matrix, could be defined. However, from our stringy point of view, the definition of an $S$-matrix is still a complicated issue, since the underlying theory is of Liouville (non-equilibrium) type [@emnsmatrix]. Whether such toy models are of relevance to realistic stringy cosmologies remains to be seen. Nevertheless, we believe that the results presented here, although preliminary, are of sufficient interest to prompt further studies along the directions suggested in this work.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank R. Szabo for discussions. The work of E.G. is supported by a King’s College London Research Studentship (KRS).
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[^1]: Here we use simplified propagators on the boundary, with the latter represented by a straight line; this means that the arguments of the logarithms are real [@kmw]. To be precise, one should use the full expression for the propagator on the disc, along the lines of [@szabo]. As shown there, and can be checked here as well, the results are unaffected.
[^2]: We note at this stage that, in our case of non-trivial cosmological RW spacetimes, the pairs of operators ${\cal D},{\cal C}$ do not represent velocity and position as in the flat space time case of ref. [@kmw], but rather velocity and acceleration. This implies that, under a finite-size scaling of the world sheet, the induced transformations of these operators do not form a representation of the Galilean transformations of the flat-space-time case.
[^3]: Notice that this rescaling becomes a trivial one in the case where $Q_0 \ne 0$.
[^4]: There is a slight point to which we would like to draw the reader’s attention. This regards the fact that such transformations depend on the recoil velocity, and thus on the energy content of the matter incident on the D-particle. In case one has a ‘foam’ situation [@recoil], in which several incident particles interact with collections of $D$-particles, which are virtual quantum excitations of the string/brane vacuum, it is unclear how the present results are modified, and hence it might be that one cannot perform simultaneous transformations to diagonalize the metric, thereby obtaining non-trivial refractive indices [@sarkar]. Such issues fall beyond the scope of the present article.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We prove that the property of a free group endomorphism being irreducible is a group invariant of the ascending HNN extension it defines. This answers a question posed by Dowdall-Kapovich-Leininger. We further prove that being irreducible and atoroidal is a commensurability invariant. The invariance follows from an algebraic characterization of ascending HNN extensions that determines exactly when their defining endomorphisms are irreducible and atoroidal; specifically, we show that the endomorphism is irreducible and atoroidal if and only if the ascending HNN extension has no infinite index subgroups that are ascending HNN extensions.'
address: |
Department of Mathematical Sciences\
University of Arkansas\
Fayetteville, AR *Web address: <https://mutanguha.com/>*
author:
- Jean Pierre Mutanguha
bibliography:
- 'refs.bib'
title: '*Irreducibility of a free group endomorphism is a mapping torus invariant*'
---
Introduction
============
Suppose S is a hyperbolic surface of finite type and $f:S \to S$ is a [*pseudo-Anosov*]{} homeomorphism, then the [*mapping torus*]{} $M_f$ is a complete finite-volume hyperbolic 3-manifold; this is Thurston’s hyperbolization theorem for 3-manifolds that fiber over a circle [@Thu82]. It is remarkable fact since the hypothesis is a dynamical statement about surface homeomorphisms but the conclusion is a geometric statement about 3-manifolds. In particular, since the converse holds as well, i.e., a hyperbolic 3-manifold that fibers over a circle will have a pseudo-Anosov [*monodromy*]{}, the property of a fibered manifold having a pseudo-Anosov monodromy is in fact a geometric invariant: if $f:S\to S$ and $f':S' \to S'$ are homeomorphisms whose mapping tori have [*quasi-isometric*]{} (q.i.) fundamental groups, then $f$ is pseudo-Anosov if and only if $f'$ is pseudo-Anosov.
There are three types of invariants that we study in geometric group theory: group invariants, which contain virtual/commensurability invariants, which contain geometric/q.i.-invariants; the geometric invariants are the most important and difficult to prove. In this paper, we exhibit geometric and commensurability invariants for free-by-cyclic groups inspired by Thurston’s hyperbolization theorem and our arguments will be general enough to also apply to ascending HNN extensions of free groups.
There is a rough correspondence between the study of the outer automorphism group of a free group $\operatorname{Out}(F)$ and the study of the mapping class group of a hyperbolic surface $\operatorname{MCG}(S)$. Under this correspondence, surface groups are paired with free groups, surfaces with graphs, and 3-manifolds that fiber over a circle with free-by-cyclic groups. However, this correspondence is not perfect; pseudo-Anosov mapping classes have three possible analogous properties for free group automorphisms: [*induced by a pseudo-Anosov (on a punctured surface)*]{}, [*atoroidal*]{}, and [*irreducible*]{} (Section \[defs\]). We originally set out to prove that irreducibility was a group invariant of the automorphism’s mapping torus and, along the way, we proved more general statements for the first property and the composite property of being both irreducible and atoroidal. Our first result is that the first property is a geometric invariant:
[Theorem]{}[geomqi]{} Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that the mapping tori $F *_\phi$ and $F' *_\psi$ are quasi-isometric. Then $\phi$ is induced by a pseudo-Anosov if and only if $\psi$ is induced by a pseudo-Anosov.
Thus starting with just a free group automorphism $\phi$ induced by a pseudo-Anosov and a quasi-isometry between $F \rtimes_\phi \mathbb Z$ and $F' *_\psi$, we find that $\psi$ is induced by a surface homeomorphism too. The proof is short but uses deep geometric theorems: Thurston’s hyperbolization [@Thu82] and Schwartz rigidity [@Sch95]. Since pseudo-Anosovs have dynamics that are very similar to those of irreducible and atoroidal automorphisms, it is likely that the latter property is a geometric invariant too.
Suppose $\phi : F \to F$ and $\psi : F' \to F'$ are free group automorphisms such that $F \rtimes_\phi \mathbb Z$ and $F' \rtimes_\psi \mathbb Z$ are quasi-isometric. Then $\phi$ is irreducible and atoroidal if and only if $\psi$ is irreducible and atoroidal.
Our main result is that being irreducible and atoroidal is a commensurability invariant, which lends credence to the conjecture; again, the argument works for endomorphisms.
[Theorem]{}[nongeomcomm]{} Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that $F *_\phi$ and $F' *_\psi$ are commensurable and neither one of the endomorphisms has an image contained in a proper free factor of their domain. Then $\phi$ is irreducible and atoroidal if and only if $\psi$ is irreducible and atoroidal.
The hypothesis on the images is necessary: Let $\phi:F_2\to F_2$ be the endomorphism on a free group of rank $2$ given by $\phi(a) = ab$ and $\phi(b) = ba$. Then $\phi$ is irreducible and atoroidal [@JPM Example 1.2]. Now let $F_2$ be a proper free factor of the free group $F_3$ generated by $\{ a,b,c \}$. Extend $\phi$ to $\psi:F_3 \to F_3$ by setting $\psi(c) \in F_2$; then $F_3*_{\psi} \cong F_2*_\phi$, but $\psi$ is reducible.
The proof of Theorem \[nongeomcomm\] follows immediately from an algebraic characterization of $F*_\phi$ that detects exactly when $\phi: F \to F$ is irreducible and atoroidal.
[Theorem]{}[grpinv]{} Suppose $\phi: F \to F$ is a free group injective endomorphism whose image is not contained in a proper free factor of $F$ and let $G = F*_\phi$. Then $\phi$ is irreducible and atoroidal if and only if $G$ has no finitely generated noncyclic subgroups with infinite index and vanishing Euler characteristic.
These results imply that irreducibility is a group invariant, our original motivation:
[Corollary]{}[irredgrp]{} Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that $F *_\phi \cong F' *_\psi$ and neither one of the endomorphisms has an image contained in a proper free factor. Then $\phi$ is irreducible if and only if $\psi$ is irreducible.
That irreducibility is a group invariant was an open problem [@DKL17c Question 1.4]. In a series of papers [@DKL15; @DKL17b; @DKL17c], Dowdall-Kapovich-Leininger studied the dynamics of ([*word-hyperbolic*]{}) free-by-cyclic groups and the main result of the third paper answered this problem under an extra condition that we now discuss:
Fix a free-by-cyclic group $G$. The [*BNS-invariant*]{} $\mathcal C(G)$ is an open cone (with convex components) in $H^1(G; \mathbb R) \cong \operatorname{Hom}(G, \mathbb R)$. By rational rays in $H^1(G; \mathbb R)$, we refer to projective classes of homomorphisms $G \to \mathbb R$ with discrete/cyclic image. Without defining the BNS-invariant, we shall state its most relevant property for our purposes: a rational ray in $H^1(G; \mathbb R)$ is [*symmetric*]{}, i.e., is in $- \mathcal C(G) \cap \mathcal C(G)$ if and only if the corresponding class of homomorphisms $[p] : G \to \mathbb R$ have finitely generated kernel $K$; in this case, $ K $ is free for cohomological reasons [@FH99; @Bie81; @St68], $G \cong K \rtimes_{\phi} \mathbb Z$ for some free group automorphism $\phi: K \to K$, and the natural projection $K \rtimes_{\phi} \mathbb Z \to \mathbb Z$ is in the projective class $[p]$. Fix a symmetric rational ray $r_0$ in $\mathcal C(G)$, and let $\phi_0: K_0 \to K_0$ be the corresponding free group automorphism. The presentation complex for $K_0 \rtimes_{\phi_0} \mathbb Z$ has a natural semi-flow with respect to the [*stable direction*]{} $\mathbb Z_+$. Dowdall-Kapovich-Leininger show in [@DKL17b] that getting from $r_0$ to any symmetric rational ray in the same component of $\mathcal C(G)$ amounts to reparametrizing this semi-flow (hence the convexity of the component) and with a careful analysis of this semi-flow, they are able to relate the monodromy stretch factors and rank of kernels for all symmetric rays in the same component. In the third paper [@DKL17c], they conclude this analysis by showing that being irreducible and atoroidal is an invariant of a component of the BNS-invariant; that is, if $\phi_1: K_1 \to K_1$ and $\phi_2: K_2 \to K_2$ are free group automorphisms corresponding to symmetric rays in the same component of $\mathcal C(G)$, then $\phi_1$ is irreducible and atoroidal if and only if $ \phi_2 $ is too. Since this result relied heavily on the analysis of the semi-flow for a component of the BNS-invariant, the technique cannot be extended to work for symmetric rational rays in separate components. Furthermore, their result does not apply to any asymmetric rational rays, i.e., it does not apply to nonsurjective injective endomorphisms. Theorem \[nongeomcomm\] addresses both of these concerns.
Masai-Mineyama have also proven a different special case of Theorem \[nongeomcomm\] that they call [*fibered commensurability*]{} [@MM17]: suppose $\phi: F \to F$ and $\psi: F' \to F'$ are free group automorphisms and let $K \le F, K' \le F'$ be finite index subgroups with an isomorphism $\epsilon: K \to K'$ such that there are the outer classes of the isomorphisms $\epsilon\left.\phi^i\right|_{K}: K \to K'$ and $\left.\psi^j \epsilon\right|_{K}:K \to K'$ are the same for some $i, j \ge 1$; a priori, we restrict ourselves to $i, j$ such that $\phi^i(K) = K$ and $\psi^j(K') = K'$, i.e., the maps $\epsilon\left.\phi^i\right|_{K}$ and $\left.\psi^j \epsilon\right|_{K}$ make sense. Masai-Mineyama prove that in this case, $\phi$ is irreducible and atoroidal if and only if $\psi$ is too. In other words, being irreducible and atoroidal is a fibered commensurability invariant. However, compared to commensurability, this equivalence is very restrictive since a typical isomorphism of finite index subgroups of free-by-cylic groups will not preserve the fibers one starts with.
Feighn-Handel’s theorem that mapping tori of free group endomorphisms are coherent is the main tool that allows us to avoid the obstacles in the two approaches discussed above. We shall explicitly use the [*preferred presentation*]{} that their algorithm produces for any finitely generated subgroup of a mapping torus (Theorem \[rewrite\]).
We conclude the introduction by noting that very little is known about geometric invariants of free-by-cyclic groups in relation to their monodromies. Here are the geometric invariants we know of: Brinkmann [@Bri00; @Bri02] showed 1) a free-by-cyclic group is word-hyperbolic (a geometric invariant) if and only if the monodromy is atoroidal and 2) a word-hyperbolic free-by-cyclic group has a menger curve [*Gromov boundary*]{} if and only if the monodromy fixes no free splitting of the fiber; Macura [@Mac02] proved that two polynomially-growing free group automorphisms that have quasi-isometric mapping tori must have polynomial growth of the same degree and, conjecturally, these mapping tori cannot be quasi-isometric to a mapping torus of an exponentially growing free group automorphism.
[**Outline.**]{} We give the standard definitions and statements of results that are most relevant to the rest of the article in Section \[defs\]. Section \[geom\] contains the proof of Theorem \[geomqi\] while Section \[nongeom\] contains that of Theorem \[nongeomcomm\]. Finally, we briefly discuss the q.i.-invariance conjecture in Section \[qi\]. In Appendix \[app\], we prove a folk theorem and its converse: a free group endomorphism is irreducible with a nontrivial periodic conjugacy class if and only if it is induced by a pseudo-Anosov on a punctured surface with one orbit of punctures.
**Acknowledgments:** I want to thank Ilya Kapovich for patiently checking my initial proof (and its numerous iterations) and Chris Leininger for sharing with me his work-in-progress in collaboration with Spencer Dowdall and I. Kapovich as it relates to my result. The appendix was born out of an insightful discussion with Saul Schleimer on a bus ride and it would not have been written up without Mladen Bestvina’s encouragement. Last but not least, I thank my advisor Matt Clay for his constant support.
Definitions and Preliminaries {#defs}
=============================
In this paper, free groups $F$ are assumed to have finite rank at least $2$. We will study the ascending HNN extension of a free group and how its properties relate to those of the defining endomorphism. Let $A \le F$ be a subgroup of a free group and $\phi : A \to F$ be an injective homomorphism, then we define the [**HNN extension**]{} of $F$ over $A$ to be: $$F*_A = \left\langle\, F, t~|~t^{-1} a t = \phi(a), \forall a \in A \,\right\rangle$$ An HNN extension has a natural map $F*_A \to \mathbb Z$ that maps $F \mapsto 0$ and $ t \mapsto 1$; we shall refer to this map as the [**natural fibration**]{}. For the rest of this paper, we restrict ourselves to HNN extensions defined over free factors. When A = F , then we call $F*_F = F*_\phi$ an [**ascending HNN extension**]{} or a [**mapping torus**]{} of $\phi : F \to F$. The latter terminology stems from the fact that the injective endomorphism $\phi$ can be topologically represented by a graph map on the rose whose topological mapping torus has a fundamental group isomorphic to $F*_\phi$ . Following this analogy, we shall call $F$ the [**fiber**]{} and $\phi$ the [**monodromy**]{} of the mapping torus. Finally, when $\phi : F \to F$ is an automorphism, we call $F*_\phi = F \rtimes_\phi \mathbb Z$ a [**free-by-cyclic group**]{}.
The following are the properties of monodromies that we will study. An endomorphism $\phi: F \to F$ is [**reducible**]{} if there is a free factorization $A_1 * \cdots * A_k * B $ of $F$ where $B$ may be trivial if $k \ge 2$, and a sequence of elements, $(x_i)_{i=1}^k$, in $F$ such that $\phi(A_i) \le x_i A_{i+1} x_i^{-1}$ where the indices are considered$\mod k$; the collection $\{A_1, \ldots, A_k \}$ is a [**$\boldsymbol{\phi}$-invariant proper free factor system**]{}. An endomorphism is [**irreducible**]{} if it is not reducible. It is [**atoroidal**]{} if there does not exist a nontrivial element $a \in F$, element $x \in F$, and integer $n \ge 1$ such that $\phi^n(a) = x a x^{-1}$, equivalently, $i_x \phi^n(a) = a$ for some inner automorphism $i_x$.
Feighn-Handel used the following proposition to show the coherence of ascending HNN extensions of free groups; the proposition and the next lemma allow us to rewrite presentations of ascending HNN extension subgroups of ascending HNN extension groups so that fibers of the subgroup lie in fibers of the ambient group.
\[rewrite\] Suppose $\Phi:\mathbb F \to \mathbb F$ is an injective endomorphism of a free group (of possibly infinite rank) and $G = \mathbb F*_\Phi = \left\langle\, \mathbb F, t~|~t^{-1} x t = \Phi(x), \forall x\in\mathbb F \,\right\rangle$. Let $p: G \to \mathbb Z$ be the natural fibration that maps $\mathbb F \mapsto 0$ and $t \mapsto 1$.
If $H \le G$ is finitely generated and $\left.p\right|_H$ is not trivial and not injective, then there is an isomorphism $\iota: F*_A \to H$, natural fibration $\pi: F*_A \to \mathbb Z$, and injective map $n:\mathbb Z \to \mathbb Z$ such that $n(1) \ge 1$ and $\left.p\right|_H \iota = n \pi$, where $F*_A$ is an HNN extension of a finitely generated free group $F$ over a free factor $A \le F$.
The next lemma will be used to show certain subgroups of a mapping torus are also mapping tori if they have vanishing Euler characteristic.
\[hwlem\] Let $F$ be a free group of finite rank $m$ and $A \le F$ a free factor of rank $n$. Then $\chi(F*_{A}) = n - m$.
Choose a basis for $A$ and extend it to a basis for $F$. Then the natural finite presentation complex $X$ is aspherical, i.e., $X$ is a $K(\pi,1)$ space for $F*_{A}$, and $$\chi(F*_{A}) = \chi(X) = 1 - (1+m) + n = n-m. \qedhere$$
Finally, we give a characterization of irreducible endomorphisms with nontrivial periodic conjugacy classes. As the techniques used in the proof of the following theorem are not related to the rest of the paper, we postpone the proof to the appendix (Proposition \[eg\] and Theorem \[thmBH2\]).
\[thmBH\] Let $\phi:F \to F$ be an infinite-order endomorphism. Then $\phi$ is irreducible and has a periodic nontrivial conjugacy class if and only if it is induced by a pseudo-Anosov homeomorphism of a surface $\Sigma_g^b$ that acts transitively on the boundary components.
This theorem allows us to partition injective endomorphisms of free groups with interesting dynamics into two categories:
- automorphisms induced by pseudo-Anosov maps.
- irreducible and atoroidal endomorphisms.
Pseudo-Anosov Monodromies {#geom}
=========================
The first result in this section is a straightforward application of Stallings’ fibration theorem and Nielsen-Thurston classification; it is the first half of the proof that irreducibility of the monodromy is a group invariant of the mapping torus. The subsequent generalizations are given to motivate the q.i.-invariance conjecture.
\[geomgrp\] Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that $F *_\phi \cong F' *_\psi$. Then $\phi$ is induced by a pseudo-Anosov if and only if $\psi$ is induced by a pseudo-Anosov.
Without loss of generality, suppose $\phi$ is induced by a pseudo-Anosov on a compact surface with boundary. Therefore, $G = F \rtimes_\phi \mathbb Z$ is the fundamental group of a compact $3$-manifold that fibers over a circle. Bieri-Neumann-Strebel [@BNS] showed that the BNS-invariant of such a compact $3$-manifold group $G$ is symmetric, which implies that $\psi$ is an automorphism and $G \cong F' \rtimes_\psi \mathbb Z$. By Stallings’ fibration theorem [@St61], $\psi$ is induced by a homeomorphism of a compact surface with boundary. Any invariant essential multicurve of the $\psi$-inducing homeomorphism would determine a non-peripheral $\mathbb Z^2$ subgroup of $G$. But since $\phi$ was induced by a pseudo-Anosov, the only $\mathbb Z^2$ subgroups of $G$ are the peripheral ones. Thus $\psi$ is induced by an infinite-order irreducible homeomorphism. By Nielsen-Thurston classification, $\psi$ is induced by a pseudo-Anosov.
Since the number of orbits of boundary components for a pseudo-Anosov is the number of boundary components in the mapping torus, this proposition combines with Theorem \[thmBH\] to give:
\[irredtorgrp\] Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that $F *_\phi \cong F' *_\psi$. Then $\phi$ is irreducible and has a periodic nontrivial conjugacy class if and only if $\psi$ is irreducible and has a periodic nontrivial conjugacy class.
Surprisingly, the analogous statement for commensurable groups is much harder to prove. The difficulty lies in showing that if the restriction of an iterate $\psi^n$ to a finite index subgroup is induced by a pseudo-Anosov, then so is $\psi$. We could use the theory of train tracks to adapt the argument in Appendix \[app\] and get a comparatively elementary proof; we opt to use Thurston’s hyperbolization [@Thu82] and Mostow’s rigidity [@Mar74] to keep the exposition short.
\[geomcomm\] Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are free group injective endomorphisms such that $F *_\phi$ and $F' *_\psi$ are commensurable. Then $\phi$ is induced by a pseudo-Anosov if and only if $\psi$ is induced by a pseudo-Anosov.
Let $H \le F*_\phi$ and $H' \le F'*_\psi$ be isomorphic finite index subgroups. Without loss of generality, suppose $\phi$ is induced by a pseudo-Anosov homeomorphism on a punctured surface. Then $\phi$ is an automorphism and $F*_\phi = F \rtimes_\phi \mathbb Z$. By Thurston’s hyperbolization theorem, $F \rtimes_\phi \mathbb Z$, and hence $H' \cong H$, is a fundamental group of a complete finite-volume hyperbolic 3-manifold $M$. Assume $H' \trianglelefteq F' *_\psi$, then Mostow’s rigidity implies the finite group $Q = (F' *_\psi)/H'$ acts on $M$ by isometries. The action is free since $F' *_\psi$ is torsion-free. Thus $F' *_\psi$ is the fundamental group of the hyperbolic 3-manifold $M/Q$. By symmetry of the BNS-invariant and Stallings’ fibration theorem, $\psi$ is induced by a surface homeomorphism. As $\psi$ has a hyperbolic mapping torus, it is induced by a pseudo-Anosov.
As mentioned before the proposition, the proof can be replaced by an argument using the theory of train tracks. However, the next theorem is a geometric statement and there is no apparent way around Thurston’s hyperbolization theorem. In fact, we will also need Schwartz’ rigidity [@Sch95] to reduce the theorem to the previous proposition.
\[geomqi\]Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are injective endomorphisms such that $F *_\phi$ and $F' *_\psi$ are quasi-isometric. Then $\phi$ is induced by a pseudo-Anosov if and only if $\psi$ is induced by a pseudo-Anosov.
Suppose $\phi$ is induced by a pseudo-Anosov. By Thurston’s hyperbolization theorem, $F*_\phi$ is the fundamental group of a complete finite-volume hyperbolic 3-manifold with cusps. In particular, Schwartz proved such groups are [*q.i.-rigid*]{}. As $F*_\phi$ and $F'*_\psi$ are quasi-isometric torsion-free groups, q.i.-rigidity of $F*_\phi$ implies they are commensurable. Thus $\psi$ is induced by a pseudo-Anosov by Proposition \[geomcomm\].
This proof underscores how difficult it is to prove the q.i.-invariance conjecture since there is no common model like $\mathbb H^3$ when studying irreducible and atoroidal endomorphisms.
Irreducible and Atoroidal Monodromies {#nongeom}
=====================================
The goal of this section is to prove that being irreducible and atoroidal is a commensurability invariant. We proved the following result in previous work [@JPM]; it essentially characterizes being an irreducible and atoroidal endomorphism.
\[invSbgrp\] Suppose $\phi : F \to F$ is an irreducible and atoroidal endomorphism. If nontrivial $K \le F$ is finitely generated and $i_x\phi^n(K) \le K$ for some $n \ge 1$ and inner automorphism $i_x$, then $[F : (i_x \phi^n)^{-k}(K)] < \infty$ for some $k \ge 0$.
The key idea in this section is to use this proposition to characterize in terms of the mapping torus exactly when a monodromy is irreducible and atoroidal. To this end, we need the following property to deal with nonsurjective monodromies.
An injective endomorphism $\phi:F \to F$ is [**minimal**]{} if the image $\phi(F)$ is not contained in a proper free factor of $F$. Automorphisms and irreducible endomorphisms are clearly minimal.
Minimality is preserved by taking powers and composing with (inner) automorphisms (See [@JPM Proposition 5.4]). We now have enough to state and prove the main result:
\[grpinv\] Suppose $\phi: F \to F$ is a minimal injective endomorphism and let $G = F*_\phi$. Then $\phi$ is irreducible and atoroidal if and only if $G$ has no finitely generated noncyclic subgroups with infinite index and vanishing Euler characteristic.
If $\phi$ is not atoroidal, then $G$ has a $\mathbb Z^2$ subgroup that necessarily has infinite index. If $\phi$ is not irreducible, then there exists a proper free factor $A \le F$, $x \in F$, and $n \ge 1$ such that $\phi^n(A) \le xAx^{-1}$. Then, using normal forms, $A *_{i_x \phi^n} \cong \langle A, t^n x \rangle \le G$. Suppose $[G: \langle A, t^n x \rangle] < \infty$, then $[F: F \cap \langle A, t^n x \rangle] < \infty$. Set $K = F \cap \langle A, t^n x \rangle = \cup_k(i_x \phi^n)^{-k}(A)$. As $[F: K] < \infty$, K is finitely generated and there exists a $k_0 \ge 1$ such that $K = (i_x \phi^n)^{-k_0}(A)$. The statements $[F:K] < \infty$, $K = (i_x \phi^n)^{-k_0}(A)$, and $A$ is a proper free factor of $F$ imply $F = (i_x \phi^n)^{-k_0}(A)$, which contradicts the minimal assumption on $\phi$. Therefore, $[G: \langle A, t^n x \rangle] = \infty$ as needed. This concludes the reverse direction.
For the forward direction, suppose $\phi$ is irreducible and atoroidal and let $H \le G$ be a finitely generated noncyclic group with $\chi(H) = 0$. We need to show $[G: H] < \infty$. Let $p:G=F*_\phi \to \mathbb Z$ be the natural fibration. Note that $\ker p = \cup_i t^i F t^{-i}$. Then $\left.p\right|_H$ is not trivial since $\ker p$ is locally free yet $H$ is finitely generated but not free: it is not cyclic and $\chi(H) = 0$. Also $\left.\ker p\right|_H = H \cap \ker p$ is not trivial as $H \not \cong \mathbb Z$.
As $H$ is finitely generated and $\left.p\right|_H$ is not trivial and not injective, by Proposition \[rewrite\], there is an isomorphism $\iota: F_m*_A \to H$, natural fibration $\pi: F_m*_A \to \mathbb Z$, and injective map $n:\mathbb Z \to \mathbb Z$ such that $n(1) \ge 1$ and $\left.p\right|_H \iota = n \pi$, where $F_m*_A$ is an HNN extension of a finitely generated free group $F_m$ over a free factor $A \le F_m$. As $\chi(H) = 0$, $A$ is not a proper free factor by Lemma \[hwlem\]. Therefore, $H \cong F_m*_{F_m}$. As $F_m$ is finitely generated, there is an $i \ge 0$ such that $K = \iota(F_m) \le t^i F t^{-i} \le \ker p$. Fix large enough $i$, then $K \le t^i F t^{-i}$ is a finitely generated nontrivial subgroup such that $i_x\bar\phi^n(K) \le K$, where $n = n(1)$, $\bar \phi$ is the natural extension of $\phi$ to $t^i F t^{-i}$, and $x \in t^i F t^{-i}$. As $\phi$ is irreducible and atoroidal, so is $\bar \phi$. By Proposition \[invSbgrp\], $(i_x \bar\phi^n)^{-k}(K)$ has finite index in $t^i F t^{-i}$ for some $k \ge 0$. Therefore, $H = \langle (t^n x)^k K (t^n x)^{-k}, t^n x \rangle$ has finite index in $G = \langle t^i F t^{-i}, t \rangle$.
The following lemma shows that the property of having a finitely generated noncyclic subgroups with infinite index and vanishing Euler characteristic is a commensurability invariant. This implies the invariance of irreducibility for atoroidal endomorphisms.
Let $G$ be a finitely generated torsion-free group and $G' \le G$ be a finite index subgroup. Then $G$ has an infinite index finitely generated noncyclic subgroup with vanishing Euler characteristic if and only if $G'$ has such a subgroup.
The reverse direction is obvious. Suppose $H \le G$ is an infinite index finitely generated noncyclic subgroup with vanishing Euler characteristic. Then $H \cap G'$ has finite index in $H$. In particular, it is finitely generated with infinite index in $G$ and hence $G'$. Furthermore, $[ H: H \cap G'] < \infty$ implies $H \cap G'$ has vanishing Euler characteristic too and it is noncyclic since the only virtually cyclic torsion-free group is $\mathbb Z$, yet $H$ is not cyclic.
\[nongeomcomm\]Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are minimal injective endomorphisms such that $F *_\phi$ and $F' *_\psi$ are commensurable. Then $\phi$ is irreducible and atoroidal if and only if $\psi$ is irreducible and atoroidal.
Finally, we combine this theorem with Corollary \[irredtorgrp\] to get:
\[irredgrp\]Suppose $\phi: F \to F$ and $\psi:F' \to F'$ are minimal injective endomorphisms such that $F *_\phi \cong F' *_\psi$. Then $\phi$ is irreducible if and only if $\psi$ is irreducible.
Q.I.-Invariance Conjecture {#qi}
==========================
Very little is known about the geometry of mapping tori whose monodromies are irreducible and atoroidal. Brinkmann proved that atoroidal automorphisms of free groups have word-hyperbolic mapping tori [@Bri00] and gave an explicit description for when such mapping tori have non-trivial splittings over cyclic subgroups [@Bri02]: an atoroidal automorphism of $F$ has a mapping torus that splits over $\mathbb Z$ if and only if the (outer) automorphism fixes a [*free splitting*]{} of $F$; in particular, mapping tori of irreducible and atoroidal automorphisms never split over $\mathbb Z$.
By work of Kapovich-Kleiner [@KK00], word-hyperbolic free-by-cyclic groups that do not split over $\mathbb Z$ have Gromov boundary homeomorphic to a menger curve. Unfortunately, the topology of the boundary is not sufficient to detect irreducible and atoroidal monodromies as there are reducible and atoroidal automorphisms that do not fix a free splitting.
The Gromov boundary of a word-hyperbolic group can be given a [*visual metric*]{} that is unique up to [*quasi-symmetry*]{} and its quasi-symmetry class is a complete geometric invariant of the group, i.e., word-hyperbolic groups are quasi-isometric if and only their visual boundaries are quasi-symmetric [@Pau96]. Although a reducible and an irreducible automorphism can have word-hyperbolic mapping tori with homeomorphic boundaries, the conjecture asserts that the visual boundaries cannot be quasi-symmetric!
{#app}
The following is a folk theorem that was used to construct examples of [*fully irreducible*]{} automorphisms, i.e., [*irreducible with irreducible powers (iwip)*]{}, along with examples that are infinite-order irreducible but not fully irreducible [@BH92 Example 1.4]. At the end of the appendix, we will show that the latter examples are complete/exhaustive (Corollary \[irredIWIP\]).
\[eg\] If $\phi:F \to F$ is an automorphism induced by a pseudo-Anosov homeomorphism of a surface $\Sigma_g^{b \ge 1}$ that acts transitively on the boundary components, then $\phi$ is irreducible and it is fully irreducible if and only if $b = 1$.
Let $f : \Sigma \to \Sigma$ be the inducing pseudo-Anosov homeomorphism. Suppose $\phi$ was reducible, i.e., there exists a $\phi$-invariant proper free factor system $A_1 * \cdots * A_k$ of $F$. Thus $i_j \phi^k(A_j) \le A_j$ for some inner automorphisms $i_j$. Let $\hat \Sigma\to \Sigma$ be the cover corresponding to $A_j \le F \cong \pi_1(\Sigma)$; the inclusion implies $f^k$ lifts to a map $\hat f: \hat \Sigma \to \hat \Sigma$. Furthermore, up to homotopy, $\hat f$ preserves the core of $\hat \Sigma$, a compact subsurface that supports $A_j$. Let $\gamma \subset \hat \Sigma$ be a peripheral simple closed curve of the core. After replacing $f$ with a power, we can assume $\gamma$ is an $f$-invariant simple closed curve. However, the projection of $\gamma$ to $\Sigma$ may not be a simple closed curve. Using the [*LERF*]{} property of $F \cong \pi_1(\Sigma)$, construct a finite cover $\bar \Sigma$ such that the projection of $\gamma$ to $\Sigma$ lifts to a simple closed curve $\bar \gamma$ in $\bar \Sigma$.
Since $\phi$ is an automorphism, after passing to a power, we can assume $f$ lifts to a homeomorphism $\bar f: \bar \Sigma \to \bar \Sigma$. This map is pseudo-Anosov since the $f$-invariant measured foliations on $\Sigma$ lift to $\bar f$-invariant measured foliations on $\bar \Sigma$. After passing to power again, we can assume $\bar \gamma$ is $\bar f$-invariant. But the only invariant nontrivial simple closed curves of a pseudo-Anosov homemorphism are the peripheral curves. Thus, the projection of $\gamma$ to $\Sigma$ is a (power of a) peripheral curve. Since this holds for any boundary component $\gamma$ of the core of $\hat \Sigma$, it must be that $A_j$ is a proper free factor corresponding to a boundary component of $\Sigma$. But $f$ acts transitively on the boundary components, hence there is a one-to-one correpondence between the proper free factors $A_1, \ldots, A_k$ and the boundary components of $\Sigma$. This is a contradiction, as all boundary components of a surface can not be simultaneously realized as proper free factors of the surface group. Therefore, $\phi$ is irreducible.
If $b=1$, then all powers of $f$ are pseudo-Anosovs that act transitively on the boundary component. Thus, all powers of $\phi$ are irreducible. If $b \ge 2$, then any boundary component of $\Sigma$ determines a periodic proper free factor.
For the rest of the appendix, we will prove the converse. Bestvina-Handel use the following proposition to prove that fully irreducible automorphisms with periodic nontrivial conjugacy classes are induced by pseudo-Anosov homeomorphisms of surfaces with one boundary component [@BH92 Proposition 4.5].
Suppose $\phi:F \to F$ is fully irreducible and there exists $k \ge 1$ and nontrivial conjugacy class $[c]$ of $F$ such that $[\phi^k(c)] = [c]$.
Then some iterate of $\phi$ has an irreducible train track representative with exactly one (unoriented) indivisible Nielsen loop which covers each edge of the graph twice.
In particular, if $k$ is minimal, then $k \le 2$ with equality if and only if $[\phi(c)] = [c^{-1}]$.
We start by extending this proposition to irreducible endomorphisms. For brevity, we assume familiarity with Bestvina-Handel’s argument [@BH92 Section 3]. The main change is we work with periodic indivisible Nielsen paths (of a fixed period) rather than indivisible Nielsen paths. The argument in the final paragraph is different too: unlike Bestvina-Handel’s argument where periodic proper free factors are enough to contradict fully irreducibility, we need to construct a fixed proper free factor system to contradict irreducibility.
\[propStab\] Suppose endomorphism $\phi:F \to F$ is irreducible, it has infinite-order, and there exists $k \ge 1$ and nontrivial conjugacy class $[c]$ of $F$ such that $[\phi^k(c)] = [c]$.
Then there exists an irreducible train track representative $f:\Gamma \to \Gamma$ with exactly one $f$-orbit of (unoriented) periodic indivisible Nielsen paths that make up an $f$-orbit of (unoriented) periodic Nielsen loops that collectively cover each edge of $\Gamma$ twice.
Represent $\phi$ by an expanding irreducible train track map $f:\Gamma \to \Gamma$. Let $\sigma$ be a loop representing $[c]$. By hypothesis, $f^k(\sigma) \simeq \sigma$. Break $\sigma$ into maximal legal segments $\sigma_0, \ldots, \sigma_s$, and note that, since $f$ is a train track, tightening the loop $f^k(\sigma)$ produces a cyclic permutation of the maximal legal segments $\sigma_i, \ldots \sigma_{s+i}$ for some $i$. Then $f^{ks}(\sigma)$ will give the original maximal legal segments $\sigma_0, \ldots, \sigma_s$. In particular, as $f$ is expanding, each segment $\sigma_j$ has a $f^{ks}$-fixed point and $\sigma$ is an $f^{ks}$-Nielsen loop, or equivalently, $\sigma$ is an $f$-periodic Nielsen loop.
The goal is to replace $f$ with another train track representative such that the $f$-orbit of $\sigma$ is a set of (unoriented) loops that collectively cover each edge of $\Gamma$ twice.
For the rest of the proof, a periodic indivisible Nielsen path (piNp) will refer to an indivisible Nielsen path of the fixed $ks$-iterate. It follows from the bounded cancellation lemma that there are finitely many piNps.
The $f$-orbit of $\sigma$ breaks into piNps; just like indivisible Nielsen paths, a piNp $\rho$ can be uniquely written as a concatenation of two legal paths $\rho = \alpha \beta$. By an $f$-orbit of unoriented piNps, we mean the minimal sequence $\{ \rho_0 = \rho, \rho_1 = f(\rho), \ldots, \rho_m = f^m(\rho) \}$ such that $f^{m+1}(\rho) = \rho \text{ or } \bar \rho$ for some piNp $\rho$. If we write each $\rho_i$ in the $f$-orbit of $\rho$ as $\rho_i = \alpha_i \beta_i$, then we get $f(\alpha_{i-1}) = \alpha_i \tau_i$ and $f(\beta_{i-1}) = \bar \tau_i \beta_i$, with the exception at the end where possibly (due to reversal of orientation) $f(\alpha_m) = \bar \beta_0 \tau_0$ and $f(\beta_m) = \bar \tau_0 \bar \alpha_0$. Since $f$ is expanding, at least one of the paths $\tau_i$ is nontrivial.
We will now describe the process of folding an $f$-orbit of piNps. Given an $f$-orbit of piNps, $\{\alpha_0\beta_0, \ldots, \alpha_m\beta_m\}$, we know that at least one of the corresponding $\tau_i$ is nontrivial. Then we can fold at the turn between $\alpha_{i-1}$ and $\beta_{i-1}$. As more than one of the $\tau_i$ may be nontrivial, the full folds take precedence over the partial folds.
We say $f$ is [**stable**]{} if the process of folding its $f$-orbits of piNps can always be done by full folds. If $f$ has no piNps, then it is vacuously stable. Since full folds do not increase the number of vertices in the graph, the process of folding a stable representative produces finitely many [*projective (equivalences) classes*]{} of graphs.
$\phi$ has a stable representative.
Suppose $f$ was not stable. Then after doing some preliminary full folds, $f$ has an $f$-orbit of piNps whose only possible folds are partial folds. Fold this orbit enough times so that every turn ${\bar \alpha_i, \beta_i}$ is at a trivalent vertex. Now apply a homotopy so that the segments $\{\alpha_0\beta_0, \ldots, \alpha_m\beta_m\}$ are isometrically and cyclically permuted. Since $\phi$ is irreducible, these segments form an invariant forest which we can collapse to create a representative with strictly fewer piNps. As there were finitely many piNps to begin with, this process will terminate with a stable representative. This ends the claim.
A stable representative $f:\Gamma \to \Gamma$ has at most one $f$-orbit of piNps.
Assume has at least one $f$-orbit of piNps $\{\rho_i\}_{i=0}^m$. Let $\Gamma$ have a [*Perron-Frobenius eigenmetric*]{} and denote with $\operatorname{vol}(\Gamma)$ the sum of all the edge lengths. Folding an $f$-orbit of piNps $\{\rho_i\}_{i=0}^m$ produces a graph $\Gamma'$ with $\operatorname{vol}(\Gamma') = \operatorname{vol}(\Gamma) - x$ and an $f'$-orbit of piNps $\{\rho_i'\}_{i=0}^m$ of $\Gamma'$ with $\operatorname{vol}(\{\rho_i'\}_{i=0}^m) = \operatorname{vol}(\{\rho_i\}_{i=0}^m) - 2x$ for some $x > 0$; the volume of a collection of paths is the sum of their lengths. Since there are finitely many projective classes of graphs, the graph $\Gamma$ and $f$-orbit $\{\rho_i\}_{i=0}^m$ must satisfy the [**critical equation**]{} $\operatorname{vol}\left(\{\rho_i\}_{i=0}^m\right) = 2 \operatorname{vol}(\Gamma)$. Note that this equation holds for any $f$-orbit of piNps. Fix one such $f$-orbit $\{\rho_i\}_{i=0}^m$ and suppose there were an $f$-orbit $\{r_i\}_{i=0}^n$. If $\{\rho_i\}_{i=0}^m$ and $\{r_i\}_{i=0}^n$ did not share all their illegal turns, then folding $\{\rho_i\}_{i=0}^m$ would eventually decrease $\operatorname{vol}(\Gamma)$ while leaving $\operatorname{vol}(\{r_i\}_{i=0}^n)$ the same. This would break the critical equation for $\{r_i\}_{i=0}^n$, contradicting stability. Therefore, all $f$-orbits of piNps have the same set of illegal turns. Since each fold in a stable representative is a full fold, there cannot be two distinct $f$-orbits of piNps that share the same illegal turns and maintain the critical equation throughout all the folds. This ends the second claim.
Stable representative $f:\Gamma \to \Gamma$ has exactly one $f$-orbit of piNps that make up an $f$-orbit of periodic Nielsen loops that collectively cover each edge of $\Gamma$ twice.
The process of folding an $f$-orbit of piNps preserves periodic Nielsen loops. Since we started with a representative in which $\sigma$ is a periodic Nielsen loop, we know $f$ has a periodic Nielsen loop in $\Gamma$. In particular, the loop splits into piNps and therefore $f$ has exactly one $f$-orbit of piNps by the previous claim. This orbit makes up an $f$-orbit of periodic Nielsen loops containing $\sigma$. Let $\{s_i\}_{i=0}^n$ be a minimal length $f$-orbit of periodic Nielsen loops formed by concatenating paths in the $f$-orbit of piNps. To maintain the critical equation, folding $\{s_i\}_{i=0}^n$ must eventually reduce the lengths of all edges; therefore, every edge of $\Gamma$ appears at least once in $\{s_i\}_{i=0}^n$.
Suppose some edge of $\Gamma$ appeared exactly once in $\{s_i\}_{i=0}^n$, say in $s_n$. Then the loop $s_n$ determines a cyclic free factor $C_n$ of $F$ such that $F = B * C_n$ where all the other loops $s_i~(i \neq n)$ determine elements in $B$. As $\phi$ is injective, $s_1^{\pm1} = [f(s_n)]$ determines a cyclic free factor of $\phi(F) \cap B$; since $s_n$ and $s_1$ can be simultaneously realized as free factors of $\phi(F)$ and $\phi$ is injective, the loops $s_{n-1}$ and $s_n$ determine cyclic free factors $C_{n-1}, C_n$ of $F$ that can be simultaneously realized, i.e., $F = B' * C_{n-1} * C_n$. Note that we use preimages to get the free factors of $F$ since we did not assume $\phi$ was an automorphism. Iterate this process to show $\{s_i\}_{i=0}^n$ determines a $\phi$-fixed proper free factor system $\{C_0, \ldots C_n \}$. This contradicts the irreducibility assumption on $\phi$. Thus every edge of $\Gamma$ appears at least twice in $\{s_i\}_{i=0}^n$ and the critical equation implies every edge appears exactly twice. This concludes the third claim and proof of the proposition.
The proof of the next theorem follows that of Bestvina-Handel [@BH92 Proposition 4.5]. We give an outline with a modification that extends the argument to injective endomorphisms.
\[thmBH2\] Suppose an endomorphism $\phi:F \to F$ is irreducible, it has infinite-order, and there exists $k \ge 1$ and nontrivial conjugacy class $[c]$ of $F$ such that $[\phi^k(c)] = [c]$.
Then $\phi$ is an automorphism induced by a pseudo-Anosov homeomorphism of a surface $\Sigma_g^{b \ge 1}$ that acts transitively on the boundary components.
Apply Proposition \[propStab\] to get an irreducible train track $f:\Gamma \to \Gamma$ with an $f$-orbit of (unoriented) periodic indivisible Nielsen loops that collectively cover each edge of $\Gamma$ twice. Let $b \ge 1$ be the number of these periodic Nielsen loops. For each periodic Nielsen loop, attach an annulus by gluing one end along the loop and call the resulting space $M$; $M$ contains $\Gamma$ as a deformation retract, so $\pi_1(M) \cong F$.
Since $f$ transitively permutes the periodic Nielsen loops (possibly reversing orientation, up to homotopy), the map $f$ extends to a map $g:M \to M$ that transitively permutes the components of $\partial M$ such that $g_* = f_* = \phi$; so $g$ is $\pi_1$-injective. Since the loops collectively covered each edge twice, the space $M$ is a surface except at finitely many singularities. Use the blow-up trick and the irreducibility of $\phi$ to conclude $M$ is a surface.
Thus $g$ is a $\pi_1$-injective map of a surface that transitively permutes the boundary components. Let $D(M)$ be the closed hyperbolic surface obtained by gluing two copies of $M$ along their boundary components. The map $g$ induces a $\pi_1$-injective map $g \cup_\partial g : D(M) \to D(M)$ such that $(g \cup_\partial g)_* = \phi *_\partial \phi$. But closed hyperbolic surfaces have coHopfian fundamental group (classification of surfaces), therefore $\phi *_\partial \phi$ is an automorphism. This implies $\phi$ is an automorphism and the map $g$, a homotopy equivalence, is homotopic to a homeomorphism.
Assume $g$ is a homeomorphism. Any $g$-invariant collection of disjoint essential simple closed curves of $g$ determines a reduction of $\phi$, thus the irreducibility of $\phi$ implies $g$ is an infinite-order irreducible homeomorphism. By Nielsen-Thurston classification, the map $g$ is isotopic to a pseudo-Anosov homeomorphism of $M = \Sigma_g^b$ that acts transitively on the boundary components.
\[irredIWIP\] If $\phi:F \to F$ is an infinite-order irreducible endomorphism that is not fully irreducible, then $\phi$ is induced by a pseudo-Anosov homeomorphism of a surface $\Sigma_g^{b \ge 2}$ that acts transitively on the boundary components.
It follows from Proposition \[invSbgrp\] that for atoroidal endomorphisms, irreducible implies fully irreducible (see also [@DKL15; @JPM]). So $\phi$ has a nontrivial periodic conjugacy class. By Theorem \[thmBH2\], $\phi$ is induced by a pseudo-Anosov homeomorphism of a surface $\Sigma_g^{b \ge 1}$ that acts transitively on the boundary components. If $\phi$ is not fully irreducible, then $b \ge 2$ by Proposition \[eg\].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We discuss the partitioning of a quantum system by subsystem separation through unitary block-diagonalization (SSUB) applied to a Fock operator. For a one-particle Hilbert space, this separation can be formulated in a very general way. Therefore, it can be applied to very different partitionings ranging from those driven by features in the molecular structure (such as a solute surrounded by solvent molecules or an active site in an enzyme) to those that aim at an orbital separation (such as core-valence separation). Our framework embraces recent developments of Manby and Miller as well as older ones of Huzinaga and Cantu. Projector-based embedding is simplified and accelerated by SSUB. Moreover, it directly relates to decoupling approaches for relativistic four-component many-electron theory. For a Fock operator based on the Dirac one-electron Hamiltonian, one would like to separate the so-called positronic (negative-energy) states from the electronic bound and continuum states. The exact two-component (X2C) approach developed for this purpose becomes a special case of the general SSUB framework and may therefore be viewed as a system-environment decoupling approach. Moreover, for SSUB there exists no restriction with respect to the number of subsystems that are generated — in the limit, decoupling of all single-particle states is recovered, which represents exact diagonalization of the problem. The fact that a Fock operator depends on its eigenvectors poses challenges to all system-environment decoupling approaches and is discussed in terms of the SSUB framework. Apart from improved conceptual understanding, these relations bring about technical advances as developments in different fields can immediately cross-fertilize one another. As an important example we discuss the atomic decomposition of the unitary block-diagonalization matrix in X2C-type approaches that can inspire approaches for the efficient partitioning of large total systems based on SSUB.'
author:
- 'Adrian H. Mühlbach'
- Markus Reiher
date: 17 October 2018
title: 'Quantum System Partitioning at the Single-Particle Level'
---
Introduction
============
The quantum mechanical study of isolated molecular systems has been an important endeavor. Examples range from scrutinizing our understanding of fundamental physical theory (as highlighted, for instance, by the high resolution results available for the dihydrogen binding energy [@Cheng18; @PhysRevA.97.060501; @puch18]) to analyzing vast amounts of experimental (gas-phase) data in great detail (examples can be found in astrochemistry [@Barone15; @Puzzarini18] as well as in atmospheric and combustion chemistry [@Glowacki12]). However, the majority of experiments in chemistry considers molecules in some specific environment (in solution, on surfaces, in solid bulk, in enzymes and so forth), which poses huge challenges for their theoretical description.
Naturally, a plethora of approximations has been developed to cope with situations in which a local phenomenon, i.e., one that can be described by studying only a subsystem, is embedded into some environment that more or less strongly interacts with the subsystem. Some of these embedding approaches were driven by chemical and physical insights resting on rather ad hoc theoretical bases of which quantum-mechanics molecular-mechanics (QMMM) coupling [@Warshel76; @Singh86; @Field90; @Lin06; @Senn07b; @Senn07a; @Senn09] is the most prominent example including its sophisticated variants such as polarizable embedding theories [@Olsen10; @Olsen11; @Sneskov11]. Various fragmentation and embedding approaches were conceived to enhance computational efficiency by reducing the number of one-particle basis functions or by fragmenting the system, which also make calculations amenable to massive parallelization; examples can be found in Refs. .
From the more formal point of view of quantum theory, nesting a subsystem into an environment of one or more subsystems requires the adoption of open-system quantum mechanics, [@Breuer02; @Amann11] which in principle can cope with any such situation. For an open quantum system, many-particle basis states defined on a subsystem may not necessarily conserve particle number as they can be combined with states from the environment to produce a total state of, in most practical cases, fixed particle number. The total state may then be expanded in terms of a (tensor) product basis where the double sum runs over indices that refer to subsystem (sub)states and to environment (sub)states. Such a partitioning of a system can be directly exploited to optimize basis states on a subsystem in numerical procedures. The density matrix renormalization group algorithm [@White92; @White93] is an example, where in each iteration step a total many-particle state may be viewed as being decomposed into a product basis of substates defined on a system and an environment of orbitals.
A very special decomposition is the Schmidt decomposition[@Schmidt07; @Schollwoeck11], which restricts the double sum over product states to a single sum by connecting each state on a system to exactly one (specially prepared, e.g., contracted) many-particle state of the environment. It is this decomposition that has prompted Knizia and Chan to define an efficient embedding model called density matrix embedding theory (DMET)[@Knizia12; @Knizia13]. DMET exploits the fact that a potentially small number of relevant system states couples, by virtue of the Schmidt decomposition, to only the same number of states in the environment, no matter how large the latter is. Obviously, the optimization of such environment states might be considered as complicated as solving the full quantum problem for the total system (i.e., for subsystem and environment). To arrive at a practical DMET approach, Knizia and Chan proposed a mean-field approximation for the environment states.[@Knizia12; @Knizia13] The mean-field approximation to the general DMET has been studied in detail by them,[@Knizia12; @Knizia13] by Scuseria and co-workers[@Bulik14; @Bulik14a], and by van Voorhis and co-workers[@Welborn16; @Ricke17].
Mean-field environments had been considered for system-environment partitioning before the introduction of DMET. The motivation for this has always been the observation that a part of a total system may be subject to strong quantum correlations whereas for the rest a mean-field approach can be chosen, which is usually taken to be Kohn–Sham density functional theory (KS-DFT)[@Hohenberg64; @Kohn65]. Within DFT, it is possible to define density-based formulations of a system-environment embedding. [@Gordon72; @Kim74; @Senatore86; @Cortona91; @Wesolowski93; @Cortona94; @Neugebauer05; @Iannuzzi06; @Jacob08; @Fux10; @Elliott10; @Goodpaster10; @Jacob14; @Fornace15; @Wesolowski15; @Ding17] The strongly correlated part of a molecule, i.e., the system, may also be described by an accurate wave-function-theory approach [@Huzinaga71; @Govind99; @Kluener02; @Huang06; @Gomes08; @Huang11; @Manby12; @Hoefener12; @Goodpaster14; @Daday14; @Dresselhaus15; @Hegely16] if deemed necessary to allow for better error control.
Mean-field approximations lend themselves to studying quantum system partitioning at the single-particle level, i.e., at the level of the one-particle equations of motion that describe the dynamics of an electron in a mean-field potential. Obviously, Hartree–Fock and Kohn–Sham equations are the most popular targets for such a decomposition. In this work, we will present a general unitary-transformation-based partitioning approach for single-particle equations. We would like to emphasize, however, that these single-particle equations do not need to be of the mean-field type. Our unitary decoupling approach will apply to any single-particle equation, which could, for instance, be of a multi-configuration self-consistent-field type, in which configuration-interaction state parameters enter the electron-electron interaction at the one-particle level.
We briefly mention that other embedding theories exploit different formulations of the quantum mechanical equations of motion. Examples are the self-energy embedding theory of Zgid that starts from a Green’s function formalism[@Kananenka15; @Lan15; @Lan17], the dynamical mean-field theory of Georges and Kotliar for the description of impurities[@Georges92; @Georges96; @Georges04; @Kotliar06], and work that allows one to nest different quantum formalisms into one another [@Fromager15; @Senjean17; @Senjean18]. Also active-orbital space methods [@Roos80; @Roos80a; @Ruedenberg82; @Olsen88; @Fleig03; @Ivanic03] can be viewed as embedding approaches nesting a set of strongly statically correlated orbitals considered for exact diagonalization into the complementary space of all other less correlated orbitals, as recently exploited by Shiozaki and co-workers in what they call the active space decomposition method[@Parker13; @Parker14; @Parker14a].
Whereas all general open-quantum-system methods operate, as they should, on the many-particle state level, such separations of a system into subsystems can be leveraged by a properly prepared one-particle basis from which the many-particle states are then constructed (either in the usual way by tensor products or in a mean-field sense for KS-DFT). This was recently demonstrated in the work of Manby, Miller, and co-workers on different variants of embedding[@Manby12; @Goodpaster14; @Fornace15] and we come back to their embedding approaches later in this work.
In this work, we consider the separation of a quantum system by a suitable linear combination of one-electron basis states that allows us to provide a separation according to any desired target, which may be defined in terms of an underlying nuclear framework or by exploiting a separation of one-particle states based on some energy criterion (producing, for instance, core-valence separation within an atomic or molecular structure). Our approach is designed to be efficient and generally applicable. It even relates to exact two-component relativistic theories. However, the fact that a Fock operator usually depends on the solution of a one-particle equation (as, in general, the 4-current or the electron density (matrix) is required to represent the interaction) will pose difficulties for all such embedding approaches, but at the same time, facilitates the proposition of suitable approximations. Within our general framework, we will show that approximations developed for exact two-component approaches may cross-fertilize developments in embedding theories by Miller and Manby.
One-Electron Hilbert space {#sec:MF}
==========================
We discuss the partitioning of a system at the level of a one-electron equation and assume that one can construct a Fock matrix for the total system and eventually diagonalize it. Clearly, if the system is very large, this will become a problem. For such cases, it will be necessary to introduce focused methods to construct and diagonalize Fock matrices, which are, for instance, known in plane-wave calculations. QMMM may be viewed as a radical solution that treats part of a system classical so that it does not at all contribute any basis states for the representation of the Fock operator, whereas semiempirical methods[@Thiel88; @Dewar92; @Clark93; @Thiel96; @Thiel98; @Clark00; @Bredow05; @Stewart07; @Lewars10; @Clark11; @Thiel14; @Bredow17; @Husch18] allow one to approximate fairly large Fock matrices.
For the sake of brevity, we focus on mean-field equations and consider the restricted formalism only. An extension to an unrestricted formalism is straightforward. We first give a concise overview on the general formalism to introduce a unified notation and keep our account self-contained.
The Fock matrix $\mathbf{F}$ is the representation of a Fock operator $\hat{F}$ within a basis $B$ consisting of $N_B$ basis functions, $$\begin{aligned}
B = \{\phi_i: i \in [1; N_B]\}.\end{aligned}$$ The Fock matrix $\mathbf{F}$ depends on the density matrix $\mathbf{P}$, $\mathbf{F}{{\left[{\scriptstyle \mathbf{P}}\right]}}$. It is the sum of the one-electron matrix $\mathbf{H}$, which does not depend on the density matrix $\mathbf{P}$, and the two-electron matrix $\mathbf{V} = \mathbf{V}{{\left[{\scriptstyle \mathbf{P}}\right]}}$, $$\begin{aligned}
\mathbf{F}{{\left[{\scriptstyle \mathbf{P}}\right]}} = \mathbf{H} + \mathbf{V}{{\left[{\scriptstyle \mathbf{P}}\right]}},\label{eq:F_MF}\end{aligned}$$ both of which are hermitian. The (closed-shell) density matrix $\mathbf{P}$, $$\begin{aligned}
\mathbf{P} = 2 \, \mathbf{C}_\text{occ} \left(\mathbf{C}_\text{occ}\right)^\mathsf{T}, \label{eq:P_def}\end{aligned}$$ is calculated from the molecular orbitals $\psi_i^\text{occ}$, which occupy the Hartree–Fock (HF) Slater determinant, in the chosen basis, $$\begin{aligned}
\psi_i^\text{occ} = \sum_{k=1}^{N_B} c^{(k)}_{i_\text{occ}} \phi_k\end{aligned}$$ where the $c^{(k)}_{i_\text{occ}}$ are elements of a vector $c_{i_\text{occ}}$ representing the $i$-th occupied orbital in this basis. All vectors $c_{i_\text{occ}}$ enter $\mathbf{C}_\text{occ}$ as column vectors. Their determination requires diagonalization of the Fock matrix $\mathbf{F}$.
The one-electron matrix $\mathbf{H}$ consists of contributions from the kinetic energy of an electron and the potential energy arising from the attractive Coulomb interaction between an electron and the $N_\text{nuc}$ nuclei of the system. We write its matrix elements $H_{ij}$ in Hartree atomic units (used throughout) as $$\begin{aligned}
H_{ij} = -\frac{1}{2}\langle\phi_i|\Delta|\phi_j\rangle - \sum_{I}^{N_\text{nuc}} Z_I \langle\phi_i|\frac{1}{r_I}|\phi_j\rangle,\label{eq:H_MF}\end{aligned}$$ with the Laplace operator in three dimensions $\Delta$, the nuclear charge number of the $I$-th nucleus $Z_I$, and the Euclidean distance $r_I$ between the integration coordinate and the position of nucleus $I$.
In a general framework that considers Hartree–Fock and Kohn–Sham density functional theory on the same algorithmic footing, the two-electron matrix $\mathbf{V}{{\left[{\scriptstyle \mathbf{P}}\right]}}$ may be thought of as consisting of the two-electron Coulomb matrix $\mathbf{J}{{\left[{\scriptstyle \mathbf{P}}\right]}}$, the two-electron exchange matrix $\mathbf{K}{{\left[{\scriptstyle \mathbf{P}}\right]}}$ and the Kohn–Sham exchange-correlation matrix $\mathbf{V}_\text{xc}{{\left[{\scriptstyle \mathbf{P}}\right]}}$, $$\begin{aligned}
\mathbf{V}{{\left[{\scriptstyle \mathbf{P}}\right]}} = \mathbf{J}{{\left[{\scriptstyle \mathbf{P}}\right]}} + \alpha \mathbf{K}{{\left[{\scriptstyle \mathbf{P}}\right]}} + \beta \mathbf{V}_\text{xc}{{\left[{\scriptstyle \mathbf{P}}\right]}}, \label{eq:V_MF}\end{aligned}$$ where $\alpha$ mixes in exact (Hartree–Fock) exchange that needs to be corrected for in $\mathbf{V}_\text{xc}$ (not shown). In Hartree–Fock theory, $\alpha$=1 and $\beta$=0 in Eq. . In Kohn–Sham density functional theory, $\beta$=1 and $\alpha$ controls the amount of exact exchange admixture.
Each component of $\mathbf{V}$ depends on the one-electron density matrix $\mathbf{P}$. The elements of the Coulomb matrix $\mathbf{J}{{\left[{\scriptstyle \mathbf{P}}\right]}}$ and the exchange matrix $\mathbf{K}{{\left[{\scriptstyle \mathbf{P}}\right]}}$ are calculated from the two-electron repulsion integrals evaluated in the chosen basis as $$\begin{aligned}
J_{ij}{{\left[{\scriptstyle \mathbf{P}}\right]}} = \sum^{N_B}_{kl} P_{kl} \langle\phi_i(1)\phi_k(2)|\frac{1}{r_{12}}|\phi_j(1)\phi_l(2)\rangle\label{eq:J_MF}\end{aligned}$$ and $$\begin{aligned}
K_{ij}{{\left[{\scriptstyle \mathbf{P}}\right]}} = -\frac{1}{2}\sum^{N_B}_{kl} P_{kl} \langle\phi_i(1)\phi_k(2)|\frac{1}{r_{12}}|\phi_l(1)\phi_j(2)\rangle,\label{eq:K_MF}\end{aligned}$$ respectively. Here, $r_{12}$ denotes the Euclidean distance between the two integration coordinates. In Kohn–Sham density functional theory [@Hohenberg64; @Kohn65], the exchange-correlation matrix $\mathbf{V}_\text{xc}{{\left[{\scriptstyle \mathbf{P}}\right]}}$ is calculated from the exchange-correlation potential $v_\text{xc}{\left[\rho{{\left[{\scriptstyle \mathbf{P}}\right]}}\right]}$, $$\begin{aligned}
V_{\text{xc},ij}{{\left[{\scriptstyle \mathbf{P}}\right]}} = \langle\phi_i|v_\text{xc}{\left[\rho{{\left[{\scriptstyle \mathbf{P}}\right]}}\right]}|\phi_j\rangle, \label{eq:VXC_MF}\end{aligned}$$ with the electron density $\rho{{\left[{\scriptstyle \mathbf{P}}\right]}}$, $$\begin{aligned}
\rho{{\left[{\scriptstyle \mathbf{P}}\right]}} = \sum_{ij}^{N_B} P_{ij} \phi_i \phi_j, \label{eq:RHO_MF}\end{aligned}$$ where we assume real orbitals.
The electronic energy $E_\text{el}{{\left[{\scriptstyle \mathbf{P}}\right]}}$ is the sum of the Coulomb energy of all nuclei $E_\text{nuc}$, the one-electron energy $E_H{{\left[{\scriptstyle \mathbf{P}}\right]}}$, the Coulomb and exchange energies $E_{JK}{{\left[{\scriptstyle \alpha,\mathbf{P}}\right]}}$, and the exchange-correlation functional $E_\text{xc}{{\left[{\scriptstyle \mathbf{P}}\right]}}$, $$\begin{aligned}
E_\text{el}{{\left[{\scriptstyle \mathbf{P}}\right]}} = E_\text{nuc} + E_H{{\left[{\scriptstyle \mathbf{P}}\right]}} + E_{JK}{{\left[{\scriptstyle \alpha,\mathbf{P}}\right]}} + \beta E_\text{xc}{{\left[{\scriptstyle \mathbf{P}}\right]}}.\end{aligned}$$ The Coulomb energy of all nuclei is calculated as $$\begin{aligned}
E_\text{nuc} = \sum^{N_\text{nuc}}_{i<j} \frac{Z_i Z_j}{r_{ij}}.\end{aligned}$$ The one-electron energy $E_H{{\left[{\scriptstyle \mathbf{P}}\right]}}$ collects the density matrix weighted contributions from the corresponding one-electron matrix $\mathbf{H}$, $$\begin{aligned}
E_H{{\left[{\scriptstyle \mathbf{P}}\right]}} = \sum^{N_B}_{ij} H_{ij} P_{ji} = \text{Tr}{\left(\mathbf{H}\mathbf{P}\right)}.\end{aligned}$$ The energy contribution $E_{JK}{{\left[{\scriptstyle \alpha,\mathbf{P}}\right]}}$ is evaluated from the Coulomb and exchange matrices $\mathbf{J}$ and $\mathbf{K}$, respectively, $$\begin{aligned}
E_{JK}{{\left[{\scriptstyle \mathbf{P}}\right]}} = \frac{1}{2} \sum^{N_B}_{ij} \left(J_{ij} + \alpha K_{ij}\right) P_{ji} = \frac{1}{2} \text{Tr}{\left(\left(\mathbf{J} + \alpha \mathbf{K}\right)\mathbf{P}\right)}.\end{aligned}$$
The self-consistent-field procedure
===================================
To obtain the ground-state energy, the electronic energy is minimized in a self-consistent manner through the self-consistent field (SCF) procedure. The following description will only consider a basic formulation that is required to later discuss all options of self-consistent and approximate embedding schemes.
Quantities that do not depend on the density matrix $\mathbf{P}$ are evaluated before the iterative part of the SCF procedure starts. Apart from the one-electron matrix $\mathbf{H}$ whose elements were defined in Eq. , this is the overlap matrix $\mathbf{S}$, with elements $$\begin{aligned}
S_{ij} = \langle\phi_i|\phi_j\rangle\end{aligned}$$ Depending on hardware constraints, the two-electron repulsion integrals in the atomic-orbital basis, which are required for the evaluation of the Coulomb and exchange matrices in Eqs. and , may be precalculated as well. As a starting point for the minimization, an initial density matrix $\mathbf{P}^{(0)}$ is to be determined, for which various options exist (e.g., superposition of atomic densities, extended Hückel theory guess, basis set projection, or the diagonalization of the one-electron matrix $\mathbf{H}$). Then, in the $n$-th iteration step, the Fock matrix $\mathbf{F}^{(n)} = \mathbf{F}{{\left[{\scriptstyle \mathbf{P}^{(n-1)}}\right]}}$ is calculated from the density matrix of the previous step, $\mathbf{P}^{(n-1)}$.
The generalized eigenvalue problem of the Roothaan–Hall equation[@Roothaan51; @Hall51] is solved to obtain the $n$-th approximation to the eigenvector matrix $\mathbf{C}^{(n)}$ and to the diagonal matrix of eigenvalues $\boldsymbol{\epsilon}^{(n)}$, $$\begin{aligned}
\mathbf{F}^{(n)}\mathbf{C}^{(n)} = \mathbf{S}\mathbf{C}^{(n)}\boldsymbol{\epsilon}^{(n)}. \label{eq:RHE}\end{aligned}$$ This can be achieved by converting it to the ordinary eigenvalue problem $$\begin{aligned}
\mathbf{S}^{-1}\mathbf{F}^{(n)}\mathbf{C}^{(n)} = \mathbf{C}^{(n)}\boldsymbol{\epsilon}^{(n)}.\end{aligned}$$
The molecular orbitals are defined by the eigenvectors in the atomic-orbital basis $B$ chosen, $$\begin{aligned}
\xi_i^{(n)} = \sum_{j=1}^{N_B} c_{i,j}^{(n)} \phi_j = \sum_{j=1}^{N_B} \mathbf{C}_{ij}^{(n)} \phi_j.\end{aligned}$$ The orbital energy of the molecular orbital $\xi_i^{(n)}$ corresponds to the diagonal entry $\epsilon_{ii}^{(n)}$ in the eigenvalue matrix $\boldsymbol{\epsilon}^{(n)}$. The $N_\text{occ}$ molecular orbitals with the lowest orbital energies enter the Hartree–Fock determinant (in the restricted formalism, $N_\text{occ}$ is equal to half the number of electrons $N_\text{el}$). From the matrix $\mathbf{C}_\text{occ}^{(n)}$ of occupied eigenvectors, $$\begin{aligned}
\mathbf{C}^{(n)}_\text{occ} =
\begin{pmatrix}
c_{i_1}^{(n)} & c_{i_2}^{(n)} & \cdots & c_{i_{N_\text{occ}}}^{(n)}
\end{pmatrix},\end{aligned}$$ a new density matrix $\mathbf{P}^{(n)}$, $$\begin{aligned}
\mathbf{P}^{(n)} = 2 \, \mathbf{C}^{(n)}_\text{occ} \left(\mathbf{C}^{(n)}_\text{occ}\right)^\mathsf{T}, \label{eq:P_construction}\end{aligned}$$ is calculated.
As convergence criteria can serve the Frobenius norm of the difference between two consecutive density matrices, $$\begin{aligned}
\delta_\mathbf{P}^{(n)} = \left\|\mathbf{P}^{(n-1)} - \mathbf{P}^{(n)}\right\|, \label{eq:D_P}\end{aligned}$$ and between total electronic energies calculated from them, $$\begin{aligned}
\delta_{E_\text{el}}^{(n)} = \left| E_\text{el}{{\left[{\scriptstyle \mathbf{P}^{(n-1)}}\right]}} - E_\text{el}{{\left[{\scriptstyle \mathbf{P}^{(n)}}\right]}} \right|, \label{eq:D_E}\end{aligned}$$ to be below a predefined threshold. If the procedure has not yet converged, a new iteration step will begin where the new density matrix $\mathbf{P}^{(n)}$ is injected into the calculation of the next Fock matrix $\mathbf{F}^{(n+1)} = \mathbf{F}{{\left[{\scriptstyle \mathbf{P}^{(n)}}\right]}}$.
We implemented all procedures discussed in this paper into a local version of PySCF 1.5b[@Sun18]. The Def2-SVP[@Weigend05] basis set was chosen for all calculations carried out in this work, which are all carried out in the Hartree–Fock approximation (all molecular structures are provided in the supporting information). At the example of formaldehyde, Fig. \[fig:FCPS\] shows the structure of the Fock matrix $\mathbf{F}$, eigenvector matrix $\mathbf{C}$, density matrix $\mathbf{P}$, and overlap matrix $\mathbf{S}$, to which we later compare transformed matrices emerging in the embedding approaches.
{width="\textwidth"}
Fock matrix block-diagonalization {#sec:FBD}
=================================
In this section, we introduce a general decoupling approach that eliminates interactions between subsystems of a molecular system at the matrix level. This then allows for separate treatments of subsystems. Decoupling is achieved through two subsequent matrix transformations: first an orthogonalization of the basis, then a block-diagonalization of the orthogonalized Fock matrix. In comparison with orbital localization schemes, which prepare the one-electron basis and then evaluate the operator matrices in this tailored basis, we manipulate the operator matrices to achieve exact block-diagonalization and then obtain basis states that are localized on either subsystem, but not necessarily localized within this subsystem.
Transforming the basis {#sec:change_of_basis}
----------------------
We first elaborate on the implications of a modified basis for the SCF procedure. As we will consider cases where the transformation would be required in each iteration step, we include the superscript ’($n$)’ for the SCF iteration steps.
A transformation matrix $\mathbf{W}$ transforms the initial basis set $B = \{\phi_i\}$ into the new basis set $\tilde{B} = \{\tilde{\phi}_i\}$, $$\begin{aligned}
\tilde{\phi}_i = \sum_{j=1}^{N_B} W_{ij} \phi_j. \label{eq:basis_transformation}\end{aligned}$$ A matrix representation $\mathbf{A}$ of an operator $\hat{A}$ in the new basis $\tilde{B}$ can then be expressed through the congruent transformation, $$\begin{aligned}
\tilde{\mathbf{A}} = \mathbf{W} \mathbf{A} \mathbf{W}^\mathsf{T}. \label{eq:transformation}\end{aligned}$$ It might be not convenient to evaluate the expressions for the transformed Fock matrix in terms of the transformed density matrix $\tilde{\mathbf{P}}$ in the basis $\tilde{B}$ as it is necessary to introduce additional transformation steps into the SCF iterations:
With a Fock matrix $\tilde{\mathbf{F}}^{(n)}$ in basis $\tilde{B}$ that is evaluated from the transformation of the original Fock matrix $\mathbf{F}^{(n)}$, $$\begin{aligned}
\tilde{\mathbf{F}}^{(n)} = \mathbf{W} \mathbf{F}^{(n)} \mathbf{W}^\mathsf{T},\end{aligned}$$ the Roothaan–Hall equation becomes $$\begin{aligned}
\begin{split}
\mathbf{W} \mathbf{F}^{(n)} \mathbf{W}^\mathsf{T} \mathbf{W}^{-\mathsf{T}} &\mathbf{C}^{(n)} = \\
&\mathbf{W} \mathbf{S} \mathbf{W}^\mathsf{T} \mathbf{W}^{-\mathsf{T}} \mathbf{C}^{(n)} \boldsymbol{\epsilon}^{(n)},\label{eq:RHE_transform}
\end{split}\end{aligned}$$ or expressed in terms of the transformed matrix quantities, $$\begin{aligned}
\tilde{\mathbf{F}}^{(n)} \tilde{\mathbf{C}}^{(n)} = \tilde{\mathbf{S}} \tilde{\mathbf{C}}^{(n)} \boldsymbol{\epsilon}^{(n)},\end{aligned}$$ with $\tilde{\mathbf{C}}^{(n)} = \mathbf{W}^{-\mathsf{T}} \mathbf{C}^{(n)}$. Solving this new generalized eigenvalue equation yields the transformed molecular orbital coefficient matrix $\tilde{\mathbf{C}}^{(n)}$ and the eigenvalue matrix $\boldsymbol{\epsilon}^{(n)}$.
The occupied eigenvector matrix $\tilde{\mathbf{C}}^{(n)}_\text{occ}$ is constructed as before and the transformed density matrix $\tilde{\mathbf{P}}^{(n)}$ is calculated from the transformed eigenvector matrix $\tilde{\mathbf{C}}^{(n)}$, $$\begin{aligned}
\tilde{\mathbf{P}}^{(n)} = 2 \hspace{0.1cm} \tilde{\mathbf{C}}^{(n)}_\text{occ} \left(\tilde{\mathbf{C}}^{(n)}_\text{occ}\right)^\mathsf{T} \label{eq:P_T_construction}\end{aligned}$$ The density matrix $\mathbf{P}^{(n)}$ in the original basis $B$ can be recovered by a back-transformation of $\tilde{\mathbf{P}}^{(n)}$, $$\begin{aligned}
\mathbf{P}^{(n)} = \mathbf{W}^\mathsf{T} \tilde{\mathbf{P}}^{(n)} \mathbf{W}. \label{eq:p_backtransform}\end{aligned}$$ This back-transformation is necessary because it would be inefficient to evaluate a new transformed Fock matrix $\tilde{\mathbf{F}}^{(n+1)}$ from the transformed density matrix $\tilde{\mathbf{P}}^{(n)}$ directly because of the 4-index transformation required for Eqs. and . Hence, the calculation of the new transformed Fock matrix $\tilde{\mathbf{F}}^{(n+1)}$ may be more efficiently achieved by a sequence of backward and forward transformations, $$\begin{aligned}
\tilde{\mathbf{F}}^{(n+1)} = \mathbf{W} \mathbf{F}{{\left[{\scriptstyle \mathbf{W}^\mathsf{T} \tilde{\mathbf{P}}^{(n)} \mathbf{W}}\right]}} \mathbf{W}^\mathsf{T}.\end{aligned}$$
Partitioning of the system {#sec:part}
--------------------------
In most of the following, we consider a subdivision of a molecular system into two parts, denoted *subsystem* ($\mathcal{S}$) and *environment* ($\mathcal{E}$). However, an extension to an arbitrary number of subsystems (including hierarchical subsystem nesting) is also discussed.
In this work, subsystem and environment are chosen according to a partitioning of the atom-centered basis set $B$ into the subsets $B_\mathcal{S}$ and $B_\mathcal{E}$, such that $$\begin{aligned}
B_\mathcal{S} \cup B_\mathcal{E} &= B,\\
B_\mathcal{S} \cap B_\mathcal{E} &= \varnothing.\end{aligned}$$ The subsets $B_\mathcal{S}$ and $B_\mathcal{E}$ consist of $n_\mathcal{S}$ and $n_\mathcal{E}$ basis functions, respectively. It is convenient to order the basis functions $\phi_i$, assigning the index $i$ such that they consist of two contiguous groups, pertaining to the subsystem and the environment. This leads to the following definition for the subsets $B_\mathcal{S}$ and $B_\mathcal{E}$, $$\begin{aligned}
B_\mathcal{S} &= \left\lbrace\phi_i: i \in [1; N_\mathcal{S}]\right\rbrace,\\
B_\mathcal{E} &= \left\lbrace\phi_i: i \in [N_\mathcal{S}+1; N_B]\right\rbrace.
\label{eq:basis_subsets}\end{aligned}$$ This ordering ensures that every matrix representation $\mathbf{A}$ of an operator $\hat{A}$ can be split into distinct subblocks, $$\begin{aligned}
\mathbf{A} = \begin{pmatrix}
\mathbf{A}_{11} & \mathbf{A}_{12}\\
\mathbf{A}_{21} & \mathbf{A}_{22}
\end{pmatrix}.\end{aligned}$$ The subblocks $\mathbf{A}_{11}$, $\mathbf{A}_{12}$, $\mathbf{A}_{21}$, and $\mathbf{A}_{22}$ are of size $N_\mathcal{S}{\times}N_\mathcal{S}$, $N_\mathcal{S}{\times}N_\mathcal{E}$, $N_\mathcal{E}{\times}N_\mathcal{S}$, and $N_\mathcal{E}{\times}N_\mathcal{E}$, respectively. Whenever we encounter block-diagonal matrices where it is possible to assign each block to either the subsystem or the environment, we will denote the diagonal blocks as $\mathbf{A}_\mathcal{S}$ and $\mathbf{A}_\mathcal{E}$, $$\begin{aligned}
\mathbf{A} = \begin{pmatrix}
\mathbf{A}_\mathcal{S} & \mathbf{0}\\
\mathbf{0} & \mathbf{A}_\mathcal{E}
\end{pmatrix}.\end{aligned}$$ It should be noted that whilst we pursue an atom-wise partitioning in most of this work, it is possible to have any kind of partition of the basis set $B$ – even basis functions centered on the same atom can be partitioned into both $B_\mathcal{S}$ and $B_\mathcal{E}$ if deemed useful (consider, for example, core-valence separations). If the ordering of the basis functions differs from the specification above, it will be trivial to construct permutation matrices which will produce a block structure according to this partitioning scheme.
Orthogonalization of the overlap matrix {#sec:FBD_orth}
---------------------------------------
We now transform the generalized eigenvalue problem in Eq. into an ordinary eigenvalue equation. This can be done with Löwdin’s symmetric orthogonalization[@Loewdin70], a congruent transformation, $$\begin{aligned}
\mathbf{X} \mathbf{S} \mathbf{X}^\mathsf{T} = \mathbf{I},\end{aligned}$$ with $$\begin{aligned}
\mathbf{X} = \mathbf{S}^{-\frac{1}{2}}.\end{aligned}$$ As it is well known, we arrive at a transformed eigenvalue equation, $$\begin{aligned}
\mathbf{X} \mathbf{F} \mathbf{X}^\mathsf{T} \mathbf{X}^{-\mathsf{T}} \mathbf{C} = \mathbf{X} \mathbf{S} \mathbf{X}^\mathsf{T} \mathbf{X}^{-\mathsf{T}} \mathbf{C} \boldsymbol{\epsilon},\end{aligned}$$ which can be rewritten as, $$\begin{aligned}
\breve{\mathbf{F}} \breve{\mathbf{C}} &= \breve{\mathbf{C}} \boldsymbol{\epsilon}, \label{eq:f_eig_orth}\end{aligned}$$ with the definition of the transformed Fock matrix $\breve{\mathbf{F}} = \mathbf{X} \mathbf{F} \mathbf{X}^\mathsf{T}$ and its eigenvector matrix $\breve{\mathbf{C}} = \mathbf{S}^{\frac{1}{2}} \mathbf{C}$. The transformed Fock matrix $\breve{\mathbf{F}}$ is hermitian. The diagonal matrix $\boldsymbol{\epsilon}$ containing the eigenvalues is invariant under this transformation. The structure of the matrices in Fig. \[fig:FCPS\] after orthogonalization is depicted in Fig. \[fig:FCPS\_orth\].
{width="\textwidth"}
Block-diagonalization of the Fock matrix {#sec:FBD_bd}
----------------------------------------
At the heart of our decoupling method is a transformation matrix $\mathbf{Q}$ separating subsystem and environment at the matrix level. To this end, we seek to transform the Fock matrix $\breve{\mathbf{F}}$ to a basis $\tilde{B}$ in which it assumes a block-diagonal form, $$\begin{aligned}
\tilde{\mathbf{F}} = \mathbf{Q} \breve{\mathbf{F}} \mathbf{Q}^\mathsf{T} =
\begin{pmatrix}
\tilde{\mathbf{F}}_{\mathcal{S}} & \mathbf{0}\\
\mathbf{0} & \tilde{\mathbf{F}}_{\mathcal{E}}\\
\end{pmatrix}. \label{eq:BD}\end{aligned}$$ Here, the block-diagonalization matrix $\mathbf{Q}$ needs to be unitary. Otherwise, the eigenvalues will not be invariant and the resulting block-diagonal Fock matrix will not be hermitian.
Such a block-diagonalization is not unique. Given that a matrix $\mathbf{Q}$ block-diagonalizes the matrix $\breve{\mathbf{F}}$, then any matrix $\mathbf{D} \mathbf{Q}$, with a unitary block-diagonal matrix $\mathbf{D}$, also transforms $\breve{\mathbf{F}}$ into a block-diagonal form, $$\begin{aligned}
\mathbf{D} \mathbf{Q} \breve{\mathbf{F}} (\mathbf{D} \mathbf{Q})^\mathsf{T}
&= \mathbf{D} \mathbf{Q} \breve{\mathbf{F}} \mathbf{Q}^\mathsf{T} \mathbf{D}^\mathsf{T} \\
&\hspace{-1cm}=
\begin{pmatrix}
\mathbf{D}_{11} & \mathbf{0}\\
\mathbf{0} & \mathbf{D}_{22}\\
\end{pmatrix}
\begin{pmatrix}
\tilde{\mathbf{F}}_{11} & \mathbf{0}\\
\mathbf{0} & \tilde{\mathbf{F}}_{22}\\
\end{pmatrix}
\begin{pmatrix}
\mathbf{D}^\mathsf{T}_{11} & \mathbf{0}\\
\mathbf{0} & \mathbf{D}^\mathsf{T}_{22}\\
\end{pmatrix} \\
&\hspace{-1cm}=
\begin{pmatrix}
\mathbf{D}_{11} \tilde{\mathbf{F}}_{11} \mathbf{D}^\mathsf{T}_{11} & \mathbf{0}\\
\mathbf{0} & \mathbf{D}_{22} \tilde{\mathbf{F}}_{22} \mathbf{D}^\mathsf{T}_{22}\\
\end{pmatrix}.\end{aligned}$$ This means that the transformation matrix $\mathbf{Q}$ can, in principle, only be determined up to a unitary block-diagonal matrix.
In addition to the non-uniqueness of the block-diagonalization matrix $\mathbf{Q}$, there also exist multiple algorithms to determine such a matrix. For example, one may subject the Fock matrix $\breve{\mathbf{F}}$ to subsequent Givens rotations [@Givens58] to eliminate those off-diagonal matrix elements which represent the system-environment coupling. This is essentially equivalent to the Jacobi eigenvalue algorithm[@Jacobi46], targeting only elements in the off-diagonal blocks. However, this iterative procedure is tedious and may have difficulties to achieve off-diagonal blocks to be sufficiently close to zero. Another example is the Householder block-diagonalization algorithm in which block reflectors are applied in an iterative fashion to eliminate off-diagonal blocks.[@Robbe05] However, it suffers from the same shortcomings as the Jacobi algorithm. Also, all of these algorithms do not offer any kind of control and physical insight into how the subsystem and environment are separated from one another.
In an attempt to obtain a well-defined block-diagonalization matrix $\mathbf{Q}$ we require it to fulfill additional constraints. For practical purposes it may be most convenient to have a transformed basis $\tilde{B}$ that is as close as possible to the initial basis $\breve{B}$. This means that the block-diagonalization matrix $\mathbf{Q}$ is as close as possible to the identity matrix, $$\begin{aligned}
\mathbf{Q} = \mathop{\mathrm{argmin}}_\mathbf{Q} \left( \left\| \mathbf{Q} - \mathbf{I} \right\| \right).
\label{eq:Q_cond}\end{aligned}$$ Under these circumstances, matrices will be subject to minimal changes upon such a transformation, leaving their matrix structure mostly intact. Subsequent approximations that are based on such a minimal-effect separation scheme can be expected to feature smallest errors.
In the following, we discuss how such a minimal-effect block-diagonalization matrix $\mathbf{Q}$ can be constructed for which the condition in Eq. has been proven to be fulfilled by Cederbaum et al.[@Cederbaum89]
First, we express the block-diagonalization in Eq. in terms of the eigenvectors $\breve{\mathbf{C}}$ and $\tilde{\mathbf{C}}$, $$\begin{aligned}
\tilde{\mathbf{F}} =
\tilde{\mathbf{C}} \boldsymbol{\epsilon} \tilde{\mathbf{C}}^\mathsf{T} =
\mathbf{Q} \breve{\mathbf{C}} \boldsymbol{\epsilon} \breve{\mathbf{C}}^\mathsf{T} \mathbf{Q}^\mathsf{T},\end{aligned}$$ with $\tilde{\mathbf{C}} = \mathbf{Q} \breve{\mathbf{C}}$. Since the transformed Fock matrix $\tilde{\mathbf{F}}$ is block-diagonal, we may also write the eigenvector matrix $\tilde{\mathbf{C}}$ in a block-diagonal form, $$\begin{aligned}
\tilde{\mathbf{C}} = \mathbf{Q} \breve{\mathbf{C}} =
\begin{pmatrix}
\tilde{\mathbf{C}}_{\mathcal{S}} & \mathbf{0}\\
\mathbf{0} & \tilde{\mathbf{C}}_{\mathcal{E}}\\
\end{pmatrix}.\end{aligned}$$ This means that the first $N_\mathcal{S}$ eigenvectors of $\breve{\mathbf{C}}$ are transformed into a basis $\tilde{B}$ in which they are located entirely on the transformed subsystem basis functions of $\tilde{B}_\mathcal{S}$. Accordingly, the last $N_\mathcal{E}$ eigenvectors of $\breve{\mathbf{C}}$ are transformed such that they are located entirely on the transformed environment basis functions of $\tilde{B}_\mathcal{E}$. This assignment of eigenvectors implies that the eigenvector matrix $\breve{\mathbf{C}}$ may be written in terms of $\breve{\mathbf{C}}_\mathcal{S}$ and $\breve{\mathbf{C}}_\mathcal{E}$, such that $$\begin{aligned}
\breve{\mathbf{C}} =
\begin{pmatrix}
\breve{\mathbf{C}}_\mathcal{S} & \breve{\mathbf{C}}_\mathcal{E}
\end{pmatrix}. \label{eq:C_part}\end{aligned}$$ The partitioning of the basis $\tilde{B}$ into the subsets $\tilde{B}_\mathcal{S}$ and $\tilde{B}_\mathcal{E}$ is defined analogously to Eq. , $$\begin{aligned}
\tilde{B}_\mathcal{S} &=
\left\lbrace\tilde{\phi}_i: i \in [1; N_\mathcal{S}]\right\rbrace,\\
\tilde{B}_\mathcal{E} &=
\left\lbrace\tilde{\phi}_i: i \in [N_\mathcal{S}+1; N_B]\right\rbrace.\end{aligned}$$ Rewriting the block-diagonalization of the Fock matrix in terms of the eigenvectors greatly simplifies the problem of determining the block-diagonalization matrix $\mathbf{Q}$. It also offers an additional layer of control considering the composition of subsystem and environment. Since the order of eigenvectors in an eigenvector matrix is arbitrary, we can freely choose which eigenvectors of $\breve{\mathbf{C}}$ are projected into the subsystem or the environment basis.
To proceed with a unique construction of the block-diagonalization matrix $\mathbf{Q}$, we need to specify how eigenvectors from $\breve{\mathbf{C}}$ are assigned to either the subsystem $\breve{\mathbf{C}}_\mathcal{S}$ or the environment $\breve{\mathbf{C}}_\mathcal{E}$. In this work, eigenvectors are assigned according to a simple localization scheme. For each eigenvector $c_i$ there exists an associated localization function $f_i$ describing by how much the first $N_\mathcal{S}$ basis functions of the orthogonalized basis contribute to the corresponding molecular orbital. The localization function employed in this work is given by, $$\begin{aligned}
f_i = \sum_{j = 1}^{N_\mathcal{S}} \breve{c}_{i,j}^{2}.
\label{eq:cost_function}\end{aligned}$$ It is, of course, possible to construct other localization functions that could be used to assign eigenvectors to either the subsystem or the environment. Then, the eigenvector matrix $\breve{\mathbf{C}}$ is constructed as, $$\begin{aligned}
\breve{\mathbf{C}} =
\begin{pmatrix}
\breve{c}_1 & \breve{c}_2 & \cdots & \breve{c}_{N_B}
\end{pmatrix}.\end{aligned}$$ where the eigenvectors are sorted in a descending order according to their localization function, such that $$\begin{aligned}
f_{1} \geq f_{2} \geq \ldots \geq f_{N_B}.\end{aligned}$$ This eigenvector ordering ensures that the $N_\mathcal{S}$ eigenvectors with the highest localization function are present in $\breve{\mathbf{C}}_\mathcal{S}$ and the remaining eigenvectors are present in $\breve{\mathbf{C}}_\mathcal{E}$.
Now that the eigenvector assignment has been discussed, we can continue with the construction of the unitary block-diagonalization matrix $\mathbf{Q}$. In the approach by Cederbaum et al. [@Cederbaum89], it is constructed as a product of two matrices $\mathbf{Q}_\text{R}$ and $\mathbf{Q}_\text{BD}$, $$\begin{aligned}
\mathbf{Q} = \mathbf{Q}_\text{R} \mathbf{Q}_\text{BD}.\end{aligned}$$ This splits the procedure into two steps. First, matrix $\mathbf{Q}_\text{BD}$ block-diagonalizes the eigenvector matrix $\breve{\mathbf{C}}$. Then, the matrix $\mathbf{Q}_\text{R}$ renormalizes the transformation, guaranteeing that the total block-diagonalization matrix $\mathbf{Q}$ is unitary. The matrix $\mathbf{Q}_\text{BD}$ takes on the form, $$\begin{aligned}
\mathbf{Q}_\text{BD} =
\begin{pmatrix}
\mathbf{I} & -\mathbf{U}^\mathsf{T} \\
\mathbf{U} & \mathbf{I}
\end{pmatrix}.\end{aligned}$$ It block-diagonalizes the eigenvector matrix $\breve{\mathbf{C}}$, $$\begin{aligned}
\mathbf{Q}_\text{BD} \breve{\mathbf{C}} =
\begin{pmatrix}
- \breve{\mathbf{C}}_{11} + \mathbf{U}^\mathsf{T} \breve{\mathbf{C}}_{21} & - \breve{\mathbf{C}}_{12} + \mathbf{U}^\mathsf{T} \breve{\mathbf{C}}_{22} \\
\mathbf{U} \breve{\mathbf{C}}_{11} + \breve{\mathbf{C}}_{21} & \mathbf{U} \breve{\mathbf{C}}_{12} + \breve{\mathbf{C}}_{22}
\end{pmatrix}.\end{aligned}$$ Inspection of the off-diagonal blocks which are supposed to vanish, yields the following solution for $\mathbf{U}$, $$\begin{aligned}
\mathbf{U} =
- \breve{\mathbf{C}}_{21} \breve{\mathbf{C}}_{11}^{-1} =
\left( \breve{\mathbf{C}}_{12} \breve{\mathbf{C}}_{22}^{-1} \right)^\mathsf{T}.
\label{eq:cederbaumU}\end{aligned}$$ Note that this matrix $\mathbf{U}$ can be constructed from either the subsystem eigenvectors $\breve{\mathbf{C}}_\mathcal{S}$ (as $\breve{\mathbf{C}}_{11}$ and $\breve{\mathbf{C}}_{21}$) or the environment eigenvectors $\breve{\mathbf{C}}_\mathcal{E}$ (as $\breve{\mathbf{C}}_{12}$ and $\breve{\mathbf{C}}_{22}$). A proof of this equality can be found in Appendix \[app:u\]. The renormalization matrix $\mathbf{Q}_\text{R}$ is then given by, $$\begin{aligned}
\mathbf{Q}_\text{R}
&=
\left( \mathbf{Q}_\text{BD} \mathbf{Q}_\text{BD}^\mathsf{T} \right)^{-\frac{1}{2}} \label{eq:QR} \\
&=
\begin{pmatrix}
\left( \mathbf{I} + \mathbf{U}^\mathsf{T} \mathbf{U} \right)^{-\frac{1}{2}} & \mathbf{0}
\\
\mathbf{0} & \left( \mathbf{I} + \mathbf{U} \mathbf{U}^\mathsf{T} \right)^{-\frac{1}{2}}
\end{pmatrix}\end{aligned}$$ Finally, the block-diagonalization matrix $\mathbf{Q}$ reads, $$\begin{aligned}
\mathbf{Q} =
\begin{pmatrix}
(\mathbf{I} + \mathbf{U}^\mathsf{T} \mathbf{U})^{-\frac{1}{2}} &
- (\mathbf{I} + \mathbf{U}^\mathsf{T} \mathbf{U})^{-\frac{1}{2}} \mathbf{U}^\mathsf{T} \\
(\mathbf{I} + \mathbf{U} \mathbf{U}^\mathsf{T})^{-\frac{1}{2}} \mathbf{U} &
(\mathbf{I} + \mathbf{U} \mathbf{U}^\mathsf{T})^{-\frac{1}{2}}
\end{pmatrix}. \label{eq:Q}\end{aligned}$$ In the literature concerning this kind of block-diagonalization,[@Cederbaum89; @Sikkema09; @Peng12; @Seino12] there exist multiple notations for the actual form of the block-diagonalization matrix $\mathbf{Q}$. In Appendix \[app:q\] we show that all of these different representations are in fact identical.
Whereas Löwdin’s transformation is a minimal orthogonalization, the Cederbaum scheme represents a minimal transformation in the sense that the new basis is as similar as possible to the old one. It is therefore no surprise that orbital localization schemes have been considered in the literature [@Zilkowski09; @Li14] that are based on a Cederbaum-type transformation, which relates to our approach here, but focuses on the preparation of particular basis states rather than on the block-diagonalization of the Fock matrix. Interestingly, Ref. notes that the Cederbaum scheme can be efficiently evaluated with techniques developed for the exact decoupling of the Dirac Hamiltonian, however without considering further implications on the relation of relativistic exact decoupling to a general embedding approach such as SSUB introduced here.
The total transformation matrix $\mathbf{W}$ transforming the original Fock matrix $\mathbf{F}$ into the block-diagonal form $\tilde{\mathbf{F}}$ then takes the form $$\begin{aligned}
\mathbf{W} = \mathbf{Q} \mathbf{S}^{-\frac{1}{2}}.\end{aligned}$$ Note that this transformation matrix $\mathbf{W}$ is, in general, not unitary. The diagonal blocks $\tilde{\mathbf{F}}_\mathcal{S}$ and $\tilde{\mathbf{F}}_\mathcal{E}$ are given by $$\begin{aligned}
\begin{split}
\tilde{\mathbf{F}}_\mathcal{S} =&
\mathbf{W}_{11} \mathbf{F}_{11} \mathbf{W}_{11}^\mathsf{T} +
\mathbf{W}_{12} \mathbf{F}_{21} \mathbf{W}_{11}^\mathsf{T} \\
&+\mathbf{W}_{11} \mathbf{F}_{12} \mathbf{W}_{12}^\mathsf{T} +
\mathbf{W}_{12} \mathbf{F}_{22} \mathbf{W}_{12}^\mathsf{T},
\end{split}
\label{eq:F_T_S}
\\
\begin{split}
\tilde{\mathbf{F}}_\mathcal{E} =&
\mathbf{W}_{21} \mathbf{F}_{11} \mathbf{W}_{21}^\mathsf{T} +
\mathbf{W}_{22} \mathbf{F}_{21} \mathbf{W}_{21}^\mathsf{T} \\
&+\mathbf{W}_{21} \mathbf{F}_{12} \mathbf{W}_{22}^\mathsf{T} +
\mathbf{W}_{22} \mathbf{F}_{22} \mathbf{W}_{22}^\mathsf{T}.
\end{split}
\label{eq:F_T_E}\end{aligned}$$\
The one-electron matrix $\tilde{\mathbf{H}}$ and the two-electron matrix $\tilde{\mathbf{V}}$ are transformed in the same way as the Fock matrix $\tilde{\mathbf{F}}$. However, these matrices are in general not block-diagonal, only their sum $\tilde{\mathbf{F}}$ is. Fig. \[fig:FCPStrafo\] provides a graphical representation of the transformed matrices from Fig. \[fig:FCPS\].
{width="\textwidth"}
We already note at this stage that the approach by Cederbaum et al. closely relates to relativistic exact two-component methods, which we discuss later in this work. We will also show how the system-environment separation according to this scheme can be generalized toward an arbitrary number of subsystems. Because of these facts, we underline the general applicability of the unitary block-diagonalization approach by assigning the general term ’subsystem separation by unitary block-diagonalization’ or SSUB (to be pronounced ’sub’) to this approach.
Partitioning into an arbitrary number of subsystems {#sec:multsub}
---------------------------------------------------
In the previous section, we considered a separation of a Fock matrix into two subsystems. Starting from one of these subsystems, we may apply another unitary block-diagonalization step, which separates the subsystem into another two subsystems. Clearly, this allows one to split a total system into any number of subsystems. This process is only limited by the finite number of basis functions, which, in the limit, corresponds to the exact diagonalization of the Fock matrix, which has been the starting point for the construction of the unitary transformation matrices.
In the following, we discuss the technicalities of the separation of the system into multiple subsystems, following the formalism elaborated above. The system shall be separated into $k$ subsystems, denoted by $\mathcal{S}_i$ each. Each subsystem consists of $N_{\mathcal{S}_i}$ basis functions. This generalization of the bipartition in section \[sec:part\] leads to the following partitioning of the basis, $$\begin{aligned}
B_{\mathcal{S}_i} = \left\lbrace
\phi_i: i \in [1 + \sum_{j = 1}^{i-1} N_{\mathcal{S}_j}; \sum_{j = 1}^{i} N_{\mathcal{S}_j}]
\right\rbrace.\end{aligned}$$
The eigenvectors in eigenvector matrix $\breve{\mathbf{C}}$ are localized such that they are located on the respective parts of the system. Unfortunately, it is not trivial to assign each eigenvector to a single subsystem when partitioning the system into multiple subsystems. A detailed description of how this was done in this work can be found in Appendix \[app:localization\]. Analogously to the partitioning in Eq. , this means that the eigenvector matrix can be written as, $$\begin{aligned}
\breve{\mathbf{C}} =
\begin{pmatrix}
\breve{\mathbf{C}}_{\mathcal{S}_1} & \breve{\mathbf{C}}_{\mathcal{S}_2} & \cdots & \breve{\mathbf{C}}_{\mathcal{S}_n}
\end{pmatrix},\end{aligned}$$ where each of these eigenvector matrices $\breve{\mathbf{C}}_{\mathcal{S}_i}$ consists of $N_{\mathcal{S}_i}$ eigenvectors.
This ordered eigenvector matrix is then block-diagonalized into $k$ subsystems by a sequential application of a block-diagonalization matrix. The total block-diagonalization matrix $\mathbf{Q}$ can be written as the product of each of these individual transformations, $$\begin{aligned}
\mathbf{Q} = \overset{\curvearrowleft}{\prod^{k-1}_{i=1}} \mathbf{Q}^{(i)} = \mathbf{Q}^{(k-1)} \mathbf{Q}^{(k-2)} \cdots \mathbf{Q}^{(2)} \mathbf{Q}^{(1)}\end{aligned}$$ Here, each block-diagonalization matrix $\mathbf{Q}^{(i)}$ produces the diagonal block $\tilde{\mathbf{C}}_{\mathcal{S}_i}$. For the intermediary eigenvector matrices $\check{\mathbf{C}}^{(i)}$, we can write a recursion relation, $$\begin{aligned}
\check{\mathbf{C}}^{(i + 1)} = \mathbf{Q}^{(i + 1)} \check{\mathbf{C}}^{(i)},\end{aligned}$$ with $\check{\mathbf{C}}^{(0)} = \breve{\mathbf{C}}$ and $\check{\mathbf{C}}^{(k-1)} = \tilde{\mathbf{C}}$. Each intermediary eigenvector matrix $\check{\mathbf{C}}^{(i)}$ can also be written as, $$\begin{aligned}
\tilde{\mathbf{C}}^{(i)} =
\begin{pmatrix}
\tilde{\mathbf{C}}_{\mathcal{S}_1} & & \mathbf{0} & \mathbf{0} & & \mathbf{0} \\
& \ddots & & & & \\
\mathbf{0} & & \tilde{\mathbf{C}}_{\mathcal{S}_i} & \mathbf{0} & & \mathbf{0} \\
\mathbf{0} & & \mathbf{0} & \check{\mathbf{C}}^{(i)}_{\mathcal{S}_{i+1}} & \cdots & \check{\mathbf{C}}^{(i)}_{\mathcal{S}_{k}}
\end{pmatrix}.\end{aligned}$$
From this representation of the intermediary eigenvector matrices, we can construct the transformation matrix $\mathbf{Q}^{(i)}$ as, $$\begin{aligned}
\mathbf{Q}^{(i)} =
\begin{pmatrix}
\mathbf{I}^{(i)} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{Q}^{(i)}_{11} & \mathbf{Q}^{(i)}_{12} \\
\mathbf{0} & \mathbf{Q}^{(i)}_{21} & \mathbf{Q}^{(i)}_{22}
\end{pmatrix},\end{aligned}$$ with the matrix subblocks $\mathbf{Q}^{(i)}_{11}$, $\mathbf{Q}^{(i)}_{12}$, $\mathbf{Q}^{(i)}_{21}$ and $\mathbf{Q}^{(i)}_{22}$ being constructed exactly as in Eq. . The dimension $d^{(i)}$ of the identity matrix subblock $\mathbf{I}^{(i)}$ is given by, $$\begin{aligned}
d^{(i)} = \sum_{j=1}^{i-1} N_{\mathcal{S}_j}.\end{aligned}$$ The matrix $\mathbf{U}^{(i)}$, which is required for the construction of the transformation matrix $\mathbf{Q}^{(i)}$ can be expressed (in analogy to Eq. ) in terms of the intermediary subsystem eigenvector matrix $\check{\mathbf{C}}^{(i-1)}_{\mathcal{S}_{i}}$, $$\begin{aligned}
\mathbf{U}^{(i)} = -
\left( \check{\mathbf{C}}^{(i-1)}_{\mathcal{S}_{i}} \right)_\text{L}
\left( \check{\mathbf{C}}^{(i-1)}_{\mathcal{S}_{i}} \right)_\text{D}^{-1},\end{aligned}$$ where the subscripts ’$\text{D}$’ and ’$\text{L}$’ denote the $N_{\mathcal{S}_i}{\times}N_{\mathcal{S}_i}$ diagonal block and the remaining lower off-diagonal block of the matrix $\check{\mathbf{C}}^{(i-1)}_{\mathcal{S}_{i}}$, respectively.
With this recursive scheme, it is now possible to construct a block-diagonalization matrix $\mathbf{Q}$ which block-diagonalizes the Fock matrix into $k$ subsystems.\
Two such sequential steps were carried out to arrive at a decomposition of formaldehyde into three subsystems: the oxygen atom, the carbon atom, and the two hydrogen atoms (see Fig. \[fig:3subsys\]). As another example, we considered a core-valence separation. The subsystems consist of the eigenvectors that describe the two 1$s$-like molecular orbitals of the carbon and oxygen atoms and of all other orbitals (shown in Fig. \[fig:corevalence\]).
{width="\textwidth"}
{width="\textwidth"}
Exploiting the block-diagonal structure in an approximate SCF procedure {#sec:BD-SCF}
=======================================================================
We now proceed to introduce an approximate SCF scheme that avoids repeatedly solving the eigenvalue problem for the whole system. The scheme is based on the block-diagonalization of the Fock matrix $\tilde{\mathbf{F}}$, leading to an advantageous representation of the eigenvalue problem encountered in the Roothaan–Hall equations.
With the initial density matrix $\mathbf{P}^{(0)}$ an initial Fock matrix $\mathbf{F}^{(1)}$ can be constructed. Following the steps laid out in a previous section concerning block-diagonalization, we obtain the total transformation matrix $\mathbf{W} = \mathbf{Q} \mathbf{S}^{-\frac{1}{2}}$ from this Fock matrix $\mathbf{F}^{(1)}$. In general, this step will be computationally expensive, since it involves (i) solving the eigenvalue problem for the whole system and (ii) diagonalizing the overlap matrix for the calculation of the inverse of its square-root required for $\mathbf{Q}$ in Eq. . Clearly, computational advantages will only emerge for subsequent steps performed for a subsystem only rather than for the total system.
In principle, this SCF procedure follows the same steps as the SCF procedure with a transformed basis in section \[sec:change\_of\_basis\]. However, the computational disadvantages arising from the additional forward and backward transformations in the exact formulation are alleviated by an advantageous representation of the eigenvalue problem, in which the dimension of the problem is significantly reduced:
An approximate SCF procedure
----------------------------
The transformed Fock matrix $\tilde{\mathbf{F}}^{(n)}$ is obtained by transforming the Fock matrix $\mathbf{F}^{(n)}$ with the transformation matrix $\mathbf{W}$. Since the Fock matrix varies throughout the SCF procedure, exact block-diagonalization would require the construction of a block-diagonalization matrix $\mathbf{Q}$ in each iteration step. This requires the solution of the full eigenvalue problem and subsequent matrix inversions and therefore the repeated evaluation of the block-diagonalization matrix $\mathbf{Q}$ would be computationally inefficient. However, if we can construct a sufficiently good initial density matrix (such as one taken from a calculation on a very similar molecular structure as, for instance, encountered in structure optimizations or first-principles molecular dynamics trajectories), we may expect changes in the Fock matrix over the course of the SCF procedure to be comparatively small. Then, the initial transformation matrix $\mathbf{W}$ can be kept over the course of a calculation and still block-diagonalize the resulting Fock matrices sufficiently well. If the initial density matrix $\mathbf{P}^{(0)}$ deviates too much from the converged one, the proposed scheme will fail to find the self-consistent solution.
The diagonal blocks $\tilde{\mathbf{F}}^{(n)}_{\mathcal{S}}$ and $\tilde{\mathbf{F}}^{(n)}_{\mathcal{E}}$ can be evaluated as in Eqs. and , respectively. However, if we assume the environment to change much less than the subsystem, we may leave $\tilde{\mathbf{F}}^{(n)}_{\mathcal{E}}$ constant, $$\begin{aligned}
\tilde{\mathbf{F}}^{(n)}_{\mathcal{E}} = \tilde{\mathbf{F}}^{(1)}_{\mathcal{E}}.\end{aligned}$$ Note that such an assumption may later be lifted in alternating freeze and thaw cycles.
Transforming the Roothaan–Hall equation with the transformation matrix $\mathbf{W} = \mathbf{Q} \mathbf{S}^{-\frac{1}{2}}$ according to Eq. requires us to solve an ordinary eigenvalue problem, $$\begin{aligned}
\tilde{\mathbf{F}}^{(n)} \tilde{\mathbf{C}}^{(n)} &= \tilde{\mathbf{C}}^{(n)} \boldsymbol{\epsilon}^{(n)}.\end{aligned}$$ Since the transformed Fock matrix $\tilde{\mathbf{F}}^{(n)}$ is block-diagonal, the whole eigenvalue problem assumes a block-diagonal form, $$\begin{aligned}
\begin{split}
\begin{pmatrix}
\tilde{\mathbf{F}}^{(n)}_{\mathcal{S}} & \mathbf{0}\\
\mathbf{0} & \tilde{\mathbf{F}}^{(n)}_{\mathcal{E}}
\end{pmatrix}
&\begin{pmatrix}
\tilde{\mathbf{C}}^{(n)}_{\mathcal{S}} & \mathbf{0}\\
\mathbf{0} & \tilde{\mathbf{C}}^{(n)}_{\mathcal{E}}
\end{pmatrix}
=\\
&\begin{pmatrix}
\tilde{\mathbf{C}}^{(n)}_{\mathcal{S}} & \mathbf{0}\\
\mathbf{0} & \tilde{\mathbf{C}}^{(n)}_{\mathcal{E}}
\end{pmatrix}
\begin{pmatrix}
\boldsymbol{\epsilon}^{(n)}_{\mathcal{S}} & \mathbf{0}\\
\mathbf{0} & \boldsymbol{\epsilon}^{(n)}_{\mathcal{E}}
\end{pmatrix}.
\end{split}\end{aligned}$$ This allows us to separate the eigenvalue problem into two smaller eigenvalue problems $$\begin{aligned}
\tilde{\mathbf{F}}^{(n)}_{\mathcal{S}} \tilde{\mathbf{C}}^{(n)}_{\mathcal{S}} = \tilde{\mathbf{C}}^{(n)}_{\mathcal{S}} \boldsymbol{\epsilon}^{(n)}_{\mathcal{S}} \label{eq:f_eig_sub}\end{aligned}$$ and $$\begin{aligned}
\tilde{\mathbf{F}}^{(n)}_{\mathcal{E}} \tilde{\mathbf{C}}^{(n)}_{\mathcal{E}} = \tilde{\mathbf{C}}^{(n)}_{\mathcal{E}} \boldsymbol{\epsilon}^{(n)}_{\mathcal{E}}, \label{eq:f_eig_env}\end{aligned}$$ both of which can be solved separately.\
However, if we set $\tilde{\mathbf{F}}^{(n)}_{\mathcal{E}} = \tilde{\mathbf{F}}^{(1)}_{\mathcal{E}}$, the second eigenvalue problem need not be solved as its solution has already been obtained for the construction of $\mathbf{W}$. The eigenvector matrix $\tilde{\mathbf{C}}^{(1)}_{\mathcal{E}}$ and the eigenvalue matrix $\boldsymbol{\epsilon}^{(1)}_{\mathcal{E}}$ can be stored for later use. Therefore, it is sufficient to solve the subsystem eigenvalue problem in Eq. .
Selecting the occupied eigenvectors follows the standard SCF procedure. $N_\text{occ}$ eigenvectors with the lowest corresponding eigenvalues are occupied with electrons. As the eigenvalues and eigenvectors of the block-diagonal Fock matrix $\tilde{\mathbf{F}}^{(n)}$ can be assigned to either the subsystem or the environment, it is possible to construct the occupied eigenvector matrices $\tilde{\mathbf{C}}^{(n)}_{\mathcal{S},\text{occ}}$ and $\tilde{\mathbf{C}}^{(n)}_{\mathcal{E},\text{occ}}$ from the eigenvector matrix blocks $\tilde{\mathbf{C}}^{(n)}_\mathcal{S}$ and $\tilde{\mathbf{C}}^{(n)}_\mathcal{E}$, respectively.
The transformed density matrix $\tilde{\mathbf{P}}^{(n)}$ is calculated from the occupied eigenvectors according to Eq. . This matrix is block-diagonal since the eigenvector matrix $\tilde{\mathbf{C}}^{(n)}_\text{occ}$ takes a block-diagonal form. In terms of the occupied eigenvector matrices $\tilde{\mathbf{C}}^{(n)}_{\mathcal{S},\text{occ}}$ and $\tilde{\mathbf{C}}^{(n)}_{\mathcal{E},\text{occ}}$, $\tilde{\mathbf{P}}^{(n)}$ reads $$\begin{aligned}
\hspace{0.5cm}
\tilde{\mathbf{P}}^{(n)} = 2 \hspace{0.1cm}
\begin{pmatrix}
\tilde{\mathbf{C}}^{(n)}_{\mathcal{S},\text{occ}} \left(\tilde{\mathbf{C}}^{(n)}_{\mathcal{S},\text{occ}}\right)^\mathsf{T} & \mathbf{0}\\
\mathbf{0} & \tilde{\mathbf{C}}^{(n)}_{\mathcal{E},\text{occ}} \left(\tilde{\mathbf{C}}^{(n)}_{\mathcal{E},\text{occ}}\right)^\mathsf{T}
\end{pmatrix}.\end{aligned}$$
The density matrix $\mathbf{P}^{(n)}$ is obtained through back-transformation of the transformed density matrix $\tilde{\mathbf{P}}^{(n)}$. Since the transformed density matrix $\tilde{\mathbf{P}}^{(n)}$ is block-diagonal, the back-transformation in Eq. can be simplified to yield the density matrix $\mathbf{P}^{(n)}$, $$\begin{aligned}
\mathbf{P}^{(n)}
=
\begin{pmatrix}
\begin{matrix}
\mathbf{W}^\mathsf{T}_{11} \tilde{\mathbf{P}}^{(n)}_\mathcal{S} \mathbf{W}_{11} +\\
\mathbf{W}^\mathsf{T}_{21} \tilde{\mathbf{P}}^{(n)}_\mathcal{E} \mathbf{W}_{21}
\end{matrix}
&
\hspace{0.1cm}
\begin{matrix}
\mathbf{W}^\mathsf{T}_{11} \tilde{\mathbf{P}}^{(n)}_\mathcal{S} \mathbf{W}_{12} +\\
\mathbf{W}^\mathsf{T}_{21} \tilde{\mathbf{P}}^{(n)}_\mathcal{E} \mathbf{W}_{22}
\end{matrix}
\vspace{0.25cm}
\\
\begin{matrix}
\mathbf{W}^\mathsf{T}_{12} \tilde{\mathbf{P}}^{(n)}_\mathcal{S} \mathbf{W}_{11} +\\
\mathbf{W}^\mathsf{T}_{22} \tilde{\mathbf{P}}^{(n)}_\mathcal{E} \mathbf{W}_{21}
\end{matrix}
&
\hspace{0.1cm}
\begin{matrix}
\mathbf{W}^\mathsf{T}_{12} \tilde{\mathbf{P}}^{(n)}_\mathcal{S} \mathbf{W}_{12} +\\
\mathbf{W}^\mathsf{T}_{22} \tilde{\mathbf{P}}^{(n)}_\mathcal{E} \mathbf{W}_{22}
\end{matrix}
\end{pmatrix}. \label{eq:P_bwd_block}\end{aligned}$$ Note that it will be possible to precompute the terms arising from $\tilde{\mathbf{P}}^{(n)}_\mathcal{E}$, if this part of the transformed density matrix is kept constant throughout the SCF procedure (for a frozen environment).
As an example, we chose a simple structure that represents a typical embedding situation, i.e., a solute surrounded by solvent molecules; here, an acetonitrile molecule surrounded by seven water molecules. A standard HF calculation is carried out on the initial equilibrium structure to obtain a converged density matrix. Then, one of the C–H bonds is elongated. For the subsequent structures with the elongated C–H bond, both a standard SCF calculation (as a reference for the exact electronic energy) and a calculation in the approximate SCF procedure exploiting the block-diagonal form of the transformed Fock matrix with a frozen environment of seven water molecules are performed. For the approximate calculation, the initial density matrix is taken from the converged calculation on the equilibrium structure performed initially. This allows us to demonstrate how large a structural perturbation may be in order for the approximate SCF to be sufficiently accurate. The results of the calculations for the perturbed structures are summarized in Fig. \[fig:perturbation\]. Up to an elongation of around 0.3 Å, the error is within chemical accuracy of 1 kcal mol$^{-1}$.
{width="\textwidth"}
Reducing the number of integrals in the two-electron matrix {#sec:V_approx}
-----------------------------------------------------------
The Fock matrix $\mathbf{F}^{(n)} = \mathbf{F}{{\left[{\scriptstyle \mathbf{P}^{(n-1)}}\right]}}$ is calculated from the density matrix $\mathbf{P}^{(n-1)}$. When evaluating the two-electron matrix $\mathbf{V}^{(n)} = \mathbf{V}{{\left[{\scriptstyle \mathbf{P}^{(n-1)}}\right]}}$ it is possible to introduce approximations that make it possible to drastically reduce the number of required two-electron integrals.
If subsystem and environment are well separated (either with respect to molecular structure in real space or in Hilbert space indicated by some energy gap), it will be possible to reduce the number of two-electron integrals required. This may facilitate the evaluation of the two-electron matrix $\mathbf{V}^{(n)} = \mathbf{V}{{\left[{\scriptstyle \mathbf{P}^{(n-1)}}\right]}}$ after the first SCF iteration step.
First, improvements in efficiency may be achieved by leaving the coupling and the pure-environment blocks of the density matrix $\mathbf{P}$ unchanged, $$\begin{aligned}
\mathbf{P}^{(n)} \approx
\begin{pmatrix}
\mathbf{P}_{11}^{(n)} & \mathbf{P}_{12}^{(0)} \\
\mathbf{P}_{21}^{(0)} & \mathbf{P}_{22}^{(0)}
\end{pmatrix}.\end{aligned}$$ If $\mathbf{P}^{(n)}$ is approximated as shown above, it is not necessary to re-evaluate certain terms of the two-electron matrix in each SCF iteration step. For example, the Coulomb matrix $\mathbf{J}^{(n)}$ we may approximate as $$\begin{aligned}
J_{ij}{{\left[{\scriptstyle \mathbf{P}^{(n-1)}}\right]}} \approx \sum^{N_\mathcal{S}}_{kl} P^{(n-1)}_{kl} \langle\phi_i\phi_j|\frac{1}{r_{12}}|\phi_k\phi_l\rangle\ + \delta J_{ij},\end{aligned}$$ with the correction $\delta J_{ij}$ collecting all terms that are not purely subsystem-dependent and may be evaluated only once in the beginning of the SCF procedure, $$\begin{aligned}
\delta J_{ij} = J_{ij}{{\left[{\scriptstyle \mathbf{P}^{(0)}}\right]}} - \sum^{N_\mathcal{S}}_{kl} P^{(0)}_{kl} \langle\phi_i\phi_j|\frac{1}{r_{12}}|\phi_k\phi_l\rangle.\end{aligned}$$ An analogous expression can be written for the exchange matrix $\mathbf{K}^{(n)}$. For the exchange-correlation matrix $\mathbf{V}_\text{xc}$ in a KS-DFT framework, the approximation will take a slightly different form as only parts of the electron density do not need to be reevaluated, $$\begin{aligned}
\rho{{\left[{\scriptstyle \mathbf{P}^{(n-1)}}\right]}} \approx \sum^{N_\mathcal{S}}_{kl} P^{(n-1)}_{kl} \phi_k \phi_l + \delta\rho,\end{aligned}$$ with $$\begin{aligned}
\delta\rho = \rho{{\left[{\scriptstyle \mathbf{P}^{(0)}}\right]}} - \sum^{N_\mathcal{S}}_{kl} P^{(0)}_{kl} \phi_k \phi_l.\end{aligned}$$
In addition, the two-electron matrix $\mathbf{V}^{(n)}$ can be approximated by restricting the iterative re-evaluation to certain matrix elements of $\mathbf{V}^{(n)}$ only, most conveniently those in the $\mathbf{V}_{11}^{(n)}$ block, $$\begin{aligned}
\mathbf{V}^{(n)} \approx
\begin{pmatrix}
\mathbf{V}_{11}^{(n)} & \mathbf{V}_{12}^{(1)} \\
\mathbf{V}_{21}^{(1)} & \mathbf{V}_{22}^{(1)}
\end{pmatrix},\end{aligned}$$ which even further reduces the number of two-electron integrals to be evaluated.
Projector-based embedding theory {#sec:MillerManby}
================================
We are now in a position to discuss the embedding of a subsystem described by a high-level theory within the low-level mean-field method. This allows us to directly relate our work to the recent work by Miller, Manby, and co-workers\cite{} as well as to Huzinaga and Cantu[@Huzinaga71] and Hégely et al.[@Hegely16]. These embedding methods are based on an augmentation of the Fock operator with a projection operators. Therefore, we refer to them as projector-based embedding methods.
For projector-based embedding methods, an initial calculation on the whole system with a low-level DFT method is usually carried out. Quantities from this initial calculation are denoted with a superscript $'(0)'$. Then, the resulting eigenvectors in the eigenvector matrix $\mathbf{C}^{(0)}$ from this initial calculation are localized in such a way that they can be assigned to either the subsystem or the environment. This produces $\mathbf{C}^{(0)}_\mathcal{S}$ and $\mathbf{C}^{(0)}_\mathcal{E}$, which introduce a corresponding separation in the density matrix $\mathbf{P}^{(0)}$, being the direct sum of the two density matrices $\mathbf{P}^{(0)}_\mathcal{S}$ and $\mathbf{P}^{(0)}_\mathcal{E}$, $$\begin{aligned}
\mathbf{P}^{(0)}_\mathcal{S} = 2 \hspace{0.1cm} \mathbf{C}^{(0)}_\mathcal{S,\text{occ}} \left( \mathbf{C}^{(0)}_\mathcal{S,\text{occ}} \right)^\mathsf{T}\end{aligned}$$ and $$\begin{aligned}
\mathbf{P}^{(0)}_\mathcal{E} = 2 \hspace{0.1cm} \mathbf{C}^{(0)}_\mathcal{E,\text{occ}} \left(\mathbf{C}^{(0)}_\mathcal{E,\text{occ}} \right)^\mathsf{T},\end{aligned}$$ which implicitly assigns a number of electrons to the subsystem and the environment.
In the following, we seek the self-consistent solution of a composite Fock matrix $\mathbf{F}_\text{comp}$, mixing the two mean-field methods, $$\begin{aligned}
\mathbf{F}_\text{comp} = \mathbf{F}^\text{Low}{{\left[{\scriptstyle \mathbf{P}^{(0)}}\right]}} + \mathbf{F}^\text{High}{{\left[{\scriptstyle \mathbf{P}_\mathcal{S}}\right]}} - \mathbf{F}^\text{Low}{{\left[{\scriptstyle \mathbf{P}_\mathcal{S}^{(0)}}\right]}}.\end{aligned}$$ In projector-based embedding methods, it is required that the new subsystem eigenvectors in $\mathbf{C}_\mathcal{S}$ are orthogonal to the environment eigenvectors in $\mathbf{C}^{(0)}_\mathcal{E}$ (which are kept constant), $$\begin{aligned}
\mathbf{C}_\mathcal{S}^\mathsf{T} \mathbf{S} \mathbf{C}^{(0)}_\mathcal{E} = \mathbf{0}. \label{eq:orthogonality}\end{aligned}$$ The expression for the electronic energy of such a composite calculation is given by, $$\begin{aligned}
E_\text{comp} = E^\text{Low}{{\left[{\scriptstyle \mathbf{P}^{(0)}}\right]}} + E^\text{High}{{\left[{\scriptstyle \mathbf{P}_\mathcal{S}}\right]}} - E^\text{Low}{{\left[{\scriptstyle \mathbf{P}_\mathcal{S}^{(0)}}\right]}}.\end{aligned}$$
To ensure that orthogonality is preserved, the Fock matrix has to be modified even further such that its eigenvectors include the eigenvectors in $\mathbf{C}^{(0)}_\mathcal{E}$. This is done by augmenting the Fock operator with projection operators. In that case, solving the eigenvalue equation for this new Fock matrix $\mathbf{F}'$ will lead to subsystem eigenvectors $\mathbf{C}_\mathcal{S}$ that are necessarily orthogonal to those of the environment. Such modified Fock matrices $\mathbf{F}'$ were proposed by Miller, Manby, and co-workers[@Manby12; @Goodpaster14], $$\begin{aligned}
\mathbf{F}'_\text{MM} = \mathbf{F}_\text{comp} +
\mu \mathbf{S} \mathbf{C}^{(0)}_\mathcal{E} {\mathbf{C}^{(0)}_\mathcal{E}}^\mathsf{T} \mathbf{S},\end{aligned}$$ and also by Huzinaga and Cantu[@Huzinaga71; @Hegely16], $$\begin{aligned}
\mathbf{F}'_\text{HC} = \mathbf{F}_\text{comp} -
\mathbf{S} \mathbf{C}^{(0)}_\mathcal{E} {\mathbf{C}^{(0)}_\mathcal{E}}^\mathsf{T} \mathbf{F}_\text{comp} -
\mathbf{F}_\text{comp} \mathbf{C}^{(0)}_\mathcal{E} {\mathbf{C}^{(0)}_\mathcal{E}}^\mathsf{T} \mathbf{S}.\end{aligned}$$ Whereas the former is approximate, the latter is exact.\
The factor $\mu$ introduced in the Miller–Manby Fock matrix $\mathbf{F}'_\text{MM}$ can be understood as an energy shift which is applied to the molecular orbitals of the environment. In principle, in the limit of $\mu$ tending to infinity, exact results can be obtained. However, due to numerical issues, it is recommended to use values for $\mu$ of a few thousand Hartree.\
The Huzinaga–Cantu matrix $\mathbf{F}'_\text{HC}$ commutes with the matrix $\mathbf{S} \mathbf{C}^{(0)}_\mathcal{E} {\mathbf{C}^{(0)}_\mathcal{E}}^\mathsf{T} \mathbf{S}$ and therefore shares its eigenvectors $\mathbf{C}^{(0)}_\mathcal{E}$.
Here, we sketch how the additional term $\mu \mathbf{S} \mathbf{C}^{(0)}_\mathcal{E} {\mathbf{C}^{(0)}_\mathcal{E}}^\mathsf{T} \mathbf{S}$ in the Miller–Manby Fock matrix $\mathbf{F}'_\text{MM}$ forces the composite Fock matrix $\mathbf{F}_\text{comp}$ to assume the eigenvectors $\mathbf{C}^{(0)}_\mathcal{E}$. We present an intuitive (but by no means rigorous) way to rationalize how these environment eigenvectors are recovered and how the energetic shift can be understood.
First, we realize that the generalized eigenvalue problem for $\mathbf{F}_\text{comp}$ in the Roothaan–Hall equation in Eq. can also be written as, $$\begin{aligned}
\mathbf{F}_\text{comp} = \mathbf{S} \mathbf{C} \boldsymbol{\epsilon} \mathbf{C}^\mathsf{T} \mathbf{S}.\end{aligned}$$ Splitting this representation of $\mathbf{F}_\text{comp}$ into its subsystem and environment parts yields, $$\begin{aligned}
\mathbf{F}_\text{comp} =
\mathbf{S} \left(\mathbf{C}_\mathcal{S} \boldsymbol{\epsilon}_\mathcal{S} \mathbf{C}^\mathsf{T}_\mathcal{S} + \mathbf{C}_\mathcal{E} \boldsymbol{\epsilon}_\mathcal{E} \mathbf{C}^\mathsf{T}_\mathcal{E}\right) \mathbf{S}.\end{aligned}$$ By adding the term $\mu \mathbf{S} \mathbf{C}^{(0)}_\mathcal{E} {\mathbf{C}^{(0)}_\mathcal{E}}^\mathsf{T} \mathbf{S}$, the modified Fock matrix $\mathbf{F}'_\text{MM}$ is obtained, $$\begin{aligned}
\mathbf{F}'_\text{MM} =
\mathbf{S} \left(
\mathbf{C}_\mathcal{S} \boldsymbol{\epsilon}_\mathcal{S} \mathbf{C}^\mathsf{T}_\mathcal{S} + \mathbf{C}_\mathcal{E} \boldsymbol{\epsilon}_\mathcal{E} \mathbf{C}^\mathsf{T}_\mathcal{E} +
\mu \mathbf{C}^{(0)}_\mathcal{E} {\mathbf{C}^{(0)}_\mathcal{E}}^\mathsf{T}
\right) \mathbf{S}.\end{aligned}$$ Assuming that $\mathbf{C}^{(0)}_\mathcal{E} \approx \mathbf{C}_\mathcal{E}$ and that for all eigenvalues $|\epsilon_{ii}| \ll \mu$ holds, we arrive at $$\begin{aligned}
\mathbf{F}'_\text{MM} \approx
\mathbf{S} \left(
\mathbf{C}_\mathcal{S} \boldsymbol{\epsilon}_\mathcal{S} \mathbf{C}^\mathsf{T}_\mathcal{S} +
\mathbf{C}^{(0)}_\mathcal{E} (\boldsymbol{\epsilon}_\mathcal{E} + \mu \mathbf{I}) {\mathbf{C}^{(0)}_\mathcal{E}}^\mathsf{T}
\right) \mathbf{S}.\end{aligned}$$ Since $\mu$ is such a large number (usually a few orders of magnitude larger than the largest eigenvalues), the eigenvectors $\mathbf{C}^{(0)}_\mathcal{E}$ dominate the matrix $\mathbf{F}'_\text{MM}$. In this representation, it is easy to see why $\mu$ is also referred to as an energy shift applied to the environment eigenvectors.
Interestingly, the projector-based embedding approach can be used in conjunction with our approximate SCF procedure. As noted in section \[sec:BD-SCF\], in the approximate SCF procedure the same transformation matrix $\mathbf{W}$ may be applied throughout the whole procedure. Since the Fock matrix changes during the optimization, the block-diagonalization is not exact, introducing errors. However, this is not the case for the modified Fock matrices $\mathbf{F}'_\text{MM}$ and $\mathbf{F}'_\text{HC}$. Since the environment eigenvectors $\mathbf{C}^{(0)}_\mathcal{E}$ are kept constant, exact block-diagonalization can be achieved for the whole SCF procedure. This implies that once the transformation matrix $\mathbf{W}$ has been calculated from the eigenvectors $\mathbf{C}^{(0)}_\mathcal{S}$ according to Eq. , projector-based embedding methods can be applied, solving the eigenvalue problem for the subsystem only.
We may now compare the Miller–Manby embedding method with the approximate block-diagonalized variant suggested in this section. We chose again formaldehyde and partitioned it by cutting through the C=O double bond. This leaves us with a subsystem containing a single oxygen atom and the environment containing a carbon and two hydrogen atoms. The eigenvectors were localized according to the cost function in Eq. . For the high- and low-level mean-field schemes we simply chose Hartree–Fock and density functional theory with the BP86 functional[@Perdew86; @Becke88], respectively. We varied the parameter $\mu$ between 100 and 1000000 Hartree (values of at least a few thousand Hartree are recommended in Refs. ). The results are shown in Table \[tab:MillerManby\]. For values of $\mu$ above 10000 Hartree, it is possible to obtain the reference energy up to within five decimals with the approximate method.
$\mu$ MM BD-MM
--------- -------------- --------------
100 -113.7078797 -113.7080317
500 -113.7106458 -113.7106769
1000 -113.7109922 -113.7110081
5000 -113.7112694 -113.7112732
10000 -113.7113041 -113.7113063
100000 -113.7113353 -113.7113362
1000000 -113.7113384 -113.7113391
: Electronic energies obtained with the Miller–Manby projector based embedding method and their approximate subsystem SCF counterparts for different values for parameter $\mu$ (in Hartree) for a BP86-in-HF calculation of formaldehyde.\[tab:MillerManby\]
Relation to embedded mean-field theory
======================================
The embedded mean-field theory[@Fornace15] (EMFT) also aims at embedding a subsystem in an environment. EMFT is applied to describe the subsystem with a computationally expensive, high-level density functional within an environment described with a cheaper, lower-level density functional. Most importantly, in comparison to the projector-based embedding schemes, it is not necessary to calculate a self-consistent solution with the low-level method in advance.
Embedded mean-field theory {#sec:EMFT}
--------------------------
The EMFT Fock matrix is a composite Fock matrix combining high- and low-level mean-field theories, $$\begin{aligned}
\mathbf{F}^\text{EMFT}{{\left[{\scriptstyle \mathbf{P}}\right]}} = \mathbf{F}^\text{Low}{{\left[{\scriptstyle \mathbf{P}}\right]}} + \mathbf{F}^\text{High}_{11}{{\left[{\scriptstyle \mathbf{P}_{11}}\right]}} - \mathbf{F}^\text{Low}_{11}{{\left[{\scriptstyle \mathbf{P}_{11}}\right]}}.\label{eq:F_EMFT}\end{aligned}$$ While this seems somewhat reminiscent of the Fock matrix expression in the projector-based embedding approach introduced earlier, it is different. Here, $\mathbf{F}_{11}{{\left[{\scriptstyle \mathbf{P}_{11}}\right]}}$ denotes the $\mathbf{F}_{11}$ block of a Fock matrix where only the subsystem block of the density matrix was used to evaluate all contributions, $$\begin{aligned}
F_{ij} =
\begin{cases}
F_{ij}\left[ \left(
\begin{smallmatrix}
\mathbf{P}_{11} & \mathbf{0} \\
\mathbf{0} & \mathbf{0}
\end{smallmatrix}
\right) \right]
&: i,j \leq N_\mathcal{S} \\
0 &: \rm else
\end{cases}.\end{aligned}$$ This Fock matrix can be used in the SCF procedure as usual to obtain a self-consistent solution. In distinction to the projector-based embedding approaches in the previous section, no initial calculation on the whole system is required. Instead, both high- and low-level mean-field methods are converged to a self-consistent solution simultaneously. The EMFT energy expression takes the composite form, $$\begin{aligned}
E_\text{el}{{\left[{\scriptstyle \mathbf{P}}\right]}} =
E^\text{Low}_\text{el}{{\left[{\scriptstyle \mathbf{P}}\right]}} +
E^\text{High}_\text{el}{{\left[{\scriptstyle \mathbf{P}_{11}}\right]}} -
E^\text{Low}_\text{el}{{\left[{\scriptstyle \mathbf{P}_{11}}\right]}}. \label{eq:E_EMFT}\end{aligned}$$
EMFT with block-orthogonalized partitioning
-------------------------------------------
After the introduction of EMFT, it became apparent that some EMFT calculations exhibit an unphysical collapse of the self-consistent solution.[@Ding17] This has been attributed to a mismatch in the functional forms of the high- and low-level density functionals employed.[@Ding17]
The unphysical collapse manifests itself in a lowering of the electronic energy of several thousand Hartree. Also, electron population analysis reveals that the collapse is accompanied by extraordinarily high electron populations in the subsystem and environment blocks, $\text{Tr}\left(\mathbf{P}_{11}\mathbf{S}_{11}\right)$ and $\text{Tr}\left(\mathbf{P}_{22}\mathbf{S}_{22}\right)$, respectively. Since the total number of electrons is constant, large negative populations in the coupling blocks $\text{Tr}\left(\mathbf{P}_{12}\mathbf{S}_{21}\right)$ and $\text{Tr}\left(\mathbf{P}_{21}\mathbf{S}_{12}\right)$ are generated. As a consequence, other observables are also affected and can be wrong by several orders of magnitude (consider, for example, the dipole moment).
To prevent this collapse of the self-consistent EMFT solution it was modified to operate on a different partitioning of the system.[@Ding17] Instead of partitioning the system based on atomic orbitals, the scheme is applied in a block-orthogonalized basis (BOEMFT). This means that all matrix operators must be transformed accordingly. The transformation matrix $\mathbf{W}$, which block-orthogonalizes the atomic orbital basis, is given by[@Ding17] $$\begin{aligned}
\mathbf{W} =
\begin{pmatrix}
\mathbf{I} & \mathbf{0}\\
- \mathbf{S}_{21} \mathbf{S}_{11}^{-1} & \mathbf{I}
\end{pmatrix}.\end{aligned}$$ Transforming a basis $B$ into the basis $\tilde{B}$ with this transformation matrix $\mathbf{W}$ according to Eq. leaves the basis functions of the subsystem invariant. Therefore, the subsystem subblock $\mathbf{A}_{11}$ is equal to the transformed subsystem block $\tilde{\mathbf{A}}_{11}$ of the transformed matrix $\tilde{\mathbf{A}}$.
Since the transformation is a block-orthogonalization, the transformed overlap matrix $\tilde{\mathbf{S}}$ is block-diagonal, $$\begin{aligned}
\tilde{\mathbf{S}} =
\begin{pmatrix}
\mathbf{S}_{11} & \mathbf{0}\\
\mathbf{0} & \tilde{\mathbf{S}}_{22}
\end{pmatrix},\end{aligned}$$ with $$\begin{aligned}
\tilde{\mathbf{S}}_{22} = \mathbf{S}_{22} - \mathbf{S}_{21} \mathbf{S}_{11} \mathbf{S}_{12}.\end{aligned}$$ In analogy to the EMFT Fock matrix in Eq. , the transformed BOEMFT Fock matrix $\tilde{\mathbf{F}}^\text{BOEMFT}$ is constructed according to $$\begin{aligned}
\tilde{\mathbf{F}}^\text{BOEMFT}{{\left[{\scriptstyle \mathbf{P}}\right]}} = \tilde{\mathbf{F}}^\text{Low}{{\left[{\scriptstyle \mathbf{P}}\right]}} + \tilde{\mathbf{F}}^\text{High}_{11}{{\left[{\scriptstyle \tilde{\mathbf{P}}_{11}}\right]}} - \tilde{\mathbf{F}}^\text{Low}_{11}{{\left[{\scriptstyle \tilde{\mathbf{P}}_{11}}\right]}}.\label{eq:F_boemft}\end{aligned}$$ Instead of the subblock $\mathbf{P}_{11}$ of the density matrix, the transformed density matrix subblock $\tilde{\mathbf{P}}_{11}$ is used. This is an approximation, producing a Fock matrix evaluated for unphysical subsystem electron densities.
The BOEMFT idea seems to resemble our approximate SCF procedure, with a basis transformation partitioning subsystem and environment. However, it is fundamentally different. In BOEMFT the transformation is only applied to prevent the unphysical collapse of the self-consistent field procedure, whereas in our case it is the decisive starting point.
Relation to relativistic decoupling theories
============================================
The block-diagonalization of Dirac-based Fock operators has been a desire for physical, formal, and numerical reasons as one wants to decouple electronic bound states from the negative-energy (continuum) states in order to arrive at variational electrons-only Hamiltonians suitable for applications in molecular physics and chemistry.[@Reiher09]
SSUB can be directly related to this block-diagonalization of the one-electron Dirac operator[@Heully86], which is very efficiently formulated in basis-set representation. In 2005, Jensen [@Jensen05] proposed a scheme to decouple the negative-energy states (sometimes called ’positronic states’) of the Dirac operator that avoids the involved algebra of Douglas–Kroll–Hess transformations.[@Hess86; @wolf02b; @reih04] This work was driven by the desire of rewriting the Dirac Hamiltonian for electrons-only problems without introducing approximations [@bary97; @bary01; @bary02; @fila03; @fila03b; @reih04; @reih04b; @reih06]. Jensen started from a free-particle Foldy–Wouthuysen transformed Dirac Hamiltonian, which in Douglas–Kroll–Hess theory is the mandatory first step [@reih04]. However, his central idea then was to construct the unitary matrix for the block-diagonalization of the four-component Dirac-based Fock operator from its eigenvectors, $$\begin{aligned}
\label{x2c}
\mathbf{U}^\text{X2C} = - \mathbf{C}_\text{SL} \mathbf{C}_\text{LL}^{-1},\end{aligned}$$ (where $S$ and $L$ refer to the small- and large-components of the molecular 4-spinors, respectively). It was then realized that the free-particle Foldy–Wouthuysen transformation can be skipped and Eq. (\[x2c\]) can be applied directly to the Dirac-based Fock operator [@Kutzelnigg05; @Ilias07; @Saue11] (written in terms of Dyall’s modified Dirac equation [@dyal97]). This approach was later called exact two-component (X2C) decoupling.[@Kutzelnigg05; @Liu06; @Kutzelnigg06; @Ilias07; @Liu07; @Sikkema09; @Saue11; @Peng12a]
As such, the X2C approach is identical to the block-diagonalization approach used by Cederbaum et al. [@Cederbaum89] earlier, but in a different context (cf. Eq. ). Caution is advised when comparing different notations for the block-diagonalization matrices of Eq. with those encountered in the literature.[@Cederbaum89; @Sikkema09; @Peng12; @Seino12] At first glance, all of them seem to be slightly different, but they are, in fact, all equivalent.
X2C is usually viewed as a way to eliminate negative-energy states that are separated from the ’electronic’ states by twice the electron rest energy per electron in the system. This energy gap is huge because it depends, according to Einstein, on the squared speed of light times the rest mass of an electron. The reduction of 4-spinors to 2-spinors essentially requires the elimination of all small-component basis functions. The contribution of these basis functions to ’electronic’, i.e., positive-energy molecular 4-spinors are usually atom centered and mostly conserved in chemical reactions.[@Visscher97] Accordingly, one is not surprised that, for physical reasons, X2C works so well at producing an effective electrons-only Hamiltonian for 2-component calculations. However, in view of SSUB this separation of electronic and positronic states is nothing but a system-environment embedding, in which the effect of the positronic states is folded into the one-electron two-component X2C Fock operator. Hence, no energy criteria need to be invoked to justify the separation of the negative-energy states, they are only required to identify the one-particle eigenstates whose spinor energies are smaller than twice the rest energy of an electron.
In fact, our analysis of SSUB shifts the focus from the physical picture to the actual mechanism in one-particle Hilbert space, which allows us to better understand the decisive approximations that are in operation in practical X2C implementations. SSUB applied to a four-component Dirac-based Fock operator produces the X2C Fock operator, but requires all eigenvectors of the original four-component operator. This implies that the solution must already be known for the construction of a unitarily transformed operator (note the relation to the formal projection operators built from the eigenstates of the Dirac Hamiltonian proposed by Mittleman [@mitt72; @mitt81] and Sucher [@such80]; in particular, see Ref. ). In relativistic quantum chemistry this procedure is nevertheless advantageous as subsequent electron-correlation methods benefit largely from the elimination of the negative-energy spinor states in the (preparatory) four-index transformation, which also holds true for general embedding schemes based on SSUB-like ideas.
Approximate approaches are usually applied that restrict and/or model the contributions to the potential. Various such approximations were proposed to obtain approximate eigenvectors for the construction of the unitary matrix in Eq. . The most popular one, which is also applied in standard applications of sequential Douglas–Kroll–Hess decoupling transformations[@Hess86], is the complete neglect of all electron-electron interaction terms, which alleviates the problem of obtaining a transformed form of these terms. In other words, the iterative construction of the unitary matrix discussed in the context of the modified SCF scheme in section \[sec:BD-SCF\] is then avoided. The unitary transformation is then constructed from eigenvectors obtained by diagonalizing an approximate Roothaan–Hall equation with the external electron-nuclei potential as the only potential. Naturally, attempts were made to improve on this restricted model by introducing mean-field potentials which affect the eigenvectors and, hence, the unitary block-diagonalizing matrix.[@hess96; @schi98b; @Sikkema09; @Autschbach12] As such, these approximate and popular versions of the X2C approach fall well into the class of approximate solutions that we discussed in section \[sec:BD-SCF\] above. In Fig. \[fig:coreapproxU\], we demonstrate how relying only on the one-electron contributions of Eq. , i.e., neglecting all other potential-energy contributions to the Fock operator, affects the accuracy of the decoupling in general embedding schemes. In comparison to the relativistic case, we see that this approximation does not work very well. In the relativistic case, the eigenvectors are mainly dominated by the one-electron contributions (because of the huge energy gap separating positive- and negative-energy states). In mean-field theory, this is not the case. The eigenvectors of the eigenvector matrix $\mathbf{H}$ and the Fock matrix $\mathbf{F}$ are fundamentally different. For this reason, the approximation fails when applied to the general case, where no large energy gap sustains it.
{width="\textwidth"}
Approximations for huge single-particle spaces
==============================================
We have mentioned in the beginning that we will not address the issue of how to construct large Fock matrices that can then be subjected to SSUB. However, given that one can construct very large Fock matrices (consider, for example, those in semi-empirical methods applied to molecular systems with thousands of atoms), the issue of diagonalizing them will be pressing. Whereas one may apply subspace diagonalization to such large Fock matrices (e.g., Lanczos or Davidson) and construct an approximate unitary matrix of reduced dimension, we may as well exploit approaches developed to cope with such situations for X2C approaches.
The preparation of the X2C operator for large molecules has prompted the development of approximate decoupling schemes at the level of the Hamiltonian itself.[@Peralta04; @Peralta05; @Thar09] A more accurate approximation is obtained at the level of the unitary transformation itself, which decomposes it into atomic blocks,[@Peng12; @Seino12] $$\begin{aligned}
\mathbf{Q} = \bigoplus_I \mathbf{Q}_{II} , \label{eq:DLU}\end{aligned}$$ where $I$ denotes a subsystem (typically an atom) for which one-particle eigenstates have been determined. which was called *diagonal local approximation to the unitary decoupling transformation* (DLU) in Ref. and *local unitary transformation* (LUT) in Ref. .
Local SSUB
----------
Here, we introduce the concept of an approximative local construction of the block-diagonalization matrix $\mathbf{Q}$, similar to the DLU and LUT schemes. Given the fact that two names, DLU and LUT, are in use in relativistic quantum chemistry for the same idea and that this idea is important in the more general context of SSUB we may propose to call it L-SSUB for local approximation to SSUB.
### Fragmenting the Basis
The local construction of the block-diagonalization matrix $\mathbf{Q}$ is based on the partitioning of the basis $B$ into $k$ fragment bases $B^{\mathcal{F}_i}$, such that $$\begin{aligned}
B = \bigcup\limits_{i=1}^{k} B^{\mathcal{F}_i},\end{aligned}$$ with $$\begin{aligned}
B^{\mathcal{F}_i} \cap B^{\mathcal{F}_j} &= \varnothing.\end{aligned}$$ Each fragment basis $B^{\mathcal{F}_i}$ consists of $N^{\mathcal{F}_i}$ basis functions. It can be partitioned into $B^{\mathcal{F}_i}_\mathcal{S}$ and $B^{\mathcal{F}_i}_\mathcal{E}$, both of which are subsets of $B_\mathcal{S}$ and $B_\mathcal{E}$, respectively, $$\begin{aligned}
B^{\mathcal{F}_i}_\mathcal{S} &= B^{\mathcal{F}_i} \cap B_\mathcal{S},\\
B^{\mathcal{F}_i}_\mathcal{E} &= B^{\mathcal{F}_i} \cap B_\mathcal{E}.\end{aligned}$$ The fragment basis should be chosen such that it captures the dominant interactions between subsystem and environment.
### Constructing the local block-diagonalization matrix
A fragment Fock matrix $\mathbf{F}^{\mathcal{F}_i}$ is constructed from a converged mean-field calculation employing the fragment basis $B^{\mathcal{F}_i}$ only. For this mean-field calculation, a modified Fock operator may be applied, considering a subset of nuclei only, not taking those nuclei into account on which the basis functions are not located. This fragment Fock matrix is then block-diagonalized with the block-diagonalization matrix $\mathbf{Q}^{\mathcal{F}_i}$ as outlined in section \[sec:FBD\]. If a fragment basis $B^{\mathcal{F}_i}$ consists exclusively of subsystem or environment basis functions, the matrix $\mathbf{Q}^{\mathcal{F}_i}$ becomes the identity matrix.
With all fragment block-diagonalization matrices $\mathbf{Q}^{\mathcal{F}_i}$, the local block-diagonalization matrix $\mathbf{Q}^\mathcal{F}$ can be constructed in analogy to Eq. , $$\begin{aligned}
\mathbf{Q}^\mathcal{F} = \bigoplus_{i=1}^{k} \mathbf{Q}^{\mathcal{F}_i}.\end{aligned}$$ However, this local block-diagonalization matrix $\mathbf{Q}^\mathcal{F}$ cannot be applied to the Fock matrix $\mathbf{F}$ directly.
The matrix $\mathbf{Q}^\mathcal{F}$, is implicitly set up in a basis in which the order of basis functions is permuted. Let $\hat{O}$ be an operator that reveals the order of a set in terms of its subsets. While the basis order of the basis $B$, in which the Fock matrix $\mathbf{F}$ is represented, is given by $$\begin{aligned}
\hat{O} B = \left[ B_\mathcal{S}, B_\mathcal{E} \right],\end{aligned}$$ the fragmented basis $B^\mathcal{F}$ has the following order, $$\begin{aligned}
\hat{O} B^\mathcal{F} = \left[
B^{\mathcal{F}_1}_\mathcal{S}, B^{\mathcal{F}_1}_\mathcal{E},
B^{\mathcal{F}_2}_\mathcal{S}, B^{\mathcal{F}_2}_\mathcal{E},
\cdots,
B^{\mathcal{F}_k}_\mathcal{S}, B^{\mathcal{F}_k}_\mathcal{E}
\right].\end{aligned}$$ While it seems to be complicated to convert the Fock matrix $\mathbf{F}$ to its fragment basis representation, it can actually be done with a rather trivial permutation. Therefore, in order to block-diagonalize the Fock matrix $\mathbf{F}$ with the local block-diagonalization matrix $\mathbf{Q}^\mathcal{F}$, we first need to transform the Fock matrix $\mathbf{F}$ into the fragment basis representation with the permutation matrix $\mathbf{K}$. Afterwards, the block-diagonal Fock matrix $\tilde{\mathbf{F}}$ is recovered in the original basis $B$ by an inverse permutation, $$\begin{aligned}
\tilde{\mathbf{F}} = \mathbf{K}^\mathsf{T} \mathbf{Q}^\mathcal{F} \mathbf{K} \mathbf{F} \mathbf{K}^\mathsf{T} \left(\mathbf{Q}^\mathcal{F}\right)^\mathsf{T} \mathbf{K}.\end{aligned}$$ This implies that the total block-diagonalization matrix $\mathbf{Q}$ in the basis $B$ can be constructed as, $$\begin{aligned}
\mathbf{Q} = \mathbf{K}^\mathsf{T} \mathbf{Q}^\mathcal{F} \mathbf{K},\end{aligned}$$ which block-diagonalizes the Fock matrix directly.
### Quantifying approximate block-diagonalizations
Since the local block-diagonalization is only approximate, we need to introduce some sort of measure that allows us to analyze the quality of the resulting block-diagonalization. A naive approach to this is to simply inspect the off-diagonal blocks of the approximately block-diagonal Fock matrix $\tilde{\mathbf{F}}'$. However, such a measure would depend on the nature of the particular system since the magnitude of the matrix elements in the Fock matrix depend on the potentials in the Fock operator. Here, we choose a diagonostic that has a more universal scope. It is based on a comparison between the eigenvectors of the approximately block-diagonalized Fock matrix $\tilde{\mathbf{F}}'$ and the exact block-diagonalized Fock matrix $\tilde{\mathbf{F}}$. I.e., the measure $d_i$ quantifies the similarity of the eigenvectors $\tilde{c}_i$ and $\tilde{c}'_i$, $$\begin{aligned}
d_i = \left| \tilde{c}_i^\mathsf{T} \tilde{c}'_i \right| ,\end{aligned}$$ which is 1 for identical one-particle states. To quantify the adequateness of a block-diagonalization for the whole system, we introduce the measure $D$, which is simply the average of all measures $d_i$ over all eigenvectors, $$\begin{aligned}
D = N_B^{-1} \sum_{i=1}^{N_B} d_i.\end{aligned}$$ The closer this measure $D$ is to $1$, the higher the accuracy of the approximate block-diagonalization.
Example I: Structural separation
--------------------------------
Results of the local SSUB scheme for the formaldehyde molecule in which the oxygen atom is separated from the other atoms are shown in Fig. \[fig:local\_ssub\]. The fragment basis for which the block-diagonalization matrix was determined consists solely of the basis functions centered on the oxygen and the carbon atom.
While this block-diagonalization is not exact, the decoupling scheme appears to decouple the most significant interactions and yields a Fock matrix $\tilde{\mathbf{F}}$ that for the most part is block-diagonal. However, the associated eigenvector matrix $\tilde{\mathbf{C}}$ and density matrix $\tilde{\mathbf{P}}$ show that there are interactions that could not be decoupled by this approach. This observation is also supported by a relatively low measure $D~=~0.9222$. For certain eigenvectors, the measure $d$ is even below $0.7$, which indicates that the separation is not perfect.
{width="\textwidth"}
Example II: Core-Valence Separation
-----------------------------------
Results of the local SSUB scheme for a core-valence separation applied to the formaldehyde molecule, in which the carbon and oxygen 1$s$-like molecular orbitals are separated from all other orbitals, are shown in Fig. \[fig:core\_valence\_local\]. The basis was split into three fragment bases. The first fragment basis consisted of the basis functions centered on the oxygen atom, the second one of the basis functions centered on the carbon atom. The remaining hydrogen basis functions constitute the third fragment basis. In this example, the transformed Fock matrix $\tilde{\mathbf{F}}$ shows a perturbed block-diagonal structure. However, the relatively high measure $D~=~0.9998$ implies that the separation was in fact successful.
{width="\textwidth"}
Conclusions
===========
In this work, we described a block-diagonalization approach directed toward an arbitrary separation of a single-particle basis to define subsystems in a very general way, which we call subsystem separation by unitary block-diagonalization (SSUB). The approach requires one calculation on the full system and as such it shares this disadvantage with embedding theories proposed by Manby and Miller and also with the relativistic exact two-component approaches. However, the disadvantage is made up for by exploiting computational efficiency for a subsystem in computational protocols that build upon a single-determinant solution. For instance, these subsequent steps of a computational protocol may involve a four-index transformation from the atomic orbital to a molecular orbital basis before a correlation calculation is considered. The latter is then significantly simplified for the subsystem (and usually would not have been possible for the total system, which is one reason for adopting an embedding approach).
The general nature of SSUB allows us to view the block-orthogonalization correction of EMFT from a more general perspective. Moreover, projector-based embedding can be simplified significantly without compromising the inherent exactness of the method in the limit for large energy-shift parameter $\mu$. It also relates the field of embedding to very different ones such as electron-positron separation in relativistic quantum chemistry and purely electronic partitionings such as core-valence separation. As such, SSUB can be viewed as a framework for studying quantum system partitioning at the single-particle level in a general way.
We emphasize that a system separation by SSUB allows one to proceed with further calculations on the separated parts of the system. In particular, it is possible to consider more accurate and, hence, computationally more expensive ab initio calculations for a subsystem in the same spirit as in projector embedding and X2C theories. Apart from this, one may exploit SSUB to accelerate standard protocols (e.g., for the solution of the SCF general eigenvalue problem). In future work, we will therefore investigate a change of parameters for which the single-particle equations are formulated. In particular, for accelerating calculations on a subsystem, a change of the nuclear framework — as it occurs in structure optimizations, first-principles molecular dynamics, and interactive reactivity studies — represents such a change of parameters. This will require a thorough study of the transferability of the system-separating matrix transformations in order to assess when one may exploit a system separation, which has been obtained for some configuration, for related configurations. Moreover, a generalization to time-dependent quantum dynamics should also be promising.
We gratefully acknowledge financial support by ETH Zurich (Grant Number ETH-46 16-2). Part of this work was presented at the ICQC 2018 in Menton, France, in June 2018. We thank Professor Trond Saue for remarks on the historical development of Jensen’s original block-diagonalization idea, mean-field spin–orbit Hamiltonians, and on two orbital localization schemes related to SSUB.
Note concerning the construction of $\mathbf{U}$ {#app:u}
================================================
Here, we show that the equality in Eq. , $$\begin{aligned}
\label{A1}
- \breve{\mathbf{C}}_{21} \breve{\mathbf{C}}_{11}^{-1} = \left( \breve{\mathbf{C}}_{12} \breve{\mathbf{C}}_{22}^{-1} \right)^\mathsf{T},\end{aligned}$$ holds by considering the orthogonality of the eigenvector matrix $\breve{\mathbf{C}}$, $$\begin{aligned}
\mathbf{I} = \breve{\mathbf{C}}^\mathsf{T} \breve{\mathbf{C}}.\end{aligned}$$ This implies that subsystem and environment eigenvectors are orthogonal, leading to the expression, $$\begin{aligned}
\mathbf{0}
&= \breve{\mathbf{C}}_\mathcal{E}^\mathsf{T} \breve{\mathbf{C}}_\mathcal{S}\\
&= \breve{\mathbf{C}}_{12}^\mathsf{T} \breve{\mathbf{C}}_{11} + \breve{\mathbf{C}}_{22}^\mathsf{T} \breve{\mathbf{C}}_{21} \\
&= \breve{\mathbf{C}}_{22}^{-\mathsf{T}} \breve{\mathbf{C}}_{12}^\mathsf{T} + \breve{\mathbf{C}}_{21} \breve{\mathbf{C}}_{11}^{-1} \\
&= \left( \breve{\mathbf{C}}_{12} \breve{\mathbf{C}}_{22}^{-1} \right)^\mathsf{T} + \breve{\mathbf{C}}_{21} \breve{\mathbf{C}}_{11}^{-1},\end{aligned}$$ from which Eq. immediately follows.
Representation of the block-diagonalization matrix $\mathbf{Q}$ {#app:q}
===============================================================
In the literature,[@Cederbaum89; @Sikkema09; @Peng12; @Seino12] there exist multiple representations of the block-diagonalization matrix $\mathbf{Q}$. These representations are very similar to the expression in Eq. and it is trivial to show that they achieve the exact same block-diagonalization. The different representations arise from the following different definitions:
1. The definition of the block-diagonalization as either $$\begin{aligned}
\tilde{\mathbf{F}} = \mathbf{Q} \mathbf{F} \mathbf{Q}^\mathsf{T},\end{aligned}$$ or $$\begin{aligned}
\tilde{\mathbf{F}} = \bar{\mathbf{Q}}^\mathsf{T} \mathbf{F} \bar{\mathbf{Q}},\end{aligned}$$ with $\bar{\mathbf{Q}} = \mathbf{Q}^\mathsf{T}$.
2. The definition of the block-diagonalization matrix $\mathbf{Q}$ as either $$\begin{aligned}
\mathbf{Q} = \mathbf{Q}_\text{R} \mathbf{Q}_\text{BD},\end{aligned}$$ or $$\begin{aligned}
\mathbf{Q} = \mathbf{Q}_\text{BD} \mathbf{Q}_\text{R}.\end{aligned}$$ This is possible since $\mathbf{Q}_\text{R}$ and $\mathbf{Q}_\text{BD}$ commute.
3. The definition of the block-diagonalization matrix $\mathbf{Q}_\text{BD}$ as either $$\begin{aligned}
\mathbf{Q}_\text{BD} =
\begin{pmatrix}
\mathbf{I} & -\mathbf{U}^\mathsf{T} \\
\mathbf{U} & \mathbf{I}
\end{pmatrix},\end{aligned}$$ or $$\begin{aligned}
\mathbf{Q}_\text{BD} =
\begin{pmatrix}
\mathbf{I} & -\bar{\mathbf{U}} \\
\bar{\mathbf{U}}^\mathsf{T} & \mathbf{I}
\end{pmatrix},\end{aligned}$$ with $\bar{\mathbf{U}} = \mathbf{U}^\mathsf{T}$. Since it is also possible to swap the sign of both $\bar{\mathbf{U}}$ and $\mathbf{U}$, this leaves us with four different definitions for the block-diagonalization matrix $\mathbf{Q}_\text{BD}$.
All of these different definitions may be combined, which leads to the large number of possible representations of the block-diagonalization matrix $\mathbf{Q}$.
Eigenvector assignment for multiple subsystems {#app:localization}
==============================================
Assigning eigenvectors to multiple subsystems is not trivial when considering multiple subsystems. When assigning an eigenvector to a subsystem with a localization function such as the one in Eq. , we must consider that this eigenvector may also contribute to another subsystem. Ranking these contributions may prove to be hard and can lead to the assignment of a single eigenvector to multiple subsystems. It is also possible that an eigenvector may not be chosen for any subsystem and is subsequently excluded from the block-diagonalization procedure. This must not occur, and each eigenvector can be assigned to exactly one subsystem only. Here, we introduce a scheme for an optimal and unique assignment of a set of eigenvectors to multiple subsystems.
We start by defining the localization function $f^{\mathcal{S}_i}_j$ for each eigenvector $\breve{c}_{j}$ for each subsystem $\mathcal{S}_i$, $$\begin{aligned}
f^{\mathcal{S}_i}_j = \sum_{r = 1 + \sum_{t = 1}^{i-1} N_{\mathcal{S}_t}}^{\sum_{t = 1}^{i} N_{\mathcal{S}_t}} \breve{c}_{j,r}^{2}.\end{aligned}$$ As in Eq. , this is simply a measure of the contribution of the subsystem basis functions to the corresponding molecular orbital. After evaluation of the localization functions, the $N_{\mathcal{S}_i}$ eigenvectors with the highest localization function $f^{\mathcal{S}_i}_j$ are assigned to the subsystem $\mathcal{S}_i$.
However, this choice of eigenvectors suffers from the aforementioned problems concerning the assignment of eigenvectors to multiple subsystems. This ambiguity can be resolved by the following iterative scheme:
1. For each pair of subsystems $\mathcal{S}_r$ and $\mathcal{S}_s$, the assignment of eigenvectors is probed for a duplicate assignment. If no collision is detected, the procedure is completed. Otherwise, we continue with the next step.
2. One of the detected collisions is chosen at random. Let $\breve{c}_{i}$ be an eigenvector that was assigned to both subsystems $\mathcal{S}_r$ and $\mathcal{S}_s$, but with a larger contribution to $\mathcal{S}_r$ than to $\mathcal{S}_s$, $$\begin{aligned}
f^{\mathcal{S}_r}_i > f^{\mathcal{S}_s}_i.\end{aligned}$$ In this case, the eigenvector $\breve{c}_{i}$ is removed from the choice of eigenvectors for subsystem $\mathcal{S}_s$.
3. Now that the subsystem $\mathcal{S}_s$ is missing an eigenvector, we need to assign a new eigenvector to this subsystem for it to be considered complete. This new eigenvector $\breve{c}_{j}$ is chosen as the eigenvector with the highest localization function $f^{\mathcal{S}_s}_j$ which has not yet been chosen for subsystem $\mathcal{S}_s$.
4. The procedure is repeated from the first step to probe for new collisions since the new choice of eigenvectors is not guaranteed to be free of multiple assignments.
Unless there is a case in which two localization functions for an eigenvector $\breve{c}_{i}$ are equal, $$\begin{aligned}
f^{\mathcal{S}_r}_i = f^{\mathcal{S}_s}_i,\end{aligned}$$ this scheme is guaranteed to produce a unique and correct assignment of the eigenvectors to multiple subsystems. However, it should also be noted that in the case of an assignment of an eigenvector to multiple subsystems, caution is advised since it may be an indication of a particularly challenging partitioning.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Strongly gravitational lensing systems (SGL) encodes cosmology information in source/lens distance ratios $\mathcal{D}_{\rm
obs}=\mathcal{D}_{\rm ls}/\mathcal{D}_{\rm s}$, which can be used to precisely constrain cosmological parameters. In this paper, based on future measurements of 390 strong lensing systems from the forthcoming LSST survey, we have successfully reconstructed the distance ratio $\mathcal{D}_{\rm obs}$ (with the source redshift $z_s\sim 4.0$), directly from the data without assuming any parametric form. A recently developed method based on model-independent reconstruction approach, Gaussian Processes (GP) is used in our study of these strong lensing systems. Our results show that independent measurement of the matter density parameter ($\Omega_m$) could be expected from such strong lensing statistics. More specifically, one can expect $\Omega_m$ to be estimated at the precision of $\Delta\Omega_m\sim0.015$ in the concordance $\Lambda$CDM model, which provides comparable constraint on $\Omega_m$ with Planck 2015 results. In the framework of the modified gravity theory (DGP), 390 detectable galactic lenses from future LSST survey would lead to stringent fits of $\Delta\Omega_m\sim0.030$. **Finally, we have discussed three possible sources of systematic errors (sample incompleteness, the determination of length of lens redshift bin, and the choice of lens redshift shells), and quantified their effects on the final cosmological constraints. Our results strongly indicate that future strong lensing surveys, with the accumulation of a larger and more accurate sample of detectable galactic lenses, will considerably benefit from the methodology described in this analysis.**
author:
- 'Tonghua Liu, Shuo Cao, Jia Zhang, Shuaibo Geng, Yuting Liu, Xuan Ji, and Zong-Hong Zhu'
title: 'Implications from simulated strong gravitational lensing systems: constraining cosmological parameters using Gaussian Processes '
---
Introduction
============
During the last decades, one of the most important issues of modern cosmology is the accelerating expansion of the universe, which has been discovered and verified by several observational probes including the type Ia supernova (SNe Ia) [@Riess; @Perlmutter; @Riess04; @Knop], baryon acoustic oscillation (BAO) [@Percival], and precise measurements of the spectrum of cosmic microwave background (CMB) [@Balbi; @Jaffe; @Spergel03; @Spergel07]. Currently, the detailed dynamics of the accelerated expansion is still not well known. The origin of this acceleration may be attributed to dark energy with negative pressure, based on the cosmological principles (homogeneous, isotropic) and Einstein¡¯s general relativity (GR). In the framework of the current standard model, the so-called $\Lambda$CDM model, the accelerated cosmological expansion is powered by Einstein’s cosmological constant, $\Lambda$, a spatially homogeneous fluid with equation of state parameter $w=p/\rho=-1$ (with $p$ and $\rho$ being the fluid pressure and energy density). However, one should note that the $\Lambda$CDM model, although providing a reasonable fit to most observational constraints, is still confronted with the well-known coincidence problem and fine-tuning problem [@Weinberg]. See @Cao11a [@Cao14] and references therein for recent discussions about more dark energy models under discussion [@Cao11b; @Cao13; @Cao15a; @Qi18] .
On the other hand, dark energy is not the only possible explanation of the present cosmic acceleration, and it is argued that the observed accelerated expansion should instead be viewed as the possible deviation from Einstein’s theory of gravity on large cosmological length scales. For instance, some unknown physical processes involving modifications of gravity theory can also account for this apparently unusual phenomenon. Some modifications are related to the possible existence of extra dimensions, which gives rise to the so-called braneworld cosmology. In this paper we investigate constraints on one interesting braneworld cosmological model proposed by @Dvali00a [@Arkani; @Dvali00b], the Dvali-Gabadadze-Porrati (DGP) braneworld, which is often used to describe a gravity spilling over large scales and into higher dimensions. So far, both models derived from introducing an exotic component like dark energy and those established by modifying Einstein’s theory of gravity can survive the above-mentioned observations. Actually, the investigation of the expected constraints on DGP braneworld cosmology has been performed from different astrophysical observations [@Xu10; @Giannantonio; @Lombriser; @Wang08]. However, it is interesting to note that based on different theoretical basis, the determination of the same cosmological parameter in different cosmological models are clearly different. The normal branch of DGP gravity is confronted by the currently available cosmic observations from the geometrical and dynamical perspectives. For instance, ref. @Xu14 made a joint analysis of the DGP braneworld cosmology with the Supernova Legacy Survey (SNLS) data, first released CMB data from Planck, and redshift space distortion (RSD) data ($\Omega_m=0.286\pm 0.008$). While comparing the results with those obtained from Planck 2018 data (TT, TE, EE+lowE+lensing) based $\Lambda$CDM model $\Omega_m=0.315\pm0.007$ [@Aghanim], differences in central values of the best-fit cosmological parameter were clearly reported. Similar analyses were carried out by @Ma19. If one wants to place more comprehensive cosmological constraints on a possible model or distinguish between dark energy and modified gravity theories, it is crucial to measure the expansion rate of universe at many different redshifts.
The power of modern cosmology lies in building up consistency rather than in single, precise, crucial experiments, which implies that every alternative method of restricting cosmological parameters is desired. In particular, a new cosmological window would open if we could measure the cosmic expansion directly within the “redshift desert”, roughly corresponding to redshifts $2<z<5$. As one of the successful predictions of general relativity in the past decades, strong gravitational lensing has become a very important astrophysical tool allowing us to use individual lensing galaxies to measure cosmological parameters [@Treu06]. When the source, lens, and observer are sufficiently well aligned, the deflection of light forms an Einstein ring, from which the source/lens distance ratios can be obtained. @Biesiada06 first proposed the possible application of this kind of observation as a cosmological tool, the importance of which method was stressed again by @Grillo [@Biesiada]. The idea of using such systems for measuring the cosmic equation of state was discussed in @Cao12JC and also in a more recent paper by @Cao15Ap. The angular diameter distance ratios may also be used to constrain different cosmological parameters in various cosmological models [@Futamase; @Treu640; @Melia]. On the one hand, in order to achieve high precision constraints on the cosmological parameters, it is still necessary to develop new complementary techniques bridging the redshift gap of current data, and furthermore increase the depth and quality of observational data sets. In this paper, we will use the model-independent method Gaussian processes (GP) to reconstruct one-dimensional function of the angular diameter distance ratios, with fixed lens (or source) redshift. An obvious benefit of this approach is that GP allow one to reconstruct a function from data directly without any parametric assumption, which has been widely used in various studies [@Seikel12a; @Seikel12b; @Cai; @Yennapureddy; @Melia18b]. The first (to our knowledge) formulations of this approach can be traced back to @Yennapureddy, which revisited the most recent and significantly improved observations of early-type gravitational lenses (158 combined systems) to distinguish $\Lambda$CDM another Friedmann-Robertson-Walker (FRW) cosmology known as the $R_h=ct$ universe. Their results showed that, the probability of $R_h=ct$ (which is characterized by a total equation of state $w=-1/3$) being the correct cosmology is higher than that of $\Lambda$CDM, with a degree of significance that grows with the number of sources considered. Therefore, although the differentiation of competing cosmologies is already quite competitive compared with those from other methods, it still suffer from the small number of lenses in the statistical sample.
In the near future, the next generation of wide and deep sky surveys, with improved depth, area and resolution may increase the current galactic-scale lens sample sizes by orders of magnitude. The purpose of our paper is to investigate the constraining capability of SGL on some fundamental cosmological parameters, using the simulated SGL sample based on the forthcoming Large Synoptic Survey Telescope (LSST) survey. More importantly, compared with the previous procedure of carrying out the reconstruction within thin redshift-shells of sources [@Yennapureddy], we turn $D_{ls}/D_s$ into a one-dimensional function of source redshift ($z_s$) for what is essentially a fixed lens redshift ($z_l$). The advantage of this work lies in the fact that, we could achieve reasonable constraints on cosmological parameters at much higher redshifts ($z\sim4$), when the sample is large enough to yield enough statistics to warrant this approach. As can clearly seen from the previous analysis [@Yennapureddy], the current SGL sample is not sufficient enough to extend our investigation to $z\sim 4$ (the data are less dispersed in the lens plane, and scattered much more in the source plane).
This paper is organized as follows. In Sect. 2 we briefly describe the methodology and the simulated strong lensing data from LSST. In Sect. 3 we introduce our improved Gaussian processes and the area minimization statistic. Two prevalent cosmologies and the fitting results on the relevant cosmological parameters are presented in Sect. 4. Finally, we summarize our conclusions in Sect. 5.
Simulated strong lensing systems
================================
For a specific strong lensing system with the intervening galaxy acting as a lens (at redshift $z_l$), the multiple image separation of the source (at redshift $z_s$) depends only on angular diameter distances to the lens and to the source, as long as one has a reliable model for the mass distribution within the lens. Moreover, compared with late-type and unknown-type counterparts, early-type galaxies are more likely to serve as intervening lenses for the background sources (quasars or galaxies). This is because such galaxies contain most of the cosmic stellar mass of the Universe. The recently released large sample include 118 galaxy-scale strong gravitational lensing systems discovered and observed in SLACS, BELLS, LSD, and SL2S surveys [@Cao15Ap], which can be used to place stringent constrains on cosmological parameters in alternative cosmological models [@Li16], and to study the mass density distribution in early-type galaxies [@Cao16]. Recent analytical work has forecast the number of galactic-scale lenses to be discovered in the forthcoming photometric surveys [@Collett15]. Such a significant increase of the number of strong lensing systems will considerably improve the constraints on the cosmological parameters. With a large increase to the known strong lens population, current work could be extended to a new regime: what kind of cosmological results one could obtain from $\sim$10,000 discoverable lens population in the forthcoming Large Synoptic Survey Telescope (LSST) survey?
Using the simulation programs publicly available [^1], we carry out a Monte Carlo simulation of the lens and source populations to forecast the yields of LSST. In our simulation, 10000 SGL systems has been obtain with the proper inputs in the following three assumptions: (i) early-type galaxies act as lenses; (ii) mass distribution of lens is approximated by the power law model; (iii) the normalization and shape of the velocity dispersion function of early-galaxies are not varying with redshift. The assumptions are well consistent with the previous studies on lensing statistics if all galaxies are early-type [@Chae; @Mitchell; @Capelo]. Moreover, we assume a flat concordance $\Lambda$CDM model with $\Omega_m=0.30$ as a fiducial cosmology.
{width="0.9\linewidth"}
Motivated by several previous studies supporting that early-type galaxies are well described by power-law mass distributions in regions covered by the X-ray and lensing observations [@Humphrey10; @Koopmans06], we model the lens galaxy with a power-law mass distribution ($\rho \sim r^{- \gamma}$). The main idea of our method is that formula for the Einstein radius in a power-law lens expresses as $$\label{Einstein} \theta_E = 4 \pi
\frac{\sigma_{ap}^2}{c^2} \frac{D_{ls}}{D_s} \left(
\frac{\theta_E}{\theta_{ap}} \right)^{2-\gamma} f(\gamma),$$ based on which the ratio of angular-diameter distances between lens and source ($D_{ls}$) and between observer and source ($D_{s}$) can be obtained $$\label{Einstein} \frac{D_{ls}}{D_s} = \frac{\theta_E} {4 \pi}
\frac{c^2}{\sigma_{ap}^2} \left( \frac{\theta_E}{\theta_{ap}}
\right)^{\gamma-2} f(\gamma)^{-1},$$ $f(\gamma)$ represents a function of the radial mass profile slop [@Koopmans] and $\sigma_{ap}$ is the luminosity averaged line-of-sight velocity dispersion of the lens inside the aperture radius, $\theta_{ap}$ (more precisely, luminosity averaged line-of-sight velocity dispersion). It is obvious that combining $\sigma_{ap}$, $\theta_E$, $\theta_{ap}$ and $\gamma$ obtained from the observations will introduce the measurement of the distance ratio of $D_{ls}/D_s$. Current observational techniques allow the redshifts of the lens $z_l$ and the source $z_s$ to be measured precisely. Moreover, imaging and spectroscopy from the Hubble Space Telescope (HST) and ground-based observatories make it possible to derive two key ingredients for individual lenses: stellar velocity dispersion, high-resolution images of the lensing systems. We take the fractional uncertainty of the Einstein radius at a level of 1%, which is reasonable for the future LSST survey to obtain high-resolution imaging with different stacking strategies for combining multiple exposures [@Collett16]. Following the Lens Structure and Dynamics (LSD) survey and the more recent Sloan Lenses ACS (SLACS) survey, we take the fractional uncertainty of the observed velocity dispersion at a level of 5%, which can be assessed from the spectroscopic data for central parts of lens galaxies. More importantly, it was shown that the power-law mass profile is still a useful assumption in gravitational lensing studies and should be accurate enough as first-order approximations to the mean properties of galaxies relevant to statistical lensing (lenses observed in different surveys with the following median values of the lens redshifts: $z_l = 0.215$ for SLACS, $z_l = 0.517$ for BELLS, $z_l = 0.81$ for LSD and $z_l = 0.456$ for SL2S [@Cao15Ap]). In our fiducial model, the average logaritmic density slope is modeled as $\gamma=2.085$ with 10% intrinsic scatter, the results from SLACS strong-lens early-type galaxies with direct total-mass and stellar-velocity dispersion measurements [@Koopmans09].
{width="0.9\linewidth"}
Following the LSST observation simulator with the assumed survey parameters summarized in Table 1 of @Collett15, we have generated a realistic population of galaxies lensed by early-type galaxies, assuming distributions of velocity dispersions and Einstein radii similar to the SL2S sample [@Sonnenfeld13]. The velocity dispersion function of the lenses in the local Universe follows the modified Schechter function [@Sheth03] $$\label{stat2} \frac{d n}{d \sigma}= n_*\left(
\frac{\sigma}{\sigma_*}\right)^\alpha \exp \left[ -\left(
\frac{\sigma}{\sigma_*}\right)^\beta\right] \frac{\beta}{\Gamma
(\alpha/\beta)} \frac{1}{\sigma} \, ,$$ where $\alpha$ is the low-velocity power-law index, $\beta$ is the high-velocity exponential cut-off index, $n_*$ is the integrated number density of galaxies, and $\sigma_*$ is the characteristic velocity dispersion. In this paper, we use the measurement of velocity dispersion function (VDF) for local early-type galaxies, based on the much larger SDSS Data Release 5 data set [@Choi07]. See @Cao12b for discussion about such choice in view of other data on velocity dispersion distribution functions. Currently, it was found that simple evolutions do not significantly affect the the appealing results based on lensing statistics, especially those from the early-type galaxy number counts [@Im02] and the redshift distribution of early-type lens galaxies [@Ofek03]. Therefore, it is assumed in our analysis that the normalization and shape of the velocity dispersion function of early-galaxies are not varying with redshift. The population of strong lenses is dominated by galaxies with velocity dispersion of $\sigma_{ap}=210\pm50$ km/s, while the lens redshift distribution is well approximated by a Gaussian with mean 0.40. Although discovering strong lenses in these surveys will require the development of new methods and algorithms [@Gavazzi14], which is beyond the scope of this paper, we are confident that the simulated population of lenses is a good representation of what the future LSST survey might yield [@Sonnenfeld13; @Gavazzi14].
The scatter plot of the simulated lensing systems is shown in Fig. 1, from which one can see the LSST lenses result in a fair coverage of lenses and sources redshifts.
{width="0.9\linewidth"}
Gaussian Processes
==================
In order to reconstruct the evolution of angular diameter distances from simulated SGL data sets, we should find a model-independent method to reconstruct $D=D_{ls}/D_s$. Although there are several methods such as principle component analysis [@Huterer03] and Gaussian smoothing [@Shafieloo06], in this paper we will reconstruct $D_{ls}/D_s$ more precisely by using the Gaussian processes (GP) method.
A model-independent method of Gaussian processes [@Seikel12a], can be employed to reconstruct the angular-diameter distance ratio from the strong lensing data straightforwardly, without any parametric assumption regarding cosmological model. Such approach has been used in various studies in the literature [@Qi19a; @Cao19]. The distribution over functions provided by GP is suitable to describe the observed data. Not that for each lens-source pairing, $D_{ls}/D_s$, two angular diameter distances are involved. Therefore, a reconstruction of $D_{ls}/D_s$ in two dimensions is required, which is very difficult to handle with GP. Following the methodology proposed by @Yennapureddy, one interesting solution of this problem is to consider small redshift ranges (with fixed source/lens redshift), which may effectively reduce the problem to a one-dimensional reconstruction. In this work, we choose to carry out the reconstruction within thin redshift-shells of lenses, which makes it possible to turn $D_{ls}/D_s$ into a one-dimensional function of source redshift ($z_s$) and thus achieve reasonable constraints on cosmological parameters at much higher redshifts ($z\sim4$). Note that for the purpose of GP reconstruction in one dimension, we assume that all the lenses in redshift bin $(z_l,z_l+\Delta z)$ have the same average redshift $z_l+\Delta z_l/2$. In order to minimize the scatter in lens redshifts, we use a bin size less than 0.02 ($\Delta
z_l=0.02$). We also find that the choice of $\Delta z_l$ may play an important role in the accuracy and reliability of our test, which will be discussed in Section 4. More importantly, in order to guarantee the precision of GP reconstruction, the selected sub-sample should be large enough to yield enough statistics. In our simulated sample of strong-lensing systems, these criteria therefore allow us to assemble a sub-sample including 390 strong-lensing systems, with the lensing galaxies covering the redshift shell of $0.3<z_l<0.32$. **Fig. 2 shows the lens redshift distribution of galactic-scale lenses discoverable in forthcoming LSST surveys. It is apparent that, compared to the current surveys with the following median values of the lens redshifts: SLACS – $z_l =
0.215$, BELLS – $z_l = 0.517$, LSD – $z_l = 0.81$ and SL2S – $
z_l = 0.456$ [@Cao15Ap], the future LSST survey is particularly promising to discover more lenses covering the redshift range of $0.25 - 0.50$. Therefore, the thin shell of $0.3 < z_l < 0.32$ is a good statistical representation of the simulated population of lenses what the future LSST survey might yield.**
At each point $z$, the reconstructed function $f(z)$ is also a Gaussian distribution with a mean value and Gaussian error. In this process, the values of the reconstructed function evaluated at any two different points $z$ and $\tilde{z}$, are connected by a covariance function $k(z,\tilde{z})$. In this paper, we take the Matérn ($\nu = 9/2$) covariance function $$\begin{aligned}
k(z,\tilde z) = &~{\sigma _f}^2\exp\left( - \frac{{3\left| {z - \tilde z} \right|}}{\ell }\right) \nonumber \\
&~~\times\Big[1 + \frac{{3\left| {z - \tilde z} \right|}}{\ell } + \frac{{27{{(z - \tilde z)}^2}}}{{7{\ell ^2}}} \nonumber \\
&~~ + \frac{{18{{\left| {z - \tilde z} \right|}^3}}}{{7{\ell ^3}}} + \frac{{27{{(z - \tilde z)}^4}}}{{35{\ell ^4}}}\Big],\end{aligned}$$ where $\ell$ provides a measure of the coherence length of the correlation in $x$-direction and $\sigma_f$ is the overall amplitude of the correlation in the $y$-direction. The values of the two hyper parameters $\sigma_f$ and $\ell$ will be optimized by GP with the observed data set. This implies that the reconstructed function is not dependent on the initial hyper-parameter settings, which guarantees the reliability of the reconstructed function. Compared with the squared exponential form for covariance function, which has been widely used in the literature [@Seikel12a; @Seikel12b; @Cai; @Yennapureddy; @Melia18b], the Matérn ($\nu=9/2$) covariance function can lead to more reliable results when applying GP to reconstructions using distance measurements [@Yang15]. Using this covariance function, values of data points at other redshifts which have not be observed can also be obtained, which could effectively bridges the redshift gap between current data. Following @Seikel12a in which the detailed technical description of GP can be found, we use the Gaussian processes in Python (GaPP) [^2] to execute the model-independent method and derive our GP results. The reconstructed function $\mathcal{D}_{\rm obs}(\langle z_l\rangle,
z_s)$, as well as the estimation of the $1\sigma$ confidence region with the 390 simulated strong lensing systems is shown in Fig. 3.
In order to demonstrate how the reconstructed function $\mathcal{D}_{\rm obs}(\langle z_l\rangle, z_s)$ works, we have constrained two simple cosmological models: the $\Lambda$CDM and DGP models under assumption of spatially flat Universe. On the other hand, in the face of different competing cosmological scenarios, it is important to find an effective way to decide which one is most favored by the data. Following the analysis of @Yennapureddy, a new type of model comparison statistics, Area Minimization Statistics, will be used for this purpose.
Competing cosmological models and statistical analysis
======================================================
Flatness of the Friedmann-Robertso-Walker (FRW) metric is assumed in our analysis, which is strongly supported by the recent Planck results [@Ma19] and independently supported by the observations of milliarcsecond compact structure of radio quasars at $z\sim 3.0$ [@Cao17; @Cao19]. In a zero-curvature universe filled with ordinary pressureless matter (cold dark matter plus baryons), dark energy, and negligible radiation, the Friedmann equation reads $$H^2(z)=H_0^2[\Omega_m(1+z)^3+(1-\Omega_{m})],$$ where $\Omega_{m}$ is the current density fraction of matter component. In the framework of this standard cosmological model, the angular diameter distance between redshifts $z_1$ and $z_2$ becomes $$\begin{aligned}
D_A^{\Lambda CDM}(z_1,z_2)&=&\frac{c}{H_0 (1+z_2)}\times\nonumber\\
\null&\null&\hskip-0.3in
\int^{z_2}_{z_1}\frac{dz}{\sqrt{\Omega_m(1+z)^3+1-\Omega_m}}.\nonumber\end{aligned}$$ Over the past decades, the importance of modified gravity theories was stressed again. The DGP model [@Dvali00b], which accounts for the cosmic acceleration without dark energy, arises from a class of brane world theories in which gravity leaks out into the bulk at large distances. More specifically, this leaking of gravity takes place only above a certain cosmological scale $r_c$. In the framework of a spatially flat DGP model, the Friedmann equation can be expressed as $$H^2-\frac{H}{r_c}=\frac{8\pi G}{3}\rho_m,$$ where the length at which the leaking occurs can be associated with the density parameter: $\Omega_{r_c}=1/(4r_c^2H_0^2)$. It is also straightforward to check the validity of the relation $\Omega_{r_c}=\frac{1}{4}(1-\Omega_m)^2$ in the flat DGP model. Thus, in the framework of a spatially flat DGP model, we can directly rewrite the above equation and obtain the angular diameter distance between redshifts $z_1$ and $z_2$ $$\begin{aligned}
D_A^{DGP}(z_1,z_2)&=&\frac{c}{H_0 (1+z_2)}\times\nonumber\\
\null&\null&\hskip-0.3in
\int^{z_2}_{z_1}[\sqrt{\Omega_m(1+z)^3+\Omega_{r_c}}+\sqrt{\Omega_{r_c}}]^{-1}
dz.\nonumber\end{aligned}$$
Now we introduce a new statistic, the “Area Minimization Statistic" to constrain cosmological parameters, which has been recently proposed to test the evolution of the Universe, and then applied to the investigation of dynamical properties of dark energy [@Yennapureddy17; @Melia18c]. It should be noted that our reconstructed distance ratio $\mathcal{D}_{\rm obs}$, with the corresponding theoretical model value $\mathcal{D}^{\rm \Lambda{\rm CDM}}$ or $\mathcal{D}^{\rm DGP}$, is an continuous function. Therefore, the discrete sampling statistics (such as the $\chi^2$ statistic) is not sufficient enough to provide an effective way to make a comparison between different models, because the sampling at random points to obtain the squares of differences between model and reconstructed curve would lose information between these points.
![[*Top panel:*]{} The distribution of frequency versus area differential $\Delta A$ for a mock sample with lens bin $0.3<z_l<0.32$; [*Bottom panel:*]{} its corresponding cumulative probability distribution.](fig4_1.eps "fig:"){width="0.9\linewidth"}\
0.2in ![[*Top panel:*]{} The distribution of frequency versus area differential $\Delta A$ for a mock sample with lens bin $0.3<z_l<0.32$; [*Bottom panel:*]{} its corresponding cumulative probability distribution.](fig4_2.eps "fig:"){width="0.9\linewidth"}
The most important assumption of “Area Minimization Statistic" is that the measurement errors are Gaussian, which should be satisfied by the mock sample with GP reconstructed curves and possible variation of $\mathcal{D}$ away from $\mathcal{D}_{\rm obs}$. More specifically, such statistic is realized by a Gaussian randomized value $$\mathcal{D}_{i,\,{\rm mock}}(\langle z_l\rangle, z_s)=\mathcal{D}_{i,\,{\rm obs}}(\langle z_l\rangle, z_s)+r
\sigma_{\mathcal{D}_{i,obs}}\;,$$ where $r$ is characterized by a Gaussian distribution $r=0.0\pm1.0$, and $\mathcal{D}_{i,\,{\rm obs}}(\langle z_l\rangle, z_s)$ represents the actual measurement at source redshift $z_s$, with 1$\sigma$ error denoted by $\sigma_{\mathcal{D}_{i,obs}}$. Therefore, the function $\mathcal{D}_{\rm mock}(\langle z_l\rangle,
z_s)$ corresponding to mock sample could be straightforwardly obtained. Finally, a normalized absolute area difference between $\mathcal{D}_{\rm mock}(\langle z_l\rangle, z_s)$ and the GP reconstructed function of the actual data can be defined as $$\Delta A= \int_{z_{\rm min}}^{z_{\rm
max}}dz_s\bigg(\frac{\big|\mathcal{D}_{\rm mock}(\langle z_l\rangle,
z_s)-
\mathcal{D}_{\rm obs}(\langle z_l\rangle, z_s)\big|}{\sigma_{\mathcal{D}}}\bigg)\;,$$ where $z_{\rm min}$ and $z_{\rm max}$ are the minimum and maximum redshifts of the mock sample. This process is repeated 10000 times in order to guarantee unbiased final results, from which one could derive a distribution of frequency versus area differential $\Delta
A$ and the cumulative probability distribution. The results are shown in Fig. 4, in which one can clearly see a 1-to-1 mapping between the value of $\Delta A$ and the corresponding frequency. More importantly, the cumulative distribution for a given $\Delta A$ quantifies the fraction of the randomized realizations whose differential area is smaller than this value.
In the framework of a specific cosmological model, we can calculate a normalized absolute area difference between the GP reconstructed function of the actual data and its theoretical counterpart $$\Delta A= \int_{z_{\rm min}}^{z_{\rm
max}}dz_s\bigg(\frac{\big|\mathcal{D}_{\rm mock}(\langle z_l\rangle,
z_s)-\mathcal{D}_{\rm th}(\langle z_l\rangle,
z_s)\big|}{\sigma_{\mathcal{D}}}\bigg)\;.$$ Therefore, based on the assumption that a curve with a smaller $\Delta A$ is a better match to $\mathcal{D}_{\rm obs}$, the cumulative distribution can be directly used to estimate the probability (i.e., the p-value) that a cosmological model is well consistent with the observations. More specifically, in order to decide which cosmology is favored by the observational data, we perform model comparison statistics by calculating its $\Delta A$ and apply the 1-to-1 mapping to determine the probability that it is inconsistent with the SGL sample. The results for different cosmological scenarios on the reconstructed $\mathcal{D}$ observations are listed in Table 1 and discussed as follows. We stress here that the observational distance ratio $\mathcal{D}$ has the advantage that the Hubble constant $H_0$ gets cancelled, hence it does not introduce any uncertainty to the results.
{width="0.9\linewidth"}
{width="0.9\linewidth"}
Cosmological model Cosmological parameter Probability
-------------------- ------------------------ -------------
$\Lambda$CDM $\Omega_m=0.285$ $10.00\%$
$\Omega_m=0.300$ $99.99\%$
$\Omega_m=0.315$ $10.00\%$
DGP $\Omega_m=0.260$ $10.00\%$
$\Omega_m=0.275 $ $99.99\%$
$\Omega_m=0.305 $ $10.00\%$
: Summary of the cosmological constraints using strong gravitational lenses with Gaussian Processes.
We start our analysis with the $\Lambda$CDM model with constant dark energy density and constant cosmic equation of state $w=-1$. The corresponding cumulative probability distributions are plotted in Fig. 5, which also locate the $\Delta A$ values with different matter density parameter: $\Omega_m=0.300$ (black), $\Omega_m=0.285$ (blue), and $\Omega_m=0.315$ (red). The probabilities associated with these differential areas are summarized in Table 1. As can be clearly seen from the results, the probability of the matter density parameter $\Omega_m=0.300$ being consistent with the GP reconstructed $\mathcal{D}_{\rm obs}$ function is $99.99\%$, while the probabilities that the matter density parameter $\Omega_m=0.285$ and $\Omega_m=0.315$ being inconsistent with the current SGL observations are $90\%$. Considering the additional assumption that a cumulative probability of $90\%$ is considered strong evidence against the model, we demonstrate that with 390 well-observed galactic strong lensing systems, one can expect the matter density parameter to be estimated with the precision of $\Delta \Omega_m
\sim 0.015$.
Now one issue which should be discussed is the comparison of our cosmological results with those of earlier studies done using alternative probes. First of all, based on the Planck temperature data combined with Planck lensing, Planck Collaboration XIII (2015) gave the best-fit parameters: $\Omega_m=0.308\pm0.012$ and $H_0=67.8\pm0.9$ (at 68.3% confidence level). More recently, the best-fit values of the cosmological parameters in the flat $\Lambda$CDM model were obtained as: ${\Omega_m}=0.255\pm0.030$ and $H_0=70.4\pm2.5 \; \rm{kms}^{-1} \; \rm{Mpc}^{-1}$, based on the latest observations of 41 Hubble parameter $H(z)$ at different redshifts, which were determined from the radial BAO size method and the differential ages of passively evolving galaxies [@Aghanim]. Let us note that the matter density parameter inferred from CMB and OHD data are highly dependent on the value of the Hubble constant. Considering the well known strong degeneracy between $\Omega_m$ and $H_0$. Therefore independent measurement of $\Omega_m$ from strong lensing statistics, with the precision comparable to *Planck* observations of the CMB radiation, could be expected and indeed is revealed here.
Working on the DGP model, the cumulative probability distributions are plotted in Fig. 6, which also locate the $\Delta A$ values with different matter density parameter. The probabilities associated with differential areas are also summarized in Table 1. Similarly, the probability of the matter density parameter $\Omega_m=0.275$ being consistent with the GP reconstructed function $\mathcal{D}_{\rm obs}$ is $99.99\%$, while the probabilities that the matter density parameter $\Omega_m=0.260$ and $\Omega_m=0.305$ being inconsistent with the current SGL observations are $90\%$. Therefore, in the framework of this modified gravity theory, the matter density parameter could be determined at the precision of of $\Delta \Omega_m \sim 0.015-0.03$. More interestingly, benefit from the redshift coverage of background sources in the lensing systems, the methodology proposed in this analysis may provide improved constraints on the DGP model, which was ruled out observationally considering the precision cosmological observational data. Such issue has been extensively discussed in many previous works [@Wang08; @Maartens10].
However, there are several sources of systematics we do not consider in the above analysis and which remain to be clarified for this methodology. **As a final remark, we discuss several possible sources of systematic errors, including sample incompleteness, the determination of length of lens redshift bin, and the choice of lens redshift shells, in order to verify their effect on the cosmological constraints.** Firstly, based on the flat $\Lambda$CDM with the full sample ($N=390$ lenses), we now estimate the systematic errors due to statistical sample incompleteness, which could directly affect the reconstructed function of the observable $\mathcal{D}_{\rm
obs}$. Fig. 7 shows the precision of the $\Omega_m$ parameter assessment as a function of SGL sample size and Table 2 shows more detailed results. One can see that, even with 50 SGL systems one can effectively place stringent fits on the matter density in the Universe ($\Delta \Omega_m \sim 0.05$), which furthermore strengthens the probative power of our method to inspire new observing programs or theoretical work in the moderate future. **Secondly, after identifying the constraints on $\Omega_m$ obtained with the minimum acceptable $\Delta z_l=0.02$ and the errors that it introduced, we should consider different values of $\Delta z_l$ for examining the role $\Delta z_l$ plays in cosmological constraints. It should be noted that the selected length of lens redshift $\Delta z_l$ not only directly determines the selection of simulated lensing systems, but also introduces systematical uncertainties in estimating cosmological model parameters. For the selection criteria of $\Delta z_l=0.01$, we unbiasedly select a sub-sample including 150 strong lensing systems out of the whole catalog of $N=390$ lenses. Based on this restricted sub-sample, the constraints on $\Omega_m$ as a function of $\Delta
z_l$ are shown in Fig. 8. The results are summarized in Table 3. It is apparent that the choice of the length of lens redshift bin, $\Delta z_l$, which affects the derived average lens redshift for the sample, will also play an important role in the $\mathcal{D}_{\rm obs}$ reconstruction and thus the effectiveness of this model-independent test. Such issue has been noted and extensively discussed in the previous analysis, concerning the most recent observations of early-type gravitational lenses [@Yennapureddy]. Thirdly, in order to investigate the impact of different lens shells on cosmological parameter distribution, we also work on two additional different redshift shells at lower redshift ($0.16<z_l<0.18$) and higher redshift ($0.73<z_l<0.75$), which respectively generate 220 lenses in the following analysis. As can be seen from the results illustrated in Table 4, the matter density parameter can be estimated at the precision of $\Delta
\Omega_m\sim0.03$ and $\Delta \Omega_m\sim0.025$, respectively. We remark here that the choice of lens redshift shells will slightly affect the constraints on the model parameter $\Omega_m$, due to the sample size difference between the selected sub-samples. Therefore, our results strongly suggest that larger and more accurate sample of the strong lensing data can become an important complementary probe in the next decade.**
{width="0.9\linewidth"}
{width="0.9\linewidth"}
Number of lensing systems Cosmological parameter
--------------------------- --------------------------- --
$N=50$ $\Omega_m=0.345\pm0.057$
$N=100$ $\Omega_m=0.330\pm0.040$
$N=150$ $\Omega_m=0.317\pm0.023$
$N=200$ $\Omega_m=0.313\pm0.020$
$N=390$ $\Omega_m=0.300\pm0.015$
: Summary of the cosmological constraints on $\Lambda$CDM model with different number of lensing systems, based on 390 SGL systems covering the redshift shell of $0.30<z_l<0.32$.
Length of lens redshift bin Cosmological parameter
----------------------------- --------------------------- --
$\Delta z_l=0.01$ $\Omega_m=0.295\pm0.018$
$\Delta z_l=0.02$ $\Omega_m=0.317\pm0.023$
$\Delta z_l=0.03$ $\Omega_m=0.320\pm0.040$
$\Delta z_l=0.04$ $\Omega_m=0.300\pm0.045$
: **Summary of the cosmological constraints on $\Lambda$CDM model with different lens redshift bin $\Delta z_l$, based on 150 SGL systems covering the redshift shell of $0.30<z_l<0.32$.**
Lens redshift shell Cosmological parameter Probability
--------------------- ------------------------ ----------------------------
$\Omega_m=0.270$ $10.00\%$
$0.16<z_l<0.18$ $\Omega_m=0.305$ $\,\,\,\,\,\,\,\ 99.99\%$
$\Omega_m=0.335$ $\,\,\,\,\,\,\,\ 10.00\%$
$\Omega_m=0.280$ $10.00\%$
$0.73<z_l<0.75$ $\Omega_m=0.300 $ $99.99\%$
$ \Omega_m=0.325 $ $10.00\%$
: **Summary of the cosmological constraints on $\Lambda$CDM model, based on two additional lens redshift shells.**
Conclusions
===========
In this paper, based on future measurements of 390 strong lensing systems from the forthcoming Large Synoptic Survey Telescope (LSST) survey, combined with the recently developed method based on model-independent reconstruction approach, Gaussian Processes (GP), we have successfully reconstructed the distance ratio $\mathcal{D}_{\rm obs}$ reaching the source redshift $z_s\sim 4.0$. Moreover, benefit from the Area Minimization Statistic, our results show that independent measurement of the matter density parameter could be expected from such strong lensing statistics at high redshifts. Therefore, one may say that the approach initiated in @Yennapureddy can be further developed. Here we summarize our main conclusions in more detail:
- Compared with the previous statistic focusing on individual data points, GP provides the $1\sigma$ confidence regions for the reconstructed $D_{obs}$ function more in line with the whole sample, which greatly restricts the possibility of cosmological models inadequately consistent with the observational data due to otherwise large measurement errors. However, considering the fact that our reconstructed distance ratio $\mathcal{D}_{\rm obs}$ is an continuous function, we apply a new statistic, the “Area Minimization Statistic" to constrain cosmological parameters, which provides an effective way to make a comparison between different models, compared with the discrete sampling statistics such as the $\chi^2$ statistic.
- Considering the additional assumption that a cumulative probability of 90% is considered strong evidence against the model, we demonstrate that with 390 well-observed galactic strong lensing systems, one can expect the matter density parameter to be estimated with the precision of $\Delta \Omega_m
\sim 0.015$. Such constraint is comparable to that derived from the recent Planck 2015 results.
- In the framework of the modified gravity theory (DGP), 390 detectable galactic lenses from future LSST survey would lead to stringent fits of $\Delta\Omega_m\sim0.030$. More importantly, benefit from the redshift coverage of the lensing systems, the methodology proposed in this analysis may provide improved constraints on the DGP model, which was ruled out observationally considering the precision cosmological observational data. Finally, the advantage of our method lies in the benefit of being independent of the Hubble constant. Therefore independent measurement of $\Omega_m$ from strong lensing statistics could be expected and indeed is revealed here.
- **We discuss several possible sources of systematic errors, including sample incompleteness, the determination of length of lens redshift bin, and the choice of lens redshift shells, in order to verify their effect on the cosmological constraints. More specifically, our findings indicate that the choice of the length of lens redshift bin, $\Delta z_l$, which affects the derived average lens redshift for the sample, plays an important role in the $\mathcal{D}_{\rm obs}$ reconstruction and thus the effectiveness of this model-independent test. Meanwhile, due to the sample size difference between different selected sub-samples, the choice of lens redshift shells will slightly affect the constraints on the cosmological parameters.**
- Our analysis could be extended to quantify the ability of future measurements of strong lensing systems from the Dark Energy Survey (DES) [@Frieman04], the Joint Dark Energy Mission (JDEM) [@Tyson], and the Square Kilometer Array (SKA) [@McKean], which encourages us to probe cosmological parameters at much higher accuracy. Moreover, we also pin hope on future observational data such as galactic-scale strong gravitational lensing systems with Type Ia supernovae acting as background sources [@Cao18], strongly lensed repeating fast radio bursts [@Li18], and strongly lensed gravitational waves (GWs) from compact binary coalescence and their electromagnetic (EM) counterparts systems [@Liao17; @Cao19b; @Qi19b; @Qi19c]. With more detectable galactic-scale lenses from the forthcoming surveys, the scheme proposed in this paper can eventually be used to carry out stringent tests on various cosmological models.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by National Key R&D Program of China No. 2017YFA0402600; the National Natural Science Foundation of China under Grants Nos. 11690023 and 11633001; Beijing Talents Fund of Organization Department of Beijing Municipal Committee of the CPC; the Fundamental Research Funds for the Central Universities and Scientific Research Foundation of Beijing Normal University; and the Opening Project of Key Laboratory of Computational Astrophysics, National Astronomical Observatories, Chinese Academy of Sciences.
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[^1]: github.com/tcollett/LensPop
[^2]: http://www.acgc.uct.ac.za/seikel/GAPP/index.html
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abstract: 'The trigger system of the Surface Detector (SD) of the Pierre Auger Observatory is described, from the identification of candidate showers ($E>1$ EeV) at the level of a single station, among a huge background (mainly single muons), up to the selection of real events and the rejection of random coincidences at a higher central trigger level (including the reconstruction accuracy). Efficiency of a single station trigger is evaluated using real data , and the high performance of event selection hierarchy will be demonstrated.'
author:
- |
D. Allard, E. Armengaud, I. Allekotte, P. Allison, J. Aublin, M. Ave, P. Bauleo, J. Beatty, T. Beau, X. Bertou, P. Billoir, C. Bonifazi, A. Chou, J. Chye, S. Dagoret-Campagne, A. Dorofeev, P.L. Ghia, M. Gómez Berisso, A. Gorgi, J.C. Hamilton, J. Harton, R. Knapik, C. Lachaud, I. Lhenry-Yvon, A. Letessier-Selvon, J. Matthews, C. Medina, R. Meyhandan, G. Navarra, D. Nitz, E.Parizot, B. Revenu, Z. Szadkowski, T. Yamamoto for the Pierre Auger Collaboration$^a$\
(a) Pierre Auger Observatory, Av San Mart[í]{}n Norte 304,(5613) Malargüe, Argentina
title: 'The trigger system of the Pierre Auger Surface Detector: operation, efficiency and stability'
---
Introduction
============
The Pierre Auger Surface Array will consist of 1600 Water Cherenkov detectors sampling ground particles of atmospheric air showers produced by a single energetic particle. The Cherenkov light detected is read out by three large photomultipliers and finally digitized at 40 MHz by Flash Analog Digital Converters (FADC). The detector is extensively described in these proceedings [@be05]. The trigger system has been designed to allow the SD of Auger to operate at a wide range of primary energies, for both vertical and very inclined showers with full efficiency for cosmic rays above $10^{19}$ eV. It should select events of interest and reject background or uninteresting events, while keeping the rate constraints imposed by the communication and data acquisition system. The trigger for the Surface Detector is hierarchical with local triggers at levels 1 and 2 (called T1 and T2), whereas level 3 (T3) is formed at the observatory campus based upon the spatial and temporal correlation of the level 2 triggers. All data satisfying the T3 trigger are stored. Additional level of triggers are implemented offline in order to select physical events (T4 physics trigger) and accurate events (T5 quality trigger), with the core inside the array. In section 2 the two levels of the local trigger are described. The efficiency is derived from real data using two different methods. Section 3 is devoted to the selection of physics events (T3 and T4) and the efficiency of the experiment is discussed. In section 4 the quality trigger adopted is presented.
Local triggers characteristics
==============================
Two different trigger modes are currently implemented at the T1 level. The first uses a Time over Threshold (ToT) trigger, requiring that 13 bins in a 120 bins window are above a threshold of 0.2 $I_{VEM}^{est}$ in coincidence on 2 PMTs [@ni01]. The estimated current for a Vertical Equivalent Muon ($I_{VEM}^{est}$) is the reference unit for the calibration of FADC traces signals [@al05]. This trigger has a relatively low rate of about 1.6 Hz, which is the expected rate for double muons for an Auger tank. It is extremely efficient for selecting small but spread signals, typical for high energy distant EAS or for low energy showers, while ignoring single muons background. The second trigger is a 3-fold coincidence of a simple 1.75 $I_{VEM}^{est}$ threshold. This trigger is more noisy, with a rate of about 100 Hz, but it is needed to detect fast signals ($<$ 200 ns) corresponding to the muonic component generated by horizontal showers.\
The T2 trigger is applied in the station controller to select from the T1 signals those likely to have come from EAS and to reduce to about 20 Hz the rate of events to be sent to the central station. All ToT triggers are directly promoted T2 whereas T1 threshold triggers are requested to pass a higher threshold of 3.2 $I_{VEM}^{est}$ in coincidence for 3 PMTs. Only T2 triggers are used for the definition of a T3.\
The probability for a station to pass the trigger requirements strongly depends on the integrated signal. We define this probability, P(S), as the ratio of stations that trigger divided by the number of stations for a given integrated signal. We have measured this probability directly from the data by two different methods. The first method is based on the existence of two pairs of detectors separated by 11 m from each other. The double sampling of signals in near locations provides a way to estimate the number of signals that did not cause a trigger. This method allows direct comparison of P(s) for individual stations. In figure \[fig1\] we show that the two pairs of stations used in this study have the same P(s) within uncertainties. The second method uses this result and assumes that a single P(s) can describe the behavior of all the stations in the surface array. For each event a Lateral Distribution Function (LDF) is fitted to all the stations that have signal. Here the LDF used is a parabola on a log-log scale. The systematic due to the use of a LDF form different from the one described in this conference is of a few percent, within the statistical uncertainties of the method [@ba05]. The LDF is assumed to be cylindrically symmetric so for each event there are regions of constant signal. In each event the stations that did and did not trigger in a given constant signal region can be identified and P(s) computed.
![\[fig1\] The points represent the two pairs of stations from method one with statistical error bars. For method 1 all showers with $\theta$ less then 60 degrees and S(1000) greater than 2 VEM have been used. The dotted line is the fit from method 2 using the same large theta and S(1000) bins. The functional form of the fit is $P(S) = s^{N}/(s^{N}+ s_{50\%}^{N}).$](ICRC_lhenry_fig1.eps){width="60.00000%"}
The agreement between both methods is shown in figure 1. Method one has low statistics so any further dependencies on the trigger probability can not be identified. Method two has high statistics and can further parameterize P(s) into zenith angle ($\theta$) and shower size parameter S(1000) bins. Correct knowledge of the trigger probability is needed for the acceptance estimation [@pa05].\
The stability of the trigger rates is of great importance for a good estimation of the acceptance of the array. The threshold T2 rates are uniform over the present array within a few percent. The ToT T2rates are more spread, since they are sensitive to the charge of the signal that depends on the characteristics of the water in the tank. The decay time of the pulses is a good estimator of the water quality in the tank. It has been shown [@he05] that each tank needs a few months after installation where this decay time slowly decreases then stabilizes to an average value around 65 ns. Once the tanks are stable, the average ToT rate over the array is 1.6 $\pm1$ Hz. The ToT rate is also dependent on temperature, its variation is carefully studied [@be05] and the influence on the higher level triggers is shown in the next section.
Event selection
===============
The main Auger T3 trigger requires the coincidence of 3 tanks which have passed the ToT conditions and meeting the requirement of a minimum of compactness (one of the tanks must have one of its closest neighbors and one of its second closest neighbors triggered). Since the ToT as a local trigger has already very low background (mainly double muons), this so-called 3ToT trigger selects mostly physical events. The rate of this T3 with the present number of working detectors in the array is around 600 events per day, or 1.3 events per triangle of 3 neighboring working stations . This trigger is extremely relevant since 90% of the selected events are showers and is mostly efficient for vertical showers. The other implemented trigger is more permissive. It requires a four-fold coincidence of any T2 with a moderate compactness requirement (among the 4 fired tanks, one can be as far as 6 km away from others within appropriate time window). Such a trigger is absolutely needed to allow for the detection of horizontal showers that generate fast signals and have wide-spread topological patterns. This trigger selects about 400 events per day, but only 2% are real showers.\
A physical trigger (T4) is needed to select only showers from the set of stored T3 data. An official physical trigger is applied offline to select events for zenith angles below 60 degrees. The chosen criteria use two main characteristics of vertical showers. The first one is the compactness of the triggered tanks, the second one is the fact that most FADC traces are spread enough in time to satisfy a ToT condition. It was shown that requiring a 3 ToT compact configuration in an event ensures that more than 99% are showers. The present physics trigger is dual and requires either a compact 3 ToT or a compact configuration of any local trigger called 4C1 (at least one fired station has 3 triggered tanks out of its 6 first neighbours). The tanks satisfying the 3 ToT or 4C1 condition must have their trigger time compatible with the speed of light (with a tolerance of 200 ns to keep very horizontal rare events). With the 3ToT T4 trigger, less than 5% of showers below 60 degrees are lost. The 4C1 trigger , whose event rate is about 2% of the previous one , ensures to keep the 5% of the showers below 60 degrees lost by the other T4 and also selects low energy events above 60 degrees. This can be seen in figure 2 where is shown on the left the distribution in angle of events selected by the different T4s and on the right the energy distribution.
![\[fig2\] Zenith angle (left plot) and energy (right plot) distribution for events satisfying the adopted T4. The red shaded area correspond to events passing 3ToT and the blue dashed area to events passing the 4C1. Those that are both 3ToT and 4C1 are counted in the 3ToT distribution](ICRC_lhenry_fig2.eps){width=".95\textwidth"}
In most selected events, a number of accidental tanks need to be removed. A complete procedure has been defined to fulfill this task. It is based on the definition of a seed that is an elementary triangle (one station with 2 neighbors in a non-aligned configuration). The seed with the highest total signal is used to define a plane front of the shower. Then all stations are examined, and are defined as accidental if their time delay to the front plane propagation is outside a defined time window. Isolated tanks are also removed. For the period of March 2005 for example, the average number of discarded tanks per event is 4 which is compatible with the expected number of accidental tanks for the corresponding number of working tanks, $N_{\mathrm{total}}$, where\
$N_{\mathrm{accidental}} =N_{\mathrm{total}} *\mathrm{ T1\_rate}* \mathrm{T3\_timewindow} = 667 * 100\mathrm{Hz}* 60~\mu \mathrm{s} = 3.96$ per event.\
99% of the obtained events are reconstructed. The definition of a T4 for horizontal showers is more challenging and is still under study [@ne05]. The effect of the temperature on the T4 rate was studied. The number of T4 events per day divided by the number of active triangles of stations has a dependence on temperature of about 1% per degree. This has to be taken into account for the estimation of the acceptance of the array when it is not saturated .
T5 quality Trigger
==================
One further step is needed to compute the acceptance of the detector and build the spectrum. Among the events having passed the T4 trigger, only those that can be reconstructed with a known energy and angular accuracy will be used. This is the task of the T5 quality trigger. Various studies have been performed to identify under which conditions events could satisfy this requirement. The main problem to solve is that if an event is close to the border of the array, a part of the shower is probably missing, and the real core could be outside of the existing array, whereas the reconstructed core will be by construction inside the array, in particular for low multiplicity events. Such events will have wrong core positions, so wrong energies and typically should not pass the quality trigger.\
The adopted T5 requires that the tank with highest signal must have at least 5 working tanks among its 6 closest neighbors at the time of the event and more over, the reconstructed core must be inside an equilateral triangle of working stations. This represents an efficient quality cut by guaranteeing that no crucial information is missed for the shower reconstruction. This study described in [@gh05] evaluates the effect of T5 on the accuracy of the reconstruction and in particular on the signal at 1000 m from shower axis S(1000). The maximum systematic uncertainty in the reconstructed S(1000) due to event sampling into the array or to the effect of a missing internal tank is around 8%.
Conclusion
==========
The hierarchical trigger of the Pierre Auger Observatory has been fully described. This trigger chain allows to decrease the event rate in a single tank from 3 kHz due to mainly background muons up to 3 per day, due to real showers, corresponding to a rejection factor of $10^{8}$.
[99]{} Pierre Auger Collaboration, these proceedings, arg-bertou-X-abs1-he14-oral. D. Nitz for the Pierre Auger Collaboration, ICRC 2001. M. Aglietta [*et al*]{}, these proceedings, usa-allison-PS-abs1-he14-poster. D. Barnhill [*et al*]{}, these proceedings usa-bauleo-PM-abs2-he14-poster. D. Allard [*et al*]{}, these proceedings fra-parizot-E-abs1-he14-poster. I. Allekotte [*et al*]{}, these proceedings, usa-arisaka-K-abs1-he15-poster. Pierre Auger Collaboration, these proceedings,mex-nellen-L-abs1-he14-oral. Pierre Auger Collaboration, these proceedings, ita-ghia-P-abs1-he14-oral. Pierre Auger Collaboration,these proceedings,usa-sommers-P-he14-oral.
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**HAUSDORFF OPERATORS ON LEBESGUE SPACES WITH POSITIVE DEFINITE PERTURBATION MATRICES ARE NON-RIESZ**
**A. R. Mirotin**
[email protected]
<span style="font-variant:small-caps;">Abstract.</span> [We consider generalized Hausdorff operators with positive definite and permutable perturbation matrices on Lebesgue spaces and prove that such operators are not Riesz operators provided they are non-zero.]{}
Key words and phrases. Hausdorff operator, Riesz operator, quasinilpotent operator, compact operator.
Introduction and preliminaries
==============================
The one-dimensional Hausdorff transformation $$(\mathcal{H}_1f)(x) =\int_\mathbb{R} f(xt)d\chi(t),\eqno(1)$$ where $\chi$ is a measure on $\mathbb{R}$ with support $[0,1]$, was introduced by Hardy [@H Section 11.18] as a continuous variable analog of regular Hausdorff transformations (or Hausdorff means) for series. Its modern $n$-dimensional generalization looks as follows: $$(\mathcal{H}f)(x) =\int_{\mathbb{R}^m} \Phi(u)f(A(u)x)du, \eqno(2)$$ where $\Phi:\mathbb{R}^m\to \mathbb{C}$ is a locally integrable function, $A(u)$ stands for a family of non-singular $n\times n$-matrices, $x\in \mathbb{R}^n$, a column vector. See survey articles [@Ls], [@CFW] for historical remarks and the state of the art up to 2014.
To justify this definition the following approach may be suggested. Hardy [@H Theorem 217] proved that (if $\chi$ is a probability measure) the transformation (1) gives rise to a regular generalized limit at infinity of the function $f$ in a sense that if $f$ is continuous on $\mathbb{R},$ and $f(x) \to l$ then $\mathcal{H}_1f(x) \to l$ when $x\to \infty.$ Note that the map $x\mapsto xt$ ($t\ne 0$) is the general form of automorphisms of the additive group $\mathbb{R}$. This observation leads to the definition of a (generalized) Hausdorff operator on a general group $G$ via the automorphisms of $G$ that was introduced and studied by the author in [@JMAA], and [@AddJMAA]. For the additive group $\mathbb{R}^n$ this definition looks as follows.
**Definition 1.** Let $(\Omega,\mu)$ be some $\sigma$-compact topological space endowed with a positive regular Borel measure $\mu,$ $\Phi$ a locally integrable function on $\Omega,$ and $(A(u))_{u\in \Omega}$ a $\mu$-measurable family of $n\times n$-matrices that are nonsingular for $\mu$-almost every $u$ with $\Phi(u) \ne 0.$ We define the *Hausdorff operator* with the kernel $\Phi$ by ($x\in\mathbb{R}^n$ is a column vector) $$(\mathcal{H}_{\Phi, A}f)(x) =\int_\Omega \Phi(u)f(A(u)x)d\mu(u).$$
The general form of a Hausdorff operator given by definition 1 (with an arbitrary measure space $(\Omega,\mu)$ instead of $\mathbb{R}^m$) gives us, for example, the opportunity to consider (in the case $\Omega=\mathbb{Z}^m$) discrete Hausdorff operators [@Forum], [@faa].
As was mentioned above Hardy proved that the Hausdorff operator (1) possesses some regularity property. For the operator given by the definition 1 the multidimensional version of his result is also true as the next proposition shows.
**Proposition 1.** [@faa] *Let the conditions of definition* 1 *are fulfilled. In order that the transformation $\mathcal{H}_{\Phi, A}$ should be regular, i.e. that $f$ is measurable and locally bounded on $\mathbb{R}^n,$ $f(x) \to l$ when $x\to \infty$ should imply $\mathcal{H}_{\Phi, A}f(x) \to l$, it is necessary and sufficient that $\int_\Omega \Phi(u)d\mu(u)=1.$*
So, as for the classic transformation considered by Hardy the Hausdorff transformation in the sense of the definition 1 gives rise to a new family (for various $(\Omega,\mu)$, $\Phi$, and $A(u)$) of regular generalized limits at infinity for functions of $n$ variables.
(For a different approach to justify the definition (2) see [@LK].)
The problem of compactness of Hausdorff operators was posed by Liflyand [@L] (see also [@Ls]). There is a conjecture that nontrivial Hausdorff operator in $L^p(\mathbb{R}^n)$ is non-compact. For the case $p=2$ and for commuting $A(u)$ this hypothesis was confirmed in [@Forum] (and for the diagonal $A(u)$ — in [@JMAA]). Moreover, we conjecture that every nontrivial Hausdorff operator in $L^p(\mathbb{R}^n)$ is non-Riesz.
Recall that a *Riesz operator* $T$ is a bounded operator on some Banach space with spectral properties like those of a compact operator; i. e., $T$ is a non-invertible operator whose nonzero spectrum consists of eigenvalues of finite multiplicity with no limit points other then $0$. This is equivalent to the fact that $T-\lambda$ is Fredholm for every $\lambda \ne 0$ [@Ruston]. For example, a sum of a quasinilpotent and compact operator is Riesz [@Dow Theorem 3.29]. Other interesting characterizations for Riesz operators one can also find in [@Dow].
In this note we prove the above mentioned conjecture for the case where the family $A(u)$ consists of permutable and positive (negative) definite matrices.
The main result
===============
We shall employ three lemmas to prove our main result.
**Lemma 1** [@JMAA] (cf. [@H (11.18.4)], [@BM]). *Let $|\det A(u)|^{-1/p}\Phi(u)\in L^1(\Omega).$ Then the operator $\mathcal{H}_{\Phi, A}$ is bounded in $L^p(\mathbb{R}^n)$ ($1\leq p\le \infty$) and* $$\|\mathcal{H}_{\Phi, A}\|\leq \int_\Omega |\Phi(u)||\det A(u)|^{-1/p}d\mu(u).$$ This estimate is sharp (see theorem 1 in [@faa]).
**Lemma 2** [@faa] (cf. [@BM]). *Under the conditions of Lemma* 1 *the adjoint for the Hausdorff operator in $L^p(\mathbb{R}^n)$ has the form* $$(\mathcal{H}_{\Phi, A}^*f)(x) =\int_\Omega \overline{\Phi(v)}|\det A(v)|^{-1}f(A(v)^{-1}x)d\mu(v).$$ *Thus, the adjoint for a Hausdorff operator is also Hausdorff.*
**Lemma 3.** *Let $S$ be a boll in $\mathbb{R}^n$, $q\in [1,\infty)$, and $R_{q,S}$ denotes the restriction operator $L^q(\mathbb{R}^n)\to L^q(S)$, $f\mapsto f|S$. If we as usual identify the dual of $L^q$ with $L^p$ ($1/p+1/q=1$), then the adjoint $R_{q,S}^*$ is the operator of natural embedding $L^p(S)\hookrightarrow L^p(\mathbb{R}^n)$*.
Proof. For $g\in L^p(S)$ let $$g^*(x)=\begin{cases}
g(x)\ \mathrm{for}\ x\in S,\\
0 \quad \quad \mathrm{for}\ x\in \mathbb{R}^n\setminus S.
\end{cases}$$ Then the map $g\mapsto g^*$ is the natural embedding $L^p(S)\hookrightarrow L^p(\mathbb{R}^n)$.
By definition, the adjoint $R_{q,S}^*: L^q(S)^*\to L^q(\mathbb{R}^n)^*$ acts according to the rule $$(R_{q,S}^*\Lambda)(f)=\Lambda(R_{q,S}f)\quad \ (\Lambda\in L^q(S)^*, f\in L^q(\mathbb{R}^n)).$$ If we (by the Riesz theorem) identify the dual of $L^q(S)$ with $L^p(S)$ via the formula $\Lambda\leftrightarrow g$, where $$\Lambda(h)=\int_S g(x)h(x)dx\quad \ (g\in L^p(S), h\in L^q(S)),$$ and analogously for the the dual of $L^q(\mathbb{R}^n)$, then the definition of $R_{q,S}^*$ takes the form $$\int_{\mathbb{R}^n}(R_{q,S}^*g)(x)f(x)dx=\int_S g(x)(f|S)(x)dx.$$ But $$\int_S g(x)(f|S)(x)dx=\int_{\mathbb{R}^n}g^*(x)f(x)dx\quad (f\in L^q(\mathbb{R}^n)).$$ The right-hand side of the last formula is the linear functional from $L^q(\mathbb{R}^n)^*$. If we (again by the Riesz theorem) identify this functional with the function $g^*$, the result follows.$\Box$
**Theorem 1.** *Let $A(v)$ be a commuting family of real positive definite $n\times n$-matrices ($v$ runs over the support of $\Phi$), and $(\det A(v))^{-1/p}\Phi(v)\in L^1(\Omega).$ Then every nontrivial Hausdorff operator $\mathcal{H}_{\Phi, A}$ in $L^p(\mathbb{R}^n)$ ($1\leq p\le \infty$) is a non-Riesz operator (and in particular it is non-compact).*
Proof. Assume the contrary. Since $A(u)$ form a commuting family, there are an orthogonal $n\times n$-matrix $C$ and a family of diagonal non-singular real matrices $A'(u)$ such that $A'(u) = C^{-1}A(u)C$ for $u\in \Omega.$ Consider the bounded and invertible operator $\widehat{C}f(x):=f(Cx)$ in $L^p(\mathbb{R}^n).$ Because of the equality $\widehat{C}\mathcal{H}_{\Phi, A}\widehat{C}^{-1}=\mathcal{H}_{\Phi, A'},$ operator $\mathcal{H}:=\mathcal{H}_{\Phi, A'}$ is Riesz and nontrivial, too.
Note that each open hyperoctant in $\mathbb{R}^n$ is $A(u)$-invariant. Chose such an open $n$-hyperoctant $U$ that $\mathcal{K}:=\mathcal{H}|L^p(U)\ne 0.$ Then $L^p(U)$ is a closed $\mathcal{K}$-invariant subspace of $L^p(\mathbb{R}^n)$ and $\mathcal{K}$ is a nontrivial Riesz operator in $L^p(U)$ by [@Dow p. 80, Theorem 3.21].
Let $1\leq p<\infty.$ To get a contradiction, we shall use the modified $n$-dimensional Mellin transform for the $n$-hyperoctant $U$ in the form
$$(\mathcal{M}f)(s):=\frac{1}{(2\pi)^{n/2}}\int_{U}|x|^{-\frac{1}{q}+is}f(x)dx,\quad s\in \mathbb{R}^n$$ Here and below we assume that $|x|^{-\frac{1}{q}+ is}$ $:= \prod_{j=1}^n |x_j|^{-\frac{1}{q}+ is_j}$ where $|x_j|^{-\frac{1}{q}+ is_j}:=$ $\exp((-\frac{1}{q}+ is_j)\log |x_j|)$. The map $\mathcal{M}$ is a bounded operator between $L^p(U)$ and $L^q(\mathbb{R}^n)$ for $1\leq p\leq 2$ ($1/p+1/q=1).$ It can be easily obtained from the Hausdorff–Young inequality for the $n$-dimensional Fourier transform by using the exponential change of variables (see [@BPT]). Let $f\in L^p(U).$ First assume that $|y|^{-1/q}f(y)\in L^1(U).$ Then as in the proof of theorem 1 from [@Forum], using the Fubini–Tonelli’s theorem and integrating by substitution $x=A(u)'^{-1}y,$ yield the following $$(\mathcal{MK}f)(s)=
\varphi(s)(\mathcal{M}f)(s)\ (s\in \mathbb{R}^n),$$ where the function $\varphi$ (“the symbol of the the Hausdorff operator” [@Forum]) is bounded and continuous on $\mathbb{R}^n.$
Thus, $$\mathcal{MK}f=\varphi \mathcal{M}f. \eqno(3)$$ By continuity the last equality is valid for all $f\in L^p(U).$
Let $1\leq p\leq 2.$ There exists a constant $c> 0,$ such that the set $\{s\in \mathbb{R}^n: |\varphi(s)|>c\}$ contains an open ball $S.$ Formula (3) implies that $$M_{\psi}R_{q,S}\mathcal{M}\mathcal{K}=R_{q,S}\mathcal{M},$$ where $\psi=(1/\varphi)|S,$ $M_{\psi}$ denotes the operator of multiplication by $\psi$, and $R_{q,S}: L^q(\mathbb{R}^n)\to L^q(S),$ $f\mapsto f|S$ — the restriction operator. Let $T=R_{q,S}\mathcal{M}.$ Passing to the conjugates gives $$\mathcal{K}^*T^*M_{\psi}^*=T^*.$$ By [@FNR3 Theorem 1] this implies that the operator $T^*=\mathcal{M}^*R_{q,S}^*$ has finite rank. By Lemma 3 $R_{q,S}^*$ is the operator of natural embedding $L^p(S)\hookrightarrow L^p(\mathbb{R}^n)$.
For $g\in L^p(\mathbb{R}^n)$ consider the operator $$(\mathcal{M}'g)(x):=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^n}|x|^{-\frac{1}{q}+is}g(s)ds,\ x\in U.$$ This is a bounded operator taking $L^p(\mathbb{R}^n)$ into $L^q(U).$ Indeed, since $$|x|^{-\frac{1}{q}+ is}=\prod_{j=1}^n |x_j|^{-\frac{1}{q}}\exp(is_j\log |x_j|),$$ we have $$(\mathcal{M}'g)(x)=|x|^{-\frac{1}{q}}\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^n}\exp(is\cdot \log|x|)g(s)ds,\ x\in U,$$ where $|x|:=|x_1|\dots |x_n|$, $\log |x|:=(\log |x_1|,\dots,\log |x_n|)$, and the dot denotes the inner product in $\mathbb{R}^n$. Thus, we can express the function $\mathcal{M}'g$ via the Fourier transform $\widehat{g}$ of $g$ as follows: $(\mathcal{M}'g)(x)=|x|^{-1/q}\widehat{g}(-\log |x|)$, ($x\in U$) and therefore $$\|\mathcal{M}'g\|_{L^q(U)}=\left(\int_U |x|^{-1}|\widehat{g}(-\log |x|)|^q dx\right)^{1/q}.$$ Putting here $y_j:=-\log|x_j|$ ($j=1,\dots,n$) and taking into account that the Jacobian of this transformation is $$\frac{\partial(x_1,\dots,x_n)}{\partial(y_1,\dots,y_n)}=\det \mathrm{diag}(e^{-y_1},\dots,e^{-y_n})=\exp\left(-\sum_{j=1}^n y_j \right),$$ we get by the Hausdorff–Young inequality that $$\|\mathcal{M}'g\|_{L^q(U)}=\|\widehat{g}\|_{L^q(\mathbb{R}^n)}\le \|g\|_{L^p(\mathbb{R}^n)}.$$
If $f\in L^p(U)$, and $f(x)|x|^{-1/q}\in L^1(U)$, $g\in L^p(\mathbb{R}^n)\cap L^1(\mathbb{R}^n)$ the Fubini–Tonelli’s theorem implies $$\int_{\mathbb{R}^n}(\mathcal{M}f)(s)g(s)ds=\int_{U}f(x)(\mathcal{M}'g)(x)dx.$$ Since the bilinear form $(\varphi,\psi)\mapsto\int \varphi \psi d\mu$ is continuous on $L^p(\mu)\times L^q(\mu)$, the last equality is valid for all $f\in L^p(U)$, $g\in L^p(\mathbb{R}^n)$ by continuity. So, $\mathcal{M}'=\mathcal{M}^*$.
It was shown above that the restriction of the operator $\mathcal{M}^*$ to $L^p(S)$ has finite rank. Since $\mathcal{M}^*$ can be easily reduced to the Fourier transform, this is contrary to the Paley–Wiener theorem on the Fourier image of the space $L^2(S)$ ($L^2(S)\subseteq L^p(S)$) (see, i. g., [@SW Theorem III.4.9]).
Finally, if $2<p\le \infty$ one can use duality arguments. Indeed, by lemma 2 the adjoint operator $\mathcal{H}_{\Phi, A'}^*$ (as an operator in $L^q(\mathbb{R}^n)$) is also of Hausdorff type. More precisely, it equals to $\mathcal{H}_{\Psi, B},$ where $B(u)= A(u)'^{-1}= \mathrm{diag}(1/a_1(u),\dots,1/a_n(u))$, $\Psi(u)=\Phi(u)|\det A(u)'^{-1}|=$ $\Phi(u)/a(u)$. It is easy to verify that $\mathcal{H}_{\Psi, B}$ satisfies all the conditions of theorem 1 (with $q,$ $\Psi$ and $B$ in place of $p,$ $\Phi$ and $A$ respectively). Since $1\le q<2$, the operator $\mathcal{H}_{\Psi, B}$ is not a Riesz operator in $L^q(\mathbb{R}^n),$ and so is $\mathcal{H}_{\Phi, A}$, because $T$ is a Riesz operator if only if its conjugate $T^*$ is a Riesz operator [@Dow p. 81, Theorem 3.22].$\Box$
Corollaries and examples
========================
For the next corollary we need the following
**Lemma 4**. *Let $J:X\to X$ be a linear isometry of a Banach space $X$. A bounded operator $T$ on $X$ which commutes with $J$ is a Riesz operator if and only if such is $JT$.*
Proof. We use the fact that an operator $T$ is a Riesz operator if and only if it is asymptotically quasi-compact [@Ruston] (see also [@Dow Theorem 3.12]). This means that $$\lim\limits_{n\to\infty}\left(\inf\limits_{C\in \mathcal{K}(X)}\|T^n-C\|^{1/n}\right)=0,$$ where $\mathcal{K}(X)$ denotes the ideal of compact operators in $X$ (Ruston condition). Since $(UT)^n=U^nT^n$ and $$\inf\limits_{C\in \mathcal{K}(X)}\|(UT)^n-C\|^{1/n}=\inf\limits_{C\in \mathcal{K}(X)}\|T^n-U^{-n}C\|^{1/n}=\inf\limits_{C'\in\mathcal{ K}(X)}\|T^n-C'\|^{1/n},$$ the result follows.$\Box$
**Corollary 1**. *Let $A(v)$ be a commuting family of real negative definite $n\times n$-matrices ($v$ runs over the support of $\Phi$), and $(\det A(v))^{-1/p}\Phi(v)\in L^1(\Omega).$ Then every nontrivial Hausdorff operator $\mathcal{H}_{\Phi, A}$ in $L^p(\mathbb{R}^n)$ ($1\leq p\le\infty$) is non-Riesz (and in particular it is non-compact).*
Proof. Let $Jf(x):=f(-x)$. Since $-A(v)$ form a commuting family of real positive definite $n\times n$-matrices, and $\mathcal{H}_{\Phi, A}=J\mathcal{H}_{\Phi, -A}$, this corollary follows from lemma 4 and theorem 1. $\Box$
**Corollary 2.** *Under the conditions of theorem 1 or corollary 1 Hausdorff operator $H_{\Phi, A}$ is not the sum of the quasinilpotent and compact operators.*
Indeed, as was mentioned in the introduction, the sum of the quasinilpotent and compact operators is a Riesz operator.
**Corollary 3.** *Let $n=1$, $\phi:\Omega\to \mathbb{C}$ and let $a(v)$ be a real and positive (negative) function on $\Omega$ ($v$ runs over the support of $\phi$), and $|a(v)|^{-1/p}\phi(v)\in L^1(\Omega).$ Then every nontrivial Hausdorff operator $$\mathcal{H}_{\phi, a}f(x)=\int_{\Omega} \phi(u)f(a(u)x)d\mu(u)\ (x\in \mathbb{R})$$ in $L^p(\mathbb{R})$ ($1\leq p\le\infty$) is a non-Riesz operator (and in particular it is non-compact).*
**Example 1**. Let $t^{-1/q}\psi(t)\in L^1(0,\infty).$ Then by corollary 3 the operator $$\mathcal{H}_\psi f(x)=\int_0^\infty\frac{\psi(t)}{t}f\left(\frac{x}{t}\right)dt$$ is a non-Riesz operator in $L^p(\mathbb{R})$ ($1\leq p\le\infty$) provided it is non-zero.
**Example 2**. Let $(t_1t_2)^{-1/p}\psi_2(t_1,t_2)\in L^1(\mathbb{R}_+^2).$ Then by theorem 1 the operator $$\mathcal{H}_{\psi_2} f(x_1,x_2)=\frac{1}{x_1x_2}\int_0^\infty\!\int_0^\infty\psi_2\left(\frac{t_1}{x_1}, \frac{t_2}{x_2}\right) f(t_1,t_2)dt_1dt_2$$ is a non-Riesz operator in $L^p(\mathbb{R}_+^2)$ ($1\leq p\le\infty$) provided it is non-zero.
[99]{}
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We consider the complexity of equivalence and learning for multiplicity tree automata, i.e., weighted tree automata over a field. We first show that the equivalence problem is logspace equivalent to polynomial identity testing, the complexity of which is a longstanding open problem. Secondly, we derive lower bounds on the number of queries needed to learn multiplicity tree automata in Angluin’s exact learning model, over both arbitrary and fixed fields.
@habrardlearning give an exact learning algorithm for multiplicity tree automata, in which the number of queries is proportional to the size of the target automaton and the size of a largest counterexample, represented as a tree, that is returned by the Teacher. However, the smallest tree-counterexample may be exponential in the size of the target automaton. Thus the above algorithm does not run in time polynomial in the size of the target automaton, and has query complexity exponential in the lower bound.
Assuming a Teacher that returns minimal DAG representations of counterexamples, we give a new exact learning algorithm whose query complexity is quadratic in the target automaton size, almost matching the lower bound, and improving the best previously-known algorithm by an exponential factor.
author:
- |
Ines Marusic [email protected]\
James Worrell [email protected]\
Department of Computer Science\
University of Oxford\
Parks Road, Oxford OX1 3QD, UK
bibliography:
- 'references.bib'
title: |
Complexity of Equivalence and Learning for\
Multiplicity Tree Automata
---
exact learning, query complexity, multiplicity tree automata, Hankel matrices, DAG representations of trees
| {
"pile_set_name": "ArXiv"
} |
---
author:
-
title: Amorphous Dynamic Partial Reconfiguration with Flexible Boundaries to Remove Fragmentation
---
/\#1[[**[\#1.]{}**]{}]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We analyse aspects of symmetry breaking for Moyal spacetimes within a quantisation scheme which preserves the twisted Poincaré symmetry. Towards this purpose, we develop the LSZ approach for Moyal spacetimes. The latter gives a formula for scattering amplitudes on these spacetimes which can be obtained from the corresponding ones on the commutative spacetime. This formula applies in the presence of spontaneous breakdown of symmetries as well. We also derive Goldstone’s theorem on Moyal spacetime. The formalism developed here can be directly applied to the twisted standard model.'
author:
- 'A. P. Balachandran[^1]'
- 'T. R. Govindarajan[^2]'
- 'Sachindeo Vaidya[^3]'
title: |
[IISc-CHEP/01/09]{}\
[IMSC-2009/01/01]{}\
[SU-4252-883]{}\
Spontaneous Symmetry Breaking In Twisted Noncommutative Quantum Theories
---
Introduction
============
Spontaneous breaking of a continuous symmetry involves a subtle interplay between an infinite number of degrees of freedom, local and spacetime symmetries, dimension of spacetime, and the notion of (non-)locality of interaction. Naturally one would suspect that the phenomenon of spontaneous symmetry breaking (SSB) leads to different physics in the context of quantum field theories on the Groenewold-Moyal (GM) plane, when the idea of locality is altered, albeit in a very precise sense: pointwise multiplication of two functions is replaced by [*star*]{}-multiplication, which is no longer commutative, and is in addition non-local. New phases and soliton solutions appear making the dynamics richer[@gubser; @stripe; @trg1; @trg2].
Writing quantum field theories on such spaces requires some care, if one wishes to discuss questions related to Poincaré invariance. To give up this spacetime symmetry almost entirely (which is what conventional quantization does) seems too heavy a price, as it affects the notion of identity of particles (“of two identical particles in one frame should describe two identical particles in [*all*]{} reference frames”), and leads to unacceptable coupling between UV- and IR- degrees of freedom as well[@minwalla]. The program of twisted quantization initiated in [@bmpv; @replyto] on the the other hand, avoids these pitfalls: Poincaré invariance can be maintained, a generalized notion of identical particles can be defined, and UV and IR degrees of freedom decouple nicely [@bpq], thus rekindling the hope that phenomenologically interesting models can be constructed. Indeed one can construct quantum gauge theories with arbitrary gauge groups consistently [@bpqv1].
In this paper, we address the issue of SSB and Higgs phenomenon in twisted quantum theories, and demonstrate signatures of noncommutativity. Our general formulation applies to an arbitrary group $G$ breaking to a subgroup $H$. The extension to the (noncommutative) Standard Model and beyond-Standard Model physics is conceptually straightforward, and will be discussed as well. Such physics merits a more elaborate investigation which we reserve for later work.
This paper is organised as follows. In Section 2, we will review twisted quantization on noncommutative spaces and gauge theories based on this formalism. Section 3 will elaborate the LSZ formalism for twisted quantisation and discuss in detail the Gell-Mann-Low formula and its modifications on the GM plane. In Section 4, we will summarise our rules for twisted quantum field theories followed by application of the same to spontaneously broken theories and Higgs mechanism on the GM plane in Sec.5. Our conclusions and future outlook are in Section 6.
Twisted Quantization
====================
The algebra of functions ${\cal A}_\theta({\mathbb R}^N)$ on the GM plane consists of smooth functions on ${\mathbb R}^N$, with the multiplication map $$\begin{aligned}
m_\theta: {\cal A}_\theta ({\mathbb R}^N) \otimes {\cal A}_\theta
({\mathbb R}^N) &\rightarrow& {\cal A}_\theta ({\mathbb R}^N)\,,
\nonumber \\
\alpha \otimes \beta &\rightarrow& \alpha \;e^{\frac{i}{2}
\overleftarrow{\partial}_\mu \theta^{\mu \nu}
\overrightarrow{\partial}_\nu} \ \beta := \alpha \ast \beta
\label{starmult}\end{aligned}$$ where $\theta^{\mu \nu}$ is a constant antisymmetric tensor.
Let $$F_\theta = e^{\frac{i}{2} \partial_\mu \otimes \theta^{\mu \nu}
\partial_\nu} = {\rm ``Twist \; element"}.
\label{twistelt}$$ Then $$m_\theta (\alpha \otimes \beta) = m_0 [F_\theta \alpha \otimes
\beta] \label{starmult1}$$ where $m_0$ is the point-wise multiplication map, also defined by (\[starmult\]).
The usual action of the Lorentz group ${\cal L}$ is not compatible with $\ast$-multiplication: transforming $\alpha$ and $\beta$ separately by an arbitrary group element $\Lambda \in {\cal L}$ and then $\ast$-multiplying them is not the same as transforming their $\ast$-product. In other words, the usual coproduct $\Delta_0(\Lambda)
= \Lambda \otimes \Lambda$ on the group algebra ${\mathbb C}\cal{L}$ of $\cal{L}$ is not compatible with $m_\theta$. But a new coproduct $\Delta_\theta$ obtained using the twist is compatible, where $$\Delta_\theta(\Lambda)\,=\,F_\theta^{-1}\,\Delta_0 (\Lambda) F_\theta.
\label{twistedcoprod}$$ Here $\partial_\mu \otimes \theta^{\mu \nu}\partial_\nu$ in $F_\theta$ is to be replaced by $~-~ P_\mu \otimes \theta^{\mu \nu} P_\nu$ where $P_\mu$ are translation generators: we are dealing with ${\cal{P}}_\theta \otimes {\cal{P}}_\theta$ where ${\cal{P}}_\theta$ is a Hopf algebra associated with the Poincaré group algebra ${\mathbb C}{\cal{P}}$ with the coproduct (\[twistedcoprod\]).
This twisted coproduct does not preserve standard (anti-)symmetrization, because it does not commute with the usual flip operator $\tau_0$ defined by $\tau_0:(\alpha \otimes \beta) ~=~
(\beta\otimes\alpha)$:
$$\Delta_\theta(\Lambda) \tau_0 \neq \tau_0 \Delta_\theta (\Lambda).$$
On the other hand, it does preserve twisted (anti-)symmetrization, defined using a new flip operator $\tau_\theta = F_\theta^{-1}\,\tau_0
(\Lambda) F_\theta$: $$\Delta_\theta(\Lambda) \tau_\theta = \tau_\theta \Delta_\theta (\Lambda).$$
Thus in noncommutative quantum theory, the usual fermions/bosons do not make sense, but twisted ones do. They are obtained from the projectors $S_\theta, A_\theta$: $$S_\theta~=~\frac{{\bf 1}~+~\tau_\theta}{2}, \qquad A_\theta~=~
\frac{{\bf 1}~-~\tau_\theta}{2}.$$
Quantum Fields
--------------
A quantum field $\chi$ on evaluation at a spacetime point (or more generally on pairing with a test function) gives an operator acting on a Hilbert space. A field at $x_1$ acting on the vacuum gives a one-particle state centered at $x_1$. When we write $\chi(x_1)\,\chi(x_2)$ we mean $(\chi\otimes\chi)(x_1,x_2)$. Acting on the vacuum we generate a two-particle state, where one particle is centered at $x_1$ and the other at $x_2$.
If $a_p$ is the annihilation operator of the free second-quantized field $\phi_\theta$ on ${\cal A}_\theta({\mathbb R}^N)$, we want, as in standard quantum field theory, $$\begin{aligned}
\langle 0 |\phi_\theta(x) a^\dagger_p |0\rangle &=& e_p(x), \\
\frac{1}{2}\langle 0 |\phi_\theta(x_1) \phi_\theta(x_2) a^\dagger_q
a^\dagger_p |0\rangle &=& \left(\frac{{\bf 1} \pm
\tau_\theta}{2}\right)(e_p \otimes e_q)(x_1,x_2) \nonumber \\
&\equiv& (e_p \otimes_{S_\theta,A_\theta} e_q)(x_1,x_2)
\label{tbasis} \end{aligned}$$ where $e_p(x) = e^{-i p \cdot x}$.
This compatibility between twisted statistics and Poincaré invariance has profound consequences for commutation relations. For example when the states are labeled by momenta, we have, from exchanging $p$ and $q$ in (\[tbasis\]), $$|p, q\rangle_{S_\theta,A_\theta} =\ \pm\,e^{ i \theta^{\mu\nu}p_\mu
q_\nu}\,|q,p \rangle_{S_\theta,A_\theta}.$$ This is the origin of the commutation relations $$\begin{aligned}
a_p^\dagger\,a_q^\dagger\,&=& \pm e^{ i \theta^{\mu\nu}p_\mu
q_\nu}\,a_q^\dagger\,a_p^\dagger \, , \\
a_p a_q &=& \pm e^{ i \theta^{\mu \nu} p_\mu q_\nu} a_q a_p \, .\end{aligned}$$
Gauge Theories
--------------
The algebra ${\cal A}_\theta({\mathbb R}^N)$, regarded as a vector space, is a module for ${\cal A}_0({\mathbb R}^N)$. This observation is of central importance to us, as it allows us to write gauge theories based on arbitrary gauge groups (as opposed to just $U(N)$). We can show this as follows.
For any $\alpha \in {\cal A}_\theta({\mathbb R}^N)$, we can define two representations $\hat{\alpha}^{L,R}$ acting on ${\cal
A}_\theta({\mathbb R}^N)$: $$\hat{\alpha}^L \xi = \alpha * \xi, \quad \hat{\alpha}^R \xi = \xi
* \alpha \quad {\rm for} \quad \xi \in {\cal A}_\theta({\mathbb
R}^N) \ ,$$ where $*$ is the GM product defined by Eq.(\[starmult\]) (or, equivalently, by Eq.(\[starmult1\])). The maps $ \alpha \rightarrow
\hat{\alpha}^{L,R}$ have the properties $$\begin{aligned}
\hat{\alpha}^L \hat{\beta}^L &=& (\hat{\alpha}\hat{\beta})^L, \\
\hat{\alpha}^R \hat{\beta}^R &=& (\hat{\beta}\hat{\alpha})^R, \label{right}\\
{[}\hat{\alpha}^L, \hat{\beta}^R] &=& 0. \label{LRcommute}\end{aligned}$$ The reversal of $\hat{\alpha}, \hat{\beta}$ on the right-hand side of (\[right\]) means that for position operators, $${[}\hat{x}^{\mu L}, \hat{x}^{\nu L}] = i \theta^{\mu \nu} =
-[\hat{x}^{\mu R}, \hat{x}^{\nu R}].$$ Hence in view of (\[LRcommute\]), $$\hat{x}^{\mu c} = \frac{1}{2} \left( \hat{x}^{\mu L} + \hat{x}^{\mu R}
\right)$$ generates a representation of the commutative algebra ${\cal
A}_0({\mathbb R}^N)$: $${[}\hat{x}^{\mu c}, \hat{x}^{\nu c}] = 0.$$
We regard elements of the gauge group ${\cal G}$ as maps from ${\cal
A}_0({\mathbb R}^N)$ to the Lie group $G$. $$g:~\hat{x^c}\longrightarrow g(\hat{x^c})~\in G$$ In cases of interest, $G$ is a compact connected Lie group . For convenience, we also think of $G$ concretely in terms of the defining finite-dimensional faithful representation by linear operators.
Fields which transform non-trivially under ${\cal G}$ are modules over ${\cal A}_\theta ({\mathbb R}^N)$. If a $d$-dimensional representation of $G$ is involved, they can be elements of ${\cal A}_\theta ({\mathbb
R}^N) \otimes {\mathbb C}^d$. Compatibility of gauge transformations (on these modules) with the $\ast$-product requires us to twist the coproduct on the gauge group too. The new coproduct is $$\Delta_\theta (g(\hat{x}^c)) = F_\theta^{-1} [g(\hat{x}^c) \otimes
g(\hat{x}^c)] F_\theta \;.
\label{twistedgauge}$$
This deformation of the coproduct for gauge transformations is neccessary if we want to be able to construct gauge scalars, and other composite operators (see [@bpqv1] for details).
Finally, we need to understand how to define covariant derivative $D_\mu$. To this end, consider for simplicity a free charged scalar field $\phi_\theta(x)$, $$\phi_\theta(x) = \int d\mu(p) (a_p e^{-i p \cdot x} + b^\dagger_p
e^{i p \cdot x}),\quad {\rm where} \quad d \mu(p) \equiv \frac{d^3
p}{2p_0}, \quad p_0=\sqrt{\vec{p}^2 + m^2}.$$ We require that the field $\phi_\theta$ obeys twisted statistics in Fock space: $$\begin{aligned}
a_p a_q &=& e^{i p \wedge q} a_q a_p , \quad {\rm where} \quad p
\wedge q \equiv p_\mu \theta^{\mu\nu} q_\nu, \nonumber \\
a_p a^\dagger _q &=& e^{-i p \wedge q} a^\dagger _q a_p + 2p_0
\delta^{(3)}(p-q)\end{aligned}$$ and similarly for $b(p), b^\dagger(p)$. The twisted operators $a(p),
a^\dagger(p),b(p),b^\dagger(p)$ can be realized in terms of untwisted Fock space operators $c(p),d(p)$ and their adjoints through the well-known “dressing transformations" [@grossefaddeev] $$\begin{aligned}
a_p &=& c_p e^{-\frac{i}{2} p \wedge P}, \qquad \qquad a^\dagger _p= c^\dagger _p
e^{\frac{i}{2} p \wedge P}, \nonumber \\
b_p &=& d_p e^{-\frac{i}{2} p \wedge P}, \qquad \qquad b^\dagger _p= d^\dagger _p
e^{\frac{i}{2} p \wedge P}, {\rm where} \\
P_\mu &=& \int d\mu(p) p_\mu \left(a^\dagger_p a_p + b^\dagger_p
b_p \right) = \int d\mu(p) p_\mu \left(c^\dagger_p c_p + d^\dagger_pd_p \right)\nonumber \\
&=& {\rm total \,
momentum\, operator}.\end{aligned}$$ Then $\phi_\theta(x)$ may be written in terms of the ordinary or commutative fields $\phi_0$ as $$\phi_\theta(x) = \phi_0(x) e^{\frac{1}{2}\overleftarrow{\partial} \wedge P},
\quad \overleftarrow{\partial} \wedge P \equiv
\overleftarrow{\partial}_\mu \theta^{\mu\nu} P_\nu
\label{twistphi}$$
As ${\cal A}_\theta({\mathbb R}^N)$ is a module for ${\cal
A}_0({\mathbb R}^N)$, we require that the covariant derivative respects this property. In addition, we also require that in quantum theory, the covariant derivative preserves statistics, and also Poincaré and gauge symmetries. The only possibility that satisfies these requirements is $$D_\mu \phi_\theta = ((D_\mu)_0 \phi_0)e^{\frac{1}{2} \overleftarrow{\partial}
\wedge P} .
\label{twistD}$$ where $(D_\mu)_0 = \partial_\mu + (A_\mu)_0$ and $(A_\mu)_0$ is the commutative gauge field, depending on $\hat{x}^c$ only.
The commutator of two covariant derivatives gives us the curvature: $$[D_\mu , D_\nu]\phi = ([(D_\mu)_0, (D_\nu)_0])e^{\frac{1}{2}
\overleftarrow{\partial} \wedge P} = [(F_{\mu\nu})_0 \phi_0]e^{\frac{1}{2}
\overleftarrow{\partial} \wedge P}.$$
The field strength $(F_{\mu\nu})_0$ transforms correctly (i.e. covariantly) under gauge transformations, so we can use it to construct the Hamiltonian of the quantum gauge theory. Pure gauge theories on the GM plane are thus identical to their commutative counterparts.
Below we will outline an approach to scattering theory of twisted fields based on the LSZ formalism. In that approach, (\[twistphi\]) and (\[twistD\]) are true in the fully interacting case as well. Thus these equations are valid with $P_\mu$ being the [*total*]{} four momentum of all fields including interactions.
We will discuss spontaneous symmetry breakdown in the presence of twists later. Here we remark only the following. If the connected, compact Lie group $G$ is the gauged group and it is spontaneously broken to the gauge theory of the subgroup $H$, then the vector fields acquiring mass are to be twisted. Only the gauge fields of $H$ escape the twist.
The LSZ approach to the scattering theory of such interactions appears to be more streamlined and non-perturbative as compared to our earlier treatments[@bpq]. There we used the interaction representation. In this representation, the free Hamiltonian is not twisted whereas the interaction Hamiltonian is $$H_\theta^I = \int d^3 x [{\cal H}_\theta^{M-G} + {\cal H}_\theta^{G}]$$ Here ${\cal H}_\theta^{M-G}$ and ${\cal H}_\theta^{G}$ correspond to the interaction Hamiltonian densities with matter and gauge fields, and with gauge fields alone respectively. The $\theta$- dependence of the interaction representation S-matrix disappears when ${\cal{H}}^G_\theta~=~0$, but that is not the case when both ${\cal{H}}_\theta ^M$ and ${\cal{H}}_\theta^G$ are present. Scattering processes that involve cross terms between ${\cal
H}_\theta^{M-G}$ and ${\cal H}_\theta^{G}$, like the $qg \rightarrow
qg$ scattering in QCD, show effects of noncommutativity.
As we will later point out, for reasons not well understood, for $\theta ^{\mu\nu}\neq 0$, the LSZ S-matrix differs from the interaction representation S-matrix and leads to different cross sections.
The LSZ Formalism for Twisted Quantum Theories
==============================================
In the next two subsections below, we review scattering theory, including the LSZ formalism for standard (untwisted fields). We then generalise the discussion to the twisted case.
Formal scattering theory
------------------------
In standard scattering theory, the Hamiltonian $H$ is split into a “free” Hamiltonian $H_0$ and an “interaction” piece $H_I$, $$H = H_0 + H_I,$$ and $H_0$ is used to define the states in the infinite past and future. Then the states at $t=0$ which in the infinite past (future) become states evolving by $H_0$ are the in (out) states: $$\begin{aligned}
e^{-i H T_-}|\psi; {\rm in}\rangle &\stackrel{T_- \rightarrow -\infty}
\longrightarrow& e^{-i H_0 T_-} |\psi; {\rm F}\rangle, \quad {\rm F}
\equiv {\rm free} \\
e^{-i H T_+}|\psi; {\rm out}\rangle &\stackrel{T_+ \rightarrow +\infty}
\longrightarrow& e^{-i H_0 T_+} |\psi; {\rm F}\rangle.\end{aligned}$$
Hence $$\begin{aligned}
|\psi; {\rm in}\rangle &=& \Omega_+ |\psi;{\rm F}\rangle, \\
|\psi; {\rm out}\rangle &=& \Omega_- |\psi;{\rm F}\rangle, \\
\Omega_\pm &\equiv& e^{i H T_\mp} e^{-i H_0 T_\mp}, \quad {\rm as} \quad
T_\mp \rightarrow \mp \infty, \\
&=& {\rm M\phi ller}~{\rm operators}.\nonumber\end{aligned}$$
We see that $$\begin{aligned}
e^{i H \tau} \Omega_\pm &=& \lim_{T_\mp \rightarrow \mp \infty} e^{i H
T_\mp } e^{-i H_0 (T_\mp~-~\tau)} , \\
&=& \Omega_\pm e^{i H_0 \tau}\end{aligned}$$ and $$|\psi; {\rm out} \rangle = \Omega_- \Omega_+^\dagger |\psi; {\rm in}\rangle$$ If the incoming state is $|k_1, k_2, \cdots k_N; {\rm F}\rangle$, it follows that $$|k_1, k_2, \cdots k_N; {\rm in}\rangle = \Omega_+ |k_1, k_2, \cdots
k_N; {\rm F}\rangle$$ has eigenvalue $\sum k_{i0}$ for the [*total*]{} Hamiltonian $H$. A similar result is true for $$|k_1, k_2, \cdots k_N; {\rm out}\rangle = \Omega_- |k_1, k_2, \cdots
k_N; {\rm F}\rangle .$$ We note that the scattering amplitude is $$\langle \xi; {\rm out} | \psi; {\rm in}\rangle = \langle \xi; {\rm
in}|\Omega_+ \Omega_-^\dagger |\psi; {\rm in}\rangle.$$ In other words, the LSZ $S$-matrix is $$S = \Omega_+ \Omega_-^\dagger, \quad |\psi; {\rm out} \rangle =
S^\dagger |\psi; {\rm in}\rangle.$$ Between the “free” states, the $S$-operator is different: $$\langle \xi; {\rm out} | \psi; {\rm in}\rangle = \langle \xi; {\rm
F}|\Omega_-^\dagger \Omega_+ |\psi; {\rm F}\rangle.$$
The LSZ formalism works exclusively with the in- and out-states, as Haag’s theorem shows that $\Omega_\pm$ do not exist in quantum field theories. The creation-annihilation operators $c_k^{\rm in (out)
\dagger}, c_k^{\rm in (out)}$ are introduced to create states $|k_1,k_2,\cdots k_N; {\rm in(out)}\rangle$ from the vacuum. The in- and out- fields $\phi_{\rm in(out)}$ are then defined using superposition. They look like free fields, but are not, since for the [*total*]{} four-momentum $P_\mu$, we have $$P_\mu |k_1,k_2,\cdots k_N; {\rm in(out)}\rangle = (\sum_i k_{i \mu})
|k_1,k_2,\cdots k_N; {\rm in(out)}\rangle .$$
It is also assumed that
\(a) The vacuum and single particle states are unique up to a phase. Then after a phase choice, there is only one vacuum $|0\rangle$, $\langle 0 | 0\rangle =1$, and $$S|0\rangle = |0\rangle.$$
\(b) There exists an interpolating field $\phi$ in the Heisenberg representation such that matrix elements of $\phi(x,t)$ between in- and out- states go to those of $\phi_{\rm in, out}(x,t)$ in the infinite past and future, $$\phi(x,t)~ -~ \phi_{\rm in, out}(x,t) \rightarrow 0 \quad {\rm as} \quad t
\rightarrow \mp \infty$$ in weak topology. (We treat the case of just one scalar field for simplicity.)
Then LSZ formalism shows that $$\langle k'_1,k'_2,\cdots k'_N; {\rm out}|k_1,k_2,\cdots k_M; {\rm
in}\rangle = \int {\cal{I}}~~G_{N+M}(x_1',\cdots,x_N';x_1\cdots x_M)$$ where $${\cal{I}}~=~\prod d^4x'_i\prod d^4x_j~e^{i(k_i'\cdot
x_i'~-~k_j\cdot x_j)} i(\partial_i'^2+m^2)\cdot i(\partial_j^2+m^2)
\label{I}$$ and $$G_{N+M} (x_1',\cdots,x_N';x_1\cdots x_M)\equiv \langle
0|T(\phi(x_1')\cdots \phi(x_N')\phi(x_1)\cdots \phi(x_M))|0\rangle$$
It is now convenient to regard all the momenta as ingoing, relabel them as $q_1,q_2,\cdots q_{N+M}$ and write $$\langle -q_1,q_2, \cdots -q_N; {\rm out} \mid q_{N+1},\cdots q_{N+M}; {\rm in}\rangle~=~\int {\cal{I}}~G_{N+M},$$ where $${\cal{I}}~=~\prod^{N+M}_{i=1} d^4x_i~e^{-iq_i\cdot x_i}~i(\partial_i^2+m^2)
\label{II}$$ and $$G_{N+M} (x_1,\cdots,x_{N+M})~=~\langle 0\mid T(\phi(x_1)\cdots \phi(x_{N+M})\mid 0 \rangle.
\label{LSZformula}$$
We will later see the differences in the scattering amplitude on the GM plane through an analysis of the Gell-Mann-Low formula.
The Gell-Mann-Low Formula
-------------------------
The Heisenberg fields $\phi$ and the free fields $\phi_F$ at time $t~=~0$ fulfill the [*same*]{} canonical algebra if the interaction has no time derivatives. We assume that to be the case.
Then in perturbation theory, we choose the same representation of the canonical algebra at time 0, namely that of the free field $\phi_F$: $$\phi(\cdot, 0) = \phi_F(\cdot,0) .$$ This implies that $$\begin{aligned}
\phi_F(\cdot, t) &=& e^{iH_0 t}\phi_F(\cdot,0)e^{-i H_0 t}, \\
\phi(\cdot,t) &=& e^{i H t}\phi(\cdot,0)e^{-i H t} = e^{i H
t}\phi_F(\cdot,0)e^{-i H t} \end{aligned}$$ or $$\phi(\cdot,t) = (e^{i H t}e^{-i H_0 t} \phi_F(\cdot,t)(e^{i H_0 t}
e^{-i H t})$$ Let us define $$U(t_1,t_2) = e^{i t_1 H_0}e^{-i(t_1-t_2)H}e^{-it_2 H_0}$$ Then $$\begin{aligned}
U(t,t) &=& 1, \\
i \frac{\partial U}{\partial t_1}(t_1,t_2) &=& H_I(t_1) U(t_1,t_2),\end{aligned}$$ where $$H_I(t) = e^{it_1 H_0} H_I(0) e^{-i t_1 H_0}, \quad H_I(0) = H_I$$ is the interaction representation Hamiltonian.
Thus $$U(t_1,t_2) = T \exp \left( -i \int_{t_1}^{t_2} dt H_I (t) \right)$$ and $$\phi(\cdot,t) = U(0,t) \phi_F(\cdot,t) U(t,0)
\label{heisenberg}$$
Gell-Mann and Low show that $$G_{N+M}(x_1,\cdots x_{N+M})= \frac{\langle 0,{\rm F}|T \left(\phi_F(x_1) \cdots
\phi_F(x_{N+M}) e^{i\int d^4 x {\cal L}_I (x)}\right) |0,{\rm
F}\rangle}{\langle 0,{\rm F}| e^{i\int d^4 x {\cal L}_I (x)}|0,{\rm
F}\rangle}$$ where $|0,{\rm F}\rangle$ is the vacuum of the free Hamiltonian $H_0$: $$H_0 |0, {\rm F}\rangle = 0 .$$
The proof is standard and will be omitted here.
The twisted quantum fields
--------------------------
Let us look at the case when $\theta^{\mu\nu} \neq 0$, first focussing on the situation with no gauge fields. Gauge fields will be treated later.
Our assumption is that the noncommutative field theory comes from the commutative one by the replacement $$\phi_\theta = \phi_0 e^{\frac{1}{2} \overleftarrow{\partial} \wedge
P}.
\label{twistfield}$$ for [*matter*]{} fields, whereas gauge fields are [*not*]{} twisted (See also (\[twistD\])). As $t\rightarrow ~\pm\infty$, the Heisenberg field $\phi_0$ for $\theta_{\mu\nu}~=~0$ becomes the corresponding in- and out- fields $\phi_0^{in,out}$. As for $P_\mu$, they are not affected by these limits, being constants of motion. Hence formally, we find, for the in- and out- fields $\phi^{in,out}_\theta$ of $\phi_\theta$, the results $$\phi_\theta \rightarrow \phi_\theta^{\rm in,out}~:~= \phi_{\rm in, out}
e^{\frac{1}{2}\overleftarrow{\partial} \wedge P}
\quad {\rm as} \quad t \rightarrow \mp \infty.
\label{phiin}$$ For the twisted in and out annihilation and creation operators $a_k^{in,out},a_k^{\dagger in,out}$ we thus find $$a_k^{in,out}~=~c_k^{in,out}e^{-\frac{i}{2}k_\mu\theta^{\mu\nu}P_\nu},\qquad
a_k^{\dagger in,out}~=~c_k^{\dagger in,out}e^{\frac{i}{2}k_\mu\theta^{\mu\nu}P_\nu}.\label{inout}$$
There is a further convention we want to explain. For consistency with our notation for the coproduct on the Poincaré group[@07081379], we define $$a_k^{\dagger in}|k_r,k_{r-1},\cdots ,k_1\rangle_{\rm in}~=~ |k_r,k_{r-1},\cdots ,k_1,k\rangle_{\rm in}$$ and similarly for the action of $a_k^{\dagger out}$. Thus for example $$|k_r,k_{r-1},\cdots ,k_1\rangle_{\rm in}~=~a_{k_1}^{\dagger in}a_{k_2}^{\dagger in}\cdots
a_{k_r}^{\dagger in}|0\rangle_{\rm in} .$$ and $$\langle -q_N,-q_{N-1},\cdots -q_1;out|q_{N+M},q_{N+M-1},\cdots ,q_{N+1}\rangle
~=~\int {\cal{I}}~G_{N+M}^\theta(x_1,x_2,\cdots , x_{N+M})
\label{scatter}$$ $G_{N+M}~\equiv G_{N+M}^0$ has got changed to $G_{N+M}^\theta $ for $\theta_{\mu\nu}\neq 0$ where $$\begin{aligned}
G_{N+M}^\theta (x_1, \cdots x_{N+M}) &=& T e^{\frac{i}{2}\sum_{I<J}\partial_{x_I}
\wedge \partial_{x_J}} W_{N+M}^0 (x_1, \cdots x_{N+M}) \nonumber \\
:&=&~T~W_{N+M}^\theta (x_1, \cdots x_{N+M})
\label{GNtheta}\end{aligned}$$ and $W_{N+M}^0$ are the standard Wightman functions for untwisted fields: $$W_{N+M}^0 (x_1, \cdots x_{N+M}) = \langle 0 | \phi_0 (x_1) \cdots \phi_0(x_{N+M})
|0\rangle.$$ It is important that because of translational invariance, the $W_{N+M}^\theta$ (and hence the $G_{N+M}^\theta$) depend only on coordinate differences.
For simplicity, we have included only matter fields, and that too of one type, in (\[scatter\]). Gauge fields can be included too, but they are not acted on by the twist exponential in (\[GNtheta\]).
The scattering matrix element is thus: $$_\theta \langle -q_N,-q_{N-1},\cdots -q_1; {\rm out}|q_{N+M},q_{N+M-1},\cdots ,q_{N+1};{\rm
in}\rangle _\theta ~=~\int {\cal{I}} G_{N+M}^\theta(x_1,x_2,\cdots,x_{M+N})
\label{ncLSZ}$$ where ${\cal{I}}$ is as defined in (\[II\]).
On Fourier transforming as in (\[LSZformula\]), the $\theta_{ij}$ (space-space) part of the twist can be partially integrated. It gives the usual phase $e^{\frac{i}{2} q_I^i \theta_{ij} q_J^j}$. The time step-function (in the time-ordering T) in (\[GNtheta\]) does not affect this manipulation.
To handle the $\theta_{0i}$ part, consider a typical term $$\begin{aligned}
g_{N+M}^\theta (x_1 \cdots x_{N+M}) &=&\theta(x_1^0 - x_2^0) \theta(x_2^0 -
x_3^0) \cdots \theta(x_{N+M-1}^0 - x_{N+M}^0) \nonumber \\
&& e^{\frac{i}{2} \sum_{I<J}
\partial_{x_I} \wedge \partial_{x_J}} W_{N+M}^0 (x_1, \cdots x_{N+M}).
\label{ncLSZ1}\end{aligned}$$ which occurs on expanding the time-ordered product in terms of retarded products.
The twist here is the product of terms $$e^{\frac{i}{2} [\partial_{x^0_I} \theta \cdot \nabla_J - (\theta \cdot
\nabla_I) \partial_{x^0_J}]}, \quad I<J, \quad \theta \cdot
\nabla_J \equiv \theta_{0i} \partial_{x_J^i}
\label{twist}$$ on retaining just $\theta_{0i}$. The coefficient of $\partial_{x^0_I}$ in the exponential is thus $$\sum_{J>I} \theta \cdot \nabla_J - \sum_{J<I} \theta \cdot \nabla_J$$ On partial integration in eq(\[ncLSZ\]) $\nabla_J$ becomes $iq_J$ and $$e^{\frac{i}{2} \partial_{x_I^0} (\sum_{J>I} - \sum_{J<I})\theta \cdot
\nabla_J} \rightarrow e^{-\frac{1}{2} (\sum_{J>I} \theta \cdot q_J -
\sum_{J<I} \theta\cdot q_J) \partial_{x_I^0}}$$ This translates the $x_I^0$’s according to $$\begin{aligned}
x_{I-1}^0 &\rightarrow& x_{I-1}^0 - \frac{1}{2} \left(\sum_{J>I-1}
\theta \cdot q_J - \sum_{J<I-1} \theta\cdot q_J\right), \\
x_I^0 &\rightarrow& x_I^0 - \frac{1}{2} \left(\sum_{J>I}
\theta \cdot q_J - \sum_{J<I} \theta\cdot q_J\right), \\
x_{I+1}^0 &\rightarrow& x_{I+1}^0 - \frac{1}{2} \left(\sum_{J>I+1}
\theta \cdot q_J - \sum_{J<I+1} \theta\cdot q_J\right)\end{aligned}$$ Or $$x_{I-1}^0 - x_I^0 \rightarrow (x_{I-1}^0 - x_I^0) - \frac{1}{2} \theta
\cdot q_J + \frac{1}{2} \sum_{J \neq I-1,I} \theta \cdot q_J.$$ But $\sum \vec{q}_J = 0$, so that $$x_{I-1}^0 - x_I^0 \rightarrow x_{I-1}^0 - x_I^0~ - ~\frac{1}{2} \theta
\cdot q_{I-1}~-~ \frac{1}{2} \theta \cdot q_I
\label{diff1}$$ Similarly, $$x_I^0 - x_{I+1}^0 \rightarrow x_I^0 - x_{I+1}^0 ~-~ \frac{1}{2} \theta
\cdot q_I ~-~ \frac{1}{2} \theta \cdot q_{I+1}
\label{diff2}$$ From (\[diff1\],\[diff2\]), we see that each $x_I^0$ is (time) shifted to $$x_I^0 + \delta x_I^0, \quad \delta x_I^0 = \delta x_I^0 (q_1, \cdots q_N).$$ where the $\delta x_I^0$ actually depend on the ordering on $x_I^0$. No further simplification seems possible.
We emphasize the following important observations.
- Firstly, (\[ncLSZ\]) involves only the $\theta^{\mu\nu}=0$ fields in $W_N^0$. So it can be used to map any commutative theory to noncommutative one, including also the standard model. But special care is needed to treat gauge fields. Gauge fields are [*[not]{}*]{} twisted unlike matter fields. As explained elselwhere, this means that the Yang-Mills tensor is not twisted, $F^{\mu\nu}_\theta~=~F^{\mu\nu}_0$. But covariant derivatives of matter fields $\phi_\theta$ are twisted: $(D_\mu\phi)_\theta~=~
(D_\mu\phi)_0~e^{\frac{1}{2}{\overleftarrow{\partial}}\wedge
P}$. where $(D_\mu\phi)_0$ is the untwisted covariant derivative of the untwisted $\phi_0$. Thus in correlators $W_N^\theta$, we must use $(D_\mu\phi)_\theta$ for matter fields, $F^{\mu\nu}_0$ for Yang-Mills tensor.
- There are ambiguities in formulating scattering theory. Thus if we substitute (\[phiin\]) directly in the LHS of (\[LSZformula\]), we get our earlier result [@07081379]. This corresponds to putting the twist factor in (\[twist\]) outside the symbol T in (\[GNtheta\]). At this moment, lacking a rigorous scattering theory, we do not know which of these is the correct answer. In this connection, we must mention the important work of Buchholz and Summers[@buchholz] which rigorously develops the wedge localisation ideas of Grosse and Lechner to establish a scattering theory for two incoming and two outgoing particles. The result resembles those in our earlier approach[@bpq], but there seem to be descrepencies in the signs of the momenta in the overall phases.
For calculating (\[ncLSZ\]), we need a formalism for doing perturbation theory to calculate Wightman functions. Once we have that, we can calculate the time-ordered product by writing it in terms of Wightman functions and twist factors. We show such a calculation below.
Perturbation Theory for Wightman Functions
------------------------------------------
Perturbation theory for Wightman functions is available [@ostendorf]. We can also construct this formalism directly. For free fields, Wightman functions are Gaussian correlated. For example, $$\langle \phi_F(x_1)\phi_F(x_2)\phi_F(x_3)\phi_F(x_4) \rangle= \langle
\phi_F(x_1)\phi_F(x_2)\rangle \langle\phi_F(x_3)\phi_F(x_4) \rangle
+{\rm permutations}
\label{wightman}$$ while for Dirac fields, there are signs attached to the succesive terms reflecting the signature of the permutations in $x_1,x_2,x_3,x_4$. The two-point functions here are well-known. For example, $$\langle \phi_F(x_1)\phi_F(x_2)\rangle = \Delta_+(x_1~-~x_2,m^2).$$ Now to calculate Wightman functions for interacting fields, we can expand Heisenberg fields $\phi$ in terms of free fields using (\[heisenberg\]) and express the resultant free field correlators in terms of two point functions.
As an illustration, consider $$\begin{aligned}
\lefteqn{\langle T(\phi_F(x_1)\phi_F(x_2)\phi_F(x_3)\phi_F(x_4))
\rangle =} \nonumber \\
&&\theta(x_1^0-x_2^0)\theta(x_2^0-x_3^0) \theta(x_3^0-x_4^0)
\langle \phi_F(x_1)\phi_F(x_2)\phi_F(x_3)\phi_F(x_4) \rangle + \cdots
\label{wightman1}\end{aligned}$$ The Wightman function is then given by (\[wightman\]). We can similarly calculate the remaining terms in (\[wightman1\]).
Renormalisation theory for Wightman functions has also been developed [@ostendorf].
Summary of our Rules for Twisted Quantum Field Theories
=======================================================
Our rules for transition from the $\theta^{\mu\nu}=0$ to the $\theta^{\mu\nu}\neq 0$ theory are simple and definite. Let us focus on scattering amplitudes. They are given by reduction formulae as in (\[ncLSZ\]). They show that to compute scattering amplitudes for $\theta^{\mu\nu}\neq 0$, we need a formula for twisted Wighman functions $W_{N+M}^\theta$ in (\[GNtheta\]) in terms of the untwisted Wightman functions $W_{N+M}^0$. We have already explained this formula: the passage from $W_{N+M}^0$ to $W_{N+M}^\theta$ is achieved by twisting all fields except the gauge fields for unbroken gauge symmetries $H$. Further if $D(V)^{\theta = 0}$ is the connection with gauge potential $V$ for the unbroken group $H$, then the covariant derivative for a twisted matter field $\psi^\theta$ is to be defined as $D_\mu^\theta(V)\psi^\theta :
= (D_\mu(V)^0\psi^0) e^{\frac{1}{2} \overleftarrow{\partial} \wedge
P}$.
This rule preserves the asymptotic conditions and shows that the spectrum of the theory is unaltered by changing $\theta^{\mu\nu}$. It covers theories with spontaneous symmetry breakdown as well provided we have a scheme for treating it for $\theta^{\mu\nu} = 0$. We remark in this connection that since twist factors with time derivatives occur in (\[GNtheta\]) within the time-ordering symbol and the amount of time-translation they generate depend on external momenta, the Gell-Mann-Low formula has to be modified significantly.
Spontaneous Symmetry Breaking
=============================
This topic is of sufficient importance that we discuss it separately. The consistency of this discussion with what we discussed earlier will be apparent.
Let us first consider the case of spontaneously broken global symmetries. Suppose we have a multiplet of quantum fields $\phi_{i}(x)$ that transforms under the action of (some representation $D(g)$ of) a symmetry group $G$ according to $$\phi_i(x) \rightarrow \phi^g_i(x) = D(g)_{ij} \phi_j(x)$$ If this is a symmetry of the theory, then the quantum charges $Q^a_0$ commute with the full Hamiltonian: $$[Q^a_0, H] =0, \quad a= 1,2, \cdots {\rm dim}\,\, G.$$ These conventionally arise from quantum currents $J^{a,\mu}(x)$ which are conserved: $$\partial_\mu J^{a,\mu}_0 =0$$ where the currents $J^{a,\mu}_0$ are constructed from the quantum fields $\phi_{i,0}(x)$ and its derivatives.
Given a commutative quantum theory with conserved currents $J^{a,
\mu}_0$, it is easy to see that in the corresponding noncommutative theory (obtained by replacing $\phi_{i,0}(x) \rightarrow
\phi_{i,\theta}(x) = \phi_{i,0}(x) e^{\frac{1}{2}
\overleftarrow{\partial} \wedge P}$), the noncommutative currents $J^{a,\mu}_\theta (x) = J^{a,\mu}_0(x)e^{\frac{1}{2}
\overleftarrow{\partial} \wedge P}$ are also conserved: $$\partial_\mu J^{a,\mu}_\theta (x) = \partial_\mu J^{a,\mu}_0(x)e^{\frac{1}{2}
\overleftarrow{\partial} \wedge P} = (\partial_\mu
J^{a,\mu}_0(x))e^{\frac{1}{2} \overleftarrow{\partial} \wedge P} =
0.$$ Interestingly, the charges $Q^a_\theta$ in the noncommutative theory $$Q^a_\theta = \int d^3 x J^{a,0}_\theta (x) = \int d^3 x J^{a,0}_0 (x)
e^{\frac{1}{2} \overleftarrow{\partial} \wedge P} = \int d^3 x
J^{a,0}_0 (x) = Q^a_0$$ are the same as in the commutative case. The last equation follows using integration by parts for terms involving $\theta_{ij}$ and using the time independence of the charges of the commutative theory for the rest. The charges $Q_0^a, Q_\theta^a$ generate the infinitesimal symmetry transformations: $$\begin{aligned}
[Q^a_0, \phi_{i,0}(x)] &=& \sum_j T^a_{ij}\phi_{j,0}(x), \label{cQ}\\
{[}Q^a_\theta, \phi_{i,\theta }(x)] &=& \sum_j T^a_{ij} \phi_{j,\theta }(x). \label{ncQ}\end{aligned}$$
Goldstone’s theorem
-------------------
Consider the vacuum expectation value of the commutator of the currents $J^{a,\mu}_\theta(y)$ and the quantum field $\phi_{i,\theta}(x)$: $$\langle 0|[J^{a,\mu}_\theta(y), \phi_{i,\theta}(x)]|0\rangle = \langle
0| [e^{\frac{1}{2}\overrightarrow{\partial_y} \wedge P} J^{a,\mu}_0(y),
\phi_0(x)e^{\frac{1}{2} \overleftarrow{\partial_x} \wedge P}] |0\rangle.
%= \langle 0|[J^{a,\mu}_0(y), \phi_{i,0}(x)]|0\rangle.$$ The first term in the commutator here is $$e^{\frac{i}{2}\partial_y\wedge \theta \partial_x} \langle
0|J_0^{a,\mu}(y)\phi_0(x)|0 \rangle =
e^{\frac{i}{2}\partial_y\wedge \theta \partial_x} \langle
0|J_0^{a,\mu}(0)\phi_0(x-y)|0 \rangle$$ where we have used translational invariance and the fact that $$\partial_y\wedge \partial_x \langle 0|J_0^\mu(0)\phi_0(x-y)| 0\rangle ~=~-~
\partial_x \wedge \partial_x \langle 0|J_0^\mu(0)\phi_0(x-y) | 0\rangle~=~0$$ in the second step. The $\theta^{\mu\nu}$ dependence in the second term as well disappears in the same manner so that $$\langle 0|[J^{a,\mu}_\theta(y), \phi_{i,\theta}(x)]|0\rangle = \langle
0|[J^{a,\mu}_0(y), \phi_{i,0}(x)]|0\rangle.$$ This commutator being the same as the one for the corresponding commutative case, the standard arguments using spectral density and Lorentz invariance [@gsw] can be used to argue for the existence of massless bosons in the symmetry-broken phase. Following [@weinberg], we reproduce this argument below.
Summing over intermediate states, and using Lorentz invariance, the vacuum expectation value of the commutator may be expressed as $$\langle 0|[J^{a,\mu}_0(y), \phi_{i,0}(x)]|0\rangle = \int d^4 p
\left( \rho^{a,\mu}_i(p) e^{-i p \cdot (y-x)} -
\tilde{\rho}^{a,\mu}_i(p) e^{ip \cdot
(y-x)}\right),$$ where the spectral densities $\rho^a_i(p), \tilde{\rho}^a_i(p)$ are defined as $$\begin{aligned}
\rho^{a,\mu}_i(p) &=& \sum_N \langle 0| J^{a,\mu}_0(0)|N\rangle \langle
N| \phi_{i,0}(0)|0\rangle \delta^4(p-p_N), \\
\tilde{\rho}^{a,\mu}_i(p) &=& \sum_N \langle 0|\phi_{i,0}(0)|N\rangle \langle
N| J^{a,\mu}_0(0)|0\rangle \delta^4(p-p_N),\end{aligned}$$ and $p_N$ is the total four-momentum in the state $\mid N \rangle$.
By Lorentz invariance and non-negativity of energy, these densities are of the form $$\begin{aligned}
\rho^{a,\mu}_i(p)&=& p^\mu \rho^a_i (p^2)\theta(p^0), \\
\tilde{\rho}^{a,\mu}_i(p)&=& p^\mu \tilde{\rho}^a_i (p^2)\theta(p^0),\end{aligned}$$
which implies that $$\begin{aligned}
\lefteqn{\langle 0|[J^{a,\mu}_0(y), \phi_{i,0}(x)]|0\rangle =}
\nonumber \\
&& i \frac{\partial}{\partial y_\mu} \int dM^2 \left( \rho^a_i (M^2)
\Delta_+(y-x;M^2) +\tilde{\rho}^a_i (M^2) \Delta_+(x-y;M^2) \right)\end{aligned}$$ where $\Delta_+(x;M^2)$ is the standard two-point Wightman function: $$\Delta_+(x;M^2) = \int d\mu(p) e^{-i p \cdot x}, \quad {\rm where}
\quad d\mu(p) = \frac{d^3 p}{2p_0}, \quad p_0 = \sqrt{\vec{p}^2 + M^2}.$$
Since $\Delta_+(x;M^2)$ and $\Delta_+(-x;M^2)$ are equal for $x$ spacelike, we can write, for such $x-y$, $$\langle 0|[J^{a,\mu}_0(y), \phi_{i,0}(x)]|0\rangle =
i \frac{\partial}{\partial y_\mu} \int dM^2 \left( \rho^a_i (M^2)+
\tilde{\rho}^a_i (M^2) \right) \Delta_+(y-x;M^2).$$
For spacelike separations, the commutator vanishes, so that $$\rho^a_i (M^2) = -\tilde{\rho}^a_i (M^2)$$ which gives us $$\langle 0|[J^{a,\mu}_0(y), \phi_{i,0}(x)]|0\rangle =
i \frac{\partial}{\partial y_\mu} \int dM^2 \rho^a_i(M^2) (\Delta_+(y-x;M^2) -
\Delta_+(x-y;M^2))$$ Now, since the current $J^{a,\mu}_0$ is conserved, we can act by $\partial/\partial y_\mu$ to get $$0 = \int dM^2 M^2 \rho^a_i(M^2)(\Delta_+(y-x;M^2) - \Delta_+(x-y;M^2))$$ and thus we get $$M^2 \rho^a_i(M^2) = 0. \label{eq1}$$
Now consider the situation when the symmetry is broken. For $\mu =0,
x^0=y^0=t$, $$\langle 0|[J^{a,0}_0(\vec{y},t), \phi_{i,0}(\vec{x},t)]|0\rangle = i
\delta(\vec{y}-\vec{x}) \int dM^2 \rho^a_i(M^2).$$
Integrating and using (\[cQ\]), we get $$\sum_j T^a_{ij}< \phi_{j,0}(x)> = i\int \rho^a_i(M^2). \label{eq2}$$
Eqs. (\[eq1\]) and (\[eq2\]) are compatible only if $$\rho^a_i(M^2) = i \delta(M^2) \sum_j T^a_{ij}\langle 0|
\phi_{j,0}(0)|0 \rangle$$ As long as the symmetry is broken, the spectral density $\rho^a_i$ is proportional to $\delta(M^2)$. Since such a term can arise only in a theory with massless particles, we are forced to conclude that a broken symmetry with $T^a_{ij}\langle 0|\phi_{j,\theta}(0)|0\rangle~\neq 0$ requires the existence of a massless particle with the same quantum numbers as $J^{a,0}_\theta$. These are nothing but the Goldstone bosons.
Spontaneously Broken Local Symmetries & twisted standard model
--------------------------------------------------------------
Now given the map between the twisted fields and untwisted ones (eq. \[twistfield\]) and our earlier established rules for getting the correlation functions for the case of $\theta_{\mu\nu}\neq 0$ eqs. [(\[GNtheta\])]{} and [(\[ncLSZ1\])]{} we can easily see that the Higgs mechanism will follow with the mass of the gauge boson being identical to the untwisted case. We can readily understand this result from the fact that the [*in*]{} and [*out*]{} fields completely determine the mass spectrum and they remain independent of $\theta_{\mu\nu}$ because of formulae like (\[phiin\]) and (\[inout\]).
The Hamiltonian $P_0~=~H$ and the spatial translation generator for the twisted standard model are the same as for the case $\theta^{\mu\nu}~=~0$ What is changed in our LSZ approach are the in and out fields which are twisted as discussed. Hence scattering calculations can be based on appropriately modified Wightman functions as we have already explained.
We will elsewhere calculate specific twisted standard model cross-sections and examine the new features coming from non-commutativity.
Final Remarks
=============
In this paper, we have outlined an approach for calculating the scattering amplitudes in twisted qft’s from untwisted ones using LSZ formalism. It works in gauge theories with or without spontaneous breakdown. Implications for the standard model will be presented in a forthcoming paper.
As remarked earlier, the results for scattering matrix in this approach differs from the interaction representation perturbation theory. The reasons for this difference remain to be pinpointed.
In our judgement, since the LSZ approach works with fully interacting fields and total momentum $P_\mu$ (including also interactions), it is probably superior to the results based on interaction representation perturbation theory. It does not change $P_\mu$ in the process of twisting, but changes just the in- and out- fields appropriately to account for the twisted statistics. This change is forced on us when the coproduct of the Poincaré-Hopf algebra is twisted.
In the presence of matter and gauge fields, the coproduct for the Poincaré algebra becomes non-associative and gives rise to a Poincaré - quasi Hopf algebra [@BALBABAR]. We will discuss this quasi- Hopf algebra in detail in another paper.
[**Acknowledgments:**]{} The work of APB is supported in part by US-DOE under grant number DE-FG02-85ER40231 and the Universidad Carlos III de Madrid. The work of APB and TRG are supported by the DST CP-STIO program.
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[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We define the notion of characteristic rank, $\mathrm{charrank}_X(\xi)$, of a real vector bundle $\xi$ over a connected finite $CW$-complex $X$. This is a bundle-dependent version of the notion of characteristic rank introduced by Július Korbaš in 2010. We obtain bounds for the cup length of manifolds in terms of the characteristic rank of vector bundles generalizing a theorem of Korbaš and compute the characteristic rank of vector bundles over the Dold manifolds, the Moore spaces and the stunted projective spaces amongst others.'
address: 'Stat-Math UNit Indian Statistical Institute 8th Mile, Mysore Road, RVCE Post Bangalore 560059 INDIA.'
author:
- 'Aniruddha C. Naolekar'
- Ajay Singh Thakur
title: Note on the characteristic rank of vector bundles
---
Introduction
============
Recently, J. Korbaš [@korbas] has introduced a new homotopy invariant, called the characteristic rank, of a connected closed smooth manifold $X$. The characteristic rank of a connected closed smooth $d$-manifold $X$, denoted by $\mathrm{charrank}(X)$, is the largest integer $k$, $0\leq k \leq d$, such that every cohomology class $x\in H^j(X;\mathbb Z_2)$, $0\leq j\leq k$ is a polynomial in the Stiefel-Whitney classes of (the tangent bundle of) $X$.
Apart from being an interesting question in its own right, part of the motivation for computing the characteristic rank comes from a theorem of Korbaš ([@korbas], Theorem 1.1), where the author has described a bound for the $\mathbb Z_2$-cup-length of (unorientedly) null cobordant closed smooth manifolds in terms of their charateristic rank. More specifically, Korbaš has proved the following.
\[korbastheorem\] [*([@korbas], Theorem1.1)*]{} Let $X$ be a closed smooth connected $d$-dimensional manifold unorientedly cobordant to zero. Let $\widetilde{H}^r(X;\mathbb Z_2)$, $r < d$, be the first nonzero reduced cohomology group of $X$. Let $z$ ($z < d -1$) be an integer such that for $j\leq z$ each element of $H^j(X;\mathbb Z_2)$ can be expressed as a polynomial in the Stiefel-Whitney classes of the manifold $X$. Then we have that $$\mathrm{cup}(X)\leq 1+\frac{d-z-1}{r}.$$
Recall that the $\mathbb Z_2$-cup-length, denoted by $\mathrm{cup}(X)$, of a space $X$ is the largest integer $t$ such that there exist classes $x_i\in H^*(X;\mathbb Z_2)$, $\mathrm{deg}(x_i)\geq 1$, such that the cup product $x_1\cdot x_2\cdots x_t\neq 0$. We mention in passing that the $\mathbb Z_2$-cup-length is well known to have connections with the Lyusternik-Shnirel’man category of the space.
With the computation of the characteristic rank in mind, Balko and Korbaš [@balkokorbas] obtained bounds for the characteristic rank of manifolds $X$ which occur as total spaces of smooth fiber bundles with fibers totally non-homologous to zero, and also in the situation where, additionally, $X$ itself is null cobordant (see [@balkokorbas], Theorems 2.1 and 2.2).
It is useful to think of the characteristic rank of a manifold as the characteristic rank “with respect to the tangent bundle” and introduce bundle-dependency as in the definition below.
\[def\] Let $X$ be a connected, finite $CW$-complex and $\xi$ a real vector bundle over $X$. The characteristic rank of the vector bundle $\xi$ over $X$, denoted by $\mathrm{charrank}_X(\xi)$, is by definition the largest integer $k$, $0\leq k\leq \mathrm{dim}(X)$, such that every cohomology class $x\in H^j(X;\mathbb Z_2)$, $0\leq j\leq k$, is a polynomial in the Stiefel-Whitney classes $w_i(\xi)$ of $\xi$. The upper characteristic rank of $X$, denoted by $\mathrm{ucharrank}(X)$, is the maximum of $\mathrm{charrank}_X(\xi)$ as $\xi$ varies over all vector bundles over $X$.
Thus, if $X$ is a connected closed smooth manifold, then $\mathrm{charrank}_X(TX)=\mathrm{charrank}(X)$ where $TX$ is the tangent bundle of $X$. Note that if $X$ and $Y$ are homotopically equivalent closed connected smooth manifolds, then $\mathrm{ucharrank}(X)=\mathrm{ucharrank}(Y)$.
In this note we discuss some general properties of $\mathrm{charrank}(\xi)$ and give a complete description of $\mathrm{charrank}_X(\xi)$ of vector bundles $\xi$ over $X$ when $X$ is: a product of spheres, the real and complex projective spaces, the Dold manifold $P(m,n)$, the Moore space $M(\mathbb Z_2,n)$ and the stunted projective spaces $\mathbb R\mathbb P^n/\mathbb R\mathbb P^m$. We now briefly describe the contents of this note.
For a connected finite $CW$-complex $X$, let $r_X$ denote the smallest positive integer such that $\widetilde{H}^{r_X}(X;\mathbb Z_2)\neq 0$. In the case that such an integer does not exist, that is, all the reduced cohomology groups $\widetilde{H}^i(X;\mathbb Z_2)=0$, $1\leq i\leq \dim (X)$, we set $r_X=\dim (X)+1$. In any case, $r_X\geq 1$.
Making the definition of the characteristic rank bundle-dependent gives the following theorem which is a straighforward generalisation of Theorem\[korbastheorem\]. In this form the theorem yields sharper bounds on the cup-length in certain cases (see Examples\[4.6\] and \[4.7\] below). We shall prove the following.
\[zeroththeorem\] Let $X$ be a connected closed smooth $d$-manifold. Let $\xi$ be a vector bundle over $X$ satisfying the following:
- there exists $k$, $k\leq \mathrm{charrank}_X(\xi)$, such that every monomial $$w_{i_1}(\xi)\cdots w_{i_r}(\xi), 0\leq i_t\leq k,$$ of total degree $d$ is zero.
Then, $$\mathrm{cup}(X)\leq 1+\frac{d-k-1}{r_X}.$$
We note that if $X$ is an unoriented boundary, then $\xi=TX$ satisfies the conditions of the theorem above with $k=\mathrm{charrank}_X(TX)$. In this theorem we do not assume that $X$ is an unoriented boundary.
If $X$ is an unoriented boundary and there exists a vector bundle $\xi$ over $X$ with $k$ satisfying the conditions of the above theorem, such that $$\label{equation}
\mathrm{charrank}(X)=\mathrm{charrank}_X(TX)<k\leq \mathrm{charrank}_X(\xi),$$ then the bound for $\mathrm{cup}(X)$ using $k$ is sharper than that obtained from Theorem\[korbastheorem\]. We note that over the null cobordant manifold $S^d\times S^m$, $d=2,4,8$, and $m\neq 2,4,8$, there exists a vector bundle $\xi$ and an integer $k$ satisfying the conditions of Theorem\[zeroththeorem\] and equation \[equation\] (see Examples\[4.6\], \[4.7\] below).
If $X$ is a connected closed smooth manifold with $\mathrm{ucharrank}(X)=\mathrm{dim}(X)$, it turns out that the cup-length $\mathrm{cup}(X)$ of $X$ can be computed as the maximal length of a non-zero product of the Stiefel-Whitney classes of a suitable bundle over $X$. We prove the following.
\[cuplength\] Let $X$ be a connected closed smooth $d$-manifold. If $$\mathrm{ucharrank}(X)=\mathrm{dim}(X),$$ then there exists a vector bundle $\xi$ over $X$ such that $$\mathrm{cup}(X)=\max\{k\mid \mbox{there exist $ i_1,\ldots , i_k\geq 1$ with $w_{i_1}(\xi)\cdots w_{i_k}(\xi)\neq 0$}\}.$$
Making the definition of characteristic rank bundle-dependent allows us, under certain conditions, to construct an epimorphism $\widetilde{KO}(X)\longrightarrow \mathbb Z_2$. It is clear from the definition that $\mathrm{charrank}_X(\xi)=\mathrm{charrank}_X(\eta)$ if $\xi$ and $\eta$ are (stably) isomorphic. Let $\mathrm{Vect}_{\mathbb R}(X)$ denote the semi-ring of isomorphism classes of real vector bundles over $X$. We then have a function $$f:\mathrm{Vect}_{\mathbb R}(X)\longrightarrow \mathbb Z_2$$ defined by $f(\xi)=\mathrm{charrank}_X(\xi) ~~~~~(\mbox{mod}~~~~2)$. We observe that under certain restrictions on the values of $\mathrm{charrank}_X(\xi)$ the function $f$ is actually a semi-group homomorphism. More precisely we prove the following.
\[maintheorem\] Let $X$ be a connected finite $CW$-complex with $r_X=1$. Assume that for any vector bundle $\xi$ over $X$, $\mathrm{charrank}_X(\xi)$ is either $r_X-1=0$ or an odd integer. Assume that $\mathrm{ucharrank}(X)\geq 1$. Then the function $$f:\mathrm{Vect}_{\mathbb R}(X)\longrightarrow \mathbb Z_2$$ defined by $f(\xi)=\mathrm{charrank}_X(\xi)~~~~~\mbox{\em (mod $2$)}$ is a surjective semi-group homomorphism and hence gives rise to a surjective group homomorphism $\widetilde{f}:KO(X)\longrightarrow \mathbb Z_2$. Furthermore, this restricts to an epimorphism $\tilde{f}:\widetilde{KO}(X)\longrightarrow \mathbb Z_2$.
The function $f$ defined in the theorem above is in general not a semi-ring homomorphism (see Remark\[importantremark\]). There is a large class of spaces that satisfy the conditions of this theorem. We prove the following.
\[mainproposition\]
1. Let $X = \mathbb R\mathbb P^n$. Then $\mathrm{ucharrank}(X)= n$ and for any vector bundle $\xi$ over $X$, the characteristic rank $\mathrm{charrank}_X(\xi)$ is either $r_X-1=0$ or is $n$.
2. Let $X=S^1\times \mathbb C\mathbb P^n$. Then $\mathrm{ucharrank}(X)= 2n+1$ and for any vector bundle $\xi$ over $X$, the characteristic rank $\mathrm{charrank}_X(\xi)$ either is $r_X-1=0$, $1$ or $2n+1$.
3. Let $X$ be the Dold manifold $P(m,n)$. Then $\mathrm{ucharrank}(X)= 2n +m$ and for any vector bundle $\xi$ over $X$, the characteristic rank $\mathrm{charrank}_X(\xi)$ is either $r_X-1=0$, $1$ or $2n+m$.
Recall that the Dold manifold $P(m,n)$ is the quotient of $S^m\times \mathbb C\mathbb P^n$ by the fixed point free involution $(x,z)\mapsto (-x,\bar{z})$.
In this note we concentrate on the computational part of characteristic rank of vector bundles. We compute the characteristic rank of vector bundles over products of spheres $S^d\times S^m$, the real and complex projective spaces, the spaces $S^1\times \mathbb C\mathbb P^n$, the Dold manifold $P(m,n)$, the Moore space $M(\mathbb Z_2,n)$ and the stunted projective space $\mathbb R\mathbb P^n/\mathbb R\mathbb P^m$. We also prove some general facts about characteristic rank of vector bundles.
The paper is organized as follows. In Section 2 we prove some general facts about $\mathrm{charrank}(\xi)$. In Section 3 we prove Theorems\[zeroththeorem\], \[cuplength\] and \[maintheorem\]. Finally, in Section 4, we compute $\mathrm{charrank}_X(\xi)$ where $X$ is one of the following spaces: the product of spheres $S^d\times S^m$, the real and complex projective spaces, the product $S^1\times \mathbb C\mathbb P^n$, the Dold manifold $P(m,n)$, the Moore space $M(\mathbb Z_2,n)$ and the stunted projective space.
[**Convention.**]{} By a space we shall mean a connected finite $CW$-complex. All vector bundles are real unless otherwise stated.
Generalities
============
In this section we make some general observations about $\mathrm{charrank}(\xi)$. Recall that, for a space $X$, $r_X$ denotes the smallest positive integer for which the reduced cohomology group $\widetilde{H}^{r_X}(X;\mathbb Z_2)\neq 0$, and if such an $r_X$ does not exist, then we set $r_X=\mathrm{dim}(X)+1$. Then for any vector bundle $\xi$ over $X$ we have $$r_X-1\leq \mathrm{charrank}_X(\xi)\leq \mathrm{ucharrank}(X).$$ We begin with some easy observations.
\[firstlemma\] Let $\xi$ and $\eta$ be any two vector bundles over a space $X$.
1. If $w_{r_X}(\xi)=0$, then $\mathrm{charrank}_X(\xi)=r_X-1$;
2. If $w(\xi)=1$, then $\mathrm{charrank}_X(\xi)=r_X-1$.
3. If $w(\eta)=1$, then $\mathrm{charrank}_{X}(\xi\oplus\eta)=\mathrm{charrank}_X(\xi)$. Hence if $\widetilde{KO}(X)=0$, then $\mathrm{charrank}_X(\xi)=r_X-1$ for any vector bundle over $X$;
4. If $\xi$ and $\eta$ are stably isomorphic, then $\mathrm{charrank}_X(\xi)=\mathrm{charrank}_X(\eta)$;
5. There exists a vector bundle $\theta$ over $X$ such that $\mathrm{charrank}_{X}(\xi\oplus \theta)=r_X-1$.
\(1) follows from the definition. Clearly, (2) follows from (1). To prove (3) we note that since $w(\xi\oplus\eta)=w(\xi)$, we have $\mathrm{charrank}_{X}(\xi\oplus\eta)=\mathrm{charrank}_X(\xi)$. As $\widetilde{KO}(X)=0$, we have $\xi\oplus\varepsilon\cong\varepsilon'$. Hence $$\mathrm{charrank}_X(\xi)=\mathrm{charrank}_X(\xi\oplus\varepsilon)=\mathrm{charrank}_X(\varepsilon')=r_X-1.$$ This completes the proof of (3). Next, if $\xi$ and $\eta$ are stably isomorphic, we have $\xi\oplus \varepsilon\cong \eta\oplus\varepsilon'$ where $\varepsilon$ and $\varepsilon'$ are trivial vector bundles. Hence (4) follows from (3). Finally, as $X$ is compact, given $\xi$ we can find a vector bundle $\theta$ such that $\xi\oplus\theta\cong \varepsilon$. Hence (5) follows from (4) and (2).
\[secondlemma\] Let $X$ be a space and $1 \leq r_X \leq \dim(X)$
1. If $\mathrm{ucharrank}(X)\geq r_X$, then $\mathrm{dim}_{\mathbb Z_2}H^{r_X}(X;\mathbb Z_2)= 1$.
2. If $r_X$ is not a power of $2$, then $\mathrm{ucharrank}(X)=r_X-1$.
If $\xi$ is a vector bundle over $X$ with $\mathrm{charrank}_X(\xi)\geq r_X$, then by Lemma\[firstlemma\] (1), $w_{r_X}(\xi)\neq 0$. This forces the equality $\mathrm{dim}_{\mathbb Z_2}H^{r_X}(X;\mathbb Z_2)= 1$ and proves (1). It is known that for any vector bundle $\xi$, the smallest integer $k$ such that $w_k(\xi)\neq 0$ is always a power of $2$ (see, for example, [@ms], page 94). Lemma\[firstlemma\] (1) now completes the proof of (2).
Let $Y$ be a space and and let $X=\Sigma Y$ be the suspension of $Y$. Then any cup-product of elements of positive degree in $H^*(X;\mathbb Z_2)$ is zero. The following lemma is an easy consequence of this fact and we omit the proof.
\[thirdlemma\] Let $Y$ be a space and $X=\Sigma Y$. Let $k_X$ be an integer defined by $$k_X=\mathrm{max}\{k\mid \mathrm{dim}_{\mathbb Z_2}H^j(X;\mathbb Z_2)\leq 1, 0\leq j\leq k, k\leq \mathrm{dim}(X)\}.$$ Let $\xi$ be any vector bundle over $X$. Then, $\mathrm{charrank}_X(\xi)\leq k_X$. In particular, $\mathrm{ucharrank}(X)\leq k_X$.
\[inequalitylemma\] Let $f:X\longrightarrow Y$ be a map between spaces. If $f^*:H^*(Y;\mathbb Z_2)\longrightarrow H^*(X;\mathbb Z_2)$ is surjective, then $$\mathrm{charrank}_{X}(f^*\xi)\geq \min\{\mathrm{charrank}_{Y}(\xi), \mathrm{dim}(X)\}$$ for any vector bundle $\xi$ over $Y$.
As $w_i(f^*\xi)=f^*(w_i(\xi))$, the surjectivity of $f^*$ implies that every cohomology class in $H^*(X;\mathbb Z_2)$ of degree at most $\mathrm{charrank}_{Y}(\xi)$ is a polynomial in the Stiefel-Whitney classes of $f^*\xi$. If $\mathrm{charrank}_{Y}(\xi)\geq \mathrm{dim}(X)$, then $$\mathrm{charrank}_{X}(f^*\xi)=\mathrm{dim}(X).$$ If $\mathrm{charrank}_{Y}(\xi)\leq \mathrm{dim}(X)$, then $\mathrm{charrank}_{Y}(\xi)\leq \mathrm{charrank}_{X}(f^*\xi)\leq \mathrm{dim}(X)$.
Before mentioning further general properties of the characteristic rank we record the characteristic rank of vector bundles over the sphere. The description of the characteristic rank of vector bundles over the spheres is an easy consequence of the following theorem due to Atiyah-Hirzebruch ([@atiyah], Theorem1), (see also [@milnor]).
\[atiyah\][*([@atiyah], Theorem1)*]{} There exists a real vector bundle $\xi$ over the sphere $S^d$ with $w_d(\xi)\neq 0$ only for $d=1,2,4$, or $8$.
For the Hopf bundle $\nu_d$ over $S^d$ ($d=1,2,4,8$), the Stiefel-Whitney class $w_d(\nu_d)$ is not zero. Thus, $$\mathrm{ucharrank}(S^d)=\left\{\begin{array}{cl}
d & \mbox{if $d=1,2,4$, or $8$}\\
d-1 & \mbox{otherwise.}\end{array}\right.$$ Note that $\mathrm{charrank}(S^d)=d-1$. We shall use the above description of characteristic rank of vector bundles over the spheres in the sequel without explicit reference.
Suppose that $\pi:S^d\longrightarrow X$ is a $k$-sheeted covering with $k>1$ odd. Since $X\cong S^d/G$, where $G$ is a finite group with $|G|=k$, we have that $d$ is odd. By Proposition3G.1 of [@hatcher], the homomorphism $\pi^*:H^i(X;\mathbb Z_2)\longrightarrow H^i(S^d;\mathbb Z_2)$ is a monomorphism with image the $G$-invariant elements for all $i\geq 0$. In particular, $H^i(X;\mathbb Z_2)=0$, $0<i<d$ and $\pi^*:H^d(X;\mathbb Z_2)\longrightarrow H^d(S^d;\mathbb Z_2)\cong\mathbb Z_2$ is an isomorphism. Thus we have the following corollary to Theorem \[atiyah\].
\[lens1\] Assume that $\pi:S^d\longrightarrow X$ is a $k$-sheeted covering with an odd $k>1$ and $d\neq 1$. Then $w(\xi)=1$ for any vector bundle $\xi$ over $X$ and we have $\mathrm{ucharrank}(X)=d-1$.
If $0 < i < d$, then obviously $w_i(\xi)= 0$. In addition, for any $\xi$ we have now $\pi^*(w_d(\xi)) = w_d(\pi^*\xi) = 0$ by Theorem \[atiyah\]. Since, $\pi^*$ is injective, we thus have $w_d(\xi)= 0$. We know that $H^d(X,\mathbb Z_2) \cong \mathbb Z_2$; this implies that $\mathrm{charrank}_X(\xi)\leq d-1$ for any $\xi$. The inequality $\mathrm{charrank}_X(\xi)\geq d-1$ for any $\xi$ is clear.
Let $L=L_m(\ell_1,\ldots ,\ell_n)$ denote the lens space which is a quotient of $S^{2n-1}$ by a free action of the cyclic group $\mathbb Z_m$ (see [@hatcher], page 144). Then, we have an $m$-sheeted covering $\pi: S^{2n-1}\longrightarrow L$. If $n >1$ and $m$ is odd, then for any vector bundle $\xi$ over $L$, the total Stiefel-Whitney class $w(\xi)=1$. In particular, $\mathrm{ucharrank}(L)= 2n-2$.
There are conditions under which one can obtain a natural upper bound on the upper characteristic rank of a space. One such condition is the existence of a spherical class. Recall that a cohomology class $x\in H^k(X;\mathbb Z_2)$ is spherical if there exists a map $f:S^k\longrightarrow X$ with $f^*(x)\neq 0$. Note that a spherical class $x\in H^k(X;\mathbb Z_2)$ is indecomposable as an element of the cohomology ring. We shall show that the upper characteristic rank of a space is bounded above by the degree of a spherical class in most cases.
\[covering\] Let $X$ be a space and assume that $x\in H^k(X;\mathbb Z_2)$ is spherical, $k\neq 1,2,4,8$. Then there does not exist a vector bundle $\xi$ over $X$ with $w_k(\xi)=x$ and we have $\mathrm{charrank}_X(\xi) < k$ for any $\xi$. As a consequence, for any covering $\pi:E\longrightarrow X$, we have $\mathrm{ucharrank}(E)<k$ (in particular, $\mathrm{ucharrank}(X)<k$ ).
Assume that $\xi$ is a vector bundle over $X$ with $w_k(\xi)= x\neq 0$. Let $f:S^k\longrightarrow X$ be a map with $f^*(x)\neq 0$. Then one has $w_k(f^*\xi)=f^*(w_k(\xi))\neq 0$, which is impossible by Theorem \[atiyah\]. Hence there is no such $\xi$. Now since there is no $\xi$ with $w_k(\xi) = x$, and $x$ is indecomposable, we see that $\mathrm{charrank}_X(\xi) < k$ for any $\xi$. The rest of the claim follows from the fact that $f$ factors through the covering projection $\pi: E \longrightarrow X$. Indeed, we have $f = \pi \circ g$ for some $g : S^k \longrightarrow E$, and then $g^*(\pi^*(x))=f^*(x)\neq 0$, which means that the class $\pi^*(x)$ is spherical. The proof is finished by taking $E$ in the role of $X$ in the preceding considerations.
When a spherical class has degree $k=1,2,4$, or $8$, there can exist vector bundles of characteristic rank greater than or equal to the degree of the spherical class. For example, the sphere $S^k$ with $k=1,2,4$, or $8$ has upper characteristic rank equal to $k$. The complex projective space $\mathbb C\mathbb P^n$ has a spherical class in degree $2$, however $\mathrm{ucharrank}(\mathbb C\mathbb P^n)=2n$ (see Example\[complexprojective\]). When a spherical class exists in degree $1,2,4$ or $8$, we have the following observation:
[**Observation:**]{} Let $X$ be a space and assume that $x\in H^k(X;\mathbb Z_2)$ is spherical, where $k=1,2,4,8$. Let $f:S^k\longrightarrow X$ be a map with $f^*(x)\neq 0$. Then for a vector bundle $\xi$ over $X$ with $\mathrm{charrank}_X(\xi)\geq k$, we can express $x$ as a polynomial $P(w_1(\xi),w_2(\xi),\ldots w_k(\xi))$. But then $0 \neq f^*(x) = f^*(P(w_1(\xi),w_2(\xi),\ldots w_k(\xi)))$ = $ P(f^*(w_1(\xi)),f^*(w_2(\xi)),\ldots,f^*(w_k(\xi)))$. Hence $f^*(w_k(\xi)) \neq 0$. Thus for any vector bundle $\xi$ over $X$ with $\mathrm{charrank}_X(\xi)\geq k$, we have $w_k(\xi)\neq 0$.
When $X$ is a connected closed smooth $d$-manifold, the characteristic rank, $\mathrm{charrank}_X(\xi)$, of $\xi$ takes values in a certain specific range. We prove the following.
\[minusone\] Let $X$ be a connected closed smooth $d$-manifold. Assume that $2r_X\leq d$. Then, for any vector bundle $\xi$ over $X$, $\mathrm{charrank}_X(\xi)$ is either $d$ or less than $d-r_X$.
Let $\xi$ be a vector bundle over $X$ with $\mathrm{charrank}_X(\xi)\geq d-r_X$. We shall show that $\mathrm{charrank}_X(\xi)=d$. Since, by Poincaré duality, the groups $H^j(X;\mathbb Z_2)=0$ for $d-r_X<j<d$, the proof will be complete if the non-zero element in $H^d(X;\mathbb Z_2)$ is a polynomial in the Stiefel-Whitney classes of $\xi$. As $\mathrm{charrank}_X(\xi)\geq d-r_X\geq r_X$, then by Lemma\[secondlemma\], $H^{r_X}(X;\mathbb Z_2)\cong \mathbb Z_2$. Hence $H^{d-r_X}(X;\mathbb Z_2)\cong \mathbb Z_2$. Let $a,b,x$ denote the non-zero cohomology classes in degrees $r_X$, $d-r_X$ and $d$ respectively. The non-degeneracy of the pairing $$H^{r_X}(X;\mathbb Z_2)\otimes H^{d-r_X}(X;\mathbb Z_2)\longrightarrow H^d(X;\mathbb Z_2)$$ implies that $a\cdot b=x$. As $\mathrm{charrank}_X(\xi)\geq d-r_X\geq r_X$ we have, by Lemma\[firstlemma\] (1), $w_{r_X}(\xi)\neq 0$ and hence $w_{r_X}(\xi)=a$ and $b=p(w_1(\xi), w_2(\xi), \ldots)$ is a polynomial in the Stiefel-Whitney classes of $\xi$. This shows that $$x=w_{r_X}(\xi)\cdot p(w_1(\xi),w_2(\xi), \ldots)$$ is a polynomial in the Stiefel-Whitney classes of $\xi$. This completes the proof of the theorem.
Let $X$ be a connected closed smooth $d$-manifold. If $X$ is an unoriented boundary, then any monomial in the Stiefel-Whitney classes of $X$ of total degree $d$ is zero (see [@ms], Theorem 4.9). Hence the non-zero element in $H^d(X;\mathbb Z_2)$ is never a polynomial in the Stiefel-Whitney classes of $X$. We thus have the following corollary.
\[unnecessarycorollary\] Let $X$ be a connected closed smooth $d$-manifold. Assume that $2r_X\leq d$. If $X$ is an unoriented boundary, then $\mathrm{charrank}(TX)<d-r_X$.
\[korbasremark\] Balko and Korbaš [@balko] showed independently the following stronger version of Corollary \[unnecessarycorollary\]: For any connected closed smooth $d$-dimensional manifold $X$ that is an unoriented boundary, if $s$, $s\leq \frac{d}{2}$, is (the biggest) such that $H^s(X;\mathbb Z_2)\neq 0$, then $\mathrm{charrank}(X)<d-s$.
Proof of Theorems\[zeroththeorem\], \[cuplength\] and \[maintheorem\]
=====================================================================
In this section we prove Theorems\[zeroththeorem\], \[cuplength\], and \[maintheorem\]. The proof of Theorem\[zeroththeorem\] is essentially the same as the proof of Theorem\[korbastheorem\]. We reproduce it here for completeness.
[**Proof of Theorem\[zeroththeorem\]**]{} Let $x=x_1 \cdot x_2\cdots x_s\neq 0$ be a non-zero product of cohomology classes of positive degree and of maximal length. Then $x\in H^d(X;\mathbb Z_2)$. If not, then by Poincaré duality one can find some $y$ in complementary dimension such that $x\cdot y\neq 0$ contradicting the maximality of $s$. By rearranging, we write $$x= \alpha_1\cdots \alpha_m\cdot\beta_1 \cdots \beta_n$$ where $\mathrm{deg}(\alpha_i)\leq k$ and $\mathrm{deg}(\beta_j)\geq k+1$. We note that $n\neq 0$. For otherwise the product $\alpha=\alpha_1\cdots \alpha_m$ which is now a polynomial in $w_1(\xi),\ldots, w_k(\xi)$, would be a non-zero element of total degree $d$ contradicting the assumption on $\xi$. Therefore, if $\beta = \beta_1\cdots \beta_n$, then $\mathrm{deg}(\beta)\geq k+1$. Thus $\mathrm{deg}(\alpha)\leq d- (k+1)$. Thus $$\begin{array}{rcl}
\mathrm{cup}(X) & = & m+n\\
& &\\
& \leq & \frac{\mathrm{deg}(\alpha)}{r_X} + \frac{\mathrm{deg}(\beta)}{(k+1)}\\
& &\\
& = & \frac{\mathrm{deg}(\alpha)}{r_X} + \frac{(d-\mathrm{deg}(\alpha))}{(k+1))}\\
& &\\
& = & \frac{((k+1-r_X)\mathrm{deg}(\alpha)+dr_X)}{ r_X(k+1)}\\
& & \\
& \leq & \frac{((k+1-r_X)(d-(k+1))+dr_X)}{r_X(k+1)}\\
& &\\
&= & 1+\frac{d-k-1}{r_X}.\end{array}$$ This completes the proof.
[**Proof of Theorem\[cuplength\].**]{} Let $\xi$ be any vector bundle over $X$ with $$\mathrm{charrank}_X(\xi)=\mathrm{ucharrank}(X)=\mathrm{dim}(X).$$ Let $\mathrm{cup}(X)=k$. We shall show that some product of the Stiefel-Whitney classes of $\xi$ of length $k$ is non-zero. Let $$x=x_1\cdot x_2\cdots x_k\neq 0$$ be a non-zero product of cohomology classes $x_i\in H^*(X;\mathbb Z_2)$ with $\mathrm{deg}(x_i)\geq 1$. As $\mathrm{charrank}_X(\xi)=\mathrm{dim}(X)$, each $x_i$ can be written as a sum of monomials in the Stiefel-Whitney classes of $\xi$. Thus $x$ can be written as a sum of monomials in the Stiefel-Whitney classes of $\xi$, each of length at least $k$. Note that the monomials of length greater than $k$ are zero by hypothesis. As $x\neq 0$, it follows that some monomial in the Stiefel-Whitney classes of $\xi$ of length $k$ is non-zero. This completes the proof of the theorem.
1. The proof of Theorem\[cuplength\] actually shows that if some product $x=x_1\cdots x_t\neq 0$ with $1\leq \mathrm{deg}(x_i)\leq \ell$, then for any vector bundle $\xi$ over $X$ with $\mathrm{charrank}_X(\xi)\geq\ell$ some product of the Stiefel-Whitney classes of $\xi$ of length greater than or equal to $t$ is non-zero.
2. The conclusion of Theorem\[cuplength\] is not true if $\mathrm{ucharrank}(X)<\mathrm{dim}(X)$. If $X=S^k$, $k\neq 1,2,4,8$, then $\mathrm{ucharrank}(X)=k-1<k$, $\mathrm{cup}(X)=1$ however $w(\xi)=1$ for any vector bundle $\xi$ over $X$.
[**Proof of Theorem\[maintheorem\].**]{} First note that the assumption $\mathrm{ucharrank}(X)\geq 1$ is odd clearly implies that the function $$f:\mathrm{Vect}_{\mathbb R}(X)\longrightarrow \mathbb Z_2$$ defined by $$f(\xi)=\mathrm{charrank}_X(\xi) ~~~~~(\mathrm{mod}~~~~~ 2)$$ is surjective. We shall now check that $f$ is a semi-group homomorphism. To see this, let $\xi$ and $\eta$ be two bundles over $X$. We have the following cases.
If $\xi$ and $\eta$ are both orientable, then so is $\xi\oplus\eta$. Hence $w_1(\xi\oplus\eta)=0$. As $r_X=1$, it follows that $\mathrm{charrank}_X(\xi\oplus\eta)=0$. The same argument shows that $\mathrm{charrank}_X(\xi)=0=\mathrm{charrank}_X(\eta)$. Thus in this case we have $f(\xi\oplus\eta)=f(\xi)+f(\eta)$.
Next suppose that both $\xi$ and $\eta$ are non-orientable. Then, on the one hand, $\xi\oplus\eta$ is orientable and hence $f(\xi\oplus\eta)=0$ as $r_X=1$. On the other hand, as $\xi$ and $\eta$ are non-orientable, we have $$f(\xi)= 1 =f(\eta).$$ Thus, we have the equality $f(\xi\oplus\eta)=f(\xi)+f(\eta)$.
Finally, assume that $\xi$ is orientable and $\eta$ is not. Then $\xi\oplus \eta$ is not orientable and hence $f(\xi\oplus \eta)=1$, $f(\xi)=0$ and $f(\eta)=1$. So in this case we have $f(\xi\oplus\eta)=f(\xi)+f(\eta)$. This completes the proof that $f$ is a semi-group homomorphism.
This gives rise to a surjective homomorphism $$\widetilde{f}: KO(X)\longrightarrow \mathbb Z_2$$ defined by $\widetilde{f}(\xi-\eta)=f(\xi)-f(\eta)$. It is now clear that $\tilde{f}$ is zero on the $\mathbb Z$ summand of $KO(X)=\mathbb Z\oplus \widetilde{KO}(X)$ and restricts to an epimorphism $\tilde{f}:\widetilde{KO}(X)\longrightarrow \mathbb Z_2$. This completes the proof.
Computations and examples
=========================
In this section we give a proof of Theorem\[mainproposition\] and compute the characteristic rank of vector bundles over $X$, where $X$ is one of the following: the product of spheres $S^d\times S^m$, the real or complex projective space, the product space $S^1\times \mathbb C\mathbb P^n$, the Moore space $M(\mathbb Z_2,n)$ and the stunted projective space $\mathbb R\mathbb P^n/\mathbb R\mathbb P^m$.
We begin by describing the characteristic rank of vector bundles over $X=S^d\times S^m$. First note that if $d=m$, then as $r_X=d$ and $\mathrm{dim}_{\mathbb Z_2}H^d(X;\mathbb Z_2)=2$, it follows from Lemma\[secondlemma\] (1) that $\mathrm{ucharrank}(X)=r_X-1=d-1$.
Let $X=S^d\times S^m$ with $d<m$. Then, $$\mathrm{ucharrank}(X)=\left\{ \begin{array}{cl}
d-1 & \mbox{if $d\neq 1, 2,4, 8$,}\\
m-1 & \mbox{if $d=1,2,4, 8$, $m\neq 2,4,8$}\\
d+m & \mbox{if $d,m=1,2,4,8$.}
\end{array}\right.$$
The lemma follows from the observations made after Theorem\[atiyah\]. We note that $r_X=d$ and consider the maps $$S^d\stackrel{i}\longrightarrow S^d\times S^m\stackrel{\pi_1}\longrightarrow S^d,$$ $$S^m\stackrel{j}\longrightarrow S^d\times S^m\stackrel{\pi_2}\longrightarrow S^m,$$ where $i$ is the map $x\mapsto (x,y)$ for a fixed $y\in S^m$ and $\pi_1$ and $\pi_2$ are projections onto the the first and second factors. The map $j$ is similarly defined. The homomorphisms $i^*$ and $j^*$ are isomorphisms (with inverses $\pi_1^*$ and $\pi_2^*$ respectively) in degree $d$ and $m$ respectively.
Assume that $d\neq 1,2,4,8$ and let $\xi$ be a vector bundle over $X$. Then as $w_d(i^*\xi)=0$, it follows that $w_d(\xi)=0$. Thus by Lemma\[firstlemma\] (1) we have $\mathrm{charrank}_X(\xi)=r_X-1=d-1$.
Next assume that $d=1,2,4,8$ and $m\neq 2,4,8$. Let $\nu_d$ denote the Hopf bundle over $S^d$. As $w_d(\nu_d)\neq 0$, it follows that $w_d(\pi_1^*\nu_d)\neq 0$. Thus $\mathrm{charrank}_{\pi_1^*\nu_d}(X)\geq m-1$. Since $m\neq 1,2,4,8$, for any vector bundle $\xi$ over $X$ we must have $w_m(\xi)=0$. This completes the proof that $\mathrm{charrank}_{\pi_1^*\nu_d}(X)=m-1$ and that $\mathrm{ucharrank}(X)=m-1$.
Finally, let $d=1,2,4,8$ and $m=1,2,4,8$. Let $\nu_d$ and $\nu_m$ denote the Hopf bundles over $S^d$ and $S^m$ respectively. Then, clearly $w_d(\pi_1^*\nu_d\oplus \pi_2^*\nu_m)\neq 0$, $w_m(\pi_1^*\nu_d\oplus \pi_2^*\nu_m)\neq 0$ and $w_{d+m}(\pi_1^*\nu_d\oplus \pi_2^*\nu_m)\neq 0$. This shows that in this case $\mathrm{charrank}(X)=d+m$. This completes the proof of the lemma.
We now come to the proof of Theorem\[mainproposition\]. First recall that the Dold manifold $P(m,n)$ is an $(m+2n)$-dimensional manifold defined as the quotient of $S^m\times\mathbb C\mathbb P^n$ by the fixed point free involution $(x,z)\mapsto (-x,\bar{z})$. This gives rise to a two-fold covering $$\mathbb Z_2\hookrightarrow S^m\times \mathbb C\mathbb P^n\longrightarrow P(m,n),$$ and via the projection $S^m\times \mathbb C\mathbb P^n\longrightarrow S^m$, a fiber bundle $$\mathbb C\mathbb P^n\hookrightarrow P(m,n)\longrightarrow \mathbb R\mathbb P^m$$ with fiber $\mathbb C\mathbb P^n$ and structure group $\mathbb Z_2$. In particular, for $n=1$, we have a fiber bundle $$S^2\hookrightarrow P(m,1)\longrightarrow \mathbb R\mathbb P^m.$$
The $\mathbb Z_2$-cohomology ring of $P(m,n)$ is given by [@do] $$H^*(P(m,n);\mathbb Z_2) = \mathbb Z_2[c,d]/(c^{m+1}, d^{n+1})$$ where $c\in H^1(P(m,n);\mathbb Z_2)$ and $d\in H^2(P(m,n);\mathbb Z_2)$.
We shall make use of the following result which shows the existence of certain bundles with suitable Stiefel-Whitney classes.
[([@stong], page 86) ]{} \[ust\] Over $P(m,n)$,
1. there exists a line bundle $\xi$ with total Stiefel-Whitney class $w(\xi)=1+c$;
2. there exists a $2$-plane bundle $\eta$ with total Stiefel-Whitney class $w(\eta)=1+c+d$.
[**Proof of Theorem\[mainproposition\].**]{} Let $X=\mathbb R\mathbb P^n$ be the real projective space. Then $r_X=1$. Let $\xi$ be a vector bundle over $X$. If $\xi$ is orientable, then $w_1(\xi)=0$ and hence, by Lemma\[firstlemma\] (1), $\mathrm{charrank}_X(\xi)=0$. On the other hand if $\xi$ is non-orientable, then $w_1(\xi)\neq 0$ and hence $\mathrm{charrank}_X(\xi)=n$ as $H^*(X;\mathbb Z_2)$ is polynomially generated by the non-zero element in $H^1(X;\mathbb Z_2)$. This proves (1).
To prove (2), let $X=S^1\times \mathbb C\mathbb P^n$, then $r_X=1$. The $\mathbb Z_2$-cohomology ring of $X$ is given by $$H^*(X;\mathbb Z_2)=H^*(S^1;\mathbb Z_2)\otimes H^*(\mathbb C\mathbb P^n;\mathbb Z_2)\cong\mathbb Z_2[a,b]/(a^2, b^{n+1}),$$ where $a$ is of degree one and $b$ is of degree two. Let $\xi$ be a vector bundle over $X$. Evidently, $\mathrm{charrank}_X(\xi)$ is completely determined by the first two Stiefel-Whitney classes of $\xi$.
We look at several cases. If $w_1(\xi)$ and $w_2(\xi)$ are both non-zero, then the description of the cohomology ring $H^*(X;\mathbb Z_2)$ forces $\mathrm{charrank}_X(\xi)=2n+1$. If $w_1(\xi)=0$, we have $\mathrm{charrank}_X(\xi)=0$. If $w_1(\xi)\neq 0$ and $w_2(\xi)=0$, then $\mathrm{charrank}_X(\xi)=1$. This completes the proof of (2).
Finally, the proof of (3) is similar to the case (2) above in view of Proposition\[ust\]. Indeed, if $w_1(\eta)=c\neq 0$ and $w_2(\eta)=d\neq 0$ (there exists such an $\eta$; see Proposition \[ust\]), then we have $\mathrm{charrank}_X(\eta)=2n+m$. If $w_1(\xi)=c\neq 0$ and $w_2(\xi)=0$ (there exists such a $\xi$; see Proposition 4.2), we have $\mathrm{charrank}_X(\xi)=1$, as $c^2\neq
d$. For other possible vector bundles, the situation is clear. This completes the proof of (3) and the theorem.
\[importantremark\] (1) We remark that, in the case (2) of the theorem above, there exists a line bundle $\gamma$ over $X$ such that $w_1(\gamma)\neq 0$. Thus, $\mathrm{charrank}_X(\gamma)=1$. We also can find a $2$-plane bundle $\eta$ over $X$ such that $w_1(\eta)=0$ and $w_2(\eta)\neq 0$. Thus $\mathrm{charrank}_X(\eta)=0$. Then for the Whitney sum $\gamma\oplus\eta$ we have $w_1(\gamma\oplus\eta)=w_1(\gamma)\neq 0$ and $w_2(\gamma\oplus\eta)=w_2(\eta)\neq 0$ and hence $\mathrm{charrank}_{\gamma\oplus\eta}(X)=2n+1$. The bundles $\gamma$ and $\eta$ can be obtained as the pull backs of suitable canonical bundles over $S^1=\mathbb R\mathbb P^1$ and $\mathbb C\mathbb P^n$ via the projections. Thus, over $X=S^1\times \mathbb C\mathbb P^n$, there exist vector bundles having all the three possible characteristic ranks.
\(2) The function $f:\mathrm{Vect}_{\mathbb R}(X)\longrightarrow \mathbb Z_2$ constructed in the proof of Theorem\[maintheorem\] is in general not a semi-ring homomorphism. For example, let $\gamma$ denote the canonical line bundle over $X=\mathbb R\mathbb P^n$ ($n$ odd). Then $w_1(\gamma)\neq 0$ and hence $f(\gamma)=1\in\mathbb Z_2$. Now, as $\gamma\otimes \gamma$ is a trivial bundle, we have $w_1(\gamma\otimes \gamma)=0$ and therefore, $f(\gamma\otimes\gamma)=0\in\mathbb Z_2$. Clearly, $0=f(\gamma\otimes\gamma)\neq f(\gamma)\cdot f(\gamma)=1$.
\[complexprojective\] Let $X=\mathbb C\mathbb P^n$ be the complex projective space. Then $r_X=2$. Let $\xi$ be a vector bundle over $X$. Then $\mathrm{charrank}_X(\xi)=1$ if $w_2(\xi)=0$ and $\mathrm{charrank}_X(\xi)=2n$ if $w_2(X)\neq 0$. For the canonical (complex) line bundle $\gamma$ over $X$ we have $\mathrm{charrank}_X(\gamma)=2n$.
We now give some examples where the bound for the cup length given by Theorem\[zeroththeorem\] is sharper than that given by Theorem\[korbastheorem\].
\[4.6\] Let $X= S^2\times S^6$ and let $\pi_1:X\longrightarrow S^2$ be the projection. Let, as usual, $\nu_2$ denote the Hopf bundle over $S^2$. Then, $\mathrm{charrank}_{TX}(X)=1$, and $\mathrm{charrank}_X(\xi)=5$ where $\xi=\pi_1^*\nu_2$. The bundle $\xi$ satisfies the condition of Theorem\[zeroththeorem\] with $k=5$. Then the bound for the cup length, $\mathrm{cup}(X$), of $X$ given by Theorem\[korbastheorem\] is $4$ and that given by Theorem\[zeroththeorem\] is $2$.
\[4.7\] Let $X = S^4\times S^8$. Let $\xi=\pi_1^*\nu_4\oplus \pi_2^*\nu_8$. Then, $\mathrm{charrank}_{TX}(X)=3$ and $\mathrm{charrank}_X(\xi)= 12$. Then $\xi$ satisfies the condition of Theorem\[zeroththeorem\] with $k=7$. Then the bound for the cup length, $\mathrm{cup}(X$), of $X$ given by Theorem\[korbastheorem\] is $3$ and that given by Theorem\[zeroththeorem\] is $2$.
These sharper estimates of Examples \[4.6\] and \[4.7\] can also be obtained from Theorem A [@korbas2].
We now compute $\mathrm{charrank}_X(\xi)$ where $X$ is the Moore space $M(\mathbb Z_2,n)$, $n>1$, and $\xi$ a vector bundle over $X$. We recall that $X$ is an $(n-1)$-connected $(n+1)$-dimensional $CW$-complex. Note that $M(\mathbb Z_2, 1)$ is the real projective space $\mathbb R\mathbb P^2$ and $M(\mathbb Z_2,n)$ is the iterated suspension $\Sigma^nM(\mathbb Z_2,1)$. We refer to [@hatcher] for basic properties of Moore spaces. We prove the following.
\[secondlastprop\] Let $X$ denote the Moore space $M(\mathbb Z_2,n)$ with $n>1$. Then, $$\mathrm{ucharrank}(X)= \left\{\begin{array}{cl}
n-1 & \mbox{if $n\neq 2$}\\
3 & \mbox{if $n=2$}
\end{array}\right.$$
The Moore space $X$ is an $(n+1)$-dimensional $CW$-complex with $n$-skeleton $S^n$. Let $i:S^n\hookrightarrow X$ denote the inclusion map. Using the cellular chain complex, for example, it is easy to see that the homomorphism $$i^*:H^n(X;\mathbb Z_2)\longrightarrow H^n(S^n;\mathbb Z_2)$$ in degree $n$ is an isomorphism and hence the non-zero element in $H^n(X;\mathbb Z_2)$ is spherical.
Assume that $n\neq 2,4,8$. Since $X$ is $(n-1)$-connected it follows from Proposition\[covering\] that $\mathrm{charrank}_X(\xi)=n-1$ for any $\xi$ over $X$. This proves the first equality for $n\neq 2, 4,8$.
Next, for $X=M(\mathbb Z_2,n)$, we observe that there is a cofiber sequence $$S^n\stackrel{f}\longrightarrow S^n \longrightarrow X\longrightarrow S^{n+1}\longrightarrow S^{n+1}$$ where $f$ is a degree $2$ map. This gives rise to an exact sequence $$\widetilde{KO}(S^{n+1})\longrightarrow \widetilde{KO}(X)\longrightarrow \widetilde{KO}(S^n)\stackrel{f^*}\longrightarrow \widetilde{KO}(S^n).$$ When $n=4,8$ the homomorphism $f^*$ is injective and hence the homomorphism $\widetilde{KO}(S^{n+1})\longrightarrow \widetilde{KO}(X)$ is surjective. When $n=2$, the homomorphism $f^*$ is the zero homomorphism and hence the homomorphism $\widetilde{KO}(X)\longrightarrow \widetilde{KO}(S^n)$ is surjective. These obeservations follow from the fact that $\widetilde{KO}(S^4)=\mathbb Z=\widetilde{KO}(S^8)$ and $\widetilde{KO}(S^2)=\mathbb Z_2$ together with the fact that $f$ is a degree $2$ map.
Thus when $n=4,8$ we have by Theorem\[atiyah\] that $w(\xi)=1$ for any vector bundle over $X=M(\mathbb Z_2,n)$. This completes the proof of the first equality when $n=4,8$.
Finally let $X=M(\mathbb Z_2,2)$. Then $X$ is a simply connected $3$-dimensional $CW$-complex. We shall show that there exists a bundle $\xi$ over $X$ with $w_2(\xi)\neq 0$ and $w_3(\xi)\neq 0$. As the homomorphism $\widetilde{KO}(X)\longrightarrow \widetilde{KO}(S^2)$ is surjective and $w_2(\nu_2)\neq 0$, there exists a bundle $\xi$ over $X$ with $w_2(\xi)\neq 0$. For this vector bundle $\xi$ over $X$ the Stiefel-Whitney class $w_3(\xi)\neq 0$. To see this we observe that if $a\in H^1(\mathbb R\mathbb P^2;\mathbb Z_2)= H^1(M(\mathbb Z_2,1);\mathbb Z_2)$ is the unique non-zero element, then $Sq^1(a)=a^2\neq 0$. Thus, by Wu’s formula and the fact that the Steenrod squares commute with the suspension homomorphism we see that $Sq^1(w_2(\xi))=w_1(\xi)w_2(\xi)+w_3(\xi)=w_3(\xi)\neq 0$. This completes the proof of the second equality.
\[lastprop\] Let $X$ denote the stunted projective space $\mathbb R\mathbb P^n/\mathbb R\mathbb P^m$ with $1\leq m\leq n-2$. Then $$\mathrm{ucharrank}(X)=\left\{\begin{array}{cl}
m & \mbox{if $m+1\neq 2,4,8$}\\
m+1 & \mbox{if $m+1=2,4,8$}
\end{array}\right.$$
The stunted projective space $X$ is $m$-connected with $(m+1)$-skeleton $X^{(m+1)}=S^{m+1}$. If $f:S^{m+1}=X^{(m+1)}\longrightarrow X$ denotes the inclusion map, then it is easy to check that the homomorphism $$f^*:H^{m+1}(X;\mathbb Z_2)\longrightarrow H^{m+1}(S^{m+1};\mathbb Z_2)$$ is an isomorphism. Thus, the non-zero element in $H^{m+1}(X;\mathbb Z_2)$ is spherical. The first equality of the proposition now follows from Proposition\[covering\].
Let $X=\mathbb R\mathbb P^n/\mathbb R\mathbb P^m$ with $m+1=2,4,8$. It is clear that the inclusion map $$\mathbb R\mathbb P^{m+2}/\mathbb R\mathbb P^m \longrightarrow \mathbb R\mathbb P^n/\mathbb R\mathbb P^m$$ where $n\geq m+2$ induces isomorphism in $\mathbb Z_2$-cohomology in degree $i$ for all $i\leq m+2$. Since $(m+2)$ is odd we have a splitting $$\mathbb R\mathbb P^{m+2}/\mathbb R\mathbb P^m = S^{m+2}\vee S^{m+1}.$$ It follows that $X$ has a spherical class in degree $(m+2)$ and hence by Proposition\[covering\] we have $\mathrm{ucharrank}(X)\leq m+1$. We shall prove the equality by showing that there exists a bundle $\xi$ over $X$ with $w_{m+1}(\xi)\neq 0$.
As $\mathbb R\mathbb P^{m+2}/\mathbb R\mathbb P^m=S^{m+1}\vee S^{m+2}$, the Hopf bundle $\nu_{m+1}$ over $S^{m+1}$ extends over $S^{m+1}\vee S^{m+2}$ to give a vector bundle $\xi$ with $w_{m+1}(\xi)\neq 0$. It is well known [@adams] that for any $n\geq m+2$ the inclusion map $$\mathbb R\mathbb P^{m+2}/\mathbb R\mathbb P^m\hookrightarrow \mathbb R\mathbb P^n/\mathbb R \mathbb P^m$$ induces an epimorphism in reduced $KO$-groups. Thus there is a vector bundle over $\mathbb R\mathbb P^n/\mathbb R \mathbb P^m$ with the required property.
We are indebted to Professor J. Korbaš for his detailed and helpful comments on an earlier draft of this manuscript. In particular, we thank him for showing us the proof of Corollary\[lens1\]. The original statement of the corollary only contained the conclusion that $\mathrm{ucharrank}(X)<d$, under the assumption that $X$ is orientable and $d\neq 1,2,4,8$. We also thank him for sending us a copy of his paper [@korbas]. We would like to thank the anonymous referee for his detailed suggestions. In particular, we thank him for showing us the proof of Proposition\[secondlastprop\]. This is shorter and stronger than proof given by the authors.
[99]{}
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Balko, L’.—Korbaš, J.: *A note on the characteristic rank of a smooth manifold*, Group actions and homogeneous spaces, Fak. Mat. Fyziky Inform. Univ. Komenského, Bratislava, 2010, pp. 1-8.
Balko, L’.—Korbaš, J.: *A note of the characteristic rank of null-cobordant manifolds*, To appear in Acta. Math. Hungar, 2011/2012.
Borel, A.: *Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts*, Ann. of Math. **57** (1953), 115-207.
Dold, A.: *Erzeugende der Thomschen Algebra $\mathfrak{N}$*, Math. Z. **65** (1956), 25-35.
Hatcher, A.: *Algebraic Topology*, Cambridge Univ. Press, 2002.
Husemoller, D.: *Fibre Bundles*, Springer-Verlag, New York, 1966.
Korbaš, J.: *Bounds for the cup-length of Poincaré spaces and their applications*, Topology Appl. **153** (2006), 2976-2986.
Korbaš, J.: *The cup-length of the oriented Grassmannians vs a new bound for zero cobordant manifolds*, Bull. Belg. Math. Soc.-Simon Stevin **17** (2010), 69-81.
Milnor, J.: *Some consequences of a theorem of Bott*, Ann. of Math. **68** (1958), 444-449.
Milnor, J.—Stasheff, J.: *Characteristic Classes*, Princeton Univ. Press, Princeton, 1974.
Stong, R. E.: *Vector bundles over Dold manifolds*, Fund. Math. **169** (2001), 85-95.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the central diffractive production of the (three neutral) Higgs bosons, with a rapidity gap on either side, in an MSSM scenario with CP-violation. We consider the $b\bar{b}$ and $\tau\bar{\tau}$ decay for the light $H_1$ boson and the four $b$-jet final state for the heavy $H_2$ and $H_3$ bosons, and discuss the corresponding backgrounds. A direct indication of the existence of CP-violation can come from the observation of either an azimuthal asymmetry in the angular distribution of the tagged forward protons (for the exclusive $pp\to p+H+p$ process) or of a sin$2\varphi$ contribution in the azimuthal correlation between the transverse energy flows in the proton fragmentation regions for the process with the diffractive dissociation of both incoming protons ($pp\to X+H+Y$). We emphasise the advantage of reactions with the rapidity gaps (that is production by the pomeron-pomeron fusion) to probe CP parity and to determine the quantum numbers of the produced central object.'
---
IPPP/03/84\
DCPT/03/168\
12 January 2004\
[**Hunting a light CP-violating Higgs via diffraction at the LHC**]{}
<span style="font-variant:small-caps;">V.A. Khoze$^{a,b}$, A.D. Martin$^a$ and M.G. Ryskin$^{a,b}$</span>\
$^a$ Department of Physics and Institute for Particle Physics Phenomenology,\
University of Durham, DH1 3LE, UK\
$^b$ Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg, 188300, Russia\
Introduction
============
It is known that third generation squark loops can introduce sizeable CP violation in the Higgs potential of the Minimal Supersymmetric Standard Model (MSSM), if the soft-supersymmetry-breaking mass parameters of the third generation are complex; see, for example, [@AP]. As a result, the neutral Higgs bosons will mix to produce three physical mass eigenstates with mixed CP parity, which we denote $H_1,H_2$ and $H_3$ in order of increasing mass. A benchmark scenario of maximal CP violation, called CPX, was introduced in Ref. [@CEPW]. In this scenario |A\_t|=|A\_b|=2 M\_[SUSY]{},||=4 M\_[SUSY]{}, M\_[\_3,\_3,\_3]{}=M\_[SUSY]{},|M\_3|=1 [TeV]{}, \[eq:jan5a\] where $A_f$ are are the soft-supersymmetry-breaking trilinear parameters of the third generation squarks and $\mu$ is the supersymmetric higgsino mass parameter. The phenomenological consequences of this model may be quite spectacular. In particular, the $H_1ZZ$ coupling of the lightest Higgs boson can be significantly suppressed; see, for example, [@CEPW] and references therein. In this case, it was shown that the LEP2 data do not exclude the existence of a light Higgs boson with mass $M_H<60$ GeV (40 GeV) in the minimal SUSY model with $\tan\beta\sim3$–4 (2–3) and CP-violating phase \_[CPX]{} (A\_t) = [arg]{}(A\_b) = [arg]{}(A\_)=[arg]{} (m\_[g]{}) = 90\^ (60\^). \[eq:A1\] Since the $H_1$ couplings to the $W$ and $Z$ gauge bosons become rather small, it would be hard to detect the light Higgs via the processes $e^+e^- \to Z^\star\to ZH_i$ or $e^+e^- \to Z^\star\to H_iH_j$.
It is therefore interesting to consider the possibility of observing a light Higgs boson at the LHC or Tevatron collider. However, in general, it will be hard to observe a light Higgs at hadron colliders via the $\bb$ decay mode because, in particular, the transverse momenta of the outgoing $b$ and $\bar b$ jets are not large. As a consequence the signal is swamped by the QCD $\bb$ background[^1]. Therefore it was proposed [@cox] to search for a CP-violating light Higgs boson in the [*exclusive*]{} process $pp\to p + H + p$ at hadron colliders, where the $+$ signs denote the presence of large rapidity gaps. Over the past few years such exclusive diffractive processes have been considered as a promising way to search for manifestations of New Physics in high energy proton-proton collisions; see, for instance, [@KMRcan; @INC; @cox; @KKMRCentr; @DKMOR; @CR]. These processes have both unique experimental and theoretical advantages in hunting for Higgs bosons as compared to the traditional non-diffractive approaches. In particular, in the exclusive diffractive reactions the $\bb$ background is suppressed [@Liverpool; @KMRItal; @KMRmm; @DKMOR], and it may be feasible to isolate the signal.
In the present paper we discuss the central [*exclusive*]{} diffractive production (CEDP) in more detail. We compare the signal and the background for observing a light neutral Higgs boson via $H_1\to\bb$ and $H_1\to\tau\tau$ decay modes. Then we evaluate the asymmetry arising from the interference of the P-even and P-odd production amplitudes. Note that this asymmetry is the most direct manifestation of CP-violation in the Higgs sector. Finally we consider the exclusive diffractive production of the heavier neutral Higgs bosons, $H_2$ and $H_3$, followed by the decays $H_2$ or $H_3\to H_1H_1\to 4 b$-jets.\
For numerical estimates, we use the formalism to describe central production in diffractive exclusive processes of [@INC], and the parameters (that is the masses, width and couplings of the Higgs bosons) given by the code “CPsuperH” [@Lee], where we choose $\phi_{\rm CPX}=90^\circ$, $\rm tan\beta=4$, $M_{\rm SUSY}=0.5$ TeV, (that is $|A_f| = 1$ TeV, $|\mu| =
2$ TeV, $|M_{\tilde g}|=1$ TeV) and the charged Higgs boson mass $M_{H^\pm}=135.72$ GeV so that the mass of the lightest Higgs boson, $H_1$, is $M_{H_1}=40$ GeV.[^2]
The exclusive process is shown schematically in Fig. \[fig:1\].
The cross section may be written[@INC] as the product of the effective gluon–gluon luminosity ${\cal L}$, and the square of the matrix element of the subprocess $gg\to H$. Note that the hard subprocess is mediated by the quark/squark triangles. For a CP-violating Higgs, there are two different vertices of the Higgs–quark interaction: the scalar Yukawa vertex and the vertex containing the $\gamma_5$ Dirac matrix. Therefore the $gg\to
H$ matrix element contains two terms:[^3] = g\_S(e\_1\^e\_2\^) - g\_P \^ e\_[1]{}e\_[2]{}p\_[1]{}p\_[2]{}/(p\_1p\_2) \[eq:1\] where $e^\perp$ are the gluon polarisation vectors and $\varepsilon^{\mu\nu\alpha\beta}$ is the antisymmetric tensor. In (\[eq:1\]) we have used a simplified form of the matrix element which already accounts for gauge invariance, assuming that the gluon virtualities are small in comparison with the Higgs mass. In forward exclusive central production, the incoming gluon polarisations are correlated, in such a way that the effective luminosity satisfies the P-even, $J_z=0$ selection rule [@INC; @KMRmm]. Therefore only the first term contributes to the strictly forward cross section. However, at non-zero transverse momenta of the recoil protons, $p_{1,2}^\perp\neq0$, there is an admixture of the P-odd $J_z=0$ amplitude of order $p_1^\perp p_2^\perp / Q_\perp^2$, on account of the $g_P$ term becoming active. Thus we consider non-zero recoil proton transverse momenta, and demonstrate that the interference between the CP-even ($g_S$) and CP-odd ($g_P$) terms leads to left-right asymmetry in the azimuthal distribution of the outgoing protons. First, we consider the background. Unfortunately, even in the exclusive process, we show below that the QCD $\bb$ background is too large. However, we shall see that it may be possible to observe such a CP-violating light Higgs boson in the $H\to \tau\tau$ decay mode, where the QED background can be suppressed by selecting events with relatively large outgoing proton transverse momenta, say, $p_{1,2}^\perp>300$ MeV.
Exclusive diffractive $H_1$ production followed by $\bb$ decay
================================================================
First, we consider the exclusive double-diffractive process ppp+(H)+p \[eq:A2\] The signal-to-background ratio is given by the ratio of the cross sections for the hard subprocesses, since the effective gluon–gluon luminosity ${\cal L}$ cancels out. The cross section for the $gg\to H$ subprocess[^4] [@INC] (ggH) = (1-) \~(1-), \[eq:2\]as the width[^5], $\Gamma(H\to gg)$, behaves as $\Gamma\sim \alpha_S^2 G_F M_H^3$, where $G_F$ is the Fermi constant. On the other hand, at leading order, the QCD background is given by the $gg\to \bb$ subprocess \~ , \[eq:3\] where $E_T$ is the transverse energy of the $b$ and $\bar b$ jets. At leading order (LO), the cross section is suppressed by the $J_z=0$ selection rule (which gives rise to the $m_b^2/E_T^2$ factor) in comparison with the inclusive process. The extra factor was crucial to suppress the background. It was shown in [@DKMOR] that it is possible to achieve a signal-to-background ratio of about 3 for the detection of a Standard Model Higgs with mass $M_H\sim 120$ GeV, by selecting $\bb$ exclusive events where the polar angle $\theta$ between the outgoing jets lies in the interval $60^\circ<\theta<120^\circ$ if the missing mass resolution $\Delta m_{\rm missing} =
1$ GeV. The situation is much worse for a light Higgs, since the signal-to-background ratio behaves as \~ M\_\^5 \[eq:4\] where we have used $\Delta\ln M_\bb^2 = 2\Delta M_\bb/M_\bb$. The $M^5$ behaviour comes just from dimensional counting. As the experimental resolution $\Delta M_\bb$ is larger than the width of the Higgs, $\Gamma_H$, the Higgs cross section (in the numerator) is driven by $G_F^2$, while the QCD background is proportional to $m_b^2$ and the size of the $\Delta M_\bb$ interval. To restore the dimensions we have to divide $m_b^2\Delta M_\bb$ by $M_\bb^5$. Thus, in going from $M_H\sim 120$ GeV to $M_H\sim40$ GeV, the expected leading-order QCD $\bb$ background increases by a factor of 240 in comparison with that for $M_\bb=120$ GeV.
Strictly speaking, there are other sources of background [@DKMOR]. There is the possibility of the gluon jet being misidentified as either a $b$ or a $\bar b$ jet, or a contribution from the NLO $gg\to \bb g$ subprocess, where the extra gluon is not separated from either a $b$ or a $\bar b$ jet. These contributions have no $m_b^2/M_\bb^2$ suppression, and hence increase only as $M_H^{-3}$, and not as $M_H^{-5}$, with decreasing $M_H$. For $M_H\sim 120$ GeV, the LO $\bb$ QCD production was only about 30% of the total background. However, for $M_{H_1}\sim40$ GeV, the LO $\bb$ contribution dominates. Finally, with the cuts of Ref. [@DKMOR], we predict that the cross section of the $H_1$ signal is[^6] $$\sigma^{\rm CEDP}(pp\to p+(H_1\to\bb)+p)\simeq 14~{\rm fb}$$ as compared to the QCD background cross section, with the same cuts[^7], of $$\sigma^{\rm CEDP}(pp\to p+(\bb)+p)\simeq 1.4\frac{\Delta M}
{1~{\rm GeV}}~{\rm pb}.$$ That is the signal-to-background ratio is only $S/B\sim 1/100$, and so even for an integrated luminosity ${\cal L} = 300~{\rm fb}^{-1}$ for $\Delta M = 1$ GeV the significance of the signal is only $3.7\sigma$. Here we have taken a $K$ factor of $K = 1.5$ for the QCD $\bb$ background, and again used the cuts and efficiencies quoted in Ref.[@DKMOR]. Therefore, to identify a light Higgs, it is desirable to study a decay mode other than $H_1\to\bb$. The next largest mode is $H_1\to\tau\tau$, with a branching fraction of about 0.07.
The dependence of the results on the mass of the $H_1$ Higgs boson is illustrated in Table 1. Clearly the cross section decreases with increasing mass. On the other hand the signal-to-background ratio increases. Therefore for the case $M_{H_1} = 50$ GeV we see a slightly improved statistical significance of $4.4\sigma$ for the $\bb$ decay mode.
$M(H_1)$ GeV cuts 30 40 50
--------------------------------------------- ------- ------- ------- -------
$\sigma(H_1){\rm Br}(\bb)$ $a$ 45 14 6
$\sigma^{\rm QCD}(\bb)$ $a$ 16000 1400 200
$A_{\bb}$ 0.14 0.07 0.04
$\sigma(H_1){\rm Br}(\tau\tau)$ $a,b$ 1.9 0.6 0.3
$\sigma^{\rm QED}(\tau\tau)$ $a,b$ 0.2 0.1 0.04
$A_{\tau\tau}$ $b$ 0.2 0.1 0.05
$M(H_2)$ GeV 103.4 104.7 106.2
$\sigma\dot {\rm Br} (H_2 \to 2H_1 \to 4b)$ $c$ 0.5 0.5 0.5
$\sigma\dot {\rm Br} (H_2 \to 2b)$ $a$ 0.1 0.1 0.2
$M(H_3)$ GeV 141.9 143.6 146.0
$\sigma\dot {\rm Br} (H_3 \to 2H_1 \to 4b)$ $c$ 0.14 0.2 0.18
$\sigma\dot {\rm Br}(H_3 \to 2b)$ $a$ 0.04 0.07 0.1
: The cross sections (in fb) of the central [*exclusive*]{} diffractive production of $H_i$ neutral Higgs bosons, together with those of the QCD($\bb$) and QED($\tau\tau$) backgrounds. The acceptance cuts applied are (a) the polar angle cut $60^\circ<\theta(b~{\rm or}~\tau)<120^\circ$ in the Higgs rest frame, (b) $p_i^\perp>300$ MeV for the forward outgoing protons and (c) the polar angle cut $45^\circ < \theta (b) <
135^\circ$. The azimuthal asymmetries $A_i$ are defined in eq.(12).
The $\tau\tau$ decay mode
=========================
At the LHC energy, the expected cross section for exclusive diffractive $H_1$ production, followed by $\tau\tau$ decay, is (ppp+(H)+p) \~1.1 [fb]{}, \[eq:5\] where the $60^\circ<\theta<120^\circ$ polar angle cut has already been included. Despite the low Higgs mass, we note that the exclusive cross section is rather small. As we already saw in (\[eq:2\]), the cross section of the hard subprocess $\hat\sigma(gg\to H)$ is approximately independent of $M_H$. Of course, we expect some enhancement from the larger effective gluon–gluon luminosity ${\cal L}$ for smaller $M_H$. Indeed, it may be approximated by [@INC; @KKMRext] 1[/]{} (M\_H + 16 [GeV]{})\^[3.3]{}, \[eq:5a\] and gives an enhancement of about 18.8 (for $M_H=40$ GeV in comparison with that for $M_H=120$ GeV).
On the other hand, in the appropriate region of SUSY parameter space, the CP-even $H\to gg$ vertex, $g_S$, is almost 2 times smaller[@cox; @Lee] than that of a Standard Model Higgs, giving a suppression of 4. Also the ratio $B(H\to \tau\tau)/B(H\to\bb)$ gives a further suppression of about 12. Although the $\tau\tau$ signal has the advantage that there is practically no QCD background[^8], exclusive $\tau^+\tau^-$ events may be produced by $\gamma\gamma$ fusion, see Fig. \[fig:2\].
The cross section for this latter QED process is appreciable. It is enhanced by two large logarithms, $\ln^2(t_{\rm min}R_p^2)$, arising from the integrations over the transverse momenta of the outgoing protons (that is of the exchanged photons). The lower limit of the logarithmic integrals is given by t\_[min]{} -(xm\_p)\^2 -(m\_p)\^2, \[eq:6\] while the upper limit is specified by the slope $R_p^2$ of the proton form factor. To suppress the QED background, one may select events with relatively large transverse momenta of the outgoing protons. For example, if $p_{1,2}^\perp > 300$ MeV, then the cross section for the QED background, for $M_{\tau\tau}=40$ GeV, is about[^9] \_[QED]{}(ppp + + p) 0.1 [fb]{}, \[eq:7\] while the signal (\[eq:5\]) contribution is diminished by the cuts, $p^\perp_{1,2}>300$ MeV, down to 0.6 fb. Thus, assuming an experimental missing mass resolution of $\Delta M\sim 1$ GeV, we obtain a healthy signal-to-background ratio of $S/B \sim 6$ for $M_{H_1} \sim
40$ GeV.
Note that in all the estimates given above, we include the appropriate soft survival factors $S^2$—that is the probabilities that the rapidity gaps are not populated by the secondaries produced in the soft rescattering. The survival factors were calculated using the formalism of Ref. [@KMRsoft]. Moreover, here we account for the fact that only events with proton transverse momenta $p_{1,2}^\perp>300$ MeV were selected. In particular, for the QED process, we have $S^2\simeq 0.7$, rather than the value $S^2\simeq 0.9$, which would occur in the absence of the cuts on the proton momenta[^10].
Azimuthal asymmetry of the outgoing protons
============================================
A specific prediction, in the case of a CP-violating Higgs boson, is the asymmetry in the azimuthal $\varphi$ distribution of the outgoing protons, caused by the interference of the CP-odd and CP-even vertices, that is between the two terms in (\[eq:1\]). The polarisations of the incoming active gluons are aligned along their respective transverse momenta, $Q_\perp - p_1^\perp$ and $Q_\perp + p_2^\perp$. Hence the contribution caused by the second term, $g_P$, is proportional to the vector product $$\vec{n}_0 \cdot (\vec{p}_1^\perp \times \vec{p}_2^\perp) \sim \sin\varphi,$$ where $\vec{n}_0$ is a unit vector in the beam direction, $\vec{p}_1$. The sign of the angle $\varphi$ is fixed by the four-dimensional structure of the second term in (\[eq:1\]); see [@KKMRCentr] for a detailed discussion. Of course, due to the P-even, $J_z=0$ selection rule, this (P-odd) contribution is suppressed in the amplitude by $p_1^\perp p_2^\perp/Q_\perp^2$, in comparison with that of the P-even $g_S$ term. Note that there is a partial compensation of the suppression due to the ratio $g_P/g_S \sim 2$. Also the soft survival factors $S^2$ are higher for the pseudoscalar and interference terms, than for the scalar term.
An observation of the azimuthal asymmetry may therefore be a direct indication of the existence of CP-violation (or P-violation in the case of CEDP) in the Higgs sector[^11]. Neglecting the absorptive effects (of soft rescattering), we find, for example, an asymmetry A= = 2[Re]{}(g\_S g\_P\^\*) r\_[S/P]{} (2/)/(|g\_S|\^2 + |r\_[S/P]{} g\_P|\^2/2). Here (numerically small) parameter $r_{S/P}$ reflects the suppression of the P-odd contribution due to the selection rule discussed above.
At the LHC energy in the absence of rescattering effects $A\simeq 0.09$ for $M_{H_1}=40$ GeV. However we find soft rescattering tends to wash out the azimuthal distribution, and to weaken the asymmetry. Besides this the real part of the rescattering amplitude multiplied by the imaginary part of the pseudoscalar vertex $g_P$ (with respect to $g_S$) gives some negative contribution. So finally we predict $A\simeq 0.07$. For the lower Tevatron energy, the admixture of the P-odd amplitude is larger, while the probability of soft rescattering is smaller. Therefore, at $\sqrt s=2$ TeV, we find that asymmetry is twice as large, $A\sim 0.17$. On the other hand the effective $gg^{PP}$ luminosity ${\cal L}$ and the corresponding cross section of $H_1$ (CEDP) production is 10 times smaller (for $M_{H_1}=40$ GeV).
The asymmetries expected at the LHC, with and without the cut $p_{1,2}^\perp>300$ MeV on the outgoing protons, are shown for different $H_1$ masses in Table 1. The asymmetry decreases with increasing Higgs mass, first, due to the decrease of $|g_P|/|g_S|$ ratio in this mass range and, second, due to the extra suppression of the P-odd amplitude arising from the factor $p_1^\perp p_2^\perp/Q_\perp^2$ in which the typical value of $Q_\perp$ in the gluon loop increases with mass.
Heavy $H_2$ and $H_3$ Higgs production with $H_1H_1$ decay
===========================================================
Another possibility to study the Higgs sector in the CPX scenario is to observe central exclusive diffractive production (CEDP) of the heavy neutral $H_2$ and $H_3$ Higgs bosons, using the $H_2,H_3\to H_1 + H_1$ decay modes. For the case we considered above ($\rm tan\beta=4$, $\phi_{\rm CPX}=90^\circ$, $M_{H_1}=40$ GeV), the masses of the heavy bosons bosons are $M_{H_2}=104.7$ GeV and $M_{H_3}=143.6$ GeV. At the LHC energy, the CEDP cross sections of the $H_2$ and $H_3$ bosons are not too small – $\sigma^{\rm CEDP}=1.5$ and $0.9\ {\rm fb}$ respectively. When the branching fractions, Br$(H_2\to H_1H_1)=0.84$, Br$(H_3\to H_1H_1)=0.54$ and Br$(H_1\to\bb)=0.92,$ are included, we find $$\sigma(pp\to p+(H\to\bb\ \bb)+p)=1.1\ {\rm and}\ 0.4\ {\rm fb}$$ for $H_2$ and $H_3$ respectively. Thus there is a chance to observe, and to identify, the central exclusive diffractive production of all three neutral Higgs bosons, $H_1, H_2$ and $H_3$, at the LHC.
The QCD background for exclusive diffractive production of four $b$-jets is significantly less than the signal. Other decay channels are also worth mentioning. For a very light boson, say $M_{H_1} = 30$ GeV, it is also possible to produce four $b$-jets via the cascade $H_3\to H_2H_1\to 4b$-jets. However, the expected cross section is about 0.02 fb, which looks too low to be useful. A larger cross section is expected for the direct $H_2\to\bb$ decay, where the branching fraction Br$(H_2\to\bb)=0.14$ for $M_{H_1}=40$ GeV leads to the cross section $\sigma(p+(H_2\to\bb)+p)$ = 0.2 fb. Note that in this case, we only need to tag two, and not four, $b$-jets. So the detection efficiency is about a factor of 1/0.6 larger. The situation is even better for $M_{H_1} = 50$ GeV, where Br$(H_2\to\bb)=0.25$ and $\sigma(p+(H_2\to\bb)+p)$ = 0.4 fb. If it is possible to compare the $4b$- and $2b$-jet signals, then it will allow a probe of the nature of the $H_2$ boson. Finally, for the heaviest boson, $H_3$, the decay mode $H_3\to H_1+Z$ is not small, with a branching fraction of Br$(H_3\to H_1+Z)=0.27$ for $M_{H_1}=40$ GeV.
Central Higgs production with double diffractive dissociation
=============================================================
To enhance the Higgs signal we study a less exclusive reaction than $pp\to p + H + p$, and allow both of the incoming protons to dissociate. In Ref.[@INC] it was called double diffractive [*inclusive*]{} production, and was written ppX + H + Y. \[eq:CIDP\] Now there is no form factor suppression as the initial protons are destroyed. Also the cross section is larger due to the increased $p_i^\perp$ phase space. Moreover the cross section is also enhanced because we no longer have the P-even selection rule, and so the pseudoscalar $gg\to H$ coupling, $g_P$, becomes active. The cross section for inclusive production, via central double dissociation (CDD) process, is found by using (i) the effective $gg^{PP}$ luminosity of Ref.[@INC], (ii) the probability, $S^2$, that the gaps survive soft rescattering, calculated using model II of [@KKMR], and (iii) the opacity of the proton given in [@KMRsoft]. Typical results, for the LHC energy, are shown in Table 2. For the Tevatron energy, the cross section appears too small, and even for a light boson of mass $M_{H_1}=30$ GeV we have Br$(H_1\to \tau\tau)\sigma<1.5$fb, while the QED background is about 15 fb.
$M(H_1)$ GeV 30 40 50
--------------------------------------------- ----------- ----------- -----------
$\sigma(H_1){\rm Br}(\tau\tau)$ 19 (4) 6 (2) 2.6 (0.8)
$\sigma^{\rm QED}(\tau\tau)$ 66 (2.2) 30 (1.5) 15 (0.9)
$M(H_2)$ GeV 103.4 104.7 106.2
$\sigma\dot {\rm Br} (H_2 \to 2H_1 \to 4b)$ 4 (2) 4 (2) 3.5 (2)
$M(H_3)$ GeV 141.9 143.6 146.0
$\sigma\dot {\rm Br} (H_3 \to 2H_1 \to 4b)$ 1.5 (0.8) 2.2 (1.2) 2 (1.1)
: The cross sections (in fb) for the central production of $H_i$ neutral Higgs bosons by [*inclusive*]{} double diffractive dissociation, together with that of the QED($\tau\tau$) background. A polar angle acceptance cuts of $60^\circ<\theta(b~{\rm or}~\tau)<120^\circ$ ($45^\circ<\theta(b)<145^\circ$) in the Higgs rest frame is applied for the case of $H_1$ ($H_2,H_3$) bosons. The numbers in brackets correspond to the imposition of the additional cut of $E^\perp_i>7$ GeV for the proton dissociated systems.
Of course, the missing mass method cannot be used to measure the mass of the Higgs boson for central production with double dissociation (CDD). Therefore the mass resolution will be not so good as for CEDP; we evaluate the background for $\Delta M$ = 10 GeV. Moreover, with the absence of the $J_z=0$ selection rule, the LO QCD $\bb$-background is not suppressed. Hence we study only the $\tau\tau$ decay mode for the light boson, $H_1$, and the four $b$-jet final state for the heavy $H_2$ and $H_3$ bosons.
The background to the $H_1\to\tau\tau$ signal arises from the $\gamma\gamma\to\tau\tau$ QED process. It is evaluated in the equivalent photon approximation. The photon flux, N\_= F\_2(x,q\^2), \[eq:35a\] was calculated using LO MRST2001 partons[@MRST01], with the integral over the photon transverse momentum running from $q=m_\rho$ up to $q=M_{\tau\tau}/2$. The lower limit is approximately where the $\gamma^*p$ cross section becomes flat and loses its $\sigma(\gamma^*p) \sim 1/q^2$ behaviour. The upper limit reflects the dependence of the $\gamma\gamma\to\tau\tau$ matrix element on the virtuality of the photon. From Table 2 we see that the $H_1$ signal for inclusive diffractive production, (\[eq:CIDP\]), exceeds the exclusive signal by more than a factor of ten. On the other hand the signal-to-background ratio is worse; $S/B_{QED}$ is about 1/5. Moreover there could be a huge background due the misidentification of a gluon dijet as a $\tau\tau$-system. To make this QCD background satisfy $B_{QCD}<S$, would require the probability of misidentifying a gluon as a $\tau$ lepton to be $P_{g/\tau}<1/1500$.
For the four $b$-jet signals of the heavy $H_2$ and $H_3$ bosons, the QCD background can be suppressed by requiring each of the four $b$-jets to have polar angle in the interval $(45^\circ,135^\circ)$, in the frame where the four $b$-jet system has zero rapidity. However in the absence of a good mass resolution, that is with only[^12] $\Delta M=10$ GeV, we expect the four $b$-jet background to be 3-5 times the signal. Nevertheless these signals are still feasible, with cross sections of the order of a few fb. For example, with an integrated luminosity of ${\cal L}
= 300~{\rm fb}^{-1}$ and an efficiency of $4b$-tagging of $(0.6)^2$ [@DKMOR], we predict about 400 $H_2$ events and 200 $H_3$ events. Taking the background-to-signal ratio to be $B/S =4$, we then have a statistical significance of about $10\sigma$ for $H_2$ and $6\sigma$ for $H_3$.
The inclusive CDD kinematics allow a study of CP-violation, and the separation of the contributions coming from the scalar and pseudoscalar $gg\to H$ couplings, $g_S$ and $g_P$ of (\[eq:1\]), respectively. Indeed, the polarizations of the incoming active gluons are aligned along their transverse momenta, $\vec{Q}_\perp-\vec{p}^\perp_1$ and $\vec{Q}_\perp+\vec{p}^\perp_2$. Hence the $gg\to H$ fusion vertices take the forms V\_S = (\_-\^\_1) (\_+\^\_2) g\_S V\_P = \_0 g\_P, where $g_S$ and $g_P$ are defined in (\[eq:1\]).
For the exclusive (CEDP) process the momenta $p^\perp_{1,2}$ were limited by the proton form factor, and typically $Q^2\gg p^2_{1,2}$. Thus V\_S = g\_S Q\^2\_ V\_P = g\_P (\_0). On the contrary, for double diffractive dissociation production (CDD) $Q^2 < p^2_{1,2}$. In this case V\_S = g\_S p\^\_1 p\^\_2 [cos]{} V\_P = g\_P p\^\_1 p\^\_2 [sin]{}. Moreover we can select events with large outgoing transverse momenta of the dissociating systems, say $p^\perp_{1,2}> 7$ GeV, in order to make reasonable measurements of the directions of the vectors $\vec{p}_1^\perp=\vec{E}^\perp_1$ and $\vec{p}_2^\perp=\vec{E}^\perp_2$. Here $E^\perp_{1,2}$ are the transverse energy flows of the dissociating systems of the incoming protons. At LO, this transverse energy is carried mainly by the jet with minimal rapidity in the overall centre-of-mass frame. The azimuthal angular distribution has the form[^13] = \_0 (1+ a [sin]{}2+ b [cos]{}2), where the coefficients are given by a= b=. Note that the coefficient $a$ arises from scalar-pseudoscalar interference, and reflects the presence of a T-odd effect. Its observation would signal an explicit CP-violating mixing in the Higgs sector. On the other hand, in the absence of CP-violation,the sign of the coefficient b reveals the CP-parity of the new boson[^14].
The predictions for the coefficients are given in Table 3 for different values of the Higgs mass, namely $M_{H_1}$ = 30, 40 and 50 GeV. The coefficients are of appreciable size and, given sufficient luminosity, may be measured at the LHC. Imposing the cuts $E^\perp_i > 7$ GeV reduces the cross sections by about a factor of two, but does not alter the signal-to-background ratio, $S/B_{QCD}$. However the cuts do give increased suppression of the QED $\tau\tau$ background and now, for the light $H_1$ boson, the ratio $S/B_{QED}$ exceeds one. We emphasize here that, since we have relatively large $E^\perp$, the angular dependences are quite insensitive to the soft rescattering corrections.
$M(H_1)$ GeV
-------------- --------- --------- --------- --------- --------- ---------
$H_1$ $-0.53$ $-0.73$ $-0.56$ $-0.55$ $-0.53$ $-0.33$
$H_2$ 0.44 0.90 0.41 0.91 0.37 0.92
$H_3$ $-0.38$ 0.92 $-0.40$ 0.91 $-0.42$ 0.90
: The coefficients in the azimuthal distribution $d\sigma/d\varphi = \sigma_0 (1+ a\sin 2\varphi + b \cos 2\varphi)$, where $\varphi$ is the azimuthal angle between the $E^\perp$ flows of the two proton dissociated systems. If there were no CP-violation, then the coefficients would be $a=0$ and $|b|=1$.
Conclusions
===========
We have evaluated the cross sections, and the corresponding backgrounds, for the central double-diffractive production of the (three neutral) CP-violating Higgs bosons at the LHC. This scenario is of interest since even a very light boson of mass about 30 GeV is not experimentally ruled out for some range of the MSSM parameters.
We have studied the production of the three states, $H_1, H_2, H_3$, both with exclusive kinematics, $pp\to p + H + p$ which we denoted CEDP, and in double-diffractive reactions where both the incoming protons may be destroyed, $pp\to X + H + Y$ which we denoted CDD. Recall that a + sign denotes the presence of a large rapidity gap. Proton taggers are required in the former processes, but not in the latter. Typical results are summarised in Tables 1 and 2, respectively. The cross sections are not large, but should be accessible at the LHC. The uncertainties in the calculation of the exclusive cross sections were discussed in Refs.[@KKMRCentr; @KKMRext]. For the light $H_1$ boson, where the contribution from the low $Q_\perp$ region is more important, the uncertainty is much larger. Recall that for the semi-inclusive CDD processes the effective gluon-gluon ($gg^{PP}$) luminosity is calculated using the LO formula. Thus we cannot exclude rather large NLO corrections. On the other hand, for CDD, the values of the cross sections are practically insensitive to the contributions from the infrared domain. Moreover, with the [*skewed*]{} CDD kinematics, the NLO BFKL corrections are expected to be much smaller than in the forward (CEDP) case. So we may expect an uncertainty of the predictions to be about a factor of 3 to 4, or even better.
It would be very informative to measure the azimuthal angular dependence of the outgoing proton systems, for both the CEDP and CDD processes. Such measurements would reveal explicitly any CP-violating effect, via the interference of the scalar and pseudoscalar $gg\to H$ vertices.
Finally, we recall the advantages of diffractive, as compared to the non-diffractive, production of Higgs bosons:
i\) a much better mass Higgs resolution is obtained by the missing mass method for exclusive events,
ii\) a clean environment, which may be important to identify four $b$-jets with transverse momenta $p_T\sim M_{H_1}/2\sim 20$ GeV (for the non-diffractive process, at the LHC energy, the QCD backgroud may be too large),
iii\) a possibility to measure CP-property of the Higgs boson and to detect CP-violation (note that the asymmetries $A_{\bb}$ and $A_{\tau\tau}$ are explicit manifestations of CP violation at the quark level),
Next, assuming that P and C parities are conserved,
iv\) the existence of the P-even, $J_z=0$ selection rule for LO central exclusive diffractive production, which means that we observe an object of natural parity (most probably $0^+$); the analysis of the azimuthal angular distribution of the outgoing protons may give additional information about the spin of the centrally produced object [@KKMRCentr],
v\) in addition we know that an object produced by the diffractive process (that is by Pomeron-Pomeron fusion) has positive C-parity, is an isoscalar and a colour singlet[^15].
The properties listed above should help to distinguish the $H_2$ and $H_3$ four-jet decay channels from the production of a SUSY particle, followed by a ‘cascade’-like decay.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Jeff Forshaw, Risto Orava, Albert de Roeck, Sasha Nikitenko, Apostolos Pilaftsis and, especially, Brian Cox and Jae Sik Lee for useful discussions. ADM thanks the Leverhulme trust for an Emeritus Fellowship and MGR thanks the IPPP at the University of Durham for hospitality. This work was supported by the UK Particle Physics and Astronomy Research Council, by grant RFBR 04-02-16073 and by the Federal Program of the Russian Ministry of Industry, Science and Technology SS-1124.2003.2.
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[^1]: The prospects for observing such a light Higgs in conventional search channels, at the Tevatron and the LHC, were studied in [@CHL; @CEMPW].
[^2]: The values are chosen to provide an ’optimistic’ scenario for the observation of a CP-violating Higgs boson in CEDP.
[^3]: For calculations of $g_S$ and $g_P$ in the MSSM with CP-violation see, for example, [@DMCL].
[^4]: In [@INC] we denoted the initial state by $gg^{PP}$ to indicate that each of the incoming gluons belongs to colour-singlet Pomeron exchange. Here this notation is assumed to be implicit.
[^5]: Strictly speaking, we should consider CP-even and CP-odd contributions to the width separately, but it does not change the conclusion qualitatively.
[^6]: Note that our CEDP cross section is about two times larger than that quoted in [@cox]. This difference occurs mainly because we use an improved approximation for the unintegrated gluon densities. To be specific, we use eq.(26) of [@MR01], rather than the simplified formula (4) of Ref.[@DKMOR] used in [@cox]. In addition we allow for the transverse momenta $p^\perp_{1,2}$ of the recoil protons in the gluon loop of Fig.1. For smaller boson masses, $M_H\sim 40$ GeV, this leads to a steeper $p^\perp_{1,2}$ dependence of the amplitude, which emphasizes larger values of the impact parameter, $b_\perp$, where the absorptive effects are weaker. Therefore we obtain a larger soft survival factor, $S^2\simeq 0.029$, at the LHC energy. However, recall that a factor of 2 difference is within the accuracy of the approach[@DKMOR; @KKMRCentr].
[^7]: Here and in what follows we assume that the proton and $b$-tagging efficiencies and the missing mass resolution in the case of a light Higgs boson are the same as for the case of $M_{\rm Higgs}=120$ GeV [@DKMOR]. Likely, this assumption is not well justified. In particular, the missing mass resolution and proton tagging efficiency may worsen at lower masses.
[^8]: There may be background caused by a pair of high $E_T$ ($\sim 15$ GeV) gluons being misidentified as a $\tau\tau$ pair. To suppress such a background down to the level of $S/B\sim 1$, the probability, $P_{g/\tau}$, that a gluon is misidentified as a $\tau$ must be less than about 1/750, assuming that the missing mass resolution is $\Delta M=1$ GeV. In [@Zepp], for an inclusive event, the probability $P_{g/\tau}$ was evaluated as 1/500. Thus it seems reasonable to suppose that the probability $P_{g/\tau}<1/750$ can be achieved in the much cleaner environment of an exclusive diffractive (CEDP) event.
[^9]: As we consider sizeable $p_{1,2}^\perp$, we account for both the $F_1$ and $F_2$ electromagnetic proton form factors.
[^10]: Without the momenta cuts, the main QED contribution comes from small $p_{1,2}^\perp$, that is from large impact parameters $b^\perp\gg R_p$, where the probability of soft rescattering is already small, see [@KMRphot] for details.
[^11]: In Ref. [@CL] (see also [@GG; @KKSZ; @GT]) a suggestion, along the same lines, was made for the explicit observation of CP-violating effects. There, various polarization asymmetries in two-photon fusion Higgs production processes were discussed. In the absence of absorptive effects, the azimuthal asymmetry $A$ may be expressed, via gluon helicity amplitudes, in the same way as the quantity $A_2$ of [@CL], written in terms of photon helicities.
[^12]: However this resolution is still sufficient to separate the $H_2$ and $H_3$ bosons.
[^13]: In the CP-conserving case, an idea similar in spirit was considered in Ref.[@DKOSZ], where it was suggested to measure the azimuthal correlations of the two tagged jets in inclusive Higgs production. However the proof of the feasibility of such an approach in non-diffractive processes requires further detailed studies of the possible dilution of the effect due to the parton showers in the inclusive environment of the jets.
[^14]: Note that we may search for any new pseudoscalar boson produced by the CDD process by looking for the corresponding azimuthal distribution, $d\sigma/d\varphi \sim {\rm sin}^2\varphi$.
[^15]: An instructive topical example, which illustrates the power of CEDP as a spin-parity analyser, concerns the determination of the quantum numbers of the recently discovered $X(3872)$ resonance[@chi3872]. A knowledge of its C-parity is important to understand its nature. If it is a C = +1 state with spin-parity $0^+$ or $2^+$ then it may be even seen in CDD production with a large rapidity gap on either side of its J/$\psi~\pi^+\pi^-$ decay. Forward proton tagging would, of course, allow a better spin-parity analysis.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Probability density functions are determined from new stellar parameters for the distance moduli of stars for which the RAdial Velocity Experiment (RAVE) has obtained spectra with $S/N\ge10$. Single-Gaussian fits to the pdf in distance modulus suffice for roughly half the stars, with most of the other half having satisfactory two-Gaussian representations. As expected, early-type stars rarely require more than one Gaussian. The expectation value of distance is larger than the distance implied by the expectation of distance modulus; the latter is itself larger than the distance implied by the expectation value of the parallax. Our parallaxes of Hipparcos stars agree well with the values measured by Hipparcos, so the expectation of parallax is the most reliable distance indicator. The latter are improved by taking extinction into account. The effective temperature absolute-magnitude diagram of our stars is significantly improved when these pdfs are used to make the diagram. We use the method of kinematic corrections devised by Schönrich, Binney & Asplund to check for systematic errors for general stars and confirm that the most reliable distance indicator is the expectation of parallax. For cool dwarfs and low-gravity giants $\ex{\varpi}$ tends to be larger than the true distance by up to 30 percent. The most satisfactory distances are for dwarfs hotter than $5500\K$. We compare our distances to stars in 13 open clusters with cluster distances from the literature and find excellent agreement for the dwarfs and indications that we are over-estimating distances to giants, especially in young clusters.'
author:
- |
J. Binney$^1$[^1], B. Burnett$^1$, G. Kordopatis$^2$, P.J. McMillan$^1$, S. Sharma$^3$, T. Zwitter$^4$, O. Bienaymé$^6$, J. Bland-Hawthorn$^3$, M. Steinmetz$^7$, G. Gilmore$^2$, M.E.K. Williams$^7$, J. Navarro$^8$, E.K. Grebel$^{9}$, A. Helmi$^{10}$, Q. Parker$^{11}$, W.A. Reid$^{11}$, G. Seabroke$^{12}$, F. Watson$^{13}$, R.F.G. Wyse$^{14}$\
$^1$ Rudolf Peierls Centre for Theoretical Physics, Keble Road, Oxford OX1 3NP, UK\
$^2$ Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK\
$^3$ Sydney Institute for Astronomy, University of Sydney, School of Physics A28, NSW 2006, Australia\
$^4$ University of Ljubljana, Faculty of Mathematics and Physics, Jadranska 19, 1000 Ljubljana, Slovenia and\
Center of Excellence SPACE-SI, Aškerčeva cesta 12, 1000, Ljubljana, Slovenia\
$^5$ Research School of Astronomy and Astrophysics, Australian National University, Cotter Rd., ACT, Canberra, Australia\
$^6$ Observatoire Astronomique de Strasbourg, 11 rue de l’Université, Strasbourg, France\
$^7$ Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany\
$^8$ Department of Physics & Astronomy, University of Victoria, 3800 Finnerty Rd., Victoria, Canada V8P 5C2\
$^{9}$ Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstr 12-14,\
D-69120, Heidelberg, Germany\
$^{10}$ Kapteyn Astronomical Institut, University of Groningen, Landleven 12, 9747 AD, Groningen, The Netherlands\
$^{11}$ Macquarie University, Sydney, Australia\
$^{12}$ Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, RH5 6NT, UK\
$^{13}$ Australian Astronomical Observatory, P.O. box 296, Epping, NSW 1710, Australia\
$^{14}$Johns Hopkins University, Departement of Physics and Astronomy, 366 Bloomberg center, 3400 N. Charles St.,\
Baltimore, MD 21218, USA\
date: 'Draft, September 26, 2013'
title: New distances to RAVE stars
---
\[firstpage\]
Introduction
============
Surveys of the stellar content of our Galaxy are key to the elucidation of the Galaxy’s structure and history. Consequently, over the last decade considerable observational resources have been devoted to such surveys. Three surveys are particularly worthy of note: the 2MASS survey [@2MASS], the Sloan Digital Sky Survey (SDSS) [@SDSS; @SEGUE] and the RAdial Velocity Experiment (RAVE) [@RAVE; @DR3]. The 2MASS survey was an all-sky, near infrared photometric survey, while the SDSS survey combined a photometric survey in the $ugriz$ system with spectroscopy for a subset of objects with spectral resolution $R=2500$. The RAVE survey has taken spectra at resolution $R\simeq7500$ of $\sim500\,000$ stars that have 2MASS photometry. The RAVE and SDSS surveys are complementary in that SDSS worked at apparent magnitudes $r\gta18$ so faint that it catalogued mainly dwarf stars that lie more than $500\pc$ from the Sun, while RAVE operates at apparent magnitudes $I\approx9-13$ and observes both nearby dwarfs and giants at distances up to $\sim4\kpc$ [@Burnettetal].
Although the ideal way to extract science from a survey is to project models into the space of observables, i.e., sky coordinates, line-of-sight velocity, apparent magnitudes, etc., and fit the projected models to the data [e.g. @BinneyBangalore], in practice one generally assigns a distance to each star and uses this distance to place the star in the space in which physics applies, namely phase space complemented with luminosity, colour, chemical composition, etc. Since RAVE’s targets overwhelmingly lie beyond the range of Hipparcos and include both dwarfs and giants, the task of assigning distances to these stars is complex. To date three papers [@Breddels; @Zwitter10; @Burnettetal] address this task with techniques of increasing sophistication. Results presented in those papers are based on stellar parameters produced by the pipeline that was developed for analysis of the RAVE spectra. This pipeline was described in the papers that accompanied the second and third releases of RAVE data [@DR2; @DR3]. Between those two data releases changes were made to the pipeline’s parameters that were designed to improve the accuracy of the derived metallicities, but the parameters from neither version of the pipeline were entirely satisfactory [@Burnettetal hereafter B11].
On account of residual internal and external inconsistencies in the parameters, a completely new pipeline has been developed for the analysis of RAVE spectra. This pipeline and the stellar parameters it produces are described in [@DR4]. The new stellar parameters form a much more compelling and consistent database than the old ones, and their arrival prompts us to revisit the assignment of distances using the new parameters as inputs.
We use the Bayesian framework described by [@BurnettB] but modified to allow for the impact of interstellar dust. Two other significant novelties are (i) the production of multi-Gaussian fits to each star’s probability density function (pdf) in distance modulus and (ii) the use of the kinematic correction factors introduced by [@SBA] to check for systematic errors in our distances. We have derived distances for all stars that have spectra to which the new pipeline assigns a signal-to-noise ratio of 10 or higher. When a star has more than one spectrum in the database, the catalogued distance is that derived from the highest S/N spectrum.
The plan of the paper is as follows. In Section \[sec:method\] we recapitulate the principles of Bayesian distance determination and describe how we take extinction into account. In Section \[sec:pdfs\] we discuss typical pdfs in distance modulus and explain how we produce multi-Gaussian fits to them. In Section \[sec:Hipp\] we compare our spectrophotometric parallaxes to Hipparcos parallaxes and ask how these comparisons are affected by neglecting extinction. In Section \[sec:all\] we analyse our distances to the generality of stars, using kinematic correction factors to test for systematic biases in distances as functions of surface gravity or effective temperature, and to modify distance pdfs (Section \[sec:SBA\]). In Section \[sec:clusters\] we compare our distances to cluster stars with the established distances to their clusters. In Section \[sec:repeat\] we examine the scatter in the distances to the same star obtained from different spectra. In Section \[sec:Av\] we examine the distribution of extinctions to stars. Section \[sec:discuss\] sums up.
Methodology {#sec:method}
===========
As in B11 we start from the trivial Bayesian statement $$\pr(\hbox{model}|\hbox{data})={\pr(\hbox{data}|\hbox{model})\pr(\hbox{model})
\over\pr(\hbox{data})},$$ where “data” comprises the observed parameters and photometry of an individual star and “model” comprises a star of specified initial mass $\cM$, age $\tau$, metallicity $\hbox{[M/H]}$, and location. We use $p({\rm
model|data})$ either to calculate expectation values $\ex{x}$ and dispersions $\sigma_x$ of quantities of interest, such as the stars’s distance $x=s$ and parallax $x=\varpi$, by integrating $P({\rm model|data})$ times an appropriate power of $x$ through the space spanned by the model parameters $\hbox{[M/H]},\tau,\cM,\ldots$, or the pdf in distance modulus by marginalising $P({\rm model|data})$ over all model parameters other than distance.
A key role is played by the prior probability $\pr(\hbox{model})$, which reflects our prior knowledge of the Galaxy: massive young stars are rarely found far from the plane, while a star far from the plane is likely to be old and have sub-solar abundances. We have used the same three-component prior used in B11: $$\label{eq:priorofx}
p(\hbox{model}) = p(\cM) \sum_{i=1}^3 p_i(\mh) \, p_i(\tau) \, p_i(\mathbf{r}),$$ where $i=1,2,3$ correspond to a thin disc, thick disc and stellar halo, respectively. We assumed an identical Kroupa-type IMF for all three components and distinguish them as follows:
#### Thin disc ($i=1$): {#thin-disc-i1 .unnumbered}
$$\begin{aligned}
\label{eq:thindisc}
p_1(\mh) &=& G(\mh, 0.2), \nonumber \\
p_1(\tau) &\propto& \exp(0.119 \,\tau/\mbox{Gyr}) \quad \mbox{for $\tau \le 10$\,Gyr,} \\
p_1(\mathbf{r}) &\propto& \exp\left(-\frac{R}{R_d^{\rm{thin}}} - \frac{|z|}{z_d^{\rm{thin}}} \right); \nonumber\end{aligned}$$
#### Thick disc ($i=2$): {#thick-disc-i2 .unnumbered}
$$\begin{aligned}
\label{eq:thickdisc}
p_2(\mh) &=& G(\mh+0.6, 0.5), \nonumber \\
p_2(\tau) &\propto& \mbox{uniform in range $8 \le \tau \le 12$\,Gyr,} \\
p_2(\mathbf{r}) &\propto& \exp\left(-\frac{R}{R_d^{\rm{thick}}} - \frac{|z|}{z_d^{\rm{thick}}} \right); \nonumber\end{aligned}$$
#### Halo ($i=3$): {#halo-i3 .unnumbered}
$$\begin{aligned}
p_3(\mh) &=& G(\mh+1.6, 0.5), \nonumber \\
p_3(\tau) &\propto& \mbox{uniform in range $10 \le \tau \le 13.7$\,Gyr,} \\
p_3(\mathbf{r}) &\propto& r^{-3.39}; \nonumber\end{aligned}$$
where $R$ signifies Galactocentric cylindrical radius, $z$ cylindrical height and $r$ spherical radius, and $G(x,y)$ is a Gaussian distribution in $x$ of zero mean and dispersion $y$. The parameter values were taken as in Table \[table:params\]; the values are taken from the analysis of SDSS data in [@Juric_cut]. The metallicity and age distributions for the thin disc come from [@Haywood] and [@Aumer], while the radial density of the halo comes from the ‘inner halo’ detected in [@Carollo]. The metallicity and age distributions of the thick disc and halo are influenced by [@Reddy] and [@Carollo].
The normalizations were then adjusted so that at the solar position, taken as $R_0=$ 8.33kpc (@Gillessen), $z_0=$ 15pc [@BinneyGS; @Juric_cut], we have number density ratios $n_2 /n_1 = 0.15$ (@Juric_cut), $n_3 /n_1 = 0.005$ (@Carollo).
Parameter Value (pc)
---------------------------- ------------
$R_d^{\rm{thin}}$ 2600
\[3pt\] $z_d^{\rm{thin}}$ 300
\[3pt\] $R_d^{\rm{thick}}$ 3600
\[3pt\] $z_d^{\rm{thick}}$ 900
: Values of disc parameters used.\[table:params\]
$Z$ $Y$ $\mh$
-------- ------- ----------
0.0022 0.230 $-0.914$
0.003 0.231 $-0.778$
0.004 0.233 $-0.652$
0.006 0.238 $-0.472$
0.008 0.242 $-0.343$
0.010 0.246 $-0.243$
0.012 0.250 $-0.160$
0.014 0.254 $-0.090$
0.017 0.260 0.000
0.020 0.267 0.077
0.026 0.280 0.202
0.036 0.301 0.363
0.040 0.309 0.417
0.045 0.320 0.479
0.050 0.330 0.535
0.070 0.372 0.727
: Metallicities of isochrones used, taking $(Z_\odot , Y_\odot) = (0.017,0.260)$.\[table:Zs\]
The IMF chosen follows the form originally proposed by [@Kroupa], with a minor modification following [@Aumer], being $$p(\cM) \propto \cases{\cM^{-1.3}&if $\cM<0.5\,$M$_\odot$,\cr
0.536 \, \cM^{-2.2}&if $0.5\,$M$_\odot \le \cM<1\, $M$_\odot$,\cr
0.536 \, \cM^{-2.519}&otherwise.}$$
We predicted the photometry of stars from the isochrones of the Padova group (@Bertelli), which provide tabulated values for the observables of stars with metallicities ranging upwards from around $\mh \approx -0.92$, ages in the range $\tau \in [0.01, 19]$Gyr and masses in the range $\cM \in
[0.15, 20]$M$_\odot$. We used isochrones for 16 metallicities as shown in Table \[table:Zs\], selecting the helium mass fraction $Y$ as a function of metal mass fraction $Z$ according to the relation used in [@Aumer], i.e.$Y \approx 0.225 + 2.1 Z$ and assuming solar values of $(Y_\odot , Z_\odot) =
(0.260,0.017)$. The metallicity values were selected by eye to ensure that there was not a great change in the stellar observables between adjacent isochrone sets.
In B11 no correction was made for the differences between the Johnson-Cousins-Glass photometric system used for the Padova stellar models that we use and the 2MASS system. Here we use the transformations of [@Koen07] to transform the 2MASS magnitudes $J_2,\ldots$ to the Johnston-Cousins-Glass magnitudes $J,\ldots$: $$\begin{aligned}
J&=&0.029+J_2+0.07(J_2-K_2)-0.045(J_2-H_2)^2\nonumber\\
H&=&H_2+0.555(H_2-K_2)^2-0.441(H_2-K_2)\nonumber\\
&&\qquad+0.089(J_2-H_2)\\
K&=&0.009+K_2+0.195(J_2-H_2)^2-0.156(J_2-H_2)\nonumber\\
&&\qquad+0.304(H_2-K_2)-0.615(H_2-K_2)^2.\nonumber\end{aligned}$$ Unless explicitly stated to the contrary, we will state $JHK$ magnitudes in the Johnston-Cousins-Glass system.
Dust both dims and reddens stars. Let the column of dust between us and a given star produce optical extinction $A_V$, then from [@RiekeL85] we take the extinctions to be $$\begin{aligned}
A_J&=&0.282A_V\nonumber\\
A_H&=&0.175A_V\\
A_K&=&0.112A_V.\nonumber\end{aligned}$$
In B11 $A_V$ was set identically to zero and the $H$ magnitude was not employed. Here we include the $H$ magnitude in the set of observations so we have three constraints on the star’s spectral distribution: the spectroscopically derived $\Teff$ and two IR colours. Consequently, we should be able to constrain the extinction to some extent. We integrate over all possible values of $A_V$. We include $A_V$ in the prior by multiplying the prior (\[eq:priorofx\]) by the probability density of $A_V$. Since $A_V$ is an intrinsically non-negative quantity, a completely flat prior would be one uniform in $a\equiv\ln(A_V)$. We do not want a flat prior but one that reflects increasing extinction with distance and higher extinction towards the Galactic centre than towards the poles. Let $a_{\rm prior}(\vx)$ be the expected value of $\ln(A_V)$ for the location $\vx$. Then a natural choice for the probability of extinctions associated with the interval $(a,a+\d a)$ is $$\begin{aligned}
\d P&=&(2\pi\sigma^2)^{-1/2}\e^{-(a-a_{\rm prior})^2/2\sigma^2}\,\d a\nonumber\\
&=&(2\pi\sigma^2)^{-1/2}\e^{-\ln^2(A_V/A_{V{\rm prior}})/2\sigma^2}\,\d a.\end{aligned}$$ The dispersion $\sigma$ reflects the random fluctuation of the extinction from one sight-line to the next on account of the cloudy nature of the interstellar medium. We have rather arbitrarily set $2\sigma^2=1$.
$A_{V{\rm prior}}$ is related to distance by $$A_{V{\rm prior}}(b,\ell,s)=A_{V\infty}(b,\ell){\int_0^s\d
s'\,\rho[\vx(s')] \over \int_0^\infty\d s'\,\rho[\vx(s')]}$$ where $\vx(s)$ is the position-vector of the point that lies distance $s$ down the line of sight $(b,\ell)$, $A_{V\infty}(b,\ell)$ is defined below and $\rho(\vx)$ is a model of the density of extincting material. Following [@Sharma11] we adopt $$\label{eq:dmodel}
\rho(\vx)=\exp\left[{R_0-R\over h_R}-{|z-z_{\rm w}|\over
k_{\rm fl}h_z}\right],$$ where $k_{\rm fl}(R)$ and $z_{\rm w}(R)$ describe the flaring and warping of the gas disc: $$\begin{aligned}
k_{\rm fl}(R)&=&1+\gamma_{\rm fl}\min(R_{\rm fl},R-R_{\rm fl})\nonumber\\
z_{\rm w}(R,\phi)&=&\gamma_{\rm w}\min(R_{\rm w},R-R_{\rm w})\sin\phi.\end{aligned}$$ Here $\phi$ is the Galactocentric azimuth that increases in the direction of Galactic rotation and places the Sun at $\phi=0$. Table \[tab:dust\] gives the values of the parameters that occur in these formulae.
We take the extinction to infinity, $A_{V\infty}(b,\ell)$, from observation: except along exceptionally obscured lines of sight, $A_{V\infty}$ is 3.1 times the reddening estimated by [@Schlegel]. However, [@ArceGoodman] pointed out that the Schlegel et al. over-estimate the reddening in regions with $E(B-V)>0.15$. Following [@Sharma11] we correct for this effect by multiplying the Schlegel et al. values of $E~(B-V)$ by the correction factor $$f(E(B-V))=0.6+0.2\left[1-\tanh\left({E(B-V)-0.15\over0.3}\right)\right],$$ which has the effect of leaving $E(B-V)$ invariant for $E(B-V)\lta0.16$ and multiplying large values of $E(B-V)$ by a factor 0.6.
The function $A_{V{\rm prior}}(s)$ is tabulated on a non-uniform grid in $s$ before each star is analysed so $A_{V{\rm prior}}$ can be subsequently obtained quickly by linear interpolation.
Given a model star characterised by (\[M/H\],$\tau,\cM$), a first estimate of the distance to the star is made under the assumption $A_V=0$. Then $A_{V{\rm
prior}}$ is evaluated for this distance and a second estimate of distance obtained, and $A_{V{\rm prior}}$ is evaluated at this improved distance and stored as $A_{V{\rm model}}$. The reddened $J-K$ colour of the star is now predicted and compared with the observed colour. The given model star is considered sufficiently plausible to be worth considering further only if both its colour reddened by e times $A_{V{\rm model}}$ is redder than the blue end of the $3\sigma$ range around the measured colour and the star’s colour reddened by $1/\e$ times $A_{V{\rm model}}$ is bluer that the red end of the measured $3\sigma$ range. If these conditions are satisfied, we consider values of $A_V$ that lie the range $(\e^{-1.5},\e^{1.5})A_{V{\rm
model}}$. For each value of $A_V$ all plausible distances are considered.
We calculate the expectation $\ex{a}$ of $a\equiv\ln A_V$ and use $\widetilde
A_V\equiv\exp(\ex{a})$ as our final estimate of the extinction to each star.
---------- ------- ------- -------------- ------------------- ------------- ------------------
$A_V(0)$ $h_R$ $h_z$ $R_{\rm fl}$ $\gamma_{\rm fl}$ $R_{\rm w}$ $\gamma_{\rm w}$
1.67 4.2 0.088 $1.12R_0$ 0.0054 8.4 0.18
---------- ------- ------- -------------- ------------------- ------------- ------------------
: Parameters of the model of the dust distribution. Distances are in kiloparsecs.[]{data-label="tab:dust"}
{width=".3\hsize"} {width=".3\hsize"} {width=".3\hsize"}
{width=".3\hsize"} {width=".3\hsize"} {width=".3\hsize"}
{width=".3\hsize"} {width=".3\hsize"} {width=".3\hsize"}
PDFs for distance {#sec:pdfs}
=================
The Bayesian argument yields the five-dimensional probability density function (pdf) that each star has a given mass, metallicity, age, line-of-sight extinction and distance, but [@BurnettB] and [@Burnettetal] reported only the implied means and standard deviations of distance and parallax. Hence they had two logically independent measures of the distance to a star: $\ex{s}$ and $1/\ex{\varpi}$. A third natural distance measure is provided by the expectation of the distance modulus $\mu=5\log_{10}(s/10\pc)$. We shall show that these three measures yield systematically different distances and conclude that $1/\ex{\varpi}$ is the most reliable estimate.
A logical next step is to inspect the pdfs we obtain for $s$, etc., after marginalising over the star’s other properties. If any of these pdfs is well approximated by a Gaussian, it can be fully characterised by its mean and dispersion. In this section we show that the pdfs often deviate significantly from a Gaussian, and in this case it is important to know more than the pdf’s mean and dispersion.
Fig. \[fig:badpdfs\] shows pdfs in distance modulus for three stars. The red curves show Gaussian distributions in distance modulus $\mu\equiv m-M$, while the green curves show distributions that are Gaussian in distance $s$ and the blue curves show distributions that are Gaussian in parallax $\varpi$. Given how strongly these three curves differ from one another, especially in the left and centre panels, it is clear that a very particular assumption is being made if one supposes that a star’s distribution of either $\mu$, $s$ or $\varpi$ is Gaussian, and if one of these distributions *is* Gaussian, the other two cannot be.
In each panel of the black curve shows the computed marginalised pdf in distance modulus $\mu$, while the red curve shows Gaussian with the same mean and standard deviation as the computed pdf. The green curve shows the pdf which is a Gaussian in distance and has the mean and standard deviation of the computed pdf in distance, while the blue curve shows the pdf which is a Gaussian in parallax and has the mean and dispersion of the computed pdf in parallax. None of the coloured curves can be considered a reasonable representation of the computed pdf. The clear message of is that it is dangerous to quantify the distance to these stars in the form $x\pm y\kpc$ because this notation implies that a Gaussian pdf adequately approximates the true pdf.
We have derived multi-Gaussian approximations to the pdf in $\mu$ since this variable is physically meaningful for any real number. We write $$\label{eq:defsfk}
P(\mu) = \sum_{k=1}^N {f_k\over \sqrt{2\pi\sigma_k^2}}
\exp\bigg(-{(\mu-\mu_k)^2\over2\sigma_k^2}\bigg),$$ where $N$, the means $\mu_k$, weights $f_k$, and dispersions $\sigma_i$ are to be determined. We take bins in distance modulus of width $w_i = 0.2$, containing a fraction $p_i$ of the total probability taken from the computed pdf, and a fraction $P_i$ of the total probability taken from the multi-Gaussian approximation and consider the statistic $$\label{eq:defsF}
F = \sum_i \left(\frac{p_i}{w_i}-\frac{P_i}{w_i}\right)^2\tilde{\sigma} w_i$$ where the weighted dispersion $$\tilde{\sigma}^2 \equiv \sum_{k=1,N} f_k \sigma_k^2$$ is a measure of the overall width of the pdf. Our definition of $F$ includes the factor $\tilde{\sigma}$ to ensure that $F$ is unchanged when the width of both the true pdf and our approximation are increased by the same factor: this condition ensures that $F$ is a measure of how well the shape of the distribution is fitted. We use $\tilde{\sigma}$ in equation (\[eq:defsF\]) rather than the dispersion of the pdf because in some circumstances (double or triple peaked distributions) the dispersion is dominated by the distance between peaks, rather than the widths of the individual peaks themselves, and it is the peaks that should set the scale. A practical difficulty is that $F$ is minimised by letting every $\sigma_k\rightarrow0$. Hence instead of minimising $F$, we minimise the alternative statistic $$F' = \sum_i \left(\frac{p_i}{w_i}-\frac{P_i}{w_i}\right)^2\; w_i$$ and only use $F$ to measure whether the fit is a sufficiently accurate description of the data.
If the value of $F$ for a Gaussian with the same mean and dispersion in $\mu$ as that taken from the computed pdf is less than a threshold value $F_t=0.04$, we accept this as an adequate description of the data. This condition holds for around 45 per cent of the RAVE stars. When it fails, we use the Levenberg-Marquardt algorithm to minimise $F'$ with $N=2$ and several different initial choices for the parameters. We accept this description of the data if it gives $F<F_t$ *and* the dispersion of the model is within 20 per cent of that of the complete pdf. The latter condition ensures that we do not accept models that provide an excellent fit to a significant component of the probability but ignore a small but non-negligible component at a different distance. If the two-Gaussian description fails, we fit a three-Gaussian approximation. We reach this stage for around 5 per cent of the RAVE stars because the double-Gaussian approximation is accepted in $\sim50$ per cent of cases. shows the multi-Gaussian models fitted to the pdfs shown in .
Any fits for which the dispersion of the fitted model differs by more than 20 per cent from that of the data is flagged as possibly inadequate. Approximately four per cent of the models are flagged for this reason. In we show some typical examples of the flagged models. We see that the problems are in fact minor ones.
Hipparcos stars {#sec:Hipp}
===============
As in B11, the primary test of the validity of our spectrophotometric distances is provided by Hipparcos stars that are likely to be single stars because in the [@vanLeeuwen] catalogue they have ${\tt soln}<10$. There are 5614 distinct stars of this type for which we have RAVE parameters, and the mean S/N ratio of their spectra is 84.
The quoted errors on the stellar parameters play a big role in the Bayesian algorithm, and good results are obtainable only with accurate error estimates. When the data were first processed using only the internal error estimates produced by the spectral-reduction pipeline, manifestly inconsistent results for Hipparcos stars were produced. The results were dramatically improved by adding to the internal errors the external errors for various classes of star derived by [@DR4] and listed in Table \[tab:ext\]. The quadrature sums of the internal and external errors prove to be quite similar to the errors adopted by B11, which could not be founded on star-specific error estimates from the old pipeline.
The black points in show histograms of the discrepancies between Hipparcos parallaxes $\varpi_{\rm H}$ and expectation values of parallaxes obtained from $P({\rm model|data})$ for three groups of stars: giants ($\log g<3.5$), hot dwarfs ($\Teff>5500\K$) and cool dwarfs. The parallax differences are normalised by the quadrature sum of the formal errors in the Hipparcos data and our adopted errors, so if our procedure were sound and the central limit theorem applied to the data, the histograms would be Gaussians of unit dispersion. This expectation is met to a pleasing extent for hot dwarfs and giants – for the hot dwarfs the mean of the distribution is $0.143$ and the dispersion is $1.061$ and for the giants they are $-0.057$ and $1.077$. Thus on average the parallaxes of the hot dwarfs are slightly too large, while those of the giants are slightly too small and our error estimates are only a shade too small. The results for the smaller number of cool dwarfs are less clear-cut: the mean and dispersion are $0.123$ and $1.314$ implying that our parallaxes are slightly too large and our errors are materially too small.
[lcccc]{} stellar type&N&$\sigma(\Teff)$&$\sigma(\log g)$&$\sigma(\hbox{[M/H])}$\
\
hot, metal-poor& 28& 314& 0.466& 0.269\
hot, metal-rich & 104& 173& 0.276& 0.119\
cool, metal-poor& 97& 253& 0.470& 0.197\
cool, metal-rich& 138& 145& 0.384& 0.111\
\
hot & 8& 263& 0.423& 0.300\
cool, metal-poor& 273& 191& 0.725& 0.217\
cool, metal-rich& 136& 89& 0.605& 0.144\
$\overline{\ex{s}/s_{\ex{\mu}}}$ $\overline{s_{\ex{\mu}}\ex{\varpi}}$ $\overline{\ex{\varpi}/\varpi_{\rm H}}$ $\overline{\ex{s}\varpi_{\rm H}}$
------------- ---------------------------------- -------------------------------------- ----------------------------------------- -----------------------------------
Hot dwarfs 1.045 1.040 0.958 1.042
Cool dwarfs 1.116 1.094 1.132 1.447
Giants 1.111 1.093 1.115 1.386
: Mean distance ratios for Hipparcos stars. Ideally all entries would be unity.[]{data-label="tab:rSD"}
It is interesting to compute means of the distances ratios. Let $$\begin{aligned}
r_{s\mu}&\equiv&\overline{\ex{s}/s_{\ex{\mu}}}\quad
r_{\mu\varpi}\equiv \overline{s_{\ex{\mu}}\ex{\varpi}}\nonumber\\
r_{\varpi{\rm H}}&\equiv&\overline{\varpi_{\rm H}/\ex{\varpi}}\quad
r_{s{\rm H}}\equiv\overline{\ex{s}\varpi_{\rm H}},\end{aligned}$$ where overbars imply averages of a group of stars and $s_{\ex{\mu}}$ is the distance implied by the expectation value of the distance modulus. Table \[tab:rSD\] gives these ratios for hot dwarfs, cool dwarfs and giants. For the hot dwarfs all ratios are pleasingly close to unity, but for both the cool dwarfs and the giants we see that $\ex{s}$ gives a systematically larger distance than $s_{\ex{\mu}}$, which in turn gives a bigger distance than $1/\ex{\varpi}$, which itself gives a bigger distance than $1/\varpi_{\rm
H}$, which we take to be the most reliable distance estimator. These biases are easily understood in terms of the weights that each estimator attaches to possibilities of long or short distances. The comparisons with the Hipparcos parallaxes clearly indicates that for stars with wide distance pdfs (cool dwarfs and giants), $1/\ex{\varpi}$ performs much better than either $\ex{s}$ or $s_{\ex{\mu}}$.
The red points show histograms of discrepancies between the Hipparcos parallaxes and parallaxes based on the multi-Gaussian fits to the distance moduli as follows. When a single Gaussian has been fitted, we convert the mean and dispersion of this Gaussian into a parallax and its error by standard formulae. If two or three Gaussians have been fitted, we choose the Gaussian that makes the Hipparcos parallax most probable and convert the mean and dispersion of this Gaussian to a parallax and its error as before. The red histogram for the hot dwarfs is an almost perfect realisation of the unit Gaussian while that for the giants is only marginally less satisfactory than the corresponding black histogram. The red histogram for the cool dwarfs is both significantly displaced to the right and broader than it should be.
clarifies the situation by splitting the histogram of the cool dwarfs into those with pdfs that have been fitted with a single Gaussian (lower panel) and those with multi-Gaussian fits (upper panel). We see that for the latter stars the crude mean of possible parallaxes is smaller than it should be, and a more satisfactory distribution of spectrophotometric parallaxes is obtained if Hipparcos is used to choose between the Gaussians. The lower panel in shows that when a cool dwarf has a single-Gaussian pdf, its parallax is systematically over-estimated. When the single- and multi-Gaussian samples are aggregated in , the over-estimated parallaxes of the single-Gaussian stars combine with the under-estimated parallaxes of the multi-Gaussian stars to produce a deceptively satisfactory black histogram. The mean S/N ratio of the Hipparcos stars with single-Gaussian fits is lower than that of the stars with multi-Gaussian fits (51.0 versus 66.5), so one suspects that with poorer data the system loses track of the possibility that the star has left the main sequence.
We test the soundness of the probabilities assigned to each Gaussian component of the pdf by calculating the sums $s_k=\sum_{\rm stars}1/f_k$, where $k=1,2,3$ depending on which Gaussian component the Hipparcos data points to, and $f_k$ is the weight of that component. Given a large and sample of stars with accurate parallaxes (so the true component is always chosen), $s_k$ should be independent of $k$ because when $f_k$ is small, that component will be rarely chosen so $s_k$ will have a small number of large contributions, while a component with large $f_k$ will be chosen often, but each contribution to $s_k$ will be modest. When we compute mean values of $1/f_k$ for our Hipparcos stars, we find 441/2807 hot dwarfs with two Gaussians fitted, and for these stars we find $s_k=(444,458)$. Similarly, 615/970 cool dwarfs have two Gaussians and for these stars we find $s_k=(577,2100)$, while 100 cool dwarfs have three Gaussians and for these stars $s_k=(94,126,476)$. 934/2015 giants have two Gaussians and these stars yield $s_k=(779,3593)$ while 492 giants have three Gaussians and for these stars $s_k=(350,759,748)$. These results suggest that the probabilities assigned the various Gaussians are broadly correct although there is a tendency for too little probability to be assigned to the weakest components.
The likely explanation of the neglect of weaker components is that the Hipparcos stars are biased towards nearer stars because stars thought to be near, usually on account of having large proper motions, preferentially entered the Hipparcos Input Catalogue. Consequently, we have tested the constancy of the $s_k$ for a sample in which distant options will have been rather rarely chosen. For the giants the distant option is the more probable one, so it is natural that for these stars Hipparcos chooses the less probable Gaussian more often than one would expect if we had parallaxes for every star in our sample.
shows the effect of setting $A_V=0$ for all stars. With reddening neglected, dwarfs must be moved to lower masses to match the observed colours, and the consequent diminution of their luminosities causes them to be brought closer to match the observed magnitudes. The overall effect is to increase the spectrophotometric parallaxes of hot dwarfs by $\sim0.05\sigma$, so those of the hot dwarfs are now on average too large by $\sim0.19\sigma$, while those of the cool dwarfs are too large by $\sim0.14\sigma$. With extinction neglected, giants need to be moved away to diminish their brightnesses so their histogram of $\ex{\varpi}-\varpi_{\rm
H}$ moves leftward, and our parallaxes become too small by $0.12\sigma$ on average. Thus the Hipparcos stars convincingly validate our procedure for taking into account the effects of dust.
compares the distribution in the fractional errors in Hipparcos parallaxes (shown in green) with the corresponding errors in our parallaxes: the black points are for the straightforward expectation values of $\varpi$ while the red points are for the parallaxes computed from the multi-Gaussian fits to the pdfs in distance modulus. For hot dwarfs the black and red histograms are similar because few of these stars have multi-modal pdfs. They show error distributions that are materially narrower than that from Hipparcos, with most values of $\sigma_\pi/\ex{\varpi}$ falling in the range $(0.18,0.38)$ with a median value of $0.26$.
For the cool dwarfs the black and red histograms are quite different in that the red histogram shows a substantial population with spectrophotometric parallaxes in error by less that 10% and essentially no stars with errors greater than 35%. The stars with $\sigma_\varpi/\ex{\varpi}<0.1$ are stars that the spectrophotometry cannot securely assign to dwarfs or giants until astrometric data become available – in the present case a Hipparcos parallax. There will probably be many stars of this type in the Gaia Catalogue. The red histogram for the giants shows a similar if smaller population of stars.
For now we must live with dwarf/giant confusion and the black histograms of parallax errors are most relevant. These show that the spectrophotometric parallaxes of cool dwarfs are not competitive with Hipparcos parallaxes, in contrast to the case of some hot dwarfs and a number of giants, which do have more precise spectrophotometric parallaxes than Hipparcos parallaxes. Thus the competitiveness of the spectrophotometric parallaxes vis a vis Hipparcos parallaxes increases along the sequence cool dwarfs to hot dwarfs to giants in parallel with the increase in the luminosities and thus typical distances of these stars.
$N_\star$ $\overline{\ex{s}/s_{\ex{\mu}}}$ $\overline{s_{\ex{\mu}}\ex{\varpi}}$ $\overline{\ex{s}\ex{\varpi}}$
--------------------- ------------------------------ ---------------------------------- -------------------------------------- --------------------------------
3[Giants ($\log g<3.5$)]{}
$\log g>2.4$ 69008 1.11 1.13 1.26
Red Clump 39900 1.04 1.04 1.09
$\log g<1.7$ 28472 1.06 1.05 1.11
3[Dwarfs ($\log g\ge3.5$)]{}
$\Teff>6500$ 22701 1.04 1.03 1.07
$5500<\Teff\le6500$ 71641 1.04 1.04 1.08
$5200<\Teff\le5500$ 19697 1.08 1.08 1.17
$\Teff\le5200$ 27408 1.13 1.12 1.29
: Ratios of distance measures for general stars with $s<2\kpc$[]{data-label="tab:gen_ratios"}
Distances to all stars {#sec:all}
======================
We have examined the statistics of distances to RAVE stars as functions of a cutoff in the S/N of the analysed spectrum and found that dependence on the cutoff S/N is weak. Below we report results obtained for stars with $\hbox{S/N}\ge10$ – the mean S/N ratio for such stars that lie closer than $1.3\kpc$ is 33.
We have investigated the sensitivity of our distances to the model of the disc used in the prior (eqs \[eq:thindisc\] and \[eq:thickdisc\]) by re-evaluating the distances to every twentieth star in the catalogue with the scale radii and scale heights of both discs multiplied by a factor $1.5$. The resulting histogram of ratios $\ex{\varpi}_2/\ex{\varpi}_1$ of the parallax with the revised prior to the parallax with the standard prior peaks sharply at $1.02$ but has a long tail to values $\sim1.2$ with the consequence that the mean of this ratio is $1.045$. This result shows that, as one would hope, our results are not sensitive to the prior.
Table \[tab:gen\_ratios\] shows the ratios of the available distance measures for ordinary stars, broken down into giants and dwarfs, with the giants subdivided into stars with lower surface gravity than the red clump ($1.7<\log g<2.4$ and $0.55\le J-K\le0.8$), the red clump itself and stars with higher gravities. We see that in every case the distances are ordered $\ex{s}>s_{\ex{\mu}}>1/\ex{\varpi}$. Moreover, $\ex{s}$ and $1/\ex{\varpi}$ are discrepant at the 26% level for the highest-gravity giants and coolest dwarfs, while for moderately cool dwarfs these measures are discrepant at the 17% level.
Kinematic distance corrections {#sec:SBA}
------------------------------
Schönrich et al. (2012; hereafter SBA) describe a technique that uses the kinematics of stellar populations to identify and correct systematic errors in distances, and we can use this technique to determine which of our discrepant distance estimates is most reliable, and potentially to correct the most reliable measure for any systematic bias.
The corrections of SBA are based on the assumption that one knows roughly how the velocity ellipsoid is oriented at each point in the Galaxy, and that the only mean-streaming motion is azimuthal circulation at a speed $v(R,z)=\Theta
g(R,z)$, where $\Theta$ is an unspecified constant and $g(R,z)$ is a function one chooses. We adopt $$g=\sqrt{1-(2\psi/\pi)^2}, \hbox{ where }\psi\equiv\arctan(z/R),$$ which has an appropriate form, but the results are very insensitive to the choice of $g$: essentially unchanged results are obtained with $g=1$. The algorithm involves converting heliocentric velocities to Galactocentric velocities and thus requires assumptions regarding the Galactocentric velocity of the Sun and the distance $R_0$ to the Galactic centre. We assume that $R_0=8.33\kpc$, that the local circular speed is $\Theta_0=230\kms$, and that the Sun’s velocity with respect to the Local Standard of Rest is $(U_0,V_0,W_0)=(11.1,12.24,7.25)\kms$ [@SchoenrichBD]. There is very little sensitivity to the value of $\Theta_0$. The azimuthal direction is assumed to be a principal axis of the velocity ellipsoid, while the latter’s longest axis is tilted with respect to the plane by angle $\beta=a_0\arctan(z/R)$, where $a_0$ is a parameter.
The corrections exploit pattern on the sky of correlations between the local Cartesian velocity components $U,V,W$ that are introduced by distance errors. To assess the magnitude of these correlations one has first to correct the raw correlations for contributions from sources other than distance errors. The most important such source is observational errors in the proper motions, so knowledge of the magnitude of these errors is needed for the correction.
Proper motions for RAVE stars can be drawn from several catalogues. [@Williams13] compares results obtained with different proper-motion catalogues, and on the basis of this discussion we decided to work with the PPMX proper motions [@Roeseretal] because these are available for all our stars and they tend to minimise anomalous streaming motions. shows a histogram of the errors for RAVE stars given in the PPMX catalogue. It shows that there is a fat tail in the error distribution, and one may show that this tail should not to be taken at face value because when one calculates the velocity dispersions of all the RAVE stars in spatial bins that are further than $\sim0.5\kpc$ from the Sun, the dispersions are often smaller than the contribution expected from proper-motion errors alone. This paradox disappears if one cuts stars with errors in one component of proper motion greater than $8\mas\yr^{-1}$, and we impose this cut throughout the SBA analysis. The only class of stars that is significantly depleted by this cut is that of the very cool dwarfs, which shrinks from $38\,330$ stars to $27\,332$ stars. This cut reduces the rms error in one component of proper motion to $2.5\mas\yr^{-1}$.
A second source of correlations that complicate the SBA analysis is rotation of the velocity ellipsoid’s principal axes as one moves around the Galaxy, and a model of the velocity ellipsoid is used to correct for this effect. The final product is the factor $1+f$ by which all distances must be contracted (or expanded if $f<0$) for all correlations between $U$, $V$ and $W$ to be accounted for by a combination of observational errors and rotation of the principal axes of the velocity ellipsoid.
SBA give two formulae for corrections, one, $f_U$, involving “targeting” $U$ and one, $f_W$, using $W$ as a target. Because the latter is independent of azimuthal streaming, it is the simpler and more reliable. Their equations (19) and (38) give the $W$ and $U$ correction factors, respectively, after the raw covariances have been corrected for observational errors using their equations (22) and (25).
$f_W(T)$ $f_U(T)$ $f_W$ $f_U$
--------------------- ------------------------------ ---------- -------- --------
3[Giants ($\log g<3.5$)]{}
$\log g>2.4$ 0.304 0.323 0.134 0.248
Red Clump 0.311 0.332 0.160 0.249
$\log g<1.7$ 0.310 0.348 0.453 0.676
3[Dwarfs ($\log g\ge3.5$)]{}
$\Teff>6500$ 0.295 0.295 -0.270 -0.210
$5500<\Teff\le6500$ 0.312 0.312 -0.081 -0.037
$5200<\Teff\le5500$ 0.286 0.286 -0.064 -0.027
$\Teff\le5200$ 0.306 0.306 -0.026 0.043
: Kinematic correction factors for general stars at $s<2\kpc$. The first two columns give results of a test in which all stars were recorded to be further from the Sun than their true locations by a factor $1.3$. The last two columns are computed from the real RAVE catalogue.[]{data-label="tab:tests"}
$f_W(T)$ $f_U(T)$ $f_W$ $f_U$
--------------------- ------------------------------ ---------- -------- --------
3[Giants ($\log g<3.5$)]{}
$\log g>2.4$ 0.203 0.203 0.066 0.185
Red Clump 0.157 0.157 0.114 0.148
$\log g<1.7$ 0.100 0.130 0.210 0.334
3[Dwarfs ($\log g\ge3.5$)]{}
$\Teff>6500$ 0.220 0.207 -0.270 -0.210
$5500<\Teff\le6500$ 0.217 0.217 -0.081 -0.037
$5200<\Teff\le5500$ 0.227 0.247 -0.064 -0.027
$\Teff\le5200$ 0.217 0.217 -0.050 0.041
: Kinematic correction factors for general stars at $s<1.3\kpc$. The first two columns report results from a test in which the recorded locations of stars were further than their true locations by a factor $1+f$ where $f$ is a random variable with mean and dispersion $0.2$.[]{data-label="tab:tests2"}
From the RAVE data we have extracted correction factors to the distance estimator $1/\ex{\varpi}$ for the three types of giants and four types of dwarf listed in Table \[tab:gen\_ratios\]. The code used to determine the corrections was tested as follows. For each star in a class, the measured $U,V,W$ velocities were replaced by values chosen from a triaxial Gaussian velocity ellipsoid that has dispersions $\sigma_i=(40,40/\surd2,30)\kms$ around systematic rotation at $200\kms$. Most tests were run with the orientation of the principal axes determined by setting $a_0=0.8$, but excellent results are obtained with other plausible values of $a_0$, including zero. Likewise, the outcome of the code tests is not sensitive to the adopted dispersions $\sigma_i$. Next proper motions and line-of-sight velocities are calculated from the model velocities, and Gaussian observational errors are added with the dispersions that are given in the PPMX catalogue. Then the stars are moved along their lines of sight to points more distant by a factor $1+f$ and their $U,V,W$ components are re-evaluated from the proper motions. In this way we obtain a catalogue of phase-space positions for a population of objects whose distances have been over-estimated by a factor $1+f$. The SBA algorithms are then used to infer from this catalogue the value of $f$.
The first two numerical columns of Table \[tab:tests\] show the fractional distance excesses $f_W$ and $f_U$ obtained by targeting $W$ and $U$ when distances to the stars have been over-estimated by a factor $1+f$ with $f=0.3$. Consequently, ideally we would have $f_W=f_U=0.3$ for all star classes. For $f_W$ this expectation is borne out for all classes to better than $5\%$, and for the dwarfs it is similarly for $f_U$. For the giants $f_U$ is up to $16\%$ larger than it should be, a result which reflects the breakdown of the approximations made by SBA when dealing with more distant stars.
The final two columns of Table \[tab:tests\] show the fractional distance excesses $f_W$ and $f_U$ for the seven classes of RAVE stars using the measured distances and velocities when $1/\ex{\varpi}$ is used as the distance measure. For the giants the differences between $f_W$ and $f_U$ are in the same sense ($f_U>f_W$) as in the tests but they are larger than in the tests. The cause of this difference is not obvious, but one suspects a major contributor is the well-known existence of clumps of stars in the $(U,V)$ plane [@Dehnen98; @Famaey; @Antoja], which conflict with the assumption of simple azimuthal streaming that is fundamental to SBA’s derivation of the formula for $f_U$. Since prominent clumps are absent from the distribution of Hipparcos stars in the $(U,W)$ and $(V,W)$ planes [@Dehnen98], $f_W$ is expected to be a more reliable diagnostic of distance errors than $f_U$. Table \[tab:tests\] then suggests that $1/\ex{\varpi}$ over-estimates distances to high-gravity giants and red-clump stars by $\sim15\%$, and gives distances to dwarfs that are too small by factors that rise from $\sim5\%$ at the cool end rising to $\sim25\%$ at the hot end.
In selecting stars for inclusion in the SBA analysis we have imposed a limit $s_{\rm max}$ on the reported distance, and the results one obtains for both the test and with the real data depend on the value chosen for $s_{\rm max}$. Table \[tab:tests\] is based on the choice $s_{\rm max}=2\kpc$. Table \[tab:tests2\] is based on $s_{\rm max}=1.3\kpc$ and the results of tests reported in the first two numerical columns of this table differ from those reported in the corresponding columns of Table \[tab:tests\] in that the distances to stars were increased by a factor $1+f$ where $f$ is now a Gaussian random variable with mean and dispersion $0.2$. The test results are fairly satisfactory for the dwarfs in that both $f_W$ and $f_U$ have values within $\sim10\%$ of the true value, $0.2$. The test results for the giants are decidedly less satisfactory in that the $f$ values are too small by an amount which increases with the typical luminosity within a class. It is easy to understand why this is so: stars that happen to get a large fractional distance increase are liable to be pushed beyond $s_{\rm max}$ whilst stars that have their distances decreased can enter the sample from beyond $s_{\rm
max}$, and the SBA algorithm correctly infers that on average the stars *in the analysed sample* have small distance over-estimates even though in the population as a whole stars have larger distance over-estimates. Clearly, for this phenomenon to be important the catalogue needs to contain many stars that really are at distances $\sim s_{\rm max}$. The dwarfs do not satisfy this condition, but the low-gravity giants very much straddle the $1.3\kpc$ distance cut.
Comparing columns 3 and 4 of Table \[tab:tests2\] with the corresponding column of Table \[tab:tests\] we see that reducing $s_{\rm max}$ from $2\kpc$ to $1.3\kpc$ has only a modest effect on the $f$ values for dwarfs and a significant effect on giants. The $f$ values of giants decrease significantly for all three classes, but the final $f_W$ factors still increase with decreasing gravity contrary to the tendency seen in the test, so we really must be over-estimating distances to the lowest-gravity (and most luminous) giants. A possible explanation is that we are using stellar parameters obtained under the assumption of Local Thermodynamic Equilibrium (LTE). The validity of LTE decreases with $\log g$, and when non-LTE effects are taken into account, the recovered gravity of a giant star increases [@Ruchti], and the predicted luminosity decreases, bringing the star closer.
The squares in show the values of $1+f_W$ obtained when the giants are grouped by $\log g$ and the dwarfs are grouped by $\Teff$ – in each case the SBA algorithm is used on 15 bins of stars at $1/\ex{\varpi}<1.3\kpc$ with equal numbers of stars in each bin, and all bins statistically independent. The triangles show the analogous ratios $\varpi_{\rm H}/\ex{\varpi}$ of our distance to that implied by the Hipparcos parallax. The curves show fifth-order polynomial fits to all the points. The squares and triangles tell the same story from a qualitative perspective: along the sequence of giants there is a steady increase in the tendency to over-estimate distances as one moves to lower gravity (and higher luminosity), while the dwarfs show a clear trend towards distance over-estimation with falling $\Teff$ with the exception of the coolest bin, which shows marked distance under-estimates. The SBA points for dwarfs tend to lie below those from Hipparcos, so SBA and Hipparcos disagree about the value of $\Teff$ at which our distances are unbiased.
Our tests suggest that $f_W$ should be a reliable guide to any systematic errors in the distances to our dwarf stars. The situation regarding the giant stars is less clear because the $f$ values are biased low unless $s_{\rm
max}$ is large enough to encompass most of the stars in the catalogue. Unfortunately, the more distant stars are, the more sensitive the returned value of $f_U$ becomes to restrictive assumptions regarding the pattern of mean-streaming and random velocities in the Galaxy and some approximations. The value of $f_W$ is less sensitive to these issues and therefore more reliable, but its sensitivity to $s_{\rm max}$ is worrying. A further blow to the credibility of $f_W$ will emerge below from an analysis of the red-clump stars.
### Kinematic corrections to multi-Gaussian pdfs
SBA assume that one is working with a simple distance estimator, while in Section \[sec:pdfs\] we saw that our most complete information is contained in a distance pdf. Can we use a kinematic analysis to refine these pdfs?
The SBA algorithm involves several sample averages such as $\ex{Wy}$, where $W$ and $y$ are quantities that depend on the distance to each star. In our analysis above we evaluated these for just one distance, but given a pdf $P(\mu)$ it is straightforward to replace $Wy$ by the expectation value of $Wy$: $$\overline{Wy}\equiv\int\d\mu\,P(\mu)W(\mu)y(\mu).$$ These expectation values are then averaged over the sample to produce the sample averages $\ex{Wy}$, etc., that appear in the SBA formalism. Thus is straightforward to use the pdfs to calculate a kinematic correction factor such as $f_y$.
It is less clear how one should modify the pdf in light of a non-zero value of $f_y$. We have experimented with two possibilities.
- Move the centres of all the Gaussians to larger or smaller distance moduli until, $f_y=0$. This procedure produces results that are rather similar to, but slightly less convincing than, those obtained without the pdfs.
- When a star has more than one Gaussian in its pdf, modify the probabilities $f_k$ (eqn. \[eq:defsfk\]) associated with the two most probable Gaussians. This procedure is appropriate if the Bayesian algorithm has correctly identified the two model stars that an observed star could be, but, perhaps driven by a faulty prior, has assigned inappropriate odds to the options. We now report results obtained with this procedure.
We make the probabilities $f_1$ and $f_2$ in equation (\[eq:defsfk\]) a function of a variable $\theta$ through $$\label{eq:defstheta}
f_1=A\cos^2(\theta),\quad f_2=A\sin^2(\theta),$$ where at the outset we fix $A\equiv f_1+f_2$ to be the total probability associated with the two most probable options. Then we make $\theta$, which is confined to the range $(0,\pi/2)$, a function of a variable $\xi$ that can span the whole real line, through $$\label{eq:defsxi}
\theta=\arctan(\e^\xi).$$ The original values of $f_i$ determine starting values for $\theta$ and $\xi$. If the kinematic analysis has returned $f_y>0$, implying that distances need to be shortened and the first Gaussian describes a nearer option than the second, then we lower $\theta$ by subtracting $5f_y$ from $\xi$ – the factor 5 is arbitrary: smaller values lead to slower convergence of the iterations but larger values can cause the iterations to undergo diverging oscillations. If, conversely, $f_y<0$, we need to increase $\theta$ and $\xi$ so we add $5f_y$ to $\xi$.
$\overline{|\xi|}$ $f_y(\i)$ $f_y({\rm f})$
--------------------- ------------------------------ ----------- ----------------
2[Giants ($\log g<3.5$)]{}
$\log g>2.4$ 2.16 0.403 0.004
Red Clump 24.85 0.906 0.858
$\log g<1.7$ 5.55 0.290 0.134
2[Dwarfs ($\log g\ge3.5$)]{}
$\Teff>6500$ 8.22 -0.252 -0.247
$5500<\Teff\le6500$ 0.39 -0.015 -0.009
$5200<\Teff\le5500$ 0.55 0.030 0.004
$\Teff\le5200$ 1.99 0.247 0.005
: Kinematic corrections to the pdfs. A large average value of the parameter $\xi$ defined by equation (\[eq:defsxi\]) implies that all the probability has been drive into one Gaussian. The second and third numerical columns give the initial and final values of the kinematic error estimator, which is ideally zero.[]{data-label="tab:pdfsba"}
In Table \[tab:pdfsba\] shows results obtained by iterating up to six times or until $|f_y|<0.01$. The first numerical column gives the mean of $|\xi|$ for all stars that have more than one Gaussian. A value greater than $\sim3$ implies that all available probability has been driven into whichever Gaussian will reduce $|f_y|$. For the giants this condition is reached after about four iterations and is signalled by successive values of $f_y$ becoming nearly identical. The second column gives the initial value of $f_y$ and the third column gives the value of $f_y$ at the end of the iterations. We see that in the case of the highest-gravity giants, adjusting the $f_i$ has reduced $f_y$ to the target value, but that there is insufficient ambiguity in the nature of the clump stars and the low-gravity giants to get $f_W$ below the target value.
There is very little ambiguity in the nature of the hottest dwarfs, so the procedure makes no significant progress in eliminating the tendency for their distances to be under-estimated.
The procedure succeeds with the remaining dwarfs: for all three classes $|f_y|$ is reduced to below the target value, and the modest values of $\overline{|\xi|}$ given in the first numerical column show that this is achieved without driving all the probability into one option.
From this analysis we conclude that there is sufficient ambiguity in the nature of stars that are cooler that $\Teff=5500\K$ and have $\log g>2.4$ to account for non-zero values of the SBA factor $f_W$ but too little ambiguity in the nature of hotter dwarfs and low-gravity giants to account for non-zero $f_W$.
Absolute magnitude of the red clump
-----------------------------------
Helium-burning stars in the red clump have frequently been used as standard candles [e.g. @Cannon70; @Pietrzynski]. Recently [@Williams13] used clump stars in the RAVE survey to analyse the velocity field around the Sun, and reviewed our knowledge of the absolute magnitudes of these objects and the possibility that they depend on age and metallicity. They identified $78\,019$ clump stars as those satisfying the cuts $0.55\le J-K\le0.8$ and $1.8\le\log g\le3$, where $\log g$ was taken from the vDR3 pipeline [@DR3]. We use the same colour range but a narrower band $(1.7,2.4)$ in $\log g$ and with gravity taken from the vDR4 pipeline [@DR4].
shows the distributions of $H$- and $K$-band absolute magnitudes for distance $1/\ex{\varpi}$ of clump stars. The distributions are satisfyingly narrow – each has a standard deviation of $0.20\mag$ – but they are skew, so while their means lie at $M_H=-1.39$ and $M_K=-1.49$ their peaks lie at $M_H=-1.42$ and $M_K=-1.53$. These magnitudes are in the SAAO system: using the formulae of [@Koen07] to convert to the 2MASS system we find the mean of $M_K$ to be $M_K=-1.51$. The sample was restricted by $1/\ex{\varpi}<1.3\kpc$ but increasing the distance cutoff to $2\kpc$ only changes the mean absolute magnitudes to $M_H=-1.36$ and $M_K=-1.46$. For comparison [@Laney12] determined $M_H=-1.49\pm0.022$ and $M_K=-1.61\pm0.022$ from a sample of 191 Hipparcos stars, and [@Williams13] used calibrations in which the 2MASS absolute magnitudes were $M_K=-1.65$, $-1.54$ and $-1.64+0.0625z/\hbox{kpc}$. In the last calibration the decrease in luminosity with increasing distance from the plane reflects the expected increasing age and decreasing metallicity of clump stars. However, the age-metallicity sensitivity of the absolute magnitude is expected to be smallest in the K band [e.g. @Salaris]. Several issues require discussion when considering why our values are $\sim0.1\,$mag fainter than those of Laney et al.
- One might argue that the figures given above actually under-estimate the scale of the conflict with [@Laney12] (and many similar values in the literature) because we ought to have corrected our values for the systematic distance over-estimates implied by the upper panel of . When this is done (using the red curve) we obtain $M_H=-1.21$ and $M_K=-1.32$; since we have moved the stars nearer, we conclude that they are less luminous.
- The study of [@Laney12] involved obtaining new $J,H,K$ photometry for their Hipparcos stars because the 2MASS photometry of Hipparcos red clump stars, which have bright apparent magnitudes, is affected by saturation, which makes them appear fainter than they really are. Unfortunately, only four of our stars were measured by Laney et al. For these stars Laney et al.obtained $J$ magnitudes brighter than the 2MASS values by amounts in the range $(-0.006,0.457)$, but their $H$ and $K$ values are not clearly brighter than the 2MASS values, which suggests that saturation in 2MASS is mainly confined to the $J$ band. Interestingly, the Bayesian algorithm assigns an anomalous extinction ($A_V=0.633$) to the star (Hipp 32222) that shows by far the strongest saturation effects, presumably because a high extinction can explain the unexpectedly faint $J$ magnitude given the spectroscopically determined $\Teff$. From this rather fragmentary evidence we infer that the effects of saturation on the 2MASS magnitudes might cause us to make the nearest clump stars under-luminous by $\sim0.1\,$mag. The triangles in suggest on the contrary that we have found these stars to be over-luminous by $\sim0.4\,$mag.
- Are the red clump stars in our sample correctly identified? shows the density of stars in the $(J-K,\log g)$ plane for two metallicity ranges. In both panels peaks in density are apparent near the theoretical locations of core helium-burning stars, These peaks are captured by our selection criteria $1.7<\log g<2.4$ and $0.55\le J-K\le0.8$. The core helium-burning model star that sits at the centre of the red circle has $\Teff=4485$, $\log g=2.37$ and $M_K=-1.60$, in agreement with the empirical data of [@Laney12].
This discussion explains why our raw distances imply absolute magnitudes for clump stars that differ little from the empirical value of Laney et al., and why these distances are only slightly larger than the Hipparcos parallaxes imply. The puzzle remains that the SBA kinematic analysis points to our distances being too large. For the SBA analysis to be correct, we would require *both* that the stellar models were too luminous *and* the Hipparcos stars to be misleading, perhaps because they are nearby and therefore anomalously young and have atypical chemistry. Consequently, we set the SBA correction factors aside for the moment but in a companion paper [@Binneyetal13] we will return to this issue in the context of dynamical Galaxy models.
Table \[tab:gen\_ratios\] shows that $1/\ex{\varpi}$ is always the shortest of our distance measures, and given the suggestion from the SBA analysis that even this measure might be too long, we do not present an SBA analysis of distances based on $\ex{s}$ or $s_{\ex{\mu}}$. However, such analyses do confirm that these measures over-estimate distances to all classes of star by even larger factors than $1/\ex{\varpi}$ does, so there is no case to be made for using them.
Effective temperature absolute magnitude diagrams
-------------------------------------------------
shows effective temperature absolute-magnitude diagrams for high-latitude ($|b|>40^\circ$) stars created either (a) using $\ex{\varpi}$ to assign a single distance to each stars (left panel) or (b) spreading each star in $M_K$ according to the multi-Gaussian fit to its pdf in distance modulus. The red octagon centred on $(\Teff,M_K)=(5780,3.28)$ shows the location of the Sun in the effective temperature absolute magnitude diagram.
The red clump is prominent in both panels but the horizontal branch extends further to the blue when the pdfs are used as a consequence of eliminating the messy scatter of stars in the left panel between the horizontal branch and the main sequence. Using the pdfs similarly eliminates the unphysical scatter of stars inside the turn-off curve. In both diagrams vertical stripes are evident, especially at the coolest temperatures: these are a legacy of the use by the pipeline of the DEGAS decision-tree routine to identify template spectra [@DR4]. This artifact is enhanced because we have smeared stars in $M_K$ but not in $\Teff$, as we should have done for consistency.
shows effective temperature absolute-magnitude diagrams for two slices through the Galaxy: $|z|<0.2\kpc$ or $0.4<|z|/\!\kpc<0.9$. For these plots we used the multi-Gaussian representations of pdfs to spread stars in distance modulus and thus in $z$. At $|z|<0.2\kpc$ the main sequence, subgiant and giant branches show up nicely, and the red clump is extremely sharp. More than $0.4\kpc$ away from the plane the lower main sequence has disappeared and giant branch becomes more strongly populated because the volume surveyed is much larger.
Cluster $\sigma_{\rm cl}$ $E(B-V)$ $\log\tau$ $\overline{\log\tau}$ $s_{\rm cl}$ $N_{\rm g}$ $\overline{s}_{\rm g}/s_{\rm cl}$ $N_{\rm d}$ $\overline{s}_{\rm d}/s_{\rm cl}$ $\overline{s}_{\rm all}/s_{\rm cl}$ $\overline{s}_{\rm all,nE}/s_{\rm cl}$ $\overline{s}_{\rm all,age}/s_{\rm cl}$
-------------------------- ------------------- ---------- ------------ ----------------------- -------------- ------------- ----------------------------------- ------------- ----------------------------------- ------------------------------------- ---------------------------------------- -----------------------------------------
Blanco 1 5.5 0.01 7.80 9.59 269 4 1.61 23 1.07 1.15 1.13 0.87
NGC 2422 3.6 0.07 7.86 8.82 490 0 $-$ 13 1.09 1.09 1.08 0.85
Alessi 34 15.4 0.18 7.89 9.58 1100 24 1.20 0 $-$ 1.20 1.82 3.71
ASCC 69 14.0 0.17 7.91 9.51 1000 30 1.63 2 0.84 1.58 2.11 4.87
NGC 6405 2.8 0.14 7.97 8.89 487 0 $-$ 12 0.94 0.94 0.90 0.71
Melotte 22 (Pleiades) 4.6 0.03 8.13 9.39 133 2 1.11 35 1.09 1.09 1.11 0.93
NGC 3532 7.1 0.04 8.49 8.95 486 1 1.71 17 1.23 1.26 1.24 0.97
NGC 2477 (M93) 5.7 0.24 8.78 9.29 1300 45 0.91 3 1.02 0.92 1.13 1.27
Hyades 5.7 0.01 8.80 9.70 46 0 $-$ 31 1.08 1.08 1.08 1.00
NGC 2632 (Praesepe, M44) 3.8 0.01 8.86 9.48 187 0 $-$ 34 1.14 1.14 1.12 1.03
NGC 2423 2.7 0.10 8.87 8.96 766 3 1.36 17 0.99 1.05 0.99 1.17
IC 4651 2.6 0.12 9.06 9.30 888 7 1.03 5 0.68 0.87 0.87 1.00
NGC 2682 (M67) 6.6 0.06 9.41 9.74 908 32 0.74 12 0.65 0.72 0.74 0.78
Cluster stars {#sec:clusters}
=============
By searching for stars that have suitable sky coordinates and line-of-sight velocities that agree with a cluster convergence point, we have identified RAVE stars in 15 open clusters. NGC 3680 has just one RAVE star so we cannot analyse its statistics. Table \[tab:cluster\_s\] lists the remaining clusters with RAVE stars in order of increasing age, giving for each cluster the values of several quantities from the literature. The values given are taken from [@Dias02] with the exception of the Hyades, where we used [@Perryman98].
Columns 7 and 8 give the number of giants in our sample and the ratio of their mean value of $1/\ex{\varpi}$ to the distance listed in Table \[tab:cluster\_s\]. Columns 9 and 10 give the same data for dwarfs, and column 11 gives the overall mean of $1/\ex{\varpi}$ for cluster stars divided by the literature distance. A tendency for the giants to over-estimate distances is evident, particularly in the younger clusters such as Alessi 34 and ASCC 69. The distances inferred for dwarfs are generally in good agreement with the literature values, but significant under-estimates are evident in the cases of the oldest clusters, IC 1651 and NGC 2682 (M67). The penultimate column gives the mean value of $1/\ex{\varpi}$ divided by the literature distance when extinction is assumed to be zero. Setting $A_V=0$ shortens distances to dwarfs and lengthens those to giants and for a few clusters the results with no dust are markedly worse but neglecting dust has little impact on most clusters.
Column 5 gives the the mean inferred value of the logarithm of age (in years) and comparing these values with the literature values in column 4 we see little sign of correlation with the result that stars in younger clusters are being presumed much older than they really are. This phenomenon reflects the fact that dating an isolated star is enormously harder than dating a cluster of coeval stars. Clearly poor ages will bias the recovered distances so in the last column of Table \[tab:cluster\_s\] we give the mean values of $1/\ex{\varpi}$ divided by the literature distance when distances are determined under the strong age prior $$P(\tau)\propto\exp\left[-\log_{10}^2(\tau/\tau_{\rm cl})/2(0.1)^2\right],$$ This cluster-specific age prior improves the accuracy of mean distances to stars in clusters older than $100\Myr$, but has an unfortunate effect on the distances to stars in younger clusters.
shows histograms of distances to stars in 12 of the 13 clusters listed in Table \[tab:cluster\_s\]; the red histograms are for our standard distances and the blue histograms are for distances obtained under the strong cluster-specific prior. The numbers in brackets after the cluster names in the top left corner of each panel give the number of giants and dwarfs in that cluster. The top panel of shows the corresponding plot for NGC 2682 (M67). We see that the strong age prior shortens distances to dwarfs and lengthens those to giants in a way that is moderate and beneficial in clusters as old as the Melotte 22 (Pleiades) but unhelpful in younger clusters. The red histograms are generally quite satisfactory.
Repeat observations {#sec:repeat}
===================
We have more than one spectrum for $12\,012$ stars and can form $8\,526$ independent pairs of measurements for the same dwarf star and $11\,868$ independent pairs of measurements for the same giant star. shows histograms of the discrepancies between these measurements when normalised in two ways. In the upper panel the difference in $\ex{\varpi}$ is divided by the mean parallax, while in the lower panel it is divided by the quadrature-sum of the uncertainties of the measurements. The median fractional parallax discrepancy is $0.063$ for giants and $0.069$ for dwarfs – it is easy to show that these values apply also to the discrepancies in distances $1/\ex{\varpi}$. The dispersions of the parallax discrepancies normalised by the formal uncertainties is $0.295$ for giants and $0.348$ for dwarfs. That these numbers are significantly smaller than unity emphasises that much of the error is external and does not derive from noise in the spectrum.
Estimated extinctions {#sec:Av}
=====================
As with distances, the Bayesian algorithm determines a probability distribution for possible extinctions to each star, and one has to consider how best to reduce this distribution to a single value for the extinction. For the reasons given in Section \[sec:method\] the code marginalises over extinctions by integrating with respect to $a\equiv\ln(A_V)$ rather than integrating with respect to $A_V$ directly. Consequently a natural quantity to output is $\ex{a}$, and we use $\widetilde A_V\equiv\e^{\ex{a}}$ as our estimator of the extinction. $\widetilde A_V$ places less weight on high extinctions than does $\ex{A_V}$.
shows that different spectra yield the same value for $\widetilde A_V$ to high precision: the dispersion in the differences divided by the quadrature sum of the uncertainties is only $0.117$ for giants and $0.097$ for dwarfs. This result is to be expected because $\widetilde A_V$ depends strongly on the photometry, and we only change the spectrum between determinations of $\widetilde A_V$.
shows in red the distribution of extinctions to Hipparcos stars; the blue points show the distribution of the prior values of the extinction to the final locations $1/\ex{\varpi}$ of the stars. Since the red and blue points follow very similar distributions, on average our recovered extinctions coincide well with our priors. This finding could indicate either that our priors are accurate guesses of the actual extinction, or that the extinction to an individual star cannot be determined from the data we have. We know that the data [*are*]{} adequate because when we took the priors from the smooth model (\[eq:dmodel\]) normalised in an average sense by the [Schlegel]{} et al. reddenings, the recovered values of $\widetilde A_V$ were systematically smaller than the prior values. Thus the data suffice to shift the recovered values away from a poor prior. Presumably the Hipparcos stars lie in directions of anomalously low extinction, an effect that is captured when the extinction is estimated to be the fraction of the measured extinction to infinity that is expected to lie within distance $s$.
For hot dwarfs most values of $\widetilde A_V$ lie in $(0.1,0.25)$ \[so $\widetilde A_J$ lies in $(0.03,0.07)$\], while a significant fraction of cool dwarfs have $\widetilde
A_V<0.1$ as we would expect given that some of these stars are quite close. The distribution of values of $\widetilde A_V$ for giants peaks around $0.2$ but has a long tail extending out to $\sim0.6$ as we expect for stars that can be quite distant.
shows histograms of the differences between our estimated extinctions $\widetilde A_V$ to stars in the complete sample and the value of the prior on extinction to the star’s proposed location. The red, blue and black histograms are for stars that lie in three ranges of Galactic latitude $b$. The means of all the two highest-latitude histograms are satisfyingly close to zero. The mean of the histogram for $|b|<20\,$deg is negative ($-0.2\sigma$) implying that the dust model slightly over-estimates extinctions to low-latitude stars.
shows the relationship between extinction and distance for hot dwarfs ($\Teff>5500\K$), cool dwarfs and giants ($\log g<3.5$) in the full RAVE sample. In addition to showing the extent of the relation between distance and extinction, these plots show how the three classes of star are distributed in distance. The ridge line through the distribution of giants has a slope $\simeq0.19\,$mag/kpc, while that through the distribution of cool dwarfs has a slope $\simeq0.78\,$mag/kpc. For comparison, the traditional relation for paths near the mid-plane is $A_V\simeq1.6 s/$kpc [e.g. @BinneyM]. Since most of our sight lines move away from the mid-plane, they naturally have lower values of extinction per unit length. Moreover, our samples are subject to the already-noted observational bias against stars high extinctions, and this bias particularly concentrates the giants at high latitudes, where extinction per unit distance is low.
The red points in show for each cluster the distribution of $\log_{10}(\widetilde A_V/A_{V{\rm cl}})$, where $A_{V{\rm
cl}}$ is 3.1 times the cluster’s literature value of $E(B-V)$. The blue points show the corresponding distributions of the values obtained by replacing $\widetilde A_V$ by the prior extinction $A_{V{\rm prior}}$ at $1/\ex{\varpi}$. For all clusters the red and blue points have similar distributions, which suggests that the priors are reasonable. In light of this result, it is striking (a) how broad the distributions are, and (b) that in four clusters (Melotte 22, Hyades, NGC 2632 and NGC 2682) the literature extinction lies off one wing or the other of the distribution. These findings call into question the very concept of a cluster-wide characteristic extinction, and suggest that if one must choose a single characteristic extinction, the literature value may be a poor choice.
Conclusions {#sec:discuss}
===========
We have extended the Bayesian approach to distance determination of [@BurnettB] to allow for extinction and reddening and to deliver pdfs in distance modulus in addition to expectation values of three distance measures, distance $s$, distance modulus $\mu$ and parallax $\varpi$.
We have fitted each star’s pdf in distance modulus with a sum of up to three Gaussians. A single Gaussian provides a good fit to about 45 per cent of the pdfs, two Gaussians provide a good fit to most of the remaining pdfs, so just 5 per cent of the pdfs require three Gaussians for a good fit. When these Gaussian decompositions are used to make Hess diagrams by splitting each star’s contribution to the density into one, two or three parts at the luminosity associated with the centre of each Gaussian component, the diagram becomes significantly sharper as the man-sequence turnoff and the horizontal branch emerge clearly. This phenomenon indicates that multi-modal pdfs are associated with stars that could be upper main-sequence stars or blue horizontal-branch stars, or could be lower main-sequence stars or subgiants.
For every class of star examined, we find that $\ex{s}>s_{\ex{\mu}}>1/\ex{\varpi}$, a phenomenon that arises because these distance measures weight differently the possibilities that a given star is far or near. The differences between these distance measures are least for hot dwarfs $\Teff>5500\K$ and red-clump stars, and greatest for very cool dwarfs ($\Teff<5200\K$) high-gravity giants ($\log g>2.4$) because hot dwarfs and red-clump stars have quite narrow pdfs in distance while the dwarf/giant ambiguity causes cool dwarfs and high-gravity giants to have broad pdfs in distance.
The RAVE survey encompasses $\sim5000$ Hipparcos stars. Histograms of the difference between our values of $\ex{\varpi}$ and the Hipparcos parallaxes normalised by the quadrature sum of our errors and the Hipparcos errors come close to the ideal of a unit Gaussian of zero mean in the cases of warm dwarfs ($\Teff>5500\K$) and giants ($\logg<3.5$), so not only are our parallax estimates fairly reliable, but our error estimates are reasonable. The situation regarding the smaller sample of cool dwarfs is unsatisfactory. The majority of these stars require multi-Gaussian fits to their pdfs. When a Hipparcos parallax is available, it agrees within the errors with one of the Gaussians as one would wish. But the single Gaussians fitted to a minority of cool dwarfs yield parallaxes that are significantly larger than the Hipparcos parallaxes. Thus our ability to determine distances to cool dwarfs is rather limited.
For giants our parallaxes are competitive with those of Hipparcos, but for cool dwarfs errors on Hipparcos parallaxes are smaller than the errors on ours by a factor $\sim3$.
The good agreement between our parallaxes and the Hipparcos parallaxes, suggests that $1/\ex{\varpi}$ is our most reliable estimator of distance, a conclusion we were able to confirm subsequently. Hence we have concentrated on assessing the accuracy of the distance estimator $1/\ex{\varpi}$.
The Hipparcos stars in the RAVE survey reveal () a tendency for our distances to the hottest dwarfs to be $\sim15\%$ too small, while our distances to dwarfs with $\Teff\sim5000\K$ are too large by about the same amount. Our distances to the coolest dwarfs are 20–30% too small. The Hipparcos stars reveal that our distances to giants are too large by a factor that increases smoothly with decreasing $\log g$ from unity at $\log
g=3.5$ to $\sim1.2$ at the lowest gravities. This phenomenon may reflect our use of stellar parameters obtained under the assumption of LTE. However, it should be noted that [@DR4] excise the cores of strong lines, where non-LTE effects will be most prominent.
The values of the kinematic corrections obtained by the method of [@SBA] for all the giants and dwarfs in the RAVE sample confirm the results from the Hipparcos stars: $1/\ex{\varpi}$ is a more reliable distance estimator for cool stars than $\ex{s}$ and for dwarfs the ratio of $1/\ex{\varpi}$ to the true distance increases with decreasing $\Teff$ except below $\Teff\sim4500\K$, where it drops abruptly. For dwarfs the SBA kinematic indicators agree moderately with each other and suggest that our distances tend to be too short by an amount that decreases with $\Teff$ from $\gta20\%$ at the hot end to perfection at $\Teff\simeq5000\K$. The shape of the plot of the ratios of our distance to true distance agrees perfectly with the Hipparcos results, but there is a small vertical offset between the curves.
For giants $1/\ex{\varpi}$ has a tendency to be too large, by an amount that emerges equally from the Hipparcos results and the SBA kinematic corrector $f_W$. The ratio of our distance to the true distance increases with decreasing $\log g$ from $\sim1.05$ at the high-gravity end to $\sim1.2$ at the low-gravity end. Unfortunately, the Hipparcos results are of course confined to $s\lta0.15\kpc$ and the SBA analysis proves sensitive to the upper limit on the distances of stars we use in the analysis. Moreover, for stars with $s\gta2\kpc$ the two SBA factors disagree with each other. Therefor it is hard to assess the accuracy of our distances to stars at $s>2\kpc$, which tend to be luminous low-gravity giants. However, the indications are that we are over-estimating these distances by $\gta20\%$.
We have identified red-clump stars by cuts in the $(J-K,\log g)$ plane and find that a histogram of these stars’ values of $M_K$ is narrow and peaks $\sim0.1\,$mag fainter than the standard magnitude. The origin of this offset is unclear. If we accept the indications from both the Hipparcos stars and the SBA analysis that we systematically over-estimate distances to giants, the offset is made significantly larger: $0.3\,$mag under-luminous.
We have identified 364 RAVE stars in 15 open clusters. Our standard distances generally form a satisfyingly narrow distribution with the cluster’s literature distance almost always within one standard deviation of the distribution’s mean. There is a clear tendency for the giants in any cluster to be assigned distances that are larger than the distances assigned to the cluster’s dwarfs. In the oldest clusters, IC 4651 and NGC 2682 (M67), the dwarf distances are only $\sim67\%$ of the cluster distance, but in the other clusters the dwarf distances appear about right.
The data barely constrain the ages of stars. Consequently, our standard distances are based the assumption that stars are quite old, older than the ages of many of the clusters we have studied. Curiously, using a prior on ages that enforces the cluster’s literature age produces a more satisfying histogram of distances only for clusters older than Melotte 22 (the Pleiades).
The data do contain sufficient information to place significant constraints on the extinctions of stars – we know this because the extinctions we first derived were systematically lower than the priors we then employed. This phenomenon led to improved priors and our extinctions now scatter nearly randomly around the prior values. Since extinction varies discontinuously from one line of sight to the next on account of the fractal nature of the ISM, and we do not have a sample of stars with accurately determined extinctions, it is hard to test the validity of our extinctions. Our results for clusters indicate that different stars in the same cluster generally have significantly different extinctions, and that the mean extinction of stars in a given cluster often differs significantly from the cluster’s literature value.
The distances we derive from different spectra of the same star are entirely consistent with one another and imply that noise in the spectrum contributes less that half the uncertainty in the derived distance.
This work could and should be significantly improved in three ways. First, photometry in optical bands is now available for most of our stars from the APASS survey [@Henden]. Use of this photometry would sharpen constraints on some combination of $A_V$ and $\Teff$. Second, the stellar models used here are now a few years old and should be updated and extended. Inclusion of $\alpha$-enhanced stars with lower metallicities should improve accuracy for stars that are far from the plane. Moreover, we could now use models for which magnitudes in the 2MASS system have been directly computed rather than obtained by transformation of magnitudes in the Johnson-Cousins system. Third, stellar parameters that include corrections for non-LTE effects as discussed by [@Ruchti] may yield improved distances, especially to luminous giants. Distances based on extended photometry and models will be made available on the RAVE website as soon as possible.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the referee for a meticulous reading of the submitted version and many useful suggestions for improvement.
Funding for RAVE has been provided by: the Australian Astronomical Observatory; the Leibniz-Institut für Astrophysik Potsdam (AIP); the Australian National University; the Australian Research Council; the French National Research Agency; the German Research Foundation (SPP 1177 and SFB 881); the European Research Council (ERC-StG 240271 Galactica); the Istituto Nazionale di Astrofisica at Padova; The Johns Hopkins University; the National Science Foundation of the USA (AST-0908326); the W. M. Keck foundation; the Macquarie University; the Netherlands Research School for Astronomy; the Natural Sciences and Engineering Research Council of Canada; the Slovenian Research Agency; the Swiss National Science Foundation; the Science & Technology Facilities Council of the UK; Opticon; Strasbourg Observatory; and the Universities of Groningen, Heidelberg and Sydney. The RAVE web site is at http://www.rave-survey.org.
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[^1]: E-mail: [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A brief introduction is given to rotating black holes in more than four spacetime dimensions.'
---
**[Higher Dimensional Generalizations of the Kerr Black Hole[^1]]{}**
Gary T. Horowitz
*Department of Physics, UCSB, Santa Barbara, CA 93106*
*[email protected]*
Introduction
============
When I was a graduate student at the University of Chicago in the late 1970’s, I often heard Chandrasekhar raving about the Kerr solution [@Kerr:1963ud]. He was amazed by all of its remarkable properties and even its mere existence. As he said at the time: “In my entire scientific life...the most shattering experience has been the realization that an exact solution of general relativity, discovered by the New Zealand mathematician Roy Kerr, provides the absolutely exact representation of untold numbers of massive black holes that populate the Universe" [@Chandra].
It took me a while to understand Chandra’s fascination, but I have come to agree. One can plausibly argue that the black hole solution discovered by Roy Kerr is the most important vacuum solution ever found to Einstein’s equation. To honor Kerr’s $70^{th}$ birthday, I would like to describe some recent generalizations of the Kerr solution to higher spatial dimensions.
Before I begin, let me say a word about the motivation for this work. There are two main reasons for studying these generalizations. The first comes from string theory, which is a promising approach to quantum gravity. String theory predicts that spacetime has more than four dimensions. For a while it was thought that the extra spatial dimensions would be of order the Planck scale, making a geometric description unreliable, but it has recently been realized that there is a way to make the extra dimensions relatively large and still be unobservable. This is if we live on a three dimensional surface (a “brane") in a higher dimensional space. String theory contains such higher dimensional extended objects, and it turns out that nongravitational forces are confined to the brane, but gravity is not. In such a scenario, all gravitational objects such as black holes are higher dimensional. The second reason for studying these solutions has nothing to do with string theory. Four dimensional black holes have a number of remarkable properties. It is natural to ask whether these properties are general features of black holes or whether they crucially depend on the world being four dimensional. We will see that many of them are indeed special properties of four dimensions and do not hold in general.
Nonrotating black holes in $D>4$
================================
To become familiar with black holes in higher dimensions, I will start by discussing nonrotating black holes. (For a more extensive reviews of the material in this section, see [@Kol:2004ww; @Harmark:2005pp].) For simplicity, we will focus on $D=5$. There are two possible boundary conditions to consider: asymptotically flat in five dimensions, or the Kaluza-Klein choice – asymptotically $M_4\times S^1$. In the asymptotically flat case, the only static black hole is the five dimensional Schwarzschild-Tangherlini solution [@Gibbons:2002av] \[5dsch\] ds\^2 = - (1-[r\_0\^2r\^2]{}) dt\^2 +(1-[r\_0\^2r\^2]{})\^[-1]{} dr\^2 + r\^2 d\_3 In the Kaluza-Klein case, there are more possibilities. Let $L$ be the length of the circle at infinity. The simplest solution with an event horizon is just the product of four dimensional Schwarzschild with radius $r_0$ and $S^1$: \[blackstring\] ds\^2 = - (1-[r\_0r]{}) dt\^2 +(1-[r\_0r]{})\^[-1]{} dr\^2 + r\^2 d+ dz\^2 This has horizon topology $S^2\times S^1$ and is sometimes called a black string, since it looks like a one dimensional extended object surrounded by an event horizon. Gregory and Laflamme (GL) showed that this spacetime is unstable to linearized perturbations with a long wavelength along the circle [@Gregory:vy]. More precisely, there is a critical size for the circle, $L_0$, of order $r_0$ such that black strings with $L\le L_0$ are stable and those with $L>L_0$ are unstable. The unstable mode is spherically symmetric, but causes the horizon to oscillate in the $z$ direction. Gregory and Laflamme also compared the total entropy of the black string with that of a five dimensional spherical black hole with the same total mass, and found that when $L>L_0$, the black hole had greater entropy. They thus suggested that the full nonlinear evolution of the instability would result in the black string breaking up into separate black holes which would then coalesce into a single black hole. Classically, horizons cannot bifurcate, but the idea was that under classical evolution, the event horizon would pinch off and become singular. When the curvature became large enough, it was plausible that quantum effects would smooth out the transition between the black string and spherical black holes.
However, it turns out that an event horizon cannot pinch off in finite time [@Horowitz:2001cz]. In particular, if one perturbs (\[blackstring\]), an $S^2$ on the horizon cannot shrink to zero size in finite affine parameter. The reason is the following. Hawking’s famous area theorem [@Hawking:1971tu] is based on a local result that the divergence $\theta$ of the null geodesic generators of the horizon cannot become negative, i.e., the null geodesics cannot start to converge. If an $S^2$ on the horizon tries to shrink to zero size, the null geodesics on that $S^2$ must be converging. The total $\theta$ can stay positive only if the horizon is expanding rapidly in the circle direction, but this produces a large shear. If the $S^2$ were to shrink to zero size in finite time, one can show this shear would drive $\theta$ negative. When it was realized that the black string cannot pinch off in finite time, it was suggested that the solution should settle down to a static nonuniform black string.
A natural place to start looking for these new solutions is with the static perturbation of the uniform black string that exists with wavelength $L_0$. Gubser [@Gubser:2001ac] did a perturbative calculation and found evidence that the nonuniform solutions with small inhomogeneity could not be the endpoint of the GL instability. Recent numerical work has found vacuum solutions describing static black strings with large inhomogeneity [@Wiseman:2002zc]. Surprisingly, all of these solutions have a mass which is larger than the unstable uniform black strings. So they cannot be the endpoint of the GL instability[^2]. Solutions describing topologically spherical black holes in Kaluza-Klein theory have also been found numerically [@Kudoh:2003ki; @Sorkin:2003ka]. When the black hole radius is much less than $L$, it looks just like (\[5dsch\]). As you increase the radius one finds that the size of the fifth dimension near the black hole grows and the black hole remains approximately spherical. It then reaches a maximum mass. Remarkably, one can continue past this point and find another branch of black hole solutions with lower mass and squashed horizons. It was conjectured by Kol [@Kol:2002xz] that the nonuniform black strings should meet the squashed black holes at a point corresponding to a static solution with a singular horizon, and this appears to be the case [@Kudoh:2004hs]. This yields a nice consistent picture of static Kaluza-Klein solutions with horizons, but it doesn’t answer the question of what is the endpoint of the GL instability. An attempt to numerically evolve a perturbed black string is underway. An earlier attempt could not be followed far enough to reach the final state [@Choptuik:2003qd].
It was suggested by Wald [@Wald] that the black string horizon might pinch off in infinite affine parameter (avoiding the above no-go theorem), but still occur at finite advanced time as seen from the outside. This is possible since the spacetime is singular when the horizon pinches off, and some evidence for this has been found [@Marolf:2005vn]. If this were the case, then the original suggestion of Gregory and Laflamme that the black string will break up into spherical black holes might still be correct.
Rotating black holes in $D>4$
=============================
With Kaluza-Klein boundary conditions, the only known solution is the rotating black string obtained by taking the product of the Kerr metric and a circle. Most of the recent work on higher dimensional rotating black holes has been in the context of asymptotically flat spacetimes, so from now on we will focus on this case.
The direct generalization of the Kerr metric to higher dimensions was found by Myers and Perry in 1986 [@Myers:1986un]. In more than three spatial dimensions, black holes can rotate in different orthogonal planes, so the general solution has several angular momentum parameters. The general solution, with all possible angular momenta nonzero is known explicitly. Like the Kerr metric, these solutions are all of the Kerr-Schild form [@KerrS] g\_ = \_ + h k\_k\_where $k_\mu$ is null.
If we set all but one of the angular momentum parameters to zero, we can write the metric in Boyer-Lindquist like coordinates. In $D$ spacetime dimensions, the solution is $$ds^2 = - dt^2 + \sin^2\theta (r^2 + a^2 ) d\vp^2 + {\mu\over r^{D-5} \rho^2} (dt - a\sin^2\theta d\vp)^2$$ + \^[-1]{} dr\^2 + \^2 d\^2 + r\^2 \^2d\_[D-4]{} where $\rho^2 = r^2 + a^2 \cos^2\theta$ and = [r\^2 + a\^2\^2]{} - [r\^[D-5]{}\^2]{} Like the Kerr metric, this solution has two free parameters $\mu,a$ which determine the mass and angular momentum: \[MJ\] M = [(D-2) \_[D-2]{}16]{} , J = [2D-2]{} Ma where $\Omega_{D-2}$ is the area of a unit $S^{D-2}$. One of the most surprising properties of these solutions is that for $D>5$, there is a regular horizon for all $M,J$! There is no extremal limit. This follows from the fact that the horizon exists where $\Psi=0$, but for $D>5$ this equation always has a solution. However, when the angular momentum is much bigger than the mass (using quantities of the same dimension, this is $J^{D-3} \gg M^{D-2}$) the horizon is like a flat pancake: it is spread out in the plane of rotation, but very thin in the orthogonal directions. Locally, it looks like the product of $D-2$ dimensional Schwarzschild and $R^2$. It is probably subject to a GL instability [@Emparan:2003sy]. If so, there would be an effective extremal limit for stable black holes in all dimensions.
In five dimensions, there is an extremal limit, but the horizon area goes to zero in this limit. The extremal limit corresponds to $\mu=a^2$. Setting $D=5$ in (\[MJ\]), the mass and angular momentum are M=[38]{}J=[23]{} Ma so in the extremal limit J\^2 = [32 27 ]{} M\^3
These solutions have recently been generalized to include a cosmological constant [@Hawking:1998kw; @Gibbons:2004js].
Black rings
===========
The above solutions all have a horizon which is topologically spherical $S^{D-2}$. There is a qualitatively new type of rotating black hole that arises in $D=5$ (and possibly higher dimensions). One can take a black string, wrap it into a circle, and spin it along the circle direction just enough so that the gravitational attraction is balanced by the centrifugal force. Remarkably, an exact solution has been found describing this [@Emparan:2001wn]. It was not found by actually carrying out the above proceedure, but by a trick involving analytically continuing a Kaluza-Klein C-metric. The solution is independent of time, $t$, and two orthogonal rotations parameterized by $\varphi$ and $\psi$, so the isometry group is $R\times U(1)^2$. Introducing two other spatial coordinates, $-1\le x\le 1$ and $y\le -1$, the metric is: $$ds^{2} =
-\frac{F(y)}{F(x)}\left [dt +
C(\nu)R \frac{1+y}{F(y)} d\psi\right ]^2$$ $$\label{blackring}
+ \frac{R^{2}}{(x-y)^{2}}F(x)
\Bigg [-\frac{G(y)}{F(y)} d\psi^{2}
-\frac{dy^2}{G(y)} +\frac{dx^2}{G(x)}
+\frac{G(x)}{F(x)} d\varphi^{2} \Bigg]$$ where $$F(\xi) = 1 + {2\nu\over 1+\nu^2} \xi, \qquad G(\xi) =
(1-\xi^{2})(1+\nu \xi),$$ C() = [1+\^2]{}\^[1/2]{} and the angular variables are periodically identified with period = = [2]{}
Although it is not obvious in these coordinates, this solution is asymptotically flat. The asymptotic region corresponds to $x\rightarrow -1, \ y\rightarrow -1$. The solution depends on two parameters $R>0$ and $0<\nu<1$ which determine the mass and angular momentum M= [3R\^22]{}[(1-)(1+\^2)]{}, J = [R\^32]{} [C()(1-)]{} The coordinates $(x,\vp)$ parameterize two-spheres, so on a constant $t$ slice, surfaces of constant $y<-1$ are topologically $S^2\times S^1$. The limiting value $y=-1$ corresponds to the axis for the $\psi$ rotations. The solution has an event horizon at $y=-1/\nu$ (where $G(y)=0$) with topology $S^2\times S^1$ and a curvature singularity at $y=-\infty$. It has no inner horizon, but it does have an ergoregion given by $-1/\nu < y < -(1+\nu^2)/2\nu$. This is just like the solution one would obtain by boosting the black string (\[blackstring\]) in the $z$ direction. This solution is called a [*black ring*]{} rather than a black string since it is wrapped around a topologically trivial circle in space.
Not surprisingly, the angular momentum now has a lower bound, but no upper bound. Solutions exist only when $J^2 \ge M^3/\pi \equiv J_{min}^2$. Something interesting happens when $J$ is near its minimum value. If $J_{min}^2 < J^2 < {32\over 27} J_{min}^2$ there are two stationary black ring solutions with different horizon area. In addition, there is a Myers-Perry black hole! So for $M,J$ in this range, there are three black holes clearly showing the black hole uniqueness theorems do not extend to higher dimensions. Near $J_{min}$, the Myers-Perry black hole has the largest horizon area. Near $(32/27)^{1/2} J_{min}$, one of the black rings has the largest area.
It is not yet known if these solutions are stable, although the solutions with large $J$ are likely to be unstable. This is because in this limit, the ring becomes very long and thin, so one expects an analog of the GL instability. It cannot settle down to an inhomogeneous ring, since the rotation would cause the system to lose energy due to gravitational waves. The outcome of this instability (assuming it exists) is unknown.
Einstein-Maxwell solutions
==========================
So far, we have considered only vacuum solutions. The higher dimensional generalization of the Kerr-Newman solution is still not known analytically (but special cases have been found numerically [@Kunz:2005nm]). However exact solutions describing generalizations of the neutral black ring have been found. In fact, the discrete nonuniqueness discussed above becomes a continuously infinite nonuniqueness when we add a Maxwell field. Suppose we consider Einstein-Maxwell theory in five dimensions S=d\^5x (R- [14]{}F\_F\^) In this theory, there is a global electric charge Q =[14]{} \_[S\^3]{} \^\*F but no global magnetic charge since one cannot integrate the two-form $F$ over the three-sphere at infinity. However, there is a local magnetic charge carried by a string. Given a point on a string, one can surround it by a two-sphere and define q= [14]{} \_[S\^2]{} F If the string is wrapped into a circle, there is no net magnetic charge at infinity. The asymptotic field resembles a magnetic dipole.
It was suggested in [@Reall:2002bh] that there should exist a generalization of (\[blackring\]) which has a third independent parameter labeling the magnetic dipole moment $q$. The explicit solution was found about a year later [@Emparan:2004wy]. Note that unlike four dimensional black holes, the dipole moment is an independent adjustible parameter. Since the only global charges are $M$ and $J$, there is continuous nonuniqueness. Solutions with $q\ne 0$ have a smooth inner horizon as well as an event horizon. There is now an upper bound on the angular momentum as well as a lower bound $J^2_{min} \le J^2 \le J^2_{max}$. Solutions with $J^2= J^2_{max}$ are extremal in that the inner and outer horizons coincide. The extremal solution has a smooth degenerate horizon, with zero surface gravity.
This dipole charge enters the first law in much the same way as an ordinary global charge. There is a corresponding potential $\phi$ and one can show [@Copsey:2005se] that for any perturbation satisfying the linearized constraints M = \_[H]{} + \_[i]{} \^[i]{} + Q+ q where, as usual, $\kappa$ is the surface gravity, $A_H$ is the horizon area, $\Omega_i$ ($i=1,2$) are the angular velocities in the two orthogonal planes, and $\Phi$ is the electrostatic potential. We have included the possibility of a global electric charge and rotation in both orthogonal planes even though the dipole ring found in [@Emparan:2004wy] has zero electric charge and only one nonzero angular momentum.
Supersymmetry
=============
Let us first recall the situation with supersymmetric black holes in four dimensions. In $D=4$, supersymmetry requires $M=Q$, so the only supersymmetric black holes in Einstein-Maxwell theory are extremal Reissner-Nordström and superpositions of them. The Kerr-Newman solution with $M=Q$, and $J\ne 0$ is supersymmetric but describes a naked singularity. (If one adds additional matter fields, there exist supersymmetric multi black hole solutions with angular momentum [@Bates:2003vx].)
The situation is different in $D=5$. To begin, the bosonic sector of the minimal supergravity theory is not just Einstein-Maxwell, but also contains a Chern-Simons term $$\label{sugra}
S = \int d^5 x\sqrt{-g} \Big(R -\frac{1}{4} F_{\mu \nu}
F^{\mu \nu} -{1\over 12\sqrt 3}
\epsilon^{\mu \nu \rho \sigma \eta} F_{\mu \nu} F_{\rho \sigma}
A_{\eta} \Big)$$ Surprisingly, the addition of the Chern-Simons term makes it easier to solve the field equations[^3], and analytic solutions describing charged, rotating black holes in this theory are known [@Breckenridge:1996sn]. Their extremal limit provides a two parameter family of rotating supersymmetric black holes [@Breckenridge:1996is]. One parameter determines the angular momenta: the angular momenta in both orthogonal planes are nonzero and equal, $J_1=J_2$. The second determines the electric charge. The mass is determined in terms of the charge $M=(\sqrt 3/2)Q$. As expected, the surface gravity vanishes for these extremal solutions, but surprisingly, the angular velocity also vanishes. The fact that there is no ergoregion follows from the fact that supersymmetric, asymptotically flat solutions must have a Killing field that remains timelike (or null) everywhere outside the horizon [@Gauntlett:1998fz]. These horizons are topologically $S^3$, and Reall has shown that these are the only supersymmetric solutions with spherical horizons [@Reall:2002bh].
However, there are also supersymmetric black rings with $S^2\times S^1$ horizons [@Elvang:2004rt]. The dipole rings discussed above are also solutions to (\[sugra\]) since the Chern-Simons term does not contribute in this case, but they are not supersymmetric. To obtain supersymmetric solutions, one must add electric charge and take a limit so that $M=(\sqrt 3/2)Q$. It turns out that the supersymmetric solutions can have two independent angular momenta, so there are three free parameters $Q,J_1,J_2$. There is also a magnetic dipole charge but it is not independent.
The near horizon limit of a $D=4$ extreme Reissner-Nordstrom black hole is $AdS_2\times S^2$ (where AdS denotes anti de Sitter spacetime). Similarly, the near horizon limit of the supersymmetric black ring is $AdS_3\times S^2$. Interestingly enough, the near horizon limit of the extreme Kerr metric and some Myers-Perry solutions also resemble (warped) products of AdS and a sphere [@Bardeen:1999px].
One of the advantages of considering supersymmetric solutions is that they minimize the mass for given charges. Since there is no lower mass solution to decay to, these supersymmetric solutions are expected to be stable. Another advantage is that it is easier to count their microstates in string theory and compare with the Hawking-Bekenstein entropy. This was done successfully for the topologically spherical black hole [@Breckenridge:1996is] and has also been discussed for the black rings [@Cyrier:2004hj; @Bena:2004tk].
In this example of five dimensional minimal supergravity, the supersymmetric solution is uniquely determined by the conserved charges at infinity. However, in ten dimensional supergravity, supersymmetric solutions have been found with three types of electric charges and three local magnetic charges but only one constraint [@EEMR; @Gauntlett:2004wh; @Bena:2004de]. So together with the two angular momenta there are a total of seven parameters, but only five global charges. So even among supersymmetric solutions, there is continuous nonuniqueness.
Discussion
==========
In four dimensions, black holes enjoy a number of special properties including:
1\) They are uniquely specified by $M,J,Q$
2\) The are topologically spherical
3\) They are stable
We have seen that none of these properties are preserved in higher dimensions. The solutions can be labelled by local charges rather than global charges. In five dimensions, black hole horizons can have topology $S^2\times S^1$ as well as $S^3$. Finally, we have seen that the straight black string is unstable, and highly rotating black rings and black holes in more than five dimensions are likely to be also. With regard to the nonuniqueness, it should be noted that there is still a finite number of parameters characterizing the solution. When this is compared with the vast number of ways of forming these objects, one can take the view that the spirit of the no hair theorem is preserved[^4].
A surprisingly large number of higher dimensional generalizations of the Kerr solution have been found. But the list is far from complete, and a great deal remains to be understood. For example, we do not yet have good restrictions on the topology of stationary black holes in $D>4$. Clearly $S^n$ and $S^2\times S^1$ are possible. Are there others? There should also be more general black ring solutions in five dimensions. The vacuum solution given in (\[blackring\]) has angular momentum only along the circle. One expects that it should also be possible to have angular momentum on the two-spheres. This would be analogous to first taking the product of the $D=4$ Kerr metric and a line to obtain a rotating black string, and then wrapping the string into a circle and spinning it to obtain another stationary black ring. Finally, all of the known higher dimensional black holes have a great deal of symmetry: in addition to time translation invariance, they are all invariant under rotations in mutually orthogonal planes. It was pointed out in [@Reall:2002bh] that on general grounds, one only expects rotating black holes to have one additional Killing field. So there may well exist much more general black holes in $D>4$ with only two Killing fields.
In short, the space of black holes is much richer in more than four spacetime dimensions. It remains to be seen whether nature takes advantage of this richness.
**Acknowledgements**
.5cm I would like to thank the organizers of the Kerr Fest (Christchurch, New Zealand, Aug. 26-28, 2004) for a stimulating meeting. I also want to thank H. Elvang and H. Reall for their explanations of some the results discussed here. This work was supported in part by NSF grant PHY-0244764.
[99]{}
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[^1]: To appear in [*Kerr Spacetime: Rotating Black Holes*]{}, eds. S. Scott, M. Visser, and D. Wiltshire (Cambridge University Press).
[^2]: In dimensions greater than 13 this changes and the nonuniform black string can be the endpoint of the instability [@Sorkin:2004qq].
[^3]: This is because one can employ various solution generating techniques known for supergravity.
[^4]: I thank B. Carter for stressing this point.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Given a set $U$ of alternatives, a choice (correspondence) on $U$ is a contractive map $c$ defined on a family $\Omega$ of nonempty subsets of $U$. Semantically, a choice $c$ associates to each menu $A \in \Omega$ a nonempty subset $c(A) \subseteq A$ comprising all elements of $A$ that are deemed selectable by an agent. A choice on $U$ is total if its domain is the powerset of $U$ minus the empty set, and partial otherwise. According to the theory of revealed preferences, a choice is rationalizable if it can be retrieved from a binary relation on $U$ by taking all maximal elements of each menu. It is well-known that rationalizable choices are characterized by the satisfaction of suitable axioms of consistency, which codify logical rules of selection within menus. For instance, (Weak Axiom of Revealed Preference) characterizes choices rationalizable by a transitive relation. Here we study the satisfiability problem for unquantified formulae of an elementary fragment of set theory involving a choice function symbol ${\mathtt{c}}$, the Boolean set operators and the singleton, the equality and inclusion predicates, and the propositional connectives. In particular, we consider the cases in which the interpretation of ${\mathtt{c}}$ satisfies any combination of two specific axioms of consistency, whose conjunction is equivalent to . In two cases we prove that the related satisfiability problem is [NP]{}-complete, whereas in the remaining cases we obtain [NP]{}-completeness under the additional assumption that the number of choice terms is constant.
**Keywords:** Decidability; NP-completeness; choice; axioms of choice consistency; .
author:
- 'Domenico Cantone[^1]$\:$, Alfio Giarlotta[^2]$\:$, Stephen Watson[^3]'
title: |
**The satisfiability problem for Boolean set theory\
with a choice correspondence[^4]\
**
---
Introduction {#SECT:intro}
============
In this paper we examine the decidability of the satisfiability problem connected to *rational choice theory*, which is a framework to model social and economic behavior. A choice on a set $U$ of alternatives is a correspondence $B \mapsto c(B)$ associating to “feasible menus" $B \subseteq U$ nonempty “choice sets" $c(B) \subseteq B$. This choice can be either *total* (or *full*) – i.e, defined for all nonempty subsets of the ground set $U$ of alternatives – or *partial* – i.e., defined only for suitable subsets of $U$.
According to the *Theory of Revealed Preferences* pioneered by the economist Paul Samuelson [@Sam38], preferences of consumers can be derived from their purchasing habits: in a nutshell, an agent’s choice behavior is observed, and her underlying preference structure is inferred. The preference revealed by a primitive choice is typically modeled by a binary relation on $U$. The asymmetric part of this relation is informative of a “strict revealed preference" of an item over another one, whereas its symmetric part codifies a “revealed similarity" of items. Then a choice is said to be *rationalizable* when the observed behavior can be univocally retrieved by maximizing the relation of revealed preference.
Since the seminal paper of Samuelson, a lot of attention has been devoted to notions of rationality within the framework of choice theory: see, among the many contributions to the topic, the classical papers [@Hou50; @Arr59; @Ric66; @Han68; @Sen71]. (See also the book [@AleBouMon07] for the analysis of the links among the theories of choice, preference, and utility. For a very recent contribution witnessing the fervent research on the topic, see [@ChaEchShm17].) Classically, the rationality of an observed choice behavior is connected to the satisfaction of suitable *axioms of choice consistency*: these are rules of selections of items within menus, codified by means of sentences of second-order monadic logic, universally quantified over menus. Among the several axioms introduced in the specialized literature, let us recall the following:
$\bullet$ *standard contraction consistency* $(\alpha)$, introduced by Chernoff [@Che54];
$\bullet$ *standard expansion consistency* $(\gamma)$, and *binary expansion consistency* $(\beta)$, both due to Sen [@Sen71];
$\bullet$ *the weak axiom of revealed preference* (), due to Samuelson [@Sam38].\
It is well-known that, under suitable assumptions on the domain, a choice is rationalizable if and only if the two standard axioms of consistency $(\alpha)$ and $(\gamma)$ hold. Further, the rationalizing preference satisfies the property of transitivity if and only if axioms $(\alpha)$ and $(\beta)$ hold if and only if holds: in this case, we speak of a *transitively rationalizable* choice. Section \[SECT:preliminaries\] provides the background to choice theory.
Although the mathematical economics literature on the topic is quite large, there are no contributions which deal with related decision procedures in choice theory. In this paper we start filling this gap. Specifically, we study the satisfiability problem for unquantified formulae of an elementary fragment of set theory (denoted ${\mathsf{BSTC}}$) involving a choice function symbol ${\mathtt{c}}$, the Boolean set operators $\cup$, $\cap$, $\setminus$ and the singleton $\{\cdot\}$, the predicates equality $=$ and inclusion $\subseteq$, and the propositional connectives $\land$, $\lor$, $\neg$, $\implies$, etc. Here we consider the cases in which the interpretation of ${\mathtt{c}}$ is subject to any combination of the axioms of consistency $(\alpha)$ and $(\beta)$, whose conjunction is equivalent to . In two cases we prove that the related satisfiability problem is [NP]{}-complete, whereas in the remaining cases we obtain [NP]{}-completeness only under the additional assumption that the number of choice terms is constant.
By depriving the ${\mathsf{BSTC}}$-language of the choice function symbol ${\mathtt{c}}$, we obtain the fragment (here denoted ${\mathsf{BSTC}}^{-}$) whose decidability was known since the birth of *Computable Set Theory* in the late 70’s. In Section \[appendixDecProc\] we rediscover such result as a by-product of the solution to the satisfiability problem of ${\mathsf{BSTC}}$ under the -semantics: the latter is based on a novel term-oriented non-clausal approach. The reader can find extensive information on Computable Set Theory in the monographs [@CanFerOmo89a; @CanOmoPol01; @SchCanOmo11; @CanUrs17].
For our purposes, it will be relevant to solve the following *lifting problem*: Given a partial choice satisfying some axioms of consistency, can we suitably characterize whether it is extendable to a total choice satisfying the same axioms? The lifting problem for the various combinations of axioms $(\alpha)$ and $(\beta)$ is addressed in depth in Section \[SECT:liftings\]. In particular, in the case of finite choice correspondences, our characterizations turn out to be effective and, with only one exception, expressible in the same ${\mathsf{BSTC}}$-language. This facilitates the design of effective procedures for the solution of the satisfiability problems of our concern. The syntax and semantics of the ${\mathsf{BSTC}}$-language, as well as the solutions of the satisfiability problem for ${\mathsf{BSTC}}$-formulae under the various combinations of axioms $(\alpha)$ and $(\beta)$ are presented in Section \[SECT:satProb\]. Finally, in Section \[SECT:Conclusions\], we draw our conclusions and hint at future developments.
Preliminaries on choice theory {#SECT:preliminaries}
==============================
Hereafter, we fix a nonempty set $U$ (the “universe"). Let ${\mathrm{Pow}}(U)$ be the family of all subsets of $U$, and ${{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$ the subfamily ${\mathrm{Pow}}(U) \setminus \{\emptyset\}$. The next definition collects some basic notions in choice theory.
\[DEF:preliminary deff on choice\] Let $\Omega \subseteq {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$ be nonempty. A map $f \colon \Omega \to {\mathrm{Pow}}(U)$ is *contractive* if $f(B) \subseteq B$ for each $B \in \Omega$. A *choice correspondence* on $U$ is a contractive map that is never empty-valued, i.e., $$c \colon \Omega \to {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}\quad \hbox{such that} \quad c(B) \subseteq B \quad \hbox{for each} \;\; B \in \Omega\,.$$ In this paper, we denote a choice correspondence on $U$ by $c \colon \Omega \rightrightarrows U$, and simply refer to it as a *choice*. The family $\Omega$ is the *choice domain* of $c$, sets in $\Omega$ are *(feasible) menus*, and elements of a menu are *items*. Further, we say that $c \colon \Omega \rightrightarrows U$ is *total* (or *full*) if $\Omega = {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$, and *partial* otherwise. The *rejection map* associated to $c$ is the contractive function ${\overline{c}}\colon \Omega \to {\mathrm{Pow}}(U)$ defined by ${\overline{c}}(B) := B \setminus c(B)$ for all $B \in \Omega$.
Given a choice $c\colon \Omega \rightrightarrows U$, the *choice set* $c(B)$ of a menu $B$ collects the elements of $B$ that are deemed selectable by an economic agent. Thus, in case $c(B)$ contains more than one element, the selection of a single element of $B$ is deferred to a later time, usually with a different procedure (according to additional information or “subjective randomization", e.g., flipping a coin). Notice that the rejection map associated to a choice may fail to be a choice, since the rejection set of some menu can be empty.
The next definition recalls the classical notion of a rationalizable choice.
\[DEF:rationalizable choice\] A choice $c \colon \Omega \rightrightarrows U$ is *rationalizable* (or *binary*) if there exists a binary relation $\precsim$ on $U$ such that the equality[^5] $c(B) = \max_{\precsim} B$ holds for all menus $B \in \Omega$.
The revealed preference theory approach postulates that preferences can be derived from choices. The preference revealed by a primitive choice is modeled by a suitable binary relation on the set of alternatives. Then a choice is rationalizable whenever the observed behavior can be fully explained (i.e., retrieved) by constructing a binary relation of revealed preference.
The rationalizability of choice is traditionally connected to the satisfaction of suitable *axioms of choice consistency*. These axioms codify rules of coherent behavior of an economic agent. Among the several axioms that are considered in the literature, the following are relevant to our analysis (a universal quantification on all the involved menus is implicit):
--------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------- --
**axiom $(\alpha)$** \[*standard contraction consistency*\]: $A \subseteq B \;\; \Longrightarrow \;\; A \cap c(B) \subseteq c(A)$
\[.1cm\] **axiom $(\gamma)$** \[*standard expansion consistency*\]: $c(A) \cap c(B) \subseteq c(A \cup B)$
\[.1cm\] **axiom $(\beta)$** \[*symmetric expansion consistency*\]: $\big(A \subseteq B \: \wedge \: c(A) \cap c(B) \neq \emptyset \big) \;\; \Longrightarrow \;\; c(A) \subseteq c(B)$
\[.1cm\] **axiom $(\rho)$** \[*standard replacement consistency*\]: $c(A) \setminus c(A \cup B) \neq \emptyset \;\; \Longrightarrow \;\; B \cap c(A \cup B) \neq \emptyset$
\[.1cm\] **** \[*weak axiom of revealed preference*\]: $\big(A \subseteq B \: \wedge \: A \cap c(B) \neq \emptyset \big) \;\; \Longrightarrow \;\; c(A) = A \cap c(B)$.
--------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------- --
\
Axiom $(\alpha)$ was studied by Chernoff [@Che54], whereas axioms $(\gamma)$ and $(\beta)$ are due to Sen [@Sen71]. was introduced by Samuelson in [@Sam38]. Axiom $(\rho)$ has been recently introduced in [@CanGiaGreWat16], in connection to the transitive structure of the relation of revealed preference.
Upon reformulating these properties in terms of items, their semantics becomes clear. Chernoff’s axiom $(\alpha)$ states that any item selected from a menu $B$ is still selected from any submenu $A \subseteq B$ containing it. Sen’s axiom $(\gamma)$ says that any item selected from two menus $A$ and $B$ is also selected from the menu $A \cup B$ (if feasible). The expansion axiom $(\beta)$ can be equivalently written as follows: if $A \subseteq B$, $x,y \in c(A)$ and $y \in c(B)$, then $x \in c(B)$. In this form, $(\beta)$ says that if two items are selected from a menu $A$, then they are simultaneously either selected or rejected in any larger menu $B$. Axiom $(\rho)$ can be equivalently written as follows: if $y \in c(B) \setminus c(B \cup \{x\})$, then $x \in c(B \cup \{x\})$. In this form, $(\rho)$ says that if an item $y$ is selected from a menu $B$ but not from the larger menu $B \cup \{x\}$, then the new item $x$ is selected from $B \cup \{x\}$. summarizes features of contraction and expansion consistency in a single – and rather strong, despite its name – axiom, in fact it is equivalent to the conjunction of $(\alpha)$ and $(\beta)$ [@Sen71].
Liftings {#SECT:liftings}
========
In this section we examine the “lifting problem": this corresponds to finding necessary and sufficient conditions such that a partial choice satisfying some axioms of consistency can be extended to a total choice satisfying the same axioms. We shall exploit such conditions in the decision results to be presented in Section \[SECT:satProb\]. The next definition makes the notion of lifting formal.
\[DEF:lifting\] Let $c \colon \Omega \rightrightarrows U$ be a choice. Given a nonempty set $\F$ of sentences of second-order monadic logic, we say that $c$ has the $\F$*-lifting property* if there is a total choice ${c^{\hbox{\tiny{+}}}}\colon {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}\rightrightarrows U$ extending $c$ (i.e., ${c^{\hbox{\tiny{+}}}}{\raisebox{-.5ex}{$|$}_{\Omega}} = c$) and satisfying all formulae in $\F$. In this case, ${c^{\hbox{\tiny{+}}}}$ is called an $\F$*-lifting* of $c$. (Of course, we are interested in cases such that $\F$ is a family of axioms of choice consistency.) Whenever $\F$ is a single formula, we simplify notation and write, e.g., $(\alpha)$-lifting, $\textsf{WARP}$-lifting, etc. Similarly, we say that $c$ has the *rational lifting property* if there is a total choice ${c^{\hbox{\tiny{+}}}}$ that is rational and extends $c$.
Notice that whenever $\F$ is a nonempty set of axioms of choice consistency (which are formulae in prenex normal form where all quantifiers are universal), if a choice has the $\F$-lifting property, then it automatically satisfies all axioms in $\F$. The same reasoning applies for the rational lifting property, since it is based on the existence of a binary relation of revealed preference that is fully informative of the choice.
On the other hand, it may happen that a partial choice satisfies some axioms in $\F$ but there is no lifting to a total choice satisfying the same axioms. The next examples exhibit two instances of this kind. (To simplify notation, we underline all items that are selected within a menu: for instance $\underline{x}\,y$ and $\underline{x}\,y\,\underline{z}$ stand for, respectively, $c(\{x,y\})=\{x\}$ and $c(\{x,y,z\})=\{x,z\}$. Obviously, we always have $\underline{x}$ for any $\{x\} \in \Omega$, so we can safely omit defining $c$ for singletons.)
\[EX:no lifting 1\] Let $U= \{x,y,z\}$ and $\Omega = \{B \subseteq U : 1 \leq \vert B \vert \leq 2\}$. Define a partial choice $c \colon \Omega \rightrightarrows U$ by $\underline{x}\,y$, $\underline{y}\,z$, and $x\,\underline{z}$. This choice is rationalizable by the (cyclic) preference $\precsim$ defined by $x \prec y \prec z \prec x$. However, $c$ does not admit any rational lifting to a total choice ${c^{\hbox{\tiny{+}}}}$, since we would have ${c^{\hbox{\tiny{+}}}}(U) = \max_{\precsim} U = \emptyset$.
\[EX:no lifting 2\] Let $U= \{x,y,z,w\}$ and $\Omega = {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}\setminus \{\{x,w\},\{y,z\},\{y,w\},\{z,w\},U\}$. Define a partial choice $c \colon \Omega \rightrightarrows U$ by $
\underline{x}\,\underline{y}\,,\; \underline{x}\,\underline{z}\,,\; x\,\underline{y}\,\underline{z}\,,\;
\underline{x}\,y\,\underline{w}\,,\; \underline{x}\,\underline{z}\,w\,,\; \underline{y}\,z\,\underline{w}\,.
$ One can easily check that $c$ satisfies axiom $(\alpha)$ (but it fails to be rationalizable). On the other hand, $c$ admits no $(\alpha)$-lifting, since ${c^{\hbox{\tiny{+}}}}(U) \neq \emptyset$ violates axiom $(\alpha)$ for any choice ${c^{\hbox{\tiny{+}}}}$ extending $c$ to the full menu $U$.
Lifting of axiom $(\alpha)$ {#SECT:lifting alpha}
---------------------------
In this section we characterize the choices that are $(\alpha)$-liftable. To that end, it is convenient to reformulate axiom $(\alpha)$ in terms of the monotonicity of the rejection map. We need the following preliminary result, whose simple proof is omitted, and a technical definition.
\[lemmaEquiv\] Let $A \subseteq B \subseteq U$. For any pair of sets $A',B' \subseteq U$, we have $\:A \cap B' \subseteq A' \;\Longleftrightarrow \; A \setminus A' \subseteq B \setminus B'\,.$
\[DEF:relativization\] Let $c \colon \Omega \rightrightarrows U$ be a choice. Given a menu $A \in {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$, the *relativized choice domain $\Omega_{A}$ w.r.t. $A$* is the collection of all submenus of $A$, that is, $B \in \Omega$ such that $B \subseteq A$; in symbols, $$\label{relativizedDomain}
\Omega_{A} {\coloneqq}\{B \in \Omega : B \subseteq A\}\,.$$ A set $\mathcal{B} \subseteq \Omega$ of menus is *$\subseteq$-closed w.r.t. $\Omega$* if $B \in \mathcal{B}$, for every $B \in \Omega$ such that $B \subseteq \bigcup \mathcal{B}$.
In view of Lemma \[lemmaEquiv\], axiom $(\alpha)$ can be equivalently rewritten as follows: $$\label{equivAlpha}
A \subseteq B \quad \Longrightarrow \quad {\overline{c}}(A) \subseteq {\overline{c}}(B)\,.$$ In this form, axiom $(\alpha)$ just asserts that enlarging the set of alternatives may only cause the set of neglected members to grow. As announced, we have:
\[THM:lifting alpha\] A partial choice $c \colon \Omega \rightrightarrows U$ has the $(\alpha)$-lifting property if and only if the following two conditions hold:
1. \[b\] $A \subseteq B \;\; \Longrightarrow \;\; A \cap c(B) \subseteq c(A)$, for all $A,B \in \Omega\,$;
2. \[c\] $\bigcup \mathcal{B} \setminus \bigcup_{B \in \mathcal{B}} \overline{c}(B) \neq \emptyset$, for every $\emptyset \neq \mathcal{B} \subseteq \Omega$ such that $\mathcal{B}$ is $\subseteq$-closed w.r.t. $\Omega\,$.
For necessity, assume that $c \colon \Omega \rightrightarrows U$ can be extended to a total choice ${c^{\hbox{\tiny{+}}}}$ on $U$ satisfying axiom $(\alpha)$, and let ${\overline{c}}$ be the associated rejection map of $c$. Since $c = {c^{\hbox{\tiny{+}}}}{\raisebox{-.5ex}{$|$}_{\Omega}}$, condition \[b\] follows immediately from axiom $(\alpha)$ for ${c^{\hbox{\tiny{+}}}}$. To prove that $c$ satisfies condition \[c\] as well, let $\mathcal{B}$ be a nonempty $\subseteq$-closed subset of $\Omega$. By the equivalent formulation (\[equivAlpha\]) of axiom $(\alpha)$, we obtain ${\overline{{c^{\hbox{\tiny{+}}}}}}(B) \subseteq {\overline{{c^{\hbox{\tiny{+}}}}}}(\bigcup \mathcal{B})$ for every $B \in \mathcal{B}$, where ${\overline{{c^{\hbox{\tiny{+}}}}}}$ is the associated rejection map of ${\overline{c}}$. Hence $$\bigcup_{B \in \mathcal{B}} {\overline{{c^{\hbox{\tiny{+}}}}}}(B) \; \subseteq \; {\overline{{c^{\hbox{\tiny{+}}}}}}\big(\bigcup \mathcal{B}\big) \; = \; \bigcup \mathcal{B} \setminus {c^{\hbox{\tiny{+}}}}\big(\bigcup \mathcal{B}\big)$$ holds. It follows that $$\emptyset \; \neq \; {c^{\hbox{\tiny{+}}}}\big(\bigcup \mathcal{B}\big) \; \subseteq \; \bigcup \mathcal{B} \setminus \bigcup\nolimits_{B \in \mathcal{B}} {\overline{{c^{\hbox{\tiny{+}}}}}}(B) \; = \; \bigcup \mathcal{B} \setminus \bigcup\nolimits_{B \in \mathcal{B}} {\overline{c}}(B)\,,$$ thus showing that \[c\] holds. This completes the proof of necessity.
For sufficiency, assume that \[b\] and \[c\] hold for the choice $c \colon \Omega \rightrightarrows U$. For each $A \in {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$, define $${c^{\hbox{\tiny{+}}}}(A) \: {\coloneqq}\: A \setminus \bigcup\nolimits_{B \in \Omega_{A}} {\overline{c}}(B)\,,$$ where we recall that $\Omega_{C}$ is the relativized choice domain w.r.t. to $C$ (cf. Definition \[DEF:relativization\]). In what follows we prove that the map ${c^{\hbox{\tiny{+}}}}\colon {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}\to {\mathrm{Pow}}(U)$ is a well-defined choice, which extends $c$ and satisfies axiom $(\alpha)$.
Since the map ${c^{\hbox{\tiny{+}}}}$ is obviously contractive by definition, to prove that it is a well-defined choice it suffices to show that it is never empty-valued. Toward a contradiction, assume that ${c^{\hbox{\tiny{+}}}}(A) = \emptyset$ for some $A \in {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$. The definition of ${c^{\hbox{\tiny{+}}}}$ readily yields $
A \; = \; \bigcup\nolimits_{B \in \Omega_{A}} {\overline{c}}(B) \; \subseteq \; \bigcup \Omega_{A} \; \subseteq \; A\,,
$ which implies $\Omega_{A} \neq \emptyset$ and $\bigcup_{B \in \Omega_{A}} {\overline{c}}(B) = \bigcup \Omega_{A}$, since $\Omega_{A}$ is $\subseteq$-closed w.r.t. $\Omega$. However, this contradicts \[c\]. Thus, ${c^{\hbox{\tiny{+}}}}\colon {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}\rightrightarrows U$ is a well-defined (total) choice.
Next, we show that ${c^{\hbox{\tiny{+}}}}$ extends $c$. Let $B \in \Omega$. Plainly, we have $B \in \Omega_{B}$ and $A \subseteq B$ for each $A \in \Omega_{B}$. Property \[b\] yields $A \cap c(B) \subseteq c(A)$, and so ${\overline{c}}(A) \subseteq {\overline{c}}(B)$ by Lemma \[lemmaEquiv\]. Thus, we obtain $${c^{\hbox{\tiny{+}}}}(B) \: = \: B \setminus \bigcup\nolimits_{A \in \Omega_{B}} {\overline{c}}(A) \: = \: B \setminus {\overline{c}}(B) \: = \: c(B)\,,$$ which proves the claim.
Finally, we show that ${c^{\hbox{\tiny{+}}}}$ satisfies $(\alpha)$. Let $\emptyset \neq B \subseteq C \subseteq U$. Since $B \subseteq C$ and $\Omega_{B} \subseteq \Omega_{C}$, we have:\
$$B \cap {c^{\hbox{\tiny{+}}}}(C) = \; B \cap \left(C \setminus \bigcup\nolimits_{A \in \Omega_{C}} {\overline{c}}(A)\right) = \; B \setminus \bigcup\nolimits_{A \in \Omega_{C}} {\overline{c}}(A)
\subseteq \; B \setminus \bigcup\nolimits_{A \in \Omega_{B}}
= \; {c^{\hbox{\tiny{+}}}}(B)\,.$$ This proves that axiom $(\alpha)$ holds for ${c^{\hbox{\tiny{+}}}}$, and the proof is complete.
Lifting of axiom $(\beta)$ {#SECT:lifting beta}
--------------------------
Here we prove that any partial choice satisfying axiom $(\beta)$ can be always lifted to a total choice still satisfying axiom $(\beta)$. To that end, we need the notion of the *intersection graph* associated to a family of sets $\mathcal{S}$: this is the undirected graph whose nodes are the sets belonging to $\mathcal{S}$, and whose edges are the pairs of distinct intersecting sets $B,B' \in \mathcal{S}$ (i.e., such that $B \cap B' \neq \emptyset$).
\[THM:lifting beta\] A partial choice has the $(\beta)$-lifting property if and only it satisfies axiom $(\beta)$.
Clearly, axiom $(\beta)$ holds for any choice that admits an extension to a total choice satisfying $(\beta)$. Thus, it suffices to prove that any choice $c \colon \Omega \rightrightarrows U$ satisfying $(\beta)$ has the $(\beta)$-lifting property. For every $A \in {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$, pick an element $u_{A} \in A$, subject only to the condition that $u_{A} \in c(A)$ whenever $A \in \Omega$. If $u_{A} \in \bigcup c[\Omega_{A}]$ (where $\Omega_{A} {\coloneqq}\{ B \in \Omega : B \subseteq A\}$), then let $\mathcal{C}_{A} \subseteq c[\Omega_{A}]$ be the connected component of the intersection graph associated to the family $c[\Omega_{A}]$ such that $u_{A} \in \bigcup \mathcal{C}_{A}$. Then, for $A \in {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$, set $${c^{\hbox{\tiny{+}}}}(A) {\coloneqq}\begin{cases}
\bigcup \mathcal{C}_{A} & \text{if } u_{A} \in \bigcup c[\Omega_{A}]\\
\{u_{A}\} & \text{otherwise}.
\end{cases}$$ By definition, ${c^{\hbox{\tiny{+}}}}$ is a total contractive map on ${{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$ that is never empty-valued. In addition, if $A \in \Omega$, then ${c^{\hbox{\tiny{+}}}}(A) = \bigcup \mathcal{C}_{A} = c(A)$. It follows that ${c^{\hbox{\tiny{+}}}}$ is a well-defined total choice that extends $c$.
To complete the proof, we only need to show that ${c^{\hbox{\tiny{+}}}}$ satisfies $(\beta)$. Let $D,E \in {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$ be such that $D \subseteq E$ and ${c^{\hbox{\tiny{+}}}}(D) \cap {c^{\hbox{\tiny{+}}}}(E) \neq \emptyset$. If $|{c^{\hbox{\tiny{+}}}}(D)| = 1$, then plainly ${c^{\hbox{\tiny{+}}}}(D) \subseteq {c^{\hbox{\tiny{+}}}}(E)$. On the other hand, if $|{c^{\hbox{\tiny{+}}}}(D)| > 1$, then ${c^{\hbox{\tiny{+}}}}(D) = \mathcal{C}_{D}$, hence ${c^{\hbox{\tiny{+}}}}(E) = \mathcal{C}_{E}$ and $\mathcal{C}_{D} \subseteq \mathcal{C}_{E}$. Thus, we obtain again ${c^{\hbox{\tiny{+}}}}(D) \subseteq {c^{\hbox{\tiny{+}}}}(E)$, as claimed.
Lifting of {#SECT:lifting of WARP}
-----------
Finally, we characterize choices that have the -lifting property. This characterization will be obtained in terms of the existence of a suitable Noetherian total preorder on the collection of the Euler’s regions of the union of the choice domain with its image under the given choice. (Recall that a *preorder* is a binary relation that is reflexive and transitive. Further, a relation $R$ on $X$ is *Noetherian* if the converse relation $R^{-1}$ is well-founded, i.e., if every nonempty subset of $X$ has an $R$-maximal element.)
Thus, let $c \colon \Omega \rightrightarrows U$ be a partial choice. Denote by $\mathcal{E}$ the Euler’s diagram of the family $
\Omega^{^{+}} {\coloneqq}\Omega \cup c[\Omega]\,,
$ namely the partition\
$
\mathcal{E} {\coloneqq}\left\{\bigcap \Gamma \:\setminus\: \bigcup (\Omega^{^{+}} \setminus \Gamma) : \emptyset \neq \Gamma \subseteq \Omega^{^{+}} \right\} \setminus \{\emptyset\}
$
of $\bigcup \Omega^{^{+}}$ formed by all the nonempty sets of the form $\bigcap \Gamma \:\setminus\: \bigcup (\Omega^{^{+}} \setminus \Gamma)$, for $\emptyset \neq \Gamma \subseteq \Omega^{^{+}}$. Further, for each $A \in {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$, denote by ${\mathsf{env}_{\mathcal{E}}(A)}$ the *envelope of $A$ in $\mathcal{E}$*, namely, the collection of regions in $\mathcal{E}$ intersecting $A$; formally, $
{\mathsf{env}_{\mathcal{E}}(A)} {\coloneqq}\{E \in \mathcal{E} : E \cap A \neq \emptyset\}\,.
$ Observe that, for each $B \in \Omega$, we have ${\mathsf{env}_{\mathcal{E}}(B)} = \{E \in \mathcal{E} : E \subseteq B\}$. It turns out that the choice $c$ can be lifted to a total choice satisfying if and only if there exists a suitable Noetherian total preorder $\lesssim$ on $\mathcal{E}$ such that\
$
E \subseteq B \text{ and } E' \subseteq c(B) \text{ ~~~(for some $B \in \Omega$)} \qquad \Longrightarrow \qquad E \lesssim E' \,.
$
More precisely, we have:
\[THM:lifting WARP\] A partial choice $c \colon \Omega \rightrightarrows U$ has the -lifting property if and only if there exists a total Noetherian preorder $\lesssim$ on the collection $\mathcal{E}$ of Euler’s regions of $\Omega \cup c[\Omega]$ such that, for all $B \in \Omega$ and $E,E' \in \mathcal{E}$, the following conditions hold:
1. \[aTheorem3\] if $E \subseteq B$ and $E' \subseteq c(B)$, then $E \lesssim E'$;
2. \[bTheorem3\] if $E$ is $\lesssim$-maximal in ${\mathsf{env}_{\mathcal{E}}(B)}$, then $E \subseteq c(B)$.
*(Necessity)* Assume that $c$ can be extended to a total choice ${c^{\hbox{\tiny{+}}}}$ on $U$ satisfying . We shall show that there exists a total Noetherian preorder $\lesssim$ on $\mathcal{E}$ satisfying conditions \[aTheorem3\] and \[bTheorem3\] of the theorem.
For $E,E' \in \mathcal{E}$, set $$\label{defLesssim}
E \lesssim E' \qquad \overset{\text{\it\tiny Def}}{\Longleftrightarrow} \qquad {c^{\hbox{\tiny{+}}}}(E\cup E') \cap E' \neq \emptyset\,.$$ In what follows we show that (i) $\lesssim$ is a total preorder, (ii) $\lesssim$ is Noetherian, (iii) $\lesssim$ satisfies condition \[aTheorem3\], and (iv) $\lesssim$ satisfies condition \[bTheorem3\].
\(i) To prove that $\lesssim$ is a total preorder, observe that $\lesssim$ is reflexive and total by construction. For transitivity, let $E,E',E'' \in \mathcal{E}$ be such that $E \lesssim E'$ and $E' \lesssim E''$ hold, i.e., $$\begin{aligned}
{c^{\hbox{\tiny{+}}}}(E \cup E') \cap E' &\neq \emptyset \label{first_eq}\\
{c^{\hbox{\tiny{+}}}}(E' \cup E'') \cap E'' &\neq \emptyset \,.\label{second_eq}\end{aligned}$$ We need to show that ${c^{\hbox{\tiny{+}}}}(E \cup E'') \cap E'' \neq \emptyset$ holds, too. Plainly, either ${c^{\hbox{\tiny{+}}}}(E \cup E' \cup E'') \cap E'' \neq \emptyset$, or ${c^{\hbox{\tiny{+}}}}(E \cup E' \cup E'') \cap (E \cup E') \neq \emptyset$ holds. In the former case, we have $$\begin{aligned}
\emptyset &\neq {c^{\hbox{\tiny{+}}}}(E \cup E' \cup E'') \cap E''\\
&= {c^{\hbox{\tiny{+}}}}(E \cup E' \cup E'') \cap (E \cup E'') \cap E'' \\
&= {c^{\hbox{\tiny{+}}}}(E \cup E'') \cap E'' & \text{[by \textsf{WARP}]}\,.\end{aligned}$$ In the latter case, and inequality (\[first\_eq\]) yield $${c^{\hbox{\tiny{+}}}}(E \cup E' \cup E'') \cap (E \cup E') \cap E' = {c^{\hbox{\tiny{+}}}}(E \cup E') \cap E' \neq \emptyset\,,$$ so that $$\label{tempDeriv}
{c^{\hbox{\tiny{+}}}}(E \cup E' \cup E'') \cap E' \neq \emptyset\,.$$ Thus, using and the two inequalities (\[tempDeriv\]) and (\[second\_eq\]), we obtain $${c^{\hbox{\tiny{+}}}}(E \cup E' \cup E'') \cap (E' \cup E'') \cap E'' = {c^{\hbox{\tiny{+}}}}(E' \cup E'') \cap E'' \neq \emptyset\,,$$ hence ${c^{\hbox{\tiny{+}}}}(E \cup E' \cup E'') \cap E'' \neq \emptyset$, again concluding that ${c^{\hbox{\tiny{+}}}}(E \cup E'') \cap E'' \neq \emptyset$. Thus, in any case, $E \lesssim E''$ holds, as claimed.
\(ii) To prove that $\lesssim$ is Noetherian, let $\emptyset \neq \mathcal{A} \subseteq \mathcal{E}$. Since $\emptyset \neq {c^{\hbox{\tiny{+}}}}(\bigcup \mathcal{A}) \subseteq \bigcup \mathcal{A}$, there exists an $\overline{E} \in \mathcal{A}$ such that ${c^{\hbox{\tiny{+}}}}(\bigcup \mathcal{A}) \cap \overline{E} \neq \emptyset$. Thus, it is enough to prove that $E \lesssim \overline{E}$ for each $E \in \mathcal{A}$. Let $E \in \mathcal{E}$. We have $$\begin{aligned}
\emptyset &\neq {c^{\hbox{\tiny{+}}}}\left(\bigcup \mathcal{A}\right) \cap \overline{E}\\
&= {c^{\hbox{\tiny{+}}}}\left(\bigcup \mathcal{A}\right) \cap (E \cup \overline{E}) \cap \overline{E}\\
&= {c^{\hbox{\tiny{+}}}}(E \cup \overline{E}) \cap \overline{E} & \text{[by \textsf{WARP}]\,,}\end{aligned}$$ i.e., $E \lesssim \overline{E}$. This shows that $\overline{E}$ is a maximum of $\mathcal{A}$, proving that the total preorder $\lesssim$ is Noetherian.
\(iii) Concerning \[aTheorem3\], let $E \subseteq A$ and $E' \subseteq c(A)$. Hence, $E' \subseteq {c^{\hbox{\tiny{+}}}}(A)$. Two applications of yield $$\emptyset \neq {c^{\hbox{\tiny{+}}}}(E') = {c^{\hbox{\tiny{+}}}}(A) \cap E' = {c^{\hbox{\tiny{+}}}}(A) \cap (E \cup E') \cap E' = {c^{\hbox{\tiny{+}}}}(E \cup E') \cap E'\,,$$ and so $E \lesssim E'$ holds, proving that \[aTheorem3\] is satisfied.
\(iv) Finally, we prove that also condition \[bTheorem3\] is satisfied. Let $A \in \Omega$, and let $E$ be $\lesssim$-maximal in ${\mathsf{env}_{\mathcal{E}}(A)}$. We need to show that $E \subseteq c(A)$. Let $E' \subseteq c(A)$. By the $\lesssim$-maximality of $E$, we have $E' \lesssim E$, i.e., by (\[defLesssim\]), ${c^{\hbox{\tiny{+}}}}(E \cup E') \cap E \neq \emptyset$. By , $$c(A) \cap E = {c^{\hbox{\tiny{+}}}}(A) \cap E = {c^{\hbox{\tiny{+}}}}(A) \cap (E \cup E') \cap E = {c^{\hbox{\tiny{+}}}}(E \cup E') \cap E \neq \emptyset\,,$$ which plainly yields $E \subseteq c(A)$. This completes the proof of necessity.
*(Sufficiency)* Let $c \colon \Omega \rightrightarrows U$ be a partial choice and assume that there exist a total Noetherian preorder $\lesssim$ on the collection $\mathcal{E}$ of the Euler’s regions of $\Omega \cup c[\Omega]$ such that conditions \[aTheorem3\] and \[bTheorem3\] hold.
For every $B \in {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$, set $$\label{defCorrespondence}
\textstyle
{c^{\hbox{\tiny{+}}}}(B) {\coloneqq}\begin{cases}
B \setminus \bigcup \mathcal{E} & \text{if } B \setminus \bigcup \mathcal{E} \neq \emptyset\\
\bigcup \displaystyle\max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(B)}\big) \cap B & \text{otherwise,}
\end{cases}
$$ where, for any nonempty $\mathcal{A} \subseteq \mathcal{E}$, $\max_{\lesssim} \mathcal{A}$ stands for the collection of the $\lesssim$-maximal members of $\mathcal{A}$. To complete the proof, we shall show that (i) ${c^{\hbox{\tiny{+}}}}$ is a well-defined total choice on $U$, (ii) ${c^{\hbox{\tiny{+}}}}$ extends $c$, and (iii) ${c^{\hbox{\tiny{+}}}}$ satisfies .
\(i) By definition, ${c^{\hbox{\tiny{+}}}}$ is a contraction on ${{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$. Thus, to prove the claim, it suffices to show that ${c^{\hbox{\tiny{+}}}}(B) \neq \emptyset$, for every $B \in {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$. Let $B \in {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$. If $B \nsubseteq \bigcup \mathcal{E}$, then the result holds trivially. Otherwise, let $\emptyset \neq B \subseteq \bigcup \mathcal{E}$. Then we have ${c^{\hbox{\tiny{+}}}}(B) = \bigcup \max_{\lesssim}\big({\mathsf{env}_{\mathcal{E}}(B)}\big) \cap B$ and ${\mathsf{env}_{\mathcal{E}}(B)} \neq \emptyset$. Since $\lesssim$ is Noetherian, we obtain $\max_{\lesssim} \mathcal{E}_{B} \neq \emptyset$. Since all members of $\max_{\lesssim}\big({\mathsf{env}_{\mathcal{E}}(B)}\big)$ intersect $B$, so does their union, proving that ${c^{\hbox{\tiny{+}}}}(B) \neq \emptyset$ holds in all cases.
\(ii) To show that ${c^{\hbox{\tiny{+}}}}$ extends $c$, let $A \in \Omega$. Since $A \subseteq \bigcup \mathcal{E}$, we have ${c^{\hbox{\tiny{+}}}}(A) = \bigcup \max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(A)}\big) \cap A$. In fact, since $A \in \Omega \cup c[\Omega]$, then ${c^{\hbox{\tiny{+}}}}(A) = \bigcup \max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(A)}\big)$. Thus, in order to prove that ${c^{\hbox{\tiny{+}}}}(A) = c(A)$, we have to show that $\bigcup \max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(A)}\big) = c(A)$. It is then enough to prove that, for $E \in \mathcal{E}$, $$\label{conditionMax}
E \subseteq c(A) \quad \Longleftrightarrow \quad E \in \max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(A)}\big)\,.$$ Let $E \in \mathcal{E}$ be such that $E \subseteq c(A)$ and let $E' \in {\mathsf{env}_{\mathcal{E}}(A)}$, so that $E' \subseteq A$ and $E' \in \mathcal{E}$. Thus, by \[aTheorem3\], $E' \lesssim E$. The arbitrarity of $E'$ yields that $E \in \max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(A)}\big)$, proving the right-implication of (\[conditionMax\]). The converse implication follows at once by condition \[bTheorem3\].
\(iii) Finally, we show that ${c^{\hbox{\tiny{+}}}}$ satisfies . Let $A,B \in \Omega$ be such that $A \subseteq B$ and $A \cap c(B) \neq \emptyset$. To prove the claim, we need to show that the equality $$\label{goal}
{c^{\hbox{\tiny{+}}}}(A) = A \cap {c^{\hbox{\tiny{+}}}}(B)$$ holds. If $A \setminus \bigcup \mathcal{E} \neq \emptyset$, then $B \setminus \bigcup \mathcal{E} \neq \emptyset$. Hence, by (\[defCorrespondence\]), equation (\[goal\]) becomes $$\textstyle
A \setminus \bigcup \mathcal{E} = A \cap (B \setminus \bigcup \mathcal{E})\,,$$ which plainly holds, since $A \subseteq B$. On the other hand, suppose $A \subseteq \bigcup \mathcal{E}$. It follows that $B \subseteq \bigcup \mathcal{E}$, since otherwise (\[defCorrespondence\]) would yield ${c^{\hbox{\tiny{+}}}}(B) = B \setminus \bigcup \mathcal{E}$, hence $A \cap {c^{\hbox{\tiny{+}}}}(B) = \emptyset$, a contradiction. Thus, we obtain $$\label{cAcB}
{c^{\hbox{\tiny{+}}}}(A) = \textstyle\bigcup \displaystyle\max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(A)}\big) \cap A
\qquad \text{and} \qquad
{c^{\hbox{\tiny{+}}}}(B) = \textstyle\bigcup \displaystyle\max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(B)}\big) \cap B\,,$$ so that to prove (\[goal\]) it suffices to show that $$\label{lastGoal}
\textstyle\bigcup \displaystyle\max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(A)}\big) \cap A ~=~
\textstyle\bigcup \displaystyle\max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(B)}\big) \cap A\,.$$ From our hypothesis $A \cap {c^{\hbox{\tiny{+}}}}(B) \neq \emptyset$ and (\[cAcB\]), it follows that $ \max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(B)}\big) \cap {\mathsf{env}_{\mathcal{E}}(A)} \neq \emptyset$. Let $E \in \max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(B)}\big) \cap {\mathsf{env}_{\mathcal{E}}(A)}$. Since ${\mathsf{env}_{\mathcal{E}}(A)} \subseteq {\mathsf{env}_{\mathcal{E}}(B)}$, we have $E \in \max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(A)}\big)$ too. But then, for $E' \in \max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(B)}\big) \cap {\mathsf{env}_{\mathcal{E}}(A)}$, we have $E \lesssim E'$, and therefore $E' \in \max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(A)}\big)$, so that $\max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(B)}\big) \cap {\mathsf{env}_{\mathcal{E}}(A)} \subseteq \max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(A)}\big)$. In addition, for $E'' \in \max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(A)}\big)$, we have $E \lesssim E''$, and therefore $E'' \in \max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(B)}\big) \cap {\mathsf{env}_{\mathcal{E}}(A)}$, so that $\max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(A)}\big) \subseteq \max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(B)}\big) \cap {\mathsf{env}_{\mathcal{E}}(A)}$. The last two set inclusions yield $\max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(B)}\big) \cap {\mathsf{env}_{\mathcal{E}}(A)} = \max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(A)}\big)$, and therefore $$\textstyle\bigcup \displaystyle\max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(B)}\big) \cap A
= \textstyle\bigcup (\displaystyle\max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(B)}\big) \cap {\mathsf{env}_{\mathcal{E}}(A)})
= \textstyle\bigcup \displaystyle\max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(A)}\big)
= \textstyle\bigcup \displaystyle\max_{\lesssim} \big({\mathsf{env}_{\mathcal{E}}(A)}\big) \cap A\,,$$ thus proving (\[lastGoal\]), and in turn completing the proof that ${c^{\hbox{\tiny{+}}}}$ satisfies .
The satisfiability problem in presence of a choice correspondence {#SECT:satProb}
=================================================================
We are now ready to define the syntax and semantics of the Boolean set-theoretic language extended with a choice correspondence, denoted by ${\mathsf{BSTC}}$, of which we shall study the satisfiability problem.
Syntax of ${\mathsf{BSTC}}$
---------------------------
The language ${\mathsf{BSTC}}$ involves
- denumerable collections $\mathcal{V}_{0}$ and $\mathcal{V}_{1}$ of *individual* and *set* variables, respectively;
- the constant $\varnothing$ (empty set);
- operation symbols: $\cdot\cup\cdot$, $\cdot\cap\cdot$, $\cdot\setminus\cdot$ , $\{\cdot\}$, ${\mathtt{c}}(\cdot)$ (choice map);
- predicate symbols: $\cdot=\cdot$, $\cdot\subseteq\cdot$, $\cdot\in\cdot$.
*Set terms* of ${\mathsf{BSTC}}$ are recursively defined as follows:
- set variables and the constant $\varnothing$ are set terms;
- if $T, T_{1},T_{2}$ are set terms and $x$ is an individidual variable, then $
T_{1} \cup T_{2},~~ T_{1} \cap T_{2},~~ T_{1} \setminus T_{2},~~ {\mathtt{c}}(T),~~ \{x\}
$ are set terms.
The *atomic formulae* (or *atoms*) of ${\mathsf{BSTC}}$ have one of the following two forms $
T_{1} = T_{2}, ~~ T_{1} \subseteq T_{2}, $ where $T_{1},T_{2}$ are set terms. Atoms and their negations are called *literals*.\
Finally, *${\mathsf{BSTC}}$-formulae* are propositional combinations of ${\mathsf{BSTC}}$-atoms by means of the usual logical connectives $\land$, $\lor$, $\lnot$, $\implies$, $\iff$.
We regard $\{x_{1},\ldots,x_{k}\}$ as a shorthand for the set term $\{x_{1}\} \cup \ldots \cup \{x_{k}\}$. Likewise, $x \in T$ and $x = y$ are regarded as shorthands for $\{x\} \subseteq T$ and $\{x\} = \{y\}$, respectively.
*Choice terms* are ${\mathsf{BSTC}}$-terms of type ${\mathtt{c}}(T)$, whereas *choice-free terms* are ${\mathsf{BSTC}}$-terms which do not involve the choice map ${\mathtt{c}}$ (at any level of nesting). We refer to ${\mathsf{BSTC}}$-formulae containing only choice-free terms as ${\mathsf{BSTC}}^{-}$-formulae.[^6]
Semantics of ${\mathsf{BSTC}}$
------------------------------
We first describe the *unrestricted semantics of ${\mathsf{BSTC}}$*, when the choice operator is not required to satisfy any particular consistency axiom.
A *set assignment* is a pair ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}=(U,M)$, where $U$ is any nonempty collection of objects, called the *domain* or *universe* of ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}$, and $M$ is an assignment over the variables of ${\mathsf{BSTC}}$ such that
- ${x^{\scriptscriptstyle M}} \in U$, for each individual variable $x \in \mathcal{V}_{0}$;
- ${\varnothing^{\scriptscriptstyle M}} {\coloneqq}\emptyset$;
- ${X^{\scriptscriptstyle M}} \subseteq U$, for each set variable $X \in \mathcal{V}_{1}$;
- ${{\mathtt{c}}^{\scriptscriptstyle M}}$ is a total choice correspondence over $U$.
Then, recursively, we put
- ${(T_{1} \otimes T_{2})^{\scriptscriptstyle M}} {\coloneqq}{T^{\scriptscriptstyle M}}_{1} \otimes {T^{\scriptscriptstyle M}}_{2}$, where $T_{1},T_{2}$ are set terms and $\otimes \in \{\cup,\cap,\setminus\}$;
- ${\{x\}^{\scriptscriptstyle M}} {\coloneqq}\{{x^{\scriptscriptstyle M}}\}$, where $x$ is an individual variable;
- ${({\mathtt{c}}(T))^{\scriptscriptstyle M}} {\coloneqq}{{\mathtt{c}}^{\scriptscriptstyle M}}({T^{\scriptscriptstyle M}})$, where $T$ is a set term.
Satisfiability of any ${\mathsf{BSTC}}$-formula $\psi$ by ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}$ (written ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}\models \psi$) is defined by putting $$\begin{array}{rclrcr}
{\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}&\models& T_{1} \star T_{2} &\qquad\text{iff}\qquad& {T^{\scriptscriptstyle M}}_{1} \star {T^{\scriptscriptstyle M}}_{2}\,,
\end{array}$$ for ${\mathsf{BSTC}}$-atoms $T_{1} \star T_{2}$ (where $T_{1},T_{2}$ are set terms and $\star \in \{=,\subseteq\}$), and by interpreting logical connectives according to their classical meaning.
For a ${\mathsf{BSTC}}$-formula $\psi$, if ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}\models \psi$ (i.e., ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}$ *satisfies* $\psi$), then ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}$ is said to be a *${\mathsf{BSTC}}$-model for $\psi$*. A ${\mathsf{BSTC}}$-formula is said to be *satisfiable* if it has a ${\mathsf{BSTC}}$-model. Two ${\mathsf{BSTC}}$-formulae $\varphi$ and $\psi$ are *equivalent* if they share exactly the same ${\mathsf{BSTC}}$-models; they are *equisatisfiable* if one is satisfiable if and only if so is the other (possibly by different ${\mathsf{BSTC}}$-models).
The *satisfiability problem* (or *decision problem*) for ${\mathsf{BSTC}}$ asks for an effective procedure (or *decision procedure*) to establish whether any given ${\mathsf{BSTC}}$-formula is satisfiable or not.
We shall also address the satisfiability problem for ${\mathsf{BSTC}}$ under other semantics: specifically, the *$(\alpha)$-semantics*, the *$(\beta)$-semantics*, and the -*semantics* (whose satisfiability relations are denoted by $\models_{\alpha}$, $\models_{\beta}$, and $\models_{\textsf{WARP}}$, respectively). These differ from the unrestricted semantics in that the interpreted choice map ${{\mathtt{c}}^{\scriptscriptstyle M}}$ is required to satisfy axiom $(\alpha)$ in the first case, axiom $(\beta)$ in the second case, and axioms $(\alpha)$ and $(\beta)$ conjunctively (namely ) in the latter case.
The decision problem for ${\mathsf{BSTC}}$-formulae {#appendixDecProc}
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The satisfiability problem for ${\mathsf{BSTC}}^{-}$- and ${\mathsf{BSTC}}$-formulae under the various semantics are [NP]{}-hard, as the satisfiability problem for propositional logic can readily be reduced to any of them (in linear time). In the cases of $(\alpha)$- and -semantics, we shall prove [NP]{}-completeness only under the additional hypothesis that the number of choice terms is constant, otherwise, in both cases, we have to content ourselves with a [NEXPTIME]{} complexity. As a by-product, it will follow that the satisfiability problem for ${\mathsf{BSTC}}^{-}$ is [NP]{}-complete. On the other hand, we shall prove that the satisfiability problem for ${\mathsf{BSTC}}$-formulae under the unrestricted and the $(\beta)$-semantics can be reduced polynomially to the satisfiability problem for ${\mathsf{BSTC}}^{-}$-formulae, thereby proving their [NP]{}-completeness.
Let $\varphi$ be a ${\mathsf{BSTC}}$-formula, $\mathsf{V}_{0} \subseteq \mathcal{V}_{0}$ and $\mathsf{V}_{1} \subseteq \mathcal{V}_{1}$ the collections of individual and set variables occurring in $\varphi$, respectively, and ${\mathcal{T}_{\varphi}}$ the collection of the set terms occurring in $\varphi$. For convenience, we shall assume that $\varnothing \in {\mathcal{T}_{\varphi}}$. Let also $$\label{choiceLiterals}
{\mathtt{c}}(T_{1}),~\ldots,~{\mathtt{c}}(T_{k})$$ be the distinct choice terms occurring in $\varphi$, with $k \geqslant 0$ (when $k=0$, $\varphi$ is a ${\mathsf{BSTC}}^{-}$-formula).
Without loss of generality, we may assume that $\varphi$ is in *choice-flat form*, namely that all the terms $T_{1},\ldots,T_{k}$ in (\[choiceLiterals\]) are choice-free. In fact, if this were not the case, then, for each choice term ${\mathtt{c}}(T)$ in $\varphi$ occurring inside the scope of a choice symbol and such that $T$ is choice-free, we could replace in $\varphi$ all occurrences of ${\mathtt{c}}(T)$ by a newly introduced variable $X_{T}$ and add the conjunct $X_{T} = {\mathtt{c}}(T)$ to $\varphi$, until no choice term is left which properly contains a choice subterm. It is an easy matter to check that the resulting formula is in choice-flat form, it is equisatisfiable with $\varphi$ (under any of our semantics), and its size is linear in the size of $\varphi$.
Without disrupting satisfiability, we may add to $\varphi$ the following formulae:
*choice conditions*:
: $\varnothing \neq {\mathtt{c}}(T_{i}) \;\land\; {\mathtt{c}}(T_{i}) \subseteq T_{i}$, for $i=1,\ldots,k$;
*single-valuedness conditions*:
: $T_{i} = T_{j} \implies {\mathtt{c}}(T_{i}) = {\mathtt{c}}(T_{j})$, for all distinct $i,j = 1,\ldots,k$,
since they are plainly true in any ${\mathsf{BSTC}}$-assignment. In this case the size of $\varphi$ could have up to a quadratic increase. However, the total number of terms remains unchanged.[^7] For the sake of simplicity, we shall assume that $\varphi$ includes its choice and single-valuedness conditions, and thereby say that it is *complete*.
Notice that the above considerations hold irrespectively of the semantics adopted.
In the sections which follow, we study the satisfiability problem for complete ${\mathsf{BSTC}}$-formulae under the various semantics described earlier. We start our course with the -semantics.
### -semantics {#WARPsemantics}
We first derive some necessary conditions for $\varphi$ to be satisfiable and later prove their sufficiency. Hence, to begin with, let us assume that $\varphi$ is satisfiable under the -semantics and let ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}= (U,M)$ be a model for it. Let ${\mathcal{R}_{\varphi}^{\scriptscriptstyle M}}$ be the Euler’s diagram of ${{\mathcal{T}_{\varphi}}^{\scriptscriptstyle M}} {\coloneqq}\{{T^{\scriptscriptstyle M}} : T \in {\mathcal{T}_{\varphi}}\}$. Notice that, for each region $\rho \in {\mathcal{R}_{\varphi}^{\scriptscriptstyle M}}$ and term $T \in {\mathcal{T}_{\varphi}}$, either $\rho \subseteq {T^{\scriptscriptstyle M}}$ or $\rho \cap {T^{\scriptscriptstyle M}} = \emptyset$. Thus, to each $\rho \in {\mathcal{R}_{\varphi}^{\scriptscriptstyle M}}$, there corresponds a Boolean map $\pi_{\rho} \colon {\mathcal{T}_{\varphi}}\rightarrow \{{\ensuremath{\mbox{$\mathsf{1}$}}\xspace},{\ensuremath{\mbox{$\mathsf{0}$}}\xspace}\}$ over ${\mathcal{T}_{\varphi}}$ (where we have identified the truth values and with ${\ensuremath{\mbox{$\mathsf{1}$}}\xspace}$ and ${\ensuremath{\mbox{$\mathsf{0}$}}\xspace}$, respectively) such that $$\label{defPlaces}
\pi_{\rho}(T) = {\ensuremath{\mbox{$\mathsf{1}$}}\xspace}\iff \rho \subseteq {T^{\scriptscriptstyle M}} \iff \rho \cap {T^{\scriptscriptstyle M}} \neq \emptyset\,,~~~~\text{for } T \in {\mathcal{T}_{\varphi}}\,.$$ Let ${\Pi_{\varphi}^{\scriptscriptstyle M}} {\coloneqq}\{\pi_{\rho} : \rho \in {\mathcal{R}_{\varphi}^{\scriptscriptstyle M}}\}$. Hence, we have:
1. \[zeroPlace\] $\pi(\varnothing) = {\ensuremath{\mbox{$\mathsf{0}$}}\xspace}$, for each $\pi \in {\Pi_{\varphi}^{\scriptscriptstyle M}}$;
2. \[firstPlace\] $\pi(T_{1} \cup T_{2}) = \pi(T_{1}) \lor \pi(T_{2})$, for each map $\pi \in {\Pi_{\varphi}^{\scriptscriptstyle M}}$ and set term $T_{1} \cup T_{2}$ in $\varphi$;
3. \[secondPlace\] $\pi(T_{1} \cap T_{2}) = \pi(T_{1}) \land \pi(T_{2})$, for each map $\pi \in {\Pi_{\varphi}^{\scriptscriptstyle M}}$ and set term $T_{1} \cap T_{2}$ in $\varphi$;
4. \[thirdPlace\] $\pi(T_{1} \setminus T_{2}) = \pi(T_{1}) \land \lnot \pi(T_{2})$, for each map $\pi \in {\Pi_{\varphi}^{\scriptscriptstyle M}}$ and set term $T_{1} \setminus T_{2}$ in $\varphi$.
In addition, we have ${T^{\scriptscriptstyle M}} = \bigcup \{\rho \in {\mathcal{R}_{\varphi}^{\scriptscriptstyle M}} : \pi_{\rho}(T) = {\ensuremath{\mbox{$\mathsf{1}$}}\xspace}\}$, for every $T \in {\mathcal{T}_{\varphi}}$.
By uniformly replacing the atomic formulae in $\varphi$ with propositional variables, in such a way that different occurrences of the same atomic formula are replaced by the same propositional variable and different atomic formulae are replaced by distinct propositional variables, we can associate to $\varphi$ its *propositional skeleton* $\mathsf{P}_{\varphi}$ (up to variables renaming). For instance, the propositional skeleton of $$\label{prop-example}
((X = Y \setminus X \:\land\: Y = X \cup {\mathtt{c}}(X_{1})) \Longrightarrow Z \neq \varnothing) \Longleftrightarrow (X = Y \setminus X \Longrightarrow (Y = X \cup {\mathtt{c}}(X_{1})) \Longrightarrow Z = \varnothing))$$ is the propositional formula $$((P_{1} \land P_{2}) \Longrightarrow \neg P_{3}) \Longleftrightarrow (P_{1} \Longrightarrow (P_{2} \Longrightarrow P_{3}))\,.$$ Plainly, a necessary condition for $\varphi$ to be satisfiable (by a ${\mathsf{BSTC}}$-model) is that its skeleton $\mathsf{P}_{\varphi}$ is propositionally satisfiable (however, the converse does not hold in general).
A collection $\mathcal{A}$ of atoms of $\varphi$ is said to be *promising* for $\varphi$ if the valuation which maps to the propositional variables corresponding to the atoms in $\mathcal{A}$ and to the remaining ones satisfies the propositional skeleton $\mathsf{P}_{\varphi}$. For instance, in the case of (\[prop-example\]), all collections of its atoms not containing both $X = Y \setminus X$ and $Y = X \cup {\mathtt{c}}(X_{1})$ are promising for (\[prop-example\]).
Let $\mathcal{A}_{\varphi}^{+}$ be the collection of the atoms in $\varphi$ satisfied by ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}$ and $\mathcal{A}_{\varphi}^{-}$ the collection of the remaining atoms in $\varphi$, namely those that are disproved by ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}$. It can be easily checked that $\mathcal{A}_{\varphi}^{+}$ is promising. In addition, for every atom $T_{1} = T_{2}$ in $\varphi$ and $\pi \in {\Pi_{\varphi}^{\scriptscriptstyle M}}$, we have $\pi(T_{1}) = \pi(T_{2})$ if and only if $T_{1} = T_{2}$ is in $\mathcal{A}_{\varphi}^{+}$. Likewise, for every atom $T_{1} \subseteq T_{2}$ in $\varphi$ and $\pi \in {\Pi_{\varphi}^{\scriptscriptstyle M}}$, we have $\pi_{\rho}(T_{1}) \leqslant \pi_{\rho}(T_{2})$ if and only if $T_{1} \subseteq T_{2}$ is in $\mathcal{A}_{\varphi}^{+}$. Thus, in particular, for every atom $T_{1} = T_{2}$ in $\mathcal{A}_{\varphi}^{-}$, there exists a map $\pi \in {\Pi_{\varphi}^{\scriptscriptstyle M}}$ such that $\pi(T_{1}) \neq \pi(T_{2})$. Likewise, for every atom $T_{1} \subseteq T_{2}$ in $\mathcal{A}_{\varphi}^{-}$, there exists a map $\pi \in {\Pi_{\varphi}^{\scriptscriptstyle M}}$ such that $\pi(T_{1}) > \pi(T_{2})$.
\[places\]Any Boolean map $\pi$ on ${\mathcal{T}_{\varphi}}$ for which the above properties \[zeroPlace\]–\[thirdPlace\] hold for each set term in $\varphi$ is called a *place for $\varphi$*.
For a given set $\mathcal{A}$ of atoms occurring in $\varphi$, a set $\Pi$ of places for $\varphi$ is $\mathcal{A}$-*ample* if
1. \[fourthPlace\] $\pi(T_{1}) = \pi(T_{2})$ for each $\pi \in \Pi$, provided that the atom $T_{1} = T_{2}$ is not in $\mathcal{A}$;
2. \[fifthPlace\] $\pi(T_{1}) \leqslant \pi(T_{2})$ for each $\pi \in \Pi$, provided that the atom $T_{1} \subseteq T_{2}$ is not in $\mathcal{A}$;
3. \[sixthPlace\] $\pi(T_{1}) \neq \pi(T_{2})$ for some $\pi \in \Pi$, if the atom $T_{1} = T_{2}$ belongs to $\mathcal{A}$;
4. \[seventhPlace\] $\pi(T_{1}) > \pi(T_{2})$ for some $\pi \in \Pi$, if the atom $T_{1} \subseteq T_{2}$ belongs to $\mathcal{A}$.
Notice that conditions \[zeroPlace\]–\[fifthPlace\] are universal, whereas conditions \[sixthPlace\]–\[seventhPlace\] are existential.
The considerations made just before Definition \[places\] yield that ${\Pi_{\varphi}^{\scriptscriptstyle M}}$ is an $\mathcal{A}_{\varphi}^{-}$-ample set of places for $\varphi$. However, in order to later establish some tight complexity results, it is convenient to enforce a polynomial bound for the cardinality of the set of places in terms of the size $|\varphi|$ of $\varphi$ (where, for instance, $|\varphi|$ could be defined as the number of nodes in the syntax tree of $\varphi$). We do this as follows: for each atom $T_{1} = T_{2}$ (resp., $T_{1} \subseteq T_{2}$) in $\mathcal{A}_{\varphi}^{-}$, we select a place $\pi_{\rho}$, with $\rho \in {\mathcal{R}_{\varphi}^{\scriptscriptstyle M}}$ such that $\rho \subseteq {T_{1}^{\scriptscriptstyle M}} \Longleftrightarrow \rho \nsubseteq {T_{2}^{\scriptscriptstyle M}}$ (resp., $\rho \subseteq {T_{1}^{\scriptscriptstyle M}} \setminus {T_{2}^{\scriptscriptstyle M}}$) holds, and call their collection $\Pi_{1}$. Plainly, we have $|\Pi_{1}| \leq |\mathcal{A}_{\varphi}^{-}|$. Notice that $\Pi_{1}$ is $\mathcal{A}_{\varphi}^{-}$-ample.[^8]
Conditions \[zeroPlace\]–\[thirdPlace\] take care of the structure of set terms in $\varphi$ but those of the form $\{x\}$ or ${\mathtt{c}}(T)$, conditions \[fourthPlace\] and \[fifthPlace\] take care of the atoms in $\varphi$ deemed to be positive, whereas conditions \[sixthPlace\] and \[seventhPlace\] take care of the remaining atoms in $\varphi$, namely those deemed to be negative.
To take care of set terms of the form $\{x\}$ in $\varphi$, we observe that, for every $x \in \mathsf{V}_{0}$, there exists a unique Euler’s region $\rho_{x} \in {\mathcal{R}_{\varphi}^{\scriptscriptstyle M}}$ such that ${x^{\scriptscriptstyle M}} \in \rho_{x}$. Let $\pi^{x}$ be the place corresponding to $\rho_{x}$ according to (\[defPlaces\]), namely $\pi^{x} {\coloneqq}\pi_{\rho_{x}}$, and put $\Pi_{2} {\coloneqq}\{\pi^{x} : x \in \mathsf{V}_{0}\}$.
\[placeAtVariables\]Let $x$ be an individual variable occurring in $\varphi$. A *place (for $\varphi$) at the variable $x$* is any place $\pi$ for $\varphi$ such that $\pi(\{x\}) = {\ensuremath{\mbox{$\mathsf{1}$}}\xspace}$.
Next, we take care of choice terms. Thus, let $\Omega {\coloneqq}\{{T_{1}^{\scriptscriptstyle M}},\ldots,{T_{k}^{\scriptscriptstyle M}}\}$ and let ${\mathcal{E}}$ be the Euler’s diagram of $\Omega \cup {{\mathtt{c}}^{\scriptscriptstyle M}}[\Omega]$. Notice that each region in ${\mathcal{E}}$ is a disjoint union of regions in ${\mathcal{R}_{\varphi}^{\scriptscriptstyle M}}$; moreover, the partial choice ${{\mathtt{c}}^{\scriptscriptstyle M}}{\raisebox{-.5ex}{$|$}_{\Omega}}$ over the choice domain $\Omega$ enjoys the $\textsf{WARP}$-lifting property. Thus, by Theorem \[THM:lifting WARP\], there exists a total Noetherian preorder $\lesssim$ on ${\mathcal{E}}$ such that, for all $\sigma',\sigma'' \in {\mathcal{E}}$ and ${T^{\scriptscriptstyle M}} \in \Omega$, we have:
1. \[Aabove\] if $\sigma' \subseteq {T^{\scriptscriptstyle M}}$ and $\sigma'' \subseteq {{\mathtt{c}}^{\scriptscriptstyle M}}({T^{\scriptscriptstyle M}})$, then $\sigma' \lesssim \sigma''$;
2. \[Babove\] if $\sigma'$ is $\lesssim$-maximal in $\mathsf{env}_{{\mathcal{E}}}({T^{\scriptscriptstyle M}})$, then $\sigma' \subseteq {{\mathtt{c}}^{\scriptscriptstyle M}}({T^{\scriptscriptstyle M}})$.
For each region $\sigma \in \mathcal{E}$, let us select a place $\pi_{\rho}$, such that $\rho \in {\mathcal{R}_{\varphi}^{\scriptscriptstyle M}}$ and $\rho \subseteq \sigma$, and call their collection $\Pi_{3}$. Set $\Pi {\coloneqq}\Pi_{1} \cup \Pi_{2} \cup \Pi_{3}$. Plainly, $|\Pi| \leqslant |\mathcal{A}| + |\mathsf{V}_{0}| + 2^{k}$, $\Pi$ is $\mathcal{A}_{\varphi}^{-}$-ample, and $\pi^{x}$ is the sole place in $\Pi$ at the variable $x$, for each individual variable $x$ in $\varphi$. To ease notation, for $\pi \in \Pi$, $\Pi' \subseteq \Pi$, and $T \in {\mathcal{T}_{\varphi}}$, we shall also write (i) $\pi \subseteq T$ for $\pi(T) = {\ensuremath{\mbox{$\mathsf{1}$}}\xspace}$, (ii) $\Pi' \subseteq T$ for $\pi' \subseteq T$, for every $\pi' \in \Pi'$, (iii) $\Pi' \ni\in T$ for $\pi' \subseteq T$, for some $\pi' \in \Pi'$.
For each $\sigma \in \mathcal{E}$, let $\Pi_{\sigma} {\coloneqq}\{\pi_{\rho} : \rho \in {\mathcal{R}_{\varphi}^{\scriptscriptstyle M}} \text{ and } \rho \subseteq \sigma\}$, and call ${\mathcal{P}}$ their collection. Then, by \[Aabove\] and \[Babove\] above, there exists a total Noetherian preorder $\precsim$ on ${\mathcal{P}}$ such that, for all $\Pi',\Pi'' \in {\mathcal{P}}$ and $i \in \{1,\ldots,k\}$, the following conditions hold:
1. \[AabovePrime\] if $\Pi' \subseteq T_{i}$ and $\Pi'' \subseteq {\mathtt{c}}(T_{i})$, then $\Pi' \precsim \Pi''$;
2. \[BabovePrime\] if $\Pi'$ is $\precsim$-maximal in $\{\Pi^{*} \in {\mathcal{P}}:
\Pi^{*} \ni\in T_{i}\}$, then $\Pi' \subseteq {\mathtt{c}}(T_{i})$.
Summing up, we have the following result:
\[necessity2LSS\] Let $\varphi$ be a ${\mathsf{BSTC}}$-formula in choice-flat form, $\mathsf{V}_{0}$ the set of individual variables occurring in it, and ${\mathtt{c}}(T_{1}),~\ldots,~{\mathtt{c}}(T_{k})$ the choice terms occurring in it. If $\varphi$ is satisfiable under the -semantics, then there exist an $\mathcal{A}$-ample set $\Pi$ of places for $\varphi$ such that $|\Pi| \leq |\mathcal{A}| + |\mathsf{V}_{0}| + 2^{k}$, for some promising set $\mathcal{A}$ of atoms in $\varphi$, and a map $x \mapsto \pi^{x}$ from $\mathsf{V}_{0}$ into $\Pi$ such that $\pi^{x}$ is the sole place in $\Pi$ at the variable $x$, for $x \in \mathsf{V}_{0}$. In addition, if $
\Pi_{{\mathtt{c}}} {\coloneqq}\big\{\pi \in \Pi : \pi \subseteq T_{i} \text{ or } \pi \subseteq {\mathtt{c}}(T_{i})\text{, for some } i \in \{1,\ldots,k\} \big\}\,,
$ $\sim_{{\mathtt{c}}}$ is the equivalence relation on $\Pi_{{\mathtt{c}}}$ such that\
$
\pi \sim_{{\mathtt{c}}} \pi' \quad \Longleftrightarrow \quad \pi(T_{i}) = \pi'(T_{i}) \text{ ~and~ } \pi({\mathtt{c}}(T_{i})) = \pi'({\mathtt{c}}(T_{i}))\text{\,,~ for every } i = 1,\ldots,k\,,
$
and ${\mathcal{P}}{\coloneqq}\Pi_{{\mathtt{c}}}/\sim_{{\mathtt{c}}}$, then there exists a total Noetherian preorder $\precsim$ on ${\mathcal{P}}$ such that conditions \[AabovePrime\] and \[BabovePrime\] are satisfied for all $\Pi',\Pi'' \in {\mathcal{P}}$ and $i \in \{1,\ldots,k\}$.
Next we show that the conditions in the preceding lemma are also sufficient for the satisfiability of our ${\mathsf{BSTC}}$-formula $\varphi$ under the -semantics. Thus, let $\Pi$, $\mathcal{A}$, $x \mapsto \pi^{x}$, $\Pi_{{\mathtt{c}}}$, $\sim_{{\mathtt{c}}}$, and ${\mathcal{P}}$ be such that the conditions in Lemma \[necessity2LSS\] are satisfied. Let $U_{_{\Pi}}$ be any set of cardinality $|\Pi|$, and $\pi \mapsto a_{\pi}$ any injective map from $\Pi$ onto $U_{_{\Pi}}$. We define an interpretation $M_{_{\Pi}}$ over the variables in $\mathsf{V}_{0} \cup \mathsf{V}_{1}$ and the choice terms ${\mathtt{c}}(T_{1}),~\ldots,~{\mathtt{c}}(T_{k})$ occurring in $\varphi$, as if they were set variables, by putting:
$$\label{finalModel}
\begin{array}{rcll}
{x^{\scriptscriptstyle M_{\Pi}}} &{\coloneqq}& a_{\pi^{x}}&\text{for } x \in \mathsf{V}_{0}\\
{X^{\scriptscriptstyle M_{\Pi}}} &{\coloneqq}& \{a_{\pi} \in U_{_{\Pi}} : \pi \in \Pi \land \pi \subseteq X \}~~~~~&\text{for } X \in \mathsf{V}_{1}\\
{({\mathtt{c}}(T_{i}))^{\scriptscriptstyle M_{\Pi}}} &{\coloneqq}& \{a_{\pi} \in U_{_{\Pi}} : \pi \in \Pi \land \pi \subseteq {\mathtt{c}}(T_{i}) \}~~~~~&\text{for } i =1,\ldots,k\,.
\end{array}$$
Notice that the choice map ${\mathtt{c}}$ is not interpreted by $M_{_{\Pi}}$. However, it is not hard to check that the set assignment ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}_{_{\Pi}} {\coloneqq}(U_{_{\Pi}},M_{_{\Pi}})$ satisfies exactly the atoms in $\mathcal{A}$, provided that choice terms in $\varphi$ are regarded as set variables with no internal structure, rather than as compound terms. Indeed, after putting ${\pi^{\scriptscriptstyle M_{\Pi}}} {\coloneqq}\{a_{\pi}\}$, for $\pi \in \Pi$, we have ${T^{\scriptscriptstyle M_{\Pi}}} = \bigcup_{\pi \subseteq T} {\pi^{\scriptscriptstyle M_{\Pi}}}$, for every $T \in {\mathcal{T}_{\varphi}}$. Thus, if $T_{1} = T_{2}$ does not belong to $\mathcal{A}$, then by condition \[fourthPlace\] we have ${T_{1}^{\scriptscriptstyle M_{\Pi}}} = {T_{2}^{\scriptscriptstyle M_{\Pi}}}$, whereas if $T_{1} = T_{2}$ belongs to $\mathcal{A}$, then by \[sixthPlace\] we have ${T_{1}^{\scriptscriptstyle M_{\Pi}}} \neq {T_{2}^{\scriptscriptstyle M_{\Pi}}}$. Analogously for atoms $T_{1} \subseteq T_{2}$ in $\varphi$. Hence, by the promisingness of $\mathcal{A}$, ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}_{_{\Pi}}$ satisfies also $\varphi$.
Next we define $M_{_{\Pi}}$ also over the choice map ${\mathtt{c}}$ in such a way that ${{\mathtt{c}}^{\scriptscriptstyle M_{\Pi}}}$ is a total choice over $U$ that satisfies , and ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}_{_{\Pi}} \models \varphi$ holds. Let $\Omega_{_{\Pi}} {\coloneqq}\{{T_{1}^{\scriptscriptstyle M_{\Pi}}},\ldots,{T_{k}^{\scriptscriptstyle M_{\Pi}}}\}$. We begin by defining ${{\mathtt{c}}^{\scriptscriptstyle M_{\Pi}}}$ over $\Omega_{_{\Pi}}$ in the most natural way, namely by putting ${{\mathtt{c}}^{\scriptscriptstyle M_{\Pi}}}({T_{i}^{\scriptscriptstyle M_{\Pi}}}) {\coloneqq}{({\mathtt{c}}(T_{i}))^{\scriptscriptstyle M_{\Pi}}}$, for $i = 1,\ldots,k$. By the choice and the single-valuedness conditions in $\varphi$, ${{\mathtt{c}}^{\scriptscriptstyle M_{\Pi}}}$ so defined is a choice over the domain $\Omega_{_{\Pi}}$. In addition, it can be checked that the existence of a total Noetherian preorder on ${\mathcal{P}}$ satisfying conditions \[AabovePrime\] and \[BabovePrime\] yield the existence of a total Noetherian preorder on the collection of the Euler’s regions of $\Omega_{_{\Pi}} \cup {{\mathtt{c}}^{\scriptscriptstyle M_{\Pi}}}[\Omega_{_{\Pi}}]$ satisfying conditions \[aTheorem3\] and \[bTheorem3\] of Theorem \[THM:lifting WARP\]. Thus, the latter theorem readily implies that ${{\mathtt{c}}^{\scriptscriptstyle M_{\Pi}}}$ has the -lifting property, so that it can be extended to a total choice on $U$ satisfying , proving that $\varphi$ is satisfiable under the -semantics.
Together with Lemma \[necessity2LSS\], the above argument yields the following result:
\[WARPTheorem\] Let $\varphi$ be a complete ${\mathsf{BSTC}}$-formula in choice-flat form, $\mathsf{V}_{0}$ the set of individual variables occurring in it, and ${\mathtt{c}}(T_{1}),~\ldots,~{\mathtt{c}}(T_{k})$ the choice terms occurring in it. Then $\varphi$ is satisfiable under the -semantics if and only if there exist (i) an $\mathcal{A}$-ample set $\Pi$ of places for $\varphi$ such that $|\Pi| \leq |\mathcal{A}| + |\mathsf{V}_{0}| + 2^{k}$, for some promising set $\mathcal{A}$ of atoms in $\varphi$, (ii) a map $x \mapsto \pi^{x}$ from $\mathsf{V}_{0}$ into $\Pi$, such that $\pi^{x}$ is the sole place in $\Pi$ at the variable $x$, for $x \in \mathsf{V}_{0}$, and (iii) a total Noetherian preorder $\precsim$ on ${\mathcal{P}}$, where ${\mathcal{P}}{\coloneqq}\Pi_{{\mathtt{c}}}/\sim_{{\mathtt{c}}}$ (with $\Pi_{{\mathtt{c}}} {\coloneqq}\big\{\pi \in \Pi : \pi \subseteq T_{i} \text{ or } \pi \subseteq {\mathtt{c}}(T_{i})\text{, for some } i \in \{1,\ldots,k\} \big\}\,,$ and $\sim_{{\mathtt{c}}}$ the equivalence relation on $\Pi_{{\mathtt{c}}}$ such that $\pi \sim_{{\mathtt{c}}} \pi' \Longleftrightarrow \pi(T_{i}) = \pi'(T_{i}) \text{ ~and~ } \pi({\mathtt{c}}(T_{i})) = \pi'({\mathtt{c}}(T_{i}))\text{\,,~ for } i = 1,\ldots,k$) such that the above conditions \[AabovePrime\] and \[BabovePrime\] are satisfied for all $\Pi',\Pi'' \in {\mathcal{P}}$ and $i \in \{1,\ldots,k\}$.
We have already observed that the satisfiability problem for ${\mathsf{BSTC}}$-formulae under the -semantics is [NP]{}-hard. In addition, the previous theorem implies at once that it belongs to [NEXPTIME]{}. However, if we restrict to ${\mathsf{BSTC}}$-formulae with a constant number of choice terms, the disequality $|\Pi| \leq |\mathcal{A}| + |\mathsf{V}_{0}| + 2^{k}$ in Theorem \[WARPTheorem\] yields $|\Pi| = \mathcal{O}(|\mathcal{A}| + |\mathsf{V}_{0}|) =\mathcal{O}(|\varphi|)$, thereby providing the following complexity result:
Under the -semantics, the satisfiability problem for ${\mathsf{BSTC}}$-formulae with $\mathcal{O}(1)$ distinct choice terms is [NP]{}-complete.
As a by-product, Theorem \[WARPTheorem\] yields a solution to the satisfiability problem for $BSTC^{-}$-formulae. Indeed, in the case of $BSTC^{-}$-formulae, Theorem \[WARPTheorem\] becomes:
\[BSTC-Theorem\] Let $\varphi$ be a ${\mathsf{BSTC}}^{-}$-formula, and $\mathsf{V}_{0}$ the set of individual variables occurring in it. Then $\varphi$ is satisfiable if and only if there exist (i) an $\mathcal{A}$-ample set $\Pi$ of places for $\varphi$ such that $|\Pi| \leq |\mathcal{A}| + |\mathsf{V}_{0}|$, for some promising set $\mathcal{A}$ of atoms in $\varphi$, and (ii) a map $x \mapsto \pi^{x}$ from $\mathsf{V}_{0}$ into $\Pi$, such that $\pi^{x}$ is the sole place in $\Pi$ at the variable $x$, for $x \in \mathsf{V}_{0}$.
Hence, we also have:
\[NPcompl2LSS\] The satisfiability problem for ${\mathsf{BSTC}}^{-}$-formulae is [NP]{}-complete.
### Unrestricted semantics
As above, let $\varphi$ be a complete ${\mathsf{BSTC}}$-formula in choice flat-form. Plainly, if $\varphi$ is satisfiable under unrestricted semantics, so is its *${\mathsf{BSTC}}^{-}$-reduction* obtained from $\varphi$ by regarding the choice terms as set variables with no internal structure.
Conversely, let us assume that the ${\mathsf{BSTC}}^{-}$-reduction $\varphi_{1}$ of $\varphi$ is satisfiable, and let ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}_{1} = (U,M_{1})$ be a model for $\varphi_{1}$. We define a total choice correspondence $c \colon {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}\rightrightarrows U$ on $U$ by putting, for every $A \in {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}$, $$\label{defMapc}
c(A) {\coloneqq}\begin{cases}
A & \text{if } A \notin \{ {T^{\scriptscriptstyle M}}_{1},\ldots, {T^{\scriptscriptstyle M}}_{k}\}\\
{({\mathtt{c}}(T_{i}))^{\scriptscriptstyle M}} & \text{if } A = {T^{\scriptscriptstyle M}}_{i}, \text{ for some } i = 1,\ldots,k.
\end{cases}$$ Observe that, by the single-valuedness conditions present in $\varphi_{1}$, if ${T^{\scriptscriptstyle M}}_{i} = {T^{\scriptscriptstyle M}}_{j}$, for distinct $i$ and $j$, then ${({\mathtt{c}}(T_{i}))^{\scriptscriptstyle M}} = {({\mathtt{c}}(T_{j}))^{\scriptscriptstyle M}}$. Hence, the map $c$ is well-defined. In addition, the choice conditions yield that $c$ is indeed a total choice on $U$.
Let ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}= (U,M)$ be the set assignment differing from ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}_{1}$ only on the interpretation of the choice map symbol ${\mathtt{c}}$, for which we have ${{\mathtt{c}}^{\scriptscriptstyle M}} {\coloneqq}c$. Plainly, ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}$ coincides with ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}_{1}$ on all the choice-free terms in $\varphi$. In addition, since $c({T^{\scriptscriptstyle M}}_{i}) = {({\mathtt{c}}(T_{i}))^{\scriptscriptstyle M}}$ for $i=1,\ldots,k$, the assignment ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}$ coincides with ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}_{1}$ on the remaining terms in $\varphi$ as well. Thus ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}\models \varphi$, proving that $\varphi$ is satisfiable when it admits a satisfiable ${\mathsf{BSTC}}^{-}$-reduction.
We have thus proved:
Under unrestricted semantics, a complete ${\mathsf{BSTC}}$-formula in choice-flat form is satisfiable if and only if it admits a satisfiable ${\mathsf{BSTC}}^{-}$-reduction.
In view of Theorem \[NPcompl2LSS\], we can conclude:
Under unrestricted semantics, the satisfiability problem for ${\mathsf{BSTC}}$-formulae is [NP]{}-complete.
### $(\beta)$-semantics
Let us now assume that our complete ${\mathsf{BSTC}}$-formula in choice-flat form $\varphi$ is satisfiable under the $(\beta)$-semantics, and let ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}=(U,M)$ be a model for it, where now ${{\mathtt{c}}^{\scriptscriptstyle M}}$ is a choice satisfying the axiom $(\beta)$. Then the model ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}$ satisfies all the following instances of the axiom $(\beta)$:
$(\beta)$*-conditions*:
: $\big(T_{i} \subseteq T_{j} \: \wedge \: {\mathtt{c}}(T_{i}) \cap {\mathtt{c}}(T_{j}) \neq \emptyset \big) \;\; \Longrightarrow \;\; {\mathtt{c}}(T_{i}) \subseteq {\mathtt{c}}(T_{j})$, for $i,j = 1,\ldots,k$.
Let $\varphi_{\beta}$ be the ${\mathsf{BSTC}}^{-}$-formula obtained by adding the $(\beta)$-conditions to the ${\mathsf{BSTC}}^{-}$-$\beta$-reduction of $\varphi$, while regarding the choice terms in it just as set variables (with no internal structure). We call the formula $\varphi_{\beta}$ the ${\mathsf{BSTC}}^{-}$-*$\beta$-reduction of $\varphi$*. Notice that $|\varphi_{\beta}| = \mathcal{O}(|\varphi|^{2})$. In addition, $\varphi_{\beta}$ is plainly satisfiable.
Conversely, let $\varphi_{\beta}$ be satisfiable and let ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}_{\beta} = (U,M_{\beta})$ be a model for $\varphi_{\beta}$. Let $\Omega_{\beta} {\coloneqq}\{{T_{i}^{\scriptscriptstyle M_{\beta}}} : i = 1, \ldots,k\}$ and $c_{\beta}$ be a map over $\Omega_{\beta}$ such that $c_{\beta}({T_{i}^{\scriptscriptstyle M_{\beta}}}) = {({\mathtt{c}}(T_{i}))^{\scriptscriptstyle M_{\beta}}}$, for $i = 1, \ldots,k$. From the choice and single-valuedness conditions, $c_{\beta}$ is a choice over the domain $\Omega_{\beta}$. In addition, by the $\beta$-conditions present in $\varphi_{\beta}$, the choice $c_{\beta}$ satisfies the $(\beta)$-axiom. Hence, Theorem \[THM:lifting beta\] yields that $c_{\beta}$ has the $(\beta)$-lifting property, i.e., there is a total choice ${c^{\hbox{\tiny{+}}}}_{\beta} \colon {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}\rightrightarrows U$ extending $c_{\beta}$ and satisfying the $(\beta)$-axiom. Let ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}= (U,M)$ be the set assignment differing from ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}_{\beta}$ only on the interpretation of the choice symbol ${\mathtt{c}}$, for which we have ${{\mathtt{c}}^{\scriptscriptstyle M}} = {c^{\hbox{\tiny{+}}}}_{\beta}$. It is not hard to check that ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}\models_{\beta} \varphi$.
Thus, we have:
A complete ${\mathsf{BSTC}}$-formula in choice-flat form is satisfiable under the $(\beta)$-semantics if and only if it admits a satisfiable ${\mathsf{BSTC}}^{-}$-$\beta$-reduction.
Since the size of the ${\mathsf{BSTC}}^{-}$-$\beta$-reduction of a given ${\mathsf{BSTC}}$-formula $\psi$ is at most quadratic in the size of $\psi$, in view of Theorem \[NPcompl2LSS\] we can conclude:
Under the $(\beta)$-semantics, the satisfiability problem for ${\mathsf{BSTC}}$-formulae is [NP]{}-complete.
### $(\alpha)$-semantics
Finally, we assume that our complete ${\mathsf{BSTC}}$-formula in choice-flat form $\varphi$ is satisfiable under the $(\alpha)$-semantics. Let ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}=(U,M)$ be a model for it, where now ${{\mathtt{c}}^{\scriptscriptstyle M}}$ is a choice satisfying the axiom $(\alpha)$. Then, the model ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}$ satisfies also the following instances of the axiom $(\alpha)$:
$(\alpha)$*-conditions*:
: $T_{i} \subseteq T_{j} \;\; \Longrightarrow \;\; T_{i} \cap {\mathtt{c}}(T_{j}) \subseteq {\mathtt{c}}(T_{j})$, for $i,j = 1,\ldots,k$.
In addition, let $\Omega {\coloneqq}\{{T_{i}^{\scriptscriptstyle M}} : i = 1, \ldots, k\}$. Then, plainly, the partial choice correspondence ${{\mathtt{c}}^{\scriptscriptstyle M}}{\raisebox{-.5ex}{$|$}_{\Omega}}$ over the choice domain $\Omega$ has the $(\alpha)$-lifting property. Hence, by Theorem \[THM:lifting alpha\]\[c\], for every $\emptyset \neq \mathcal{B} \subseteq \Omega$ such that $\mathcal{B}$ is $\subseteq$-closed w.r.t. $\Omega$, we have $$\label{nonEmptynessSem}
\bigcup \mathcal{B} \setminus \bigcup\nolimits_{B \in \mathcal{B}} \overline{c}(B) \neq \emptyset\,.$$ In fact, (\[nonEmptynessSem\]) holds for every $\emptyset \neq \mathcal{B} \subseteq \Omega$, irrespectively of whether $\mathcal{B}$ is $\subseteq$-closed w.r.t. $\Omega$ or not, as can be easily checked. Thus, ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}$ satisfies also the following further conditions:
*nonemptiness conditions*:
: $\bigcup_{i \in I} T_{i} \setminus \bigcup_{i \in I}(T_{i} \setminus {\mathtt{c}}(T_{i})) \neq \varnothing$, for each $\emptyset \neq I \subseteq \{1,\ldots,k\}$.
Let $\varphi_{\alpha}$ be the ${\mathsf{BSTC}}^{-}$-formula obtained by adding the $(\alpha)$- and the nonemptiness conditions to the ${\mathsf{BSTC}}^{-}$-reduction of $\varphi$, while regarding the choice terms in it just as set variables (with no internal structure). We call the formula $\varphi_{\alpha}$ the ${\mathsf{BSTC}}^{-}$-*$\alpha$-reduction of $\varphi$*. Notice that $|\varphi_{\alpha}| = \mathcal(|\varphi|^{2} + k\cdot 2^{k})$. In addition, $\varphi_{\alpha}$ is plainly satisfiable.
Conversely, let us assume that $\varphi_{\alpha}$ is satisfiable and let ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}_{\alpha} = (U,M_{\alpha})$ be a model for it. Let $\Omega_{\alpha} {\coloneqq}\{{T_{i}^{\scriptscriptstyle M_{\alpha}}} : i = 1, \ldots,k\}$ and $c_{\alpha}$ be a map over $\Omega_{\alpha}$ such that $c_{\alpha}({T_{i}^{\scriptscriptstyle M_{\alpha}}}) = {X_{i}^{\scriptscriptstyle M_{\alpha}}}$, for $i = 1, \ldots,k$. As before, thanks to the choice and the single-valuedness conditions, $c_{\alpha}$ is a choice over the domain $\Omega_{\alpha}$. In addition, from the $(\alpha)$- and the nonemptiness conditions, Theorem \[THM:lifting alpha\] yields that $c_{\alpha}$ has the $(\alpha)$-lifting property, i.e., there is a total choice ${c^{\hbox{\tiny{+}}}}_{\alpha} \colon {{\mathrm{Pow}}^{\hbox{\tiny{+}}}(U)}\rightrightarrows U$ extending $c_{\alpha}$ and satisfying the $(\alpha)$-axiom. Let ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}= (U,M)$ be the set assignment differing from ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}_{\alpha}$ only on the interpretation of the choice symbol ${\mathtt{c}}$, for which we have ${{\mathtt{c}}^{\scriptscriptstyle M}} = {c^{\hbox{\tiny{+}}}}_{\alpha}$. It is routine to check that ${\ensuremath{\mbox{\boldmath $\mathcal{M}$}}\xspace}\models_{\alpha} \varphi$.
Hence, we have:
Under the $(\alpha)$-semantics, a complete ${\mathsf{BSTC}}$-formula in choice-flat form is satisfiable if and only if it admits a satisfiable ${\mathsf{BSTC}}^{-}$-$\alpha$-reduction.
As observed earlier, the size of $\varphi_{\alpha}$ is $\mathcal{O}(|\varphi|^{2} + k\cdot 2^{k})$. Thus, in general, the satisfiability problem for ${\mathsf{BSTC}}$-formulae under the $(\alpha)$-semantics is in [NEXPTIME]{}. However, if the number of distinct choice terms is restricted to be $\mathcal{O}(1)$, we have $|\varphi_{\alpha}| = \mathcal{O}(|\varphi|^{2})$ and therefore, in view of Theorem \[NPcompl2LSS\], we have:
Under the $(\alpha)$-semantics, the satisfiability problem for ${\mathsf{BSTC}}$-formulae with $\mathcal{O}(1)$ distinct choice terms is [NP]{}-complete.
Conclusions {#SECT:Conclusions}
===========
In this paper we have initiated the study of the satisfiability problem for unquantified formulae of an elementary fragment of set theory enriched with a choice correspondence symbol. Apart from the obvious theoretical reasons that motivate this approach, our analysis has its roots in applications within the field of social and individual choice theory. In fact, the satisfiability tests implicit in our results naturally yield an effective way of checking whether the observed choice behavior of an economic agent is induced by an underlying rationality on the set of alternatives.
Future research on the topic is related to the extension of the current approach to a more general setting. In this direction, it is natural to examine the satisfiability problem for semantics characterized by other types of axioms of choice consistency, which are connected to rationalizability issues. We are currently studying the lifting (and the associated satisfiability problem) of several combinations of axioms, namely, $(\alpha)$ adjoined with $(\gamma)$ and/or $(\rho)$ (see [@CanGiaWat17]). The motivation of this analysis is that the satisfaction of these axioms is connected to the (quasi-transitive) rationalizability of a choice. More generally, it appears natural to examine the lifting of $(m,n)$*-Ferrers properties* in the sense of [@GiaWat14] (see also [@GiaWat17]). In fact, these properties give rise to additional types of rationalizability – the so-called $(m,n)$*-rationalizability* – in which the relation of revealed preference satisfies structural forms of pseudo-transitivity (see [@CanGiaGreWat16]).
We also intend to find decidable extensions with choice correspondence terms of the three-sorted fragment of set theory ${\ensuremath{\mbox{$\mathrm{3LQST_{0}^{R}}$}}\xspace}$ (see [@CN16]) and of the four-sorted fragments $({\ensuremath{\mbox{$4\mathrm{LQS}^{R}$}}\xspace})^h$, with $h \in \mathbb{N}$ (see [@CanNic13a]), which admit a restricted form of quantification over individual and set variables (in the case of ${\ensuremath{\mbox{$\mathrm{3LQST_{0}^{R}}$}}\xspace}$), and also over collection variables (in the case of $({\ensuremath{\mbox{$4\mathrm{LQS}^{R}$}}\xspace})^h$. The resulting decision procedures would allow to reason automatically on very expressive properties in choice theory.
Acknowledgements {#acknowledgements .unnumbered}
================
Thanks are due to the referees for their useful comments.
[10]{} \[2\] \[1\][`#1`]{} \[2\][`#2`]{} \[2\][[\#2](#1)]{} \[1\][doi:]{} \[2\][\#2]{}
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[^1]: Department of Mathematics and Computer Science, University of Catania, Italy. Email: [email protected]
[^2]: Department of Economics and Business, University of Catania, Italy. Email: [email protected]
[^3]: Department of Mathematics and Statistics, York University, Toronto, Canada. Email: [email protected]
[^4]: This is an extended version of a paper by the same title that will appear in the proceedings of GandALF 2017, the *Eighth International Symposium on Games, Automata, Logics, and Formal Verification*, Rome (Italy), 20-21-22 September 2017.
[^5]: Recall that $\max_{\precsim} B = \{a \in B : (\nexists b \in B) (a \prec b)\}$, where $a \prec b$ means $a \precsim b$ and $\neg(b \precsim a)$.
[^6]: Up to minor syntactic differences, ${\mathsf{BSTC}}^{-}$-formulae are essentially -formulae, whose decision problem has been solved (see, for instance, [@CanOmoPol01 Exercise 10.5]).
[^7]: See Footnote \[FootnoteFiner\].
[^8]: \[FootnoteFiner\]A finer construction would yield an $\mathcal{A}_{\varphi}^{-}$-ample set of places $\Pi_{1}'$ such that $|\Pi_{1}'| \leq |{\mathcal{T}_{\varphi}}|$. Notice that $|\Pi_{1}| = \mathcal{O}(|\Pi_{1}'|^{2})$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We consider a recent coinfection model for Tuberculosis (TB), Human Immunodeficiency Virus (HIV) infection and Acquired Immunodeficiency Syndrome (AIDS) proposed in \[Discrete Contin. Dyn. Syst. 35 (2015), no. 9, 4639–4663\]. We introduce and analyze a multiobjective formulation of an optimal control problem, where the two conflicting objectives are: minimization of the number of HIV infected individuals with AIDS clinical symptoms and coinfected with AIDS and active TB; and costs related to prevention and treatment of HIV and/or TB measures. The proposed approach eliminates some limitations of previous works. The results of the numerical study provide comprehensive insights about the optimal treatment policies and the population dynamics resulting from their implementation. Some nonintuitive conclusions are drawn. Overall, the simulation results demonstrate the usefulness and validity of the proposed approach.
[**Keywords:**]{} Tuberculosis; HIV; Epidemic model; Treatment strategies; Optimal control theory; Multiobjective optimization.
[**2010 Mathematics Subject Classification:**]{} 90C29; 92C50.
author:
- |
Roman Denysiuk$^1$\
`[email protected]`
- |
Cristiana J. Silva$^2$\
`[email protected]`
- |
Delfim F. M. Torres$^2$\
`[email protected]`
date: |
$^1$Algoritmi R&D Center, University of Minho, Portugal\
$^2$Center for Research and Development in Mathematics\
and Applications (CIDMA), Department of Mathematics,\
University of Aveiro, 3810-193 Aveiro, Portugal
title: 'Multiobjective optimization to a TB-HIV/AIDS coinfection optimal control problem[^1]'
---
Introduction {#sec:intro}
============
The human immunodeficiency virus (HIV) is a retrovirus that infects cells of the immune system, destroying or impairing their function. HIV is transmitted primarily via unprotected sexual intercourse, contaminated blood transfusions, hypodermic needles, and from mother to child during pregnancy, delivery, or breastfeeding [@RomMark07]. As the infection progresses, the immune system becomes weaker, and the person becomes more susceptible to infections. The most advanced stage of HIV infection is acquired immunodeficiency syndrome (AIDS) [@WhoSiteHivAids]. There is no cure or vaccine to AIDS. However, antiretroviral (ART) treatment improves health, prolongs life, and substantially reduces the risk of HIV transmission. In both high-income and low-income countries, the life expectancy of patients infected with HIV who have access to ART is now measured in decades, and might approach that of uninfected populations in patients who receive an optimum treatment (see [@DeeksEtAl13] and references cited therein). However, ART treatment still presents substantial limitations: does not fully restore health; treatment is associated with side effects; the medications are expensive; and is not curative. Following the Joint United Nations Programme on HIV and AIDS (UNAIDS), in 2013 there were approximately 35 million people living with HIV globally. An estimated 2.1 million people became newly infected with HIV in 2013, down from 3.4 million in 2001 worldwide. The number of new HIV infection among children has declined 58% since 2001, being in 2013 approximately 240 000 worldwide. The number of AIDS-related deaths have fallen by 35% since the peak in 2005. In 2013, approximately 1.5 million people died from AIDS-related causes worldwide. In 2013, around 12.9 million people living with HIV had access to ART therapy, which represents, approximately, 37% of all people living with HIV [@UNAIDSFactSheet2014; @UNAIDSGapRep2014].
*Mycobacterium tuberculosis* is the cause of most occurrences of tuberculosis (TB) and is usually acquired via airborne infection from someone who has active TB. It typically affects the lungs (pulmonary TB) but can affect other sites as well (extrapulmonary TB). According with the World Health Organization (WHO), in 2013, an estimated 9.0 million people developed TB and 1.5 million died from the disease, 360 000 of whom were HIV-positive. TB is slowly declining each year and it is estimated that 37 million lives were saved between 2000 and 2013 through effective diagnosis and treatment. However, since most deaths from TB are preventable, the death toll from the disease is still unacceptably high and efforts to combat it must be accelerated [@WHO14].
Following WHO, the human immunodeficiency virus (HIV) and *mycobacterium tuberculosis* are the first and second cause of death from a single infectious agent, respectively [@WHO13]. Both HIV/AIDS and TB are present in all regions of the world [@Morison01; @WHO14]. Individuals infected with HIV are more likely to develop TB disease because of their immunodeficiency, and HIV infection is the most powerful risk factor for progression from TB infection to disease [@GetahunEtAl10]. In 2013, 1.1 million of 9.0 million people who developed TB worldwide were HIV-positive. The number of people dying from HIV-associated to TB has been falling since 2003. However, there were still 360 000 deaths from HIV-associated to TB in 2013, and further efforts are needed to reduce this burden [@WHO14]. ART is a critical intervention for reducing the risk of TB morbidity and mortality among people living with HIV and, when combined with TB preventive therapy, it can have a significant impact on TB prevention [@WHO14]. Collaborative TB/HIV activities (including HIV testing, ART therapy and TB preventive measures) are crucial for the reduction of TB-HIV coinfected individuals. WHO estimates that these collaborative activities prevented 1.3 million people from dying, from 2005 to 2012. However, significant challenges remain: the reduction of tuberculosis related deaths among people living with HIV has slowed in recent years; the ART therapy is not being delivered to TB-HIV coinfected patients in the majority of the countries with the largest number of TB/HIV patients; the pace of treatment scale-up for TB/HIV patients has slowed; less than half of notified TB patients were tested for HIV in 2012; and only a small fraction of TB/HIV infected individuals received TB preventive therapy [@UNAIDSRep2013]. The study of the joint dynamics of TB and HIV present formidable mathematical challenges due to the fact that the models of transmission are quite distinct [@ChavezEtAll09]. Here we focus on a recent mathematical model of optimal control for TB-HIV/AIDS coinfection proposed by [@SiTo15].
Optimal control is a branch of mathematics developed to find optimal ways to control a dynamic system [@PoBoGrMi62], e.g. a dynamic system that models infectious diseases. Optimal control has been applied to TB models, HIV models and also co-infection models (see, e.g., [@rv2:agustu; @JuLeFe02; @KirsLenSer96; @LeWo07; @MaMuChuMu09; @RoSiTo14; @SiTo13; @SiTo15] and references cited therein for TB-HIV/AIDS models and [@rv2:Okosun] for co-infection of malaria and cholera). In this paper we consider the optimal control problem for the TB-HIV/AIDS model proposed in [@SiTo15] from a multiobjective perspective. Our approach avoids the use of weight parameters and allows to obtain a wide range of optimal control strategies. These strategies offer the decision maker useful information for effective decision making.
Traditional mathematical programming methods for solving multiobjective optimization problems (MOPs) convert the original problem into a single-objective optimization problem. This is referred as to scalarization and the function to be optimized, which depends on some parameters, is termed the scalarizing function. A solution to the scalarizing function, obtained using a single-objective optimization algorithm, is expected to be Pareto optimal. For approximating multiple Pareto optimal solutions, repeated runs with different parameter settings must be performed. The weighted sum method [@GaSa55] consists in minimizing a weighted sum of multiple objectives. For problems with a convex Pareto front, this method guarantees finding solutions in the entire Pareto optimal region. However, it fails to find solutions in nonconvex regions of the Pareto front. Weighted metric methods [@Mi99] are based on minimizing a weighted distance between some reference point and the feasible objective region. The widely used approach belonging to this class of methods is the Chebyshev method [@Bo76], which consists in minimizing a weighted infinity norm. Although solutions in convex and nonconvex regions of the Pareto front can be obtained by this method, a resulting scalarizing function becomes nondifferentiable even when all the objectives are differentiable. The problem resulting from the Chebyshev method can be reformulated in the smooth form. The resulting formulation is known as the goal attainment method [@Mi99] or the Pascoletti–Serafini scalarization [@PaSe84]. In this method, a slack variable is minimized and the weighted difference for each objective is converted into a constraint. Although the problem can be solved in a differentiable form, problem complexity is augmented by adding one additional variable and $m$ constraints (where $m$ is the number of objectives). The normal boundary intersection and normal constraint methods use a hyperplane with uniformly distributed points passing through the critical points of the Pareto front. The normal boundary intersection method [@DaDe98] searches for the maximum distance from a point on the simplex along the normal pointing toward the origin. The obtained point may or may not be a Pareto optimal point, with the resulting scalarizing problem including an equality constraint that is not easy to treat for all the cases. On the other hand, the normal constraint method [@Me04] uses an inequality constraint reduction of the feasible objective space and the normalized function values to cope with disparate function scales. The method is successful in achieving a uniform distribution of approximating points, though there is no guarantee that an obtained point is Pareto optimal. Here, motivated by the results recently obtained in [@DeSiTo15] for a TB model and in [@MyID:325; @MyID:333] for the dengue disease, we adopt the $\epsilon$-constraint method [@HaLaWi71]. This method suggests optimizing one objective function and converting all other objectives as constraints, setting an upper bound to each of them. Solutions obtained using multiobjective optimization provide comprehensive insights about the optimal strategies and the diseases dynamics resulting from implementation of those strategies.
The paper is organized as follows. In Section \[sec:model\] we briefly describe the TB-HIV/AIDS model from [@SiTo15]. The multiobjective optimization theory is applied to an optimal control problem in Section \[sec:optContolProb\]: we start by formulating the optimal control problem in Subsection \[subsecOCprob\], then we consider this problem from a multiobjective perspective (Subsection \[subsecMultiobj\]) and we describe the numerical method that we use to solve the multiobjective problem (Subsection \[subsec:scalar\]). In Section \[sec:scalar\] we present and discuss numerical results for the multiobjective problem. We end with Section \[sec:conc\] of conclusions and future work.
TB-HIV/AIDS coinfection model {#sec:model}
=============================
The present study considers the population model for TB-HIV/AIDS coinfection proposed in [@SiTo15], where TB, HIV and TB-HIV infected individuals have access to respective disease treatment, and single HIV-infected and TB-HIV co-infected individuals under HIV and TB/HIV treatment, respectively, stay in a *chronic* stage of the HIV infection. The model divides the population into eleven mutually exclusive compartments: susceptible individuals ($S$); TB-latently infected individuals, who have no symptoms of TB disease and are not infectious ($L_T$); TB-infected individuals, who have active TB disease and are infectious ($I_T$); TB-recovered individuals ($R$); HIV-infected individuals with no clinical symptoms of AIDS ($I_H$); HIV-infected individuals under treatment for HIV infection ($C_H$); HIV-infected individuals with AIDS clinical symptoms ($A$); TB-latent individuals co-infected with HIV (pre-AIDS) ($L_{TH}$); HIV-infected individuals (pre-AIDS) co-infected with active TB disease ($I_{TH}$); TB-recovered individuals with HIV-infection without AIDS symptoms ($R_{H}$); and HIV-infected individuals with AIDS symptoms co-infected with active TB ($A_T$). The total population at time $t$, denoted by $N(t)$, is given by $$\begin{gathered}
N(t) = S(t) + L_T(t) + I_T(t) + R(t) + I_H(t) + A(t) + C_H(t) \\
+ L_{TH}(t) + I_{TH}(t) + R_H(t) + A_T(t).\end{gathered}$$ Susceptible individuals acquire TB infection from individuals with active TB at a rate $\lambda_T$, given by $$\lambda_T(t) = \frac{\beta_1}{N(t)} (I_T(t) + I_{TH}(t) + A_T(t)),$$ where $\beta_1$ is the effective contact rate for TB infection. Similarly, susceptible individuals acquire HIV infection, following effective contact with people infected with HIV at a rate $\lambda_H$, given by $$\lambda_H(t) = \frac{\beta_2}{N(t)} [I_H(t) + I_{TH}(t) + L_{TH}(t)
+ R_H(t) + \eta_C C_H(t) + \eta_A (A(t) + A_T(t))],$$ where $\beta_2$ is the effective contact rate for HIV transmission, $\eta_A \geq 1$ is the modification parameter that accounts for the relative infectiousness of individuals with AIDS symptoms and $\eta_C \leq 1$ is the modification parameter that translates the partial restoration of immune function of individuals with HIV infection that use correctly the antiretroviral treatment. The remaining parameters used to describe the model are presented in Table \[tab:params\].
Symbol Description Value
------------ -------------------------------------------------------------------------- -----------
$T$ Considered time in years 10
$N(0)$ Initial population size 30000
$\gamma_1$ Modification parameter 0.9
$\gamma_2$ Modification parameter 1.1
$\eta_C$ Modification parameter 0.9
$\eta_A$ Modification parameter 1.05
$\delta$ Modification parameter 1.03
$\psi$ Modification parameter 1.07
$\beta_1$ TB transmission rate 0.6
$\beta_2$ HIV transmission rate 0.1
$\mu$ Recruitment rate 430.0
$k_1$ Rate at which individuals leave $L_T$ class by becoming infectious 1.0/2.0
$k_2$ Rate at which individuals leave $L_{TH}$ class by becoming TB infectious $1.3 k_1$
$k_3$ Rate at which individuals leave $L_{TH}$ class 2.0
$\rho_1$ Rate at which individuals leave $I_H$ class to $A$ 0.1
$\rho_2$ Rate at which individuals leave $I_{TH}$ class 1.0
$\omega_1$ Rate at which individuals leave $C_H$ class 0.09
$\omega_2$ Rate at which individuals leave $R_H$ class 0.15
$\tau_1$ TB treatment rate for $L_T$ individuals 2.0
$\tau_2$ TB treatment rate for $I_T$ individuals 1.0
$\phi$ HIV treatment rate for $I_H$ individuals 1.0
$\alpha_1$ AIDS treatment rate 0.33
$\alpha_2$ HIV treatment rate for $A_T$ individuals 0.33
$r$ Fraction of $L_{TH}$ individuals that take HIV and TB treatment 0.3
$d_N$ Natural death rate 1.0/70.0
$d_T$ TB induced death rate 1.0/10.0
$d_A$ AIDS induced death rate 0.3
$d_{TA}$ AIDS-TB induced death rate 0.33
: Model parameters, borrowed from [@SiTo15].[]{data-label="tab:params"}
Two control functions, which represent prevention and treatment measures, are introduced into the model and are continuously implemented during a considered period of disease treatment: the control $u_1(t)$ represents the fraction of $I_{TH}$ individuals that takes HIV and TB treatment, simultaneously; $u_2(t)$ represents the fraction of $I_{TH}$ individuals that takes TB treatment only [@SiTo15].
The transmission dynamics of TB-HIV/AIDS coinfection is modeled by the following system of differential equations: $$\label{eq:model}
\small
\left\{
\begin{array}{l}
\dot{S}(t) = \mu - \lambda_T(t) S(t) - \lambda_H(t) S(t) - d_N S(t),\\
\dot{L}_T(t) = \lambda_T(t) S(t) + \gamma_1 \lambda_T(t) R(t) - (k_1 + \tau_1 + d_N) L_T(t),\\
\dot{I}_T(t) = k_1 L_T(t) - (\tau_2 + d_T + d_N + \delta \lambda_H(t)) I_T(t),\\
\dot{R}(t) = \tau_1 L_T(t) + \tau_2 I_T(t) - (\gamma_1 \lambda_T(t)
+ \lambda_H(t) + d_N) R(t),\\
\dot{I}_H(t) = \lambda_H(t) S(t) - (\rho_1 + \phi + \psi \lambda_T(t) + d_N) I_H(t)
+ \alpha_1 A(t) + \lambda_H(t) R(t) + \omega_1 C_H(t),\\
\dot{A}(t) = \rho_1 I_H(t) + \omega_2 R_H(t) - \alpha_1 A(t) - (d_N + d_A) A(t),\\
\dot{C}_H(t) = \phi I_H(t) + u_1(t) \rho_2 I_{TH}(t) + r k_3 L_{TH}(t)
- (\omega_1 + d_N) C_H(t),\\
\dot{L}_{TH}(t) = \gamma_2 \lambda_T(t) R_H(t) - (k_2 + k_3 + d_N) L_{TH}(t),\\
\dot{I}_{TH}(t) = \delta \lambda_H(t) I_T(t) + \psi \lambda_T(t) I_H(t)
+ \alpha_2 A_T(t) + k_2 L_{TH}(t) - (\rho_2 + d_N + d_T) I_{TH}(t),\\
\dot{R}_H(t) = u_2(t) \rho_2 I_{TH}(t) + (1 - r) k_3 L_{TH}(t)
- (\gamma_2 \lambda_T(t) + \omega_2 + d_N) R_H(t),\\
\dot{A}_T(t) = (1 - (u_1(t) + u_2(t))) \rho_2 I_{TH}(t)
- (\alpha_2 + d_N + d_{TA}) A_T(t).
\end{array}
\right.$$ The model flow is illustrated in Figure \[fig:model:flow\].
![Model for TB-HIV/AIDS transmission.[]{data-label="fig:model:flow"}](model.eps)
The initial conditions are given in Table \[tab:init:conds\].
-------------------- ------------------------------------------------------------ ----------------------
Cathegory Description Initial Value
\[0.1cm\] $S$ Susceptible $\frac{66N(0)}{120}$
\[0.1cm\] ${L}_T$ TB-Latent $\frac{37N(0)}{120}$
\[0.1cm\] $I_T$ TB-Active infected $\frac{5N(0)}{120}$
\[0.1cm\] $R$ TB-Recovered $\frac{37N(0)}{120}$
\[0.1cm\] $I_H$ HIV-Infected (pre-AIDS) $\frac{2N(0)}{120}$
\[0.1cm\] $A$ HIV-infected with AIDS symptoms $\frac{37N(0)}{120}$
\[0.1cm\] $C_H$ HIV-infected under ART therapy $\frac{N(0)}{120}$
\[0.1cm\] $L_{TH}$ TB-Latent co-infected with HIV (pre-AIDS) $\frac{2N(0)}{120}$
\[0.1cm\] $I_{TH}$ HIV-Infected (pre-AIDS) co-infected with active TB $\frac{2N(0)}{120}$
\[0.1cm\] $R_H$ TB-recovered with HIV-infection (pre-AIDS) $\frac{N(0)}{120}$
\[0.1cm\] $A_T$ HIV-Infected with AIDS symptoms co-infected with active TB $\frac{N(0)}{120}$
\[0.1cm\]
-------------------- ------------------------------------------------------------ ----------------------
: Initial conditions for the state variables of the TB-HIV/AIDS model , borrowed from [@SiTo15].[]{data-label="tab:init:conds"}
Note that the period spent in class $I_{TH}$ does not change with the control measures because the controls represent the fraction of individuals that are treated both for TB and HIV and only for TB, and not the treatment duration. Indeed, the period spent in class $I_{TH}$ is given by the constant treatment rate $\rho_2$.
Multiobjective approach to an optimal control problem {#sec:optContolProb}
=====================================================
Traditionally, the problem of finding a control law for a given system is addressed by optimal control theory [@PoBoGrMi62].
Optimal control problem {#subsecOCprob}
-----------------------
In the optimal control approach, the aim is to find the optimal values $u_1^*$ and $u_2^*$ of the controls $u_1$ and $u_2$, such that the associated state trajectories $S^*$, $L_T^*$, $I_T^*$, $R^*$, $I_H^*$, $A^*$, $C_H^*$, $L_{TH}^*$, $I_{TH}^*$, $R_{H}^*$, $A_{T}^*$, are solution of system in the time interval $[0, T]$, with the initial conditions in Table \[tab:init:conds\], and minimize an objective functional. Consider the state system of ordinary differential equations and the set of admissible control functions given by $$\begin{aligned}
\Omega = \{ (u_1(\cdot),u_2(\cdot))
\in (L^{\infty}(0,T))^2 \, | \, & 0 \leq u_1(t),u_2(t) \leq 0.95 \\
& \, \wedge \, u_1(t) + u_2(t) \leq 0.95, \, \forall t \in [0,T] \}.\end{aligned}$$ According to [@SiTo15], the objective functional can be defined as $$\label{oc:problem}
J(u_1(\cdot),u_2(\cdot))
= \int_{0}^{T}\left[ A(t) + A_T(t) + w_1u_1^2(t) + w_2u_2^2(t) \right]dt,$$ where the constants $w_1$ and $w_2$ are a measure of the relative cost of the interventions associated with the controls $u_1$ and $u_2$, respectively. Note that the objective functional is a function of state and control variables. Its minimization implies three important aspects: (i) reducing the number of individuals with AIDS symptoms, (ii) decreasing the number of individuals with AIDS symptoms and active TB disease and (iii) reducing the costs of implementing treatment policies. The optimal control problem consists in determining ($S^*$, $L_T^*$, $I_T^*$, $R^*$, $I_H^*$, $A^*$, $C_H^*$, $L_{TH}^*$, $I_{TH}^*$, $R_{H}^*$, $A_{T}^*$), associated to admissible controls $\left(u_1^*(\cdot), u_2^*(\cdot)\right) \in \Omega$ on the time interval $[0, T]$, satisfying , the initial conditions in Table \[tab:init:conds\], and minimizing the objective functional , , $$J(u_1^*(\cdot), u_2^*(\cdot)) = \min_{\Omega} J(u_1(\cdot), u_2(\cdot)).$$ The approach based on optimal control theory adopted in [@SiTo15] allows to obtain the optimal solution to the cost functional , which is defined from some decision maker’s perspective by means of the constants $w_1$ and $w_2$. However, the choice of the values of $w_1$ and $w_2$ requires some knowledge about the problem and the decision maker’s preferences, which often are not available in advance. Another disadvantage consists in the fact that a single optimal solution to does not provide all useful insights about the optimal strategies and corresponding dynamics. A large range of alternatives remain unexplored and the decision maker is limited in his/her options. In [@SiTo15], numerical simulations to the optimal control problem are performed using $(w_1, w_2)=(25,25)$ and $(w_1, w_2)=(250,25)$. Both values are larger than one, which suggests that they are adapted to the scale of the objectives. The choice of the values of these parameters is not straightforward and there is no guarantee that the best compromise solution has been obtained.
Multiobjective optimization {#subsecMultiobj}
---------------------------
Our work addresses the optimal control problem for the TB-HIV/AIDS coinfection model from a multiobjective perspective. A multiojective optimization problem is formulated by decomposing the cost functional shown in into two components, representing different aspects that must be taken into consideration when dealing with TB-HIV/AIDS. The problem of finding the optimal controls is defined as: $$\label{mo:problem}
\begin{array}{rl}
\text{minimize}
& f_1(A(\cdot),A_T(\cdot))
= \displaystyle \int_{0}^{T} \left[ A(t) + A_T(t) \right] dt, \\[0.2cm]
& f_2(u_1(\cdot),u_2(\cdot))
= \displaystyle \int_{0}^{T} \left[ u_1^2(t) + u_2^2(t) \right] dt, \\[0.2cm]
\text{subject to}
& 0 \leq u_1(t) \leq 0.95, \\
& 0 \leq u_2(t) \leq 0.95, \\
& u_1(t) + u_2(t) \leq 0.95.
\end{array}$$ In the above formulation, the weights are absent and the two objectives represent the medical and economical perspectives, respectively. This naturally reflects the conflicting nature of the underlying decision making problem, hence, solving is interesting and challenging.
Scalarization {#subsec:scalar}
-------------
A traditional mathematical programming approach to solving a multiojective optimization problem consists in transforming an original problem with multiple objectives into a number of single-objective subproblems. This is referred to as *scalarization*. The transformation is performed by means of a scalarizing function with some user-defined parameters. A single Pareto optimal solution is sought by optimizing each subproblem. Repeated runs with different parameter settings for the scalarizing function are used to approximate multiple Pareto optimal solutions.
Several approaches to scalarization have been developed. They differ in the way the scalarizing function is formulated. The weighted sum method suggests minimizing a weighted sum of the objectives [@GaSa55]. The limitation of this method is that solutions can only be obtained in convex regions of the Pareto front. On the other hand, the $\epsilon$-constraint method [@HaLaWi71] suggests optimizing one objective function and converting all other objectives into constraints by setting an upper bound to each of them. This method can find solutions in both convex and nonconvex regions of the Pareto front. The method of weighted metrics [@Mi99] seeks to minimize the distance between the feasible objective region and some reference point. This method is also known as *compromise programming* [@Zeleny1976]. For measuring the distance, a weighted $L_p$ norm is utilized. When the value of $p$ is small, the method may fail to find solutions in nonconvex regions. When $p=\infty$, the method defines the weighted Chebyshev problem [@Bo76]. This problem consists in minimizing the largest weighted deviation of one objective. By optimizing the weighted Chebyshev problem, solutions from convex and nonconvex regions can be generated. A major drawback is that even when the original MOP is differentiable, the resulting single-objective problem is nondifferentiable. Weakly Pareto optimal solutions can be also obtained [@Mi99]. A relaxed formulation of the Chebyshev problem with differentiable scalarizing function is known as the Pascoletti–Serafini scalarization [@PaSe84]. Though, this method introduces one additional variable and one constraint for each objective function. The normal boundary intersection [@DaDe98] uses a hyperplane with evenly distributed points that passes through the extreme points of the Pareto front. For each point on the hyperplane, the method searches for the maximum distance along the normal pointing toward the origin. The normal constraint method [@Me04] suggests optimizing one objective and employing an inequality constraint reduction of the feasible space using the points on the hyperplane. However, there is no guarantee that the solutions obtained by the normal boundary intersection and normal constraint methods are Pareto optimal. A comparative analysis of the different scalarization approaches on optimal control problems from epidemiology can be found in [@MyID:333; @DeSiTo15].
Motivated by the results recently obtained in [@DeSiTo15] for a TB model, we adopt here the $\epsilon$-constraint method [@HaLaWi71]. This method suggests optimizing one objective function while converting all other objectives into constraints by setting an upper bound to each of them. The problem to be solved is of the following form: $$\label{method:eps}
\begin{array}{rlll}
\underset{\boldsymbol{x} \in \Omega}{\text{minimize}} & f_l(\boldsymbol{x}) & \\
\text{subject to} & f_i(\boldsymbol{x})\leq \epsilon_i,
& \forall i \in \{1,\ldots,m\} \wedge i \neq l,
\end{array}$$ where the $l$th objective is minimized, the parameter $\epsilon_i$ represents an upper bound of the value of $f_i$ and $m$ is the number of objectives. The major reasons for using this method are as follows. The $\epsilon$-constraint method is able to find solutions in convex and nonconvex regions of the Pareto optimal front. When all the objective functions in the MOP are convex, problem is also convex and has a unique solution. For any given upper bound vector $\boldsymbol{\epsilon} = \{\epsilon_1, \ldots, \epsilon_{m-1}\}$, the unique solution of problem is Pareto optimal [@Mi99]. Moreover, when considering different scenarios in the model, the optimal solutions obtained for the same values of $\epsilon$ can be used for comparison, as they will lie on the same line in the objective space determined by the corresponding value of $\epsilon$. This characteristic is convenient and helpful for the analysis of the dynamics in the TB-HIV/AIDS model.
Numerical simulations and discussion {#sec:scalar}
====================================
This section presents and discusses numerical results for the optimal controls using the multiobjective optimization approach. Moreover, possible scenarios of applying the control strategies are investigated.
Experimental setup {#sec:setup}
------------------
The fourth-order Runge–Kutta method is used for numerically integrating system . The control and state variables are discretized using $100$ equally spaced time intervals over the period $[0,T]$. The integrals defining objective functions in are calculated using the trapezoidal rule.
Using the formulation , the first objective in is minimized and the second objective is set as the constraint bounded by the value of $\epsilon$. The Pareto front is discretized by defining 100 evenly distributed values of $\epsilon$ over the range of the second objective, $f_2$, which can be calculated as $f_2^{\min}$ for $\left(u_1(\cdot), u_2(\cdot)\right)
\equiv 0$ and $f_2^{\max}$ for $u_1(t) + u_2(t) \equiv 0.95$. It is worth noting that the $\epsilon$-constraint method does not need the information about the range of $f_1$, which is not known beforehand due to the presence of the constraint imposed on $u_1(t) + u_2(t)$, whereas the majority of methods discussed in Section \[sec:scalar\] may need this information. To solve the problems with different values of $\epsilon$, the MATLAB^^ function `fmincon` with a sequential quadratic programming algorithm is used, setting the maximum number of function evaluations to $50,000$. The behavior of the dynamics in the TB-HIV/AIDS model are investigated for the cases when both the controls $u_1$ and $u_2$ are applied separately and simultaneously. In the following, the notation for resulting multiobjective optimization problems (MOPs) will be as shown in Table \[tab:mops\]: MOP1 refers to the case when $u_1$ and $u_2$ are applied simultaneously; MOP2 refers to the case when $u_1$ is applied alone; MOP3 refers to the case when $u_2$ is applied alone. The components of the vector $\boldsymbol{f} = (f_1, f_2)^{\text{T}}$ are as shown in .
MOP1 MOP2 MOP3
------------------------------------------ ------------------------------ ------------------------------
$\boldsymbol{f}(u_1(\cdot), u_2(\cdot))$ $\boldsymbol{f}(u_1(\cdot))$ $\boldsymbol{f}(u_2(\cdot))$
: Different multiobjective optimization problems.[]{data-label="tab:mops"}
Results and discussion
----------------------
Our study investigates optimal control strategies for different treatment periods. The trade-off curves for the treatment scenarios presented in Table \[tab:mops\], considering $T \in \{10, 30, 50\}$, are displayed in Figure \[fig:pf:T\]. Overall, the larger period of study, the higher the number of individuals with AIDS and active TB. As expected, reducing the amount of control measures leads to the increase in the number of individuals with AIDS and active TB, whereas decrease in the number of individuals with AIDS and active TB can be achieved though rising expenses for treatment. These results clearly reflect the conflicting nature of the two objectives. Also, it can be seen that an efficient range of the control policies is limited, as starting from some point the reduction in the number of individuals with AIDS and active TB is possible through exponential increase in expenses for medication. Since available resources are often scarce, scenarios involving high expenses may be practically unacceptable. All curves share similar features and similar trends can be identified through analysis of the obtained solutions. Due to these facts and space limitation, in what follows the obtained results are discussed only for $T=10$.
Figure \[fig:pf\] shows the trade-off curves obtained for MOP1, MOP2 and MOP3, corresponding to $T=10$. As one can see, the three curves share a common point in the objective space. This represents an economic perspective, i.e., there is no treatment of TB and HIV/AIDS, the only focus is on saving money. Naturally, this leads to an uncontrollable spread of the diseases and higher numbers of individuals with AIDS and active TB. As control policies start being implemented, the response of $(A+A_T)$ differs for the three considered cases. The best scenarios from the medical perspective, i.e., when the maximum amounts of controls are applied, are different. Interestingly, the lowest number of $(A+A_T)$ is achieved when only implementing $u_1(\cdot)\equiv 0.95$, which corresponds to the treatment of patients for HIV/AIDS and TB together. In this case, scenarios resulting from MOP1 and MOP2 are identical, being represented by the same point in the objective space. However, when optimal strategies involve the treatment for TB, this allows to decrease the number of $(A+A_T)$ in scenarios representing trade-off between the economic and medical perspectives. It can be understood observing all the intermediate solutions for MOP1 in Figure \[fig:pf\], which give a less value of $f_1$ when comparing with solutions for MOP2 involving the same amount of control measures. Though, treating patients only for TB appears to be an ineffective approach when comparing with scenarios represented by MOP1 and MOP2, as optimal solutions give significantly larger numbers of $(A+A_T)$ along the whole Pareto optimal region.
For the extreme solutions, i.e., those corresponding to the maximum and minimum amounts of controls applied though the period of study, the dynamics of $A$, $A_T$ and $(A+A_T)$ can be observed in Figures \[fig:f0:inds\] and \[fig:max:control\]. Without applying the controls, as discussed above, the corresponding dynamics are identical for MOP1, MOP2 and MOP3 and are presented in Figures \[fig:f0:inds\]. This scenario corresponds to the natural progression of the diseases. When only the control $u_2$ is applied, the number of $A$ and $A_T$, as well as their sum, are larger than for cases when $u_1$ and $u_1+u_2$ are implemented. The dynamics for $A$ and $A_T$ are identical for MOP1 and MOP3, which are depicted in Figure \[fig:max:control:MOP12\].
To analyze intermediate scenarios, two solutions lying on intermediate regions of the trade-off curves are selected. This is done as follows. The objective space is divided by two horizontal lines corresponding to $f_2=3$ and $f_2=6$ (dashed lines in Figure \[fig:pf\]). The intersections of these lines with the trade-off curves give the three different solutions, with each of them corresponding to MOP1, MOP2 and MOP3. These solutions are selected for discussing the dynamics of the diseases. These lines can be interpreted as the constraints defining available resources for treatment. In turn, the selected solutions represent the best treatment options in such circumstances, as they allow to achieve the lowest values of $f_1$. It is worth noting that the solutions on different curves, shown in Figure \[fig:pf\], are identically distributed with respect to the values of $f_2$ due to the use of the $\epsilon$-constrain method. Since MOP1–3 were solved for the same values of $\epsilon$, the corresponding solutions can be used for a fair comparison. Figure \[fig:f3\] presents the trajectories of the control variables and the dynamics of $A$ and $A_T$ for solutions corresponding to $f_2=3$. From Figures \[fig:f3:control:MOP1\]–\[fig:f3:control:MOP3\], it can be seen that the changes of the total amount of control measures are similar. For MOP1, the total control is composed of $u_1$ and $u_2$, which change differently during the period of study. There is a peak in $u_2$, taking place between the third and forth years, after which it decreases. The dynamics of $A$ and $A_T$ have similar trajectories, as shown in Figures \[fig:f3:inds:MOP1\]–\[fig:f3:inds:MOP3\]. However, there is a minor increase in $A$ for MOP3 in the beginning of the forth year. Similar trends for the control and state variables are observed in Figure \[fig:f6\]. Though, the peak in $u_2$ occurs later with a higher value and the number of individuals with AIDS and active TB is lower due to the large amount of medication.
Solutions obtained using multiobjective optimization can provide comprehensive insights about the optimal strategies and the diseases dynamics resulting from implementation of those strategies. Since visual representations can help to better understand results and spot patterns that are not obvious at first, in what follows each of the variables $u_1$, $u_2$, $A$ and $A_T$ is defined as a function of time and the objective to which it is conflicting. By doing so, it is possible to provide the visualization of the entire optimal set of each variable. The set of optimal values defines a surface. Slicing a surface gives the trajectory of the corresponding dynamic over the period of study.
Figure \[fig:surface:controls\] shows the discrete representations of surfaces defined by the controls over the whole Pareto optimal region. On the other hand, the discrete representations of surfaces determined by the responses of $A$ and $A_T$ to optimal treatment strategies are illustrated in Figure \[fig:surface:inds\]. The plots for other classes of human population can be obtained in a similar way.
Conclusions {#sec:conc}
===========
This paper investigates a mathematical model for TB-HIV/AIDS coinfection recently proposed in [@SiTo15]. A multiobjective formulation is proposed. This approach avoids the use of weight parameters and allows to obtain a wide range of optimal control strategies, which offer useful information for effective decision making. Two clearly conflicting objectives are defined for search the optimal controls. The first objective reflects aspirations in controlling TB and HIV/AIDS diseases, whereas the second objective aims to reduce the costs of implementing control policies. The present study extends the previous work [@SiTo15] by the extensive analysis of the optimal control in the TB-HIV/AIDS coinfection model, which enriches the knowledge about the model. Indeed, it is important not just to formulate a model but also to obtain useful information about the process modeled. Our simulation results reveal the optimal treatment strategies for TB and HIV infections and exposure to medication of different fractions of the population. This can be used as an input for planning activities to fight against TB and HIV. The choice of a final solution can be made including the goals of public healthcare and available funds. The results here obtained clearly demonstrate the usefulness and advantages of a multiobjective approach. The presence of the clearly conflicting objectives gives rise to a set of optimal solutions representing different trade-offs between them. Thus, each obtained solution reveals different perspectives on coping with AIDS and active TB diseases. The treatment of individuals infected by both HIV and TB can provide the best effects, except for the extreme scenarios. As analyses showed, the set of optimal trade-off solutions can offer to the decision maker an understanding of all possible trends in applying the controls. Moreover, the dynamics of different classes of individuals in the population appears as a response to the implemented treatment measures. The ability to obtain, analyze and choose from a set of alternatives, constitutes the major advantage of the proposed approach, motivating its practical use in the process of planning intervention measures by health authorities. As future work, we intend to study the effects of the parameters in the TB-HIV/AIDS coinfection model. Also, it would be interesting to investigate the population dynamics resulting from implementation of optimal treatment policies found by optimizing various types of objectives. Considering $L^1$ objectives in the optimal control problem is also the subject of future work.
Acknowledgements {#acknowledgements .unnumbered}
================
Silva and Torres were supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (FCT), within project UID/MAT/04106/2013; and by the FCT project TOCCATA, ref. PTDC/EEI-AUT/2933/2014. Silva is also grateful to the FCT post-doc fellowship SFRH/ BPD/72061/2010. The authors would like to thank two anonymous referees for valuable comments and suggestions.
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[^1]: This is a preprint of a paper whose final and definite form is with ’Computational and Applied Mathematics’, ISSN 0101-8205 (print), ISSN 1807-0302 (electronic). Submitted 04-Feb-2016; revised 11-June-2016 and 02-Sept-2016; accepted for publication 15-March-2017.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We show by resonance effects in HgBa$_2$CuO$_{4+\delta}$ (Hg-1201) and by Zn substitutions in YBa$_2$Cu$_3$O$_{7-\delta}$ (Y-123) compounds that the fully symmetric Raman spectrum has two distinct electronic contributions. The A$_{1g}$ response consists of the superconducting pair breaking peak at the 2$\Delta $ energy and a collective mode close to the magnetic resonance energy. These experimental results reconcile the *d-wave* model to the A$_{1g}$ Raman response function in so far as a collective mode that is distinct from the pair breaking peak is present in the A$_{1g}$ channel.'
address: |
$^{1}$Laboratoire de Physique du Solide ESPCI, 10 rue Vauquelin, 75231 Paris, France\
$^{2}$Matériaux et phénom$\grave{e}$nes Quantiques (UMR 2437 CNRS), Université Paris 7, 2 place Jussieu 75251 Paris, France\
$^{3}$Service de Physique de l’Etat Condensée, CEA-Saclay, 91191 Gif-sur-Yvette, France
author:
- 'M. Le Tacon$^{1,2}$, A. Sacuto$^{1,2}$, and D. Colson$^{3}$'
title: 'Two Distinct Electronic Contributions in the Fully Symmetric Raman Response of High $T_{c}$ Cuprates'
---
.
In the last few years, it has been well established that the superconducting gap of the hole-doped cuprates at the optimal doping regime has the $d_{x^2-y^2}$ symmetry [@Dev94; @Kang96; @Sacuto00].
This symmetry manifests itself in the low energy part of the Raman spectra. In the B$_{2g}$ channel [@polar] (probing the nodal directions), the electronic continuum behaves as a linear function of the Raman shift, while it follows a cubic law in the B$_{1g}$ channel [@polar] (anti-nodal directions)(see ref. ). In the latter one, a well defined pair breaking peak near 2$\Delta$ = 8$k_{B}T_c$ is observed. However, existing theories based on the $d_{x^2-y^2}$ model, fail to reproduce the position, the intensity, and the shape of the broad electronic peak observed in the fully symmetric A$_{1g}$ channel [@polar; @Sacuto00; @Wenger97; @Strohm97]. Expansion of the Raman vertex to the second order of the Fermi surface harmonics [@Dev95] and resonant effects [@Sherman02] have been proposed to reproduce the relative A$_{1g}$ peak position and intensity with respect to that of B$_{1g}$. In these pictures, the A$_{1g}$ peak is treated as another manifestation of the pair breaking peak observed in the B$_{1g}$ channel. Unfortunately, the back flow prevents the reproduction of the location, on one hand, and on the other hand, of the sharpness and the strong intensity of the A$_{1g}$ peak. For a generic tight-binding model, the calculated screened A$_{1g}$ channel is only a tiny fraction of the B$_{1g}$ response [@Wenger97]. This is in clear contradiction to all experiments and most studies showing the magnitude of the A$_{1g}$ peak being even larger than the B$_{1g}$ [@Gallais02; @Chen94; @Gasparov97; @Sacuto98]. In this paper we show that the A$_{1g}$ response has two components: one component originating from the pair breaking close to the 2$\Delta$ energy and the other from a collective mode which tracks the magnetic resonance [@Gallais02; @MLT]. In this sense, our experimental results reconcile the A$_{1g}$ Raman response of the cuprates at the optimal doping regime with the *d-wave* model in so far as a collective mode is present in the A$_{1g}$ channel.
Electronic Raman Scattering (ERS) measurements have been carried out with a JY T64000 triple spectrometer in subtractive configuration using different lines of mixed Argon - Krypton laser gas. The Raman spectra were corrected for the spectrometer response, the Bose factor and the optical constants producing the imaginary part $\chi^{\prime\prime}(\omega)$ of the Raman response. The crystals were mounted in vacuum (10$^{-6}$ mbar) on the cold finger of a liquid-helium flow cryostat. The power density was about 10W/cm$^{2}$ on the sample surface, and the laser spot heating estimated from the Anti-Stokes/Stokes intensity ratio of the Raman responses was less than 3K.
Let’s focus first on ERS measurements of optimally doped Hg1201 single crystals ($T_c$=95K). They have been grown by flux method whose detailed procedure is described elsewhere [@Colson94]. Figure \[fig:plane\] shows the superconducting Raman responses $\chi^{\prime\prime}_{S}(\omega)$ of Hg-1201 obtained for various excitations lines in A$_{1g}$ and B$_{1g}$ channels.
![Raman responses $\chi^{\prime\prime}_{S}(\omega)$ of optimally doped Hg-1201 for different excitations lines in the A$_{1g}$ (black line) and B$_{1g}$ (gray line) channels. The insets exhibit $\chi^{\prime\prime}_{S}(\omega)$- $\chi^{\prime\prime}_{N}(\omega)$ for both A$_{1g}$ and B$_{1g}$ channels.[]{data-label="fig:plane"}](figure1.EPS){width="9.5cm"}
The Raman responses are composed of a broad electronic continuum surrounded by an assembly of narrow peaks corresponding to the well identified phonons [@Krantz94]. At first glance, the Raman responses for each excitation line (E.L.) reveal that the A$_{1g}$ continuum exhibits a strong maximum around 330 cm$^{-1}$, with an asymmetric part in its high energy side. This manifests itself as a bump for blue (488 nm) and green (514 nm) E.L., and as a “plateau” for yellow (568 nm) and red (647 nm) ones, which are around 520 cm$^{-1}$ near the maximum of the B$_{1g}$ continuum that corresponds to the pair breaking peak.
The Raman responses of the blue and green lines show strong phonon features super-imposed to the electronic continuum near 520 cm$^{-1}$ which complicates the extraction of the electronic background. On the contrary, under the yellow and red E.L., the phonon modes are out of resonance thus their structures are strongly reduced and the electronic contribution can be easily extracted. Subtractions of the normal $\chi^{\prime\prime}_{N}(\omega)$ response from the superconducting $\chi^{\prime\prime}_{S}(\omega)$ one are reported in the insets of Figure 1. The Raman responses $\chi^{\prime\prime}_{S}(\omega)$-$\chi^{\prime\prime}_{N}(\omega)$ for the yellow and red lines are almost free of phonon contribution. The broad continua in the A$_{1g}$ and B$_{1g}$ channels correspond to the electronic contributions from the superconducting state, and the sharp features show misfits between the superconducting and normal phonon structures. After substraction of the normal state contribution, the A$_{1g}$ response is still asymmetric, and for each E.L., the high energy part of this response is centered near the maximum of the B$_{1g}$ superconducting gap. The asymmetry of the A$_{1g}$ response is thus intrinsic to the superconducting state.
To go further and prove that the broad A$_{1g}$ peak consists effectively of two distinct electronic components, we have performed ERS measurements on high quality optimally doped YBCO single crystals grown by the self flux method [@Kaiser87], where copper is substituted by zinc. Zn is a divalent ion known to substitute preferentially in the CuO$_{2}$ layers without altering the carrier concentration [@Bobroff99]. In addition to the pure YBa$_{2}$Cu$_{3}$O$_{7-\delta }$(Y-123, *T*$_{c}$= 92K), we have studied YBa$_{2}$(Cu$_{1-y}$Zn$_{y})_{3}$O$_{7-\delta }$ single crystals with y=0.005 (*T*$_{c}$=87K), y=0.01 (*T*$_{c}$=83K), y=0.02 (*T*$_{c}$=73K) and y=0.03 (*T*$_{c}$= 64K). Zn concentration was verified by chemical analysis using an electron probe. $T_c$ measurements were obtained from DC- magnetization and we found $dT_c/dy \sim$ - 10K/[%]{}.
Figure \[fig:Zn\] shows the $\chi^{\prime\prime}_{S}(\omega)$-$\chi^{\prime\prime}_{N}(\omega)$ Raman responses in A$_{1g}$ and B$_{1g}$ channels in Y-123 for various Zn contents. Insets exhibit the A$_{1g}$ and B$_{1g}$ Raman responses in the normal and superconducting states before subtraction. The A$_{1g}$ and B$_{1g}$ Raman responses show a set of sharp phonon peaks lying on a strong electronic background. In the A$_{1g}$ channel, for pure YBCO, the $\chi^{\prime\prime}_{S}(\omega)$-$\chi^{\prime\prime}_{N}(\omega)$ Raman response shows a broad and strong asymmetric peak which spreads out in the high energy side and reaches its maximum close to 330 cm$^{-1}$. In the B$_{1g}$ channel, the $\chi^{\prime\prime}_{S}(\omega)$-$\chi^{\prime\prime}_{N}
(\omega)$ response for pure YBCO, exhibits a well defined and nearly symmetric peak close to 530 cm$^{-1}$. These A$_{1g}$ and B$_{1g}$ superconducting spectra are very similar to those obtained from Hg-1201 at the optimal doping. Here again, the A$_{1g}$ response exhibits a maximum around 330 cm$^{-1}$ with an asymmetric part which extends up to the pair breaking peak near 530 cm$^{-1}$. The positions of the A$_{1g}$ and B$_{1g}$ peaks are nearly the same for both Y-123 and Hg-1201. The changes in the band structure induced by two CuO$_{2}$ layers instead of one CuO$_{2}$ layer do not affect the A$_{1g}$ and B$_{1g}$ peak positions rather, the critical temperature (92 K for Y-123 and 95 K for Hg-1201) seems to govern the A$_{1g}$ and B$_{1g}$ peak energies at the optimal doping. This is observed for many cuprates where the B$_{1g}$ peak is found close to 8$k_{B}T_c$ and the A$_{1g}$ peak maximum close to 5$k_{B}T_c$ at the optimal doping (see Table 1 of Ref. ). Adding some Zn in the pure Y-123, one can see that the intensity of the pair breaking peak seen in the B$_{1g}$ channel decreases, but the peak does not disappear and is still present even in the sample with y=0.03 ($T_c$ =63K), contrary to what is suggested in Ref. . The B$_{1g}$ peak does not shift in energy, and thus does not follow $T_c$, but this effect and other related to nonmagnetic impurity substitutions in YBCO will be discuss in a next paper.
![$\chi^{\prime\prime}_{S}(\omega)-\chi^{\prime\prime}_{N}(\omega)$ A$_{1g}$ (left panel) and B$_{1g}$ (right panel) Raman responses of optimally doped YBa$_{2}$(Cu$_{1-y}$Zn$_{y})_{3}$O$_{7-\delta }$ for various Zn concentrations y. $\lambda$=514nm. In each inset are plotted the Raman response functions in the normal (gray line) and superconducting (black line) states.[]{data-label="fig:Zn"}](figure2.EPS){width="9.5cm" height="15.5cm"}
Let’s focus now on the A$_{1g}$ channel. As Zn content increases, the low energy contribution in the A$_{1g}$ response becomes broader, shifts to a lower energy (from 330 to 300 cm$^{-1}$), strongly decreases in its intensity, and finally disappears. On the contrary, the high energy contribution in the A$_{1g}$ response moderately decreases in its intensity but keeps the same position at 530 cm$^{-1}$.
For y=0.01, the $\chi^{\prime\prime}_{S}(\omega)$-$\chi^{\prime\prime}_{N}(\omega)$ A$_{1g}$ response, clearly shows two components. The first is centred at 300 cm$^{-1}$ and the second is close to 530 cm$^{-1}$. For higher Zn concentrations, the intensity ratio of the upper and the lower energy parts of the A$_{1g}$ response, is reversed in such a way that for y=0.03, the lower energy component completely disappears while the upper component persists. A straightforward comparison between the left and right panels reveals that the high energy component (530 cm$^{-1}$) in the A$_{1g}$ response tracks the B$_{1g}$ peak. For both y=0.02 and y=0.03 when the lower energy component in the A$_{1g}$ response is weak and does no longer mix with the higher energy component, the ratio of the spectral weight between the higher energy component in the A$_{1g}$ response and the pair breaking peak in the B$_{1g}$ response remains constant. In these cases the peaks observed in A$_{1g}$ and B$_{1g}$ channels are located at the same energy [@com2] and correspond both to the pair breaking peak.
This gives experimental evidence that the A$_{1g}$ response has two distinct components and that the one of higher energy corresponds to the pair breaking peak. As the B$_{1g}$ one probes the anti-nodal directions of the $d_{x^2-y^2}$ superconducting gap, and the A$_{1g}$ Raman response has no symmetry restriction, it is therefore not surprising to observe the pair breaking peak in both A$_{1g}$ and B$_{1g}$ channels. The low energy component of the A$_{1g}$ Raman response corresponding to the maximum of the electronic continuum is intrinsic to the superconducting state and disappears above $T_c$ as it was already pointed out in previous works [@Gallais02; @MLT]. The A$_{1g}$ peak is located at 5$k_BT_c$ well below the 2$\Delta$ energy gap (8$k_BT_c$) and therefore cannot be induced by individual electronic excitations which required energies beyond 2$\Delta$. As a consequence the A$_{1g}$ mode has to be a bound state of quasi-particle pairs at an energy less than 2$\Delta$ and refers to a collective mode. We have not yet identified the origin of the A$_{1g}$ mode but several scenarios can be figured out. Among them the Bogoliubov-Anderson collective mode [@Anderson] calculated in the frame work of the *d-wave* model merits to be considered as well as a double magnon with a zero spin flip energy as suggested by E. Demler [@Demler]. Zero spin flip energy is possible for a *d-wave* superconductor if we consider spin flip excitations over two nodal regions.
Zero spin flip quasi-particle excitations have already been invoked to explain the quadratic increase of the $^{17}$O spin-lattice relaxation rate under magnetic fields accross the vortex lattice NMR spectrum in YBCO [@Vesna]. In our case, the A$_{1g}$ peak tracks the acoustic magnetic resonance detected by inelastic neutron scattering [@Rossat91] at **Q**=($\pi,\pi$) as previously shown [@Gallais02; @MLT]. A double spin flip of transfered momenta **Q**=($\pi,\pi$) and **Q**=($-\pi,-\pi$) is then needed for preserving the total transfer momentum close to zero in the Raman scattering process. The first spin flip is over two anti-nodal regions and costs the magnetic resonance energy whereas the second spin flip over two nodal regions costs zero energy. In this Raman process the A$_{1g}$ mode takes the same energy as the magnetic resonance as expected experimentally. Theoretical investigations on this last scenario are in progress.
In summary, the ERS spectra of Hg-1201, free of phonon peaks, reveal that the A$_{1g}$ response and its asymmetry near the pair breaking peak are of electronic origin. Moreover, ERS in Y-123 substituted with Zn shows that the A$_{1g}$ peak has two distinct components: one at the higher energy corresponding to the pair breaking peak observed in the B$_{1g}$ channel and the other at lower energy corresponding to another electronic contribution that is distinct from the pair breaking peak. This study reconciles the A$_{1g}$ Raman response function with the *d-wave* model where the pair breaking peak manifests itself in both B$_{1g}$ and A$_{1g}$ channels. This implies the existence of a charge collective mode below the pair breaking peak energy which we have previously related to the magnetic resonance.
**Acknowledgements** We wish to thank M. Cazayous, Y. Gallais, V.F. Mitrovic, V.N. Muthukumar, E.Demler, A. Benlagra and S. Nakamae for very fruitful discussions.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We give an alternative method to that of Hardy-Ramanujan-Rademacher to derive the leading exponential term in the asymptotic approximation to the partition function $p(n,a)$, defined as the number of decompositions of a positive integer $n$ into integer summands, with each summand appearing at most $a$ times in a given decomposition. The derivation involves mapping to an equivalent physical problem concerning the quantum entropy and energy currents of particles flowing in a one-dimensional (1D) channel connecting thermal reservoirs, and which obey Gentile’s intermediate statistics with statistical parameter $a$. The method is also applied to partitions associated with Haldane’s fractional exclusion statistics.'
address: 'Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755'
author:
- 'Miles P. Blencowe[@auth] and Nicholas C. Koshnick'
title: Partition asymptotics from 1D quantum entropy and energy currents
---
= 9truein = .5 truein
{#section .unnumbered}
A classic result in the theory of partitions is the Hardy-Ramanujan-Rademacher formula for the unrestricted partition function $p(n,\infty)$, wherein the latter, combinatoric quantity is represented as a power series whose terms involve elementary functions of $n$.[@hardy; @rademacher; @andrews] This series yields the following asymptotic approximation: $$p(n,\infty)\sim\frac{1}{4\sqrt{3} n} e^{\pi\sqrt{2/3}\sqrt{n}}.
\label{asympt0}$$ A series representing $p(n,1)$, the number of decompositions of $n$ into distinct summands, has also been derived (see, e.g., Sec. 24.2.2 of Ref. ), yielding the asymptotic approximation $$p(n,1)\sim\frac{1}{4\cdot 3^{1/4}\cdot n^{3/4}}
e^{\pi\sqrt{1/3}\ \sqrt{n}}.
\label{asympt1}$$ And more recently,[@hagis] Hagis used the Hardy-Ramanujan-Rademacher method to derive a power series representation of $p(n,a)$ for arbitrary $a=1,2,\ldots$, yielding the asymptotic approximation $$p(n,a)\sim\frac{\sqrt{12}\ a^{1/4}}{(1+a)^{3/4}\ (24 n)^{3/4}}
e^{\pi\sqrt{2a/[3(1+a)]}\ \sqrt{n}},
\label{asympt2}$$ where $n\gg a$. As an example, for $a=4$ the number of partitions of $n=1000$ to five significant figures is $2.4544 \times 10^{28}$, while approximation (\[asympt2\]) gives $2.4527 \times 10^{28}$, accurate to within $0.1\%$.
In the present work, we give an alternative and more direct derivation of the asymptotic approximation to $\ln p(n,a)$ which, from Eq. (\[asympt2\]), is: $$\ln p(n,a) \sim \pi\sqrt{\frac{2 a}{3(1+a)}}\cdot\sqrt{n}.
\label{lnasympt}$$ The derivation begins by considering a 1D quantum channel which supports particles obeying Gentile’s intermediate statistics[@gentile] characterised by statistical parameter $a$, the maximum occupation number of particles in a single particle state, with $a=1$ describing fermions and $a=\infty$ bosons. The left end of the channel is connected to a particle source and the right end to a particle sink. The channel is dispersionless so that particle packets with different mean energies have the same velocity $c$ and hence transmission time ${\tau}=L/c$, where $L$ is the channel length. Imposing periodic boundary conditions on the channel length, the single-particle energies are $\epsilon_{j}=h f_{j} =h j/{\tau}$, $j=1,2,\dots$, where $h$ is Planck’s constant. The total energy $E_{n}$ of a given Fock state is $E_{n}=\sum_{j} \epsilon_{j}
n_{j} ={nh}/{\tau}$, where $n=\sum_{j=1}^{\infty}j n_{j}$, and $n_{j}\leq a$ is the occupation number of, say, the right-propagating mode $j$.
We now suppose that the source emits a finite number of particles with fixed total energy $E_{n}$. The maximum possible entropy of this collection of right-propagating particles subject to the fixed energy constraint is $S(n,a)=k_{B}\ln
p(n,a)$. Thus, the problem to determine the asymptotic approximation to $\ln p(n,a)$ is equivalent to determining the asymptotic approximation to the entropy $S(n,a)$ of the just-described physical system. (C.f. Sec. 4 of Ref. , where the same set-up restricted to bosons was considered in the problem to determine the optimum capacity for classical information transmission down a quantum channel.)
The crucial next step is to consider a slightly different set-up, in which the particle source and sink are replaced by two thermal reservoirs described by grand canonical ensembles, with the chemical potentials of the left and right reservoirs satisfying $\mu_{L}=\mu_{R}=0$, the temperature of the right reservoir $T_{R}=0$, and the temperature $T_{L}$ of the left reservoir chosen such that the thermal-averaged energy current flowing in the channel satisfies $\dot{\bar{E}}(T_{L},a)=E_{n}/{\tau}$. (Note that the chemical potentials are set to zero since there is no constraint on the thermal-averaged particle number.) With this choice, the thermal-averaged, channel entropy current $\dot{\bar{S}}(T_{L},a)$ coincides with $S(n,a)/{\tau}$ in the thermodynamic limit $E_{n}$ (equivalently $n)\rightarrow\infty$.
The advantage with using the latter, grand canonical ensemble description as opposed to the former, microcanonical ensemble description is the greater ease with which the energy and entropy currents can be calculated. The starting formula for the single channel energy current is: $$\dot{\bar{E}}(T,a)=\sum_{j=1}^{\infty} \epsilon_{j}
\left[\bar{n}_{a}(\epsilon_{j})/L\right]c,
\label{startenergy}$$ where we have dropped the subscript on $T_{L}$, and where $\bar{n}_{a}(\epsilon)$ is the intermediate statistics thermal-averaged occupation number of the right-moving state with energy $\epsilon$:[@gentile] $$\bar{n}_{a}(\epsilon)=\frac{1}{e^{\beta E}-1}-\frac{a+1}{e^{\beta E (a+1)}-1}.
\label{ioccup}$$ In the limit $L\rightarrow\infty$ (equivalently $\tau\rightarrow\infty$), we can replace the sum with an integral over $j$ and, changing integration variables $j\rightarrow \epsilon = (h/{\tau}) j=(hc/L) j$, we have \[c.f. Eq. (13) of Ref. \]: $$\dot{\bar{E}}(T,a)=\frac{1}{h}
\int_{0}^{\infty}d\epsilon \epsilon \bar{n}_{a}(\epsilon).
\label{intenergy}$$ A formula for entropy current can be derived as follows. First note that the thermal-averaged occupation energy $\bar{\epsilon}=
\epsilon\bar{n}_{a}(\epsilon)$ and the entropy $\bar{s}$ for a given mode with energy $\epsilon$ are related through the first law: $d\bar{s}/dT
=(1/T)d\bar{\epsilon}/dT$. Integrating with respect to temperature and then summing over the right propagating channel modes, we obtain $$\dot{\bar{S}}(T,a)=-\frac{k_{B}}{h}
\int_{0}^{\infty}d\epsilon \epsilon
\int_{beta}^{\infty} d\beta' \beta'\frac{\partial \bar{n}_{a}}
{\partial\beta'}.
\label{intentropy}$$
The integrals are straightforwardly carried out by noting from (\[ioccup\]) that the thermal-averaged occupation energy $\bar{\epsilon}=
\epsilon\bar{n}_{a}(\epsilon)$ of level $\epsilon$ for statistical parameter $a$ is just the difference in the thermal-averaged occupation energies of levels $\epsilon$ and $\epsilon(a+1)$ for bosons. Thus, we require only the integrals for the bosonic case: $\dot{\bar{E}}(T,\infty)=\pi^{2}(k_{B}T)^{2}/(6h)$ and $\dot{\bar{S}}(T,\infty)=\pi^{2}k_{B}^{2}T/(3h)$, giving $$\dot{\bar{E}}(T,a)=\left(1-\frac{1}{1+a}\right)\frac{\pi^{2}(k_{B}T)^{2}}{6h}
\label{intenergyf}$$ and $$\dot{\bar{S}}(T,a)=\left(1-\frac{1}{1+a}\right)\frac{\pi^{2}k_{B}^{2} T}{3h}.
\label{intentropyf}$$
Comparing powers of $T$ appearing in Eqs. (\[intenergyf\]) and (\[intentropyf\]), and recalling that $\dot{\bar{E}}(T,a)=E_{n}/{\tau}$ and $\dot{\bar{S}}(T,a) \sim S(n,a)/{\tau}$, we learn immediately that $\ln
p(n,a) \sim C(a)\sqrt{n}$, where the $n$-independent factor $C(a)$ is given by $$C(a)=\frac{\sqrt{h}\dot{\bar{S}}(T,a)}{k_{B}\sqrt{\dot{\bar{E}}(T,a)}}.
\label{theformula}$$ Substituting in the expressions (\[intenergyf\]) and (\[intentropyf\]) for $\dot{\bar{E}}$ and $\dot{\bar{S}}$, respectively, we finally obtain $C(a) = \pi\sqrt{2 a/[3(1+a)]}$, in agreement with Eq. (\[lnasympt\]).
We will now carry out the same steps as above for particles obeying Haldane’s fractional exclusion statistics[@haldane] to derive the asymptotic approximation to the logarithm of yet another type of partition function, $\tilde{p}(n,g)$, which also interpolates between the unrestricted and distinct partition functions \[Eqs. (\[asympt0\]) and (\[asympt1\]), respectively\]. Following the usual conventions, the statistics parameter is denoted by $g=1/a$ (so that $g=0$ describes bosons and $g=1$ fermions). Partitions associated with exclusion statistics are subject to additional constraints as compared with partitions associated with intermediate statistics (see below).
The energy and entropy currents for particles obeying exclusion statistics are[@rego; @krive] $$\dot{\bar{E}}(T,g)=\frac{(k_{B}T)^{2}}{h}
\int_{0}^{\infty}dx x \bar{n}_{g}(x)
\label{halenergy}$$ and $$\begin{aligned}
\dot{\bar{S}}(T,g)=&-&\frac{k_{B}^{2}T}{h}
\int_{0}^{\infty}dx
\left \{ \bar{n}_{g}\ln \bar{n}_{g}+(1-g \bar{n}_{g}) \ln(1-g \bar{n}_{g})\right.\cr
&-&\left. [1+(1-g) \bar{n}_{g}]
\ln [1+(1-g) \bar{n}_{g}]\right \},
\label{halentropy}\end{aligned}$$ where $x=\beta \epsilon$ and the thermal-averaged occupation number is[@wu] $$\bar{n}_{g}(x)=
\left[w(x)+g\right]^{-1},
\label{hoccup1}$$ with the function $w(x)$ given by the implicit equation $$w(x)^{g}[1+w(x)]^{1-g}=e^{x}.
\label{hoccup2}$$
Again, comparing powers of $T$ appearing in Eqs. (\[halenergy\]) and (\[halentropy\]), we learn immediately that $\ln
\tilde{p}(n,g) \sim \tilde{C}(g)\sqrt{n}$, where the $n$-independent factor $\tilde{C}(g)$ is given in terms of $\dot{\bar{E}}$ and $\dot{\bar{S}}$ as in Eq. (\[theformula\]). Substituting in the expressions for $\dot{\bar{E}}$ and $\dot{\bar{S}}$ and performing a change of variables from $x$ to $w$,[@rego] Eq. (\[theformula\]) becomes after some algebra $$\tilde{C}(g)=\frac{s(g)}{\sqrt{e(g)}},
\label{theformula2}$$ where $$e(g)= \int_{w_{g}(0)}^{\infty} dw
\frac{1}{w(1+w)}\left[(1-g)\ln (1+w) +g\ \ln w\right]
\label{halenergyf}$$ and $$s(g)= \int_{w_{g}(0)}^{\infty}dw
\left[\ln(1+w)/w-\ln w/(w+1)\right].
\label{halentropyf}$$ Using the identity $s(g)=2
e(g)$, Eq. (\[theformula2\]) can be further simplified to $$\tilde{C}(g)=\sqrt{2 s(g)}.
\label{theformula3}$$
Let us now describe some of the properties and consequences of result (\[theformula3\]). Integral (\[halentropyf\]) can be rewritten in terms of dilogarithms [@lee] and only for certain choices of lower integration limit do closed-form solutions exist. For example, from (\[hoccup2\]) we have $w_{g=0}(0)=0$ and $w_{g=1}(0)=1$ and solving the respective integrals, we obtain $s(0)=\pi^{2}/3$ and $s(1)=\pi^{2}/6$. Substituting these values into (\[theformula3\]), we indeed obtain the arguments of the exponentials in the asymptotic approximations to the unrestricted and distinct partition functions, Eqs. (\[asympt0\]) and (\[asympt1\]) respectively. It is tempting to speculate that closed-form solutions to the integral $s(g)$ exist only for $g=0$, $1/2$, $1/3$, $1/4$, and $1$ in the interval $[0,1]$, since it is only for these rational values that Eq. (\[hoccup2\]) can be solved analytically for the lower integration limit $w_{g}(0)$. For $g=1/2$, we have $w_{1/2}(0)=(-1+\sqrt{5})/2$ and $s(1/2)=\pi^{2}/5$, so that $$\ln \tilde{p}(n,1/2) \sim \pi\sqrt{2/5}\cdot\sqrt{n}.
\label{lnasympt2}$$ Note that $\tilde{C}_{g=1/2}\ (=\pi\sqrt{2/5})< C_{a=2}\ (=2\pi/3)$, signalling the fact that $\tilde{p}(n,g)<p(n,a=1/g)$ for $0<g<1$, a consequence of additional constraints on the allowed partitions associated with Haldane’s statistics. These constraints are discussed in Ref. . The above, closed-form solutions for $g=0$, $g=1/2$, and $1$ were obtained by solving the integral $s(g)$ numerically and then noting that the result when divided by $\pi^{2}$ was rational. This method does not work for the $g=1/3,1/4$ cases, however, owing to the complicated form of the lower limits $w_{1/3}(0)$ and $w_{1/4}(0)$ (they are roots of third and fourth degree polynomial equations, respectively). A more sophisticated method is required in order to determine whether or not closed-form solutions exist for these latter two cases.
We would like to thank Peter Hagis Jr. and George Andrews for very helpful correspondences. Discussions with Makoto Itoh and Jay Lawrence are also gratefully acknowledged.
Electronic address: [email protected] G.H. Hardy and S. Ramanujan, Proc. London Math. Soc. (2) [**17**]{}, 75 (1918). H. Rademacher, Proc. London Math. Soc. (2) [**43**]{}, 241 (1937). G.E. Andrews, [*The Theory of Partitions*]{} (Addison-Wesley, Reading, MA, 1976). M. Abramowitz and I.A. Stegun, Eds., [ *Handbook of Mathematical Functions*]{} (U.S. G.P.O., Washington, DC, 1964). P. Hagis, Trans. Amer. Math. Soc. [**155**]{}, 375 (1971). G. Gentile, Nuovo Cimento [**17**]{}, 493 (1940). See also S. Katsura, K. Kaminishi, and S. Inawashiro, J. Math. Phys. [ **11**]{}, 2691 (1970) and references therein. C.M. Caves and P.D. Drummond, Rev. Mod. Phys. [**66**]{}, 481 (1994). U. Sivan and Y. Imry, Phys. Rev. B [**33**]{}, 551 (1986). F.D.M. Haldane, Phys. Rev. Lett. [**67**]{}, 937 (1991). L.G.C. Rego and G. Kirczenow, Phys. Rev. B [**59**]{}, 13080 (1999). I.V. Krive and E.R. Mucciolo, Phys. Rev. B [**60**]{}, 1429 (1999). Y.-S. Wu, Phys. Rev. Lett. [**73**]{}, 922 (1994). Interestingly, if the chemical potential of the left reservoir is non-zero, then the lower integration range of Eq. (18) tends to zero in the degenerate limit so that the single channel entropy current is universal, i.e., independent of the particle statistics. In general, the formulation of thermodynamical quantities in terms of polylogarithms is a powerful method for revealing equivalences between different statistical systems. See M.H. Lee, Phys. Rev. E [**55**]{}, 1518 (1997); J. Math. Phys. [**36**]{}, 1217 (1994). M.V.N. Murthy and R. Shankar, Phys. Rev. B [**60**]{}, 6517 (1999).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The hierarchy problem in particle physics has recently been approached from a geometric point of view in different models. These approaches postulate the existence of extra dimensions with various geometric properties, to explain how the hierarchy between the apparent scale of gravity ${\bar M}_P \sim 10^{18}$ GeV and the weak scale $m_W \sim 100$ GeV can be generated. Generally, these models predict that the effects of gravity mediated interactions become strong at the weak scale. This fact makes the NLC a promising tool for testing such extra dimensional models.'
address: 'Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309, USA'
author:
- Hooman Davoudiasl
title: 'Probing the Geometry of the Universe at the NLC[^1]'
---
SLAC–PUB–8767\
January 2001
Introduction
============
The gravitational potential $V(r)$ of a test mass $m_t$ at a distance $r$ is observed to be
$$V(r) = - \, G_N \frac{m_t}{r},
\label{V}$$
where $G_N$ is the 4-$d$ Newton’s constant. Thus, gravitational interactions can be described by a non-renormalizable field theory, where the spin-2 graviton mediates the force, and couples to the energy momentum tensor with dimensionful coupling $1/{{\bar M}_P}$, where ${{\bar M}_P}\sim G_N^{-1/2}\sim 10^{18}$ GeV. However, the electroweak interactions have a typical scale of order the $W$ mass $m_W \sim 100$ GeV. If the Higgs boson of the Standard Model (SM) is responsible for the electroweak symmetry breaking, then we expect that the mass of the Higgs $m_H \sim m_W$. Thus, $m_H$ seems to be stable against $O({{\bar M}_P})$ quantum corrections. Explaining the origin of the large ratio ${{\bar M}_P}/m_H\sim 10^{16}$ is referred to as the hierarchy problem in particle physics.
There have been a number of proposals for solving the hierarchy problem. An interesting and theoretically appealing proposal is low energy supersymmetry. In a supersymmetric solution, new fields are added to the Lagrangian, such that every known field will have a superpartner of weak scale mass. However, there is, as yet, no experimental evidence for this and other ideas requiring the discovery of new particles around the weak scale.
A new class of ideas approaches the hierarchy problem from a geometric point of view. Instead of postulating extra fields, such as in supersymmetry, one postulates the existence of extra dimensions in the universe. Here, we present two models that approach the question of hierarchy from an extra dimensional viewpoint. The first one, due to Arkani-Hamed, Dimopoulos, and Dvali (ADD) [@ADD] uses $n > 1$ large extra dimensions; we only briefly discuss this model. The second model, due to Randall and Sundrum(RS) [@RS], assumes a warped 5-$d$ universe, and is the main subject of what follows.
Large Extra Dimensions
======================
In the ADD model, the assumption is that the fundamental scale of gravity in $(4 + n)$-$d$ is $M_F$. The gravitational potential $V(r)$ at distances $r \gg R$, where $R$ is the typical size of the extra dimensions, is given by Gauss’ law $$V(r) = - \, G_N \frac{m_t}{M_F^{(2 + n)} R^n \, r}.
\label{V4+n}$$ To recover the observed gravitational force, we must have $${{\bar M}_P}^2 \sim M_F^{(2 + n)} R^n.
\label{mp}$$ Now, if we require that the $M_F \sim m_W$, in order to eliminate the hierarchy between the two scales, we are forced to have large extra dimensions of size 1 fm$\lsim R \lsim$ 1mm , for $2 \leq n \leq 6$. The case $n = 1$ is ruled out, since it requires $R \sim 1$ AU, which would result in deviations in Newtonian gravity at the scale of the solar system.
In the ADD scenario with large extra dimensions, (i) there is a Kaluza-Klein tower of gravitons with mass $m_n \sim n/R$ with equal spacing; (ii) each KK mode couples with $1/{{\bar M}_P}$ in 4-$d$; (iii) the KK tower at energies $\sqrt s$ $\sim M_F \sim 1$ TeV interacts strongly, only suppressed by $1/M_F$, due to the KK multiplicity of $O(10^{16})$; (iv) the SM resides on a 4-$d$ wall in a $(4 + n)$-$d$ bulk; (v) the geometry is factorizable, and the $n$ extra dimensions are flat, that is the metric is of the form $$ds^2 = \eta_{\mu \nu} \, dx^\mu dx^\nu + \sum_{i = 4}^{3 + n} dx_i^2.
\label{addmetric}$$
The RS Model
============
This model [@RS] is based on a 5-$d$ spacetime of constant negative curvature, called $AdS_5$, truncated by two 4-$d$ Minkowski walls, separated by a fixed distance $L = \pi \, r_c$ with $r_c \sim {{\bar M}_P}^{-1}$ as the compactification scale; the $5^{th}$ dimension $y$ is parameterized by an angular variable $\phi \in [-\pi, \pi]$ and $y = \phi \, r_c$. The geometry is required to respect the $Z_2$ symmetry $\phi \to -\phi$. The “Planck wall” is at $\phi = 0$, whereas the “SM wall”, corresponding to the visible 4-$d$ universe, is at $\phi = \pi$. The energy density on the Planck wall $V_P$ is equal and opposite to that on the SM wall and we have $V_P \sim M_5^3 k$, where $M_5 \sim {{\bar M}_P}$ is the fundamental 5-$d$ scale, and $k \sim {{\bar M}_P}$ is the curvature scale. The 5-$d$ cosmological constant is given by $\Lambda_5 = - k V_P$. Thus, we see that the parameters of the model do not establish new hierarchies.
The geometry of this model is warped and non-factorizable, with the metric $$ds^2 = e^{-2 \sigma(\phi)} \eta_{\mu \nu} \, dx^\mu dx^\nu + r_c^2 \, d\phi^2 \, \, ; \, \,
\sigma(\phi) = k \, r_c \, |\phi|,
\label{rsmetric}$$ where $e^{-2 \sigma(\phi)}$ is the warp factor. This geometric warp factor offers a possible explanation of the hierarchy problem. Basically, if one writes down a 5-$d$ action with Planckian mass parameters $m_5 \sim {{\bar M}_P}$, after a KK reduction to 4 dimensions, the 4-$d$ fields with canonical 4-$d$ kinetic terms will have mass parameters $m_4 = m_5 \, e^{- k \, r_c \pi}$. To have $m_4 \sim m_W$, we only need to require $k r_c \sim 10$, which has been shown to be easily realized in a mechanism that stabilizes the size of the $5^{th}$ dimension [@GW2]. In this way, numbers of $O(10)$ generate large hierarchies of $O(10^{16})$
This model has features that are quite distinct from the ones of the ADD model. In the RS model [@RS; @DHR1] (i) the KK tower of gravitons starts at $m \sim 1$ TeV, the spacings between the tower masses $\Delta m \sim 1$ TeV are unequal and given by roots of Bessel functions; (ii) the zero mode (massless 4-$d$) graviton couples with $1/{{\bar M}_P}$ and the massive KK tower gravitons couple with $1/\Lambda_\pi
\sim 1$ TeV$^{-1}$; this can be understood by noting that the wavefunction of the zero mode along the $5^{th}$ dimension is localized near the Planck wall, characterized by ${{\bar M}_P}$, whereas the KK graviton wavefunctions are localized near the SM wall, characterized by $\Lambda_\pi$. The RS-type models are sometimes referred to as “Localized Gravity” models. (iii) In the original proposal by Randall and Sundrum [@RS], the SM fields are taken to reside only on the SM wall. With these features, the RS model predicts resonant production of KK gravitons at ${\sqrt s} \sim 1$ TeV at colliders such as the NLC.
There have been a number of generalizations and extensions of the original RS proposal. Some of these extensions study the possibility of having SM fields in the bulk and deriving the 4-$d$ physics from the 5-$d$ picture [@GW1], since the SM scale of order 1 TeV can be generated on the SM wall through the warp factor. For various phenomenological reasons it is least problematic to keep the Higgs field on the SM wall [@RSf; @GP; @DHR3]. However, as a first step, one can study the effect of placing the SM gauge fields in the bulk and keeping the fermions on the SM wall [@DHR2; @P]. In this case, one finds that the fermions on the wall couple to the KK gauge fields $\sqrt {2 k r_c \pi} \sim 10$ times more strongly than they couple to the zero mode gauge fields ($\gamma, g, W^{\pm}, Z$). Here one expects to get strong constraints from data, and indeed agreement with precision electroweak data requires that the lightest KK gauge boson have a mass $m_1^{(A)} \grtsim 23$ TeV. This value pushes the scale on the SM wall far above $\sim 1$ TeV, making this scenario disfavored in the context of the hierarchy problem.
The above bound can be somewhat relaxed, if the fermions also reside in the bulk [@RSf]. In fact, by introducing bulk fermion 5-$d$ masses $m_\Psi$, one can change the couplings of the fermion zero modes (observed SM fermions) to various KK fields, and place different bounds on the RS model, depending on the value of the bulk mass parameter $\nu \equiv m_\Psi/k$ [@GN; @GP; @DHR3]. The parameter $\nu$ controls the shape of the fermion zero mode wavefunction $f^{(0)} \sim e^{\nu \sigma(\phi)}$. Thus, negative values of $\nu$ localize $f^{(0)}$ near the Planck wall, whereas positive values of $\nu$ localize $f^{(0)}$ near the SM wall.
Avoiding large FCNC’s may require keeping the value of $\nu$ nearly universal for all fermions. The 4-$d$ Yukawa couplings of SM fermions depend on the value of $\nu$. It can be shown that keeping the 5-$d$ Yukawa couplings $\lambda_5 \sim 1$ requires that the $\nu
\grtsim - 0.8$ (for the lightest fermion) [@GP; @DHR3]. Avoiding the generation of a new hierarchy, on the other hand, forces us to have $\nu \lsim - 0.3$. This can be understood by noting that as $\nu$ increases, the fermions get more and more localized toward the SM wall, and this takes us back to the case where leaving the fermions on the wall gave us a large lower bound on the mass of the first KK gauge field [@DHR3]. Although placing the SM gauge and fermion fields in the 5-$d$ bulk results in a rich phenomenology [@DHR3], the range where the theory seems viable is rather narrow, and keeping the SM fields on the wall appears to be less problematic.
Conclusions
===========
The hierarchy problem can be approached from a geometric $(4 + n)$-$d$ point of view, in the ADD and the RS scenarios. Each scenario predicts a distinct set of signatures at $\sqrt s \sim 1$ TeV. Thus, an NLC with $\sqrt s \sim 1$ TeV can test these ideas and possibly yield information on the geometry of the extra dimensions of the universe by probing such features as their number, size, and curvature.
[99]{}
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[^1]: Presented at the 5th International Linear Collider Workshop (LCWS 2000), Batavia, Illinois, October 24-28, 2000. Work supported by the Department of Energy, contract DE–AC03–76SF00515.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Let $\,{\boldsymbol{L}}\,$ be a second order, uniformly elliptic operator, and consider the equation $-\,{\boldsymbol{L}}\, u=\,f \,$ under the homogeneous boundary condition $\, u=\,0 \,.$ It is well known that $\,f \in
C({{\overline{\Omega}}})\,$ does not guarantee $\,{{\nabla}}^2\,u \in C({{\overline{\Omega}}})\,$. This gap led to look for functional spaces $\,C_*({{\overline{\Omega}}})\subset\,C({{\overline{\Omega}}})\,,$ as large as possible, for which $\,f\in \,C_*({{\overline{\Omega}}})\,$ *merely* guarantees the continuity of $\,{{\nabla}}^2\,u\,$ (but nothing more, say). Hölder continuity is too restrictive to fulfill this minimal requirement since in this case $\,{{\nabla}}^2\,u\,$ inherits the whole regularity enjoyed by $\,f\,$ (we say that *full regularity* occurs). This two opposite situations led us to look for significant cases in which *intermediate regularity* (i.e., between *mere continuity* and *full regularity*) occurs. This holds for data in Log spaces $\, D^{0,\,{{\alpha}}}({{\overline{\Omega}}})\,,$ $\,0<\,{{\alpha}}<\,+\infty\,,$ simply obtained by replacing in the modulus of continuity of Hölder spaces the quantity $
\,1/|\,x-\,y|\,$ by $\, \log\,(\,1/|\,x-\,y|)\,.$ If $\,f \in
D^{0,\,{{\alpha}}},$ for some fixed $\,{{\alpha}}>\,1\,,$ then $\,{{\nabla}}^2\,u \in
D^{0,\,{{\alpha}}-\,1}\,.$ This regularity is optimal.
The above picture opened the way to further investigation. Below we study the more general problem of data $\,f\,$ in subspaces of continuous functions $\,D_{{{\overline{\omega}}}}\,$, characterized by a given *modulus of continuity* $\,{{\overline{\omega}}}(r)\,.$ Hölder and Log spaces are particular cases. A significant new, lets say curious, case is shown by the family of functional spaces $\, C^{0,\,{{\lambda}}}_{{\alpha}}({{\overline{\Omega}}})
\,,$ $\,0 \leq\,{{\lambda}}<\,1\,$, $\,{{\alpha}}\in\,{{\mathbb R}}\,$. In particular, $\,C^{0,\,{{\lambda}}}_0({{\overline{\Omega}}})=\,C^{0,\,{{\lambda}}}({{\overline{\Omega}}})\,$, and $\,C^{0,\,0}_{{\alpha}}({{\overline{\Omega}}})=\, D^{0,\,{{\alpha}}}({{\overline{\Omega}}})\,$. Main point is that full regularity occurs for $\,{{\lambda}}>\,0\,$ and arbitrary $\,{{\alpha}}\in\,{{\mathbb R}}\,$. If $\,f \in\, C^{0,\,{{\lambda}}}_{{\alpha}}({{\overline{\Omega}}}) \,$ then $\,{{\nabla}}^2\,u
\in C^{0,\,{{\lambda}}}_{{\alpha}}({{\overline{\Omega}}})\,$.
[**Mathematics Subject Classification**]{}: 35A09,35B65, 35J25.
[**Keywords.**]{} Linear elliptic boundary value problems, classical solutions, continuity properties of higher order derivatives, data spaces of continuous functions, intermediate and full regularity.
author:
- 'by H. Beirão da Veiga'
title: |
**On classical solutions to elliptic boundary value problems. The full regularity spaces $\,
C^{0,\,{{\lambda}}}_{{\alpha}}({{\overline{\Omega}}}) \,$.**
---
Introduction. {#introduction}
=============
We start by some notation. By ${{\Omega}}$ we denote an open, bounded, connected set in ${{\mathbb R}}^n\,$, locally situated on one side of its boundary $\,{{\Gamma}}\,.$ The boundary $\,{{\Gamma}}\,$ is of class $\,C^{2,\,{{\lambda}}}\,,$ for some $\,{{\lambda}}\,,$ $\,0<\,{{\lambda}}\leq \,1\,.$ Notation ${{\Omega}}_0 \subset \subset {{\Omega}}$ means that the open set ${{\Omega}}_0$ satisfies the property ${{\overline{\Omega}}}_0 \subset {{\Omega}}$.
By $\,C({{\overline{\Omega}}})\,$ we denote the Banach space of all real continuous functions $\,f\,$ defined in $\,{{\overline{\Omega}}}\,$. The “sup” norm is denoted by $ \|\,f\,\|\,. $ We also appeal to the classical spaces $\,C^k({{\overline{\Omega}}})\,$ endowed with their usual norms $ \|\,u\,\|_k\,,$ and to the Hölder spaces $\,C^{0,\,{{\lambda}}}({{\overline{\Omega}}})\,,$ endowed with the standard semi-norms and norms. The space $\,C^{0,\,1}({{\overline{\Omega}}})\,,$ is sometimes denoted by $\,Lip\,({{\overline{\Omega}}})\,,$ the space of Lipschitz continuous functions in $\,{{\overline{\Omega}}}\,.$ We set $$I(x;\,r)=\,\{\,y:\,|y-\,x| \leq\, r\,\}\,, \quad\,{{\Omega}}(x;\,r)=\, {{\Omega}}\,\cap\,I(x;\,r)\,.$$ Symbols $c\,$ and $\,C\,$ denote generical positive constants. We may use the same symbol to denote different constants.
Let us present some reasons that led us to the present study. We say that solutions to a specific boundary value problem are *classical* if all derivatives appearing in the equations and boundary conditions are continuous up to the boundary on their domain of definition. We call *“minimal assumptions problem”* the investigation of “minimal assumptions” on the data which guarantee that solutions are classical. The very starting point of these notes was reference [@BVJDE], where the main goal was to look for *minimal assumptions* on the data which guarantee classical solutions to the $\,2-D\,$ Euler equations in a bounded domain. The study of this problem led to the auxiliary problem $$\left\{
\begin{array}{l}
{\boldsymbol{L}}\,u=\,f \quad \textrm{in} \quad {{\Omega}}\,,\\
u=\,0 \quad \textrm{on} \quad {{\Gamma}}\,.
\end{array}
\right.
\label{lapnao}$$ We do not discuss here the relation between the Euler equations and problem . The interested reader is referred to the original paper [@BVJDE], and also to [@BVJP], where a complete description is presented.
Below we consider second order, uniformly elliptic operators $${\boldsymbol{L}}=\,\sum_1^n a_{i\,j}(x)\, {{\partial}}_i\,{{\partial}}_j\,.\label{elle}$$ Without loss of generality, we assume that the matrix of coefficients is symmetric. To avoid conditions depending on the single case, we assume once and for all that the operator’s coefficients are Lipschitz continuous in $\,{{\overline{\Omega}}}\,.$ Lower order terms can be considered without difficulty.
A Hölder continuity assumption on $\,f\,$ is unnecessarily restrictive to guarantee $\,{{\nabla}}^2\,u\in\,C({{\overline{\Omega}}})\,,$ where $\,u\,$ is the solution to problem . On the other hand, continuity of $\,f\,$ is not sufficient to guarantee continuity of $\,{{\nabla}}^2\,u\,.$ This situation led us to consider in [@BVJDE] a Banach space $\,\,C_*({{\overline{\Omega}}})\,$, $\, C^{0,\,{{\lambda}}}({{\overline{\Omega}}})\subset
\,C_*({{\overline{\Omega}}})\subset\,C({{\overline{\Omega}}})\,,$ for which the following result holds (Theorem 4.5, in [@BVJDE]).
Let $\,f \in \,C_*({{\overline{\Omega}}})\,$ and let $\,u\,$ be the solution of problem . Then $\,u \in\, C^2({{\overline{\Omega}}})\,,$ moreover, $$\|\,{{\nabla}}^2\,u\,\| \leq \,c\,\|\,f\,\|_*\,.\label{lapili}$$ \[laplaces\]
The above result was stated for constant coefficients operators, however the proof applies without any modification to variable coefficients case, since it is based on some properties of the Green functions, which hold in the general case.
For the readers convenience we recall definition and main properties of $\,C_*({{\overline{\Omega}}})\,$ (see [@BVJDE] and, for complete proofs, [@BVSTOKES]). Define, for $\,f \in \,C({{\overline{\Omega}}})\,,$ and for each $\,r>\,0\,$,$${{\omega}}_f(r) \equiv \, \sup_{\,x,\,y
\in\,{{\Omega}}\,;\, 0<\,|x-\,y| \leq\,r } \,|\,f(x)-\,f(y)\,|\,,\label{cinco}$$ and consider the semi-norm $$[\,f\,]_* =\,[\,f\,]_{*,\,R} \equiv \int_0^R \,{{\omega}}_f(r) \,\frac{dr}{r}\,,\label{seis}$$ where $\,R>\,0\,$ is fixed. The finiteness of the above integral is known as *Dini’s continuity condition*. We define the functional space $$C_*({{\overline{\Omega}}}) \equiv\,\{\,f \in\,C(\,{{\overline{\Omega}}}): \,[\,f\,]_*
<\,\infty\,\}\label{cstar}$$ normalized by $ \,\|\,f\,\|_* =\,[\,f\,]_*+\,\|\,f\,\|\,.$ Norms defined for two distinct values of $\,R\,$ are equivalent. We have shown that $\,C_*({{\overline{\Omega}}})\,$ is a Banach space, that the embedding $\,
C_*({{\overline{\Omega}}}) \subset \,C({{\overline{\Omega}}})\,$ is compact, and that the set $\,C^{\infty}({{\overline{\Omega}}})\,$ is dense in $\,C_*({{\overline{\Omega}}})\,.$
The regularity theorem \[laplaces\] for data in $\,C_*({{\overline{\Omega}}})\,$ raise a number of new questions. Contrary to the case of Hölder continuity, where full regularity is restored ($\,{{\nabla}}^2\,u\,$ and $\,f\,$ has the same regularity), no significant additional regularity is obtained for data in $\,C_*({{\overline{\Omega}}})\,$, besides mere continuity of $\,{{\nabla}}^2\,u\,.$ So, we are in the presence of two totally opposite behaviors. An “intermediate” situation is shown by the Log spaces $\, D^{0,\,{{\alpha}}}({{\overline{\Omega}}})\,,$ $\,0<\,{{\alpha}}<\,+\infty\,.$ In the constant coefficients case, if $\,f \in
D^{0,\,{{\alpha}}}_{loc}({{\Omega}})\,$ for fixed $\,{{\alpha}}>\,1\,,$ then $\,{{\nabla}}^2\,u
\in D^{0,\,{{\alpha}}-\,1}_{loc}({{\Omega}})\,.$ This regularity result is *optimal*. Furthermore, it holds up to “flat boundary points”. See theorem \[laplohas\] below.
The above picture leads us to consider general data spaces $\,D_{{\overline{\omega}}}({{\overline{\Omega}}})\,$, characterized by a given *modulus of continuity* function $\,{{\overline{\omega}}}(r)\,.$ These spaces are contained between $\,Lip({{\overline{\Omega}}})\,$ and $\,C_*({{\overline{\Omega}}})\,$. Hölder and Log spaces are particular cases. To each suitable $\,{{\overline{\omega}}}(r)\,$ there corresponds a $\,{{\widehat{\omega}}}(r)\,$ such that $\,{{\nabla}}^2\,u \in D_{{\widehat{\omega}}}\,$ for $\,f \in
\,D_{{\overline{\omega}}}\,,$ see theorem \[sufasvero\]. Clearly, $\,{{\overline{\omega}}}(r)\leq
\,c\,{{\widehat{\omega}}}(r)\,$, for some $c>0$. This situation occurs for data in Log spaces, see theorem \[laplolas\]. Furthermore, if a reverse inequality $\,{{\widehat{\omega}}}(r)\leq \,c\,{{\overline{\omega}}}(r)\,$ holds, then full regularity occurs, see theorem \[sufasvero-2\]. This is the situation for data in Hölder spaces. A more general, quite significant, case of full regularity concerns the new family of functional spaces $\,C^{0,\,{{\lambda}}}_{{\alpha}}({{\overline{\Omega}}})\,,$ $\,0 \leq\,{{\lambda}}<\,1\,$, $\,{{\alpha}}\in\,{{\mathbb R}}\,$, called here Hölog spaces. For $\,{{\lambda}}>\,0\,$ and $\,{{\alpha}}=\,0\,,$ $\,C^{0,\,{{\lambda}}}_0({{\overline{\Omega}}})=\,C^{0,\,{{\lambda}}}({{\overline{\Omega}}})\,,$ is a Hölder classical space. For $\,{{\lambda}}=\,0\,$ and $\,{{\alpha}}>0\,$, $\,C^{0,\,0}_{{\alpha}}({{\overline{\Omega}}})=\, D^{0,\,{{\alpha}}}({{\overline{\Omega}}})\,$ is a Log space. Main point is that, for $\,{{\lambda}}>\,0\,,$ $\,{{\nabla}}^2\,u $ and $\,f\,$ enjoy the same $\,C^{0,\,{{\lambda}}}_{{\alpha}}({{\overline{\Omega}}})\,$ regularity (full regularity). See theorem \[laplohas\].
The assumptions on the data spaces $\,D_{{\overline{\omega}}}({{\overline{\Omega}}})\,$ required in theorems \[sufasvero\] and \[sufasvero-2\] can be substantially weakened. However, explicit statements in this direction would not add particularly significant features, at the cost of more involved manipulations.
The spaces $\,D_{{\overline{\omega}}}({{\overline{\Omega}}})\,$. General properties. {#novicas}
====================================================================================
In this section we define the spaces $\,D_{{\overline{\omega}}}({{\overline{\Omega}}})\,$ and state some general properties. We consider real, *continuous*, *non-decreasing* functions $\,{{\overline{\omega}}}(r)\,$, defined for $\,0\,\leq r\,<\,R\,,$ for some $\,R>\,0\,.$ Furthermore, $\,{{\overline{\omega}}}(0)=\,0\,,$ and $\,{{\overline{\omega}}}(0)>0\,$ for $\,r>\,0\,.$ These three conditions are assumed everywhere in the sequel. Sometimes, the functions $\,{{\overline{\omega}}}(r)\,$ will be called *oscillation functions*.
Recalling , we set $$[f]_{{\overline{\omega}}}= \, \sup_{\,0<\ r\,<\,R } \,\frac{{{\omega}}_f(r)}{{{\overline{\omega}}}(r)}\,.\label{fom}$$ Hence, $${{\omega}}_f(r) \leq\, [f]_{{\overline{\omega}}}\,{{\overline{\omega}}}(r)\,, \quad \forall \, r \in(0,\,R)\,. \label{fom2}$$ Further, we define the linear space $$D_{{\overline{\omega}}}({{\overline{\Omega}}}) =\,\{\, f\in\,C({{\overline{\Omega}}}) :\, [f]_{{\overline{\omega}}}<\,\infty\,\}\,.
\label{defdom}$$ One easily shows that $\,[f]_{{\overline{\omega}}}\,$ is a semi-norm in $\,D_{{\overline{\omega}}}({{\overline{\Omega}}})\,.$ We introduce a norm by setting $$\|f\|_{{\overline{\omega}}}=\,[f]_{{\overline{\omega}}}+\, \|f\|\,.\label{omnorm}$$ Two norms with distinct values of the parameter $\,R\,$ are equivalent, due to the addition of $\,\|f\|\,$ to the semi-norms.
It is worth noting that, beyond the three conditions on $\,{{\overline{\omega}}}(r)\,$ introduced above, any other property assumed in the sequel is merely needed in an arbitrarily small neighborhood of the origin. This fact may be used without a continual reference. In the sequel, to avoid continual specification, we introduce the following definitions.
We say that $\,{{\overline{\omega}}}(r)\,$ is *concave* if it is concave in a neighborhood of the origin, and say that $\,{{\overline{\omega}}}(r)\,$ is *differentiable* if it is point-wisely differentiable (not necessarily continuously differentiable), for each $\,r>\,0\,,$ in a neighborhood of the origin.\[finacas\]
Next we establish some useful properties of the above functional spaces.
If $$0<\,k_0 \leq\,\frac{{{\overline{\omega}}}(r)}{{{\overline{\omega}}}_0(r)} \leq\,k_1<\,+\infty\,,
\label{pertinho}$$ for $\,r\,$ in some neighborhood of the origin, then $\,D_{{\overline{\omega}}}({{\overline{\Omega}}})=\,D_{{{\overline{\omega}}}_0}({{\overline{\Omega}}})\,,$ with *equivalent norms*.
\[eqnormes\]
The proof is immediate.
If $\,\|f_n\|_{{\overline{\omega}}}\leq\,C_0\,,$ and $\,f_n \rightarrow \,f\,$ in $\,C({{\overline{\Omega}}})\,$ then $\,\|f\|_{{\overline{\omega}}}\leq\,C_0\,.$\[umlemmas\]
The proof is immediate.
$D_{{\overline{\omega}}}({{\overline{\Omega}}})$ is a Banach space.\[completos\]
Let $f_n$ be a Cauchy sequence in $D_{{\overline{\omega}}}({{\overline{\Omega}}})\,.$ It follows, in particular, that $\,f_n \rightarrow \,f\,$ in $\,C({{\overline{\Omega}}})\,,$ where $\, f \in \,D_{{\overline{\omega}}}({{\overline{\Omega}}})\,.$ On the other hand, for $\,|x-\,y|=\,r\,$, $$\begin{array}{l}
\frac{|\,(f(x)-\,f_n(x)\,)
-(f(y)-\,f_n(y)\,)\,}{{{\overline{\omega}}}(r)}=\\\\
\lim_{m \rightarrow \,\infty}\, \frac{|\,(f_m(x)-\,f_n(x)\,)
-(f_m(y)-\,f_n(y)\,)\,}{{{\overline{\omega}}}(r)}\leq\,\limsup_{m \rightarrow
\,\infty} \,[\,f_m-\,f_n\,]_{{\overline{\omega}}}\,.\end{array}$$ Hence $$[\,f-\,f_n\,]_{{\overline{\omega}}}\leq\,\limsup_{m \rightarrow
\,\infty} \,[\,f_m-\,f_n\,]_{{\overline{\omega}}}\,.$$ From the Cauchy sequence hypothesis it readily follows that $$\lim_{n \rightarrow
\,\infty} \,[\,f-\,f_n\,]_{{\overline{\omega}}}=\,0\,.$$
Next we consider compact embedding properties. In the sequel, $ {{\overline{\omega}}}\,<<\, {{\overline{\omega}}}_1 $ mean that $$\lim_{r\rightarrow\,0}\,\frac{{{\overline{\omega}}}(r)}{{{\overline{\omega}}}_1(r)}=\,0\,.\label{pactos}$$
Assume that $ {{\overline{\omega}}}<< {{\overline{\omega}}}_1\,.$ Then the embedding $$D_{{\overline{\omega}}}({{\overline{\Omega}}})\,\subset\,D_{{{\overline{\omega}}}_1}({{\overline{\Omega}}})\,,$$is compact.\[compactos\]
By assumption $$\|\,f_n\,\|_{{\overline{\omega}}}=\,[\,f_n\,]_{{\overline{\omega}}}+\,\|\,f_n\,\| \leq\,C_0, \quad \forall \, n\,.$$ From it follows that ${{\overline{\omega}}}(r) \leq \,{{\overline{\omega}}}_1(r)\,$ for $\,r\in\,(\,0,\,R_0)\,,$ for some $\,R_0>\,0\,$. For $\,r \in (R_0,\,R)$ one has $\,{{\overline{\omega}}}(r)\leq\frac{{{\overline{\omega}}}(R)}{{{\overline{\omega}}}_1(R_0)}\,{{\overline{\omega}}}_1(r)\,.$ So there is a positive constant $\,C\,$ such that $${{\overline{\omega}}}(r)\leq \,C\,{{\overline{\omega}}}_1(r)\,,\quad \forall \,r\in\,(0,\,R)\,.$$ By the Ascoli-Arzela Theorem, the embedding$$D_{{\overline{\omega}}}({{\overline{\Omega}}})\subset\,C({{\overline{\Omega}}})$$ is compact. Hence, by appealing to lemma \[umlemmas\], one shows that there is a subsequence, still denoted $\,f_n\,$, which converges uniformly to some $\,f\in\,D_{{\overline{\omega}}}({{\overline{\Omega}}})\,.$ Without loss of generality, we assume that $\,f=\,0\,.$
Let $\,|\,x-\,y\,|=\,r\,.$ One has $$\frac{|\,f_n(x)-\,f_n(y)\,|}{{{\overline{\omega}}}_1(r)}=\,\frac{|\,f_n(x)-\,f_n(y)\,|}{{{\overline{\omega}}}(r)}\,\frac{{{\overline{\omega}}}(r)}{{{\overline{\omega}}}_1(r)}\,,
\quad \forall \, n\,.$$ Given $\,{{\epsilon}}>\,0\,,$ it follows from that there is $\,R_0({{\epsilon}}) >\,0\,$ such that $$0<\,r\leq\,R_0({{\epsilon}})\, \implies \frac{{{\overline{\omega}}}(r)}{{{\overline{\omega}}}_1(r)}<\,{{\epsilon}}\,. \label{evon}$$ Hence, for $\,0<\,|x-\,y|\leq\,R_0({{\epsilon}})\,,$ $$\frac{|\,f_n(x)-\,f_n(y)\,|}{{{\overline{\omega}}}_1(r)} \leq\,C_0\,{{\epsilon}}\,,\quad \forall \,n\,.\label{zed}$$ On the other hand, if $\,r \in (\,R_0({{\epsilon}}),\,R)\,,$ one has $$\frac{|\,f_n(x)-\,f_n(y)\,|}{{{\overline{\omega}}}_1(r)} \leq
\,\frac{2}{{{\overline{\omega}}}_1(R_0({{\epsilon}}))}\,\|f_n\|\,.$$ Since the sequence $\,\|f_n\|\,$ converges to zero, there is an index $\,N({{\epsilon}})$ such that, for each $\,n>\,N({{\epsilon}})\,,$ the right hand side of the last inequality is smaller than ${{\epsilon}}\,.$ This fact, together with , shows that holds for $\,0<\,|x-\,y|\leq\,R\,$ and $\,n>\,N({{\epsilon}})\,$ (increase the constant $\,C_0\,,$ if necessary). So, $$\lim_{n\rightarrow\,+\,\infty} \,[\,f_n\,]_{{{\overline{\omega}}}}=\,0\,.$$
Assume that $\,{{\overline{\omega}}}\,$ is concave. Then $${{\overline{\omega}}}(k\,r) \leq\,k\,{{\overline{\omega}}}(r)\,, \quad \forall \,k\geq\,1\,.\label{concavo}$$ \[cocas\]
The proof is immediate.
In reference [@BV-arxiv], Theorem 4.4, we claimed that $\,C^{\infty}({{\overline{\Omega}}})\,$ is dense in Log spaces, leaving the proof to the reader. This result is wrong, as shown below. It is worth noting that $\,C^{\infty}({{\overline{\Omega}}})\,$ is dense in $\,C_*({{\overline{\Omega}}})\,,$ a result that has a central role in reference [@BVSTOKES].
Assume that $\,{{\overline{\omega}}}(r)\,$ is concave and that $\,{{\overline{\omega}}}_1(r)<<\,{{\overline{\omega}}}(r)\,.$ Then $\,D_{{{\overline{\omega}}}_1}({{\overline{\Omega}}})\,$ is not dense in $\, D_{{\overline{\omega}}}({{\overline{\Omega}}})\,.$\[osned\]
We assume that the origin belongs to $\,{{\Omega}}\,,$ and argue in a neighborhood $\,I=\,I(0,\,{{\delta}}) \subset {{\Omega}}\,.$ Define $\,f\,$ by setting $\,f(x)=\,{{\overline{\omega}}}(|x|)\,.$ We show that $\,[\,f-\,g]_{\,{{\overline{\omega}}}}\geq\,1\,,$ for each $\,g \in
D_{{{\overline{\omega}}}_1}({{\overline{\Omega}}})\,.$ It is sufficient to consider the one-dimensional case. One has $$\frac{|\,(f(x)-\,g(x)\,)-\,(\,f(0)-\,g(0)\,)\,|}{{{\overline{\omega}}}(|x|)}=\,\Big|\,1-\,\frac{\,g(x)-\,g(0)\,}{{{\overline{\omega}}}(|x|)}\,\Big|\,.$$ Hence $\,[\,f-\,g]_{\,{{\overline{\omega}}}}\geq\,1\,$ follows, if we show that $$\lim_{x\rightarrow\,0} \,\frac{\,g(x)-\,g(0)\,}{{{\overline{\omega}}}(|x|)}=\,0\,.
\label{desejada}$$ Let’s prove this last inequality. One has, as $\,x
\rightarrow\,0\,,$ $$\lim\,\frac{\,g(x)-\,g(0)\,}{{{\overline{\omega}}}(|x|)}=\,\lim\,\frac{\,g(x)-\,g(0)\,}{{{\overline{\omega}}}_1(|x|)}
\cdot\,\lim\,\frac{{{\overline{\omega}}}_1(|x|)}{{{\overline{\omega}}}(|x|)}=\,0\,.\label{liminfas}$$
Note that in the above proof we did not explicitly appeal to the concavity assumption. This assumption was introduced merely to guarantee that $\,f(x)=\,{{\overline{\omega}}}(|x|)\,$ belongs to $\, D_{{\overline{\omega}}}\,$ in a neighborhood of the origin. This holds if $${{\overline{\omega}}}(s)\leq\,{{\overline{\omega}}}(r)+\,c\,{{\overline{\omega}}}(s-r)\,, \quad \textrm{for} \quad
\,0<\,r<\,s<\,{{\rho}}\,,\label{regdens}$$ for some constant $c\geq\,1\,,$ and some $\,{{\rho}}>\,0\,.$ By lemma \[cocas\], concave oscillation functions satisfy with $c=\,1\,.$
The above result shows, in particular, that $\,C^{0,\,\mu}({{\overline{\Omega}}})\,$ is not dense in $\,C^{0,\,{{\lambda}}}({{\overline{\Omega}}})\,$ for $1\geq\,\mu> {{\lambda}}>\,0\,.$ In particular $Lip\,({{\overline{\Omega}}})\,$, hence $\,C^1({{\overline{\Omega}}})\,$, is not dense in $\,C^{0,\,{{\lambda}}}({{\overline{\Omega}}})\,$.
We end this section by stating an extension theorem, where $
\,{{\Omega}}_{{{\delta}}}\equiv\,\{\,x:\, dist(x,\,{{\Omega}}\,) <\,{{\delta}}\,\}\,.$
Assume that $\,{{\Omega}}\,$ is convex or, alternatively, that $\,{{\overline{\omega}}}(r)\,$ is concave (concavity may be replaced by condition ) . Then there is a $\,{{\delta}}>0\,$ such that the following holds. There is a linear continuous map $\,T\,$ from $\,C({{\overline{\Omega}}})\,$ to $\,C({{\overline{\Omega}}}_{{{\delta}}})\,,$ and from $\,D_{{\overline{\omega}}}({{\overline{\Omega}}})\,$ to $\,D_{{\overline{\omega}}}({{\overline{\Omega}}}_{{{\delta}}})\,,$ such that $\,T\,f(x)=\,f(x)\,$, for each $\,x \in\,{{\overline{\Omega}}}\,.$\[bahois\]
The proof follows by appealing to the argument used to prove the Theorem 2.3 in [@BVSTOKES]. See reference [@BV-arxiv]. Note that the classical proof of approximation of functions on compact subsets of $\,{{\Omega}}\,$ by appealing to mollification, does not work here. Otherwise, the density property refused by theorem \[osned\] would hold.
Spaces $\,D_{{{\widehat{\omega}}}}({{\overline{\Omega}}})\,$ and regularity. The main theorems. {#domega}
===============================================================================================
In this section we state the theorems \[sufasvero\] and \[sufasvero-2\]. From now on we assume that the modulus of continuity $\,{{\overline{\omega}}}(r)\,$ satisfy the condition $$\int_0^R \,{{\overline{\omega}}}(r) \,\frac{dr}{r}\,\leq\,C_R\,,\label{massim}$$ for some constant $C_R\,.$ Assumption is equivalent to the inclusion $\, D_{{\overline{\omega}}}({{\overline{\Omega}}})\subset\,C_*({{\overline{\Omega}}})\,.$ This assumption is almost necessary to obtain $\,{{\nabla}}^2\,u \in\, C({{\overline{\Omega}}})\,.$
We put each suitable oscillation function $\,{{\overline{\omega}}}(r)\,$ in correspondence with a unique, related oscillation function $\,{{\widehat{\omega}}}(r)\,$ defined by setting ${{\widehat{\omega}}}(\,0)=\,0\,,$ and $${{\widehat{\omega}}}(\,r)=\,\int_0^r \,{{\overline{\omega}}}(s) \,\frac{ds}{s}\label{chapeu}$$ for $\,0<\,r\leq\,R\,.$ Hence, to a functional space $\,D_{{\overline{\omega}}}({{\overline{\Omega}}})\,$ there corresponds a well defined functional space $\,D_{{{\widehat{\omega}}}}({{\overline{\Omega}}})\,.$ Obviously, $\,{{\widehat{\omega}}}\,$ satisfies all the properties described in section \[novicas\] for generical oscillation functions. In particular, Banach spaces $$D_{{\widehat{\omega}}}({{\overline{\Omega}}}) =\,\{\, f\in\,C({{\overline{\Omega}}}) :\, [f]_{{\widehat{\omega}}}<\,\infty\,\}
\label{defdom2}$$ turn out to be well defined.
Next we discuss some additional restrictions on the data spaces $D_{{\overline{\omega}}}({{\overline{\Omega}}})$. We start by excluding $\,Lip({{\overline{\Omega}}})\,$ as data space since this *singular* case, largely considered in literature, is borderline. In fact, to assign $\,f \in\,Lip\,({{\overline{\Omega}}})\,$ is equivalent to assign $\,{{\nabla}}\,f\in\,L^\infty({{\Omega}})\,,$ which is the starting point of a new chapter. So, we impose the *strict* limitation $$Lip({{\overline{\Omega}}})\, \subset\, D_{{\overline{\omega}}}({{\overline{\Omega}}})\, \subset\,C_*({{\overline{\Omega}}})\,.\label{namely}$$ Exclusion of $\,Lip({{\overline{\Omega}}})\,$ means that $\,{{\overline{\omega}}}(r)\,$ does not verify $ \,{{\overline{\omega}}}(r) \leq\,c\,r\,,$ for any positive constant $\,c\,.$ Hence $\,\limsup (\,{{\overline{\omega}}}(r)/\,r\,)=\,+\,\infty\,,$ as $\,r
\rightarrow\,0\,$. We simplify, by assuming that $$\lim_{r \rightarrow\,0}\, \frac{{{\overline{\omega}}}(r)}{r}=\,+\,\infty\,.\label{simplelas}$$ In particular, the graph of $\,{{\overline{\omega}}}(r)\,$ is tangent to the vertical axis at the origin (as for Hölder and Log spaces). This picture also shows that *concavity* of the graph is here a quite natural assumption. Concavity implies that left and right derivatives are well defined, for $r>\,0\,$. By also taking into account that $\,{{\overline{\omega}}}(r)\,$ is non-decreasing, we realize that pointwise differentiability of $\,{{\overline{\omega}}}(r)\,,$ for $r>\,0\,$, is not a particularly restrictive assumption. This last claim is reenforced by the equivalence result for norms, under condition . This equivalence allows regularization of oscillation functions $\,{{\overline{\omega}}}(r)\,,$ staying inside the same original functional space $\,D_{{\overline{\omega}}}({{\overline{\Omega}}})\,.$ Summarizing, *differentiability* and *concavity* (recall definition \[finacas\]) are natural assumptions here.
If $\,{{\overline{\omega}}}(r)\,$ is concave, not flat, and differentiable, it follows that $$\frac{{{\overline{\omega}}}(r)}{r\,\,{{\overline{\omega}}}'(r)}> \,1\,,\label{flitas}$$for all $r>\,0\,.$ This justifies the assumption $$\lim_{r\rightarrow 0}\,\frac{{{\overline{\omega}}}(r)}{r\,\,{{\overline{\omega}}}'(r)}=\,C_1 > \,1\,,\label{naflitas}$$where $\,C_1=\,+\,\infty\,$ is admissible. Assumption is reenforced by the particular situation in Lipschitz, Hölder, and Log cases. The limit exists and is given by, respectively, $\,1\,$, $\,\frac{1}{{{\lambda}}}\,,$ and $\,+\,\infty\,.$ As expected, the Lipschitz case stays outside the admissible range. Note that, basically, the larger is the space, the larger is the limit.
The above consideration allow us to assume in theorems \[sufasvero\] and \[sufasvero-2\] that oscillation functions $\,{{\overline{\omega}}}(r)\,$, are concave, differentiable, and satisfies conditions , , and .
Note that, due to a possible loss of regularity, it could happen that a $\,D_{{{\widehat{\omega}}}}({{\overline{\Omega}}})\,$ space is not contained in $\,C_*({{\overline{\Omega}}})\,,$ as happens in theorem \[laplolas\], if $\,1<{{\alpha}}<2\,.$ In other words, $\,{{\widehat{\omega}}}(r)\,$ does not necessarily satisfy .
Next, we define the quantity $$B(r)=:\frac{\,r\,\int_r^{R} \,\frac{{{\overline{\omega}}}(s)}{s^2} \,ds}{\int_0^r
\,\frac{{{\overline{\omega}}}(s)}{s}\,ds}\,.\label{eeles}$$ \[assasdois\]The following result holds.
Assume that $\,{{\overline{\omega}}}(r)\,$ is concave and satisfies assumptions , and . Then $$\lim_{r\,\rightarrow \,0}\,B(r)=\,\frac{1}{C_1-\,1}\,.\label{peles}$$ In particular there is a positive constant $\,C_2\,$ such that $$B(r)\leq\,C_2\,\label{eeles-2}$$ in some neighborhood of the origin.\[beat\]
By appealing to , and to a de L’Hôpital’s rule one shows that $$\lim_{r\,\rightarrow \,0} \,\frac1r \,\int_0^{r} \,\frac{{{\overline{\omega}}}(s)}{s}
\,ds =\,+\,\infty\,.\label{lecas}$$ On the other hand $$\lim_{r\,\rightarrow \,0}\,B(r)= \,\lim_{r\,\rightarrow \,0}
\frac{\int_r^{R} \,\frac{{{\overline{\omega}}}(s)}{s^2} \,ds}{\frac1r \,\int_0^r
\,\frac{{{\overline{\omega}}}(s)}{s}\,ds}\,.\label{doente}$$ Equation shows that the denominator $\,g(r)\,$ of the fraction in the right hand side of goes to $\,+\,\infty\,$ as $\,r\,$ goes to zero. Furthermore its derivative $$g'(r)=\, \frac{1}{r^2}\,\Big(\,{{\overline{\omega}}}(r) -\,\int_0^r
\,\frac{{{\overline{\omega}}}(s)}{s}\,ds\,\Big)$$ is strictly negative for positive $\,r\,$ in a neighborhood of the origin, as follows from the inequality $\,{{\overline{\omega}}}(r) -\,\int_0^r
\,\frac{{{\overline{\omega}}}(s)}{s}\,ds\,<\,0\,,$ for $\,r>\,0\,.$ Let’s show this last inequality. Since the left hand side of the inequality goes to zero with $\,r\,,$ it is sufficient to show that its derivative is strictly negative for $\,r>\,0\,.$ This follows easily by appealing to . The above results allow us to apply to the limit one of the well known forms of de L’Hôpital’s rule. Straightforward calculations, together with , show .
Next we state our main results, theorems \[sufasvero\] and \[sufasvero-2\]. In the first theorem constant coefficients are assumed.
Assume that the oscillation function $\,{{\overline{\omega}}}(r)\,$, concave and differentiable, satisfies conditions , , and . Define $\,{{{\widehat{\omega}}}}(r)\,$ by . Let ${{\Omega}}_0 \subset \subset \,{{\Omega}}\,$, $\,f \in D_{{{\overline{\omega}}}}({{\overline{\Omega}}})\,,$ and $\,u\,$ be the solution of problem , where the operator coefficients are constant. Then $\,{{\nabla}}^2\,u \in \,D_{{{\widehat{\omega}}}}({{\Omega}}_0)\,$ and $$\|\,{{\nabla}}^2\,u\,\|_{\,{{\widehat{\omega}}},\,{{\Omega}}_0} \leq \,C\,\|\,f\,\|_{{{\overline{\omega}}}}\,,\label{hahega}$$ for some positive constant $\,C= C({{\Omega}}_0,\,{{\Omega}})\,.$ The result is optimal in the sharp sense defined in section \[optimus\] . Furthermore, the above regularity holds up to flat boundary points. \[sufasvero\]
A point $\,x\,\in\,{{\partial}}\,{{\Omega}}\,$ is said to be a *flat boundary point* if the boundary is flat in a neighborhood of the point. The meaning of *sharp optimality* is the following (our abbreviate notation seems clear).
We say that a given regularity statement of type ${{\overline{\omega}}}\rightarrow
\,{{\widehat{\omega}}}\,$ is sharp if any regularity statement ${{\overline{\omega}}}\rightarrow
\,{{\widehat{\omega}}}_0\,,$ obtained by replacing $\,{{\widehat{\omega}}}\,$ by any other $\,{{\widehat{\omega}}}_0\,,$ implies the existence of a constant $c$ for which $\,{{\widehat{\omega}}}(r)\leq\,c\,{{\widehat{\omega}}}_0(r)\,.$\[sharpas\]
The sharp regularity claimed in theorem \[sufasvero\] will be proved in section \[optimus\].
Much stronger results hold if the constant $C_1\,$ in equation is positive and finite. In this case one has $$D_{{{\widehat{\omega}}}}({{\overline{\Omega}}})=\,D_{{{\overline{\omega}}}}({{\overline{\Omega}}})\,.\label{fullas}$$ In fact, by de l’Hôpital rule, one shows that $$\lim_{r\rightarrow 0}\,\frac{{{\widehat{\omega}}}(r)}{\,{{\overline{\omega}}}(r)}=\,
\lim_{r\rightarrow 0}\,\frac{{{\overline{\omega}}}(r)}{r\,\,{{\overline{\omega}}}'(r)},$$ if the second limit exists. Hence, under this last hypothesis, the identity holds if (actually, and only if) the limit is positive and finite. Clearly, holds by merely assuming the inequality required in proposition \[eqnormes\]. We will show that if holds then the operator ${\boldsymbol{L}}$ can have variable coefficients, and full regularity occurs up to any (regular) boundary point. More precisely, one has the following result.
Assume that the oscillation function $\,{{\overline{\omega}}}(r)\,$, concave and differentiable, satisfies conditions , , and for some $\,C_1<\,+\,\infty\,$ . Further, define $\,{{{\widehat{\omega}}}}(r)\,$ by . Let $\,f \in D_{{{\overline{\omega}}}}({{\overline{\Omega}}})\,,$ and let $\,u\,$ be the solution of problem . Then $\,{{\nabla}}^2\,u \in \,D_{{{\overline{\omega}}}}({{\overline{\Omega}}})\,$ and $$\|\,{{\nabla}}^2\,u\,\|_{\,{{\overline{\omega}}}} \leq \,C\,\|\,f\,\|_{{{\overline{\omega}}}}\,,\label{hahega-2}$$ for some positive constant $\,C\,.$ Regularity in the sharp sense holds.\[sufasvero-2\]
Regularity in the sharp sense follows trivially from full regularity. But it is quite significant, even necessary, in dealing with intermediate regularity results, like in theorem \[sufasvero\]. See the example shown in section \[DOPO\], in the framework of Log spaces $\,D^{0,\,{{\alpha}}}({{\overline{\Omega}}})\,.$
The conditions imposed in the above statements can be weakened as follows. We start by replacing the concavity assumption by the existence of a constant $\,k_1>\,1\,$ such that $${{\overline{\omega}}}(k_1\,r) \leq\,c_1\,{{\overline{\omega}}}(r)\,\label{comka}$$ for some positive constant $\,c_1\,,$ and for $\,r\,$ in a neighborhood of the origin. We take into account that, if holds, then given $\,k_2>\,1\,,$ there is a positive constant $\,c_2\,$ such that $${{\overline{\omega}}}(k_2\,r) \leq\,c_2\,{{\overline{\omega}}}(r)\,,\label{comka-zw}$$ for $\,r\,$ in some $\,{{\delta}}_0-$neighborhood of the origin. The proof is obvious, by a bootstrap argument. Take into account that, if $\,k_2>\,k_1\,,$ there is an integer $m$ such that $\,k_2
\leq\,k_1^{\,m}\,.$ If $\,{{\overline{\omega}}}(r)\,$ is concave the lemma \[cocas\] shows for $\,k_2=\,c_2=\,1\,.$
Actually, in the sequel we will prove that in theorem \[sufasvero\], concavity, differentiability, and assumptions , , and , can be replaced by the more general set of assumptions , , , and . The same holds for theorem \[hahega-2\], by adding the assumption .
For previous related results we refer to [@burch] and [@shapiro]. The author is grateful to Piero Marcati who, after a seminar on our results, found the above references.
An H-K-L-G inequality. {#apotes-G}
======================
In this section we prove the Theorem \[ofundas-G\] below. The proof is an adaptation of that developed in [@JBS] to prove the so called Hölder-Korn-Lichtenstein-Giraud inequality (see [@JBS], part II, section 5, appendix 1) in the framework of Hölder spaces. Following [@JBS], we considered *singular kernels* $\,{\mathcal{K}}(x)$ of the form $${\mathcal{K}}(x)=\,\frac{{{\sigma}}(x)}{|x|^n}\,,\label{kapes}$$ where $\,{{\sigma}}(x)\,$ is infinitely differentiable for $\,x\neq\,0\,$, and satisfies the properties $\,{{\sigma}}(t\,x)=\,{{\sigma}}(x)\,,$ for $\,t>0\,,$ and $$\int_S {{\sigma}}(x) \,dS =\,0\,,$$ where $\,S=\,\{\,{x:\,|x|=1}\,\}\,$. It follows easily that, for $\,0<\,{{\rho}}_1<\,{{\rho}}_2\,,$$$\int_{{{\rho}}_1 <|x|< {{\rho}}_2} {\mathcal{K}}(x) \,dx =\,\int_{{{\rho}}_1 <|x|} {\mathcal{K}}(x)
\,dx=
\,\int {\mathcal{K}}(x) \,dx=\,0\,,\label{simp}$$ where the last integral is in the Cauchy principal value sense.
For continuous functions $\,\phi\,$ with compact support, the convolution integral $$({\mathcal{K}}\ast \phi)(x)=\,\int \,{\mathcal{K}}(x-y)\,\phi(y)\,dy\,,\label{convint}$$ extended to the whole space $\,{{\mathbb R}}^n$, exists as a Cauchy principal value and is finite.
We set $\,I({{\rho}})=\{\,x:\,|\,x\,| \leq\, {{\rho}}\,\}\,,$ $\,D_{{{\overline{\omega}}}}({{\rho}})=\,\,D_{{{\overline{\omega}}}}(I({{\rho}}))\,,$ and do the same for other functional spaces, norms, and semi-norms labeled by ${{\rho}}\,$.
Let $\,{\mathcal{K}}(x)$ be a singular kernel enjoying the properties described above. Further, assume that the oscillation function $\,{{\overline{\omega}}}\,$ satisfies , , , and . Let $\phi \in
\,D_{{{\overline{\omega}}}}({{\rho}})\,,$ vanish for $\,|x| \geq\,{{\rho}}\,.$ Then $\,{\mathcal{K}}\ast
\phi \in\, D_{{{\widehat{\omega}}}}({{\rho}})\,.$ Furthermore, in the sphere $\,I({{\rho}})\,,$ one has $$[\,({\mathcal{K}}\ast \phi)\,]_{{\widehat{\omega}}}\leq\,C\,\|\,
\phi\,\|_{{\overline{\omega}}}\,,\label{tima}$$ where $\,C=\,C(n,\,{{\overline{\omega}}}\,,|\|\,{{\sigma}}\,\||\,)\,.$ \[ofundas-G\]
Let $x_0,\,x_1 \in I({{\rho}})\,$, $0<|x_0-\,x_1|=\,{{\delta}}<\,{{\delta}}_0
\leq\,{{\rho}}\,.$ The positive constant $\,{{\delta}}_0\,$ is fixed here in correspondence to the choice $\,k_2=\,3\,$ in . In the concave case (assumed, for clearness, in the statements of theorems \[sufasvero\] and \[sufasvero-2\]), we may set $\,k_2=\,1\,$.
For convenience, we will use the simplified notation $\,{{\overline{\omega}}}(r)=\,{{\overline{\omega}}}_\phi(r)\,.$ From it follows that $$({\mathcal{K}}\ast \phi)(x)=\,\int \,\big(\,\phi(y)-\phi(x)\big)\,{\mathcal{K}}(x-\,y)
\,dy\,.$$ Hence, with abbreviated notation, $$\begin{array}{l}
({\mathcal{K}}\ast \phi)(x_0)\,-({\mathcal{K}}\ast \phi)(x_1)=\\
\\
\int\, \Big\{\,\big(\,\phi(y)-\phi(x_0)\big)\,{\mathcal{K}}(x_0-\,y)
\,-\big(\,\phi(y)-\phi(x_1)\big)\,{\mathcal{K}}(x_1-\,y)\,\Big\} \,dy=\\
\\
\int_{|y-x_0|<\,2{{\delta}}} \,\{...\} \,dy +\,\int_{2{{\delta}}<|y-x_0|<\,{{\delta}}_0}
\,\{...\} \,dy +\,\int_{{{\delta}}_0<|y-x_0|} \,\{...\} \,dy \equiv
\,I_1+I_2+I_3\,.
\end{array}
\label{decomp-G}$$ Since$$\{y:\,|y-x_1|<\,2{{\delta}}\} \subset\, \{y: \,|y-x_0|<\,3{{\delta}}\}$$ it follows that $$\begin{array}{l}
\int_{|y-x_0|<\,2{{\delta}}}
\,\big|\,\phi(y)-\phi(x_1)\big|\,|{\mathcal{K}}(x_1-\,y)| \,dy \leq\\\\
\int_{|y-x_1|<\,3{{\delta}}}
\,\big|\,\phi(y)-\phi(x_1)\big|\,|{\mathcal{K}}(x_1-\,y)| \,dy\leq \\
\\
\|\,{{\sigma}}\,\|\,\int_0^{3{{\delta}}}\,\frac{{{\overline{\omega}}}(r)}{r} \,dr\,\leq\,
\|\,{{\sigma}}\,\|\,[\,\phi\,]_{{{\overline{\omega}}}} \,\int_0^{3{{\delta}}}\,\frac{{{\overline{\omega}}}(r)}{r}
\,dr\,,
\end{array}
\label{pora3}$$ where we appealed to polar-spherical coordinates with $\,
r=\,|x_1-y|\,,$ to the fact that ${{\sigma}}$ is positive homogeneous of order zero, to , and to definition .
A similar, simplified, argument shows that equation holds by replacing $x_1$ by $x_0$ and $3{{\delta}}$ by $2{{\delta}}$. So, $$|I_1| \leq \,2\,\|\,{{\sigma}}\,\|\,[\,\phi\,]_{{{\overline{\omega}}}}
\,\int_0^{3{{\delta}}}\,\frac{{{\overline{\omega}}}(r)}{r} \,dr\,\leq
\,c\,\|\,{{\sigma}}\,\|\,[\,\phi\,]_{{{\overline{\omega}}}}\,\,\int_0^{{{\delta}}}\,\frac{{{\overline{\omega}}}(r)}{r}\,
\,dr\,$$ where we have appealed to for $\,k_2=\,3\,.$ Hence, $$|I_1| \leq \,c\,\|\,{{\sigma}}\,\|\,[\,\phi\,]_{{{\overline{\omega}}}}\,{{\widehat{\omega}}}({{\delta}})\,.\label{pora4-G}$$
On the other hand $$I_2=\,\int_{2{{\delta}}<|y-x_0|<\,{{\delta}}_0}
\,\big(\,\phi(x_1)-\phi(x_0)\big)\,{\mathcal{K}}(x_0-\,y) \,dy+$$ $$\int_{2{{\delta}}<|y-x_0|<\,{{\delta}}_0}
\,\big(\,\phi(y)-\phi(x_1)\big)\,\big({\mathcal{K}}(x_0-\,y)-\,{\mathcal{K}}(x_1-\,y)\big)
\,dy\,.$$ The first integral vanishes, due to . Hence, $$|I_2| \leq\,\int_{2{{\delta}}<|y-x_0|<\,{{\delta}}_0}
\,\big|\,\phi(y)-\phi(x_1)\big|\,\big|\,{\mathcal{K}}(x_0-\,y)-\,{\mathcal{K}}(x_1-\,y)\,
\big| \,dy\,.$$ Further, by the mean-value theorem, there is a point $x_2$, between $x_0$ and $x_1$, such that $$\big|\,{\mathcal{K}}(x_0-\,y)-\,{\mathcal{K}}(x_1-\,y)\,\big|
\leq\,\big|\,{{\nabla}}\,{\mathcal{K}}(x_2-\,y)\,\big|\,{{\delta}}\,.$$ Since $${{\partial}}_i\,{\mathcal{K}}(x)=\,\frac{1}{|x|^{n+\,1}}\,\Big[\,({{\partial}}_i\,{{\sigma}})
\Big(\frac{x}{|x|}\Big)\,-\,n\,\frac{x_i}{|x|}\,{{\sigma}}(x)\,\Big]\,,$$ it readily follows that $$\begin{array}{l}
\big|\,{\mathcal{K}}(x_0-\,y)-\,{\mathcal{K}}(x_1-\,y)\,\big| \leq \\
\\
c\,\||\,{{\sigma}}\,\|| \,\frac{{{\delta}}}{|y-\,x_2|^{n+1}}
\leq\,c\,\||\,{{\sigma}}\,\||
\,\frac{{{\delta}}}{|y-\,x_0|^{n+1}}\,,\end{array}
\label{esao2}$$ where $\,\||\,{{\sigma}}\,\||\,$ denotes the sum of the $L^\infty$ norms of ${{\sigma}}$ and of its first order derivatives on the surface of the unit sphere $I(0,1)\,.$ Note that, for $\,|x_0-\,y|>\,2\,{{\delta}}\,,$ one has $$|x_0-\,y|\leq\,2\,|x_2-\,y|\leq\,4\,|x_0-\,y|\,.$$ On the other hand, for $\,2\,{{\delta}}<|x_0-\,y|\,,$ $$|x_1-\,y|\leq\,3\,|x_0-\,y|\,.$$ So, $$|\,\phi(y)-\phi(x_1)\,| \leq\,[\,\phi\,]_{{{\overline{\omega}}}\,} \, {{\overline{\omega}}}(\,3\,|\,x_0-\,y|\,)\,.$$ The above estimates show that $$\begin{array}{l}
|\,I_2\,| \leq\,c\,\||\,{{\sigma}}\,\||\,[\,\phi\,]_{{{\overline{\omega}}}\,}
\,{{\delta}}\,\int_{2{{\delta}}}^{{{\delta}}_0} \, {{\overline{\omega}}}\big(\,3\,r\,\big)\,
r^{-2} \,dr\\
\\
\leq \,c\,\||\,{{\sigma}}\,\||\,[\,\phi\,]_{{{\overline{\omega}}}\,}
\,{{\delta}}\,\int_{2{{\delta}}}^{\,{{\delta}}_0} \,{{\overline{\omega}}}(\,r\,)\, r^{-2} \,dr\,,\end{array}\label{esao2nao}$$ where we appealed to for $\,k_2=\,3\,.$ Finally, by , it readily follows that $$|\,I_2\,|\, \leq \,c\,\||\,{{\sigma}}\,\||\,[\,\phi\,]_{{{\overline{\omega}}}\,}\,{{\widehat{\omega}}}({{\delta}})\label{esao23-G}$$ for $\,{{\delta}}\in\,(0,\,{{\delta}}_0\,)\,.$
Finally we consider $I_3$. By arguing as for $I_2$, in particular by appealing to and , one shows that $$\begin{array}{l}
|I_3| \leq\,C\,{{\delta}}\,|\|\,{{\sigma}}\,\||\,\int_{|y-x_0|>\,{{\delta}}_0} \,
\,\frac{|\,\phi(y)-\phi(x_1)\big|}{\,|y-\,x_0|^{n+1}} \; dy \leq \\
\\
C\,{{\delta}}\,|\|\,{{\sigma}}\,\||\,\|\,\phi\,\|
\leq\,C\,|\|\,{{\sigma}}\,\||\,\|\,\phi\,\|\,{{\widehat{\omega}}}({{\delta}})\,.\end{array}\label{thislast}$$ Note that, by a de l’Hôpital rule, one shows that holds with $\,{{\overline{\omega}}}(r)\,$ replaced by $\,{{\widehat{\omega}}}(r)\,.$ From equation , by appealing to , , and , one shows that $$|\,({\mathcal{K}}\ast \phi)(x_0)\,-({\mathcal{K}}\ast
\phi)(x_1)\,|\leq\,C\, \||\,{{\sigma}}\,\|| \,\|\,\phi\,\|_{{{\overline{\omega}}}\,}\,{{\widehat{\omega}}}({{\delta}})\,,\label{tresis}$$ for each couple of points $\,x_0,\,x_1 \in\,I({{\rho}})\,$ such that $\,0<\,|x_0-\,x_1\,| \leq\,{{\delta}}_0\,.$ Hence holds.
We may easily estimate $|\,({\mathcal{K}}\ast \phi)(x_0)\,-({\mathcal{K}}\ast
\phi)(x_1)\,|\,$ for pairs of points $\,x_0,\,x_1\,$ for which $\,{{\delta}}_0<\,|x_0-\,x_1\,| < {{\rho}}\,.$ However this is superfluous, since $\,{{\delta}}_0\,$ is fixed “once and for all”.
The interior regularity estimate in the constant coefficients case. {#elipse}
===================================================================
In this chapter we apply the theorem \[ofundas-G\] to prove the basic interior regularity result for solutions of the elliptic equation in the framework of $\,D_{{{\overline{\omega}}}}\,$ data spaces. In this section ${\boldsymbol{L}}$ ia a constant coefficients operator. The proof is inspired by that developed in Hölder spaces in [@JBS], part II, section 5. For convenience, assume that $n\geq\,3\,.$
By a fundamental solution of the differential operator $\,{\boldsymbol{L}}\,$ one means a distribution $\,J(x)\,$ in $\,{{\mathbb R}}^n\,$ such that $${\boldsymbol{L}}\,J(x)=\,{{\delta}}(x)\,.\label{fundas}$$ The celebrated Malgrange-Ehrenpreis theorem states that every non-zero linear differential operator with constant coefficients has a fundamental solution (see, for instance, [@yosida], Chap. VI, sec. 10). We recall that the analogue for differential operators whose coefficients are polynomials (rather than constants) is false, as shown by a famous Hans Lewy’s counter-example.
In particular, for a second order elliptic operator with constant coefficients and only higher order terms, one can construct explicitly a fundamental solution $\,J(x)$ which satisfies the properties (i), (ii), and (iii), claimed in [@JBS], namely,
\(i) $J(x)$ is a real analytic function for $\,|x| \neq\,0\,.$
\(ii) For $n\geq\,3\,$ $$J(x)=\,\frac{{{\sigma}}(x)}{|x|^{n-\,2}}\,,\label{jicas}$$ where ${{\sigma}}(x)$ is positive homogeneous of degree $\,0\,$.
\(iii) Equation holds. In particular, for every sufficiently regular, compact supported, function $\,v\,$, one has $$v(x)=\,\int \,J(x-\,y)\,({\boldsymbol{L}}\,v)(y) \,dy\,.$$ For a second order elliptic operator as above, one has$$J(x)=\,c\,\big(\,\sum\, A_{i\,j}\,x_i x_j\,\big)^{\frac{2-\,n}{2}}\,,\label{janota}$$ where $A_{i\,j}$ denotes the cofactor of $a_{i\,j}$ in the determinant $|\,a_{i\,j}\,|\,.$
Following [@JBS], we denote by ${\boldsymbol{S}}$ the operator $$({\boldsymbol{S}}\,{{\phi}})(x)=\,\int \,J(x-\,y)\,{{\phi}}(y) \,dy=\,(J\ast{{\phi}})(x)\,.\label{beesse}$$ Note that, in the constant coefficients case, the operator ${\boldsymbol{T}}$ introduced in reference [@JBS] vanishes.
Point (iii) above (see also [@JBS] “Lemma” A) shows that if $v$ is compact supported and sufficiently regular (for instance of class $\,C^2\,$), then $$v=\, {\boldsymbol{S}}{\boldsymbol{L}}\,v\,.\label{slfi}$$ Due to the structure of the function $\,{{\sigma}}(x)\,$ appearing in equation , it readily follows that second order derivatives of $({\boldsymbol{S}}\,{{\phi}})(x)$ have the form $\,{{\partial}}_i\,{{\partial}}_j
\,{\boldsymbol{S}}\,{{\phi}}=\,{\mathcal{K}}_{i\,j} \ast \phi\,,$ where the ${\mathcal{K}}_{i\,j}$ enjoy the properties described for singular kernels ${\mathcal{K}}$ in section \[apotes-G\].
We write, in abbreviated form, $${{\nabla}}^2\,{\boldsymbol{S}}\,{{\phi}}\,(x)=\,\int \,{\mathcal{K}}(x-y)\,\phi(y)\,dy\,,\label{convintos}$$ where $\,{\mathcal{K}}(x)\,$ enjoys the properties described at the beginning of section \[apotes-G\]. From it follows that $${{\nabla}}^2\,{\boldsymbol{S}}{\boldsymbol{L}}v=\,\int \,{\mathcal{K}}(x-y)\,{\boldsymbol{L}}\,v(y)\,dy\,.$$ Hence, by Theorem \[ofundas-G\], one gets $$[\,{{\nabla}}^2\,{\boldsymbol{S}}{\boldsymbol{L}}v\,]_{{{\widehat{\omega}}};\,2\,{{\rho}}}\leq\,C\,[\,
{\boldsymbol{L}}\,v\,]_{{{\overline{\omega}}};\,2\,{{\rho}}}\,.\label{masimus}$$ By appealing to we get the following result.
Assume that the differential operator $\,{\boldsymbol{L}}\,$ has constant coefficients and that the oscillation function $\,{{\overline{\omega}}}\,$ satisfies assumptions , , , and . Let $\,v\,$ be a support compact function $ \in
C^{2}(2\,{{\rho}})\,,$ such that $\,{\boldsymbol{L}}\,v \in D_{{{\overline{\omega}}}}(2\,{{\rho}})\,.$ Then $$[{{\nabla}}^2\, v\,]_{{{\widehat{\omega}}};\,2\,{{\rho}}}\leq\,C\,[\,
{\boldsymbol{L}}\,v\,]_{{{\overline{\omega}}};\,2\,{{\rho}}}\,.\label{larj}$$ \[lemitas\]
One has the following interior regularity result. For brevity we have consider two spheres of radius ${{\rho}}$ and $R$, $R>\,{{\rho}}$, in the particular case $R=\,2\,{{\rho}}.$
Assume that the hypothesis of proposition \[lemitas\] hold. Further, let $u \in C^{2}(2\,{{\rho}})\,$ be such that $\,{\boldsymbol{L}}\,u \in
D_{{{\overline{\omega}}}}(2\,{{\rho}})\,.$ Then $\,{{\nabla}}^2\,u \in \,D_{{{\widehat{\omega}}}}({{\rho}})\,,$ moreover $$[{{\nabla}}^2\,u\,]_{\,{{\widehat{\omega}}};\,{{\rho}}} \leq \,C\,[\,{\boldsymbol{L}}\,u\,]_{{{\overline{\omega}}},\,2\,{{\rho}}}+
\,c({{\theta}})\,\Big(\,\frac{\|u\|}{{{\rho}}^3}+\,\frac{\|\,{{\nabla}}\,u\|}{{{\rho}}^2}+\,\frac{\|\,{{\nabla}}^2\,u\|}{{{\rho}}}\,\Big)\,\frac{|x-\,y|}{{{\overline{\omega}}}(|x-\,y|)}\,,
\label{acasisim}$$ for some positive constant $\,C,$ independent of ${{\rho}}\,$. In particular, $$[{{\nabla}}^2\,u\,]_{\,{{\widehat{\omega}}};\,{{\rho}}} \leq \,C\,[\,{\boldsymbol{L}}\,u\,]_{{{\overline{\omega}}},\,2\,{{\rho}}}+
\,\frac{c({{\theta}})}{{{\rho}}^3} \,\|\,u\|_{C^2(2{{\rho}})}\,.
\label{acasisim-2}$$ \[ajudas\]
Fix a no-negative $C^\infty\,$ function ${{\theta}}$, defined for $\,0\leq t \leq 1\,$ such that ${{\theta}}(t)=\,1$ for $\,0\leq t \leq \frac13\,,$ and ${{\theta}}(t)=\,0$ for $\,\frac23\leq t \leq 1\,.$ Further fix a positive real $\,{{\rho}}\,$, for convenience $0<{{\rho}}<\,\frac12\,,$ and define $${{\zeta}}(x)= \left\{
\begin{array}{l}
1 \quad \textrm{for} \quad |x|\leq {{\rho}}\,,\\
\\
{{\theta}}\big(\frac{|x|-\,{{\rho}}}{{{\rho}}}\big) \quad \textrm{for} \quad {{\rho}}\leq
|x|\leq 2\,{{\rho}}\,.
\end{array}
\right.\label{zetaze}$$ Next we consider $\,{{\zeta}}(x)\,$ for points $\,x$ such that $\,{{\rho}}\leq\,|x|\leq\,2\,{{\rho}}\,,$ and leave to the reader different situations. Due to symmetry, it is sufficient to consider the one dimensional case $${{\zeta}}(t)=\,{{\theta}}\big(\frac{t-\,{{\rho}}}{{{\rho}}}\big) \quad \textrm{for} \quad {{\rho}}\leq t\leq 2\,{{\rho}}\,.$$ Hence $${{\zeta}}'(t)=\,{{\theta}}' \big(\frac{t-\,{{\rho}}}{{{\rho}}}\big)\,\frac{1}{{{\rho}}}\,,$$ and $${{\zeta}}''(t)=\,{{\theta}}'' \big(\frac{t-\,{{\rho}}}{{{\rho}}}\big)\,\frac{1}{{{\rho}}^2}\,.$$ Further, $${{\rho}}^2\, |{{\zeta}}''(t_2)-{{\zeta}}''(t_1)| \leq\,\Big| \, {{\theta}}''
\Big(\frac{t_2-\,{{\rho}}}{{{\rho}}}\Big)
-\,{{\theta}}''\Big(\frac{t_1-\,{{\rho}}}{{{\rho}}}\Big)\,\Big|\,,$$ where $$\Big|\,\frac{t_2-\,{{\rho}}}{{{\rho}}}-\,\frac{t_1-\,{{\rho}}}{{{\rho}}}\,\Big|=\,\Big|
\frac{t_2-\,t_1}{{{\rho}}}\Big|\leq\,\frac13 <\,1\,.$$ So $$|{{\zeta}}''(t_2)-{{\zeta}}''(t_1)\,| \leq \, \frac{1}{{{\rho}}^3}\,
\,[\,{{\theta}}''\,]_{Lip}\,|\,|t_2-\,t_1|\,,
\label{zedois}$$ where $\,[\,\cdot\,]_{Lip}\,$ denotes the usual Lipschitz semi-norm.
Set $$v=\,{{\zeta}}\,u\,.
\label{vezu}$$ Note that $\,{\boldsymbol{L}}\,v \in D_{{{\overline{\omega}}}}(2{{\rho}})\,,$ moreover the support of $\,v\,$ is contained in $\,|x|<\,2{{\rho}}\,.$
On the other hand, $${\boldsymbol{L}}v=\,{{\zeta}}{\boldsymbol{L}}u+\, N\,.
\label{lvn}$$ One has $$\begin{array}{l}
|\,({{\zeta}}{\boldsymbol{L}}u)(x)-\,({{\zeta}}{\boldsymbol{L}}u)(y)\,| \leq\,\|{{\zeta}}\|\,[ {\boldsymbol{L}}u ]_{{{\overline{\omega}}}}\,{{\overline{\omega}}}(|x-\,y|) +\,\|\,{{\nabla}}\,{{\zeta}}\,\|\,\| {\boldsymbol{L}}u\,\|\,|x-\,y| \\
\\
\leq \,[ {\boldsymbol{L}}u ]_{{{\overline{\omega}}}}\,{{\overline{\omega}}}(|x-\,y|) +c\,\|{{\theta}}' \|\,\frac{1}{{{\rho}}}\,\|{{\nabla}}^2\,u\|\,|x-\,y|\,.
\end{array}$$ Hence, $$[\,{{\zeta}}{\boldsymbol{L}}u\,]_{{{\overline{\omega}}}} \leq\, [ {\boldsymbol{L}}u ]_{{{\overline{\omega}}}}+c\,\|{{\theta}}' \|\,\frac{1}{{{\rho}}}\,\|{{\nabla}}^2\,u\|\,\frac{|x-\,y|}{{{\overline{\omega}}}(|x-\,y|)}\,.
\label{princ}$$ Next we prove that $$[\,N\,]_{{\overline{\omega}}}\leq\,c({{\theta}})\,\Big(\,\frac{\|u\|}{{{\rho}}^3}+\,\frac{\|\,{{\nabla}}\,u\|}{{{\rho}}^2}+\,\frac{\|\,{{\nabla}}^2\,u\|}{{{\rho}}}\,\Big)\,\frac{|x-\,y|}{{{\overline{\omega}}}(|x-\,y|}\,.
\label{eanes}$$ One has $$N\cong ({{\nabla}}^2 {{\zeta}})\,u+\,({{\nabla}}\,{{\zeta}})\,({{\nabla}}\,u)\equiv A+\,B\,.$$ By appealing in particular to , straightforward calculations show that $$\begin{array}{l}
|A(x)-A(y)|\leq\,\|{{\nabla}}u\|\,\|{{\nabla}}^2\,{{\zeta}}\|\,|x-\,y|+\,\|u\| \frac{1}{{{\rho}}^3}[{{\theta}}'']_{Lip} |x-\,y|\\
\\
\leq\,\Big( \frac{1}{{{\rho}}^2} \|{{\theta}}''\|\,\|{{\nabla}}u\|+\, \frac{1}{{{\rho}}^3} [{{\theta}}'']_{Lip} \,\| u\|\,\Big)\,|x-y|\,.
\end{array}$$ Hence $$[\,A\,]_{{\overline{\omega}}}\leq\,c({{\theta}})\,\Big(\,\frac{\|u\|}{{{\rho}}^3}+\,\frac{\|\,{{\nabla}}\,u\|}{{{\rho}}^2}\,\Big)\,\frac{|x-\,y|}{{{\overline{\omega}}}(|x-\,y|}\,.
\label{aass}$$ Similar manipulations show that $$[\,B\,]_{{\overline{\omega}}}\leq\,c({{\theta}})\,\Big(\,\frac{\|\,{{\nabla}}\,u\|}{{{\rho}}^2}+\,\frac{\|\,{{\nabla}}^2\,u\|}{{{\rho}}}\,\Big)\,\frac{|x-\,y|}{{{\overline{\omega}}}(|x-\,y|}\,.
\label{bbss}$$ Equation follows from and .
Lastly, from , , and one shows that $$[\,{\boldsymbol{L}}\,v\,]_{{\overline{\omega}}}\leq \,[\,{\boldsymbol{L}}\,u\,]_{{\overline{\omega}}}+
\,c({{\theta}})\,\Big(\,\frac{\|u\|}{{{\rho}}^3}+\,\frac{\|\,{{\nabla}}\,u\|}{{{\rho}}^2}+\,\frac{\|\,{{\nabla}}^2\,u\|}{{{\rho}}}\,\Big)\,\frac{|x-\,y|}{{{\overline{\omega}}}(|x-\,y|}\,.
\label{tormes}$$ In the following not labeled norms concern the domain $\,I(2\,{{\rho}})\,.$
From , , , and one gets $$\begin{array}{l}
[{{\nabla}}^2\,u\,]_{\,{{\widehat{\omega}}};\,{{\rho}}} \leq\,[{{\nabla}}^2\,v\,]_{\,{{\widehat{\omega}}}}
\leq\,C\,[\,{\boldsymbol{L}}\,v\,]_{{{\overline{\omega}}}}\\
\\
\leq \,C\,[\,{\boldsymbol{L}}\,u\,]_{{\overline{\omega}}}+
\,c({{\theta}})\,\Big(\,\frac{\|u\|}{{{\rho}}^3}+\,\frac{\|\,{{\nabla}}\,u\|}{{{\rho}}^2}+\,\frac{\|\,{{\nabla}}^2\,u\|}{{{\rho}}}\,\Big)\,\frac{|x-\,y|}{{{\overline{\omega}}}(|x-\,y|}\,,
\end{array}
\label{acasis}$$ where $\,0<2\,{{\rho}}<1\,.$
The interior regularity estimate in the variable coefficients case. {#muda}
===================================================================
In this section we extend the estimate to uniformly elliptic operators with variable coefficients $${\boldsymbol{L}}=\,\sum_1^n a_{i\,j}(x) {{\partial}}_i\,{{\partial}}_j\,.\label{elld}$$ To avoid non significant manipulations we assume that the coefficients $\, a_{i\,j}(x) $ are Lipschitz continuous in $\,I(2\,{{\rho}})\,,$ which Lipschitz constants bounded by a constant $\,A\,$. Following the same belief, we left to the reader the introduction of lower order terms.
We assume that $${{\overline{\omega}}}(r)\leq\,k_1\,{{\widehat{\omega}}}(r)\,,
\label{junto}$$ for some positive constant $k_1$, and $\,r\,$ in some neighborhood of the origin. This yields $\,D_{{{\overline{\omega}}}}({{\overline{\Omega}}})=\,D_{{{\widehat{\omega}}}}({{\overline{\Omega}}})\,,$ recall proposition \[eqnormes\]. Assumption holds if in equation the constant $C_1$ is finite. In fact, $$\lim_{r\rightarrow 0}\,\frac{{{\overline{\omega}}}(r)}{\,{{\widehat{\omega}}}(r)}=\lim_{r\rightarrow
0}\,\frac{r\,\,{{\overline{\omega}}}'(r)}{{{\overline{\omega}}}(r)}=\,\frac{1}{C_1}\,,$$ if the second limit exists.
In the following we appeal to the constant coefficients operator $${\boldsymbol{L}}_0=\,\sum_1^n b_{i\,j} {{\partial}}_i\,{{\partial}}_j\,,\label{ellez}$$ where $\,b_{i\,j}=\, a_{i\,j}(0)\,.$ Clearly, $${\boldsymbol{L}}_0 v(x)=\,{\boldsymbol{L}}v(x)+ (\,{\boldsymbol{L}}_0 -\,{\boldsymbol{L}}) v(x)\,.
\label{zerob}$$ One has $$\begin{array}{l}
(\,{\boldsymbol{L}}_0-\,{\boldsymbol{L}})v(x)-\,(\,{\boldsymbol{L}}_0-\,{\boldsymbol{L}})v(y)=\\
\\
\,(\,( b_{i\,j}-\, a_{i\,j}(x)\,)\,(\,{{\partial}}^2_{i\,j}\,v(x)-\,{{\partial}}^2_{i\,j}\,v(y)\,)
+\,(\,( a_{i\,j}(y)-\, a_{i\,j}(x)\,)\,(\,{{\partial}}^2_{i\,j}\,v(y)
\end{array}
\label{semn}$$ where, for convenience, summation on repeated indexes is assumed. Straightforward calculations easily lead to the following pointwise estimate $$|\,(\,{\boldsymbol{L}}_0-\,{\boldsymbol{L}})v(x)-\,(\,{\boldsymbol{L}}_0-\,{\boldsymbol{L}})v(y)\,| \leq\, c\,A
\big(\,2\,{{\rho}}\,[\,{{\nabla}}^2\,v\,]_{{\overline{\omega}}}+\,\|{{\nabla}}^2\,v\,\|\,\frac{|x-y|}{{{\overline{\omega}}}(|x-y|)}\,\big) \,{{\overline{\omega}}}(|x-y|)\,,
\label{ellez}$$ where norms and semi-norms concern the sphere $\,I(0,\,2\,{{\rho}})\,$.
Next assume that $\,v\in C^{2}(2\,{{\rho}})\,$ has compact support in $\,I(0,\,2\,{{\rho}})\,$, and $\,{\boldsymbol{L}}\,v \in D_{{{\overline{\omega}}}}(2\,{{\rho}})\,.$ Then, by , , and it follows that $$[{{\nabla}}^2\, v\,]_{{{\widehat{\omega}}};\,2\,{{\rho}}}\leq\,C\,[\, {\boldsymbol{L}}\,v\,]_{{{\overline{\omega}}};\,2\,{{\rho}}}\,+\,C\,{{\rho}}\,[\,{{\nabla}}^2\,v\,]_{{{\overline{\omega}}};\,2\,{{\rho}}}
+\, C\,\|{{\nabla}}^2\,v\,\|\,\frac{|x-y|}{{{\overline{\omega}}}(|x-y|)}\,.$$ In particular $$[{{\nabla}}^2\, v\,]_{{{\widehat{\omega}}};\,2\,{{\rho}}}\leq\,C\,[\, {\boldsymbol{L}}\,v\,]_{{{\overline{\omega}}};\,2\,{{\rho}}}\,+\,C\,{{\rho}}\,[\,{{\nabla}}^2\,v\,]_{{{\overline{\omega}}};\,2\,{{\rho}}}
+\, C\,\|{{\nabla}}^2\,v\,\|\,.\label{seigual}$$ Now, from , one gets $$(\,1-C\,k_1\,{{\rho}}\,)\,[{{\nabla}}^2\, v\,]_{{{\overline{\omega}}};\,2\,{{\rho}}}\leq\,C\,(\,[\,
{\boldsymbol{L}}\,v\,]_{{{\overline{\omega}}};\,2\,{{\rho}}} +\,\|{{\nabla}}^2\,v\,\|\,).\label{seigual}$$ Next we set $$v=\,{{\zeta}}\,u$$ and argue as done to prove . This proves the following result, in the case of variable coefficients operators.
Assume that the oscillation function $\,{{\overline{\omega}}}\,$ satisfies conditions , , , , and . Further, assume that $$0<\,{{\rho}}\leq\,\frac{1}{2 C k_1}\,,$$ and let $\,{\boldsymbol{L}}\,u \in D_{{{\overline{\omega}}}}(2\,{{\rho}})\,,$ for some $u \in
C^{2}(2\,{{\rho}})\,.$ Then $\,{{\nabla}}^2\,u \in \,D_{{{\widehat{\omega}}}}({{\rho}})\,,$ and $$[{{\nabla}}^2\,u\,]_{\,{{\widehat{\omega}}};\,{{\rho}}} \leq \,C\,[\,{\boldsymbol{L}}\,u\,]_{{{\overline{\omega}}},\,2\,{{\rho}}}+
\,\frac{C}{{{\rho}}^3} \,\|\,u\|_{C^2(2{{\rho}})}\,,\label{acasopas-2}$$ for suitable positive constants $\,C\,,$ independent of ${{\rho}}\,$.
Proof of theorems \[sufasvero\] and \[sufasvero-2\]. {#elipse2}
====================================================
The local estimates (estimates in $\,{{\Omega}}_0 \,,$ $\,{{\Omega}}_0 \subset
\subset\,{{\Omega}}\,$) claimed in theorems \[sufasvero\] and \[sufasvero-2\] follow immediately from the interior estimates, by appealing to the classical method consisting in covering ${{\overline{\Omega}}}_0\,$ by a finite number of sufficiently small spheres. For brevity, we may estimate the quantities originated by the terms $\|\,u\|_{C^2(2{{\rho}})}\,,$ see the right hand sides of equations and , simply by appealing to the theorem \[laplaces\], which shows that solutions $u$ satisfy the estimate $$\|\,u\,\|_{C^2({{\overline{\Omega}}})} \leq \,c\,\|\,f\,\|_*\,.\label{lapili-2}$$ Concerning regularity up to the boundary one proceeds as follows. The main point, the extension of the interior regularity estimate from spheres to half-spheres, is obtained by following the argument described in part II, section 5.6, reference [@JBS]. One starts by showing that the interior estimate in spheres also hold for half-spheres, under the zero boundary condition on the flat part of the boundary. One appeals to “reflection” of $\,u\,$ through the flat boundary, as an odd function, in the orthogonal direction, from the half to the whole sphere. In this way the half-sphere problem goes back to an whole-sphere problem, absolutely similar to that considered in section \[elipse\], see [@JBS]. Note tat is is sufficient, and more convenient, to show this extension to half-spheres merely for constant coefficient operators. The regularity result “up to flat boundary points” claimed in theorem \[sufasvero\] follows.
Extension of the half-sphere’s estimate, from constant coefficients to variable coefficients, is obtained exactly as done in section \[muda\] for whole spheres. Obviously, this requires assumption . Then, sufficiently small neighborhoods of boundary points are regularly mapped, one to one, onto half-spheres. This procedure allows extension of the local estimate for functions $\,u\,$ defined on sufficiently small neighborhoods of boundary points, vanishing on the boundary. A well known finite covering argument leads to the thesis of theorem \[sufasvero-2\].
The above extension to non-flat boundary points requires local changes of coordinates. This transforms constant coefficients in variable coefficients operators. So, local regularity up to non-flat boundary points for constant coefficients operators can not be claimed here. This is a challenging open problem. One may start by considering the particular case of data in Log spaces.
We note that in the proof of Theorem 1, section 5.4, part II, in reference [@JBS], density of $\,C^1\,$ in Hőlder spaces is used. The same occurs in the proof of lemma B, section 5.3.
The Log spaces $ D^{0,\,{{\alpha}}}({{\overline{\Omega}}})\,.$ An intermediate regularity result. {#DOPO}
==================================================================================================
The following is a significant example of functional space $\,D_{{{\overline{\omega}}}}({{\overline{\Omega}}})\,$ which yields intermediate (not full) regularity, based on the well known formulae $$\int \,\frac{(-\log{r})^{-\,{{\alpha}}}}{r} \,dr
=\,\frac{1}{{{\alpha}}-\,1}\,(-\log{r})^{1-\,{{\alpha}}}\,,\label{simvales}$$ where $\,0<\,{{\alpha}}<\,+\infty\,$ (for $\,{{\alpha}}=\,1\,$ the right hand side should be replaced by $\,-\log\,(-\log{r})\,).$ Equation shows that the $\,C_*({{\overline{\Omega}}})\,$ semi-norm is finite if $${{\omega}}_f(r)\leq\, C\,(-\log{r})^{-\,{{\alpha}}}\,,\label{alfas}$$ for some $\,{{\alpha}}>\,1\,$ and some constant $\,C>\,0\,.$ This led to define, for each fixed $\,{{\alpha}}>\,0\,,$ the semi-norm $$[\,f\,]_{{{\alpha}}} \equiv\,
\sup_{\,r \in (0,\,1) }\,\frac{{{\omega}}_f(r)}{{{\omega}}_{{\alpha}}(r)} \,,\label{alfas2}$$ where the *oscillation function* $\,{{\omega}}_{{\alpha}}(r)\,$ is defined by setting $${{\omega}}_{{\alpha}}(r) =\,(-\log{r})^{-\,{{\alpha}}}\,.\label{alfaerre}$$ Hence $\,[\,f\,]_{{{\alpha}}}\,$ is the smallest constant for which the estimate $$|f(x)-\,f(y)\,|\leq\, [\,f\,]_{{{\alpha}}} \,\cdot\,\Big(\,
\log\,\frac{1}{|\,x-\,y|}\,\Big)^{-\,{{\alpha}}}\label{alforreca}$$ holds for all couple $\,x,\,y \in\,{{\overline{\Omega}}}\,$ such that $\,|x-\,y|<\,1\,.$ Note that we have merely replaced, in the definition of Hölder spaces, the quantity $$\frac{1}{|\,x-\,y|} \quad \textrm{ by} \quad
\log{\frac{1}{|\,x-\,y|}}\,,$$ and allow $\,{{\alpha}}\,$ to be arbitrarily large.
For each real positive $\,{{\alpha}}\,,$ we set $$D^{0,\,{{\alpha}}}({{\overline{\Omega}}}) \equiv\,\{\,f \in\,C(\,{{\overline{\Omega}}}): \,[\,f\,]_{{{\alpha}}}
<\,\infty\,\}\,.\label{cstar}$$ A norm is introduced in $\,D^{0,\,{{\alpha}}}({{\overline{\Omega}}})\,$ by setting $
\,\|\,f\,\|_{{{\alpha}}}\equiv\,[\,f\,]_{{{\alpha}}}+\,\|\,f\,\|\,. $\[defcstar\]
We call these spaces Log spaces. We remark that in reference [@BV-arxiv] we have called these spaces H-log spaces.
The restriction $\,|x-\,y|<\,1\,$ in equation is due to the behavior of the function $\,\log{r}\,,$ for $\,r \geq\,1\,.$ Note that, by replacing $\,0<\,|x-\,y|<\,1\,$ by $\,0<\,|x-\,y|<\,{{\rho}}\,$ in equation , for some $\,0<\,{{\rho}}<\,1\,,$ it follows that $$[\,f\,]_{{{\alpha}};\,{{\rho}}} \le\,[\,f\,]_{{{\alpha}}}
\leq\,[\,f\,]_{{{\alpha}};\,{{\rho}}}+\,\frac{2}{(-\log{{{\rho}}})^{-\,{{\alpha}}}}\,\|\,f\,\|\,,
\label{clearnot}$$ where the meaning of $\,[\,f\,]_{{{\alpha}};\,{{\rho}}}\,$ seems clear. Hence, the norms $\,\|\,f\,\|_{\,{{\alpha}}}\,$ and $\,\|\,f\,\|_{{{\alpha}};\,{{\rho}}}\,$ are equivalent. We may also avoid the above $\,|x-\,y|<\,1\,$ inconvenient by replacing in the denominator of the right hand side of the quantity $\,r\,$ by $\,r/R\,$, where $\,R
=\,\textrm{diam}\,{{\Omega}}\,$, and by letting $r\in (0,\,R)\,.$ We rather prefer the first definition, since the second one implies more ponderous notation.
For $\,0<\,{{\beta}}<\,{{\alpha}}\,,$ and $\,0<\,{{\lambda}}\leq\,1\,,$ the (compact) embedding $$\quad C^{0,\,{{\lambda}}}({{\overline{\Omega}}})\subset D^{0,\,{{\alpha}}}({{\overline{\Omega}}}) \subset D^{0,\,\beta}({{\overline{\Omega}}})\, \subset \,C({{\overline{\Omega}}})\label{oitooos}$$ hold. Furthermore, for $\,1<\,{{\alpha}}\,,$ one has the (compact) embedding $\,
D^{0,\,{{\alpha}}}({{\overline{\Omega}}}) \subset \,C_*({{\overline{\Omega}}})\,.\,$ Note that $\,D^{0,\,1}({{\overline{\Omega}}}) \nsubseteq
\,C_*({{\overline{\Omega}}})\,$.
The properties proved in reference [@BV-arxiv] for $\,
D^{0,\,{{\alpha}}}({{\overline{\Omega}}})\,$ spaces follow here from that proved for $\,D_{{\overline{\omega}}}({{\overline{\Omega}}})\,$ spaces, since $\,{{\omega}}_{{\alpha}}(r)\,$ is a particular case of function $\,{{\overline{\omega}}}(r)\,.$ It is worth noting that in reference [@BV-arxiv] we claimed, and left the proof to the reader, that $\,C^{\infty}({{\overline{\Omega}}})\,$ is dense in $\,
D^{0,\,{{\alpha}}}({{\overline{\Omega}}})\,$. Actually, as shown in theorem \[osned\], this result is false.
The following result is a particular case of theorem \[sufasvero\].
Let ${{\Omega}}_0 \subset \subset \,{{\Omega}}\,$, $\,f\in\, D^{0,\,{{\alpha}}}({{\overline{\Omega}}})\,$ for some $\,{{\alpha}}>\,1\,,$ and $\,u\,$ be the solution of problem , where ${\boldsymbol{L}}$ has constant coefficients. Then $\,{{\nabla}}^2\,u \in\, D^{0,\,{{\alpha}}-\,1}({{\Omega}}_0)\,,$ moreover $$\|\,{{\nabla}}^2\,u\,\|_{\,{{\alpha}}-1,\,{{\Omega}}_0} \leq \,C\,\|\,f\,\|_{{\alpha}}\,,\label{hahega}$$ for some positive constant $\,C= C({{\alpha}},\,{{\Omega}}_0,\,{{\Omega}})\,.$ The regularity result holds up to flat boundary points. Results are optimal in the sharp sense, see section \[optimus\]. In particular,for $\,\beta >\,{{\alpha}}-1\,,$ $\,{{\nabla}}^2\,u \in\,
D^{0,\,\beta}({{\Omega}}_0)\,$ is false in general.\[laplolas\]
Theorem \[laplolas\] is a particular case of theorem \[sufasvero\]. In fact, the oscillation function $\,{{\omega}}_{{\alpha}}(r)\,$ is concave and differentiable for $\,r>\,0\,,$ satisfies for $\,{{\alpha}}> \,1\,,$ and holds. Further, condition follows from $$\lim_{r\rightarrow 0}\,\frac{{{\omega}}_{{\alpha}}(r)}{r\,\,{{\omega}}'_{{\alpha}}(r)} =\,+
\,\infty\,.\label{biri}$$ In reference [@BV-arxiv] the above regularity result was claimed up to the boundary. However the proof is not complete, since extension to non-flat boundary points requires estimates for variable coefficients operators. The reason for this requirement was explained in section \[elipse2\].
Next we apply to the results stated in theorem \[laplolas\] to illustrate, by means of a simple example, the meaning of *sharp* optimality. This concept will be discussed in a more abstract form in section \[optimus\]. Optimality of regularity results is not confined here to the particular family of spaces under consideration, but is something stronger. Let us illustrate the distinction. The theorem \[laplolas\] claims that $\,{{\nabla}}^2\,u\in\,D^{0,\,{{\alpha}}-\,1}({{\overline{\Omega}}})\,$. Optimality *restricted* to the Log spaces framework means that, given $\,\beta>\,{{\alpha}}-\,1\,,$ there is at least a data $\,f\in\,
D^{0,\,{{\alpha}}}({{\overline{\Omega}}})\,$ for which $\,{{\nabla}}^2\,u\,$ does not belong to $\,D^{0,\,\beta}({{\overline{\Omega}}})\,.$ This situation does not exclude that (for instance, and to fix ideas) for all $\,f\in\,D^{0,\,{{\alpha}}}({{\overline{\Omega}}})\,$ the oscillation $\,{{\omega}}(r)\,$ of $\,{{\nabla}}^2\,u\,$ satisfies the stronger estimate $${{\omega}}(r)\leq\, C_f\,\big[\,\log
\big(\log{\frac1r}\,\big)\big]^{-1}\,\cdot\,(-\log{r})^{-\,({{\alpha}}-1)}\,.\label{alfasin}$$ In fact, for each $\,\beta>\,{{\alpha}}-\,1\,,$ one has $$(-\log{r})^{-\,\beta} << \,\big[\,\log
\big(\log{\frac1r}\,\big)\big]^{-1}\,\cdot\,(-\log{r})^{-\,({{\alpha}}-1)}
\,<<\,(-\log{r})^{-\,({{\alpha}}-1)}\,.$$ Sharp optimality avoids the above, and similar, possibilities. This fact is significant in all cases in which full regularity is not reached, as in Theorem \[laplolas\]. This is the meaning giving here to the sharpness of a regularity result.
Concerning references, not related to our results but merely to Log spaces (mostly for $n=\,1\,$, or ${{\alpha}}=\,1$), the author is grateful to Francesca Crispo for calling our attention to the treatise [@fiorenza], to which the reader is referred. In particular, as claimed in the introduction of this volume, the space $\,D^{0,\,1}({{\overline{\Omega}}})\,$ was considered in reference [@shara]. See also definition 2.2 in reference [@fiorenza]. Other references, quoted in [@fiorenza], are [@diening], [@samko], [@zhikov1], [@zhikov2], and [@zhikov3].
Hölog spaces $\,C^{0,\,{{\lambda}}}_{{\alpha}}({{\overline{\Omega}}})\,$ and full regularity. {#hologos}
==============================================================================================
If, for some $\,{{\lambda}}>\,0\,,$ one has $\,{{\widehat{\omega}}}(r)=\,{{\lambda}}\,{{\overline{\omega}}}(r)\,$ in a neighborhood of the origin, then there is a $\,k>\,0\,$ such that $\,{{\overline{\omega}}}(r)=\,k\,r^{{\lambda}}\,.$ This fact could suggest that Hölder spaces could be the unique full regularity class inside our framework. However, *full regularity* is also enjoyed by other spaces. The following is a particularly interesting case. Consider oscillation functions $${{\omega}}_{{{\lambda}},\,{{\alpha}}}(r)=\,r^{{\lambda}}\,(-\log{r})^{-\,{{\alpha}}}\,, \quad \,r<\,1\,,\label{lalfas}$$ where $\,0\leq\,{{\lambda}}<\,1\,$ and $\,{{\alpha}}\in\,{{\mathbb R}}\,.$ For $\,{{\lambda}}=\,0\,$ and $\,{{\alpha}}>\,0\,$ we re-obtain the Log space $\,D^{0,\,{{\alpha}}}({{\overline{\Omega}}})\,$, for $\,{{\lambda}}>\,0\,$ and $\,{{\alpha}}=\,0\,$ we re-obtain $\,C^{0,\,{{\lambda}}}({{\overline{\Omega}}})\,$. Theorem \[compactos\] shows that (compact) inclusions $$C^{0,\,{{\lambda}}_2}({{\overline{\Omega}}})\subset \,C^{0,\,{{\lambda}}}_{{\alpha}}({{\overline{\Omega}}})\subset
\,C^{0,\,{{\lambda}}}_\beta({{\overline{\Omega}}}) \subset\,C^{0,\,{{\lambda}}}({{\overline{\Omega}}})\subset
\,C^{0,\,{{\lambda}}}_{-\beta}({{\overline{\Omega}}}) \subset \,C^{0,\,{{\lambda}}}_{-{{\alpha}}}({{\overline{\Omega}}})
\subset\,C^{0,\,{{\lambda}}_1}({{\overline{\Omega}}})$$ hold for $\,{{\alpha}}>\,\beta>\,0\,$ and $\,0<\,{{\lambda}}_1<\,{{\lambda}}<\,{{\lambda}}_2<\,1\,.$ The reader should note that the set $$\bigcup_{{{\lambda}},\,{{\alpha}}} \,C^{0,\,{{\lambda}}}_{{\alpha}}({{\overline{\Omega}}})\,,$$ where $\,0<{{\lambda}}<\,1\,,$ and $\,{{\alpha}}\in\,{{\mathbb R}}\,,$ is a *totally* ordered set, in the obvious way. In the totally ordered sub-chain merely consisting of classical Hőlder spaces, each $\,C^{0,\,{{\lambda}}}\,$ space can be enlarged, to became an infinite, ordered chain, $\,C^{0,\,{{\lambda}}}_{{\alpha}}({{\overline{\Omega}}})\,$, $\,{{\alpha}}\in\,{{\mathbb R}}\,.$ Clearly, the spaces $\,C^{0,\,{{\lambda}}}_{{\alpha}}({{\overline{\Omega}}})\,$ enjoy all the interesting properties described in section \[novicas\].
To abbreviate notation, in this section we set $${{\overline{\omega}}}(r)\equiv {{\omega}}_{{{\lambda}},\,{{\alpha}}}(r)\,, \quad [\,f\,]_{\,{{\overline{\omega}}}} \equiv
[\,f\,]_{{{\lambda}},\,{{\alpha}}}\,, \quad \textrm{and} \quad \|\,f\,\|_{\,{{\overline{\omega}}}}
\equiv \|\,f\,\|_{{{\lambda}},\,{{\alpha}}}\,.$$ The following full regularity result holds.
Let $\,f\in\,C^{0,\,{{\lambda}}}_{{\alpha}}({{\overline{\Omega}}})\,$ for some $\,{{\lambda}}\in
\,(\,0,\,1\,)\,$ and some $\,{{\alpha}}\in {{\mathbb R}}\,.$ Let $\,u\,$ be the solution of problem , where the differential operator $\,{\boldsymbol{L}}\,$ may have variable coefficients. Then $\,{{\nabla}}^2\,u
\in\,C^{0,\,{{\lambda}}}_{{\alpha}}({{\overline{\Omega}}})\,.$ Moreover $$\|\,{{\nabla}}^2\,u\,\|_{{{\lambda}},\,\,{{\alpha}}} \leq \,C\,\|\,f\,\|_{{{\lambda}},\,\,{{\alpha}}} \,,\label{haligas}$$ for some positive constant $\,C\,.$ The result is optimal, in the sharp sense.\[laplohas\]
Note that full regularity $\,{{\omega}}_{{{\lambda}},\,{{\alpha}}}\rightarrow\,{{\omega}}_{{{\lambda}},\,{{\alpha}}}\,$ is a little surprising here. In fact, at the light of theorem \[laplolas\], we could merely expected the intermediate regularity result $\,{{\omega}}_{{{\lambda}},\,{{\alpha}}}\rightarrow\,{{\omega}}_{{{\lambda}},\,{{\alpha}}-\,1}\,.$
We appeal to the theorem \[sufasvero\]. Assumptions and are trivially verified. Let’s prove . Set $$L(r)=\,\log{\frac1r}\,.$$ Straightforward calculations show that $${{\overline{\omega}}}'(r)=\, r^{{{\lambda}}-\,1}\,L(r)^{-\,{{\alpha}}} \,\big(\,{{\lambda}}+\,{{\alpha}}\,L(r)^{-\,1}\,\big)\label{oprimas}$$ and that $${{\overline{\omega}}}''(r)=\,-\, r^{{{\lambda}}-\,2}\,L(r)^{-\,{{\alpha}}} \,\Big(\,{{\lambda}}\,(1-\,{{\lambda}})
-\, (\,2{{\lambda}}-\,1) \,{{\alpha}}\,L(r)^{-\,1}-\,
{{\alpha}}\,({{\alpha}}+\,1)\,L(r)^{-\,2}\,\Big)\,.\label{osegu}$$ Equation shows that $\,{{\overline{\omega}}}''(r)<\,0\,$ in a neighborhood of the origin, since $\, \lim_{r\rightarrow
0}\,L(r)=\,+\,\infty\,.$ Hence $\,{{\overline{\omega}}}\,$ is concave. Furthermore holds since $$\lim_{r\rightarrow 0}\,\frac{{{\overline{\omega}}}(r)}{r\,\,{{\overline{\omega}}}'(r)} =\,\frac1{{\lambda}}>
\,1\,.\label{flita222}$$To prove full regularity we appeal to de l’Hôpital rule and to to show that $$\lim_{r\rightarrow 0}\,\frac{{{\widehat{\omega}}}(r)}{\,{{\overline{\omega}}}(r)}=\,\lim_{r\rightarrow
0}\,\frac{{{\overline{\omega}}}(r)}{r\,\,{{\overline{\omega}}}'(r)}=\,\frac1{{\lambda}}\,.\label{lambes}$$In particular holds for $\,r\,$ in some neighborhood of the origin. Hence proposition \[eqnormes\] applies.
It would be interesting to study higher order regularity results in the framework of Hölog spaces.
Sharpness of the regularity results. {#optimus}
====================================
In this section we prove the *sharpness* of our regularity results (a simple example was shown at the end of section \[DOPO\]). The proof is quite adaptable to different situations, local and global results, etc. We merely show the main argument. We construct a counter-example, which concerns constant coefficients operators (we could easily deny case by case), which shows that any stronger regularity result can not occur. We start by considering the Laplace operator ${{\Delta}}$. We remark that the argument applies to the regularity results stated in theorems \[sufasvero\] and \[sufasvero-2\]. However, in the second theorem, the conclusion is obvious, due to full regularity.
For convenience, we assume that $\,{{\overline{\omega}}}(r)\,$ is differentiable, and that there is a positive constant $C$ such that $$\frac{{{\overline{\omega}}}(r)}{r\,\,{{\overline{\omega}}}'(r)}\geq\,C>\,0\,,\label{flotas}$$for $r>\,0\,,$ in a neighborhood of the origin. Note that holds, with $\,C=\,1\,,$ if $\,{{\overline{\omega}}}(r)$ is concave.
Assume that $\,{{\overline{\omega}}}(r)\,$ satisfies the above hypothesis, and let $\,{{\widehat{\omega}}}_{0}(r)$ be a given oscillation function. Assume that the results stated in theorem \[sufasvero\] hold by replacing ${{\widehat{\omega}}}$ by ${{\widehat{\omega}}}_{0}\,$. Then there is a constant $c$ for which $\,{{\widehat{\omega}}}(r)\leq\,c\,{{\widehat{\omega}}}_0(r)\,.$\[shap\]
We may say that any regularity result better than is false.
For simplicity, we start by assuming that $\,{\boldsymbol{L}}=\,{{\Delta}}\,.$ Consider the function $$u(x)=\,{{\widehat{\omega}}}(|x|) \,x_1\,x_2\,,\label{emrn}$$ defined in ${{\mathbb R}}^n\,, n\geq\,2\,$. Actually, we are merely interested in the behavior near the origin (see below).
Straightforward calculations show that $${{\Delta}}\,u(x)=\,(n+\,2)\,\frac{x_1\,x_2}{|\,x\,|^2}\,{{\overline{\omega}}}(x)
+\,\frac{x_1\,x_2}{|\,x\,|^2}\,|\,x\,| \,{{\overline{\omega}}}'(|x|)\,.\label{eodel}$$In particular, $\,{{\Delta}}\,u(0)=\,0\,$. By appealing to one shows that $$|\,{{\Delta}}\,u(x)-\,{{\Delta}}\,u(0)\,|=\, |\,{{\Delta}}\,u(x)\,| \leq\, C\,
{{\overline{\omega}}}(|x|)\,.$$ Hence, in a neighborhood of the origin, $\,f(x)=\,{{\Delta}}\,u(x)\,$ belongs to $\,D_{{{\overline{\omega}}}}\,.$
On the other hand, straightforward calculations show that $${{\partial}}_1\,{{\partial}}_2\,u(x)=\,{{\widehat{\omega}}}(|x|)+\,\frac{1}{|x|^2}\,\big(\,x_1^2+\,x_2^2-\,2\,\frac{x_1^2\,x_2^2}{|x|^2}\,\big)
\cdot\,{{\overline{\omega}}}(|x|)+\,\frac{x_1^2\,x_2^2}{|x|^4}\,\cdot (\,|\,x\,| \,{{\overline{\omega}}}'(|x|)\,)\,.\label{papar}$$ In particular $\,{{\partial}}_1\,{{\partial}}_2\,u(0)=\,0\,,$ and $$|\,{{\partial}}_1\,{{\partial}}_2\,u(x)-\,\,{{\partial}}_1\,{{\partial}}_2\,u(0)\,|\geq\,{{\widehat{\omega}}}(|x|)$$ for $ 0<|x| <<1\,,$ since in equation the coefficients of $\,{{\overline{\omega}}}(|x|)\,$ and of $\,|\,x\,| \,{{\overline{\omega}}}'(|x|)\,$ are nonnegative. On the other hand, if $\,{{\widehat{\omega}}}_{0}(r)$ regularity holds, one has $$|\,{{\partial}}_1\,{{\partial}}_2\,u(x)-\,\,{{\partial}}_1\,{{\partial}}_2\,u(0)\,| \leq\,
(\,c\,\|f\|_{{\overline{\omega}}}\,)\,{{\widehat{\omega}}}_{0}(|x|)$$ for some $c>\,0\,$. Hence $\,{{\widehat{\omega}}}(r) \leq\,c_0\,{{\widehat{\omega}}}_{0}(r)\,,$ for $
\,r>\,0\,,$ in a neighborhood of the origin.
If $\,{\boldsymbol{L}}\,$ is given by we replace by $$u(x)=\,{{\widehat{\omega}}}(|x|) \,\sum_1^n b_{i\,j} x_i\,x_j \,,\label{emrn2}$$ where $\,B\neq\,0\,$ is symmetric and $$\,\sum_{i,\,j=\,1}^n \, a_{i\,j}\, b_{i\,j}=\,0\,.$$ In particular, if a specific coefficient $\, a_{k\,l}\,$ vanishes, we may simply choose $\,u(x)=\,{{\widehat{\omega}}}(|x|) \,x_k\,x_l\,,$ as done in .
We localize the above result as follows. Assume that $\,0 \in
{{\Omega}}\,,$ and consider the function $$u(x)=\,\psi(|x|)\, {{\widehat{\omega}}}(|x|) \,x_1\,x_2\,,\label{cotresa-2}$$ where $\psi(r)\,$ is non-negative, indefinitely differentiable, vanishes for $\,r\geq\,\rho>\,0\,,$ and is equal to $\,1\,$ for $\,|x|<\,\frac{\rho}{2}\,.$ The radius $\,\rho\,$ is such that $\,I(0,\,\rho)\,$ is contained in $\,{{\Omega}}\,.$ The above truncation allows us to assume homogeneous boundary conditions in $\,{{\Omega}}\,$ (we may consider combinations of functions as above, centered in different points in $\,{{\Omega}}\,$, with distinct radius, and distinct cut-off functions).
It is worth noting that in the above argument the specific expressions of the coefficients of $\,{{\overline{\omega}}}(|x|)\,$ and $\,|x|\,{{\overline{\omega}}}'(|x|)\,$ are secondary (even if the non-negativity of these coefficients was exploited). They are homogeneous functions of degree zero, without particular influence on the minimal regularity. The crucial point is that the second order derivative $\,{{\partial}}_1\,{{\partial}}_2\,u(x)\,,$ due to the term $\,\,x_1\,x_2\,$ in , leaves unchanged the “bad term” $\,{{\widehat{\omega}}}(|x|)\,.$ This does not occur for derivatives $\,{{\partial}}_i^2\,u(x)\,,$ hence does not occur for $\,{{\Delta}}\,u(x)\,.$
It looks interesting to note that the “bad term” $\,{{\widehat{\omega}}}(|x|)\,$ can not be eliminated by the other two terms which are present in the right hand side of . Even when full regularity occurs (like in Hölder and Hölog spaces), the “bad term” $\,{{\widehat{\omega}}}(|x|)\,$ is still not eliminated. It simply is as regular as the other two terms, $\,{{\overline{\omega}}}(|x|)$ and $\,|\,x\,| \,{{\overline{\omega}}}'(|x|)\,$. See also [@BV-ALBPAO], section 6, for some comment.
On data spaces larger then $\,{\boldsymbol{C}}_*({{\overline{\Omega}}})\,.$ {#cestrelas-mais}
============================================================================
In the context of [@BVJDE], Theorem \[laplaces\] was peripheral. Hence, the proof (written in a still existing manuscript, denoted here \[BVUN\]), remained unpublished. Actually, at that time, we proved the above result for more general elliptic boundary value problems. The proofs depend only on the behavior of the related Green’s functions. Recently, by following the same ideas, we have shown, see [@BVSTOKES], that for every $\,{\boldsymbol{f}}\in
\,{\boldsymbol{C}}_*({{\overline{\Omega}}})\,$ the solution $({\boldsymbol{u}},\,p)$ to the Stokes system $$\left\{
\begin{array}{l}
-\,{{\Delta}}\,{\boldsymbol{u}}+\,{{\nabla}}\,p=\,{\boldsymbol{f}}\quad \textrm{in} \quad {{\Omega}}\,,\\
{{\nabla}}\cdot\,{\boldsymbol{u}}=\,0 \quad \textrm{in} \quad {{\Omega}}\,,\\
{\boldsymbol{u}}=\,0 \quad \textrm{on} \quad {{\Gamma}}\end{array}
\right.\label{doistokes}$$ belongs to $\,{\boldsymbol{C}}^2({{\overline{\Omega}}})\times\,C^1({{\overline{\Omega}}})\,$.
In the manuscript \[BVUN\] we also tried to extend the result claimed in theorem \[laplaces\] to data belonging to functional spaces $\,B_*({{\overline{\Omega}}})\,$ *containing* $\,C_*({{\overline{\Omega}}})\,.$ By setting $${{\omega}}_f(x;\,r)= \, \sup_{y \in\,{{\Omega}}(x;\,r)}\,|\,f(x)-\,f(y)\,|\,,$$ we may write $$[\,f\,]_* =\,\int_0^R \,\sup_{ \,x \in\,{{\overline{\Omega}}}}\, {{\omega}}_f(x;\,r)\,
\,\frac{dr}{r}\,.\label{catriz}$$ So, together with $\,C_*({{\overline{\Omega}}})\,,$ we have considered a functional space $\,B_*({{\overline{\Omega}}})\,$ obtained by commuting *integral* and *sup* operators in the right hand side of definition : For each $\,f \in\,C({{\overline{\Omega}}})\,,$ we defined the semi-norm $$\langle\,f\,\rangle_* = \,\sup_{ \,x \in\,{{\overline{\Omega}}}}\, \int_0^R \,
{{\omega}}_f(x;\,r)\,
\,\frac{dr}{r}\,,\label{seis-bis}$$ and a related functional space$\, B_*({{\overline{\Omega}}})\,.$ We have shown that the inclusion $ \,C_*({{\overline{\Omega}}}) \subset\,B_*({{\overline{\Omega}}})\,$ is proper, by constructing strongly oscillating functions which belong to $\,B_*({{\overline{\Omega}}})\,$ but not to $\,C_*({{\overline{\Omega}}})\,.$ This construction was recently published in reference [@BV-LMS], Proposition 1.7.1. Furthermore, in \[BVUN\], we have shown that Theorem \[laplaces\] and similar results hold in a weaker form, for data $\,f
\in\,B_*({{\overline{\Omega}}})\,$, by proving that the first order derivatives of the solution $\,u\,$ are Lipschitz continuous in $\,{{\overline{\Omega}}}\,.$ The proof is published in reference [@BV-LMS], actually for data in a functional space $\,D_*({{\overline{\Omega}}})\,$ containing $\,B_*({{\overline{\Omega}}})\,$. See Theorem 1.3.1 in [@BV-LMS]. A similar extension holds for the Stokes problem, as shown in reference [@BVJP], Theorem 6.1, where we have proved that if $\,{\boldsymbol{f}}\in \,{\boldsymbol{D}}_*({{\overline{\Omega}}})\,,$ then the solution $\,({\boldsymbol{u}},\,p)\,$ of problem satisfies the estimate $\, \|\,{\boldsymbol{u}}\,\|_{1,\,1}+\,\|\,p\,\|_{0,\,1} \leq\,C\,
|\|\,{\boldsymbol{f}}\,|\|_*\,.$ Full regularity for data in $\,{\boldsymbol{B}}_*({{\overline{\Omega}}})\,$ would follow from a possible density of regular functions in this last space, a challenging open problem. The simple proof would be obtained by replacing the space $ \,{\boldsymbol{C}}_*({{\overline{\Omega}}})\,$ by $\,{\boldsymbol{B}}_*({{\overline{\Omega}}})\,$ in [@BVSTOKES], section 4. A similar remark holds for $\,{\boldsymbol{D}}_*({{\overline{\Omega}}})\,.$ However, in this last case, the desired density result looks quite unlikely.
[35]{} H. Beirão da Veiga, *On the solutions in the large of the two-dimensional flow of a nonviscous incompressible fluid*, J. Diff. Eq., 54, (1984), no.3, 373-389. H. Beirão da Veiga, Concerning the existence of classical solutions to the Stokes system. On the minimal assumptions problem, *J. Math. Fluid Mech.*, 16 (2014), 539-550. H. Beirão da Veiga, *Classical solutions to the two-dimensional Euler equations and elliptic boundary value problems, an overview*, in “Recent Progress in the Theory of the Euler and Navier-Stokes Equations”, *London Math. Soc. Lecture Notes*, Edited by J. C. Robinson, J. L. Rodrigo, W. Sadowski, and A. V. López. Cambridge University Press, 2016. H. Beirão da Veiga, *On some regularity results for the stationary Stokes system, and the $\,2-D\,$ Euler equations*, Portugaliae Math., 72 (2015) 285-307. H. Beirão da Veiga, H-log spaces of continuous functions, potentials, and elliptic boundary value problems, ArXiv 1503.04173 \[math.AP\], 13.03.2015. H. Beirão da Veiga, Elliptic boundary value problems in spaces of continuous functions, to appear in a special volume of DCDS Series S. L. Bers, F. John, and M. Schechter, *Partial Differential Equations*, John Wiley and Sons, Inc., New-York, 1964. C.C. Burch, *The Dini condition and regularity of weak solutions of elliptic equations*, J. Diff. Eq., 30, (1978), 308-323. D.V. Cruz-Uribe and A. Fiorenza, *Variable Lebesgue Spaces Foundations and Harmonic Analysis*, Springer, Basel 2013. L. Diening, *Maximal function on generalized Lebesgue spaces $\,L^{p(\cdot)}\,$* , Math. Inequal. Appl., 7(2):245�253, 2004. O.A. Ladyzenskaya, *The Mathematical Theory of Viscous Incompressible Flow*, Gordon and Breach, New-York, 1969. S. Samko, *Convolution type operators in $\,L^{p(x)}\,,$* Integral Transform, Spec. Funct., 7(1�2):123� 144, 1998. V.L. Shapiro, *Generalized and classical solutions of the nonlinear stationary Navier-Stokes equations*, Trans. Amer. Math. Soc., 216 (1976) 61-79. I. I. Sharapudinov, *The basis property of the Haar system in the space $\,L^{p(t)}[0,\,1\,]\,$, and the principle of localization in the mean*, Mat. Sb. (N.S.), 130(172)(2):275�283, 286, 1986. V.A. Solonnikov, *On estimates of Green’s tensors for certain boundary value problems*, Doklady Akad. Nauk. **130** (1960), 128-131. K. Yosida, *Functional Analysis*, Springer-Verlag, Berlin-Heidelberg, 2nd edition, 1968. V. V. Zhikov, *On the homogenization of nonlinear variational problems in perforated domains*, Russian J. Math. Phys., 2(3):393�408, 1994. V. V. Zhikov, *On Lavrentiev�s phenomenon*, Russian J. Math. Phys., 3(2):249�269, 1995. V. V. Zhikov, *On some variational problems*, Russian J. Math. Phys., 5(1):105�116 (1998), 1997.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Chien-Hsien Lin'
- 'Wing-Huen Ip'
- 'Zhong-Yi Lin'
- Fumi Yoshida
- 'Yu-Chi Cheng'
title: 'Detection of Large Color Variation of Potentially Hazardous Asteroid (297274) 1996 SK '
---
Introduction {#sect:intro}
============
The main asteroid belt located between the orbits of Mars and Jupiter is composed of a population of small bodies of primitive composition. The largest member, (1) Ceres, with a diameter of 914 km, will be visited by the DAWN spacecraft in 2015. Following (1) Ceres, (2) Pallas (544 km), (4) Vesta (525 km) and (10) Hygiea (431 km) are the most massive asteroids, which might be classified as dwarf planets. Smaller objects down to the size of km and sub-km range are mostly ejectas from impact cratering and/or catastrophic fragments via collisional process (Bottke et al. [@Bot02; @Bot05]). Yoshida et al. ([@Yos04]) discussed in detail the collisional evolution of asteroid families using the young Karin family as an example. They pointed out that photometric measurements of the asteroid family members could provide important clues to the corresponding orbital evolution, internal composition and surface effects because of space weathering process (Clark et al. [@Cla02]; Sasaki et al. [@Sas01]).
Because of the long-term gravitational perturbations of Jupiter and Saturn, some of the collisional fragments could be injected into orbits intercepting the orbits of the terrestrial planets, which potentially cause surface impact events. These scattered stray bodies are further classified as the Amor asteroids if their perihelion distances (q) are between 1.3 AU and 1.017 AU, the Apollo asteroids if their semi-major axis $a> 1.0$ AU and $q<1.017$ AU, and Aten asteroids if $a<1.0$ AU and the aphelion $Q>0.983$ AU. As shown by Bottke et al. [@Bot02], the majority of these terrestrial planets-crossing asteroids is from the inner asteroid belt even though some of them could be originated from the middle or outer asteroid belt or of cometary origin.
Among the Near-Earth asteroids (NEAs), which are the general term for the Apollo and Aten asteroids, a number of them have non-zero probability of hitting the Earth in future. For example, it has been estimated that the total number of a subgroup of NEAs called Potentially Hazardous Asteroids or PHAs with $D>100$ m is approximately 4700$\pm$1450 (Mainzer et al. [@Mai12]). A close monitoring and in-depth investigations of the basic physical properties of the PHAs like sizes, shapes and compositions are therefore important. In addition, the PHAs could also represent very valuable natural resources for space exploration and utilization because of their relatively easy access. With these key issues in mind, we have initiated a cooperative project at the Space Science Institute, Macau University of Science and Technology together with the Astronomy Institute, National Central University, to produce a photometric survey of the taxonomical types of NEAs in low inclination orbits. In this work we report the results of an interesting object (297274) 1996 SK, an Apollo asteroid and PHA, based on the observations on May 22 and 23, 2012, at Lulin Observatory, Taiwan. The observations are described in Section 2. The results of the data analysis are given in Section 3. In Section 4, a summary and discussion on the implications of the physical properties of the color variation will be given.
Observations {#sect:Obs}
============
In our first set of observational targets, the selection criteria are (1) their lack of prior measurements of the lightcurves and surface color, and (2) the suitability of their optical brightness for time-series photometry. Asteroid (297274) 1996 SK of absolute magnitude $H_v = 16.866$, with a semi-major axis $a = 2.434$ AU, eccentricity $e = 0.794$ and inclination $i = 1.962^\circ$ was close to opposition and satisfied these condition in May, 2012. With its perihelion distance $q = 0.5$ AU and low inclination, (297274) 1996 SK is classified as a PHA. It was observed on May 22 and 23, 2012 by multi-filter photometry using the LOT, one-meter telescope at Lulin Observatory. The CCD imaging camera is the PI-1300B, which has 1340 x 1300 pixels with effective pixel scale of $0.516''$.
The observational log is given in Table \[tab1\]. The filters used are broad-band Bessel $BVRI$, which have centered wavelengths of 442, 540, 647, and 786 nm, respectively. The R-band exposure time is 60 seconds per frame and the measurement sequence was made of continuous 20 frames for each run. In total, seven runs were made. However, due to the unstable weather on May 23, much fewer data were acquired. Three sets of B, V and I filter measurements were made in the first half night of May 22, and another set was made in the next night. The Landolt standard star fields used for color calibration were SA107 on May 22 and SA109 on May 23 (Landolt [@Lan92]). The calibrated absolute magnitudes and colors of each star are listed in Table \[tab2\]. The photometric accuracy is 0.044 on average. All targets were observed with airmass of ＜ 2 during the nights.
Instrument Filter Exposure Date $r^*$ $\Delta^*$ $\Phi^*(^\circ)$ Airmass
------------ -------- ------------ --------- -------- ------------ ------------------ ------------
60s/frame May 22 1.454 0.443 4.218 1.28-1.97
\[-1ex\] 60s/frame May 23 1.467 0.456 4.987 1.28-1.53
: Observation Log of (297274) 1996 SK[]{data-label="tab1"}
: The quantity on 16:00 UT of each date;
r: Heliocentric distance (A.U.);
$\Delta$: Geocentric distance (A.U.);
$\Phi$: The phase angle of Sun-target-observer.
[ccccccccc]{} Star & $V^\alpha$ &$V^\beta$ &$(B-V)^\alpha$ &$(B-V)^\beta$ &$(V-R)^\alpha$ &$(V-R)^\beta$ &$(V-I)^\alpha$ &$(V-I)^\beta$\
107 459 &12.284 &12.252 &0.900 &0.915 &0.525 &0.370 &1.045 &0.940\
107 457 &14.910 &14.887 &0.792 &0.830 &0.494 &0.507 &0.964 &0.971\
107 456 &12.919 &12.875 &0.921 &0.918 &0.537 &0.549 &1.015 &1.035\
107 592 &11.847 &11.895 &1.318 &1.204 &0.709 &0.389 &1.357 &1.050\
107 599 &14.675 &14.671 &0.698 &0.727 &0.433 &0.463 &0.869 &0.898\
107 600 &14.884 &14.863 &0.503 &0.540 &0.339 &0.358 &0.700 &0.715\
107 601 &14.646 &14.632 &1.412 &1.441 &0.923 &0.949 &1.761 &1.787\
107 602 &12.116 &12.116 &0.991 &0.934 &0.545 &0.367 &1.074 &0.962\
109 949 &12.828 &12.829 &0.806 &0.805 &0.500 &0.503 &1.020 &1.024\
109 954 &12.436 &12.435 &1.296 &1.305 &0.764 &0.756 &1.496 &1.491\
109 956 &14.639 &14.644 &1.283 &1.269 &0.779 &0.788 &1.525 &1.533\
$\alpha$: Magnitudes and color indices from Landolt [@Lan92];
$\beta$: Mean values measured from this study.
The standard data processing was performed by using IRAF program (Image Reduction and Analysis Facility, supplied by National Optical Astronomy Observatories) with $ccdproc$ package for image reduction, $apphot$ for photometry, and $photcal$ for standard stars flux calibrations.
Results {#sect:results}
=======
Figure \[Fig1\] shows the raw lightcurves of (297274) 1996 SK observed on May 22 and 23. Differential photometry was applied while the reference stars without time variability were chosen with R-band magnitude brighter than $17.0$ in the USNO-A2.0 catalog.
[cccccc]{} $UT_V$ &V &B-V &V-R &V-I &$Airmass^*$\
May 22& & & & &\
13:30:36& $16.259\pm0.006$ & $0.840\pm0.012$ & $0.520\pm0.007$ & $0.769\pm0.008$ & 1.553\
13:31:58& $16.270\pm0.006$ & $0.835\pm0.012$ & $0.511\pm0.007$ & $0.769\pm0.008$ & 1.546\
14:20:46& $16.390\pm0.007$ & $0.652\pm0.013$ & $0.459\pm0.009$ & $0.847\pm0.009$ & 1.373\
14:22:10& $16.386\pm0.006$ & $0.644\pm0.012$ & $0.475\pm0.008$ & $0.859\pm0.008$ & 1.370\
15:07:03& $16.063\pm0.005$ & $0.748\pm0.010$ & $0.472\pm0.007$ & $0.827\pm0.007$ & 1.297\
15:08:26& $16.065\pm0.005$ & $0.741\pm0.010$ & $0.478\pm0.007$ & $0.832\pm0.007$ & 1.296\
May 23& & & & &\
17:34:31 & $16.495\pm0.007$ & $0.906\pm0.017$ & $0.457\pm0.009$ & $0.707\pm0.012$ & 1.523\
Mean & $16.275\pm0.016$ & $0.767\pm0.033$ & $0.482\pm0.021$ & $0.801\pm0.023$ &\
: The airmass is displayed for the time of V-band observed because the BVRI observations in each color measurement were obtained in sequential order in a short time interval within 11 min.
Using the Plavchan algorithm (Plavchan et al. [@Pla08]) to compute periodogram, the spin period of (297274) 1996 SK was found to be $4.656\pm0.122$ hours. The uncertainty in the frequency was estimated based on the method of Horne et al. ([@Hor86]). The periodogram and the folded lightcurve from the R-band measurements along with the rotation phase are shown in Figure \[Fig2\]. The lightcurve shows that (297274) 1996 SK has a rather smooth configuration. For an ellipsoidal shape of the asteroid, the peak-to-peak variation ($\Delta m$) of the lightcurve can be used to calculate the ratio of the long axis to short axis ($a/b$) according to the formula $\Delta m = 2.5log(a/b)$. From the lightcurve of the (297274) 1996 SK, the $\Delta m$ was 0.44. It means that the $a/b$ is about 1.50. However, since the above a/b value is obtained by assuming that the asteroid was observed at an aspect angle (i.e., the angle between the line of sight and spin axis) of $90^\circ$, the actual axial ratio ($a/b$) may be more than that. The asteroid’s diameter ($D$) can be calculated by using the formula (Yoshida et al. [@Yos04]), $logD=3.130-0.5logA-0.2H$, where $H$ is the absolute magnitude and $A$ is the surface albedo (Yoshida et al., 2004). Assuming $A = 0.2$ (corresponding to the mean albedo of S-type asteroids) and $H = 16.866$ mag for (297274) 1996 SK, its diameter is 1.28 km. The long axis and the short axis can be computed to be 1.57 km and 1.05 km, respectively.
Table \[tab3\] summarizes the color measurement results obtained on March 22 and 23. The multi-wavelength observations of (297274) 1996 SK at several different times allow us to estimate the color indices and to examine the possible changes of its surface color during the rotation. Figure \[Fig3\] displays the color variations at four phases of rotation observed on the two days. It reveals that both B-V and V-I colors vary significantly, while V-R change is comparatively small. The maximum changes between the phase 0 to 0.5 for B-V, V-R and V-I are 0.258, 0.058 and 0.146, respectively. Such a large range of the color variation indicates the possible presence of surface heterogeneity on (297274) 1996 SK.
The brightness magnitudes of the B, V and I bands follow the general trend of the R-band lightcurve. The average values of $B-V = 0.767\pm 0.016$, $V-R = 0.482\pm 0.021$, and $V-I = 0.801\pm 0.025$ of (297274) 1996 SK can be compared with the known colors from different taxonomies of NEAs determined by previous observations archived in the “Data Base of Physical and Dynamical Properties of NEAs” from European Asteroid Research Node. These results are plotted in Figure \[Fig4\], which shows the B-V and V-R terms generally divided into S-group (S, Q, R-types et al.) , X-group (X, E-types et al.) and C-group (C, F, B-types et al.) of NEAs. It indicates that the surface color of (297274) 1996 SK locates on the boundary between S-group and X-group asteroids.
![Surface color variations of PHA (297274) 1996 SK over the phase of rotation.[]{data-label="Fig3"}](297274_fig3.eps){width="0.9\columnwidth"}
![The color-color diagram of NEAs with known color indices in different taxonomic types \[$\alpha$\] and (297274) 1996 SK observed from this work.($\alpha$: Betzler et al., 2010; Carbognani, 2008; Dandy et al., 2003; Hapke, 2000; Hergenrother et al., 2009; Hicks et al., 2011, 2012, 2013; Jewitt et al., 2006, 2013; Karashevich et al., 2012; Pieters et al., 2000; Ye, 2011.)[]{data-label="Fig4"}](297274_fig4.eps){width="0.9\columnwidth"}
![Relative reflectance spectrum of (297472) 1996 SK from our data (thick line) in comparison with those integrated spectra of S-type asteroids from the archived data of “Small Bodies Node”. The shaded area indicates the range of the S-type spectra.[]{data-label="Fig5"}](297274_fig5.eps){width="0.9\columnwidth"}
![A comparison of the relative reflectance spectra of (297274) 1996 SK taken at different times of March 22 and 23 with that reported by Rabinowitz (1998).[]{data-label="Fig6"}](297274_fig6.eps){width="0.9\columnwidth"}
Figure \[Fig5\] illustrates the average relative reflectance spectrum of (297274) 1996 SK obtained by subtracting the solar colors $B-V = 0.665$, $V-R = 0.367$, and $V-I = 0.705$ (Howell, [@How95]) from its colors. It falls into the spectral region of the S-type asteroids, so (297274) 1996 SK should be classified as a member of S-type objects. It is interesting to note that Rabinowitz ([@Rab98]) reported color measurements of (297274) 1996 SK in October, 1996, with $V-R = 0.430\pm 0.070$ and $V-I = 0.678\pm 0.0587$, respectively. These values are close to the corresponding results obtained on May 23 (see Table \[tab3\] and Figure \[Fig6\]) similar to shape of Q-type asteroids while still having significant differences from those taken at other times. The possible implication will be discussed later.
Summary and Discussion {#sect:summary&discussion}
======================
Our observations of PHA (297274) 1996 SK at opposition in May, 2012, lead to the following conclusion:
1. The rotation period of this asteroid is found to be $4.656\pm 0.122$ hours, i.e., well below the spin cutoff of 2.2 hours.
2. The amplitude of lightcurve variability is $\Delta m = 0.44$ indicating an elongated shape with the ratio of the long axis to the short axis $(a/b) = 1.50$ but possibly underestimated.
3. The average color indices of $B-V = 0.767\pm 0.033$, $V-R = 0.482\pm 0.021$, $V-I = 0.801\pm 0.025$ and the corresponding surface reflectance means that (297274) 1996 SK belongs to the S-type taxonomic class. With the surface albedo assumed to be 0.2, which is typical value of S-type asteroids, and $H_v = 16.866$, the projected long and short axes are 1.57 km and 1.05 km, respectively.
4. Over the rotation range of 133 degrees, (297274) 1996 SK displays significant color changes which might imply the existence of large change in mineralogical and/or compositional variation on its surface.
The detection of large color change is an important result of this work because it could mean that (297274) 1996 SK might contain various properties of its surface spectra. Because there is no information on the relation of the color measurements to the rotational phase in the work of Rabinowitz ([@Rab98]), it is difficult to analyze the cause of the color differences between our present results and his work. One thing is nearly certain; that is they could not be caused by short-term space weathering effect since the associated time scale is at least on the order of a million years (Vernazza et al. [@Ver09]). From this point of view, the existence of inhomogeneous surface composition or differential space weathering effect would be the most viable explanation. The first scenario would mean that (297274) 1996 SK might contain the interface material of some differentiated region of its parent body at impact disruption. The second scenario has been discussed by Yoshida et al ([@Yos04]) in the case of the color variation of (832) Karin - see also Sasaki, T. et al. ([@Sas04; @Sas06a]), Sasaki, S. et al. ([@Sas06b]), and Ito and Yoshida ([@Ito07]). This could have come about by micrometeoroid impact process on young and older surface areas (Clark et al. [@Cla02]; Sasaki et al. [@Sas04]). Figure 6 shows the relative reflectance spectra observed in different time varying from S-type to Q-type. It might be also related to the second scenario that the asteroid has two parts of weathered and un-weathered surfaces. It was a possibility that Rabinoiwtz ([@Rab98]) had measured the colors of the vicinity of the phase which we observed on May 23. Both possibilities mean that (297274) 1996 SK should not be covered by a homogeneous regolith layer of small particles. Could this surface cleansing be achieved by tidal breakup in previous close encounters with Earth and other terrestrial planets as proposed by Nesvorny et al ([@Nes10])? These are issues we plan to investigate in the future work.
This work was partially supported by the Project 019/2010/A2 of Macau Science and Technology Development Fund: MSAR No. 0166 and Taiwan Ministry of Education under the Aim for Top University Program NCU. The Lulin Observatory is operated by Institute of Astronomy, National Central University, Taiwan, under a grant of NSC 96-2752-M-008-011-PAE.
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\[lastpage\]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper we describe the stable and unstable leaves for the geodesic flow on the space of non-wandering geodesics of a Margulis Space Time and prove contraction properties of the leaves under the flow. We also show that mondromy of Margulis Space Times are “Anosov representations in non semi-simple Lie groups".'
address: |
Department of Mathematics\
University of Paris 11\
Orsay 91400\
France
author:
- Sourav Ghosh
title: Anosov Structure on Margulis Space Time
---
[^1]
Introduction
============
A Margulis Space Time $\mathsf{M}$ is a quotient manifold of the three dimensional affine space by a free, non-abelian group acting as affine transformations with discrete linear part. It owes its name to Grigory Margulis, who was the first to use these spaces, in [@marg1] and [@marg2], as examples to answer Milnor’s following question in the negative.
Is the fundamental group of a complete, flat, affine manifold virtually polycyclic? [@milnor]
Observe that if $\mathsf{M}$ is a Margulis Space Time then the fundamental group $\pi_1(\mathsf{M})$ does not contain any translation. By combining results of Fried, Goldman and Mess from [@fried], [@mess], a complete flat affine manifold either has a polycyclic fundamental group or is a Margulis Space Time. In this paper we will only consider Margulis Space Times whose linear part contains no parabolic, although by Drumm there exists Margulis Space Time whose linear part contains parabolics. Fried and Goldman showed in [@fried] that a conjugate of the linear part of the affine action of the fundamental group forms a subgroup of $\mathsf{SO}^0(2,1)$ in $\mathsf{GL}({\mathbb{R}}^3)$. Therefore, a Margulis Space Time comes with a parallel Lorentz metric.
The parallelism classes of timelike geodesics of $\mathsf{M}$ can be parametrized by a non-compact complete hyperbolic surface $\Sigma$. Recent work by Danciger, Gueritaud and Kassel in [@dgk] have shown that $\mathsf{M}$ is a ${\mathbb{R}}$-bundle over $\Sigma$ and the fibers are time like geodesics.
Previous works of Jones, Charette, Goldman, Labourie and Margulis in [@jones], [@labourie; @invariant] and [@geodesic] showed that the dynamics of $\mathsf{M}$ is closely related to that of $\Sigma$. Jones, Charette and Goldman showed in [@jones] that bispiralling geodesics in $\mathsf{M}$ exists and they correspond to bispiralling geodesics in $\Sigma$. Goldman and Labourie showed in [@geodesic] that non-wandering geodesics in $\mathsf{M}$ correspond to non-wandering geodesics in $\Sigma$.
In this paper, we first chalk out some preliminary notions, in order to prepare the grounds to explicitly describe the stable and unstable laminations of $\mathsf{U}_{\hbox{\tiny rec}}{\mathsf{M}}$, the space of non-wandering geodesics in $\mathsf{M}$, under the geodesic flow. We carry on to show that the stable lamination contracts under the forward flow and the unstable lamination contracts under the backward flow. More precisely, we prove the following,
\[mainthm1\] Let $\underline{\mathcal{L}}^+$ and $\underline{\mathcal{L}}^-$ be two laminations of the metric space $\mathsf{U}_{\hbox{\tiny rec}}{\mathsf{M}}$ as defined in definition \[lem\]. The geodesic flow on the space of non-wandering geodesics in $\mathsf{M}$ contracts $\underline{\mathcal{L}}^+$ exponentially in the forward direction of the flow and contracts $\underline{\mathcal{L}}^-$ exponentially in the backward direction of the flow.
Moreover, in the last section using a natural extension of the definition of Anosov representation given in section 2.0.7 of [@orilab] we define the notion of an Anosov representation in our context replacing manifolds by metric spaces. Using this definition we can restate our theorem by the following theorem:
\[geomano\] Let $\mathsf{N}$ be the space of all oriented space-like affine lines in the three dimentional affine space and let $\mathcal{L}$ be the orbit foliation of the flow $\Phi$ on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$. Then there exist a pair of foliations on $\mathsf{N}$ so that $(\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M},\mathcal{L})$ admits a geometric $(\mathsf{N}, \mathsf{SO}^0(2,1)\ltimes\mathbb{R}^3)$ Anosov structure.
In other words, mondromy of Margulis Space Times are “Anosov representations in non semi-simple Lie groups".\
**Acknowledgments:** I would like to express my gratitude towards my advisor Prof. Francois Labourie for his guidance. I would like to thank Andres Sambarino for the many helpful discussions that we had. I would also like to thank Thierry Barbot for his careful eye in finding a gap in a previous unpublished version of this work.
Background
==========
Affine Geometry
---------------
An $\textit{affine}$ $\textit{space}$ is a set $\mathbb{A}$ together with a vector space $\mathbb{V}$ and a faithful and transitive group action of $\mathbb{V}$ on $\mathbb{A}$. We call $\mathbb{V}$ the underlying vector space of $\mathbb{A}$ and refer to its elements as translations. An $\textit{affine}$ $\textit{transformation}$ $F$ between two affine spaces $\mathbb{A}_1$ and $\mathbb{A}_2$, is a map such that for all $x$ in $\mathbb{A}_1$ and for all $v$ in $\mathbb{V}_1$, $F$ satisfies the following property: $$\label{1}
F(x + v) = F(x) + \mathtt{L}(F).v$$ for some linear transformation $\mathtt{L}(F)$ between $\mathbb{V}_1$ and $\mathbb{V}_2$. Therefore, by fixing an origin $O$ in $\mathbb{A}$, one can represent an affine transformation $F$, from $\mathbb{A}$ to itself as a combination of a linear transformation and a translation. More precisely, $$\label{2}
F(O + v) = O + \mathtt{L}(F).v + \left(F(O)-O\right).$$ We denote $(F(O)-O)$ by $\mathtt{u}(F)$. Let us denote the space of affine automorphisms of $\mathbb{A}$ onto itself by $\mathsf{Aff}(\mathbb{A})$.\
Let $\mathsf{GL}(\mathbb{V})$ be the general linear group of $\mathbb{V}$. We consider the semidirect product $\mathsf{GL}(\mathbb{V})\ltimes \mathbb{V}$ of the two groups $\mathsf{GL}(\mathbb{V})$ and $\mathbb{V}$ where the multiplication is defined by $$\begin{aligned}
(g_1, v_1).(g_2, v_2) {\mathrel{\mathop:}=}(g_1g_2, v_1 + g_1.v_2)\end{aligned}$$ for $g_1, g_2$ in $\mathsf{GL}(\mathbb{V})$ and $v_1, v_2$ in $\mathbb{V}$. Using equation \[2\] we obtain that the following map: $$F\mapsto(\mathtt{L}(F),\mathtt{u}(F))$$ defines an isomorphism between $\mathsf{Aff}(\mathbb{A})$ and $\mathsf{GL}(\mathbb{V})\ltimes \mathbb{V}$.\
Let us denote the tangent bundle of $\mathbb{A}$ by $\mathsf{T}\mathbb{A}$. The tangent bundle $\mathsf{T}\mathbb{A}$ of an affine space $\mathbb{A}$ is a trivial bundle and is canonically isomorphic to $\mathbb{A} \times \mathbb{V}$ as a bundle. The geodesic flow $\tilde{\Phi}$ on $\mathsf{T}\mathbb{A}$ is defined as follows, $$\begin{aligned}
\tilde{\Phi}_t \colon \mathsf{T}\mathbb{A} &\longrightarrow \mathsf{T}\mathbb{A} \\
\notag (p,v) &\mapsto (p +tv,v).\end{aligned}$$
Hyperboloid Model of Hyperbolic Geometry
----------------------------------------
Let $\left(\mathbb{R}^{2,1}, \langle\mid\rangle\right)$ be a Minkowski Space Time where the quadratic form corresponding to the metric $\langle\mid\rangle$ is given by $$\begin{aligned}
\label{lorentz}
\mathcal{Q} {\mathrel{\mathop:}=}\begin{pmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0& -1\\
\end{pmatrix}.\end{aligned}$$ Let $\mathsf{SO}(2,1)$ denote the group of linear transformations of $\mathbb{R}^{2,1}$ preserving the metric $\langle\mid\rangle$ and $\mathsf{SO}^{0}(2,1)$ be the connected component containing the identity of $\mathsf{SO}(2,1)$.\
The cross product $\boxtimes$ associated with this quadratic form is defined as follows: $$\begin{aligned}
u \boxtimes v {\mathrel{\mathop:}=}(u_2v_3 - u_3v_2 , u_3v_1 - u_1v_3 , u_2v_1 - u_1v_2)^{t}\end{aligned}$$ where $u,v$ is denoted by $(u_1, u_2, u_3)^{t}$ and $(v_1, v_2, v_3)^{t}$ respectively. The cross product $\boxtimes$ satisfies the following properties for all $u, v$ in $\mathbb{R}^{2,1}$: $$\begin{aligned}
&\notag\langle u, v \boxtimes w \rangle = \det [u,v,w],\\
&\label{box}\langle u \boxtimes v, u \boxtimes v\rangle = \langle u, v\rangle ^2 - \langle u, u \rangle \langle v, v\rangle ,\\
&\notag u\boxtimes v = - v \boxtimes u.\end{aligned}$$ Now for all real number $k$ we define, $$\begin{aligned}
\mathsf{S}^{k} {\mathrel{\mathop:}=}\lbrace v \in \mathbb{R} \mid \langle v,v \rangle \ = k \rbrace .\end{aligned}$$ We note that $\mathsf{S}^{-1}$ has two components. We denote the component containing ($0,0,1$)$^{t}$ as $\mathbb{H}$. The quadratic form gives rise to a Riemannian metric of constant negative curvature on the submanifold $\mathbb{H}$ of $\mathbb{R}^{2,1}$. The space $\mathbb{H}$ is called the $\textit{hyperboloid}$ $\textit{model}$ $\textit{of}$ $\textit{hyperbolic}$ $\textit{geometry}$. Let $\mathsf{U}\mathbb{H}$ denote the unit tangent bundle of $\mathbb{H}$. The map $$\begin{aligned}
\label{theta}
\Theta : \mathsf{SO}^{0}(2,1) &\longrightarrow \mathsf{U}\mathbb{H}\\
\notag g &\longmapsto \left(g(0,0,1)^t, g(0,1,0)^t\right),\end{aligned}$$ gives an analytic identification between $\mathsf{SO}^{0}(2,1)$ and $\mathsf{U}\mathbb{H}$. Let $\tilde{\phi}_t$ denote the geodesic flow on $\mathsf{U}\mathbb{H} \cong \mathsf{SO}^{0}(2,1)$. We note that $\tilde{\phi_t}(g) = g.a(t)$ where $$\begin{aligned}
\label{a}
a(t) {\mathrel{\mathop:}=}\left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & \text{cosh(t)} & \text{sinh(t)} \\
0 & \text{sinh(t)} & \text{cosh(t)} \end{array}
\right).\end{aligned}$$ We also note that $\tilde{\phi}_t$ is the image of the geodesic flow on $\mathsf{PSL}(2,\mathbb{R})$ under the identification of $\mathsf{PSL}(2,\mathbb{R})$ and $\mathsf{SO}^{0}(2,1)$.
There is a canonical metric $d_{\mathsf{U}\mathbb{H}}$ on the unit tangent bundle $\mathsf{U}\mathbb{H}$ whose restriction on $\mathbb{H}$ is the hyperbolic metric. The metric $d_{\mathsf{U}\mathbb{H}}$ is unique upto the action of the maximal compact subgroup of $\mathsf{SO}^{0}(2,1)$. Let $g\in\mathsf{SO}^{0}(2,1)\cong\mathsf{U}\mathbb{H}$. We recall that the $\textit{horocycles}$ $\tilde{\mathcal{H}}_g^\pm$ for the geodesic flow $\tilde{\phi}$ passing through the point $g$ is defined as follows: $$\begin{aligned}
\tilde{\mathcal{H}}_g^+ &{\mathrel{\mathop:}=}\{h\in\mathsf{U}\mathbb{H}\mid\lim_{t\to\infty}d_{\mathsf{U}\mathbb{H}}(\tilde{\phi}_t g,\tilde{\phi}_t h)=0 \},\\
\tilde{\mathcal{H}}_g^- &{\mathrel{\mathop:}=}\{h\in\mathsf{U}\mathbb{H}\mid\lim_{t\to-\infty}d_{\mathsf{U}\mathbb{H}}(\tilde{\phi}_t g,\tilde{\phi}_t h)=0 \}.\end{aligned}$$ We note that under the identification $\Theta$, the horocycle $\tilde{\mathcal{H}}_g^{\pm}$ passing through $g$ is given by $g.u^\pm(t)$, where $u^\pm(t)$ are defined as follows: $$\begin{aligned}
\label{u1}
u^+(t) {\mathrel{\mathop:}=}\left( \begin{array}{ccc}
1 & -2t & 2t \\
2t & 1-2t^2 & 2t^2 \\
2t & -2t^2 & 1+2t^2
\end{array} \right),\\
u^-(t) {\mathrel{\mathop:}=}\label{u2}
\left( \begin{array}{ccc}
1 & 2t & 2t \\
-2t & 1-2t^2 & -2t^2 \\
2t & 2t^2 & 1+2t^2
\end{array} \right).\end{aligned}$$ We also note that $\tilde{\mathcal{H}}^\pm$ is the image of the horocycles of $\mathsf{PSL}(2,\mathbb{R})$ under the identification of $\mathsf{PSL}(2,\mathbb{R})$ and $\mathsf{SO}^{0}(2,1)$.
Let ${\nu}$ be defined as follows: $$\begin{aligned}
{\nu} \colon \mathsf{SO}^0(2,1) &\longrightarrow \mathsf{S}^{1}\\
\notag g &\longmapsto g(1,0,0)^t,\end{aligned}$$ and also let ${\nu}^\pm$ be defined as follows: $$\begin{aligned}
{\nu}{^\pm} \colon \mathsf{SO}^0(2,1) &\longrightarrow \mathsf{S}^0\\
\notag g &\longmapsto g.\left(0,\pm\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)^t.\end{aligned}$$ The map ${\nu}$ is called the $\textit{neutral section}$ and the maps ${\nu}^+$(respectively ${\nu}^-$) are called the $\textit{positive}$ (respectively $\textit{negative}$) $\textit{limit sections}$. We list a few properties of the neutral section and the limit sections as follows: $$\begin{aligned}
&{\nu}(\tilde{\phi_t}g) = {\nu}(g),\label{nu1}\\
&{\nu}(h.g) = h.{\nu}(g),\label{nu2}\\
&{\nu}^{\pm}(\tilde{\phi_t}g) = e^{\pm t}{\nu}^{\pm}(g),\label{limit1}\\
&{\nu}^{\pm}(h.g) = h.{\nu}^{\pm}(g),\label{limit2}\\
&{\nu}^{+}(g.u^+(t)) = {\nu}^{+}(g),\label{limit3}\\
&{\nu}^{-}(g.u^-(t)) = {\nu}^{-}(g).\label{limit4}\end{aligned}$$ where $t\in\mathbb{R}$ and $g, h\in\mathsf{SO}^0(2,1)$.
Let $\Gamma$ be a free, nonabelian subgroup with finitely many generators. We consider the left action of $\Gamma$ on $\mathsf{U}\mathbb{H}$. We notice that the action of $\Gamma$ being from the left and the action of $a(t)$ being from the right, the two actions commute. Furthermore, given a free and proper action of $\Gamma$ on $\mathsf{U}\mathbb{H}$, one gets an isomorphism between $\Gamma \backslash \mathsf{U}\mathbb{H}$ and $\mathsf{U}\Sigma$, where $\mathsf{U}\Sigma$ is the unit tangent bundle of the surface $\Sigma{\mathrel{\mathop:}=}\Gamma \backslash \mathbb{H}$. We note that the flow $\tilde{\phi}$ on $\mathsf{U}\mathbb{H}$ gives rise to a flow $\phi$ on $\mathsf{U}\Sigma$.
Let $x_0$ be a point in $\mathbb{H}$. Let $\Gamma .x_0$ denote the orbit of $x_0$ under the action of $\Gamma$. We denote the closure of $\Gamma .x_0$ inside the closure of $\mathbb{H}$ by $\overline{\Gamma .x}_0$. We define the $\textit{limit set}$ of the group $\Gamma$ to be the space $\overline{\Gamma .x}_0\backslash\Gamma .x_0$ and denote it by $\Lambda_{\infty}\Gamma$. We note that the collection $\overline{\Gamma .x}_0\backslash\Gamma .x_0$ is independent of the particular choice of $x_0$. We also know that $\Lambda_{\infty}\Gamma$ is compact.
A point $g\in \mathsf{U}\Sigma$ is called a $\textit{wandering point}$ of the flow $\phi$ if there exists an $\epsilon$-neighborhood $\mathcal{B}_\epsilon(g)\subset\mathsf{U}\Sigma$ around $g$ and a real number $t_0$ such that for all $t>t_0$ we have that $$\mathcal{B}_\epsilon(g)\cap\phi_t\mathcal{B}_\epsilon(g)=\emptyset .$$ Moreover, a point is called $\textit{non-wandering}$ if it is not a wandering point.
Let $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\Sigma$ be the space of all non-wandering points of the geodesic flow $\phi$ on $\mathsf{U}\Sigma$. We denote the lift of the space $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\Sigma$ in $\mathsf{U}\mathbb{H}$ by $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$. Now if the action of $\Gamma$ on $\mathbb{H}$ is free and proper and moreover $\Gamma$ contains no parabolics, then the space $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\Sigma$ is compact. We note that the subspace $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ can also be given an alternate description as follows: $$\begin{aligned}
\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H} = \left\lbrace (x,v) \in \mathsf{U}\mathbb{H} \mid \lim_{t \to \pm\infty}\tilde{\phi}^1_t x \in \Lambda_{\infty}\Gamma\right\rbrace\end{aligned}$$ where $\tilde{\phi}_t(x,v) = (\tilde{\phi}^1_tx, \tilde{\phi}^2_tv)$. Furthermore, we note that the space $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ can be identified with the space $\left(\Lambda_{\infty}\Gamma\times\Lambda_{\infty}\Gamma\setminus\{(x,x)\mid x\in \Lambda_{\infty}\Gamma\} \right)\times\mathbb{R}$.
Metric Anosov Property
----------------------
The definitions in this section, which can also be found in subsection 3.2 of [@pressure; @metric], has been included here for the sake of completeness.
\[lam\] Let $(\mathcal{X},d)$ be a metric space. A $\textit{lamination}$ $\mathcal{L}$ of $\mathcal{X}$ is an equivalence relation on $\mathcal{X}$ such that for all $x$ in $\mathcal{X}$ there exist an open neighborhood $\mathcal{U}_x$ of $x$ in $\mathcal{X}$, two topological spaces $\mathcal{U}_1$ and $\mathcal{U}_2$ and a homeomorphism $\mathfrak{f}_x$ from $\mathcal{U}_1 \times \mathcal{U}_2$ onto $\mathcal{U}_x$ satisfying the following properties,
1. for all $w, z$ in $\mathcal{U}_x\ \cap\ \mathcal{U}_y$ we have $p_2\left(\mathfrak{f}^{-1}_x(w)\right) = p_2\left(\mathfrak{f}^{-1}_x(z)\right)$ if and only if $p_2\left(\mathfrak{f}^{-1}_y(w)\right) = p_2\left(\mathfrak{f}^{-1}_y(z)\right)$ where $p_2$ is the projection from $\mathcal{U}_1 \times \mathcal{U}_2$ onto $\mathcal{U}_2$,
2. for all $w, z$ in $\mathcal{X}$ we have $w\mathcal{L}z$ if and only if there exists a finite sequence of points $w_1, w_2,..., w_n$ in $\mathcal{X}$ with $w_1 = w$ and $w_n = z$, such that $w_{i+1}$ is in $\mathcal{U}_{w_{i}}$, where $\mathcal{U}_{w_{i}}$ is a neighborhood of $w_i$ and $p_2\left(\mathfrak{f}^{-1}_{w_i}(w_i)\right) = p_2\left(\mathfrak{f}^{-1}_{w_i}(w_{i+1})\right)$ for all $i$ in $\{1,2,...,n-1\}$.
The homeomorphism $\mathfrak{f}_x$ is called a $\textit{chart}$ and the equivalence classes are called the $\textit{leaves}$.\
A $\textit{plaque open set}$ in the chart corresponding to $\mathfrak{f}_x$ is a set of the form $\mathfrak{f}_x(\mathcal{V}_1 \times \{x_2\})$ where $x = \mathfrak{f}_x(x_1,x_2)$ and $\mathcal{V}_1$ is an open set in $\mathcal{U}_1$. The $\textit{plaque topology}$ on $\mathcal{L}_x$ is the topology generated by the plaque open sets. A plaque neighborhood of $x$ is a neighborhood for the plaque topology on $\mathcal{L}_x$.
A $\textit{local product structure}$ on $\mathcal{X}$ is a pair of two laminations $\mathcal{L}_1$, $\mathcal{L}_2$ satisfying the following property: for all $x$ in $\mathcal{X}$ there exist two plaque neighborhoods $\mathcal{U}_1$, $\mathcal{U}_2$ of $x$, respectively in $\mathcal{L}_1$, $\mathcal{L}_2$ and a homeomorphism $\mathfrak{f}_x$ from $\mathcal{U}_1 \times \mathcal{U}_2$ onto a neighborhood $\mathcal{W}_x$ of $x$, such that $\mathfrak{f}_x$ defines a chart for both the laminations $\mathcal{L}_1$ and $\mathcal{L}_2$.
Now, let us assume that $\psi_t$ be a flow on $\mathcal{X}$. A lamination $\mathcal{L}$ invariant under the flow $\psi_t$ is called $\textit{transverse}$ to the flow, if for all $x$ in $\mathcal{X}$, there exists a plaque neighborhood $\mathcal{U}_x$ of $x$ in $\mathcal{L}_x$, a topological space $\mathcal{V}$, a positive $\epsilon$ and a homeomorphism $\mathfrak{f}_x$ from $\mathcal{U}_x \times \mathcal{V} \times (-\epsilon,\epsilon)$ onto an open neighborhood $\mathcal{W}_x$ of $x$ in $\mathcal{X}$ satisfying the following condition: $$\psi_t(\mathfrak{f}_x(u,v,s)) = \mathfrak{f}_x(u,v,s + t)$$ for $u$ in $\mathcal{U}_x$, $v$ in $\mathcal{V}$ and for $s, t$ in the interval $(-\epsilon,\epsilon)$. Let $\mathcal{L}^{.}$ be a lamination which is transverse to the flow $\psi_t$. We define a new lamination $\mathcal{L}^{.,0}$, called the $\textit{central lamination}$, starting from $\mathcal{L}^{.}$ as follows, we say $y, z$ in $\mathcal{X}$ belongs to the same equivalence class of $\mathcal{L}^{.,0}$ if for some real number $t$, $\psi_ty$ and $z$ belongs to the same equivalence class of $\mathcal{L}^{.}$.
A lamination $\mathcal{L}$ invariant under a flow $\psi_t$ is said to $\textit{contract}$ $\textit{under}$ $\textit{the}$ $\textit{flow}$ if there exists a positive real number $T_0$ such that for all $x$ in $\mathcal{X}$, the following holds: there exists a chart $\mathfrak{f}_x$ of an open neighbourhood $W_x$ of $x$, and for any two points $y, z$ in $W_x$ with $y, z$ being in the same equivalence class of $\mathcal{L}$, we have, $$d(\psi_ty,\psi_tz) < \frac{1}{2}d(y,z)$$ for all $t>T_0$.
We note that a lamination ‘contracts under a flow’ if and only if the lamination contracts exponentially under the flow.
A flow $\psi_t$ on a compact metric space is called $\textit{Metric Anosov}$, if there exist two laminations $\mathcal{L}^+$ and $\mathcal{L}^-$ of $\mathcal{X}$ such that the following conditions hold: $$\begin{aligned}
1&.\ (\mathcal{L}^+,\mathcal{L}^{-,0})\text{ defines a local product structure on }\mathcal{X},\\
2&.\ (\mathcal{L}^-,\mathcal{L}^{+,0})\text{ defines a local product structure on }\mathcal{X},\\
3&. \text{ the leaves of }\mathcal{L}^+\text{ are contracted by the flow,}\\
4&. \text{ the leaves of }\mathcal{L}^-\text{ are contracted by the inverse flow.}\end{aligned}$$
In such a case we call $\mathcal{L}^+$, $\mathcal{L}^-$, $\mathcal{L}^{+,0}$ and $\mathcal{L}^{-,0}$ respectively the $\textit{stable}$, $\textit{unstable}$, $\textit{central stable}$ and $\textit{central unstable}$ laminations.
Margulis Space Times and Surfaces
---------------------------------
A $\textit{Margulis Space Time}$ $\mathsf{M}$ is a quotient manifold of the three dimensional affine space $\mathbb{A}$ by a free, non-abelian group $\Gamma$ which acts freely and properly as affine transformations with discrete linear part. In [@marg1] and [@marg2] Margulis showed the existence of these spaces. Later in [@D] Drumm introduced the notion of $\textit{crooked planes}$ and constructed fundamental domains of a certain class of Margulis Space Times. In his construction the crooked planes give the boundary of appropriate fundamental domains for a certain class of Margulis Space Times. Recently, in [@dgk] Danciger–Gueritaud–Kassel showed that for any Margulis Space Time one can find a fundamental domain whose boundaries are given by union of crooked planes.
If $\Gamma$ is a subgroup of $\mathsf{GL}(\mathbb{R}^3)\ltimes\mathbb{R}^3$ such that $\mathsf{M}{\mathrel{\mathop:}=}\Gamma\backslash\mathbb{A}$ is a Margulis Space Time then by a result proved by Fried–Goldman in [@fried] we get that a conjugate of $\mathtt{L}(\Gamma)$ is a subgroup of $\mathsf{SO}^0(2,1)$. Therefore without loss of generality we can take $\Gamma\subset\mathsf{G}{\mathrel{\mathop:}=}\mathsf{SO}^0(2,1)\ltimes\mathbb{R}^3$ where $\Gamma$ is a free non-abelian group with finitely many generators. In this thesis I will only consider Margulis Space Times such that $\mathtt{L}(\Gamma)$ contains no parabolic elements.
Let $\mathsf{M}{\mathrel{\mathop:}=}\Gamma\backslash\mathbb{A}$ be a Margulis Space Time such that $\mathtt{L}(\Gamma)$ contains no parabolic elements. Then the action of $\mathtt{L}(\Gamma)$ on $\mathbb{H}$ is Schottky. Hence $\Sigma{\mathrel{\mathop:}=}\mathtt{L}(\Gamma)\backslash\mathbb{H}$ is a non-compact surface with no cusps.
Now let $\mathsf{T}\mathsf{M}$ be the tangent bundle of $\mathsf{M}$. As $\mathtt{L}(\Gamma)\subset\mathsf{SO}^0(2,1)$ we have that $\mathsf{T}\mathsf{M}$ carries a Lorentzian metric $\langle\mid\rangle$. Let $$\mathsf{U}\mathsf{M}{\mathrel{\mathop:}=}\{(X,v)\in \mathsf{T}\mathsf{M}\mid\langle v\mid v\rangle_X=1\}.$$ We note that $\mathsf{U}\mathsf{M}\cong\Gamma\backslash\mathsf{U}\mathbb{A}$ where $\mathsf{U}\mathbb{A}{\mathrel{\mathop:}=}\mathbb{A}\times\mathsf{S}^{1}$. The geodesic flow $\tilde{\Phi}$ on $\mathsf{T}\mathbb{A}$ gives rise to a flow $\Phi$ on $\mathsf{U}\mathsf{M}$.
We recall that a point $(X,v)\in \mathsf{U}\mathsf{M}$ is called a $\textit{wandering point}$ of the flow $\Phi$ if there exists an $\epsilon$-neighborhood $\mathcal{B}_\epsilon(X,v)\subset\mathsf{U}\Sigma$ around $(X,v)$ and a real number $t_0$ such that for all $t>t_0$ we have that $$\mathcal{B}_\epsilon(X,v)\cap\Phi_t\mathcal{B}_\epsilon(X,v)=\emptyset .$$ Moreover, a point is called $\textit{non-wandering}$ if it is not a wandering point.
We denote the space of all non-wandering points of the flow $\Phi$ on $\mathsf{U}\mathsf{M}$ by $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$. Moreover, we denote the lift of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$ into $\mathsf{U}\mathbb{A}$ by $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$.
In [@labourie; @invariant] Goldman–Labourie–Margulis proved the following theorem:
[\[Goldman–Labourie–Margulis\]]{} \[N\] Let $\Gamma$ be a non-abelian free discrete subgroup of $\mathsf{G}$ with finitely many generators giving rise to a Margulis Space Time and let $\mathtt{L}(\Gamma)$ contains no parabolic elements. Then there exists a map $${N}: \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}\longrightarrow\mathbb{A}$$ and a positive Hölder continuous function $${f}: \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}\longrightarrow\mathbb{R}$$ such that
1. for all $\gamma\in\Gamma$ we have ${f}\circ\mathtt{L}(\gamma) = {f}$,
2. for all $\gamma\in\Gamma$ we have ${N}\circ\mathtt{L}(\gamma) = \gamma{N}$, and
3. for all $g\in\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ and for all $t\in\mathbb{R}$ we have $${N}(\tilde{\phi_t}g) = {N}(g) + \left(\int\limits_{0}^{t}{f}(\tilde{\phi_s}(g))ds\right){\nu}(g).$$
We call ${N}$ a $\textit{neutralised section}$. Using the existence of a neutralised section Goldman–Labourie proved the following theorem in [@geodesic]:
[\[Goldman–Labourie\]]{} \[commute\] Let $\Gamma$ be a non-abelian free discrete subgroup of $\mathsf{G}$ with finitely many generators giving rise to a Margulis Space Time and let $\mathtt{L}(\Gamma)$ contains no parabolic elements. Also let $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\Sigma$ and $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$ be defined as above. Now if ${N}$ is a neutralised section, then there exists an injective map $\hat{\mathtt{N}}$ such that the following diagram commutes, $$\begin{CD}
\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H} @> \mathtt{N} >> \mathsf{U}\mathbb{A} \\
@ V{\pi} VV @ VV {\pi} V \\
{\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\Sigma} @> {\hat{\mathtt{N}}} >> {\mathsf{U} \mathsf{M}} \\
\end{CD}$$ where $\mathtt{N}{\mathrel{\mathop:}=}(N,\nu)$. Moreover, $\hat{\mathtt{N}}$ is an orbit equivalent Hölder homeomorphism onto $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$.
Metric Anosov structure on Margulis Space Time
==============================================
Let $\mathsf{M}$ be a Margulis Space Time. In this section, first we define a distance function $d$ on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$ such that $(\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M},d)$ is a metric space. Next, we define two laminations $\mathcal{L}^\pm$ on the metric space $(\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M},d)$ which are invariant under the flow $\Phi_t$ on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$. Finally, we show that the lamination $\mathcal{L}^+$ is a stable lamination and the lamination $\mathcal{L}^-$ is an unstable lamination for the flow $\Phi_t$ on $(\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M},d)$. We note that the method used in this paper to construct the distance function $d$ and to prove contraction properties of the lamination is inspired by [@pressure; @metric].
Metric space structure
----------------------
The restriction of any euclidean metric on $\mathbb{A}\times\mathbb{V}$ to the subspace $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$, defines a distance on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$. We call this distance the $\textit{euclidean distance}$ on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$. In this section we will define a distance on the space $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ such that the distance is locally bilipschitz equivalent to any euclidean distance on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ and also is $\Gamma$-invariant, so as to get a distance on the quotient space $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$.
We note that any two euclidean metric on $\mathbb{A}\times\mathbb{V}$ are bilipschitz equivalent with each other and hence any two euclidean distances on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ are also bilipschitz equivalent with each other. Fix an euclidean distance $d$ on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$. The action of $\Gamma$ on the space $\mathbb{A}\times\mathbb{V}$ gives rise to a collection of distances related to $d$ defined as follows: for any $\gamma$ in $\Gamma$ define, $$\begin{aligned}
d_{\gamma}: \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}&\times\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}\longrightarrow \mathbb{R}\\
\notag(x,y)&\longmapsto d\left(\gamma ^{-1} x,\gamma ^{-1} y\right)\end{aligned}$$ Since each element of $\Gamma$ acts as a bilipschitz automorphism with respect to any euclidean distance, any two distances in the family $\{ d_{\gamma} \} _{\gamma \in \Gamma}$ are bilipschitz equivalent with each other.
Compactness of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\Sigma$ implies that $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$ is compact and hence we can choose a pre-compact fundamental domain $D$ of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$ inside $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ with an open interior. We can also choose a suitable precompact open set $U$ which contains the closure of $D$. We note that properness of the action of $\Gamma$ on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ implies that the cover of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ by the open sets $\{ \gamma . U \} _{\gamma \in \Gamma}$, is locally finite.
A path joining two points $x$ and $y$ in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ is a pair of tuples, $$\begin{aligned}
\mathcal{P} = \left((z_0,z_1,...,z_n),(\gamma_1,\gamma_2,...,\gamma_n)\right)\end{aligned}$$ where $z_i \in \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ and $\gamma_i \in \Gamma$ such that the following two conditions hold, $$\begin{aligned}
&\text{1. }x = z_0 \in \gamma_1.U \text{ and } y = z_n \in \gamma_{n}.U,\\
&\text{2. }\text{for all } n > i > 0, z_i \in \gamma_{i}.U \cap \gamma_{i+1}.U.\end{aligned}$$
The length of a path is defined by, $$\begin{aligned}
l(\mathcal{P}) {\mathrel{\mathop:}=}\sum\limits_{i=0}^{n-1}d_{\gamma_{i+1}}\left(z_i,z_{i+1}\right)\end{aligned}$$
\[dist\] We then define, $$\begin{aligned}
\tilde{d}(x,y) {\mathrel{\mathop:}=}\text{inf } \{ l(\mathcal{P}) \mid \mathcal{P} \text{ joins } x \text{ and } y \} \end{aligned}$$
$\tilde{d}$ is a $\Gamma$-invariant pseudo-metric.
If $\mathcal{P} = ((z_0,z_1,...,z_n),(\gamma_1,\gamma_2,...,\gamma_n))$ is a path joining $\gamma x$ and $\gamma y$, then the path, $$\begin{aligned}
\gamma ^{-1}.\mathcal{P} {\mathrel{\mathop:}=}\left(\left(\gamma ^{-1}z_0, \gamma ^{-1}z_1,...,\gamma ^{-1}z_n\right),\left(\gamma ^{-1}\gamma_1,\gamma ^{-1}\gamma_2,...,\gamma ^{-1}\gamma_n\right)\right)\end{aligned}$$ is a path joining $x$ and $y$. Moreover, $$\begin{aligned}
\notag l(\mathcal{P}) &= \sum \limits _{i=0}^{n-1}d_{\gamma _{i+1}}\left(z_i,z_{i+1}\right) = \sum \limits _{i=0}^{n-1}d\left(\gamma _{i+1} ^{-1}z_i,\gamma _{i+1} ^{-1}z_{i+1}\right)\\
\notag &= \sum \limits _{i=0}^{n-1}d\left(\left(\gamma ^{-1} \gamma _{i+1}\right) ^{-1} \gamma ^{-1}z_i,\left(\gamma ^{-1} \gamma _{i+1}\right) ^{-1}\gamma ^{-1}z_{i+1}\right) \\
\notag &= \sum \limits _{i=0}^{n-1}d_{\gamma ^{-1}\gamma _{i+1}}\left(\gamma ^{-1}z_i,\gamma ^{-1}z_{i+1}\right)\\
\notag &= l\left(\gamma ^{-1}.\mathcal{P}\right).\end{aligned}$$ Hence, using the definition of $\tilde{d}$ we get $\tilde{d}(\gamma x,\gamma y)$ is equal to $\tilde{d}(x,y)$.\
We also notice that $l(\mathcal{P})$ is a sum of distances. So $l(\mathcal{P})$ is non-negative and hence $\tilde{d}$ is non-negative.
It remains to show that $\tilde{d}$ is a metric and $\tilde{d}$ is locally bilipschitz equivalent to any euclidean distance. As all euclidean distances are bilipschitz equivalent with each other, it suffices to show that $\tilde{d}$ is locally bilipschitz equivalent with $d$.
$\tilde{d}$ is a metric and $\tilde{d}$ is locally bilipschitz equivalent to $d$.
Let $z$ be a point in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$. There exists a neighbourhood $V$ of $z$ in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ such that $$A {\mathrel{\mathop:}=}\{ \gamma \mid \gamma.U \cap V \neq \emptyset \}$$ is a finite set. We fix $V$ and choose a positive real number $\alpha$ so that $$\mathop{\bigcup}_{\gamma \in A} \left\lbrace x \mid d_\gamma(z,x)\leqslant \alpha\right\rbrace \subset V.$$ We have seen that any two distances in the family $\{ d_{\gamma} \} _{\gamma \in \Gamma}$ are bilipschitz equivalent with each other. Hence $A$ being a subset of $\Gamma$, any two distances in $A$ are bilipschitz equivalent with each other. Now finiteness of $A$ implies that we can choose a constant $K$ such that for all $\beta_1,\beta_2$ in $A$ we have that $d_{\beta_1}$ and $d_{\beta_2}$ are $K$-bilipschitz equivalent with each other. We set, $$W {\mathrel{\mathop:}=}\mathop\bigcap_{\gamma \in A} \left\lbrace x \mid d_\gamma (z,x) \leqslant \frac{\alpha}{10K}\right\rbrace.$$ We note that $W$ is a subset of $V$ because $K$ is bigger than 1.
By construction, if $x,y$ is in $W$ then for all $\gamma$ in $A$ we have, $$\begin{aligned}
\label{4}
d_\gamma(x,y) \leqslant d_\gamma(x,z) + d_\gamma(z,y) \leqslant \frac{\alpha}{5K}.\end{aligned}$$ Now let $x$ be any point in $ W$, $y$ be any general point and $$\mathcal{P} = ((z_0,z_1,...,z_n),(\gamma_1,\gamma_2,...,\gamma_n))$$ be a path joining $x$ and $y$.
We notice that $x = z_0$ is in $\gamma_1 U$. On the other hand $x$ is also an element of $W$, which is a subset of $V$. Therefore, $$\begin{aligned}
\gamma_1 U \cap V \neq \emptyset\end{aligned}$$ Hence $\gamma_1$ is in $A$. If there exists $k$ such that $\gamma_{k}$ is not in $A$ then we choose $j$ to be the smallest $k$ such that $\gamma_k$ is not in $A$. $$\begin{aligned}
\label{5}
l(\mathcal{P}) = \sum \limits _{i=0}^{n-1}d_{\gamma _{i+1}}(z_i,z_{i+1})
\geqslant \sum\limits _{i=0}^{j-1}d_{\gamma _{i+1}}(z_i,z_{i+1}).\end{aligned}$$ Now using the fact that $d_{\gamma_{j-1}}$ is K-bilipschitz equivalent with $d_{\gamma_i}$ for any $\gamma_i$ in $A$ we get, $$\begin{aligned}
\sum\limits _{i=0}^{j-1}d_{\gamma _{i+1}}(z_i,z_{i+1}) \geqslant \frac{1}{K}\sum\limits _{i=0}^{j-1}d_{\gamma _{j-1}}(z_i,z_{i+1}).\end{aligned}$$ Now from the triangle inequality it follows that $$\begin{aligned}
\frac{1}{K}\sum\limits _{i=0}^{j-1}d_{\gamma _{j-1}}(z_i,z_{i+1}) &\geqslant \frac{1}{K}d_{\gamma _{j-1}}(z_0,z_j)\\
\notag &\geqslant \frac{1}{K}(d_{\gamma_{j-1}}(z,z_j)- {d_{\gamma_{j-1}}}(z,z_0)).\end{aligned}$$ The point $z_0 = x$, belongs to $W$ and $\gamma_{j-1}$ belongs to $A$. Therefore, by the definition of $W$ we get that $$\begin{aligned}
\label{6}
{d_{\gamma_{j-1}}}(z,z_0) \leqslant \frac{\alpha}{10K}.\end{aligned}$$ We also know that $\gamma_j$ is not in $A$. Hence $\gamma_j . U$ does not intersect with $V$. The point $z_j$ by definition belongs to $\gamma_j . U$ and so $z_j$ is not in $V$. Therefore by the choice of $\alpha$ it follows that $$\begin{aligned}
\label{7}
d_{\gamma_{j-1}}(z,z_j) > \alpha .\end{aligned}$$ Using the inequalities \[5\] and \[6\] we get that $$\begin{aligned}
\frac{1}{K}\left( d_{\gamma_{j-1}}(z,z_j)- {d_{\gamma_{j-1}}}(z,z_0)\right) &> \frac{1}{K}\left(\alpha - \frac{\alpha}{10K}\right).\end{aligned}$$ Now as $K$ is bigger than 1 we have, $$\begin{aligned}
\label{8}
\frac{1}{K}\left(\alpha - \frac{\alpha}{10K}\right) \geqslant \frac{1}{K}\left(\alpha - \frac{\alpha}{10}\right) > \frac{\alpha}{5K}.\end{aligned}$$ Finally, using the inequalities from \[5\] to \[8\] we get that if there exists $k$ such that $\gamma_{k}$ is not in $A$ then, $$\begin{aligned}
\label{9}
l(\mathcal{P}) > \frac{\alpha}{5K}.\end{aligned}$$\
On the other hand, if for all $k$ we have $\gamma_k$ in $A$, then for all $\gamma \in A$ we have, $$\begin{aligned}
l(\mathcal{P}) &= \sum \limits _{i=0}^{n-1}d_{\gamma _{i+1}}(z_i,z_{i+1}) \geqslant \frac{1}{K}\sum\limits_{i=0}^{n-1}d_{\gamma}(z_i,z_{i+1}).\end{aligned}$$ And using triangle inequality it follows that $$\begin{aligned}
\frac{1}{K}\sum\limits_{i=0}^{n-1}d_{\gamma}(z_i,z_{i+1}) \geqslant \frac{1}{K}d_\gamma(x,y).\end{aligned}$$ Therefore, in the case when for all $k$, $\gamma_k$ is in $A$, we have for all $\gamma$ in $A$, $$\begin{aligned}
\label{10}
l(\mathcal{P}) \geqslant \frac{1}{K}d_\gamma(x,y).\end{aligned}$$
Combining the inequalities \[9\] and \[10\] and using the definition of $\tilde{d}$ we have that for any point $x$ in $W$, any general point $y$ and for all $\gamma$ in $A$, $$\begin{aligned}
\label{11}
\tilde{d}(x,y) \geqslant \frac{1}{K} \inf\left(\frac{\alpha}{5},d_\gamma(x,y)\right). \end{aligned}$$ Therefore for any point $y$ distinct from $z$ we have, $$\begin{aligned}
\tilde{d}(z,y) > 0.\end{aligned}$$ The above is true for any arbitrary choice of $z$ and hence it follows that $\tilde{d}$ is a metric.
Moreover, if $x, y$ are points in $W$ and $\gamma$ is in $A$ then from the inequality \[4\] we get, $$\begin{aligned}
d_\gamma(x,y) \leqslant \frac{\alpha}{5K} \leqslant \frac{\alpha}{5}\end{aligned}$$ and hence for all $x$, $y$ in $W$ and $\gamma$ in $A$, $$\begin{aligned}
\label{12}
\inf \left(\frac{\alpha}{5},d_\gamma(x,y)\right) = d_\gamma(x,y).\end{aligned}$$ Therefore, from the inequalities \[11\] and \[12\] it follows that for $x$, $y$ in $W$ and for any $\gamma$ in $A$, $$\begin{aligned}
\tilde{d}(x,y) \geqslant \frac{1}{K}d_\gamma(x,y).\end{aligned}$$ We know that there exists $\gamma_a$ such that the point $z$ is inside the open set $\gamma_a . U$. We note that the above defined $\gamma_a$ is also an element of $A$. Finally, we set $W_a$ to be the intersection of of the set $W$ with the set $\gamma_a . U$. Let $x , y$ be any two points in $W_a$. We choose the path $\mathcal{P}_0 = ((x,y),(\gamma_a,\gamma_a))$ and get that $$\begin{aligned}
\tilde{d}(x,y) = \text{inf } \{ l(\mathcal{P}) \mid \mathcal{P} \text{ joins } x \text{ and } y \} \leqslant l(\mathcal{P}_0)=d_{\gamma _a}(x,y).\end{aligned}$$ Hence, $\tilde{d}$ is bilipschitz equivalent to $d_{\gamma _a}$ on $W_a$ and the distance $d$ is bilipschitz equivalent to $d_{\gamma _a}$. Therefore, $d$ is bilipschitz to $\tilde{d}$ on $W_a$. Since $z$ was arbitrarily chosen it follows that $d$ is locally bilipschitz equivalent to $\tilde{d}$.
The lamination and its lift
---------------------------
In this subsection, we explicitly describe two laminations of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ for the flow $\Phi_t$ on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ and show that the laminations are equivariant under the action of the flow and the action of $\Gamma$. We will also define the notion of a leaf lift.
Let $Z$ be a point in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$. We know from the theorem \[commute\] that for all $Z\in\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ there exists an unique $g\in\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ such that $Z = \mathtt{N}(g)$.
\[lam1\] The positive and central positive partition of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ are respectively given by, $$\begin{aligned}
\mathcal{L}^{+}_{\mathtt{N}(g)} &{\mathrel{\mathop:}=}\tilde{\mathcal{L}}^{+}_{\mathtt{N}(g)} \cap \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}\\
\mathcal{L}^{+,0}_{\mathtt{N}(g)} &{\mathrel{\mathop:}=}\tilde{\mathcal{L}}^{+,0}_{\mathtt{N}(g)} \cap \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}\end{aligned}$$ where $$\begin{aligned}
\tilde{\mathcal{L}}^{+}_{\mathtt{N}(g)} {\mathrel{\mathop:}=}\lbrace ({N}(g) &+ s_1 {\nu}^+(g), {\nu}(g) + s_2 {\nu}^+(g))\mid s_1, s_2 \in \mathbb{R} \rbrace, \\
\tilde{\mathcal{L}}^{+,0}_{\mathtt{N}(g)} {\mathrel{\mathop:}=}\{ ({N}(g) &+ s_1 {\nu}^+(g) + t{\nu}(g) , {\nu}(g) + s_2 {\nu}^+(g))\mid t, s_1, s_2 \in \mathbb{R} \}. \end{aligned}$$
\[lam2\] The negative and central negative partition of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ are respectively given by, $$\begin{aligned}
\mathcal{L}^{-}_{\mathtt{N}(g)} {\mathrel{\mathop:}=}\tilde{\mathcal{L}}^{-}_{\mathtt{N}(g)} \cap \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}\\
\mathcal{L}^{-,0}_{\mathtt{N}(g)} {\mathrel{\mathop:}=}\tilde{\mathcal{L}}^{-,0}_{\mathtt{N}(g)} \cap \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}\end{aligned}$$ where $$\begin{aligned}
\tilde{\mathcal{L}}^{-}_{\mathtt{N}(g)} {\mathrel{\mathop:}=}\{ ({N}(g) &+ s_1 {\nu}^-(g),{\nu}(g) + s_2 {\nu}^-(g))\mid s_1, s_2 \in \mathbb{R} \},\\
\tilde{\mathcal{L}}^{-,0}_{\mathtt{N}(g)} {\mathrel{\mathop:}=}\{ ({N}(g) &+ s_1 {\nu}^-(g) + t{\nu}(g), {\nu}(g) + s_2 {\nu}^-(g))\mid t, s_1, s_2 \in \mathbb{R} \}.\end{aligned}$$
\[nulimit\] Let $g, h$ be two points in $\mathsf{U}\mathbb{H}$ then $$\begin{aligned}
h\text{ is in }\bigcup_{t \in \mathbb{R}}\tilde{\mathcal{H}}_{\tilde{\phi_t}g}^+\text{ if and only if }{\nu}(h) = {\nu}(g) + \frac{\langle {\nu}(h), {\nu}^-(g) \rangle}{\langle {\nu}^+(g), {\nu}^-(g) \rangle}{\nu}^+(g).\end{aligned}$$
Let $h$ be a point of $\bigcup_{t \in \mathbb{R}}\tilde{\mathcal{H}}_{\tilde{\phi_t}g}^+$. Hence there exist real numbers $t_1, t_2$ such that $h = {\nu}(ga(t_1)u^+(t_2))$. Therefore, we have $$\begin{aligned}
{\nu}(h) &= {\nu}(ga(t_1)u^+(t_2)) = ga(t_1)u^+(t_2){
\global\colveccount3
\begin{pmatrix}
\colvecnext
}{1}{0}{0} = ga(t_1){
\global\colveccount3
\begin{pmatrix}
\colvecnext
}{1}{2t_2}{2t_2}\\
&= ga(t_1)\left({
\global\colveccount3
\begin{pmatrix}
\colvecnext
}{1}{0}{0} + {
\global\colveccount3
\begin{pmatrix}
\colvecnext
}{0}{2t_2}{2t_2}\right) = \nu(g) + 2t_2.ga(t_1){
\global\colveccount3
\begin{pmatrix}
\colvecnext
}{0}{1}{1}\\
&= \nu(g) + 2t_2(\cosh t_1 + \sinh t_1).g{
\global\colveccount3
\begin{pmatrix}
\colvecnext
}{0}{1}{1}\\
&= \nu(g) + 2\sqrt{2}\ t_2(\cosh t_1 + \sinh t_1).\nu^+(g).\end{aligned}$$ Now we notice that $$\begin{aligned}
\langle\nu(h), \nu^-(g)\rangle &= \langle\nu(g) + 2\sqrt{2}\ t_2(\cosh t_1 + \sinh t_1).\nu^+(g), \nu^-(g)\rangle\\ &= 2\sqrt{2}\ t_2(\cosh t_1 + \sinh t_1).\langle\nu^+(g), \nu^-(g)\rangle .\end{aligned}$$ Combining the above two calculations we get $$\begin{aligned}
\nu(h) = \nu(g) + \frac{\langle \nu(h), \nu^-(g) \rangle}{\langle \nu^+(g), \nu^-(g) \rangle}\nu^+(g).\end{aligned}$$ Now let $g, h$ be two points in $\mathsf{U}\mathbb{H}$ satisfying, $$\begin{aligned}
\nu(h) = \nu(g) + a_1\nu^+(g)\end{aligned}$$ for some real number $a_1$. Using the definition of $\nu$ and $\nu^+$ we observe that the above equation is equvalent to the following equation, $$\begin{aligned}
\left(g.u^+\left(\frac{a_1}{2\sqrt{2}}\right)\right)^{-1}.h{
\global\colveccount3
\begin{pmatrix}
\colvecnext
}{1}{0}{0} = \left(u^+\left(\frac{a_1}{2\sqrt{2}}\right)\right)^{-1}{
\global\colveccount3
\begin{pmatrix}
\colvecnext
}{1}{a_1/\sqrt{2}}{a_1/\sqrt{2}} = {
\global\colveccount3
\begin{pmatrix}
\colvecnext
}{1}{0}{0}.\end{aligned}$$ We know that the only elements of $\mathsf{SO}^0(2,1)$ fixing the vector ${
\global\colveccount3
\begin{pmatrix}
\colvecnext
}{1}{0}{0}$ are of the form $a(t)$ for some real number $t$. Hence there exist a real number $t_1$ such that $$\begin{aligned}
\left(g.u^+\left(\frac{a_1}{2\sqrt{2}}\right)\right)^{-1}.h = a(t_1).\end{aligned}$$ Therefore, $$\begin{aligned}
h = g.u^+\left(\frac{a_1}{2\sqrt{2}}\right).a(t_1) = g.a(t_1).u^+\left(\frac{a_1\exp(-t_1)}{2\sqrt{2}}\right)\end{aligned}$$ and the result follows.
\[nu+\] Let $g, h$ be two points in $\mathsf{U}\mathbb{H}$ and $h$ is in $\bigcup_{t \in \mathbb{R}}\tilde{\mathcal{H}}_{\tilde{\phi_t}g}^+$ then $$\begin{aligned}
\frac{\langle \nu(g), \nu^-(h) \rangle}{\langle \nu^+(h), \nu^-(h) \rangle}\nu^+(h) = -\frac{\langle \nu(h), \nu^-(g) \rangle}{\langle \nu^+(g), \nu^-(g) \rangle}\nu^+(g).\end{aligned}$$
We know that if $h$ is in $\bigcup_{t \in \mathbb{R}}\tilde{\mathcal{H}}_{\tilde{\phi_t}g}^+$ then $g$ is in $\bigcup_{t \in \mathbb{R}}\tilde{\mathcal{H}}_{\tilde{\phi_t}h}^+$. Therefore using lemma \[nulimit\] we get $$\begin{aligned}
\nu(h) = \nu(g) + \frac{\langle \nu(h), \nu^-(g) \rangle}{\langle \nu^+(g), \nu^-(g) \rangle}\nu^+(g)\end{aligned}$$ and $$\begin{aligned}
\nu(g) = \nu(h) + \frac{\langle \nu(g), \nu^-(h) \rangle}{\langle \nu^+(h), \nu^-(h) \rangle}\nu^+(h).\end{aligned}$$ Hence $$\begin{aligned}
\frac{\langle \nu(g), \nu^-(h) \rangle}{\langle \nu^+(h), \nu^-(h) \rangle}\nu^+(h) = -\frac{\langle \nu(h), \nu^-(g) \rangle}{\langle \nu^+(g), \nu^-(g) \rangle}\nu^+(g).\end{aligned}$$
For all $g$ in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ we define, $$\begin{aligned}
\mathcal{H}_g^{\pm} {\mathrel{\mathop:}=}\tilde{\mathcal{H}}_g^{\pm} \cap \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}.\end{aligned}$$
\[14\] The following equations are true for all $g$ in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$, $$\begin{aligned}
1.&\ {\mathcal{L}}^{+,0}_{\mathtt{N}(g)} = \left\lbrace\mathtt{N}(h) \mid h \in \bigcup_{t \in \mathbb{R}}{\mathcal{H}}_{\tilde{\phi_t}g}^{+}\right\rbrace,\\
2.&\ {\mathcal{L}}^{-,0}_{\mathtt{N}(g)} = \left\lbrace\mathtt{N}(h) \mid h \in \bigcup_{t \in \mathbb{R}}{\mathcal{H}}_{\tilde{\phi_t}g}^{-}\right\rbrace.\end{aligned}$$
We start with defining a function, $$\begin{aligned}
F : \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H} &\times \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H} \rightarrow \mathbb{R}\\
\notag (g,h) &\mapsto \det[({N}(g)-{N}(h)), \nu(g), \nu(h)] .\end{aligned}$$ Using equation \[nu1\] and theorem \[N\] we get that $$\begin{aligned}
\label{13}
F(\tilde{\phi _t}g, \tilde{\phi _t}h) = F(g,h)\end{aligned}$$ for all $t\in\mathbb{R}$. Again using equation \[nu2\] and theorem \[N\] we get that the neutralised section and the neutral section are equivariant under the action of $\Gamma$. Hence for all $\gamma$ in $\Gamma$ we have, $$\begin{aligned}
\label{18}
F(\gamma g,\gamma h) &= \det [({N}(\gamma g)-{N}(\gamma h)), \nu(\gamma g), \nu(\gamma h)] \\
\notag &= \det [\gamma({N}(g)-{N}(h)), \gamma \nu(g), \gamma \nu(h))] \\
\notag &= \det [\gamma] \det [({N}(g)-{N}(h)), \nu(g), \nu(h)] \\
\notag &= \det [({N}(g)-{N}(h)), \nu(g), \nu(h)] \\
\notag &= F(g, h).\end{aligned}$$ Now for a fixed real number $c_0$ we consider the space, $$\begin{aligned}
\mathfrak{K} {\mathrel{\mathop:}=}\lbrace (g_1,g_2) \mid d_{\mathsf{U}\mathbb{H}}(g_1,g_2) \leqslant c_0 \rbrace \subset \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H} \times \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}.\end{aligned}$$ Compactness of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\Sigma$ implies that $\mathfrak{K}_{\Gamma}$, the projection of $\mathfrak{K}$ in $\Gamma \backslash (\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H} \times \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H})$, is compact. Now continuity of $F$ implies that $F$ is uniformly continuous on $\mathfrak{K}_{\Gamma}$.
Let $g$ and $h$ be two points in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ such that $h$ is in $\mathcal{H}_g^+$. Given any such choice of $g$ and $h$ we can choose a sufficiently large $t_0$ such that $d_{\mathsf{U}\mathbb{H}}(\tilde{\phi}_{t_0} g,\tilde{\phi}_{t_0} h)$ is arbitrarily close to zero, hence we have $F(\tilde{\phi}_{t_0} g,\tilde{\phi}_{t_0} h)$ arbitrarily close to zero. Therefore by using equation \[13\] it follows that $F(g,h)$ is zero for all $h$ in $\mathcal{H}_g^+$.
Now using equation \[13\], equation \[nu1\] and lemma \[nulimit\] we have, $$\begin{aligned}
0 &= F(\tilde{\phi _t}g, \tilde{\phi _t}h) = \det[({N}(\tilde{\phi _t}g)-{N}(\tilde{\phi _t}h)), \nu(\tilde{\phi _t}g), \nu(\tilde{\phi _t}h)]\\
&= \det[({N}(\tilde{\phi _t}g)-{N}(\tilde{\phi _t}h)), \nu(g), \nu(h)]\\
&= \det[({N}(\tilde{\phi _t}g)-{N}(\tilde{\phi _t}h)), \nu(g), \nu(g) + \frac{\langle \nu(h), \nu^-(g) \rangle}{\langle \nu^+(g), \nu^-(g) \rangle}\nu^+(g)]\\
&= \frac{\langle \nu(h), \nu^-(g) \rangle}{\langle \nu^+(g), \nu^-(g) \rangle}\det[({N}(\tilde{\phi _t}g)-{N}(\tilde{\phi _t}h)), \nu(g), \nu^+(g)].\end{aligned}$$ Therefore for all $h$ in $\mathcal{H}_g^+$ and for all real number $t$ we have $$\begin{aligned}
\det[({N}(\tilde{\phi _t}g)-{N}(\tilde{\phi _t}h)), \nu(g), \nu^+(g)] = 0.\end{aligned}$$ Hence there exist real numbers $a_1, b_1$ such that $$\begin{aligned}
\label{19}
{N}(\tilde{\phi _t}h) &= {N}(\tilde{\phi _t}g) + a_1 \nu(g) + b_1 \nu^+(g)\\
\notag&= {N}(g) + \left(a_1 + \int\limits_{0}^{t}{f}(\tilde{\phi_s}(g))ds\right) \nu(g) + b_1 \nu^+(g).\end{aligned}$$ Combining lemma \[nulimit\] and equation \[19\] we get that $$\begin{aligned}
{\mathcal{L}}^{+,0}_{\mathtt{N}(g)} \supseteq \left\lbrace\mathtt{N}(h) \mid h \in \bigcup_{t \in \mathbb{R}}{\mathcal{H}}_{\tilde{\phi_t}g}^+\right\rbrace\end{aligned}$$ Now let $W\in{\mathcal{L}}^{+,0}_{\mathtt{N}(g)}$. By theorem \[commute\] we know that there exist $h\in\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ such that $W = \mathtt{N}(h)$. Now the choice of $W$ implies that there exist some real number $a_2$ such that $$\begin{aligned}
\nu(h) = \nu(g) + a_2 \nu^+(g).\end{aligned}$$ Using lemma \[nulimit\] we get that $h\in\bigcup\limits_{t \in \mathbb{R}}\tilde{\mathcal{H}}_{\tilde{\phi_t}g}^+$. Therefore $h$ is in $$\begin{aligned}
\bigcup\limits_{t \in \mathbb{R}}{\mathcal{H}}_{\tilde{\phi_t}g}^+ = \left(\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H} \cap \bigcup\limits_{t \in \mathbb{R}}\tilde{\mathcal{H}}_{\tilde{\phi_t}g}^+\right)\end{aligned}$$ and we have $$\begin{aligned}
{\mathcal{L}}^{+,0}_{\mathtt{N}(g)} \subseteq \left\lbrace\mathtt{N}(h) \mid h \in \bigcup_{t \in \mathbb{R}}{\mathcal{H}}_{\tilde{\phi_t}g}^+\right\rbrace .\end{aligned}$$ Similarly the other equality follows.
\[lps\] Let $\mathcal{U}_{\mathtt{N}(g)}\subset\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ be a neighborhood of a point $\mathtt{N}(g)$ in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$. Then the following map is a local homeomorphism: $$\begin{aligned}
{\amalg}_{\mathtt{N}(g)} : \mathcal{U}_{\mathtt{N}(g)} &\rightarrow (\Lambda_{\infty}\Gamma \times \Lambda_{\infty}\Gamma \setminus \Delta) \times \mathbb{R}\\
\notag \mathtt{N}(h) &\mapsto \left( h^-, h^+, \left\langle {N}(h)-{N}(g), \nu\left(g^-,h^+\right)\right\rangle \right)\end{aligned}$$ where $h^\pm{\mathrel{\mathop:}=}\lim_{t\to\pm\infty}\pi(\tilde{\phi}_t h)$ and $\pi$ is the projection from $\mathsf{U}\mathbb{H}$ onto $\mathbb{H}$.
Let $g$ be a point in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$. We note that for $g\in\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ the points $g^{\pm}$ lies in $\Lambda_{\infty}\Gamma$. We observe that $\partial\mathbb{H}\setminus\{g^+\}$ is homeomorpic to $\mathbb{R}$. Given any $g$, let $\mathcal{V}_{g^-}$ denote a connected bounded open neighborhood of $g^-$ in $\partial\mathbb{H}\setminus\{g^+\}$ and $\mathcal{V}_{g^+}$ be a connected open neighborhood of $g^+$ in $\partial\mathbb{H}\setminus\{g^-\}$ such that $\mathcal{V}_{g^-} \cap \mathcal{V}_{g^+}$ is empty and $\mathcal{V}_{g^-} \times \mathcal{V}_{g^+}$ is a subset of $\partial\mathbb{H} \times \partial\mathbb{H} \setminus \Delta$. We define $\mathcal{U}_{g^\pm} {\mathrel{\mathop:}=}\mathcal{V}_{g^\pm} \cap \Lambda_\infty\Gamma$. Let $\mathcal{U}_{g}$ be the open subset of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ corresponding to the open set $\mathcal{U}_{g^-} \times \mathcal{U}_{g^+} \times \mathbb{R}$. We consider the following continuous map, $$\begin{aligned}
{\mathfrak{N}}_g : \mathcal{U}_{g} &\longrightarrow \mathbb{A}\\
h &\longmapsto {N}(h) - \left\langle {N}(h)-{N}(g), \nu(g^-,h^+)\right\rangle \nu(h)\end{aligned}$$ We notice that $$\nu(g^-,h^+)=\frac{\nu^-(g)\boxtimes \nu^+(h)}{\langle \nu^-(g),\nu^+(h)\rangle}.$$ Hence for all real number $t$ we have $${\mathfrak{N}}_g(\tilde{\phi}_th) = {\mathfrak{N}}_g(h).$$ Now we define the following continuous map: $$\begin{aligned}
{\Pi}_g : \mathcal{U}_{g^-} \times \mathcal{U}_{g^+} \times \mathbb{R} &\longrightarrow \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}\\
\notag (h^-, h^+, t) &\longmapsto \left({\mathfrak{N}}_g + t\nu, \nu\right)(h^-, h^+, t)\end{aligned}$$ and conclude by observing that $${\amalg}_{\mathtt{N}(g)}\circ{\Pi}_g = \textsf{Id},$$ $${\Pi}_g\circ{\amalg}_{\mathtt{N}(g)} = \textsf{Id}.$$
\[lps1\] Let $\mathcal{L}^+$ be as defined in definition \[lam1\]. Then $\mathcal{L}^+$ is a lamination of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$.
We now show that the equivalence relation $\mathcal{L}^+$ on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ satisfy properties (1) and (2) of definition \[lam\] for the local homeomorphism $\amalg$.
*Property* (1): Let $g_1, g_2$ be two points in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$, $h_1, h_2$ be two points in the intersection $\mathcal{U}_{g_1} \cap$ $\mathcal{U}_{g_2}$ and $p^{+,0}$ be the projection from $\mathcal{U}_{g^-}\times\mathcal{U}_{g^+}\times\mathbb{R}$ onto $\mathcal{U}_{g^+} \times \mathbb{R}$. We notice that if $$\begin{aligned}
p^{+,0}\circ{\amalg}_{\mathtt{N}(g_1)}(\mathtt{N}(h_1)) = p^{+,0}\circ{\amalg}_{\mathtt{N}(g_1)}(\mathtt{N}(h_2))\end{aligned}$$ then $h_1^+ = h_2^+$ and $$\begin{aligned}
\left\langle {N}(h_1)-{N}(g_1), \nu(g_1^-,h_1^+)\right\rangle = \left\langle {N}(h_2)-{N}(g_1), \nu(g_1^-,h_2^+) \right\rangle .\end{aligned}$$ Now using proposition \[14\], corollary \[nu+\] and the fact that $h_1^+ = h_2^+$ we get $$\nu^+(h_2) = c\nu^+(h_1)$$ $${N}(h_2) = {N}(h_1) + s\nu^+(h_1) + t\nu(h_1)$$ where $c, s, t \in\mathbb{R}$. Hence for $i\in\{1,2\}$ $$\nu(g_i^-,h_2^+)=\nu(g_i^-,h_1^+).$$ Finally using the fact that $$\left\langle {N}(h_2)-{N}(h_1), \nu(g_1^-,h_1^+) \right\rangle = 0$$ and $$\nu(g^-,h^+)=\frac{\nu^-(g)\boxtimes \nu^+(h)}{\langle \nu^-(g),\nu^+(h)\rangle}$$ we get $t = 0$. Therefore $$\left\langle {N}(h_2)-{N}(h_1), \nu(g_2^-,h_1^+)\right\rangle = \left\langle s\nu^+(h_1), \frac{\nu^-(g_2)\boxtimes \nu^+(h_1)}{\langle \nu^-(g_2),\nu^+(h_1)\rangle}\right\rangle = 0.$$ Hence $$\begin{aligned}
\left\langle {N}(h_1)-{N}(g_2), \nu(g_2^-,h_1^+)\right\rangle = \left\langle {N}(h_2)-{N}(g_2), \nu(g_2^-,h_2^+) \right\rangle\end{aligned}$$ and it follows that $$\begin{aligned}
p^{+,0}\circ{\amalg}_{\mathtt{N}(g_2)}(\mathtt{N}(h_1)) = p^{+,0}\circ{\amalg}_{\mathtt{N}(g_2)}(\mathtt{N}(h_2)).\end{aligned}$$ Similarly if we have $$\begin{aligned}
p^{+,0}\circ{\amalg}_{\mathtt{N}(g_2)}(\mathtt{N}(h_1)) = p^{+,0}\circ{\amalg}_{\mathtt{N}(g_2)}(\mathtt{N}(h_2))\end{aligned}$$ then $$\begin{aligned}
p^{+,0}\circ{\amalg}_{\mathtt{N}(g_1)}(\mathtt{N}(h_1)) = p^{+,0}\circ{\amalg}_{\mathtt{N}(g_1)}(\mathtt{N}(h_2)).\end{aligned}$$
*Property* (2): Let $\{\mathtt{N}(h_i)\}_{i\in\{1, 2,..., n\}}$ be a sequence of points such that for all $i\in\{1,2,...,n-1\}$ we have $$\mathtt{N}(h_{i+1})\in\mathcal{U}_{\mathtt{N}(h_i)}$$ and $$\begin{aligned}
p^{+,0}\circ{\amalg}_{\mathtt{N}(h_i)}(\mathtt{N}(h_i)) = p^{+,0}\circ{\amalg}_{\mathtt{N}(h_i)}(\mathtt{N}(h_{i+1})).\end{aligned}$$ Hence we have $h_i^+ = h_{i+1}^+$ and $$0 = \left\langle {N}(h_i)-{N}(h_i), \nu(h_i^-,h_i^+)\right\rangle = \left\langle {N}(h_{i+1})-{N}(h_i), \nu(h_i^-,h_{i+1}^+) \right\rangle .$$ Now using proposition \[14\], corollary \[nu+\] and $h_i^+ = h_{i+1}^+$ we get that $$\nu^+(h_{i+1}) = c_i\nu^+(h_i),$$ $${N}(h_{i+1}) = {N}(h_i) + s_i\nu^+(h_i) + t_i\nu(h_i)$$ for some real numbers $c_i, s_i$ and $t_i$. Hence $$\nu(h_i^-,h_i^+)=\nu(h_i^-,h_{i+1}^+).$$ Now using the fact that $$\begin{aligned}
\left\langle {N}(h_{i+1})-{N}(h_i), \nu(h_i^-,h_{i+1}^+)\right\rangle = 0\end{aligned}$$ we get $t = 0$. Hence we have $$\begin{aligned}
{\mathcal{L}}^{+}_{\mathtt{N}(h_i)} = {\mathcal{L}}^{+}_{\mathtt{N}(h_{i+1})}.\end{aligned}$$ Therefore we conclude that $$\begin{aligned}
{\mathcal{L}}^{+}_{\mathtt{N}(h_1)} = {\mathcal{L}}^{+}_{\mathtt{N}(h_n)}.\end{aligned}$$ Now we show the other direction. Let $h\in\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ such that $\mathtt{N}(h)\in{\mathcal{L}}^{+}_{\mathtt{N}(g)}$. Using proposition \[14\] we get that $h^+ = g^+$. Let $\mathcal{V}_{g^-}$ be a connected bounded open neighborhood of $g^-$ in $\partial_\infty\mathbb{H}\setminus\{g^+\}$ containing the point $h^-$ and let $\mathcal{V}_{g^+}$ be a connected open neighborhood of $g^+$ in $\partial_\infty\mathbb{H}\setminus\{g^-\}$ such that the intersection $\mathcal{V}_{g^+}\cap\mathcal{V}_{g^-}$ is empty. We denote the sets $\mathcal{V}_{g^\pm} \cap \Lambda_\infty\Gamma$ respectively by $\mathcal{U}_{g^\pm}$, the open subset of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ corresponding to the open set $\mathcal{U}_{g^-} \times \mathcal{U}_{g^+} \times \mathbb{R}$ by $\mathcal{U}_{g}$ and the open set $\mathtt{N}(\mathcal{U}_{g})$ around $\mathtt{N}(g)$ by $\mathcal{U}_{\mathtt{N}(g)}$. Now we consider the chart $\left(\mathcal{U}_{\mathtt{N}(g)}, {\amalg}_{\mathtt{N}(g)}\right)$ and notice that $$p^{+,0}\circ{\amalg}_{\mathtt{N}(g)}(\mathtt{N}(g)) = \left( g^+, 0 \right).$$ Since $\mathtt{N}(h)\in\mathcal{L}^+_{\mathtt{N}(g)}$, using the definition of $\mathcal{L}^+_{\mathtt{N}(g)}$ we get $$\begin{aligned}
\left\langle {N}(h)-{N}(g), \nu(g^-,g^+)\right\rangle = 0.\end{aligned}$$ Now using corollary \[nu+\] and the fact that $h^+ = g^+$ we obtain $$\nu(g^-,g^+)=\nu(g^-,h^+).$$ Hence $$\begin{aligned}
\left\langle {N}(h)-{N}(g), \nu(g^-,h^+)\right\rangle = 0\end{aligned}$$ and we finally have $$p^{+,0}\circ{\amalg}_{\mathtt{N}(g)}(\mathtt{N}(g)) = p^{+,0}\circ{\amalg}_{\mathtt{N}(g)}(\mathtt{N}(h)).$$ Therefore we conclude that ${\mathcal{L}}^{+}$ defines a lamination with plaque neighborhoods given by the image of the open sets $\mathcal{U}_{g^-}$ for $g^-$ in $\Lambda_\infty\Gamma\setminus\{g^+\}$.
\[lps2\] Let $\mathcal{L}^{-,0}$ be as defined in definition \[lam2\]. Then $\mathcal{L}^{-,0}$ is a lamination of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$. Moreover, it is the central lamination corresponding to the lamination $\mathcal{L}^-$.
We show that the equivalence relation $\mathcal{L}^{-,0}$ on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ satisfy properties (1) and (2) of definition \[lam\] for the local homeomorphism $\amalg$.
*Property* (1): Let $g_1, g_2$ be two points in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$, $h_1, h_2$ be two points in the intersection $\mathcal{U}_{g_1} \cap$ $\mathcal{U}_{g_2}$ and $p^{+,0}$ be the projection from $\mathcal{U}_{g^-}\times\mathcal{U}_{g^+}\times\mathbb{R}$ onto $\mathcal{U}_{g^+}\times\mathbb{R}$. We see that $$p^{-}\circ{\amalg}_{\mathtt{N}(g_1)}(\mathtt{N}(h_1)) = p^{-}\circ{\amalg}_{\mathtt{N}(g_1)}(\mathtt{N}(h_2))$$ if and only if $$p^{-}\circ{\amalg}_{\mathtt{N}(g_2)}(\mathtt{N}(h_1)) = p^{-}\circ{\amalg}_{\mathtt{N}(g_2)}(\mathtt{N}(h_2))$$ since we have $$p^{-}\circ{\amalg}_{\mathtt{N}(g_1)}(\mathtt{N}(h_1)) = h_1^- = p^{-}\circ{\amalg}_{\mathtt{N}(g_2)}(\mathtt{N}(h_1))$$ and $$p^{-}\circ{\amalg}_{\mathtt{N}(g_1)}(\mathtt{N}(h_2)) = h_2^- = p^{-}\circ{\amalg}_{\mathtt{N}(g_2)}(\mathtt{N}(h_2)).$$
*Property* (2): Let $\{\mathtt{N}(h_i)\}_{i\in\{1, 2,..., n\}}$ be a sequence of points such that for all $i\in\{1,2,...,n-1\}$ we have $$\mathtt{N}(h_{i+1})\in\mathcal{U}_{\mathtt{N}(h_i)}$$ and $$p^{-}\circ{\amalg}_{\mathtt{N}(h_i)}(\mathtt{N}(h_i)) = p^{-}\circ{\amalg}_{\mathtt{N}(h_i)}(\mathtt{N}(h_{i+1})).$$ Hence for all $i\in\{1,2,...,n-1\}$ we have $h_i^- = h_{i+1}^-$. Now using proposition \[14\] we get that $${\mathcal{L}}^{-,0}_{\mathtt{N}(h_i)} = {\mathcal{L}}^{-,0}_{\mathtt{N}(h_{i+1})}$$ for all $i$ in $\{1,2,...,n-1\}$. Hence $${\mathcal{L}}^{-,0}_{\mathtt{N}(h_{1})} = {\mathcal{L}}^{-,0}_{\mathtt{N}(h_{n})}.$$ Now we show the other direction. Let $h\in\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ such that $\mathtt{N}(h)\in{\mathcal{L}}^{-,0}_{\mathtt{N}(g)}$. Using proposition \[14\] we get that $h^- = g^-$. Let $\mathcal{V}_{g^+}$ be a connected bounded open neighborhood of $g^+$ in $\partial_\infty\mathbb{H}\setminus\{g^-\}$ containing the point $h^+$ and let $\mathcal{V}_{g^-}$ be a connected open neighborhood of $g^-$ in $\partial_\infty\mathbb{H}\setminus\{g^+\}$ such that $\mathcal{V}_{g^+}\cap\mathcal{V}_{g^-}$ is empty. We denote the sets $\mathcal{V}_{g^\pm} \cap \Lambda_\infty\Gamma$ respectively by $\mathcal{U}_{g^\pm}$, the open subset of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ corresponding to the open set $\mathcal{U}_{g^-} \times \mathcal{U}_{g^+} \times \mathbb{R}$ by $\mathcal{U}_{g}$ and the open set $\mathtt{N}(\mathcal{U}_{g})$ around $\mathtt{N}(g)$ by $\mathcal{U}_{\mathtt{N}(g)}$. Now we consider the chart $\left(\mathcal{U}_{\mathtt{N}(g)}, {\amalg}_{\mathtt{N}(g)}\right)$ and notice that $$p^{-}\circ{\amalg}_{\mathtt{N}(g)}(\mathtt{N}(g)) = g^- = h^- = p^{-}\circ{\amalg}_{\mathtt{N}(g)}(\mathtt{N}(h)).$$ Therefore we conclude that ${\mathcal{L}}^{-,0}$ defines a lamination with plaque neighborhoods given by the image of the open sets $\mathcal{U}_{g^+}\times\mathbb{R}$ for $g^+$ in $\Lambda_\infty\Gamma\setminus\{g^+\}$.
Now the fact that ${\mathcal{L}}^{-,0}$ is the central lamination corresponding to the lamination $\mathcal{L}^-$ follows from definition \[lam2\].
The laminations $({\mathcal{L}}^{+},{\mathcal{L}}^{-,0})$ and $({\mathcal{L}}^{-},{\mathcal{L}}^{+,0})$ define a local product structure on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$.
Using proposition \[lps\], \[lps1\] and \[lps2\] we get that $({\mathcal{L}}^{+},{\mathcal{L}}^{-,0})$ defines a local product structure. In a similar way one can show that $({\mathcal{L}}^{-},{\mathcal{L}}^{+,0})$ also defines a local product structure.
The laminations are equivariant under the action of $\Gamma$.
Let $Z$ be in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ such that $Z = \mathtt{N}(g)$ for some $g\in\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ and $W\in{\mathcal{L}}_Z^{+}$. Therefore there exist real numbers $s_1$, $s_2$ such that $$W = (\tilde{{N}}(g) + s_1 \nu^+(g), \nu(g) + s_2 \nu^+(g)).$$ Now for all $\gamma$ in $\Gamma$ we get, $$\begin{aligned}
\gamma . Z & = \gamma . \mathtt{N}(g)\\
&= \mathtt{N}(\gamma . g)\end{aligned}$$ and $$\begin{aligned}
\gamma . W &= \gamma . (\tilde{{N}}(g) + s_1 \nu^+(g), \nu(g) + s_2 \nu^+(g))\\
&= (\gamma . \tilde{{N}}(g) + s_1 . \gamma . \nu^+(g), \gamma . \nu(g) + s_2 . \gamma . \nu^+(g))\\
&= (\tilde{{N}}(\gamma . g) + s_1 \nu^+(\gamma . g), \nu(\gamma . g) + s_2 \nu^+(\gamma . g)).\end{aligned}$$ Therefore $\gamma . W\in\tilde{\mathcal{L}}^+_{\gamma . Z}$ and $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ is invariant under the action of $\Gamma$ implies that $\gamma . W\in{\mathcal{L}}^+_{\gamma . Z}$. Hence we get that for all $\gamma$ in $\Gamma$, $${\mathcal{L}}^+_{\gamma . Z} = \gamma . {\mathcal{L}}^+_Z.$$ Similarly one can show that for all $\gamma$ in $\Gamma$, $${\mathcal{L}}^-_{\gamma . Z} = \gamma . {\mathcal{L}}^-_Z.$$
The laminations are equivariant under the geodesic flow.
Let $Z$ be in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ such that $Z = \mathtt{N}(g)$ for some $g\in\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ and $W\in{\mathcal{L}}_Z^{+}$. Therefore there exist real numbers $s_1$, $s_2$ such that $$W = (\tilde{{N}}(g) + s_1 \nu^+(g), \nu(g) + s_2 \nu^+(g)).$$ We have for all real number $t$, $$\begin{aligned}
\tilde{\Phi}_t Z &= \tilde{\Phi}_t \mathtt{N}(g)\\
&= ({N}(g) + t\nu(g), \nu(g))\end{aligned}$$ and $$\begin{aligned}
\tilde{\Phi}_t W &= \tilde{\Phi}_t ({N}(g) + s_1 \nu^+(g), \nu(g) + s_2 \nu^+(g))\\
&= ({N}(g) + s_1 \nu^+(g) + t . (\nu(g) + s_2 \nu^+(g)), \nu(g) + s_2 \nu^+(g))\\
&= (({N}(g) + t\nu(g)) + (s_1 + ts_2) \nu^+(g), \nu(g) + s_2 \nu^+(g)).\end{aligned}$$ Therefore for all real number $t$ we have $\tilde{\Phi}_t . W\in\tilde{\mathcal{L}}^+_{\tilde{\Phi}_t . Z}$ and $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ is invariant under the geodesic flow implies that $\tilde{\Phi}_t . W\in{\mathcal{L}}^+_{\tilde{\Phi}_t . Z}$. Hence we get that for all real number $t$, $$\begin{aligned}
{\mathcal{L}}^+_{\tilde{\Phi}_t . Z} = \tilde{\Phi}_t . {\mathcal{L}}^+_Z.\end{aligned}$$ Similarly one can show that for all real number $t$, $$\begin{aligned}
{\mathcal{L}}^-_{\tilde{\Phi}_t . Z} = \tilde{\Phi}_t . {\mathcal{L}}^-_Z.\end{aligned}$$
\[lem\] We denote the projection of $\mathcal{L}^{\pm},\mathcal{L}^{\pm,0}$ on the space $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$ by $\underline{\mathcal{L}}^{\pm}, \underline{\mathcal{L}}^{\pm,0}$ respectively.
Now we define the notion of a leaf lift. The leaf lift is a map from the leaves of the lamination through a point, to the tangent space of $\mathsf{U}\mathbb{A}$ at that point. We will use this leaf lift to compare distance between the metric $\tilde{d}$ and the norm on the tangent space on any point of the leaves. We define the leaf lift as follows:\
The $\textit{positive}$ $\textit{leaf}$ $\textit{lift}$ is the map, $$\begin{aligned}
i^+_{\mathtt{N}(g)} &: \tilde{\mathcal{L}}^{+}_{\mathtt{N}(g)} \longrightarrow \mathsf{T}_{\mathtt{N}(g)}\mathsf{U}\mathbb{A}\\
\notag({N}(g) + s_1 \nu^+(g) &,\nu(g) + s_2 \nu^+(g))
\longmapsto (s_1 \nu^+(g), s_2 \nu^+(g)).\end{aligned}$$ and the $\textit{negative}$ $\textit{leaf}$ $\textit{lift}$ is the map, $$\begin{aligned}
i^-_{\mathtt{N}(g)} &: \tilde{\mathcal{L}}^{-}_{\mathtt{N}(g)} \longrightarrow \mathsf{T}_{\mathtt{N}(g)}\mathsf{U}\mathbb{A}\\
\notag({N}(g) + s_1 \nu^-(g) &,\nu(g) + s_2 \nu^-(g))
\longmapsto (s_1 \nu^-(g), s_2 \nu^-(g)).\end{aligned}$$ where we identify $\mathsf{T}_{\mathtt{N}(g)}\mathsf{U}\mathbb{A}$ with $\mathsf{T}_{{N}(g)}\mathbb{A} \times \mathsf{T}_{\nu(g)}\mathsf{S}^{1}$.
Contraction Properties
----------------------
In this subsection we will prove that the leaves denoted by $\mathcal{L}^+$ contracts in the forward direction of the geodesic flow and the leaves denoted by $\mathcal{L}^-$ contracts in the backward direction of the geodesic flow. We will prove it only for the forward direction of the flow. The other case will follow similarly. We start with the following construction whose raison d’$\hat{\text{e}}$tre would be apparent in proposition \[lip\].
\[prelip\] There exists a $\Gamma$-invariant map from $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ into the space of euclidean metrics on $\mathbb{R}^3\times\mathbb{R}^3$ sending $Z$ to $\|.\|_Z$ such that for all positive integer $n$, there exists a positive real number $t_n$ satisfying the following property: if $t>t_n$, $Z\in\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ and $W\in\tilde{\mathcal{L}}^+_\text{Z}$ then $$\begin{aligned}
{\| i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t W) - i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t Z) \|}_{\tilde{\Phi}_t Z} \leqslant \frac{1}{2^n}{\| i^+_Z(W) - i^+_Z(Z)\|}_Z.\end{aligned}$$
Let $\langle | \rangle_{\mathtt{N}(g)}$ be a positive definite bilinear form on the tangent space $\mathsf{T}_{\mathtt{N}(g)}(\mathbb{A} \times \mathbb{V})$ satisfying the following properties, $$\begin{aligned}
&\text{1. }\langle (\nu^{\alpha}(g),0)| (\nu^{\beta}(g),0) \rangle_{\mathtt{N}(g)} = \langle (0,\nu^{\alpha}(g))| (0,\nu^{\beta}(g)) \rangle_{\mathtt{N}(g)} = {\delta}_{\alpha\beta},\\
&\text{2. }\langle (\nu^{\alpha}(g),0)| (0,\nu^{\beta}(g)) \rangle_{\mathtt{N}(g)} = \langle (0,\nu^{\alpha}(g))| (\nu^{\beta}(g),0) \rangle_{\mathtt{N}(g)} = 0.\end{aligned}$$ where ${\delta}_{\alpha\beta}$ is the dirac delta function with $\alpha,\beta$ in $\{., +, -\}$. We define the map $\|.\|$ as follows, $$\begin{aligned}
\| X \|_{\mathtt{N}(g)} {\mathrel{\mathop:}=}\sqrt{\langle X | X \rangle}_{\mathtt{N}(g)},\end{aligned}$$ where $X$ is in $\mathsf{T}_{\mathtt{N}(g)}(\mathbb{A} \times \mathbb{V})$. Now from equation \[nu2\], equation \[limit2\] and theorem \[N\] we get that $\|.\|$ is $\Gamma$-invariant, that is, $$\begin{aligned}
\| \gamma X \|_{\gamma\mathtt{N}(g)} = \| X \|_{\mathtt{N}(g)}.\end{aligned}$$ Let $Z = \mathtt{N}(g)$ be in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ and $W\in\tilde{\mathcal{L}}^+_\text{Z}$. Therefore there exists real numbers $s_1$ and $s_2$ such that $$W = ({N}(g) + s_1\nu^+(g) ,\nu(g) + s_2\nu^+(g) ).$$ Hence the norm is $$\begin{aligned}
\label{15}
&{\| i^+_Z(W) - i^+_Z(Z)\|}_Z = {\| (s_1\nu^+(g), s_2\nu^+(g)) \|}_Z = \sqrt{s_1^2 + s_2^2}.\end{aligned}$$ We note that $\tilde{\Phi}_t Z = ({N}(g) + t\nu(g),\nu(g))$ and using theorem \[N\] we get that there exists a positive real number $t_1$ such that $${N}(g) + t\nu(g)={N}(\tilde{\phi}_{t_1}g).$$ Moreover $t$ and $t_1$ are related by the following formula, $$\begin{aligned}
t = \int\limits_{0}^{t_1} {f}(\tilde{\phi}_s g)ds.\end{aligned}$$ Therefore we have $$\begin{aligned}
\| i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t W) &- i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t Z)\|_{\tilde{\Phi}_t Z} = {\| ((s_1 + ts_2)\nu^+(g), s_2\nu^+(g)) \|}_{\tilde{\Phi}_t Z}\\
&= {\| ((s_1 + ts_2)\nu^+(g), s_2\nu^+(g)) \|}_{\mathtt{N}(\phi_{t_1}g)}\\
&= {\sqrt{( s_1 + ts_2 )^2 + s_2^2}} \text{ . } {\|(\nu^+(g),0) \|}_{\mathtt{N}(\phi_{t_1}g)}\\
&= {\sqrt{( s_1 + ts_2 )^2 + s_2^2}} \text{ . } {\| e^{-t_1}(\nu^+(\phi_{t_1}g),0) \|}_{\mathtt{N}(\phi_{t_1}g)}\end{aligned}$$ Hence the norm is $$\begin{aligned}
\label{16}
\| i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t W) - i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t Z)\|_{\tilde{\Phi}_t Z} \notag &= {\sqrt{( s_1 + ts_2 )^2 + s_2^2}}\text{ . }e^{-t_1}\\
&\leqslant \sqrt{2}\sqrt{s_1^2 + s_2^2}(1+t)e^{-t_1}.\end{aligned}$$ We also know that ${\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}} \Sigma}$ is compact. Hence ${f}$ is bounded on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$. Therefore there exists a constant $c_1$ such that $$t = \int\limits_{0}^{t_1} {f}(\tilde{\phi}_s g)ds \leqslant \int\limits_{0}^{t_1} c_1 ds = c_1t_1.$$ We choose a constant $c$ bigger than $\max\{1,2c_1\}$ and get that $$\begin{aligned}
\label{17}
(1 + t) e^{-t_1} \leqslant c e^{-\frac{t}{2c_1}}.\end{aligned}$$ Now by combining equation \[15\], inequalities \[16\] and \[17\] we get that $$\begin{aligned}
{\| i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t W) - i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t Z) \|}_{\tilde{\Phi}_t Z} \leqslant \sqrt{2}c e^{-\frac{t}{2c_1}}{\| i^+_Z(W) - i^+_Z(Z)\|}_Z.\end{aligned}$$ Hence for all positive integer $n$, there exists $t_n\in\mathbb{R}$ such that if $t>t_n$, $Z\in\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ and $W\in\mathcal{L}^+_\text{Z}$ then $$\begin{aligned}
{\| i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t W) - i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t Z) \|}_{\tilde{\Phi}_t Z} \leqslant \frac{1}{2^n}{\| i^+_Z(W) - i^+_Z(Z)\|}_Z.\end{aligned}$$
\[lip\] Let $d$ be a $\Gamma$-invariant distance on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ which is locally bilipschitz equivalent to an euclidean distance and let $\|.\|$ be the $\Gamma$-invariant map from $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ to the space of euclidean metrics on $\mathbb{R}^3 \times \mathbb{R}^3$ as constructed in the proof of proposition \[prelip\]. There exist positive constants $K$ and $\alpha$ such that for any $Z\in\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ and for any $W\in\mathcal{L}^+_{Z}$, the following statements are true, $$\begin{aligned}
&\text{1. If }d{(W,Z)} \leqslant \alpha, \text{ then }{\| i^+_Z(Z) - i^+_Z(W) \|}_{Z} \leqslant Kd{(W,Z)},\\
&\text{2. If } {\| i^+_Z(Z) - i^+_Z(W) \|}_{Z} \leqslant \alpha, \text{ then } d{(W,Z)} \leqslant K{\| i^+_Z(Z) - i^+_Z(W) \|}_{Z}.\end{aligned}$$
Since $\Gamma$ acts cocompactly on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ and both $d$ and $\|.\|$ are $\Gamma$-invariant, it suffices to prove the above assertion for $Z$ in a compact subset $D$ of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$, where $D$ is the closure of a suitably chosen fundamental domain.\
We can define an euclidean distance $d_Z$ on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$, uniquely using the euclidean metric $\|.\|_Z$ on $\mathbb{R}^3 \times \mathbb{R}^3$, by taking the embedding of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ in $\mathbb{A} \times \mathbb{R}^3$. We notice that for any $Z$ in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ and for any $W$ in $\mathcal{L}^+_{Z}$, $d_Z(W,Z)$ is equal to $\|i^+_Z(W)-i^+_Z(Z)\|_Z$. Now, any two euclidean distances are bilipschitz equivalent with each other and by our hypothesis, $d$ is locally bilipschitz equivant to an euclidean distance. Therefore, in particular, $d$ is locally bilipschitz equivalent with $d_Z$ for $Z$ in $D$, that is, there exist constants $K_Z$ depending on $Z$, and open sets $U_Z$ around $Z$, such that the distance $d_Z$ and $d$ are $K_Z$ bilipschitz equivalent with each other on $U_Z$.\
Let $C_{(X,Y)}$ for any $X$ and $Y$ in $D$, be a constant such that the distance $d_X$ and $d_Y$ are $C_{(X,Y)}$ bilipschitz equivalent with each other. It follows from the construction of the norm $\|.\|$, as done in proposition \[prelip\], that we can choose the constants $C_{(X,Y)}$ in such a way that $C_{(X,Y)}$ vary continuously on $(X,Y)$. As $D$ is compact it follows that $C_{(X,Y)}$ is bounded above by some constant $C$. Hence, for all $X$ and $Y$ in $D$, $d_X$ and $d_Y$ are $C$ bilipschitz equivalent with each other.\
Now, we consider the open cover of $D$ by the open sets $U_Z$. As $D$ is compact, there exist points $Z_1, Z_2,..., Z_n$ in $D$ such that $U_{Z_1}, U_{Z_2},..., U_{Z_n}$ covers $D$. Let $\beta$ be the Lebesgue number of this cover for the distance $d$ and $K_0$ be the maximum of $K_{Z_1}, K_{Z_2},..., K_{Z_n}$. Therefore, for any $Z$ in $D$, the open ball of radius $\beta$ around $Z$ for the metric $d$, denoted by $B_d(Z,\beta)$, lies inside $U_{Z_j}$ for some $j$ in $\lbrace 1,2,...,n \rbrace$. Hence, $d$ and $d_{Z_j}$ are $K_0$ bilipschitz equivalent with each other on $B_d(Z,\beta)$. As $d_Z$ and $d_{Z_j}$ are $C$ bilipschitz equivalent with each other, it follows that $d$ and $d_Z$ are $CK_0$ bilipschitz equivalent with each other on $B_d(Z,\beta)$. Moreover, we note that the constants $\beta$, $C$, $K_0$ and hence also $CK_0$, does not depend on $Z$. Therefore, $d$ and $d_Z$ are $CK_0$ bilipschitz equivalent with each other on $B_d(Z,\beta)$, for all $Z$ in $D$.\
As any two distances $d_X$ and $d_Y$, for all $X$, $Y$ in $D$ are $C$ bilipschitz equivalent with each other. Without loss of generality we can choose a point $X$ in $D$ and consider the distance $d_X$. The note that the set $\{B_d(Z,\beta): Z \in D\}$ is an open cover of $D$. Let $\beta_1$ be a Lebesgue number for this cover for the metric space $(D,d_X)$. Therefore, the open ball $B_{d_X}(Y_1,\beta_1)$ for any $Y_1$ in $D$, lies inside an open ball $B_d(Y_2,\beta)$ for some point $Y_2$ in $D$. Now, as $d$ and $d_Z$ are $CK_0$ bilipschitz equivalent with each other on the ball $B_d(Z,\beta)$ for all $Z$ in $D$, it follows that $d$ and $d_X$ are $CK_0$ bilipschitz equivalent with each other on the ball $B_{d_X}(Y_2,\beta_1)$. As $Y_2$ was chosen arbitrarily we have that $d$ and $d_X$ are $CK_0$ bilipschitz equivalent with each other on the ball $B_{d_X}(Y,\beta_1)$, for all $Y$ in $D$.\
Now, we know that $d_X$ and $d_Z$ are $C$ bilipschitz equivalent with each other. Therefore we get that $d$ and $d_Z$ are $CK_0$ bilipschitz equivalent with each other on the ball $B_{d_Z}(Y,\frac{\beta_1}{C})$, for all $Y$ in $D$. In particular one has, $d$ and $d_Z$ are $CK_0$ bilipschitz equivalent with each other on the ball $B_{d_Z}(Z,\frac{\beta_1}{C})$. Finally, set $\alpha$ to be $\min \lbrace \frac{\beta_1}{C}, \beta\rbrace$ and $K$ to be $CK_0$ to get that for any $Z$ in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ and $W$ in $\mathcal{L}^+_\text{Z}$ we have, $$\begin{aligned}
&\text{1. If }d{(W,Z)} \leqslant \alpha, \text{ then }{\| i^+_Z(Z) - i^+_Z(W) \|}_{Z} \leqslant Kd{(W,Z)},\\
&\text{2. If } {\| i^+_Z(Z) - i^+_Z(W) \|}_{Z} \leqslant \alpha, \text{ then } d{(W,Z)} \leqslant K{\| i^+_Z(Z) - i^+_Z(W) \|}_{Z}.\end{aligned}$$.
Let $\mathcal{L}^{\pm}$ be two laminations on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ as defined in definitions \[lam1\], \[lam2\] and let $\tilde{d}$ be the $\Gamma$ invariant metric, as defined in definition \[dist\]. Under these assumptions, for the metric $\tilde{d}$ on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ we have that $$\begin{aligned}
&\text{1. }\mathcal{L}^{+} \text{ is contracted in the forward direction of the geodesic flow, and, }\\
&\text{2. }\mathcal{L}^{-} \text{ is contracted in the backward direction of the geodesic flow.}\end{aligned}$$
Let $\|.\|$ be the $\Gamma$-invariant map from $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ to the space of euclidean metrics on $\mathbb{R}^3 \times \mathbb{R}^3$ as constructed in the proof of proposition \[prelip\] and let $K$ and $\alpha$ be as in the proposition \[lip\] for the distance $\tilde{d}$. We choose a positive integer $n$ such that $$\begin{aligned}
\frac{K}{2^n} < 1 \text{ , } \frac{K^2}{2^n} \leqslant \frac{1}{2}.\end{aligned}$$ Let $t_n$ be the constant as in proposition \[prelip\] for our chosen $n$. Also let $Z$ be in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ and $W$ be in $\mathcal{L}^+_{Z}$, so that $\tilde{d}(W,Z) \leqslant \alpha$. Then using proposition \[lip\] we get $$\begin{aligned}
{\| i^+_Z(W) - i^+_Z(Z) \|}_{Z} \leqslant K\tilde{d}{(W,Z)}.\end{aligned}$$ Furthermore, using proposition \[prelip\] we get for all $t>t_n$ that $$\begin{aligned}
&{\| i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t W) - i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t Z) \|}_{\tilde{\Phi}_t Z} \leqslant \frac{1}{2^n}{\| i^+_Z(W) - i^+_Z(Z)\|}_Z.\end{aligned}$$ It follows that $$\begin{aligned}
{\| i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t W) - i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t Z) \|}_{\tilde{\Phi}_t Z} \leqslant \frac{K\alpha}{2^n} \leqslant \alpha.\end{aligned}$$ Hence again using proposition \[lip\] we have $$\begin{aligned}
\tilde{d}(\tilde{\Phi}_t W, \tilde{\Phi}_t Z) \leqslant K {\| i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t W) - i^+_{\tilde{\Phi}_t Z}(\tilde{\Phi}_t Z) \|}_{\tilde{\Phi}_t Z}.\end{aligned}$$ Combining the above inequalities, for all $t>t_n$ we get $$\begin{aligned}
\label{convergence}
\tilde{d}(\tilde{\Phi}_t W, \tilde{\Phi}_t Z) &\leqslant \frac{K^2}{2^n} \tilde{d}{(W,Z)} \leqslant \frac{1}{2}\ \tilde{d}{(W,Z)}.\end{aligned}$$ Hence $\mathcal{L}^+$ is contracted in the forward direction of the geodesic flow. The proof of the contraction of $\mathcal{L}^-$ follows similarly.
Metric Anosov structure on the quotient
---------------------------------------
Let us now consider what happens in the quotient, that is, $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$. Let $Z$ be in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ and $\epsilon$ be a positive real number. We define, $$\begin{aligned}
\mathcal{L}^{\pm}_\epsilon(Z) {\mathrel{\mathop:}=}\mathcal{L}^{\pm}_Z \cap B_{\tilde{d}}(Z, \epsilon),\end{aligned}$$ and $$\begin{aligned}
\mathcal{K}_\epsilon(Z) {\mathrel{\mathop:}=}{\Pi}_{Z} \left(\mathcal{L}^+_\epsilon(Z)\times \mathcal{L}^-_\epsilon(Z) \times (-\epsilon,\epsilon)\right) \subset \mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}\end{aligned}$$ where ${\Pi}_{Z}$ is the local product structure at $Z$ defined by the stable and unstable leaves.
We know that there exists a positive real number $\epsilon_0$ such that for any non identity element $\gamma$ of $\Gamma$ and for $Z$ in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ we have, $$\begin{aligned}
\gamma (\mathcal{K}_{\epsilon_0}(Z)) \cap \mathcal{K}_{\epsilon_0}(Z) = \emptyset .\end{aligned}$$
Let us fix $\alpha$ as in proposition \[lip\] and let $\epsilon_1$ be from the open interval $\left(0,\min\left\lbrace\alpha,\frac{\epsilon_0}{2}\right\rbrace\right)$. Now let $z$ be any point of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$ and let $Z$ be a point in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ in the preimage of $z$. Our choice of $\epsilon_1$ gives us that the inequality \[convergence\] holds for the geodesic flow on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ for the points in the chart $\mathcal{K}_{\epsilon_1}(Z)$. Hence the inequality \[convergence\] also holds for the geodesic flow on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$ for points in the chart which is in the projection of $\mathcal{K}_{\epsilon_1}(Z)$.
Therefore $\underline{\mathcal{L}}^+$, the projection of $\mathcal{L}^{+}$ in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$, is contracted in the forward direction of the geodesic flow on $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$. A similar proof holds for $\underline{\mathcal{L}}^-$, the projection of $\mathcal{L}^{-}$ in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$, too.
Anosov representations
======================
In this section we define the notion of an Anosov representation in the context of the non-semisimple Lie group $\mathsf{G} {\mathrel{\mathop:}=}\mathsf{SO}^0(2,1)\ltimes\mathbb{R}^3$.
Pseudo-Parabolic subgroups
--------------------------
Let $\mathbb{X}$ be the space of all affine null planes. We observe that $\mathsf{G}$ acts transitively on $\mathbb{X}$. Hence for all $P\in\mathbb{X}$ we have $$\mathbb{X} = \mathsf{G}.P \cong \mathsf{G}/\mathsf{Stab}_{\mathsf{G}}(P).$$
\[levi\] If $P\in\mathbb{X}$ then we define $$\mathsf{P}_P{\mathrel{\mathop:}=}\mathsf{Stab}_{\mathsf{G}}(P).$$ We call $\mathsf{P}_P$ a $\textit{pseudo-parabolic}$ subgroup of $\mathsf{G}$.
Let $\mathsf{V}(P)$ denote the vector space underlying a null plane $P$, let $v_0{\mathrel{\mathop:}=}(1,0,0)^t$ and $v_0^\pm{\mathrel{\mathop:}=}(0,\pm1,1)^t$ and let $\mathcal{C}$ be the upper half of $\mathsf{S}^0\backslash\{0\}$. Now we consider the space $$\mathcal{N}{\mathrel{\mathop:}=}\{(P_1,P_2)\mid P_1,P_2\in\mathbb{X}, \mathsf{V}(P_1)\neq \mathsf{V}(P_2)\}$$ and define the following map $$\begin{aligned}
v : \mathcal{N} &\longrightarrow \mathsf{S}^{1}\\
(P_1,P_2) &\longmapsto v(P_1,P_2)\end{aligned}$$ where $v(P_1,P_2)\in\mathsf{V}(P_1)\cap\mathsf{V}(P_2)\cap\mathsf{S}^{1}$ is such that if $v(Q_1)\in\mathsf{V}(Q_1)\cap\mathcal{C}$ and $v(Q_2)\in\mathsf{V}(Q_2)\cap\mathcal{C}$ then $(v(Q_1),v(Q_1,Q_2),v(Q_2))$ gives the same orientation as $(v_0^+,v_0,v_0^-)$. We observe that $$v(P_1,P_2)=-v(P_2,P_1).$$
\[open\] The space $\mathcal{N}$ is the unique open $\mathsf{G}$ orbit in $\mathbb{X}\times\mathbb{X}$ for the diagonal action of $\mathsf{G}$ on $\mathbb{X}\times\mathbb{X}$.
We start by observing that $\mathcal{N}$ is open and dense in $\mathbb{X}\times\mathbb{X}$. Now let $(P_1,P_2)$ and $(Q_1,Q_2)$ be two arbitrary points in $\mathcal{N}$. We consider the vector $v(P_1,P_2)\in\mathsf{S}^{1}$ corresponding to the point $(P_1,P_2)$ and the vector $v(Q_1,Q_2)\in\mathsf{S}^{1}$ corresponding to the point $(Q_1,Q_2)$. Now as $\mathsf{SO}^0(2,1)$ acts transitively on $\mathsf{S}^{1}$ we get that there exist $g\in\mathsf{SO}^0(2,1)$ such that $$v(Q_1,Q_2)=g.v(P_1,P_2).$$ We choose $X(Q_1,Q_2)\in Q_1\cap Q_2$ and $X(P_1,P_2)\in P_1\cap P_2$ and observe that $$(e,X(Q_1,Q_2)-O)\circ(g,0)\circ(e,X(P_1,P_2)-O)^{-1}.P_1=Q_1,$$ $$(e,X(Q_1,Q_2)-O)\circ(g,0)\circ(e,X(P_1,P_2)-O)^{-1}.P_2=Q_2,$$ where $e$ is the identity element in $\mathsf{SO}^0(2,1)$. Therefore $\mathcal{N}$ is an open $\mathsf{G}$ orbit in $\mathbb{X}\times\mathbb{X}$. Now as $\mathbb{X}\times\mathbb{X}$ is connected the result follows.
Let $\mathsf{N}$ be the space of oriented space like affine lines. We think of $\mathsf{N}$ as the space $\mathsf{U}\mathbb{A}/\sim$ where $(X,v)\sim(X_1,v_1)$ if and only if $(X_1,v_1)=\tilde{\Phi}_t(X,v)$ for some $t\in\mathbb{R}$. We denote the equivalence class of $(X,v)$ by $[(X,v)]$. Now let us consider the following map $$\begin{aligned}
\imath^\prime : \mathcal{N} &\longrightarrow \mathsf{N}\\
(P_1,P_2) &\longmapsto [(X(P_1,P_2),v(P_1,P_2))]\end{aligned}$$ where $X(P_1,P_2)$ is any point in $P_1\cap P_2$. We observe that $\imath^\prime$ gives a $\mathsf{G}$ equivariant map.
Let us denote the plane passing through $X$ with underlying vector space generated by the vectors $w_1$ and $w_2$ by $P_{X,w_1,w_2}$. Now we consider another map $$\begin{aligned}
\imath : \mathsf{U}\mathbb{A} &\longrightarrow \mathcal{N}\\
(X,v) &\longmapsto (P_{X,v,v^+},P_{X,v,v^-})\end{aligned}$$ where $v^\pm\in\mathcal{C}$ such that $\langle v^\pm\mid v\rangle = 0$ and $(v^+,v,v^-)$ gives the same orientation as $(v_0^+,v_0,v_0^-)$. We observe that $\imath$ is a $\mathsf{G}$ equivariant map. Now as $P_{X+tv,v,v^+}=P_{X,v,v^+}$ and $P_{X+tv,v,v^-}=P_{X,v,v^-}$ we get that the map $\imath$ gives rise to a map, which we again denote by $\imath$, $$\imath :\mathsf{N} \longrightarrow \mathcal{N}.$$ Moreover, we observe that $\imath\circ\imath^\prime=\mathsf{Id}$ and $\imath^\prime\circ\imath=\mathsf{Id}$.
Geometric Anosov structure
--------------------------
Geometric Anosov structures were first intoduced by Labourie in [@orilab]. In this subsection we give an appropriate definition of geometric Anosov property and show that $(\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M},\mathcal{L})$ admits a geometric Anosov structure.
Let $(P^+,P^-)\in\mathcal{N}$ such that $P^+{\mathrel{\mathop:}=}P_{O,v_0,v_0^+}$ and $P^-{\mathrel{\mathop:}=}P_{O,v_0,v_0^-}$. We denote $\mathsf{Stab}_\mathsf{G}(P^\pm)$ respectively by $\mathsf{P}^\pm$. We note that the pair $\mathbb{X}^\pm{\mathrel{\mathop:}=}\mathsf{G}/\mathsf{P}^\pm$ gives a pair of continuous foliations on the space $\mathsf{N}$ whose tangential distributions $\mathsf{E}^\pm$ satisfy $$\mathsf{TN} = \mathsf{E}^+\oplus\mathsf{E}^-.$$
We say that a vector bundle $\mathsf{E}$ over a compact topological space whose total space is equipped with a flow $\{\varphi_t\}_{t\in\mathbb{R}}$ of bundle automorphisms is $\textit{contracted}$ by the flow as $t\to\infty$ if for any metric $\|.\|$ on $\mathsf{E}$, there exists positive constants $t_0$, $A$ and $c$ such that for all $t>t_0$ and for all $v$ in $\mathsf{E}$ we have $$\begin{aligned}
\|\varphi_t(v)\| \leqslant Ae^{-ct}\|v\|.\end{aligned}$$
Let $\mathcal{L}$ denote the orbit foliation of $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$ under the flow $\Phi$. We say that $(\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M},\mathcal{L})$ admits a $\textit{geometric}$ $(\mathsf{N}, \mathsf{G})$-$\textit{Anosov structure}$ if there exist a map $$F: \widetilde{\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}}\longrightarrow\mathsf{N}$$ such that the following holds:
1. For all $\gamma\in\Gamma$ we have $F\circ\gamma=\gamma\circ F$,
2. For all $t\in\mathbb{R}$ we have $F\circ\tilde{\Phi}_t=F$,
3. By the flow invariance, the bundles $F^\pm{\mathrel{\mathop:}=}F^*\mathsf{E}^\pm$ are equipped with a parallel transport along the orbits of $\tilde{\Phi}$. The bundle $F^+$ gets contracted by the lift of the flow $\tilde{\Phi}_t$ as $t\to\infty$ and $F^-$ gets contracted by the lift of the flow $\tilde{\Phi}_t$ as $t\to-\infty$.
Let us define the map $F$ as follows: $$\begin{aligned}
F: \widetilde{\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}} &\longrightarrow\mathsf{N}\\
(X,v) &\longmapsto [(X,v)]\end{aligned}$$ We note that the map $F$ is clearly $\Gamma$-equivariant and is also invariant under the flow $\tilde{\Phi}$. Now we observe that $$\mathsf{T}_{\imath\left([(X,v)]\right)}\mathsf{G}/\mathsf{P}^-\cong\mathbb{R}.v^+\oplus\mathbb{R}.v^+$$ and $$\mathsf{T}_{\imath\left([(X,v)]\right)}\mathsf{G}/\mathsf{P}^+\cong\mathbb{R}.v^-\oplus\mathbb{R}.v^-$$ where $v^+,v^-\in\mathcal{C}$ such that $\langle v^\pm\mid v\rangle=0$ and $(v^+,v,v^-)$ gives the same orientation as $(v_0^+,v_0,v_0^-)$.
Now using proposition \[prelip\] we notice that $F^+$ gets contracted by the lift of the flow $\tilde{\Phi}_t$ as $t\to\infty$ and $F^-$ gets contracted by the lift of the flow $\tilde{\Phi}_t$ as $t\to-\infty$. Moreover, as $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathsf{M}$ is compact we have that the convergence is independent of the choice of the metric.
[99]{}
BRIDGEMAN, M., CANARY, R., LABOURIE, F. and SAMBARINO, A.: *“The pressure metric for convex homomorphisms."* arXiv preprint arXiv:1301.7459 (2013). CHARETTE, V., GOLDMAN, W. M. and JONES, C. A.: *“Recurrent Geodesics in Flat Lorentz 3-Manifolds."* (2001). DANCIGER, J, GUERITAUD, F. and KASSEL, F.: *“Geometry and Topology of Complete Lorentz SpaceTimes of Constant Curvature."* (2013). DRUMM, T.: *“Fundamental polyhedra for Margulis space-times"*, Doctoral Dissertation, University of Maryland (1990). FRIED, D. and GOLDMAN, W. M.: *“Three-dimensional affine crystallographic groups."* Adv. in Math. 47, 1-49, (1983). GOLDMAN, W. M. and LABOURIE, F.: *“Geodesics in Margulis spacetimes."* Ergodic Theory and Dynamical Systems, 32, 643-651, (2012). GOLDMAN, W. M., LABOURIE, F. and MARGULIS, G.: *“Proper affine actions and geodesic flows of hyperbolic surfaces."* Annals of mathematics 170.3, 1051-1083, (2009). GUICHARD, O. and WIENHARD, A.: *“Anosov representations: Domains of discontinuity and applications."* Inventiones Mathematicae, Volume 190, Issue 2, 357-438, (2012). LABOURIE, F.: *“Anosov Flows, Surface Groups and Curves in Projective Space."* Inventiones Mathematicae, Volume 165, Issue 1, 51-114, (2006). MARGULIS, G. A.: *“Free completely discontinuous groups of affine transformations."* Dokl. Akad. Nauk SSSR 272, 785-788, (1983). MARGULIS, G. A.: *“Complete affine locally flat manifolds with a free fundamental group."* J. Soviet Math. 134, 129-134, (1987). MESS, G.: *“Lorentz spacetimes of constant curvature."* Geom. Dedicata 126, 3-45, (2007). MILNOR.J: *“On fundamental groups of complete affinely flat manifolds."* Adv. in Math.25, 178-187, (1977).
[^1]: The research leading to these results has received funding from the European Research Council under the [*European Community*]{}’s seventh Framework Programme (FP7/2007-2013)/ERC [*grant agreement*]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this contribution, we introduce an efficient method for solving the optimal control problem for an unconstrained nonlinear switched system with an arbitrary cost function. We assume that the sequence of the switching modes are given but the switching time in between consecutive modes remains to be optimized. The proposed method uses a two-stage approach as introduced by [@xu04] where the original optimal control problem is transcribed into an equivalent problem parametrized by the switching times and the optimal control policy is obtained based on the solution of a two-point boundary value differential equation. The main contribution of this paper is to use a Sequential Linear Quadratic approach to synthesize the optimal controller instead of solving a boundary value problem. The proposed method is numerically more efficient and scales very well to the high dimensional problems. In order to evaluate its performance, we use two numerical examples as benchmarks to compare against the baseline algorithm. In the third numerical example, we apply the proposed algorithm to the Center of Mass control problem in a quadruped robot locomotion task.'
address:
- |
Agile and Dexterous Robotics Lab, ETH Zürich, Switzerland\
(e-mail: {farbodf, depardo, buchlij}@ethz.ch),
- |
Automatic Control Laboratory, ETH Zürich, Switzerland\
(e-mail: [email protected]. ethz.ch).
author:
- Farbod Farshidian
- Maryam Kamgarpour
- Diego Pardo
- Jonas Buchli
bibliography:
- 'bibliography/references.bib'
title: Sequential Linear Quadratic Optimal Control for Nonlinear Switched Systems
---
Control design for hybrid systems, Switching stability and control, Optimal control of hybrid systems, Optimal control theory, Real-time control, Riccati equations, and Mobile robots.
Introduction
============
Switched systems are a subclass of a general family known as hybrid systems. Hybrid system model consists of a finite number of dynamical subsystems subjected to discrete events which cause transition between these subsystems. This transition is either triggered by an external input, or through the intersection of the continuous states trajectory to a certain manifolds known as the switching surfaces. Switched systems are usually characterized by systems that have continuous state transition during these switches. Switched system models are encountered in many practical applications such as automobiles and locomotives with different gears, DC-DC converters, manufacturing processes, biological systems, and robotics.
Our interest in switched systems originates from an application on a legged robot where we model the Center of Mass (CoM) as a switched system. The control goal is to synthesize a controller which, for a given gait, stabilizes the robot while it minimizes a cost function. To fulfill this task, the robot can manipulate the ground reaction forces at the stance feet and adjust the switching times between different stance leg configurations. For instance, assume the problem of controlling the walking gait for a quadruped robot. In this task, the gait is fixed, thus the sequence of mode switches are known. The control task is to modulate the contact forces of the stance legs and to determine the switching times between each mode.
The optimal control problem for the switched systems involves synthesizing the optimal controller for the continuous inputs and finding a mode sequence and the switching times between the modes. In general, the procedure of synthesizing an optimal control law for a switched system can be divided into three subtasks [@giua01; @xu04]: (1) finding the optimal sequence of the modes, (2) finding the optimal switching times between consecutive modes, (3) finding the optimal continuous control inputs. Given the switching sequence and times, the third subtask is a regular optimal control problem with a finite number of discontinuities in the system vector field. The necessary condition of optimality in the context of hybrid systems has been derived from Pontryagin’s maximum principle [@branicky98; @sussmann99; @riedinger03] and subsequently, various computational techniques have been developed to solve this problem [@shaikh07; @soler12; @pakniyat14]. Based on the Pontryagin’s maximum principle, the optimal solution should satisfy a two-point boundary value problem (BVP). However, similar to the classical control problem, the difficulties related to numerical solution of the necessary condition of optimality limits the application of this approach. In [@riedinger99], it has been shown that for a Linear-Quadratic (LQ) problem it is sufficient to solve a sequence of Riccati equations with proper transversality conditions at the switching times in order to optimize the continuous inputs but the mode switches should be still calculated based on the enumerations of all the possible switches at each time step. In order to ease the computational burden of finding the optimal switching behavior, in [@bengea05], the switched system is embedded in larger family of systems defined by the convex hull of the switched subsystems. It has been shown that if the sufficient and necessary conditions for optimality in the embedded systems exists, the bang-bang optimal solution of the embedded problem is also the optimal solution of the switched system; otherwise a sub-optimal solution can be derived.
[@borrelli05] propose an off-line method to synthesize an optimal control law for a discrete linear hybrid system with linear inequality constraints. The proposed method is a combination of dynamic programming and multi-parametric quadratic programming which designs a feedback law for continuous and discrete inputs in the feasible regions. A simpler approach in [@bemporad99] uses a mixed integer linear/ quadratic program to solve the optimal control problem for mixed logical dynamical systems.
Optimizing the cost function with respect to the switching times has been studied for autonomous systems by [@egerstedt03; @johnson11; @wardi12] and for non-autonomous systems by [@kamgarpour12]. By using the derivative of the cost function with respect to the switching time, these methods use nonlinear programming techniques to optimize the cost function. However, in general these methods do not consider the sensitivity of the continuous inputs’ control law to the switching times.
While many of the aforementioned approaches are computationally demanding for real-time robotic applications, there is a class of efficient optimal control algorithms known as Sequential Linear Quadratic (SLQ) methods which can be applied to real-time, complex robotic applications [@neunert16]. An SLQ algorithm sequentially solves the extremal problem around the latest estimation of the optimal trajectories and improves these optimal trajectories using the extremal problem solution [@mayne66; @todorov05; @sideris05]. Motivated by their efficiency in solving regular optimal control problems, in this paper we have extended an SLQ algorithm to solve the optimal control problem for nonlinear switched system with predefined mode sequence. To this end, we adopt an approach introduced by [@xu04] where the primary switched problem is transcribed into an equivalent problem. We then introduce a two-stage optimization method to optimize the continuous inputs and the switching times. While [@xu04] use a computationally demanding approach based on solving a set of two-point BVPs, we propose a new SLQ algorithm to efficiently solve the optimal control problem for nonlinear switched systems.
The main contributions of this paper are: (1) it uses an efficient SLQ algorithm to synthesize the optimal control law for the continuous inputs. (2) it calculates the cost function derivative with respect to switching times using an LQ approximation of the problem. This approximation is obtained without any additional computation from the SLQ solution. (3) it introduces a new practical application of the optimal control for switched systems in the field of motion planning of legged robots.
Problem Formulation {#sec:problem}
===================
In this section, we briefly introduce the optimal control problem based on the parameterization of switching times. We assume that the switched system dynamics consist of $I$ subsystems where the system dynamics for the $i$th subsystem (${ i \in \{1,2,\dots,I\} }$) is as follows $$\label{eq:system_dynamics}
\dot{\vx}(t) = \vf_i \left( \vx(t),\vu(t) \right) \qquad \text{\textit{for }} \, t_{i-1} \leq t < t_{i},$$ where $\vx(t) \in \mathbb{R}^{n_x}$ is the continuous state, $\vu(t) \in \mathbb{R}^{n_u}$ is the piecewise continuous control input, and $\vf_i: \mathbb{R}^{n_x} \times \mathbb{R}^{n_u} \rightarrow \mathbb{R}$ is the vector field of subsystem $i$ which is continuously differentiable and Lipschitz up to the first order derivatives. $t_i$ is the switching time between subsystem $i$ and $i+1$. $t_0$ and $t_I$ are respectively the given initial time and the final time. The initial state is $\vx_0$ and $\vx(t_{i}^-) = \vx(t_{i}^+)$ at the switching moments because of the state continuity condition. The optimal control problem for the switched system in Equation is defined as $$\label{eq:general_opt}
\min\limits_{\scriptstyle \begin{matrix}
\vt\!\in\!\mathbb{T}, \vu(\cdot)
\end{matrix}
} \Phi(\vx(t_I))+ \sum_{i=1}^I{\int_{t_{i-1}}^{t_{i}} { L_i(\vx,\vu)dt}},$$ where $\Phi(\cdot)$ and $L_i(\cdot,\cdot)$ are the final cost and the running cost (for subsystem $i$) which are continuously differentiable and Lipschitz up to the second order derivatives. $\mathbb{T}$ is a polytope in $\mathbb{R}^{^{I-1}}$ defined as $\mathbb{T} = \left\{ (t_1, \dots, t_{I-1}) | t_o \leq t_1 \leq \dots \leq t_{I-1} \leq t_I \right\}$.
The optimal control problem based on the parameterization of switching times can be defined as the following two-stage optimization problem $$\begin{aligned}
\label{eq:two_stage_opt}
\min_{\vt \in \mathbb{T}} J[\vt,\vx^*,\vu^*] \quad \text{s.t. } \{\vx^*,\vu^*\} = \argmin J[\vt,\vx,\vu]. \end{aligned}$$ with $$\begin{aligned}
J[\vt,\vx,\vu] &= \Phi(\vx_{I})+ \sum_{i=1}^I{(t_{i}-t_{i-1}) \int_{i-1}^{i} {L_i(\vx(z),\vu(z))dz}}, \label{eq:equivalent_cost} \\
\frac{d\vx(z)}{dz} &= (t_{i}-t_{i-1}) \vf_i \left(\vx(z),\vu(z) \right) \qquad \text{\textit{for }} i-1 \leq z < i
\label{eq:equivalent_system} \\
t &= (t_{i}-t_{i-1})(z-i) + t_{i}, \label{eq:z}\end{aligned}$$ in which we have replaced the independent time variable $t$ with a normalized time variable $z$ defined by . With this change of variable, while the switching times are still part of the decision variables, they are fixed parameters for the bottom-level optimization in . Therefore, the reformulated optimal control problem does not have variable switching times and it reduces to a conventional optimal control problem parameterized over the switching times (Theorem 1 in [@xu04]). In the next section, we introduce our algorithm for calculating the optimized cost function and its gradient.
Solution Approach {#sec:ocs2}
=================
In this section, we use a gradient-based method to solve the top-level optimization introduced in and a dynamic programming approach to synthesize the continuous inputs control law. In each iteration of the gradient-based method for finding the optimal switching time, we first solve a continuous-time optimal control problem for the system with a fixed switching times. Then, the cost function gradient with respect to the switching times is calculated in order to determine the descent direction for the switching times update. This approach is similar to [@xu04]. However, instead of using the two-point BVP solver for optimizing $J(\vt)$ with respect to $\vu$ and calculating its gradient with respect to $\vt$, we use a more efficient approach. This facilitates implementing this algorithm in real-time on complex systems such as legged robots.
Our **O**ptimal **C**ontrol for **S**witched **S**ystems algorithm (OCS2 algorithm) consists of two main steps: a method which synthesizes the continuous input controller and a method which calculates the parameterized cost function derivatives with respect to the switching times. For synthesizing the continuous input controller, OCS2 uses the SLQ algorithm in Algorithm \[alg:slq\]. As a dynamic programming approach, SLQ uses the Bellman equation of optimality to locally estimate the value function and consequently the optimal control law. In the second step of OCS2, we use the approximated problem for calculating the value function from the first step to efficiently compute the value function gradient with respect to switching times.
Using the SLQ algorithm to calculate the optimal control law for the continuous inputs has two major advantages. First, as discussed by [@sideris10], this algorithm has a linear time complexity with respect to the optimization time horizon in contrast to many standard discretization-based algorithms which scale cubically (such as the direct collocation methods). Second, since in each iteration of SLQ, an LQ problem is optimized, we can efficiently calculate the value function derivatives by obtaining derivatives of an LQ subproblem’s value function. However, in contrast to a regular time-variant LQ problem in which the system dynamics and the cost function are independent of the switching times, in this problem the cost and the system dynamics of the approximated LQ subproblem are functions of the switching times. Therefore, in order to calculate the cost function gradient, we should also consider the LQ subproblem variations to the switching times. We tackle this problem more rigorously in Theorem 1.
Here, we first briefly introduce the SLQ algorithm. We consider an intermediate iteration of the algorithm. We assume that $\{\bar{\vx}(z)\}_{z= 0}^{I}$ and $\{\bar{\vu}(z)\}_{z= 0}^{I}$ are the nominal state and input trajectories which are obtained by forward integrating system dynamics in (performing a rollout) using the latest estimation of the optimal control law with a fixed switching time vector $\bar{\vt}$. For simplicity of notation, in the followings we have dropped the dependencies of the nominal trajectories and consequently the approximated LQ problem with respect to $\bar{\vt}$. The linearized system dynamics of the equivalent system in around these nominal trajectories are defined as $$\begin{aligned}
& \cfrac{d(\delta\vx)}{dz} = (t_{i}-t_{i-1}) \left( \vA_i(z)\delta\vx + \vB_i(z)\delta\vu \right) \quad i-1 \leq z < i \notag \\
& \vA_i(z) = \frac{\partial \vf_i(\bar{\vx}(z),\bar{\vu}(z))}{\partial\vx}, \quad \vB_i(z) = \frac{\partial \vf_i(\bar{\vx}(z),\bar{\vu}(z))}{\partial\vu} ,
\label{eq:dynamics_linear_approximation}\end{aligned}$$ where $\delta \vx(z) = \vx(z)-\bar{\vx}(z)$, $\delta\vu(z) = \vu(z)-\bar{\vu}(z)$. The quadratic approximation of the cost function in is $$\begin{aligned}
& \tilde{J} = \tilde{\Phi}(\vx_{t_f})+ \sum_{i=1}^I{\int_{i-1}^{i} { (t_{i}-t_{i-1})\tilde{L}_i(z,\vx,\vu)dz}} \notag \\
&\tilde{\Phi}_f(\vx) = \ q_f + \vq_f^\top \delta\vx + \frac{1}{2} \delta\vx^\top \vQ_f \delta\vx \notag \\
& \tilde{L}_i(z,\vx,\vu) = q_i(z) + \delta\vx^\top \vq_i(z) + \delta\vu^\top \vr_i(z) + \delta\vx^T \vP_i(z) \delta\vu \notag \\
& \hspace{15mm} + \frac{1}{2} \delta\vx^\top \vQ_i(z) \delta\vx + \frac{1}{2} \delta\vu^\top \vR_i(z) \delta\vu. \label{eq:cost_quadratic_approximation}\end{aligned}$$ In the above, $q(z)$, $\vq(z)$, $\vr(z)$, $\vQ(z)$, $\vP(z)$, and $\vR(z)$ are the coefficients of the Taylor expansion of the cost function in evaluated at the nominal trajectories. The optimal control law for this LQ extremal subproblem can be derived by solving the following Riccati equations [@bryson75] $$\begin{aligned}
& -\frac{d\vS(z)}{dz} = (t_{i}-t_{i-1}) \vW(z), \hspace{3mm} \vS(i^-)=\vS(i^+), \hspace{2mm} \vS(I)=\vQ_{f} \label{eq:riccati_Sm} \\
& -\frac{d\vs(z)}{dz} = (t_{i}-t_{i-1}) \vw(z), \hspace{5mm} \vs(i^-)=\vs(i^+), \hspace{2mm} \vs(I)=\vq_{f} \label{eq:riccati_Sv} \\
& -\frac{ds(z)}{dz} = (t_{i}-t_{i-1}) w(z), \hspace{5mm} s(i^-)=s(i^+), \hspace{2mm} s(I)=q_{f} \label{eq:riccati_s} ,\end{aligned}$$ where $\vS(z)$ and $\vW(z)$ are in $\mathbb{R}^{^{n_x \times n_x}}$, $\vs(z)$ and $\vw(z)$ are in $\mathbb{R}^{^{n_x}}$, $s(z)$ and $w(z)$ are in $\mathbb{R}$. These matrices are defined as $$\begin{aligned}
&\vW(z) = \vQ(z) + \vA(z)\!^\top \vS(z) + \vS(z) \vA(z) - \vL(z)\!^\top \vR(z) \; \vL(z) \label{eq:riccati_Sm_o} \\
& \vw(z) = \vq(z) + \vA(z)^\top \vs(z) - \vL(z)^\top \vR(z) \; \vl(z) \label{eq:riccati_Sv_o} \\
& w(z) = q(z) - 0.5\alpha (2-\alpha)\; \vl(z)^\top \vR(z) \; \vl(z) \label{eq:riccati_s_o} \\
&\vl(z) = - \vR(z)^{-1} \big( \vr(z) + \vB(z)^\top \vs(z) \big) \label{eq:optimal_control_l}\\
&\vL(z) = - \vR(z)^{-1} \big( \vP(z)^\top + \vB(z)^\top \vS(z) \big) . \label{eq:optimal_control_L} \end{aligned}$$ The updated optimal control law, the value function, and the total cost are defined as $$\begin{aligned}
& \vu(z,\vx) = \bar{\vu}(z) + \alpha \vl(z) + \vL(z) \delta\vx \label{eq:slq_policy}\\
& V(z,\vx) = s(z) + \delta\!\vx^\top \vs(z) + \frac{1}{2}\delta\!\vx^\top \vS(z) \delta\!\vx \label{eq:value_function} \\
& \tilde{J} = V(0,\vx_0) = s(0),\end{aligned}$$ where $z$ is defined in . $\alpha \in [0,1]$ is the learning rate for backtracking line-search [@armijo96]. In each iteration of SLQ, the line-search parameter controls the maximum step to move along the feedforward component of the control law update. This parameter is chosen by starting with a full step in the direction of the update ($\alpha=1$), and then iteratively shrinking the step size until the cost associated to the updated controller rollout is lower than the current nominal trajectories cost. Before proceeding to the main theorem and its proof, we state the following lemma which will be used in Theorem 1.
$H: \mathbb{R}^n \rightarrow \mathbb{R}$ and $\vy: \mathbb{R} \rightarrow \mathbb{R}^n$ as twice continuously differentiable functions. If $|\tau-\bar{\tau}|<\varepsilon_1$ and $|\delta\!\vy|<\varepsilon_2$, for small enough $\varepsilon_1$ and $\varepsilon_2$, $H(\vy(\tau)+\delta\!\vy)$ can be approximated as $$\begin{aligned}
H(\vy(\tau)+\delta\!\vy) &\simeq H(\bar{\vy}) + \delta\!\vy^\top \nabla\!H(\bar{\vy}) + \partial\bar{\vy}^\top \nabla\!H(\bar{\vy}) (\tau-\bar{\tau}) \notag \\
& + \frac{1}{2} \delta\!\vy^\top \nabla^2\!H(\bar{\vy}) \delta\!\vy + \delta\!\vy^\top \nabla^2\!H(\bar{\vy}) \partial\bar{\vy} (\tau-\bar{\tau}) \notag \\
& + \frac{1}{2} \Big( \partial\bar{\vy}^\top \nabla^2\!H(\bar{\vy}) \partial\bar{\vy} + \partial^2\bar{\vy}^\top \nabla\!H(\bar{\vy}) \Big) (\tau-\bar{\tau})^2 , \notag\end{aligned}$$ where $\bar{\vy} = \vy(\bar{\tau})$, $\partial\bar{\vy} = \frac{d\vy}{d\tau}|_{\bar{\tau}}$, and $\partial^2\bar{\vy} = \frac{d^2\vy}{d\tau^2}|_{\bar{\tau}}$. Furthermore, $\nabla\!H$ and $\nabla^2\!H$ are the gradient and the Hessian of $H$ respectively.
This lemma can be easily proven using the Taylor series expansion of $H$ and few steps of simplification.
The partial derivative of the value function in and the partial derivative of the total cost with respect to the switching time $t_j$ can be derived as $$\begin{aligned}
\partial_{t_j}V(z,\vx) =& \partial_{t_j}s(z) + \delta\!\vx^\top \partial_{t_j}\vs(z) + \frac{1}{2}\delta\!\vx^\top \partial_{t_j}\vS(z) \delta\!\vx - \partial_{t_j}\bar{\vx}^\top \vs(z) \notag \\
& - \frac{1}{2} \partial_{t_j}\bar{\vx}^\top \vS(z) \delta\!\vx - \frac{1}{2} \delta\!\vx^\top \vS(z) \partial_{t_j}\bar{\vx} \\
\partial_{t_j}\tilde{J} =& \partial_{t_j}s(0),\end{aligned}$$ where $z$ is defined by , $\partial_{t_j}$ is the partial derivative operator with respect to switching time $t_j$. Then $\partial_{t_j}\bar{\vx}$ and $\partial_{t_j}\bar{\vu}$ are respectively the state and input sensitivity to switching time $t_j$ which are calculated by the following differential equation $$\begin{aligned}
\frac{d (\partial_{t_j}\bar{\vx})}{dz} =& ( \delta_{i,j} - \delta_{i-1,j} ) \vf_i(\bar{\vx}(z),\bar{\vu}(z)) \notag \\
& + (t_{i}-t_{i-1}) \left( \vA_i(z) \partial_{t_j}\bar{\vx} + \vB_i(z) \partial_{t_j}\bar{\vu} \right) \label{eq:state_sensitivity} \\
\partial_{t_j}\bar{\vu}(z,\partial_{t_j}\bar{\vx}) &= -\vL(z) \partial_{t_j}\bar{\vx} + \partial_{t_j}\vL(z) \delta\bar{\vx} + \partial_{t_j}\vl(z) ,
\label{eq:input_sensitivity}\end{aligned}$$ with the initial condition $\partial_{t_j}\bar{\vx}(0) = \mathbf{0}$ and the transversality condition $\partial_{t_j}\bar{\vx}(i^-) = \partial_{t_j}\bar{\vx}(i^+)$. $\delta_{i,j}$ is the Kronecker delta which is one if the variables are equal, and zero otherwise. Furthermore, $\partial_{t_j}\vS(z)$, $\partial_{t_j}\vs(z)$, and $\partial_{t_j}s(z)$ can be calculated from the following set of linear differential equations $$\begin{aligned}
-\frac{d (\partial_{t_j}\vS(z))}{dz} &= ( \delta_{i,j} - \delta_{i-1,j} ) \vW(z) + (t_{i}-t_{i-1}) \partial_{t_j}\!\vW(z) \label{eq:riccati_nablaSm} \\
-\frac{d (\partial_{t_j}\vs(z))}{dz} &= ( \delta_{i,j} - \delta_{i-1,j} ) \vw(z) + (t_{i}-t_{i-1}) \partial_{t_j}\!\vw(z) \label{eq:riccati_nablaSv} \\
-\frac{d (\partial_{t_j}s(z))}{dz} &= ( \delta_{i,j} - \delta_{i-1,j} ) w(z) + (t_{i}-t_{i-1}) \partial_{t_j}\!w(z) , \label{eq:riccati_nablas}\end{aligned}$$ where we have $$\begin{aligned}
&\partial_{t_j}\!\vW(z) = \vA(z)^\top \partial_{t_j}\vS(z) + \partial_{t_j}\vS(z) \vA(z) - \partial_{t_j}\vL(z)^\top \vR(z) \vL(z) \notag \\
& \hspace{14mm} - \vL(z)^\top \vR(z) \partial_{t_j}\vL(z) \\
&\partial_{t_j}\!\vw(z) = \partial_{t_j}\vq(z) + \vA(z)^\top \partial_{t_j}\vs(z) - \partial_{t_j}\vL(z)^\top \vR(z) \vl(z) \notag \\
& \hspace{14mm} - \vL(z)^\top \vR(z) \partial_{t_j}\vl(z) \\
&\partial_{t_j}\!w(z) = \partial_{t_j}q(z) - 0.5\alpha (2-\alpha) \big( \partial_{t_j}\vl(z)^\top \vR(z) \vl(z) \notag \\
& \hspace{14mm} + \vl(z)^\top \vR(z) \partial_{t_j}\vl(z) \big) \\
&\partial_{t_j}\vl(z) = - \vR(z)^{-1} \big( \partial_{t_j}\vr(z) + \vB(z)^\top \partial_{t_j}\vs(z) \big) \\
&\partial_{t_j}\vL(z) = - \vR(z)^{-1} \vB(z)^\top \partial_{t_j}\vS(z) .\end{aligned}$$ The derivative of the cost coefficients with respect to $t_j$ is $$\begin{bmatrix} \partial_{t_j}\vq(z) \\ \partial_{t_j}\vr(z) \\ \partial_{t_j}q(z) \end{bmatrix} =
\begin{bmatrix} \vQ(z) & \vP(z) \\ \vP(z)^\top & \vR(z) \\ \vq(z)^\top & \vr(z)^\top \end{bmatrix}
\begin{bmatrix} \partial_{t_j}\vx(z) \\ \partial_{t_j}\vu(z) \end{bmatrix} .
\label{eq:intermidiate_cost_function_sensitivity}$$ Finally, the terminal condition are defined as $$\partial_{t_j}\!\vS(I)=\mathbf{0}, \hspace{2mm} \partial_{t_j}\!\vs(I)=\vQ_f \partial_{t_j}\!\vx(I), \hspace{2mm} \partial_{t_j}\!s(I)=\vq_f^\top \partial_{t_j}\!\vx(I),
\label{eq:final_cost_function_sensitivity}$$ and the transversality conditions are $$\partial_{t_j}\!\vS(i^-) = \partial_{t_j}\!\vS(i^+), \hspace{2mm}
\partial_{t_j}\!\vs(i^-) = \partial_{t_j}\!\vs(i^+), \hspace{2mm}
\partial_{t_j}\!s(i^-) = \partial_{t_j\!}s(i^+).$$
**Given** - Mode switch sequence and switching times - Cost function and system dynamics **Initialization** - Initialize the controller with a stable control law, $\vu_0(\cdot)$ - Forward integrate system dynamics: $\{\ \bar{\vx}(z),\bar{\vu}(z)\}_{z=0}^{I},$ - Compute the LQ approximation of the problem along the nominal trajectory, and . - Solve the final value differential equations, (\[eq:riccati\_Sm\]-\[eq:riccati\_s\]). - Line search for the optimal $\alpha$ with policy - Update control law: $\vu^*(z,\vx) = \bar{\vu}(z) + \alpha^* \vl(z) + \vL(z) \delta\vx $
**Given** - Mode switch sequence and initial switching times, $\vt_0$ - The optimal control problem in equations and **Initialization** - Empty the solution bag - Initialize SLQ policy, $\vu_0(\cdot)$ - Initialize the switching times, $\vt_0$ - Compute the equivalent cost function in Equation - Compute the equivalent system dynamics in Equation - Initialized the SLQ policy with the controller in the solution bag that has the most similar switching vector - Initialized the SLQ policy, $\vu_0(z,\vx)$. - Run the SLQ algorithm - Get the optimal control $\vu^*(z,\vx;\vt^*)$ - Memorize the pair $\left(\vt^*,\vu^*(z,\vx;\vt^*)\right)$ in the solution bag - Calculate $\partial_{t_j}\vS(0)$, $\partial_{t_j}\vs(0)$, and $\partial_{t_j}s(0)$ using (\[eq:riccati\_nablaSm\]-\[eq:riccati\_nablas\]). - Calculate the cost function gradient $\nabla_{\vt}J = [\partial_{t_j}s(0)]_j$ - Use a gradient-descent method to update $\vt^*$\
**Return** the optimal $\vt^*$, and $\vu^*(z,\vx;\vt^*)$.
Equations (\[eq:riccati\_nablaSm\]-\[eq:riccati\_nablas\]) can be derived by directly differentiating the Riccati equations in (\[eq:riccati\_Sm\]-\[eq:riccati\_s\]), where based on the continuity condition, the order of differentiation with respect to time, $z$, and $t_j$ has been changed. For calculating $\partial_{t_j}\!\vW$, $\partial_{t_j}\!\vw$, and $\partial_{t_j}\!w$, we need to calculate the derivative of the linearized dynamics and the quadratic approximation of the cost function with respect to switching time $t_j$ as well. To do so, we need to examine the impact of $t_j$ variation on the system dynamics and cost function of the approximated LQ subproblem.
In each approximated LQ subproblem, we use a quadratic approximation of the cost function components $L$ and $\Phi$ around the nominal trajectories $\bar{\vx}(z)$ and $\bar{\vu}(z)$ of the equivalent system . We can readily derive the differential equation which determines the sensitivity of the state trajectory with respect to switching time $t_j$ by differentiating both side of the equivalent systems dynamics in with respect to $t_j$. $$\partial_{t_j} \frac{d \bar{\vx}}{dz} = ( \delta_{i,j} - \delta_{i-1,j} ) \vf_i(\bar{\vx},\bar{\vu}) \notag + (t_{i}-t_{i-1}) \left( \vA_i \partial_{t_j}\!\bar{\vx} + \vB_i \partial_{t_j}\!\bar{\vu} \right).$$ For simplicity in notation, we drop this dependencies on time, $z$. Using the continuity condition of the state trajectory, the order of derivatives on the right hand side of the equation is changed which is resulted in . Furthermore, since the control input trajectory in SLQ comprises a time-varying feedforward term and a time-varying linear state feedback, its sensitivity can be calculated as . Moreover, for the initial condition of this equation we have $\partial_{t_j}\bar{\vx}(z=0) = \mathbf{0}$ due to the fixed initial state. Based on Lemma 1, the second order approximation of the intermediate cost $L(\vx,\vu)$ around the nominal trajectories (which are in turn a function of $t_j$) can be written as $$\begin{aligned}
& L(z,\bar{\vx}(s)+\delta\!\vx,\bar{\vu}(s)+\delta\!\vu) \simeq
q(z) +
\begin{bmatrix} \vq(z) \\ \vr(z) \end{bmatrix}^\top
\begin{bmatrix} \partial_{t_j}\vx(z) \\ \partial_{t_j}\vu(z) \end{bmatrix} (t_j-\bar{t}_j)
\notag \\
& \hspace{2mm}
+\begin{bmatrix} \delta\!\vx \\ \delta\!\vu \end{bmatrix}^\top\!
\left(
\begin{bmatrix} \vq(z) \\ \vr(z) \end{bmatrix} +
\begin{bmatrix} \vQ(z) & \vP(z) \\ \vP(z)^\top & \vR(z) \end{bmatrix}
\begin{bmatrix} \partial_{t_j}\vx(z) \\ \partial_{t_j}\vu(z) \end{bmatrix}
(t_j-\bar{t}_j) \right) \notag \\
& \hspace{2mm}
+\frac{1}{2}
\begin{bmatrix} \delta\!\vx \\ \delta\!\vu \end{bmatrix}^\top\!\!
\begin{bmatrix} \vQ(z) & \vP(z) \\ \vP(z)^\top & \vR(z) \end{bmatrix}
\begin{bmatrix} \delta\!\vx \\ \delta\!\vu \end{bmatrix}
+ \frac{(t_j-\bar{t}_j)^2}{2}
\begin{bmatrix} \vq(z) \\ \vr(z) \end{bmatrix}^\top\!
\begin{bmatrix} \partial_{t_j}^2\vx(z) \\ \partial_{t_j}^2\vu(z) \end{bmatrix}
\notag \\
& \hspace{2mm}
+ \frac{(t_j-\bar{t}_j)^2}{2}
\begin{bmatrix} \partial_{t_j}\vx(z) \\ \partial_{t_j}\vu(z) \end{bmatrix}^\top\!
\begin{bmatrix} \vQ(z) & \vP(z) \\ \vP(z)^\top & \vR(z) \end{bmatrix}
\begin{bmatrix} \partial_{t_j}\vx(z) \\ \partial_{t_j}\vu(z) \end{bmatrix}\end{aligned}$$
In the above, we have combined the like terms of the state and input increments. Based on this equation the sensitivity of the cost function components will be $$\begin{aligned}
\partial_{t_j} \vr(z) =& \vP(z)^\top \partial_{t_j}\vx(z) + \vR(z) \partial_{t_j}\vu(z), \hspace{3mm} \partial_{t_j} \vR(z) = \mathbf{0}, \\
\partial_{t_j} \vq(z) =& \vQ(z) \partial_{t_j}\vx(z) + \vP(z) \partial_{t_j}\vu(z), \hspace{5mm} \partial_{t_j} \vQ(z) = \mathbf{0}, \\
\partial_{t_j} q(z) =& \vq(z)^\top \partial_{t_j}\vx(z) + \vr(z)^\top \partial_{t_j}\vu(z), \hspace{2mm} \partial_{t_j} \vP(z) = \mathbf{0},\end{aligned}$$ which can be written in matrix form as . By the same process, we can derive for the final cost.
In order to find the linearized system dynamics sensitivity with respect to the switching time $t_j$, we can use the result from Lemma 1 where we only keep the terms up to the first order. $$\begin{aligned}
& \frac{d}{dz} \left( \bar{\vx} + \partial_{t_j}\!\vx(t_j-\bar{t}_j) + \delta\!\vx \right) \simeq (\bar{t}_i-\bar{t}_{i-1}) \big( \vf(\bar{\vx},\bar{\vu}) + \vA_i \delta\!\vx + \vB_i \delta\!\vu \big) \notag \\
& \hspace{0mm}
+ (t_j-\bar{t}_j) \big( ( \delta_{i,j} - \delta_{i-1,j} ) \vf_i(\bar{\vx},\bar{\vu}) + (\bar{t}_{i}-\bar{t}_{i-1}) \left( \vA_i \partial_{t_j}\bar{\vx} + \vB_i \partial_{t_j}\bar{\vu} \right) \big) ,\end{aligned}$$ where for simplicity in notation, we drop the dependencies on time, $z$. By equating the coefficients, we get $$\begin{aligned}
&\frac{d \bar{\vx}}{dz} \simeq (\bar{t}_i-\bar{t}_{i-1})f(\bar{\vx},\bar{\vu}) \\
&\frac{d \delta\!\vx}{dz} \simeq
(\bar{t}_{i}-\bar{t}_{i-1}) \big( \vA_i \delta\!\vx + \vB_i \delta\!\vu \big) \\
&\frac{d (\partial_{t_j}\vx)}{dz} \simeq ( \delta_{i,j} - \delta_{i-1,j} ) \vf_i(\bar{\vx},\bar{\vu}) + (\bar{t}_{i}-\bar{t}_{i-1}) \big(\vA_i \partial_{t_j}\!\bar{\vx} + \vB_i \partial_{t_j}\!\bar{\vu} \big) ,\end{aligned}$$ which are respectively the nominal trajectory equation, the linear approximation of system dynamics, and trajectory sensitivity. Based on this approximation, the linear part is not a function of $t_j$, so we get $\partial_{t_j}\!\vA_i(z) = \mathbf{0}$ and $\partial_{t_j}\!\vB_i(z) = \mathbf{0}$ (note that the effect of the switching time variations on the approximated system dynamics would have been appeared if we had used a second order or a higher order approximation).
OCS2 algorithm
==============
In Section \[sec:ocs2\], we have discussed the technical details behind the OCS2 algorithm. Here, we highlight the main steps of the algorithm for synthesizing the optimal continuous control law and the optimal switching times (refer to Algorithm \[alg:ocs2\]). Each iteration of the OCS2 algorithm has three main steps namely (1) using SLQ algorithm to find the continuous inputs’ optimal control, (2) calculating the cost function gradient based on the LQ approximation of the problem which has been already calculated by the SLQ algorithm, (3) using a gradient descent method to update the switching times where we use the Frank-Wolfe method [@jaggi13]. An interesting aspect of our algorithm is its linear-time computational complexity with respect to the optimization time-horizon both in SLQ algorithm and calculating cost gradient. As we will show in the next section, this characteristic results in a superior performance of our algorithm in comparison to the two-point BVP approach originally introduced by [@xu04].
As discussed, OCS2 uses the SLQ algorithm for solving a nonlinear optimal control problem on the equivalent system with fixed switching times (refer to Algorithm \[alg:slq\]). The SLQ algorithm is based on an iterative scheme where in each iteration it forward integrates the controlled system and then approximates the nonlinear optimal control problem with a local LQ problem. In general, the SLQ algorithm requires an initial stable controller for the first forward integration. For deriving the initial controller, we define a set of operating points in each switching mode (normally one point) and approximate the optimal control problem around these operating points with an LQ approximation. Then, we follow the same process described in the SLQ algorithm to design an initial controller.
A good initialization can often increase the convergence speed of the SLQ algorithm. One interesting characteristic of our algorithm is the warm starting scheme for the initial policy of SLQ. Here, we use a memorization scheme where we store the solutions of the different runs of SLQ in a solution bag and later initialize the policy of a new run of SLQ with the most similar switching time’s policy (refer to Algorithm \[alg:ocs2\]). The similarity between two switching time sequences is measured as a sum of squared differences between corresponding times in two sequences.
Case Studies {#sec:numerical_example}
============
In this section, we evaluate the performance of the proposed method on three numerical examples. The first two examples are provided as illustrative cases to compare the performance of the proposed algorithm to the baseline algorithm introduced by [@xu04] and the third one is about the application of our method in motion control of legged robots. We here refer to the baseline algorithm as the BVP method since it is based on the solution of two-point BVPs.
#### Example 1: {#example-1 .unnumbered}
The first example is a nonlinear switched system with three mode switches which are defined as followings $$\begin{aligned}
& 1: \left\{ \begin{array}{ll}
\dot{x}_1 = x_1 + u_1\sin{x_1}\\
\dot{x}_2 = -x_2 + u_1\cos{x_2}
\end{array}
\right., \quad
2: \left\{ \begin{array}{ll}
\dot{x}_1 = x_2 + u_1\sin{x_2}\\
\dot{x}_2 = -x_1 - u_1\cos{x_1}
\end{array}
\right. \notag \\
& 3: \left\{ \begin{array}{ll}
\dot{x}_1 = -x_1 - u_1\sin{x_1}\\
\dot{x}_2 = x_2 + u_1\cos{x_2}
\end{array}
\right.\end{aligned}$$ with the initial condition $\vx_0 = \begin{bmatrix} 2, 3 \end{bmatrix}^\top$. The optimization goal is to calculate the optimal switching times $t_1$ (from subsystem 1 to 2), $t_2$ (from subsystem 2 to 3), and the continuous control input $u_1$ such that the following cost function is minimized $$\label{eq:exp1_cost}
J = 0.5 \| \vx(3)-\vx_g \|^2 + 0.5 \int_0^3{ \left( \| \vx(t)-\vx_g \|^2 + \|\vu(t)\|^2 \right) dt } ,$$ where $\vx_g = \begin{bmatrix} 1,& -1\end{bmatrix} ^\top$. We apply both OCS2 and BVP to this system with uniformly distributed initial switching times. The optimized switching times, the optimized cost, the number of the iterations, and number of function calls (bottom-level optimization calls) for both algorithms are presented in Table \[tab:performance\]. As it illustrates, OCS2 and BVP converge to the same solution within a comparable number of iterations and function calls. However, the consumed CPU times are significantly different. The BVP method utilizes about 235 seconds on an Intel’s core i7 2.7-GHz processor while the OCS2 algorithm uses only 31 seconds which is roughly 7.5 times faster (Figure \[fig:cpu\_time\]).
#### Example 2: {#example-2 .unnumbered}
In order to examine the scalability of both algorithms to higher dimensions, we augment each subsystem in Example 1 with two more states and one additional control input. The augmented states have the following system dynamics $$\begin{aligned}
& 1': \left\{ \begin{array}{ll}
\dot{x}_3 = -x_3 + 2 x_3 u_2 \\
\dot{x}_4 = x_4 + x_4 u_2
\end{array}
\right., \quad
2': \left\{ \begin{array}{ll}
\dot{x}_3 = x_3 - 3 x_3 u_2 \\
\dot{x}_4 = 2 x_4 - 2 x_4 u_2
\end{array}
\right. \notag \\
& 3': \left\{ \begin{array}{ll}
\dot{x}_3 = 2 x_3 + x_3 u_2 \\
\dot{x}_4 = -x_4 + 3 x_4 u_2
\end{array}
\right.\end{aligned}$$ with the initial condition $\vx_0 = \begin{bmatrix} 2, 3, 1, 1 \end{bmatrix}^{\top}$. The cost function is the same as Example 1 with $\vx_g = \begin{bmatrix} 1, -1, 2, 2\end{bmatrix} ^{\top}$. As Example 1, the optimized solutions are comparable for both algorithms (refer to Table \[tab:performance\]). However the CPU times are drastically different. For the BVP method, the CPU time is about 1500 seconds while the OCS2 algorithm uses only 76 seconds which is 19.5 times more efficient. This manifests the efficiency of the OCS2 algorithm in higher dimensional problems where computational time of the BVP algorithm prohibitively increases.
![Comparison between the CPU time consumption of the BVP algorithm (blue) and the OCS2 algorithm (orange). The evaluation is done on an Intel’s core i7 2.7-GHz processor. The BVP method run-time on Example 3 is missing since the algorithm failed to terminate.[]{data-label="fig:cpu_time"}](cpu_time_comparison_compressed.pdf){width="\columnwidth"}
Alg. $J$ $s_1$ $s_2$ Itr. FC.
-- ------ --------- -------- -------- ------ -----
BVP 5.4498 0.2235 1.0198 9 13
OCS2 5.4438 0.2324 1.0236 7 14
BVP 10.3797 0.2754 1.6076 9 13
OCS2 10.3888 0.2973 1.5978 8 20
: Comparison between the performance of the BVP and OCS2 algorithms. In the table Itr. is the number of iterations in the gradient-descent method until it converges. FC. is the number of requests for cost function and its derivative evaluation in the gradient-descent algorithm.[]{data-label="tab:performance"}
#### Example 3: Quadruped’s CoM motion control {#example-3-quadrupeds-com-motion-control .unnumbered}
In this example, we apply OCS2 to the quadruped’s CoM control problem. In a quadruped robot, based on the stance legs configuration the system shows different dynamical behavior. Furthermore, at the switching instances the non-elastic nature of contacts introduces a jump in the state trajectory. Therefore, basically a legged robot have to be modeled as a nonlinear hybrid system with discontinuous state trajectory.
The computational burden of solving the optimal control problem on such a nonlinear and hybrid system has encouraged researchers to use CoM dynamics instead of the complete system dynamics. In addition to optimizing a lower dimension problem, this approach has another important advantage: the CoM state trajectory is more smooth and the impact force does not cause a noticeable jump in states. Therefore the CoM dynamics can be modeled as a switched system which facilitates solving the optimal control problem. The CoM system has in total $12$ states consisting of: $3$ states for orientation, $\vth$, $3$ states for position, $\vp$, and $6$ states for linear and angular velocities in body frame, $\vv$ and $\vom$. The control inputs of the CoM system are an input which triggers the mode switch and the contact forces of the stance legs $\{\vlambda^i(t)\}_{i=1}^4$. The simplified CoM equation can be derived from the Newton-Euler equation as $$\left\{
\begin{array}{ll}
\dot{\vth} = \vR(\vth) \vom, \hspace{6mm}
\dot{\vom} = \vI^{-1} \big( -\vom \times \vI\vom + \sum_{i=0}^4{\sigma_i \vJ_{\omega,i}^T \vlambda_i} \big), \\
\dot{\vp} = \vR(\vth) \vv, \hspace{8mm}
\dot{\vv} = \frac{1}{m} \big( m\vg + \sum_{i=0}^4{\sigma_i \vJ_{v,i}^T\vlambda_i} \big)
\end{array}
\right.$$ where $\vR(\vth)$ is the rotation matrix, $\vg$ is gravitational acceleration in body frame, $\vI$ and $m$ are moment of inertia about the CoM and the total mass respectively. $\vJ_{\omega,i}$ and $\vJ_{v,i}$ are the Jacobian matrices of the $i$th foot with respect to CoM which depends on the foot position and CoM’s position and orientation (for technical specifications of this quadruped robot refer to [@semini11]). In this example, we study the trotting gait.
In a quadruped trotting gait, the pairwise alternating diagonal legs are the stance legs. In each phase of the motion only two opposing feet are on the ground. Since we assume that our robot has point feet, it looses controllability in one degree of freedom, namely the rotation around the connecting line between the two stance feet. Therefore, the subsystems in each phase of the trotting gait are not controllable individually. However, the ability to switch from one diagonal pair to the other, allows the robot for gaining control over that degree of freedom. Therefore, for a successful trot, the robot requires to plan and control the contact forces as well as the switching times.
The cost function in this example is defined as . The control effort weighting matrix is a diagonal matrix which penalizes all continuous control inputs (contact forces) equally. Furthermore, our cost function penalizes intermediate orientation and z-offsets during the entire trajectory. In the final cost, we penalize deviations from a target point $p = [3, 0, 0]^\top$ which lies 3 m in front of the robot. For the following evaluation, we assume that the x direction points to the front, the y direction points to the left and the z direction is orthogonal to ground. Furthermore, we penalize the linear and angular velocities to ensure the robot comes to a stop towards the end of the trajectory.
Figure \[fig:switching\_times\] shows the difference between the initial and optimized switching times. The initial switching times have been chosen uniformly over the three-second period. During optimization the controller both extends and shortens different switching times. More interestingly, it set the time periods of two phases to zero. Therefore, although we initially asked the robot to take 8 steps the optimized trot has 6 steps.
Figure \[fig:contact\_forces\] shows the optimized contact forces of the trotting experiment. The vertical contact force $\lambda_z$ which needs to compensate for gravity is fairly equally distributed between the legs. However, we can see that the plots are not perfectly symmetric. This results from the quadruped being modeled after the hardware, which has unsymmetrical inertia. Additionally, we can see that the force profiles are non-trivial and thus would be difficult to derive manually. However, the contact forces are smooth within each step sequence which makes tracking easy.
Figure \[fig:hyq\] shows a few snapshots of the quadruped during the optimized trotting gait. The CoM motion is controlled by the OCS2-designed controller. The swing feet’s trajectories (the ones that are not in contact) are designed based on a simple heuristic that the distance between the CoM and a swing foot at the moment of take off (breaking contact) and at the moment of touch down (establishing contact) should be the same.
![Comparison between the initial (blue) and optimized (yellow) switching times sequence. After optimization mode 6 and 7 switching times are set to zero and the number of total steps reduced to 6 form the initial 8 steps.[]{data-label="fig:switching_times"}](switching_times.pdf){width="\columnwidth"}
![Contact forces as optimized by the trotting gait controller. For visibility, contact forces in z-direction have been scaled by a factor of 1/100. The gray dotted vertical lines indicate the switching times. The non-symmetric pattern results from the fact, that the quadruped is not perfectly symmetric and thus also not modeled as such.[]{data-label="fig:contact_forces"}](contact_forces.pdf){width="\columnwidth"}
![Few snapshots of the modeled quadruped during optimized trotting gait. The CoM motion is controlled by the controller synthesized by OCS2 algorithm. The swing foot trajectories (the ones that are not in contact) are designed based on a simple heuristic over the CoM linear velocity.[]{data-label="fig:hyq"}](HyQGaitCropped.png){width="\columnwidth"}
CONCLUSIONS AND FUTURE WORK
===========================
In this paper, we have presented an efficient method to solve the optimal control problem for nonlinear switched systems with predefined mode sequence. The proposed method is based on a two-stage optimization scheme which optimizes the continuous inputs as well as the switching times. In order to obtain an accurate estimation of the cost function derivatives, [@xu04] have introduced a set of two-point BVPs. Although there exist many numerical methods for solving two-point BVP (e.g. collocation method), many of them do not scale properly to high dimensional problems such as controlling quadruped locomotion. To tackle this issue, we have proposed the OCS2 algorithm which is based on an SLQ algorithm. In order to obtain cost function derivatives, OCS2 takes into account the sensitivity of the approximated LQ models with respect to the switching times. OCS2 obtains an approximation of the LQ model sensitivity, only by using the LQ approximation of the problem which has been already calculated by SLQ. Therefore, the algorithm can calculate values of the derivatives with no further computational cost for evaluating the higher order derivatives of the system dynamics and cost function. This feature increases the proposed algorithm efficiency in comparison to other methods such as direct collocation which rely on higher order derivatives.
In order to demonstrate the computational efficiency of the algorithm, we have compared the CPU time used by the OCS2 algorithm with the baseline method introduced by [@xu04]. We observe that as the dimensions of state and input spaces increase, the difference in computational time between the algorithms becomes significant to the point that for the quadruped CoM control task, the run-time of the baseline algorithm becomes prohibitively long.
| {
"pile_set_name": "ArXiv"
} |
[**** ]{}\
Arnab Laha^1^, Abhijit Biswas^2^, Somnath Ghosh^1,\*^\
**[1]{} Department of Physics, Indian Institute of Technology Jodhpur, Rajasthan 342011, India\
**[2]{} Institute of Radio Physics and Electronics, University of Calcutta, Kolkata-700009, India\
[email protected]****
Abstract {#abstract .unnumbered}
========
We report a specially configured non-Hermitian optical microcavity, imposing spatially imbalanced gain-loss profile, to host an exclusively proposed next nearest neighbor resonances coupling scheme. Adopting scattering matrix ($S$-matrix) formalism, the effect of interplay between such proposed resonance interactions and the incorporated non-Hermiticity in the microcavity is analyzed drawing a special attention to the existence of hidden singularities, namely exceptional points ($EP$s); where at least two coupled resonances coalesce. We establish adiabatic flip-of-states phenomena of the coupled resonances in the complex frequency plane ($k$-plane) which is essentially an outcome of the fact that the respective $EP$ is being encircled in system parameter plane. Encountering such multiple $EP$s, the robustness of flip-of-states phenomena have been analyzed via continuous tuning of coupling parameters along a special hidden singular line which connects all the $EP$s in the cavity. Such a numerically devised cavity, incorporating the exclusive next neighbor coupling scheme, have been designed for the first time to study the unconventional optical phenomena in the vicinity of $EP$s.
Introduction
============
Over the years, resonance interaction phenomena in open quantum systems have been attracted enormous attention in various research field of modern physics. Various interesting interaction phenomena exploiting local and non-local interdependence between the resonance states have reported in literature. Specifically, in the photonics domain, interesting techniques have developed for modeling and simulation of different specially configured coupled optical systems to study such interactions between the states. This paper present a specially configured coupled optical system with discrete resonances where interesting effects of next nearest neighbor interaction between them have been topologically explored. In the contemporary research field, next nearest neighbor interaction between the resonances has always been a great physical insight because it is a pivotal feature in many natural and artificial physical phenomena. Statistically, 1D Ising model, a mathematical model of ferromagnetism in solid state physics, gives a clear interpretations of next nearest neighbor interaction, while considering the physical effect of superimposition of very long range spin interaction with conventional nearest neighbor short range interaction on a 1D crystalline lattice [@Kijewski]. Lately, next nearest neighbor interactions have also been explored in the contexts of QCD through three states Potts model (a generalization of the Ising model) [@Bernaschi], Betts lattice considering extended Hubbard model to study pairing enhancement [@Fang]. Influence of such interactions phenomena have also attracted considerable attention to study various physical applications like entanglement of the Heisenberg chain [@Gu], thermal transportation in low dimensional lattice [@Santhosh], spectrum of plasmon excitations in graphene (considering next-nearest-neighbor tight-binding model) [@Kadirko] etc. In the optical context lately, effect of next nearest neighbor coupling have widely discussed on optically pumped nanodevice arrays [@Csaba], Bose–Einstein condensation in optical lattices [@Zaleski], photonic superlattice to implements 1D random mass Dirac equation on a chip [@Keil] etc.
Apart from the previous studies, corroborating the analogy between non-Hermitian open quantum system and counterpart open optical geometries with suitable pumping, we explore an innovative unconventional scheme to study a nontrivial special next nearest neighbor interaction between discrete resonance states in a coupled optical microcavity. The cavity is partially pumped via spatially distributed inhomogeneous gain-loss profile. In such a cavity the resonances are appeared in complex energy plane. Here the coupling phenomena between the resonances are entirely controlled internally i.e. by system topology and internal gain-loss variation.
![(Color online) Schematic diagram of the proposed coupling scheme between the resonances appeared in complex energy plane. The resonances labeled by green colors are being interacted whereas the resonances labeled by red colors remain isolated. []{data-label="figure_1"}](figure1.eps){width="7cm"}
The special non-trivial coupling scheme, schematically shown in figure \[figure\_1\], is proposed in such a way that, a specific resonance is strictly allowed to interact with its next nearest neighbor only with a restriction of one-to-one coupling when the intermediate resonances between each of the two coupled states remains isolated. Proposed coupling scheme can be mathematically demonstrated by the following Hamiltonian function described below.
Consider a quantum mechanical coupled system subjected by an external field $h_n$ with discrete resonances characterized by the parameters $\Lambda_n (n=1,2,3...)$. Now the Hamiltonian can be written as- $${H}=p\sum\limits_{i,j} C_{2i-1,2j+1}\Lambda_{2i-1}\Lambda_{2j+1} +q\sum\limits_{k,l} C_{2k-1,2l+1}\Lambda_{2k-1}\Lambda_{2l+1}+r\sum\limits_{n}h_n\Lambda_n
\label{equation_1}$$ Here, $i,j=1,3,5....2n+1$ and $k,l=2,4,6....2n$ $(i,j,k,l\in n)$. The coefficients $C$ indicate the interaction of one resonance to its next nearest neighbor. $p$, $q$ and $r$ are the real dimensionless parameters. For our proposed scheme, there must be one isolated resonance between two coupled resonances. i.e. the resonances appears in sites labeled by odd integers ($i$ and $j$) are interacting, whereas the resonances appears in sites labeled by even integers ($k$ and $l$) remain isolated. So, purposely choosing the parameter $q$ as 0 we neglect the second term of Eqn. \[equation\_1\]. Here we also deliberately neglect all possible coupling phenomena of the resonances with external field and henceforth set the parameter $r$ at 0. So, according to the proposed coupling scheme the Hamiltonian function (Eqn. \[equation\_1\]) must be reformed as $${H}=p\sum\limits_{i,j} C_{2i-1,2j+1}\Lambda_{2i-1}\Lambda_{2j+1};\quad i,j=1,3,5..
\label{equation_2}$$
Towards the topological studies based on resonance interactions in such a non-Hermitian open optical microcavity the phenomena of avoided resonance crossing ($ARC$) play a key role [@Heiss1; @Cartarius3; @Laha2; @Ghosh; @Laha1]. Usually, $ARC$ occur in complex energy plane where two interacting resonances repel each other via crossing/anticrossing of their energies and widths i.e. essentially their real and imaginary parts. Such $ARC$ phenomena between two interacting resonance states have been referred the presence of a specific spectral singularity where they are very close to a special type of degeneracy which is rather different form genuine Hermitian degeneracy. These specific hidden spectral singularities, usually appeared in parameter space with at least either two real valued parameters or a complex parameter, are named as [*exceptional points*]{} ($EP$s). At these $EP$s the system Hamiltonian becomes defective and two coupled levels coalesce [@Heiss1; @Cartarius3; @Laha2; @Ghosh].
An $EP$ leads to crucial modifications on associated coupled eigenvalues’ behavior under the influence of coupling parameters; where the phenomenon of flipping of states in the complex eigenvalue plane is the most significant in the context of optical mode converter [@Laha2; @Ghosh]. In parameter space, adiabatically a moderate variation of the chosen coupling parameters along a closed contour around an $EP$ results in the permutation between the corresponding coupled eigenvalues (exchanging their positions) in complex energy plane exhibiting $EP$ as a second order branch point [@Cartarius3; @Laha2; @Ghosh; @Menke]. Consequently, the corresponding eigenstates are also permuted exhibiting $EP$ as a forth order branch point followed by an additional phase change after each round [@Heiss1] in a manner like $\{\psi_1,\psi_2\}\rightarrow\{\psi_2,-\psi_1\}$. By contrast, for an $EP$ which is not being enclosed by the parametric contour; associated eigenvalues make individual loop and avoid any kind of permutation. This unique features of [*flip-of-state*]{} phenomenon in the vicinity of $EP$s have theoretically been explored in various non-Hermitian systems like atomic [@Cartarius3; @Menke] as well as molecular [@Lefebvre] spectra, partially pumped optical microcavity [@Laha2], laser [@Berry], optical waveguide [@Ghosh] etc. and also verified experimentally [@Dembowski2]. Technologically, this unconventional phenomena leads a key feature towards sensor operation [@Wiersig1] in the context of single particle detection in microcavity [@Wiersig2] and also for mode management in dark-state laser [@Hodaei].
In this paper for the first time to the best of our knowledge, we explore $EP$s with their unconventional specific features in a non-Hermitian optical microcavity operating under the proposed non-trivial next-nearest-neighbor resonance coupling condition. A specially configured non $\mathcal{PT}$-symmetric Fabry-Perot type optical microcavity with spatially unbalanced gain-loss profile has been reported for this specific purpose. Recent advanced development in fabrication technology for growth of such Fabry-Perot type optical micro-resonators with enhanced precision and control on output coupling without proper phase matching (which is not possible for other geometries) have resulted in extensive contemporary research attention towards easier technological implementation. Numerically designing such a microcavity, we encounter at least three second order $EP$s in the functional parameter space of the cavity via internally coupled resonances situated in next nearest neighbor positions with rigid one-to-one coupling restriction which is entirely controlled by topology of the microcavity. We also establish a formation of special hidden singular line, namely [*exceptional line*]{}, to correlate the identified $EP$s in parameter plane. Very recently such correlation have reported by the authors in the similar form of the optical microcavity operating under nearest neighbor resonance interaction only [@Laha2]. Unconventional cascaded flip-of-state mechanism in complex frequency plane ($k$-plane) with its robustness against parameter fluctuations/ deformations has also reported in the vicinity of $EP$s by encircling multiple $EP$s in parameter plane via continuous tuning of coupling parameters along exceptional line. Overall, exploiting $EP$s the optical performances of the microcavity have judicially tailored under restricted operating condition.
Matrix formulation towards appearance of exceptional points incorporating next nearest neighbor resonance coupling {#matrix_formulation}
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Mathematically, to elaborate all the essential aspects of a second order $EP$ under specially proposed next nearest neighbor coupling scheme as described by Eqn. \[equation\_2\], a matrix formulation should be needed. The proposed scheme demands at least a real $3 \times 3$ Hamiltonian for this specific purpose. The following Hamiltonian $H$ with the form $H_0+\lambda H_p$ describes a passive quantum system $H_0$ with three discrete energy eigenvalues $\varepsilon_i (i = 1,2,3)$, which is subjected by a perturbation $H_p$. i.e. $${H}=\left(\begin{array}{ccc}\varepsilon_1 & 0 & 0 \\0 & \varepsilon_2 & 0 \\0 & 0 & \varepsilon_3\end {array}\right)+\lambda U\left(\begin{array}{ccc}\omega_1 & 0 & 0 \\0 & \omega_2\ & 0 \\ 0 & 0 & \omega_3\end {array}\right)U^\dagger
\label{equation_3}$$ where, $${U(\xi)}=\left(\begin{array}{ccc}\cos\xi & 0 & -\sin\xi \\0 & 1 & 0 \\\sin\xi & 0 & \cos\xi\end {array}\right)
\label{equation_4}$$
This matrix form can be trivially extended for specific higher dimensional applications. Here, $\lambda$ is a real/ complex tunable constant and an unitary transformation across the parameter $\lambda$ is executed by the matrix $U$, parametrized by $\xi$. The form of the unitary matrix $U$ is intentionally chosen to explore the proposed coupling scheme only. In the perturbation part, $\omega_i (i = 1,2,3)$, represent the coupling terms. Now the eigenvalues of $H$ are given by
$$\begin{aligned}
& E_{1,3}(\lambda)=\frac{\varepsilon_1+\varepsilon_3+\lambda\left(\omega_1+\omega_3\right)}{2}\pm {C}\\
& E_{2}(\lambda)=\varepsilon_2+\lambda\omega_2
\end{aligned}$$
\[equation\_5\]
where, $${C}=\biggl[\left(\frac{\varepsilon_1-\varepsilon_3}{2}\right)^2+\left(\frac{\lambda\left(\omega_1-\omega_3\right)}{2}\right)^2
+\frac{\lambda}{2}\left(\varepsilon_1-\varepsilon_3\right)\left(\omega_1-\omega_3\right)\cos(2\xi)\biggr]^{1/2}
\label{equation_6}$$ From the eigenvalue expressions (Eqns. \[equation\_5\]), it is clearly observed that the coupling term $C$ (given by (Eqn. \[equation\_6\])) appears only in the expressions of $E_1$ and $E_3$ but not in the expression of $E_2$. i.e. the states $E_1$ and $E_3$ are being interacted keeping $E_2$ as an isolated state. Clearly, at $\xi = 0$ two interacting levels $E_1$ and $E_3$ are degenerate at the point $\lambda = -(\varepsilon_1-\varepsilon_3)/(\omega_1-\omega_3)$. To lift this degeneracy one need to couple them by switching on $\xi$ and then avoided resonance crossing ($ARC$) occur between $E_1$ and $E_3$ with variation of the parameter $\lambda$. Now, to explore $EP$ with pertinent connection to this phenomena of $ARC$, the parameter $\lambda$ is chosen as a complex variable with form $\lambda = \lambda_R + i\lambda_I$. For a specific set of parameters all the three eigenvalues are plotted in Fig. \[figure\_2\] as a function of $\lambda_R$ for two distinct values of $\lambda_I$. Interestingly, two different behavior of $ARC$s between $E_1$ and $E_3$ are clearly observed with anti-crossing and crossing between $\Re(E)$ and $\Im(E)$ in the top panel; whereas, crossing and anti-crossing between $\Re(E)$ and $\Im(E)$ in the bottom panel of Fig. \[figure\_2\] respectively.
Accordingly, it is also observed that the change in $\lambda_I$ does not effect the intermediate state $E_2$ i.e. it remains isolated and varies as a function of $\lambda_R$ only. So in complex $\lambda$ plane two types of $ARC$s between $E_1$ and $E_3$ must be connected by a square root branch point singularity, whereas the intermediate state $E_2$ must be unaffected by such type of singularities. At this singular point (i.e. essentially where $C$ should be vanished) two interacting levels are being coalesced. Such singular point are called hidden singularity namely exceptional point ($EP$). So the $EP$ of the defined Hamiltonian is situated at a complex conjugate point given by $${\lambda_{EP}}=-\frac{\varepsilon_1-\varepsilon_3}{\omega_1-\omega_3}\exp(\pm 2i\xi)
\label{equation_7}$$ Now, the coupled energy eigenvalues can be expressed in terms of the characteristics of identified $EP$ as $$E_{1,3}(\lambda)=E_{EP} \pm c_1\sqrt{\lambda-\lambda_{EP}}
\label{equation_8}$$ In complex $\lambda$-plane, the values of $\sqrt{\lambda-\lambda_{EP}}$ on two different Riemann surfaces specify two distinct coupled energy levels, where the cross-joint of them represents the approximate $EP$ location. In similar way without losing any generality for higher dimensional situation, introducing a Hamiltonian matrix of the order $3n \times 3n$ $(n \in I)$ with appropriate coupling elements we should be able to establish the existence of such multiple second order $EP$s validating the proposed robust interaction restriction.
Design of the Fabry-Perot type microcavity
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Cavity specifications with operating parameters
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![**(a)** (Color online) $3D$ schematic diagram of the designed Fabry-Perot type optical microcavity with nonuniform back ground refractive index; **(b)** $2D$ cross-sectional view of the same microcavity occupying the region $0\le x\le L$ with $L=10$ $\mu m$. Here $L_G=3$ $\mu m$ and $L_R=7$ $\mu m$ . The real background refractive indices are as $n_{R1}=1.5$ and $n_{R2}=4.5$. The eigenstates $\psi_L^+$ and $\psi_R^-$ indicate the incident waves with complex amplitudes $A$ and $D$ whereas the eigenstates $\psi_L^-$ and $\psi_R^+$ indicate the scattered waves with complex amplitudes $B$ and $C$ respectively; **(c)** Schematic non-linear distribution of $S$-matrix poles in complex $k$-plane of the microcavity under operating condition. The poles indicated by green circles represent the pair of interacted states whereas the poles indicated by red circles represent the isolated states. []{data-label="figure_3"}](figure3.eps){width="8.5cm"}
In order to achieve our goals, we design a two port Fabry-Perot type open optical microcavity with one dimensionally nonuniform background refractive index i.e. $n_R(x)$, as schematically shown ($3D$ view) in Fig. \[figure\_3\](a). Fig. \[figure\_3\](b) represents the $2D$ cross-section of the same microcavity which occupies the region $0\le x\le L$. Along length scale the distribution of $n_R(x)$ is given as follows. $$n_R(x)=
\left\{
\begin{array}{ll}
n_{R1}, & 0\le x\le L_G\\
n_{R2}, & L_G\le x\le L_R\\
n_{R1}, & L_R\le x\le L\\
\end{array}
\right.
\label{equation_9}$$ Now to add non-hermiticity, the cavity is pumped partially by introducing spatially unbalanced gain (with co-efficient $\gamma$) and loss profile in the two halves i.e. in the regions $0\le x\le L_G$ and $L_R\le x\le L$ respectively maintaining a fixed loss-to-gain ratio $\tau$. Hence, for $\gamma=0$ the cavity behaves like an Hermitian system. Under operating condition the refractive indices of the gain and loss regions are denoted by $n_G$ and $n_L$ respectively which can be expressed as
$$\begin{aligned}
n_G & = n_R-i\gamma,\quad 0\le x\le L_G\\
n_L & = n_R+i\tau\gamma,\quad L_R\le x\le L
\end{aligned}$$
\[equation\_10\]
Specifically, for $\gamma \not=0$ the parameter $\tau$ can adjust the incorporated non-hermiticity independently in terms of system openness and coupling strength. For a fixed value of $\tau=1$, $\mathcal{PT}$-symmetry is conserved. But during operation, we set the parameter $\tau \not= 1$ to avoid $\mathcal{PT}$-symmetry constraint deliberately.
Scattering matrix formalism for calculation of eigenvalues
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Numerically, to study the resonance interaction phenomena in the designed microcavity we adopt a established method of scattering matrix ($S$-matrix) formalism where virtual states of resonances of the Hamiltonian corresponding to the real system are calculated in terms of poles of the associated $S$-matrix [@Laha2; @Laha1]. Using electro-magnetic scattering theory, here the matrix elements are analytically calculated as function of real system parameters. Now, for the designed cavity associated $S$-matrix can be defined through the input and output eigenstates relation given by $$\left(\begin{array}{c}B\\C\end {array}\right)=S(n(x),\omega)\left(\begin{array}{c}A\\D\end {array}\right)
\label{equation_11}$$ Exploiting numerical root finding method, the poles of the defined $S$-matrix are calculated by solving the equation $$\frac{1}{max[eig S(\omega)]}=0
\label{equation_12}$$ Here, the denominator in L.H.S. of Eqn. \[equation\_12\] gives the maximal-modulus eigenvalues of the matrix $S(\omega)$.
Obeying current conservation and causality conditions, the $S$-matrix poles are calculated only at the lower half of the complex frequency plane ($k$-plane) for physical acceptability. Distribution of the calculated poles are schematically shown in Fig. \[figure\_3\](c). Interestingly, a nonlinear pattern have been observed in the pole distribution. Contextually, the equidistant linear distribution of $S$-matrix poles have already shown by choosing a same form of Fabry-Perot type optical microcavity with uniform background refractive index along length scale [@Laha2]. But to establish the proposed next nearest neighbor coupling scheme, such nonlinearity in pole distribution is deliberately introduced by choosing nonuniform background refractive index along cavity-length, where only tuning such non-uniformity the distribution in $S$-matrix poles may be controlled as required for specific purposes. Associated interaction phenomena between the matrix poles (as shown in Fig. \[figure\_3\](c)) are topologically controlled through the spatial variation of unbalanced gain-loss profile with tunable parameters $\gamma$ and $\tau$. Accordingly, three pairs of interacting poles are deliberately identified keeping an intermediate isolated pole between each pairs. During operations, cavity is accompanied by avoided crossings between the interacting $S$-matrix poles.
Numerical results towards encounter of hidden singularities with associated optical performances
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Identifying the hidden singular points
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![(Color online) Trajectories of three chosen poles (dotted blue, red and black line) exhibiting $ARC$s (clearly shown in upper panel for both **(a)** and **(b)**) followed by the partial pumping in terms of unbalanced spatial gain-loss distribution in the cavity. In the passive cavity two green circles indicate the position of two poles which are interacting and the red circle denotes the position of intermediate isolated pole. The loss-to-gain ration is set at $\tau = 5.32$ in **(a)** and $\tau = 5.33$ in **(b)** respectively. The crossing/ anticrossing behavior of $\Re(k)$ and $\Im(k)$ are separately depicted as a function of $\gamma$ for both the $\tau$ values in lower panels. The red crosses in the top right panels represent the approximate positions of branch point singularities.[]{data-label="figure_4"}](figure4.eps){width="13cm"}
To encounter an $EP$, the mathematical concept of $ARC$s between eigenvalues of the matrix Hamiltonian (Eqn. \[equation\_3\]) as delineated in section \[matrix\_formulation\], has been exploited where the cavity resonances (i.e. the eigenvalues) are treated as associated $S$-matrix poles. Accordingly, in the passive cavity three distinguish poles are precisely chosen over a particular frequency range. Then introducing the spatially unbalanced gain-loss profile by tuning the parameters $\gamma$ and $\tau$ the chosen poles are forced to interact mutually. It has been observed that with introduction of non-uniform gain-loss in the cavity, a pole belonging to the chosen set is being coupled with the pole situated at the next-nearest-neighbor position, whereas the intermediate pole remains unaffected. Now, for two distinct values of $\tau$, the evolution of resonance energies and widths with increasing amount of the parameter $\gamma$ are plotted in Fig. \[figure\_4\] in terms of the $\Re(k)$ and $\Im(k)$ (i.e. real and imaginary part of the frequencies) of the chosen S-matrix poles. At first, we have set the parameter $\tau=5.32$ and accordingly slowly tuned the gain-coefficient $\gamma$ from $0$ to $0.1$. In this situation the trajectories of $S$-matrix poles are depicted in Fig. \[figure\_4\](a). The level repulsion phenomenon between the poles appearing in the next neighbor position are clearly observed in the upper panel with a zoomed in view. With increase in $\gamma$, the $\Re(k)$ experiences crossing whereas the $\Im(k)$ undergoes anti-crossing as shown in the lower panel. But for slight increase in $\tau=5.33$, a different behavior of level repulsion phenomenon has occurred as shown in Fig. \[figure\_4\](b); where the exchange in identities between the coupled poles (i.e. change in evolution direction from the previous case) have clearly observed in the upper panel. In this situation $ARC$ occurs with $\Re(k)$ undergoing anti-crossing and $\Im(k)$ experiencing crossing as shown in lower panel. Now, in both cases it is clearly observed that the intermediate pole is not effected by the incorporated non-hermiticity in the resonator. Even in significant change of the parameter $\tau$ it remains unaffected by other coupled poles and behaves as an isolated pole with change in the parameter $\gamma$.
Thus the behavior of $ARC$s between the coupled poles for two different $\tau$ values as depicted in Fig. \[figure\_4\](a) and \[figure\_4\](b) are topologically dissimilar. So, there must be an abrupt transition between two $\tau$-values where the coupled poles coalesce at a critical square root singular point in $(\gamma, \tau)$-plane; at which the associated eigenfunctions loose their identities. In complex $k$-plane, the positions approximately indicated by red crosses in upper panel (right side) of both Fig. \[figure\_4\](a) and \[figure\_4\](b) respectively are identified as the appearance of such singular point. For our chosen specific set of cavity parameters, in $(\gamma, \tau)$-plane the position of this singular point have found at $\sim(0.055, 5.328)$.
Cascaded state flipping around the identified singularity
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![(Color online) Trajectories of the three poles in complex $k$-plane (initial positions are marked by the brown circles) associated with the identified singularity denoted by red cross in $(\gamma, \tau)$-plane at inset for an encircling process (blue circular path at inset) centering it with $a=0.04\ a.u.$. In $k$-plane, dotted red and blue lines represent the trajectories of the coupled poles whereas dotted black line represent the trajectories of the intermediate isolated poles after one round encirclement around the singularity in $(\gamma, \tau)$-plane. The dynamics of coupled poles are depicted with respect to the $\Im(k)$ axis where the ticks labels are shown in the right sides, while the left side distribution in ticks labels correspond to the $\Im(k)$ axis to depict the dynamics of isolated pole. Such two different distribution in ticks labels corresponding to the $\Im(k)$ axis is considered for clear visibility. A zoomed in view around the passive position of the intermediate pole is also shown for clear visibility in loop formation.[]{data-label="figure_5"}](figure5.eps){width="8.5cm"}
In this part we have explored an unique feature of the designed microcavity in the vicinity of the identified hidden singularity towards flipping of cavity resonances in the context of optical mode converter. Accordingly, we study the effect of encircling around this singular point. We choose circle with center at $(\gamma_0, \tau_0)$ as a closed loop in $2D$ ($\gamma, \tau$)-plane which can be expressed by the following parametric equation [@Cartarius3; @Laha2; @Ghosh; @Menke]
$$\begin{aligned}
&\gamma(\phi)=\gamma_{0}\left[1+a\,cos(\phi)\right]\\
&\tau(\phi)=\tau_{0}\left[1+a\,sin(\phi)\right]
\end{aligned}$$
\[equation\_13\]
where, $a$ $(\in [0,1])$ represents a certain small characteristics parameter (equivalent to radius of the circle) and $\phi$ $(\in [0, 2\pi])$ is a tunable angle. This method opens up a possibility to scan a large area around the singularity at once. Choosing enough small steps on the enclosing loop, motion of resonances can be properly traced. Once the described circle in parameter place is traced, successively the position exchanging behavior between the eigenvalues gives the proof about existence of an exceptional point [@Cartarius3; @Laha2; @Ghosh; @Menke].
Now choosing the identified singularity as the center, a closed contour is patterned in ($\gamma, \tau$)-plane with $a=0.04$ $a.u.$ as shown at the inset of Fig. \[figure\_5\]. The parameter $a$ have chosen in such a way that the described contour should rightly enclose the identified singularity. An anticlockwise operation has been performed along this closed contour. Now Fig. \[figure\_5\] shows how the coupled pair of poles (appear in next nearest neighbor positions) are associated with the singularity and the intermediate isolated pole is affected by such encircling process in complex $k$-plane. Here, in $k$-plane each point on the red blue and black trajectories indicate the point-to-point evolution of $S$-matrix poles from their starting positions (represented by the brown circles) with associated encircling process (following green circle at the inset) around the respective singularity (denoted by red cross at the inset) in ($\gamma, \tau$)-plane. Interestingly, as the result of one round encircling process in parameter plane around the singularity, two coupled poles have exchanged their positions in complex $k$-plane. However, the intermediate pole has completed an individual loop in same plane i.e. after complete of encircling process it returns to its initial position. Thus the intermediate pole remains unaffected by the presence of singularity inside the enclosing loop. Accordingly, another encirclement around the singularity (i.e. total two successive rounds) results that the pair of coupled poles regain their initial positions with formation of a complete loop; whereas, intermediate pole makes an extra loop exactly along the previous path in $k$-plane. Such position exchanging behavior between the coupled poles may be called as the *flip-of-states*. Trajectories marked by dotted red, blue and black curves are associated with the pair of coupled poles and intermediate isolated pole respectively. Arrows indicate the direction of progression in both $k$-plane and ($\gamma, \tau$)-plane.
Now, we demonstrate a numerical observation behind the reason for isolation of intermediate pole even in presence of sufficiently effective pumping in terms of spatially unbalanced gain-loss profile. The complex cavity resonances are listed in the table \[table\_1\] for both passive and initial pumped conditions.
------------------------ ---------- ---------- ---------- ----------
$\Re(k)$ $\Im(k)$ $\Re(k)$ $\Im(k)$
State-1 8.683 - 0.078 8.731 0.021
State-2 (intermediate) 8.726 - 0.118 8.306 - 1.765
State-3 8.770 - 0.078 8.730 0.068
------------------------ ---------- ---------- ---------- ----------
: Complex resonances of the microcavity for both passive and initial pumped conditions. All values are given in absolute unit.
\[table\_1\]
It has been observed that in passive cavity the imaginary part of the resonance frequencies ($\Im(k)$) are almost equal for two poles appear in next nearest neighbor position whereas the intermediate pole appears at a lower $\Im(k)$. Now when the encircling process around the singularity starts, certain amount of gain and proportionate loss (which depends on position of the respective singularity in ($\gamma, \tau$)-plane and the characteristics parameter $a$ of the enclosing loop) is imposed instantly on each of the three poles; which results in starting their movement. So after this instant initial pumping, it has been significantly noticed that there is a huge change in the value of $\Im(k)$ of the poles appearing in the next nearest neighbor positions, however there is a very small change in $\Im(k)$ of the intermediate pole. Due to this anamolous behavior of poles in $\Im(k)$, we display results in Fig. \[figure\_5\] in a different manner for clear visibility; where in same $k$-plane the trajectories of two coupled pole and the intermediate pole are depicted with two different distribution in $\Im(k)$-axis as shown in the right and left sides respectively. For clear visibility in the loop formation for the case of intermediate pole, we also present a zoomed out view around its position in $k$-plane. Essentially, the factor $\Im(k)$ physically represents the resonance width. Thus from this numerical observation it can be said that the resonance width may be responsible for this anamolous interaction process [@Menke]. Because of sufficient change in $\Im(k)$, the poles appearing in the next nearest neighbor positions are coupled as we increase the pumping, and depicts the phenomena of flip-of-sates following the adiabatic encirclement around associated singularity. Whereas due to exact opposite behavior of $Im(k)$ (i.e. very little change due to initial pumping) of the intermediate state, it remains unaffected by the presence of singularity even in addition of sufficient pumping.
![Approximate locations of three embedded $EP$s denoted by three red crosses in the ($\gamma, \tau$)-plane. Blue dashed line represents the exceptional line with negative tangent.[]{data-label="figure_6"}](figure6.eps){width="7cm"}
![(Color online) Trajectories of the coupled eigenvalues (initial positions are marked as the brown circles) corresponding to all three consecutive $EP$s in complex $k$-plane (denoted as three red crosses at the inset) for a common encircling process in $(\gamma, \tau)$-plane around the center at $\sim(0.056, 5.325)$ (marked as blue dot in inset) with $a=0.065\ a.u.$. in presence of modest random fluctuations on the enclosing contour (described as blue fluctuated circular path at inset). Here the described prametric path encloses both $EP_1$ and $EP_2$, except $EP_3$.[]{data-label="figure_7"}](figure7.eps){width="13cm"}
Thus the state flipping behavior between the pair of coupled poles associated with the identified singularity clearly establishes the fact that their dynamics is entirely controlled by the exception point ($EP$) even in presence of intermediate isolated pole. Here the $EP$ conventionally exhibits as a second order branch point for eigenvalues. This is the direct observation of exceptional singular behavior of a hidden branch point [@Cartarius3; @Laha2; @Ghosh; @Menke].
Formation of hidden singular line
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Similarly, tuning the factor $\tau$ over the amount of gain-coefficient $\gamma$, we have encountered at least three different $EP$s in $(\gamma,\tau)$-plane with deliberate identification of three different set of poles, where each set must contain three distinguished poles according to the proposed scheme. To correlate all the identified $EP$s, we plot them in $(\gamma, \tau)$-plane as shown in Fig. \[figure\_6\] where each red cross denotes each of the embedded $EP$s in the cavity. Here, blue dotted line gives the best fitting which indicates that the identified $EP$s follow a special straight line in parameter plane which may be called as hidden singular line. Here such hidden singular line is termed as [*exceptional line*]{}. Previously, this formation of exceptional line has been reported for the first time by the authors while considering the nearest neighbor coupling situations between the consecutive poles to explore $EP$s in a different class of Fabry-Perot microcavities [@Laha2]. As all the identified second order $EP$s in the designed cavity are correlated by an single exceptional line, successive state switching between the $EP$s has been achieved straight forwardly i.e. simply by tuning the coupling parameters $\gamma$ and $\tau$ adiabatically. Towards the exploration of unconventional optical effects associated with $EP$s, the formation of such exceptional line brings in a new degree of freedom for manipulation of cavity resonances.
Stable optical performance towards cascaded state-flipping mechanism
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Exploiting the special feature of exceptional line, we explore the stable optical performance of the designed microcavity in the vicinity of identified $EP$s via numerical exemplification of robustness of associated flip-of-state phenomena. A common encircling process in parameter plane has been chosen to explore such flip-of-state phenomena corresponding to all the identified $EP$s in complex $k$-plane at a time. Associated results are displayed in Fig. \[figure\_7\]. To consider all the $EP$s with respect to the same enclosing process, the center of described loop has chosen at any arbitrary point following exceptional line in ($\gamma, \tau$)-plane say at $\sim(0.056, 5.325)$ (denoted by black dot at the inset); i.e. none of the $EP$s represent the center of described contour in $(\gamma, \tau)$-plane. The characteristics parameter $a$ has chosen as $0.075$ $a.u.$ to enclose $EP_1$ and $EP_2$ only; except $EP_3$ (as shown in inset of Fig. \[figure\_7\]). So we can observe the effect of encircling on $EP$s for both inside and outside the enclosing loop in same $k$-plane. Here exceptional line has been exploited to shift the encircling parameter set around the particularly chosen arbitrary center between three identified $EP$s and accordingly the dynamics of the poles corresponding to each $EP$ have been analyzed. We have also added some deliberate random fluctuations on the enclosing loop to substantiate the rigidity of described state-flipping behavior associated with each $EP$ against unwanted fabrication tolerances of state-of-the-art techniques due to various real natural effects.
Now following the one round encircling process along the described contour in $(\gamma, \tau)$-plane, it has been observed that the coupled poles associated with $EP_1$ and $EP_2$ are exchanging their positions in a very generic fashion; while the coupled poles corresponding to $EP_3$ are constructing the individual loops in complex $k$-plane. Accordingly, for two successive encirclement along the contour in parameter plane the eigenvalues corresponding to the first two $EP$s have formed complete loop (two individual complete loop corresponding to different $EP$) in $k$-plane after second permutation; whereas eigenvalues corresponding to $EP_3$ traversed the exact previous path once again avoiding any kind of permutations. We also study the trajectories of intermediate isolated poles associated with each $EP$. However, here also they have behaved in previous manner as expected i.e. all of them are unaffected by presence of other singularities and remain isolated making individual loops followed by the described encirclement process. Hence purposely we exclude the trajectories of isolated poles form Fig. \[figure\_7\] for clear visibility of trajectories of the coupled poles associate with the described encirclement process around the $EP$s. Thus the flip-of-states phenomena around $EP$s is omnipresent until the one-to-one coupling restriction is topologically preserved and the identified $EP$s must be untouched by the deformations in parametric loop. Contextually, due to substantial modifications in the encircling process the microcavity may support secondary states unconventionally which can interact with both scattering as well as isolated states in the microcavity [@Laha2]. As a result, described one-to-one coupling restriction between the resonances with an intermediate isolated resonance may be interrupted and then the singular behavior of the identified $EP$s may be destroyed i.e. state-flipping behavior should no longer be stabled.
Thus from the results described in Fig. \[figure\_7\] the following conclusions should be drawn. The state-flipping behavior in $k$-plane corresponding to each identified $EP$ is independent as unaffected by the presence of any other $EP$ inside or outside the contour in $(\gamma, \tau)$-plane. Such optical performances of the designed cavity present the robust behavior even in presence of parameter fluctuations/ deformations. Hence, robust behavior of flip-of-state phenomena is extremely promising for device level implementation using any state-of-the-art fabrication technique with modest tolerances.
Conclusions
===========
In summary, using $S$-matrix formalism we have modeled a non $\mathcal{PT}$-symmetric two port open Fabry-Perot styled optical microcavity to explore a non-trivial next nearest neighbor resonance interaction which is entirely controlled by the system topology i.e. geometry of the cavity and spatial distribution of unbalanced gain-loss. Non-uniform variation in background real refractive index has adopted purposely to introduce an inherent non-linearity in distribution of $S$-matrix poles in complex eigenvalue plane. Adjusting the factor $\tau$ over gain co-efficient $\gamma$, three different second order $EP$s have been embedded in operating parameter plane of the cavity under strict restriction of one-to-one coupling. Unveiling the formation of special exceptional line in the parameter plane supported by each identified $EP$s, we explore unique cascaded state-flipping mechanism between the coupled poles corresponding to the encircled $EP$s with successive state switching between them along the reported special line. We have established that if an $EP$ is rightly encircled by the parametric loop which may centered either at that $EP$ or any arbitrary point following the exceptional line, then associated flip-of-state phenomena is ubiquitous. Moreover, this occurs irrespective from presence of any other $EP$ inside the described parametric loop. Such state-flipping mechanism is evident even in presence of moderate deformation/ fluctuations on parameter variation during encircling process. Hence exploring the special next-nearest-neighbor coupling scheme and exploiting the concept of exceptional line, stable optical performance of such degenerate microcavity has been achieved by establishing the robustness of unique state-flipping behavior in the vicinity of $EP$s even in presence of an intermediate isolated resonance between two coupled resonances. Recent developments in the fabrication technology for growth of such optical microcavities with high precision and control may open up a new platform to implement high performances integrated photonic devices on chip.
Funding Information {#funding-information .unnumbered}
===================
This work acknowledges the financial support by Department of Science and Technology (DST), India under the INSPIRE Faculty Fellow grant \[IFA-12; PH-23\].
[1]{}
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| {
"pile_set_name": "ArXiv"
} |
High-$T_c$ superconductivity occurs when the parent antiferromagnetic (AF) insulator with the CuO$_2$ plane is doped with holes or electrons. In the [*p*]{}-type materials, the long-range AF order vanishes for a slight amount of hole doping whereas in the [*n*]{}-type materials, the AF order persists up to a high doping concentration of ${\sim}$0.14 electrons per Cu and the superconducting (SC) doping range is much narrower [@Tokura]. The [*p*]{}-type materials show $T$-linear in-plane electrical resistivity [@Takagi] and split neutron peaks around ${\bf q} = (\pi,\pi)$ indicating incommensurate spin fluctuations [@LSCO_neu1] whereas the [*n*]{}-type materials show $T^2$ dependence of the in-plane resistivity [@NCCO_res] and $(\pi,\pi)$ commensurate spin fluctuations [@NCCO_neu]. In order to elucidate the mechanism of high-$T_c$ superconductivity, it is very important to clarify the origin of the similarities and the differences between the [*p*]{}-type and the [*n*]{}-type materials.
In this Letter, we report on a study of the chemical potential shift in Nd$_{2-x}$Ce$_{x}$CuO$_{4}$ (NCCO) as a function of doped electron concentration. The shift can be deduced from the core-level shifts in photoemission spectra because the binding energy of each core level is measured relative to the chemical potential $\mu$. In a previous study [@LSCO_mu], we found that in La$_{2-x}$Sr$_{x}$CuO$_{4}$ (LSCO) the chemical potential shift is unusually suppressed in the underdoped region and attributed this observation to the strong stripe fluctuations which exist in this system. As for the chemical potential jump between La$_2$CuO$_4$ and Nd$_2$CuO$_4$, which would represent the band gap of the parent insulator, it was estimated to be at most 300 meV in previous valence-band photoemission studies [@NCCO_AIPES1; @NCCO_AIPES2], which is much smaller than the 1.5–2.0 eV charge-transfer (CT) gap of the parent insulator estimated from optical studies [@Uchida].
High-quality single crystals of NCCO ($x=$ 0, 0.05, 0.125 and 0.15) were grown by the traveling-solvent floating-zone method as described elsewhere [@Onose]. Uncertainties in the Ce concentration were ${\pm}0.01$. For $x=0.15$, both as-grown and reduced samples were measured while for the other compositions only as-grown samples were measured. The as-grown samples were all antiferromagnetic and did not show superconductivity. Only the $x=0.15$ sample showed superconductivity after reduction in an Ar atmosphere and its $T_c$ was ${\sim}25$ K.
X-ray photoemission spectroscopy (XPS) measurements were performed using both the Mg $K{\alpha}$ (${\it h}{\nu} = 1253.6~$eV) and Al $K{\alpha}$ (${\it h}{\nu} = 1486.6~$eV) lines and a hemispherical analyzer. All the spectra were taken at liquid-nitrogen temperature (${\sim}80$ K) within $40$ minutes after scraping. We did not observe a shoulder on the higher binding energy side of each O 1[*s*]{} peak, indicating the high quality of the sample surfaces free from degradation. Although the energy resolution was about $0.8$ eV for both $K{\alpha}$ lines, we could determine the core-level shifts with an accuracy of about ${\pm}50$ meV because most of the spectral line shapes did not change with $x$. In XPS measurements, a high voltage of $>1$ kV is used to decelerate photoelectrons, and it is usually difficult to stabilize the high voltage with the accuracy of $\ll 100$ meV. In order to overcome this difficulty, we directly monitored the voltage applied to the outer hemisphere and the retarding fringe, and confirmed that the uncertainty could be reduced to less than $10$ meV. To eliminate other unexpected causes of errors, we measured the $x=0.05$ sample as a reference just after the measurement of each sample.
Figure 1 shows the XPS spectra of the O 1[*s*]{}, Nd 3$d_{5/2}$ and Cu 2$p_{3/2}$ core levels taken with the Al $K{\alpha}$ line. Here, the integral background has been subtracted and the intensity has been normalized to the peak height [@rem]. The Nd 3$d_{5/2}$ spectra are composed of the 3$d_{5/2}$4$f^4$ and 3$d_{5/2}$4$f^3$ final-state components, where denotes a ligand hole, and O [*KLL*]{} Auger signals overlap them. The Cu 2$p_{3/2}$ spectra are composed of the 2$p_{3/2}$3$d^{10}$ and 2$p_{3/2}$3$d^{9}$ components, but only the 2$p_{3/2}$3$d^{10}$ peaks are shown in the figure. One can see the obvious doping dependent shifts of O $1s$ and Nd $3d$ core levels from both the displaced and overlayerd plots in Fig. 1. To deduce the amount of the core-level shifts reliably, we used the peak position for the Nd 3[*d*]{} spectra and the mid point of the lower binding energy slope for the O 1[*s*]{} spectra. We used the mid-point position rather than the peak position for O $1s$ because the line shape on the higher binding energy side of the O 1[*s*]{} peak was sensitive to a slight surface degradation or contamination. The Cu 2[*p*]{} core-level line shape was not identical between different $x$’s, and becomes broader as $x$ increases. This is because the doped electrons in the CuO$_{2}$ plane produce Cu$^{1+}$ sites on the otherwise Cu$^{2+}$ background, which yield an overlapping chemically shifted component located on the lower binding side of the Cu$^{2+}$ peak. Therefore, it was difficult to uniquely determine the shift of the Cu 2[*p*]{} core level and we only take its peak positions in the following.
Figure 2 shows the binding energy shift of each core level relative to the as-grown $x=0.05$ sample. Here, we have assumed that the change of the electron concentration caused by the oxygen reduction was ${\sim}0.04$ per Cu (oxygen reduction being ${\sim}0.02$) as reported previously [@oxygen_reduction]. One can see that the Nd 3[*d*]{} and O 1[*s*]{} levels move toward higher binding energies with electron doping. The shift of Cu 2[*p*]{} is defined by the shift of the peak position, and is in the opposite direction to Nd 3[*d*]{} and O 1[*s*]{} because of the Cu$^{1+}$ components mentioned above. We also measured the shifts of the core levels using the Mg $K{\alpha}$ line and almost the same results were obtained as shown in Fig. 2.
While the shift of the chemical potential changes the core-level binding energy, there is another factor that could affect the binding energy, that is, the change in the Madelung potential due to Ce$^{4+}$ substitution for Nd$^{3+}$. However, the identical shifts of the O 1[*s*]{} and Nd 3[*d*]{} core levels indicate that the change in the Madelung potential has negligible affects on the core-level shifts because it would shift the core levels of the O$^{2-}$ anion and Nd$^{3+}$ cation in the opposite directions. Moreover, as the shifts of the O $1s$ and Nd $3d$ core levels toward higher binding energies with electron doping are opposite to what would be expcted from increasing core-hole screening capability with $x$, excluding the core-hole screening mechanism as the main cause of the core-level shifts. Therefore, we conclude that the shifts of the O $1{\it s}$ and Nd $3{\it d}$ core levels are largely due to the chemical potential shift ${\Delta}{\mu}$. We have evaluated ${\Delta}{\mu}$ in NCCO by taking the average of the shifts of the two core levels.
Figure 3(a) shows ${\Delta}{\mu}$ in NCCO as well as ${\Delta}{\mu}$ in LSCO [@LSCO_mu] as a function of electron or hole carrier concentration. In order to obtain the jump in $\mu$ between Nd$_{2}$CuO$_4$ and La$_{2}$CuO$_4$, we also measured the O $1{\it s}$ and Cu $2{\it p}$ levels in LSCO as shown in Fig. 4, and found that the O $1{\it s}$ and Cu $2{\it p}$ levels in Nd$_{2}$CuO$_{4}$ lie at ${\sim}150$ meV and ${\sim}400$ meV higher binding energies than those in La$_{2}$CuO$_{4}$, respectively. The fact that the observed jump is different between O 1[*s*]{} and Cu 2[*p*]{} is not surprising because Nd$_{2}$CuO$_{4}$ and La$_{2}$CuO$_{4}$ are different materials with different crystal structures. Thus the chemical potential jump between Nd$_{2}$CuO$_{4}$ and La$_{2}$CuO$_{4}$ cannot be uniquely determined from those data but it should be much smaller than the CT gap of about $1.5$ eV for Nd$_{2}$CuO$_{4}$ and $2.0$ eV for La$_{2}$CuO$_{4}$ estimated from the optical measurements [@Uchida]. This small jump is in accordance with the early valence-band photoemission studies of LSCO and NCCO [@NCCO_AIPES1; @NCCO_AIPES2].
Figure 3(a) demonstrates the different behaviors of ${\Delta}{\mu}$ between LSCO and NCCO. In LSCO, ${\Delta}{\mu}$ is suppressed in the underdoped region $x {\le}0.13$, whereas in NCCO ${\Delta}{\mu}$ monotonously increases with electron doping in the whole concentration range. Figure 3(c) represents the phase diagram of LSCO and NCCO drawn against the chemical potential ${\mu}$ and the temperature $T$. One can see that the ${\mu}-T$ phase diagram is rather symmetric between the hole doping and electron doping unlike the widely used ${\it x}-T$ phase diagram. That is, in both the electron- and hole-doped cases, the SC region is adjecent to the AF region, as proposed by Zhang [@HTSC_phase] based on SO(5) symmetry. The present phase diagram implies that as a function of $\mu$, $T_c$ increases up to the point where the superconductivity is taken over by the AF ordering. Such a behavior is reminiscent of the superfluid-solid transition in the $p-T$ phase diagram of $^3$He [@tesa].
The monotonous increase of $\mu$ in NCCO may be understood within the simple rigid-band model. The chemical potential of Nd$_{2}$CuO$_{4}$ lies near the bottom of the conduction band. When electrons are doped, ${\mu}$ moves upward into this band as long as the AF ordering and hence the AF band structure persist as in NCCO. This behavior is contrasted with the suppression of ${\Delta}{\mu}$ in underdoped LSCO, where the AF ordering is quickly destroyed and the electronic structure is dramatically reorganized by a small amount of hole doping. It has been suggested [@LSCO_mu] that the suppression of ${\Delta}{\mu}$ is related to the charge fluctuations in the form of stripes in LSCO. Indeed in La$_{2-x}$Sr$_{x}$NiO$_{4}$ (LSNO), where static stripe order is stable at $x\simeq$ 0.33 [@LSNO_exp], ${\Delta}{\mu}$ is anomalously suppressed in the underdoped region $x{\le}0.33$ [@LSNO_mu]. In LSCO, dynamical stripe fluctuations have been implied by inelastic neutron scattering studies [@LSCO_neu1; @LSCO_neu2] whereas any sign of stripes is absent in the neutron study of NCCO [@NCCO_neu]. In Fig. 3, we compare the chemical potential shift ${\Delta}{\mu}$ with the incommensurability ${\epsilon}$ which was deduced from the neutron experiments both for LSCO [@LSCO_neu1] and NCCO [@LSCO_neu2]. This figure shows that in the region where $\mu$ does not move ($x{\le}0.13$ in LSCO), ${\epsilon}$ linearly increases with $x$. In NCCO, where the chemical potential ${\mu}$ monotonously moves upward, the incommensurability does not change with $x$ (remains zero), in other words, static nor dynamical stripes do not exist. In the overdoped region of LSCO, the number of stripes saturates, doped holes overflow into the interstripe region and $\mu$ moves fast with hole doping.
The smallness of the chemical potential jump between Nd$_{2}$CuO$_{4}$ and La$_{2}$CuO$_{4}$ indicates that $\mu$ lies within the CT gap of Nd$_{2}$CuO$_{4}$ ($\sim$1.5 eV) and La$_{2}$CuO$_{4}$ ($\sim$2.0 eV). This behavior was clearly observed in an angle-resolved photoemission (ARPES) study of LSCO [@LSCO_ARPES], where $\mu$ is located $\sim$0.4 eV above the top of the valence band of La$_{2}$CuO$_{4}$. Such a behavior is quite peculiar from the view point of the rigid-band model, and therefore a dramatic change in the electronic structure should occur with hole doping. In NCCO, however, the doping-induced change is not so dramatic as in LSCO as mentioned above, and the chemical potential pinning well below the bottom of the conduction band is not very likely. Another possible cause of the small chemical potential jump is that the CT gap is indirect and is smaller than that estimated from the optical studies. In optical conductivity spectra, only the direct transition can be unambiguously measured, and if the gap is indirect, the gap would be estimated much larger than the CT gap. This idea is consistent with the recent resonant inelastic x-ray scattering study [@RIXS] and its theoretical analysis using $t$-$t^{'}$-$t^{''}$-$U$ model [@RIXS_the], which has yielded an indirect gap that is smaller than the direct one by $\sim 0.5$ eV.
In summary, we have experimentally determined the doping dependence of the chemical potential shift ${\Delta}{\mu}$ in NCCO and observed a monotonous shift with doping. Comparison with LSCO indicates that the change in the electronic structure with carrier doping is more moderate in NCCO. The monotonous shift is consistent with the observation that spin fluctuations are commensurate in NCCO. The small chemical potential jump between the $n$-type and $p$-type materials is confirmed and is attributed to the indirect CT gap and the chemical potential pinning within the CT gap in LSCO.
The authors would like to thank S. Tesanovic, A. Ino and T. Mizokawa for enlightening discussions. Collaboration with G. A. Sawatzky and J. van Elp in the early stage of this work is gratefully acknowledged. This work was supported by a Grant-in-Aid for Scientific Research in Priority Area “Novel Quantum Phenomena in Transition Metal Oxides” and a Special Coordination Fund for the Promotion of Science and Technology from the Ministry of Education and Science and by New Energy and Industrial Technology Development Organization (NEDO).
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- '[**Mikhail Kamenskii, Oleg Makarenkov, Paolo Nistri**]{}'
title: |
\
[**Applications to Periodically Perturbed Autonomous Systems**]{}\
[**(Dedicated to Prof. R. Johnson on the occasion of his 60th birthday)**]{}
---
[**Abstract.**]{} [By means of a linear scaling of the variables we convert a singular bifurcation equation in ${{\mathbb{R}}}^n$ into an equivalent equation to which the classical implicit function theorem can be directly applied. This allows to deduce the existence of a unique branch of solutions as well as a relevant property of the spectrum of the derivative of the singular bifurcation equation along the branch. We use these results to show the existence, uniqueness and the asymptotic stability of periodic solutions of a $T$-periodically perturbed autonomous system bifurcating from a $T$-periodic limit cycle of the autonomous unperturbed system. This problem is classical, but the novelty of the method proposed is that it allows us to solve the problem without any reduction of the dimension of the state space as it is usually done in the literature by means of the Lyapunov-Schmidt method.]{}
[***AMS Subject Classification:***]{} 37G15, 34E10, 34C25.
[***Key words:***]{} Bifurcation equation, autonomous system, limit cycle, periodic perturbations, Poincaré map, periodic solutions, stability.
Introduction
============
In Section 2 we consider an abstract bifurcation equation of the form $$\label{1}
\Phi(v,{\varepsilon}):=P(v)+{\varepsilon}Q(v,{\varepsilon})=0$$ where $P\in C^2({{\mathbb{R}}}^n, {{\mathbb{R}}}^n), Q\in C^1({{\mathbb{R}}}^n\times[0,1], {{\mathbb{R}}}^n)$ and, for ${\varepsilon}>0$ sufficiently small, we look for the existence of zeros $v_{\varepsilon}$ of the map $\Phi$. Here it is assumed the existence of a $v_0\in {{\mathbb{R}}}^n$ such that $P(v_0)=0$ with the matrix $P'(v_0)$ singular. In other words, we deal with an abstract singular bifurcation problem in ${{\mathbb{R}}}^n$ with a small bifurcation parameter ${\varepsilon}>0$. Due to the singularity of $P'(v_0)$ it is not possible to use directly to (\[1\]) the classical implicit function theorem to show the existence and uniqueness of a branch $\{v_{\varepsilon}\}$, ${\varepsilon}>0$ small, of solutions of the equation $\Phi(v,{\varepsilon})=0$.
In this paper, by means of a linear scaling of the variables $v\in {{\mathbb{R}}}^n$ we convert the problem of finding zeros of (\[1\]) to the problem of finding zeros of a map $\Psi(w,{\varepsilon})$ for which there exists a unique $w_0\in {{\mathbb{R}}}^n$ such that $\Psi(w_0,0)=0$ and $\Psi'_w(w_0,0)$ is not singular. Therefore, the new bifurcation equation $\Psi(w,{\varepsilon})=0$ can be solved by means of the classical implicit function theorem to conclude the existence and uniqueness of a branch of zeros $\{w_{\varepsilon}\}$, for ${\varepsilon}>0$ small. The advantage and the novelty of the approach is that getting the equation $\Psi(w,{\varepsilon})=0$ does not require solving any implicit equations which is usually done when applying the Lyapunov-Schmidt reduction approach (see [@ch], Ch. 2, § 4).
Our bifurcation equation $\Psi(w,{\varepsilon})=0$ is, therefore, formally different from that given by Lyapunov-Schmidt reduction (see e.g. [@l]). That is why we show in Section 3 that applying our general result to the perturbed autonomous system $$\label{2}
\dot x=f(x)+{\varepsilon}g(t,x,{\varepsilon}).$$ where $f\in C^2({{\mathbb{R}}}^n, {{\mathbb{R}}}^n), g\in C^1({{\mathbb{R}}}\times {{\mathbb{R}}}^n \times
[0,1], {{\mathbb{R}}}^n)$ is $T$-periodic and ${\varepsilon}>0$ is small, leads to the same classical Malkin-Loud (or sometimes called Melnikov) bifurcation function. We end up, therefore, with the statement that a well known classical result on the existence, uniqueness and asymptotic stability of a family of $T$-periodic solution of (\[2\]) bifurcating from the $T$-periodic limit cycle $x_0$ of the autonomous system $\dot x=f(x)$ (see Malkin [@m], Loud [@l], Blekhman [@ble]) follows from our bifurcation theorem, while avoiding the Lyapunov-Schmidt reduction reduces the analysis significantly.
A first result in this direction has been obtained by the authors in [@enoc08] by means of a version of the implicit function theorem for directionally continuous functions, see [@bressan]. The idea of using the linear scaling has been, therefore, already reported at the conference [@enoc08]. But the approach in [@enoc08] is based on the employ of isochronous surfaces of the Poincaré map transversally intersecting the limit cycle $x_0$ that requires a non-trivial information about smoothness of these surfaces, while the considerations in this paper rely on very basic facts of analysis only.
The paper is organized as follows. In Section 2 we first reduce the abstract singular bifurcation equation (\[1\]) to an equivalent non-singular bifurcation equation, then in Theorem 1 we provide conditions under which the non-singular problem satisfies the assumptions of the classical implicit function theorem. Furthermore, in Theorem 2 we establish a relevant property of the spectrum of the derivative of the singular bifurcation equation along the branch which permits to study the asymptotic stability of the bifurcating zeros. In Section 3, under the standard assumption that the Malkin’s bifurcation function associated to (\[2\]) has non-degenerate zeros, the results stated in Section 2 permit to show (Theorem 3) the existence of a parametrized family of $T$-periodic solutions of (\[2\]) bifurcating from the $T$-periodic limit cycle of the unperturbed system as well as their asymptotic stability. The main tools to prove Theorem 3 consist in a representation formula for the Malkin’s bifurcation function in terms of the $T$-periodic perturbation of the autonomous system and of a formula for its derivative. These formulas are stated in Lemma 2 and Lemma 3 respectively.
Variables scaling to transform a singular bifurcation problem into a non-singular one
=====================================================================================
Consider the function $\Phi: {{\mathbb{R}}}^n\times[0,1]\to {{\mathbb{R}}}^n$ defined by $$\label{bif}
\Phi(v,{\varepsilon})=P(v)+{\varepsilon}Q(v,{\varepsilon})$$ where $P\in C^2({{\mathbb{R}}}^n, {{\mathbb{R}}}^n), Q\in C^1({{\mathbb{R}}}^n\times[0,1], {{\mathbb{R}}}^n)$ and ${\varepsilon}>0$ is a small parameter.
In this Section, assuming the existence of $v_0\in {{\mathbb{R}}}^n$ such that $P(v_0)=0$ with $P'(v_0)$ singular, we provide a method to show the existence and the uniqueness of the solution $v_{\varepsilon}$ of the equation $$\Phi(v,{\varepsilon})=0$$ for ${\varepsilon}>0$ sufficiently small, without using the usual Lyapunov-Schmidt reduction approach. To this aim we assume the existence of a linear projector $\Pi:{{\mathbb{R}}}^n\to{{\mathbb{R}}}^n$ such that $\mbox{Im}\,\Pi \bigoplus \mbox{Ker}\,\Pi={{\mathbb{R}}}^n,$ $\mbox{Im}\,\Pi$ and $\mbox{Ker}\,\Pi$ are invariant subspaces under $P'(v_0)$ and $\Pi P'(v_0)=\Pi Q(v_0,0)=0$.
Since $P'(v_0)$ is singular we cannot apply the classical implicit function theorem, see e.g. [@impl-book], to study the existence of connected components of zeros of $\Phi$ emanating from $(v_0,0)$. Observe that, in general, as it is shown in [@l] and [@mn1], there could exist several branches of zeros of $\Phi$ emanating from $(v_0,0)$. In this paper we provide conditions (which are apparently generic when applying the result to differential equations, see Section 3) under which the branch is unique. In particular in Section 3, such conditions are expressed in terms of the Malkin bifurcation function associated to (\[2\]), see [@m]. More precisely, in Section 3 we have $v_0=x_0(\theta_0)$, where $x_0$ is a one parameter curve of zeros of $P$ and $\theta_0$ is a non-degenerate simple zero of the Malkin bifurcation function. The approach to achieve this result is commonly based on the classical Lyapunov-Schmidt reduction method. In the infinite dimensional case, see [@henry] and more recently [@banach-kmn].
In this paper we propose a different approach based on an equivalent formulation of the problem. More precisely, by means of a scaling of the variables, we rewrite the problem of finding zeros of $\Phi(v,{\varepsilon})$, for ${\varepsilon}>0$ small, as a non-singular bifurcation problem to which apply the classical implicit function theorem. Namely, we associate to the map $\Phi$ the following function $$\label{phi}
\Psi(w,{\varepsilon})=\dfrac{1}{{\varepsilon}}\left(\Phi(v_0+{\varepsilon}w,{\varepsilon})-\Pi\Phi(v_0+{\varepsilon}w,{\varepsilon})+\dfrac{1}{{\varepsilon}}\Pi \Phi(v_0+{\varepsilon}w,{\varepsilon})\right),$$ for any $w\in\mathbb{R}^n$ and any ${\varepsilon}>0$, and we look for zeros of $\Psi$ branching from some $(w_0,0).$ Indeed, as it is easy to see, $(v,{\varepsilon})\in\mathbb{R}^n\times[0,1]$ is a zero of $\Phi$ if and only if $\left(\dfrac{v-v_0}{{\varepsilon}},{\varepsilon}\right)$ is a zero of $\Psi.$
0.2truecm In the sequel the vector space of linear operators $L:{{\mathbb{R}}}^n\to{{\mathbb{R}}}^n$ will be denoted by $\mathcal{L}(\mathbb{R}^n)$. Next Lemma provides the main properties of the function $\Psi$.
\[lem1\] Assume that $P\in
C^2(\mathbb{R}^n,\mathbb{R}^n)$ and $Q\in
C^1(\mathbb{R}^n\times[0,1], \mathbb{R}^n)$. Let $v_0\in\mathbb{R}^n$ be such that $P(v_0)=0$ and $P'(v_0)$ singular. Let $\Pi:\mathbb{R}^n\to\mathbb{R}^n$ be a linear projector invariant with respect to $P'(v_0)$ such that $\Pi
P'(v_0)=\Pi Q(v_0,0)=0.$ Define $\Psi(w,0)$ as follows $$\label{psi}
\Psi(w,0)=\dfrac{1}{2}
\Pi P''(v_0)ww+\Pi Q'_v(v_0,0)w+\Pi
Q'_{\varepsilon}(v_0,0)+(I-\Pi)P'(v_0)w+(I-\Pi)Q(v_0,0)$$ with $$\label{psiprime}
\Psi'_w(w,0)=
\Pi P''(v_0)w+\Pi Q'_v(v_0,0)+(I-\Pi)P'(v_0).$$ Then $\Psi\in C^0(\mathbb{R}^n\times\mathbb{R},\mathbb{R}^n)$ and $\Psi'_w\in
C^0(\mathbb{R}^n\times\mathbb{R},\mathcal{L}(\mathbb{R}^n)).$
[**Proof.**]{} From (\[phi\]) the Taylor expansion with the rest in the Lagrange’s form leads to $$\begin{aligned}
\Pi \Psi(w,{\varepsilon})&=&\frac{1}{{\varepsilon}^2}\Pi \Phi(v_0+{\varepsilon}w,{\varepsilon})=\frac{1}{{\varepsilon}^2}\Pi(P(v_0+{\varepsilon}w)+{\varepsilon}Q(v_0+{\varepsilon}w,{\varepsilon}))=\\
&=&\frac{1}{{\varepsilon}^2}\Pi\left(P(v_0)+{\varepsilon}P'(v_0)w+\frac{1}{2}{\varepsilon}^2 P''\left(v_0+\widehat{{\varepsilon}}(w,{\varepsilon})w\right)ww\
+{\varepsilon}Q(v_0,0)+\right.\\
& &+\left.{\varepsilon}^2
Q'_v\left(v_0+\widetilde{{\varepsilon}}(w,{\varepsilon})w,\widetilde{{\varepsilon}}(w,{\varepsilon})\right)w+{\varepsilon}^2
Q'_{\varepsilon}\left(v_0+\widetilde{{\varepsilon}}(w,{\varepsilon})w,\widetilde{{\varepsilon}}(w,{\varepsilon})\right)\right)\end{aligned}$$ and $$\begin{aligned}
(I-\Pi)\Psi(w,{\varepsilon})&=&\frac{1}{{\varepsilon}}(I-\Pi)(P(v_0+{\varepsilon}w)+{\varepsilon}Q(v_0+{\varepsilon}w,{\varepsilon}))=\\
&=&\frac{1}{{\varepsilon}}(I-\Pi)\left(P(v_0)+{\varepsilon}P'\left(v_0+\overline{{\varepsilon}}(w,{\varepsilon})w\right)w+{\varepsilon}Q(v_0+{\varepsilon}w,{\varepsilon})\right),\end{aligned}$$ where $\widehat{{\varepsilon}}(w,{\varepsilon}),\widetilde{{\varepsilon}}(w,{\varepsilon}),\overline{{\varepsilon}}(w,{\varepsilon})\in[0,{\varepsilon}].$ Using the fact that $P(v_0)=\Pi P'(v_0)=\Pi Q(v_0,0)=0$ we get $$\begin{aligned}
\Psi(w,{\varepsilon})&=&\frac{1}{2}\Pi P''\left(v_0+\widehat{{\varepsilon}}(w,{\varepsilon})w\right)ww
+\Pi
Q'_v\left(v_0+\widetilde{{\varepsilon}}(w,{\varepsilon})w,\widetilde{{\varepsilon}}(w,{\varepsilon})\right)w+\\
& & +
\Pi
Q'_{\varepsilon}\left(v_0+\widetilde{{\varepsilon}}(w,{\varepsilon})w,\widetilde{{\varepsilon}}(w,{\varepsilon})\right)+
(I-\Pi)P'\left(v_0+\overline{{\varepsilon}}(w,{\varepsilon})w\right)w+(I-\Pi)Q(v_0+{\varepsilon}w,{\varepsilon}).\end{aligned}$$ From this formula we conclude that $\Psi\in
C^0(\mathbb{R}^n\times\mathbb{R},\mathbb{R}^n).$
Let us now prove that $\Psi'_w\in
C^0(\mathbb{R}^n\times\mathbb{R},\mathcal{L}(\mathbb{R}^n)).$ The Taylor expansion applied to $P'(v_0+{\varepsilon}w)$ permits to write $$\begin{aligned}
\Pi \Psi'_w(w,{\varepsilon})&=& \frac{1}{{\varepsilon}^2} \Pi({\varepsilon}P'(v_0+{\varepsilon}w) +{\varepsilon}^2 Q'_v(v_0+{\varepsilon}w,{\varepsilon}))= \\&=&\frac{1}{{\varepsilon}^2}\Pi\left({\varepsilon}P'(v_0)+{\varepsilon}^2 P''(v_0+\widetilde{{\varepsilon}}(w,{\varepsilon})w)w+
{\varepsilon}^2 Q'_v(v_0+{\varepsilon}w,{\varepsilon})\right),\\
(I-\Pi)\Psi'_w(w,{\varepsilon})&=& \frac{1}{{\varepsilon}}(I-\Pi)\left({\varepsilon}P'(v_0+{\varepsilon}w)+{\varepsilon}^2 Q'_v(v_0+{\varepsilon}w,{\varepsilon})\right),\end{aligned}$$ where $\widetilde{{\varepsilon}}(w,{\varepsilon})\in[0,{\varepsilon}].$ Taking into account that $\Pi P'(v_0)=0$ we have $$\Psi'_w(w,{\varepsilon})=\Pi P''(v_0+\widetilde{{\varepsilon}}(w,{\varepsilon})w)w+\Pi
Q'_v(v_0+{\varepsilon}w,{\varepsilon})+(I-\Pi)P'(v_0+{\varepsilon}w)+{\varepsilon}(I-\Pi)Q'_v(v_0+{\varepsilon}w,{\varepsilon})$$ and so $\Psi'_w(w,{\varepsilon})\to \Psi'_w(w_0,0)$ as $w\to w_0$ and ${\varepsilon}\to 0.$ This concludes the proof.
An example of linear projector which is invariant with respect to $P'(v_0)$ is the Riesz projector $\Pi_R:{{\mathbb{R}}}^n\to{{\mathbb{R}}}^n$ given by $$\Pi_R:=\dfrac{1}{2\pi i}\int_{\Gamma} (\lambda I- P'(v_0))^{-1}\,
d\lambda,$$ where $\Gamma$ is a circumference centered at $0$ and containing in its interior the only zero eigenvalue of $P'(v_0)$. In fact, by the Riesz decomposition theorem the subspaces $\mbox{Im}\,\Pi_R$ and $\mbox{Ker}\,\Pi_R$ are invariant with respect to $P'(v_0),$ $\mbox{Im}\,\Pi_R \bigoplus \mbox{Ker}\,\Pi_R={{\mathbb{R}}}^n$ and $\Pi_R
P'(v_0)=0$.
We can now prove the following.
\[th1\] Assume that $P\in C^2(\mathbb{R}^n,\mathbb{R}^n)$ and $Q\in C^1(\mathbb{R}^n\times[0,1], \mathbb{R}^n)$. Let $v_0\in\mathbb{R}^n$ be such that $P(v_0)=0$ and $P'(v_0)$ is singular. Let $\Pi:\mathbb{R}^n\to\mathbb{R}^n$ be a linear projector (not necessary one-dimensional) invariant with respect to $P'(v_0)$ with $P'(v_0)$ invertible on $(I-\Pi)\mathbb{R}^n.$ Finally, assume that $\Pi Q(v_0,0)=0,$ $\Pi P''(v_0)\Pi\,r\;
\Pi\, s=0 \quad \mbox{for any} \quad r,s\in\mathbb{R}^n,$ and that $$\label{INV}
-\Pi
P''(v_0)(I-\Pi)\left(P'(v_0)|_{(I-\Pi)\mathbb{R}^n}\right)^{-1}Q(v_0,0)+\Pi
Q'_v(v_0,0)$$ is invertible on $\Pi\mathbb{R}^n.$ Then there exists a unique $w_0\in\mathbb{R}^n$ such that $\Psi(w_0,0)=0$ and $\Psi'_w(w_0,0)$ is non-singular.
[**Proof.**]{} We start by showing the existence of a $w_0\in\mathbb{R}^n$ such that $\Psi(w_0,0)=0.$ First, observe that applying $(I-\Pi)$ to (\[psi\]) we obtain the map $w \to
(I-\Pi) P'(v_0)w + (I-\Pi) Q(v_0,0)$ and the equation $$\label{eqn1}
(I-\Pi)P'(v_0)w +(I-\Pi)Q(v_0,0)=(I-\Pi)P'(v_0)(I-\Pi)w +(I-\Pi)Q(v_0,0)=0$$ is solvable with respect to $(I-\Pi)w$; in fact by our assumptions $$w_1=-\left(\left.P'(v_0)\right|_{(I-\Pi)\mathbb{R}^n}\right)^{-1}Q(v_0,0).$$ is the solution of (\[eqn1\]) with $w_1\in (I-\Pi){{\mathbb{R}}}^n$. Now, we solve the equation $$\label{eqn2}
\frac{1}{2}\Pi P''(v_0)(\Pi w+w_1)(\Pi
w+w_1)+\Pi Q'_v(v_0,0)(\Pi w+w_1)+\Pi
Q'_{\varepsilon}(v_0,0)=0$$ with respect to $\Pi w.$ By assumption $\Pi P''(v_0)\Pi\,r\; \Pi\,
s=0 \;\; \mbox{for any} \;\; r,s\in\mathbb{R}^n,$ moreover $P''(v_0)ab=P''(v_0)ba,$ hence we can rewrite equation (\[eqn2\]) as follows $$\begin{array}{l}
\Pi P''(v_0)w_1\Pi w+\Pi Q'_v(v_0,0)\Pi w=
-\dfrac{1}{2}\Pi P''(v_0)w_1\,w_1 -\Pi
Q'_v(v_0,0))w_1-\Pi Q'_{\varepsilon}(v_0,0).
\end{array}$$ Since by assumption the operator $\Pi P''(v_0)w_1+\Pi Q'(v_0,0)$ is invertible, the last equation has a unique solution $ w_2$ with $w_2 \in \Pi\,{{\mathbb{R}}}^n$. Hence $w_0=w_2 + w_1$ is a zero of $\Psi(w,0)$.
From Lemma \[lem1\] we have that $\Psi$ is continuous at $(w_0,0),$ $\Psi'_w$ exists and is continuous at $(w_0,0).$ To apply the classical implicit function theorem it remains to show that $\Psi'_w(w_0,0)$ is non-singular. We argue by contradiction assuming that there exists $h\not= 0$ such that $$\label{ST1}
\Psi'_w(w_0,0)h=\Pi P''(v_0)w_0 h+\Pi
Q'_v(v_0,0)h+(I-\Pi)P'(v_0)h=0.$$ Applying $(I-\Pi)$ to (\[ST1\]) we obtain $(I-\Pi)P'(v_0)h=0$ that is $(I-\Pi)h=0$ and so $h=\Pi h.$ Therefore, $$\begin{array}{l}
\Pi P''(v_0)w_0h=\Pi P''(v_0)\Pi w_0 \Pi h+\Pi
P''(v_0)(I-\Pi) w_0 \Pi h=\\
\qquad\qquad\quad\;\;\;=-\Pi
P''(v_0)\left(P'(v_0)|_{(I-\Pi)\mathbb{R}^n}\right)^{-1}Q(v_0,0)\Pi
h
\end{array}$$ and applying $\Pi$ to (\[ST1\]) we obtain $$-\Pi
P''(v_0)\left(\left.P'(v_0)\right|_{(I-\Pi)\mathbb{R}^n}\right)^{-1}Q(v_0,0)\Pi
h+\Pi Q'_v(v_0,0)\Pi h=0.$$ This contradicts our assumption and the proof is completed.
The conclusions of Theorem \[th1\] permit to apply the classical implicit function theorem to obtain the existence of a $\delta>0$ such that the equation $\Psi(w,{\varepsilon})=0$ has, for any ${\varepsilon}\in [0, \delta]$, a unique solution $w_{\varepsilon}$ such that $\|w_0-w_{{\varepsilon}}\|\le \delta$. Therefore, for ${\varepsilon}>0$ small, there exists a family $\{w_{{\varepsilon}}\}$ of zeros of the map $\Psi$ such that $w_{{\varepsilon}}\to w_0$ as ${\varepsilon}\to 0$.
Moreover, under our regularity assumptions ${\varepsilon}\to
\Phi'_v(v_0+{\varepsilon}w_{{\varepsilon}}, {\varepsilon})$ is a continuous map; thus, for any ${\varepsilon}>0$ sufficiently small, there exists an eigenvalue $\lambda_{{\varepsilon}}$ of $\Phi'_v(v_0+{\varepsilon}w_{{\varepsilon}}, {\varepsilon})$ with the property that $\lambda_{{\varepsilon}}\to 0$ as ${\varepsilon}\to 0$.
We are now in the position to formulate the following result.
\[th2\] Assume all the conditions of Theorem \[th1\] and that zero is a simple eigenvalue of $P(v_0)$. Let $v_0=x(\theta_0)$, where $\theta\to x(\theta)$ is a $C^2$-parametrized curve of zeros of the map $P$. Let $\{w_{{\varepsilon}}\}$ and $\{\lambda_{{\varepsilon}}\}$ as in Remark 2. Let $\lambda_*\in\mathbb{R}$ be the eigenvalue of the operator $\left.\Pi P''(v_0)w_0\right|_{\Pi\mathbb{R}^n}+\left.\Pi
Q'_v(v_0,0)\right|_{\Pi\mathbb{R}^n}.$ Then $$\lambda_{\varepsilon}={\varepsilon}\lambda_*+o({\varepsilon}).$$
[**Proof.**]{} Let $l_{\varepsilon}$ be the unitary eigenvector of $\Phi'_v(v_0+{\varepsilon}w_0,{\varepsilon})$ associated to the eigenvalue $\lambda_{\varepsilon},$ namely $$\label{TI}
\Phi'_v(v_0+{\varepsilon}w_{\varepsilon},{\varepsilon})l_{\varepsilon}=\lambda_{\varepsilon}l_{\varepsilon}.$$ Clearly, $$\label{clear}
l_{\varepsilon}\to\dfrac{\dot x_0(\theta_0)}{\left\|\dot
x_0(\theta_0)\right\|}\quad{\rm as}\quad {\varepsilon}\to 0.$$ Now we observe that $$\Psi'_w(w,{\varepsilon})=\frac{1}{{\varepsilon}}\left({\varepsilon}\Phi'_v(v_0+{\varepsilon}w,{\varepsilon})-{\varepsilon}\Pi\Phi'_v(v_0+{\varepsilon}w,{\varepsilon})+\Pi\Phi'_v(v_0+{\varepsilon}w,{\varepsilon})\right)$$ and using (\[TI\]) we get $$\label{FI}
\Pi\Psi'_w(w_{\varepsilon},{\varepsilon})l_{\varepsilon}=\frac{1}{{\varepsilon}}\Pi\Phi'_v(v_0+{\varepsilon}w_{\varepsilon},{\varepsilon})l_{\varepsilon}=\frac{1}{{\varepsilon}}\lambda_{\varepsilon}\Pi l_{\varepsilon}$$ for any ${\varepsilon}>0$ sufficiently small. By Lemma \[lem1\] as ${\varepsilon}\to 0$ we have $$\begin{aligned}
\Pi\Psi'_w(w_{\varepsilon},{\varepsilon})l_{\varepsilon}&\to&\Pi P''(v_0)w_0\dfrac{\dot
x_0(\theta_0)}{\left\|\dot x_0(\theta_0)\right\|}+\Pi
Q'_v(v_0,0)\dfrac{\dot
x_0(\theta_0)}{\left\|\dot x_0(\theta_0)\right\|}\label{FIbis}.
\end{aligned}$$ From this, by (\[FI\]) we have that $\dfrac{\lambda_{\varepsilon}}{{\varepsilon}}\to a\in\mathbb{R}$ as ${\varepsilon}\to 0$ and $$\Pi
P''(v_0)w_0\dot
x_0(\theta_0)+\Pi
Q'_v(v_0,0)\dot
x_0(\theta_0)=a \dot x_0(\theta_0).$$ Therefore, $a=\lambda_*,$ and the proof is completed.
An application to periodically perturbed autonomous equations
=============================================================
In this Section we show that the results of the previous Section can be straight apply to the problem of bifurcation of asymptotically stable $T$-periodic solutions to $T$-periodically perturbed autonomous systems. Specifically, by showing that our function (\[INV\]) is nothing else than the Malkin’s bifurcation function, as far as periodically perturbed autonomous systems are concerned, we prove the existence of a unique branch of asymptotically stable periodic solutions emanating from the family of periodic solutions represented by limit cycle $x_0$ of the unperturbed system.
The system under consideration is the following $$\label{ps}
\dot x=f(x)+{\varepsilon}g(t,x,{\varepsilon}).$$ where $f\in C^2({{\mathbb{R}}}^n, {{\mathbb{R}}}^n),\, g\in C^1({{\mathbb{R}}}\times {{\mathbb{R}}}^n \times
[0,1], {{\mathbb{R}}}^n)$ is $T$-periodic and ${\varepsilon}>0$ is the bifurcation parameter. We assume that the unique solution of any Cauchy problem associated to (\[ps\]) is defined on $[0,T]$.
We associate to the unperturbed autonomous system $$\label{us}
\dot x=f(x)$$ the Malkin’s bifurcation function [@m] $$M(\theta)=\int_0^T\left<g(t,x_0(t+\theta),0),z_0(t+\theta)\right>dt$$ where $\left<\cdot, \cdot\right>$ denotes the usual scalar product in ${{\mathbb{R}}}^n$ and $z_0$ is the $T$-periodic function of the adjoint system $$\dot z=-(f'(x_0(t)))^*z$$ of the linearized system of $$\dot y=f'(x_0(t))y$$ of autonomous system (\[us\]). Let $\theta\in[0,T],$ we define the projector $\Pi:{{\mathbb{R}}}^n\to{{\mathbb{R}}}^n$ as follows $$\Pi\xi=\dot x_0(\theta)\left<\xi,z_0(\theta)\right>.$$ Finally, we convert the problem of finding $T$-periodic solutions to (\[ps\]) into the fixed point problem for the associated Poincaré map $\mathcal{P}_{{\varepsilon}}$ as illustrated in the following. We consider the function $x:[0,T]\times{{\mathbb{R}}}^n\times[0,1]\to {{\mathbb{R}}}^n$ given by $$x(t,v,{\varepsilon})=x(t)$$ for all $t\in[0,T],$ where $x(t)$ is the solution of systems equation (\[ps\]). The Poincaré map for system (\[ps\]) is defined by $$\mathcal{P}_{\varepsilon}(v)=x(T,v,{\varepsilon}).$$ The functions $P$ and $Q$ of the previous section are defined as $P(v)=\mathcal{P}_0(v)-v,$ $Q(v,{\varepsilon})=\dfrac{\mathcal{P}_{\varepsilon}(v)-\mathcal{P}_0(v)}{{\varepsilon}}$ that leads to $$\mathcal{P}_{\varepsilon}(v)-v=P(v)+{\varepsilon}Q(v,{\varepsilon}).$$ Observe that, since $P(x_0(\theta))=0$ for any $\theta$, we have that $P'(x_0(\theta))\,\dot x_0(\theta)=0$ and so $$(\mathcal{P}_0)'(x_0(\theta))-I=P'(x_0(\theta))$$ is a singular $n\times n$ matrix for any $\theta\in [0,T]$.
0.2truecm With $x_0, z_0, \Pi, P, Q$ as introduced before we have the following two results. The first one provides a representation formula for the Malkin’s bifurcation function, the second one a formula for its derivative.
\[lem3\] For any $\theta\in [0,T]$ the limit $Q(v,0):=\lim_{{\varepsilon}\to
0}Q(v,{\varepsilon})$ exists and $$M(\theta)=\left<Q(x_0(\theta),0),z_0(\theta)\right>.$$ Moreover, $Q\in C^1(\mathbb{R}^n\times[0,1],\mathbb{R}^n).$
[**Proof.**]{} Differentiating with respect to time one can see that the function $y(t)=\dfrac{\partial}{\partial {\varepsilon}}
x(t,x_0(\theta),{\varepsilon})$ evaluated at ${\varepsilon}=0$ solves, for any $\theta\in [0,T]$, the Cauchy problem $$\dot y=f'(x_0(t+\theta))y+g(t,x_0(t+\theta),0),\quad y(0)=0.$$ A direct computation shows that $$\frac{d}{dt}\left<y(t),z_0(t+\theta)\right>=\left<g(t,x_0(t+\theta),0),z_0(t+\theta)\right>$$ and, integrating over the period, yields $$M(\theta)=\left<y(T),z_0(\theta)\right>=\left<Q(x_0(\theta),0),z_0(\theta)\right>.$$
\[lem4\] For any $\theta\in [0,T]$ we have $$\label{R}
M'(\theta)=\left<-P''(x_0(\theta))(I-\Pi)\left(\left.P'(x_0(\theta))\right|_{(I-\Pi)\mathbb{R}^n}\right)^{-1}Q(x_0(\theta),0)\dot
x_0(\theta)+Q'_v(x_0(\theta),0)\dot
x_0(\theta),z_0(\theta)\right>.$$
[**Proof.**]{} By Perron’s Lemma [@perron] we have that $$\left<\dot x(\theta), z_0(\theta)\right>=\left<\dot x(0),
z_0(0)\right>$$ for any $\theta\in [0,T]$. Without loss of generality we may assume that $\left<\dot x(0), z_0(0)\right>=1.$ As a consequence, by the definition of the projector $\Pi$, we get $$\label{pi}
\left<\xi, z_0(\theta)\right>=\left<\Pi\,\xi, z_0(\theta)\right>,$$ for any $\theta\in [0,T]$. Therefore $$\left<P'(x_0(\theta))h,
z_0(\theta)\right>=\left<\Pi\,P'(x_0(\theta))(I-\Pi)h,
z_0(\theta)\right>=0,$$ for any $\theta\in [0,T]$ and any $h\in {{\mathbb{R}}}^n$. Then, by deriving with respect to $\theta$, we obtain $$\left<P'(x_0(\theta))h,\dot
z_0(\theta)\right>= \left<-P''(x_0(\theta))\dot
x_0(\theta)h,z_0(\theta)\right>,$$ for any $\theta\in [0,T]$ and any $h\in {{\mathbb{R}}}^n$. Therefore, we can rewrite the left hand side of (\[R\]) with $(I-\Pi)\left(\left.P'(x_0(\theta))\right|_{(I-\Pi)\mathbb{R}^n}\right)^{-1}Q(x_0(\theta),0)=h$ as follows $$\left<P'(x_0(\theta))(I-\Pi)\left(\left.P'(x_0(\theta))\right|_{(I-\Pi)\mathbb{R}^n}\right)^{-1}Q(x_0(\theta),0),\dot
z_0(\theta)\right>+\left<Q'_v(x_0(\theta),0)\dot
x_0(\theta),z_0(\theta)\right>$$ or equivalently, $$\left<Q(x_0(\theta),0),\dot z_0(\theta)\right>+\left<Q'_v(x_0(\theta),0)\dot
x_0(\theta),z_0(\theta)\right>,$$ which is the derivative of $M(\theta)$ at any $\theta\in [0,T]$ according to the formula given by Lemma \[lem3\].
0.2truecm Finally, we can prove the following.
\[th3\] Assume that there exists $\theta_0\in [0,T]$ such that $(\mathcal{P}_0)'(x_0(\theta_0))$ has $n-1$ eigenvalues with negative real parts, $M(\theta_0)=0$ and $M'(\theta_0)<0.$ Then, for ${\varepsilon}>0$ sufficiently small, equation (\[ps\]) has a unique $T$-periodic solution $x_{\varepsilon}$ such that $x_{\varepsilon}(t)\to
x_0(t+\theta_0)$ as ${\varepsilon}\to 0$ uniformly in $[0,T]$. Moreover the solutions $\{x_{\varepsilon}\}$ are asymptotically stable.
[**Proof.**]{} Let $v_0=x_0(\theta_0)$, from Lemma \[lem3\] we have $$\Pi Q(x_0(v_0),0)=\dot
x_0(\theta_0)\left<Q(v_0,0),z_0(\theta_0)\right>=\dot
x_0(\theta_0)M(\theta_0)=0.$$ By (\[pi\]) we obtain $$M'(\theta_0)=\left<-\Pi
P''(v_0)(I-\Pi)\left(\left.P'(v_0)\right|_{(I-\Pi)\mathbb{R}^n}\right)^{-1}Q(v_0,0)\dot
x_0(\theta_0) + \Pi Q'_v(v_0,0)\dot x_0(\theta_0),
z_0(\theta_0)\right>\not= 0,$$ and so (\[INV\]) is invertible on $\Pi {{\mathbb{R}}}^n$. Moreover, from the fact that $P(x_0(\theta))=0$ for any $\theta\in [0,T]$, we obtain that $$P''(v_0)\dot x_0(\theta_0)\dot x_0(\theta_0) + P'(v_0)
x''_0(\theta_0)=0$$ Since $\Pi P'(v_0) x''_0(\theta_0)= \Pi P'(v_0) \Pi
x''_0(\theta_0)=0$ we have that $\Pi P''(v_0)\;\Pi\,r\; \Pi\,s=0$ for any $r,s\in {{\mathbb{R}}}^n$. Therefore, all the conditions of Theorem \[th1\] are satisfied and so, compare Remark 2, equation (\[ps\]) has a unique $T$-periodic solution $x_{\varepsilon}$ satisfying $$\left\|w_0-\frac{x_{\varepsilon}(0)-v_0}{{\varepsilon}}\right\|\le\delta,$$ with $\Psi(w_0,0)=0$. Moreover $$\Pi P''(v_0) w_0 \dot x_0(\theta_0) + \Pi Q'_v(v_0,0) \dot
x_0(\theta_0)= \lambda_*\, \dot x_0(\theta_0).$$ But $$\mbox{sign}\,\lambda_*=\mbox{sign}\,\left<\Pi P''(v_0) w_0 \dot
x_0(\theta_0) + \Pi Q'_v(v_0,0)
\dot
x_0(\theta_0),z_0(\theta_0)\right>=\mbox{sign}\,M'(\theta_0)=-1$$ Therefore, from Theorem \[th2\] there exists $\lambda_{\varepsilon}={\varepsilon}\lambda_* +o({\varepsilon})$ eigenvalue of $(\mathcal{P}_{\varepsilon})'(x_{\varepsilon}(0))-I$. This implies that $${\rm det}\left((\mathcal{P}_{\varepsilon})'(x_{\varepsilon}(0))-I-\lambda_{\varepsilon}I\right)=0.$$ Hence, $\rho_{\varepsilon}=1+\lambda_{\varepsilon}=1+\lambda_*{\varepsilon}+o({\varepsilon})$ is an eigenvalue of $(\mathcal{P}_{\varepsilon})'(x_{\varepsilon}(0))$ converging to 1 as ${\varepsilon}\to 0.$ Since $\lambda_*<0,$ then $|\rho_{\varepsilon}|<1$ for ${\varepsilon}>0$ sufficiently small. This ends the proof.
0.4truecm[**Acknowledgments.**]{} The first author acknowledge the support by RFBR 09-01-92429 and 06-01-72552. The second author is supported by the Grant BF6M10 of Russian Federation Ministry of Education and U.S. CRDF (BRHE), by RFBR Grant 09-01-00468, by the President of Russian Federation Young PhD Student grant MK-1620.2008.1 and by Marie Curie grant PIIF-GA-2008-221331. The third author is supported by INdAM-GNAMPA.
Finally, we would like to acknowledge that the simple proof of Lemma 2 was suggested by Rafael Ortega during personal communications and it is taken from an his own unpublished manuscript.
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Henry, D. (1981). Geometric theory of nonlinear parabolic equations, Lecture Notes in Mathematics, [**840**]{}, Springer-Verlag, Berlin-New York.
Kamenskii M., Makarenkov O. and Nistri P. (2008). State variables scaling to solve the Malkin’s problem on periodic oscillations in perturbed autonomous systems, 6th European Nonlinear Dynamics Conference, ENOC 2008, June 30-July 4, 2008. Saint Petersburg, Russia. (http://lib.physcon.ru).
Kamenskii M., Makarenkov O. and Nistri P. (2008). Periodic bifurcation from families of periodic solutions for semilinear differential equations with Lipschitzian perturbation in Banach spaces. Adv. Nonlinear Stud. [**8**]{}, 271-288.
Krantz S.G. and Parks H.R. (2003). The Implicit Function Theorem, History, Theory and Applications, Birkauser Boston.
Loud W. S. (1959). Periodic solutions of a perturbed autonomous system. Ann. Math. [**70**]{}, 490-529.
Makarenkov O. and Nistri P. (2008). “Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations. J. Math. Anal. Appl. [**338**]{}, 1401-1417.
Malkin I. G. (1949). On Poincaré’s theory of periodic solutions. Akad. Nauk SSSR. Prikl. Mat. Meh. [**13**]{}, 633-646. (Russian).
Perron O. (1930). Die Ordnungszahlen der Differentialgleichungssysteme. Math. Zeitschr. [**31**]{}, 748-766.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Although the currently popular deep learning networks achieve unprecedented performance on some tasks, the human brain still has a monopoly on general intelligence. Motivated by this and biological implausibility of deep learning networks, we developed a family of biologically plausible artificial neural networks (NNs) for unsupervised learning. Our approach is based on optimizing principled objective functions containing a term that matches the pairwise similarity of outputs to the similarity of inputs, hence the name - similarity-based. Gradient-based online optimization of such similarity-based objective functions can be implemented by NNs with biologically plausible local learning rules. Similarity-based cost functions and associated NNs solve unsupervised learning tasks such as linear dimensionality reduction, sparse and/or nonnegative feature extraction, blind nonnegative source separation, clustering and manifold learning.'
author:
- 'Cengiz Pehlevan[^1]'
- 'Dmitri B. Chklovskii[^2]'
title: 'Neuroscience-inspired online unsupervised learning algorithms'
---
Introduction
============
Inventors of the original artificial neural networks (NNs) derived their inspiration from biology [@rosenblatt1958perceptron]. However, today, most artificial NNs such as, for example, backpropagation-based convolutional deep learning networks, resemble natural NNs only superficially. Given that, on some tasks, such artificial NNs achieve human or even superhuman performance, why should one care about such dissimilarity with natural NNs? The algorithms of natural NNs are relevant if one’s goal is not just to outperform humans on certain tasks but to develop general-purpose artificial intelligence rivaling human. As contemporary artificial NNs are far from achieving this goal and natural NNs, by definition, achieve it, natural NNs must contain some “secret sauce” that artificial NNs lack. This is why we need to understand the algorithms implemented by natural NNs.
Motivated by this argument, we have been developing artificial NNs that could plausibly model natural NNs on the algorithmic level. In our artificial NNs, we do not attempt to reproduce many biological details, as in existing biophysical modeling work, but rather develop algorithms that respect major biological constraints.
For example, biologically plausible algorithms must be formulated in the [**online**]{} (or streaming), rather than offline (or batch), setting. This means that input data are streamed to the algorithm sequentially, and the corresponding output must be computed before the next input sample arrives. Moreover, memory accessible to a biological algorithm is limited so that no significant fraction of previous input or output can be stored.
Another key constraint is that in biologically plausible NNs, learning rules must be [**local**]{}: a biological synapse can update its weight based on the activity of only the two neurons that the synapse connects. Such “locality” of the learning rule is violated by most artificial NNs including backpropagation-based deep learning networks. In contrast, our NNs employ exclusively local learning rules. Such rules are also helpful for hardware implementations of artificial NNs in neuromorphic chips [@davies2018loihi; @poikonen2017mixed].
We derive the algorithms performed by our NNs from optimization objectives. In addition to deriving learning rules for synaptic weights, as is done in existing artificial NNs, we also derive the architecture, activation functions, and dynamics of neural activity from the same objectives. To do this, we postulate only a cost function and an optimization algorithm, which in our case is alternating stochastic gradient descent-ascent [@olshausen1996emergence]. The steps of this algorithm map to a NN, specifying its architecture, activation functions, dynamics, and learning rules. Viewing both weight and activity updates as the steps of an online optimization algorithm allows us to predict the output of our NNs to a wide range of stimuli without relying on exhaustive numerical simulation.
To derive local learning rules we employ optimization objectives operating with pairwise similarities of inputs and outputs of a NN rather than individual data points. Typically, our objectives favor similar outputs for similar inputs. Hence, the name - similarity matching objectives. The transformation of dissimilar inputs in the NN depends on the optimization constraints. Despite using pairwise similarities we still manage to derive [*online*]{} optimization algorithms.
Our focus is on [**unsupervised**]{} learning. This is not a hard constraint, but rather a matter of priority. Whereas humans are clearly capable of supervised learning, most of our learning tasks lack big labeled datasets. On the mechanistic level, most neurons lack a clear supervision signal.
This paper is organized as follows. We start by presenting the conventional approach to deriving unsupervised NNs (Section 2). While the conventional approach generates a reasonable algorithmic model of a single biological neuron, multi-neuron networks violate biological constraints. To overcome this difficulty, in Section 3, we introduce similarity-based cost functions and show that linear dimensionality reduction NNs derived from such cost functions are biologically plausible. In Section 4, we introduce a sign-constrained similarity-matching objective and discuss algorithms for sparse feature extraction and nonnegative independent component analysis. In Section 5, we discuss other sign-constrained networks for clustering and manifold learning. We conclude by discussing potential applications of our work to neuromorphic computing and charting future directions.
Background
==========
Single-neuron online Principal Component Analysis (PCA)
-------------------------------------------------------
In the seminal 1982 paper [@oja1982simplified], Oja proposed that a biological neuron can be viewed as an implementation of a mathematical algorithm solving a computational objective. He proposed to model a neuron by an online Principal Component Analysis (PCA) algorithm. As PCA is a workhorse of data analysis used for dimensionality reduction, denoising, and latent factor discovery, Oja’s model offers an algorithmic-level description of biological NNs.
Oja’s single-neuron online PCA algorithm works as follows. At each time step, $t$, it receives an input data sample, $\x_t\in\R^n$. As our focus is on the online setting, we use the same variable, $t$, to measure time and index the data points. Then, the algorithm computes and outputs the corresponding top principal component value, $y_t\in\R$: $$\begin{aligned}
\label{oja_neuron}
y_t \longleftarrow \w_{t-1}^\top\x_t, \end{aligned}$$ where $\w_{t-1}\in\R^n$ is the feature vector computed at time step, $t-1$. Here and below lowercase italic letters are scalar variables and lowercase boldfaced letters designate vectors.
At the same time step, $t$, after computing the principal component, the algorithm updates the (normalized) feature vector with a learning rate, $\eta$, $$\begin{aligned}
\label{oja_update}
\w_t\longleftarrow \w_{t-1}+\eta\left(\x_t-\w_{t-1} y_t\right)y_t.\end{aligned}$$ If data are drawn i.i.d. from a stationary distribution with a mean vector of zero, the feature vector, $\w$, converges to the eigenvector corresponding to the largest eigenvalue of input covariance [@oja1982simplified].
The steps of the Oja algorithm , naturally correspond to the operations of a biological neuron. Assuming that the components of the input vector are represented by the activities (firing rates) of the upstream neurons, describes a weighted summation of the inputs by the output neuron. Such weighted summation can be naturally implemented by storing the components of feature vector, $\bf w$, in the corresponding synaptic weights. If the activation function is linear, the output, $y_t$, is simply the weighted sum.
The weight update is a biologically plausible local synaptic learning rule. The first term of the update, $\x_t y_t$, is proportional to the correlation of the pre- and postsynaptic neurons’ activities and the second term, $\w_t y_t^2$, also local, asymptotically normalizes the synaptic weight vector to one. In neuroscience, synaptic weight updates proportional to the correlation of the pre- and postsynaptic neurons’ activities are called Hebbian.
Minimization of the reconstruction error yields biologically implausible multi-neuron networks
----------------------------------------------------------------------------------------------
Next, we would like to build on Oja’s insightful identification of biological processes with the steps of the online PCA algorithms by computing multiple principal components using multi-neuron NNs. Instead of trying to extend the Oja model heuristically, we will derive them by using optimization of a principled objective function. Specifically, we postulate that the algorithm minimizes the reconstruction error, derive an online algorithm optimizing such objective, and map the steps of the algorithm onto biological processes.
In the conventional reconstruction error minimization approach, each data sample, $\x_t\in\R^n$, is approximated as a linear combination of each neuron’s feature vector weighted by its activity [@olshausen1996emergence]. Then the minimization of the reconstruction (or coding) error can be expressed as follows: $$\begin{aligned}
\label{rE}
\min_{\W } \sum_{t=1}^T \min_{\y_t}\left\Vert \x_t -\W \y_t\right\Vert_2^2,\end{aligned}$$ where matrix $\W\in\R^{n\times k}$, $k<n$, is a concatenation of feature column-vectors and $T$ is both the number of data samples and (in the online setting) the number of time steps.
In the offline setting, a solution to the optimization problem is PCA: the columns of optimum $\W$ are a basis for the $k$-dimensional principal subspace [@udell2016generalized]. Elements of $\W$ could be constrained to avoid unreasonably low or high values.
In the online setting, can be solved by alternating minimization [@olshausen1996emergence]. After the arrival of data sample, $\x_t$, the feature vectors are kept fixed while the objective is minimized with respect to the principal components by running the following gradient-descent dynamics until convergence: $$\begin{aligned}
\label{oja_multineuron}
\dot \y_t = \W^\top_{t-1}\x_t - \W^\top_{t-1}\W_{t-1}\y_t,\end{aligned}$$ where $\dot{}$ is a derivative with respect to a continuous time variable which runs within a time step, $t$. Unlike a closed-form output of a single Oja neuron , is iterative.
After the output, ${\bf y}_t$ converges, at the same time step, $t$, the feature vectors are updated according to the following gradient-descent step, with respect to $\bf W$ on the total objective: $$\begin{aligned}
\label{oja_multiupdate}
\W_t \longleftarrow \W_{t-1} + \eta \left(\x_t - \W_{t-1}\y_t\right)\y_t^\top.\end{aligned}$$ If there was a single output channel, the algorithm , would reduce to ,, provided that the scalar $\W^\top_{t-1}\W_{t-1}$ is rescaled to unity.
In NN implementations of algorithm ,, feature vectors are represented by synaptic weights and components by the activities of output neurons. Then can be implemented by a single-layer NN, Fig. \[fig:SMnet\]A, in which activity dynamics converges faster than the time interval between the arrival of successive data samples. The lateral connection weights, $-\W^\top_{t-1}\W_{t-1}$, decorrelate neuronal feature vectors by suppressing activities of correlated neurons.
However, implementing update in the single-layer NN architecture, Fig. \[fig:SMnet\]A, requires nonlocal learning rules making it biologically implausible. Indeed, the last term in implies that updating the weight of a synapse requires the knowledge of output activities of all other neurons which are not available to the synapse. Moreover, the matrix of lateral connection weights, $-\W^\top_{t-1}\W_{t-1}$, in the last term of is computed as a Grammian of feedforward weights, clearly a nonlocal operation. This problem is not limited to PCA and arises in networks of nonlinear neurons as well [@olshausen1996emergence; @lee1999learning].
To respect the local learning constraint, many authors constructed biologically plausible single-layer networks using heuristic local learning rules that were not derived from an objective function [@foldiak1989adaptive; @diamantaras1996principal]. However, we think that abandoning the optimization approach creates more problems than it solves. Alternatively, NNs with local learning rules can be derived if one introduces a second layer of neurons [@olshausen1997sparse]. However, such architecture does not map naturally on biological networks.
We have outlined how the conventional reconstruction approach fails to generate biologically plausible multi-neuron networks for online PCA. In the next section, we will introduce an alternative approach that overcomes this limitation. Moreover, this approach suggests a novel view of neural computation leading to many interesting extensions.
Similarity-based approach to linear dimensionality reduction
============================================================
In this section, we propose a different objective function for the optimization approach to constructing PCA NNs, which we term similarity matching. From this objective function, we derive an online algorithm implementable by a NN with local learning rules. Then, we introduce other similarity-based algorithms for linear dimensionality reduction which include more biological features such as different neuron classes.
Similarity-matching objective function
--------------------------------------
We start by stating an objective function that will be used to derive NNs for linear dimensionality reduction. The similarity of a pair of inputs, $\x_t$ and $\x_{t'}$, both in $\R^n$, can be defined as their dot-product, $\x_t^\top \x_{t'} $. Analogously, the similarity of a pair of outputs, which live in $\R^k$ with $k<n$, is $\y_{t}^\top \y_{t'}$. Similarity matching, as its name suggests, learns a representation where the similarity between each pair of outputs matches that of the corresponding inputs: $$\begin{aligned}
\label{CMDS}
\min_{ \y_1,\ldots,\y_T} \frac{1}{T^2} \sum_{t=1}^T \sum_{t'=1}^T \left(\x_t^\top \x_{t'} - \y_t^\top\y_{t'}\right)^2.\end{aligned}$$ This offline objective function, previously employed for multidimensional scaling, is optimized by the projections of inputs onto the principal subspace of their covariance, i.e. performing PCA up to an orthogonal rotation. Moreover, has no local minima other than the principal subspace solution [@pehlevan2015normative; @GeLeeMa].
The similarity-matching objective may seem like a strange choice for deriving an online algorithm implementable by a NN. In , pairs of inputs and outputs from different time steps interact with each other. Yet, in the online setting, an output must be computed at each time step without accessing inputs or outputs from other time steps. In addition, synaptic weights do not appear explicitly in seemingly precluding mapping onto a NN.
Variable substitution trick
---------------------------
Both of the above concerns can be resolved by a simple math trick akin to completing the square [@pehlevan2018similarity]. We first focus on the cross-term in the expansion of the square in , which we call similarity alignment. By introducing a new variable, ${\bf W} \in \R^{k\times n}$, we can re-write the cross-term: $$\begin{aligned}
\label{cross}
&- \frac{1}{T^2}\sum_{t=1}^T \sum_{t'=1}^T \y_{t}^\top \y_{t'} \x_t^\top \x_{t'} = \min_{\W\in \R^{k\times n}} \, -\frac{2}{T} \sum_{t=1}^T\y_t^\top \W \x_t + \Tr \, \W^\top \W.\end{aligned}$$ To prove this identity, find optimal ${\bf W}$ by taking a derivative of the expression on the right with respect to ${\bf W}$ and setting it to zero, and then substitute the optimal ${\bf W}^* = \frac 1T\sum_{t=1}^T \y_t\x_t^\top$ back into the expression. Similarly, for the quartic $\y_t$ term in : $$\begin{aligned}
\label{quartic}
&\frac 1{T^2}\sum_{t=1}^T \sum_{t'=1}^T \y_{t}^\top \y_{t'} \y_t^\top \y_{t'} = \max_{\M \in \R^{k\times k}} \, \frac{2}{T} \sum_{t=1}^T\y_t^\top \M \y_t - \Tr \, \M^\top \M.\end{aligned}$$ By substituting and into we get: $$\begin{aligned}
\label{SMMW2t}
\min_{{\bf W}\in \mathbb{R}^{k\times n}}\max_{{\bf M}\in \mathbb{R}^{k\times k}} \, \frac{1}{T} \sum_{t=1}^T \left[2\, {\rm Tr}\left({\bf W}^\top{\bf W}\right) - {\rm Tr}\left({\bf M}^\top{\bf M}\right) + \min_{{\bf y}_t\in \mathbb{R}^{k\times 1}} l_t({\bf W},{\bf M},{\bf y}_t)\right],\end{aligned}$$ where $$\begin{aligned}
\label{lyapunov}
l_t({\bf W},{\bf M},{\bf y}_t)=-4{\bf x}_t^\top{\bf W}^\top{\bf y}_t + 2{\bf y}_t^\top{\bf M}{\bf y}_t.\end{aligned}$$ In the resulting objective function, ,, optimal outputs at different time steps can be computed independently, making the problem amenable to an online algorithm. The price paid for this simplification is the appearance of the minimax optimization problem in variables, [**W**]{} and [**M**]{}. Minimization over $\bf W$ aligns output channels with the greatest variance directions of the input and maximization over $\bf M$ diversifies the output by decorrelating output channels similarly to the Grammian, $\W^\top\W$, used previously. This competition in a gradient descent/ascent algorithm results in the principal subspace projection which is the only stable fixed point of the corresponding dynamics [@pehlevan2015MDS].
Online algorithm and neural network
-----------------------------------
We are ready to derive an algorithm for optimizing online. First, we minimize with respect to the output variables, ${\bf y}_t$, while holding ${\bf W}$ and ${\bf M}$ fixed using gradient-descent dynamics: $$\begin{aligned}
\label{grad}
\dot\y_t=\W \x_t-\M \y_t.\end{aligned}$$ As before, dynamics converges within a single time step, $t$, and outputs $\y_t$. After the convergence of $\y_t$, we update ${\bf W}$ and ${\bf M}$ by gradient descent of and gradient ascent of respectively: $$\begin{aligned}
\label{Hebb}
W_{ij} \leftarrow W_{ij} + \eta \left(y_ix_j-W_{ij}\right), \qquad M_{ij} \leftarrow M_{ij} + \frac{\eta}2\left(y_iy_j-M_{ij}\right).\end{aligned}$$
Algorithm ,, first derived in [@pehlevan2015MDS], can be naturally implemented by a biologically plausible NN, Fig. \[fig:SMnet\]B. Here, activity of the upstream neurons corresponds to input variables. Output variables are computed by the dynamics of activity in a single layer of neurons. Variables ${\bf W}$ and ${\bf M}$ are represented by the weights of synapses in feedforward and lateral connections respectively. The learning rules are local, i.e. the weight update, $\Delta W_{ij}$, for the synapse between $j^{\rm th}$ input neuron and $i^{\rm th}$ output neuron depends only on the activities, $x_j$, of $j^{\rm th}$ input neuron and, $y_i$, of $i^{\rm th}$ output neuron, and the synaptic weight. In neuroscience, learning rules for synaptic weights ${\bf W}$ and ${\bf -M}$ (here minus indicates inhibitory synapses, see Eq.) are called Hebbian and anti-Hebbian respectively.
To summarize this Section so far, starting with the similarity-matching objective, we derived a Hebbian/anti-Hebbian NN for dimensionality reduction. The minimax objective can be viewed as a zero-sum game played by the weights of feedforward and lateral connections [@pehlevan2018similarity; @seung2017correlation]. This demonstrates that synapses with local updates can still collectively work together to optimize a global objective. A similar, although not identical, NN was proposed by Földiak [@foldiak1989adaptive] heuristically. The advantage of our optimization approach is that the offline solution is known.
Although no proof of convergence exists in the online setting, algorithm , performs well on large-scale data. A recent paper [@giovannucci2018efficient] introduced an efficient, albeit non-biological, modification of the similarity-matching algorithm, Fast Similarity Matching (FSM), and demonstrated its competitiveness with the state-of-the-art principal subspace projection algorithms in both processing speed and convergence rate[^3], Figure \[fig:SMnet\]D. FSM produces the same output $\y_t$ for each input $\x_t$ as similarity-matching by optimizing by matrix inversion, $\y_t = \M^{-1}\W\x_t$. It achieves extra computational efficiency by keeping in memory and updating the $\M^{-1}$ matrix rather than $\M$. We refer the reader to [@giovannucci2018efficient] for suggestions on the implementation of these algorithms.
Other similarity-based objectives and linear networks
-----------------------------------------------------
As the algorithm , and the NN in Fig.\[fig:SMnet\]B were derived from the similarity-matching objective , they project data onto the principal subspace but do not necessarily recover principal components [*per se*]{}. To derive PCA algorithms we modified the objective function to encourage orthogonality of $\W$ [@pehlevan2015optimization; @minden2018biologically]. Such algorithms are implemented by NNs of the same architecture as in Fig.\[fig:SMnet\]B but with slightly different local learning rules.
Although the similarity-matching NN in Fig. \[fig:SMnet\]B relies on biologically plausible local learning rules, it lacks biological realism in several other ways. For example, computing output requires recurrent activity that must settle faster than the time scale of the input variation, which is unlikely in biology. To respect this biological constraint, we modified the dimensionality reduction algorithm to avoid recurrency [@minden2018biologically].
Another non-biological feature of the NN in Fig.\[fig:SMnet\]B is that the output neurons compete with each other by communicating via lateral connections. In biology, such interactions are not direct but mediated by interneurons. To reflect these observations, we modified the objective function by introducing a whitening constraint: $$\begin{aligned}
\label{NIPS3}
&\min_{\y_1,\ldots,\y_T} - \frac{1}{T^2}\sum_{t=1}^T \sum_{t'=1}^T \y_{t}^\top \y_{t'} \x_t^\top \x_{t'}, \qquad {\rm s.t.} \quad \frac 1T \sum_{t}\y_t\y_t^\top = \I_k,\end{aligned}$$ where $\I_k$ is the $k$-by-$k$ identity matrix. Then, by implementing the whitening constraint using the Lagrange formalism, we derived NNs where interneurons appear naturally - their activity is modeled by the Lagrange multipliers, $\z_t^\top \z_{t'}$ (Fig. \[fig:SMnet\]C), [@pehlevan2015normative]: $$\begin{aligned}
\label{NIPS3_minmax}
\min_{\y_1,\ldots,\y_t} \max_{\z_1,\ldots,\z_T } &- \frac{1}{T^2}\sum_{t=1}^T \sum_{t'=1}^T \y_{t}^\top \y_{t'} \x_t^\top \x_{t'} + \frac{1}{T^2}\sum_{t=1}^T \sum_{t'=1}^T \z_{t}^\top \z_{t'} \left(\y_{t}^\top \y_{t'} -\delta_{t,t'}\right),\end{aligned}$$ where $\delta_{t,t'}$ is the Kronecker delta. Notice how contains the $\y$-$\z$ similarity-alignment term similar to . We can now derive learning rules for the $\y$-$\z$ connections using the variable substitution trick, leading to the network in Figure \[fig:SMnet\]C. For details of this and other NN derivations, see [@pehlevan2015normative].
Nonnegative similarity-matching objective and nonnegative independent component analysis
========================================================================================
So far we considered similarity-based NNs comprising linear neurons. But many interesting computations require nonlinearity and biological neurons are not linear. To construct more realistic and powerful similarity-based NNs, we note that the output of biological neurons is nonnegative (firing rate cannot be below zero). Hence, we modified the optimization problem by requiring that the output of the similarity-matching cost function is nonnegative: $$\begin{aligned}
\label{nonnegative}
\min_{ \y_1,\ldots,\y_T\ge 0} \frac{1}{T^2} \sum_{t=1}^T \sum_{t'=1}^T \left(\x_t^\top \x_{t'} - \y_t^\top\y_{t'}\right)^2.\end{aligned}$$ Here, the number of output dimensions, $k$, may be greater than the number of input dimensions, $n$, leading to a dimensionally expanded representation. Eq. can be solved by the same online algorithm as except that the output variables are projected onto the nonnegative domain. Such algorithm maps onto the same network and same learning rules as in Eq., Fig. \[fig:SMnet\]B, but with rectifying neurons (ReLUs) [@pehlevan2014NMF; @pehlevan2017blind; @pehlevan2019spiking], Fig. \[fig:Gabor\]A.
Nonnegative similarity-matching network learns features that are very different from PCA. For example, when the network is trained on whitened natural scenes it extracts edge filters [@pehlevan2014NMF] (Fig. \[fig:Gabor\]) as opposed to Fourier harmonics expected for a translationally invariant dataset. Motivated by this observation, Bahroun and Soltoggio [@bahroun2017online] developed a convolutional nonnegative similarity matching network with multiple resolutions, and used it as an unsupervised feature extractor for subsequent linear classification on CIFAR-10 dataset. They found that nonnegative similarity matching NNs are superior to other single-layer unsupervised techniques [@bahroun2017online; @bahroun2017building], Table \[Tab:Multi\_resolution\].
As edge filters emerge also in the independent component analysis (ICA) of natural scenes [@bell1997independent] we investigated a connection of nonnegative similarity matching with nonnegative independent component analysis (NICA) used for blind source separation. The NICA problem is to recover independent, nonnegative and well-grounded (finite probability density function in any positive neighborhood of zero) sources, $\s_t\in\R^d$, from observing only their linear mixture, $\x_t={\bf A}\s_t$, where ${\bf A}\in \R^{n\times d}$, and $n\geq d$.
Our solution of NICA is based on the observation that NICA can be solved in two steps [@plumbley2002conditions], Fig. \[fig:NICA\]A. First, whiten the data and reduce it to $d$ dimensions to obtain an orthogonal rotation of the sources (assuming that the mixing matrix is full-rank). Second, find an orthogonal rotation of the whitened sources that yields a nonnegative output, Fig. \[fig:NICA\]A. The first step can be implemented by the whitening network in Fig. \[fig:SMnet\]C. The second step can be implemented by the nonnegative similarity-matching network (Fig. \[fig:Gabor\]A) because an orthogonal rotation does not affect dot-product similarities [@pehlevan2017blind]. Therefore, NICA is solved by stacking the whitening and the nonnegative similarity-matching networks, Fig. \[fig:NICA\]B. This algorithm performs well compared to other popular NICA algorithms [@pehlevan2017blind], Fig. \[fig:NICA\]C.
Non-negative similarity-based networks for clustering and manifold tiling
=========================================================================
Nonnegative similarity-matching can also cluster well-segregated data [@kuang2012symmetric; @pehlevan2014NMF] and, for data concentrated on manifolds, it can tile them [@sengupta2018manifold]. To understand this behavior, we analyze the optimal solutions of nonnegative similarity-based objectives. Finding the optimal solution for a constrained similarity-based objective is rather challenging as has been observed for the non-negative matrix factorization problem. Here, we introduce a simplified similarity-based objective that allows us to make progress with the analysis and admits an intuitive interpretation. First, we address the simpler clustering task which, for highly segregated data, has a straightforward optimal solution. Second, we address manifold learning by viewing it as a soft-clustering problem.
A similarity-based cost function and NN for clustering
------------------------------------------------------
The key to our analysis is formulating a similarity-based cost function, an optimization of which will yield an online algorithm and a NN for clustering. The algorithm should assign inputs $\x_t$ to $k$ clusters based on pairwise similarities and output cluster assignment indices $\y_{t}$.
To arrive at a cost function, consider first a single pair of data points, $\x_1$ and $\x_2$. If their similarity, $\x_1^\top \x_{2}<\alpha$, where $\alpha$ is a pre-set threshold, then the points should be assigned to separate clusters, i.e. $\y_1 = [1,0]^\top$ and $\y_2 = [0,1]^\top$, setting output similarity, ${\bf y}_1^\top{\bf y}_{2}$ to 0. If $\x_1^\top \x_{2}>\alpha$, then the points are assigned to the same cluster, e.g. ${\bf y}_1={\bf y}_{2} = [1,0]^\top$. Such $\y_1$ and $\y_2$ are optimal solutions (although not unique) to the following optimization problem: $$\begin{aligned}
\label{Eq_single}
\min_{{\bf y}_1\geq 0,{\bf y}_2\geq 0} \left(\alpha-\x_1^\top \x_2\right){\bf y}_1^\top{\bf y}_{2}, \qquad {\rm s.t.} \quad \left\Vert{\bf y}_1\right\Vert_2\leq 1, \, \left\Vert{\bf y}_{2}\right\Vert_2\leq 1.\end{aligned}$$
To obtain an objective function that would cluster the whole dataset of $T$ inputs we simply sum over all possible input pairs: $$\begin{aligned}
\label{Eq_obj}
\min_{{\bf y}_1\geq 0,\ldots,{\bf y}_T\geq 0} \sum_{t=1}^T\sum_{t'=1}^T\left(\alpha-\x_t^\top \x_{t'}\right){\bf y}_t^\top{\bf y}_{t'} \qquad
{\rm s.t.}\quad \left\Vert{\bf y}_1\right\Vert_2\leq 1, \quad\ldots\quad ,\left\Vert{\bf y}_T\right\Vert_2\leq 1.\end{aligned}$$ Does optimization of produce the desired clustering output? This depends on the dataset. If a threshold, $\alpha$, exists such that the similarities of all pairs within the same cluster are greater and similarities of pairs from different clusters are less than $\alpha$, then the cost function is minimized by the desired hard-clustering output, provided that $k$ is greater than or equal to the number of clusters.
To solve the objective in the online setting, we introduce the constraints in the cost via Lagrange multipliers and using the variable substitution trick, we can derive a NN implementation of this algorithm [@sengupta2018manifold] (Fig. \[fig:MNets\]A). The algorithm operates with local Hebbian and anti-Hebbian learning rules, whose functional form is equivalent to .
![\[fig:MNets\] Biologically-plausible NNs for clustering and manifold learning. A) A biologically-plausible excitatory-inhibitory NN implementation of the algorithm. In this version, anti-Hebbian synapses operate at a faster time scale than Hebbian synapses [@sengupta2018manifold]. B) Hard and soft $k$-means networks. Rectified neurons are perfect (hard $k$-means) or leaky (soft $k$-means) integrators. They have learned (homeostatic) activation thresholds and ephaptic couplings. C) When augmented with a hidden nonlinear layer, the presented networks perform clustering in the nonlinear feature space. Shown is the NN of [@bahroun2017neural], where the hidden layer is formed of Random Fourier Features [@rahimi2008random] to obtain a low-rank approximation to a Gaussian kernel. The two-layer NN operates as an online kernel clustering algorithm, and D) performs on par to other state-of-the-art kernel clustering algorithms [@bahroun2017neural]. Shown is performance (Normalized Mutual Information - NMI) on Forest Cover Type dataset. Figure modified from [@bahroun2017neural].](Figure4.pdf){width="100.00000%"}
Manifold-tiling solutions
-------------------------
In many real-world problems, data points are not well-segregated but lie on low-dimensional manifolds. For such data, the optimal solution of the objective effectively tiles the data manifold [@sengupta2018manifold].
We can understand such optimal solutions using soft-clustering, i.e. clustering where each stimulus may be assigned to more than one cluster and assignment indices are real numbers between zero and one. Each output neuron is characterized by the weight vector of incoming synapses which defines a centroid in the input data space. The response of a neuron is maximum when data fall on the centroid and decays away from it. Manifold-tiling solutions for several datasets are shown in Fig. \[fig:tiling\].
![\[fig:tiling\] Analytical and numerical manifold-tiling solutions of for representative datasets provide accurate and useful representations. A) A circular manifold (left) is tiled by overlapping localized receptive fields (right). In the continuum limit ($k\rightarrow \infty$), receptive fields are truncated cosines of the polar angle, $\theta$ [@sengupta2018manifold]. Similar analytical and numerical results are obtained for a spherical 3D manifold and SO(3) (not shown, see [@sengupta2018manifold]). B) Learning the manifold of the 0 digit from the MNIST dataset by tiling the manifold with overlapping localized receptive fields. Left: Two-dimensional linear embedding (PCA) of the outputs. The data gets organized according to different visual characteristics of the hand-written digit (e.g., orientation and elongation). Right: Sample receptive fields over the low-dimensional embedding.](Figure5.pdf){width="\textwidth"}
We can prove this result analytically by taking advantage of the convex relaxation in the limit of infinite number of output dimensions, i.e. $k\to\infty$. Indeed, if we introduce Gramians ${\bf D}$, such that $D_{tt'}=\x_t^\top \x_{t'}$, and ${\bf Q}$, such that $Q_{tt'}=\y_t^\top \y_{t'}$ and do not specify the dimensionality of $\y$ by leaving the rank of ${\bf Q}$ open, we can rewrite as: $$\begin{aligned}
\label{optim}
\min_{\substack{ {\bf Q}\in \mathcal{CP}^T \\ {\rm diag}{\bf Q} \le{\bf 1} }}
-\Tr(({\bf D}-\alpha {\bf E}){\bf Q}),\end{aligned}$$ where ${\bf E}$ is a matrix whose elements are all ones, and the cone of [*completely positive*]{} $T\times T$ matrices, i.e. matrices ${\bf Q}\equiv{\bf Y}^\top{\bf Y}$ with ${\bf Y}\ge 0$, is denoted by $\mathcal{CP}^T$ [@berman2003completely]. Redefining the variables makes the optimization problem convex. For arbitrary datasets, optimization problems in $\mathcal{CP}^T$ are often intractable for large $T$ [@berman2003completely], despite the convexity. However, for symmetric datasets, i.e. circle, 2-sphere and SO(3), we can optimize by analyzing the Karush–Kuhn–Tucker conditions [@sengupta2018manifold] (Fig. \[fig:tiling\]A).
Other similarity-based NNs for clustering and manifold-tiling
-------------------------------------------------------------
A related problem to objective is the previously studied convex semidefinite programming relaxation of community detection in graphs [@cai2015robust], which is closely related to clustering. The semidefinite program is related to by requiring the nonnegativity of ${\bf Q}$ instead of the nonnegativity of ${\bf Y}$:$$\begin{aligned}
\min_{ {\bf Q} \succeq 0,\, {\bf Q}\geq 0, \, {\rm diag}{\bf Q} \le{\bf 1} }
-\Tr(({\bf D}-\alpha {\bf E}){\bf Q}).\end{aligned}$$ While we chose to present our similarity-based NN approach to clustering and manifold-tiling through the cost function in , similar results can be obtained for other versions of similarity-based clustering objective functions. The nonnegative similarity-matching cost function Eq. and the NN derived from it (Fig. \[fig:Gabor\]A) can be used for clustering and manifold learning as well [@kuang2012symmetric; @pehlevan2014NMF; @sengupta2018manifold]. The $K$-means cost function can be cast into a similarity-based form and a NN (Fig. \[fig:MNets\]B) can be derived for its online implementation [@pehlevan2017clustering]. We introduced a soft-$K$-means cost, also a relaxation of another semidefinite program for clustering [@kulis2009semi], and an associated NN (Fig. \[fig:MNets\]B) [@pehlevan2017clustering], and showed that they can perform manifold tiling [@tepper2017clustering].
The algorithms we discussed operate with the dot product as a measure of similarity in the inputs. By augmenting the presented NNs by an initial random, nonlinear projection layer (Fig. \[fig:MNets\]C), it is possible to implement nonlinear similarity measures associated with certain kernels [@bahroun2017neural]. A clustering algorithm using this idea is shown to perform on par with other online kernel clustering algorithms [@bahroun2017neural], Fig. \[fig:MNets\]D.
Discussion
==========
To overcome the non-locality of the learning rule in NNs derived from the reconstruction error minimization, we proposed a new class of cost functions called similarity-based. To summarize, the first term in the similarity-based cost functions, $$\begin{aligned}
\label{sbc}
\min_{\forall t, \y_t\in \Omega} \left[- \sum_{t=1}^T \sum_{t'=1}^T \y_{t}^\top \y_{t'} \x_t^\top \x_{t'}+ f(\y_1,...,\y_T)\right],\end{aligned}$$ is the covariance of the similarity of the outputs and the similarity of the inputs. Hence, the name “similarity-based" cost functions. Previously, such objectives were used in linear kernel alignment [@cristianini2002kernel]. Our key observation is that optimization of objective functions containing such term in the online setting gives rise to local synaptic learning rules [@pehlevan2018similarity].
To derive biologically plausible NNs from , one must choose not just the first term but also the function, $f$, and the optimization constraints, $\Omega$, so that the online optimization algorithm is implementable by biological mechanisms. We and others have identified a whole family of such functions and constraints (Table \[table:feat\]), some of which were reviewed in this article. As a result, we can relate many features of biological NNs to different terms and constraints in similarity-based cost functions and, hence, give them computational interpretations.
Our framework provides a systematic procedure to design novel NN algorithms by formulating a learning task using similarity-based cost functions. As evidenced by the high-performing algorithms discussed in this paper, our procedure of incorporating biological constraints does not impede but rather facilitates the design process by limiting the algorithm search to a useful part of the NN algorithm space.
OPTIMIZATION FEATURE BIOLOGICAL FEATURE
--------------------------------- --------------------------------------------------------------------------
Similarity (anti-)alignment (Anti-)Hebbian plasticity [@pehlevan2015MDS; @pehlevan2018similarity]
Nonnegativity constraint ReLU activation function [@pehlevan2014NMF; @seung2017correlation]
Sparsity regularizer Adaptive neural threshold [@hu2014SMF]
Adaptive lateral weights [@pehlevan2015normative; @seung2017correlation]
Anti-Hebbian interneurons [@pehlevan2015normative]
Copositive output Grammian Anti-Hebbian inhibitory neurons [@sengupta2018manifold]
Constrained activity $l_1$-norm Giant interneuron [@pehlevan2017clustering]
: Current list of objectives, regularizers and constraints that define a similarity-based optimization problem and solvable by a NN with local learning.
\[table:feat\]
The locality of learning rules in similarity-based NNs makes them naturally suitable for implementation on adaptive neuromorphic systems, which have already been explored in custom analog arrays [@poikonen2017mixed]. For broader use in the rapidly growing world of low-power, spike-based hardware with on-chip learning [@davies2018loihi], similarity-based NNs were missing a key ingredient: spiking neurons. Very recent work [@pehlevan2019spiking] developed a spiking version of the nonnegative similarity matching network and took a step towards neuromorphic applications.
Despite the successes of similarity-based NNs, many interesting challenges remain. 1) Whereas numerical experiments indicate that our online algorithms perform well, most of them lack global convergence proofs. Even for PCA networks we can only prove linear stability of the desired solution in the stochastic approximation setting. 2) Motivated by biological learning, which is mostly unsupervised, we focused on unsupervised learning. Yet, supervision, or reinforcement, does take place in the brain. Therefore, it is desirable to extend our framework to supervised, semi-supervised and reinforcement learning settings. Such extensions may be valuable as general purpose machine learning algorithms. 3) Whereas most sensory stimuli are correlated time series, we assumed that data points at different times are independent. How are temporal correlations analyzed by NNs? Solving this problem is important both for modeling brain function and developing general purpose machine learning algorithms. 4) Another challenge is stacking similarity-based NNs. Heuristic approach to stacking yields promising results [@bahroun2017online]. Yet, except for the Nonnegative ICA problem introduced in Section 4, we do not have a theoretical understanding of how and why to stack similarity-based NNs. 5) Finally, neurons in biological NNs signal each other using all-or-none spikes, or action potentials, as opposed to real-valued signals we considered. Is there an optimization theory accounting for spiking in biological NNs?
Acknowledgments {#acknowledgments .unnumbered}
---------------
We thank our collaborators Anirvan Sengupta, Mariano Tepper, Andrea Giovannucci, Alex Genkin, Victor Minden, Sreyas Mohan, Yanis Bahroun for their contributions, Andrea Soltoggio for discussions, and Siavash Golkar and Alper Erdogan for commenting on the manuscript. This work was in part supported by a gift from the Intel Corporation.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[ l@\#1 =l@\#1 \#2]{}]{}
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[^1]: [email protected]
[^2]: [email protected]
[^3]: A package with implementations of these algorithms is on <https://github.com/flatironinstitute/online_psp> and <https://github.com/flatironinstitute/online_psp_matlab>.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Recently we have shown that the reduction of the Carruthers-Nieto symmetric quantum phase fluctuation parameter $(U)$ with respect to its coherent state value corresponds to an antibunched state, but the converse is not true. Consequently reduction of $U$ is a stronger criterion of nonclassicality than the lowest order antibunching. Here we have studied the possibilities of reduction of $U$ in intermediate states by using the Barnett Pegg formalism. We have shown that the reduction of phase fluctuation parameter $U$ can be seen in different intermediate states, such as binomial state, generalized binomial state, hypergeometric state, negative binomial state, and photon added coherent state. It is also shown that the depth of nonclassicality can be controlled by various parameters related to intermediate states. Further, we have provided specific examples of antibunched states, for which $U$ is greater than its poissonian state value.'
---
[Reduction of Quantum Phase Fluctuations in Intermediate States]{}
[Amit Verma]{}[^1] [and Anirban Pathak]{}[^2]
Jaypee Institute of Information Technology University, A-10, Sector-62, Noida, UP-201307
**INDIA**
**PACS number(s): 42.50.Lc, 42.50.Ar, 42.50.-p**
**Keywords:** quantum phase, nonclassical state, quantum fluctuation, intermediate states.
Introduction
============
A state which does not have any classical analogue is known as nonclassical state. For example, squeezed state and antibunched state are nonclassical. of a quantum state is a measure of the total fluctuations of the amplitude Particular parameters, which are essentially combination of standard deviations of some function of quantum phase, were introduced by Carruthers and Nieto [\[]{}\[carutherrs\]\] as a measure of quantum phase fluctuations. In recent past people have used Carruthers Nieto parameters to study quantum phase fluctuations of coherent light interacting with a nonlinear nonabsorbing medium of inversion symmetry [\[]{}\[Gerry\]-\[enu:pathak\]\]. But unfortunately any discussion regarding the physical meaning of these parameters were missing since recent past. Recently we have shown that the reduction of the Carruthers-Nieto symmetric quantum phase fluctuation parameter $(U)$ with respect to its poissonian state value corresponds to an antibunched state, but the converse is not true [\[]{}\[the:phase-prakash\]\]. Consequently reduction of $U$ is a stronger criterion of nonclassicality than the lower order antibunching.
The intermediate states are nonclassical in general. It has also been observed that almost all the intermediate states satisfy the condition of higher order antibunching [\[]{}\[A-Verma\]\]. As the condition of higher order antibunching is stronger than that of usual antibunching in the sense that a state which is antibunched in the $lth$ order has to be antibunched in $(l-1)th$ order too but the converse is not true [\[]{}\[enu:garcia\]\]. Therefore, it seems quite reasonable to check whether the intermediate states satisfy the stronger condition of reduction of $U$ or not. Present study reveals that the intermediate states may satisfy this stronger nonclassical criterion (i.e. reduction of $U$ criterion).
The introduction of hermitian phase operators have some ambiguities (interested readers can see the review [\[]{}\[Lynch2\]\]) which lead to many different formalisms [\[]{}\[enu:L.-Suskind-and\]-\[enu:bp\] and references there in\] of quantum phase. Among the different formalisms, Susskind Glogower (SG) [\[]{}\[enu:L.-Suskind-and\]\], Pegg Barnett [\[]{}\[enu:D.-T.-Pegg\]\] and Barnett Pegg (BP) [\[]{}\[enu:bp\]\] formalisms played most important role in the studies of phase properties and the phase fluctuations of various physical systems. For example, SG formalism has been used by Fan [\[]{}\[Fan\]\], Sanders [\[]{}\[Sander\]\], Yao [\[]{}\[Yao\]\], Gerry [\[]{}\[Gerry\]\], Carruthers and Nieto [\[]{}\[carutherrs\]\] and many others to study the phase properties and phase fluctuations. On the other hand Lynch [\[]{}\[Lynch\],\[Lynch1\]\], Vacaro [\[]{}\[Vacaro\]\], Tsui [\[]{}\[Y.-K.-Tsui,\]\], Pathak and Mandal [\[]{}\[enu:pathak\]\] and others have used the BP formalism for the same purpose. The physical interpretation of reduction of $U$ is valid in both SG and BP formalism of quantum phase [\[]{}\[the:phase-prakash\]\]. Here we have studied the possibilities of observing reduction of $U$ with respect to its poissonian state value (for intermediate state) in BP formalism.
The importance of a systematic study of quantum phase fluctuation of intermediate state has also increased with the recent observations of quantum phase fluctuations in quantum computation [\[]{}\[qutrit\], \[L.-L.-Sanchez-Soto\]\] and superconductivity [\[]{}\[Y.-K.-Tsui,\], \[Nature, supercond\]\] and with the success in experimental production of photon added coherent state [\[]{}\[the:photonadded-experiment\]\]. These observations along with the fact that intermediate states satisfy stronger criterion of nonclassicality (namely the criterion of HOA ) have motivated us to study quantum phase fluctuation of intermediate states. In next section we briefly introduce quantum phase fluctuation parameter ($U$) and the meaning of reduction of $U$. In section 3, it is shown that the reduction of phase fluctuation parameter $U$ can be seen in different intermediate state, such as binomial state, hypergeometric state, generalized binomial state, negative binomial state and photon added coherent state. Role of various parameters in controlling the depth of nonclassicality is also discussed. Finally in section 4 we conclude.
Measures of quantum phase fluctuations: Understanding their physical meaning
=============================================================================
Dirac [\[]{}\[enu:dirac\]\] introduced the quantum phase operator in 1926. Immediately after Dirac’s introductory work it was realized that the uncertainty relation $\Delta N\Delta\phi\ge\frac{1}{2}$ associated with Dirac’s quantum phase has many problems [\[]{}\[Lynch2\]\]. Later on Louisell [\[]{}\[enu:W.-H.-Louisell,\]\] had shown that most of the problems can be solved if instead of bare phase operator we consider sine $(S)$ and cosine $(C)$ operators which satisfy $$\begin{array}{c}
[N,C]=-iS\end{array}\label{eq:phase5.1}$$ and $$[N,S]=iC.\label{eq:phase5.2}$$ Therefore, the uncertainty relations associated with them are $$\Delta N\Delta C\ge\frac{1}{2}\left|\langle S\rangle\right|\label{eq:phase5.3}$$ and $$\Delta N\Delta S\ge\frac{1}{2}\left|\langle
C\rangle\right|.\label{eq:phase5.4}$$ There are several formalism of quantum phase, and each formalism defines sine and cosine in an unique way. The sine and cosine operators in Susskind Glogower formalism is essentially originated due to a rescaling of the photon annihilation and creation operators with the photon number operator. Another convenient way is to rescale an appropriate quadrature operator with the averaged photon number. Barnett and Pegg followed this convention and defined the exponential of phase operator $E$ and its Hermitian conjugate $E^{\dagger}$ as [\[]{}\[enu:bp\]\] $$\begin{array}{lcl}
E & = & \left(\overline{N}+\frac{1}{2}\right)^{-1/2}a(t)\\
E^{\dagger} & = & \left(\overline{N}+\frac{1}{2}\right)^{-1/2}a^{\dagger}(t)\end{array}\label{taro}$$ where $\overline{N}$ is the average number of photons present in the radiation field after interaction. The usual cosine and sine of the phase operator are defined as $$\begin{array}{lcl}
C & = & \frac{1}{2}\left(E+E^{\dagger}\right)\\
S & = &
-\frac{i}{2}\left(E-E^{\dagger}\right)\end{array}\label{chauddo}$$ which satisfy $$\langle C^{2}\rangle+\langle
S^{2}\rangle=1.\label{eq:bp2}$$ Squaring and adding (\[eq:phase5.3\]) and (\[eq:phase5.4\]) we obtain $$(\Delta N)^{2}\left[(\Delta S)^{2}+(\Delta
C)^{2}\right]\left/\left[<S>^{2}+<C>^{2}\right]\right.\geq\frac{1}{4}.\label{eq:babu}$$ Carruthers and Nieto [\[]{}\[carutherrs\]\] introduced (\[eq:babu\]) as measure of quantum phase fluctuation and named it as $U$ parameter. To be precise, Carruthers and Nieto defined following parameter as a measure of phase fluctuation[^3]: $$U\left(\theta,t,|\alpha|^{2}\right)=(\Delta N)^{2}\left[(\Delta S)^{2}+(\Delta C)^{2}\right]\left/\left[\langle S\rangle^{2}+\langle C\rangle^{2}\right]\right.\label{kuri}$$ where, $\theta$ is the phase of the input coherent state $|\alpha\rangle$, $t$ is the interaction time and $|\alpha|^{2}$ is the mean number of photon prior to the interaction. Later on this parameter draw more attention and many groups [\[]{}\[Gerry\]-\[enu:pathak\]\] have used these parameters as a measure of quantum phase fluctuation.
The total noise of a quantum state is a measure of the total fluctuations of the amplitude. For a single mode quantum state having density matrix $\rho$ it is defined as [\[]{}\[the:orlowski\]\]$$\begin{array}{lcl}
T(\rho) & = & (\Delta X)^{2}+(\Delta\dot{X})^{2}\end{array}.\label{eq:total noise}$$ In analogy to it we can define the total phase fluctuation as $$T=(\Delta S)^{2}+(\Delta
C)^{2}.\label{eq:totalnoise-phase}$$ Now using the relations (\[eq:phase5.3\]), (\[eq:phase5.4\]), (\[eq:bp2\]), (\[eq:babu\]) and (\[eq:totalnoise-phase\]) we obtain $$\left(\Delta N\right)^{2}\left(\Delta S\right)^{2}+\left(\Delta
N\right)^{2}\left(\Delta C\right)^{2}\geq\frac{1}{4}\left(\langle
S\rangle^{2}+\langle C\rangle^{2}\right)=\frac{1}{4}\left(\langle
S^{2}\rangle+\langle C^{2}\rangle-\left(\left(\Delta
S\right)^{2}+\left(\Delta C\right)^{2}\right)\right)$$ or, $$\frac{1}{4}\left(1-\left(\left(\Delta S\right)^{2}+\left(\Delta
C\right)^{2}\right)\right)\le\left(\left(\Delta S\right)^{2}+\left(\Delta C\right)^{2}\right)\left(\Delta
N\right)^{2}$$ or, $$U=\frac{\left(\left(\Delta S\right)^{2}+\left(\Delta
C\right)^{2}\right)\left(\Delta N\right)^{2}}{\left(1-\left(\left(\Delta S\right)^{2}+\left(\Delta
C\right)^{2}\right)\right)}=\frac{T\left(\Delta N\right)^{2}}{\left(1-T\right)}\ge\frac{1}{4}.\label{eq:total
noise2}$$ and$$[C,S]=\frac{i}{2}\left(\overline{N}+\frac{1}{2}\right)^{-\frac{1}{2}}.\label{eq:bp3}$$ Therefore, $$(\Delta C)^{2}(\Delta
S)^{2}\geq\frac{1}{16}\frac{1}{\left(\overline{N}+\frac{1}{2}\right)}.\label{eq:bp4}$$ Now we can write,$$T=(\Delta C)^{2}+(\Delta S)^{2}\ge(\Delta
C)^{2}+\frac{1}{16\left(\overline{N}+\frac{1}{2}\right)(\Delta
C)^{2}}.$$ The function $T=(\Delta
C)^{2}+\frac{1}{16\left(\overline{N}+\frac{1}{2}\right)(\Delta
C)^{2}}$ has a clear minima at $(\Delta
C)^{2}=\frac{1}{4\left(\overline{N}+\frac{1}{2}\right)^{\frac{1}{2}}}$, which corresponds to a coherent state and thus the total fluctuation in quantum phase variables $\left((\Delta C)^{2}+(\Delta
S)^{2}\right)$ can not be reduced below its coherent state value $\frac{1}{2\left(\overline{N}+\frac{1}{2}\right)^{\frac{1}{2}}}$. Now since $(\Delta N)^{2}$ is positive and the $U=\frac{T(\Delta
N)^{2}}{\left(1-T\right)}=b(\Delta N)^{2}\geq\frac{1}{4}$, therefore $b=\frac{T}{\left(1-T\right)}$ is positive. Under these conditions $b$ increases monotonically with the increase in $T$. Thus the minima of $T$ corresponds to the minima of $b$ too and consequently, $b$ is minimum for coherent state. In other words $b$ can not be reduced below its coherent state value. Therefore any reduction in $U=b(\Delta N)^{2}$ with respect to its poissonian state value will mean a decrease in $(\Delta N)^{2}$ with respect to its poissonian state counter part. Thus the reduction of $U$ with respect to its poissonian state value implies antibunching but the converse is not true. Earlier we have reported reduction of $U$ with respect to coherent state in some simple optical processes and verified that the states are antibunched for the corresponding parameters. But the fact that every antibunched state is not associated with the reduction of $U$ was not verified in the earlier work. Here we have shown that reduction of $U$ is possible for various intermediate states and have also provided examples of intermediate states, which are antibunched for specific values of parameters but do not show reduction of $U$ for the same values of the parameters. The intermediate states are studied under BP formalism because of the inherent computational simplicity of these formalism over the others. As the value of $U$ in coherent (poissonian) state is $\frac{1}{2}$ our requirement of strong nonclassicality reduces to $$d_{u}=U-\frac{1}{2}<0,\label{eq:du1}$$ further simplification of the criterion (\[eq:du1\]) is possible in BP formalism since the symmetric phase fluctuation parameter in BP formalism reduces to $U=[<a^{\mathit{\dagger\mathrm{2}}}a^{2}>+<a^{\mathit{\dagger}}a>-<a^{\mathit{\dagger}}a>^{2}][\frac{<a^{\mathit{\dagger}}a>-<a^{\dagger}><a>+\frac{1}{2}}{<a^{\dagger}><a>}]$ and consequently our requirement for strong nonclassicality is $$d_{u}=[<a^{\mathit{\dagger\mathrm{2}}}a^{2}>+<a^{\mathit{\dagger}}a>-<a^{\mathit{\dagger}}a>^{2}][\frac{<a^{\mathit{\dagger}}a>-<a^{\dagger}><a>+\frac{1}{2}}{<a^{\dagger}><a>}]-\frac{1}{2}<0.\label{eq:du2}$$ Now in light of this criterion we would like to study the nonclassical behavior of intermediate states.
Quantum phase fluctuations in intermediate states
=================================================
Actually, an intermediate state is a quantum state which reduces to two or more distinguishably different states (normally, distinguishable in terms of photon number distribution) in different limits. In 1985, such a state was first time introduced by Stoler *et al.* [\[]{}\[stoler\]\]. To be precise, they introduced Binomial state (BS) as a state which is intermediate between the most nonclassical number state $|n\rangle$ and the most classical coherent state $|\alpha\rangle$. They defined BS as $$\begin{array}{lr}
|p,M\rangle=\sum_{n=0}^{M} & B_{n}^{M}\end{array}|n\rangle=\sum_{n=0}^{M}\sqrt{\left(\begin{array}{c}
M\\
n\end{array}\right)p^{n}(1-p)^{M-n}}|n\rangle\,\,\,\:0\leq p\leq1.\label{eq:binomial1}$$ This state[^4] is called intermediate state as it reduces to number state in the limit $p\rightarrow0$ and $p\rightarrow1$ (as $|0,M\rangle=0$ and $|1,M\rangle=|M\rangle$) and in the limit of $M\rightarrow\infty,p\rightarrow1$, where $\alpha$ is a real constant, it reduces to a coherent state with real amplitude. Since the introduction of BS as an intermediate state it was always been of interest to quantum optics, nonlinear optics, atomic physics and molecular physics community. Consequently, different properties of binomial state have been studied [\[]{}\[the:hong-chenfu-genralized-BS\]-\[the:OEBS\]\]. In these studies it has been observed that the nonclassical phenomena (such as, antibunching, squeezing and higher order squeezing) can be seen in BS. This trend of search for nonclassicality in Binomial state, continued in nineties and in one hand, several version of generalized BS has been proposed [\[]{}\[the:hong-chenfu-genralized-BS\]-\[the:Hong-yi-Fan-generalized-BS\]\] and in the other hand people went beyond binomial states and proposed several other form of intermediate states (such as The studies in the nineties were mainly limited to theoretical predictions but the recent developments in the experimental techniques made it possible to verify some of those theoretical predictions. For example, we can note that, as early as in 1991 Agarwal and Tara [\[]{}\[the:agarwal-photonadded\]\] introduced photon added coherent state as $$|\alpha,m\rangle=\frac{a^{\dagger
m}|\alpha\rangle}{\langle\alpha|a^{m}a^{\dagger
m}|\alpha\rangle},\label{eq:photonadded1}$$ (where $m$ is an integer and $|\alpha\rangle$ is coherent state) but the experimental generation of the state has happened only in recent past when Zavatta, Viciani and Bellini [\[]{}\[the:photonadded-experiment\]\] succeed to produce it in 2004. It is easy to observe that this is an intermediate state, since it reduces to coherent state in the limit $m\rightarrow0$ and to number state in the limit $\alpha\rightarrow0$. It is also been found that most of these intermediate states show antibunching, squeezing, higher order squeezing, subpoissonian photon statistics etc. Inspired by these observations, many schemes to generate intermediate states have been proposed in recent past [\[]{}\[the:photonadded-experiment\],\[the:Valverde\],\[the:RLoFranco1\]\]. Thus the intermediate states provide a perfect test bed to test the satisfiability of any new criterion of nonclassicality. Keeping this in mind we will investigate the possibility of satisfication of (\[eq:du2\]) for different intermediate states in the following subsections.
Binomial state
--------------
Binomial state is originally defined as (\[eq:binomial1\]), from which it is straight forward to show that $$\begin{array}{lcl}
a|p,M\rangle & = & [Mp]^{\frac{1}{2}}|p,M-1\rangle.\end{array}\label{eq:eigen1}$$ Similarly, we can write, $$\langle M,p|a^{\dagger}=\langle M-1,p|\left[Mp\right]^{\frac{1}{2}}.\label{eq:bino1}$$ Consequently, we obtain,
$$\begin{array}{lcl}
\left\langle M,p|a^{\dagger}a|p,M\right\rangle & = & Mp\end{array},\label{eq:bino2}$$
$$\left\langle M,p|a^{\dagger2}a^{2}|p,M\right\rangle =M(M-1)p^{2}\label{eq:bino3}$$
and $$\left\langle a^{\dagger}\right\rangle \left\langle a\right\rangle =Mp(\sum_{n=0}^{M-1}B_{n}^{M-1}B_{n}^{M})^{2}.\label{eq:bino4}$$ Now using equations (\[eq:du2\]) and (\[eq:eigen1\]-\[eq:bino4\]) one can obtain,
$$\begin{array}{ccc}
d_{U(BS)} & =[\frac{Mp(1-p)}{(\sum_{n=0}^{M-1}B_{n}^{M-1}B_{n}^{M})^{2}} & [\frac{1}{2Mp}+1-(\sum_{n=0}^{M-1}B_{n}^{M-1}B_{n}^{M})^{2}]-\frac{1}{2}]\end{array}\label{eq:du-binomial}$$
\[h\]
From the Fig. 1, it is clear that the binomial state shows reduction of fluctuation of quantum phase with respect to its coherent state counter part and thus it satisfies this stronger criterion of nonclassicality. But it does not satisfy the criterion for higher values of $p$. In [\[]{}\[A-Verma\]\] we have shown that Binomial state is always antibunched up to any order. For higher values of $p$ it is antibunched in every order and thus satisfies the other strong criterion of nonclassicality but do not satisfy the criterion laid down on the basis of quantum phase fluctuations. Earlier we had reported [\[]{}\[the:phase-prakash\]\] that reduction of quantum phase fluctuation means antibunching but the converse is not true. This is the first time when an example of such a state which is antibunched but reduction of quantum phase fluctuation with respect to coherent state does not happen, is found.
Generalized binomial state
--------------------------
As we have mentioned earlier there are different form of generalized binomial states [\[]{}\[the:hong-chenfu-genralized-BS\]-\[the:Hong-yi-Fan-generalized-BS\]\], in the present section we have chosen generalized binomial state introduced by Roy and Roy [\[]{}\[the:broy&proy-generalized-bs\]\] for our study. Roy and Roy have introduced the generalized binomial state as $$|N,\alpha,\beta\rangle=\sum_{n=0}^{N}\sqrt{\omega(n,N,\alpha,\beta)}|n\rangle\label{eq:gen-bino1}$$ where, $$\omega(n,N,\alpha,\beta)=\frac{N!}{(\alpha+\beta+2)_{N}}\frac{(\alpha+1)_{n}(\beta+1)_{N-n}}{n!(N-n)!}\label{eq:gen-bino2}$$ where $(x)_{r}$ is conventional Pochhammer symbol and $\alpha,\beta>-1$, $n=0,1,....,N$. Now with the help of properties of Pochhammer symbol and operator algebra we can obtain following relations:$$\begin{array}{lcl}
a|N,\alpha,\beta\rangle & = & \left\{ \frac{N(\alpha+1)}{(\alpha+\beta+2)}\right\} ^{\frac{1}{2}}\sum_{l=0}^{N-1}\left\{ \frac{(N-1)!(\alpha+2)_{l}(\beta+1)_{N-1-l}}{(\alpha+2+\beta+1)_{N-1}l!(N-1-l)!}\right\} ^{\frac{1}{2}}|l\rangle\\
& = & \left\{ \frac{N(\alpha+1)}{(\alpha+\beta+2)}\right\} ^{\frac{1}{2}}\sum_{n=0}^{N-1}\sqrt{\omega(n,N-1,\alpha+1,\beta)}|n\rangle,\end{array}\label{eq:gb1}$$ $$\langle N,\alpha,\beta|a^{\dagger}a|N,\alpha,\beta\rangle=\frac{N(\alpha+1)}{(\alpha+\beta+2)},\label{eq:gb2}$$ $$\langle N,\alpha,\beta|a^{\dagger2}a^{2}|N,\alpha,\beta\rangle=\frac{N(N-1)(\alpha+1)(\alpha+2)}{(\alpha+\beta+2)(\alpha+\beta+3)}\label{eq:gb3}$$ and $$\begin{array}{ccc}
\left\langle a^{\dagger}\right\rangle \left\langle a\right\rangle & = & \frac{N^{2}(\alpha+1)^{2}}{(\alpha+\beta+2)^{2}}[\sum_{n=0}^{N-1}\sqrt{\omega(n,N-1,\alpha+1,\beta)}\end{array}]^{2}.\label{eq:gb4}$$ Therefore, $$\begin{array}{lcl}
d_{U(GBS)} & = & \left[\frac{(\beta+1)(\alpha+\beta+N+2)}{N^{3}(\alpha+1)(\alpha+\beta+3)[\frac{(N-1)!(\alpha+2)_{n}(\beta+1)_{N-n}}{(\alpha+\beta+3)_{N-1}(N-n-1)!}]}\right.\left[\frac{N(N-1)(\alpha+1)(\alpha+2)}{(\alpha+\beta+2)(\alpha+\beta+3)}\right.\\
& - & \left.\begin{array}{c}
\left.\frac{N^{4}(\alpha+1)^{2}(N-1)!(\alpha+2)_{n}(\beta+1)_{N-n}}{(\alpha+\beta+2)^{2}(\alpha+\beta+3)_{N-1}(N-n-1)!}+\frac{1}{2}\right]\end{array}-\frac{1}{2}\right]\end{array}\label{eq:gb5}$$
\[h\]
From Fig. 2 it is clear that the reduction of quantum phase fluctuation happens for Roy and Roy generalized binomial state. It is also observed that the depth of nonclassicality reduces with the increase in $\alpha$ and $\beta$. But as far as the higher (second) order antibunching is concerned, the depth of nonclassicality associated with it decreases with increase in $\alpha$ and increases with increase in $\beta$ (see Fig. 2 and 3 of [\[]{}\[A-Verma\]\]). Farther it had been observed in [\[]{}\[A-Verma\]\] that for particular values of $\alpha,\beta$ and $N$ the state does not show second order antibunching. But here it is found that for the same values of $\alpha,\beta$ and $N$ we can obtain reduction of quantum phase fluctuation parameter with respect to its coherent state counterpart. Consequently we can say that, reduction of quantum phase fluctuation means antibunching but does not essentially mean higher order antibunching and therefore, it is not essential that these two stronger conditions of nonclassicality would appear simultaneously.
Photon added coherent state:
----------------------------
Photon added coherent state (PACS) is defined as [\[]{}\[the:agarwal-photonadded\]\]) $$\begin{array}{lcl}
|\alpha,m> & = &
\frac{exp(-|\alpha|^{2}/2)}{[L_{m}(-|\alpha|^{2})m!]^{1/2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}\sqrt{(n+m)!}}{n!}|n+m>\end{array}\label{eq:pacs1}$$ where $L_{m}(x)$$\begin{array}{cc} = & \sum_{n=0}^{m}\frac{(-x)^{n}m!}{(n!)^{2}(m-n)!}\end{array}$is Lauguere polynomial. Rigorous operator algebra yields $$<a^{\dagger}a>=\frac{exp(-|\alpha|^{2})}{L_{m}(-|\alpha|^{2})m!}\sum_{n=0}^{\infty}\frac{(n+m)!\alpha^{2(n+1)}(m+n+1)^{2}}{(n+1)!^{2}},\label{eq:pacs2}$$ $$<a^{\dagger2}a^{2}>=\frac{exp(-|\alpha|^{2})}{L_{m}(-|\alpha|^{2})m!}\sum_{n=0}^{\infty}\frac{(n+m)!\alpha^{n+2}(m+n+1)^{2}(m+n+2)^{2}}{(n+2)!^{2}}\label{eq:pacs3}$$ and $$<a^{\dagger}>=<a>=\frac{exp(-|\alpha|^{2})}{L_{m}(-|\alpha|^{2})m!}\sum_{n=0}^{\infty}\frac{(n+m)!\alpha^{2n+1}(m+n+1)}{(n+1)(n!)^{2}}.\label{eq:pacs4}$$ By substituting equations (\[eq:pacs2\]-\[eq:pacs4\]) in (\[eq:du2\]) we can easily obtain a long expression of $d_{U}$. Essential characteristic of $d_{U}$ of PACS can be seen in Fig 3. It is easy to observe that the reduction of quantum phase fluctuation is possible in photon added coherent state. Since the depth of nonclassicality increases with the increase in $m$. So we can conclude, the more photon are added to coherent state the more nonclassical it is as far as the depth of nonclassicality associated with quantum phase fluctuation is concerned. This particular characteristic is also been reflected in higher order antibunching [\[]{}\[A-Verma\]\].
\[h\]
Other intermediate states
-------------------------
As it is mentioned in the earlier sections, there exist several different intermediate states. For the systematic study of possibility of reduction of quantum phase fluctuation in intermediate states, we have studied all the well known intermediate states. Since the procedure followed for the study of different states is similar, mathematical detail has not been shown in the subsections below. But from the expression of $d_{U}$ and the corresponding plots it would be easy to see that the reduction of quantum phase fluctuation can be observed in all the intermediate states studied below.
### Negative binomial state
Negative Binomial State(NBS) can be defined as $$|p,M\mbox{>}=\sum_{n=M}^{\infty}[\left(\begin{array}{c}
n\\
M\end{array}\right)p^{M+1}(1-p)^{n-M}]^{1/2}|n>.\label{eq:nbs}$$ Similar operator algebra yields
$$d_{U(NBS)}=\left[\frac{(M+1)(1-p)^{2M+1}}{p^{2(M+2)}}\left\{ \frac{\frac{(M+1-p)}{p}-\frac{p^{2(M+1)}}{(1-p)^{2M}}\left(\sum_{n=M}^{\infty}\sqrt{\left(\begin{array}{c}
n+1\\
M\end{array}\right)\left(\begin{array}{c}
n\\
M\end{array}\right)(1-p)^{(n+1)}(n+1)}\right)^{2}+\frac{1}{2}}{\left(\sum_{n=M}^{\infty}\sqrt{\left(\begin{array}{c}
n+1\\
M\end{array}\right)\left(\begin{array}{c}
n\\
M\end{array}\right)(1-p)^{(n+1)}(n+1)}\right)^{2}}\right\}
-\frac{1}{2}\right].\label{eq:du-2}$$
Variation of quantum phase fluctuation parameter with respect to the probability $p$ for NBS is shown in the Fig. 4 and it has been observed that the state is more nonclassical for lower values of $p$ and higher values of $M$ [\[]{}\[A-Verma\]\]. This is consistent with the earlier observations on higher order antibunching of NBS.
\[h\]
Hyper geometric state
---------------------
Hyper geometric state (HS) is defined as [\[]{}\[the:hypergeomtric\]\] $$|L,M,p>=\sum_{n=0}^{M}H_{n}^{M}(p,L)|n>\label{eq:hs1}$$ where, $H_{n}^{M}=\left[\left(\begin{array}{c}
Lp\\
n\end{array}\right)\left(\begin{array}{c}
L(1-p)\\
M-n\end{array}\right)\right]^{1/2}\left(\begin{array}{c}
L\\
M\end{array}\right)^{-1/2}$. For this intermediate state we obtain $$\begin{array}{lcl}
d_{U(HS)} & = & \left[\frac{pM(1-p)(L-M)\left(\begin{array}{c}
L\\
M\end{array}\right)^{2}}{(L-1)\sqrt{Lp}\left(\sum_{n=0}^{M-1}\sqrt{\left(\begin{array}{c}
Lp\\
n\end{array}\right)\left(\begin{array}{c}
L(1-p)\\
M-n\end{array}\right)\left(\begin{array}{c}
Lp-1\\
n\end{array}\right)\left(\begin{array}{c}
L(1-p)\\
M-n-1\end{array}\right)}\right)^{2}}\right.\\
& \times & \left.\left\{ Mp-\frac{\sqrt{Lp}}{\left(\begin{array}{c}
L\\
M\end{array}\right)^{2}}\left(\sum_{n=0}^{M-1}\sqrt{\left(\begin{array}{c}
Lp\\
n\end{array}\right)\left(\begin{array}{c}
L(1-p)\\
M-n\end{array}\right)\left(\begin{array}{c}
Lp-1\\
n\end{array}\right)\left(\begin{array}{c}
L(1-p)\\
M-n-1\end{array}\right)}\right)^{2}+\frac{1}{2}\right\} -\frac{1}{2}\right].\end{array}\label{eq:du-hs}$$
\[h\]
Fig. 5 depicts the characteristics of quantum phase fluctuation of HS. From this figure we can clearly observe that HS does not satisfy the reduction of quantum phase fluctuation criterion for higher values of $p$ but it satisfies the condition of higher order antibunching and consequently the condition of antibunching for those values of $p.$ This is consistent with our theoretical prediction that reduction of quantum phase fluctuation means antibunching but the converse is not true.
Conclusions
===========
In essence, all the intermediate states described above [^5] that the binomial state can show reduction of fluctuation of quantum phase with respect to its coherent state counter part and thus it can satisfy this stronger criterion of nonclassicality. But it does not satisfy the criterion for higher values of $p$ (see Fig. 1). In [\[]{}\[A-Verma\]\] we have shown that Binomial state is always antibunched up to any order. Thus for higher values of $p$ it is higher order antibunched and consequently satisfies the other strong criterion of nonclassicality but do not satisfy the criterion laid down on the basis of quantum phase fluctuations. Similar phenomenon is also observed in HS. Earlier we had reported [\[]{}\[the:phase-prakash\]\] that reduction of quantum phase fluctuation means antibunching but the converse is not true. This is the first time when an example of such a state which is antibunched but does not show reduction of quantum phase fluctuation with respect to coherent state, is found. Further from the study of phase properties of Roy Roy GBS and HS we have learnt that the reduction of quantum phase fluctuation mean antibunching but does not essentially mean higher order antibunching and therefore, it is not essential that these two stronger conditions of nonclassicality appear simultaneously. In connection to PACS we have observed that the more photon are added to coherent state the more nonclassical the PACS, is as far as the depth of nonclassicality associated with quantum phase fluctuation is concerned. This particular characteristic has also been reflected in higher order antibunching [\[]{}\[A-Verma\]\]. Further we have seen that the NBS is more nonclassical for lower values of $p$. This is also similar with the earlier observations on higher order antibunching of NBS.
**Acknowledgement**: AP thanks to DST, India for partial financial support through the project grant SR\\FTP\\PS-13\\2004.
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[^1]: [email protected]
[^2]: [email protected]
[^3]: They had also introduced two more parameters $S$ and $Q$ for the purpose of calculation of the phase fluctuations. But these parameters are not relevant for the present work.
[^4]: The state is named as binomial state because the photon number distribution associated with this state $\left(i.e.\,|B_{n}^{M}|^{2}\right)$is simply a binomial distribution.
[^5]: In reciprocal binomial state, we have not observed this phenomenon.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Anomalous transport in one-dimensional translation invariant Hamiltonian systems with short range interactions, is shown to belong in general to the KPZ universality class. Exact asymptotic forms for density-density and current-current time correlation functions and their Fourier transforms are given in terms of the Prähofer-Spohn scaling functions, obtained from their exact solution for the polynuclear growth model. The exponents of corrections to scaling are found as well, but not so the coefficients. Mode coupling theories developed previously are found to be adequate for weakly nonlinear chains, but in need of corrections for strongly anharmonic interparticle potentials. A simple condition is given under which KPZ behavior does not apply, sound attenuation is only logarithmically superdiffusive and heat conduction is more strongly superdiffusive than under KPZ behavior.'
author:
- Henk van Beijeren
title: 'Exact results for anomalous transport in one-dimensional Hamiltonian systems'
---
Since the discovery by Alder and Wainwright[@alderw] of long-time tails in the Green-Kubo current-current time correlations, such as the velocity autocorrelation function it has been clear that transport in one and two dimensional Hamiltonian systems must be anomalous in most cases. One-dimensional systems have been studied extensively in the past decades, both by mode coupling techniques[@delfini; @delfini2; @lietal; @leedadswell] and dynamical scaling[@rama], and also by computer simulations[@delfini; @delfini2; @lietal; @leedadswell; @grassberger]. Most studied are the exponents $\alpha$ describing the divergence of the coefficients of heat conduction and sound damping with system size $L$ as $L^{\alpha}$, and $\delta$ describing the power law $t^{-(1-\delta)}$ by which the corresponding current-current time correlation functions decay. For both exponents various values have been proposed, with $\delta
=1/3$ for both heat conduction and sound attenuation and $\alpha=1/3$ for heat conduction but 1/2 for sound attenuation being the most common ones in recent publications.
Here I will argue that for generic Hamiltonian systems the long time behavior of the dynamics can be obtained [*exactly*]{} in terms of the scaling functions obtained by Prähofer and Spohn[@PS] for the polynuclear growth model, which is in the KPZ universality class. The values of $\delta$ and $\alpha$ mentioned above are confirmed. But also the [*coefficients*]{} of size dependent transport coefficients and long-time current-current correlation functions are obtained exactly, as well as the scaling functions describing, among other things the asymptotic behaviors of the various density-density time correlation functions and their Fourier transforms. These results hold in all generality, for generic short ranged 1d Hamiltonians, from weakly anharmonic chains up to mixtures of hard points. They establish a rare example of exact results that may be obtained for non-integrable Hamiltonian systems out of equilibrium. In addition the special conditions under which such systems do not belong to the KPZ universality class will be formulated simply and sharply, together with the consequences for long time and short wave length behavior.
More specifically, I will discuss classical one-dimensional N-particle systems described by a translation invariant Hamiltonian with short range interactions and periodic boundary conditions. Following one of the ground-laying papers by Ernst, Hauge and Van Leeuwen[@ehvl] I will assume that all slow variables of relevance for the long time behavior of hydrodynamics and related time correlation functions are the long-wave length Fourier components of the densities of conserved quantities, i.e. particle number, momentum and energy, plus products of these. This is a crucial assumption. It is not satisfied for most exactly solvable models, which have additional slow modes, such as solitons[@toda]. For one- and two-dimensional systems the method of EHvL has to be generalized somewhat: instead of assuming that the time correlation functions of hydrodynamic modes decay exponentially with time, one has to write down the mode coupling equations as a set of coupled nonlinear equations for these correlation functions that must be solved self-consistently[@delfini; @delfini2; @lietal].
EHvL define the hydrodynamic modes, to leading order in the wave number $k$ as linear combinations of the Fourier transforms of the microscopic densities of particles, momentum and energy[^1], $\rho^{\mu}(k,t)=\sum_{j=1}^N M^{\mu}_j \exp(-ikx_j)-\delta_{k0}\langle \hat{M}(k=0)\rangle$, with $M^{\mu}_j=1,p_j,e_j$ for the particle density $n(k,t)$, the momentum density $g(k,t)$ and the energy density $e(k,t)$ respectively [^2]. The hydrodynamic modes are two sound modes[^3] $a_1(k,t)$ and $a_{-1}(k,t)$ and a heat mode $a_H(k,t)$, given respectively, to leading order in $k$ by $$\begin{aligned}
&a_{\sigma}(k,t)=\left(\frac{\beta}{2\rho}\right)^{1/2}\left(c_0^{-1}p(k,t)+\sigma g(k,t)\right),\label{asigma}\\
&a_H(k,t)=\left(\frac{\beta}{nTC_p}\right)^{1/2}(e(k,t)-hn(k,t)).\label{aH}\end{aligned}$$ Here, $\sigma$=$\pm1$, $T$ is the equilibrium temperature, $n$ the equilibrium number density and $\rho$=$nm$; $C_p$=$T(\partial s/\partial T)_p$ is the specific heat per particle at constant pressure $p$, with $s$ the equilibrium entropy per particle; $c_0$=$(\partial p/\partial \rho)_s^{1/2}$ is the adiabatic sound velocity in the limit of zero wave number and $h$ is the equilibrium enthalpy per particle. Furthermore, $$\begin{aligned}
p(k,t)&=(\partial p/\partial e)_n e(k,t) +(\partial p/\partial n)_e n(k,t),\label{asound}\\
&=\frac{\gamma-1}{\alpha T}e(k,t) +(\partial p/\partial n)_e n(k,t),\nonumber\end{aligned}$$ where $\gamma=C_p/C_v$ is the specific heat ratio and $\alpha=-n^{-1}\left(\partial n/\partial T\right)_p$ the thermal expansion coefficient. The allowed values of $k$ are of the form $k=\frac{2\pi n} L$. To leading order in $k$ the hydrodynamic modes are normalized under the inner product $(f,g)=\frac 1 L \langle f^*g\rangle,$ with $\langle \rangle$ a grand canonical equilibrium average.
The time correlation functions of the hydrodynamic modes satisfy linear equations involving memory kernels, viz.$$\begin{aligned}
&\frac{\partial \hat{S}_{\sigma}(k,t)}{\partial\ t}=-i\sigma c_0 k\hat{S}_{\sigma}(k,t)-k^2 \int_0^t d\tau \hat{M}_{\sigma}(k,\tau)\hat{S}_{\sigma}(k,t-\tau),\label{soundmem}\\
&\frac{\partial \hat{S}_{H}(k,t)}{\partial\ t}=-k^2 \int_0^t d\tau \hat{M}_{H}(k,\tau)\hat{S}_{H(k,t-\tau)}.\label{heatmem}\end{aligned}$$ Here $\hat{S}_{\sigma}(k,t)=(a_{\sigma}(k,0),a_{\sigma}(k,t))$ etc. The memory kernels may be expressed through a diagrammatic mode coupling expansion as a sum of irreducible skeleton diagrams[@skeleton]. These consist of propagators, representing stationary density correlation functions $\hat{S}_{\zeta}(\ell,t_{\alpha})$, and vertices representing the coupling of one propagator $\hat{S}(\ell,t_{\alpha})$ to two propagators $\hat{S}_{\mu}(q,t_{\alpha'})$ and $\hat{S}_{\nu}(\ell-q,t_{\alpha''})$, with coupling strength $\ell W_{\zeta}^{\mu\nu}$. For the long time dynamics only a few of these 27 couplings are important; only couplings to two sound modes of the same sign or to two heat modes may give rise to long-lived perturbations, all other combinations of pairs of modes rapidly die out through oscillations. From EHvL[@ehvl] the relevant non-vanishing coupling strengths to leading order in $k$ can be obtained as $$\begin{aligned}
W_{\sigma}^{\sigma\sigma}&=\frac{\sigma}{(2\rho\beta)^{1/2}c_0}\left(\frac{\partial c_0n}{\partial n}\right)_s\label{wsss}\\
W_{\sigma}^{-\sigma-\sigma}&=\frac{\sigma}{(2\rho\beta)^{1/2}}\left[\frac 1{c_0}\left(\frac{\partial c_0n}{\partial n}\right)_s-2\frac{\gamma-1}{\alpha T}\right]\label{wsss'}\\
W_{\sigma}^{HH}&=
\frac{-\sigma(\gamma -1)
}{(2\rho\beta )^{1/2}nC_p}\left(\frac{\partial\left[\frac{n C_p}{\alpha}\right]}{\partial T}\right)_p\label{wshh}\\
W_{H}^{\sigma\sigma}&=\frac{\sigma k_B^{1/2}c_0}{(n C_p)^{1/2}}.\label{whss}\end{aligned}$$
Now a central observation is the following: due to the first term on the right-hand side of Eq. (\[soundmem\]) the sound-sound correlation functions will have their weights centered around the positions $x(t)=x(0)\pm c_0t$, in other words, these functions will assume the forms $\hat{S}_{\sigma}(k,t)=\exp(-i\sigma c_0kt)\hat{\Sigma}_{\sigma}(k,t)$, with $\hat{\Sigma}_{\sigma}(k,t)$ to a first approximation real non-oscillating functions. As a consequence the mode coupling contributions to $\hat{M}^{\sigma}$ are dominated by those diagrams in which all vertices are of the type $V_{\sigma}^{\sigma\sigma}$ (but, only in the limit $k=0$ diagrams in which the first and last vertex are of type $V_{\sigma}^{-\sigma-\sigma}$ and all the other ones of type $V_{-\sigma}^{-\sigma-\sigma}$ also contribute to the leading order). All other contributions for at least some time will oscillate out of phase with the angular frequency $\sigma c_0k$ of the sound mode under consideration. The remaining contributions, especially so if described in a coordinate frame comoving at the speed of sound, can be identified with the terms in a similar mode coupling expansion for the fluctuating Burgers equation[@mcburgers], $$\begin{aligned}
&\frac{\partial \rho(x,t)}{\partial t}=\frac{\kappa} 2\frac{\partial \rho^2}{\partial x} +\frac D{A}\frac{\partial^2 \rho}{\partial x^2} + \frac{\partial \eta}{\partial x},
\label{FB} \end{aligned}$$ with $A$=$\hat{S}_B(0,0)$, with the density-density time correlation function $\hat{S}_B(k,t)$ defined, in the limit $L\to\infty$ as $$\begin{aligned}
\hat{S}_B(k,t)=\int_{-\infty}^{\infty} dx e^{-ikx} S_B(x,t)\equiv \int_{-\infty}^{\infty} dx e^{-ikx}\langle\rho(0,0)\rho(x,t)\rangle\nonumber\end{aligned}$$ and $\eta(x,t)$ representing gaussian white noise with $\langle\eta(x,t)\eta(x',t')\rangle$=$2D\delta(x-x')\delta(t-t')$. The brackets denote an average over the stationary distribution of the density field. This is similar to the hydrodynamic equations, but simpler because there is only one conservation law. The function $\hat{S}_B(k,t)$ satisfies an equation similar to Eqs. (\[soundmem\],\[heatmem\]), of the form $$\begin{aligned}
&\frac{\partial\hat{S}_B(k,t)}{\partial t}=-k^2\int_0^t d\tau \hat{M}_B(k,\tau)\hat{S}_B(k,t-\tau).
\label{burgmem}\end{aligned}$$ The mode coupling expansion for this memory kernel has exactly the same structure as the set of dominant terms for the sound mode memory kernel; all propagators correspond to the same type of correlation function and all vertices carry the same weight factor $W$, in the case of the Burgers equation given by $W_B=\kappa\sqrt{A} $.
From their exact solution of the polynuclear growth model[@PS] Prähofer and Spohn obtained exact expressions for the long time, respectively small frequency behavior of the function $\hat{S}_B(k,t)$ and its temporal Fourier transform $\tilde{S}_B(k,\omega)$. In the infinite system limit, $L\to\infty$ these are of the form[@sasa] $$\begin{aligned}
&\hat{S}_B(k,t)=A\hat{f}_{PS}\left((2A\kappa^2t^2)^{1/3}k\right)\label{hatfPS}\\
&\tilde{S}_B(k,\omega)=\sqrt\frac{A}{2\kappa^2|k|^3}\mathring{f}_{PS}\left(\frac{\omega}{(2A\kappa^2)^{1/2}|k|^{3/2}}\right)\label{mathringfPS},\end{aligned}$$ with the functions $\hat{f}_{PS}$ and $\mathring{f}_{PS}$ defined in Eqs. ((5.3) and (5.7) of Ref.[@PS]. From Eq. \[burgmem\] one may obtain expressions for the memory kernel in terms of these scaling functions. For the full Fourier transform one obtains $$\begin{aligned}
\tilde{M}_B(k,\omega)&=\sqrt{ 2A\kappa^2}M_{PS}\left(k,\frac{\omega}{\sqrt{2A\kappa^2}}\right),\label{Mkomega}\end{aligned}$$ with $$\begin{aligned}
\tilde{M}_{PS}(k,\omega)&=\frac {i\omega}{k^2} + \left(\sqrt{k}\mathring{f}^+_{PS}\left(\frac{\omega}{|k|^{3/2}}\right)\right)^{-1},
\label{mem}\end{aligned}$$ where $\mathring{f}_{PS}^+(w)\equiv\int_0^{\infty} d\tau\exp(iw \tau)\hat{f}_{PS}(\tau^{2/3})$.
The corresponding expressions for the long time behavior of the sound modes at non-vanishing $k$ are $$\begin{aligned}
&\hat{S}_{\sigma}(k,t)=\exp(-i\sigma c_0kt)\hat{f}_{PS}\left((\sqrt{2}V_st)^{2/3}k\right),
\label{Ssigma}\\
&\hat{M}_{\sigma}(k,t)=2V_s^2\exp(-i\sigma c_0kt)\hat{M}_{PS}(k,\sqrt{2}V_st),
\label{Msigma}\end{aligned}$$ with $V_s=W_{\sigma}^{\sigma\sigma}$.
Next, I consider the wave number dependent sound damping constant $\Gamma(k)\equiv2\tilde{M}_{\sigma}(k,0)$ and define the sound currents as $$\hat{J}_{\sigma}(k,t)=\left(\frac{\beta}{2\rho}\right)^{1/2}\left[\sigma\hat{J}_l(k,t)+\frac {\gamma -1} {\alpha T c_0}\hat{J}_H(k,t) \right],$$ where $\hat{J}_l(k,t)$ and $\hat{J}_H(k,t)$ are the longitudinal current and the heat current[@ehvl], denoted by EHvL as $J_l$ and $J_{\lambda}$ respectively. Eq. (5.11) of Ref.[@PS] can now be used to obtain the leading small-$k$ behavior of $\Gamma(k)$ and long time behavior of $\langle\hat{J}_{\sigma}(0,0)\hat{J}_{\sigma}(0,t)\rangle$ as $$\begin{aligned}
&\Gamma(k)=\frac {8} {19.444} \sqrt\frac{2V_s^2}{|k|}\label{gammak}\\
&\frac 1 L \langle \hat{J}_{\sigma}(0,t)\hat{J}_{\sigma}(0,0))\rangle= \frac{2.1056[V_s^2 +V_{s'}^2]}{2\sqrt{3}\Gamma_E(1/3)}\left(\frac1 {\sqrt{2}V_s|t|}\right)^{2/3},
\label{gksoundsound}\end{aligned}$$ with $\Gamma_E$ Euler’s gamma function[^4]and $V_{s'}=W_{\sigma}^{-\sigma-\sigma}$.
The leading higher order corrections are obtained by replacing in the diagrammatic expansion of the memory kernel just one pair of vertices of type $V_{\sigma}^{\sigma\sigma}$ by vertices of type $V_{\sigma}^{-\sigma-\sigma}$ or $V_{\sigma}^{HH}$. One easily shows that all these terms add contributions proportional to $|k|^{-1/3}$ to $\Gamma(k)$ and contributions proportional to $t^{-7/9}$ to the current-current correlation function. Since there are infinitely many such contributions, there seems to be no straightforward way of determining the coefficients exactly. However, estimates based on the simplest contributing diagrams can be made[@tbp]. Further corrections obtain from terms with $4,6,\cdots$ vertices of type $V_{\sigma}^{-\sigma-\sigma}$ or $V_{\sigma}^{HH}$. Each of these appears to be of the form $Ck^{-\mu}$ for $\Gamma(k)$ and $Dt^{-\nu}$ for the current correlation function, with $C$ and $D$ constants and $\mu$ and $\nu$ of the form $\mu=1/3-\sum_{j=2}^{\infty}m_j(2/3)^j$ and $\nu=2/3+\sum_{j=2}^{\infty}2n_j(2/3)^j$ respectively, with $m_j$ and $n_j$ natural numbers. Again, for each exponent there is an infinity of contributing terms.
The leading long time behavior of $\hat{S}_{H}(k,t)$ is determined in similar way by the sum of all contributions to $\hat{M}^{H}(k,t)$ where the first and last vertex are of type $V_H^{\sigma\sigma}$ and all other vertices are of type $V_{\sigma}^{\sigma\sigma}$, all with the same value of $\sigma$. These terms do contain an oscillating factor $\exp(-i\sigma c_0kt)$, but these oscillations are much slower than the oscillations in any of the other terms. Since we have to include the contributions to $\hat{M}^{H}$ of either sign of $\sigma$, we cannot express $\hat{S}_{H}$ directly in terms of the Prähofer-Spohn scaling functions, but we can do so immediately for the memory kernel. A simple analysis yields to leading order $$\begin{aligned}
\hat{M}_{H}(k,t)=2{V_{H}^2}\cos(\sigma c_0kt)
\hat{M}_{PS}(k,\sqrt{2}V_s t) ,
\label{Mheat}\end{aligned}$$ with $V_{H}=|W_{H}^{\sigma\sigma}|$. For the $k$-dependent heat conduction coefficient and the heat current time correlation function this leads to the expressions $$\begin{aligned}
\lambda(k)&=nC_p D_T(k)=nC_p\frac{2.1056}{2} \frac{V_H^2}{V_s}\left( \frac{V_s}{2c_0|k|}\right)^{1/3}, \\
\frac 1 L &\langle \hat{J}_{H}(0,t)\hat{J}_{H}(0,0)\rangle=
\frac{nC_p}{k_B\beta^2}{V_H}^2 \frac{2.1056}{\sqrt{3}\Gamma_E(1/3)}\left(\frac1 {\sqrt{2}V_s|t|}\right)^{2/3}.
\label{heatheat}\end{aligned}$$
The heat mode correlation function in Fourier representation is given to leading order by $$\begin{aligned}
&\tilde{S}_{H}(k,\omega)=
\frac 1{-i\omega+2k^2 V_H^2\sum_{\sigma}\tilde{M}_{PS}\left(k,\frac{\omega-\sigma c_0k}{\sqrt{2}V_s}\right) } + cc.\end{aligned}$$ From this expression and the asymptotic forms of the Prähofer-Spohn scaling functions[@PS] one immediately finds the long-time behavior of the heat-heat correlation function for fixed $k$ as $$\begin{aligned}
\hat{S}_{H}(k,t)=\exp\left[-
%k^{5/3}1.0528\frac{V_H^2}{V_s}\left(\frac {V_s}{c_0}\right)^{1/3}
k^2D_T(k)|t| \right].
\label{hm}\end{aligned}$$ Higher order corrections may be obtained in similar way as for the sound modes.
The dynamic structure factor $\tilde{S}(k,\omega)$, i.e. the spatio-temporal Fourier transform of the density-density time correlation function exhibits Brillouin peaks at $\omega=\pm c_0k$ and a central Rayleigh peak, as usual, but, due to the anomalous transport the width and inverse height of the Brillouin peaks scale with $k$ as $|k|^{3/2}$[@delfini; @delfini2] and those of the Rayleigh peak with $|k|^{5/3}$[^5], in contrast to the usual scaling with $k^2$. Also, the shape of these peaks is not Lorentzian, but is given through the Prähofer-Spohn scaling functions as $$\begin{aligned}
&\tilde{S}(k,\omega)=\sum_{\sigma}(\hat{n}(0),\hat{a}_{\sigma}(0))^2\tilde{S}_{\sigma}(k,\omega)\nonumber\\
&\ \ \ +(\hat{n}(0),\hat{a}_{H}(0))^2\tilde{S}_{H}(k,\omega), \end{aligned}$$ with $\tilde{S}_{\sigma}(k,\omega)$ the Fourier transform of ${S}_{\sigma}(k,t)$.
All results contained in Eqs. (\[hatfPS\]-\[hm\]) hold in the limit $L\to\infty$. For finite periodic systems they apply for $k=2\pi n/L$ and for times or inverse frequencies small compared to the sound mode traversal time $L/c_0$. For correlation functions at $k=0$, like in Eqs. (\[gksoundsound\]) and (\[heatheat\]) this time range may be extended to a value proportional to $L^{3/2}$.
It seems fair to pose that the leading long time dynamics of 1d hydrodynamic systems belong to the KPZ universality class. The sound-sound correlation functions to leading order are identical to the density-density correlation function of the fluctuating Burgers equation, while the heat-heat correlation functions are directly expressible in terms of KPZ memory functions in coordinate systems moving at the speed of sound. But notice that the correction terms decay only slightly faster with time and in most cases will not be negligible up to very large times.
As discussed by Prähofer and Spohn[@PS] a self consistent one-loop mode coupling approximation comes remarkably close to the exact solution to the Burgers equation, although there are deviations in the scaling functions of up to 10% and not all details of the functional behavior are captured correctly. Similar results are to be expected for the one-loop mode coupling approximation to 1d hydrodynamics. Some care is required, however with using published results. In previous analyses Delfini et al.[@delfini] and Wang and Li[@lietal] assumed the sound modes were linear combinations of momentum density and displacement field (equivalent to number density in the absence of transversal modes), without contributions from the energy density. As can be seen from Eq. (\[asound\]) this is justified if $(\partial p/\partial e)_n=0$ or equivalently, if $C_p=C_v$. This is the case for harmonic chains, so it will be a good approximation for weakly anharmonic chains. For general potentials corrections are needed.
It has been remarked in several places that the characteristics of heat conduction and sound dissipation change markedly under certain special conditions, such as having an anharmonic nearest neighbor potential symmetric in the deviations from the average nearest neighbor distance under zero pressure[@delfini; @delfini2] (this includes the FPU-$\beta$ model), or zero pressure in a system of constrained hard points[@edhs]. This can be understood as resulting from a vanishing mode coupling amplitude $V_s$. In other words, the condition for having non-KPZ behavior quite simply and generally is $\frac n {c_0}\left(\frac{\partial c_0 }{\partial n}\right)_s=-1$. It is satisfied indeed for the classes of systems mentioned above, but in general it does not require any of the criteria quoted above, nor the condition $C_p=C_v$ as conjectured in Ref.[@leedadswell]. As discussed by Delfini et al.[@delfini2], the mode coupling under these conditions is dominated by the coupling of a sound mode to three sound modes of the same type. Sound damping becomes almost normal. In contrast to what is stated in Ref.[@delfini2] it is still superdiffusive, but only logarithmically so (see Ref.[@Devillard] for the equivalent case of a growth model with leading nonlinearity of cubic order). The $k$-dependent heat conduction coefficient diverges roughly as $|k|^{-1/2}$ and the heat current time correlation function decays as $t^{-1/2}$, both up to logarithmic corrections. The latter must be responsible for the gradual increase with time of the exponent $\delta$ for heat conduction from roughly 2/5 to 1/2, which has been reported for simulation results[@delfini2; @leedadswell].
In stationary states the $k^{\alpha-2}$ behavior of the Rayleigh peak implies a nonlinear temperature profile with, for large systems a cusp of form $|x-x_0|^{1-\alpha}$ (the inverse Fourier transform) near a boundary located at $x_0$. Such nonlinear profiles have been observed regularly in simulations, but so far I have nowhere seen mention of this simple interpretation.
Like in the case of the fluctuating Burgers equation, the mode coupling equations can also be obtained from fluctuating nonlinear hydrodynamic equations[@bm]. Since their structure remains exactly the same, all the results obtained above hold for all one-dimensional systems satisfying the usual Landau-Lifshitz fluctuating hydrodynamic equations with [*finite*]{} transport coefficients. The asymptotic long time behavior of density-density and current-current time correlation functions is independent of the values of the transport coefficients, again like for the fluctuating Burgers equation. Obviously, transport coefficients in nonlinear transport equations may be finite even if the corresponding Green-Kubo integrals are divergent. Whether this actually will happen for one and two dimensional Hamiltonian systems, as far as I know is an open question.
More detailed derivations of the results presented here will be published elsewhere. Besides comparisons to existing numerical results, new Molecular Dynamics simulations will be performed. Applications to quantum systems will also be studied. They look feasible but will require careful consideration of all quantum aspects.
The author acknowledges the generous support of the Humboldt foundation as well as additional support by NSF Grant No. DMR 08-02120 and AFOSRGrant No.AF-FA49620-01-0154 of J. L. Lebowitz. He much appreciated the hospitality of the Technische Universität München, where an important part of this work was done. He especially enjoyed many clarifying discussions with Herbert Spohn.
[89]{} B. J. Alder, T. E. Wainwright, Phys. Rev. Lett. **18**, 988 (1967) L. Delfini, S. Lepri, R. Livi, A. Politi, Phys. Rev. E **73**, 060201 (2006) L. Delfini, S. Lepri, R. Livi, A. Politi, J. Stat. Mech. P02007 (2007) J. -S. Wang, B. Li, Phys. Rev. Lett. **92**, 074302 (2004); Phys. Rev. E **70** 021204 (2004) G. R. Lee-Dadswell, B. G. Nickel and C. G. Gray, J. Stat. Phys. **132**, 1 (2008) O. Narayan, S. Ramaswamy, Phys. Rev. Lett. **89**, 200601 (2002) P. Grassberger, W. Nadler, L. Yang, Phys. Rev. Lett. **89**, 180601 (2002) M. Prähofer, H. Spohn, J. Stat. Phys. **115**, 255 (2004) M. H. Ernst, E. H. Hauge, J. M. J. van Leeuwen, J. Stat. Phys. **15**, 7 (1976) An example is the Toda lattice, see e.g. N. Theodorakopoulos, M. Peyrard, Phys. Rev. Lett. **83**, 2293 (1999) A. Khuranat, J. Phys. A: Math. Gen. **18**, 2415 (1985); H. van Beijeren, M. H. Ernst, J. Stat. Phys. **21**, 125 (1979) sec. 7 H. van Beijeren, R. Kutner, H. Spohn, Phys. Rev. Lett. **54**, 2026 (1985); E. Frey, U. C. Täuber, T. Hwa, Phys. Rev. E **53**, 4424 (1996) T. Sasamoto, H. Spohn, J. Stat. Phys. **137**, 917 (2009) H. van Beijeren, to be published L. Delfini, S. Denisov, S. Lepri, R. Livi, P. K. Mohanty, A. Politi, Eur. Phys. J. Special topics [**1**46]{}, 21 (2007) P. Devillard and H. Spohn, J. Stat. Phys. [**66**]{}, 1089 (1992) D. Bedeaux, P. Mazur, Physica **73**, 431 (1974); D. Bedeaux, P. Mazur, Physica **75**, 79 (1974)
[^1]: The sign convention used here for the spatial Fourier transforms is opposite to that used by Prähofer and Spohn, but the same as that used by Ernst, Hauge and Van Leeuwen[@ehvl]. Since the scaling functions obtained by Prähofer and Spohn are even in $k$ this is of no severe consequence.
[^2]: Note that the energies and momenta of the particles are localized at the actual positions of the particles. For chains on which particles cannot pass each other, a natural alternative is localizing these densities at the average positions of the particles[@delfini; @delfini2; @lietal]. However, one can show that both approaches are equivalent, at least so in their predictions of the long time dynamics[@tbp]
[^3]: I use $\sigma=\pm1$ for right respectively left moving sound modes, rather than positive respectively negative frequency, as is conventional.
[^4]: Similar expressions may be obtained for the wave number dependent diffusion coefficient and for the current-current time correlation function of the fluctuating Burgers equation.
[^5]: This corresponds to the well-known value $\alpha=1/3$ for the size dependence of the heat conduction coefficient[@delfini; @delfini2; @leedadswell; @rama; @grassberger]
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"pile_set_name": "ArXiv"
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[**Quantum Scattering Theoretical Description of Thermodynamical Transport Phenomena**]{}
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> We give a method of describing thermodynamical transport phenomena, based on a quantum scattering theoretical approach. We consider a quantum system of particles connected to thermodynamical reservoirs by leads. The effects of the reservoirs are imposed as an asymptotic condition at the end of the leads. We derive an expression for a current of a conserved quantity, which is independent of the details of the Hamiltonian operator. The Landauer formula and its generalizations are derived from this method.
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[**$ \; \langle$ 1. Introduction $ \; \rangle$** ]{} Statistical mechanical description of thermodynamical responses has been one of the important subjects of nonequilibrium statistical mechanics. Some methods have been proposed for this purpose (for example, see Ref. \[1\]). Of special interest is a method pioneered by Landauer \[2\], who heuristically derived an expression for an electric current, employing a scattering theoretical approach. His method was generalized to some cases; for example, a multichannel case \[3-5\], a case of a finite temperature \[4,6\], a case of a heat current \[6,7\] and a case of an inelastic scattering process caused by a random potential \[8\]. These methods describe linear responses of a system to thermodynamical gaps of reservoirs which induce Fermi distributions in the system. However, these have not explicitly treated an effect of inelastic scattering processes caused by scatterers, and applications of these methods have been mainly restricted to mesoscopic phenomena. Moreover, there have not existed a unified statistical mechanical derivation of all the generalizations of Landauer formula \[9\].
The purpose of the present Letter is to give a statistical mechanical method for descriptions of thermodynamical responses, based on a quantum scattering theoretical approach. We show that Landauer formula and its generalizations are derived by this method. This method covers all the cases which have been contained in the generalizations of Landauer formula. Moreover, it can be applied to some new cases; nonlinear responses to thermodynamical gaps of more general reservoirs inducing non-Fermi distributions in a system, inelastic processes caused by scatterers, and currents other than an electric or a heat current, etc. So, this method can give new generalizations of Landauer formula.
[**$ \; \langle$ 2. Set-up $ \; \rangle$** ]{} We consider a quantum system of particles in a three-dimensional region $\Omega$. The system consists of two kinds of particles, which we call ‘transport particles’ and ‘scatterers’. The transport particles are in a scattering state, and the scatterers are in a bound state. The region $\Omega$ consists of a finite region $\Omega_0$ and $N$ semi-infinite columned regions $\Omega_j, \, j=1,2,\cdots,N$. The semi-infinite columned region $\Omega_j$ connects the region $\Omega_0$ to infinity. We call the columned region the ‘lead’.
For simplicity, we treat a system of only two particles; one transport particle and one scatterer. We assume the Hamiltonian operator $\hat{H}$ of this system to be of the form
$$\hat{H} \equiv \frac{1}{2m}\left
\{\hat{{\mbox{\boldmath $p$}}}-\frac{q}{c}{\mbox{\boldmath $A$}}(\hat{{\mbox{\boldmath $x$}}})\right \}^2 +
\frac{1}{2M}\left \{ \hat{{\mbox{\boldmath $P$}}}-\frac{Q}{c}{\mbox{\boldmath $A$}}(\hat{{\mbox{\boldmath $X$}}})\right
\}^2 + U(\hat{{\mbox{\boldmath $x$}}}, \hat{{\mbox{\boldmath $X$}}})
\label{Hamil.1}$$
where $c$ is the velocity of light, $m$ and $M$ are the masses of the transport particle and the scatterer, respectively, $q$ and $Q$ are the charges of the transport particle and the scatterer, respectively, $\hat{{\mbox{\boldmath $x$}}}$ and $\hat{{\mbox{\boldmath $X$}}}$ are the coordinate operators of the transport particle and the scatterer, respectively, $\hat{{\mbox{\boldmath $p$}}}$ and $\hat{{\mbox{\boldmath $P$}}}$ are the momentum operators of the transport particle and the scatterer, respectively, ${\mbox{\boldmath $A$}}(\hat{{\mbox{\boldmath $x$}}})$ and ${\mbox{\boldmath $A$}}(\hat{{\mbox{\boldmath $X$}}})$ are the vector potential operators acting on the transport particle and the scatterer, respectively, $U(\hat{{\mbox{\boldmath $x$}}},
\hat{{\mbox{\boldmath $X$}}})$ is the potential operator of the transport particle and the scatterer. Here the square of a vector means the inner product of the vector with itself.
The state of this system at time $t$ is described by a density operator $\hat{\rho}(t)$ which obeys the Liouville-von Neuman equation
$$i \hbar \frac{d\hat{\rho}(t)}{dt} = [\hat{H},
\hat{\rho}(t)]
\label{Liouv}$$
where $2\pi\hbar$ is the Planck constant.
We introduce $\mid {\mbox{\boldmath $x$}}, {\mbox{\boldmath $X$}}\; \rangle$ as the eigenstate of the operators $\hat{{\mbox{\boldmath $x$}}}$ and $\hat{{\mbox{\boldmath $X$}}}$ with eigenvalues ${\mbox{\boldmath $x$}}$ and ${\mbox{\boldmath $X$}}$, respectively. We introduce the unit vectors ${{\mbox{\boldmath $e$}}}_k^{\scriptscriptstyle (j)}$, $k=1,2,3$ as a basis of ${\mbox{\boldmath $R$}}^3$ such that ${{\mbox{\boldmath $e$}}}_1^{\scriptscriptstyle (j)}$ is parallel to the $j$-th columned region and is pointing to the finite region $\Omega_0$. We define $\hat{x}_k^{\scriptscriptstyle (j)}$ by $\hat{x}_k^{\scriptscriptstyle (j)} \equiv
{{\mbox{\boldmath $e$}}}_k^{\scriptscriptstyle (j)} \cdot \hat{{\mbox{\boldmath $x$}}}$, and introduce $x_k^{\scriptscriptstyle (j)}$ as a eigenvalue of $\hat{x}_k^{\scriptscriptstyle (j)}$ ($j=1,2,\cdots,N$, $k=1,2,3$). We assume the functions ${\mbox{\boldmath $A$}}({\mbox{\boldmath $x$}})$ and $U({\mbox{\boldmath $x$}}, {\mbox{\boldmath $X$}})$ to have the asymptotic forms satisfying
$${\mbox{\boldmath $A$}}({\mbox{\boldmath $x$}}) \stackrel{x _1^{\scriptscriptstyle
(j)}\rightarrow-\infty}{\sim} {\mbox{\boldmath $A$}}^{\scriptscriptstyle
(j,\infty)}(x_2^{\scriptscriptstyle (j)}, x_3^{\scriptscriptstyle
(j)}), \;\;\;\;\; \mbox{in} \;\;{\mbox{\boldmath $x$}}\in \Omega_j,$$
$$U({\mbox{\boldmath $x$}}, {\mbox{\boldmath $X$}}) \stackrel{x_1^{(j)}\rightarrow-\infty}{\sim}
U^{\scriptscriptstyle (j,\infty)}(x_2^{\scriptscriptstyle (j)},
x_3^{\scriptscriptstyle (j)}, {\mbox{\boldmath $X$}}), \;\;\;\;\; \mbox{in} \;\;{\mbox{\boldmath $x$}}\in \Omega_j$$
where ${\mbox{\boldmath $A$}}^{\scriptscriptstyle
(j,\infty)}(x_2^{\scriptscriptstyle (j)}, x_3^{\scriptscriptstyle
(j)})$ is a function of $x_2^{\scriptscriptstyle (j)}$ and $x_3^{\scriptscriptstyle (j)} $, and $U^{\scriptscriptstyle
(j,\infty)}(x_2^{\scriptscriptstyle (j)}, x_3^{\scriptscriptstyle
(j)}, {\mbox{\boldmath $X$}})$ is a function of $x_2^{\scriptscriptstyle (j)}$, $x_3^{\scriptscriptstyle (j)} $ and ${\mbox{\boldmath $X$}}$ only. We consider the operator $\hat{H}^{\scriptscriptstyle (j,\infty)}$ defined by
$$\begin{aligned}
\hat{H}^{\scriptscriptstyle (j,\infty)} \equiv
&& \!\!\!\!\!\!\!\!\!
\frac{1}{2m}\left\{\hat{{\mbox{\boldmath $p$}}}-\frac{q}{c}{\mbox{\boldmath $A$}}^{\scriptscriptstyle
(j,\infty)}(\hat{x}_2^{\scriptscriptstyle (j)},
\hat{x}_3^{\scriptscriptstyle (j)})\right\}^2 \nonumber \\
&& +
\frac{1}{2M}\left\{\hat{{\mbox{\boldmath $P$}}}-\frac{Q}{c}{\mbox{\boldmath $A$}}(\hat{{\mbox{\boldmath $X$}}})\right\}^2 +
U^{\scriptscriptstyle (j,\infty)}(\hat{x}_2^{\scriptscriptstyle (j)},
\hat{x}_3^{\scriptscriptstyle (j)}, \hat{{\mbox{\boldmath $X$}}})
\label{Hamil.2}
\end{aligned}$$
The eigenstate $\mid \Phi_{k n}^{\scriptscriptstyle
(j,\infty)} \; \rangle$ of the operator $\hat{H}^{\scriptscriptstyle
(j,\infty)}$ can be represented as
$$\begin{aligned}
\mid \Phi_{k n}^{\scriptscriptstyle (j,\infty)} \; \rangle =
\; \mid \phi_{k}^{\scriptscriptstyle (j)} \; \rangle \; \otimes \mid
\varphi_{k n}^{\scriptscriptstyle (j,\infty)} \; \rangle.
\label{eigen.1}
\end{aligned}$$
Here, $\mid \phi_{k}^{\scriptscriptstyle (j)} \; \rangle$ is the eigenstate of the operator ${\mbox{\boldmath $e$}}_1^{\scriptscriptstyle (j)}
\cdot
\hat{{\mbox{\boldmath $p$}}}$ with the eigenvalue $\hbar k$ and have an orthonormality
$$\begin{aligned}
\langle \; \phi_{k}^{\scriptscriptstyle (j)} \; \mid
\phi_{k'}^{\scriptscriptstyle (j)} \; \rangle = 2\pi \delta(k - k').
\end{aligned}$$
And $\mid \varphi_{k n}^{\scriptscriptstyle (j,\infty)} \;
\rangle$ is introduced as the eigenstate of the operator $\hat{\cal{H}}_k^{\scriptscriptstyle (j,\infty)}$ which is defined by $\langle \; \phi_{k}^{\scriptscriptstyle
(j)}\mid\hat{H}^{\scriptscriptstyle (j,\infty)}\mid
\phi_{k'}^{\scriptscriptstyle (j)} \; \rangle \equiv
\hat{\cal{H}}_k^{\scriptscriptstyle (j,\infty)}\langle \;
\phi_{k}^{\scriptscriptstyle (j)} \; \mid
\phi_{k'}^{\scriptscriptstyle (j)} \; \rangle$. We introduce $E_{kn}^{\scriptscriptstyle (j)}$ as the eigenvalue of the operator $\hat{H}^{\scriptscriptstyle (j,\infty)}$ corresponding to the eigenstate $\mid \Phi_{kn}^{\scriptscriptstyle
(j,\infty)} \; \rangle$. And $\mid x_2^{\scriptscriptstyle (j)}, x_3^{\scriptscriptstyle
(j)}, {\mbox{\boldmath $X$}}\; \rangle$ is introduced as the eigenstate of the operators $\hat{x}_2^{\scriptscriptstyle (j)}$, $\hat{x}_3^{\scriptscriptstyle (j)}$ and $\hat{{\mbox{\boldmath $X$}}}$ with eigenvalues $x_2^{\scriptscriptstyle (j)}$, $x_3^{\scriptscriptstyle
(j)}$ and ${\mbox{\boldmath $X$}}$, respectively. We assume an orthonormality
$$\int_{S_j}dx_2^{\scriptscriptstyle
(j)}dx_3^{\scriptscriptstyle (j)}\int_{\Omega}d{\mbox{\boldmath $X$}}\langle \;
\varphi_{kn}^{\scriptscriptstyle (j,\infty)} \mid
x_2^{\scriptscriptstyle (j)}, x_3^{\scriptscriptstyle (j)}, {\mbox{\boldmath $X$}}\;
\rangle \, \langle \; x_2^{\scriptscriptstyle (j)},
x_3^{\scriptscriptstyle (j)}, {\mbox{\boldmath $X$}}\mid
\varphi_{kn'}^{\scriptscriptstyle (j,\infty)} \; \rangle =
\delta_{nn'}
\label{ortho.1}$$
of the states $\{ \mid \varphi_{kn}^{\scriptscriptstyle
(j,\infty)} \; \rangle\}_{n}$, where $S_j$ represents the projection of the cross section of the $j$-th columned region onto the $x_2^{\scriptscriptstyle (j)}x_3^{\scriptscriptstyle (j)}$ plane.
We assume that there exists an eigenstate $\mid
\Psi_{kn}^{\scriptscriptstyle (j)} \; \rangle$ of the Hamiltonian operator $\hat{H}$ with the eigenvalue $E_{kn}^{\scriptscriptstyle
(j)}$. Here, the eigenstate $\mid \Psi_{kn}^{\scriptscriptstyle (j)} \;
\rangle$ satisfies the following asymptotic condition:
$$\begin{aligned}
\langle \; {\mbox{\boldmath $x$}}, {\mbox{\boldmath $X$}}\mid\Psi_{kn}^{\scriptscriptstyle (j)}
\; \rangle \stackrel{\mid{\mbox{\boldmath $x$}}\mid \rightarrow \infty}{\sim}
\hspace{8.5cm} \nonumber
\end{aligned}$$
$$\begin{aligned}
\hspace{0.5cm} \left\{ \begin{array}{c} \langle \; {\mbox{\boldmath $x$}},
{\mbox{\boldmath $X$}}\mid \Phi_{kn}^{\scriptscriptstyle (j,\infty)} \; \rangle \;\; +
{\displaystyle { \!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{\begin{array}{c} {\scriptstyle k'n'} \\
\left(\begin{array}{c} {\scriptstyle kk'<0} \\
{\scriptstyle E_{k'n'}^{\scriptscriptstyle
(j)}=E_{kn}^{\scriptscriptstyle (j)}} \end{array}\right) \end{array} }
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! }} r_{\scriptscriptstyle
(k',n':k,n)}^{\scriptscriptstyle (j)}\langle \; {\mbox{\boldmath $x$}},
{\mbox{\boldmath $X$}}\mid\Phi_{k'n'}^{\scriptscriptstyle (j,\infty)} \; \rangle, \\
\hspace{6cm} \mbox{in} \;\; {\mbox{\boldmath $x$}}\in\Omega_{j}
\vspace{0.8cm} \\
{\displaystyle
{ \!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{\begin{array}{c} {\scriptstyle k'n'} \\
\left(\begin{array}{c} {\scriptstyle kk'<0} \\
{\scriptstyle E_{k'n'}^{\scriptscriptstyle
(l)}=E_{kn}^{\scriptscriptstyle (j)}} \end{array}\right) \end{array} }
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! }} t_{\scriptscriptstyle
(k',n':k,n)}^{\scriptscriptstyle (l,j)}\langle \; {\mbox{\boldmath $x$}},
{\mbox{\boldmath $X$}}\mid\Phi_{k'n'}^{\scriptscriptstyle (l,\infty)} \; \rangle,
\hspace{3cm} \\
\hspace{7.1cm} \mbox{in} \;\; {\mbox{\boldmath $x$}}\in\Omega_{l}, \;\; l
\neq j \end{array}\right.
\label{asymp.3}
\end{aligned}$$
where $r_{\scriptscriptstyle
(k',n':k,n)}^{\scriptscriptstyle (j)}$ and $t_{\scriptscriptstyle
(k',n':k,n)}^{\scriptscriptstyle (l,j)}$ are constants determined by the fact that $\mid \Psi_{kn}^{\scriptscriptstyle (j)} \; \rangle$ is an eigenstate of the operator $\hat{H}$. It should be noted that the eigenstate $\mid
\Psi_{kn}^{\scriptscriptstyle (j)} \; \rangle$ may not be determined uniquely by the asymptotic condition (\[asymp.3\]). So we should introduce a new suffix in order to distinguish the eigenstates having the same asymptotic form. But in order to avoid a complicated notation we do not write such a suffix explicitly, but distinguish such degenerate states by different values of $n$ from now on.
We assume that the condition
$$\lim_{\mid {\small {\mbox{\boldmath $X$}}} \mid \rightarrow \infty} \langle \;
{\mbox{\boldmath $x$}}, {\mbox{\boldmath $X$}}\mid \hat{\rho}(t) \mid {\cal{X}} \; \rangle = 0
\label{bound}$$
is satisfied for an arbitrary state $\mid \cal{X} \;
\rangle$. The condition (\[bound\]) may be satisfied by the assumption that the scatterer is in a bound state. The transport particle is assumed to be in a scattering state which satisfies the asymptotic condition
$$\begin{aligned}
\langle \; {\mbox{\boldmath $x$}},{\mbox{\boldmath $X$}}\mid && \!\!\!\!\!\!\!\!\!\!
\hat{\rho}(t)\mid{\mbox{\boldmath $x$}}',{\mbox{\boldmath $X$}}' \; \rangle \stackrel{\mid{\mbox{\boldmath $x$}}\mid
\rightarrow\infty, \mid{\mbox{\boldmath $x$}}'\mid\rightarrow\infty}{\sim} \nonumber \\
&& \sum_{j=1}^{N}{ \!\! \sum_{\begin{array}{c} {\scriptstyle kn} \\ {\scriptstyle (k>0)}
\end{array} } \!\! } F_j(k, n) \langle \; {\mbox{\boldmath $x$}},
{\mbox{\boldmath $X$}}\mid\Psi_{kn}^{\scriptscriptstyle {(j)}} \; \rangle \langle \;
\Psi_{kn}^{\scriptscriptstyle (j)}\mid{\mbox{\boldmath $x$}}',{\mbox{\boldmath $X$}}' \; \rangle
\label{asymp}
\end{aligned}$$
where $F_j(k, n)$ is a positive function of $k$ and $n$. We interpret that the function $F_j(k, n)$ represents a distribution induced in the system by an interaction with the reservoir at the end of the $j$-th lead.
[**$ \; \langle$ 3. Current of a Conserved Quantity $ \;
\rangle$** ]{} We consider a conserved quantity. This quantity corresponds to a Hermitian operator $\hat{G}$, which satisfies the relation $[\hat{H}, \hat{G}] = 0$.
We introduce the symmetrized product $\hat{\cal{X}}\star\hat{\cal{Y}}$ of two arbitrary operators $\hat{\cal{X}}$ and $\hat{\cal{Y}}$ as
$$\begin{aligned}
\hat{\cal{X}}\star\hat{\cal{Y}} \equiv \{\hat{\cal{X}}
\hat{\cal{Y}}+(\hat{\cal{X}} \hat{\cal{Y}})^{\dagger}\}/2
\label{product}
\end{aligned}$$
where $(\hat{\cal{X}} \hat{\cal{Y}})^{\dagger}$ means the Hermitian conjugate operator of the operator $\hat{\cal{X}} \hat{\cal{Y}}$. By Eqs. (\[Liouv\]), (\[bound\]), (\[product\]) and Gauss’ divergence theorem we obtain an equation of continuity
$$\begin{aligned}
\frac{\partial}{\partial t} \int_{\Omega} d{\mbox{\boldmath $X$}}&&
\!\!\!\!\!\!\!\!\!\! \langle \; {\mbox{\boldmath $x$}}, {\mbox{\boldmath $X$}}\mid
(\hat{\rho}(t)\star\hat{G}) \mid {\mbox{\boldmath $x$}}, {\mbox{\boldmath $X$}}\; \rangle \nonumber \\
&&
+ \frac{\partial}{\partial {\mbox{\boldmath $x$}}} \cdot \int_{\Omega} d{\mbox{\boldmath $X$}}\langle \; {\mbox{\boldmath $x$}}, {\mbox{\boldmath $X$}}\mid ((\hat{\rho}(t)\star\hat{G}) \star
\hat{{\mbox{\boldmath $v$}}}) \mid {\mbox{\boldmath $x$}}, {\mbox{\boldmath $X$}}\; \rangle = 0
\label{conti.2}
\end{aligned}$$
where $\hat{{\mbox{\boldmath $v$}}}$ is defined by $\hat{{\mbox{\boldmath $v$}}} \equiv
- [\hat{H}, \hat{{\mbox{\boldmath $x$}}}] /(i\hbar)$. By Eq. (\[conti.2\]) we interpret the quantity $\int_{\Omega}
d{\mbox{\boldmath $X$}}\langle \; {\mbox{\boldmath $x$}}, {\mbox{\boldmath $X$}}\mid ((\hat{\rho}(t)\star\hat{G}) \star
\hat{{\mbox{\boldmath $v$}}}) \mid {\mbox{\boldmath $x$}}, {\mbox{\boldmath $X$}}\; \rangle$ as a current density of the conserved quantity due to the transport particle at time $t$.
We consider the current $J_j$ in the $j$-th lead, defined by
$$\begin{aligned}
\!\!\! J_{j}\equiv\lim_{x_1^{\scriptscriptstyle
(j)} \rightarrow -\infty}\int_{S_j}dx_2^{\scriptscriptstyle
(j)}dx_3^{\scriptscriptstyle (j)} \; {\mbox{\boldmath $e$}}_1^{\scriptscriptstyle
(j)}\cdot &&
\!\!\!\!\!\!\!\!\!\! \int_{\Omega} d{\mbox{\boldmath $X$}}\langle \; {\mbox{\boldmath $x$}},
{\mbox{\boldmath $X$}}\mid ((\hat{\rho}(t)\star\hat{G}) \star \hat{{\mbox{\boldmath $v$}}}) \mid {\mbox{\boldmath $x$}},
{\mbox{\boldmath $X$}}\; \rangle.
\label{flow}
\end{aligned}$$
We assume that the eigenstate $\mid\Psi_{kn}^{\scriptscriptstyle (j)} \; \rangle$ is also the eigenstate of the operator $\hat{G}$ \[14\]. And we introduce $G_{kn}^{\scriptscriptstyle (j)}$ as the eigenvalue of the operator $\hat{G}$ corresponding to the eigenstate $\mid\Psi_{kn}^{\scriptscriptstyle (j)} \; \rangle$. From Eqs. (\[asymp\]) and (\[flow\]) we derive
$$\begin{aligned}
J_{j} && \!\!\!\!\!\!\!\!\!\! =
\frac{1}{\hbar}{ \!\! \sum_{\begin{array}{c} {\scriptstyle kn} \\ {\scriptstyle (k>0)}
\end{array} } \!\! }F_{j}(k, n) \,
G^{\scriptscriptstyle (j)}_{kn}\left\{ \frac{\partial
E_{kn}^{\scriptscriptstyle (j)}}{\partial k}\right. -
{ \!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{\begin{array}{c} {\scriptstyle k'n'} \\
\left(\begin{array}{c} {\scriptstyle k'>0} \\
{\scriptstyle E_{-k'n'}^{\scriptscriptstyle
(j)}=E_{kn}^{(j)}} \end{array}\right) \end{array} }
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! }\mid r_{\scriptscriptstyle (-k', n',:k,
n)}^{\scriptscriptstyle (j)}\mid^2\left.\frac{\partial
E_{-k'n'}^{\scriptscriptstyle (j)}}{\partial k'}\right\}
\hspace{0.3cm} \nonumber \\
&& - { \!\! \sum^{N}_{\begin{array}{c} {\scriptstyle l=1} \\ {\scriptstyle (l\neq
j)}
\end{array} } \!\! }\frac{1}{\hbar}{ \!\! \sum_{\begin{array}{c} {\scriptstyle kn} \\ {\scriptstyle (k>0)}
\end{array} } \!\! }F_{l}(k, n) \,
G^{\scriptscriptstyle (l)}_{kn}
{ \!\!\!\!\!\!\!\!\!\!\!\!\!\sum_{\begin{array}{c} {\scriptstyle k'n'} \\
\left(\begin{array}{c} {\scriptstyle k'>0} \\
{\scriptstyle E_{-k'n'}^{\scriptscriptstyle
(j)}=E_{kn}^{\scriptscriptstyle (l)}} \end{array}\right) \end{array} }
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! }\mid t_{\scriptscriptstyle (-k',
n':k, n)}^{\scriptscriptstyle (j,l)}\mid^2\frac{\partial
E_{-k'n'}^{\scriptscriptstyle (j)}}{\partial k'}
\label{resul}
\end{aligned}$$
where we used the following fact \[15\]:
$$\begin{aligned}
\mbox{If} && \!\!\! E_{kn}^{\scriptscriptstyle
(j)}=E_{k'n'}^{\scriptscriptstyle (j)} \; \mbox{and} \; kk'>0, \;
\mbox{then} \nonumber \\
&& \int_{S_j}dx_2^{\scriptscriptstyle
(j)}dx_3^{\scriptscriptstyle (j)}\int_{\Omega} d{\mbox{\boldmath $X$}}\nonumber \\
&& \hspace{1cm} \frac{1}{2}\left\{ \langle \;
{\mbox{\boldmath $x$}},{\mbox{\boldmath $X$}}\mid({\mbox{\boldmath $e$}}_{1}^{\scriptscriptstyle
(j)}\cdot{\hat{{\mbox{\boldmath $v$}}}})\mid\Phi_{kn}^{\scriptscriptstyle (j,\infty)}
\; \rangle \langle \; \Phi_{k'n'}^{\scriptscriptstyle
(j,\infty)}\mid{\mbox{\boldmath $x$}},{\mbox{\boldmath $X$}}\; \rangle\right. \nonumber \\
&& \hspace{2cm} + \left.\langle \;
{\mbox{\boldmath $x$}},{\mbox{\boldmath $X$}}\mid\Phi_{kn}^{\scriptscriptstyle (j,\infty)} \; \rangle
\langle \; \Phi_{k'n'}^{\scriptscriptstyle
(j,\infty)}\mid({\mbox{\boldmath $e$}}_{1}^{\scriptscriptstyle
(j)}\cdot{\hat{{\mbox{\boldmath $v$}}}})\mid{\mbox{\boldmath $x$}},{\mbox{\boldmath $X$}}\; \rangle\right\} \nonumber \\
&& \hspace{3.5cm} \stackrel{\mid x_{1}^{\scriptscriptstyle
(j)}\mid\rightarrow\infty}{\sim} \frac{1}{\hbar}\frac{\partial
E_{kn}^{\scriptscriptstyle (j)}}{\partial k} \, \delta_{n' n} \,
\delta_{k' k}, \;\;\;\;\; \mbox{in} \; {\mbox{\boldmath $x$}}\in\Omega_j.
\label{ortho.2}
\end{aligned}$$
By the equation of continuity (\[conti.2\]), the current of the quantity $G$ which flows through any cross section of the $j$-th columned region takes the value $J_j$ in any steady state. It is important to note that the effect of interference among the incident wave, its reflective waves and its transmitted waves does not appear in the quantity $J_j$, because of Eq. (\[ortho.2\]). Eq. (\[resul\]) is one of the main results in the present Letter.
[**$ \; \langle$ 4. Derivation of Landauer Formula and its Generalizations $ \; \rangle$** ]{} As a special case, we consider a case satisfying the following conditions: (A) The scattered wave has the same wave number as its incident wave, that is, when $r_{\scriptscriptstyle (-k', n':k,
n)}^{\scriptscriptstyle (j)} \neq 0$, $kk'>0$ and $E_{-k'n'}^{\scriptscriptstyle (j)}=E_{kn}^{\scriptscriptstyle (j)}$ in $ n'$, $k$, $ n$ on ${\cal F}_{j}$, or $t_{\scriptscriptstyle
(-k', n':k, n)}^{\scriptscriptstyle (l,j)}\neq 0$, $kk'>0$ and $E_{-k'n'}^{\scriptscriptstyle (l)}=E_{kn}^{\scriptscriptstyle (j)}$ in $k$, $ n$ on ${\cal F}_{j}$ and in $ n'$ on ${\cal F}_{l}$, the relation $k'= k$ is satisfied, where ${\cal F}_j$ is the support of the function $F_{j}(k, n)$. It implies that the scattering processes are almost elastic. (B) The distribution function $F_j(k, n)$ is a function only of the energy eigenvalue $E_{kn}^{\scriptscriptstyle (j)}$. So we put a function $\tilde{F}_j(E_{kn}^{\scriptscriptstyle
(j)})$ of $E_{kn}^{\scriptscriptstyle (j)}$ instead of $F_j(k, n)$. (C) The operator $\hat{G}$ is a function only of the operator $\hat{H}$. So we put an operator $\tilde{G} (\hat{H})$, which is a function of $\hat{H}$, instead of $\hat{G}$. (D) The coefficients $ r_{\scriptscriptstyle (k', n':k,
n)}^{\scriptscriptstyle (j)}$ and $ t_{\scriptscriptstyle (k', n':k,
n)}^{\scriptscriptstyle (l,j)}$ are dependent on the suffixes $k$ and $n$ on ${\cal F}_{j}$ only through the energy eigenstate $E_{kn}^{\scriptscriptstyle (j)}$. So we put the coefficients $ r_{\scriptscriptstyle (k', n')}^{\scriptscriptstyle (j)}
(E_{kn}^{\scriptscriptstyle (j)})$ and $t_{\scriptscriptstyle (k', n')}
^{\scriptscriptstyle (l,j)}(E_{kn}^{\scriptscriptstyle (j)})$ instead of the coefficients $ r_{\scriptscriptstyle (k', n':k,
n)}^{\scriptscriptstyle (j)}$ and $ t_{\scriptscriptstyle (k', n':k,
n)}^{\scriptscriptstyle (l,j)}$, respectively. (E) The function $E_{kn}^{\scriptscriptstyle (j)}$ of the variable $k$ projects the domain $(0, \infty)$ of the variable $k$ to the domain $\Lambda_j$ of the variable $\varepsilon$. The domain $\Lambda_j$ is independent of the value of the suffix $n$. (F) The suffix $n$ takes $\tilde{n}$ of values on ${\cal F}_j$. Under the conditions (A)-(F), Eq. (\[resul\]) becomes
$$\begin{aligned}
J_{j} = \frac{\tilde{n}}{2\pi\hbar} \left\{\int_{\Lambda_j}
\right. d\varepsilon && \!\!\!\!\!\!\! \tilde{F}_j(\varepsilon) \,
\tilde{G} (\varepsilon)\left ( 1- R_j(\varepsilon) \right ) \nonumber
\\
&& - { \!\! \sum^{N}_{\begin{array}{c} {\scriptstyle l=1} \\ {\scriptstyle (l\neq
j)}
\end{array} } \!\! } \left.\int_{\Lambda_l}d\varepsilon \tilde{F}_{l}(\varepsilon)
\, \tilde{G} (\varepsilon) T_{\scriptscriptstyle jl}(\varepsilon)\right\}.
\label{resul.2}
\end{aligned}$$
Here $R_j(\varepsilon)$ and $T_{\scriptscriptstyle jl}
(\varepsilon)$ are defined by $R_j(\varepsilon)\equiv\sum_{k'n'}\mid r_{\scriptscriptstyle (-k', n')}
^{\scriptscriptstyle (j)} (\varepsilon) \mid^2 $ and $T_{\scriptscriptstyle jl}(\varepsilon)
\equiv\sum_{k'n'}\mid t_{\scriptscriptstyle (-k', n')}
^{\scriptscriptstyle (j,l)}(\varepsilon) \mid^2$, respectively where the sums over the suffixes $k'$ and $n'$ are taken only over values satisfying the condition $E_{-k'n'}^{\scriptscriptstyle (j)}=\varepsilon$ and $E_{-k'n'}
^{\scriptscriptstyle (l)}=\varepsilon$, respectively. And we used the fact that the sum over the suffix $k$ in $k>0$ corresponds to the integral over the suffix $k$ in $k>0$ multiplied the factor $1/(2\pi)$.
From Eq. (\[resul.2\]) we can derive Landauer formula and its generalizations which have already been proposed. For example, if $\tilde{G}(\hat{H})=q$, $\tilde{F}_j(\varepsilon)=\lim_{T\rightarrow
0}\{\exp\{(\varepsilon-\mu_j)/(k_B
T)\}+1\}^{-1}=\theta(\varepsilon-\mu_j)$ (where $k_B$, $T$ and $\mu_j$ are positive constants), $\mid (\mu_j-\bar{\mu})/
\bar{\mu}\mid<<1$ (where $\bar{\mu}\equiv(\mu_1+\mu_2+\cdots+\mu_N)/N$), $\tilde{n}=2$ and $\Lambda_j=(0,\infty)$, then Eq. (\[resul.2\]) become
$$\begin{aligned}
J_{j} \approx \frac{q}{\pi\hbar}\left\{\left ( 1-
R_j(\bar{\mu}) \right )\mu_j\right. + { \!\! \sum^{N}_{\begin{array}{c} {\scriptstyle l=1} \\ {\scriptstyle (l\neq
j)}
\end{array} } \!\! } \left. T_{\scriptscriptstyle jl}(\bar{\mu}) \;
\mu_l\right\} \label{resul.3}
\end{aligned}$$
in the first approximation of $(\mu_j-\bar{\mu})/\bar{\mu}$. Eq. (\[resul.3\]) is one of the generalizations of Landauer formula, and was proposed by M. Büttiker \[5\].
[**$ \; \langle$ 5. Conclusion and Remarks $ \; \rangle$** ]{} In the present Letter, we have described thermodynamical nonlinear responses of a quantum system to thermodynamical gaps of reservoirs. Here, the thermodynamical reservoirs were connected to the system with leads, and effects of the reservoirs were introduced as an asymptotic condition at the ends of the leads. Thermodynamical gaps of the reservoirs cause thermodynamical transport phenomena. We derived an expression of a current of a conserved quantity, which is independent of the details of the Hamiltonian operator. For example, this method can describe an energy current which is caused inside a system connected to heat reservoirs having different temperatures.
This method leads to Landauer formula and its generalizations which have already been proposed. Moreover, we can also give further generalizations of Landauer formula by this method. For example, in this method we can deal with nonlinear responses, reservoirs inducing non-Fermi distribution in a system, inelastic processes caused by scatterers and currents other than an electric or a heat current, etc., which have not been treated earlier. The method in the present Letter can be generalized to the case of many transport particles and many scatterers \[16\]. (However the two-particles system, which we discussed in the present Letter, can be interpreted as a mean field approximation for a system consisting of many transport particles and many scatterers.) This method can also be generalized to the case where interactions of the particles are given by a complex potential only in a finite region.
We interpreted that the function $F_j(k, n)$ of $k$ and $n$ represents a distribution induced in the system by an interaction with thermodynamical reservoir at the end of the $j$-th lead, because it represents a distribution of the incident wave at the end of the $j$-th lead, and we can use an equilibrium distribution function as the stationary distribution function $F_j(k, n)$. But these reservoirs do not cause a dissipation of the system in this method. And as the distribution function $F_j(k, n)$ we can select a distribution function which is not an equilibrium distribution function. In this sense, we can describe non-thermodynamical transport phenomena by this method.
One may notice that this method does not treat distributions of thermodynamic quantities inside a system, because we do not assume the local equilibrium assumption, etc. in this method. For example, this method does not treat temperature distribution inside a system connected to heat reservoirs having different temperatures. To treat such a distribution by a generalization of this method remains as a future problem.
[**$\langle \; $ Acknowledgements $ \; \rangle$** ]{} I wish to express my gratitude to K. Kitahara and T. Hara for a careful reading of the manuscript and for valuable comments. I acknowledge S. Komiyama for a stimulating lecture on Landauer formula.
——————————————————–
0.55cm
\[1\] D. N. Zubarev, Nonequiliblium Statistical Mechanics, Consultant Bureau, New York (1971)
\[2\] R. Landauer, Philos. Mag. 21 (1970) 863
\[3\] M. Y. Azbel, J. Phys. C14 (1981) L225
\[4\] M. Büttiker, Y. Imry, R. Landauer and S. Pinhas, Phys. Rev. B31 (1985) 6207
\[5\] M. Büttiker, Phys. Rev. Lett. 57 (1986) 1761
\[6\] H. L. Engquist and P. W. Anderson, Phys. Rev. B24 (1981) 1151
\[7\] U. Sivan and Y. Imry, Phys. Rev. B33 (1986) 551
\[8\] Y. Gefen and G. Shön, Phys. Rev. B30 (1984) 7323
\[9\] There [*is*]{} a statistical mechanical derivation of a generalization of Landauer formula for an electric current in a restricted situation where the system consists of non-interacting electrons, reservoirs connected to the system do not have temperature gaps, and the system causes only elastic scattering processes \[10-13\]. But it should be noted that the methods used in Refs. \[10-13\] are not same with our method. Refs. \[10-12\] used the linear response theory to an external disturbance derived from the potential energy, and Ref. \[13\] used a different asymptotic condition with our method.
\[10\] E. N. Economou and C. M. Soukoulis, Phys. Rev. Lett. 46 (1981) 618
\[11\] D. S. Fisher and P. A. Lee, Phys. Rev. B23 (1981) 6851
\[12\] H. U. Baranger and A. D. Stone, Phys. Rev. B40 (1989) 8169
\[13\] S. Komiyama and H. Hirai, Phys. Rev. B54 (1996) 2067
\[14\] The exchangeable operators $\hat{H}$ and $\hat{G}$ have a common eigenstate, and the state $\mid\Psi_{kn}^{\scriptscriptstyle
(j)}\rangle$ is a eigenstate of the operator $\hat{H}$. But the common eigenstate of operators $\hat{H}$ and $\hat{G}$ may not have the asymptotic form (\[asymp.3\]). So, this is a new assumption.
\[15\] The outline of the derivation of Eq. (\[ortho.2\]) is as follows. We first note that the divergence of the integrated function of $x_2^{\scriptscriptstyle (j)}$ and $x_3^{\scriptscriptstyle (j)}$ in the left hand side of Eq. (\[ortho.2\]) must be zero when $E_{kn}^{\scriptscriptstyle
(j)}=E_{k'n'}^{\scriptscriptstyle (j)}$. So the left hand side of Eq. (\[ortho.2\]) must be zero except for the case $k=k'$ in $\mid x_1^{\scriptscriptstyle (j)} \mid
\rightarrow \infty$, ${\mbox{\boldmath $x$}}\in\Omega_j$ when $E_{kn}^{\scriptscriptstyle
(j)}=E_{k'n'}^{\scriptscriptstyle (j)}$. Besides, we have $ \langle
\; {\mbox{\boldmath $x$}},{\mbox{\boldmath $X$}}\mid({\mbox{\boldmath $e$}}_{1}^{\scriptscriptstyle
(j)}\cdot{\hat{{\mbox{\boldmath $v$}}}})\mid\Phi_{kn}^{\scriptscriptstyle (j,\infty)}
\; \rangle \sim \langle \; {\mbox{\boldmath $x$}},{\mbox{\boldmath $X$}}\mid
\frac{1}{\hbar}(\partial{\hat{\cal{H}}}_{k}^{\scriptscriptstyle
(j,\infty)} / \partial k)\mid\Phi_{kn}^{\scriptscriptstyle (j,\infty)}
\; \rangle$ as $\mid x_{1}^{\scriptscriptstyle
(j)}\mid\rightarrow\infty$ in ${\mbox{\boldmath $x$}}\in\Omega_j$. By these and Eq. (\[ortho.1\]), we obtain Eq. (\[ortho.2\]).
\[16\] For such a generalization we must deal with energy eigenstates of the system consisting of the many transport particles and many scatterers, which must be required to satisfy the Pauli principle.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
In this paper, we pursue the study of second order BSDEs with jumps (2BSDEJs for short) started in our accompanying paper [@kpz3]. We prove existence of these equations by a direct method, thus providing complete wellposedness for 2BSDEJs. These equations are a natural candidate for the probabilistic interpretation of some fully non-linear partial integro-differential equations, which is the point of the second part of this work. We prove a non-linear Feynman-Kac formula and show that solutions to 2BSDEJs provide viscosity solutions of the associated PIDEs.
[**Key words:**]{} Second order backward stochastic differential equation, backward stochastic differential equation with jumps, model uncertainty, PIDEs, viscosity solutions.
[**AMS 2000 subject classifications:**]{} 60H10, 60H30
author:
- 'Nabil [Kazi-Tani]{}[^1]'
- 'Dylan [Possamaï]{}[^2]'
- 'Chao [Zhou]{}[^3]'
title: 'Second Order BSDEs with Jumps: Existence and probabilistic representation for fully-nonlinear PIDEs'
---
Introduction
============
Motivated by duality methods and maximum principles for optimal stochastic control, Bismut studied in [@bis] a linear backward stochastic differential equation (BSDE). In their seminal paper [@pardpeng], Pardoux and Peng generalized such equations to the non-linear Lipschitz case and proved existence and uniqueness results in a Brownian framework. Since then, a lot of attention has been given to BSDEs and their applications, not only in stochastic control, but also in theoretical economics, stochastic differential games and financial mathematics. Given a filtered probability space $(\Omega,\mathcal F,\left\{\mathcal F_t\right\}_{0\leq t\leq T},\mathbb P)$ generated by an $\mathbb R^d$-valued Brownian motion $B$, solving a BSDE with generator $g$, and terminal condition $\xi$ consists in finding a pair of progressively measurable processes $(Y,Z)$ such that $$\begin{aligned}
Y_t=\xi +\int_t^T g_s(Y_s,Z_s)ds-\int_t^T Z_s dB_s,\text{ }\mathbb P-a.s, \text{ } t\in [0,T]. \label{def_bsde}\end{aligned}$$
The process $Y$ defined this way is a possible generalization of the conditional expectation of $\xi$, since when $g$ is the null function, we have $Y_t=\E^{\P}\left[\xi | \Fc_t\right]$, and in that case, $Z$ is the process appearing in the $(\Fc_t)$-martingale representation property of $(\E^{\P}\left[\xi | \Fc_t\right])_{t\geq 0}$. In the case of a filtered probability space generated by both a Brownian motion $B$ and a Poisson random measure $\mu$ with compensator $\nu$, the martingale representation for $(\E^{\P}\left[\xi | \Fc_t\right])_{t\geq 0}$ becomes $$\begin{aligned}
\E^{\P}[\xi | \Fc_t] =\xi+ \int_0^t Z_s dB_s + \int_0^t \int_{\R^d\backslash \{0\}} U_s(x)(\mu-\nu)(dx,ds),\ \mathbb P-a.s.,\end{aligned}$$ where $U$ is a predictable function. This leads to the following natural generalization of equation (\[def\_bsde\]) to the case with jumps. We will say that $(Y,Z,U)$ is a solution of the BSDE with jumps (BSDEJ in the sequel) with generator $g$ and terminal condition $\xi$ if for all $t \in [0,T]$, we have $\mathbb P-a.s.$ $$\begin{aligned}
Y_t=\xi +\int_t^T g_s(Y_s,Z_s,U_s)ds-\int_t^T Z_s dB_s -\int_t^T \int_{\R^d\backslash \{0\}} U_s(x)(\mu-\nu)(dx,ds). \label{def_bsdej}\end{aligned}$$
Tang and Li [@tangli] were the first to prove existence and uniqueness of a solution for (\[def\_bsdej\]) in the case where $g$ is Lipschitz in $(y,z,u)$. In the continuous framework, Soner, Touzi and Zhang [@stz] generalized the BSDE to the second order case. Their key idea in the definition of the second order BSDEs (2BSDEs) is that the equation has to hold $\P$-almost surely, for every $\P$ in a class of non-dominated probability measures. Furthermore, they proved a uniqueness result using a representation result of the 2BSDEs as essential supremum of standard BSDEs.
Our aim in this paper is to pursue the study undertaken in [@kpz3]. More precisely, we prove existence of a solution to equation by a direct approach. Inspired by the representation obtained in Theorem $4.1$ of [@kpz3], we construct a solution by using the tool of regular conditional probability distributions. This gives a complete wellposedness theory for 2BSDEJs.
The last part of our study is to establish a connection with partial integro-differential equations (PIDEs for short). Indeed, Soner, Touzi and Zhang proved in [@stz] that Markovian 2BSDEs, are connected in the continuous case to a class of parabolic fully non-linear PDEs. On the other hand, we know that solutions to standard Markovian BSDEJs provide viscosity solutions to some parabolic partial integro-differential equations whose non-local operator is given by a quantity similar to $\langle \widetilde{v},\nu\rangle$ defined in (see [@bbp] for more details). Then in the Markovian case, 2BSDEJs are the natural candidates for the probabilistic interpretation of fully non-linear PIDEs. This is the purpose of the second part of this article. During the revision of this paper, in two beautiful articles, Neufeld and Nutz [@nn2; @nn3] constructed so-called non-linear Lévy processes, and showed that they provided probabilistic representations for viscosity solutions to a certain class of fully non-linear PIDEs. These objects are related to 2BSDEJs in the sense that they roughly correspond to the case of generator equal to $0$. However, the method they used for their construction (which is actually and extension of Nutz and van Handel [@nvh] to the Skorohod space of càdlàg functions) allows them to do not assume any strong pathwise regularity, unlike in our approach. Nonetheless, an extension of their method to the case of a non-zero generator is far from trivial, as it would require to study measurability of fully non-linear (and not only sub-linear) stochastic kernels.
The rest of the paper is organized as follows. In Section \[section.1\], in order to introduce our readers to the theory, we provide several definitions and results on the set of probability measures on the Skorohod space $\D$ that we will work with. In Section \[sec.2BSDE\], we introduce the generator of our 2BSDEJs and the assumptions under which we will be working, we recall from [@kpz3] the natural spaces and norms for the solution of a 2BSDEJ, and give the formulation of the 2BSDEJs. Section \[section.3\] is devoted to the proof of our existence result. Finally, in Section \[sec.PIDE\], we study the links between solutions to some fully-nonlinear PIDEs and 2BSDEJs. The Appendix is dedicated to the proof of some important technical results needed throughout the paper.
Preliminaries on probability measures {#section.1}
=====================================
The stochastic basis
--------------------
Let $\Omega:= \D([0,T],\mathbb R^d)$ be the space of càdlàg paths defined on $[0,T]$ with values in $\R^d$ and such that $w(0)=0$, equipped with the Skorohod topology, so that it is a complete, separable metric space (see [@bil] for instance).
We denote $B$ the canonical process, $\mathbb F:=\left\{\mathcal F_t\right\}_{0\leq t\leq T}$ the filtration generated by $B$, $\mathbb F^+:=\left\{\mathcal F_t^+\right\}_{0\leq t\leq T}$ the right limit of $\mathbb F$ and for any $\mathbb P$, $\overline{\mathcal F}_t^\mathbb P:=\mathcal F_t^+\vee\mathcal N^\mathbb P(\mathcal F_t^+)$ where $$\mathcal N^\mathbb P(\mathcal G):=\left\{E\in\Omega,\text{ there exists $\widetilde E\in\mathcal G$ such that $E\subset\widetilde E$ and $\mathbb P(\widetilde E)=0$}\right\}.$$
We then define as in [@stz] a local martingale measure $\mathbb P$ as a probability measure such that $B$ is a $\mathbb P$-local martingale. We then associate to the jumps of $B$ a counting measure $\mu_{B}$, which is a random measure on $\mathbb R^+\times E$ equipped with its Borel $\sigma$-field $\mathcal B(\R^+)\times\mathcal B(E)$ (where $E:=\mathbb R^d\backslash \{0\}$), defined pathwise by $$\mu_{B}(A,[0,t]) := \sum_{0<s\leq t} \mathbf{1}_{\{\Delta B_s \in A\}}, \; \forall t \geq 0, \; \forall A \in\mathcal B(E).$$
We recall that (see for instance Theorem I.4.18 in [@jac]) under any local martingale measure $\P$, we can decompose $B$ uniquely into the sum of a continuous local martingale, denoted by $B^{\P,c}$, and a purely discontinuous local martingale, denoted by $B^{\P,d}$. Then, we define $\overline{\mathcal P}_W$ as the set of all local martingale measures $\mathbb P$, such that $\mathbb P$-a.s.
- The quadratic variation of $B^{\P,c}$ is absolutely continuous with respect to the Lebesgue measure $dt$ and its density takes values in $\mathbb S^{>0}_d$, which is the space of all $d\times d$ real valued positive definite matrices.
- The compensator $\lambda^\mathbb P_t(dx,dt)$ of the jump measure $\mu_B$ exists under $\mathbb P$ and can be decomposed as follows $$\lambda^\P_t(dx,dt)=\nu^\P_t(dx)dt,$$ for some $\F$-predictable random measure $\nu^\P$ on $E$.
We will denote by $\widetilde\mu_{B}^\mathbb P(dx,dt)$ the corresponding compensated measure, and for simplicity, we will often call $\nu^\P$ the compensator of the jump measure associated to $B$. Finally, as shown in [@kpz3], it is possible, using results of Bichteler [@bich] to give a pathwise definition of the density (with respect to the Lebesgue measure) of the continuous part of $[B,B]$, which we denote by $\widehat a$.
Martingale problems and probability measures
--------------------------------------------
In this section, we recall the families of probability measures introduced in [@kpz3]. Let $\mathcal{N}$ be the set of $\mathbb F$-predictable random measures $\nu$ on $\mathcal{B}(E)$ satisfying $$\int_0^t \int_{E}(1\wedge {\left|x\right|}^2)\nu_s(\omega,dx)ds <+\infty \text{ and } \int_0^t\int_{ {\left|x\right|}>1 } x \nu_s(\omega,dx)ds <+\infty,\ \text{for all } \omega \in \Omega, \label{hyp_nu}$$ and let $\Dc$ be the set of $\mathbb F$-predictable processes $\alpha$ taking values in $\mathbb S_d^{>0}$ with $\int_0^T|\alpha_t(\omega)|dt<+\infty$, for all $\omega \in \Omega$. We define a martingale problem as follows
For $\mathbb F$-stopping times $\tau_1\leq\tau_2$, for $(\alpha,\nu)\in\mathcal D\times\mathcal N$ and for a probability measure $\mathbb P_1$ on $\mathcal F_{\tau_1}$, we say that $\P$ is a solution of the *martingale problem* $(\P_1,\tau_1,\tau_2,\alpha,\nu)$ if
- $\P = \P_1$ on $\Fc_{\tau_1}$.
- The canonical process $B$ on $[\tau_1,\tau_2]$ is a semimartingale under $\P$ with characteristics $$\begin{aligned}
\left( -\int_{\tau_1}^{\cdot} \int_E x \mathbf{1}_{{\left|x\right|}>1}\nu_s(dx)ds, \int_{\tau_1}^{\cdot} \alpha_s ds,\ \nu_s(dx)ds \right).
\end{aligned}$$
We say that the martingale problem associated to $(\alpha,\nu)$ has a unique solution if, for every stopping times $\tau_1, \tau_2$ and for every probability measure $\P_1$, the martingale problem $(\P_1,\tau_1,\tau_2,\alpha,\nu)$ has a unique solution.
Let now $\overline{\Ac}_W$ be the set of $(\alpha, \nu) \in \Dc \times \Nc$, such that there exists a solution to the martingale problem $(\P^0,0,+\infty,\alpha,\nu)$, where $\P^0$ is such that $\P^0(B_0=0)=1$. We also denote by $\Ac_W$ the set of $(\alpha, \nu) \in \overline{\Ac}_W$ such that there exists a unique solution to the martingale problem $(\P^0,0,+\infty,\alpha,\nu)$. We denote $\P^{\alpha}_{\nu}$ this unique solution and set $$\begin{aligned}
\Pc_W := \left\{ \P^{\alpha}_{\nu}, \ (\alpha, \nu) \in \Ac_W \right\}.\end{aligned}$$
Our main interest in this paper will be a particular sub-class of $\Pc_W$, which can be defined as follows. First, we define $$\mathcal A_{\rm det}:=\left\{(I_d,F),\ F\in\mathcal N \text{ and $F$ is deterministic}\right\},$$ and we let $\widetilde{\Ac}_{\rm det}$ be the separable class of coefficients generated by $\Ac_{\rm det}$ (to avoid unnecessary technicalities, we will refrain from giving the precise definition here, and we refer instead the reader to Definition $A.2$ in [@kpz3] for more details). We emphasize that thanks to Proposition $A.1$ in [@kpz3], we have $\widetilde{\Ac}_{\rm det}\subset \Ac_W$. For simplicity, we let $\mathcal V$ designate the measure $F\in\mathcal N$ such that $(I_d,F)\in\widetilde{\mathcal A}_{\rm det}$. Moreover, we will still denote $\mathbb P_{0,F}:=\mathbb P_F^{I_d}$, for any $F\in\mathcal V$.
Next, we introduce the following set $\mathcal R_F$ of $\F$-predictable functions $\beta:E\longmapsto \R$ such that for Lebesgue almost every $s\in[0,T]$ $${\left|\beta_s\right|}(\omega,x)\leq C(1\wedge{\left|x\right|}),\ F_s(\omega,dx)-a.e.,\text{ for every }\omega\in\Omega,$$ and such that for every $\omega\in\Omega$, $x\longmapsto\beta_s(\omega,x)$ is strictly monotone on the support of the law of $\Delta B_s$ under $\P_{0,F}$.
Next, for each $F\in\mathcal V$ and for each $(\alpha,\beta)\in\mathcal D\times\mathcal R_F$, we define $$\mathbb P^{\alpha,\beta}_F:=\mathbb P_{0,F}\circ\left(X^{\alpha,\beta}_.\right)^{-1},$$ where $$\label{X_alphabeta}
X^{\alpha,\beta}_t:=\int_0^t\alpha_s^{1/2}dB_s^{\P_{0,F},c}+\int_0^t\int_E\beta_s(x)\left(\mu_B(dx,ds)-F_s(dx)ds\right),\ \mathbb P_{0,F}-a.s.$$
Finally, we let $$\overline{\mathcal P}_S:=\underset{F\in\mathcal V}{\bigcup}\left\{\mathbb P^{\alpha,\beta}_F, \ (\alpha,\beta)\in\mathcal D\times\mathcal R_F\right\}.$$
We recall the following results from [@kpz3]
Every probability measure in $\overline{\mathcal P}_S$ satisfies the predictable martingale representation property and the Blumenthal $0-1$ law.
Preliminaries on 2BSDEJs {#sec.2BSDE}
========================
The Non-linear Generator
------------------------
In this subsection we will introduce the function which will serve as the generator of our 2BSDEJ. Let us define the following spaces for $p\geq 1$ $$\hat{L}^p:= \left\{\xi,\ \mathcal F_T\text{-measurable, s.t. }\xi\in L^p(\nu),\text{ for every $\nu\in\Nc$}\right\}.$$
We then consider a map $$\begin{aligned}
H_t(\omega,y,z,u,\gamma,\tilde{v}):[0,T]\times\Omega\times\mathbb{R}\times\mathbb{R}^d\times \hat{L}^2 \times D_1\times D_2\rightarrow \mathbb{R},\end{aligned}$$ where $D_1 \subset \mathbb{R}^{d\times d}$ is a given subset containing $0$ and $D_2 \subset \hat{L}^1$ is the domain of $H$ in $\tilde{v}$.
Define the following conjugate of $H$ with respect to $\gamma$ and $\tilde{v}$ by $$\begin{aligned}
F_t(\omega,y,z,u,a,\nu):=\underset{\{\gamma,\tilde{v}\} \in D_1 \times D_2}{\Sup}\Big\{\frac12\trace(a\gamma)+\int_{E} \tilde{v}(e) \nu(de)-H_t\big(\omega,y,z,u,\gamma,\tilde{v}\big)\Big\}, \label{F_fenchel}\end{aligned}$$ for $a \in \mathbb S_d^{>0}$ and $\nu \in \mathcal{N}$.
In the remainder of this paper, we formulate the needed hypothesis for the generator directly on the function $F$, and the BSDEs we consider also include the case where $F$ does not take the form . Nonetheless, this particular form allows to retrieve easily the framework of the standard BSDEs or of the $G$-stochastic analysis on the one hand (see sections 3.4 and 3.5 in [@kpz3]), and to establish the link with the associated PDEs on the other hand. In the latter cases, $H$ is evaluated at $\tilde{v}(\cdot) = Av(\cdot)$, where $A$ is the following non local operator, defined for any $\mathcal{C}^2$ function $v$ on $\R^d$ with bounded gradient and Hessian, and $y \in \R^d$ by: $$\label{def_v_nu}
(Av)(y,e):= v(e+y) - v(y)- e. (\nabla v)(y), \text{ for } e \in E.$$
The assumptions on $v$ ensure that $(Av)(y,\cdot)$ is an element of $\hat{L}^1$.
The operator $A$ applied to $v$ will only appear again in Section \[sec.PIDE\], when we explore the links between 2BSDEJs and solutions to fully-nonlinear PIDEs. For the time being, we only want to insist on the fact that this particular non local form comes from the intuition that the 2BSDEJs is an essential supremum of standard BSDEJs. Indeed, solutions to Markovian BSDEJs provide viscosity solutions to some parabolic partial integro-differential equations with similar non-local operators (see [@bbp] for more details).
We define next $\widehat{F}^{\mathbb P}_t(y,z,u):=F_t(y,z,u,\widehat{a}_t,\nu^{\mathbb P}_t) \text{ and } \widehat{F}_t^{\mathbb P,0}:=\widehat{F}^{\mathbb P}_t(0,0,0).$ We also denote by $D^1_{F_t(y,z,u)}$ the domain of $F$ in $a$ and by $D^2_{F_t(y,z,u)}$ the domain of $F$ in $\nu$, for a fixed $(t,\omega,y,z,u)$. As in [@stz] we fix a constant $\kappa \in (1,2]$ and restrict the probability measures in $\mathcal{P}_H^\kappa\subset \mathcal{P}_{\widetilde{\mathcal A}}$
\[def\] $\mathcal{P}_H^\kappa$ consists of all $\mathbb P \in \overline{\mathcal{P}}_{S}$ such that
- $\displaystyle\mathbb E^\mathbb P\left[\int_0^T\int_E {\left|x\right|}^2{\nu}^{\mathbb P}_t(dx)dt\right]<+\infty.$
- $\underline{a}^\mathbb P \leq \widehat{a}\leq \bar{a}^\mathbb P, \text{ } dt\times d\mathbb P-as \text{ for some } \underline{a}^\mathbb P, \bar{a}^\mathbb P \in \mathbb{S}_d^{>0}, \text{ and } \mathbb{E}^{\mathbb{P}}\left[\left(\int_0^T{\left|\widehat{F}_t^{\mathbb P,0}\right|}^\kappa dt\right)^{\frac2\kappa}\right]<+\infty.$
The above conditions assumed on the probability measures in $\mathcal P^\kappa_H$ ensure that under any $\mathbb P\in\mathcal P^\kappa_H$, the canonical process $B$ is actually a true square integrable càdlàg martingale. This will be important when we will define standard BSDEJs under each of these probability measures.
We now state our main assumptions on the function $F$
\[assump.href\] [(i)]{} The domains $D^1_{F_t(y,z,u)}=D^1_{F_t}$, $D^2_{F_t(y,z,u)}=D^2_{F_t}$ are independent of $(\omega,y,z,u)$.
[(ii)]{} For fixed $(y,z,u,a,\nu)$, $F$ is $\mathbb{F}$-progressively measurable in $D^1_{F_t} \times D^2_{F_t} $.
[(iii)]{} The following uniform Lipschitz-type property holds. For all $(y,y',z,z',u,t,a,\nu,\omega)$ $$\begin{aligned}
&{\left| F_t(\omega,y,z,u,a,\nu)- F_t(\omega,y',z',u,a,\nu)\right|}\leq C\left({\left|y-y'\right|}+{\left| a^{1/2}\left(z-z'\right)\right|}\right).\end{aligned}$$
[(iv)]{} For all $(t,\omega,y,z,u^1,u^{2},a,\nu)$, there exist two processes $\gamma$ and $\gamma'$ such that $$\begin{aligned}
\int_{E}\delta^{1,2} u(x)\gamma'_t(x)\nu(dx)\leq F_t(\omega,y,z,u^1,a,\nu)- F_t(\omega,y,z,u^2,a,\nu)&\leq \int_{E}\delta^{1,2} u(x)\gamma_t(x)\nu(dx),\end{aligned}$$ where $\delta^{1,2} u:=u^1-u^2$ and $c_1(1\wedge {\left|x\right|}) \leq \gamma_t(x) \leq c_2(1\wedge {\left|x\right|})$ with $-1+\delta\leq c_1\leq0, \; c_2\geq 0,$ and $c_1'(1\wedge {\left|x\right|}) \leq \gamma'_t(x) \leq c_2'(1\wedge {\left|x\right|})$ with $-1+\delta\leq c_1'\leq0, \; c_2'\geq 0,$ for some $\delta >0$.
[(v)]{} $F$ is uniformly continuous in $\omega$ for the Skorohod topology, that is to say that there exists some modulus of continuity $\rho$ such that for all $(t,\omega,\omega',y,z,u,a,\nu)$ $${\left|F_t(\omega,y,z,u,a,\nu)-F_t(\omega',y,z,u,a,\nu)\right|}\leq \rho\left(d_S(\omega_{.\wedge t},\omega'_{.\wedge t})\right),$$ where $d_S$ is the Skorohod metric and where $\omega_{.\wedge t}(s):=\omega(s\wedge t)$.
The Spaces and Norms
--------------------
We now define as in [@stz], the spaces and norms which will be needed for the formulation of the 2BSDEJs.
For $p\geq 1$, $L^{p,\kappa}_H$ denotes the space of all $\mathcal F_T$-measurable scalar r.v. $\xi$ with $${\left\|\xi\right\|}_{L^{p,\kappa}_H}^p:=\underset{\mathbb{P} \in \mathcal{P}_H^\kappa}{\sup}\mathbb E^{\mathbb P}\left[|\xi|^p\right]<+\infty.$$
$\mathbb H^{p,\kappa}_H$ denotes the space of all $\mathbb F^+$-predictable $\mathbb R^d$-valued processes $Z$ with $${\left\|Z\right\|}_{\mathbb H^{p,\kappa}_H}^p:=\underset{\mathbb{P} \in \mathcal{P}_H^\kappa}{\sup}\mathbb E^{\mathbb P}\left[\left(\int_0^T|\widehat a_t^{1/2}Z_t|^2dt\right)^{\frac p2}\right]<+\infty.$$
$\mathbb D^{p,\kappa}_H$ denotes the space of all $\mathbb F^+$-progressively measurable $\mathbb R$-valued processes $Y$ with $$\mathcal P^\kappa_H-q.s. \text{ c\`adl\`ag paths, and }{\left\|Y\right\|}_{\mathbb D^{p,\kappa}_H}^p:=\underset{\mathbb{P} \in \mathcal{P}_H^\kappa}{\sup}\mathbb E^{\mathbb P}\left[\underset{0\leq t\leq T}{\sup}|Y_t|^p\right]<+\infty.$$
$\mathbb J^{p,\kappa}_H$ denotes the space of all $\mathbb F^+$-predictable functions $U$ with $${\left\|U\right\|}_{\mathbb J^{p,\kappa}_H}^p:=\underset{\mathbb{P} \in \mathcal{P}_H^\kappa}{\sup}\mathbb E^{\mathbb P}\left[\left(\int_0^T\int_{E}{\left|U_t(x)\right|}^2 \nu^{\mathbb P}_t(dx)dt\right)^{\frac p2}\right]<+\infty.$$
For each $\xi \in L^{1,\kappa}_H$, $\mathbb P\in \mathcal P^\kappa_H$ and $t \in [0,T]$ denote $$\mathbb E_t^{\mathcal P^\kappa_H,\mathbb P}[\xi]:=\underset{\mathbb P^{'}\in \mathcal P^\kappa_H(t^{+},\mathbb P)}{\esup^{\mathbb P}}\mathbb E^{\mathbb P^{'}}_t[\xi],\ \mathbb P-a.s., \text{ where } \mathcal P^\kappa_H(t^{+},\mathbb P):=\left\{\mathbb P^{'}\in\mathcal P^\kappa_H:\mathbb P^{'}=\mathbb P \text{ on }\mathcal F_t^+\right\}.$$
Then we define for each $p\geq \kappa$, $$\mathbb L_H^{p,\kappa}:=\left\{\xi\in L^{p,\kappa}_H:{\left\|\xi\right\|}_{\mathbb L_H^{p,\kappa}}<+\infty\right\} \text{ where } {\left\|\xi\right\|}_{\mathbb L_H^{p,\kappa}}^p:=\underset{\mathbb P\in\mathcal P^\kappa_H}{\sup}\mathbb E^{\mathbb P}\left[\underset{0\leq t\leq T}{\esup}^{\mathbb P}\left(\mathbb E_t^{\mathcal P^\kappa_H,\mathbb P}[|\xi|^\kappa]\right)^{\frac{p}{\kappa}}\right].$$
Next, we denote by $\mbox{UC}_b(\Omega)$ the collection of all bounded and uniformly continuous maps $\xi:\Omega\rightarrow \mathbb R$ with respect to the Skorohod topology, and we let $$\mathcal L^{p,\kappa}_H:=\text{the closure of $\mbox{UC}_b(\Omega)$ under the norm ${\left\|\cdot\right\|}_{\mathbb L^{p,\kappa}_H}$, for every $1<\kappa \leq p$}.$$
For a given probability measure $\mathbb P\in\mathcal P^\kappa_H$, the spaces $L^p(\mathbb P)$, $\mathbb D^p(\mathbb P)$, $\mathbb H^p(\mathbb P)$ and $\mathbb J^p(\mathbb P)$ correspond to the above spaces when the set of probability measures is reduced to the singleton $\left\{\mathbb P\right\}$. Finally, $\mathbb H^{p}_{loc}(\mathbb P)$ (resp. $\mathbb J^{p}_{loc}(\mathbb P)$) denotes the space of all $\mathbb F^+$-predictable $\mathbb R^d$-valued processes $Z$ (resp. $\mathbb F^+$-predictable functions $U$) with $$\left(\int_0^T{\left|\widehat a_t^{1/2}Z_t\right|}^2dt\right)^{\frac p2}<+\infty,\ (\text{resp. }\ \mathbb P-a.s.$$
$\mathbb J^{p}_{loc}(\mathbb P)$ denotes the space of all $\mathbb F^+$-predictable functions $U$ with $$\left(\int_0^T\int_{E}{\left|U_t(x)\right|}^2 \nu^{\mathbb P}_t(dx)dt\right)^{\frac p2}<+\infty,\ \mathbb P-a.s.$$
Formulation {#formulation:2bsdej}
-----------
We shall consider the following 2BSDEJ, which must hold for $0\leq t\leq T$ and $\mathcal{P}^\kappa_H\text{-q.s.}$ $$Y_t=\xi +\int_t^T\widehat{F}^{\mathbb P}_s(Y_s,Z_s,U_s)ds -\int_t^T Z_sdB^{\mathbb P,c}_s-\int_t^T \int_{E} U_s(x) {\mu}^{\mathbb P}_B(dx,ds) + K^{\mathbb P}_T-K^{\mathbb P}_t.
\label{2bsdej}$$
We say $(Y,Z,U)\in \mathbb D^{2,\kappa}_H \times \mathbb H^{2,\kappa}_H \times \mathbb J^{2,\kappa}_H$ is a solution to the $2$BSDEJ if
- $Y_T=\xi$, $\mathcal{P}^\kappa_H$-q.s.
- For all $\mathbb P \in \mathcal{P}^\kappa_H$ and $0\leq t\leq T$, the process $K^{\mathbb P}$ defined below is predictable and has non-decreasing paths $\mathbb P-a.s.$ $$K_t^{\mathbb P}:=Y_0-Y_t - \int_0^t\widehat{F}^{\mathbb P}_s(Y_s,Z_s,U_s)ds+\int_0^tZ_sdB^{\mathbb P,c}_s + \int_0^t \int_{E} U_s(x) {\mu}^{\mathbb P}_B(dx,ds).
\label{2bsdej_K}$$
- The family $\left\{K^{\mathbb P}, \mathbb P \in \mathcal P_H^\kappa\right\}$ satisfies the minimum condition $$K_t^{\mathbb P}=\underset{ \mathbb{P}^{'} \in \mathcal{P}_H^\kappa(t^+,\mathbb{P}) }{ \einf^{\mathbb P} }\mathbb{E}_t^{\mathbb P^{'}}\left[K_T^{\mathbb{P}^{'}}\right], \text{ } 0\leq t\leq T, \text{ } \mathbb P-a.s., \text{ } \forall \mathbb P \in \mathcal P_H^\kappa.
\label{2bsdej.minK}$$
If the family $\left\{K^{\mathbb P}, \mathbb P \in \mathcal P_H^\kappa\right\}$ can be aggregated into a universal process $K$, we call $(Y,Z,U,K)$ a solution of the $2$BSDEJ .
Following [@stz], in addition to Assumption \[assump.href\], we will always assume
\[assump.h2ref\]
[(i)]{} $\mathcal P_H^\kappa$ is not empty.
[(ii)]{} The process $F $ satisfies the following integrability condition $$\phi^{2,\kappa}_H:=\underset{\mathbb P\in\mathcal P^\kappa_H}{\sup}\mathbb E^{\mathbb P}\left[\underset{0\leq t\leq T}{\esup}^{\mathbb P}\left(\mathbb E_t^{\mathcal P^\kappa_H,\mathbb P}\left[\int^T_0|\Fh^{\mathbb P,0}_s|^\kappa ds\right]\right)^{\frac{2}{\kappa}}\right]<+\infty$$
We recall the uniqueness result proved in [@kpz3].
\[uniqueref\] Let Assumptions \[assump.href\] and \[assump.h2ref\] hold. Assume $\xi \in \mathbb{L}^{2,\kappa}_H$ and that $(Y,Z,U)$ is a solution to the $2$BSDEJ . Then, for any $\mathbb{P}\in\mathcal{P}^\kappa_H$ and $0\leq t_1< t_2\leq T$, $$\begin{aligned}
\label{representationref}
Y_{t_1}&=\underset{\mathbb{P}^{'}\in\mathcal{P}^\kappa_H(t_1^+,\mathbb{P})}{\esup^\mathbb{P}}y_{t_1}^{\mathbb{P}^{'}}(t_2,Y_{t_2}), \text{ }\mathbb{P}-a.s.,\end{aligned}$$ where, for any $\mathbb{P}\in\mathcal{P}^\kappa_H$, $\mathbb{F}^+$-stopping time $\tau$, and $\mathcal{F}^+_{\tau} $-measurable r.v. $\xi\in\mathbb{L}^2({\mathbb P})$, we denote by $(y^{\mathbb{P}},z^{\mathbb{P}},u^\P):=(y^{\mathbb{P}}(\tau,\xi),z^{\mathbb{P}}(\tau,\xi),u^\P(\tau,\xi))$ the solution to the following standard BSDEJ on $0\leq t\leq \tau$ $$\label{bsdej}
y^{\mathbb{P}}_t=\xi + \int_t^{\tau}\widehat{F}^{\mathbb P}_s(y^{\mathbb{P}}_s,z^{\mathbb{P}}_s,u^{\mathbb{P}}_s)ds-\int_t^{\tau}z^{\mathbb{P}}_sdB^{\mathbb P,c}_s - \int_t^{\tau} \int_{E} u^{\mathbb{P}}_s(x) {\mu}^{\mathbb P}_B(dx,ds), \text{ } \mathbb P-a.s.$$
\[rem:bsde\] We first emphasize that existence and uniqueness results for the standard BSDEJs are not given directly by the existing literature, since the compensator of the counting measure associated to the jumps of $B$ is not deterministic. However, since all the probability measures we consider satisfy the martingale representation property and the Blumenthal $0-1$ law, it is clear that we can straightforwardly generalize the proof of existence and uniqueness of Tang and Li [@tangli] $($see also [@bech] and [@crep] for related results$)$. Furthermore, the usual *a priori* estimates and comparison theorems will also hold.
A direct existence argument {#section.3}
===========================
The aim of this section is to prove the following result, which is our first main theorem.
\[mainref\] Let $\xi\in\mathcal L^{2,\kappa}_H$. Under Assumptions \[assump.href\] and \[assump.h2ref\], there exists a unique solution $(Y,Z,U)\in\mathbb D^{2,\kappa}_H\times\mathbb H^{2,\kappa}_H\times\mathbb J^{2,\kappa}_H$ of the $2\rm{BSDEJ}$ .
In the article [@stz], the main tool to prove existence of a solution is the so-called regular conditional probability distributions of Stroock and Varadhan [@str]. Indeed, these tools allow to give a pathwise construction for conditional expectations. Since, at least when the generator is null, the $y$ component of the solution of a BSDE can be written as a conditional expectation, the r.c.p.d. allows us to construct solutions of BSDEs pathwise. Earlier in the paper, we have identified a candidate solution to the 2BSDEJ as an essential supremum of solutions of classical BSDEJs (see ). However, those BSDEJs are written under mutually singular probability measures. Hence, being able to construct them pathwise allows us to avoid the problems related to negligible-sets. In this section we will generalize the approach of [@stz] to the jump case.
Notations {#parag.notations}
---------
For the convenience of the reader, we recall below some of the notations introduced in [@stz]. Remember that we are working in the Skorohod space $\Omega=\mathbb{D}\left([0,T],\mathbb R^d\right)$ endowed with the Skorohod metric, denoted $d_S$, which makes it a complete and separable space.
$\bullet$ For $0\leq t\leq T$, we denote by $\Omega^t:=\left\{\omega\in \mathbb D\left([t,T],\mathbb R^d\right)\right\}$ the shifted canonical space of càdlàg paths on $[t,T]$ which are null at $t$, $B^t$ the shifted canonical process. $\mathbb F^t$ is the filtration generated by $B^t$. For any local martingale measure $\P$ on $(\Omega^t,\mathcal B(\Omega^t))$, we let $B^{t,\P,c}$, be the continuous local martingale part of $B^t$ and $B^{t,\P,d}$ its discontinuous martingale part. We again associate tot he jumps of $B^t$ a counting measure $\mu_{B^t}$, and we restrict ourselves to the set $\overline{P}^t_W$ of local martingale measures $\P$ such that
- The quadratic variation of $B^{t,\P,c}$ is absolutely continuous with respect to the Lebesgue measure $dt$ and its density takes values in $\mathbb S^{>0}_d$. Using Bichteler integration theory [@bich], we can once more define this density pathwise and we denote it by $\widehat a^t$
- The compensator $\lambda^{t,\mathbb P}_s(dx,ds)$ of the jump measure $\mu_{B^t}$ exists under $\mathbb P$ and can be decomposed as follows $$\lambda^{t,\P}_s(dx,ds)=\nu^{t,\P}_s(dx)ds,$$ for some $\F^t$-predictable random measure $\nu^{t,\P}$ on $E$.
We will denote by $\widetilde\mu_{B^t}^{t,\mathbb P}(dx,ds)$ the corresponding compensated measure. Let $\mathcal{N}^t$ be the set of $\mathbb F^t$-predictable random measures $\nu$ on $\mathcal{B}(E)$ satisfying $$\int_t^T \int_{E}(1\wedge {\left|x\right|}^2)\nu_s(\widetilde\omega,dx)ds <+\infty \text{ and } \int_t^T\int_{ {\left|x\right|}>1 } x \nu_s(\widetilde\omega,dx)ds <+\infty, \; \forall \ \widetilde\omega \in \Omega^t,$$
and let $\mathcal D^t$ be the set of $\mathbb F^t$-predictable processes $\alpha$ taking values in $\mathbb S_d^{>0}$ with $\int_t^T|\alpha_s(\widetilde\omega)|ds<+\infty$, for every $\widetilde\omega \in \Omega^t$. Exactly as in Section \[section.1\], we can define semimartingale problems and the corresponding probability measures. Define then $$\mathcal A^t_{\rm det}:=\left\{(I_d,F),\ F\in\mathcal N^t \text{ and $F$ is deterministic}\right\},$$ let $\widetilde{\Ac}^t_{\rm det}$ be the separable class of coefficients generated by $\Ac_{\rm det}$, let $\mathcal V^t$ designate the measures $F\in\mathcal N^t$ such that $(I_d,F)\in\widetilde{\mathcal A}^t_{\rm det}$ and denote, for any $F\in\mathcal V^t$, by $\mathbb P_{t,F}$ the unique solution to the martingale problem associated to the couple $(I_d,F)$. Next, we define exactly as in Section \[section.1\] a set $\mathcal R_F^t$ of $\F^t$-predictable functions $\beta:E\longmapsto \R$, and we define for each $F\in\mathcal V^t$ and for each $(\alpha,\beta)\in\mathcal D^t\times\mathcal R_F^t$ $$\mathbb P^{t,\alpha,\beta}_F:=\mathbb P_{t,F}\circ\left(X^{\alpha,\beta}_.\right)^{-1}.$$
Finally, we let $$\overline{\mathcal P}^t_S:=\underset{F\in\mathcal V^t}{\bigcup}\left\{\mathbb P^{t,\alpha,\beta}_F, \ (\alpha,\beta)\in\mathcal D^t\times\mathcal R_F^t\right\},$$ and we emphasize that this set enjoys the same properties as $\overline{\Pc}_S$. We next define important operations on the shifted spaces and their paths.
$\bullet$ For $0\leq s\leq t\leq T$ and $\omega\in \Omega^s$, we define the shifted path $\omega^t\in \Omega^t$ by $$\omega^t_r:=\omega_r-\omega_t,\text{ }\forall r\in [t,T].$$
$\bullet$ For $0\leq s\leq t\leq T$ and $\omega\in \Omega^s$, $\widetilde \omega\in\Omega^t$ we define the concatenation path $\omega\otimes_t\widetilde \omega\in\Omega^s$ by $$(\omega\otimes_t\widetilde \omega)(r):=\omega_r1_{[s,t)}(r)+(\omega_{t}+\widetilde\omega_r)1_{[t,T]}(r),\text{ }\forall r\in[s,T].$$
$\bullet$ For $0\leq s\leq t\leq T$ and a $\mathcal F^s_T$-measurable random variable $\xi$ on $\Omega^s$, for each $\omega \in\Omega^s$, we define the shifted $\mathcal F^t_T$-measurable random variable $\xi^{t,\omega}$ on $\Omega^t$ by $$\xi^{t,\omega}(\widetilde\omega):=\xi(\omega\otimes_t\widetilde \omega),\text{ }\forall \ \widetilde\omega\in\Omega^t.$$ Similarly, for an $\mathbb F^s$-progressively measurable process $X$ on $[s,T]$ and $(t,\omega)\in[s,T]\times\Omega^s$, we can define the shifted process $\left\{X_r^{t,\omega},r\in[t,T]\right\}$, which is $\mathbb F^t$-progressively measurable.
$\bullet$ For a $\mathbb F$-stopping time $\tau$, we use the same simplification as [@stz] $$\omega\otimes_\tau\widetilde \omega:=\omega\otimes_{\tau(\omega)}\widetilde \omega,\text{ }\xi^{\tau,\omega}:=\xi^{\tau(\omega),\omega},\text{ }X^{\tau,\omega}:=X^{\tau(\omega),\omega}.$$
$\bullet$ We define the “shifted” generator by $$\widehat F^{t,\omega,{\mathbb P}}_s(\widetilde\omega,y,z,u):=F_s(\omega\otimes_t\widetilde\omega,y,z,u,\widehat a^t_s(\widetilde\omega), \nu{t,{\mathbb P}}_s(\widetilde\omega)), \text{ }\forall (s,\widetilde\omega)\in[t,T]\times\Omega^t.$$ Then note that since $F$ is assumed to be uniformly continuous in $\omega$ for the Skorohod topology, then so is $\widehat F^{t,\omega}$. Notice that this implies that for any $\mathbb P\in\overline{\mathcal P}^t_{S}$ $$\mathbb E^\mathbb P\left[\left(\int_t^T{\left|\widehat F^{t,\omega,{\mathbb P}}_s(0,0,0)\right|}^{\kappa}ds\right)^{\frac2\kappa}\right]<+\infty,$$ for some $\omega$ if and only if it holds for all $\omega\in\Omega$.
$\bullet$ We also extend Definition \[def\] in the shifted spaces
\[set\_proba\_shift\] $\mathcal P^{t,\kappa}_H$ consists of all $\mathbb P:=\mathbb P^{t,\alpha,\beta}_F\in\overline{\mathcal P}^t_{S}$ such that
- $\underline{a}^\mathbb P\leq\widehat a^t_s\leq\overline{a}^\mathbb P, \ ds\times d\mathbb P-a.s.\text{ for some }\underline{a}^\mathbb P,\overline{a}^\mathbb P\in\mathbb S^{>0}_d$ and $\displaystyle\mathbb E^\mathbb P\left[\int_t^T\int_E {\left|x\right|}^2{\nu}^{t,\mathbb P}_s(dx)ds\right]<+\infty.$
- The following integrability condition holds $$\mathbb E^\mathbb P\left[\left(\int_t^T{\left|\widehat F^{t,\omega,{\mathbb P}}_s(0,0,0)\right|}^{\kappa}ds\right)^{\frac2\kappa}\right]<+\infty,\text{ for all $\omega\in\Omega$}.$$
$\bullet$ Finally, we define the so-called regular conditional probability distributions (r.c.p.d. in the sequel). For given $\omega\in \Omega$, $\mathbb F$-stopping time $\tau$ and $\mathbb P\in\mathcal P^\kappa_H$, the r.c.p.d. of $\mathbb P$ is a probability measure $\mathbb P^\omega_\tau$ on $\mathcal F_T$ such that for every bounded $\mathcal F_T$-measurable random variable $\xi$ $$\mathbb E^\mathbb P_\tau\left[\xi\right](\omega)=\mathbb E^{\mathbb P^\omega_\tau}[\xi],\text{ for $\mathbb P$-a.e. $\omega$.}$$
Besides, $\mathbb P^\omega_\tau$ naturally induces a probability measure $\mathbb P^{\tau,\omega}$ on $\mathcal F_T^{\tau(\omega)}$ such that the $\mathbb P^{\tau,\omega}$-distribution of $B^{\tau(\omega)}$ is equal to the $\mathbb P^\omega_\tau$-distribution of $\left\{B_t-B_{\tau(\omega)},\ t\in[\tau(\omega),T]\right\}.$ Besides, we have $$\mathbb E^{\mathbb P^\omega_\tau}[\xi]=\mathbb E^{\mathbb P^{\tau,\omega}}[\xi^{\tau,\omega}].$$
We emphasize that the above notations correspond to the ones used in [@stz] when we consider the subset of $\Omega$ consisting of all continuous paths from $[0,T]$ to $\mathbb R^d$ whose value at time $0$ is $0$.
We now prove that there exists a relation between $(\widehat a^{t,\omega},\left(\nu^{\mathbb P}\right)^{t,\omega})$ and $(\widehat a^t,\nu^{t,\mathbb P^{t,\omega}})$.
\[relationhata\] Let $\mathbb P\in\mathcal P^\kappa_H$ and $\tau$ be an $\mathbb F$-stopping time. Then, for $\mathbb P$-a.e. $\omega\in\Omega$, we have for $ds\times d\mathbb P^{\tau,\omega}$-a.e. $(s,\widetilde\omega)\in[\tau(\omega),T]\times\Omega^{\tau(\omega)}$ $$\begin{aligned}
&\widehat a_s^{\tau,\omega}(\widetilde\omega)=\widehat a_s^{\tau(\omega)}(\widetilde\omega)\text{, and } (\nu_ s^{{\mathbb P}})^{\tau,\omega}(\widetilde\omega,A)=\nu_s^{\tau(\omega),\mathbb P^{\tau,\omega}}(\widetilde\omega,A)\text{ for every $A\in\mathcal B(E)$.}
$$
This result is important for us, because it implies that for $\mathbb P$-a.e. $\omega\in\Omega$ and for $ds\times d\mathbb P^{t,\omega}-a.e. \ (s,\widetilde\omega)\in[t,T]\times\Omega^t$ $$F_s\left(\omega\otimes_t\widetilde\omega,y,z,u,\widehat a_s(\omega\otimes_t\widetilde\omega),\nu^{\mathbb P}_s(\omega\otimes_t\widetilde\omega)\right)=F_s\left(\omega\otimes_t\widetilde\omega,y,z,u,\widehat a_s^t(\widetilde\omega),\nu^{t,\mathbb P^{t,\omega}}_s(\widetilde\omega)\right),$$ which justifies the choice we made for the “shifted” generator.
The proof of the equality for $\widehat a$ is the same as in Lemma $4.1$ of [@stz2], so we omit it. Now, for $s\geq \tau$ and for any $A\in\mathcal B(E)$, we know by the Doob-Meyer decomposition that there exist a $\mathbb P$-local martingale $M$ and a $\mathbb P^{\tau,\omega}$-martingale $N$ such that $$\begin{aligned}
\mu_B([0,s],A)&=M_s+\int_0^s\nu^\mathbb P_r(A)dr,\ \mathbb P-a.s.\\
\mu_{B^{\tau(\omega)}}([\tau(\omega),s],A)&=N_s+\int_{\tau}^s\nu^{\tau(\omega),\mathbb P^{\tau,\omega}}_r(A)dr,\ \mathbb P^{\tau,\omega}-a.s.\end{aligned}$$
Then, we can rewrite the first equation above for $\mathbb P$-a.e. $\omega\in\Omega$ and for $\mathbb P^{\tau,\omega}$-a.e. $\widetilde\omega\in\Omega^{\tau(\omega)}$ $$\mu_B(\omega\otimes_\tau\widetilde\omega,[0,s],A)=M_s^{\tau,\omega}(\widetilde\omega)+\int_0^s\nu_r^{\mathbb P,\tau,\omega}(\widetilde\omega,A)dr.
\label{eq:trilili}$$
Now, by definition of the measures $\mu_B$ and $\mu_{B^{\tau(\omega)}}$, we have $$\mu_B(\omega\otimes_\tau\widetilde\omega,[0,s],A)=\mu_B(\omega,[0,\tau],A)+\mu_{B^{\tau(\omega)}}(\widetilde\omega,[\tau,s],A).$$
Hence, we obtain from that for $\mathbb P$-a.e. $\omega\in\Omega$ and for $\mathbb P^{\tau,\omega}$-a.e. $\widetilde\omega\in\Omega^{\tau(\omega)}$ $$\begin{aligned}
\mu_B(\omega,[0,\tau],A)-\int_0^\tau\nu^\mathbb P_r(\omega,A)dr+N_s(\widetilde\omega)-M_s^{\tau,\omega}(\widetilde\omega)=\int_\tau^s\left(\nu_r^{\mathbb P,\tau,\omega}(\widetilde\omega,A)-\nu^{\tau(\omega),\mathbb P^{\tau,\omega}}_r(\widetilde\omega,A)\right)dr\end{aligned}$$
On the left-hand side above, the terms which are $\mathcal F_\tau$-measurable are constants in $\Omega^{\tau(\omega)}$ and using the same arguments as in Step $1$ of the proof of Lemma \[lemme.technique\], we can show that $M^{\tau,\omega}$ is a $\mathbb P^{\tau,\omega}$-local martingale for $\mathbb P$-a.e. $\omega\in\Omega$. This means that the left-hand side is a $\mathbb P^{\tau,\omega}$-local martingale while the right-hand side is a predictable finite variation process. By the martingale representation property which still holds in the shifted canonical spaces, we deduce that for $\mathbb P$-a.e. $\omega\in\Omega$ and for $ds\times d\mathbb P^{\tau,\omega}$-a.e. $(s,\widetilde\omega)\in[\tau(\omega),T]\times\Omega^{\tau(\omega)}$ $$\int_\tau^s\left(\nu_r^{\mathbb P,\tau,\omega}(\widetilde\omega,A)-\nu^{\tau(\omega),\mathbb P^{\tau,\omega}}_r(\widetilde\omega,A)\right)dr=0,$$ which is the desired result.
Existence when $\xi$ is in $\rm{UC_b}(\Omega)$ {#sec.existtt}
-----------------------------------------------
When $\xi$ is in $\rm{UC_b}(\Omega)$, we know that there exists a modulus of continuity function $\rho$ for $\xi$ and $F$ in $\omega$. Then, for any $0\leq t \leq s \leq T,\ (y,z,u)\in \mathbb R \times \mathbb{R}^d\times\hat L^2$ and $\omega,\omega'\in \Omega,\ \widetilde{\omega}\in\Omega^t,\ {\mathbb P\in\mathcal P^{t,\kappa}_H}$, $$\begin{aligned}
\left|\xi^{t,\omega}\left(\widetilde{\omega}\right)-\xi^{t,\omega'}\left(\widetilde{\omega}\right)\right| \leq \rho\left(d_{S,t}(\omega,\omega')\right) \text{, } \left|\widehat{F}_s^{t,\omega,\mathbb P}\left(\widetilde{\omega},y,z,u\right)-\widehat{F}_s^{t,\omega',\mathbb P}\left(\widetilde{\omega},y,z,u\right)\right| \leq \rho\left(d_{S,t}(\omega,\omega')\right),\end{aligned}$$ where $d_{S,t}(\omega,\omega'):=d_S(\omega_{.\wedge t},\omega'_{.\wedge t}),$ $d_S$ being the Skorohod distance. We then define for all $\omega\in\Omega$ $$\Lambda\left(\omega\right):=\underset{0\leq s\leq t}{\sup}\Lambda_t\left(\omega\right):=\underset{0\leq s\leq t}{\sup}\ \underset{\mathbb P\in\mathcal P^{t,\kappa}_H}{\sup}\left(\mathbb E^\mathbb P\left[{\left|\xi^{t,\omega}\right|}^2+\int_t^T|\widehat F^{t,\omega,\mathbb P}_s(0,0,0)|^2ds\right]\right)^{1/2}.$$
Now since $\widehat F^{t,\omega,\mathbb P}$ is also uniformly continuous in $\omega$ for the Skorohod topology, it is easily verified that $\Lambda\left(\omega\right)<\infty \text{ for some } \omega\in\Omega \text{ iff it holds for all } \omega\in\Omega.$ Moreover, when $\Lambda$ is finite, it is uniformly continuous in $\omega$ for the Skorohod topology and is therefore $\mathcal F_T$-measurable. Now, by Assumption \[assump.h2ref\], we have $\Lambda_t\left(\omega\right)<\infty \text{ for all } \left(t,\omega\right)\in\left[0,T\right]\times\Omega.$ To prove existence, we define the following value process $V_t$ pathwise $$\label{sol}
V_t(\omega):=\underset{\mathbb P\in\mathcal P^{t,\kappa}_H}{\sup}\mathcal Y^{\mathbb P,t,\omega}_t\left(T,\xi\right), \text{ for all } \left(t,\omega\right)\in\left[0,T\right]\times\Omega,$$ where, for any $\left(t_1,\omega\right)\in\left[0,T\right]\times\Omega,\ \mathbb P\in\mathcal P^{t_1,\kappa}_H, t_2\in\left[t_1,T\right]$, and any $\mathcal F_{t_2}$-measurable $\eta\in\mathbb L^{2}\left(\mathbb P\right) $, we denote $\mathcal Y^{\mathbb P,t_1,\omega}_{t_1}\left(t_2,\eta\right):= y^{\mathbb P,t_1,\omega}_{t_1}$, where $\left(y^{\mathbb P,t_1,\omega},z^{\mathbb P,t_1,\omega},u^{\mathbb P,t_1,\omega}\right) $ is the solution of the following BSDEJ on the shifted space $\Omega^{t_1} $ under $\mathbb P$ $$\begin{aligned}
\label{eq.bsdeeeeref}
\nonumber y^{\mathbb P,t_1,\omega}_{s}&=\eta^{t_1,\omega}+\int^{t_2}_{s}\widehat{F}^{t_1,\omega,\mathbb P}_{r}\left(y^{\mathbb P,t_1,\omega}_{r},z^{\mathbb P,t_1,\omega}_{r},u_r^{\mathbb P,t_1,\omega}\right)dr-\int^{t_2}_{s}z^{\mathbb P,t_1,\omega}_{r}dB^{t_1,\mathbb P,c}_{r}\\
&\hspace{0.9em}-\int_s^{t_2}\int_{E}u_r^{\mathbb P,t_1,\omega}(x)\widetilde\mu^\mathbb P_{B^{t_1}}(dx,dr), \ \mathbb P-a.s.,\ s\in[t_1,t_2],\end{aligned}$$ where as usual $\widetilde\mu^\mathbb P_{B^{t_1}}(dx,ds):=\mu_{B^{t_1}}(dx,ds)-\nu_s^{t_1,\mathbb P}(dx)ds$. In view of the Blumenthal $0-1$ law, $y^{\mathbb P,t,\omega}_{t}$ is constant for any given $\left(t,\omega\right)$ and $\mathbb P\in\mathcal P^{t,\kappa}_H$, and therefore the value process $V$ is well defined. Let us now show that $V$ inherits some properties from $\xi$ and $F$.
\[unifcont\] Let Assumptions \[assump.href\] and \[assump.h2ref\] hold and consider some $\xi$ in $\rm{UC_b}(\Omega)$. Then for all $\left(t,\omega\right)\in\left[0,T\right]\times\Omega$ we have $\left|V_t\left(\omega\right)\right|\leq C\Lambda_t\left(\omega\right) $. Moreover, for all $\left(t,\omega,\omega'\right)\in\left[0,T\right]\times\Omega^2$, $\left|V_t\left(\omega\right)-V_t\left(\omega'\right)\right|\leq C\rho\left(d_{S,t}(\omega,\omega')\right) $. Consequently, $V_t$ is $\mathcal F_t$-measurable for every $t\in\left[0,T\right]$.
$\rm{(i)}$ For each $\left(t,\omega\right)\in\left[0,T\right]\times\Omega $ and $\mathbb P\in\mathcal P^{t,\kappa}_H $, let $\alpha$ be some positive constant which will be fixed later and let $\eta\in(0,1)$. Since $F$ is uniformly Lipschitz in $(y,z)$ and satisfies Assumption \[assump.href\]$\rm{(iv)}$, we have $${\left|\widehat F_s^{t,\omega,\mathbb P}(y,z,u)\right|}\leq {\left|\widehat F_s^{t,\omega,\mathbb P}(0,0,0)\right|}+C\left({\left|y\right|}+|\left(\widehat a^t_s\right)^{1/2}z|+\left(\int_{E}{\left|u(x)\right|}^2\nu^{t,\mathbb P}_s(dx)\right)^{1/2}\right).$$
Now apply Itô’s formula. We obtain $$\begin{aligned}
&e^{\alpha t}{\left|y_t^{\mathbb P,t,\omega}\right|}^2+\int_t^Te^{\alpha s}{\left|(\widehat a^t_s)^{1/2}z_s^{\mathbb P,t,\omega}\right|}^2ds+\int_t^T\int_{E}e^{\alpha s}{\left|u_s^{\mathbb P,t,\omega}(x)\right|}^2\nu^{t,\mathbb P}_s(dx)ds\\
&= e^{\alpha T}{\left|\xi^{t,\omega}\right|}^2+2\int_t^Te^{\alpha s}y_s^{\mathbb P,t,\omega}\widehat F_s^{t,\omega,\mathbb P}(y_s^{\mathbb P,t,\omega},z_s^{\mathbb P,t,\omega},u_s^{\mathbb P,t,\omega})ds\\
&\hspace{0.9em}-\alpha\int_t^Te^{\alpha s}{\left|y_s^{\mathbb P,t,\omega}\right|}^2ds-2\int_t^Te^{\alpha s}y^{\mathbb P,t,\omega}_{s^-}z_s^{\mathbb P,t,\omega}dB^{t,\mathbb P,c}_s\\
&\hspace{0.9em}-\int_t^T\int_{E}e^{\alpha s}\left(2y^{\mathbb P,t,\omega}_{s^-}u_s^{\mathbb P,t,\omega}(x)+{\left|u_s^{\mathbb P,t,\omega}(x)\right|}^2\right)\widetilde\mu^{\mathbb P}_{B^{t}}(dx,ds)\\
&\leq e^{\alpha T}{\left|\xi^{t,\omega}\right|}^2+\int_t^Te^{\alpha s}{\left|\widehat F_s^{t,\omega,\mathbb P}(0,0,0)\right|}^2ds+\left(1+2C+\frac{2C^2}{\eta}-\alpha\right)\int_t^Te^{\alpha s}{\left|y_s^{\mathbb P,t,\omega}\right|}^2ds\\
&\hspace{0.9em}+\eta\int_t^Te^{\alpha s}{\left|(\widehat a^t_s)^{1/2}z_s^{\mathbb P,t,\omega}\right|}^2ds+\eta\int_t^T\int_{E}e^{\alpha s}{\left|u_s^{\mathbb P,t,\omega}(x)\right|}^2\nu^{t,\mathbb P}_s(dx)ds\\
&\hspace{0.9em}-2\int_t^Te^{\alpha s}y^{\mathbb P,t,\omega}_{s^-}z_s^{\mathbb P,t,\omega}dB^{t,\mathbb P,c}_s-\int_t^T\int_{E}e^{\alpha s}\left(2y^{\mathbb P,t,\omega}_{s^-}u_s^{\mathbb P,t,\omega}(x)+{\left|u_s^{\mathbb P,t,\omega}(x)\right|}^2\right)\widetilde\mu^{\mathbb P}_{B^{t}}(dx,ds).\end{aligned}$$
Now choose $\eta=1/2$ and $\alpha$ large enough. By taking expectation we obtain easily ${\left|y_t^{\mathbb P,t,\omega}\right|}^2\leq C{\left|\Lambda_t(\omega)\right|}^2.$ The result then follows from the arbitrariness of $\mathbb P$.
$\rm{(ii)}$ The proof is exactly the same as above, except that one has to use uniform continuity in $\omega$ of $\xi^{t,\omega}$ and $ F^{t,\omega}$. Indeed, for each $\left(t,\omega\right)\in\left[0,T\right]\times\Omega $ and $\mathbb P\in\mathcal P^{t,\kappa}_H $, let $\alpha$ be some positive constant which will be fixed later and let $\eta\in(0,1)$. By Itô’s formula we have, since $ F$ is uniformly Lipschitz $$\begin{aligned}
&e^{\alpha t}{\left|y_t^{\mathbb P,t,\omega}-y_t^{\mathbb P,t,\omega'}\right|}^2+\int_t^T\scriptstyle e^{\alpha s}\left({\left|(\widehat a^t_s)^{1/2}(z_s^{\mathbb P,t,\omega}-z_s^{\mathbb P,t,\omega'})\right|}^2+\int_{E}\scriptstyle e^{\alpha s}(u_s^{\mathbb P,t,\omega}-u_s^{\mathbb P,t,\omega'})^2(x)\nu^{t,\mathbb P}_s(dx)\right)ds\\
&\leq e^{\alpha T}{\left|\xi^{t,\omega}-\xi^{t,\omega'}\right|}^2+2C\int_t^Te^{\alpha s}{\left|y_s^{\mathbb P,t,\omega}-y_s^{\mathbb P,t,\omega'}\right|}^2ds\\
&\hspace{0.9em}+2C\int_t^T{\left|y_s^{\mathbb P,t,\omega}-y_s^{\mathbb P,t,\omega'}\right|}{\left|(\widehat a_s^t)^{1/2}(z_s^{\mathbb P,t,\omega}-z_s^{\mathbb P,t,\omega'})\right|}ds\\
&\hspace{0.9em}+2C\int_t^Te^{\alpha s}{\left|y_s^{\mathbb P,t,\omega}-y_s^{\mathbb P,t,\omega'}\right|}\left(\int_{E}{\left|u_s^{\mathbb P,t,\omega}(x)-u_s^{\mathbb P,t,\omega'}(x)\right|}^2\nu^{t,\mathbb P}_s(dx)\right)^{1/2}ds\\
&\hspace{0.9em}+2C\int_t^Te^{\alpha s}{\left|y_s^{\mathbb P,t,\omega}-y_s^{\mathbb P,t,\omega'}\right|}{\left|\widehat F^{t,\omega,\mathbb P}_s(y_s^{\mathbb P,t,\omega},z_s^{\mathbb P,t,\omega},u_s^{\mathbb P,t,\omega})-\widehat F^{t,\omega',\mathbb P}_s(y_s^{\mathbb P,t,\omega},z_s^{\mathbb P,t,\omega},u_s^{\mathbb P,t,\omega})\right|}ds\\
&\hspace{0.9em}-\alpha\int_t^Te^{\alpha s}{\left|y_s^{\mathbb P,t,\omega}-y_s^{\mathbb P,t,\omega'}\right|}^2ds-2\int_t^Te^{\alpha s}(y^{\mathbb P,t,\omega}_{s^-}-y_{s^-}^{\mathbb P,t,\omega'})(z_s^{\mathbb P,t,\omega}-z_s^{\mathbb P,t,\omega'})dB^{t,\mathbb P,c}_s\\
&\hspace{0.9em}-\int_t^T\int_{E}e^{\alpha s}\left(2(y^{\mathbb P,t,\omega}_{s^-}-y^{\mathbb P,t,\omega'}_{s^-})(u_s^{\mathbb P,t,\omega}-u_s^{\mathbb P,t,\omega'})+(u_s^{\mathbb P,t,\omega}-u_s^{\mathbb P,t,\omega'})^2\right)(x)\widetilde\mu^{\mathbb P}_{B^{t}}(dx,ds).\end{aligned}$$
We then deduce $$\begin{aligned}
&e^{\alpha t}{\left|y_t^{\mathbb P,t,\omega}-y_t^{\mathbb P,t,\omega'}\right|}^2+\int_t^T\scriptstyle e^{\alpha s}\left({\left|(\widehat a^t_s)^{\1/2}(z_s^{\mathbb P,t,\omega}-z_s^{\mathbb P,t,\omega'})\right|}^2+\int_{E}\scriptstyle e^{\alpha s}(u_s^{\mathbb P,t,\omega}-u_s^{\mathbb P,t,\omega'})^2(x)\nu^{t,\mathbb P}_s(dx)\right)ds\\
&\leq e^{\alpha T}{\left|\xi^{t,\omega}-\xi^{t,\omega'}\right|}^2+\int_t^Te^{\alpha s}{\left|\widehat F^{t,\omega,\mathbb P}_s(y_s^{\mathbb P,t,\omega},z_s^{\mathbb P,t,\omega},u_s^{\mathbb P,t,\omega})-\widehat F^{t,\omega',\mathbb P}_s(y_s^{\mathbb P,t,\omega},z_s^{\mathbb P,t,\omega},u_s^{\mathbb P,t,\omega})\right|}^2ds\\[0.3em]
&\hspace{0.9em}+\eta\int_t^Te^{\alpha s}{\left|(\widehat a^t_s)^{1/2}(z_s^{\mathbb P,t,\omega}-z_s^{\mathbb P,t,\omega'})\right|}^2ds+\eta\int_t^T\int_{E}e^{\alpha s}{\left|u_s^{\mathbb P,t,\omega}(x)-u_s^{\mathbb P,t,\omega'}(x)\right|}^2\nu^{t,\mathbb P}_s(dx)ds\\
&\hspace{0.9em}+\left(2C+C^2+\frac{2C^2}{\eta}-\alpha\right)\int_t^Te^{\alpha s}{\left|y_s^{\mathbb P,t,\omega}-y_s^{\mathbb P,t,\omega'}\right|}^2ds\\
&\hspace{0.9em}-2\int_t^Te^{\alpha s}(y^{\mathbb P,t,\omega}_{s^-}-y_{s^-}^{\mathbb P,t,\omega'})(z_s^{\mathbb P,t,\omega}-z_s^{\mathbb P,t,\omega'})dB^{t,{\mathbb P},c}_s\\
&\hspace{0.9em}-\int_t^T\int_{E}e^{\alpha s}\left(2(y^{\mathbb P,t,\omega}_{s^-}-y^{\mathbb P,t,\omega'}_{s^-})(u_s^{\mathbb P,t,\omega}-u_s^{\mathbb P,t,\omega'})+(u_s^{\mathbb P,t,\omega}-u_s^{\mathbb P,t,\omega'})^2\right)(x)\widetilde\mu^{\mathbb P}_{B^{t}}(dx,ds).\end{aligned}$$
Now choose $\eta=1/2$ and $\alpha$ such that $\iota:=\alpha -2C-C^2-\frac{2C^2}{\eta}\geq 0$. We obtain the desired result by taking expectation and using the uniform continuity in $\omega$ of $\xi$ and $F$.
The next proposition is a dynamic programming property verified by the value process, which will be crucial when proving that $V$ provides a solution to the 2BSDEJ with generator $F$ and terminal condition $\xi$. The result and its proof are intimately connected to Proposition $4.7$ in [@stz2] and use the same type of arguments.
\[progdyn\] Under Assumptions \[assump.href\], \[assump.h2ref\] and for $\xi\in \rm{UC_b}(\Omega)$, we have for all $0\leq t_1<t_2\leq T$ and for all $\omega \in \Omega$ $$V_{t_1}(\omega)=\underset{\mathbb P\in \mathcal P^{t_1,\kappa}_H}{\sup}\mathcal Y_{t_1}^{\mathbb P,t_1,\omega}(t_2,V_{t_2}^{t_1,\omega}).$$
Let us emphasize here that all the regularity in $\omega$ we assumed so far is because it is a sufficient condition in order to obtain the measurability and the dynamic programming property for . It is however clear that such assumptions are too restrictive and we hope to be able to weaken them in a future work. In fact, in a recent paper, Nutz and van Handel [@nvh] showed the required regularity and dynamic programming for conditional non-linear expectations $($corresponding roughly to 2BSDEs with a generator equal to $0)$ for terminal conditions which were only upper semi-analytic. An extension of their result to our framework would allow us to get rid off our regularity assumptions, and therefore off the limitations induced by the continuity with respect to the Skorohod topology. Indeed, even for fairly regular functions $f$, the random variable $f(B_t)$ is continuous for the Skorohod distance only for almost every $t\in[0,T]$[^4]
The proof is almost the same as the proof in [@stz2], with minor modifications due to the introduction of jumps. We therefore relegate it to the appendix.
Now we are facing the problem of the regularity in $t$ of $V$. Indeed, if we want to obtain a solution to the 2BSDEJ, then it has to be at least càdlàg, $\mathcal P^\kappa_H-q.s.$ To this end, we define now for all $(t,\omega)$, the $\mathbb F^+$-progressively measurable process $$V_t^+:=\underset{r\in\mathbb Q\cap(t,T],r\downarrow t}{\overline \lim}V_r.$$
\[lem.cadlag\] Under the conditions of the previous Proposition, we have $$V_t^+=\underset{r\in\mathbb Q\cap(t,T],r\downarrow t}{\lim}V_r,\text{ }\mathcal P^\kappa_H-q.s.$$ and thus $V^+$ is càdlàg, $\mathcal P^\kappa_H-q.s.$
The proof is relegated to the appendix.
We follow now Remark $4.9$ in [@stz2], and for a fixed $\mathbb P\in\mathcal P^\kappa_H$, we introduce the following reflected BSDE with jumps (RBSDEJ for short) and with lower obstacle $V^+$ under $\mathbb P$ $$\begin{aligned}
&\widetilde{Y}_t^{\mathbb P}=\xi+\int_t^T\widehat{F}^\mathbb P_s(\widetilde{Y}_s^{\mathbb P},\widetilde{Z}_s^{\mathbb P},\widetilde{U}_s^{\mathbb P},\nu)ds-\int_t^T\widetilde{Z}_s^{\mathbb P}dB_s^{\mathbb P,c}-\int_t^T\int_{E}\widetilde{U}_s^{\mathbb P}(x)\widetilde{\mu}^{\mathbb P}_B(dx,ds)+\widetilde{K}^{\mathbb P}_T-\widetilde{K}^{\mathbb P}_t\\
&\widetilde{Y}_t^{\mathbb P}\geq V^+_t,\ 0\leq t\leq T,\ \mathbb P-a.s.\\
&\int_0^T\left(\widetilde{Y}_{s^-}^{\mathbb P}-V^{+}_{s^{-}}\right)d\widetilde{K}^{\mathbb P}_s=0,\text{ } \mathbb P-a.s.,\end{aligned}$$ where we emphasize that the process $\widetilde{K}^\mathbb P$ is predictable.
Existence and uniqueness of the above RBSDEJ under our Assumptions, with the restrictions that the compensator is not random, have been proved by Hamadène and Ouknine [@hamaou] or Essaky [@ess]. However, their proofs can be easily generalized to our context.
Let us now argue by contradiction and suppose that $\widetilde Y^\mathbb P$ is not equal $\mathbb P-a.s.$ to $V^+$. Then we can assume without loss of generality that $\widetilde Y^\mathbb P_0>V^+_0$, $\mathbb P-a.s.$. Fix now some $\eps>0$ and define the following stopping time $$\tau^\eps:=\inf\left\{t\geq 0,\ \widetilde Y^\mathbb P_t\leq V^+_t+\eps\right\}.$$
Then $\widetilde Y^\mathbb P$ is strictly above the obstacle before $\tau^\eps$, and therefore $\widetilde K^\mathbb P$ is identically equal to $0$ in $[0,\tau^\eps]$. Hence, we have $$\widetilde{Y}_t^{\mathbb P}=\widetilde Y^\mathbb P_{\tau^\eps}+\int_t^{\tau^\eps}\widehat{F}^{\mathbb P}_s(\widetilde{Y}_s^{\mathbb P},\widetilde{Z}_s^{\mathbb P},\widetilde{U}_s^{\mathbb P})ds-\int_t^{\tau^\eps}\widetilde{Z}_s^{\mathbb P}dB_s^{\mathbb P,c}-\int_t^{\tau^\eps}\int_{E}\widetilde{U}_s^{\mathbb P}(x)\widetilde{\mu}^{\mathbb P}_B(dx,ds),\ \mathbb P-a.s.$$
Let us now define the following BSDEJ on $[0,\tau^\eps]$ $$y^{+,\mathbb P}_t=V^+_{\tau^\eps}+\int_t^{\tau^\eps}\widehat{F}^{\mathbb P}_s(y_s^{+,\mathbb P},z_s^{+,\mathbb P},u_s^{+,\mathbb P})ds-\int_t^{\tau^\eps}z_s^{+,\mathbb P}dB_s^{\mathbb P,c}-\int_t^{\tau^\eps}\int_{E}u_s^{+,\mathbb P}(x)\widetilde{\mu}^{\mathbb P}_B(dx,ds),\ \mathbb P-a.s.$$
By the standard *a priori* estimates already used in this paper, we obtain that $$\widetilde Y_0^\mathbb P\leq y^{+,\mathbb P}_0+C{\left|V^+_{\tau^\eps}-\widetilde Y^\mathbb P_{\tau^\eps}\right|}\leq y^{+,\mathbb P}_0+C\eps,$$ by definition of $\tau^\eps$. Following the arguments in Step $1$ of the proof of Theorem $4.5$ in [@stz2], we can show that $y^{+,\mathbb P}_0\leq V^+_0$ which in turn implies $\widetilde Y_0^\mathbb P\leq V^+_0+C\eps,$ hence a contradiction by arbitrariness of $\eps$. Therefore, we have obtained the following decomposition $$V_t^+=\xi+\int_t^T\widehat{F}^{\mathbb P}_s(V_s^+,\widetilde{Z}_s^{\mathbb P},\widetilde{U}_s^{\mathbb P})ds-\int_t^T\widetilde{Z}_s^{\mathbb P}dB_s^{\mathbb P,c}-\int_t^T\int_{E}\widetilde{U}_s^{\mathbb P}(x)\widetilde{\mu}^{\mathbb P}_B(dx,ds)+\widetilde{K}^{\mathbb P}_T-\widetilde{K}^{\mathbb P}_t,\ \mathbb P-a.s.$$
Finally, since $V^+$ and $B$ are càdlàg, we can use the result of Karandikar [@kar] to give a pathwise definition of $[B,V^+]$. We then have $$d[Y,B]^c_t=d\langle Y^{\mathbb P,c},B^{\mathbb P,c}\rangle_t=\widetilde Z^\P_td\langle B^{\mathbb P,c},B^{\mathbb P,c}\rangle_t=\widetilde Z_t^\P d[B,B]^c_t=\widehat a_t\widetilde Z_t^\P dt, \text{ } \mathbb P-a.s., \ \forall{\mathbb P}\in\mathcal{P}^\kappa_H,$$ $$\Delta[Y,B]_t=\widetilde U_t^\P(\Delta B_t)\Delta B_t,\text{ } \mathcal{P}^\kappa_H-q.s.,$$ so that we can define aggregators $Z$ and $U$ for the the families $\{\widetilde Z^\mathbb P,\ \mathbb P\in\mathcal P^\kappa_H\},$ and $\{\widetilde U^\mathbb P,\ \mathbb P\in\mathcal P^\kappa_H\}.$
We next prove the representation for $V$ and $V^+$, and that, as shown in Proposition $4.11$ of [@stz2], we actually have $V=V^+$, $\mathcal P^{\kappa}_H-q.s.$, which shows that in the case of a terminal condition in ${\rm UC_b}(\Omega)$, the solution of the $2$BSDEJ is actually $\mathbb F$-progressively measurable.
\[prop.repref\] Assume that $\xi\in {\rm UC_b}(\Omega)$ and that Assumptions \[assump.href\] and \[assump.h2ref\] hold. Then we have $$V_t=\underset{\mathbb P^{'}\in\mathcal P_H(t,\mathbb P)}{\esup^\mathbb P}\mathcal Y_t^{\mathbb P^{'}}(T,\xi)\text{ and } V_t^+=\underset{\mathbb P^{'}\in\mathcal P_H^\kappa(t^+,\mathbb P)}{\esup^\mathbb P}\mathcal Y_t^{\mathbb P^{'}}(T,\xi), \text{ }\mathbb P-a.s., \text{ }\forall \mathbb P\in\mathcal P_H^\kappa.$$
Besides, we also have for all $t$, $V_t=V_t^+, \text{ }\mathcal P_H^\kappa-q.s.$
The proof for the representations is the same as the proof of proposition $4.10$ in [@stz2], since we also have a stability result for BSDEJs under our assumptions. For the equality between $V$ and $V^+$, we also refer to the proof of Proposition $4.11$ in [@stz2].
Therefore, in the sequel we will use $V$ instead of $V^+$. Finally, we have to check that the minimum condition holds. Fix $\mathbb P$ in $\mathcal P^\kappa_H$ and $\mathbb P^{'}\in\mathcal P^\kappa_H(t^+,\mathbb P)$. Then, proceeding exactly as in Step $2$ of the proof of Theorem $4.1$ in [@kpz3], but introducing the process $\gamma'$ of Assumption \[assump.href\]$\rm{(iv)}$ instead of $\gamma$, we can similarly obtain $$\begin{aligned}
\label{eq:M'}
V_t-y_t^{\mathbb P^{'}}&\geq\mathbb E^{\mathbb P^{'}}_t\left[\int_t^TM^{' \mathbb P^{'}}_sd\widetilde K_s^{\mathbb P^{'}}\right]
\geq \mathbb E^{\mathbb P^{'}}_t\left[\underset{t\leq s\leq T}{\inf}M^{' \mathbb P^{'}}_s\left(\widetilde K_T^{\mathbb P^{'}}-\widetilde K_t^{\mathbb P^{'}}\right)\right],\end{aligned}$$ where $M^{' \mathbb P^{'}}$ is defined as $M^{\mathbb P^{'}}$ but with $\gamma'$ instead of $\gamma$. Now let us prove that for any $n>1$ $$\mathbb E^{\mathbb P^{'}}_t\left[\left(\underset{t\leq s\leq T}{\inf}M^{' \mathbb P^{'}}_s\right)^{-n}\right]<+\infty,\ \mathbb P^{'}-a.s.
\label{eq:momentsexp}$$
First we have $$\begin{aligned}
M^{' \mathbb P^{'}}_s&=\exp\left(\int_t^s\lambda_rdr+\int_t^s\eta_r\widehat a_r^{-1/2}dB_s^{\mathbb P^{'},c}-\frac12\int_t^s{\left|\eta_r\right|}^2dr+\int_t^s\int_E\gamma'_r(x)\widetilde\mu^{\mathbb P^{'}}_B(dx,dr)\right)\\
&\hspace{0.9em}\times\prod_{t\leq r\leq s}(1+\gamma'_r(\Delta B_r))e^{-\gamma'_r(\Delta B_r)}.\end{aligned}$$
Define, then $A^{\mathbb P^{'}}_s=\mathcal E\left(\int_t^s\eta_r\widehat a_r^{-1/2}dB_s^{\mathbb P^{'},c}\right)$ and $C^{\mathbb P^{'}}_s=\mathcal E\left(\int_t^s\int_E\gamma'_r(x)\widetilde\mu^{\mathbb P^{'}}_B(dx,dr)\right)$. Notice that both these processes are strictly positive martingales, since $\eta$ and $\gamma'$ are bounded and we have assumed that $\gamma'$ is strictly greater than $-1$. We have $$M^{' \mathbb P^{'}}_s=\exp\left(\int_t^s\lambda_rdr\right)A^{\mathbb P^{'}}_s C^{\mathbb P^{'}}_s.$$
Since the process $\lambda$ is bounded, we have $$\begin{aligned}
\left(\underset{t\leq s\leq T}{\inf}M^{' \mathbb P^{'}}_s\right)^{-n}&\leq C\left(\underset{t\leq s\leq T}{\inf}A^{\mathbb P^{'}}_sC^{\mathbb P^{'}}_s\right)^{-n}=C\left(\underset{t\leq s\leq T}{\sup}\left\{(A^{\mathbb P^{'}}_sC^{\mathbb P^{'}}_s)^{-1}\right\}\right)^{n}.\end{aligned}$$
Using the Doob inequality for the submartingale $(A_sC_s)^{-1}$, we obtain $$\begin{aligned}
\mathbb E^{\mathbb P^{'}}_t\left[\left(\underset{t\leq s\leq T}{\inf}M^{' \mathbb P^{'}}_s\right)^{-n}\right]&\leq C\mathbb E^{\mathbb P^{'}}_t\left[(C^{\mathbb P^{'}}_T A^{\mathbb P^{'}}_T)^{-n}\right]\leq C\left(\mathbb E^{\mathbb P^{'}}_t\left[(C^{\mathbb P^{'}}_T)^{-2n}\right]\mathbb E^{\mathbb P^{'}}_t\left[(A^{\mathbb P^{'}}_T)^{-2n}\right]\right)^{1/2}<+\infty,\end{aligned}$$ where we used the fact that since $\eta$ is bounded, the continuous stochastic exponential $A^{\mathbb P^{'}}$ has negative moments of any order, and where the same result holds for the purely discontinuous stochastic exponential $C^{\mathbb P^{'}}$ by Lemma A.$4$ in [@kpz3].
Then, we have for any $p>1$ $$\begin{aligned}
&\mathbb E^{\mathbb P^{'}}_t\left[\widetilde K_T^{\mathbb P^{'}}-\widetilde K_t^{\mathbb P^{'}}\right]\\
&=\mathbb E^{\mathbb P^{'}}_t\left[\left(\underset{t\leq s\leq T}{\inf}M^{' \mathbb P^{'}}_s\right)^{1/p}\left(\widetilde K_T^{\mathbb P^{'}}-\widetilde K_t^{\mathbb P^{'}}\right)\left(\underset{t\leq s\leq T}{\inf}M^{' \mathbb P^{'}}_s\right)^{-1/p}\right]\\
&\leq \left(\mathbb E^{\mathbb P^{'}}_t\left[\underset{t\leq s\leq T}{\inf}M^{' \mathbb P^{'}}_s\left(\widetilde K_T^{\mathbb P^{'}}-\widetilde K_t^{\mathbb P^{'}}\right)\right]\right)^{1/p}\left(\mathbb E^{\mathbb P^{'}}_t\left[\underset{t\leq s\leq T}{\inf}(M^{' \mathbb P^{'}}_{s})^{-\frac{2}{p-1}}\right]\mathbb E^{\mathbb P^{'}}_t\left[\left(\widetilde K_T^{\mathbb P^{'}}-\widetilde K_t^{\mathbb P^{'}}\right)^2\right]\right)^{\frac{p-1}{2p}}\\
&\leq C\left(\underset{\mathbb P^{'}\in\mathcal P^{\kappa}_H(t^+,\mathbb P)}{\esup^\mathbb P}\mathbb E^{\mathbb P^{'}}\left[\left(\widetilde K_T^{\mathbb P^{'}}-\widetilde K_t^{\mathbb P^{'}}\right)^2\right]\right)^{\frac{p-1}{2p}}\left(V_t-y_t^{\mathbb P^{'}}\right)^{1/p},\ \mathbb P-a.s.,\end{aligned}$$ where we used . Arguing as in Step $\rm{(iii)}$ of the proof of Theorem \[uniqueref\], the above inequality along with Proposition \[prop.repref\] shows that we have $$\underset{\mathbb P^{'}\in\mathcal P^{\kappa}_H(t^+,\mathbb P)}{\einf^\mathbb P}\mathbb E^{\mathbb P^{'}}\left[\widetilde K_T^{\mathbb P^{'}}-\widetilde K_t^{\mathbb P^{'}}\right]=0,\ \mathbb P-a.s.,$$ that is to say that the minimum condition \[2bsdej.minK\] is satisfied.
Finally, when the terminal condition is in $\Lc^{2,\kappa}_H$, it suffices to approximate it by a sequence $(\xi_n)_{n\geq 0}\subset {\rm UC_b}(\Omega)$, and to pass to the limit using the a priori estimates obtained in [@kpz3]. The proof is similar to step (ii) of the proof of Theorem 4.6 in [@stz], we therefore omit it.
Fully non-linear PIDEs {#sec.PIDE}
======================
Markovian 2BSDEJs
-----------------
In this section, we specialize our discussion on 2BSDEJs by considering the so-called Markovian case for which the generator $F$ and the function $H$ have a specified deterministic form $$H_t(\omega, y,z,u,\gamma,v)=:h(t,B_t(\omega),y,z,u,\gamma,v),\text{ and } F_t(\omega, y,z,u,a,\nu)=: f(t,B_t(\omega),y,z,u,a,\nu),$$ for some function $f:[0,T]\times\R^d\times\R\times\R^d\times \hat L^2\times\mathbb S^{>0}_d\times\Nc\rightarrow \R$ and some function $h:[0,T]\times\R^d\times\R\times\R^d\times \hat L^2\times D_1\times D_2$. We also denote by $D^1_{f(t,x,y,z,u)}$ the domain of $F$ in $a$ and by $D^2_{f(t,x,y,z,u)}$ the domain of $f$ in $\nu$, for a fixed $(t,x,y,z,u)$
Moreover, we also need to define Markovian counterparts of the set $\Vc$ of predictable compensators, and of the sets $\Rc_F$ of predictable functions. More precisely, we let $\Vc^m$ be the set of measures in $\Nc$ which does not depend on $t$ and $\omega$ (i.e. the Lévy measures in $\Nc$), and for any $F\in\Vc^m$, let $\Rc_F^m$ to be the set of functions $\beta:E\rightarrow E$ which verify $${\left|\beta\right|}(x)\leq C(1\wedge{\left|x\right|}), \ F(dx)-a.e.$$ and such that $x\longmapsto \beta(x)$ is strictly monotone on the support of $F$. Finally, we let $\mathfrak V^m$ to be the set of measures $\widetilde F:=F\circ\left(\beta\right)^{-1}$ for some $F\in\Vc^m$ and some $\beta\in\Rc^m_F$.
We can now define the Legendre-Fenchel transform of the generator $f$ as follows $$\hat h(t,x,y ,z,u,\gamma,v):=\underset{(a,\nu)\in\mathbb S^{>0}_d\times\mathfrak V^m}{\sup}\left\{\frac12{\rm Tr}(a\gamma)+\int_E(Av)(x,e)\nu(de)-f(t,x,y,z,u,a,\nu)\right\},$$ for any $(t,x,y,z,u,\gamma,v)\in[0,T]\times \R^d\times\R\times\R^d\times \hat L^2\times C^2_b(E)$, where $C^2_b(E)$ denotes the set of functions from $E$ to $E$ which are $C^2$ with a bounded gradient and Hessian, which we endow with the topology of uniform convergence on compact sets, and where we recall that the non-local operator $A$ is defined in .
For simplicity, we abuse notations and let $\mathcal P^\kappa_h:=\mathcal P^\kappa_H$, as well as $\mathcal P^{\kappa,t}_h:=\mathcal P^{\kappa,t}_H$ for any $t\in[0,T]$. Assumptions \[assump.href\] and \[assump.h2ref\] now take the following (stronger) form
\[assump.href3\] [(i)]{} $\Pc^\kappa_h$ is not empty and the domains $D^1_{f(t,x,y,z,u)}=D^1_{f(t)}$ and $D^2_{f(t,y,z,u)}=D^2_{f(t)}$ are independent of $(x,y,z,u)$.
[(ii)]{} The following uniform Lipschitz-type property holds. For all $(y,y',z,z',u,t,a,\nu,x)$ $$\begin{aligned}
&{\left| f(t,x,y,z,u,a,\nu)- f(t,x,y',z',u,a,\nu)\right|}\leq C\left({\left|y-y'\right|}+{\left| a^{1/2}\left(z-z'\right)\right|}\right).\end{aligned}$$
\(iii) The map $t\longmapsto f(t,x,y,z,u,a,\nu)$ has left-limits and is uniformly continuous from the right, uniformly in $(a,\nu)\in D^1_{f(t)}\times D^2_{f(t)}$.
[(iv)]{} For all $(t,x,y,z,u^1,u^{2},a,\nu)$, there exist two functions $\gamma$ and $\gamma'$ such that $$\begin{aligned}
\int_{E}\delta^{1,2} u(x)\gamma'_t(x)\nu(dx)\leq f(t,x,y,z,u^1,a,\nu)- f(t,x,y,z,u^2,a,\nu)&\leq \int_{E}\delta^{1,2} u(x)\gamma_t(x)\nu(dx),\end{aligned}$$ where $\delta^{1,2} u:=u^1-u^2$ and $c_1(1\wedge {\left|x\right|}) \leq \gamma_t(x) \leq c_2(1\wedge {\left|x\right|})$ with $-1+\delta\leq c_1\leq0, \; c_2\geq 0,$ and $c_1'(1\wedge {\left|x\right|}) \leq \gamma'_t(x) \leq c_2'(1\wedge {\left|x\right|})$ with $-1+\delta\leq c_1'\leq0, \; c_2'\geq 0,$ for some $\delta >0$.
[(v)]{} $F$ is uniformly continuous in $x$, that is to say that there exists some modulus of continuity $\rho$ such that for all $(t,x,x',y,z,u,a,\nu)$ $${\left|f(t,x,y,z,u,a,\nu)-f(t,x',y,z,u,a,\nu)\right|}\leq \rho\left({\left|x-x'\right|}\right).$$
\[rem.linear\] With the exception of $(iii)$, the above assumptions are simple restatements of Assumptions \[assump.href\] and \[assump.h2ref\]. We emphasize that $(iii)$ above will be important in the proof of the Feynman-Kac representation formula. Notice also that since $\R^d$ is a convex space, it is always possible to choose the modulus $\rho$ to be both concave and with linear growth.
We consider a given Borel-measurable function $g:\R^d\longrightarrow\R$. The rest of this section will be devoted to relationships existing between the following 2BSDEJ $$\label{2bsdejmark}
Y_t=g(B_T)-\int_t^Tf\left(s,B_s,Y_s,Z_s,U_s,\widehat a_s,\nu^\P_s\right)ds-\int_t^TZ_sdB_s+K_T^\P-K_t^\P,\ \Pc^\kappa_h-a.s.,$$ and the following fully non-linear PIDE $$\label{pide}
\begin{cases}
-\partial_tu(t,x)-\hat h(t,x,u(t,x),Du(t,x),\Kc u(t,x,\cdot),D^2u(t,x),u(t,x+\cdot))=0,\ (t,x)\in[0,T)\times\R^d\\
u(T,x)=g(x),\ x\in\R^d,
\end{cases}$$ where $y\longmapsto \Kc v(t,x,y)$ is a function from $E$ to $E$ defined by $$\Kc v(t,x,y):=v(t,x+y)-v(t^-,x).$$
Smooth solutions of and Feynman-Kac formula
-------------------------------------------
We start by showing that a smooth solution to provides a solution to the 2BSDEJ . As was already showed in [@stz], and unlike what happens for classical Markovian BSDEJs, the proof is not a simple application of Itô’s formula. Indeed, verifying the minimal condition creates unavoidable technical difficulties.
Let Assumption \[assump.href3\] hold and assume in addition that $h$ is continuous $($where continuity with respect to its fifth and seventh argument are to be understood w.r.t. the topology of uniform convergence on compact sets$)$, that $D^1_{f}:=D^1_{f(t)}$ and $D^2_{f}:=D^2_{f(t)}$ are actually independent of $t$, that $D^1_{f}$ is bounded from above and away from $0$, that $D^2_f$ is such that $$\label{condition}\underset{\nu\in D^2_{f}\cap\mathfrak V^m}{\sup}\int_E\left({\left|x\right|}^2{\bf 1}_{{\left|x\right|}<1}+{\left|x\right|}{\bf 1}_{{\left|x\right|}\geq 1}\right)\nu(dx)<+\infty,$$ and that $g$ is bounded. Let $u\in C^{1,2}([0,T),\R^d)$ be a classical solution of with bounded gradient and Hessian, such that in addition $$\{(u,Du,\Kc u)(t,B_{t^-}), \ t\in[0,T]\}\in\D^{2,\kappa}_H\times\H^{2,\kappa}_H\times\mathbb J^{2,\kappa}_H.$$ Then, if we define $$Y_t:=u(t,B_t), \;ÊZ_t:=Du(t,B_{t^-}),\ U_t(\cdot):=\Kc u(t,B_{t^-},\cdot),\ \Gamma_t:=D^2u(t,B_{t^-}),\ K_t^\P:=\int_0^tk_s^\P ds,$$ $$k_t^\P:=\hat h (t, B_{t^-},Y_t,Z_t, U_t,\Gamma_t,u(t,B_{t^-}+\cdot))-\frac12{\rm Tr}[\widehat a_t\Gamma_t]-\int_E(Au)(B_{t^-},x)\nu^\P_t(dx)+f(t,B_t,Y_t,Z_t,U_t,\widehat a_t,\nu^\P_t),$$ $(Y,Z,U)$ is the unique solution of .
The condition above seems difficult to avoid with our approach here, and basically demands uniform moments for the small and large jumps of the canonical process over the whole uncertainty set. Moreover, we would like to point out that this condition also appears in the recent work [@nn3], when the authors look at viscosity solution to PIDE when $f=0$.
A simple application of Itô’s formula shows that $(Y,Z,U)$ does satisfy the equation . Since in addition $Y_T=g(B_T)\in\L^{2,\kappa}_H$ (because $g$ is bounded), it only remains to verify that for any $\P\in\Pc^\kappa_h$ and for any $t\in[0,T]$ $$\underset{\P^{'}\in\Pc^\kappa_h(t^+,\P)}{\einf^\P}\E^{\P^{'}}_t\left[\int_t^Tk_s^{\P^{'}}ds\right]=0,\ \P-a.s.$$
Towards this goal, we follow the proof of Theorem $5.3$ in [@stz] and we adapt it to our jump framework. Let us outline the proof for the sake of clarity. The main idea is, as is common in stochastic control problems, to find an $\eps$-optimal control in the definition of $\hat h$, which means here both a volatility process and a jump measure. The main difficulty after that is to be able to find a probability measure in $\Pc^\kappa_h$ such that the characteristics of the canonical process $B$ under this measure coincide with the $\eps$-optimal controls. We emphasize that even though it may be possible to find such a measure in the larger set $\overline{\Pc}_W$ (and even in this case it may prove impossible, see Remark 2.3 in [@stz]), it is not clear at all that this measure will be in $\overline{\Pc}_S$, and thus in $\Pc^\kappa_h$.
We now start the proof. By a classical measurable selection argument, for any $\eps>0$, we can find a predictable process $a^\eps$ taking values in $D^1_f$ and a predictable random measure $\nu^\eps$, taking values in $D^2_f\cap\mathfrak V^m$ (i.e. there exist $(F^\eps,\beta^\eps)\in\Vc^m\times\Rc^m_F$ such that $\nu^\eps=F^\eps\circ (\beta^\eps)^{-1}$) with $$\hat h (t, B_{t^-},Y_t,Z_t, U_t,\Gamma_t,u(t,B_{t^-}+\cdot))\leq \frac12{\rm Tr}[a^\eps_t\Gamma_t]+\int_E(Au)(B_{t^-},x)\nu^\eps_t(dx)-f(t,B_t,Y_t,Z_t,U_t,a^\eps_t,\nu^\eps_t)+\eps.$$
Fix now some $\P:=\P^{\alpha,\beta}_F\in\Pc^\kappa_H$ and some $t\in[0,T]$. We will now show that we can find some $(\alpha^\eps,b^\eps,\widetilde F^\eps)$ such that $\P^{\alpha^\eps,b^\eps}_{\widetilde F^\eps}\in\Pc^\kappa_H(t^+,\P)$ and for $s\in[t,T]$ $$\widehat a_s=a^\eps,\ \nu^{\P^{\alpha^\eps,b^\eps}_{\widetilde F^\eps}}_s=\nu^\eps,\ ds\times d\P^{\alpha^\eps,b^\eps}_{\widetilde F^\eps}-a.e.$$
For notational simplicity, let us define for $s\geq r\geq t$ $$\begin{aligned}
X_s^r:=&\ h(s,B_{s^-},Y_s,Z_s, U_s,\Gamma_s,u(s,B_{s^-}+\cdot))- \frac12{\rm Tr}[a^\eps_r\Gamma_s]-\int_E(Au)(B_{s^-},x)\nu^\eps_r(dx)\\
&+f(s,B_s,Y_s,Z_s,U_s,a^\eps_r,\nu^\eps_r).\end{aligned}$$ We next define a sequence of $\F$-stopping times. Let $$\begin{aligned}
\tau_0^\eps:&=\inf\Big\{s\geq t,\ X_s^t\geq 2\eps\text{ or }X_{s^-}^t\geq 2\eps\Big\}\wedge T\\
\tau_{n+1}^\eps:&=\inf\Big\{s\geq \tau_n^\eps,\ X_s^{\tau_n^\eps}\geq 2\eps\text{ or }X_{s^-}^{\tau_n^\eps}\geq 2\eps\Big\}\wedge T,\ n\geq 0.\end{aligned}$$ Notice that since by definition $X^s_s\leq \eps$ for any $s\geq t$, we always have $\tau_{n+1}^\eps>\tau_n^\eps$, for any $n\geq 0$, and $\tau_0^\eps>t$. Besides, since $B,Y,Z,U,\Gamma, u$ are all càdlàg, it is a classical result that the $\tau^\eps_n$ are indeed $\F$-stopping times.
Next, for any $\omega\in\Omega$, the maps $t\mapsto \hat h(t,x,y,z,u,\gamma,v)$ and $t\mapsto f(t,x,y,z,u,a,\nu)$ are respectively uniformly continuous from the right (see [@applebaum], Appendix $2.8$ for more details) and uniformly continuous from the right uniformly in $(a,\nu)\in D^1_f\times D^2_f$ (since they are càdlàg on the compact $[0,T]$). Moreover, since we assumed that $D^1_f$ was bounded from above and away from $0$, that $$\underset{\nu\in D^2_{f}\cap\mathfrak V^m}{\sup}\int_E\left((1\wedge {\left|x\right|}^2)+{\left|x\right|}{\bf 1}_{{\left|x\right|}\geq 1}\right)\nu(dx)<+\infty,$$ and that $Du$ and $D^2u$ were bounded, we can deduce that for any omega, the function $$\begin{aligned}
h(s,B_{s^-}(\omega),Y_s(\omega),Z_s(\omega), U_s(\omega),\Gamma_s(\omega),u(s,B_{s^-}(\omega)+\cdot))- \frac12{\rm Tr}[a\Gamma_s(\omega)]-\int_E(Au)(B_{s^-}(\omega),x)\nu(dx)\\+f(s,B_s(\omega),Y_s(\omega),Z_s(\omega),U_s(\omega),a,\nu),
\end{aligned}$$ is uniformly continuous from the right in $s$, uniformly in $(a,\nu)$. This implies that the $\tau_n^\eps$ cannot accumulate and that it is possible to find some $\delta(\eps,\omega)>$, independent of $n$, such that $\tau_{n+1}^\eps(\omega)\geq\tau_n^\eps(\omega)+\delta(\eps,\omega)$. In particular, this also implies that there exists some finite $N\in\N$ such that $\tau_n^\eps=T$ for any $n\geq N$.
Let us now define for $s\geq \tau_0^\eps$ $$\tilde a^\eps_s:=\sum_{n=0}^{+\infty}a^\eps_{\tau^\eps_n}{\bf 1}_{s\in[\tau_n^\eps,\tau_{n+1}^\eps)},\ \tilde F^\eps_s:=F_t{\bf 1}_{0\leq s\leq \tau_0^\eps}+{\bf 1}_{s\geq \tau_0^\eps}\sum_{n=0}^{+\infty}F^\eps_{\tau^\eps_n}{\bf1}_{s\in[\tau_n^\eps,\tau_{n+1}^\eps)},\ \tilde \beta^\eps_s:=\sum_{n=0}^{+\infty}\beta^\eps_{\tau^\eps_n}{\bf 1}_{s\in[\tau_n^\eps,\tau_{n+1}^\eps)}.$$ Consider next the following SDE on $[\tau_0^\eps,T]$ $$\label{sde}
dZ_s= \left(a^\eps_s(Z_\cdot)\right)^{1/2}dB^{\P_{0,\tilde F^\eps},c}_s+\int_E\tilde\beta_s(Z_\cdot,x)\left(\mu_B(dx,ds)-\tilde F^\eps_s(dx)ds\right),\ \P_{0,\tilde F^\eps}-a.s.$$
It is proved in Lemma \[lemma.sde\] that the above SDE has a unique strong solution $Z^\eps$ on $[\tau_0^\eps,T]$ such that $Z^\eps_{\tau_0^\eps}=0.$ Define then $$\alpha^\eps_t:=\alpha_t{\bf 1}_{0\leq t\leq \tau_0^\eps}+{\bf 1}_{t\geq \tau_0^\eps}\tilde a^\eps(Z^\eps_\cdot),\ b^\eps_t(x):=\beta_t(x){\bf 1}_{0\leq t\leq \tau_0^\eps}+{\bf 1}_{t\geq \tau_0^\eps}\tilde \beta^\eps(Z^\eps_\cdot,x).$$ It is immediate to verify that $\alpha^\eps\in\mathcal D$, $\tilde F^\eps\in\nu$ and $b^\eps\in\mathcal R_{\tilde F^\eps}$ (see the proof of Lemma A.3 in [@kpz3] for similar arguments). We can therefore define the probability measure $\P_{\tilde F^\eps}^{\alpha^\eps,b^\eps}\in\mathcal P^\kappa_h$, which by definition, coincides with $\P$ on $\mathcal F_{t^+}$ (since $\tau_0^\eps>t$). Using $(2.6)$ and $(2.7)$ in [@kpz3], we then deduce that $$\widehat a_s= \tilde a^\eps_s,\ \nu^{\P^{\alpha^\eps,b^\eps}_{\tilde F^\eps}}_s=\nu_s^{\tilde F^\eps,\tilde\beta^\eps},\ \P^{\alpha^\eps,b^\eps}_{\tilde F^\eps}-a.s.\text{ on }[\tau_0^\eps,T].$$
This implies that, $\P^{\alpha^\eps,b^\eps}_{\tilde F^\eps}-a.s$, $$\begin{aligned}
\hat h (s, B_{s^-},Y_s,Z_s, U_s,\Gamma_s,u(s,B_{s^-}+\cdot))\leq &\ \frac12{\rm Tr}[\widehat a_s\Gamma_s]+\int_E(Au)(B_{s^-},x)\nu^{\P^{\alpha^\eps,b^\eps}_{\tilde F^\eps}}_s(dx)\\
&-f\left(s,B_s,Y_s,Z_s,U_s,\widehat a_s,\nu^{\P^{\alpha^\eps,b^\eps}_{\tilde F^\eps}}_s\right)+\eps,\text{ for a.e. $s\in[\tau_0^\eps,T]$}.
\end{aligned}$$ Finally,
$$\underset{\P^{'}\in\Pc^\kappa_h(t^+,\P)}{\einf^\P}\E^{\P^{'}}_t\left[\int_t^Tk_s^{\P^{'}}ds\right]\leq 2\eps(T-t)+\E^{\mathbb P^{\alpha^\eps,b^\eps}_{\tilde F^\eps}}\left[\int_{\tau_0^\eps}^Tk_s^{\mathbb P^{\alpha^\eps,b^\eps}_{\tilde F^\eps}}ds\right]\leq 4\eps(T-t).$$ Since $\eps>0$ was arbitrary, this ends the proof.
2BSDEJs and viscosity solutions to fully non-linear PIDEs
---------------------------------------------------------
### Time-space regularity of Markovian solutions to 2BSDEJs
In this section, we specialize the discussion and notations of Section \[sec.existtt\] to the Markovian framework and obtain additional regularity results.
For simplicity, let us denote $$B_s^{t,x}:=x+B^t_s,\ \text{for all $(t,s,x)\in[0,T]\times[t,T]\times\R^d,$}$$ and for any $(t,x)\in[0,T]\times\R^d$, any $\F^t$-stopping time $\tau$, any $\P\in\Pc^{\kappa,t}_h$, and r.v. $\eta\in L^2(\P)$ which is $\Fc^t_\tau$-measurable, we let $(\Yc^{\P,t,x},\Zc^{\P,t,x},\Uc^{\P,t,x}):=(\Yc^{\P,t,x}(\tau,\eta),\Zc^{\P,t,x}(\tau,\eta),\Uc^{\P,t,x}(\tau,\eta))$ be the unique solution to the following BSDEJ $$\begin{aligned}
\label{bsdejmark}
\nonumber \Yc^{\mathbb P,t,x}_{s}&=\eta+\int^{\tau}_{s}f\left(r,B_r^{t,x},\Yc^{\mathbb P,t,x}_{r},\Zc^{\mathbb P,t,x}_{r},\Uc_r^{\mathbb P,t,x},\widehat a^t_r,\nu^{t,\P}_r\right)dr-\int^{\tau}_{s}\Zc^{\mathbb P,t,x}_{r}dB^{t,\mathbb P,c}_{r}\\
&\hspace{0.9em}-\int_s^{\tau}\int_{E}\Uc_r^{\mathbb P,t,x}(e)\widetilde\mu^\mathbb P_{B^{t}}(de,dr), \ \mathbb P-a.s.,\ s\in[t,\tau].\end{aligned}$$
Then, exactly as the process $V$ defined in , we consider the value function $$u(t,x):=\underset{\P\in\Pc^{t,\kappa}_h}{\sup}\Yc^{\P,t,x}_t\left(T,g(B_T^{t,x})\right),\ (t,x)\in[0,T]\times \R^d,$$ which is indeed deterministic due to the Blumenthal $0-1$ law, which holds true for any $\P\in\Pc^{t,\kappa}_h$.
Before stating the next result, we need to consider in addition the following assumption.
\[assump.markovian\] The map $x\longmapsto g(x)$ has linear growth and $$\begin{aligned}
\Lambda(t,x):=\underset{\P\in\Pc^{t,\kappa}_h}{\sup}\left(\E^\P\left[{\left|g\left(B^{t,x}_T\right)\right|}+\int_t^T{\left|f\left(s,B_s^{t,x},0,0,0,\widehat a^t_s,\nu^{t,\P}_s\right)\right|}^\kappa\right]\right)^{\frac1\kappa}, \label{grandlambda.def}\end{aligned}$$ is such that $$\underset{\P\in\Pc^{t,\kappa}_h}{\sup}\E^\P\left[\underset{t\leq s\leq T}{\sup}\left(\Lambda\left(s,B^{t,x}_s\right)\right)^2\right]<+\infty,\text{ for any $(t,x)\in[0,T]\times\R^d$}.$$
We emphasize that we could make the weaker assumption that $g$ has polynomial growth, say of integer order $p$ but, to compensate this, we would need the additional assumption that the compensators $\nu$ that we consider have moments of order $p$, thus reducing the set of probability measures we allow for. The wellposedness of our 2BSDEJ \[2bsdej\] would still hold. For the sake of simplicity we will directly ask that $g$ has linear growth.
We refer to Remark 5.8 in [@stz] for sufficient conditions ensuring that Assumption \[assump.markovian\] holds true. The next result generalizes Theorem 5.9 and Proposition 5.10 of [@stz].
\[prop.reg\] We have the following results
- Let Assumptions \[assump.href3\] and \[assump.markovian\] hold, and assume furthermore that $g$ is uniformly continuous. Then the 2BSDEJ has a unique solution $(Y,Z,U)\in\D^{2,\kappa}_H\times\H^{2,\kappa}_H\times\mathbb J^{2,\kappa}_H$. Moreover, we have the identity $Y_t=u(t,B_t)$ and the function $u$ is uniformly continuous in $x$, uniformly in $t$, and right-continuous in $t$.
- Let Assumptions \[assump.href3\] and \[assump.markovian\] hold, and assume furthermore that $g$ is lower semi-continuous. Then $u$ is lower semicontinuous in $(t,x)$, from the right in $t$, that is to say that for any $(t,x)\in[0,T]\times \R^d$ and any sequence $(t_n,x_n)_{n\geq 0}$ such that $$(t_n,x_n)\underset{n\rightarrow +\infty}{\longrightarrow}(t,x)\text{ and } t_n> t,\text{ for all $n\geq 0$},$$ we have $\underset{n\rightarrow +\infty}{\underline{\lim}}u(t_n,x_n)\geq u(t,x).$
In order to prove this Proposition, we will need the following weak dynamic programming property, in the spirit of the work by Bouchard and Touzi [@bt]. It is very closely related to the proof of the dynamic programming property of Proposition \[progdyn\], with the additional difficulty that less regularity is assumed on $g$. Moreover, its proof is very close to the proofs of Proposition 5.14 and Lemmas 6.2 and 6.4 in [@stz]. Hence, we will only sketch some parts of its proof, which is relegated to the appendix.
\[partial.dpp\] Under assumptions \[assump.href3\] and \[assump.markovian\], for any family of $\F^t$-stopping times $\{\tau^{\P}, \mathbb P\in\mathcal P^{t,\kappa}_h\}$: $$\begin{aligned}
u(t,x) \leq \underset{\mathbb P\in\mathcal P^{t,\kappa}_h}{\sup} \mathcal Y^{\mathbb P,t,x}_t\left(\tau^{\P},X\right), \text{ for all } \left(t,x\right)\in\left[0,T\right]\times\R^d \label{partial.dpp1}\end{aligned}$$ and for any $\mathcal{F}^t_{\tau^{\P}}$-measurable r.v. $X$ such that $X\geq u(\tau^{\P},B_{\tau^{\P}}^{t,x})$, $\P$-a.s. Moreover, when the function $g$ is lower semi-continuous, $$\begin{aligned}
u(t,x) = \underset{\mathbb P\in\mathcal P^{t,\kappa}_h}{\sup} \mathcal Y^{\mathbb P,t,x}_t\left(\tau^{\P},u(\tau^{\P},B_{\tau^{\P}}^{t,x})\right), \text{ for all } \left(t,x\right)\in\left[0,T\right]\times\R^d, \label{partial.dpp2}\end{aligned}$$
Proposition \[prop.reg\] (i) can be proved exactly as in [@stz], using Theorem \[mainref\], Lemmas \[unifcont\] and \[lem.cadlag\] and Proposition \[prop.repref\].
\(ii) We follow [@stz]. We start by introducing the functional $$\mathfrak J(\P,t,x):=\mathbb E^\P\left[\mathfrak y_t^\P\left(t,x\right))\right],\ (\P,t,x)\in \Pc^\kappa_h\times[0,T]\times\R^d,$$ where $\mathfrak y ^{\P}(t,x)$ is the first component of the solution to the BSDEJ under $\P$ with terminal condition $g(x+B_T-B_t)$ and generator $f(s,x+B_s-B_t,y,z,u,a,\nu)$. The first step of the proof is to show the following identity $$\label{identity}
u(t,x)=\underset{\P\in\Pc^\kappa_h}{\sup}\mathfrak J(\P,t,x).$$
First, by , we have for any $\P\in\mathcal P^\kappa_h$ and for $\P-a.e.$ $\omega\in\Omega$ $$\mathfrak y_t^\P(t,x)(\omega)=\Yc_t^{\P^{t,\omega},t,x}\left(T,g\left(B_T^{t,x}\right)\right)\leq u(t,x).$$ This implies, that $\mathfrak J(\P,t,x)\leq u(t,x)$. The other inequality can be proved exactly as in the proof of Proposition 5.10 in [@stz]. Then, it is clearly sufficient to show that the map $(t,x)\longmapsto J(\P,t,x)$ is lower semi-continuous, from the right in $t$, for any $\P\in\mathcal P^\kappa_h$.
Consider thus some $(t,x)\in[0,T]\times \R^d$, some $\P\in\Pc^\kappa_h$ and a sequence $(t_n,x_n)_{n\geq 0}$ such that $(t_n,x_n)\rightarrow (t,x)$ and $t_n>t$ for any $n\geq 0$. Consider the following $\underline{\lim}$ $$\xi_n:=\underset{k\geq n}{\inf} g(x_k+B_T-B_{t_k}),\ f^{n,\P}(s,y,z,u):=\underset{k\geq n}{\inf} f(s,x_k+B_s-B_{t_k},y,z,u,\widehat a_s,\nu^\P_s),$$ $$\overline\xi:=\underset{n\rightarrow+\infty}{\lim}\xi_n,\ \overline{f}^\P:=\underset{n\rightarrow+\infty}{\lim}f^{n,\P},$$ and let $(\Yc^n,\Zc^n,\Uc^n)$ denote the solution to the BSDEJ under $\P$ with terminal condition $\xi_n$ and generator $f^{n,\P}$. Since $g$ and the modulus of uniform continuity of $f$ have linear growth in $x$ (remember Remark ), since $f$ is uniformly Lipschitz continuous, and since under any of the measure considered the canonical process $B$ is a square-integrable martingale (see Definition \[def\]), it can be checked directly that this BSDEJ has a unique solution, and by stability for BSDEJ that $$\underset{n\rightarrow +\infty}{\lim}\E^\P[\Yc^n_t]=\E^\P[\overline{\Yc}_t],$$ where $\overline{\Yc}$ denotes the first component of the solution of the BSDEJ with terminal condition $\overline\xi$ and generator $\overline{f}^\P$.
Since $g$ is lower semi-continuous, $f$ is uniformly continuous in $x$, $B$ is càdlàg and $t_n>t$ for any $n\geq 0$, we deduce that $$\overline\xi\geq g(x+B_T-B_t),\text{and }\overline f^\P(s,y,z,u)=f(s,x+B_s-B_t,y,z,u,\widehat a_s,v_s^\P).$$ Hence $$\begin{aligned}
\underset{n\rightarrow +\infty}{\underline \lim} \mathfrak J(\P,t_n,x_n)\geq \underset{n\rightarrow +\infty}{\lim}\E^\P[\Yc^n_t]=\E^\P[\overline{\Yc}_t]\geq \E^\P[\mathfrak y^\P_t(t,x)]=\mathfrak J(\P,t,x),\end{aligned}$$ which ends the proof.
### Viscosity solution of PIDE
$(i)$ A bounded lower-semicontinuous function $v$ is called a viscosity super-solu-tion of the PIDE if $v(T,\cdot)\geq g(\cdot)$ and if for every function $\varphi\in C^{3}_b([0,T)\times\R^d)$ $($i.e. the space of functions from $[0,T)\times\R^d$ which are thrice continuously differentiable with bounded derivatives$)$ such that $$0=v(t_0,x_0)-\varphi(t_0,x_0)=\underset{(t,x)\in[0,T)\times \R^d}{\min}(v-\varphi)(t,x),$$ we have $$-\partial_tv(t_0,x_0)-\hat h(t_0,x_0,\varphi(t_0,x_0),D\varphi(t_0,x_0),\Kc \varphi(t_0,x_0,\cdot),D^2\varphi(t_0,x_0),\varphi(t_0,x_0+\cdot))\geq 0.$$ $(ii)$ A bounded upper-semicontinuous function $v$ is called a viscosity sub-solution of the PIDE if $v(T,\cdot)\leq g(\cdot)$ and if for every function $\varphi\in C^{3}_b([0,T)\times\R^d)$ such that $$0=v(t_0,x_0)-\varphi(t_0,x_0)=\underset{(t,x)\in[0,T)\times \R^d}{\max}(v-\varphi)(t,x),$$ we have $$-\partial_tv(t_0,x_0)-\hat h(t_0,x_0,\varphi(t_0,x_0),D\varphi(t_0,x_0),\Kc \varphi(t_0,x_0,\cdot),D^2\varphi(t_0,x_0),\varphi(t_0,x_0+\cdot))\leq 0.$$ $(iii)$ A continuous function $v$ is a viscosity solution of if it is both a viscosity sub and supersolution.
This section is devoted to the proof of the following result, which generalizes to the case of 2BSDEJs Proposition 5.4 of [@nn3], which considers the case $f=0$. The arguments are classical, as soon as one has at disposition a dynamic programming principle. We however give a detailed proof, since the presence of the non-linearity $f$ complicates the estimates.
\[th:visco\] Let Assumption \[assump.href3\] hold and assume in addition that $h$ is continuous, that $t\longmapsto f(t,x,y,z,a,\nu)$ is uniformly continuous from the right, uniformly in all the other variables, that $D^1_{f}:=D^1_{f(t)}$ and $D^2_{f}:=D^2_{f(t)}$ are actually independent of $t$, that $D^1_{f}$ is bounded from above and away from $0$, that $D^2_f$ is such that $$\underset{\nu\in D^2_{f}\cap\mathfrak V^m}{\sup}\int_E\left({\left|x\right|}^2+{\left|x\right|}{\bf 1}_{{\left|x\right|}\geq 1}\right)\nu(dx)<+\infty.$$ and that $g$ is uniformly continuous and bounded. Then, $u$ is a viscosity solution of the PIDE .
We would to point out two differences with [@nn3]. First, the set of test functions that we consider is not the same, since we assume more regularity. However, as is well-known in viscosity theory, this is actually without loss of generality by simple density arguments. Then, the integrability assumptions on the jumps of the canonical process under the measures considered is not the same. Indeed, we assume a uniform control for the second moment of both the small and large jumps, while [@nn3] only does it for the small jumps. This seems unavoidable in our setting since we want the canonical process to be square-integrable under every measures, and we want to have a uniform control on its norm. Nonetheless, the added assumption allows us to get rid off the assumption of [@nn3] on the limit as $\epsilon$ goes to $0$ of the first order moment of small jumps $($see their condition $(5.2))$.
Of course, Theorem \[th:visco\] should be complemented with a comparison theorem which would then imply uniqueness of viscosity solutions to . Given the length of this paper, we will refrain from studying this problem here, and we refer instead the reader to Proposition $5.5$ in [@nn3] and the references therein for examples of assumptions under which such a result holds.
Before proving this theorem, we will need the following lemmas, which notably insure that the function $u$ is jointly continuous, which is needed if we want to prove that it is a (continuous) viscosity solution of the PIDE .
\[lemma:estimates\] Let the assumptions of Theorem hold. Then, for some constant $C>0$, we have that for any $(t,t')\in[0,T]^2$ $$\underset{\P\in\Pc^{t',\kappa}_h}{\sup}\E^\P\left[\underset{t'\leq s\leq t}{\sup}{\left|B_s^{t'}\right|}^2\right]\leq C(t-t'),\ \underset{\P\in\Pc^{t',\kappa}_h}{\sup}\E^\P\left[\underset{t'\leq s\leq t}{\sup}{\left|B_s^{t'}\right|}\right]\leq C\sqrt{{\left|t-t'\right|}}.$$
First of all, by BDG inequality, we have for any $\P\in\Pc^{t',\kappa}_h$ and for some constant $C>0$ which may vary from line to line $$\E^\P\left[\underset{t'\leq s\leq t}{\sup}{\left|B_s^{t'}\right|}^2\right]\leq C\E^\P\left[\int_{t'}^t{\left|\widehat a^{t'}_s\right|}ds+\int_{t'}^t\int_E{\left|x\right|}^2\nu^{t',\P}_s(dx)ds\right]\leq C(t-t'),$$ where we have used the fact that $D^1_f$ is bounded and that $\underset{\nu\in D^2_{f}\cap\mathfrak V^m}{\sup}\int_E{\left|x\right|}^2\nu(dx)$ is finite. The second term can be treated similarly.
\[lemma:unifcontt\] Let the assumptions of Theorem hold. Then, the map $t\longmapsto u(t,x)$ is uniformly continuous. More precisely, if $\rho$ denotes the modulus of continuity in $x$ of $g$ and $f$, we have for some constant $C>0$ $${\left|u(t,x)-u(t',x)\right|}\leq C\left(\rho\left(C\sqrt{{\left|t-t'\right|}}\right)^{\frac12}+(1+{\left|x\right|})\sqrt{{\left|t-t'\right|}}\right), \ (t,t',x)\in[0,T]^2\times \R^d.$$
By , we have $$\begin{aligned}
{\left|u(t,x)-u(t',x)\right|}&\leq \underset{\P\in\Pc^\kappa_h}{\sup}\E^\P\left[{\left|\mathfrak y_t^\P(t,x)-\mathfrak y_{t'}^\P(t',x) \right|}\right]\\
&\leq \underset{\P\in\Pc^\kappa_h}{\sup}\E^\P\left[{\left|\mathfrak y_t^\P(t,x)-\mathfrak y_{t}^\P(t',x) \right|}\right]+\underset{\P\in\Pc^\kappa_h}{\sup}\E^\P\left[{\left|\mathfrak y_t^\P(t',x)-\mathfrak y_{t'}^\P(t',x) \right|}\right].\end{aligned}$$ For the first term on the right-hand side, we have by classical linearization arguments for Lipschitz BSDEJs, following the same line as Step $2$ of the proof of Theorem $4.1$ in [@kpz3] that $$\underset{\P\in\Pc^\kappa_h}{\sup}\E^\P\left[{\left|\mathfrak y_t^\P(t,x)-\mathfrak y_{t}^\P(t',x) \right|}\right]\leq C\underset{\P\in\Pc^\kappa_h}{\sup}\E^\P\left[\underset{0\leq s\leq T}{\sup}{\left|M_s\right|}\rho\left({\left|B_t-B_{t'}\right|}\right)\right],$$ where $M$ is a process such that $$\underset{\P\in\Pc^\kappa_h}{\sup}\E^\P\left[\underset{0\leq s\leq T}{\sup}{\left|M_s\right|}^p\right]<+\infty, \ \text{for any $p\geq 1$.}$$ Using twice Cauchy-Schwarz inequality and the fact that $\rho$ has linear growth, we deduce that $$\begin{aligned}
\underset{\P\in\Pc^\kappa_h}{\sup}\E^\P\left[{\left|\mathfrak y_t^\P(t,x)-\mathfrak y_{t}^\P(t',x) \right|}\right]\leq &\ \underset{\P\in\Pc^\kappa_h}{\sup}\E^\P\left[\underset{0\leq s\leq T}{\sup}{\left|M_s\right|}^4\right]^{\frac14}C\left(1+\underset{\P\in\Pc^{\kappa}_h}{\sup}\E^\P\left[\underset{0\leq s\leq T}{\sup}{\left|B_s\right|}^2\right]^{\frac14}\right)\\
&\times\underset{\P\in\Pc^\kappa_h}{\sup}\E^\P\left[\rho\left({\left|B_t-B_{t'}\right|}\right)\right]^{\frac12}.\\
\leq &\ C\rho\left(C\sqrt{{\left|t-t'\right|}}\right)^{\frac12},\end{aligned}$$ where we used in the last line the fact the supremum of $M$ has moments of any order, Lemma \[lemma:estimates\] as well as Jensen’s inequality (remember that $\rho$ is concave).
We then have, assuming w.l.o.g. that $t\leq t'$ and denoting by $(\mathfrak z^\P(t',x),\mathfrak u^\P(t',x))$ the second and third components of the solution of the BSDEJ associated to $\mathfrak y^\P(t',x)$ $$\begin{aligned}
&\underset{\P\in\Pc^\kappa_h}{\sup}\E^\P\left[{\left|\mathfrak y_t^\P(t',x)-\mathfrak y_{t'}^\P(t',x) \right|}\right]\\
&\leq \underset{\P\in\Pc^\kappa_h}{\sup}\E^\P\left[\int_t^{t'}{\left|f\left(s,x+B_s-B_{t'},\mathfrak y^\P(t',x),\mathfrak z^\P(t',x),\mathfrak u^\P(t',x),\widehat a_s,\nu^\P_s\right)\right|}ds\right]\\
&\leq C\sqrt{t'-t}\left(1+{\left|x\right|}+\underset{\P\in\Pc^\kappa_h}{\sup}\left\{{\left\|B\right\|}_{\D^2(\P)}^2+{\left\|\mathfrak y_s^\P(t',x)\right\|}_{\D^2(\P)}^2+{\left\|\mathfrak z_s^\P(t',x)\right\|}^2_{\H^2(\P)}+{\left\|\mathfrak u^\P(t',x)\right\|}_{\mathbb J^2(\P)}\right\}\right.\\
&\left.\hspace{0.9em}+\underset{\P\in\Pc^\kappa_h}{\sup}\mathbb E^\P\left[\int_0^T{\left|f(s,0,0,0,0,\widehat a_s,\nu^\P_s)\right|}^2ds\right]\right)\\
&\leq C(1+{\left|x\right|})\sqrt{t'-t},\end{aligned}$$ where we used the fact that $f$ is uniformly continuous in $x$, uniformly Lipschitz in $(y,z,u)$ and that $g$ is bounded and $f$ is sufficiently integrable. Hence the desired result.
We can now proceed to the
[**Viscosity super-solution**]{}: First of all, by Proposition \[prop.reg\] and Lemma \[lemma:unifcontt\], we know that the map $(t,x)\longmapsto u(t,x)$ is continuous. Let us now prove the viscosity super-solution property. Let $\varphi\in C^{3}_b([0,T)\times\R^d)$ and $(t_0,x_0)\in[0,T)\times\R^d$ be such that $$0=u(t_0,x_0)-\varphi(t_0,x_0)=\underset{(t,x)\in[0,T)\times \R^d}{\min}(u-\varphi)(t,x).$$ Fix some $\eta>0$ such that $t_0+\eta<T$. By with the constant family of stopping times $t_0+\eta$, we know that $$\begin{aligned}
\label{ineq:visco}
\nonumber\varphi(t_0,x_0)=u(t_0,x_0) &= \underset{\mathbb P\in\mathcal P^{t_0,\kappa}_h}{\sup} \mathcal Y^{\mathbb P,t_0,x_0}_{t_0}\left(t_0+\eta,u(t_0+\eta,B_{t_0+\eta}^{t_0,x_0})\right)\\
&\geq \underset{\mathbb P\in\mathcal P^{t_0,\kappa}_h}{\sup} \mathcal Y^{\mathbb P,t_0,x_0}_{t_0}\left(t_0+\eta,\varphi(t_0+\eta,B_{t_0+\eta}^{t_0,x_0})\right),\end{aligned}$$ where we used the comparison theorem for BSDEJs in the last inequality.
For notational simplicity, we denote by $(y^{\P,t_0,x_0,\varphi},z^{\P,t_0,x_0,\varphi},u^{\P,t_0,x_0,\varphi})$ the solution to the BSDEJ associated to $\mathcal Y^{\mathbb P,t_0,x_0}_{t_0}\left(t_0+\eta,\varphi(t_0+\eta,B_{t_0+\eta}^{t_0,x_0})\right)$. We also define for any $(\P,\omega,s,a,\nu)\in\Pc^{t_0,\kappa}_h\times\Omega\times[t_0,t_0+\eta]\times D^1_f\times (D^2_f\cap\mathfrak V^m)$ $$\begin{aligned}
&\mathcal L^{t_0,x_0}\varphi(\omega,s,a,\nu):= \varphi_t(s,B_s^{t_0,x_0}(\omega))+\frac12\Tr{aD^2\varphi(s,B_s^{t_0,x_0}(\omega)}+\int_E(A\varphi)(B_{s^-}^{t_0,x_0}(\omega),e)\nu(de)\\
&-f\left(s,B_s^{t_0,x_0}(\omega),\varphi(s,B_s^{t_0,x_0}(\omega)),D\varphi(s,B_s^{t_0,x_0}(\omega)),\mathcal K\varphi(s,B_{s^-}^{t_0,x_0}(\omega),\cdot),a,\nu\right).\end{aligned}$$ Let us then define $$\delta y_s^{\P,t_0,x_0,\varphi}:=y_s^{\P,t_0,x_0,\varphi}-\varphi(s,B_s^{t_0,x_0}),\ \delta z_s^{\P,t_0,x_0,\varphi}:=z_s^{\P,t_0,x_0,\varphi}-D\varphi(s,B_s^{t_0,x_0})$$ $$\delta u_s^{\P,t_0,x_0,\varphi}(\cdot):=u_s^{\P,t_0,x_0,\varphi}(\cdot)-\mathcal K\varphi(s,B_s^{t_0,x_0},\cdot)$$ $$\begin{aligned}
\delta f^{\P,t_0,x_0,\varphi}_s:=&\ f\left(s,B_s^{t_0,x_0},y_s^{\P,t_0,x_0,\varphi},z_s^{\P,t_0,x_0,\varphi},u_s^{\P,t_0,x_0,\varphi},\widehat a^{t_0}_s,\nu^{\P,t_0}_s\right)
\\
&-f\left(s,B_s^{t_0,x_0},\varphi(s,B_s^{t_0,x_0}),D\varphi(s,B_s^{t_0,x_0}),\mathcal K\varphi(s,B_{s^-}^{t_0,x_0},\cdot),\widehat a^{t_0}_s,\nu^{\P,t_0}_s\right).\end{aligned}$$
By a simple application of Itô’s formula under $\P$, we deduce that $$\begin{aligned}
\nonumber \delta y^{\P,t_0,x_0,\varphi}_{t_0}=&\ \int_{t_0}^{t_0+\eta}\mathcal L^{t_0,x_0}\varphi(s,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)ds-\int_{t_0}^{t_0+\eta}\delta f^{\P,t_0,x_0,\varphi}_sds-\int_{t_0}^{t_0+\eta}\delta z^{\P,t_0,x_0,\varphi}_sd(B^{t_0})^{c,\P}_s\\
&-\int_{t_0}^{t_0+\eta}\int_E\delta u^{\P,t_0,x_0,\varphi}_s(e)\left(\mu_{B^{t_0}}(de,ds)-\nu^{\P,t_0}_s(de)ds\right).\end{aligned}$$ Using the same arguments that lead us to , and in particular using the Lipschitz properties of $f$, we can define a positive càdlàg process $M'$, whose supremum has finite moments of any order such that $$\begin{aligned}
\label{eq:visco1}
\nonumber\delta y^{\P,t_0,x_0,\varphi}_{t_0}&\geq \E^\P\left[\int_{t_0}^{t_0+\eta}M'_s\mathcal L^{t_0,x_0}\varphi(s,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)ds\right]\\
\nonumber&\geq \E^\P\left[\int_{t_0}^{t_0+\eta}M'_s\left(\mathcal L^{t_0,x_0}\varphi(s,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)-\mathcal L^{t_0,x_0}\varphi(t_0,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)\right)ds\right]\\
&\hspace{0.9em}+\E^\P\left[\int_{t_0}^{t_0+\eta}M'_s\mathcal L^{t_0,x_0}\varphi(t_0,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)ds\right].\end{aligned}$$ By , we know that the left-hand side of is non-positive for every $\P\in\Pc^{t_0,\kappa}_h.$ Let us therefore fix some $(a,\nu)\in D^1_f\times (D^2_f\cap\mathfrak V^m)$ and consider the measure $\P(a,\nu)$ under which $\widehat a^{t_0}$ and $\nu^{\P(a,\nu),t_0}$ are equal to $a$ and $\nu$ respectively[^5]. We deduce that $$\begin{aligned}
\label{eq:visco2}
\nonumber&\Lc^{\P(a,\nu),t_0,x_0}\varphi(t_0,a,\nu)\int_{t_0}^{t_0+\eta}\E^{\P(a,\nu)}[M'_s]ds\\
&\leq \E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s\int_{t_0}^{t_0+\eta}{\left|\mathcal L^{t_0,x_0}\varphi(s,a,\nu)-\mathcal L^{t_0,x_0}\varphi(t_0,a,\nu)\right|}ds\right]
$$ Let us now estimate the right-hand side of . We first have, using the fact that $\varphi\in C^{3}_b([0,T)\times\R^d)$ and Lemma $$\begin{aligned}
\label{phit}
&\nonumber\E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s\int_{t_0}^{t_0+\eta}{\left|\varphi_t(s, B_s^{t_0,x_0})-\varphi_t(t_0,x_0)\right|}ds\right]\\
\nonumber&\leq C \E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s\int_{t_0}^{t_0+\eta}(s-t_0+{\left|B_s^{t_0}\right|})ds\right]\\
\nonumber &\leq C \left(\eta^2+\int_{t_0}^{t_0+\eta}\left(\E^{\P(a,\nu)}\left[\underset{t_0\leq t\leq T}{\sup}(M'_t)^2\right]\right)^{1/2}\left(\E^{\P(a,\nu)}\left[{\left|B_s^{t_0}\right|}^2\right]\right)^{1/2}ds\right)\\
&\leq C(\eta^2+\eta^{3/2}).\end{aligned}$$ Similarly, we obtain that $$\begin{aligned}
\label{phixx}
\E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s\int_{t_0}^{t_0+\eta}{\left|\frac12\Tr{aD^2\varphi(s,B_s^{t_0,x_0})}-\frac12\Tr{aD^2\varphi(t_0,x_0)}\right|}ds\right]\leq C(\eta^2+\eta^{3/2}).\end{aligned}$$ We also have $$\begin{aligned}
\label{phisaut}
\nonumber&\mathbb E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s\int_{t_0}^{t_0+\eta}{\left|\int_E(A\varphi)(B_{s^-}^{t_0,x_0}(\omega),e)\nu(de)-\int_E(A\varphi)(x_0,e)\nu(de)\right|}ds\right]\\
&\leq C\underset{\nu\in D^2_{f}\cap\mathfrak V^m}{\sup}\left\{\int_E{\left|e\right|}^2\nu(de)\right\}(\eta^2+\eta^{3/2}).\end{aligned}$$ Finally, for the term involving the generator $f$, we have, using that $f$ is uniformly continuous in $(t,x)$, and uniformly Lipschitz in $(y,z,u)$ $$\begin{aligned}
\label{phif}
\nonumber&\mathbb E^{\P(a,\nu)}\Big[\underset{t_0\leq s\leq T}{\sup}M'_s\int_{t_0}^{t_0+\eta}\Big|f(s,B_s^{t_0,x_0},\varphi(s,B_s^{t_0,x_0}),D\varphi(s,B_s^{t_0,x_0}),\Kc\varphi(s,B_s^{t_0,x_0},\cdot),a,\nu)\\
\nonumber &\hspace{3.5em}-f(t_0,x_0,\varphi(t_0,x_0),D\varphi(t_0,x_0),\Kc\varphi(t_0,x_0,\cdot),a,\nu)\Big|ds\Big]\\
\nonumber&\leq C\int_{t_0}^{t_0+\eta}\left(\rho(s-t_0)+\mathbb E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s\rho({\left|B_s^{t_0}\right|})\right]+{\left|s-t_0\right|}+\mathbb E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s{\left|B_s^{t_0}\right|}\right]\right)ds\\
&\hspace{0.9em}+C\int_{t_0}^{t_0+\eta}\int_E\E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s{\left|\Kc\varphi(s,B_s^{t_0,x_0},e)-\Kc\varphi(t_0,x_0,e)\right|}\right](1\wedge{\left|e\right|})\nu(de)ds\end{aligned}$$ Since $\rho$ is concave, by Jensen’s inequality, the first term on the right-side verifies $$\begin{aligned}
&\int_{t_0}^{t_0+\eta}\left(\rho(s-t_0)+\mathbb E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s\rho({\left|B_s^{t_0}\right|})\right]+{\left|s-t_0\right|}+\mathbb E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s{\left|B_s^{t_0}\right|}\right]\right)ds\\
&\leq C\left(\eta\rho(\eta)+\eta^2+\eta^{3/2}+\int_{t_0}^{t_0+\eta}\mathbb E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s\rho\left({\left|B_s^{t_0}\right|}\right)\right]ds\right).\end{aligned}$$ Then, we have by using twice the Cauchy-Schwarz inequality and using the fact that $\rho$ is concave and has linear growth $$\begin{aligned}
\mathbb E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s\rho\left({\left|B_s^{t_0}\right|}\right)\right]\leq &\ C\left(\E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}(M'_s)^4\right]\right)^{\frac14}\left(1+\E^{\P(a,\nu)}\left[{\left|B_s^{t_0}\right|}^2\right]\right)^{\frac14}\\
&\times\left(\E^{\P(a,\nu)}\left[\rho\left({\left|B_s^{t_0}\right|}\right)\right]\right)^{\frac12}\\
\leq &\ C\rho^{1/2}\left(\E^{\P(a,\nu)}\left[{\left|B_s^{t_0}\right|}\right]\right).\end{aligned}$$ Since $\rho^{1/2}$ is also concave, we obtain by Jensen’s inequality that $$\begin{aligned}
&\int_{t_0}^{t_0+\eta}\left(\rho(s-t_0)+\mathbb E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s\rho({\left|B_s^{t_0}\right|})\right]+{\left|s-t_0\right|}+\mathbb E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s{\left|B_s^{t_0}\right|}\right]\right)ds\\
&\leq C\left(\eta\rho(\eta)+\eta^2+\eta^{3/2}+\eta\rho^{1/2}(\sqrt{\eta})\right).\end{aligned}$$ For the second term, we have similarly $$\begin{aligned}
&\int_{t_0}^{t_0+\eta}\E^{\P(a,\nu)}\left[\underset{t_0\leq s\leq T}{\sup}M'_s\int_E{\left|\Kc\varphi(s,B_s^{t_0,x_0},e)-\Kc\varphi(t_0,x_0,e)\right|}\right](1\wedge{\left|e\right|})\nu(de)ds\\
&\leq C\underset{\nu\in D^2_{f}\cap\mathfrak V^m}{\sup}\left\{\int_E({\left|e\right|}\wedge{\left|e\right|}^2)\nu(de)\right\}(\eta^2+\eta^{3/2}),\end{aligned}$$ so that becomes $$\begin{aligned}
\label{phif2}
\nonumber&\mathbb E^{\P(a,\nu)}\Big[\underset{t_0\leq s\leq T}{\sup}M'_s\int_{t_0}^{t_0+\eta}\Big|f(s,B_s^{t_0,x_0},\varphi(s,B_s^{t_0,x_0}),D\varphi(s,B_s^{t_0,x_0}),\Kc\varphi(s,B_s^{t_0,x_0},\cdot),a,\nu)\\
\nonumber &\hspace{3.5em}-f(t_0,x_0,\varphi(t_0,x_0),D\varphi(t_0,x_0),\Kc\varphi(t_0,x_0,\cdot),a,\nu)\Big|ds\Big]\\
&\leq C(\eta\rho(\eta)+\eta\rho^{1/2}(\sqrt{\eta})+\eta^2+\eta^{3/2}).\end{aligned}$$ Hence, using , , and in , we obtain $$\begin{aligned}
&\Lc^{\P(a,\nu),t_0,x_0}\varphi(t_0,a,\nu)\int_{t_0}^{t_0+\eta}\E^{\P(a,\nu)}[M'_s]ds\\
&\leq C\underset{\nu\in D^2_{f}\cap\mathfrak V^m}{\sup}\left\{\int_E({\left|e\right|}{\bf 1}_{{\left|e\right|}>1}+{\left|e\right|}^2)\nu(de)\right\}\eta\left(\rho(\eta)+\rho^{1/2}(\sqrt{\eta})+\eta+\eta^{1/2}\right).\end{aligned}$$ Dividing both sides by $\eta$, letting $\eta$ go to $0$ and using the fact that $M'$ is càdlàg, we deduce that $$\E^{\P(a,\nu)}\left[M'_{t_0}\right]\Lc^{\P(a,\nu),t_0,x_0}\varphi(t_0,a,\nu)\leq 0,$$ which is the desired result since $M'$ is positive.
: Sub-solution property can be treated similarly, so we only detail the steps different from the proof for super-solution property. Let $\varphi\in C^{3}_b([0,T)\times\R^d)$ and $(t_0,x_0)\in[0,T)\times\R^d$ be such that $$0=u(t_0,x_0)-\varphi(t_0,x_0)=\underset{(t,x)\in[0,T)\times \R^d}{\max}(u-\varphi)(t,x).$$ Fix some $\eta>0$ such that $t_0+\eta<T$. By with the constant family of stopping times $t_0+\eta$, we know that $$\begin{aligned}
\label{ineq:viscosub}
\nonumber\varphi(t_0,x_0)=u(t_0,x_0) &= \underset{\mathbb P\in\mathcal P^{t_0,\kappa}_h}{\sup} \mathcal Y^{\mathbb P,t_0,x_0}_{t_0}\left(t_0+\eta,u(t_0+\eta,B_{t_0+\eta}^{t_0,x_0})\right)\\
&\leq \underset{\mathbb P\in\mathcal P^{t_0,\kappa}_h}{\sup} \mathcal Y^{\mathbb P,t_0,x_0}_{t_0}\left(t_0+\eta,\varphi(t_0+\eta,B_{t_0+\eta}^{t_0,x_0})\right),\end{aligned}$$ where we used the comparison theorem for BSDEJs in the last inequality.
By the same arguments that lead us to , and in particular using the Lipschitz properties of $f$ as Step $2$ of the proof of Theorem $4.1$ in [@kpz3], we can define a positive càdlàg process $M$, whose supremum has finite moments of any order such that $$\begin{aligned}
\nonumber\delta y^{\P,t_0,x_0,\varphi}_{t_0}&\leq \E^\P\left[\int_{t_0}^{t_0+\eta}M_s\mathcal L^{t_0,x_0}\varphi(s,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)ds\right]\\
\nonumber&\leq \E^\P\left[\int_{t_0}^{t_0+\eta}M_s\left(\mathcal L^{t_0,x_0}\varphi(s,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)-\mathcal L^{t_0,x_0}\varphi(t_0,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)\right)ds\right]\\
\nonumber&\hspace{0.9em}+\E^\P\left[\int_{t_0}^{t_0+\eta}M_s\mathcal L^{t_0,x_0}\varphi(t_0,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)ds\right].\\
\nonumber&\leq \E^\P\left[\int_{t_0}^{t_0+\eta}M_s\left(\mathcal L^{t_0,x_0}\varphi(s,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)-\mathcal L^{t_0,x_0}\varphi(t_0,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)\right)ds\right]\\
&\hspace{0.9em}+\underset{(a,\nu)\in\mathbb S^{>0}_d\times\mathfrak V^m}{\sup}\mathcal L^{t_0,x_0}\varphi(t_0,a,\nu)\eta\E^\P\left[\underset{t_0\leq s\leq T}{\sup}M_s\right].\end{aligned}$$
Taking supremum on both sides of the above equation and by , we deduce that $$\begin{aligned}
&-\frac1\eta\underset{\mathbb P\in\mathcal P^{t_0,\kappa}_h}{\sup} \E^{\P}\left[\underset{t_0\leq s\leq T}{\sup}M_s\int_{t_0}^{t_0+\eta}{\left|\mathcal L^{t_0,x_0}\varphi(s,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)-\mathcal L^{t_0,x_0}\varphi(t_0,\widehat a^{t_0}_s,\nu^{\P,t_0}_s)\right|}ds\right]\\
&\leq \underset{(a,\nu)\in\mathbb S^{>0}_d\times\mathfrak V^m}{\sup}\mathcal L^{t_0,x_0}\varphi(t_0,a,\nu)\underset{\mathbb P\in\mathcal P^{t_0,\kappa}_h}{\sup}\E^\P\left[\underset{t_0\leq s\leq T}{\sup}M_s\right].\end{aligned}$$ Exactly as in the proof of the super-solution property, we can show that the term on the l.h.s. of the above inequality tends to $0$ as $\eta$ goes to $0$, which finishes the proof by the positivity of $M$.
Appendix
========
Technical proofs
----------------
W.l.o.g., we assume that $t_1=0$ and $t_2=t$. Thus, we have to prove $$V_0(\omega)=\underset{\mathbb P\in \mathcal P^\kappa_H}{\sup}\mathcal Y_0^\mathbb P(t,V_t).$$
Denote $(y^\mathbb P,z^\mathbb P,u^\mathbb P):=(\mathcal Y^\mathbb P(T,\xi),\mathcal Z^\mathbb P(T,\xi),\mathcal U^\mathbb P(T,\xi))$
[(i)]{} For any $\mathbb P\in \mathcal P^\kappa_H$, we know by Lemma \[lemme.technique\] in the Appendix, that for $\mathbb P-a.e.$ $\omega\in\Omega$, the r.c.p.d. $\mathbb P^{t,\omega}\in\mathcal P^{t,\kappa}_H$. Now thanks to the paper of Tang and Li [@tangli], we know that the solution of BSDEJs on the Wiener-Poisson space with Lipschitz generator can be constructed via Picard iteration. Thus, it means that at each step of the iteration, the solution can be formulated as a conditional expectation under $\mathbb P$. By the properties of the r.p.c.d. and Proposition \[relationhata\], this entails that $$\label{eq.picard}
y_t^\mathbb P(\omega)=\mathcal Y_t^{\mathbb P^{t,\omega},t,\omega}(T,\xi), \text{ for } \mathbb P-a.e.\text{ } \omega\in\Omega.$$
Hence, by definition of $V_t$ and the comparison principle for BSDEJs, we get that $y_0^\mathbb P\leq \mathcal Y_0^\mathbb P(t,V_t)$. By arbitrariness of $\mathbb P$, this leads to $$V_0(\omega)\leq\underset{\mathbb P\in \mathcal P^\kappa_H}{\sup}\mathcal Y_0^\mathbb P(t,V_t).$$
[(ii)]{} For the other inequality, we proceed as in [@stz2]. Let $\mathbb P\in\mathcal P_H^\kappa$ and $\epsilon>0$. By separability of $\Omega$, there exists a partition $(E_t^i)_{i\geq 1}\subset \mathcal F_t$ such that $d_{S,t}(\omega,\omega')\leq \epsilon$ for any $i$ and any $\omega,\omega'\in E_t^i$.
Now for each $i$, fix a $\widehat \omega_i\in E_t^i$ and let $\mathbb P^i_t$ be an $\epsilon-$optimizer of $V_t(\widehat\omega_i)$. If we define for each $n\geq 1$, $\mathbb P^n:=\mathbb P^{n,\epsilon}$ by $$\begin{aligned}
\mathbb P^n(E):=\mathbb E^\mathbb P\left[\sum_{i=1}^n\mathbb E^{\mathbb P^i_t}\left[1_E^{t,\omega}\right]1_{E_t^i}\right]+\mathbb P(E\cap\widehat E^n_t),\text{ where } \widehat E^n_t:=\bigcup_{i>n}E^i_t, \label{multi_proba}\end{aligned}$$
then, by Lemma \[lemma\_multi\_proba\], we know that $\mathbb P^n\in \mathcal P^\kappa_H$. Besides, by Lemma \[unifcont\] and its proof, we have for any $i$ and any $\omega\in E_t^i$ $$\begin{aligned}
V_t(\omega)&\leq V_t(\widehat\omega_i)+C\rho(\epsilon)\leq\mathcal Y_t^{\mathbb P^i_t,t,\widehat\omega_i}(T,\xi)+\epsilon+C\rho(\epsilon)\\
&\leq \mathcal Y_t^{\mathbb P^i_t,t,\omega}(T,\xi)+\epsilon+C\rho(\epsilon)=\mathcal Y_t^{(\mathbb P^n)^{t,\omega},t,\omega}(T,\xi)+\epsilon+C\rho(\epsilon),\end{aligned}$$ where we used successively the uniform continuity of $V$ in $\omega$, the definition of $\mathbb P^i_t$, the uniform continuity of $\mathcal Y_t^{\mathbb P,t,\omega}$ in $\omega$ and finally the definition of $\mathbb P^n$.
Then, it follows from that $$\label{eq.ggggg}
V_t\leq y_t^{\mathbb P^n}+\epsilon+C\rho(\epsilon),\text{ }\mathbb P^n-a.s. \text{ on } \bigcup_{i=1}^nE_t^i.$$
Let now $(y^n,z^n,u^n):=(y^{n,\epsilon},z^{n,\epsilon},u^{n,\epsilon})$ be the solution of the following BSDEJ on $[0,t]$ $$\begin{aligned}
\label{grougrouref}
\nonumber y_s^n&=\left[y_t^{\mathbb P^n}+\epsilon+C\rho(\epsilon)\right]1_{\cup_{i=1}^nE_t^i}+V_t1_{\widehat E_t^n}+\int_s^t\widehat F^{\mathbb P^n}_r(y^n_r,z^n_r,u^n_r)dr-\int_s^tz^n_rdB^{\mathbb P^n,c}_r\\
&\hspace{0.9em}-\int_s^t\int_{E}u_r^n(x)\widetilde\mu^{\mathbb P^n}_B(dx,dr),\text{ }\mathbb P^n-a.s.\end{aligned}$$
By the comparison principle for BSDEJs, we know that $\mathcal Y^\mathbb P_0(t,V_t)\leq y_0^n$. Then since $\mathbb P^n=\mathbb P$ on $\mathcal F_t$, the equality also holds $\mathbb P-a.s.$ Using the same arguments and notations as in the proof of Lemma \[unifcont\], we obtain $${\left|y_0^n-y_0^{\mathbb P^n}\right|}^2\leq C\mathbb E^\mathbb P\left[\epsilon^2 +\rho(\epsilon)^2+{\left|V_t-y_t^{\mathbb P^n}\right|}^21_{\widehat E_t^n}\right].$$
Then, by Lemma \[unifcont\], we have $$\begin{aligned}
\mathcal Y_0^\mathbb P(t,V_t)\leq y_0^n&\leq y_0^{\mathbb P^n}+C\left(\epsilon+\rho(\epsilon)+\scriptstyle\left(\mathbb E^\mathbb P\left[\Lambda_t^2 1_{\widehat E_t^n}\right]\right)^{\frac12}\right)\leq V_0+C\left(\epsilon+\rho(\epsilon)+\scriptstyle\left(\mathbb E^\mathbb P\left[\Lambda_t^2 1_{\widehat E_t^n}\right]\right)^{\frac12}\right).\end{aligned}$$
Then it suffices to let $n$ go to $+\infty$, use the dominated convergence theorem, and let $\epsilon$ go to $0$.
For each $\mathbb P$, we define $\widetilde V^\mathbb P:=V-\mathcal Y^\mathbb P(T,\xi).$ Then, we recall that we have $$\widetilde V^\mathbb P\geq 0,\ \mathbb P-a.s.$$
Now for any $0\leq t_1< t_2\leq T$, let $(y^{\mathbb P,t_2},z^{\mathbb P,t_2},u^{\mathbb P,t_2}):=(\mathcal Y^\mathbb P(t_2,V_{t_2}),\mathcal Z^\mathbb P(t_2,V_{t_2}),\mathcal U^\mathbb P(t_2,V_{t_2}))$. Once more, we remind that since solutions of BSDEJs can be defined by Picard iterations, we have by the properties of the r.p.c.d. that $$\mathcal Y^\mathbb P_{t_1}(t_2,V_{t_2})(\omega)=\mathcal Y_{t_1}^{\mathbb P^{t_1,\omega},t_1,\omega}(t_2,V_{t_2}^{t_1,\omega}),\text{ for $\mathbb P-a.e.$ $\omega$}.$$
Hence, we conclude from Proposition \[progdyn\] that $V_{t_1}\geq y_{t_1}^{\mathbb P,t_2},\text{ }\mathbb P-a.s.$ Denote $$\widetilde y_t^{\mathbb P,t_2}:=y_t^{\mathbb P,t_2}-\mathcal Y^{\mathbb P}_t(T,\xi),\text{ }\widetilde z_t^{\mathbb P,t_2}:=\widehat a_t^{-1/2}(z_t^{\mathbb P,t_2}-\mathcal Z^{\mathbb P}_t(T,\xi)),\text{ }\widetilde u_t^{\mathbb P,t_2}:=u_t^{\mathbb P,t_2}-\mathcal U^{\mathbb P}_t(T,\xi).$$
Then $\widetilde V_{t_1}^\mathbb P\geq \widetilde y_{t_1}^{\mathbb P,t_2}$ and $(\widetilde y^{\mathbb P,t_2},\widetilde z^{\mathbb P,t_2},\widetilde u^{\mathbb P,t_2})$ satisfies the following BSDEJ on $[0,t_2]$ $$\widetilde y^{\mathbb P,t_2}_t=\widetilde V_{t_2}^\mathbb P+\int_t^{t_2}f_s^\mathbb P(\widetilde y^{\mathbb P,t_2}_s,\widetilde z^{\mathbb P,t_2}_s,\widetilde u_s^{\mathbb P,t_2})ds-\int_t^{t_2}\widetilde z^{\mathbb P,t_2}_sdW_s^\mathbb P-\int_t^{t_2}\int_{\mathbb R^d}\widetilde u^{\mathbb P,t_2}_s(x)\widetilde\mu^\mathbb P_B(dx,ds),$$ where $$\begin{aligned}
f_t^\mathbb P(\omega,y,z,u):&=\widehat F^\mathbb P_t(\omega,y+\mathcal Y^{\mathbb P}_t(\omega),\widehat a_t^{-1/2}(\omega)(z+\mathcal Z^{\mathbb P}_t(\omega)),u+{\mathcal U}^{\mathbb P}_t(\omega))\\
&\hspace{0.9em}-\widehat{F}^\mathbb P_t(\omega,\mathcal Y^{\mathbb P}_t(\omega),\mathcal Z^{\mathbb P}_t(\omega),\mathcal U^{\mathbb P}_t(\omega)).\end{aligned}$$
By the definition given in Royer [@roy], we conclude that $\widetilde V^\mathbb P$ is a positive $f^\mathbb P$-supermartingale under $\mathbb P$. Since $f^\mathbb P(0,0,0)=0$, we can apply the downcrossing inequality proved in [@roy] to obtain classically that for $\mathbb P-a.e.$ $\omega$, the limit $$\underset{r\in\mathbb Q\cup(t,T],r\downarrow t}{\lim}\widetilde V^\mathbb P_r(\omega)$$ exists for all $t$. Finally, since ${\mathcal Y}^{\mathbb P}$ is càdlàg, we obtain the desired result.
\[lemma.sde\] The SDE with initial condition $Z_{\tau^0_\eps}=0$ has a unique solution on $[\tau_0^\eps,T]$.
The proof follows the line of the proof of Example 4.5 in [@stz3], and we provide it for comprehensiveness. For simplicity, we only prove the result for $\tau_0^\eps=0$. This does not pertain any loss of generality, since the general result can proved similarly by working on shifted spaces instead. We proceed by induction and let $Z^{0,\eps}$ be the solution of the SDE $$Z^{\eps,0}_t=\int_0^t\left(a_0^\eps(Z^{\eps,0}_\cdot)\right)^{1/2}dB_s^{\P_{0,\tilde F^\eps},c}+\int_E\beta^\eps_0(Z^{\eps,0}_\cdot,x)\left(\mu_B(dx,ds)-\tilde F^\eps_s(dx)ds\right),\ \P_{0,\tilde F^\eps}-a.s.$$ Since $a_0^\eps$ and $\beta_0^\eps$ are actually $\mathcal F_0$-measurable, they are deterministic and thus $Z^{\eps,0}$ is indeed well-defined. Let then $\tilde \tau_0^\eps:=0$ and $\tilde \tau_1^\eps:=\tau_1^\eps(Z^{\eps,0})$. By Lemma $9.4$ in [@stz3], $\tilde \tau_1^\eps$ is still an $\F$-stopping time. We pursue the construction by setting $Z^{\eps,1}_t:=Z^{\eps,0}_t$ for $t\in[0,\tilde\tau_1^\eps]$ as well as, for $t\geq \tilde\tau_1^\eps$ $$Z^{\eps,1}_t=Z^{\eps,0}_{\tilde\tau_1^\eps}+\int_{\tilde\tau_1^\eps}^t\left(a_1^\eps(Z^{\eps,1}_\cdot)\right)^{1/2}dB_s^{\P_{0,\tilde F^\eps},c}+\int_E\beta^\eps_1(Z^{\eps,1}_\cdot,x)\left(\mu_B(dx,ds)-\tilde F^\eps_s(dx)ds\right),\ \P_{0,\tilde F^\eps}-a.s.$$
Using the fact that $a_1^\eps$ and $\beta_1^\eps$ are $\Fc_{\tau_1}$-measurable, we can then argue as in [@stz3] to obtain that $a_1^\eps(Z^{\eps,1})=a_1^\eps(Z^{\eps,0})$ and $\beta_1^\eps(Z^{\eps,1},x)=\beta_1^\eps(Z^{\eps,0},x)$. Therefore $Z^{\eps,1}$ is also well defined. By repeating the procedure for $n\geq 2$, and since we know that there exists $N\in\N$ such that $\tau_n^\eps=T$ for $n\geq N$, after a finite number of steps we have constructed the unique strong solution $Z^\eps$ to the SDE on $[0,T)$. Since it is càdlàg, we extend it at time $T$ by setting $Z_T^\eps:=\underset{t\uparrow T}{\lim}\ Z_t^\eps$, which finishes the construction.
The measures $\P^{\alpha,\beta}_F$
----------------------------------
\[stopping\_time\] Let $\tau$ be an $\F$-stopping time. Let $\omega \in \Omega$, $s\geq \tau(\omega)$ and $H$ be a $\Fc^{\tau(\omega)}_s$-measurable random variable. There exists a $\Fc_s$-measurable random variable $\widetilde H$, such that $$\begin{aligned}
H = \widetilde H^{\tau,\omega}. \label{def_S_tilde}\end{aligned}$$
means that for all $\widetilde \omega \in \Omega^{\tau(\omega)},$ $H(\widetilde \omega) =\widetilde H^{\tau,\omega}(\widetilde \omega) = \widetilde H(\omega\otimes_{\tau(\omega)}\widetilde \omega).$ We set $$\begin{aligned}
\forall \ \omega_1 \in \Omega, \; \widetilde H(\omega_1):= H(\omega_1^{\tau(\omega)}). \end{aligned}$$ Using the fact that for $s \geq \tau(\omega)$ $$\begin{aligned}
(\omega\otimes_{\tau(\omega)}\widetilde \omega)^{\tau(\omega)}(s)=\omega(\tau(\omega))+\widetilde \omega(s)-\omega(\tau(\omega))-\widetilde\omega(\tau(\omega))=\widetilde \omega(s),\end{aligned}$$ we have that $\widetilde H$ satisfies by construction. Indeed $$\begin{aligned}
\widetilde H^{\tau,\omega}(\widetilde \omega)= H\left[ \left( \omega\otimes_{\tau(\omega)}\widetilde \omega\right)^{\tau(\omega)}\right]=H(\widetilde \omega).\end{aligned}$$
Notice now that for any $\omega \in \Omega$, $\widetilde H: \Omega \rightarrow [\tau(\omega),T]$ is (Borel) measurable as a composition of measurable mappings. Finally, let us prove that $\widetilde H$ is $\Fc_s$-measurable. Since $H$ is $\Fc^{\tau(\omega)}$-measurable, there exists some measurable function $\phi$ such that for any $\widetilde \omega\in\Omega^{\tau(\omega)}$ $$H(\widetilde\omega)=\phi\left(B_t^{\tau(\omega)}(\widetilde\omega),\ \tau(\omega)\leq t\leq s\right).$$ Therefore, we have for any $\omega^1\in\Omega$ $$\widetilde H(\omega^1)=\phi\left(B_t(\omega^1)-B_\tau(\omega_1),\ \tau(\omega)\leq t\leq s\right),$$ which clearly implies that $\widetilde H$ is indeed $\Fc_s$-measurable.
\[lemme.technique\] Let $\P \in \overline{\mathcal P}_{S}$ and $\tau$ be an $\F$-stopping time. Then $$\P^{\tau,\omega} \in \overline{\mathcal P}_{S}^{\tau(\omega)} \text{ for } \; \P-a.e. \; \omega \in \Omega.$$
: Let us first prove that $$\begin{aligned}
(\P_{0,F})^{\tau,\omega} = \P_{\tau(\omega),F^{\tau,\omega}}, \;\; \P_{0,F}\text{-a.s. on } \Omega, \label{egalite.probas}\end{aligned}$$ where $(\P_{0,F})^{\tau,\omega}$ denotes the probability measure on $\Omega^{\tau}$, constructed from the regular conditional probability distribution (r.c.p.d.) of $\P_{0,F}$ for the stopping time $\tau$, evaluated at $\omega$, and $ \P_{\tau(\omega),F^{\tau,\omega}}$ is the unique solution of the martingale problem $(\P^1,\tau(\omega),T,I_d,F^{\tau,\omega})$, where $\P^1$ is such that $\P^1(B^{\tau}_{\tau}=0)=1$.
We recall that thanks to Remark 2.2 in [@kpz3], it is enough to show that outside a $\P_{0,F}$-negligible set, the shifted processes $M^{\tau}, J^{\tau},Q^{\tau}$ (which are defined in this Remark) are $(\P_{0,F})^{\tau,\omega}$-local martingales. In order to show this, for any $\omega$ in $\Omega$, and any $t\geq s\geq \tau(\omega)$, take any $\Fc_s^{\tau(\omega)} $-measurable random variable $H$. By Lemma \[stopping\_time\], there exists a $\Fc_s $-measurable random variable $\widetilde{H}$ such that $H=\widetilde H^{\tau,\omega}$. Then, following the definitions in Subsection \[parag.notations\], we have $$\begin{aligned}
\Delta B_t^{\tau,\omega}(\widetilde{\omega}) &= \Delta B_t (\omega \otimes_{\tau} \widetilde{\omega}) = \Delta (\omega \otimes_{\tau} \widetilde{\omega})(t) = \Delta \omega_t \mathbf{1}_{\{t \leq \tau \}} + \Delta \widetilde{\omega}_t \mathbf{1}_{\{t > \tau \}},\end{aligned}$$ and for $t \geq \tau$ $$\begin{aligned}
B_t (\omega \otimes_{\tau} \widetilde{\omega}) &= (\omega \otimes_{\tau} \widetilde{\omega})(t) = \omega_{\tau} +\widetilde{\omega}_t = B_{\tau}(\omega)+B_t^{\tau}(\widetilde{\omega}).\end{aligned}$$
From this we obtain $$\begin{aligned}
M_t^{\tau,\omega}(\widetilde{\omega}) = M_t (\omega \otimes_{\tau} \widetilde{\omega})=& \ B_t (\omega \otimes_{\tau} \widetilde{\omega}) - \sum_{u \leq t} \mathbf{1}_{{\left|\Delta B_u (\omega \otimes_{\tau} \widetilde{\omega}) \right|}>1} \Delta B_u (\omega \otimes_{\tau} \widetilde{\omega})\\
&+ \int_{0}^t\int_E x \mathbf{1}_{{\left|x\right|}>1}F_u(\omega \otimes_{\tau} \widetilde{\omega},dx) \,du\\
=& \ B_t^{\tau}(\widetilde{\omega}) + B_t(\omega) - \sum_{u \leq \tau} \mathbf{1}_{{\left|\Delta \omega_u\right|}>1} \Delta \omega_u - \sum_{\tau < u \leq t} \mathbf{1}_{{\left|\Delta B_u^{\tau}(\widetilde{\omega})\right|}>1} \Delta B_u^{\tau}(\widetilde{\omega})\\
&+ \int_{0}^{\tau}\int_E x \mathbf{1}_{{\left|x\right|}>1}F_u(\omega,dx)\,du +\int_{\tau}^t\int_E x \mathbf{1}_{{\left|x\right|}>1}F_u^{\tau,\omega}(\widetilde{\omega},dx)\,du\\
=& \ M_t^{\tau}(\widetilde{\omega}) + M_{\tau}(\omega), \;\; \forall \omega \in \Omega.\end{aligned}$$ Localizing if necessary, we can now compute $$\begin{aligned}
\E^{(\P_{0,F})^{\tau,\omega}} \left[ HM_t^{\tau} \right] &= \E^{(\P_{0,F})^{\tau,\omega}} \left[ H(M_t^{\tau,\omega} - M_{\tau}(\omega)) \right]\nonumber \\
&=\E^{(\P_{0,F})^{\tau,\omega}} \left[ \widetilde H^{\tau,\omega}M_{t}^{\tau,\omega} \right] - \E^{(\P_{0,F})^{\tau,\omega}} \left[ \widetilde H^{\tau,\omega}\right] M_{\tau}(\omega)\nonumber\\
&= \E^{\P_{0,F}}_{\tau}[\widetilde HM_{t}](\omega)- \E^{\P_{0,F}}_{\tau}[\widetilde H](\omega)M_{\tau}(\omega), \text{for $\mathbb P_{0,F}$-a.e. $\omega$}\nonumber\\
&=\E^{\P_{0,F}}_{\tau}[\widetilde HM_{s}](\omega)- \E^{\P_{0,F}}_{\tau}[\widetilde H](\omega)M_{\tau}(\omega), \text{for $\mathbb P_{0,F}$-a.e. $\omega$}\nonumber\\
&=\E^{(\P_{0,F})^{\tau,\omega}} \left[ HM_s^{\tau} \right]
, \text{for $\mathbb P_{0,F}$-a.e. $\omega$,} \label{eq1_lemme_technique}\end{aligned}$$
where we use the fact that $M$ is a $\P_{0,F}$-local martingale. Since $H$ is arbitrary, we have that $M^{\tau}$ is a $(\P_{0,F})^{\tau,\omega}$-local martingale for $\mathbb P_{0,F}$-a.e. $\omega$.
We treat the case of the process $J^{\tau}$ analogously and write $$\begin{aligned}
J_t^{\tau,\omega}(\widetilde{\omega}) =& \ \left(M_t^{\tau,\omega}(\widetilde{\omega}) \right)^2 - t - \int_0^{\tau} \int_E x^2 F_u(\omega,dx)du - \int_{\tau}^t \int_E x^2 F_u^{\tau,\omega}(\widetilde{\omega},dx)du \\
=& \ \left(M_t^{\tau}(\widetilde{\omega})\right)^2 + \left(M_{\tau}(\omega)\right)^2 +2 M_t^{\tau}(\widetilde{\omega}) M_{\tau}(\omega) - (t-\tau) - \int_{\tau}^t \int_E x^2 F_u^{\tau,\omega}(\widetilde{\omega},dx)du\\
&- \int_0^{\tau} \int_E x^2 F_u(\omega,dx)du - \tau\\
=& \ J_t^{\tau}(\widetilde{\omega}) + J_{\tau}(\omega) + 2 M_t^{\tau}(\widetilde{\omega}) M_{\tau}(\omega).\end{aligned}$$
Then we can compute the expectation, for $\P_{0,F}$-a.e. $\omega$ $$\begin{aligned}
\E^{(\P_{0,F})^{\tau,\omega}} \left[ HJ_t^{\tau} \right] &= \E^{(\P_{0,F})^{\tau,\omega}} \left[ HJ_t^{\tau,\omega} - 2H M_t^{\tau} M_{\tau}(\omega) \right] - \E^{(\P_{0,F})^{\tau,\omega}} \left[ H\right]J_{\tau}(\omega)\\
&=\E^{\P_{0,F}}_\tau \left[\widetilde HJ_t\right](\omega)-\E^{(\P_{0,F})^{\tau,\omega}} \left[ H M_s^\tau M_\tau(\omega)\right]-\E^{(\P_{0,F})^{\tau,\omega}} \left[ H\right]J_{\tau}(\omega)\\
&=\E^{\P_{0,F}}_\tau \left[\widetilde HJ_s\right](\omega)-\E^{(\P_{0,F})^{\tau,\omega}} \left[ H M_s^\tau M_\tau(\omega)\right]-\E^{(\P_{0,F})^{\tau,\omega}} \left[ H\right]J_{\tau}(\omega)\\
&=\E^{(\P_{0,F})^{\tau,\omega}} \left[ HJ_s^{\tau} \right].\end{aligned}$$
$J^{\tau}$ is then a $(\P_{0,F})^{\tau,\omega}$-local martingale $\text{for $\mathbb P_{0,F}$-a.e. $\omega$}$. Finally, we do the same kind of calculation for $Q^{\tau}$, and we obtain $$\begin{aligned}
Q_t^{\tau,\omega}(\widetilde{\omega}) =&\ \int_0^t \int_E g(x) \mu_B(\omega \otimes_{\tau} \widetilde{\omega},dx,du) - \int_0^t \int_E g(x) F_u^{\tau,\omega}(\widetilde{\omega},dx)\,du\\
=& \ \int_0^{\tau} \int_E g(x) \mu_B(\omega,dx,du) + \int_{\tau}^t \int_E g(x) \mu_{B^{\tau}}(\widetilde{\omega},dx,du) \\
&- \int_0^{\tau} \int_E g(x) F_u(\omega,dx)du - \int_{\tau}^t \int_E g(x) F_u^{\tau,\omega}(\widetilde{\omega},dx)\,du\\
=& \ Q_t^{\tau}(\widetilde{\omega}) + Q_{\tau}(\omega).\end{aligned}$$
And again we compute the expectation over the $\widetilde{\omega} \in \Omega^{\tau}$, under the measure $(\P_{0,F})^{\tau,\omega}$ $$\begin{aligned}
\E^{(\P_{0,F})^{\tau,\omega}} \left[ HQ_t^{\tau} \right] &= \E^{(\P_{0,F})^{\tau,\omega}} \left[ \widetilde H^{\tau,\omega}Q_t^{\tau,\omega} - HQ_{\tau}(\omega) \right]\\
&= \E^{\P_{0,F}}_{\tau}[\widetilde HQ_t](\omega)- \E^{(\P_{0,F})^{\tau,\omega}} \left[ H\right] Q_{\tau}(\omega),\text{ for $\mathbb P_{0,F}$-a.e. $\omega$}\\
&= \E^{(\P_{0,F})^{\tau,\omega}} \left[ HQ_s^{\tau} \right], \text{ for $\mathbb P_{0,F}$-a.e. $\omega$.}\end{aligned}$$
We have the desired result, and conclude that (\[egalite.probas\]) holds true. We can now deduce that for any $(\alpha,\beta)\in\mathcal D\times \Rc_F$ $$\P^{\tau(\omega),\alpha^{\tau,\omega},\beta^{\tau,\omega}}_{F^{\tau,\omega}} \in \overline{\mathcal P}_{S}^{\tau(\omega)}, \; \P_{0,F}\text{-a.s. on } \Omega. \label{step1}$$
Indeed, if $(\alpha,\beta) \in \Dc \times \Rc_F$, then $(\alpha^{\tau,\omega}, \beta^{\tau,\omega}) \in \Dc^{\tau(\omega)} \times \Rc_{F^{\tau(\omega)}}^{\tau(\omega)}$, because for $\P_{0,F}-a.e.\ \omega$ and for Lebesgue almost every $s\in [\tau(\omega),T]$ $$\int_{\tau(\omega)}^T {\left|\alpha_s^{\tau,\omega}(\widetilde{\omega})\right|}ds < \infty, (\P_{0,F})^{\tau,\omega}-a.s. \ (\text{and thus } \P_{\tau(\omega),F^{\tau,\omega}}-a.s.),$$ $${\left|\beta^{\tau,\omega}_s\right|}(\widetilde \omega,x)\leq C(1\wedge {\left|x\right|}), F^{\tau,\omega}(\widetilde \omega,dx)-a.e.,\ \text{for } \P_{\tau(\omega),F^{\tau,\omega}}-a.e. \ \widetilde \omega,$$ $$x\longmapsto \beta^{\tau,\omega}_s(\widetilde \omega, x) \text{ is strictly monotone for } F^{\tau,\omega}(\widetilde \omega,dx)-a.e. \ x,\ \text{for } \P_{\tau(\omega),F^{\tau,\omega}}-a.e. \ \widetilde \omega.$$
: We define $\widetilde{\tau}:= \tau \circ X^{\alpha,\beta}$, $\widetilde{\alpha}^{\tau,\omega} := \alpha^{\widetilde{\tau},\zeta_{\alpha,\beta}(\omega)}$, $\widetilde{F}^{\tau,\omega} := F^{\widetilde{\tau},\zeta_{\alpha,\beta}(\omega)}$ and $\widetilde{\beta}^{\tau,\omega} := \beta^{\widetilde{\tau},\zeta_{\alpha,\beta}(\omega)}$ where $\zeta_{\alpha,\beta}$ is a measurable map such that $B=\zeta_{\alpha,\beta}(X^{\alpha,\beta})$, $\P_{0,F}$-a.s. We refer to Lemma $2.2$ in [@stz2] (which can be proved similarly in our setting) for the existence of $\zeta_{\alpha,\beta}$. Moreover, $\widetilde{\tau}$ is an $\mathbb F$-stopping time and we have $\tau = \widetilde{\tau} \circ \zeta_{\alpha,\beta}$, $\P_{F}^{\alpha,\beta}-a.s.$ by definition. Using (\[step1\]), we deduce $$\P^{\tau(\omega),\widetilde{\alpha}^{\tau,\omega},\widetilde{\beta}^{\tau,\omega}}_{\widetilde F^{\tau,\omega}} \in \overline{\mathcal P}_{S}^{\tau(\omega)}, \ \P^{\alpha, \beta}_F\text{-a.s. on } \Omega.$$
: We show that $$\E^{\P^{\alpha, \beta}_F} \left[ \phi\left(B_{t_1\wedge \tau}, \dots, B_{t_n\wedge \tau}\right)\psi\left(B_{t_1},\dots,B_{t_n}\right)\right] = \E^{\P^{\alpha, \beta}_F} \left[ \phi\left(B_{t_1\wedge \tau}, \dots, B_{t_n\wedge \tau}\right)\psi_{\tau}\right]$$ for every $0<t_1<\dots<t_n\leq T$, every continuous and bounded functions $\phi$ and $\psi$ and where $$\psi_{\tau}(\omega) = \E^{\P^{\tau(\omega),\widetilde{\alpha}^{\tau,\omega},\widetilde{\beta}^{\tau,\omega}}_{\widetilde{F}^{\tau,\omega}}} \left[\psi(\omega(t_1), \dots, \omega(t_k), \omega(t)+B_{t_{k+1}}^t, \dots, \omega(t)+B_{t_{n}}^t) \right],$$ for $t:=\tau(\omega) \in [t_k, t_{k+1})$.
Recall that $\P^{\tau(\omega),\widetilde{\alpha}^{\tau,\omega},\widetilde{\beta}^{\tau,\omega}}_{\widetilde{F}^{\tau,\omega}}$ is defined by $\P^{\tau(\omega),\widetilde{\alpha}^{\tau,\omega},\widetilde{\beta}^{\tau,\omega}}_{\widetilde{F}^{\tau,\omega}} = \P_{\tau(\omega),\widetilde F^{\tau,\omega}} \circ \left(X^{\widetilde{\alpha}^{\tau,\omega},\widetilde{\beta}^{\tau,\omega}}\right)^{-1}$. For simplicity, we denote $$\widetilde \P:=\P_{\tau(\omega),\widetilde F^{\tau,\omega}}.$$ We then have $$\begin{split}
& \psi_{\tau}(\omega) = \E^{\widetilde{\P}} \Big[\psi\Big(\omega(t_1), \dots, \omega(t_k), \omega(t)+\int_t^{t_{k+1}} \left(\alpha_s^{\widetilde{\tau},\zeta_{\alpha,\beta}(\omega)}\right)^{1/2}d(B_s^{\tau(\omega)})^{\widetilde \P, c} \\
&\hspace{0.5em}+ \int_t^{t_{k+1}}\int_{E} \beta_s^{\widetilde\tau,\zeta_{\alpha,\beta}(\omega)}(x) \left (\mu_{B^{\tau(\omega)}}(dx,ds) - F_s^{\widetilde{\tau},\zeta_{\alpha,\beta}(\omega)}(dx)ds\right) , \dots, \omega(t)\\
&\hspace{0.5em}+\int_t^{t_{n}} \left(\alpha_s^{\widetilde{\tau},\beta_{\alpha}(\omega)}\right)^{1/2}d(B_s^{\tau(\omega)})^{\widetilde \P,c}+ \int_t^{t_{n}}\int_{E} \beta_s^{\widetilde\tau,\zeta_{\alpha,\beta}(\omega)}(x) \left (\mu_{B^{\tau(\omega)}}(dx,ds) - F_s^{\widetilde{\tau},\zeta_{\alpha,\beta}(\omega)}(dx)ds\right) \Big) \Big].
\end{split}$$
Denote $$\widehat \P:= \P_{\widetilde \tau,F^{\widetilde{\tau},\omega}}.$$ Then, $\forall \ \omega \in \Omega$, if $t:= \widetilde{\tau}(\omega) = \tau\left(X^{\alpha,\beta}(\omega)\right) \in [t_k, t_{k+1}[$, $$\begin{aligned}
\label{eq_psi_2}
\nonumber &\psi_{\tau}\left(X^{\alpha,\beta}(\omega)\right)=\E^{\widehat \P} \Big[ \psi\Big( X_{t_1}^{\alpha,\beta}(\omega), \dots, X_{t_k}^{\alpha,\beta}(\omega), X_{t}^{\alpha,\beta}(\omega) +\int_t^{t_{k+1}} \left(\alpha_s^{\widetilde{\tau},\omega}\right)^{1/2}d(B_s^{\widetilde \tau(\omega)})^{\widehat \P,c} \\
\nonumber
&\hspace{0.5em}+ \int_t^{t_{k+1}}\int_{E} \beta_s^{\widetilde\tau,\omega}(x) \left (\mu_{B^{\widetilde\tau(\omega)}}(dx,ds) - F_s^{\widetilde{\tau},\omega}(dx)ds\right) , \dots, X_{t}^{\alpha,\beta}(\omega)+\int_t^{t_{n}} \left(\alpha_s^{\widetilde{\tau},\omega}\right)^{1/2} d(B_s^{\widetilde{\tau}(\omega)})^{\widehat \P, c} \\
&\hspace{0.5em}+ \int_t^{t_{n}}\int_{E} \beta_s^{\widetilde\tau,\omega}(x)\left (\mu_{B^{\widetilde\tau(\omega)}}(dx,ds) - F_s^{\widetilde{\tau},\omega}(dx)ds\right) \Big) \Big].\end{aligned}$$
We remark that for every $\omega \in \Omega$, $$\begin{aligned}
&\alpha_s(\omega) = \alpha_s\left(\omega \otimes_{\widetilde{\tau}(\omega)} \omega^{\widetilde{\tau}(\omega)}\right) = \alpha_s^{\widetilde{\tau},\omega}\left(\omega^{\widetilde{\tau}(\omega)}\right),\end{aligned}$$ and similar relations hold for both $F$ and $\beta$.
By definition, the $(\P_{0,F})^{\widetilde{\tau},\omega}$-distribution of $B^{\widetilde{\tau}(\omega)}$ is equal to the $(\P_{0,F})^{\omega}_{\widetilde{\tau}}$-distribution of $(B_{\cdot} - B_{\widetilde{\tau}(\omega)})$. Therefore since by , $(\P_{0,F})^{\widetilde \tau,\omega} = \P_{\widetilde\tau(\omega),F^{\widetilde \tau,\omega}}, \;\; \P_{0,F}\text{-a.s. on } \Omega$, (\[eq\_psi\_2\]) then becomes $$\begin{split}
\psi_{\tau}\left(X^{\alpha,\beta}(\omega)\right)=\ &\E^{(\P_{0,F})^{\omega}_{\widetilde{\tau}}} \Big[ \psi\Big( X_{t_1}^{\alpha,\beta}(\omega), \dots, X_{t_k}^{\alpha,\beta}(\omega), X_{t}^{\alpha,\beta}(\omega) +\int_t^{t_{k+1}} \alpha_s^{1/2}(B)d(B_s^{(\P_{0,F})^\omega_{\widetilde \tau},c}) \\
&\hspace{-2.7em}+ \int_t^{t_{k+1}}\int_{E} \beta_s(x) (\mu_{B}(dx,ds) - F_s(dx)ds) , \dots, X_{t}^{\alpha,\beta}(\omega)+\int_t^{t_{n}} \alpha_s^{1/2}(B)d(B_s^{(\P_{0,F})^\omega_{\widetilde \tau},c}) \\
&\hspace{-2.7em}+ \int_t^{t_{n}}\int_{E} \beta_s(x) (\mu_{B}(dx,ds) - F_s(dx)ds) \Big) \Big] \\
=\ & \E^{(\P_{0,F})^{\omega}_{\widetilde{\tau}}} \Big[ \psi\Big( X_{t_1}^{\alpha,\beta}, \dots, X_{t_k}^{\alpha,\beta}, X_{t_{k+1}}^{\alpha,\beta}, \dots, X_{t_n}^{\alpha,\beta} \Big) \Big] \\
=\ & \E^{\P_{0,F}} \left[ \psi\Big( X_{t_1}^{\alpha,\beta}, \dots, X_{t_k}^{\alpha,\beta}, X_{t_{k+1}}^{\alpha,\beta}, \dots, X_{t_n}^{\alpha,\beta} \Big) | \mathcal{F}_{\widetilde{\tau}} \right](\omega), \; \; \P_{0,F}\text{-a.s. on } \Omega.
\end{split}$$
Then we have $$\begin{split}
& \E^{\P^{\alpha, \beta}_F} \left[ \phi\left(B_{t_1\wedge \tau}, \dots, B_{t_n\wedge \tau}\right)\psi_{\tau}\right] = \E^{\P_{0,F}} \left[ \phi\Big( X_{t_1\wedge \widetilde{\tau}}^{\alpha,\beta}, \dots, X_{t_n\wedge \widetilde{\tau}}^{\alpha,\beta} \Big)\psi_{\widetilde{\tau}}(X^{\alpha,\beta}) \right] \\
& = \E^{\P_{0,F}} \left[ \phi\Big( X_{t_1\wedge \widetilde{\tau}}^{\alpha,\beta}, \dots, X_{t_n\wedge \widetilde{\tau}}^{\alpha,\beta} \Big) \E^{\P_{0,F}} \left[ \psi\Big( X_{t_1}^{\alpha,\beta}, \dots, X_{t_k}^{\alpha,\beta}, X_{t_{k+1}}^{\alpha,\beta}, \dots, X_{t_n}^{\alpha,\beta} \Big) | \mathcal{F}_{\widetilde{\tau}} \right] \right] \\
&= \E^{\P_{0,F}} \left[ \phi\Big( X_{t_1\wedge \widetilde{\tau}}^{\alpha,\beta}, \dots, X_{t_n\wedge \widetilde{\tau}}^{\alpha,\beta} \Big) \psi\Big( X_{t_1}^{\alpha,\beta}, \dots, X_{t_k}^{\alpha,\beta}, X_{t_{k+1}}^{\alpha,\beta}, \dots, X_{t_n}^{\alpha,\beta} \Big) \right] \\
& = \E^{\P^{\alpha, \beta}_F} \left[ \phi\left(B_{t_1\wedge \tau}, \dots, B_{t_n\wedge \tau}\right) \psi\left(B_{t_1}, \dots, B_{t_n}\right)\right].
\end{split}$$
: Now we prove that $\P^{\tau,\omega} = \P^{\widetilde{\alpha}^{\tau,\omega},\widetilde{\beta}^{\tau,\omega}}_{\widetilde{F}^{\tau,\omega}}$, $\P$-a.s. on $\Omega$.
By definition of the conditional expectation, $$\psi_{\tau}(\omega) = \E^{\P^{\tau,\omega}} \left[\psi(\omega(t_1), \dots, \omega(t_k), \omega(t)+B_{t_{k+1}}^t, \dots, \omega(t)+B_{t_{n}}^t) \right], \text{ $\P^{\alpha,\beta}_F$-a.s.},$$ where $t:= \tau(\omega) \in [t_k, t_{k+1}[$, and where the $\P^{\alpha,\beta}_F$-null set can depend on $(t_1, \dots, t_n)$ and $\psi$, but we can choose a common null set by standard approximation arguments.
Then by a density argument we obtain $$\E^{\P^{\tau,\omega}}\left[\eta\right] = \E^{\P^{\widetilde{\alpha}^{\tau,\omega},\widetilde{\beta}^{\tau,\omega}}_{\widetilde{F}^{\tau,\omega}}} \left[\eta\right], \text{ for $\mathbb P^{\alpha,\beta}_F$-a.e. $\omega$},$$ for every bounded and $\mathcal{F}_T^{\tau(\omega)}$-measurable random variable $\eta$. This implies $\P^{\tau,\omega} = \P^{\widetilde{\alpha}^{\tau,\omega},\widetilde{\beta}^{\tau,\omega}}_{\widetilde{F}^{\tau,\omega}}$, $\P$-a.s. on $\Omega$. And from the Step 1 we deduce that $\P^{\tau,\omega} \in \overline{\mathcal P}_{S}^{\tau(\omega)}$.
\[lemma\_multi\_proba\] We have $\P^n \in \Pc^{\kappa}_H$, where $\P^n$ is defined by (\[multi\_proba\]).
Since by definition, $\P^i_t \in \Pc^{t,\kappa}_H$ and $\P \in \Pc^\kappa_H$, we have $\P^i_t = \P^{\alpha^i, \beta^i}_{F^i}$ and $\P = \P^ {\alpha, \beta}_F$, for $F^i\in\Vc^t $, $(\alpha^i, \beta^i) \in \Dc^t\times\Rc_{F^i}^t$ and $(F,\alpha,\beta) \in \Vc\times\Dc\times\Rc_F$, $i=1, \dots, n$. Next we define $$\begin{aligned}
\overline{\alpha}_s &:= \alpha_s \mathbf{1}_{[0,t)}(s) + \left[\sum_{i=1}^n \alpha^i_s \mathbf{1}_{E^i_t}(X^{\alpha,\beta}) + \alpha_s \mathbf{1}_{\Eh^n_t}(X^{\alpha,\beta}) \right] \mathbf{1}_{[t,T]}(s), \text{ and}\\
\overline{F}_s &:= F_s \mathbf{1}_{[0,t)}(s) + \left[\sum_{i=1}^n F^i_s \mathbf{1}_{E^i_t}(X^{\alpha,\beta}) + F_s \mathbf{1}_{\Eh^n_t}(X^{\alpha,\beta}) \right] \mathbf{1}_{[t,T]}(s), \text{ and}\\
\overline{\beta}_s &:= \beta_s \mathbf{1}_{[0,t)}(s) + \left[\sum_{i=1}^n \beta^i_s \mathbf{1}_{E^i_t}(X^{\alpha,\beta}) + \beta_s \mathbf{1}_{\Eh^n_t}(X^{\alpha,\beta}) \right] \mathbf{1}_{[t,T]}(s).\end{aligned}$$ Now following the arguments in the proof of step 3 of Lemma \[lemme.technique\], we can prove that for any $0<t_1< \dots < t_k=t<t_{k+1}<t_n$ and any continuous and bounded functions $\phi$ and $\psi$, $$\begin{aligned}
&\E^{\P^{\alpha,\beta}_F} \left[ \phi(B_{t_1}, \dots, B_{t_k}) \sum_{i=1}^n \E^{\P^{\alpha^i,\beta^i}_{F^i}} \left[\psi(B_{t_1}, \dots, B_{t_k},B_t + B^t_{t_{k+1}}, \dots, B_t + B^t_{t_n})\right]\mathbf{1}_{E^i_t} \right]\\
&= \E^{\P^{\overline{\alpha},\overline{\beta}}_{\overline{F}}} \left[ \phi(B_{t_1}, \dots, B_{t_k})\psi(B_{t_1}, \dots, B_{t_n}) \right].\end{aligned}$$ This implies that $\P^n = \P^{\overline{\alpha},\overline{\beta}}_{\overline{F}} \in \overline{\Pc}_{S}$. And since all the probability measures $\P^i$ satisfy the requirements of Definition \[set\_proba\_shift\], we have $\P^n \in \Pc^{\kappa}_H$.
A weak dynamic programming principle
------------------------------------
The proof follows closely the steps of Proposition $5.14$ and Lemma $6.2$ and $6.4$ in [@stz].
Let us first fix $\P$ and $X$ and denote $\tau = \tau^{\P}$ for simplicity. By the a priori estimates and recalling definition \[grandlambda.def\], we have ${\left|u(t,x)\right|} \leq C \Lambda(t,x)$ and then we can assume w.l.o.g that ${\left|X\right|} \leq C \Lambda(\tau,B_{\tau}^{t,x})$, $\P$-a.s. Then by assumption \[assump.markovian\], $X$ is in $\L^2(\P)$ and hence $\mathcal Y^{\mathbb P,t,x}_t\left(\tau,X\right)$ is well defined. Now by , $$\begin{aligned}
\mathcal Y^{\mathbb P,t,x}_t\left(T,g(B_T^{t,x})\right) = \mathcal Y^{\mathbb P,t,x}_t\left(\tau,\mathcal Y^{\mathbb P^{\tau,\omega},\tau(\omega),B_{\tau}^{t,x}(\omega)}_{\tau}\left( T,g(B_T^{\tau(\omega),B_{\tau}^{t,x}(\omega)})\right)\right).\end{aligned}$$ By Lemma \[lemme.technique\], $\P^{\tau,\omega} \in \mathcal{P}_h^{\kappa,\tau(\omega)}$, for $\P$-a.e. $\omega \in \Omega^t$. Next by definition of $u$ and $X$, $$\begin{aligned}
\mathcal Y^{\mathbb P^{\tau,\omega},\tau(\omega),B_{\tau}^{t,x}(\omega)}_{\tau}\left( T,g(B_T^{\tau(\omega),B_{\tau}^{t,x}(\omega)})\right) &\leq u\left(\tau(\omega),B_{\tau}^{t,x}(\omega) \right)\leq X(\omega), \; \text{ for $\P$-a.e. $\omega \in \Omega^t$.}\end{aligned}$$ Finally, the comparison theorem for standard BSDEJs implies that $$\begin{aligned}
\mathcal Y^{\mathbb P,t,x}_t\left(T,g(B_T^{t,x})\right) \leq \mathcal Y^{\mathbb P,t,x}_t\left(\tau,X\right),\end{aligned}$$ and taking the supremum over $\P$ yields inequality \[partial.dpp1\].
Let us now prove equality when we know in addition that $g$ lower semi-continuous . By Proposition \[prop.reg\], $u$ is lower semi-continuous in the variables $(t,x)$, from the right in $t$, and therefore $u$ is measurable and $u(\tau,B_{\tau}^{t,x})$ is $\mathcal{F}_{\tau}$-measurable. This implies that inequality \[partial.dpp1\] holds for the particular choice $X=u(\tau,B_{\tau}^{t,x})$.
We now prove the reverse inequality. The first step is to show that $$\begin{aligned}
\mathcal Y^{\mathbb P,s,x}_s\left(t,\phi(B_t^{s,x})\right) \leq u(s,x),\label{dpp_step1}\end{aligned}$$ for any $s<T$ and fixed $t\in (s,T]$ and for any continuous real-valued function $\phi$ such that $-\Lambda(t,\cdot)\leq \phi(\cdot)\leq \Lambda(t,\cdot)$. Following step 2 of the proof of Theorem $3.5$ in [@bt], for any $\epsilon >0$, we can find sequences $(x_i,r_i)_{i\geq 1} \subset \R^d \times (0,T]$ and $\P_i\in \mathcal{P}_h^{\kappa,t}$ such that $$\begin{aligned}
\mathcal Y^{\mathbb \P_i,t,\cdot}_t\left(T,g(B_T^{t,\cdot})\right)\geq \phi(\cdot)-\epsilon \text{ on } B(x_i,r_i),\end{aligned}$$ where $B(x_i,r_i)$ denote the open balls centered at $x_i$ with radius $r_i$, which form a countable cover of $\R^d$. From this we can build a partition $(A_i)_{i\geq 1}$ of $\R^d$ defined by $A_i:=B(x_i,r_i) \backslash \cup_{j<i}B(x_j,r_j)$ and a partition of $\Omega$ defined in the following way $$\begin{aligned}
E^i:=\{B_t^{s,x} \in A_i\}, \; i\geq 1, \text{ and } \, \widehat E^n:= \cup_{i>n}E^i, \; n\geq 1.\end{aligned}$$ Then $$\begin{aligned}
\Omega = \left(\cup_{i=1}^n E^i \right)\cup \widehat E^n \;\text{ and } \, \lim_{n\to +\infty}\P(\widehat E^n)=0.\end{aligned}$$ Exactly as in , we define $$\begin{aligned}
\mathbb P^n(E):=\mathbb E^\mathbb P\left[\sum_{i=1}^n\mathbb E^{\mathbb P^i}\left[1_E^{t,\omega}\right]1_{E^i}\right]+\mathbb P(E\cap\widehat E^n),\text{ for all } E \in \mathcal F_T^s.\end{aligned}$$ It follows from Lemma \[lemma\_multi\_proba\] and the definition of $\P^n$ that $\P^n$ coincide with $\P$ on $\mathcal F_t$, $\P^n \in \mathcal P_h^{s,\kappa}$ and $(\P^n)^{s,\omega}=\P^i$, for $\P$-a.e. $\omega$ in $E_i$, $1\leq i\leq n$. Then we have $$\begin{aligned}
\mathcal Y^{\P^n,s,x}_t\left(T,g(B_T^{s,x})\right)(\omega) &= \mathcal Y^{\P^i,t,B_t^{s,x}(\omega)}_t\left(T,g(B_T^{t,B_t^{s,x}(\omega)})\right)\\
&\geq \phi(B_t^{s,x}(\omega))-\epsilon \;\;\;\; \text{ for $\P$-a.e. $\omega$ in $E_i$, $1\leq i\leq n$. }\end{aligned}$$ We use now the comparison theorem for BSDEs: $$\begin{aligned}
u(s,x) &\geq \mathcal Y^{\P^n,s,x}_s\left(T,g(B_T^{s,x})\right) = \mathcal Y^{\mathbb \P,s,x}_s\left(t,\mathcal Y^{\P^n,s,x}_t(T,g(B_T^{s,x}))\right)\\
&\geq \mathcal Y^{\P,s,x}_s\left(t,\phi(B_t^{s,x})-\epsilon \right) \mathbf{1}_{(\widehat E^n)^c} + \mathcal Y^{\P^n,s,x}_t(T,g(B_T^{s,x}))\mathbf{1}_{(\widehat E^n)}\end{aligned}$$ Since $\epsilon$ was arbitrary, we can use the stability results for BSDEs to conclude that inequality holds true.
The rest of the proof is exactly the same as the end of the proof of Lemma $6.4$ in [@stz]: we use the lower semi-continuity of $u(t,\cdot)$ to approximate it by an increasing sequence of continuous functions and we use inequality to prove the desired inequality for constant stopping times. Then we have the inequality for stopping times taking finitely many values by a simple backward induction and for stopping times taking countably many values by a limiting argument. Finally we approximate an arbitrary stopping time by a decreasing sequence of stopping times taking countably many values, and then we only need the function $u$ to be lower semi-continuous in $(t,x)$ from the right in $t$, to proceed exactly as in [@stz].
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[^1]: CMAP, Ecole Polytechnique, Paris, [email protected].
[^2]: CEREMADE, Université Paris-Dauphine, Paris, [email protected]. Part of this work was carried out while the author was working at CMAP, Ecole Polytechnique, whose financial support is kindly acknowledged.
[^3]: Department of Mathematics, National University of Singapore, Singapore, [email protected]. Part of this work was carried out while the author was working at CMAP, Ecole Polytechnique, whose financial support is kindly acknowledged.
[^4]: As mentioned in the introduction, the results of [@nvh] have actually been extended very recently to a jump setting in [@nn2; @nn3]. The authors do manage, in the case of a null generator, to construct what is actually a solution to the corresponding 2BSDEJ, without any regularity assumptions on the terminal condition.
[^5]: Notice that since $a$ and $\nu$ are deterministic, such a measure does exist and is in $\Pc^{t_0,\kappa}_h$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Over the past 50 years, electron-nuclear double resonance (ENDOR) has become a fairly ubiquitous spectroscopic technique, allowing the study of spin transitions for nuclei which are coupled to electron spins. However, the low spin number sensitivity of the technique continues to pose serious limitations. Here we demonstrate that signal intensity in a pulsed Davies ENDOR experiment depends strongly on the nuclear relaxation time T$_{1n}$, and can be severely reduced for long T$_{1n}$. We suggest a development of the original Davies ENDOR sequence that overcomes this limitation, thus offering dramatically enhanced signal intensity and spectral resolution. Finally, we observe that the sensitivity of the original Davies method to T$_{1n}$ can be exploited to measure nuclear relaxation, as we demonstrate for phosphorous donors in silicon and for endohedral fullerenes N@C$_{60}$ in CS$_2$.'
author:
- 'Alexei M. Tyryshkin'
- 'John J. L. Morton'
- Arzhang Ardavan
- 'S. A. Lyon'
bibliography:
- 'bib.bib'
title: |
Davies ENDOR revisited:\
Enhanced sensitivity and nuclear spin relaxation
---
Introduction
============
Electron-nuclear double resonance (ENDOR) belongs to a powerful family of polarization transfer spectroscopic methods and permits the measurement of small energy (nuclear spin) transitions at the much enhanced sensitivity of higher energy (electron spin) transitions [@feher56]. ENDOR is thus an alternative to NMR methods, with the benefits of improved spin-number sensitivity and a specific focus on NMR transitions of nuclei coupled to paramagnetic species (reviewed in Refs [@kevan76; @schweiger01]).
In an ENDOR experiment, the intensity of an electron paramagnetic resonance (EPR) signal (e.g. an absorption signal in continuous wave EPR, or a spin echo signal in pulsed EPR) is monitored while strong RF irradiation is applied to excite nuclear spin transitions of the nuclei that are coupled to the electron spin. Although the EPR signal may be strong, the RF-induced changes are often rather weak and therefore it is quite common to find the ENDOR signal to constitute only a few percent of the total EPR signal intensity. Many different ENDOR schemes have been developed to improve sensitivity and spectral resolution of the ENDOR signal and to aid in analysis of congested ENDOR spectra [@kevan76; @schweiger01; @gemperle91]. However, low visibility of the ENDOR signal remains a common problem to all known ENDOR schemes, and long signal averaging (e.g. hours to days) is often required to observe the ENDOR spectrum at adequate spectral signal/noise.
A low efficiency in spin polarization transfer (and thus low intensity of the ENDOR response) is inherent to continuous wave ENDOR experiments, which depend critically on accurate balancing of the microwave and RF powers applied to saturate the electron and nuclear spin transitions, and various spin relaxation times within the coupled electron-nuclear spin system, including the electron and nuclear spin-lattice relaxation times, T$_{1e}$ and T$_{1n}$, and also the cross-relaxation (flip-flop) times, T$_{1\rm{x}}$ [@dalton72]. The ENDOR signal is measured as a partial de-saturation of the saturated EPR signal and generally constitutes a small fraction of the full EPR signal intensity [@kevan76]. Since spin relaxation times are highly temperature dependent, balancing these factors to obtain a maximal ENDOR response is usually only possible within a narrow temperature range.
Pulsed ENDOR provides many improvements over the continuous wave ENDOR methods [@gemperle91; @schweiger01] and most importantly eliminates the dependence on spin relaxation effects by performing the experiment on a time scale which is short compared to the spin relaxation times. Furthermore, combining microwave and RF pulses enables 100$\%$ transfer of spin polarization, and therefore the pulsed ENDOR response can in principle approach a 100$\%$ visibility (we define the ENDOR visibility as change in the echo signal intensity induced by the RF pulse, normalized to the echo intensity in the absence of the pulse [@schweiger01; @epel01]). In practice, the situation is far from perfect and it is common to observe a pulsed ENDOR response of the level of a few percent, comparable to continuous wave ENDOR. In this paper we discuss the limitations of the pulsed ENDOR method, and specifically Davies ENDOR [@davies74]. We suggest a modification to the pulse sequence which dramatically enhances the signal/noise and can also improve spectral resolution. We also show how traditional Davies ENDOR may be used to perform a measurement of the nuclear relaxation time, T$_{1n}$. While not discussed in this manuscript, a similar modification is also applicable to Mims ENDOR method [@mims65].
Materials and Methods
=====================
We demonstrate the new ENDOR techniques using two samples: phosphorus $^{31}$P donors in silicon, and endohedral fullerenes $^{14}$N@C$_{60}$ (also known as *i*-NC$_{60}$) in CS$_2$ solvent. Silicon samples were epitaxial layers of isotopically-purified $^{28}$Si (a residual $^{29}$Si concentration of $\sim 800$ ppm as determined by secondary ion mass spectrometry [@itoh04]) grown on p-type natural silicon (Isonics). The epi-layers were 10 $\mu$m thick and doped with phosphorus at $1.6\cdot 10^{16}$ P/cm$^{3}$. Thirteen silicon pieces (each of area 9$\times$3 mm$^2$) were stacked together to form one EPR sample. This sample is referred as $^{28}$Si:P in the text.
N@C$_{60}$ consists of an isolated nitrogen atom in the $^4$S$_{3/2}$ electronic state incarcerated in a C$_{60}$ fullerene cage. Our production and subsequent purification of N@C$_{60}$ is described elsewhere [@mito]. High-purity N@C$_{60}$ powder was dissolved in CS$_{2}$ to a final concentration of 10$^{15}$/cm$^3$, freeze-pumped to remove oxygen, and finally sealed in a quartz tube. Samples were 0.7 cm long, and contained approximately $5\cdot 10^{13}$ N@C$_{60}$ molecules.
Both $^{28}$Si:P and N@C$_{60}$ can be described by a similar isotropic spin Hamiltonian (in angular frequency units): $$\label{Hamiltonian}
\mathcal{H}_0=\omega_e S_z - \omega_I I_z + a \!\cdot\! \vec{S}
\!\cdot\! \vec{I},$$ where $\omega_e=g\beta B_0/\hbar$ and $\omega_I=g_I\beta_n
B_0/\hbar$ are the electron and nuclear Zeeman frequencies, $g$ and $g_I$ are the electron and nuclear g-factors, $\beta$ and $\beta_n$ are the Bohr and nuclear magnetons, $\hbar$ is Planck’s constant and $B_0$ is the magnetic field applied along $z$-axis in the laboratory frame. In the case of $^{28}$Si:P, the electron spin S=1/2 (g-factor = 1.9987) is coupled to the nuclear spin I=1/2 of $^{31}$P through a hyperfine coupling $a=117$ MHz (or 4.19 mT) [@fletcher54; @feher59]. The X-band EPR signal of $^{28}$Si:P consists of two lines (one for each nuclear spin projection $M_I = \pm 1/2$). Our ENDOR measurements were performed at the high-field line of the EPR doublet corresponding to $M_I=-1/2$. In the case of N@C$_{60}$, the electron has a high spin S=3/2 (g-factor = 2.0036) that is coupled to a nuclear spin I=1 of $^{14}$N through an isotropic hyperfine coupling $a=15.7$ MHz (or 0.56 mT) [@murphy96]. The N@C$_{60}$ signal comprises three lines and our ENDOR experiments were performed on the central line ($M_I=0$) of the EPR triplet.
Pulsed EPR experiments were performed using an X-band Bruker EPR spectrometer (Elexsys 580) equipped with a low temperature helium-flow cryostat (Oxford CF935). The temperature was controlled with a precision greater than $0.05$ K using calibrated temperature sensors (Lakeshore Cernox CX-1050-SD) and an Oxford ITC503 temperature controller. This precision was needed because of the strong temperature dependence of the electron spin relaxation times in the silicon samples (T$_{1e}$ varies by five orders of magnitude between 7 K and 20 K) [@alexeisi]. Microwave pulses for $\pi$/2 and $\pi$ rotations of the electron spin were set to 32 and 64 ns for the $^{28}$Si:P sample, and to 56 and 112 ns for the N@C$_{60}$ sample, respectively. In each case the excitation bandwidth of the microwave pulses was greater than the EPR spectral linewidth (e.g. 200 kHz for $^{28}$Si:P [@alexeisi], and 8.4 kHz for N@C$_{60}$ [@eseem05]) and therefore full excitation of the signal was achieved. RF pulses of 20-50 $\mu$s were used for $\pi$ rotations of the $^{31}$P nuclear spins in $^{28}$Si:P and the $^{14}$N nuclear spins in N@C$_{60}$.
Standard Davies ENDOR Sequence
==============================
Figure \[fig1\]A shows a schematic of the Davies ENDOR sequence [@davies74], while Figure \[fig2\]A shows the evolution of the spin state populations during the sequence (for illustration purposes we consider a simple system of coupled electron S=1/2 and nuclear I=1/2 spins, however the same consideration is applicable to an arbitrary spin system). In the *preparation* step of the pulse sequence, a selective microwave $\pi$ pulse is applied to one of the electron spin transitions to transfer the initial thermal polarization (i) of the electron spin to the nuclear spin polarization (ii). In the *mixing* step a resonant RF pulse on the nuclear spin further disturbs the electron polarization to produce (iii), which can be *detected* using a two-pulse (Hahn) echo pulse sequence [@schweiger01]. A side result of the detection sequence is to equalize populations of the resonant electron spin states (iv). There then follows a delay, $t_r$, before the experiment is repeated (e.g. for signal averaging). Analysis of this *recovery* period has hitherto been limited (although the effect of $t_r$ has been discussed with respect to ENDOR lineshape [@epel01] and stochastic ENDOR acquisition [@epel03]), yet it is this recovery period which is crucial in optimizing the sequence sensitivity.
![Pulse sequences for Davies ENDOR experiments. (A) The traditional Davies experiment requires long recovery time t$_r \gg$ T$_{1e}$ and t$_r
\gg$ T$_{1n}$ to allow the spin system to fully recover to a thermal equilibrium before the experiment can be repeated (e.g. for signal averaging). (B) An additional RF pulse applied after echo detection helps the spin system to recover to a thermal equilibrium in a much shorter time limited only by T$_{1e}$. Thus, signal averaging can be performed at a much faster rate and an enhanced signal/noise can be achieved in a shorter experimental time. t$_{\rm{opt}}$ represents an optional delay of several T$_{1e}$ which can be inserted for a secondary improvement in signal/noise and to avoid overlapping with electron spin coherences in case of long T$_{2e}$.[]{data-label="fig1"}](fig1_comp.eps){width="3.5in"}
{width="7in"}
Nuclear spin relaxation times T$_{1n}$ (and also cross-relaxation times T$_{1x}$) are usually very long, ranging from many seconds to several hours, while electron spin relaxation times T$_{1e}$ are much shorter, typically in the range of microseconds to milliseconds. In a typical EPR experiment, t$_r$ is chosen to be several T$_{1e}$ (i.e. long enough for the electron spin to fully relax, but short enough to perform the experiment in a reasonable time). Thus, in practice it is generally the case that T$_{1n}
\gg$ t$_r \gg$ T$_{1e}$, i.e. t$_r$ is short on the time scale of T$_{1n}$ while long on the time scale of T$_{1e}$. With this choice of t$_r$, during the recovery period only the electron spin (and not the nuclear spin) has time to relax before the next experiment starts. As shown in Figure \[fig2\]A, the second and all subsequent shots of the experiment will start from initial state (v), and not from the thermal equilibrium (i). While the first shot yields a 100$\%$ ENDOR visibility, subsequent passes give strongly suppressed ENDOR signals. Upon signal summation over a number of successive shots, the overall ENDOR response is strongly diminished from its maximal intensity and fails to achieve the theoretical 100$\%$ by a considerable margin.
One obvious solution to overcoming this limitation is to increase the delay time $t_r$ so that it is long compared to the nuclear spin relaxation time T$_{1n}$ (Figure \[fig2\]B). In other words, t$_r \gg$ (T$_{1n}$, T$_{1x}$)$\gg$ T$_{1e}$, so that the entire spin system (including electron and nuclear spins) has sufficient time between successive experiments to fully relax to thermal equilibrium. However, this can make the duration of an experiment very long, and the advantage of an enhanced per-shot sensitivity becomes less significant. From calculations provided in the Appendix, it can be seen that an optimal trade-off between signal/noise and experimental time is found at t$_r \approx
5/4$T$_{1n}$.
A better solution to this problem involves a modification of the original Davies ENDOR sequence which removes the requirement for t$_r$ to be greater than T$_{1n}$, permitting enhanced signal/noise at much higher experimental repetition rates, limited only by T$_{1e}$.
Modified Davies ENDOR Sequence
==============================
Our modified Davies ENDOR sequence is shown in Figure \[fig1\]B. An additional RF pulse is introduced at the end of the sequence, after echo signal formation and detection. This second RF pulse is applied at the same RF frequency as the first RF pulse and its sole purpose is to re-mix the spin state populations in such a way that the spin system relaxes to thermal equilibrium on the T$_{1e}$ timescale, independent on T$_{1n}$. The effect of this second RF pulse is illustrated in Figure \[fig2\]C. After echo signal detection, the spin system is in state (iv) and the second RF pulse converts it to (ix). This latter state then relaxes to thermal equilibrium (i) within a short t$_r$ ($> 3$T$_{1e}$). In this modified sequence each successive shot is identical and therefore adds the optimal ENDOR visibility to the accumulated signal.
The discussion in Figure \[fig2\]C assumes an ideal $\pi$ rotation by the RF pulses. However, in experiment the RF pulse rotation angle may differ from $\pi$, and such an imperfection in either RF pulse will lead to a reduction in the ENDOR signal. Errors in the first pulse have the same effect as in a standard Davies ENDOR experiment, reducing the ENDOR signal by a factor $(1-\cos\theta)/2$, where $\theta$ the actual rotation angle. Errors in the second RF pulse (and also accumulated errors after the first pulse) cause incomplete recovery of spin system back to the thermal equilibrium state (i) at the end of each shot, thus reducing visibility of the ENDOR signal in the successive shots. The pulse rotation errors can arise from inhomogeneity of the RF field in the resonator cavity (e.g. spins in different parts of the sample are rotated by different angle) or from off-resonance excitation of the nuclear spins (when the excitation bandwidth of the RF pulses is small compared to total width of the inhomogeneously-broadened ENDOR line). It is desirable to eliminate (or at least partially compensate) some of these errors in experiment.
We find that introducing a delay t$_{\rm{opt}}$, to allow the electron spin to fully relax before applying the second RF pulse (Figure \[fig1\]B), helps to counter the effect of rotation errors. In numerical simulations, using the approach developed in ref. [@epel01; @bowman00] and taking into account electron and nuclear spin relaxation times and also a finite excitation bandwidth of the RF pulses, we observed that introducing t$_{\rm{opt}} \gg T_{1e}$ produces about 30% increase in the ENDOR signal visibility (however, at cost of a slower acquisition rate with repetition time t$_{\rm{opt}}+$ t$_{r}$). In the following sections, we demonstrate the capabilities of this modified Davies ENDOR sequence, using two examples of phosphorous donors in silicon and N@C$_{60}$ in CS$_2$.
Application of the modified Davies ENDOR
========================================
Improved Sensitivity
--------------------
Figure \[fig3\]A shows the effect of experimental repetition time, t$_r$, on the ENDOR visibility, using a standard Davies ENDOR sequence applied to $^{28}$Si:P. Although t$_r$ is always longer than the electron spin relaxation time (T$_{1e} = 1$ ms for $^{28}$Si:P at 10 K [@alexeisi]), increasing the repetition time from 13 ms to 1 second improves the visibility by an order of magnitude. As shown below, T$_{1n} = 288$ ms for the $^{31}$P nuclear spin at 10 K, and therefore we observe that the ENDOR signal visibility is weak ($\sim 2$%) when t$_r = 13$ ms is shorter than T$_{1n}$ but the visibility increases to a maximum 22% when t$_r = 1$ s is longer than T$_{1n}$. The observed maximal visibility 22% does not reach a theoretical 100% limit because of the finite excitation bandwidth of the applied RF pulses ($t_{RF}=50$ $\mu$s in these experiments) which is smaller than total linewidth of the inhomogeneously-broadened $^{31}$P ENDOR peak.
![Davies ENDOR spectra for $^{28}$Si:P, showing the low-frequency $^{31}$P nuclear transition line at 10 K. The spectra are normalized with respect to the spin echo intensity with no RF pulse applied. (A) Three spectra measured with the traditional Davies ENDOR pulse sequence (see Figure \[fig1\]A) using different repetition times as labeled. The same number of averages (n=20) was applied for each spectrum and therefore the spectral acquisition times were approximately proportional to the repetition times (i.e. 5000, 210 and 65 seconds, respectively). (B) The spectrum measured with our modified Davies pulse sequence (see Figure \[fig1\]B) using short repetition time (13 ms) shows a comparable signal/noise to the spectrum measured with a standard Davies pulse sequence at much longer repetition time (1 s). t$_{\rm{opt}}$=6.5 ms was used in the modified Davies ENDOR experiment.[]{data-label="fig3"}](fig3_comp.eps){width="3in"}
Through the use of the modified Davies ENDOR sequence proposed above, the same order of signal enhancement is possible at the faster 13 ms repetition time (e.g. at t$_r \ll$ T$_{1n}$), as shown in Figure \[fig3\]B. This is an impressive improvement indeed, considering that the acquisition time was almost 100 times shorter in the modified Davies ENDOR experiment. The signal is slightly smaller in the modified Davies ENDOR spectrum because of the imperfect $\pi$ rotation of the recovery RF pulse (e.g. due to inhomogeneity of the RF field as discussed above). Figure \[fig5\]A shows a similar signal enhancement effect for N@C$_{60}$.
Improved Spectral Resolution
----------------------------
Spectral resolution in a traditional Davies ENDOR experiment is determined by the duration of the RF pulse inserted between the preparation and detection microwave pulses (see Figure \[fig1\]). The electron spin relaxation time T$_{1e}$ limits the maximum duration of this RF pulse, and in turn, the achievable resolution in the ENDOR spectrum. However, there is no such limitation on the duration of the second (recovery) RF pulse in the modified Davies ENDOR sequence, as it is applied after the electron spin echo detection. Thus, in the case where the duration of the first RF pulse limits the ENDOR resolution, applying a longer (and thus, more selective) second RF pulse can offer substantially enhanced spectral resolution. In this scheme, the first RF pulse is short and non-selectively excites a broad ENDOR bandwidth, however the second RF pulse is longer and selects a narrower bandwidth from the excited spectrum. Note that both RF pulses correspond to $\pi$ rotations, hence the power of the second pulse must be reduced accordingly.
![Spectral resolution enhancement in the modified Davies ENDOR experiment can be achieved by increasing the spectral selectivity of the second (recovery) RF pulse, as illustrated for the $M_S=-3/2$ ENDOR line of $^{14}$N@C$_{60}$, in CS$_2$ at 190 K. (A) A comparison of the traditional Davies ENDOR with a 9 $\mu$s RF pulse and the modified Davies ENDOR with an additional 9 $\mu$s recovery RF pulse demonstrates a significant enhancement in signal intensity. If the second RF pulse is lengthened (to 180 $\mu$s in this case), the selectivity of the recovery pulse increases and the enhanced component of the ENDOR line becomes better resolved. (B) The oscillating background is identical in all spectra in (A) and it can be removed from the modified Davies ENDOR spectra by subtracting the spectrum obtained with a traditional Davies ENDOR.[]{data-label="fig5"}](fig5_comp.eps){width="3.2in"}
Figure \[fig5\] illustrates this effect for N@C$_{60}$, in which the intrinsic $^{14}$N ENDOR lines are known to be very narrow ($<1$ kHz). Increasing the duration of the recovery RF pulse from 9 $\rm{\mu}$s to 180 $\rm{\mu}$s dramatically increases the resolution and reveals two narrow lines, at no significant cost in signal intensity or experiment duration. In Figure \[fig5\]B, what appears to be a single broad line is thus resolved into two, corresponding to two non-degenerate $\Delta M_I=1$ spin transitions of $^{14}$N $I=1$ nuclear spin at electron spin projection $M_S = -3/2$. We notice the presence of a broad oscillating background in the modified Davies ENDOR spectra in Figure \[fig5\]A. This background matches the signal detected using a standard Davies ENDOR, where it is clearly seen to have a recognizable *sinc*-function shape (i.e. its modulus) and thus corresponds to the off-resonance excitation profile of the first RF pulse. As shown in Figure \[fig5\]B, this background signal can be successfully eliminated from the modified Davies ENDOR spectra by subtracting the signal measured with a standard Davies ENDOR.
Measuring Nuclear Spin Relaxation Times T$_{1n}$
================================================
As already indicated in Figure \[fig3\]A, the signal intensity in a traditional Davies ENDOR increases as the repetition time t$_r$ is made longer, as compared to the nuclear spin relaxation time T$_{1n}$. It is shown in the Appendix that, in case when T$_{1n}\sim$ t$_r \gg$ T$_{1e}$, the ENDOR signal intensity varies as: $$\label{ENDOR_tr_dependence}
I_{\rm{ENDOR}} \sim 1-\exp{\left(-t_r/T_{1n} \right)}.$$ Thus, measuring the signal intensity in a traditional Davies ENDOR as a function of t$_r$ yields a measure of T$_{1n}$, as illustrated in Figure \[fig4\]A for $^{28}$Si:P and in Figure \[fig4\]B for N@C$_{60}$. In both cases, T$_{1n}$ is found to be much longer than T$_{1e}$ (cp. T$_{1n} = 280$ ms and T$_{1e} = 1$ ms for $^{28}$Si:P at 10 K [@alexeisi], and T$_{1n} = 50$ ms and T$_{1e} = 0.32$ ms for $^{14}$N@C$_{60}$ in CS$_2$ at 190 K [@relaxcs2]), as must be expected because nuclear spins have a smaller magnetic moment and are therefore less prone to fluctuating magnetic fields in the host environment.
![Intensity of the traditional Davies ENDOR signal as a function of repetition time measured for (A) $^{28}$Si:P (52.33 MHz line) at 10 K, and (B) $^{14}$N@C$_{60}$ (22.626 MHz line) in CS$_2$ at 190 K. An exponential fit (dashed line) yields the respective nuclear spin relaxation time T$_{1\rm{n}}$. In (A) the signal intensity at short times ($< 0.1$ s) deviates from the exponential fit due to the transient effects arising from a finite electron spin T$_{1e}$ time, as described in ref. [@epel01].[]{data-label="fig4"}](fig4_comp.eps){width="3.5in"}
Using Davies ENDOR to measure nuclear spin relaxation times, T$_{1n}$ and T$_{2n}$, has been already proposed, however the applicability of suggested pulse schemes has been limited to cases where T$_{1n}$ (or T$_{2n}$) $<$ T$_{1e}$ [@hofer86; @hofer94]. Herein, we extend the method to (more common) cases where T$_{1n}$ is greater than T$_{1e}$.
Conclusions
===========
We have shown that signal intensity in the traditional Davies ENDOR experiment is strongly dependent on the experimental repetition time and that the addition of the second (recovery) RF pulse at the end of the pulse sequence eliminates this dependence. This modification to the Davies pulse sequence dramatically enhances the signal/noise (allowing signal acquisition at much faster rate without loss of the signal intensity), and can also improve the spectral resolution. We also demonstrate that the sensitivity of the Davies ENDOR to nuclear relaxation time can be exploited to measure T$_{1n}$. The technique of adding an RF recovery pulse after electron spin echo detection can be applied to the general family of pulsed ENDOR experiments, in which a non-thermal nuclear polarization is generated, including the popular technique of Mims ENDOR [@schweiger01; @mims65].
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Kyriakos Porfyrakis for providing the N@C$_{60}$ material. We thank the Oxford-Princeton Link fund for support. Work at Princeton was supported by the NSF International Office through the Princeton MRSEC Grant No. DMR-0213706 and by the ARO and ARDA under Contract No. DAAD19-02-1-0040. JJLM is supported by St. John’s College, Oxford. AA is supported by the Royal Society.
Appendix {#appendix .unnumbered}
========
Here we describe how a compromise can be reached, using the traditional Davies ENDOR, between maximal ENDOR ‘per-shot’ signal and overall experiment duration. The equations below show the evolution of state populations — a quantitative equivalent of those shown in Figure 2 in the main text, with the difference that a partial nuclear relaxation is considered during the repetition time $t_r$. Thus, we assume T$_{1n}\sim$ t$_r \gg$ T$_{1e}$.
*Legend*: $$\left(%
\begin{array}{c}
\uparrow_e \uparrow_n \\
\uparrow_e \downarrow_n \\
\downarrow_e \downarrow_n \\
\downarrow_e \uparrow_n \\
\end{array}%
\right),~~~~a= g \beta_e
B/2kT~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ 1st shot: $$\left(%
\begin{array}{c}
-a \\
-a \\
+a \\
+a \\
\end{array}%
\right) \rightarrow \pi_e \rightarrow
\left(%
\begin{array}{c}
-a \\
+a \\
-a \\
+a \\
\end{array}%
\right) \rightarrow \pi_n \rightarrow
\left(%
\begin{array}{c}
+a \\
-a \\
-a \\
+a \\
\end{array}%
\right) \rightarrow e.s.e \rightarrow
\left(%
\begin{array}{c}
+a \\
-a \\
-a \\
+a \\
\end{array}%
\right) \rightarrow {\rm delay}~t_r \rightarrow
\left(%
\begin{array}{c}
-a (1-\exp{\left(-t_r/T_{1n} \right)}) \\
-a (1+\exp{\left(-t_r/T_{1n} \right)})\\
+a (1-\exp{\left(-t_r/T_{1n} \right)})\\
+a (1+\exp{\left(-t_r/T_{1n} \right)})\\
\end{array}%
\right)$$ 2nd and subsequent shots: $$\left(%
\begin{array}{c}
-a (1-\exp{\left(-t_r/T_{1n} \right)})\\
-a (1+\exp{\left(-t_r/T_{1n} \right)})\\
+a (1-\exp{\left(-t_r/T_{1n} \right)})\\
+a (1+\exp{\left(-t_r/T_{1n} \right)})\\
\end{array}%
\right) \rightarrow \pi_e \rightarrow
\left(%
\begin{array}{c}
-a (1-\exp{\left(-t_r/T_{1n} \right)})\\
+a (1-\exp{\left(-t_r/T_{1n} \right)})\\
-a (1+\exp{\left(-t_r/T_{1n} \right)})\\
+a (1+\exp{\left(-t_r/T_{1n} \right)})\\
\end{array}%
\right) \rightarrow \pi_n \rightarrow
\left(%
\begin{array}{c}
+a (1-\exp{\left(-t_r/T_{1n} \right)})\\
-a (1-\exp{\left(-t_r/T_{1n} \right)})\\
-a (1+\exp{\left(-t_r/T_{1n} \right)})\\
+a (1+\exp{\left(-t_r/T_{1n} \right)})\\
\end{array}%
\right) \rightarrow$$ $$\rightarrow e.s.e. \rightarrow
\left(%
\begin{array}{c}
+a (1-\exp{\left(-t_r/T_{1n} \right)})\\
-a \\
-a \\
+a (1+\exp{\left(-t_r/T_{1n} \right)})\\
\end{array}%
\right) \rightarrow {\rm delay}~t_r \rightarrow
\left(%
\begin{array}{c}
-a (1-\exp{\left(-t_r/T_{1n} \right)}) \\
-a (1+\exp{\left(-t_r/T_{1n} \right)})\\
+a (1-\exp{\left(-t_r/T_{1n} \right)})\\
+a (1+\exp{\left(-t_r/T_{1n} \right)})\\
\end{array}%
\right)$$
The intensity of the ENDOR signal is therefore: $$2a(1-\exp{\left(-t_r/T_{1n} \right)}).$$ As the signal-to-noise is proportional to the square root of the number of samples, and thus to $\sqrt{1/t_r}$, we can define a signal efficiency of: $$2a(1-\exp{\left(-t_r/T_{1n} \right)})/\sqrt{t_r}.$$ This figure is maximized when $t_r \approx 1.25$ T$_{1n}$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Two-dimensional carbon, or graphene, is a semi-metal that presents unusual low-energy electronic excitations described in terms of Dirac fermions. We analyze in a self-consistent way the effects of localized (impurities or vacancies) and extended (edges or grain boundaries) defects on the electronic and transport properties of graphene. On the one hand, point defects induce a finite elastic lifetime at low energies with the enhancement of the electronic density of states close to the Fermi level. Localized disorder leads to a universal, disorder independent, electrical conductivity at low temperatures, of the order of the quantum of conductance. The static conductivity increases with temperature and shows oscillations in the presence of a magnetic field. The graphene magnetic susceptibility is temperature dependent (unlike an ordinary metal) and also increases with the amount of defects. Optical transport properties are also calculated in detail. On the other hand, extended defects induce localized states near the Fermi level. In the absence of electron-hole symmetry, these states lead to a transfer of charge between the defects and the bulk, the phenomenon we call self-doping. The role of electron-electron interactions in controlling self-doping is also analyzed. We also discuss the integer and fractional quantum Hall effect in graphene, the role played by the edge states induced by a magnetic field, and their relation to the almost field independent surface states induced at boundaries. The possibility of magnetism in graphene, in the presence of short-range electron-electron interactions and disorder is also analyzed.'
author:
- 'N. M. R. Peres$^{1,2}$, F. Guinea$^{1,3}$, and A. H. Castro Neto$^{1}$'
bibliography:
- 'graphite0\_1.bib'
title: 'Electronic Properties of Disordered Two-Dimensional Carbon'
---
Introduction
============
Carbon is a life sustaining element that, due to the versatility of its bonding, is present in nature in many allotropic forms. Besides being an element that is fundamental for life on the planet, it has been explored recently for basic science and technology in the form of three-dimensional graphite, [@BCP88] one-dimensional nanotubes, [@Retal03] zero-dimensional fullerenes, [@Setal02] and more recently in the form of two-dimensional Carbon, also known as graphene. Experiments in graphene-based devices have shown that it is possible to control their electrical properties by the application of external gate voltage, [@Netal04; @Betal04; @Netal05; @Netal05b; @Zetal05; @Betal05; @Zetal05c; @Zetal05b] opening doors for carbon-based nano-electronics. In addition, the interplay between disorder and magnetic field effects leads to an unusual quantum Hall effect predicted theoretically [@PGN05; @NGP05; @GS05] and measured experimentally [@Eetal03; @Netal05; @Zetal05]. These systems can be switched from n-type to p-type carriers and show entirely new electronic properties. We show that their physical properties can be ascribed to their low dimensionality, and the phenomenon of self-doping, that is, the change in the bulk electronic density due to the breaking of particle-hole symmetry, and the unavoidable presence of structural defects. Our theory not only provides a description of the recent experimental data, but also makes new predictions that can be checked experimentally. Our results have also direct implication in the physics of Carbon based materials such as graphite, fullerenes, and carbon nanotubes.
Graphene is the building block for many forms of Carbon allotropes. Its structure consists of a Carbon honeycomb lattice made out of hexagons (see Fig. \[honey\]). The hexagons can be thought of Benzene rings from which the Hydrogen atoms were extracted. Graphite is obtained by the stacking of graphene layers that is stabilized by weak van der Waals interactions. [@P72] Carbon nanotubes are synthesized by graphene wrapping. Depending on the direction in which graphene is wrapped, one can obtain either metallic or semiconducting electrical properties. Fullerenes can also be obtained from graphene by modifying the hexagons into pentagons and heptagons in a systematic way. Even diamond can be obtained from graphene under extreme pressure and temperatures by transforming the two-dimensional sp$^2$ bonds into three-dimensional sp$^3$ ones. Therefore, there has been enormous interest over the years in understanding the physical properties of graphene in detail. Nevertheless, only recently, with the advances in material growth and control, that one has been able to study truly two-dimensional Carbon physics.
One of the most striking features of the electronic structure of perfect graphene planes is the linear relationship between the electronic energy, $E_{{\bm k}}$, with the two-dimensional momentum, ${\bm k} =(k_x,k_y)$, that is: $E_{{\bm k}} = {v_{\rm F}}|{\bm k}|$, where ${v_{\rm F}}$ is the Dirac-Fermi velocity. This singular dispersion relation is a direct consequence of the honeycomb lattice structure that can be seen as two interpenetrating triangular sub-lattices. In ordinary metals and semiconductors the electronic energy and momentum are related quadratically via the so-called effective mass, $m^*$, ($E_{{\bm k}} = \hbar^2 {\bm k}^2/(2 m^*)$), that controls much of their physical properties. Because of the linear dispersion relation, the effective mass in graphene is zero, leading to a unusual electrodynamics. In fact, graphene can be described mathematically by the two-dimensional Dirac equation, whose elementary excitations are particles and holes (or anti-particles), in close analogy with systems in particle physics. In a perfect graphene sheet the chemical potential, $\mu$, crosses the Dirac point and, because of the dimensionality, the electronic density of states vanishes at the Fermi energy. The vanishing of the effective mass or density of states has profound consequences. It has been shown, for instance, that the Coulomb interaction, unlike in an ordinary metal, remains unscreened [@mele] and gives rise to an inverse quasi-particle lifetime that increases linearly with energy or temperature [@GGV96], in contrast with the usual metallic Fermi liquid paradigm, where the inverse lifetime increases quadratically with energy.
The fact that graphene is a two-dimensional system has also serious consequences in terms of the positional order of the Carbon atoms. Long-range Carbon order in graphene is only really possible at zero temperature because thermal fluctuations can destroy long-range order in two-dimensions (the so-called, Hohenberg-Mermin-Wagner theorem [@MW66]). At a finite temperature $T$, topological defects such as dislocations are always present. Furthermore, because of the particular structure of the honeycomb lattice, the dynamics of lattice defects in graphene planes belong to the generic class of kinetically constrained models[@DS00; @RS03], where defects are never completely annealed since their number decreases only as a logarithmic function of the annealing time [@DS00]. Indeed, defects are ubiquitous in carbon allotropes with sp$^2$ coordination and have been observed in these systems [@Hetal04b]. As a consequence of the presence of topological defects, the electronic properties discussed previously, are significantly modified leading to qualitatively new physics. As we show below, extended defects can lead to the phenomenon of self-doping with the formation of electron or hole pockets close to the Dirac points. We show, however, that the presence of such defects can still lead to long electronic mean free paths. We present next an analysis of the physical properties of graphene as a function of the density of defects, at zero and finite temperature, frequency, and magnetic field. The defects analyzed here, like boundaries (edges), dislocations, vacancies, can be considered strong distortions of the perfect system. In this respect, our work complements the studies of defects and interactions in systems described by the two-dimensional Dirac equation [@r1].
The role of disorder on the electronic properties of coupled graphene planes shows also its importance on the unexpected appearance of ferromagnetism in proton irradiated graphite [@Ketal00; @Eetal02; @MHM02; @Ketal03b; @Eetal03b; @MP05]. In a recent publication, the role of the exchange mechanism on a disordered graphene plane was addressed [@PGN05b]. It was found that disorder can stabilizes a ferromagnetic phase in the presence of long-range Coulomb interactions. On the other hand, the effect of disorder on the density of states of a single graphene plane amounts to the creation of a finite density of states at zero energy. Therefore, a certain amount of screening should be present and the question of whether the interplay of disorder and short-range Coulomb interaction may stabilize a ferromagnetic ground state has to be addressed as well.
Moreover, with the current experimental techniques, it is possible to study not only a single layer of graphene but also graphene multi-layers (bilayers, trilayers, etc). Recent experiments provide direct evidence that while the high-energy physics of graphene multi-layers (for energies above around 100 meV from the Dirac point) is quite different from that of single layer graphene, the low-energy physics seems to be universal, two-dimensional, independent of the number of layers, and dominated by disorder [@Betal04; @Zetal05; @Zetal05b]. Hence, the work described here maybe fundamental for the understanding of this low-energy behavior. There is still an interesting question whether this universal low-energy physics survives in bulk graphite.
In this paper we present a comprehensive and unabridged study of the electronic properties of graphene in the presence of defects (localized and extended), and electron-electron interaction, as a function of temperature, external frequency, gate voltage, and magnetic field. We study the electronic density of states, the electron spectral function, the frequency dependent conductivity, the magneto-transport, and the integer and fractional quantum Hall effect. We also discuss the possibility of a magnetic instability of graphene due to short-range electron-electron interactions and disorder (the problem of ferromagnetism in the presence of disorder and [*long-range*]{} Coulomb interactions was discussed in a previous publication [@PGN05b]).
The paper is organized as follows: in Sec. \[tmatrix\] a formal solution for the single impurity and many impurities $T-$matrix calculation is given. Details of the position averaging procedure are given in Sec. \[landau\], in connection with the same problem, but in a magnetic field. In Sec. \[sdos\] the problem of Dirac fermions in a disordered honeycomb lattice is studied within the full Born approximation (FBA) and the full self-consistent Born approximation (FSBA) for the density of states. Using the results of Sec. \[sdos\], the spectral and transport properties of Dirac fermions are computed in Sec. \[sspec\]. In Sec. \[smag\] we address the question of magnetism and the interplay between short-range electron-electron interactions and disorder. The density of states of Dirac fermions in a magnetic field perpendicular to a graphene plane is studied in Sec. \[landau\] and the magneto-transport properties of this system are computed both at zero and finite frequencies, using the FSBA. The quantization values for the integer quantum Hall effect and for Jain’s sequence of the fractional quantum Hall effect are discussed. Finally, Sec. \[conclusions\] contains our conclusions. We have also added appendices with the details of the calculations.
Impurities and vacancies.
=========================
The honeycomb lattice can be described in terms of two triangular sub-lattices, $A$ and $B$ (see Fig. \[honey\]). The unit vectors of the underlying triangular sub-lattice are $$\begin{aligned}
\bm a_1 &=& \frac a 2 (3,\sqrt 3,0)\,,
\nonumber \\
\bm a_2 &=& \frac a 2 (3,-\sqrt 3,0)\,,\end{aligned}$$ where $a$ is the lattice spacing (we use units such that $K_B=1= \hbar$). The reciprocal lattice vectors are: $$\bm b_1= \frac {2\pi}{3a} (1,\sqrt 3,0)\,,
\hspace{.2cm}
\bm b_2= \frac {2\pi}{3a} (1,-\sqrt 3,0)\,.
\label{recv}$$ The vectors connecting any $A$ atom to its nearest neighbors are: $$\begin{aligned}
\bm \delta_1 &=& \frac a 2 (1,\sqrt 3,0),
\nonumber \\
\bm \delta_2 &=& \frac a 2 (1,-\sqrt 3,0),
\nonumber \\
\bm \delta_3 &=& a (1,0,0)
\label{nnv}\end{aligned}$$ and the vectors connecting to next-nearest neighbors are: $$\begin{aligned}
{\bf n}_1 &=& - {\bf n}_2 = {\bf a}_1 \, ,
\nonumber
\\
{\bf n}_3 &=& - {\bf n}_4 = {\bf a}_2 \, ,
\nonumber
\\
{\bf n}_5 &=& - {\bf n}_6 = \bm a_1-\bm a_2 \, .
\label{nnnv}\end{aligned}$$
![\[honey\] (color on line) A honeycomb lattice with vacancies, showing its primitive vectors.](Fig_BZ.eps){width="6cm"}
In what follows we use a tight-binding description for the $\pi$-orbitals of Carbon with a Hamiltonian given by: $$\begin{aligned}
H_{{\rm t.b.}} &=& -t \sum_{\langle i,j \rangle ,\sigma} (a^{\dag}_{i,\sigma} b_{j,\sigma}
+ {\rm h.c.}) \nonumber \\
&+& t' \sum_{\langle \langle i,j \rangle \rangle ,\sigma} (a^{\dag}_{i,\sigma} a_{j,\sigma}
+ b^{\dag}_{i,\sigma} b_{j,\sigma} + {\rm h.c.}) \, ,
\label{Htb}\end{aligned}$$ where $a^{\dag}_{i,\sigma}$ ($a_{i,\sigma}$) creates (annihilates) and electron on site ${\bf R}_i$ with spin $\sigma$ ($\sigma = \uparrow,\downarrow$) on sub-lattice $A$ and $b^{\dag}_{i,\sigma}$ ($b_{i,\sigma}$) creates (annihilates) and electron on site ${\bf R}_i$ with spin $\sigma$ ($\sigma = \uparrow,\downarrow$) on sub-lattice $B$. $t$ is the nearest neighbor ($\langle i, j \rangle$) hopping energy ($t \approx 2.7$ eV), and $t'$ is the next-nearest neighbor ($\langle \langle i,j \rangle \rangle$) hopping energy ($t'/t \approx 0.1$). We notice [*en passant*]{} that in earlier studies of graphite [@M64] it has been assumed that $t'=0$. This assumption, however, is not warranted since there is overlap between Carbon $\pi$-orbitals in the same sub-lattice. In fact, we will show that $t'$ plays an important role in graphene since it breaks the particle-hole symmetry and is responsible for various effects observed experimentally.
Translational symmetry is broken by the presence of disorder. Localized defects such as vacancies and impurities are included in the tight-biding description by the addition of a local energy term: $$H_{{\rm imp.}} = \sum_{i,\sigma} V_{i} \left(a^\dag_{i,\sigma}a_{i,\sigma} +
b^\dag_{i+\bm\delta_3,\sigma}b_{i+\bm\delta_3,\sigma} \right)
\, ,
\label{impurities}$$ where $V_{i}$ is a random potential at site ${\bf R}_i$. In momentum space we define: $$a^\dag_{i,\sigma}=\frac {1}{\sqrt {N_A}}
\sum_{\bm k}e^{i\bm k\cdot \bm R_i}
a^\dag_{\bm k,\sigma}\,,\hspace{.2cm}
b^\dag_{i,\sigma}=\frac {1}{\sqrt {N_B}}
\sum_{\bm k}e^{i\bm k\cdot \bm R_i}
b^\dag_{\bm k,\sigma}\,,$$ where $N_A=N_B=N$, and the non-interacting Hamiltonian, $H_1 = H_{{\rm
t.b.}}+H_{{\rm imp.}}$, reads: $$\begin{aligned}
H_1&=&\sum_{\bm k,\sigma}[\phi(\bm k)a^\dag_{\bm k,\sigma}b_{\bm k,\sigma}+
\phi^\ast(\bm k) b^\dag_{\bm k,\sigma}a_{\bm k,\sigma}]\nonumber\\
&+&\sum_{\bm k,\sigma} \tilde{\phi}(\bm k)
(a^\dag_{\bm k,\sigma}a_{\bm k,\sigma}+
b^\dag_{\bm k,\sigma}b_{\bm k,\sigma})
\nonumber \\
&+&
\sum_{\bm q,\bm k,\sigma} V_{{\bf q}}
[a^\dag_{\bm k+\bm q,\sigma}a_{\bm k,\sigma}
+b^\dag_{\bm k+\bm q,\sigma}b_{\bm k,\sigma}] \, ,
\label{h1}\end{aligned}$$ where $$\begin{aligned}
\phi(\bm k) &=& -t\sum_{i=1}^3 e^{i {\bf k} \cdot {\bf \delta}_i} \, ,
\nonumber
\\
{\tilde \phi}(\bm k)&=&t'\sum_{i=1}^6
e^{i {\bf k} \cdot {\bf n}_i} \, ,\end{aligned}$$ and $V_{{\bf q}}$ is the Fourier transform of the random potential due to impurities. Hamiltonian (\[h1\]) is the starting point of our approach.
The single impurity problem and the T-matrix approximation {#tmatrix}
----------------------------------------------------------
In the single impurity case one can write $V_{\bf q} = V/N$ where $V$ is the strength of the impurity potential. In what follows we use standard finite temperature Green’s function formalism [@DS88; @BF04]. Because of the existence of two sub-lattices, the Green’s function can be written as a $2\times 2$ matrix: $$\begin{aligned}
\bm G_{\sigma}(\bm k,{\bf p},\tau) = \left(\begin{array}{cc}
G_{AA,\sigma} (\bm k,{\bf p}, \tau) \hspace{0.5cm} & G_{AB,\sigma}(\bm k,{\bf p},\tau) \\
G_{BA,\sigma} (\bm k,{\bf p},\tau) \hspace{0.5cm} & G_{BB,\sigma}(\bm k,{\bf p},\tau)
\end{array}\right) \, ,\end{aligned}$$ where $$\begin{aligned}
G_{AA,\sigma}(\bm k,{\bf p}, \tau) &=& - \langle {\cal T} a_{\bm k,\sigma}(\tau) a_{\bm p,\sigma}^{\dag}(0)
\rangle \, ,
\nonumber
\\
G_{AB,\sigma}(\bm k,{\bf p}, \tau) &=& - \langle {\cal T} a_{\bm k,\sigma}(\tau) b_{\bm p,\sigma}^{\dag}(0)
\rangle \, ,
\nonumber
\\
G_{BA,\sigma}(\bm k,{\bf p},\tau) &=& - \langle {\cal T} b_{\bm k,\sigma}(\tau) a_{\bm p,\sigma}^{\dag}(0)
\rangle \, ,
\nonumber
\\
G_{BB,\sigma}(\bm k,{\bf p},\tau) &=& - \langle {\cal T} b_{\bm k,\sigma}(\tau) b_{\bm p,\sigma}^{\dag}(0)
\rangle \, ,
\label{defg}\end{aligned}$$ where $\tau$ is the “imaginary” time, and ${\cal T}$ is the time ordering operator.
For a single impurity the Green’s function can be written as $\bm G(\bm k,{\bf p},\tau) = \delta_{\bm k,\bm p} \bm G(\bm k,\tau)$, where [@DS88]: $$\bm G(\bm k,\omega_n)=\bm G^0(\bm k,\omega_n)+\bm G^0(\bm k,\omega_n)
\bm T_{{\rm imp.}} (\omega_n)\bm G^0(\bm k,\omega_n)\,,
\label{g1imp}$$ where $\omega_n = 2 \pi T (n +1/2)$ is the fermionic Matsubara frequency, $\bm G^0(\bm k,\omega_n)$ is the propagator of the tight-binding Hamiltonian (\[Htb\]) and $$\bm T_{{\rm imp.}}(\omega_n)=\frac V N [\bm 1- V \bar {\bm G}^0(\omega_n)]^{-1}\,,
\label{t1imp}$$ is the single impurity T-matrix, where: $$\bar {\bm G}^0(\omega_n)= \frac{1}{N} \sum_{\bm k}\bm G^0(\bm k,\omega_n)\,.$$
The above result is exact for a single impurity. For a finite but small density, $n_i=N_i/N$, of impurities, the Green’s function equation becomes: $$\bm G(\bm k,\omega_n)=\bm G^0(\bm k,\omega_n)+\bm G^0(\bm k,\omega_n)
\bm T(\omega_n)\bm G(\bm k,\omega_n)\,,
\label{finiteimp}$$ which is valid up to first order in $n_i$, that is, it takes only into account the multiple scattering of the electrons by a single impurity. Equation (\[finiteimp\]) can be solved as: $$\bm G(\bm k,\omega_n)=[[\bm G^0(\bm k,\omega_n)]^{-1}-\bm T(\omega_n)]^{-1}\,,
\label{gimp}$$ where $$\bm T(\omega_n)= N_i \bm T_{{\rm imp.}}(\omega_n) =
V n_i [\bm 1-V \bar {\bm G}^0(\omega_n)]^{-1}\,.
\label{timp}$$ For vacancies we take $V \to \infty$ and (\[timp\]) reduces to: $$\bm T(\omega_n)= - n_i [\bar {\bm G}^0(\omega_n)]^{-1}\,.
\label{tvac}$$
It worth stressing that Eqs. (\[g1imp\]) and (\[t1imp\]) although similar in form to Eqs. (\[gimp\]) and (\[timp\]) have a very different meaning. Whereas the first set applies to the single impurity problem, the latter set is the consequence of an assemble average over the impurity positions (see Sec. \[landau\] for details on the averaging procedure in the context of Landau levels) with a re-summation procedure, corresponding to the FBA [@BF04].
The low-energy physics and the electronic density of states {#sdos}
-----------------------------------------------------------
The results of the previous subsection are entirely general, in the sense that no approximation for the band structure was made. Consider, for simplicity, the tight-binding Hamiltonian (\[Htb\]) in the case of $t'=0$, that can be written, in momentum space, as: $$\begin{aligned}
H_{{\rm t.b.}} = \sum_{{\bf k},\sigma}
[a^{\dag}_{{\bf k},\sigma},b^{\dag}_{{\bf k},\sigma}]
\cdot \left[
\begin{array}{cc}
0 \hspace{0.5cm} & \phi({\bf k}) \\
\phi^*({\bf k}) \hspace{0.5cm} & 0
\end{array}
\right]
\cdot
\left[
\begin{array}{c}
a_{{\bf k},\sigma} \\
b_{{\bf k},\sigma}
\end{array}
\right] \, ,\end{aligned}$$ which can be diagonalized and produces the spectrum: $$\begin{aligned}
E_{\pm}({\bf k}) = \pm |\phi({\bf k})| \, ,
\label{spectra0}\end{aligned}$$ where the plus (minus) sign is related with the upper (lower) band. It is easy to show that the spectrum vanishes at the $K$ point in the Brillouin zone with wave-vector, ${\bf Q} = (2 \pi/(3 \sqrt{3} a),
2 \pi/(3 a))$, and other five points in the Brillouin zone related by symmetry. In fact, it is easy to show that: $$\begin{aligned}
\phi({\bf Q} +{\bf p})
&\simeq& \frac 3 2 ta e^{i\pi/3}(p_y-ip_x)
\nonumber\\
&+&
\frac 3 8 ta^2e^{i\pi/3}(p_x^2-p_y^2 -2 i p_x p_y) \, ,
\label{asimptf}
\\
\frac{\phi({\bf Q}+{\bf p})}{|\phi({\bf Q}+{\bf p})|} &=&
e^{i \delta({\bf Q}+{\bf p})}
\approx e^{i\pi/3} \frac{(p_y-i p_x)}{|{\bf p}|} \, ,
\label{coher}\end{aligned}$$ where $\bm p$ ($p \ll Q$) is measured relatively to the $K$ point in Brillouin zone and we have defined $e^{i\delta(\bm k)}=\phi(\bm k)/\vert\phi(\bm k)\vert$, for latter use. Using (\[asimptf\]) in (\[spectra0\]) we find: $$E_{\pm}({\bf Q}+{\bf p})
\simeq \pm \frac 3 2 ta \vert \bm p\vert =
\pm v_F\vert \bm p\vert \, ,
\label{esympt}$$ for the electron’s dispersion. Eq. (\[esympt\]) is the dispersion of a relativistic particle with “light” velocity $v_F = 3 t a/2$, that is, a Dirac fermion. Hence, at low energies (energies much lower than the bandwidth), the effective description of the tight-binding problem reduces the 6 points in the Brillouin zone to 2 Dirac cones, each one of them associated with a different sublattice. The low energy description is valid as long as the characteristic momenta (energy) of the excitations is smaller than a cut-off, $k_c$ ($D=v_F k_c$), of the order of the inverse lattice spacing. In the spirit of a Debye model, where one conserves the total number of states in the Brillouin zone, we choose $k_c$ such that $\pi k_c^2=(2\pi)^2/A_c$, where $A_c=3\sqrt{3}a^2/2$ is the area of the hexagonal unit cell. Hence, eq.(\[esympt\]) is valid for $p \ll k_c$ and $E \ll D = k_c v_F$.
So far we have discussed the case of $t'=0$. When $t' \neq 0$ the problem can also be easily diagonalized and one finds that, close to the K point, the electron dispersion changes to: $$\begin{aligned}
E_{\pm}({\bf Q}+{\bf p}) \approx -3 t' \pm v_F |{\bf p}| + \frac{9 t' a^2}{4}
{\bf p}^2 \, ,\end{aligned}$$ showing that $t'$ does not change the Dirac physics but introduces an asymmetry between the upper and lower bands, that is, it breaks the particle-hole symmetry. Hence, $t'$ affects only the intermediate to high energy behavior and preserves the low-energy physics. For many of the properties discussed in this section $t'$ does not play an important role and will be dropped. Nevertheless, we will see later that in the presence of extended defects $t'$ plays an important role and has to be introduced in order to provide a consistent physical picture of graphene.
For $t'=0$ we find: $$\begin{aligned}
{ G}_{AA}(\omega_n,{\bm k})&=& \sum_{j=\pm 1}
\frac{1/2}{i\omega_n - j \vert \phi(\bm k)\vert}
\label{gaa}\,,\\
{ G}_{AB}(\omega_n,{\bm k})&=& \sum_{j=\pm 1}
\frac{je^{i\delta(\bm k)}/2}
{i\omega_n - j \vert \phi(\bm k)\vert}\label{gab}\,,\\
{ G}_{BA}(\omega_n,\bm k)&=& \sum_{j=\pm 1}
\frac{je^{-i\delta(\bm k)}/2}
{i\omega_n - j \vert \phi(\bm k)\vert}\label{gba}\,,\\
{ G}_{BB}(\omega_n,{\bm k})
&=& { G}_{AA}(\omega_n,\bm k)\,.
\label{gbb}\end{aligned}$$ The expansion of the energy around the K point simplifies greatly the expressions in the calculation of the T-matrix, since they lead to simpler forms to Eqs. (\[gaa\])-(\[gbb\]). For the case of vacancies, Eq.(\[tvac\]), it is easy to see that at low energies the T-matrix reads: $$\bm T(\omega_n)= - n_i [\bar G_{AA}^0(\omega_n)]^{-1} \, \bm I\,,$$ where $\bm I$ a 2$\times$2 identity matrix, and $$\begin{aligned}
\bar G_{AA}^0(\omega_n) &=& \frac{1}{2N} \sum_{j=\pm 1,{\bf k}}
\frac{1}{i\omega_n - j \vert \phi(\bm k)\vert}
\nonumber
\\
&=& \frac{1}{2 \rho}\sum_{j=\pm 1} \int \frac{d^2 k}{(2 \pi)^2}
\frac{1}{i\omega_n - j v_F k}
\nonumber
\\
&=& \frac{1}{4 \pi \rho} \int_0^{k_c}
\frac{dk \, k}{i\omega_n - j v_F k}
\nonumber
\\
&=& - \frac{1}{4 \pi \rho v_F^2} \, \, i\omega_n \ln\left(D^2/\omega_n^2\right) \, ,\end{aligned}$$ where $\rho = S/V$ is the graphene planar density ($S$ is the area of the graphene layer). After a Wick rotation ($i\omega_n \to \omega + i 0^+$) one finds: $$\bar G^0_{AA}(\omega+i0^+)=-F_0(\omega) -i\pi\rho_0(\omega)\,,$$ where, $$\begin{aligned}
F_0(\omega)&=&\frac{2 \omega}{D^2} \ln\left(\frac{D}{|\omega|}\right) \, ,
\label{F}
\\
\rho_0(\omega)&=&
\frac{\vert\omega\vert}{D^2} \, ,
\label{R}\end{aligned}$$ where we have used that $\rho = 1/A_c = k_c^2/(4 \pi)$, and hence $4 \pi \rho v_F^2
= D^2$. In the above equations we always assume $|\omega| \ll D$. Notice that $\rho_0(\omega)$ is simply the density of states of two-dimensional Dirac fermions.
### A single vacancy
Assuming that a single unit cell has been diluted, we use Eqs. (\[g1imp\]) and (\[t1imp\]) to determine the correction to the Dirac fermion density of states. The actual density of states, $\rho(\omega)$, is given by: $$\begin{aligned}
\rho(\omega) &=& -\frac 1 {\pi} {\rm Im} \bar G_{AA}(\omega+i0^+)
\nonumber
\\
&=&\rho_0(\omega) - \frac {2/N}{D^2}\frac{\rho_0(\omega)}
{F_0^2(\omega)+\pi^2\rho_0^2(\omega)}
\nonumber\\
&\approx& \rho_0(\omega) -\frac {2/N}{\vert\omega\vert\log^2(D/|\omega|)}
\hspace{0.5cm}
(\omega\rightarrow 0)\,,
\label{R1imp}\end{aligned}$$ indicating that the contribution of the vacancy to the density of states is singular in the low frequency regime. The contribution is negative because one has exactly one missing state associated with the vacancy. The electronic wave function around a single impurity was computed in Ref.\[\]. The result obtained here is identical to the one obtained in the dilution problem in Heisenberg antiferromagnets [@CCC01; @CCC02]. The reason for this coincidence is easy to understand: the low energy excitations of an antiferromagnet in the ordered Néel phase are antiferromagnetic magnons with linear dispersion relation, that is, relativistic bosons with a “speed of light” given by the spin-wave velocity. Since we have been discussing a non-interacting problem, the statistics plays no role, and the effect of disorder is the same for relativistic bosons or fermions.
### The full Born approximation (FBA)
The situation is clearly different if one has a finite density of vacancies. In this case we have to deal with Eqs. (\[gimp\]) and (\[timp\]) corresponding to the FBA where all one-impurity scattering events have been considered. As before, the density of states is given by $\rho(\omega)=-{\rm Im} \bar G_{AA}(\omega+i0^+)/\pi$ and it is possible, after some tedious algebra, to obtain an analytical expression for this quantity, given by: $$\begin{aligned}
\rho(\omega) &=& \frac {\rho_0(\omega)}{D^2}
\frac {2 n_i}{a(\omega)}\ln\left(\frac{D^2}{b^2(\omega)+c^2(\omega)}\right)
\nonumber\\
&+& \frac {1}{\pi D^2}\sum_{\alpha=\pm 1} \frac{\alpha b(\omega)}{a(\omega)}
\left[\arctan\left(\frac{a(\omega)D}{c}\right) \right.
\nonumber
\\
&+& \left. \arctan\left(\frac{\alpha b(\omega)}{c(\omega)}\right)
\right] \, ,
\label{RFBA}\end{aligned}$$ with $$\begin{aligned}
a(\omega)&=&F_0^2(\omega)+\pi^2\rho_0^2(\omega) \, ,
\nonumber \\
b(\omega)&=& a(\omega)\omega-n_i F_0(\omega) \, ,
\nonumber \\
c(\omega)&=&n_i\pi\rho_0(\omega) \, ,
\label{abcfunc}\end{aligned}$$ where $F_0(\omega)$ and $\rho_0(\omega)$ are defined in (\[F\]) and (\[R\]), respectively. A plot of Eq. (\[RFBA\]) is given in Fig. [\[fig\_RFBA\]]{} for two values of the impurity concentration $n_i$. Once again, the low energy behavior of $\rho(\omega)$ is the same found in the context of diluted antiferromagnets. [@CCC01; @CCC02] We remark that the dilution procedure introduces a low energy scale proportional to $Dn_i$, as can be seen from panel (c) in Fig. [\[fig\_RFBA\]]{}.
![\[fig\_RFBA\] (color on line) Density of states obtained from the FBA. Panel [**(a)**]{} shows $\rho(\omega)$ over the entire band; panel [**(b)**]{} shows the low energy part, where it is seen that the peak in $\rho(\omega)$ has a higher value for lower $n_i$; panel [**(c)**]{} shows that the peak in $\rho(\omega)$ appears at an energy scale of the order of $n_iD/4$.](Fig_RFBA.eps){width="8cm"}
### The full self-consistent Born approximation (FSBA)
The FBA does not take into account electronic scattering from multiple vacancies, it accounts only for multiple scattering from a single one. In order to include some contributions from multiple site scattering another partial series summation can be performed by replacing the bare propagator in the expression of the T-matrix in (\[tvac\]) by full propagator, leading to the FSBA. Because the matrix elements of the scattering potential computed from two Bloch states $\vert\bm k\rangle$ and $\vert\bm p\rangle$ are assumed momentum independent, the self-energy for the electrons depends only on the frequency. The self-consistent problem requires, in general, a careful numerical solution but in this particular case it is possible to reduce the problem to a set of coupled algebraic equations. The self-consistent problem requires the solution of the equation: $$\Sigma(\omega_n)=\frac {-n_i}{\bar G_{AA}^0(\omega_n-\Sigma(\omega_n))} \, ,
\label{CPA}$$ where $\Sigma(\omega_n)$ is the electron self-energy. One can show that the self-energy can be written as: $$\Sigma (\omega+i0^+)= \frac {n_i}
{F(\omega)+i\pi\rho(\omega)}\,,
\label{elast}$$ where $F(\omega)$ and $\rho(\omega)$ are determined by the following set of coupled algebraic equations: $$\begin{aligned}
\label{SCF}
F(\omega)=\frac b{2a(\omega) D^2}\Psi(F,\rho,\omega)+\frac{c(\omega)}{a(\omega)D^2}
\Upsilon(F,\rho,\omega)\, , \\
\label{SCR}
\pi\rho(\omega)=\frac{c(\omega)}{2a(\omega)D^2}\Psi(F,\rho,\omega)-\frac{b(\omega)}{a(\omega)D^2}
\Upsilon(F,\rho,\omega)\, ,\end{aligned}$$ where we used the definitions (\[abcfunc\]) and also defined the functions $\Psi(F,\rho,\omega)$ and $\Upsilon(F,\rho,\omega)$: $$\begin{aligned}
\Psi(F,\rho,\omega) \sum_\alpha \ln \left[\frac{(\alpha a(\omega) D+
b(\omega))^2+c^2(\omega)}{b^2(\omega)+c^2(\omega)} \right] \, ,
\\
\Upsilon(F,\rho,\omega)=-2 \arctan[b(\omega)/c(\omega)]
\nonumber
\\
+
\sum_\alpha \alpha \arctan[a(\omega)D/c(\omega)-\alpha b(\omega)/c(\omega)]
\, .\end{aligned}$$ The solution of Eqs. (\[SCF\]) and (\[SCR\]) describes the effect of the vacancies on the density of states of the Dirac Fermions. $\rho(\omega)$ is the self-consistent density of states, and $F(\omega)$ corresponds to the real part of self-energy (in analogy with $\rho_0(\omega)$ and $F_0(\omega)$ defined in (\[R\]) and (\[F\])). In Fig. \[fig\_RFSCB\] we show the result of this procedure for various impurity concentrations.
![\[fig\_RFSCB\] (color on line) Density of states obtained in FSBA. Panel [**(a)**]{} shows $\rho(\omega)$; panel [**(b)**]{} shows $F({\omega})$; panel [**(c)**]{} shows $\rho(\omega)$ at low energies. ](Fig_RFSCB.eps){width="8cm"}
The low-energy behavior of the density of states, showing a parabolic enhancement of $\rho(\omega)$, has also been found in the context of heavy-fermion superconductors [@SMV86]. An exact numerical calculation of the electronic density was carried out in Ref.\[\], where it was found that besides the low energy dome-like shape of the $\rho(\omega)$ (as shown in Fig. \[fig\_RFSCB\]), a large peak appears very close to $\omega=0$. This peak is reminiscent of the single impurity result given in (\[R1imp\]). Hence, besides the peak, the FSBA gives a very good account of the density of states in this problem.
Notice that the self-energy, $\Sigma(\omega)$, in (\[elast\]) depends on $n_i$ in a non-trivial way, since both the self-consistent $F(\omega)$ and $\rho(\omega)$ also depend on $n_i$. The self-energy is depicted in Fig. \[fig\_self\] for various values of the dilution density $n_i$.
![\[fig\_self\] (color on line) Imaginary (left panel) and real (right panel) part of the self-energy obtained from the FSBA. Note that both quantities are divided by $Dn_i$.](Fig_self_energy.eps){width="8cm"}
Spectral and transport properties {#sspec}
=================================
The electronic spectral function is defined as: $$\begin{aligned}
A(\bm k,\omega)=- \frac{1}{\pi} {\rm Im} G(\bm k,\omega+i0^+) \, ,\end{aligned}$$ and can be interpreted as the probability density that an electron has momentum $\bm k$ and energy $\omega$. For a non-interacting, non-disordered, problem, the spectral function is simply a Dirac delta function at $\omega = E({\bf k})$. In the presence of disorder and/or electron-electron interactions the spectral function is broadened and its sharpness determines whether the electronic system supports quasi-particles. The spectral function can be measured directly in angle resolved photoemission experiments (ARPES) [@Lanzara05].
In terms of the self-energy, $\Sigma({\bf k},\omega)$, the spectral function reads: $$\begin{aligned}
A(\bm k,\omega)=- \frac{1}{\pi} \frac{{\rm Im} \Sigma({\bf k},\omega)}{[\omega-E({\bf k})-{\rm Re}\Sigma({\bf k},\omega)]^2+[{\rm Im}\Sigma({\bf k},\omega)]^2} \, .\end{aligned}$$ In the case of graphene, there are two contributions to the self-energy, $$\begin{aligned}
\Sigma({\bf k},\omega) = \Sigma_{{\rm e.-e.}} (\bm k) + \Sigma_{{\rm dis.}}(\omega) \, , \end{aligned}$$ where $\Sigma_{{\rm e.-e.}} (\bm k)$ is the self-energy correction due to the electron-electron interactions that was computed originally in Ref. \[\]: $$\begin{aligned}
{\rm Im} \Sigma_{{\rm e.-e.}} (\bm k) = \frac{1}{48} \left(\frac{e^2}{\epsilon_0 v_F}\right)^2 v_F |\bm k| \, ,
\label{selfee}\end{aligned}$$ where $e$ is the electron charge, and $\epsilon_0$ the dielectric constant of graphene. The other contribution, $\Sigma_{{\rm dis.}}$, is due to disorder and is given in (\[elast\]).
Notice that these two contributions to the self-energy have very different dependence with the energy: while the electron-electron self-energy decreases as the energy (momentum) decreases, the self-energy due to disorder increases as the energy decreases. Hence, electron-electron interactions are dominant at high energies while disorder is dominant at low energies. This interplay between the two self-energies leads to the prediction that there will be a [*minimum*]{} in the self-energy for some energy where the electron-electron interaction becomes of the same order of the electron-vacancy interaction. In Fig. \[fig\_spec\_func\] we plot the self-energy as a function of energy for various impurity concentrations together with the spectral function (inset). One can clearly observe the non-monotonic dependence of the self-energy with the energy. This behavior should be observable in ARPES experiments.
![\[fig\_spec\_func\] (color on line) Imaginary part of the electron’s self energy including both the effect of disordered and electron-electron interaction. The inset shows an intensity plot for the spectral function $A(\bm k,\omega)$ for $D=8.248$ $eV$, $n_i=0.0001$.](Fig_Nature_life_time_layer.eps){width="8cm"}
Assuming an electric field applied in the $x$-direction, the frequency dependent (real part) conductivity is calculated from the Kubo formula: $$\begin{aligned}
\sigma({\bf q},\omega) = \frac{1}{\omega} \int_0^{\infty} dt
e^{i \omega t} \langle [J^{\dag}_x({\bf q},t),J_x({\bf q},0)] \rangle
\label{kubo}\end{aligned}$$ where $J_x$ is the $x$-component of the current operator which, due to gauge invariance, has the form [@PK03]: $$J_x=-ite\sum_{i,\sigma,\bm \delta}\bm u_x\cdot \bm \delta
a^\dag_{i,\sigma}b_{i+\delta,\sigma}
-
\bm u_x\cdot \bm \delta
b^\dag_{i,\sigma+\delta} a_{i,\sigma}$$ (the notation $i+\delta$ means $\bm R_i+\bm \delta$). In Fourier space, and after expanding the general expression around the K-point in the Brillouin zone, we obtain: $$J_x=-iv_Fe\sum_{\bm k,\sigma}(e^{-i\pi/3}
a^\dag_{\bm k,\sigma}b_{\bm k,\sigma}
-
e^{i\pi/3}
b^\dag_{\bm k,\sigma} a_{\bm k,\sigma})\,.
\label{current}$$ Substitution of (\[current\]) into (\[kubo\]) shows that the problem depends on the Green’s functions defined in (\[defg\]). However, due to the special form of Eq. (\[coher\]) the conductivity does not have contributions coming from products of Green’s functions of the form $G_{AB}G_{BA}$. Taking into account the number of bands and the spin degeneracy, the Kubo formula for the real part of the conductivity at finite frequency and temperature has the form: $$\begin{aligned}
\sigma(\omega,T)=- \frac{4 v_F^2e^2}{N A_c \omega}
\int_{-\infty}^{\infty}
\frac {d\epsilon}{2\pi}[f(\epsilon+ \omega)-f(\epsilon)]\times
\nonumber\\
\sum_{\bm k}
{\rm Im} G_{AA}(\bm k,\epsilon+i0^+)
{\rm Im} G_{AA}(\bm k,\epsilon+ \omega+i0^+)\,,
\label{kubofinal}\end{aligned}$$ where $f(\epsilon) = 1/(e^{(\epsilon-\mu)/T}+1)$ is the Fermi-Dirac distribution function. The integral over $\bm k$ in (\[kubofinal\]) can be performed and find: $$\begin{aligned}
\sigma(\omega,T)&=&-\frac{e^2}{2 \pi^2 \omega}
\int_{-\infty}^{\infty}
d\epsilon[f(\epsilon+ \omega)-f(\epsilon)] K(\omega,\epsilon) \, ,
\label{sigmat}\end{aligned}$$ where $$\begin{aligned}
K(\omega,\epsilon) = {\rm Im} \Sigma(\epsilon+ \omega) \, \, {\rm Im} \Sigma(\epsilon) \, \, \Theta( \omega,\epsilon)\,,
\label{kern}\end{aligned}$$ ($\Theta(\omega,\epsilon)$ defined in Appendix \[ap1\]).
It is instructive to consider the zero-temperature, zero-frequency limit of the conductivity in Eq. (\[kubofinal\]) (restoring $\hbar$): $$\sigma_0 = \frac 2{\pi}\frac{e^2}{h}\left (1-
\frac {[{\rm Im} \Sigma(0)]^2}{D^2+[{\rm Im} \Sigma(0)]^2}
\right) \approx \frac 2{\pi}\frac{e^2}{h} \, .
\label{universal}$$ The result (\[universal\]) shows that as long as ${\rm Im} \Sigma(0)\ll D$, $\sigma_0$ has a universal value independent of the dilution concentration, in agreement with earlier theoretical works [@F86; @L93], and in agreement with the experimental data in graphene [@Netal05].
![\[fig\_kernel\_cond\] (color on line) Left: Kernel of the integral for $\sigma(0,T)$ as function of the energy. Product of the Fermi function derivative by $K(\epsilon)$ at two different temperatures.](Fig_kernel_cond_b.eps){width="8cm"}
At finite temperatures the integral in (\[sigmat\]) has to be evaluated numerically. Consider $\sigma(0,T)$ whose behavior is determined by $K(\epsilon) \equiv K(0,\epsilon)$. The quantities $K(\epsilon)$ and $-f'(\epsilon)K(\epsilon)$ ( $-f'(\epsilon)$ is the derivative of the Fermi function in order to $\epsilon$) are both represented in Fig. \[fig\_kernel\_cond\]. The behavior of $K(\epsilon)$ shows, “V”-like shape as the energy $\epsilon$ is varied. As a consequence, $\sigma(0,0)$ should present the same “V”-like shape as the chemical potential $\mu$ moves around $\mu=0$. Such behavior has indeed been observed in atomically thin carbon films [@Netal04; @Netal05], where the density of electrons was controled by a gate potential. The temperature dependence of the $\sigma(0,T)$, for $\mu=0$, is depicted in Fig. \[fig\_cond\_T\] for different vacancy concentrations, and it is found to follow Sommerfeld asymptotic expansion, but the number of terms needed to fit the numerical curve grows very fast as the dilution is reduced.
![\[fig\_cond\_T\] (color on line) Dependence of $\sigma(T,0)$ on the temperature and on the impurity dilution $n_i$.](Fig_condutivity_T_b.eps){width="8cm"}
In Fig. \[fig\_cond\_omega\] we plot the frequency dependence of $\sigma(\omega,T)$ obtained from numerical integration of (\[sigmat\]) with the self-energy given in (\[elast\]). At low temperature, we see that $\sigma(\omega,T)$ develops a maximum around an energy value that is dependent on the number of impurities. In fact, if plot $\sigma(\omega,T)$ as function of $\omega / \sqrt{n_i}$, the conductivity almost shows scaling behavior for all impurity dilutions (see lower left panel). As the temperature increases, and if $\sigma (0,T)$ is sufficiently large, the conductivity $\sigma(\omega,T)$ acquires a Drude-like behavior (right panel).
![\[fig\_cond\_omega\] (color on line) Dependence of $\sigma(T,\omega)$ (in units of $e^2/(\pi \hbar)$) , on the frequency $\omega$.](Fig_condutivity_omega_1.eps){width="8cm"}
Magnetic response and the role of short range Coulomb interactions {#smag}
==================================================================
The ferromagnetism measured in proton irradiated graphite opens the question whether the interplay of interactions and disorder can drive the system from a paramagnetic to a magnetic ground state [@Eetal03]. We study the effect of disorder on the magnetic susceptibility in the presence of short-range interactions, and the resulting change in the tendency towards magnetic instabilities. The problem of magnetic instabilities due to long-range exchange interactions [@B29] in the presence of small density of carriers was discussed in great detail in ref. \[\]. We do not address here the effects associated to the interplay between the long-range Coulomb interaction and different types of lattice disorder[@SGV05] and the appearance of local moments close to defects [@OS91; @H01; @Letal04; @Vetal05].
The paramagnetic susceptibility of graphene is given by: $$\begin{aligned}
\chi (T)&=& \frac {\partial m_z}{\partial h}=4\frac{\partial }{\partial h}
\frac 1 N \sum_{\bm k}\sum_nG_{AA}(\bm k,i\omega_n-h)\nonumber\\
&=&-4\int_{-\infty}^\infty d\epsilon f'(\epsilon)\rho(\epsilon)\,, \end{aligned}$$ where $$m_z(T)=2\sum_{i,\sigma}\sigma\langle
a^\dag_{i,\sigma}a_{i,\sigma}
\rangle\,,$$ is the magnetization, and $\rho(\epsilon) $ is the electronic density of states which, in the presence of disorder, is given in (\[SCR\]). Within the Stoner mechanism [@S47], ferromagnetism is possible if the local electron-electron interaction term (the so-called Hubbard term)[@Tetal98], $U$, is large than a critical value given by: $$\begin{aligned}
\frac{1}{U_F^c}= \frac{1}{4} \chi(0) \, .
\label{ufc}\end{aligned}$$ In the case of an antiferromagnetic instability the same criteria would lead to another critical value of $U$ given by: $$\frac{1}{U_{AF}^c}= \frac 2{\pi D}
\int_{-\infty}^D d\epsilon \rho(\epsilon)
\arctan \left[\frac {D}{
{\rm Im} \Sigma(\epsilon )} \right]\,.
\label{uafc}$$ Notice that in the case of antiferromagnetism one finds that $U_{AF}^c \approx D/(1-n_i)$ when ${\rm Im}\Sigma \to 0$, in agreement with Hartree-Fock calculations [@PAD04].
In Fig. \[fig\_sus\] we plot the magnetic susceptibility as function of $T$ for different values of $n_i$. The signature of the presence of Dirac fermions comes from the linear dependence on $T$ for $T/D \ll 1$. Notice that, unlike the case of an ordinary metal that has a temperature dependent Pauli susceptibility, the graphene susceptibility increases with temperatures and number of impurities. At low temperatures $\chi(T)$ presents a small upturn not visible in Fig. \[fig\_sus\]. From the value of $\chi(0)$ and (\[ufc\]) we obtain the critical interaction required for a ferromagnetic transition, which is shown in the lower left panel of Fig. \[fig\_sus\]. Notice that the critical interaction strength for ferromagnetism decreases as the vacancy concentration increases indicating that disorder favors a ferromagnetic transition.
Using (\[uafc\]) and the results of the previous sections we can also calculate the critical value for an antiferromagnetic transition. The result is shown in the lower right panel of Fig. \[fig\_sus\]. In contrast with the ferromagnetic case, the antiferromagnetic instability is suppressed by disorder, requiring a large value of the electron-electron interaction. Notice that the value of the critical ferromagnetic coupling is always bigger than the antiferromagnetic one, indicating that at half-filling the graphene lattice is more susceptible to antiferromagnetic correlations. This result is consistent with an old proposal by Linus Pauling that graphene should be a resonant valence bond (RVB) state with local singlet correlations [@P72].
![\[fig\_sus\] (color on line) Top: Dependence of $\chi(T)$ on the temperature and on the impurity dilution $n_i$. Bottom: Dependence of $U_F$ and $U_{AF}$ (in units of $D$) as function of $n_i$.](Fig_sus.eps){width="7cm"}
Hence, the Stoner criteria seems to be unable to explain the ferromagnetic behavior observed experimentally. One might ask whether additional scattering mechanisms, such that provided by long-range electron-electron interactions, can modify the critical values of the couplings. The self-energy correction due to long-range electron-electron scattering is given in (\[selfee\]) and can be added to the Dyson equation for the Green’s function and a new self-consistent density of states can be computed. This approach does not modify the value of $U_F^c$ which is determined by the low frequency behavior of the self-energy. In case of antiferromagnetism we find that indeed it leads to an increase on $U_{AF}^c$, but the result is non-conservative since the integral over the density of states gives a smaller value than $(1-n_i)$. Therefore, we find from these calculations and previous ones [@PGN05b] that graphene is not particularly susceptible to ferromagnetism.
Magneto-Transport {#landau}
=================
The description of the magneto-transport properties of electrons in a disordered honeycomb lattice is complex because of the interference effects associated with the Hofstadter problem [@GF97]. As in the previous sections, we simplify our problem by describing the electrons in the honeycomb lattice as Dirac fermions in the continuum. A similar approach was considered by Abrikosov in the quantum magnetoresistance study of non-stoichiometric chalcogenides [@A98]. In the case of graphene, the effective Hamiltonian describing Dirac fermions in a magnetic field (including disorder) can be written as: $H=H_0+H_i$ with $$H_0=-v_F\sum_{i=x,y}\sigma_i[-i \partial_i+eA_i(\bm r)]\,,
\label{llH0}$$ where, in the Landau gauge, $(A_x,A_y,A_z)=(-By,0,0)$ is the vector potential for a constant magnetic field $B$ in the $z-$direction, $\sigma_i$ is the $i=x,y,z$ Pauli matrix, and $$H_i=V {\sum_{j=1}^{N_i}\delta(\bm r-\bm r_j) \bm I}\,.
\label{llH1}$$ The formulation of the problem in second quantization requires the solution of $H_0$, which is sketched in Appendix \[ap2\]. The field operators are defined as (see Appendix \[ap2\] for notation; the spin index is omitted for simplicity): $$\begin{aligned}
\Psi(\bm r)&=&\sum_{k}\frac {e^{ikx}}{\sqrt L}
\left(
\begin{array}{c}
0\\
\phi_0(y)
\end{array}
\right)c_{k,-1} \nonumber\\
&+&
\sum_{n,k,\alpha}
\frac {e^{ikx}}{\sqrt {2L}}
\left(
\begin{array}{c}
\phi_{n}(y-kl_B^2)\\
\phi_{n+1}(y-kl_B^2)
\end{array}
\right)c_{k,n,\alpha}\,.\end{aligned}$$ The sum over $n=0,1,2,\ldots,$ is cut off at an $n_0$ given by $E(1,n_0)=D$. In this representation $H_0$ becomes diagonal, leading to Green’s functions of the form (in Matsubara representation): $$G_0(k,n,\alpha;i\omega)
=\frac 1{i\omega - E(\alpha,n)}\,,$$ is effectively $k$-independent, and $
E(\alpha,-1)=0$ is the zero energy Landau level. The part of Hamiltonian due to the impurities is written as:
$$\begin{aligned}
H_i& =& \frac V L\sum_{j=1}^{N_i}\sum_{p,k}e^{-ix_j(p-k)}
\left[\phi_0(y_j-pl_B^2)\phi_0(y_j-kl_B^2) c^\dag_{p,-1}c_{k,-1}
+\sum_{n,\alpha}\frac {\alpha}{\sqrt 2}
\phi_0(y_j-pl_B^2)\phi_{n+1}(y_j-kl_B^2) c^\dag_{p,-1}c_{k,n,\alpha}
\right.
\nonumber\\
&+&
\sum_{n,\alpha}\frac {\alpha}{\sqrt 2}
\phi_{n+1}(y_j-pl_B^2)\phi_0(y_j-kl_B^2) c^\dag_{p,n,\alpha} c_{k,-1}
\nonumber\\
&+&
\left.
\sum_{n,m,\alpha,\lambda}
\frac 1 2[\phi_n(y_j-pl_B^2)\phi_m(y_j-kl_B^2)
+ \alpha\lambda
\phi_{n+1}(y_j-pl_B^2)\phi_{m+1}(y_j-kl_B^2)]
c^\dag_{p,n,\alpha} c_{k,m,\lambda}
\right]\,.
\label{hint2}\end{aligned}$$
Equation (\[hint2\]) describes the elastic scattering of electrons in Landau levels by the impurities. It is worth noting that this type of scattering connects Landau levels of negative and positive energy.
![\[fig\_dos\_landau\] (color on line) Electronic density of states in a magnetic field for different dilutions and magnetic field. The non-disordered DOS and the position of the Landau levels in the absence of disorder are also shown. The two arrows in the upper panel show the position of the renormalized Landau levels (see Fig.\[fig\_self\_energy\]) given by the solution of Eq. (\[rpole\]). The energy is given in units of $\omega_c\equiv E(1,1)$.](Fig_dos_landau_b.eps){width="8cm"}
The full self-consistent Born approximation
-------------------------------------------
In order to describe the effect of impurity scattering on the magnetoresistance of graphene, the Green’s function for Landau levels in the presence of disorder needs to be computed. In the context of the 2D electron gas, an equivalent study was performed by Otha and Ando,[@O68; @O71; @Atot]using the averaging procedure over impurity positions of Duke[@D68]. Here the averaging procedure over impurity positions is performed in the standard way, namely, having determined the Green’s function for a given impurity configuration $(\bm r_1,\ldots\bm r_{N_i})$, the position averaged Green’s function is determined from (as in Sec. \[tmatrix\]): $$\begin{aligned}
\langle G(p,n,\alpha;i\omega;\bm r_1,\ldots\bm r_{N_i})\rangle
\equiv G(p,n,\alpha;i\omega) \nonumber\\
= L^{-2N_i}\left[\prod_{j=1}^{N_i}
\int d\bm r_j\right] G(p,n,\alpha;i\omega;\bm r_1,\ldots\bm r_{N_i})\,.\end{aligned}$$ In Sec. \[tmatrix\] the averaging involved plane wave states; in the presence of Landau levels the average over impurity positions involves the wave functions of the one-dimensional harmonic oscillator. In the averaging procedure we have used the following identities: $$\begin{aligned}
\int dy \phi_n(y-pl_B^2)\phi_m(y-pl_B^2)&=&\delta_{n,m}\,,
\\
\int dp \phi_n(y-pl_B^2)\phi_m(y-pl_B^2)&=&
\frac {\delta_{n,m}}{l^2_B}
\, .\end{aligned}$$ After a lengthy algebra, the Green’s function in the presence of vacancies, in the FSBA, can be written as: $$\begin{aligned}
\label{gn}
G(p,n,\alpha;\omega+0^+)&=&[\omega-E(n,\alpha)-\Sigma_1(\omega)]^{-1}\,,
\\
\label{g0}
G(p,-1;\omega+0^+)&=&[\omega-\Sigma_2(\omega)]^{-1}\,,
\label{greenlandau}\end{aligned}$$ where $$\begin{aligned}
\label{S1}
\Sigma_1(\omega)&=&- n_i[Z(\omega)]^{-1}\,,\\
\label{S2}
\Sigma_2(\omega)&=&- n_i[g_cG(p,-1;\omega+0^+)/2 + Z(\omega)]^{-1}\,,\\
\label{Z}
Z(\omega)&=&g_cG(p,-1;\omega+0^+)/2\nonumber\\
&+&g_c\sum_{n,\alpha}G(p,n,\alpha;\omega+0^+)/2\,,\end{aligned}$$ and $g_c=A_c/(2\pi l_B^2)$ is the degeneracy of a Landau level per unit cell. One should notice that the Green’s functions do not depend upon $p$ explicitly. The self-consistent solution of Eqs. (\[gn\]), (\[g0\]), (\[S1\]), (\[S2\]) and (\[Z\]) gives density of states, the electron self-energy, and the renormalization of Landau level energy position due to disorder.
The effect of dilution in the density of states of Dirac fermions in a magnetic field is shown in Fig. \[fig\_dos\_landau\]. For reference we note that $E(1,1)=0.14$ eV, for $B=14$ T, and $E(1,1)=0.1$ eV, for $B=6$ T. From Fig. \[fig\_dos\_landau\] we see that given an impurity concentration the effect of broadening due to vacancies is less effective as the magnetic field increases. It is also clear that the position of Landau levels is renormalized relatively to the non-disordered case. The renormalization of the Landau level position can be determined from poles of (\[greenlandau\]): $$\omega-E(\alpha,n)-{\rm Re}\Sigma(\omega)=0\,.
\label{rpole}$$ Of course, due to the importance of scattering at low energies, the solution to Eq. (\[rpole\]) does not represent exact eigenstates of system since the imaginary part of the self-energy is non-vanishing, however these energies do determine the form of the density of states, as we discuss below.
In Fig. \[fig\_self\_energy\], the graphical solution to Eq. (\[rpole\]) is given for two different energies ($E(-1,n)$, with $n=1,2$), being clear that the renormalization is important for the first Landau level. This result is due to the increase of the scattering at low energies, which is present already in the case of zero magnetic field. The values of $\omega$ satisfying Eq. (\[rpole\]) show up in density of states as the energy values where the oscillations due to the Landau level quantization have a maximum. In Fig. \[fig\_dos\_landau\], the position of the renormalized Landau levels is shown in the upper panel (marked by two arrows), corresponding to the bare energies $E(-1,n)$, with $n=1,2$. The importance of this renormalization decreases with the reduction of number of vacancies. This is clear in Fig. \[fig\_dos\_landau\] where a visible shift toward low energies is evident when $n_i$ has a small 10$\%$ change, from $n_i=10^{-3}$ to $n_i=9 \times 10^{-4}$.
![\[fig\_self\_energy\] (color on line) Self-consistent results for $\Sigma_1(\omega)$ (top) and $\Sigma_2(\omega)$(bottom). The energy is given in units of $\omega_c\equiv E(1,1)$. In the left panels we show the intercept of $\omega-E(\alpha,n)$ with ${\rm Re}\Sigma(\omega)$ as required in (\[rpole\]).](Fig_self_energy_landau_c.eps){width="8cm"}
Calculation of the transport properties
---------------------------------------
The study of the magnetoresistance properties of the system requires the calculation of the conductivity tensor. In terms of the field operators, the current density operator $\bm j$ is given by [@GGV96]: $$\bm j = v_Fe[\Psi^\dag(x,y)\sigma_x \Psi(x,y),
\Psi^\dag(x,y)\sigma_y \Psi(x,y)]\,,$$ leading to current operator in the Landau basis written as:
$$\begin{aligned}
J_x&=&v_Fe\sum_{p,\alpha}\frac 1 {\sqrt 2}\left[
c^\dag_{p,-1}c_{k,0,\alpha} + c^\dag_{p,0,\alpha}c_{p,-1}\right]
+v_Fe\sum_{p,n,\alpha,\lambda}
\frac 1 2
\left[
\lambda (1-\delta_{n,0})c^\dag_{p,n,\alpha} c_{p,n-1,\lambda}
+ \alpha c^\dag_{p,n,\alpha} c_{p,n+1,\lambda}
\right]\,,\\
J_y&=&v_Fe\sum_{p,\alpha}\frac i {\sqrt 2}\left[
c^\dag_{p,-1}c_{k,0,\alpha} - c^\dag_{p,0,\alpha}c_{p,-1}\right]
+v_Fe\sum_{p,n,\alpha,\lambda}
\frac i 2
\left[
-\lambda (1-\delta_{n,0})c^\dag_{p,n,\alpha} c_{p,n-1,\lambda}
+ \alpha c^\dag_{p,n,\alpha} c_{p,n+1,\lambda}
\right]\,.\end{aligned}$$
As in Sec. \[sspec\], we compute the current-current correlation function and from it the conductivity tensor is derived. The diagonal component of the conductivity tensor $\sigma_{xx}(\omega,T)$ is given by (with the spin included): $$\begin{aligned}
\label{s11}
\sigma_{xx}(\omega,T)&=&-\frac {4(v_Fe)^2}{2\pi l_B^2}
\frac 1 {\omega}\int_{-\infty}^{\infty}
\frac {d\epsilon}{2\pi}[f(\epsilon+ \omega)-f(\epsilon)]
\left[ \frac 1 2\sum_{\alpha_1}[{\rm Im} G(-1;\epsilon+i0^+)
{\rm Im} G(0,\alpha;\epsilon+ \omega+i0^+)\right.\nonumber\\
&+&{\rm Im} G(0,\alpha;\epsilon+i0^+)
{\rm Im} G(-1;\epsilon+ \omega+i0^+)]+
\frac 1 4 \sum_{n\ge 1,\alpha,\lambda}
{\rm Im} G(n,\alpha;\epsilon+i0^+){\rm Im} G(n-1,\lambda;\epsilon+ \omega+i0^+)
\nonumber\\
&+&
\left.
\frac 1 4 \sum_{n\ge 0,\alpha,\lambda}
{\rm Im} G(n,\alpha;\epsilon+i0^+){\rm Im} G(n+1,\lambda;\epsilon+ \omega+i0^+)
\right]\,,\end{aligned}$$ and the off-diagonal component $\sigma_{xy}(\omega,T)$ of the conductivity tensor is given by: $$\begin{aligned}
\label{s12}
\sigma_{xy}(\omega,T)&=&\frac {2(v_Fe)^2}{4\pi l_B^2}
\frac 1 {\omega}\int_{-\infty}^{\infty}
\frac {d\epsilon}{2\pi}\tanh \left(\frac{\epsilon}{2T}
\right)
\sum_{\alpha,\gamma}\left[
\gamma[{\rm Re} G(0,\alpha;\epsilon+\gamma \omega + i0^+){\rm Im} G(-1;\epsilon
+i0^+)\right.\nonumber\\
&-&{\rm Re} G(-1;\epsilon+\gamma \omega + i0^+){\rm Im} G(0,\alpha;\epsilon
+i0^+)]+\sum_{\lambda,n\ge 1}\frac {\gamma}2
[{\rm Re} G(n,\alpha;\epsilon+\gamma \omega + i0^+){\rm Im} G(n-1,\lambda;\epsilon
+i0^+)\nonumber\\
&-&
\left.{\rm Re} G(n-1,\alpha;\epsilon+\gamma \omega + i0^+)
{\rm Im} G(n,\lambda;\epsilon+i0^+)
]\right]\,.\end{aligned}$$
If we neglect the real part of the self-energy, and assume ${\rm Im}\Sigma_{i}(\omega)=\Gamma=$ constant ($i=1,2$), and let $\omega \to 0$, Eq. (\[s11\]) reduces to Eq. (85) in Ref. \[\], if we further assume the case $E(1,1)\gg \Gamma$ then Eq. (\[s12\]) reduces to Eq. (86) of the same reference.
As in Sec. \[sspec\], it is instructive to consider first the case $\omega,T\rightarrow 0$, which leads to ($\sigma_{xx}(0,0)=\sigma_0$): $$\begin{aligned}
\sigma_0&=&\frac {e^2}h \frac 2{\pi}
\left[\frac {{\rm Im} \Sigma_1(0)/{\rm Im}\Sigma_2(0)-1}{1+({\rm Im}\Sigma_1(0)/\omega_c)^2
}\right.\nonumber\\
&+&\left.\frac {n_0+1}{n_0+1 + ({\rm Im}\Sigma_1(0)/\omega_c)^2}
\right]\,.
\label{s0B}\end{aligned}$$ When ${\rm Im} \Sigma_1(0)\simeq{\rm Im}\Sigma_2(0)$ and $\omega_c\gg {\rm Im}\Sigma_1(0)$ (or $n_0\gg {\rm Im}\Sigma_1(0)/\omega_c$), with $\omega_c=E(0,1)=\sqrt 2 v_F /l^2_B$, Eq. (\[s0B\]) reduces to: $\sigma_0\simeq 2/\pi (e^2/h)$, which is identical to the result (\[universal\]) in the absence of the field found in Sec. \[sspec\]. This result was obtained previously by Ando and collaborators using the second order self-consistent Born approximation [@ST98; @AZS02]. However, in the FSBA it is required that the above conditions be satisfied for this result to hold. From Fig. \[fig\_kernel\_landau\] we see that the above conditions hold approximately over a wide ranges of field strength.
![\[fig\_kernel\_landau\] (color on line) Kernel of the conductivity (in units of $\pi h/e^2$) as function of energy for different magnetic fields and for a dilution concentration of $n_i=0.001$. The horizontal line represents the universal limit $\pi h \sigma_0/e^2=2$.](Fig_cond_kernel_landau.eps){width="8cm"}
Because the d.c. magneto-transport properties have been measured graphene samples [@Netal04] subjected to a gate potential (allowing to tune the electronic density), it is important to compute the conductivity kernel, since this has direct experimental relevance. In the the case $\omega\rightarrow 0$ we write the conductivity $\sigma_{xx}(0,T)$ as: $$\sigma_{xx}(0,T)= \frac {e^2}{\pi h}\int_{-\infty}^{\infty}
d\epsilon \frac {\partial f(\epsilon)}{\partial \epsilon}
K_B(\epsilon)\,,
\label{sxxb}$$ where the conductivity kernel $K_B(\epsilon)$ is given is Appendix \[ap1\]. The magnetic field dependence of kernel $K_B(\epsilon)$ is shown in Fig. \[fig\_kernel\_landau\]. Observe that the effect of disorder is the creation of a region where $K_B(\epsilon)$ remains constant before it starts to increase in energy with superimposed oscillations coming from the Landau levels. The same effect, but with the absence of the oscillations, was identified at the level of the self-consistent density of states plotted in Fig. \[fig\_dos\_landau\]. Together with $\sigma_{xx}(0,T)$, the Hall conductivity $\sigma_{xy}(0,T)$ allows the calculation of the resistivity tensor: $$\begin{aligned}
\rho_{xx} &=& \frac{\sigma_{xx}}{\sigma_{xx}^2+\sigma_{xy}^2} \, ,
\nonumber
\\
\rho_{xy} &=& \frac{\sigma_{xy}}{\sigma_{xx}^2+\sigma_{xy}^2} \, .\end{aligned}$$
Let us now focus on the optical conductivity, $\sigma_{xx}(\omega)$. This quantity can be probed by reflectivity experiments on the sub-THz to Mid-IR frequency range.[@B05] We depict the behavior of Eq. (\[s11\]) in Fig. \[fig\_sigomega\_landau\] for different magnetic fields. It is clear that the first peak is controlled by the $E(1,1)-E(1,-1)$, and we have checked it does not obey any particular scaling form as function of $\omega/B$. On the other hand, as the effect of scattering becomes less important the high energy conductivity oscillations start obeying the scaling $\omega/\sqrt{B}$, as we show in the lower right panel of Fig. \[fig\_sigomega\_landau\].
![\[fig\_sigomega\_landau\] (color on line) Optical conductivity (in units of $\pi h/e^2$) at 10 $K$ as function of the energy for different magnetic fields and for a dilution concentration of $n_i=0.001$. The vertical arrows in the upper left panel, labeled [**a**]{}, [**b**]{}, and [**c**]{}, represent the renormalized $E(1,1)-E(-1,0)$, $E(2,1)-E(-1,0)$, and $E(1,1)-E(1,-1)$ transitions.](Fig_condutivity_landau.eps){width="8cm"}
Extended defects
================
Self-doping in the absence of electron-hole symmetry.
-----------------------------------------------------
The standard description of a graphene sheet, following the usual treatment of the electronic band structure of graphite[@BCP88; @SW58; @M64; @DSM77], assumes that in the absence of interlayer interactions the electronic structure of graphite shows electron-hole symmetry. This can be justified using a tight binding model by considering only hopping between $\pi$ orbitals located at nearest neighbor Carbon atoms. Within this approximation it can be shown that in certain graphene edges[@WS00; @W01] one would find a flat surface band [@Metal05]. Disclinations (a pentagonal or heptagonal ring) can also lead to a discrete spectrum and states at zero energy[@GGV92; @GGV93b]. Other types of defects, like a combination of a five-fold and seven-fold ring (a lattice dislocation) or a Stone-Wales defect (made up of two pentagons and two heptagons) also lead to a finite density of states at the Fermi level[@CEL96; @MA01; @DSL04].
Band structure calculations show that the electronic structure of a single graphene plane is not strictly symmetrical around the energy of the Dirac points[@Retal02]. The absence of electron-hole symmetry shifts the energy of the states localized near impurities above or below the Fermi level, leading to a transfer of charge from/to the clean regions to the defects. Hence, the combination of localized defects and the lack of perfect electron-hole symmetry around the Dirac points leads to the possibility of self-doping, in addition to the usual scattering processes whose influence on the transport properties has been discussed in the preceding sections.
Point defects, like impurities and vacancies, can nucleate a few electronic states in their vicinity. Hence, a concentration of $n_i$ impurities per Carbon atom leads to a change in the electronic density of the regions between the impurities of order $n_i$. The corresponding shift in the Fermi energy is $\epsilon_{\rm F}
\simeq v_{\rm F} \sqrt{n_{i}}$. In addition, the impurities lead to a finite elastic mean free path, $l_{\rm elas} \simeq a n_{i}^{-1/2}$, and to an elastic scattering time $\tau_{\rm elas} \simeq ( v_{\rm F} n_i
)^{-1}$, in agreement with the FSBA calculation in the preceding sections. Hence, the regions with few impurities can be considered low-density metals in the dirty limit, as $\tau_{\rm elas}^{-1} \simeq \epsilon_{\rm F}$.
Extended lattice defects, like edges, grain boundaries, or microcracks, are likely to induce the formation of a number of electronic states proportional to their length, $L/a$, where $a$ is of the order of the lattice constant. Hence, a distribution of extended defects of length $L$ at a distance proportional to $L$ itself gives rise to a concentration of $L/a$ carriers per unit Carbon in regions of order $(L/a)^2$. The resulting system can be considered a metal with a low density of carriers, $n_{\rm carrier} \propto a/L$ per unit cell, and an elastic mean free path $l_{\rm elas} \simeq L$. Then, we obtain: $$\begin{aligned}
\epsilon_{\rm F} &\simeq &\frac{v_{\rm F}}{\sqrt{a L}} \nonumber \\
\frac{1}{\tau_{{\rm elas}}} &\simeq &\frac{v_{\rm F}}{L}\end{aligned}$$ and, therefore, $( \tau_{{\rm elas}} )^{-1} \ll \epsilon_{\rm F}$ when $a/L \ll
1$. Hence, the existence of extended defects leads to the possibility of self-doping but maintaining most of the sample in the clean limit. In this regime, coherent oscillations of the transport properties are to be expected, although the observed electronic properties will correspond to a shifted Fermi energy with respect to the nominally neutral defect–free system.
Electronic structure near extended defects.
-------------------------------------------
We describe the effects that break electron-hole symmetry near the Dirac points in terms of a finite next-nearest neighbor hopping between $\pi$ orbitals, $t'$, in (\[Htb\]). From band structure calculations[@Retal02], we expect that $| t'
/ t | \le 0.2$. We calculate the electronic structure of a ribbon of width $L$ terminated at zigzag edges, which are known to lead to surface states for $t'=0$. The translational symmetry along the axis of the ribbon allows us to define bands in terms of the wavevector parallel to this axis. In Fig. \[ribbon\_bands\], we show the bands closest to $\epsilon=0$ for a ribbon of width 200 unit cells and different values of $t' / t$. The electronic structure associated to the interior region (the continuum cone), projected in Fig. \[ribbon\_bands\] is not significantly changed by $t'$. The localized surface bands, extending from $k_\parallel = ( 2 \pi )/3$ to $k_\parallel = - ( 2 \pi )/3$, on the other hand, acquires a dispersion of order $t'$ (for a perturbative treatment of this effect, see ref. \[\]). Hence, if the Fermi energy remains unchanged at the position of the Dirac points ($\epsilon_{\rm Dirac} = - 3 t'$), this band will be filled, and the ribbon will no longer be charge neutral. In order to restore charge neutrality, the Fermi level needs to be shifted down (for the sign of $t'$ chosen in the figure) by an amount of order $t'$. As a consequence, some of the extended states near the Dirac points are filled, leading to the phenomenon of self-doping. The local charge as function of distance to the edges, setting the Fermi energy so that the ribbon is globally neutral. Note that the charge transferred to the surface states is very localized near the edges of the system.
Electrostatic effects.
----------------------
The charge transfer discussed in the preceding subsection is suppressed by electrostatic effects, as large deviations from charge neutrality have an associated energy cost. In order to study these charging effects we add to the free-electron Hamiltonian (\[Htb\]) the Coulomb energy of interaction between electrons: $$\begin{aligned}
H_I = \sum_{i,j} U_{i,j} n_i n_j \, ,
\label{interact}\end{aligned}$$ where $n_i = \sum_{\sigma} (a^{\dag}_{i,\sigma} a_{i,\sigma} +
b^{\dag}_{i,\sigma} b_{i,\sigma})$ is the number operator at site ${\bf R}_i$, and $$\begin{aligned}
U_{i,j} = \frac{e^2}{\epsilon_0 |{\bf R}_i-{\bf R}_j|} \, ,\end{aligned}$$ is the Coulomb interaction between electrons. We expect, on physical grounds, that an electrostatic potential builds up at the edges, shifting the position of the surface states, and reducing the charge transferred to/from them. The potential at the edge induced by a constant doping $\delta$ per Carbon atom is roughly, $\sim (\delta e^2/a) (W/a)$ ($\delta e^2/a$ is the Coulomb energy per Carbon), and $W$ the width of the ribbon ($W/a$ is the number of Carbons involved). The charge transfer is arrested when the potential shifts the localized states to the Fermi energy, that is, when $t' \approx (e^2/a) (W/a) \delta$. The resulting self-doping is therefore $\delta \sim ( t' a^2 ) / ( e^2 W )$.
We treat the Hamiltonian (\[interact\]) within the Hartree approximation (that is, we replace $H_I$ by $H_{{\rm M.F.}} = \sum_i V_i n_i$ where $V_i = \sum_j U_{i,j} \langle n_j \rangle$, and solve the problem self-consistently for $\langle n_i \rangle$). Numerical results for graphene ribbons of length $L = 80 \sqrt{3} a$ and different widths are shown in Fig. \[ribbon\_charge\] and Fig. \[dopingfig\] ($t'/t= 0.2$ and $e^2/a
= 0.5 t$). The largest width studied is $\sim 0.1 \mu$m, and the total number of carbon atoms in the ribbon is $\approx 10^5$. Notice that as $W$ increases, the self-doping decreases indicating that, for a perfect graphene plane ($W \rightarrow \infty$), the self-doping effect disappears. For realistic parameters, we find that the amount of self-doping is $10^{-4} - 10^{-5}$ electrons per unit cell for domains of sizes $0.1 - 1 \mu$m, in agreement with the amount of charge observed in these systems.
Edge and surface states in the presence of a magnetic field.
------------------------------------------------------------
We can analyze the electronic structure of a graphene ribbon of finite width in the presence of a magnetic field. The resulting tight binding equations can be considered as an extension of the Hofstadter problem[@H76] to a honeycomb lattice with edges. The bulk electronic structure is characterized by the Landau level structure discussed in previous sections. These states are modified at the edges, leading to chiral edge states, as discussed in relation to the Integer Hall Effect (IQHE)[@H82]. The existence of two Dirac points leads to two independent edge states, with the same chirality. In addition, Landau levels with positive energy should behave in an electron-like fashion, moving upwards in energy as their “center of gravity” approaches the edges. Landau levels with negative energy should be shifted towards lower energy near the edges.
A zig-zag edge induces also a non-chiral surface band. If the width of these states is much smaller than the magnetic field they will not be much affected by the presence of the field. The extension of the surface states is comparable to the lattice spacing for most of the range $|k_\parallel| \le ( 2 \pi ) / 3$, except near the Dirac points, so that the effect of realistic magnetic fields on these states is negligible.
The finite value of the second nearest neighbor hopping $t'$ modifies the Landau levels obtained from the analysis of the Dirac equation. Elementary calculations (as those given in Appendix \[ap2\]) lead to: $$E_{\pm}(n)=-3t'+2l_B^{-2}\alpha\left( n + \frac{1}{2} \right) \pm
\sqrt{l_B^{-4}\alpha^2+2l_B^{-2}\gamma^2 n}\,,
\label{ELL}$$ with $n=0,1,2,3,\ldots$, $\alpha = 9t'a^2/4$ and $\gamma=3ta/2$, with the single assumption that $t\gg t'$. This solution points out a number of interesting aspects, the most important of which is disappearance of the zero energy Landau level, made partially of holes and partially of electrons. With $t'$, the electron or hole nature of the energy level becomes unambiguous, and half of the original zero energy Landau level (with $t'=0$)moves down in energy (relatively to the Fermi energy) and the other half moves up. In addition, the level spacing for electron and hole levels becomes unequal.
The presence of a magnetic field acting on the ribbon does not break the translational symmetry along the direction parallel to the ribbon, that allows us to discuss the electronic structure in terms of the same bands calculated in the absence of the magnetic field. Results for the ribbon analyzed in Fig. \[ribbon\_bands\] for different magnetic fields are shown in Fig. \[boundary\_field\]. The “center of gravity” of the wavefunctions associated to the levels moves in the direction transverse to the ribbon as the momentum is increased. The results show the bulk Landau levels, and their changes as the wavefunctions approach the edges. The surface band is practically unchanged, except for small avoided crossings every time that it becomes degenerate with a bulk Landau level. The results show quite accurately the expected scaling $\epsilon_n \propto \pm \sqrt{n}$ for the eigenenergies derived from the Dirac equation, with small corrections due to lattice effects and a finite $t'$ (see Appendix \[ap2\]). The corresponding wavefunctions for different bands and momenta are shown in Fig. \[boundary\_field\_charge\]. The Landau levels move rigidly towards the edges, where one also find surface states.
We can compute the Hall conductivity from the number of chiral states induced by the field at the edges[@H82]. If we fix the chemical potential above the $n$-th level, there are $2 \times ( 2 n + 1 )$ edge modes crossing the Fermi level (including the spin degeneracy). Hence, the Hall conductivity is: $$\sigma_{xy} = \frac{e^2}{h} 2 \times ( 2 n + 1 ) = \frac{4 e^2}{h} (n + 1/2)
\, .
\label{iqhe}$$ This result should be compared with the usual IQHE in heterostructures, in which the factor of $1/2$ is absent. The presence of this $1/2$ factor is a direct consequence of the presence of the zero mode in the Dirac fermion problem. The existence of this anomalous IQHE was predicted long ago in the context of high energy physics [@jackiw84; @schakel91] and more recently in the context of graphene [@PGN05; @GS05], but was observed in graphene only recently by two independent groups [@Netal05; @Zetal05b]. An incomplete IQHE, with a finite longitudinal resistivity, was observed in HOPG graphite [@Ketal03].
Fractional quantum Hall effect
------------------------------
While the IQHE depends only of the cyclotron energy, $\omega_c$, and therefore is a robust effect, the fractional quantum Hall effect (FQHE) is a more delicate problem since it is a result of electron-electron interactions. The problem of electron-electron interactions in the presence of a large magnetic field in a honeycomb lattice is a complex problem that deserves a separate study. In this paper we make a few conjectures about the structure of the FQHE based on generic properties of Laughlin’s wave functions.
The electrons occupying the lowest Landau level are assumed to be in a many-body wavefunction written as (${\bf R}_i = (x_i,y_i)$ and $z=x+iy$) [@MS03] : $$\Psi=\exp(-i 2 m \sum_{i<j}\alpha_{i,j})
\Phi(z_1,\ldots,z_N)\,,$$ where $\alpha_{i,j}=\arctan [{\rm Im}(z_i
-z_j)/{\rm Re}(z_i-z_j)]$, $\Phi
(z_1,\ldots,z_N)$ is an anti-symmetric function of the interchange of two $z's$, and $m=0,1,2,\ldots$. The effect of the singular phase associated with the many-body wave function is to introduce an effective magnetic field $B^\ast$ given by: $$B^\ast = B - \bm\nabla \bm a(\bm r)/e=B - 2\pi (2 m) \rho(\bm r)/e\,,$$ where the gauge field $ \bm a(\bm r)$ is given by $$\bm a(\bm r_i) = 2 m \sum_{j\ne i}\bm\nabla(\bm r_i)\alpha_{i,j}\,,$$ and $\rho(\bm r)$ is the electronic density. The procedure outlined above is called flux-attachment and leads to appearance of composite fermions. These composite particles do not feel the external field $B$ but instead an effective field $B^{\ast}$. Therefore, the FQHE of electrons can be seen as an IQHE of these composite particles.
Given an electronic density $\delta $, we may define an effective filling factor $p^\ast$ for the composite particles as: $$p^\ast = \frac {2 \pi \delta}{e B^\ast}\,.
\label{pef}$$ In the lowest Landau level the electron filling factor is: $$p = \frac {2 \pi \delta} {e B}\,,
\label{p}$$ and combining Eqs. (\[pef\]) and (\[p\]) we obtain: $$p = \frac {p^\ast}{2 m p^\ast +1}\, ,$$ so, we can write: $$\begin{aligned}
B^{\ast} = B (1 - 2 m p) \, .
\label{bast}\end{aligned}$$
The crucial assumption in the case of graphene is that the effective $p^\ast$ associated with the integer quantum Hall effect of composite particles has the form given in (\[iqhe\]), that is (spin ignored): $$p^\ast =(2n+1)\,,\hspace{1cm} n=0,1,2,3,\ldots \,,
\label{ass}$$ (the effective field $B^\ast$ is such that the system has one or more filled composite particle Landau levels, and the chemical potential lies between two these) leading to a quantized Hall conductivity given by: $$\sigma_{xy}= \frac {2n+1}{2 m(2n+1)+1}\frac{2 e^2}{h} \, .
\label{fqheII}$$ For $n=0$, one obtains the so-called Laughlin sequence: $\sigma_{xy}= 1/(2 m + 1) (2 e^2/h)$, and for $m=0$ we recover (\[iqhe\]). This argument shows that Jain’s sequence is quite different from that of the 2D electrons gas [@J89].
As in the case of the IQHE, the FQHE can be thought in terms of chiral edge states, or chiral Luttinger liquids, that circulate at the edge of the sample [@wen]. One can see the IQHE and FQHE as direct consequences of the presence of these edge states. Because of their chiral nature, edge states do not localize in the presence of disorder and hence the quantization of the Hall conductivity is robust. In graphene, as we have discussed previously, zig-zag edges support surface states that are non-chiral Luttinger liquids. We have recently shown that electron-electron interactions between chiral Luttinger liquids and non-chiral surface states can lead to instabilities of the chiral edge modes leading to edge reconstruction [@NGP05] and hence to the destruction of the quantization of conductivity. We also have shown that this edge reconstruction depends strongly on the amount of disorder at the edge of the sample. While this effect is not strong in the IQHE (because the cyclotron energy is very large when compared with the other energy scales), it makes the experimental observation of the FQHE in graphene very difficult.
Concluding Remarks {#conclusions}
==================
To summarize, we have analyzed the influence of local and extended lattice defects in the electronic properties of single graphene layer. Our results show that: (1) Point defects, such as vacancies, lead to an enhancement of the density of states at low energies and to a finite density of states at the Dirac point (in contrast to the clean case where the density of states vanishes); (2) Vacancies have a strong effect in the Dirac fermion self-energy leading to a very short quasi-particle lifetime at low energies; (3) The interplay between local defects and electron-electron interaction lead to the existence of a minimum in the imaginary part of the electron self-energy (a result that can be measured in ARPES); (4) The low temperature d.c. conductivity is a universal number, independent on the disorder concentration and magnetic field; (5) The d.c. conductivity, as in the case of a semiconductor, increases with temperature and chemical potential (a result that can be observed by applying a bias voltage to the system); (6) The a.c. conductivity increases with frequency at low frequency and at very low impurity concentrations can be fitted by a Drude-like model; (7) The magnetic susceptibility of graphene increases with temperature (it is not Pauli-like, as in an ordinary metal) and is sensitive to the amount of disorder in the system (it increases with disorder); (8) Within the Stoner criteria for magnetic instabilities we find that graphene is very stable against magnetic ordering and that the phase diagram of the system is dominated by paramagnetism; (9) In the presence of a magnetic field and disorder, the electronic density of states shows oscillations due to the presence of Landau levels which are shifted from their positions because of disorder; (10) The magneto-conductivity presents oscillations in the presence of fields and that their dependence with chemical potential and frequency are rather non-trivial, showing transitions between different Landau levels; (11) Extended defects, such as edges, lead to the effect of self-doping where charge is transfered from/to the defects to the bulk in the absence of particle-hole symmetry; (12) The effect of extended defects on transport is very weak and that electron scattering is dominated by local defects such as vacancies; (13) The quantization of the Hall conductance in the IQHE is anomalous relative to the case of the 2D electron gas with an extra factor of $1/2$ due to the presence of a zero mode in the Dirac fermion dispersion; (14) We conjecture that the FQHE in graphene has a sequence of states which is very different from the sequence found in the 2D electron gas and we propose a formula for that sequence.
The results and experimental predictions made in this work are based on a careful analysis of the problem of Dirac fermions in the presence of disorder, electron-electron interactions and external fields. We use well established theoretical techniques and find results that agree quite well with a series of amazing new experiments in graphene [@Netal04; @Betal04; @Zetal05c; @Netal05; @Zetal05; @Netal05b; @Zetal05b]. The main lesson of our work is that graphene presents a completely new electrodynamics when compared to ordinary metals which are described quite well within Landau’s Fermi liquid theory. In this work, we focus on the effects of disorder and electron-electron interaction and have shown that Dirac fermion respond to these perturbations in a way which is quite different from ordinary electrons. In fact, graphene is a non-Fermi liquid material where there is no concept of an effective mass and, therefore, a system where Fermi liquid concepts are not directly applicable. A new phenomenology, beyond Fermi liquid theory, has to be developed for this system. Our work can be considered a first step in that direction.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank D. Basov, W. de Heer, A. Geim, G.-H. Gweon, P. Kim, A. Lanzara, Z.Q. Li, J. Nilsson, V. Pereira, J. L. Santos, S.-W. Tsai, and S. Zhou for many useful discussions. N.M.R.P and F. G. are thankful to the Quantum Condensed Matter Visitor’s Program at Boston University. A.H.C.N. was supported through NSF grant DMR-0343790. N. M. R. P. thanks Fundação para a Ciência e Tecnologia for a sabbatical grant partially supporting his sabbatical leave, the ESF Science Programme INSTANS 2005-2010, and FCT under the grant POCTI/FIS/58133/2004.
$\Theta(\omega,\epsilon)$, and $K_B(\epsilon)$ {#ap1}
==============================================
In the calculation of $\sigma(\omega,T)$ and $\sigma(0,T)$ we defined:
$$\begin{aligned}
\Theta(\omega,\epsilon)
&=&\sum_{s_1=\pm 1,s_2=\pm 1}
\left[
\frac {N+s_1M}B\left(
\arctan\left[\frac {D-s_1A}{B}\right]+\arctan\left[\frac {s_1A}{B}\right]
\right)+
\frac {P+s_2V}E\left(
\arctan\left[\frac {D-s_2C}{E}\right]+\arctan\left[\frac {s_2C}{E}\right]
\right)\right.
\nonumber\\
&+&
\left.
\frac M 2 \log \left[\frac {(D-s_1A)^2+B^2}{A^2+B^2}\right]
+
\frac V 2 \log \left[\frac {(D-s_1C)^2+E^2}{C^2+E^2}\right]
\right]\,,\end{aligned}$$
where $$\begin{aligned}
M&=&(C^2+E^2-A^2-B^2)/{\cal D},\hspace{0.5cm}V=-M\nonumber\,,\\
N&=&2(s_1A-s_2C)(A^2+B^2)/{\cal D}\nonumber\,,\\
P&=&-2(s_1A-s_2C)(A^2+B^2)/{\cal D}\nonumber\,,\\
{\cal D}&=&(A^2+B^2-C^2-E^2)^2\nonumber\, \\
&+&4(A^2+B^2)(C^2-s_1s_2AC)\nonumber\,,\\
&+&4(C^2+E^2)(A^2-s_1s_2AC)\,,\nonumber\\
A&=&\epsilon+\omega-{\rm Re}\Sigma(\epsilon+\omega)\nonumber\,,\\
B&=&{\rm Im} \Sigma(\epsilon+\omega)\nonumber\,,\\
C&=&\epsilon-{\rm Re}\Sigma(\epsilon)\nonumber\,,\\
E&=&{\rm Im} \Sigma(\epsilon)\,,\nonumber\\\end{aligned}$$ and $\cos\alpha=(C^2-E^2)/(C^2+E^2)$.
In the magneto-transport properties, $\sigma_{xx}(0,T)$ given by Eq. (\[sxxb\]), depends on the kernel $K_B(x)$, which is defined as:
$$\begin{aligned}
K_B(x)&=&\frac {v_F^2}{2\pi l_B^2}\sum_\alpha\left[
\frac {{\rm Im}\Sigma_2(x)}{[x -{\rm Re}\Sigma_2(x)]^2+[{\rm Im}\Sigma_2(x)]^2}
\frac {{\rm Im}\Sigma_1(x)}{[x - E(\alpha,0)-{\rm Re}\Sigma_1(x)]^2+[{\rm Im}\Sigma_1(x)]^2}
\right.\nonumber\\
&+&
\sum_{\lambda,n\ge 1}\left.
\frac {{\rm Im}\Sigma_1(x)}{[x -E(\alpha,n)-{\rm Re}\Sigma_1(x)]^2+[{\rm Im}\Sigma_1(x)]^2}
\frac {{\rm Im}\Sigma_1(x)}{[x -E(\lambda,n-1)-{\rm Re}\Sigma_1(x)]^2+[{\rm Im}\Sigma_1(x)]^2}
\right] \, .\end{aligned}$$
The Dirac equation in a magnetic field {#ap2}
======================================
The Hamiltonian (\[llH0\]) can be solved using a trial spinor of the form: $$\psi(\bm r)\left(
\begin{array}{c}
c_1\phi_1(y)\\
c_2\phi_2(y)
\end{array}
\right)\frac {e^{ik_xx}}{\sqrt{L}}\,,$$ with $L$ the size of the system in the $x$ direction. After straightforward manipulations, the eigenproblem reduces to: $${v_F \sqrt 2}{l_B}
\left(
\begin{array}{cc}
0 & a \\
a^\dag&0
\end{array}
\right)
\left(
\begin{array}{c}
c_1 \phi_1(y)\\
c_2 \phi_2(y)
\end{array}
\right)
E
\left(
\begin{array}{c}
c_1 \phi_1(y)\\
c_2 \phi_2(y)
\end{array}
\right)\,,
\label{h2}$$ where $$\begin{aligned}
a &=& \frac {1}{\sqrt 2 l_B}(y+l^2_B\partial_y)\,,
\label{a}
\\
a^\dag &=& \frac {1}{\sqrt 2 l_B}(y-l^2_B\partial_y)\,,
\label{ad}\end{aligned}$$ with the magnetic length defined as $l^2_B=1/(eB)$. For the case of $E\ne 0$, it is simple to see that the spinor $$\frac 1 {\sqrt{2}}\left(
\begin{array}{c}
\phi_n(y)\\
\alpha \phi_{n+1}(y)
\end{array}
\right)\,,$$ is an eigenfunction of (\[h2\]) with eigenvalue $E(\alpha,n)=\alpha v_F \sqrt{2}/l_B\sqrt{n+1}$, with $\alpha=\pm 1$, and $\phi_n$ ( $n=0,1,2,\ldots$) the $n$ eigenfunction of the usual 1D harmonic oscillator. In addition, there exists a zero energy mode whose eigenfunction is given by: $$\left(
\begin{array}{c}
0\\
\phi_0(y)
\end{array}
\right)\,,$$ that completes the solution of the original eigenproblem. As in the more conventional Landau level problem, the degeneracy of each level is $L^2B/\phi_0$, with $\phi_0=h/e$ the quantum of flux.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Prior knowledge on properties of a target model often come as discrete or combinatorial descriptions. This work provides a [*unified computational framework*]{} for defining norms that promote such structures. More specifically, we develop associated tools for optimization involving such norms given only the orthogonal projection oracle onto the non-convex set of desired models. As an example, we study a norm, which we term the doubly-sparse norm, for promoting vectors with few nonzero entries taking only a few distinct values. We further discuss how the K-means algorithm can serve as the underlying projection oracle in this case and how it can be efficiently represented as a quadratically constrained quadratic program. Our motivation for the study of this norm is regularized regression in the presence of rare features which poses a challenge to various methods within high-dimensional statistics, and in machine learning in general. The proposed estimation procedure is designed to perform automatic feature selection and aggregation for which we develop statistical bounds. The bounds are general and offer a statistical framework for norm-based regularization. The bounds rely on novel geometric quantities on which we attempt to elaborate as well.'
author:
- 'Amin Jalali[^1]'
- 'Adel Javanmard[^2]'
- 'Maryam Fazel[^3]'
bibliography:
- 'JJF19.bib'
date: April 2019
title: New Computational and Statistical Aspects of Regularized Regression with Application to Rare Feature Selection and Aggregation
---
Keywords: Convex geometry, Hausdorff distance, structured models, combinatorial representations, K-means, regularized linear regression, statistical error bounds, rare features.
Introduction
============
A large portion of estimation procedures in high-dimensional statistics and machine learning have been designed based on principles and methods in continuous optimization. In this pursuit, incorporating prior knowledge on the target model, often presented as discrete and combinatorial descriptions, has been of interest in the past decade. Aside from many individual cases that have been studied in the literature, a number of general frameworks have been proposed. For example, [@bach2013learning; @obozinski2016unified] define sparsity-related norms and their associated optimization tools from support-based monotone set functions. On the other hand, several unifications have been proposed for the purpose of providing estimation and recovery guarantees. A well-known example is the work of [@chandrasekaran2012convex] which connects the success of norm minimization in model recovery given random linear measurements to the notion of Gaussian width [@Gordon88]. However, many of the final results of these frameworks (excluding discrete approaches such as [@bach2013learning]) are quantities that are hard to compute; even evaluating the norm. Therefore, many a time computational aspects of these norms and their associated quantities are treated on a case by case basis. In fact, a [*unified*]{} framework for turning discrete descriptions into continuous tools for estimation, that 1) provides a [*computational*]{} suite of optimization tools, and 2) is amenable to [*statistical analysis*]{}, is largely underdeveloped.
Consider a measurement model $y = X\beta^\star + \epsilon$, where $X\in\mathbb{R}^{n\times p}$ is the [*design*]{} matrix and $\epsilon\in\mathbb{R}^n$ is the [*noise*]{} vector. Given combinatorial descriptions of the underlying model, say $\beta^\star\in{\mathcal{S}}\subset\mathbb{R}^p$, in addition to $X$ and $y$, much effort and attention has been dedicated to understanding [*constrained estimators*]{} for recovery. For example, only assuming access to the (non-convex) projection onto the set of desired models ${\mathcal{S}}$ enables devising a certain class of recovery algorithms constrained to ${\mathcal{S}}$; Iterative Hard Thresholding (IHT) algorithms, [@blumensath2008iterative Section 3] [@blumensath2011sampling] (projects onto the set of $k$-sparse vectors), [@jain2010guaranteed Section 2] (projects onto the set of rank-$r$ matrices), [@roulet2017iterative] (does 1-dimensional K-means which is projection onto the set of models with $K$ distinct values), belong to this class. However, a major subset of estimation procedures focus on norms, designed based on the non-convex structure sets, for estimation. Working with convex functions, such as norms, for promoting a structure is a prominent approach due to its flexibility and robustness. Namely, the proposed norms can be used along with different loss functions and constraints[^4]. In addition, the continuity property of these functions allows the optimization problems to take into account points that are [*near*]{} (but not necessarily inside) the structure set; a [*soft*]{} approach to specifying the model class. The seminal work of [@chandrasekaran2012convex] provides guarantees for norm minimization estimation, constrained with $X\beta=y$ or ${\|X\beta-y\|}_2\leq \delta$, using the notion of Gaussian width. Dantzig selector is another popular category of constrained estimators studied in the literature (e.g., [@chatterjee2014generalized]) but other variations also exist ( provides a list). In analyzing all of these constrained estimators, [*the tangent cone*]{}, at the target model with respect to the norm ball, is the determining factor for recoverability. Then, the notion of Gaussian width of such cone [@chandrasekaran2012convex; @Gordon88] allows for establishing high probability bounds for recovery from many random ensembles of design. In a way, the Gaussian width, or a related quantity known as the statistical dimension [@amelunxen2014living], are local quantities that can be understood as an operational method for [*gauging the model complexity with respect to the norm*]{} and determining the minimal acquisition requirements for recovery from random linear measurements.
However, regularized estimators pose further challenges for analysis. More specifically, consider $$\begin{aligned}
\label{eq:estimator}
{\widehat{\beta}}~\equiv~ {\mathop{{\operatorname{argmin}}}}_\beta ~~ \frac{1}{2n}\|y - X\beta\|_2^2 + \lambda {\|\beta\|}\end{aligned}$$ where $\lambda$ is the regularization parameter. From an optimization theory perspective, for a fixed design and noise, and a norm minimization problem constrained with ${\|X\beta-y\|}\leq \delta$ (see and ) are equivalent if a certain value of $\delta$, corresponding to $\lambda$, is being used; meaning that ${\widehat{\beta}}$ for these estimators will be equal. However, the mapping between theses problem parameters is in general complicated (e.g., see [@aravkin2016level]) which renders the aforementioned equivalence useless when studying error bounds that are expressed in terms of these problem parameters (e.g., see bounds in and their dependence on $\lambda$). Furthermore, in the study of [*expected*]{} error bounds for a family of noise vectors (or design matrices), such equivalence is in general irrelevant (e.g., fixing $\lambda$, each realization of noise will imply a different $\delta$ corresponding to the given value of $\lambda$). Nonetheless, a good understanding of regularized estimators with [*decomposable norms*]{} have been developed; see [@negahban2012unified; @candes2013simple; @foygel2014corrupted; @wainwright2014structured; @vaiter2015model] for slightly different definitions. These are norms with a special geometric structure and only a handful of examples are known (including the $\ell_1$ norm and the nuclear norm). In regularization with general norms, it is possible to provide a high-level analysis, inspired by the analysis for decomposable norms, and provide error bounds; e.g., see [@banerjee2014estimation] and follow up works. However, the proposed bounds are in a way [*conceptual*]{} and no general computational guidelines for evaluating these bounds exist. In this work, we introduce a geometric quantity for gauging model complexity with respect to a norm in [*regularized estimation.*]{} Such quantity, accompanied by a few computational guidelines and connections to the rich literature on convex geometry, then allows for principled approach towards evaluating the previous conceptual error bounds leading to our final statistical characterizations for that are sensitive to 1) norm-induced properties of design, and 2) non-local properties of the model with respect to the norm.
A motivation behind our pursuit of a computational and statistical framework for regularization is to handle the [*presence of many rare features*]{} in real datasets, which has been a challenging proving ground for various methods within high-dimensional statistics, and in machine learning in general; see for further motivation. In this work, we study an efficient estimator, namely a regularized least-squares problem, for [*automatic feature selection and aggregation*]{} and develop statistical bounds. The regularization, an atomic norm proposed by [@jalali2013convex], poses new challenges for computation (even norm evaluation) and statistical analysis (e.g., non-decomposability). We extend the computational framework provided in [@jalali2013convex] for this norm, in , and provide statistical error bounds in . We also establish advantages over Lasso (). Moreover, our estimation and prediction error bounds, rely on simple geometric notions to gauge condition numbers and model complexity with respect to the norm. These bounds are quite general and go beyond regularization for feature selection and aggregation.
Summary of Technical Contributions
----------------------------------
In this work, we consider regularized regression in the presence of rare features (presented in ) as our main case study. In our attempt to address this problem, we develop several general results for defining norms from given combinatorial descriptions and for statistical analysis of norm-regularized least-squares, as summarized in the following:
1. We adopt an approach to defining norms from given descriptions of desired models, and provide a unified machinery to derive useful quantities associated to a norm for optimization (e.g., the dual norm, the subdifferential of the norm and its dual, the proximal mapping, projection onto the dual norm ball, etc); see . Our approach relies on [*the non-convex orthogonal projection onto the set of desired models.*]{} In , we discuss how a discrete algorithm such as K-means clustering can be used to define a norm, namely the doubly-sparse norm, for promoting vectors with few nonzero entries taking only a few distinct values. Our results extend those of [@jalali2013convex] to any structure.
Complementing the existing statistical analysis approaches, for least-squares regularized with any norm, we take a variational approach, through quadratic functions, to understanding norms and provide alternative error bounds that can be easier to interpret, compute, and generalize:
2. We provide a prediction error bound in terms of [*norm-induced aggregation measures*]{} of the design matrix for when the noise satisfies the convex concentration property or is a subgaussian random vector. We do this by making a novel use of the Hanson-Wright inequality for when the dual to the norm has a concise variational representation; . The new bounds are deterministic with respect to the design matrix, are interpretable, and allow for taking detailed information on the design matrix into account, going beyond results on well-known random ensembles which might be unrealistic in real applications.
Most of the existing estimation bounds for norm-regularized regression can be unified under the notion of [*decomposability*]{}; see [@negahban2012unified; @candes2013simple; @foygel2014corrupted; @vaiter2015model] for slightly different definitions. Our results, in contrast, do not rely on such assumption:
3. In gauging model complexity with respect to the regularizer, we introduce a novel geometric measure, termed as [*the relative diameter*]{}, which then allows for simplified derivations for restricted eigenvalue constants and prediction error bounds. More specifically, we go beyond decomposability [*and*]{} we provide techniques to compute such complexity measure (). We provide calculations for a variety of norms (e.g., ordered weighted $\ell_1$ norms) used in the high-dimensional statistics literature; and . In , we provide further insight into the notion of relative diameter and compare with existing quantities in the literature. Through illustrative examples, we showcase the sensitivity of the relative diameter to the properties of the model and the norm.
Finally, we use the aforementioned developments to design and analyze a regularized least-squares estimator for regression in the presence of rare features:
4. We propose to use doubly-sparse regularization for regression in the presence of rare features (). We discuss how such choice allows for [*automatic feature aggregation*]{}. We use the insights and tools we develop in the paper for regression with rare features and establish the advantage of regularizing the least-squares regression with the doubly-sparse norm, given in , over Lasso, in .
Last but not least, we provide various characterizations related to a number of norms common in the high-dimensional statistics literature such as the ordered weighted $\ell_1$ norms (commonly used for simultaneous feature selection and aggregation; e.g., see [@figueiredo2016ordered].) which could be of independent interest. See and . Proof of technical lemmas are deferred to Appendices.
#### Notations.
Denote by ${\|A\|}$ and ${\|A\|}_F$ the operator norm and the Frobenius norm of a matrix $A$. We also represent its smallest and largest singular values by $\sigma_{\min}(A)$ and $\sigma_{\max}(A)$. For a positive integer $p$, we denote by $[p]$ the set $\{1,2,\ldots, p\}$. For a compact set $\mathcal{M}\subset \mathbb{R}^p$, the polar set is denoted by $\mathcal{M}^\star = \{x:~ \langle x,y\rangle \leq 1, ~ \forall y\in \mathcal{M} \}$. For a positive integer $p$, we denote by $\mathbb{S}^{p-1}$ the $(p-1)$-dimensional unit sphere, $\mathbb{S}^{p-1} \equiv \{ x\in {\mathbb{R}}^p:\, \|x\|_2 = 1\}$. Given a set $\mathcal{M} \subset {\mathbb{R}}^p$, we denote by ${{\operatorname{conv}}}(\mathcal{M})$ the convex hull of $\mathcal{M}$, i.e., ${{\operatorname{conv}}}(\mathcal{M})\equiv \{\sum_{i=1}^k w_i x_i:\, \sum_{i=1}^k w_i = 1,\, w_i\ge0,\, x_i\in \mathcal{M},\,k\in\mathbb{N}\}$. Moreover, define ${{\operatorname{cone}}}(\mathcal{M}) \equiv \{\alpha a:~ \alpha \in\mathbb{R}_+,~a\in \mathcal{M} \}$. In addition, given a compact set $\mathcal{M}\subset {\mathbb{R}}^p$, a point $a\in \mathcal{M}$ is an [*extreme point*]{} of $\mathcal{M}$ if $a=(b+c)/2$ for $b,c\in \mathcal{M}$ implies $a=b=c$. Denote by ${\boldsymbol{1}}_p$ and ${\boldsymbol{0}}_p$ the vectors of all ones and all zeros in $\mathbb{R}^p$, respectively. We may drop the subscripts when clear from the context. For two vectors $\beta,\theta\in\mathbb{R}^p$, their Hadamard (entry-wise) product is denoted by $\beta \circ \theta$ where $(\beta \circ \theta)_i = \beta_i\theta_i$ for $i\in[p]$. The unit simplex in $\mathbb{R}^p$ is denoted by ${\mathbf{\Delta}}_p\equiv \{u\in\mathbb{R}^p:~ u\geq {\boldsymbol{0}}_p,~ {\boldsymbol{1}}^{{\sf T}}u = 1\}$. The full unit simplex is denoted by $\widetilde{\mathbf{\Delta}}_p\equiv \{u\in\mathbb{R}^p:~ u\geq {\boldsymbol{0}}_p,~ {\boldsymbol{1}}^{{\sf T}}u \leq 1\}$. In all of this work, we assume the model ($\beta$ or $\beta^\star$) is nonzero.
Motivation: Regularized Regression for Rare Features {#sec:intro}
====================================================
Data sparsity has been a challenging proving ground for various methods. Sparse sensing matrices in the established field of compressive sensing [@berinde2008combining], the inherent sparsity of document-term matrices in text data analysis [@wang2010latent], the ubiquitous sparsity of biological data, from gut microbiota to gene sequencing data, and the sparse interaction matrices in recommendation systems, have been challenging the established methods that otherwise have provable guarantees when certain well-conditioning properties (e.g., the restricted isometry property in compressive sensing) hold. See [@yan2018rare] for further motivations.
A common approach when lots of rare features are present is to remove the very rare features in a pre-processing step (e.g., Treelets by [@lee2008]). This is not efficient as it may discard large amount of information and better approaches are needed to make use of the rare features to boost estimation and predictive power. On the other hand, there have been efforts for establishing success of $\ell_1$ minimization in case of [*certain sparse sensing matrices*]{} (e.g., see [@berinde2008combining; @berinde2008sparse; @gilbert2010sparse]) where gaps between their statistical requirements and information-theoretical limits exist. Combinatorial approaches for subset selection, through integer programming, have also been restricted to certain sparse design matrices to achieve polynomial-time recovery [@del2018subset]. Instead, a variety of ad-hoc methods, based on solving different optimization programs, have been proposed for going beyond sparse models and making use of rare features [@bondell2008simultaneous; @zeng2014ordered] and there has been a recent interest in this problem within the high-dimensional statistics community. While some of these estimators come with a statistical theory, they may require extensive prior knowledge [@li2018graph; @yan2018rare] which could be expensive or difficult to gather in real applications.
Our Approach: Doubly-Sparse Regularization {#sec:reg-est}
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We approach this problem through feature aggregation, but unlike previous works, we do so in an automatic fashion at the same time as estimation. More specifically, in learning a linear model from noisy measurements, we use the model proposed by [@jalali2013convex]: we are interested in vectors that are not only sparse (to be able to ignore unnecessary features) but also have only a few distinct values, which induces a grouping among features and allows for automatic aggregation. We refer to this prior as [*double-sparsity*]{} and elaborate on it in the sequel. Considering the [*structure norm*]{} (see [@jalali2013convex], , or ) corresponding to this prior, we study a regularized least-squares optimization program in . Since the existing machinery of atomic norms [@chandrasekaran2012convex] does not come with tools for optimization, we develop new tools in that can be used to efficiently compute and analyze the proposed estimator. Superior performance over the use of $\ell_1$ regularization (Lasso) in the presence of rare features is showcased in .
### The Prior and the Regularization
A $k$-sparse vector $\beta\in\mathbb{R}^n$ can be expressed as a linear combination of $k$ indicator functions for singletons in $\{1,\ldots,n\}$; i.e., $\beta = \sum_{t=1}^k \beta_t {\boldsymbol{1}}(\{i_t\})$ where ${{\operatorname{Supp}}}(\beta) = \{i_1,\ldots,i_k\}$. In contrast, we are interested in vectors that can be expressed as a linear combination of a few indicator functions using a coarse partitioning of $\{1,\ldots,n\}$; i.e., $\beta = \sum_{t=1}^d \beta_t {\boldsymbol{1}}(S_t)$ where $S_1,\ldots,S_d$ partition $\{1,\ldots,n\}$ and $d$ is small. Here, $\beta_t$’s can be zero; i.e., we are allowing $0$ to be one of the $d$ distinct values. To combine the two priors, for two fixed values $1\leq d \leq k \leq p$, one can consider vectors $\beta = \sum_{t=1}^d \beta_t {\boldsymbol{1}}(S_t)$ where $S_1,\ldots,S_d\subset\{1,\ldots,n\}$ are non-empty and disjoint and ${|S_1\cup\ldots\cup S_d|}=k$. Those are the vectors with at most $k$ nonzero values where the top $k$ entries have at most $d$ distinct values. Finally, to make the prior more suitable for our regression setting, we allow for arbitrary sign patterns within each part.
Given a vector $\beta$ denote by $\bar{\beta}$ the sorted version of ${|\beta|}$ in descending order; i.e., $\bar{\beta}_1\geq \bar{\beta}_2\geq \cdots \geq \bar{\beta}_p \geq 0$. Then, we consider $$\label{eq:def-Skd}
{\mathcal{S}}_{k,d} \equiv \bigl\{\beta:~ {{\operatorname{card}}}(\beta) \leq k \,,~ {|\{\bar{\beta}_1,\ldots,\bar{\beta}_k\}|} \leq d \bigr\};$$ [*the vectors with at most $k$ nonzero values whose top $k$ absolute values take at most $d$ distinct values*]{}. illustrates an example. See [@jalali2013convex] for further detail and existing works around this idea. With the aid of the machinery presented in , we can define a norm, referred to as the *doubly-sparse norm*, that can help in recovery of models from ${\mathcal{S}}_{k,d}$ in a sense characterized by our statistical error bounds. For two fixed values $1\leq d \leq k \leq p$, we refer to this norm as the $(k\square d)$-norm, denoted by ${\|\cdot\|}_{k\square d}$.
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### A Statistical Analysis
Consider a measurement model $y = X\beta^\star + \epsilon$, where $X\in\mathbb{R}^{n\times p}$ is the [*design*]{} matrix and $\epsilon\in\mathbb{R}^n$ is the [*noise*]{} vector. We then consider the following estimator, $$\begin{aligned}
\label{eq:estimator-kd}
{\widehat{\beta}}\equiv {\mathop{{\operatorname{argmin}}}}_\beta ~~ \frac{1}{2n}\|y - X\beta\|_2^2 + \lambda {\|\beta\|}_{k\square d}\,,\end{aligned}$$ where $\lambda$ is the regularization parameter. In , we analyze and provide [*prediction error*]{} bounds, namely bounds for ${\|X(\beta^\star - \hat\beta)\|}_2$.
More generally, we consider regularization with any norm. In providing a prediction error bound, we show how [*norm-specific aggregation measures*]{} can be used to bound the regularization parameter (). For estimation error, we provide a general tight analysis through the introduction of [*relative diameter*]{} (, , and ). We make partial progress in computing the relative diameter, namely we do so for ${{\|\cdot\|}}_{k\square 1}$, and its dual, but we also provide computations for some important classes of polyhedral norms to showcase possible strategies; for ordered weighted $\ell_1$ norms studied in [@figueiredo2016ordered], and, for weighted $\ell_1$ and $\ell_\infty$ norms. See and for details of computations.
### Optimization Procedures {#sec:opt-algs-kd}
In computing ${\widehat{\beta}}$ from , or more generally , one can use different optimization algorithms. While ${{\|\cdot\|}}_{k\square d}$ might seem complicated to even be evaluated, we show in that there exists an efficient procedure for computing its proximal mapping (for a definition, see , and for a characterization in the case of ${{\|\cdot\|}}_{k\square d}$, see ). Therefore, here, we only discuss two proximal-based optimization strategies to illustrate the computational efficiency of the estimator in . The optimization program in is unconstrained and its objective is convex and the sum of a smooth and a non-smooth term. Therefore, as we have access to the proximal mapping associated to the non-smooth part, proximal gradient algorithm seems like a natural choice for optimization. For $t=1,\ldots, T$, we compute $$\begin{aligned}
\label{eq:iterative}
g^t = \frac{1}{n}X^{{\sf T}}X \beta^{t} - X^{{\sf T}}y~,~~
\beta^{t+1} = {{\operatorname{prox}}}(\beta^t - \eta_t\, g^t; {{\|\cdot\|}}) \end{aligned}$$ where $\eta_t$ is the step size. The algorithm, with an appropriate choice of step size, reaches an $\delta$-accurate solution (in prediction loss) in $O(1/\delta)$ steps. See [@parikh2014proximal] for further details on proximal algorithms.
As we will see later, the proximal mapping is the solution to a convex optimization program and may not admit a closed form representation unlike simple norms such as the $\ell_1$ norm (whose proximal mapping is soft-thresholding). Therefore, it might be inevitable to work with approximate solutions. In such case, [*inexact proximal methods*]{} [@schmidt2011convergence] may be employed which allow for a controlled inexactness in computation of the proximal mapping (more specifically, inexactness in the objective) but provide similar convergence rates as in the exact case.
Alternating Direction Method of Multipliers (ADMM) may also be used to solve , similar to the discussions in Section 6.4 of [@boyd2011distributed] for the $\ell_1$ norm. The non-trivial ingredient of such strategy is the proximal mapping for the regularizer, which is available here.
While this paper is concerned with the regularized estimation, it is worth mentioning that the ability to compute the proximal mapping also enables solving the generalized Dantzig selector (defined in ) as discussed in [@chatterjee2014generalized].
Projection-based Norms {#sec:structure-norms}
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Given a compact set ${\mathcal{A}}\subset \mathbb{R}^{p}$ of desired model parameters, which is symmetric, spans $\mathbb{R}^p$, and none of its members belongs to the convex hull of the others, the [*atomic norm*]{} framework [@bonsall1991general; @chandrasekaran2012convex] defines a norm through $$\begin{aligned}
\label{eq:atomic_repr}
{\|{\beta}\|}_{\mathcal{A}}= \inf \bigl\{ \sum_{\omega\in{\mathcal{A}}} c_\omega :\; {\beta}= \sum_{\omega\in{\mathcal{A}}} c_\omega\, \omega \;,\; c_\omega \geq 0 \, \bigr\}.\end{aligned}$$ This optimization problem is hard to solve in general and one might end up with linear programs that are difficult to solve or might have to resort to discretization (e.g., [@shah2012linear]) or to case-dependent reformulations (e.g., [@tang2013compressed]).
Alternatively, one might consider the dual norm as the building block for further computations: the support function to the norm ball or to the atomic set, namely $$\begin{aligned}
\label{eq:dual-norm-lin}
{\|{\theta}\|}_{\mathcal{A}}^\star
\equiv
\sup_{ {\|{\beta}\|}_{\mathcal{A}}\leq 1} ~\langle {\beta},{\theta}\rangle
= \sup_{a\in {\mathcal{A}}} ~\langle a,{\theta}\rangle .\end{aligned}$$ Assuming ${\mathcal{A}}\subseteq \mathbb{S}^{p-1}$, using the above variational characterization, and ${{\operatorname{dist}}}^2({\theta},{\mathcal{A}})
\equiv \inf_{a\in {\mathcal{A}}} \|a-{\theta}\|_2^2$, we get $$\begin{aligned}
\label{eq:atomic-dual-dist}
{{\operatorname{dist}}}^2({\theta},{\mathcal{A}})
= 1 + {\|{\theta}\|}_2^2 - 2 {\|{\theta}\|}_{\mathcal{A}}^\star.\end{aligned}$$ While the dual norm is 1-homogeneous, the other terms above are not, which limits the uses of this expression. As evident from the result of , homogenizing the atomic set ${\mathcal{A}}$ into ${\mathcal{S}}\equiv {{\operatorname{cone}}}({\mathcal{A}}) = \{\lambda a:~ \lambda\in\mathbb{R},\, a\in{\mathcal{A}}\}$ provides a better object to work with. Next, we elaborate on this direction and provide a framework for defining norms that comes with a computational suite for computing various quantities associated to these norms.
Some of the material in and have been previously mentioned in [@jalali2013convex] without proof and restricted to the so-called [*$d$-valued models*]{}. We generalize this framework and use it for addressing the problem of interest in .
Definition and Characterizations {#sec:Snorms-def}
--------------------------------
Given a closed set ${\mathcal{S}}\subseteq \mathbb{R}^p$ that is scale-invariant (closed with respect to scaling by any $a\in\mathbb{R}$ which make it symmetric with respect to the origin as well) and spans $\mathbb{R}^p$, consider an associated convex set ${\mathcal{B}}_{\mathcal{S}}$ defined as $$\begin{aligned}
\label{eq:Snorm-gauge}
{\mathcal{B}}_{\mathcal{S}}= {{\operatorname{conv}}}\{ \beta: \; \beta\in{\mathcal{S}}\,,\; {\|\beta\|}_2 = 1 \} \,.\end{aligned}$$ Since ${\mathcal{B}}_{\mathcal{S}}$ is a symmetric compact convex body with the origin in its interior, the corresponding symmetric gauge function is defined as $$\begin{aligned}
\label{eq:gauge}
{\|\beta\|}_{\mathcal{S}}\equiv \inf \{ \gamma>0 :~ \beta \in \gamma {\mathcal{B}}_{\mathcal{S}}\},\end{aligned}$$ is a norm with ${\mathcal{B}}_{\mathcal{S}}$ as the unit norm ball. One can view ${\|\cdot\|}_{\mathcal{S}}$ as an atomic norm with atoms given by the extreme points of the unit norm ball as ${\mathcal{A}}_{\mathcal{S}}= {{\operatorname{ext}}}({\mathcal{B}}_{\mathcal{S}})$. Using atoms, we can express ${\|\cdot\|}_{\mathcal{S}}$ as in with ${\mathcal{A}}= {\mathcal{A}}_{\mathcal{S}}$. As we will see later, ${\beta}\in{\mathcal{A}}_{\mathcal{S}}$ if and only if ${\|{\beta}\|}_{\mathcal{S}}= {\|{\beta}\|}_2 = {\|{\beta}\|}_{\mathcal{S}}^\star$.
As an alternative to , provides a way to compute the dual to this norm. Denote by $${\Pi}(\theta; {\mathcal{S}}) = {\Pi}_{\mathcal{S}}(\theta)= \textstyle{\mathop{{\operatorname{argmin}}}}_\beta\{ {\|\theta - \beta\|}_2:~ \beta \in {\mathcal{S}}\}$$ the (non-convex) orthogonal projection onto ${\mathcal{S}}$. Note that the projection mapping onto a non-convex set is set-valued in general. We refer to for further details and proofs of the following statements.
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\[lem:dual-len-proj\] Given any closed scale-invariant set ${\mathcal{S}}\subseteq \mathbb{R}^p$ which spans $\mathbb{R}^p$, the dual norm to ${\|\cdot\|}_{\mathcal{S}}$ is given by $$\begin{aligned}
\label{eq:dual_norm_gen}
{\|\theta\|}_{\mathcal{S}}^\star
\equiv \sup_{{\|\beta\|}_{\mathcal{S}}\leq 1} \langle\beta,\theta \rangle
= {\| {\Pi}(\theta; {\mathcal{S}}) \|}_2
\end{aligned}$$ where ${\| {\Pi}(\theta; {\mathcal{S}}) \|}_2$ refers to the $\ell_2$ norm of any member of the set and is well-defined. Moreover, $$\label{eq:dual-pair}
\langle \theta , {\Pi}(\theta; {\mathcal{S}}) \rangle = {\|{\Pi}(\theta;{\mathcal{S}})\|}_2^2= {\|{\Pi}(\theta; {\mathcal{S}})\|}_{\mathcal{S}}\, {\|\theta\|}_{\mathcal{S}}^\star $$ which illustrates the pair of achieving vectors in the definition of dual norm and yields $$\begin{aligned}
\label{eq:Snorm-dual-dist}
({\|\theta\|}_{\mathcal{S}}^\star )^2
= {\|\theta\|}_2^2 - {{\operatorname{dist}}}^2(\theta,{\mathcal{S}}). \end{aligned}$$
illustrates . is also known as the [*alignment property*]{} in the literature. In contrast with , the expression in is 2-homogeneous in $\theta$. With the above characterization for the dual norm we get $$\begin{aligned}
\label{eq:Snorm-from-dual}
{\|\beta\|}_{\mathcal{S}}= \sup \, \{\, \langle \beta , \theta \rangle :~ {\| {\Pi}(\theta; {\mathcal{S}}) \|}_2 \leq 1 \, \}. \end{aligned}$$
Since the optimal $\beta$ in the definition of the dual norm in is known to be ${\Pi}_{\mathcal{S}}(\theta)$, we can easily characterize the subdifferential as in the following.
\[subdiff\_dualnorm\_gen\] The subdifferential of the dual norm at ${\beta}\neq 0$ is given by $$\partial{\|{\beta}\|}_{\mathcal{S}}^\star = \frac{1}{{\|{\Pi}_{\mathcal{S}}({\beta})\|}_2} {{\operatorname{conv}}}\left({\Pi}_{\mathcal{S}}({\beta})\right)$$ which in turn implies $\partial (\tfrac{1}{2}{{\|{\beta}\|}_{\mathcal{S}}^\star}^2) = {{\operatorname{conv}}}\left({\Pi}_{\mathcal{S}}({\beta}) \right)$.
Proof of is given in .
While an oracle that computes the projection enables us to carry out many computations for quantities related to the structure norm (e.g., the value of ${\|{\beta}\|}_{\mathcal{S}}\,$, the proximal operators for the norms and squared norms, as well as projection onto ${\mathcal{B}}_{\mathcal{S}}$, as discussed in the rest of this section), some properties of the structure set can highly simplify these computations. In the following, we consider the [*invariance*]{} properties of the structure (under permutations and sign changes) and in , we discuss [*monotonicity*]{} properties of the structure. is not entirely new and has been discussed in the literature in one form or another.
\[lem:inv-proj\] Consider a closed set $A\subseteq\mathbb{R}^p$, convex or non-convex, and the orthogonal projection mapping ${\Pi}(\cdot\,;A)$. Then,
- Provided that $A$ is closed under a change of signs of entries (i.e., $\beta\in A$ implies $s \circ \beta\in A$ for any sign vector $s\in\{\pm1\}^p$) then $\theta \circ \beta \geq 0$ for any $\theta\in{\Pi}(\beta; A)$.
- Provided that $A$ is closed under permutation of entries (i.e., $\beta\in A$ implies $\pi({\beta})\in A$ for any permutation operator $\pi(\cdot)$) then $\beta$ and any $\theta\in{\Pi}(\beta; A)$ have the same ordering: $\beta_i > \beta_j$ implies $\theta_i \geq \theta_j$ for all $i,j\in[p]$.
Proof of is given in .
Examples
--------
In the following, we provide a few examples of structure norms, both existing and new;
- projection of $\beta$ onto ${\mathcal{S}}= \{\lambda e_i :\lambda \in\mathbb{R},\, i\in[p]\}$, where $e_i$ is the $i$-th standard basis vector, is the set of all $\|\beta\|_\infty e_{i^*}$ with $i^* \in \arg\max\{i\in[p]:\, \beta_i = \|\beta\|_\infty\}$. The length of such projections is indeed the $\ell_\infty$ norm which is dual to $\ell_1$ norm.
- When ${\mathcal{S}}$ is the set of all rank-1 matrices, projection onto ${\mathcal{S}}$ is the principal component and its length is the largest singular value of the matrix, the operator norm.
- For structure norms defined based on ${\mathcal{S}}_{k,d}$, given in , see . provides a schematic of this family of norms, for different values of $k$ and $d$, as well as their dual norms.
- consider $w\in\mathbb{R}^p$ satisfying $w_1\geq w_2 \geq \cdots \geq w_p >0$ and ${\mathcal{S}}=\{\gamma Qw: \gamma\in\mathbb{R},~Q\in\mathcal{P}_\pm \}\subset \mathbb{R}^{p}$ where $\mathcal{P}_\pm $ is the set of signed permutation matrices. As established in , we have $$\begin{aligned}
{\|\beta\|}_{\mathcal{S}}\equiv {\|w\|}_2 \cdot {\|\beta\|}_{w}^\star\end{aligned}$$ where ${\|\beta\|}_{w}\equiv \langle w,\bar\beta\rangle$ is the ordered weighted $\ell_1$ norm associated to $w$. Projection onto ${\mathcal{S}}$ requires sorting the absolute values of the input vector.
- As another example, consider ${\mathcal{S}}=\{\gamma Q: \gamma\in\mathbb{R},~Q\in\mathcal{P}_\pm \}\subset \mathbb{R}^{p\times p}$ where $\mathcal{P}_\pm $ is the set of signed permutation matrices. Given a matrix $A$, its projection onto ${\mathcal{S}}$ can be derived by projecting ${|A|}$ onto $\{\gamma P:~ \gamma\in\mathbb{R},~P\in\mathcal{P}\}$ where $\mathcal{P}$ is the set of permutation matrices. However, we already know efficient algorithms for finding the nearest permutation matrix (without a scaling factor $\gamma$); algorithms for solving the assignment problem such as the Hungarian method. establishes that these two solutions are related.
\[lem:proj\_S\_SB2\] We have ${{\operatorname{cone}}}({\Pi}_{\mathcal{S}}({\beta})) = {{\operatorname{cone}}}({\Pi}_{{\mathcal{S}}\cap\mathbb{S}^{p-1}}({\beta}))$. In other words, one can project onto ${\mathcal{S}}\cap\mathbb{S}^{p-1}$ and later find the correct scaling of the projected point to get ${\Pi}_{\mathcal{S}}({\beta})$.
Proof of is given in . The above is also helpful in making use of ${\Pi}(\cdot\,;{\mathcal{S}})$ in place of ${\Pi}(\cdot\,; {\mathcal{S}}\cap \mathbb{S}^{p-1})$ in greedy algorithms such as the one studied in [@tewari2011greedy].
Quantities based on a Representation {#sec:repr}
------------------------------------
Note that while the dual norm (or its subdifferential, characterized in ) can be directly computed from the projection, computation of quantities such as the norm value in , or objects we discuss next, namely the projection onto the dual norm ball, the proximal mapping for the norm, or the subdifferential for the norm, could greatly benefit from a [*representation*]{} of the projection onto the structure which can then be plugged into the aforementioned optimization programs. For the structure ${\mathcal{S}}_{k,d}$ considered in , we have access to an efficient representation for the dual norm in terms of a quadratically constrained quadratic program (QCQP).
The subdifferential of a norm is useful in devising subgradient-based algorithms and can be computed via $$\begin{aligned}
\label{eq:subd-proj}
\partial {\|\beta\|} = {\mathop{{\operatorname{Argmax}}}}_\theta \bigl\{ \langle \beta,\theta \rangle : {\|\theta\|}^\star \leq 1 \bigr\}.\end{aligned}$$ Alternatively, consider the proximal mapping associated to ${{\|\cdot\|}}$ which is defined as the unique solution to the following optimization program, $$\begin{aligned}
\label{def:prox}
{{\operatorname{prox}}}({\beta;{{\|\cdot\|}}}) \equiv {\mathop{{\operatorname{argmin}}}}_\theta ~ \frac{1}{2}{\|\beta-\theta\|}_2^2 + {\|\theta\|} .\end{aligned}$$ The proximal mapping enables a wide range of optimization strategies that are commonly more efficient that subgradient-based methods; e.g., [@parikh2014proximal]. For example, in , we briefly mentioned proximal gradient descent as well as ADMM for solving the regularized least-squares problem or assuming an efficient routine for evaluating the proximal mapping.
The proximal mapping admits a closed form solutions for simple cases such as the $\ell_1$ norm or the nuclear norm; soft-thresholding. However, more generally it can be computed through projection onto the dual norm ball, namely as $$\begin{aligned}
\label{eq:prox-proj}
{{\operatorname{prox}}}({\beta;{{\|\cdot\|}}})
= \beta - {\mathop{{\operatorname{argmin}}}}_\theta \bigl\{ {\|\beta-\theta\|}_2^2
:~ {\|\theta\|}^\star \leq 1
\bigr\}.\end{aligned}$$ For computing or , one may express the dual norm ball as ${\mathcal{B}}^\star = \{\theta:~ \langle \beta, \theta \rangle \leq 1 ~\forall \beta\in{\mathcal{B}}\}$ where ${\mathcal{B}}= \{\beta:~ {\|\beta\|} \leq 1\}$. Therefore, the proximal mapping may be computed through $$\begin{aligned}
{{\operatorname{prox}}}({\beta;{{\|\cdot\|}}})
= \beta - {\mathop{{\operatorname{argmin}}}}_\theta \bigl\{ {\|\beta-\theta\|}_2^2
:~ \langle \tilde\beta,\theta \rangle \leq 1 ~~ \forall \tilde\beta\in{\mathcal{B}}\bigr\} .\end{aligned}$$ Since ${\mathcal{B}}$ may have an infinite number of elements, or exponentially-many, it is not straightforward to solve such a quadratic optimization problem especially in each iteration of another algorithm such as proximal gradient descent or ADMM described in . Therefore, a more efficient representation of the dual norm ball could enable an efficient computation of the proximal mapping, subgradients, etc.
#### Black-box versus Representable.
In the case of structure norms, namely ${{\|\cdot\|}}_{\mathcal{S}}$, we have (by assumption) an efficient routine to evaluate the projection onto ${\mathcal{S}}$ which allows us to check membership (feasibility) in $\{\theta:~{\|\theta\|}_{\mathcal{S}}^\star\leq 1\} = \{\theta:~ {\|{\Pi}(\theta;{\mathcal{S}})\|}_2\leq 1\}$. Optimization (for or ) given only a feasibility oracle is still not easy. However, in cases such as ${\mathcal{S}}_{k,d}$, it is possible to derive [*an efficient representation*]{} for the projection onto ${\mathcal{S}}$ and the dual norm, which can then replace the dual norm ball membership constraints and yield the objects of interest (subgradients or the proximal mapping) as solutions to manageable convex optimization programs. More concretely, assume we can establish $$\begin{aligned}
\label{eq:dualnorm-eff-repr}
{\|\theta\|}^\star = \min_u \bigl\{ f(\theta, u):~ (\theta, u)\in \mathcal{T} \bigr\}\end{aligned}$$ where $\mathcal{T}$ is a finite-dimensional convex set and $f$ is a convex function. Then, the proximal mapping can be expressed as $${{\operatorname{prox}}}({\beta;{{\|\cdot\|}}}) = \beta - {\mathop{{\operatorname{argmin}}}}_\theta \bigl\{ {\|\beta-\theta\|}_2^2
:~ f(\theta,u)\leq 1,~ (\theta,u)\in\mathcal{T}
\bigr\}.$$ Deriving a representation as in is the main focus of for ${{\|\cdot\|}}_{k\square d}^\star$; given in .
Doubly-sparse Norms (kd-norm) {#sec:the-norm}
-----------------------------
Here, we discuss a structure motivated by the statistical estimation problem at hand, namely regression in the presence of rare features. As we show, a fast discrete algorithm, namely the 1-dimensional K-means algorithm, can be used to define a norm for feature aggregation as well as for computing its optimization-related quantities.
For two fixed values $1\leq d \leq k \leq p$, the structure set ${\mathcal{S}}={\mathcal{S}}_{k,d}$ in is scale-invariant and spans $\mathbb{R}^p$. Therefore, we consider the structure norm associated to ${\mathcal{S}}_{k,d}$ to which we refer as the $(k\square d)$-norm and we denote by ${\|\cdot\|}_{k\square d}$, or ${\|\cdot\|}_\sq$ when clear from the context. Specifically, $$\begin{aligned}
\label{eq:kd-def}
{\|\beta\|}_{k\square d} \equiv \inf \{ \gamma>0 :~ \beta \in \gamma {\mathcal{B}}_{{\mathcal{S}}_{k,d}}\}\,,\end{aligned}$$ with ${\mathcal{B}}_{{\mathcal{S}}_{k,d}} = {{\operatorname{conv}}}\{ \beta: \; \beta\in{\mathcal{S}}_{k,d} \,,\; {\|\beta\|}_2 = 1 \}$. According to , we have ${\|\theta\|}_{k\square d}^\star(\theta) = {\| {\Pi}(\theta; {\mathcal{S}}_{k,d}) \|}_2$, and in turn, ${\|\beta\|}_{k\square d} = \sup\{\langle \theta, \beta\rangle:~ {\|\theta\|}_{k\square d}^\star \leq 1 \}$. Next, we address the computational aspects.
### Examples; for Different Values of $k$ and $d$ {#sec:kdnorms-examples}
It is clear from that ${\mathcal{S}}_{k,d_1}\subset {\mathcal{S}}_{k,d_2}$ for $d_1\leq d_2$: since $k$ is fixed, if ${|\{\bar{\beta}_1,\ldots,\bar{\beta}_k\}|} \leq d_1$ then ${|\{\bar{\beta}_1,\ldots,\bar{\beta}_k\}|} \leq d_2$. Therefore, ${\|\cdot\|}_{k\square 1} \geq \cdots \geq {\|\cdot\|}_{k\square k} $ for any $k\in\{1,\ldots,p\}$.
Note that a similar monotonicity does not hold with respect to $k$. Consider $1\leq d\leq k_1\leq k_2 \leq p$. If ${{\operatorname{card}}}(\beta)\leq k_1$ then ${{\operatorname{card}}}(\beta)\leq k_2$. However, if ${|\{\bar{\beta}_1,\ldots,\bar{\beta}_{k_1}\}|} \leq d$, the addition of elements $\bar{\beta}_{k_1+1}=\ldots=\bar{\beta}_{k_2}=0$ to the set may increase the number of distinct values by $1$. Therefore, ${\mathcal{S}}_{k_1,d}\subseteq {\mathcal{S}}_{k_2,d+1}$ for any $1\leq d\leq k_1\leq k_2 \leq p$.
However, with ${{\operatorname{val}}}(\beta)\equiv {|\{ {|\beta_i|}\neq 0:~ i\in[p]\}|}$ and $\widetilde{{\mathcal{S}}}_{k,d} \equiv \{\beta:~{{\operatorname{card}}}(\beta)\leq k,~ {{\operatorname{val}}}(\beta)\leq d\}$, the addition of the extra zero elements do not change ${{\operatorname{val}}}$, and we get $\widetilde{{\mathcal{S}}}_{k_1,d} \subseteq \widetilde{{\mathcal{S}}}_{k_2,d}$ for any $1\leq d\leq k_1\leq k_2 \leq p$. The new definition differs from in not counting zero as a separate value among the top $k$ entries. For example, the dual norm corresponding to $\widetilde{{\mathcal{S}}}_{p,1}$ is $({\|\beta\|}^\star)^2 = \max_{r\in[p]} \frac{1}{r}(\sum_{i=1}^r \bar\beta_i)^2$.
Nonetheless, we have $ {\|\cdot\|}_1 = {\|\cdot\|}_{1\square 1}\geq \cdots \geq {\|\cdot\|}_{p\square p}={\|\cdot\|}_2$. It is worth noting that for any $k\in \{1,\ldots,p\}$, ${\|\cdot\|}_{k\square k}$ coincides with the $k$-support norm [@argyriou2012sparse]. Furthermore, () establishes that $$\begin{aligned}
\label{lem:norm-k-1}
{\|\beta\|}_{k\square 1} = \max\{\frac{1}{\sqrt{k}}{\|\beta\|}_1 , \sqrt{k}{\|\beta\|}_\infty \}.\end{aligned}$$ As a corollary, we get ${\|\cdot\|}_{p\square 1} =\sqrt{p}{\|\cdot\|}_\infty$. See for a full picture for ${{\|\cdot\|}}_{k\square d}$ and ${{\|\cdot\|}}_{k\square d}^\star$.
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(10.5,8.5) (10.8,8.4)[[${{\|\cdot\|}}_2$]{}]{}
-.2in -.2in
(0,0) (2.5,1)[(1,0)[11]{}]{} (14,.8)[$k$]{} (3,.5)[(0,1)[9]{}]{} (2.3,9)[$d$]{}
(4.5,2.5) (3.6,3)[[${{\|\cdot\|}}_\infty$]{}]{} (6,2.5) (7,2.5) (7.5,2.5) (5.1,1.6)[[ ${{\|\cdot\|}}_{1,\text{top-}k} \cdot \frac{1}{\sqrt{k}}$ ]{}]{} (8,2.5) (9,2.5) (10.5,2.5) (10.8,2.4)[[${{\|\cdot\|}}_1\cdot\tfrac{1}{\sqrt{p}}$]{}]{}
(6,4) (7,4) (7.5,4) (8,4) (9,4) (8.3,4.4) (10.5,4)
(7,5) (7.5,5.5) (5.4,6.3)[[${{\|\cdot\|}}_{2,\text{top-}k}$]{}]{} (8,6)
(9,5) (9,5.5) (9,6)
(10.5,5) (10.5,5.5) (10.8,5.4)[[${{\|\cdot\|}}_{p \sq d}^\star $]{}]{} (10.5,6)
(9,7) (10.5,7)
(10.5,8.5) (10.8,8.4)[[${{\|\cdot\|}}_2$]{}]{}
### The Projection and its Combinatorial Representation
Before discussing the projection onto ${\mathcal{S}}_{k,d}$, in , we state a lemma to establish a reduction principle that allows simplifying such projection. This reduction makes use of the [*invariance*]{} and [*monotonicity*]{} properties for such projection. We established the former in . For the latter, can be thought of as an implication of the Occam razor principle. In simple terms, if the characteristic property that defines a structure ignores zero values, the projected vector will have a support included in the support of original vector; there is no need to have new values in those places when computing the projection. Similarly, if the characteristic property treats similar values as one value, there is no need to map them to distinct values in the projection. These suggest that we can always consider problems in a reduced space; only considering non-zero entries and distinct values in our structure of interest, namely ${\mathcal{S}}_{k,d}$.
\[lem:monot\] Consider a closed scale-invariant set ${\mathcal{S}}\subseteq\mathbb{R}^p$ that spans $\mathbb{R}^p$. Moreover, consider any orthogonal projection $\theta \in {\Pi}(\beta; {\mathcal{S}})$. We have:
- If $u\in{\mathcal{S}}$ implies $u-u_ie_i\in {\mathcal{S}}$ for all $i\in[p]$, then ${{\operatorname{Supp}}}(\theta) \subseteq {{\operatorname{Supp}}}(\beta)$ for any $\theta \in {\Pi}(\beta; {\mathcal{S}})$; i.e., $\beta_i=0$ implies $\theta_i = 0$ for any $i\in[p]$ and any $\theta\in {\Pi}(\beta;{\mathcal{S}})$.
More generally, consider an orthogonal projection matrix $P=P^{{\sf T}}=P^2$. If (i) $u\in{\mathcal{S}}$ implies $Pu\in{\mathcal{S}}$, and, (ii) $\beta=P\beta$, then, $\theta\in{\Pi}(\beta;{\mathcal{S}})$ implies $P\theta=\theta$.
- If $u\in{\mathcal{S}}$ implies $u - (u_i-u_j)e_i\in{\mathcal{S}}$ for all $i,j\in[p]$, then $\beta_i = \beta_j$ implies $\theta - (\theta_i-\theta_j)e_i \in{\Pi}(\beta;{\mathcal{S}})$ for any $\theta\in{\Pi}(\beta;{\mathcal{S}})$.
More generally, consider a pair $(A,B)$ of oblique projection matrices, i.e., $A^2=A$ and $B^2=B$, satisfying $A^{{\sf T}}A + B^{{\sf T}}B =2I$. Assume $A\beta=B\beta=\beta$, and that $u\in{\mathcal{S}}$ implies $Au,Bu\in{\mathcal{S}}$. Then, for any $\theta\in{\Pi}(\beta;{\mathcal{S}})$, we have $A\theta, B\theta\in {\Pi}(\beta;{\mathcal{S}})$.
Proofs for , , , , and , are given in .
\[lem:proj-cardk\] If ${\mathcal{S}}$ is sign and permutation invariant and ${\mathcal{S}}\subseteq \{\beta:~ {{\operatorname{card}}}(\beta)\leq k\}$, then for all $\theta\in{\Pi}(\beta; {\mathcal{S}})$ we have $\theta_i=0$ whenever ${|\beta_i|} < \bar\beta_k$.
\[lem:proj-bar-Skd\] For a given $\beta$, consider ${{\operatorname{sign}}}(\beta)$ (where ${{\operatorname{sign}}}(0)$ is arbitrary from $\{+1,-1\}$) and a permutation $\pi$ for which $\pi({|\beta|})$ is sorted in descending order. Then $${\Pi}(\beta;{\mathcal{S}}_{k,d}) = \bigl\{
\pi^{-1}(\theta) \circ {{\operatorname{sign}}}(\beta):~ \theta\in{\Pi}(\bar\beta; {\mathcal{S}}_{k,d})
\bigr\}~,~~
{\Pi}(\bar\beta;{\mathcal{S}}_{k,d}) = \bigl\{
\pi({|\theta|}):~ \theta\in{\Pi}(\beta; {\mathcal{S}}_{k,d})
\bigr\}$$
\[lem:proj-Skd\] The following procedure returns all of the projections of $\beta\in\mathbb{R}^p$ onto ${\mathcal{S}}_{k,d}$ defined in :
- project $\beta$ onto ${\mathcal{S}}_{k,k}$ (zero out all entries except the $k$ of the entries with largest absolute values) and consider the shortened output $\beta^{(k)}\in\mathbb{R}^k$,
- project $\beta^{(k)}$ onto ${\mathcal{S}}_{k,d} \subset \mathbb{R}^k$ (perform the 1-dimensional K-means algorithm on entries of ${|\beta^{(k)}|}$ and stack the corresponding centers with signs according to $\beta^{(k)}$),
- put the new entries back in a $p$-dimensional vector, by padding with zeros.
Repeat this procedure when there are multiple choices in steps (i) or (ii).
We will use to compute the dual norm and further derive a combinatorial representation for it. Note that while computing the projection itself can be done through K-means, we are interested in a representation for this projection which can can then be used in computing other quantities; as discussed in .
\[lem:proj-Skd-combinat-rep\] For a given vector $\theta\in {\mathbb{R}}^p$, denote by $\bar{\theta}$ the sorted version of $|\theta|$ in descending order, i.e., $\bar{\theta}_1\ge \cdots\ge \bar{\theta}_p \ge 0$. Then, $$\begin{aligned}
{\| {\Pi}(\theta; {\mathcal{S}}_{k,d}) \|}_2^2
= \max \bigl\{ \sum_{i=1}^d \frac{1}{{|{\mathcal{I}}_i|}} ({\boldsymbol{1}}^{{\sf T}}\bar{\theta}_{{\mathcal{I}}_i})^2 :~ ({\mathcal{I}}_1,\cdots,{\mathcal{I}}_d)\in{\bar{\texttt{P}}}(k,d) \bigr\} \nonumber\end{aligned}$$ where ${\bar{\texttt{P}}}(k,d)$ is the set of all partitions of $\{1,\ldots,k\}$ into $d$ groups of consecutive elements. Then, $$\begin{aligned}
\left[
\frac{{\boldsymbol{1}}^{{\sf T}}\bar{\theta}_{{\mathcal{I}}_i}}{{|{\mathcal{I}}_i|}}{\boldsymbol{1}}_{{\mathcal{I}}_i},
\cdots
\frac{{\boldsymbol{1}}^{{\sf T}}\bar{\theta}_{{\mathcal{I}}_d}}{{|{\mathcal{I}}_d|}}{\boldsymbol{1}}_{{\mathcal{I}}_d},
0, \cdots, 0
\right]^{{\sf T}}\in {\Pi}(\bar{\theta}; {\mathcal{S}}_{k,d}).\end{aligned}$$
Using , the statement of the can be alternatively represented as $$\begin{aligned}
\label{dual-norm-characterization}
({\|\beta\|}_{k\square d}^\star)^2
=({\|\bar{\beta}\|}_{k\square d}^\star)^2
= \sup_{A\in {\overline{\texttt{BD}}}(k,d)} \, \bar{\beta}^{{\sf T}}A \bar{\beta}\end{aligned}$$ where $\bar{\beta}$ is nonnegative and non-increasing, and ${\overline{\texttt{BD}}}(k,d)$ is the set of block diagonal matrices with $d$ blocks exactly covering the first $k$ rows and columns and zero elsewhere, where on each block of size $q$, all of the entries are equal to $\frac{1}{q}$. Note that if the input is not a sorted nonnegative vector, then we need to consider ${\texttt{BD}}(k,d)\equiv\{PAP^{{\sf T}}:P\in \mathcal{P}_\pm,A\in{\overline{\texttt{BD}}}(k,d)\}$, where $\mathcal{P}_\pm $ is the set of signed permutation matrices. This brings us to $$\begin{aligned}
\label{dual-norm-var}
({\|\beta\|}_{k\square d}^\star)^2
= \sup_{A\in {\texttt{BD}}(k,d)} \, \beta^{{\sf T}}A \beta.\end{aligned}$$ The aforementioned representations, in , , and , all depend on an efficient characterization of combinatorial sets such as ${\bar{\texttt{P}}}(k,d)$ or ${\texttt{BD}}(k,d)$. below shows that ${\texttt{BD}}(k,d)$ is of exponential size, which renders direct optimization inefficient.
\[lem:BD-size\] $|{\texttt{BD}}(k,d)| < (\frac{2epd}{k})^k$.
is proved in .
Next, we review a dynamic programming approach to reformulate the above in terms of a quadratic program.
### A Dynamic Program and a QCQP Representation {#sec:qcqp}
Consider a non-negative sorted vector $\bar{\beta}$ with $\bar{\beta}_1\geq \cdots \geq \bar{\beta}_p\geq 0$. A dynamic program can be used to perform 1-dimensional K-means clustering required in the second step of projection onto ${\mathcal{S}}_{k,d}$ (detailed in ) as well as in . For example, see [@wang2011ckmeans] for how a 1-dimensional K-means clustering problem can be cast as a dynamic program. Furthermore, this dynamic program can be represented as a quadratically-constrained quadratic program (QCQP) [@jalali2013convex] as discussed next. More specifically, the following two lemmas describe how projection onto ${\mathcal{S}}_{k,d}$ and the dual norm unit ball ${\mathcal{B}}^*$ can be computed as solving a QCQP. See for an illustration related to ${\bar{\texttt{P}}}(k,d)$ and the dynamic program.
\[lem:projS-qcqp\] We have $$\begin{aligned}
{\|{\Pi}(\bar{\beta}; {\mathcal{S}}_{k,d})\|}_2^2 =
\min_{\{\nu_{m,e}\}} \Bigl\{ \nu_{k,d}:~
\frac{1}{s-m+1} ( {\boldsymbol{1}}^{{\sf T}}\bar{\beta}_{[m,s]})^2 \leq \nu_{s,e} - \nu_{m-1,e-1} ~~ \forall (e,m,s)\in {\texttt{T}}(k,d)
\Bigr\},\end{aligned}$$ where ${\texttt{T}}(k,d)\; \equiv\; \{(e,m,s) : 1\leq e \leq d ,~ e \leq m \leq s \leq k-d+e\}$, and $u_{[m,s]} = [u_m, \cdots, u_s]$.
Proof for is given in .
\[lem:proj-dual-ball-qcqp\] For ${\mathcal{B}}^\star = \{u:~{\|u\|}^\star_{k\square d} \leq 1\}$, we have $$\begin{gathered}
{\Pi}(\bar{\theta}; {\mathcal{B}}^\star)
= {\mathop{{\operatorname{argmin}}}}_{u} \min_{\{\nu_{m,e}\}}
\Bigl\{ {\|\bar{\theta}-u\|}_2^2 :~
\nu_{k,d}\leq 1, ~ u_1\geq \cdots \geq u_p \geq 0,\\
\frac{1}{s-m+1} ( {\boldsymbol{1}}^{{\sf T}}u_{[m,s]})^2 \leq \nu_{s,e} - \nu_{m-1,e-1} ~~ \forall (e,m,s)\in {\texttt{T}}(k,d)
\Bigr\}\end{gathered}$$ which is a QCQP.
Proof for is given in .
The above provides us with the proximal mapping through ${{\operatorname{prox}}}_{{\|\cdot\|}_\sq} (\bar{\theta})
= \bar{\theta} - {\Pi}(\bar{\theta}; {\mathcal{B}}^\star)$. A QCQP such as the one above can be solved via interior point methods among many others.
(200,0)
(0,0)(1,0)[14]{}[(0,1)[5]{}]{} (0,0)(0,1)[6]{}[(1,0)[13]{}]{} (0,0)(.85,.85)[6]{}[(1,1)[.7]{}]{} (13,5)(-.85,-.85)[6]{}[(-1,-1)[.7]{}]{} (13.2,5)[$\nu_{13,5}$]{}
(0,0)[(4,1)[4]{}]{} (4,1) (4,1)[(4,1)[4]{}]{} (8,2) (8,2)[(2,1)[2]{}]{} (10,3)
(0,0)[(1,1)[1]{}]{} (1,1) (1,1)[(2,1)[2]{}]{} (3,2) (3,2)[(1,1)[1]{}]{} (4,3) (4,3)[(2,1)[2]{}]{} (6,4) (5.35,4.3)[$\nu_{6,4}$]{}
(0,0)[(4,1)[4]{}]{} (4,1) (0,0)[(3,1)[3]{}]{} (3,1) (3,1)[(1,1)[1]{}]{} (4,2) (4,2)[(3,1)[3]{}]{} (7,3) (7,3)[(2,1)[2]{}]{} (9,4) (9,4)[(4,1)[4]{}]{} (13,5)
(1,1)(1,0)[9]{} (2,2)(1,0)[9]{} (3,3)(1,0)[9]{} (4,4)(1,0)[9]{} (5,5)(1,0)[9]{}
The representation of the dual norm in is through a maximization. Therefore, in replacing a dual norm constraint with this representation, we will have as many as ${|{\mathcal{A}}|}$ constraints which leads to a semi-infinite optimization program in many cases of interest. The representation in is also a maximization problem ($\ell_2$ squared minus distance squared) with possibly many constraints. However, in the case of ${\mathcal{S}}_{k,d}$, the use of allows for reformulation in terms of a dynamic program which reduces the number of constraints from exponentially-many, namely ${|{\texttt{BD}}(k,d)|}$, to ${|{\texttt{T}}(k,d)|} \leq k^2d$.
Prediction Error Bound for Regularized Least-Squares {#sec:concise-var}
====================================================
Consider a measurement model $y = X\beta^\star + \epsilon$, where $X\in\mathbb{R}^{n\times p}$ is the [*design*]{} matrix and $\epsilon\in\mathbb{R}^n$ is a noise vector. For any given norm ${{\|\cdot\|}}$, and not only those studied in , we then consider the regularized estimator in with $\lambda$ as the regularization parameter. Rather standard analysis of yields [*prediction error bounds*]{}, namely bounds for ${\|X(\beta^\star - \hat\beta)\|}_2$, as well as [*estimation error bounds*]{}, namely bounds on $\|{\widehat{\beta}}-\beta^\star\|$ and $\|{\widehat{\beta}}-\beta^\star\|_2$. In this section, we review a standard prediction error bound () and then present a novel analysis for establishing bounds on the regularization parameter which is needed in such prediction error bound. Estimation error bounds will be studied in building upon the results presented here.
\[lem:oracle\] If $\lambda \ge \|\frac{2}{n} X^{{\sf T}}\epsilon\|^\star$, then ${\widehat{\beta}}$ obtained from satisfies $$\begin{aligned}
\label{prediction-error}
\frac{1}{n}{\|X(\beta^\star - {\widehat{\beta}})\|}_2^2 \leq 3\lambda {\|\beta^\star\|} \,.
\end{aligned}$$
follows from a standard oracle inequality and is proved in .
The prediction error bound in , and the estimation error bounds in , are conditioned on $\lambda \geq {\|\frac{2}{n}X^{{\sf T}}\epsilon\|}^\star$. In this section we make a novel use of the Hanson-Wright inequality to compute this bound for a broad family of noise vectors $\epsilon \in\mathbb{R}^n$ while our bounds are deterministic with respect to the design matrix. Our proof assumes a concise variational representation for the dual norm (as in ) and provides a bound in terms of novel [*aggregate measures*]{} of the design matrix [*induced by the norm*]{} (given in ). In the following, we elaborate on the variational formulation. In , we examine this property for structure norms (defined in ). In , we provide examples of norms admitting a concise representation, and finally, in , we state the bounds.
#### A Concise Variational Formulation.
Any squared vector norm can be expressed in a variational form ([@bach2012optimization] (Prop. 1.8 and Prop. 5.1) and [@jalali2017variational]): consider any norm ${\|\cdot\|}$ and its dual ${\|\cdot\|}^\star$. Then, $$\begin{aligned}
\label{eq:squared-dual-norm-var}
({\|\beta\|}^\star)^2
= \sup_{{\|\theta\|}\leq 1} \langle \theta, \beta \rangle ^2
= \sup_{{M}\in{\mathcal{M}}} \beta^{{\sf T}}{M}\beta\end{aligned}$$ where ${\mathcal{M}}= \{\theta\theta^{{\sf T}}:~ {\|\theta\|}\leq 1\}$. It is easy to see that the set ${\mathcal{M}}$ that is used in the variational representation above is not unique. For example, ${{\operatorname{conv}}}({\mathcal{M}})$ or ${\mathcal{M}}= \{\theta\theta^{{\sf T}}:~ \theta \in{{\operatorname{ext}}}({\mathcal{B}}_{{\|\cdot\|}})\}$ also work. For an atomic norm (defined in ), it is clear from the above that ${|{\mathcal{M}}|} \leq {|{\mathcal{A}}|}$. However, in cases such as ${\|\cdot\|}_{k\square d}$, one can find a set ${\mathcal{M}}$ which is much smaller than ${\mathcal{A}}$. For example, in , , as well as for ${{\|\cdot\|}}_{k\square d}$, the atomic set is infinite while we can find a small finite-size ${\mathcal{M}}$. For a norm such as the ordered weighted $\ell_1$ norm [@zeng2014ordered], which has a finite number of atoms, it seems that a smaller ${\mathcal{M}}$ cannot be found; see .
For a norm that admits a representation as in with a reasonably-sized ${\mathcal{M}}$, we can provide a prediction error bound in terms of ${|{\mathcal{M}}|}$ as well as certain [*aggregation*]{} quantities defined based on the elements in ${\mathcal{M}}$. For example, in the case of ${{\|\cdot\|}}_{k\square d}$, with a corresponding variational representation given in , we provide the prediction error bound in . As another example, in , we provide these calculations for the case of $k$-support norm as well as the $(k\square 1)$-norm.
Example: Structure Norms with Finite Unions of Subspaces {#sec:UoS}
--------------------------------------------------------
Consider a closed scale-invariant set ${\mathcal{S}}$ that spans $\mathbb{R}^p$ and the corresponding structure norm ${{\|\cdot\|}}_{\mathcal{S}}$ and unit norm ball ${\mathcal{B}}_{\mathcal{S}}= \{\beta:~{\|\beta\|}_{\mathcal{S}}\leq 1\}$. In this section, we connect a representation for ${{\|\cdot\|}}_{\mathcal{S}}^\star$ as in to a representation of ${\mathcal{S}}$ as a union of subspaces.
A closed scale-invariant set ${\mathcal{S}}$ can always be represented as a union of subspaces. However, imagine this is possible for a given set with finitely many subspaces; namely $m\geq 1 $ subspaces. For $i\in[m]$, denote by $U_i\in\mathbb{R}^{p\times d_i}$ an orthonormal basis for the $i$-th subspace. Then, $$\begin{aligned}
{\|\theta\|}_{\mathcal{S}}^\star
&= \sup\{ \langle \beta, \theta \rangle:~ {\|\beta\|}_{\mathcal{S}}\leq 1 \} \\
&= \sup\{ \langle \beta, \theta \rangle:~ \beta\in {{\operatorname{ext}}}({\mathcal{B}}_{\mathcal{S}}) \} \\
&= \sup\{ \langle \beta, \theta \rangle:~ \beta = U_iw,~ w\in\mathbb{S}^{d_i-1},~i\in[m] \} \\
&= \max_{i\in[m]}~{\|U_i^{{\sf T}}\theta\|}_2.\end{aligned}$$ Then, it is easy to see that we get a representation as in with $$\begin{aligned}
{\mathcal{M}}= \bigl\{ U_iU_i^{{\sf T}}:~ i \in[m] \bigr\}\end{aligned}$$ where each element of ${\mathcal{M}}$, namely $U_iU_i^{{\sf T}}$, is an orthogonal projector of rank $d_i$. summarizes these observations and its proof is given in .
\[lem:dualvar-UoS\] Consider a finite set of positive semidefinite matrices ${\mathcal{M}}= \{M_1, \ldots, M_m\}\subset \mathbb{R}^{p\times p}$ and $f(\beta) \equiv \sup_{{M}\in{\mathcal{M}}} \beta^{{\sf T}}{M}\beta$. Then, $\sqrt{f}$ is a semi-norm.
Suppose ${{\operatorname{conv}}}({\mathcal{M}})\cap \mathbb{S}_{++}^p\neq \emptyset$. Then, (i) $\sqrt{f}$ is a norm. (ii) if each $M_i$ is an orthogonal projector then $\sqrt{f}\equiv {{\|\cdot\|}}_{\mathcal{S}}^\star$ for ${\mathcal{S}}= \bigcup_{i\in[m]} {{\operatorname{range}}}(M_i)$.
Examples of Norms with a Concise Variational Representation {#sec:concise-examples}
-----------------------------------------------------------
In the following, we review some examples with a concise variational representation.
\[ex:group-l1-Mset\] Consider the group $\ell_1$ norm with $K$ non-overlapping groups (sum of $\ell_2$ norms over each group). Then, in the representation of the dual norm, we can use ${\mathcal{M}}= \{{M}_1, \ldots, {M}_K\}$ where ${M}_i$ is the identity matrix over rows and columns corresponding to the $i$-th group and zero elsewhere. We get ${|{\mathcal{M}}|} = K$, the number of groups.
More generally, consider the overlapping group Lasso norm [@jacob2009group] defined as $${\|\beta\|} \equiv \inf \bigl\{ \sum_{i=1}^K {\|v^{(i)}\|}_2 :~ \beta = \sum_{i=1}^K v^{(i)},~ v^{(i)}\in\mathbb{R}^p,~{{\operatorname{Supp}}}(v^{(i)})\subseteq {\mathcal{G}}_i, ~ \text{for }i\in[K] \bigr\}$$ where ${\mathcal{G}}=({\mathcal{G}}_1,\ldots, {\mathcal{G}}_K)$ is a given set of $K$ subsets of $[p]$ that may overlap; if they do not overlap and they partition $[p]$, ${{\|\cdot\|}}$ reduces to the group $\ell_1$ norm mentioned above. As characterized in Lemma 2 of [@jacob2009group], the dual norm is given by $${\|\theta\|}^\star = \max_{i\in [K]}~ {\|\theta_{{\mathcal{G}}_i}\|}_2$$ where $\theta_{{\mathcal{G}}_i}$ is the restriction of $\theta\in\mathbb{R}^p$ to the entries in ${\mathcal{G}}_i\subseteq[p]$. The above representation of ${{\|\cdot\|}}^\star$ can be used to derive a representation as in where ${\mathcal{M}}= \{M_1,\ldots, M_K\}$, and, for each $i\in[K]$, $M_i$ is the identity matrix over rows and columns corresponding to ${\mathcal{G}}_i$ and zero elsewhere.
The bound given in quickly deteriorates as $d$ gets close to $k$ or $1$. and are presented to provide improved bounds for ${{\|\cdot\|}}_{k\square k}$ and ${{\|\cdot\|}}_{k\square 1}$, respectively.
\[ex:ksupp-Mset\] Consider the $k$-support norm, denoted by ${\|\cdot\|}_{{k-{\operatorname{sp}}}}$ and defined as the symmetric gauge function corresponding to ${\mathcal{A}}= \{x:~ {\|x\|}_0 \leq k,~ {\|x\|}_2 =1\}$ [@argyriou2012sparse]. It is easy to see that the $k$-support norm coincides with the doubly-sparse norm for $k=d$. It has been shown that $({\|\theta\|}_{{k-{\operatorname{sp}}}}^\star)^2 = \sum_{i=1}^k \bar\theta_i^2$ [@argyriou2012sparse]. A representation as in through outer products of atoms of the $k$-support norm ball, namely ${\mathcal{M}}= \{\theta\theta^{{\sf T}}:~ \theta\in{{\operatorname{ext}}}({\mathcal{B}}_{{k-{\operatorname{sp}}}})\}$, leaves us with a set ${\mathcal{M}}$ with infinite number of elements. On the other hand, it is easy to verify that $$\begin{aligned}
{\mathcal{M}}= \bigl\{ {{\operatorname{diag}}}(s):~ s\in \{0,1\}^p, {\|s\|}_0 = k \bigr\}\end{aligned}$$ provides a valid expression for $({\|\theta\|}_{{k-{\operatorname{sp}}}}^\star)^2$ as in . Observe that ${|{\mathcal{M}}|} = \binom{p}{k} \leq (ep/k)^k$.
\[ex:Mset-k1\] It is shown in that
- ${{\operatorname{ext}}}({\mathcal{B}}_{k\square 1}) = {\mathcal{S}}_{k,1}\cap \mathbb{S}^{p-1} = \{Q\theta:~ Q\in \mathcal{P}_\pm,~ \theta = \frac{1}{\sqrt{k}}[{\boldsymbol{1}}_k^{{\sf T}}~,~ {\boldsymbol{0}}_{p-k}^{{\sf T}}]^{{\sf T}}\}$,
- ${{\operatorname{ext}}}({\mathcal{B}}_{k\square 1}^\star) = \{Q\theta:~\theta\in A,~Q\in\mathcal{P}_\pm\}$ where $A = \{\sqrt{k}e_1, \frac{1}{\sqrt{k}}{\boldsymbol{1}}_p\}$.
Therefore, it is easy to see that a concise representation exists,
- in the case of regularization with ${{\|\cdot\|}}_{k\square 1}$, with ${|{\mathcal{M}}|}\leq \binom{p}{k}\leq (ep/k)^k$,
- in the case of regularization with ${{\|\cdot\|}}_{k\square 1}^\star$, with ${|{\mathcal{M}}|}\leq p+1$,
for representing their dual norms.
From we know that ${{\|\cdot\|}}_{k\square 1}^\star$ is an ordered weighted $\ell_1$ norm with $w = \frac{1}{\sqrt{k}}[{\boldsymbol{1}}_k^{{\sf T}}~,~ {\boldsymbol{0}}_{p-k}^{{\sf T}}]^{{\sf T}}$. While establishes a concise variational formulation in this case, an arbitrary ordered weighted $\ell_1$ norm may not be concisely representable, as discussed next.
\[ex:Mset-dual-owl\] Here, we provide a quadratic variational representation for ${{\|\cdot\|}}_{w}$ inspired by Example 1.2 in [@chen2015structured]. Recall the atomic set for ${{\|\cdot\|}}_{w}$ from Theorem 1 of [@zeng2014ordered] and the variational representation from with $${\mathcal{M}}= \{\theta\theta^{{\sf T}}:~ \theta\in {{\operatorname{ext}}}({\mathcal{B}}_{{{\|\cdot\|}}_{w}})\}
= \bigcup_{i\in[p]} \bigcup_{S:{|S|}=i} \Bigl\{\frac{1}{(\sum_{j=1}^i w_j)^2} v_Sv_S^{{\sf T}}: v\in\{\pm1\}^p \Bigr\}.$$ It is easy to see that ${|{\mathcal{M}}|} \leq \sum_{i=1}^p \binom{p}{i}2^{i-1} = (3^p-1)/2$ which is not a good bound for problems in which $p$ is big.
Consider two arbitrary norms ${{\|\cdot\|}}_{(1)}$ and ${{\|\cdot\|}}_{(2)}$ with representations for their squared dual norms as in through ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$, respectively. Then, for the infimal convolution of the two norms, defined as $$\begin{aligned}
{\|\beta\|} \equiv \inf \bigl\{ {\|u\|}_{(1)} + {\|v\|}_{(2)}:~ \beta = u+v \bigr\}\,,\end{aligned}$$ we know (e.g., see Fact 2.21 in [@artacho2014applications]) that ${{\|\cdot\|}}^\star \equiv \max\{ {{\|\cdot\|}}_{(1)}^\star , {{\|\cdot\|}}_{(2)}^\star \} $. Therefore, we get a representation for ${{\|\cdot\|}}^\star$ as in with ${\mathcal{M}}= {\mathcal{M}}_1\cup {\mathcal{M}}_2$. See [@jalali2010dirty; @agarwal2012noisy] for applications of the infimal convolution in regularization.
The above is not an exhaustive list of norms with a concise variational representation for their dual. For example, consider $\Omega_2^\star(\cdot)$ ($p=q=2$) defined in Equation (2) of [@obozinski2016unified]. Depending on the submodular function $F$ used in this definition, one might be able to get smaller representations.
Bounds on the Regularization Parameter {#sec:bounds-reg-param}
--------------------------------------
\[def:cvx-conc-prop\] Let ${x}$ be a random vector in ${\mathbb{R}}^n$. We will say that ${x}$ has the convex concentration property with constant $K$ if for every $1$-Lipschitz convex function $h:{\mathbb{R}}^n\rightarrow{\mathbb{R}}$, we have $\mathbb{E}[h({x})]<\infty$ and for every $t>0$, $$\begin{aligned}
\mathbb{P}\big\{{|h({x})-\mathbb{E}[h({x})]|}\ge t\big\}\le 2 e^{-\frac{t^2}{2K^2}}.
\end{aligned}$$
\[lem0\] Let $u$ be a mean-zero random vector in ${\mathbb{R}}^n$. There exists a constant $c>2$, such that if $u$ has the convex concentration property with constant $K$ then for any matrix ${B}\in{\mathbb{R}}^{n\times n}$ and every $t>0$, $$\begin{aligned}
\label{eq:hanson}
\mathbb{P}\big\{{|u^{{\sf T}}Bu-\mathbb{E}[u^{{\sf T}}{B}u]|}\ge t\big\} \le 2\exp\left(-\frac{1}{c}\min\left(\frac{t^2}{2K^4\|B\|_F^2},\frac{t}{K^2\|B\|}\right)\right).
\end{aligned}$$
\[lem:noise-dual-norm-Mset\] Suppose that $\epsilon\in {\mathbb{R}}^n$ is a zero-mean random vector with covariance matrix $\Sigma \equiv\mathbb{E}[\epsilon \epsilon^{{\sf T}}] \in {\mathbb{R}}^{n\times n}$, such that $\Sigma^{-1/2} \epsilon$ satisfies the convex concentration property () with parameter at most $\eta$. Moreover, assume holds for ${{\|\cdot\|}}$ and a finite set ${\mathcal{M}}\subset\mathbb{R}^{p\times p}$. Then, for any value of $0<p_0<\frac{1}{2}$, the following holds true with probability at least $1 - 2 p_0$, $$\begin{aligned}
{\|\frac{1}{n} X^{{\sf T}}\epsilon\|}^\star \le {\Lambda}\end{aligned}$$ where $$\begin{aligned}
\begin{split}\label{eq:def-all-phi}
{\Lambda}&\equiv \frac{1}{\sqrt{n}}\left( {\Lambda}_0 + 2\eta^2 \cdot \max\left\{ {\Lambda}_2 \sqrt{\kappa} ~,~ {\Lambda}_1 \kappa \right\}
\right)^{1/2} \\
\tilde{X} &\equiv \Sigma^{1/2} X\,,\\
{\Lambda}_0 &\equiv \sup_{A\in{\mathcal{M}}}~ \frac{1}{n} {{\sf Tr}}(\tilde{X}A\tilde{X}^{{\sf T}}), \\
{\Lambda}_1 &\equiv \sup_{A\in{\mathcal{M}}}~ \frac{1}{n}{\|\tilde{X}A\tilde{X}^{{\sf T}}\|}_{\rm op},\\
{\Lambda}_2 &\equiv \sup_{A\in{\mathcal{M}}}~ \frac{1}{n}{\|\tilde{X}A\tilde{X}^{{\sf T}}\|}_F\,,\\
\kappa &\equiv \frac{c}{2}\log\frac{{|{\mathcal{M}}|}}{p_0}\,,
\end{split}
\end{aligned}$$ where $c>2$ is the constant in the Hanson-Wright inequality given in .
Define $g = \frac{1}{n} X^{{\sf T}}\epsilon$ which is a random vector. Invoking the characterization of dual norm $\|\cdot\|_{k\square d}^\star$ given by , we have $$\begin{aligned}
({\|g\|}^\star)^2
= \sup_{A\in {\mathcal{M}}} \, g^{{\sf T}}A g
= \sup_{A\in {\mathcal{M}}} \, \frac{1}{n} \epsilon^{{\sf T}}(\frac{1}{n}X A X^{{\sf T}}) \epsilon .
\end{aligned}$$ We next use a Hanson-Wright inequality to upper bound the right-hand side with high probability. More specifically, we use a result by [@adamczak2015note] on the Hanson-Wright inequality given in .
For any fixed $A\in{\mathcal{M}}$ (need not be positive semidefinite) define $B = \frac{1}{n}XAX^{{\sf T}}$. Then, $$\mathbb{E} [\epsilon^{{\sf T}}B \epsilon]
= \langle \mathbb{E} [\epsilon\epsilon^{{\sf T}}], B \rangle
= \langle \Sigma, B \rangle \leq {\Lambda}_0,$$ where ${\Lambda}_0$ is defined in . Therefore, for any $t>0$, Hanson-Wright inequality implies $$\mathbb{P}\left( \epsilon^{{\sf T}}B \epsilon \geq {\Lambda}_0 + t \right)
\leq 2 \exp \left(
-\frac{1}{c} \min\left\{ \frac{t^2}{2\eta^4 {\Lambda}_2^2} ~,~
\frac{t}{\eta^2{\Lambda}_1} \right\}
\right)$$ where ${\Lambda}_1$ and ${\Lambda}_2$ are defined in . Taking a union bound over all $A\in{\mathcal{M}}$, we get $$\begin{aligned}
\mathbb{P}\left(
\sup_{A\in {\mathcal{M}}} \, \epsilon^{{\sf T}}(\frac{1}{n}X A X^{{\sf T}}) \epsilon
\,\geq\, {\Lambda}_0 + t \right)
&\leq 2 \exp \left(
-\frac{1}{c} \min\left\{ \frac{t^2}{2\eta^4 {\Lambda}_2^2} ~,~
\frac{t}{\eta^2{\Lambda}_1} \right\}
\right) \cdot {|{\mathcal{M}}|} \\
&\leq 2p_0 \cdot \exp \left(
-\frac{1}{c} \min\left\{ \frac{t^2}{2\eta^4 {\Lambda}_2^2} ~,~
\frac{t}{\eta^2{\Lambda}_1} \right\}
+ \log \frac{{|{\mathcal{M}}|}}{p_0}
\right). \end{aligned}$$
The right-hand side will be bounded by $2p_0$ (as desired in the statement) if the argument to the exponential is non-positive. This provides a lower bound for $t$ which is consistent with the fact that we would like $t$ to be as small as possible in the left-hand side of the above chain of inequalities. Therefore, we choose $$t = \eta^2 \cdot \max\left\{
{\Lambda}_2 \sqrt{2c \log \frac{{|{\mathcal{M}}|}}{p_0}}
~,~
{\Lambda}_1 c \log \frac{{|{\mathcal{M}}|}}{p_0}
\right\}$$ which establishes the claim.
\[rem:other-HW\] In proving , we use a variation of the Hanson-Wright inequality given in , from [@adamczak2015note]. This result is particularly useful when matrices $A\in {\mathcal{M}}$ are not necessarily positive semidefinite. As an example, see Example 1 in [@jalali2017variational]. On the other hand, when ${\mathcal{M}}\subset \mathbb{S}_+^p$, other variations of the Hanson-Wright inequality may be used (a tail inequality – not necessarily a two-sided inequality – suffices) to establish variations of . These variations may allow for other classes of noise distributions. As an example, working with the Hanson-Wright inequality in [@hsu2012tail] requires ${\mathcal{M}}\subset \mathbb{S}_+^p$ but allows for $\epsilon\in\mathbb{R}^n$ to be [*a subgaussian random vector*]{}; for some $K\geq 0$, $\mathbb{E}\exp\langle \epsilon, u\rangle \leq \exp(K^2 {\|u\|}_2^2/2)$ for all $u\in\mathbb{R}^n$. This class neither covers nor is included in the class with the convex concentration property.
Finally, let us complement the bound of with an upper bound on $\lambda$. The following bound is well-known but has been provided for completeness. The proof is given in .
\[lem:lam-upper-bnd\] Consider measurements of the form $y = X\beta^\star + \epsilon$ and the estimator in . If $\lambda \geq \frac{1}{n}{\|X^{{\sf T}}y\|}^\star$, then ${\widehat{\beta}}=0$.
Existing Approaches {#sec:lambda-bnds-existing}
-------------------
[@jalali2018missing] also leverage the Hanson-Wright inequality in regularized regression where they consider a modification of Lasso for recovery of a sparse transition matrix in a vector autoregressive process with subgaussian noise and incomplete observations. In such problem, the design is constructed through the action of the transition matrix on previous innovations. Therefore, instead of aggregate quantities ${\Lambda}_0$, ${\Lambda}_1$, and ${\Lambda}_2$ here, for the design matrix, they arrive at structural summary quantities for the transition matrix (Section 1.3 in this reference) which allow for quantifying the [*dependence*]{} within design caused by autoregression. The bounds of [@jalali2018missing] in terms of these structural summary quantities can be compared with the bounds in [@melnyk2016estimating Theorem 3.3] that are agnostic to the model properties. Following a similar line of thought as that of [@jalali2018missing], combined with the general machinery provided in this section, one can derive bounds on the regularization parameter for many correlation scenarios (beyond autoregression) in the design matrix.
On the other hand, most of the existing literature for bounding the regularization parameter assume both $X$ and $\epsilon$ are drawn from well-known random ensembles for which concentration results exist. Most notably, generic chaining [@talagrand2014upper] is used leading to bounds in terms of the Gaussian width (or subgaussian width, sub-exponential width, etc) of the unit norm ball. For example, see [@banerjee2014estimation; @chen2016structured] for certain subgaussian design matrices, [@sivakumar2015beyond] for results on sub-exponential noise and design, [@melnyk2016estimating Theorem 3.3] for the case of autoregressive models, and [@johnson2016structured] for an active sampling scenario.
Even beyond the random nature of existing results, computing the Gaussian width of a norm ball is not straightforward and requires a case by case consideration; e.g., see [@chen2015structured]. General approaches for bounding this Gaussian width include bounding the Gaussian width of [*all*]{} tangent cones (Lemma 3 in [@banerjee2014estimation]) as well as careful partitioning of the extreme points of the norm ball (Lemma 2 in [@maurer2014inequality]).
Estimation Error Bounds and the Relative Diameter {#sec:est}
=================================================
Consider the setup of : a measurement model $y = X\beta^\star + \epsilon$, where $X\in\mathbb{R}^{n\times p}$ is the [*design*]{} matrix and $\epsilon\in\mathbb{R}^n$ is the noise vector. For any given norm ${{\|\cdot\|}}$, and not only those studied in , we then consider the regularized estimator in . Rather now-well-known analysis of yields [*estimation error bounds*]{}, namely bounds on $\|{\widehat{\beta}}-\beta^\star\|$ and $\|{\widehat{\beta}}-\beta^\star\|_2$. In this section, we review existing estimation error bounds (e.g., see [@wainwright2014structured] for a review) and provide proofs for the sake of completeness. Let us summarize the main ingredients in establishing these bounds:
- Optimality condition for the regularized estimator in , with $\lambda \geq {\|\frac{2}{n}X^{{\sf T}}\epsilon\|}^\star$, yields $v = {\widehat{\beta}}-\beta^\star \in {{\color{ygcolor}\Xi}}({\beta^\star;{{\|\cdot\|}}})$ where $$\begin{aligned}
\label{eq:ErrSet}
{{\color{ygcolor}\Xi}}({\beta^\star;{{\|\cdot\|}}}) &\equiv \bigl\{v :~ \frac{1}{2}\|v\| + \|\beta^\star\| \ge \|\beta^\star+v\| \bigr\}\end{aligned}$$ is in general a non-convex set and hard to characterize.
- The restricted eigenvalue (RE) constant, defined as $$\begin{aligned}
\label{eq:def-REc}
{{\color{ygcolor}{\alpha}}}(A) = \min_{u\in A\backslash \{0\}}\frac{ \frac{1}{n}{\|Xu\|}_2^2 }{ {\|u\|}_2^2},\end{aligned}$$ characterizes the effect of $X$ on the error $v$, and when evaluated positive on ${{\color{ygcolor}\Xi}}({\beta^\star;{{\|\cdot\|}}})$ allows for transforming the prediction error bound into estimation error bounds.
- The restricted norm compatibility constant [@negahban2012unified] is defined as $$\begin{aligned}
\label{eq:norm-compat}
{{\color{ygcolor}{\psi}}}(A) = \sup_{u\in A\backslash 0} \frac{{\|u\|}}{{\|u\|}_2}, \end{aligned}$$ and when evaluated on ${{\color{ygcolor}\Xi}}({\beta^\star;{{\|\cdot\|}}})$, allows for relating ${\|v\|}$ and ${\|v\|}_2$ in establishing estimation error bounds using a prediction error bound and the restricted eigenvalue condition.
\[thm:estimation\] Suppose that the sample covariance ${\widehat{\Sigma}}\equiv (X^{{\sf T}}X)/n$ satisfies the RE condition on ${{\color{ygcolor}\Xi}}$ with constant ${{\color{ygcolor}{\alpha}}}>0$. For $\lambda \ge \|\frac{2}{n}X^{{\sf T}}\epsilon\|^\star$, then, the estimator ${\widehat{\beta}}$ given by satisfies the bounds $$\begin{aligned}
\|{\widehat{\beta}}-\beta^\star\| &\le \frac{3}{{{\color{ygcolor}{\alpha}}}}\lambda{{\color{ygcolor}{\psi}}}^2\,,\label{eq:square-B}\\
\|{\widehat{\beta}}-\beta^\star\|_2 &\le \frac{3}{{{\color{ygcolor}{\alpha}}}} \lambda {{\color{ygcolor}{\psi}}}\,. \label{eq:L2-B}
\end{aligned}$$ where ${{\color{ygcolor}{\psi}}}= {{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}})$; see and .
is proved in .
However, the main point of deviation from the existing standard analysis is the introduction of a new quantity, namely [*the relative diameter of the norm ball at $\beta^\star$*]{}; see . Using this quantity, we define a superset for ${{\color{ygcolor}\Xi}}({\beta^\star;{{\|\cdot\|}}})$, in , which allows for bounding all of the above quantities and leads to concrete (as opposed to [*conceptual*]{}) bounds.
Relative Diameter
-----------------
Replacing ${{\color{ygcolor}\Xi}}$ with a more computational-friendly [*superset of ${{\color{ygcolor}\Xi}}$*]{}, in computing ${{\color{ygcolor}{\psi}}}$ and ${{\color{ygcolor}{\alpha}}}$, allows for deriving valid bounds that can be explicitly evaluated. We do so by introducing a new quantity, namely the [*relative diameter of the dual norm ball with respect to $\beta^\star$*]{}, and by providing which replaces ${{\color{ygcolor}\Xi}}$ with a simple cone defined in terms of the relative diameter. Further elaborations and discussions on the notion of relative diameter are postponed to and .
Before defining our main quantity in , let us review some definitions from convex geometry. Let $A$ and $B$ be two non-empty subsets of ${\mathbb{R}}^p$. Define the Hausdorff distance ${{\operatorname{dist}}}_H(A,B)$ by $$\begin{aligned}
{{\operatorname{dist}}}_H(A,B) \equiv \max\,\{\sup_{a\in A} {{\operatorname{dist}}}(a,B),\, \sup_{b\in B} {{\operatorname{dist}}}(b,A)\},
\end{aligned}$$ where for a given point $a$ and a set $B$, ${{\operatorname{dist}}}(a,B) = \inf_{b\in B} \|a-b\|_2$ denotes the distance of point $a$ from set $B$ in $\ell_2$ norm. For a given set $A\subset {\mathbb{R}}^p $, the corresponding support function $\sigma_A(v):{\mathbb{R}}^p \mapsto {\mathbb{R}}$ is defined as $
\sigma_A(v) \equiv \sup_{a\in A}~ {\langle}a,v{\rangle}$. Note that $B\subseteq A \subset {\mathbb{R}}^p$ if and only if $\sigma_B(v) \le \sigma_A(v)$ for all $v\in {\mathbb{R}}^p$. The Hausdorff distance can then be defined alternatively as $$\begin{aligned}
\label{eq:hausdorff}
{{\operatorname{dist}}}_H(A,B) = \sup_{\|v\|_2\le1 } |\sigma_A(v) - \sigma_B(v)|\,.\end{aligned}$$
\[def:varphi\] Given a norm ${{\|\cdot\|}}$ on $\mathbb{R}^p$, denote the unit ball in the dual norm by ${\mathcal{B}}^\star\equiv \{z\in{\mathbb{R}}^p: \|z\|^\star \le1\}$ and the subdifferential of ${{\|\cdot\|}}$ at $\beta$ by $\partial {\|\beta\|}$. We define [*a measure of complexity of $\beta\in\mathbb{R}^p \backslash \{0\}$ with respect to the norm ${{\|\cdot\|}}$*]{} denoted by $\varphi(\beta; {{\|\cdot\|}})$ as follows, $$\begin{aligned}
\label{eq:varphi}
\varphi = \varphi(\beta; {{\|\cdot\|}})\equiv {{\operatorname{dist}}}_H({\mathcal{B}}^\star, \partial{\|\beta\|}).\end{aligned}$$ Furthermore, since $\partial {\|\beta\|}$ is a subset of (in fact, a face of) ${\mathcal{B}}^\star$ we have $$\begin{aligned}
\label{eq:varphi-max}
\varphi (\beta; {{\|\cdot\|}})
= \adjustlimits\max_{z\in {\mathcal{B}}^\star} \min_{g\in \partial{\|\beta\|}}~ {\|z-g\|}_2.\end{aligned}$$
As an example, for the case of $\ell_1$ norm we have $\varphi(\beta; {{\|\cdot\|}}_1) = 2\sqrt{{\|\beta\|}_0}$. In , we present a few strategies for computing or upper bounding the relative diameter accompanied by detailed computations for a few families of norms in and . In , we provide further insights on $\varphi(\beta; {{\|\cdot\|}})$.
New Estimation Bounds {#sec:new-bounds}
---------------------
Recall the error set ${{\color{ygcolor}\Xi}}= {{\color{ygcolor}\Xi}}({\beta;{{\|\cdot\|}}})$ defined in . As it may be seen from the definition, this is generally a non-convex set with a complicated structure. Therefore, it is not in general easy to compute the associated restricted norm compatibility constant ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}})$ or the restricted eigenvalue constant ${{\color{ygcolor}{\alpha}}}({{\color{ygcolor}\Xi}})$ for a given design. Therefore, a reasonable strategy is to find a simpler set to which ${{\color{ygcolor}\Xi}}$ is a subset. Computing the two aforementioned constants for such a superset of ${{\color{ygcolor}\Xi}}$ cannot decrease ${{\color{ygcolor}{\psi}}}$ and cannot increase ${{\color{ygcolor}{\alpha}}}$. Therefore, the prediction error bound of and the estimation error bounds of cannot decrease meaning that we will have new valid error bounds.
Next, we use the notion of relative diameter to define a computationally-friendly set that covers ${{\color{ygcolor}\Xi}}$ and replaces it in the computation of ${{\color{ygcolor}{\psi}}}$ and $\alpha$.
\[lem:cones\] Consider the set ${{\color{ygcolor}\Xi}}({\beta;{{\|\cdot\|}}})$ from and the cone ${\mathcal{C}}(\varphi)$ defined as $$\begin{aligned}
\label{eq:cone-varphi}
{\mathcal{C}}(\varphi) = \bigl\{v:~ {\|v\|} \le 2\varphi {\|v\|}_2\bigr\}\end{aligned}$$ with $\varphi=\varphi({\beta;{{\|\cdot\|}}})$ defined in . Then, ${{\color{ygcolor}\Xi}}({\beta;{{\|\cdot\|}}})\subseteq {\mathcal{C}}(\varphi)$.
In the above, $\varphi = \varphi({\beta;{{\|\cdot\|}}})$ is defined based on the Hausdorff distance between the dual norm ball and the subdifferential of the norm at $\beta$. For example, for the case of $\ell_1$ norm we have $\varphi = 2\sqrt{{\|\beta^\star\|}_0}$ and hence , with ${{\color{ygcolor}\Xi}}$ replaced by ${\mathcal{C}}(2\sqrt{{\|\beta^\star\|}_0})$ recovers the classical estimation result on Lasso [@buhlmann2011statistics].
For $v\in {{\color{ygcolor}\Xi}}$, we have $$\begin{aligned}
\label{dummy1}
\frac{1}{2}\|v\| \le \|v\| + \|\beta^\star\| - \|\beta^\star+v\|\,.
\end{aligned}$$ By convexity of $\|\cdot\|$ we have $$\sup_{w\in \partial\|\beta^\star\|} {\langle}w,v{\rangle}\le \|\beta^\star+v\| - \|\beta^\star\|\,.$$ Therefore, $$\begin{aligned}
\label{Hauss1}
\|\beta^\star\| - \|\beta^\star+v\| + \|v\|
\le \sup_{\|z\|^\star\le 1} {\langle}z,v{\rangle}- \sup_{w\in \partial\|\beta^\star\|} {\langle}w,v{\rangle}\,.
\end{aligned}$$ Recall the notation ${\mathcal{B}}^\star$ for the unit ball in the dual norm. We proceed by writing the right-hand side of in terms of support functions: $$\begin{aligned}
\begin{split}\label{Hauss2}
\|\beta^\star\| - \|\beta^\star+v\| + \|v\|
&\le \|v\|_2 \left[\sigma_{{\mathcal{B}}^\star}\Big(\frac{v}{\|v\|_2}\Big) - \sigma_{\partial\|\beta^\star\|}\Big(\frac{v}{\|v\|_2}\Big) \right]
\\
&\stackrel{(a)}{=} \|v\|_2 \left|\sigma_{{\mathcal{B}}^\star}\Big(\frac{v}{\|v\|_2}\Big) - \sigma_{\partial\|\beta^\star\|}\Big(\frac{v}{\|v\|_2}\Big) \right|
\\
&\stackrel{(b)}{=} \|v\|_2\, {{\operatorname{dist}}}_H({\mathcal{B}}^\star, \partial\|\beta^\star\|) = \varphi \|v\|_2\,,
\end{split}
\end{aligned}$$ where $(a)$ follows from the characterization of subdifferential [@watson1992characterization] as $\partial\|\beta^\star\| = \{w: {\langle}w,\beta^\star{\rangle}= \|\beta^\star\|, \, \|w\|^\star = 1 \} \subset {\mathcal{B}}^\star$ and the fact that $\sigma_A (\cdot) \leq \sigma_B(\cdot)$ for $A\subseteq B$, and $(b)$ follows from the characterization of Hausdorff distance, given by . By combining and , we get $\|v\|\le 2\varphi \|v\|_2$, and hence $v\in {\mathcal{C}}(\varphi)$. This completes the proof.
Recall from above that evaluating different ingredients of the statistical error bounds on a superset of ${{\color{ygcolor}\Xi}}$ yields valid bounds. As an example, recall the restricted norm compatibility constant defined in as ${{\color{ygcolor}{\psi}}}(A) = \sup_{u\in A\backslash 0} \frac{{\|u\|}}{{\|u\|}_2}$. It is then easy to see from that $$\begin{aligned}
{{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}({\beta;{{\|\cdot\|}}})) \leq {{\color{ygcolor}{\psi}}}({\mathcal{C}}(\varphi({\beta;{{\|\cdot\|}}}))) = 2 \varphi({\beta;{{\|\cdot\|}}}).\end{aligned}$$ In the sequel, we study the RE condition for a family of subgaussian design matrices where in the proof we leverage and compute the RE constant for ${\mathcal{C}}(\varphi)$ instead of ${{\color{ygcolor}\Xi}}$.
\[thm:RE-random\] Consider
- A closed scale-invariant set ${\mathcal{S}}$, spanning $\mathbb{R}^p$, that further satisfies ${\mathcal{S}}\subseteq \{\beta:~ {{\operatorname{card}}}(\beta)\leq k\}$, and the corresponding cone ${\mathcal{C}}(\varphi)$ for $\varphi=\varphi(\beta^\star;{{\|\cdot\|}}_{\mathcal{S}})$.
- A sequence of design matrices $X\in {\mathbb{R}}^{n\times p}$, with dimensions $n\to \infty$, $p = p(n)\to \infty$ satisfying the following assumptions, for constant $\lambda_{\min}, \lambda_{\max}, \kappa$ independent of $n$. For each $n$, $\Sigma\in {\mathbb{R}}^{p\times p}$ is such that $\lambda_{\min}(\Sigma) \ge c_{\min}>0$ and $\lambda_{\max}(\Sigma)\le c_{\max}<\infty$.
- Assume that the rows of $X$ are independent subgaussian random vectors in ${\mathbb{R}}^p$ rows with second moment matrix $\Sigma$.
Then, for any fixed constant $c>0$, the empirical covariance ${\widehat{\Sigma}}\equiv (X^{{\sf T}}X)/n$ satisfies the RE condition over ${\mathcal{C}}(\varphi)$ for ${{\color{ygcolor}{\alpha}}}= \lambda_{\min}/2$, with probability at least $1-2p^{-ck}$, provided that $$\begin{aligned}
n\ge C\lambda_{\min}^{-2} \varphi^4 k\log p, \label{n-condition}
\end{aligned}$$ where $C = C(c, \lambda_{\min},\lambda_{\max},\kappa)$.
Proof of is given in . We follow a similar approach to that of [@loh2012]. However, instead of considering as many atoms as present in the target model, we only consider two atoms which allows for easy generalization to cases beyond sparsity.
\[rem:Eq\] For any ${{\color{ygcolor}q}}>1$, consider $$\begin{aligned}
\label{eq:ErrSet-q}
{{\color{ygcolor}\Xi}}^{({{\color{ygcolor}q}})}({\beta^\star;{{\|\cdot\|}}}) &\equiv \bigl\{v :~ \frac{1}{{{\color{ygcolor}q}}}\|v\| + \|\beta^\star\| \ge \|\beta^\star+v\| \bigr\},\end{aligned}$$ which for ${{\color{ygcolor}q}}=2$ yields ${{\color{ygcolor}\Xi}}^{(2)} = {{\color{ygcolor}\Xi}}$ defined in . Note that ${{\color{ygcolor}\Xi}}^{({{\color{ygcolor}q}})}$ is the whole space for $0<{{\color{ygcolor}q}}\leq 1$ which is not of interest in our discussion. An easy adaptation of yields ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}^{({{\color{ygcolor}q}})}) \leq \frac{{{\color{ygcolor}q}}}{{{\color{ygcolor}q}}-1}\varphi({\beta;{{\|\cdot\|}}})$. Notice the complicated dependence of the left-hand side on ${{\color{ygcolor}q}}$ while the right-hand side’s dependence is clear.
Define $\theta = {\|\frac{1}{n}X^{{\sf T}}\epsilon\|}^\star$. Then, for any $\lambda > \theta$ used in , the prediction error bound of and the estimation error bounds of read as $$\begin{aligned}
\frac{1}{n}{\|X(\beta^\star - {\widehat{\beta}})\|}_2^2
\leq 2\,(\lambda+\theta) {\|\beta^\star\|}\,,~
{\|{\widehat{\beta}}-\beta^\star\|}
\leq 2\,\frac{\lambda^2(\lambda+\theta)}{(\lambda-\theta)^2} \Bigl(\frac{\varphi^2}{{{\color{ygcolor}{\alpha}}}}\Bigr)\,,~
{\|{\widehat{\beta}}-\beta^\star\|}_2
\leq 2\,\frac{\lambda(\lambda+\theta)}{\lambda-\theta} \Bigl(\frac{\varphi}{{{\color{ygcolor}{\alpha}}}}\Bigr) \,,\end{aligned}$$ where ${{\color{ygcolor}{\alpha}}}= {{\color{ygcolor}{\alpha}}}({\mathcal{C}}^{(\lambda/\theta)})$. Moreover, an adaptation of yields ${{\color{ygcolor}{\alpha}}}= {{\color{ygcolor}{\alpha}}}({\mathcal{C}}^{(\lambda/\theta)}) = \lambda_{\min}/2$ for $$n \geq (36C^2 k \log p) (\lambda_{\min}^{-2} \varphi^4) (\frac{\lambda}{\lambda-\theta})^4.$$ Proof of the above statements is deferred to .
For future reference, we define ${{\color{ygcolor}\Xi}}^{(\infty)} \equiv \bigl\{v :~ \|\beta^\star\| \ge \|\beta^\star+v\| \bigr\}$ known as the set of descent directions at $\beta$ with respect to ${{\|\cdot\|}}$. The closed convex hull of ${{\color{ygcolor}\Xi}}^{(\infty)}$ is the tangent cone at $\beta$. We refer to ${{\color{ygcolor}\Xi}}^{(\infty)}$ as the [*constrained error set*]{}, as it an important object in the analysis of the Dantzig selector [@chatterjee2014generalized; @chen2015structured].
Computing the Relative Diameter {#sec:varphi}
===============================
Recall the definition of relative diameter $\varphi({\beta;{{\|\cdot\|}}})$ in . Here, we provide some tools to exactly compute or upper bound $\varphi$. The rest of this section focuses on such computations for a few major classes of norms: ordered weighted $\ell_1$ norms and their dual norms (which are polyhedral norms) as well as doubly-sparse norms and their dual norms.
Tools for Computing $\varphi$ {#sec:varphi-props}
-----------------------------
The following is easy to see from the definition.
$\varphi(\beta; {{\|\cdot\|}})$ is order-0 homogeneous with respect to its first argument and order-1 homogeneous with respect to its second argument.
\[lem:varphi-max-ext\] Denote by ${{\operatorname{ext}}}(A)$ the set of extreme points of a compact convex set $A$. Then, $$\begin{aligned}
\label{eq:varphi-max-ext}
\varphi(\beta;{{\|\cdot\|}})
= \adjustlimits\max_{z\in {{\operatorname{ext}}}{\mathcal{B}}^\star} \min_{g\in \partial{\|\beta\|}} ~{\|z-g\|}_2.\end{aligned}$$
Distance to a convex set is a continuous convex function. Moreover, ${\mathcal{B}}^\star$ is a compact convex set. Therefore, by Bauer’s Maximum Principle (e.g., see [@schirotzek2007nonsmooth Proposition 1.7.8]) a maximizer can be found among the extreme points of ${\mathcal{B}}^\star$.
Recall that $\varphi^2(\beta;{{\|\cdot\|}}_1)=4{\|\beta\|}_0$ and observe that $\varphi^2(\beta; {{\|\cdot\|}}_2) = 4$, for any $\beta\neq 0$. provides us with a procedure to compute $\varphi$ for many other common norms:
1. characterize ${{\operatorname{ext}}}({\mathcal{B}}^\star)$ as well as $\partial {\|\cdot\|}$,
2. characterize ${{\operatorname{dist}}}(z, \partial {\|\beta\|})$ for each $z\in {{\operatorname{ext}}}({\mathcal{B}}^\star)$, possibly making use of any structure in members of ${{\operatorname{ext}}}({\mathcal{B}}^\star)$,
3. possibly simplify the previous step by ignoring those $z\in {{\operatorname{ext}}}({\mathcal{B}}^\star)$ that can be seen that are sub-optimal in the final maximization over all $z\in {{\operatorname{ext}}}({\mathcal{B}}^\star)$,
4. take the maximum of all the computed distances ${{\operatorname{dist}}}(z, \partial {\|\beta\|})$ over $z\in {{\operatorname{ext}}}({\mathcal{B}}^\star)$.
We follow this procedure to exactly compute $\varphi$,
- for weighted $\ell_1$ norms in ,
- for weighted $\ell_\infty$ norms in , and directly for the $\ell_\infty$ norm in ,
- for ${{\|\cdot\|}}_{k\square 1}$ in .
Furthermore, allows for simplifying the computation of $\varphi$, when the dual norm is a structure norm; i.e., all of the extreme points of ${\mathcal{B}}^\star$ have the same $\ell_2$ norm, namely $\eta$. Then, since we are only dealing with the extreme points and not all members of ${\mathcal{B}}^\star$ as in the original definition, we get $$\begin{aligned}
\varphi^2(\beta;{{\|\cdot\|}})
= \eta^2 + \adjustlimits\max_{z\in {{\operatorname{ext}}}{\mathcal{B}}^\star} \min_{g\in \partial\|\beta\|} ~{\|g\|}_2^2 - 2{\langle}z,g{\rangle}.
$$ For example, the dual to an ordered weighted $\ell_1$ norm is a structure norm; see .
For structure norms (norms whose extreme points are all on the unit sphere), we can simplify $\varphi(\beta; {{\|\cdot\|}}_{\mathcal{S}}^\star)$ as follows. Recall that the orthogonal projection onto a non-convex set, such as ${\Pi}_{\mathcal{S}}(\beta)$, is a set-valued mapping in general. However, in the case of closed scale-invariant sets ${\mathcal{S}}$, establishes that all of the outputs have the same $\ell_2$ norm.
\[lem:varphi-dualSnorm\] Given a closed scale-invariant set ${\mathcal{S}}\subset \mathbb{R}^p$, consider the corresponding structure norm ${{\|\cdot\|}}_{\mathcal{S}}$. Then, $$\begin{aligned}
\varphi(\beta;{{\|\cdot\|}}_{\mathcal{S}}^\star)
&= \adjustlimits\max_{z} \min_{g} \left\{ {\|z-g\|}_2:~
z\in {{\operatorname{ext}}}{\mathcal{B}},~g\in \partial{\|\beta\|}_{\mathcal{S}}^\star \right\} \nonumber\\
&= \adjustlimits\max_{z} \min_{g} \left\{ {\|z-g\|}_2:~
z\in {\mathcal{S}}\cap \mathbb{S}^{p-1},~
g\in \frac{1}{{\|{\Pi}_{\mathcal{S}}(\beta)\|}_2} {{\operatorname{conv}}}\left({\Pi}_{\mathcal{S}}(\beta)\right)
\right\}\end{aligned}$$ where we used and .
#### Upper-bounding $\varphi$.
In some cases, it is not straightforward to follow the procedure we discussed before for exact computation of $\varphi$. In such cases, we upper bound $\varphi$ instead:
- Ordered weighted $\ell_1$ norms ${{\|\cdot\|}}_{w}$ in , implying an upper bound for $\ell_\infty$ norm in ,
- illustrates the doubly-sparse norms and their dual norms. We provide an upper bound for ${{\|\cdot\|}}_{k\square 1}^\star$ in .
Here is an upper bounding strategy:
\[lem:varphi-maxmin\] The max-min inequality gives $$\varphi(\beta; {{\|\cdot\|}}) \leq
\adjustlimits \min_{g\in \partial{\|\beta\|}} \max_{z\in {{\operatorname{ext}}}{\mathcal{B}}^\star} ~{\|z-g\|}_2.$$
In the following, we present the bound for $\varphi$ for ordered weighted $\ell_1$ norms as a sample of results in .
Ordered Weighted $\ell_1$ Norms {#sec:varphi-owl}
-------------------------------
Here, we provide bounds on $\varphi$ for a class of norms, namely the ordered weighted $\ell_1$ norms. The main technique is to upper bound using the max-min inequality as given in .
Given $\beta$, sort ${|\beta|}$ in descending order to get $\bar\beta$. Given $w_1\geq w_2 \geq \cdots \geq w_p \geq 0$, the ordered weighted $\ell_1$ norm is defined as ${\|\beta\|}_{w}= \sum_{i=1}^p w_i \bar\beta_i$. This norm encompasses $\ell_1$, $\ell_\infty$, and OSCAR [@bondell2008simultaneous].
\[lem:varphi-OWL\] Given $\beta\in\mathbb{R}^p$, set $d = {|\{ {|\beta_i|}\neq 0:~ i\in[p]\}|}$. Moreover, define ${\mathcal{G}}= ({\mathcal{G}}_1, \cdots, {\mathcal{G}}_d)$ as the partition of ${{\operatorname{Supp}}}(\bar\beta)$ into $d$ subsets where for any $i,j\in{{\operatorname{Supp}}}(\bar\beta)$ and any $t\in[d]$: $i,j\in {\mathcal{G}}_t$ if and only if $\bar\beta_i = \bar\beta_j$. Then, for ${\|\cdot\|}_{w}$, $$\begin{aligned}
\varphi^2(\beta; {{\|\cdot\|}}_{w})
~\leq~ {\|w_{\mathcal{G}}\|}_2^2 + 3\sum_{t=1}^d \frac{1}{{|{\mathcal{G}}_t|}} (\sum_{j\in {\mathcal{G}}_t} w_j)^2
~\leq~ 4{\|w_{\mathcal{G}}\|}_2^2\,,\end{aligned}$$ where we abuse the notation with ${\mathcal{G}}= {\mathcal{G}}_1\cup\cdots{\mathcal{G}}_d = {{\operatorname{Supp}}}(\bar\beta)$. The bounds are achieved with equality for $w={\boldsymbol{1}}$ (the $\ell_1$ norm).
Proof of is given in .
\[lem:varphi-linf\]Setting $w$ to the first standard basis vector we get ${\|\cdot\|}_{w}= {\|\cdot\|}_\infty$. Hence, $\varphi(\beta; {{\|\cdot\|}}_\infty) \leq \sqrt{1+3/t} \leq 2$ where $t = {|\{i\in[p]:~ {|\beta_i|} = {\|\beta\|}_\infty\}|} \ge 1$.
We next employ , to precisely compute $\varphi$ for $\ell_\infty$ norm.
\[lem:varphi-linf-exact\] For the $\ell_\infty$ norm and $\beta\neq 0$, $$\begin{aligned}
\varphi^2(\beta;{{\|\cdot\|}}_\infty) = 1 + \frac{1}{\max\{t-1,1/3\}}\end{aligned}$$ where $t = {|\{i\in[p]:~ {|\beta_i|} = {\|\beta\|}_\infty\}|} \ge 1$.
Proofs for and are given in .
In the case of ordered weighted $\ell_1$ norms [@zeng2014ordered], in , we provide a simple and interpretable bound on $\varphi(\beta;{{\|\cdot\|}}_{w})$ for any $\beta\in\mathbb{R}^p$. The bound relies on the clustering of values in $\beta$ as well as the sparsity pattern of $\beta$ in interaction with $w$, and is closely connected to the K-means objective for the entries of $\beta$.
On the other hand, the computations in Theorem 5 and Example 3.2 of [@chen2015structured] rely on upper bounding ${{\|\cdot\|}}_{w}$ with $\ell_1$ and $\ell_2$ norm and provide a crude bound on ${{\color{ygcolor}{\psi}}}$ for the constrained error set in terms of ${\|\beta\|}_0$, $w_1$, and the average of entries of $w$, as $\frac{2pw_1^2}{{\|w\|}_1}\sqrt{s}$ where $s = {{\operatorname{card}}}(\beta^\star)$. Note that the constrained error set is contained in ${{\color{ygcolor}\Xi}}$, hence has a smaller value for ${{\color{ygcolor}{\psi}}}$.
Since the bound in [@chen2015structured Example 3.2] is derived through upper bounding with $\ell_1$ norm (which coincides with ${{\|\cdot\|}}_{w}$ for $w={\boldsymbol{1}}_p$), it is easy to construct examples of $w$ for which the bound in is much better. For example, as an extreme case, consider the $\ell_\infty$ norm corresponding to $w=e_1$. In such case, for $\beta\neq0$, gives $\varphi(\beta; {{\|\cdot\|}}_\infty) \leq \sqrt{1+3/t}\leq 2$, for $t={|\{i\in[p]:~{|\beta_i|} = {\|\beta\|}_\infty\}|} \leq {\|\beta\|}_0$, while [@chen2015structured Example 3.2] gives a bound of $(p+1)\sqrt{{\|\beta\|}_0}$ for ${{\color{ygcolor}{\psi}}}$ evaluated on the constrained error set.
Insights on Relative Diameter {#sec:varphi-insight}
=============================
Recall the discussion in the beginning of on the complexity of the error set ${{\color{ygcolor}\Xi}}= {{\color{ygcolor}\Xi}}({\beta;{{\|\cdot\|}}})$, defined in , and how finding and working with a computationally-friendly superset of ${{\color{ygcolor}\Xi}}$ allows for simplifying the computation of the associated restricted norm compatibility constant and the restricted eigenvalue constant for a given design. In the following, we review some of the existing approaches to finding such a superset and provide comparisons with the proposed superset in .
#### When decomposable.
For example, let us consider the class of norms that satisfy the decomposability condition of [@negahban2012unified Definition 1]. More specifically, suppose that $A\subseteq \bar A \subseteq \mathbb{R}^p$ and $\bar A^\perp = \{v:~ \langle u,v\rangle=0 ~ \forall u\in \bar A \}$ are such that for all $u\in A$ and all $v\in \bar A^\perp$ we have ${\|u+v\|} = {\|u\|}+{\|v\|}$. This assumption is satisfied by the $\ell_1$ norm and the nuclear norm but is otherwise very restrictive. Relying on such assumption, namely the decomposability of ${{\|\cdot\|}}$ with respect to $(A,\bar A)$, it is easy to show that (e.g., see end of Section 2 in [@negahban2012unified]) for $\beta\in A$, $${{\color{ygcolor}\Xi}}({\beta;{{\|\cdot\|}}}) \subset \bigl\{ v:~ {\|v\|} \leq 4 {\|{\Pi}(v; \bar A)\|}\},$$ which then yields tight prediction and estimation error bounds. However, the above strategy cannot be applied to general norms; as easy examples as the $\ell_\infty$ norms or a weighted $\ell_1$ norm.
#### When the width is all we need.
As discussed above, the approximation of ${{\color{ygcolor}\Xi}}$ with a superset is being used to upper bound ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}})$ and to lower bound ${{\color{ygcolor}{\alpha}}}({{\color{ygcolor}\Xi}})$. We are not aware of any proposals in the literature for the former and one of our main contributions lies in the introduction of the relative diameter and the associated superset for ${{\color{ygcolor}\Xi}}$, provided in , that makes both of these tasks possible. However, an alternative strategy has been used in the literature to lower bound $\alpha({{\color{ygcolor}\Xi}})$ through connections to constrained estimators: $$\begin{aligned}
{\widehat{\beta}}_{\rm D} &\equiv {\mathop{{\operatorname{argmin}}}}_\beta \, \bigl\{ {\|\beta\|} :~ {\| X^{{\sf T}}(X\beta - y)\|}^\star \leq \lambda \bigr\} \,, \label{eq:estimator-dantzig} \\
{\widehat{\beta}}_{\rm E} &\equiv {\mathop{{\operatorname{argmin}}}}_\beta \, \bigl\{ {\|\beta\|} :~ X\beta = y \bigr\} \,, \label{eq:estimator-constrained} \\
{\widehat{\beta}}_{\rm T} &\equiv {\mathop{{\operatorname{argmin}}}}_\beta \, \bigl\{ {\|\beta\|} :~ {\|X\beta - y\|}_2\leq \delta \bigr\} \,, \label{eq:estimator-tube} \\
{\widehat{\beta}}_{\rm N} &\equiv {\mathop{{\operatorname{argmin}}}}_\beta \, \bigl\{ {\|X\beta - y\|}_2:~ {\|\beta\|}\leq \tau \bigr\} \,, \label{eq:estimator-ball} \end{aligned}$$ where is discussed in [@chatterjee2014generalized; @banerjee2014estimation; @chen2015structured; @cai2016geometric], and are discussed in [@chandrasekaran2012convex], and is discussed in [@li2015geometric], [and the analysis for all of them models the norm ball with its tangent cone at $\beta^\star$ and studies the interaction of the design matrix and the noise with such model (i.e., the tangent cone)]{}. More specifically, [@banerjee2014estimation] shows that the Gaussian width of the regularized error set ${{\color{ygcolor}\Xi}}({\beta;{{\|\cdot\|}}})$ and the constrained error set (namely $\{v:~ {\|\beta+v\|} \leq {\|\beta\|}\}$, whose closure is the tangent cone at $\beta$) are of the same order, which then allows for providing a sample complexity result to attain a desired RE constant (in the nature of ). See [@tropp2015convex] for general sample complexity results, in relation to RE, for independent subgaussian measurements established through tools for bounding a nonnegative empirical process as well as the notion of Gaussian width.
#### Relative diameter enables required computations.
Alternatively, in this work, we observe that the error set can be bounded as in : $${{\color{ygcolor}\Xi}}({\beta;{{\|\cdot\|}}}) \subset {\mathcal{C}}(\varphi) = \bigl\{v:~ {\|v\|} \le 2 {\|v\|}_2 \cdot \varphi({\beta;{{\|\cdot\|}}}) \bigr\}.$$ where $\varphi$, the relative diameter with respect to ${{\|\cdot\|}}$ at $\beta$, is defined in . This readily implies ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}})\leq 2\varphi$. Moreover, as illustrated through , $\varphi$ and the associated superset also allow for a straightforward lower bounding of the RE constant $\alpha({{\color{ygcolor}\Xi}})$.
#### Some Remarks.
- Let us recall implying ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}) \leq 2\varphi$ where $$\begin{aligned}
{{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}) &= \sup_v\, \Bigl\{\frac{{\|v\|}}{{\|v\|}_2} :~ \frac{1}{2}{\|v\|}+{\|\beta\|} \geq {\|\beta+v\|} \Bigr\}, \\
\varphi(\beta;{{\|\cdot\|}}) &= \adjustlimits\sup_{z} \inf_g \Bigl\{ {\|z-g\|}_2:~ {\|z\|}^\star \leq 1,~ {\|g\|}^\star\leq 1,~ \langle g,\beta\rangle ={\|\beta\|} \Bigr\} .\end{aligned}$$ On a high level, the transformation from ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}})$ to $\varphi$ can be seen as going from a primal quantity to a dual quantity.
- Note that, as clear from the definition of $\varphi$, it is not a local quantity, and as it can be seen from the example in , can change with the changes in the norm even though the tangent cone at $\beta$ is being kept the same. This hints on suitability of $\varphi$ in analyzing the regularized problem (while tangent cone is relevant for constrained problems). However, the tangent cone still affects the computation of $\varphi$ through its relation to the subdifferential: the dual to tangent cone is the cone of subdifferential.
- It is worth mentioning that [@chen2015structured] is concerned with the Dantzig selector, not the regularized estimator, and only provides strategies to bound ${{\color{ygcolor}{\psi}}}$ for the [*constrained*]{} error set.
- Several geometric quantities related to a norm have been studied in the high-dimensional statistics literature. Gaussian width [@Gordon88; @chandrasekaran2012convex] has been a prominent quantity in linear models. See [@amelunxen2014living; @foygel2014corrupted; @jalali2014minimum; @banerjee2014estimation; @chen2015structured; @vaiter2015model; @su2016slope; @figueiredo2016ordered] for other quantities.
#### An Illustrative Example.
Here, we consider a parametrized family of norms and examine the values of ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}^{(\infty)})$, $ {{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}})$, and $\varphi$, to showcase how $\varphi$ remains faithful to the true quantity ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}})$ as the norm changes, where ${{\color{ygcolor}\Xi}}^{(\infty)} \equiv \{v:~{\|\beta+v\|} \leq {\|v\|}\}$; see .
For any value $\gamma>0$, we consider the norm $$\begin{aligned}
{\|\beta\|} \equiv \max\bigl\{
{|\beta_1|}
+\frac{3}{4}{|\beta_2|} ~,~
\frac{\gamma}{\gamma+4}{|\beta_1|}
+\frac{9}{10}{|\beta_2|} ~,~
\frac{\gamma}{\gamma+5}{|\beta_1|}
+\frac{9}{2}{|\beta_2|}
\bigr\}\end{aligned}$$ in $\mathbb{R}^2$. Considering $\beta = [0,1]^{{\sf T}}= e_2$, it is easy to see that $\varphi$ has three separate modes; i.e., as $\gamma$ changes, the optimal $z\in{\mathcal{B}}^\star$ jumps among three (distinct) possible choices. The subdifferential, and hence the tangent cone, do not change with $\gamma$. However, ${{\color{ygcolor}{\psi}}}$ for the tangent cone (equal to ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}^{(\infty)})$) is not going to be a constant, as the norm changes with $\gamma$.
.1in ![Values of $\varphi(e_2; {{\|\cdot\|}})$ (solid line with warm colors), ${{\color{ygcolor}{\psi}}}= {{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}})$ (blue dash-dotted line), and ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}^{(\infty)})$ (black dotted line), evaluated numerically, for $\beta=[0,1]^{{\sf T}}$ and different values of $\gamma$. The three colors on the solid line indicate the regimes under which the achieving $z\in{\mathcal{B}}^\star$ is the same. Observe that $\varphi$ closely follows the other two, in all three regimes.[]{data-label="fig:maxWL1-quants"}](JJF19-fig1.eps "fig:"){width=".6\textwidth"}
From , we expect ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}^{(\infty)}) \leq {{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}) \leq 2\varphi$. Moreover, establishes ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}^{(q)}) \leq \frac{q}{q-1}\varphi$ for all $q>1$, which implies ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}^{(\infty)}) < \varphi$. All of these can be observed in as well. As established in , larger values of the regularization constant $\lambda$ allow for basing the analysis on ${{\color{ygcolor}\Xi}}^{(q)}$ for larger values of $q$, which in turn makes the error bounds in terms of $\varphi$ closer to those in terms of ${{\color{ygcolor}\Xi}}^{(q)}$.
#### Comparison over maximum of weighted $\ell_1$ norms.
In this experiment, we randomly generate maximum of weighted $\ell_1$ norms and compute and plot $\varphi$, ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}})$, and ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}^{(\infty)})$ for them. provides the results. As it can be seen from , $\varphi$ closely approximates ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}})$ for most cases. In generating a norm, we first pick a random integer to determine the number of weighted $\ell_1$ norms that are involved. We always include $w=[1,1]^{{\sf T}}$ (corresponding to the $\ell_1$ norm), and we choose the rest of the weight vectors as random points in the positive orthant to the right and below of $w=[1,1]^{{\sf T}}$.
As discussed in the previous experiment, we expect ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}^{(\infty)}) \leq {{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}) \leq 2\varphi$ and ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}^{(\infty)}) < \varphi$, both of which can be observed in as well. However, $\varphi$ has a lower bound as $2\,{{\operatorname{dist}}}(0,\partial {\|\beta\|}) \leq \varphi({\beta;{{\|\cdot\|}}})$ which holds generally whenever ${\Pi}(0,\partial {\|\beta\|}) \in\partial {\|\beta\|}$.
.1in ![For $100$ randomly generate maximum of weighted $\ell_1$ norms, and for $\beta=[0,1]^{{\sf T}}$, we plot $\varphi$, $2\varphi$, ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}})$, ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}^{(\infty)})$, as well as the ratio between the first and the last, which as predicted by , is always above 1. []{data-label="fig:varphi-vpsi-randnorms"}](JJF19-fig2 "fig:"){width="60.00000%"}
Doubly-Sparse Regularization; Optimization and Statistical Bounds {#sec:kd-final}
=================================================================
Prediction Error for ${{\|\cdot\|}}_{k\square d}$ {#sec:PredErr}
-------------------------------------------------
Here, we consider the linear measurement model $y = X\beta^\star+\epsilon$, with $X\in {\mathbb{R}}^{n\times p}$ the design matrix and $\epsilon\in {\mathbb{R}}^n$ a noise vector. We apply the prediction bounds established in to the case of doubly-sparse regularized estimator given by . As a result, we bound ${\|X(\beta^\star - {\widehat{\beta}})\|}_2$ in terms of the $k$ and $d$ used in defining the regularizer ${\|\cdot\|}_{k\square d}$, the properties of $\beta^\star$ (number of nonzeros and distinct values), and certain properties of $X$. As we will see, column aggregation in $X$ plays a natural role in the final bound.
\[thm:pred-err-kd\] Suppose that noise vector $\epsilon$ is zero mean Gaussian vector with covariance matrix $\Sigma \equiv \mathbb{E}[\epsilon \epsilon^{{\sf T}}]$. Define $$\label{eq:phi0-1}
\begin{aligned}
\phi_0\;&\equiv \sup_{J \subseteq [p]: |J|\le k-d+1} \, \frac{{\|\Sigma^{1/2} X_{J} {\boldsymbol{1}}\|}_2^2}{n |J|} , \\
\phi_1 \;&\equiv \sup_{J\subseteq[p]: |J|\le k} \frac{{\|\Sigma^{1/2} X_J\|}_{\rm op}^2}{n}\,,
\end{aligned}$$ and for an arbitrary fixed value of $0<p_0<1/2$, let $$\begin{aligned}
\label{eq:phi}
\phi\; \equiv\; \frac{1}{\sqrt{n}} \left(d \phi_0 + c \min(d \phi_0, \phi_1) \Big[k \log({2epd}/{k}) + \log (1/p_0)\Big] \right)^{1/2}\,,\end{aligned}$$ where $c>2$ is the numerical constant in the Hanson-Wright inequality given in . Let ${\widehat{\beta}}$ be obtained from with $\lambda \geq \phi$. Then, with probability at least $1 - 2 p_0$, it satisfies $$\begin{aligned}
\label{prediction-error-kd}
\frac{1}{n}{\|X(\beta^\star - {\widehat{\beta}})\|}_2^2 \leq 3\lambda \|\beta^\star\|_{k\square d} \,.
\end{aligned}$$
We first apply to the case of doubly-sparse regularization and show that in this case ${\Lambda}\le \phi$, where $\phi$ is given by . The result then follows readily from .
In specializing to the case of doubly-sparse regularization, it is easy to see that $\mathcal{M} = {\texttt{BD}}(k,d)$ due to the characterization . In addition, by the concentration inequality of Lipschitz function of Gaussian vectors, we have that $\Sigma^{-1/2}\epsilon$ satisfies the convex concentration with constant one.
Let $\tilde{X}\equiv \Sigma^{1/2} X$ and write $$\begin{aligned}
{\Lambda}_0 \equiv \sup_{A\in {\texttt{BD}}(k,d)} \frac{1}{n} {{\sf Tr}}(\tilde{X} A \tilde{X}^{{\sf T}}) \le d \times \sup_{J\subseteq[p], |J|\le k-d+1} \frac{{\|\tilde{X}_J{\boldsymbol{1}}\|}_2^2}{n|J|} = d \phi_0\,,\end{aligned}$$ where we uses the structure of $A\in {\texttt{BD}}(k,d)$, namely it has only a nonzero principle sub-matrix of size $k$. Further, this sub-matrix is block diagonal with $d$ blocks and for a block of size $q$, all of its entries are $1/q$.
We also have $$\begin{aligned}
{\Lambda}_1 \equiv \sup_{A\in{\texttt{BD}}(k,d)} \frac{1}{n} {\|\tilde{X}A\tilde{X}^{{\sf T}}\|}_{\rm op} \le \frac{1}{n} \sup_{A\in {\texttt{BD}}(k,d)} {\|A\|}_{\rm op}\times \sup_{J\subseteq{p}, |J|\le k} {\|\tilde{X}_J\|}_{\rm op}^2 \le \phi_1\,,\end{aligned}$$ since ${\|A\|}_{\rm op}\le 1$, for $A\in {\texttt{BD}}(k,d)$. As another bound on ${\Lambda}_1$, note that any $A\in {\texttt{BD}}(k,d)$ can be written as $A = u_1u_1^{{\sf T}}+\dotsc+ u_du_d^{{\sf T}}$, where each $u_i$ has entries $1/\sqrt{|J_i|}$ on a set $J_i\subseteq[p]$, with $|J_i|\le k-d+1$ and zero everywhere else. Hence, $$\begin{aligned}
\frac{1}{n} {\|\tilde{X} A\tilde{X}^{{\sf T}}\|}_{\rm op} = \frac{1}{n} {\|\tilde{X}u_iu_i^{{\sf T}}\tilde{X}^{{\sf T}}\|}_{\rm op} \le \frac{1}{n}\sum_{i=1}^d {\|\tilde{X}u_i\|}_2^2 =\frac{1}{n|J_i|}\sum_{i=1}^d {\|\tilde{X}_{J_i} {\boldsymbol{1}}\|}_2^2 \le d \phi_0\,.\end{aligned}$$ Combining the above two bounds we obtain ${\Lambda}_1 \le \min(d\phi_0,\phi_1)$.
By using , we have $$\begin{aligned}
\kappa \equiv \frac{c}{2} \log \frac{|{\texttt{BD}}(k,d)|}{p_0} < \frac{c}{2} \Big(k\log(2epd/k) + \log(1/p_0) \Big)\,.\end{aligned}$$ Finally, we note that $$\begin{aligned}
{\Lambda}_2 \equiv \sup_{A\in {\texttt{BD}}(k,d)} \frac{1}{n}{\|\tilde{X}A\tilde{X}^{{\sf T}}\|}_F \le \sqrt{d} \sup_{A\in {\texttt{BD}}(k,d)} \frac{1}{n}{\|\tilde{X}A\tilde{X}^{{\sf T}}\|}_{\rm op} \le \sqrt{d} {\Lambda}_1 \,,\end{aligned}$$ where in the first inequality we used the fact that the matrices in ${\texttt{BD}}(k,d)$ are at most of rank $d$. Consequently, ${\Lambda}_2< {\Lambda}_1\sqrt{\kappa}$. By plugging the above bounds on ${\Lambda}_0$, ${\Lambda}_1$, ${\Lambda}_2$, and $\kappa$ in , we obtain that ${\Lambda}\le \phi$, which completes the proof.
Examples {#sec:examples-kd}
--------
#### Lasso.
Note that for $k=d=1$, the structure norm $\|\cdot\|_{1\square 1}$ becomes exactly the $\ell_1$ norm and the estimator ${\widehat{\beta}}$ in reduces to the Lasso estimator with regularization parameter $\lambda$. We show that recovers the prediction bound of Lasso [@buhlmann2011statistics Corollary 6.1]. Suppose that the noise $\epsilon$ has i.i.d. zero mean Gaussian entries with variance at most $\sigma^2$, and the columns of $X$ are normalized so that each column has $\ell_2$-norm $\sqrt{n}$. Then, $\phi_0= \phi_1 = \sigma^2$. Setting $p_0 = 1/(2ep)$, we get $\phi = (\sigma/\sqrt{n}) (1+ 2c \log(2ep))^{1/2}$. Therefore, with $\lambda = \phi$, the bound simplifies to $$\begin{aligned}
\label{eq:pred-lasso}
\frac{1}{n}{\|X(\beta^\star - {\widehat{\beta}})\|}_2^2
\leq 3 \sigma \sqrt{\frac{1+ 2c \log(2ep)}{n}} \|\beta^\star\|_1
\lesssim \sigma \sqrt{\frac{\log p}{n} } \|\beta^\star\|_1\,.
$$ We denote the right-hand side of by ${{\sf err}^{\rm Lasso}}$. Note that the design matrix $X$ appears in the prediction error bound through the quantities $\phi_0$ and $\phi_1$, which for rare-features are expected to be small.
#### Gain over Lasso with Doubly-sparse Norms.
We next want to discuss the gain that the estimator achieves over Lasso when the true underlying parameter $\beta^\star$ is sparse and takes only a few distinct values.
\[lem:Gain\] Consider a sequence of design matrices $X\in {\mathbb{R}}^{n\times p}$, with dimension $n \to \infty$, and $p = p(n) \to \infty$, satisfying the following assumptions for constants $C_{\max}, C > 0$ independent of $n$. For each $n$, $\Psi\in {\mathbb{R}}^{p\times p}$ is such that $$\sigma_{\max} (\Psi)\le C_{\max} < \infty,\quad \sup_{J\subseteq[p], |J|\le k} \frac{1}{|J|} ({\boldsymbol{1}}^{{\sf T}}\Psi_{J,J} {\boldsymbol{1}}) \le C_* \le C_{\max}\,.$$ In addition, $X\Psi^{-1/2}$ that has i.i.d. subgaussian rows, with zero mean and subgaussian norm $\kappa=\|\Psi^{-1/2} x_1\|_{\psi_2}$, and the noise vector $\epsilon\in{\mathbb{R}}^n$ has i.i.d. Gaussian entries with variance at most $\sigma^2$. Then, there exist constants $ c_0, c, C >0$, depending on the subgaussian norm $\kappa$, such that the following holds. With probability at least $1- 2p^{-ck} - 2 p^{-c(k-d+1)}$, the following holds for $\phi_0$ and $\phi_1$ given by : $$\begin{aligned}
\phi_0 \le C_{*}\sigma^2 \left(1+ C \sqrt{\frac{(k-d+1)\log p}{n}}\right) \,,\quad \quad \phi_1\le C_{\max}\sigma^2 \left(1+ C\sqrt{\frac{k \log p}{n}}\right) \,.\end{aligned}$$ Consequently, by , if $n\ge c_0 k \log p$ we have $$\begin{aligned}
\phi\le \tilde{C} \sigma\left[\min(dC_*,C_{\max}) \frac{k}{n} \log\Big(\frac{2epd}{k}\Big) \right]^{1/2}\,,\end{aligned}$$ for a constant $\tilde{C}>0$.
We refer to for the proof of . Plugging $\lambda \asymp \phi$ in gives that with probability at least $1 - (pd/k)^{-k}$, $$\begin{aligned}
\label{eq:pred-gainDS}
\frac{1}{n}{\|X(\beta^\star - {\widehat{\beta}})\|}_2^2
\lesssim \sigma \sqrt{\frac{k \log(pd/k))}{n}}\, \|\beta^\star\|_{k\square d}\,.\end{aligned}$$ We denote the right-hand side of by ${{\sf err}^{\rm DS}}$. Comparing the bounds with the Lasso prediction bound , we get $$\begin{aligned}
\label{gain-ratioDS}
\frac{{{\sf err}^{\rm DS}}}{{{\sf err}^{\rm Lasso}}} \le C \sqrt{k- \frac{k\log(k/d)}{\log p}}\times \frac{\|\beta^\star\|_{k\square d}}{\|\beta^\star\|_1}\,.\end{aligned}$$ Note that $\|\beta^\star\|_{k\square d}/\|\beta^\star\|_1 \le 1$. To see this, note that the 1-sparse vectors are in ${\mathcal{S}}_{k,d}$ for all $k, d\ge 1$ and hence the $\ell_1$ unit ball is inside ${\mathcal{B}}_{{\mathcal{S}}_{k,d}}$, which by definition implies the claim. To show the gain over Lasso (which corresponds to $k = d =1$), we consider the following two cases:
- Assume that $\max_{i\in {{\operatorname{Supp}}}(\beta^\star)}|\beta^\star_i| / \min_{i\in {{\operatorname{Supp}}}(\beta^\star)} |\beta^\star_i|\le c_0$. For $d=1$ and a value of $1\le k\le p$, by using , we have $$\begin{aligned}
\frac{{\|\beta^\star\|}_{k\square 1}}{{\|\beta^\star\|}_1} \le \max\Big\{\frac{1}{\sqrt{k}}, c_0\frac{\sqrt{k}}{k^\star}\Big\} = \frac{1}{\sqrt{k}} \max\{1, c_0{k}/{k^\star}\}\,.\end{aligned}$$ Using this bound in , we obtain $$\begin{aligned}
\frac{{{\sf err}^{\rm DS}}}{{{\sf err}^{\rm Lasso}}} \le C \sqrt{1- \frac{\log k}{\log p}}\times \max\{1, c_0{k}/{k^\star}\}\,.\end{aligned}$$ Since $k$ can grows as large as $p$, we see that the ratio above can be made arbitrarily small, showcasing the gain over Lasso.
- Assume the doubly-sparse estimator with $k\ge k^\star$ and $d\ge d^\star$. Then, $\beta^\star \in {\mathcal{S}}_{k,d}$ and hence $\|\beta^\star\|_{k,d} = \|\beta^\star\|_2$ by definition of structured norms; see . Therefore, $\|\beta^\star\|_{k\square d}/\|\beta^\star\|_1$ can be made as small as $1/\sqrt{k^*}$ (when $d^\star =1$). Therefore, the bound in becomes $$\frac{{{\sf err}^{\rm DS}}}{{{\sf err}^{\rm Lasso}}} \le C \sqrt{k- \frac{k\log(k/d)}{\log p}}\times \sqrt{\frac{1}{k^\star}} = C \sqrt{1- \frac{\log(k/d)}{\log p}}\times \sqrt{\frac{k}{k^\star}}\,.$$
Again, as $k/k^\star\ge 1$ can get arbitrarily close to one, $k$ can grow up to $p$, and $d$ can be as small as one, this ratio can be made arbitrarily small which demonstrates the gain over Lasso in prediction error.
#### The $k$-support norm.
The $k$-support norm coincides with ${{\|\cdot\|}}_{k\square k}$ and the results of can be specialized to yield prediction error bounds for the regularized regression with the $k$-support norm. However, in setting $d$ equal to $k$ in , we can get a tighter bound on the size of the corresponding ${\mathcal{M}}$. More specifically, improves the bound ${|{\mathcal{M}}|}\leq (2ep)^k$ from , to a bound ${|{\mathcal{M}}|} \leq (ep/k)^k$. Using this bound and calculating $\phi_0$ and $\phi_1$ in Theorem \[thm:pred-err-kd\] for case of $k=d$, we obtain the following corollary which is analogous to for the $k$-support norm regularization:
Consider a sequence of design matrices $X\in {\mathbb{R}}^{n\times p}$, with dimension $n \to \infty$, and $p = p(n) \to \infty$, satisfying the following assumptions for constants $C_{\max}, C > 0$ independent of $n$. For each $n$, $\Psi\in {\mathbb{R}}^{p\times p}$ is such that $$\sigma_{\max} (\Psi)\le C_{\max} < \infty,\quad \sup_{J\subseteq[p], |J|\le k} \frac{1}{|J|} ({\boldsymbol{1}}^{{\sf T}}\Psi_{J,J} {\boldsymbol{1}}) \le C_* \le C_{\max}\,.$$ In addition, $X\Psi^{-1/2}$ that has i.i.d. subgaussian rows, with zero mean and subgaussian norm $\kappa=\|\Psi^{-1/2} x_1\|_{\psi_2}$, and the noise vector $\epsilon\in{\mathbb{R}}^n$ has i.i.d. Gaussian entries with variance at most $\sigma^2$.
Then, specializing for $k = d$, with probability at least $1- 2p^{-ck} - 2 p^{-c}$, $$\begin{aligned}
\phi_0 \le C_{*}\sigma^2 \left(1+ C \sqrt{\frac{\log p}{n}}\right) \,,\quad \quad \phi_1\le C_{\max}\sigma^2 \left(1+ C\sqrt{\frac{k \log p}{n}}\right) \,,\end{aligned}$$ In addition, by , if $n\ge c_0 k \log p$, for some constant $c_0> 0$, we obtain the following bound on $\phi$ for case of $k$-support norm $$\begin{aligned}
\phi\le \tilde{C} \sigma\left[\min(kC_*, C_{\max})\frac{k}{n} \log\Big(\frac{ep}{k}\Big) \right]^{1/2}\,,\end{aligned}$$ for a constant $\tilde{C}>0$.
Plugging $\lambda \asymp \phi$ in gives the following prediction bound for the $\|\cdot\|_{k\square k}$ regularized estimator $\hat{\beta}$: $$\begin{aligned}
\label{eq:errkk} \frac{1}{n}{\|X(\beta^\star - {\widehat{\beta}})\|}_2^2
\lesssim \sigma \sqrt{ \min(kC_*, C_{\max}) \frac{k \log(p/k)}{n}}\, \|\beta^\star\|_{k\square k}\,.\end{aligned}$$
#### The $\|\cdot\|_{k\square 1}$ norm.
Our next example is the other extreme case, namely $d =1$. We characterize the prediction error for ${\|\cdot\|}_{k\square 1}$ regularized estimator in lemma below. The next corollary follows from .
\[lem:k1\] Consider a sequence of design matrices $X\in {\mathbb{R}}^{n\times p}$, with dimension $n \to \infty$, and $p = p(n) \to \infty$, satisfying the following assumptions for constants $C_{\max}, C > 0$ independent of $n$. For each $n$, $\Psi\in {\mathbb{R}}^{p\times p}$ is such that $$\sigma_{\max} (\Psi)\le C_{\max} < \infty,\quad \sup_{J\subseteq[p], |J|\le k} \frac{1}{|J|} ({\boldsymbol{1}}^{{\sf T}}\Psi_{J,J} {\boldsymbol{1}}) \le C_* \le C_{\max}\,.$$ In addition, $X\Psi^{-1/2}$ that has i.i.d. subgaussian rows, with zero mean and subgaussian norm $\kappa=\|\Psi^{-1/2} x_1\|_{\psi_2}$, and the noise vector $\epsilon\in{\mathbb{R}}^n$ has i.i.d. Gaussian entries with variance at most $\sigma^2$.
There exist constants $C, c_0, c>0$ such that the following holds. Assume $n\ge c_0 k \log p$ and let $$\begin{aligned}
\phi = C \sigma \sqrt{C_* \frac{k \log(p/k)}{n}} \end{aligned}$$ Let ${\widehat{\beta}}$ be obtained from with $d =1$ and $\lambda \geq \phi$. Then, with probability at least $1- 2p^{-ck} - 2 (ep/k)^{-k}$, we have $$\begin{aligned}
\label{eq:pred-gain2}
\frac{1}{n}{\|X(\beta^\star - {\widehat{\beta}})\|}_2^2
\lesssim 3\lambda \|\beta^\star\|_{k\square 1}\,.\end{aligned}$$
Using $\lambda \asymp \phi$ in gives the following prediction bound for the $\|\cdot\|_{k\square 1}$ regularized estimator $\hat{\beta}$: $$\begin{aligned}
\label{eq:pred-gain3}
\frac{1}{n}{\|X(\beta^\star - {\widehat{\beta}})\|}_2^2
\lesssim \sigma \sqrt{ C_{*} \frac{k \log(p/k)}{n}}\, \|\beta^\star\|_{k\square 1}\,.\end{aligned}$$ To compare with the ${\|\cdot\|}_{k\square k}$ regularizer, we denote by $\errkk$ and $\errk1$ the right-hand side of and . We then have $$\begin{aligned}
\frac{\errk1}{\errkk} \lesssim \sqrt{\frac{C_*}{\min(kC_*, C_{\max})}} \times \frac{\|\beta^\star\|_{k\square 1}}{\|\beta^\star\|_{k\square k}}\,.\end{aligned}$$ Now suppose that $\beta^\star \in {\mathcal{S}}_{k,1}$. Then, ${\|\beta^\star\|}_{k\square 1} = \|\beta^\star\|_{k\square k} = \|\beta^\star\|_2$ and the above ratio becomes $\sqrt{\frac{C_*}{\min(kC_*, C_{\max})}}$. Recall that $C_*$ was the maximum of the quadratic forms $({\boldsymbol{1}}^{{\sf T}}\Psi_{J,J} {\boldsymbol{1}})/|J|$, over all subsets $J\subseteq[p]$, with $|J|\le k$. In addition, $C_{\max}$ is the bound on the operator norm of the covariance $\Psi$. Hence, $C_*\le C_{\max}$ and depending on $\Psi$, this ratio can be made as small as $1/\sqrt{k}$.
Discussions {#sec:disc}
===========
#### Challenges without Decomposability. {#sec:new-challenges}
Most of the existing work on norm regularization can be unified under the notion of [*decomposability*]{}; see [@negahban2012unified; @candes2013simple; @vaiter2015model] for slightly different definitions. While most of the works on statistical analysis for norm regularization, and especially the earlier works, do not explicitly mention decomposability, it is the main proof ingredient; e.g., see Lemma 4.1 in [@bickel2009simultaneous] for how decomposability comes into play. Therefore, common mechanisms established for analyzing Lasso, nuclear norm regularized estimators, or more generally those with decomposable norms, cannot be used in our case. Therefore, similar to [@banerjee2014estimation], we aim at identifying more general geometric quantities but extend beyond conceptual bounds, introducing computation-friendly quantities.
#### Algorithms Based on Non-convex Projection.
Only assuming access to the non-convex projection (onto the set of desired models) can also be used in devising algorithms. For example, Iterative Hard Thresholding algorithms [@blumensath2008iterative Section 3] [@blumensath2011sampling] (projects onto the set of $k$-sparse vectors, namely ${\mathcal{S}}_{k,k}$), [@jain2010guaranteed Section 2] (projects onto the set of rank-$r$ matrices), [@roulet2017iterative] (does K-means which is projection onto the set of $d$-valued models [@jalali2013convex]), belong to this class. However, the machinery proposed in this work allows for devising convex regularization functions (norms) which then can be combined with general loss functions and constraints; unlike the specific constrained loss minimization setups required in the aforementioned works.
#### Gaussian width of the Norm Ball and Unions of Subspaces.
Lemma 2 of [@maurer2014inequality] provides an upper bound for the Gaussian width of a norm ball by splitting the computation over subsets of extreme points. Consider a structure norm associated to a set ${\mathcal{S}}$ which is a finite union of subspaces ${\mathcal{S}}_1,\ldots,{\mathcal{S}}_m$, with dimensions $d_1,\ldots, d_m$, respectively. Then, the Gaussian width of ${\mathcal{S}}_i\cap \mathbb{S}^{p-1}$ is given by $$\omega({\mathcal{S}}_i\cap \mathbb{S}^{p-1})
= \mathbb{E} _g \,\sup_{z\in {\mathcal{S}}_i\cap \mathbb{S}^{p-1}}\, \langle z, g \rangle
= \mathbb{E} _g \,\sup_{z\in {\mathcal{S}}_i\cap \mathbb{S}^{p-1}}\, \langle z, {\Pi}(g; {\mathcal{S}}_i) \rangle
= \mathbb{E} _g {\|{\Pi}(g; {\mathcal{S}}_i)\|}_2 \leq \sqrt{d_i}$$ for $g\sim\mathcal{N}(0,I_p)$. Applying Lemma 2 of [@maurer2014inequality] to this splitting of ${\mathcal{S}}\cap \mathbb{S}^{p-1}$ yields $$\omega({\mathcal{B}}_{\mathcal{S}})
= \omega({\mathcal{S}}\cap \mathbb{S}^{p-1})
\leq \max_{i\in[m]} \sqrt{d_i} + 2 \sqrt{\log m}.$$ The Gaussian width of the unit norm ball is the quantity used in [@banerjee2014estimation; @chen2015structured] to bound $\frac{1}{n}{\|X^{{\sf T}}\epsilon\|}_{\mathcal{S}}^\star$ related to the regularization parameter. We instead make use of the Hanson-Wright inequality to get , providing a bound that is deterministic with respect to the design (and not restricted to a few random ensembles of design) and is also sensitive to norm-induced properties of the design.
#### Possible Generalizations.
Our result can be easily extended to regularized loss minimization for smooth loss functions and beyond the least-squares loss. The introduction and characterization of $\varphi$ can also be used beyond the regression setup in this paper; e.g., see [@goldstein2018structured] for a possible application domain. For least-squares with random design, results of and can be extended to many more noise distributions, as discussed in and , as well as to sub-exponential noise as remarked by [@adamczak2015note Remark 2.8].
Acknowledgements {#acknowledgements .unnumbered}
================
Adel Javanmard was partially supported by an Outlier Research in Business (iORB) grant from the USC Marshall School of Business, a Google Faculty Research award and the NSF CAREER Award DMS-1844481. Maryam Fazel was supported in part by grants NSF TRIPODS CCF 1740551, ONR N00014-16-1-2789, and NSF CCF-1409836. This work was carried out in part while the authors were visiting the Simons Institute for the Theory of Computing.
Proofs: Projection-based Norms {#app:Snorm-summary}
==============================
\[app:projection\]
Since $\omega_0\in {\Pi}_{\mathcal{S}}({\beta})$ and ${\mathcal{S}}$ is scale invariant, we have $\{\lambda\omega_0 :\; \lambda\in\mathbb{R}\}\subset {\mathcal{S}}$ which in turn implies ${\Pi}_{\{\lambda\omega_0 :\; \lambda\in\mathbb{R}\}}({\beta}) =\omega_0$. Note that projection onto a line is a singleton. Therefore, $$\label{eq:dual_val}
\langle{{\beta}},{\omega_0}\rangle = \langle{{\Pi}_{\{\lambda\omega_0 :\; \lambda\in\mathbb{R}\}}({\beta})},{\omega_0}\rangle = {\|\omega_0\|}_2^2 \,.$$ Optimality of projection yields ${\|\omega_0 - {\beta}\|}_2 \leq {\|{\|\omega_0\|}_2 \cdot {\theta}- {\beta}\|}_2$ for all ${\theta}\in{\mathcal{S}}\cap \mathbb{S}^{p-1}$. This, after algebraic manipulations and an application of , yields $$\begin{aligned}
{\|\omega_0\|}_2
= \langle \frac{\omega_0}{{\|\omega_0\|}_2}, {\beta}\rangle
\geq \sup_{{\theta}\in{\mathcal{S}}\cap \mathbb{S}^{p-1}} \langle {\theta},{\beta}\rangle
= \sup_{{\theta}\in{\mathcal{B}}_{\mathcal{S}}} \langle {\theta},{\beta}\rangle
= {\|{\beta}\|}_{\mathcal{S}}^\star .\end{aligned}$$ Since $\frac{\omega_0}{{\|\omega_0\|}_2} \in {\mathcal{S}}\cap \mathbb{S}^{p-1}$, we get equality and the proof is finished.
The above also establishes $\langle{{\beta}},{{\Pi}_{\mathcal{S}}({\beta})}\rangle = {\|{\Pi}_{\mathcal{S}}({\beta})\|}_2^2= {\|{\Pi}_{\mathcal{S}}({\beta})\|}_{\mathcal{S}}\, {\|{\beta}\|}_{\mathcal{S}}^\star$ which illustrates the pair of achieving vectors in the definition of dual norm. This has been known as the [*alignment property*]{} in the literature. As a corollary, we get the following.
\[cor:proj-nonexp\] The projection onto a closed scale-invariant set ${\mathcal{S}}$ is non-expansive; i.e., ${\|{\Pi}_{\mathcal{S}}({\beta})\|}_2 \leq {\|{\beta}\|}_2$ for all ${\beta}\,$.
The proof is by expanding ${\|{\Pi}_{\mathcal{S}}({\beta})-{\beta}\|}_2^2\geq 0$ and using .
Alternatively, since ${{\|\cdot\|}}_{\mathcal{S}}\geq {{\|\cdot\|}}_2$, we get ${{\|\cdot\|}}_{\mathcal{S}}^\star \geq {{\|\cdot\|}}_2$, which also establishes the claim.
Note that while projection onto convex sets is always non-expansive, projection onto general non-convex sets can be expansive. However, the distance to a general set is still non-expansive (e.g., see [@clarke1990optimization Proposition 2.4.1, page 50]). This should not be confused with the Kolmogorov criterion for projection onto a [*convex*]{} set ${\mathcal{C}}$, i.e., ${\theta}= {\Pi}_{\mathcal{C}}({\beta})$ if and only if ${\theta}\in{\mathcal{C}}$ and $\langle{{z}-{\theta}},{{\beta}-{\theta}}\rangle \leq 0$ for all ${z}\in{\mathcal{C}}\,$, since we are interested in projection onto a non-convex set ${\mathcal{S}}\,$.
Consider the following characterization of the subdifferential [@watson1992characterization], $$\begin{aligned}
\partial {\|{\beta}\|}_{\mathcal{S}}^\star
&= \left\{ {g}:\; \langle{{g}},{{\beta}}\rangle = {\|{\beta}\|}_{\mathcal{S}}^\star\,,\; {\|{g}\|}_{\mathcal{S}}\leq 1 \right\} \\
&= \left\{ {g}:\; \langle{{g}},{{\beta}}\rangle = {\|{\Pi}_{\mathcal{S}}({\beta})\|}_2 \,,\; {\|{g}\|}_{\mathcal{S}}\leq 1 \right\} \,.\end{aligned}$$ Using the results of , one can check that any ${\theta}\in{\Pi}_{\mathcal{S}}({\beta})/ {\|{\Pi}_{\mathcal{S}}({\beta})\|}_2$ satisfies the definition of subgradients. Since subdifferential is a convex set, we get a one-sided inclusion; i.e., $\supseteq$. Next, notice that the squared dual norm can be written as $$\begin{aligned}
\label{eq:dum0sub}
({\|{\beta}\|}_{\mathcal{S}}^\star)^2
= {\|\beta\|}_2^2 - {\|\beta - {\Pi}(\beta; {\mathcal{S}})\|}_2^2
= {\|\beta\|}_2^2 - \inf_{\theta\in {\mathcal{S}}} {\|\beta-\theta\|}_2^2
= \sup_{{\theta}\in{\mathcal{S}}}\; 2\langle{{\beta}},{{\theta}}\rangle - {\|{\theta}\|}_2^2\end{aligned}$$ where the inner function, say $f({\beta},{\theta})\,$, is continuous, linear in ${\beta}$, and concave quadratic in ${\theta}$.
On the other hand, by , we have ${\|\beta\|}_{\mathcal{S}}^\star \leq {\|\beta\|}_2$. Moreover, ${{\|\cdot\|}}_{\mathcal{S}}^\star$ is continuous and Lipschitz. Therefore, there exists a neighborhood $U \ni \beta$ such that for every $u\in U$ $$({\|u\|}_{\mathcal{S}}^\star)^2
= \sup_{{\theta}}\bigl\{ 2\langle{u},{{\theta}}\rangle - {\|{\theta}\|}_2^2 :~ {\theta}\in{\mathcal{S}},~ {\|{\theta}\|}_2 \leq 2{\|{\beta}\|}_{\mathcal{S}}^\star\bigr\}.$$ Note that the constraint set (indexing $\theta$) is compact as ${\mathcal{S}}$ is assumed to be a closed set. Moreover, it is clear from the second equality in that the optimal solutions to the above parametric minimization are the members of ${\Pi}(u; {\mathcal{S}})$. All in all, the above parametric minimization satisfies the requirements of Theorem 3 in [@yu2012differentiability] and we get equality for the subdifferential.
Consider the orthogonal projection mapping as ${\Pi}(\beta;A) = {\mathop{{\operatorname{Argmin}}}}_{u\in A} {\|\beta-u\|}_2^2 = {\mathop{{\operatorname{Argmin}}}}_{u\in A} \sum_{i=1}^p (u_i-\beta_i)^2$. Consider $\theta\in {\Pi}(\beta;A)$.
If $A$ is invariant with respect to sign flips, $\theta\in A $ implies ${|\theta|} \circ {{\operatorname{sign}}}(\beta) \in A$, where ${{\operatorname{sign}}}(0)$ can be chosen as either $+1$ or $-1$. By optimality, ${\|\theta-\beta\|}_2^2 \leq {\|{|\theta|} \circ {{\operatorname{sign}}}(\beta) - \beta\|}_2^2$ which implies $\sum_{i=1}^p \theta_i\beta_i = \langle \theta, \beta\rangle \geq \langle {|\theta|}, {|\beta|}\rangle = \sum_{i=1}^p {|\theta_i\beta_i|} \geq \sum_{i=1}^p \theta_i\beta_i$. Therefore, all inequalities hold with equality implying $\theta_i\beta_i \geq 0$ for all $i\in[p]$.
If $A$ is invariant under a permutation of the entries, $\theta \in A$ implies $\pi_{\beta}^{-1}(\pi_\theta(\theta))\in A$ where $\pi_u$ is any permutation for which $\pi_u(u)$ is sorted in non-increasing order. Optimality implies ${\|\theta-\beta\|}_2^2 \leq {\|\pi_\beta^{-1}(\pi_\theta(\theta)) - \beta\|}_2^2 = {\|\pi_\theta(\theta) -\pi_\beta(\beta)\|}_2^2$. This implies $\langle \theta, \beta \rangle \geq \langle \pi_\theta(\theta), \pi_\beta(\beta)\rangle$. The reverse inequality also holds as a result of the rearrangement inequality. Therefore, $\langle \pi_\theta(\theta), \pi_\beta(\beta)\rangle = \langle \theta, \beta \rangle$. Consider any $i,j\in[p]$ for which $\beta_i>\beta_j$. If $\theta_i<\theta_j$, define $\tilde \theta$ with all entries the same as $\theta$ except for $\tilde \theta_i = \theta_j$ and $\tilde \theta_j= \theta_i$. Then, $\langle \pi_{\tilde\theta}(\tilde\theta), \pi_\beta(\beta) \rangle \geq \langle \tilde \theta, \beta\rangle > \langle \theta,\beta \rangle = \langle \pi_\theta(\theta),\pi_\beta(\beta)\rangle $ while the first and last terms are equal. This is a contradiction which implies that the claim should hold true.
Consider any ${\theta}\in {\Pi}_{{\mathcal{S}}\cap\mathbb{S}^{p-1}}({\beta})$ and any ${z}\in {\Pi}_{{\mathcal{S}}}({\beta})\,$. The optimality of ${z}$ gives ${\|{z}\|}_2^2 - 2\langle{{\beta}},{{z}}\rangle \leq {\|{\|{z}\|}_2{\theta}\|}_2^2 - 2\langle{{\beta}},{{\|{z}\|}_2{\theta}}\rangle$ and the optimality of ${\theta}$ gives $\langle{{\beta}},{{\theta}}\rangle \geq \langle{{\beta}},{{z}/{\|{z}\|}_2}\rangle$. Combining these two inequalities proves $\langle{{\beta}},{{\theta}}\rangle = \langle{{\beta}},{{z}/{\|{z}\|}_2}\rangle = {\|{z}\|}_2$ (last equality uses ) which illustrates that ${\|{z}\|}_2{\theta}\in {\Pi}_{\mathcal{S}}({\beta})$ and ${z}/{\|{z}\|}_2\in {\Pi}_{{\mathcal{S}}\cap\mathbb{S}^{p-1}}({\beta})$. In other words, given ${\theta}\in {\Pi}_{{\mathcal{A}}_{\mathcal{S}}}({\beta})$, we have $\langle{{\beta}},{{\theta}}\rangle {\theta}\in {\Pi}_{\mathcal{S}}({\beta}) \,$.
Here is another explanation: since ${\mathcal{S}}$ is scale-invariant, one can first find the direction of projection on ${\mathcal{S}}$ and later find the correct scaling as $$\begin{aligned}
{\Pi}_{\mathcal{S}}({\beta})
&= \arg\left\{\min_{{\theta}\in{\mathcal{S}}}\, {\|{\beta}-{\theta}\|}_2^2 \right\} \\
&= \arg\left\{\min_{{\theta}\in{\mathcal{S}}}\, {\|{\theta}\|}_2^2 - 2\langle{{\beta}},{{\theta}}\rangle \right\} \\
&= \arg\left\{\min_{\tau\geq0}\, \left(\tau^2 - 2\tau \max_{{\theta}\in{\mathcal{S}}\cap\mathbb{S}^{p-1}}\, \langle{{\beta}},{{\theta}}\rangle \right)\right\} \end{aligned}$$ which shows that for finding the direction of ${\Pi}_{\mathcal{S}}({\beta})$ it suffices to project onto ${\mathcal{S}}\cap\mathbb{S}^{p-1}\,$. Yet another explanation comes from the result that says the dual norm is equal to the largest inner product with atoms. Hence, combining this with , we get $$\max_{{\theta}\in{\mathcal{S}}\cap\mathbb{S}^{p-1}} \langle{{\beta}},{{\theta}}\rangle={\|{\beta}\|}_{\mathcal{S}}^\star= \langle{{\beta}},{{\Pi}_{\mathcal{S}}({\beta})/{\|{\Pi}_{\mathcal{S}}({\beta})\|}_2}\rangle,$$ which proves our result.
Proofs: The $(k\square d)$-norm {#app:kd-norm}
===============================
For the first statement, we prove the more general version and then apply it to $P=I-e_ie_i^{{\sf T}}$. Consider $\theta\in{\Pi}(\beta;{\mathcal{S}})$ and assume $P\beta=\beta$, $P=P^{{\sf T}}=P^2$, and $Pu\in{\mathcal{S}}$ for all $u\in {\mathcal{S}}$. Then, $P\theta\in{\mathcal{S}}$ and optimality implies ${\|P\theta-\beta\|}_2^2\geq {\|\theta-\beta\|}_2^2$ which is equivalent to ${\|(I-P)\theta\|}_2^2=0$ and in turn to $P\theta=\theta$.
For the second statement, we prove the more general statement and then apply it to $A = I-e_ie_i^{{\sf T}}+ e_ie_j^{{\sf T}}$ and $B=I-e_je_j^{{\sf T}}+ e_je_i^{{\sf T}}$. Consider $\theta\in{\Pi}(\beta;{\mathcal{S}})$. By the assumption, $A\theta\in{\mathcal{S}}$. Therefore, optimality of $\theta$ implies ${\|A\theta-A\beta\|}_2^2 = {\|A\theta-\beta\|}_2^2 \geq {\|\theta-\beta\|}_2^2$ which in turn implies $(\theta-\beta)^{{\sf T}}(A^{{\sf T}}A - I)(\theta-\beta)\geq 0$. A similar argument establishes $(\theta-\beta)^{{\sf T}}(B^{{\sf T}}B - I)(\theta-\beta)\geq 0$. Adding up the two inequalities we get $0=0$ and hence all the inequalities so far have to hold with equality, implying that $A\theta$ and $B\theta$ are also optimal; i.e., $A\theta,B\theta\in{\Pi}(\beta; {\mathcal{S}})$.
Suppose ${|\beta_i|} < \bar\beta_k$ and $\theta_i\neq 0$ for some $\theta\in{\Pi}(\beta;{\mathcal{S}})$ and some $i\in[p]$. Therefore, there exists $j\in[p]$ for which ${|\beta_j|}\geq \bar\beta_k$ and $\theta_j=0$; otherwise, ${{\operatorname{card}}}(\theta)>k$. Consider a new vector $\tilde\theta$ with all entries equal to those of $\theta$ except for $\tilde\theta_i=0$ and $\tilde\theta_j = {|\theta_i|}{{\operatorname{sign}}}(\beta_j)$. Then, ${{\operatorname{dist}}}^2(\theta;{\mathcal{S}}) - {{\operatorname{dist}}}^2(\tilde\theta;{\mathcal{S}})
=(\theta_i-\beta_i)^2+(\theta_j-\beta_j)^2-(\tilde\theta_j-\beta_j)^2-(\tilde\theta_i-\beta_i)^2
=(\theta_i-\beta_i)^2+\beta_j^2-({|\theta_i|}-{|\beta_j|})^2-\beta_i^2
=2{|\theta_i|}({|\beta_j|}-\beta_i {{\operatorname{sign}}}(\theta_i))
\geq 2{|\theta_i|}({|\beta_j|}-{|\beta_i|})>0$. This contradicts the optimality of $\theta$. Therefore, the claim is established.
Observe that ${\mathcal{S}}_{k,d}$ satisfies all of the assumptions in and . Consider $\theta\in{\Pi}(\bar\beta; {\mathcal{S}}_{k,d})$. By , if two entries of $\bar\beta$ are equal, the same entries in $\theta$ are going to be equal. Therefore, while there might be several options for $\pi$, $\pi^{-1}(\theta)$ is unique for all such $\pi$. Moreover, if $\bar\beta_i=0$, implies $\theta_i=0$. Therefore, while there might be ambiguities in choosing ${{\operatorname{sign}}}(\beta)$ over its zero entries, the solution to $\pi^{-1}(\theta) \circ {{\operatorname{sign}}}(\beta)$ will be unique given a fixed $\theta\in{\Pi}(\bar\beta; {\mathcal{S}}_{k,d})$. Therefore, we can fix a choice for ${{\operatorname{sign}}}(\beta)$ and a choice for $\pi$, which in turn makes $\theta\mapsto \pi^{-1}(\theta) \circ {{\operatorname{sign}}}(\beta)$ well-defined and [*invertible*]{}. Therefore, observe that ${\|\beta - \pi^{-1}(\theta) \circ {{\operatorname{sign}}}(\beta)\|}_2
={\|{|\beta|} - \pi^{-1}(\theta)\|}_2
={\|\pi({|\beta|}) - \theta\|}_2
={\|\bar\beta - \theta\|}_2$. This, in conjunction with sign and permutation invariance of ${\mathcal{S}}_{k,d}$ establishes the optimality of $\pi^{-1}(\theta) \circ {{\operatorname{sign}}}(\beta)$.
On the other hand, consider $\gamma\in {\Pi}(\beta;{\mathcal{S}}_{k,d})$ and define $\theta = \pi(\gamma \circ {{\operatorname{sign}}}(\beta))$. Again: 1) $\gamma$ will be zero off the support of $\beta$, hence $\gamma \circ {{\operatorname{sign}}}(\beta)$ is well-defined, 2) $\gamma \circ \beta \geq 0$ by , hence $\gamma \circ {{\operatorname{sign}}}(\beta)\geq 0$, and 3) $\gamma \circ {{\operatorname{sign}}}(\beta)$ will not have different entries where ${|\beta|}$ has equal entries, therefore $\pi(\gamma \circ {{\operatorname{sign}}}(\beta))$ is well-defined. It remains to show that such vector is a projection of $\bar\beta$. Similar to the above, observe that ${\|\gamma -\beta \|}_2 = {\|\pi(\gamma \circ {{\operatorname{sign}}}(\beta)) - \pi(\beta \circ {{\operatorname{sign}}}(\beta))\|}_2 = {\|\pi(\gamma \circ {{\operatorname{sign}}}(\beta)) - \bar\beta\|}_2$, which establishes optimality.
Preliminary observation: By , ${{\operatorname{Supp}}}(\theta) \subseteq {{\operatorname{Supp}}}(\beta)$. By , $\theta \circ \beta \geq 0$ which together with the support inclusion result completely determines the sign of nonzeros in $\theta$; sign of any nonzero $\theta_i$ is ${{\operatorname{sign}}}(\beta_i)$. Since ${\mathcal{S}}_{k,d}$ is both sign and permutation invariant, and the sign of nonzero entries of the projections are determined, we will adjust the sign whenever we swap entries.
By , any projection $\theta\in{\Pi}(\beta;{\mathcal{S}}_{k,d})$ will have ${{\operatorname{card}}}(\theta)\leq k$ and $S={{\operatorname{Supp}}}(\theta)\subseteq \{i:~ {|\beta_i|}\geq \bar\beta_k\}$. Therefore, ${\|\theta-\beta\|}_2^2 = {\|\beta_{S^c}\|}_2^2 + {\|\theta- \beta_{S}\|}_2^2$. Consider $A_k = \{i:{|\beta_i|} = \bar\beta_k\}$. Then, by , there exists a projection $\tilde\theta$ which takes a single absolute value over $A_k$. Therefore, the indices in $S\cap A_k$ can be re-assigned arbitrarily (with appropriate sign adjustment) within $A_k$ without changing the distance. This validates the first step of our procedure.
Let us restrict the space to any set $A$ with ${{\operatorname{Supp}}}(\theta)\subseteq A\subseteq \{i:~ {|\beta_i|}\geq \bar\beta_k\}$ and ${|A|}=k$. Observe that ${\|\theta-\beta\|}_2^2 = {\|\beta_{A^c}\|}_2^2 + {\|\theta- \beta_{A}\|}_2^2$. Optimality of $\theta$ implies that $\theta_A$ has at most $d$ distinct absolute values and $\theta_A$ is closest to $\beta_A$ among all such vectors. This indeed is equivalent to $\theta_A = {\Pi}(\beta_A; {\mathcal{S}}_{k,d})$ where ${\mathcal{S}}_{k,d}\subseteq \mathbb{R}^k$ here. This validates the second step of our procedure.
[An alternative proof for ]{} We can work with $\bar\beta$ to simplify the presentation. Therefore, assume $\beta=\bar\beta$ for the rest of this proof. We follow the procedure discussed in proof of (given next) but instead keep track of the optimal solutions, rather than the optimal value, to characterize the projection itself.
It can be seen from the reformulations in that we can first project onto ${\mathcal{S}}_{k,k}$, namely the set of $k$-sparse vectors. This leads to zeroing out all entries except the $k$ with largest absolute values (corresponding to the first $k$ entries of $\bar\theta$).
The procedure resulting in , as discussed, is a K-means procedure into $d$ groups.
Finally, in comparing and a similar expression for $\theta$, as in , we can put the centers back into their original positions (before turning $\theta$ to $\bar\theta$), with the corresponding sign, to get the final result. This is the consequence of optimality in conjunction with the fact that minimal distance is achieved when two vectors have the same sign pattern and pattern of absolute values (rearrangement inequality.)
Denote the optimal solution (the projection) with $\gamma^\star \in {\mathcal{S}}_{k,d}$. allows for computing ${\Pi}(\theta; {\mathcal{S}}_{k,d})$ from ${\Pi}(\bar{\theta}; {\mathcal{S}}_{k,d})$ and the sign and order patterns in $\theta$. In projecting $\bar{\theta}$, the optimal $\bar{\gamma}^\star$ will be nonnegative and sorted. Therefore, from , the projection can be expressed as $$\begin{aligned}
&{\| {\Pi}(\bar{\theta}; {\mathcal{S}}_{k,d}) \|}_2^2 = {\|\bar{\theta}\|}_2^2 - \min \bigl\{ {\|\bar{\theta}-\gamma\|}_2^2:~\gamma\in {\mathcal{S}}_{k,d} \bigr\} \label{eq:dumm3}\\
&= {\|\bar{\theta}\|}_2^2 - \min \bigl\{ {\|\bar{\theta}-\gamma\|}_2^2:~\gamma\in {\mathcal{S}}_{k,d} ,~ \gamma_1\geq \cdots \geq \gamma_k \geq 0 ,~
\gamma_{k+1} = \ldots = \gamma_p = 0 \bigr\} \\
&= {\|\bar{\theta}_{1:k}\|}_2^2 - \min \bigl\{ {\|\bar{\theta}_{1:k}-\gamma_{1:k}\|}_2^2:~\gamma\in {\mathcal{S}}_{k,d} ,~
\gamma_1\geq \cdots \geq \gamma_k \geq 0,
\gamma_{k+1} = \ldots = \gamma_p = 0 \bigr\}.
\label{eq:pf-dum1}\end{aligned}$$ Considering the definition of ${\mathcal{S}}_{k,d}$ in , observe that the last minimization is indeed a K-means clustering problem. Since $\gamma_{1:k}$ can only take $d$ distinct values, we can turn the optimization problem into choosing the optimal partition of entries and then assign the optimal value to each partition separately. This yields $$\begin{aligned}
&{\| {\Pi}(\bar{\theta}; {\mathcal{S}}_{k,d}) \|}_2^2 \nonumber\\
&= {\|\bar{\theta}_{1:k}\|}_2^2 - \min \bigl\{
\sum_{i=1}^d {\|\bar{\theta}_{{\mathcal{I}}_i}- \frac{1}{{|{\mathcal{I}}_i|}}{\boldsymbol{1}}{\boldsymbol{1}}^{{\sf T}}\bar{\theta}_{{\mathcal{I}}_i} \|}_2^2
:~ ({\mathcal{I}}_1,\cdots,{\mathcal{I}}_d)\in{\bar{\texttt{P}}}(k,d) \bigr\} \nonumber\\
&= \max \bigl\{ \sum_{i=1}^d \frac{1}{{|{\mathcal{I}}_i|}} ({\boldsymbol{1}}^{{\sf T}}\bar{\theta}_{{\mathcal{I}}_i})^2 :~ ({\mathcal{I}}_1,\cdots,{\mathcal{I}}_d)\in{\bar{\texttt{P}}}(k,d) \bigr\} \label{eq:dumm2}\end{aligned}$$ as claimed.
Consider partitioning $[p]$ into $d+1$ groups, $d$ of which having a total size of $k$. This can be done by first selecting $k$ out $p$ elements and then partitioning these $k$ elements into $d$ groups. We can then get an upper bound by allowing for empty groups; hence, $|{\texttt{BD}}(k,d)| = 2^k|{\texttt{P}}(k,d)|
\leq 2^k\binom{p}{k} d^k
\leq (\frac{2epd}{k})^k$.
Consider a dynamic programming formulation of the 1-dimensional K-means clustering similar to [@wang2011ckmeans]. However, modify the formulation to align with the quantity of interest in ; namely $\sum_{i=1}^d \frac{1}{{|{\mathcal{I}}_i|}} ({\boldsymbol{1}}^{{\sf T}}\bar{\theta}_{{\mathcal{I}}_i})^2$ which is to be maximized.
More specifically, consider $$\begin{aligned}
{\| {\Pi}(\bar{\theta}; {\mathcal{S}}_{k,d}) \|}_2^2
&= \max \bigl\{ \sum_{i=1}^d \frac{1}{{|{\mathcal{I}}_i|}} ({\boldsymbol{1}}^{{\sf T}}\bar{\theta}_{{\mathcal{I}}_i})^2 :~ ({\mathcal{I}}_1,\cdots,{\mathcal{I}}_d)\in{\bar{\texttt{P}}}(k,d) \bigr\} \nonumber \\
&= \min\{t: \sum_{i=1}^d \frac{1}{{|{\mathcal{I}}_i|}} ({\boldsymbol{1}}^{{\sf T}}\bar{\theta}_{{\mathcal{I}}_i})^2 \leq t ~~ \text{for all } ({\mathcal{I}}_1,\cdots,{\mathcal{I}}_d)\in{\bar{\texttt{P}}}(k,d)\} .\end{aligned}$$ Define $$\begin{aligned}
\tilde\nu_{s,e} = \max\{ \sum_{i=1}^e \frac{1}{{|{\mathcal{I}}_i|}} ({\boldsymbol{1}}^{{\sf T}}\bar{\theta}_{{\mathcal{I}}_i})^2 : ({\mathcal{I}}_1,\cdots,{\mathcal{I}}_d)\in{\bar{\texttt{P}}}(s,e)\} \end{aligned}$$ as the optimal [*cost-to-go values*]{} and observe that they satisfy the following, $$\begin{aligned}
\tilde\nu_{s,e} = \max_{e\leq m \leq s} \{
\tilde\nu_{m-1,e-1} + \frac{1}{s-m+1}{|\bar\theta_{[m,s]}|}_1^2
\}.\end{aligned}$$ The above notation can be turned into inequalities (as in the QCQP) which finishes the proof. Observe that the optimal values for $\nu_{s,e}$ in the QCQP are equal to the values for $\tilde\nu_{s,e}$.
Observe that $$\begin{aligned}
{\Pi}(\bar{\theta}; {\mathcal{B}}^\star)
&= {\mathop{{\operatorname{argmin}}}}_u \bigl\{ {\|u-\bar{\theta}\|}_2^2:~ {\|u\|}_\sq^2 \leq 1 \bigr\} \\
&= {\mathop{{\operatorname{argmin}}}}_u \bigl\{ {\|u-\bar{\theta}\|}_2^2:~ {\|u\|}_\sq^2 \leq 1,~
u_1\geq \cdots \geq u_p \geq 0 \bigr\} \\
&\stackrel{(a)}{=} {\mathop{{\operatorname{argmin}}}}_u \Bigl\{ {\|u-\bar{\theta}\|}_2^2:~
u_1\geq \cdots \geq u_p \geq 0 ,~ \\
&\qquad \qquad \qquad \min_{\{\nu_{m,e}\}} \Bigl\{ \nu_{k,d}:~
\frac{1}{s-m+1} ( {\boldsymbol{1}}^{{\sf T}}u_{[m,s]})^2 \leq \nu_{s,e} - \nu_{m-1,e-1} ~~ \forall (e,m,s)\in {\texttt{T}}(k,d)\Bigr\} \leq 1
\Bigr\} \\
&= {\mathop{{\operatorname{argmin}}}}_u \min_{\{\nu_{m,e}\}}\bigl\{ {\|u-\bar{\theta}\|}_2^2:~
u_1\geq \cdots \geq u_p \geq 0 ,~\\
&\qquad \qquad \qquad ~ \nu_{k,d} \leq 1,~
\frac{1}{s-m+1} ( {\boldsymbol{1}}^{{\sf T}}u_{[m,s]})^2 \leq \nu_{s,e} - \nu_{m-1,e-1} ~~ \forall (e,m,s)\in {\texttt{T}}(k,d)
\bigr\}\end{aligned}$$ where in $(a)$ we uses the fact that the variable $u$ is sorted and we plugged in the representation for ${\|u\|}_\sq^2$ given in .
Proofs: Prediction Error {#app:prediction}
========================
By optimality of ${\widehat{\beta}}$ we can write $\frac{1}{2n} \|y- X{\widehat{\beta}}\|_2^2 + \lambda \|{\widehat{\beta}}\| \le \frac{1}{2n} \|y- X\beta^\star\|_2^2 + \lambda \|\beta^\star\|$. By plugging in $y = X\beta^\star + \varepsilon$ and after some algebraic calculation, we get $$\frac{1}{2n}\|X({\widehat{\beta}}-\beta^\star)\|_2^2 + \lambda \|{\widehat{\beta}}\| \le \lambda \|\beta^\star\| + \frac{1}{n}\epsilon^{{\sf T}}X ({\widehat{\beta}}- \beta^\star)\,.$$ By the choice of $\lambda$, this implies that $$\begin{aligned}
&\frac{1}{2n}\|X({\widehat{\beta}}-\beta^\star)\|_2^2 \\ &\le \frac{1}{n}\epsilon^{{\sf T}}X ({\widehat{\beta}}- \beta^\star) + \lambda \|\beta^\star\| - \lambda \|{\widehat{\beta}}\| \nonumber \\
&\le \|\frac{1}{n}\epsilon^{{\sf T}}X\|^\star \|{\widehat{\beta}}- \beta^\star\| + \lambda \|\beta^\star\| - \lambda \|{\widehat{\beta}}\| \nonumber \\
&\le \frac{\lambda}{2} \|{\widehat{\beta}}- \beta^\star\| + \lambda \|\beta^\star\| - \lambda \|{\widehat{\beta}}\|\label{eq:rearrange}\\
&\le \frac{1}{2} \left(\|{\widehat{\beta}}\| + \|\beta^\star\| \right) + \lambda \|\beta^\star\| - \lambda \|{\widehat{\beta}}\|\nonumber\\
&\le \frac{3}{2} \lambda \|\beta^\star\|\,,\nonumber
\end{aligned}$$ where we use the triangle inequality in the penultimate step. This concludes the proof.
The positive semidefiniteness assumption on $M_i$, for $i\in[p]$, makes $\beta^{{\sf T}}M_i \beta$ a convex function in $\beta$ and therefore $f$ is convex. Moreover, $f(a\beta) = a^2f(\beta)$ for any $a\in\mathbb{R}$. Therefore, [@jalali2017variational Lemma 3.5] establishes that $\sqrt{f}$ is a semi-norm.
Next, observe that $f(\beta) = \sup\{ \beta^{{\sf T}}M \beta:~ M\in{{\operatorname{conv}}}({\mathcal{M}}) \}$ as the objective is linear in $M$. Therefore, if there exists a positive definite matrix in ${{\operatorname{conv}}}({\mathcal{M}})$ then $f$ is strongly convex. Then, [@jalali2017variational Lemma 3.5] establishes that $\sqrt{f}$ is a norm.
Suppose, for each $i\in[m]$, $M_i$ is an orthogonal projector; i.e., there exists an orthonormal matrix $U_i\in\mathbb{R}^{p\times d_i}$ for some $d_i\in[p]$ where $M_i = U_iU_i^{{\sf T}}$. Then, for the compact set $A = \{ \theta:~ \langle \beta, \theta \rangle\leq \sqrt{f(\beta)} \}$ and $\sigma_A(\beta) = \sup_{\theta\in A}\langle \beta,\theta\rangle$, which denotes the support function for the set $A$, we have $$\sigma_{A}(\beta)
= \sqrt{f(\beta)}
= \max_{i\in[m]} {\|U_i^{{\sf T}}\beta\|}_2
= \max \bigl\{ \langle \beta, \theta \rangle:~ \theta = U_iw,~ w\in\mathbb{S}^{d_i-1},~i\in[m] \bigr\}
= \sigma_B(\beta)$$ where $B = \bigcup_{i\in[m]}\{U_iw:~w\in\mathbb{S}^{d_i-1},~i\in[m]\}$ is a compact set. By the above equality of support functions for the two closed sets $A$ and $B$, we have ${{\operatorname{conv}}}(A) = {{\operatorname{conv}}}(B)$. On the other hand, $B$ being a subset of $\mathbb{S}^{p-1}$ implies $B = {{\operatorname{ext}}}({{\operatorname{conv}}}(B))$. Therefore, ${{\operatorname{ext}}}({{\operatorname{conv}}}(A)) = B$. Observe that $A$ is the dual norm ball for $\sqrt{f}$. Moreover, $B = {\mathcal{S}}\cap \mathbb{S}^{p-1}$ for the given set ${\mathcal{S}}$. Piecing all these together, we establish the claim.
From the assumption, observe that $$\begin{aligned}
\lambda
\geq \frac{1}{n}{\|X^{{\sf T}}y\|}^\star
= \frac{1}{n}\sup_{\beta\neq 0} \frac{\beta^{{\sf T}}X^{{\sf T}}y}{{\|\beta\|}}
\geq \frac{1}{n}\sup_{\beta\neq 0} \frac{\beta^{{\sf T}}X^{{\sf T}}y - \frac{1}{2}{\|X\beta\|}_2^2 }{{\|\beta\|}}\end{aligned}$$ which after a rearrangement yields $$\frac{1}{2n}{\|X\beta-y\|}_2^2 +\lambda {\|\beta\|}\geq \frac{1}{2n}{\|y\|}_2^2$$ for all $\beta\neq 0$. This establishes the optimality of ${\widehat{\beta}}=0$.
We first bound $\phi_1$. Fix a subset $J\subset [p]$, with $|J|\le k$. Using the concentration bound for singular values of matrices with i.i.d. subgaussian rows (see e.g. [@vershynin2012 Equation (5.26)]), we get $$\|\frac{1}{n} X^{{\sf T}}_J X_J - \Psi_{J,J}\|\le C\sqrt{\frac{k\log p}{n}}C_{\max}\,,$$ with probability at least $1 - 2p^{-ck}$, where $c=c_\kappa$ and $C= C_\kappa$ depend on the subgaussian norm $\kappa$. By choosing $C$ large enough, we can make constant $c>0$ sufficiently large. The claim for $\phi_1$ then follows by union bounding over all subsets $J\subseteq[p]$, with $|J|\le k$.
We next bound $\phi_0$. For a random variable $Z$, denote by $\|Z\|_{\psi_1}$ and $\|Z\|_{\psi_2}$ the sub exponential and subgaussian norms of $Z$, respectively. For a random vector $Z$, these norms are defined as $\|Z\|_{\psi_1} = \sup\{\|Z^{{\sf T}}v\|_{\psi_1}:\, \|v\|_2 = 1\}$ and $\|Z\|_{\psi_2} = \sup\{\|Z^{{\sf T}}v\|_{\psi_2}:\, \|v\|_2 = 1\}$.
Fix a subset $J\subset [p]$, with $|J|\le k-d+1$ and define $Z\equiv \frac{1}{\sqrt{|J|}}(X_J{\boldsymbol{1}})$. We then have $\|Z_i\|_{\psi_2} = \frac{1}{\sqrt{|J|}} \|X_{i,J} {\boldsymbol{1}}\|_{\psi_2} \le \frac{1}{\sqrt{|J|}}\|(\Psi_{J,J})^{1/2} {\boldsymbol{1}}\|_2\times \|X_{i,J} (\Psi_{J,J})^{-1/2}\|_{\psi_2} \le \frac{\kappa}{\sqrt{|J|}} \|(\Psi_{J,J})^{1/2} {\boldsymbol{1}}\|_2$. Then, by [@vershynin2012 Lemma (5.14)] we have $\|Z_i^2\|_{\psi_1}\le 2\|Z_i\|_{\psi_2}^2\le (2\kappa^2/|J|) ({\boldsymbol{1}}^{{\sf T}}\Psi_{J,J}{\boldsymbol{1}}) \le 2\kappa^2 C_*$. We also have $$\begin{aligned}
\mathbb{E}[Z_i^2] = \frac{1}{|J|}\mathbb{E}[{\boldsymbol{1}}^{{\sf T}}X_{i,J}^{{\sf T}}X_{i,J} {\boldsymbol{1}}] = \frac{1}{|J|} ({\boldsymbol{1}}^{{\sf T}}\Psi_{J,J} {\boldsymbol{1}}) \le C_*\,.\end{aligned}$$
Employing concentration tail bound for sub-exponential random variables, see e.g. [@vershynin2012 Corollary 5.17], we obtain $$\frac{1}{n}\|Z\|_2^2 \le C_* + 2\kappa^2 C_* C \sqrt{\frac{(k-d+1) \log p}{n}}\,,$$ with probability at least $1 - 2p^{-c(k-d+1)}$, for some constant $c>0$ (depending on constants $C, C_*, \kappa>0$). By choosing $C$ large enough, we can make constant $c>0$ sufficiently large. Recall the definition of $\phi_0$, given by , specialized to i.i.d. noise entries: $$\phi_0\;\equiv \sup_{J \subseteq [p]: |J|\le k-d+1} \, \frac{\sigma^2 {\|X_{J} {\boldsymbol{1}}\|}_2^2}{n |J|}\,.$$
The claim on the $\phi_0$ bound follows by union bounding over all subsets $J\subseteq[p]$, with $|J|\le k-d+1$.
Proofs: Estimation Error {#sec:estimation}
========================
By optimality of ${\widehat{\beta}}$ we have $$\frac{1}{2n} \|X{\widehat{\beta}}-y\|_2^2+ \lambda \|{\widehat{\beta}}\| \le \frac{1}{2n} \|X\beta^\star-y\|_2^2+ \lambda \|\beta^\star\|.$$ By rearranging the terms we get $$\frac{1}{2n}\|X({\widehat{\beta}}-\beta^\star)\|_2^2 \le
\frac{1}{n} \langle X^{{\sf T}}\epsilon, {\widehat{\beta}}-\beta^\star\rangle
+ \lambda \|\beta^\star\| - \lambda \|{\widehat{\beta}}\|\,,$$ and using the choice of $\lambda$ and the Cauchy-Schwarz inequality (similar to ), we have $$\begin{aligned}
\label{eq:opt-estimation-tri}
\frac{1}{2n}\|Xv\|_2^2
\le \frac{\lambda}{2} \|v\| + \lambda \|\beta^\star\| - \lambda \|\beta^\star+v\|
\leq \frac{3}{2}\lambda {\|v\|}\,,
\end{aligned}$$ where $v = {\widehat{\beta}}-\beta^\star$ and we used the triangle inequality to get the last bound. As a consequence, $v\in {{\color{ygcolor}\Xi}}$, where ${{\color{ygcolor}\Xi}}$ is given by . Define $$\begin{aligned}
{{\color{ygcolor}{\gamma}}}({{\color{ygcolor}\Xi}}) \equiv \sup_{u\in {{\color{ygcolor}\Xi}}} \;\frac{{\|u\|}^2}{\frac{1}{n}{\|Xu\|}_2^2}.
\end{aligned}$$ Since $v\in{{\color{ygcolor}\Xi}}$, the definition of ${{\color{ygcolor}{\gamma}}}={{\color{ygcolor}{\gamma}}}({{\color{ygcolor}\Xi}})$ implies $$\begin{aligned}
\label{eq:pf-est-norm}
{\|v\|}
\leq \frac{\frac{1}{n}{\|Xv\|}_2^2}{\lambda {\|v\|}}\lambda {{\color{ygcolor}{\gamma}}}\leq 3\lambda {{\color{ygcolor}{\gamma}}}\end{aligned}$$ where the second inequality is an application of . Next, to bound ${\|v\|}_2$, recall the definition of the restricted eigenvalue constant ${{\color{ygcolor}{\alpha}}}={{\color{ygcolor}{\alpha}}}({{\color{ygcolor}\Xi}})$ from and observe that $$\begin{aligned}
\label{eq:pf-est-L2}
{{\color{ygcolor}{\alpha}}}{\|v\|}_2^2 \leq \frac{1}{n}{\|Xv\|}_2^2
\stackrel{(a)}{\leq} 3\lambda {\|v\|}
\stackrel{(b)}{\leq} 9\lambda^2 {{\color{ygcolor}{\gamma}}}\end{aligned}$$ where $(a)$ is due to and $(b)$ is by .
Next, observe that $$\begin{aligned}
{{\color{ygcolor}{\gamma}}}({{\color{ygcolor}\Xi}})
= \sup_{u\in {{\color{ygcolor}\Xi}}} \;\frac{{\|u\|}^2}{{\|u\|}_2^2} \frac{{\|u\|}_2^2}{\frac{1}{n}{\|Xu\|}_2^2}
\leq ( \sup_{u\in {{\color{ygcolor}\Xi}}} \;\frac{{\|u\|}^2}{{\|u\|}_2^2}) \cdot (\sup_{u\in {{\color{ygcolor}\Xi}}} \; \frac{{\|u\|}_2^2}{\frac{1}{n}{\|Xu\|}_2^2})
\leq \frac{{{\color{ygcolor}{\psi}}}^2({{\color{ygcolor}\Xi}})}{{{\color{ygcolor}{\alpha}}}({{\color{ygcolor}\Xi}})} \,,\end{aligned}$$ which together with and establishes the desired bounds.
By optimality of ${\widehat{\beta}}$ we have $$\frac{1}{2n} \|X{\widehat{\beta}}-y\|_2^2+ \lambda \|{\widehat{\beta}}\| \le \frac{1}{2n} \|X\beta^\star-y\|_2^2+ \lambda \|\beta^\star\|.$$ By rearranging the terms we get $$\frac{1}{2n}\|X({\widehat{\beta}}-\beta^\star)\|_2^2 \le
\frac{1}{n} \langle X^{{\sf T}}\epsilon, {\widehat{\beta}}-\beta^\star\rangle
+ \lambda \|\beta^\star\| - \lambda \|{\widehat{\beta}}\|\,,$$ and using the choice of $\lambda$ and the Cauchy-Schwarz inequality (similar to ), we have $$\begin{aligned}
\label{eq:opt-estimation}
\frac{1}{2n}\|Xv\|_2^2 \le \frac{\lambda}{2} \|v\| + \lambda \|\beta^\star\| - \lambda \|\beta^\star+v\|\,,
\end{aligned}$$ where $v = {\widehat{\beta}}-\beta^\star$. As a consequence, $v\in {{\color{ygcolor}\Xi}}$, where ${{\color{ygcolor}\Xi}}$ is given by . From the definition of the restricted norm compatibility constant in , we get ${\|v\|} \leq {{\color{ygcolor}{\psi}}}{\|v\|}_2$ for any $v\in {{\color{ygcolor}\Xi}}$ and for ${{\color{ygcolor}{\psi}}}= {{\color{ygcolor}{\psi}}}(\beta^\star;{{\|\cdot\|}})$. Therefore, by RE condition on ${{\color{ygcolor}\Xi}}$ for ${\widehat{\Sigma}}$ we obtain $$\begin{aligned}
\label{eq:dum-ap}
\frac{1}{n}\|Xv\|_2^2 \ge {{\color{ygcolor}{\alpha}}}\|v\|_2^2 \ge \frac{{{\color{ygcolor}{\alpha}}}}{{{\color{ygcolor}{\psi}}}^2} \|v\|^2\,.
\end{aligned}$$ In addition, by using triangle inequality in , we have $1/(2n)\|Xv\|_2^2\le (3/2) \lambda \|v\|$ and so $$\begin{aligned}
\label{eq:opt-estimation2}
\frac{1}{n}\|Xv\|_2^2 + \lambda \|v\| &\le 4\lambda \|v\|
\stackrel{(a)}{\le} \frac{4\lambda{{\color{ygcolor}{\psi}}}}{\sqrt{n{{\color{ygcolor}{\alpha}}}}} \|Xv\|_2
\stackrel{(b)}{\le} \frac{1}{2n} \|Xv\|_2^2 + \frac{8}{{{\color{ygcolor}{\alpha}}}}\lambda^2{{\color{ygcolor}{\psi}}}^2
\,.
\end{aligned}$$ where $(a)$ is by and $(b)$ holds true because $(\frac{1}{\sqrt{n}}{\|Xv\|}_2 - \frac{4\lambda {{\color{ygcolor}{\psi}}}}{\sqrt{{{\color{ygcolor}{\alpha}}}}} )^2\geq 0$. Therefore, $$\begin{aligned}
\label{eq:opt-estimation3}
\frac{1}{2n}\|Xv\|_2^2 + \lambda \|v\|\le \frac{8}{{{\color{ygcolor}{\alpha}}}}\lambda^2{{\color{ygcolor}{\psi}}}^2\,.
\end{aligned}$$ This implies $\|v\|\le 8\lambda{{\color{ygcolor}{\psi}}}^2/{{\color{ygcolor}{\alpha}}}$, which proves claim .
To prove claim , we again apply the RE condition to and write $${{\color{ygcolor}{\alpha}}}\|v\|_2^2\le \frac{1}{n} \|Xv\|_2^2\le \frac{1}{n} \|Xv\|_2^2 + 2\lambda\|v\| \le \frac{16}{{{\color{ygcolor}{\alpha}}}} \lambda^2 {{\color{ygcolor}{\psi}}}^2\,,$$ which gives the desired result.
For any ${{\color{ygcolor}q}}>1$, consider $$\begin{aligned}
\label{eq:ErrSet-q}
{{\color{ygcolor}\Xi}}^{({{\color{ygcolor}q}})}({\beta^\star;{{\|\cdot\|}}}) &\equiv \bigl\{v :~ \frac{1}{{{\color{ygcolor}q}}}\|v\| + \|\beta^\star\| \ge \|\beta^\star+v\| \bigr\},\end{aligned}$$ which for ${{\color{ygcolor}q}}=2$ yields ${{\color{ygcolor}\Xi}}^{(2)} = {{\color{ygcolor}\Xi}}$ defined in . Note that ${{\color{ygcolor}\Xi}}^{({{\color{ygcolor}q}})}$ is the whole space for $0<{{\color{ygcolor}q}}\leq 1$ which is not of interest in our discussion.
- An easy adaptation of yields $${{\color{ygcolor}\Xi}}^{({{\color{ygcolor}q}})} \subseteq {\mathcal{C}}^{({{\color{ygcolor}q}})} \equiv \bigl\{v:~ {\|v\|} \leq \frac{{{\color{ygcolor}q}}}{{{\color{ygcolor}q}}-1}\cdot \varphi({\beta;{{\|\cdot\|}}}) \cdot {\|v\|}_2\bigr\}$$ and implies ${{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}^{({{\color{ygcolor}q}})}) \leq \frac{{{\color{ygcolor}q}}}{{{\color{ygcolor}q}}-1}\varphi({\beta;{{\|\cdot\|}}})$; for any $q>1$.
- If $\lambda \geq {{\color{ygcolor}\widetilde{q}}}{\|\frac{1}{n}X^{{\sf T}}\epsilon\|}^\star$ for some ${{\color{ygcolor}\widetilde{q}}}>1$, then the prediction error bound of reads as $\frac{1}{n}{\|X(\beta^\star - {\widehat{\beta}})\|}_2^2 \leq 2(1+\frac{1}{{{\color{ygcolor}\widetilde{q}}}})\lambda {\|\beta^\star\|}$, and the estimation error bounds of read as: reads as ${\|{\widehat{\beta}}-\beta^\star\|} \leq 2(1+\frac{1}{{{\color{ygcolor}\widetilde{q}}}})\lambda {{\color{ygcolor}{\psi}}}^2/{{\color{ygcolor}{\alpha}}}$ and reads as ${\|{\widehat{\beta}}-\beta^\star\|}_2 \leq 2(1+\frac{1}{{{\color{ygcolor}\widetilde{q}}}})\lambda {{\color{ygcolor}{\psi}}}/{{\color{ygcolor}{\alpha}}}$ where ${{\color{ygcolor}{\psi}}}= {{\color{ygcolor}{\psi}}}({{\color{ygcolor}\Xi}}^{({{\color{ygcolor}\widetilde{q}}})})$ and ${{\color{ygcolor}{\alpha}}}= {{\color{ygcolor}{\alpha}}}({{\color{ygcolor}\Xi}}^{({{\color{ygcolor}\widetilde{q}}})})$.
- Combining the above two items, for ${{\color{ygcolor}q}}={{\color{ygcolor}\widetilde{q}}}$, we get $$\begin{aligned}
\frac{1}{n}{\|X(\beta^\star - {\widehat{\beta}})\|}_2^2
&\leq 2(1+\frac{1}{{{\color{ygcolor}\widetilde{q}}}})\lambda {\|\beta^\star\|}
\\
{\|{\widehat{\beta}}-\beta^\star\|}
&\leq 2(1+\frac{1}{{{\color{ygcolor}\widetilde{q}}}})\frac{\lambda}{{{\color{ygcolor}{\alpha}}}} (\frac{{{\color{ygcolor}\widetilde{q}}}}{{{\color{ygcolor}\widetilde{q}}}-1})^2 \varphi^2
= (\frac{2\varphi^2}{{{\color{ygcolor}{\alpha}}}})\cdot \frac{{{\color{ygcolor}\widetilde{q}}}({{\color{ygcolor}\widetilde{q}}}+1)}{({{\color{ygcolor}\widetilde{q}}}-1)^2} \lambda
\\
{\|{\widehat{\beta}}-\beta^\star\|}_2
&\leq 2(1+\frac{1}{{{\color{ygcolor}\widetilde{q}}}})\frac{\lambda}{{{\color{ygcolor}{\alpha}}}}\frac{{{\color{ygcolor}\widetilde{q}}}}{{{\color{ygcolor}\widetilde{q}}}-1}\varphi
= (\frac{2\varphi}{{{\color{ygcolor}{\alpha}}}})\cdot \frac{{{\color{ygcolor}\widetilde{q}}}+1}{{{\color{ygcolor}\widetilde{q}}}-1} \lambda\end{aligned}$$ where we now use ${{\color{ygcolor}{\alpha}}}= {{\color{ygcolor}{\alpha}}}({\mathcal{C}}^{({{\color{ygcolor}\widetilde{q}}})}) \geq {{\color{ygcolor}{\alpha}}}({{\color{ygcolor}\Xi}}^{({{\color{ygcolor}\widetilde{q}}})})$.
- Observe that we can use any ${{\color{ygcolor}\widetilde{q}}}\in (1, \frac{\lambda}{\theta}]$ we wish in our analysis. Define $\theta = {\|\frac{1}{n}X^{{\sf T}}\epsilon\|}^\star$. It is easy to see that among all ${{\color{ygcolor}\widetilde{q}}}\in (1, \frac{\lambda}{\theta}]$, largest ${{\color{ygcolor}\widetilde{q}}}$ minimizes all three bounds (ignoring the dependence of ${{\color{ygcolor}{\alpha}}}= {{\color{ygcolor}{\alpha}}}({\mathcal{C}}^{({{\color{ygcolor}\widetilde{q}}})})$ on ${{\color{ygcolor}\widetilde{q}}}$) and ${{\color{ygcolor}{\alpha}}}({\mathcal{C}}^{(\lambda/\theta)})\geq {{\color{ygcolor}{\alpha}}}({\mathcal{C}}^{({{\color{ygcolor}\widetilde{q}}})})$ for any ${{\color{ygcolor}\widetilde{q}}}\in (1, \frac{\lambda}{\theta}]$. Therefore, plugging ${{\color{ygcolor}\widetilde{q}}}=\frac{\lambda}{\theta}$ we get $$\begin{aligned}
\frac{1}{n}{\|X(\beta^\star - {\widehat{\beta}})\|}_2^2
&\leq 2(\lambda+\theta) {\|\beta^\star\|}
\\
{\|{\widehat{\beta}}-\beta^\star\|}
&\leq (\frac{2\varphi^2}{{{\color{ygcolor}{\alpha}}}})\cdot \frac{\lambda^2(\lambda+\theta)}{(\lambda-\theta)^2}
\\
{\|{\widehat{\beta}}-\beta^\star\|}_2
&\leq (\frac{2\varphi}{{{\color{ygcolor}{\alpha}}}})\cdot \frac{\lambda(\lambda+\theta)}{\lambda-\theta} \end{aligned}$$ for any $\lambda > \theta$ used in , where ${{\color{ygcolor}{\alpha}}}= {{\color{ygcolor}{\alpha}}}({\mathcal{C}}^{(\lambda/\theta)})$.
- an adaptation of yields ${{\color{ygcolor}{\alpha}}}= {{\color{ygcolor}{\alpha}}}({\mathcal{C}}^{(\lambda/\theta)}) = \lambda_{\min}/2$ for $$n \geq (C^2 k \log p) \cdot (\lambda_{\min}^{-1}\varphi^2 \cdot 6 (\frac{{{\color{ygcolor}\widetilde{q}}}}{{{\color{ygcolor}\widetilde{q}}}-1})^2 )^2
= (36C^2 k \log p) (\lambda_{\min}^{-2} \varphi^4) (\frac{\lambda}{\lambda-\theta})^4.$$
Proofs: RE for Subgaussian Designs {#app:proof-RE}
==================================
By triangle inequality we have $$v^{{\sf T}}{\widehat{\Sigma}}v \ge v^{{\sf T}}\Sigma v - |v^{{\sf T}}(\Sigma - {\widehat{\Sigma}}) v| \ge \lambda_{\min} \|v\|_2^2 - |v^{{\sf T}}(\Sigma - {\widehat{\Sigma}}) v|\,.$$
Let $\Gamma\equiv \Sigma - {\widehat{\Sigma}}$. Using the above inequality, it suffices to show that $$\begin{aligned}
\label{claim0}
|v^{{\sf T}}\Gamma v| \le \frac{1}{2}\lambda_{\min}\|v\|_2^2\,, \quad \text{for all } v\in {\mathcal{C}}(\varphi)\,.
\end{aligned}$$
\[lem:quad-upper\] Consider a closed scale-invariant set ${\mathcal{S}}$ that span $\mathbb{R}^p$ as well as the corresponding structure norm ${{\|\cdot\|}}_{\mathcal{S}}$. For a given matrix $\Gamma\in\mathbb{R}^{p\times p}$, and a given value $\delta>0$, suppose the following holds for all $v\in ({\mathcal{S}}\oplus {\mathcal{S}})$, $$\begin{aligned}
\label{eq:quad-upper-v-2s}
{|v^{{\sf T}}\Gamma v|} \leq \delta \|v\|_2^2.
\end{aligned}$$ Then, we have the following for all $v\in\mathbb{R}^p$, $$\begin{aligned}
{|v^{{\sf T}}\Gamma v|} \leq 3\delta {\|v\|}_{\mathcal{S}}^2.
\end{aligned}$$
We follow a similar approach to proof of Lemma 12 in [@loh2012].
By definition of ${{\|\cdot\|}}_{\mathcal{S}}$, there exists a set of $\alpha_i>0$ and $v_i\in {\mathcal{S}}$ for which ${v}=\sum_i \alpha_i v_i$, $\sum_i \alpha_i=1$ and ${\|v_i\|}_2 \leq \|{v}\|$. Observe that, $$v^{{\sf T}}\Gamma v = \sum_{i,j} \alpha_i \alpha_j v_i^{{\sf T}}\Gamma v_j.$$ Moreover, each of $v_i$, $v_j$, and $(v_i+v_j)/2$, are in $ ({\mathcal{S}}\oplus {\mathcal{S}})$. Therefore, using we get $$\begin{aligned}
{|v_i^{{\sf T}}\Gamma v_j|} &\leq \frac{1}{2}{|(v_i+v_j)^{{\sf T}}\Gamma (v_i+v_j)|} + \frac{1}{2}{|v_i^{{\sf T}}\Gamma v_i|} + \frac{1}{2} {|v_j^{{\sf T}}\Gamma v_j|}\\
&\leq \frac{\delta}{2} \|v_i+v_j\|_2^2 + \frac{\delta}{2} \|v_i\|_2^2 + \frac{\delta}{2} \|v_j\|_2^2\\
&\le 3 \delta\|{v}\|^2 .
\end{aligned}$$ Since $\alpha_i$’s define a convex combination, we get ${|{v}^{{\sf T}}\Gamma {v}|} \leq 3\delta\|{v}\|^2$.
By virtue of and definition of ${\mathcal{C}}(\varphi)$, in order to prove Claim it suffices to show that holds for $\delta = {\lambda_{\min}}/{(24\varphi^2)}$.
\[lem:concentration\] Under the assumptions of , for any constants $c_0, c_1>0$, there exists $C = C(\lambda_{\max},\lambda_{\min},\kappa,c_0, c_1)$ such that $$\begin{aligned}
\max\left\{\|({\widehat{\Sigma}}- \Sigma)_{A,A}\|_2:\,\; A\subseteq[p],\, |A| \le c_0 k \right\} \le C\sqrt{\frac{k\log p}{n}}\,,
\end{aligned}$$ with probability at least $1-2p^{-c_1k}$.
Fix an arbitrary $v\in {\mathcal{S}}\oplus{\mathcal{S}}$. Then, by our assumption that ${\mathcal{S}}\subseteq\{\beta:~{{\operatorname{card}}}(\beta)\leq k\}$, $v$ is $2k$ sparse. Denote by $A$ the support of $v$. Then, by employing with $c_0 =2$, we have $$\begin{aligned}
|v^{{\sf T}}\Gamma v| = |v_A^{{\sf T}}\Gamma_{A,A} v_{A}|\le \|\Gamma_{A,A}\|_2 \|v_A\|_2^2
\le C\sqrt{\frac{k\log p}{n}} \|v\|_2^2 \,,
\end{aligned}$$ with probability at least $1-2p^{-c_1k}$. Hence, for $n\ge (24C/\lambda_{\min})^2 \varphi^4 k\log p$, we obtain for $\delta = {\lambda_{\min}}/{(24\varphi^2)}$. This completes the proof.
Fix $A\subseteq[p]$, with $|A|\le c_0 k$ and let $X_A\in {\mathbb{R}}^{n\times |A|}$ be the sub-matrix containing columns of $X$ that are in the set $A$. We can write ${\widehat{\Sigma}}_{A,A} = (X_A^{{\sf T}}X_A)/n$ and $\Sigma_{A,A} = \mathbb{E}(X_A^{{\sf T}}X_A)$. By employing the result of Remark 5.40 in [@vershynin2012], for every $t\ge 0$, with probability at least $1-2 e^{-ct^2}$ the following holds: $$\|{\widehat{\Sigma}}_{A,A}-\Sigma_{A,A}\|_{2} \le \max\{\delta , \delta^2\}\,, \quad \text{ where } \delta = C\sqrt{\frac{k}{n}} + \frac{t}{\sqrt{n}}\,,$$ where $C=C(\kappa, c_0)$ and $c= c({\kappa}) > 0$ depend only on the subgaussian norms of the rows of $X$ and constant $c_0$. Choosing $t = \sqrt{\tilde{c} k \log p}$, and using , we get that with probability at least $1 - 2p^{c\tilde{c}k}$, $$\|{\widehat{\Sigma}}_{A,A}-\Sigma_{A,A}\|_2\le (C+\sqrt{\tilde{c}}) \sqrt{\frac{k\log p}{n}}\,.$$ We next define ${\mathcal{F}}\equiv \{A\subseteq[p]: \, |A|\le c_0 k\}$. Note that $|{\mathcal{F}}|\le p^{c_0 k}$. The proof is completed by taking union bound over all sets in ${\mathcal{F}}$ and choosing $\tilde{c} = (c_0+c_1)/c$.
Computing $\varphi$ for Different Families of Norms {#app:varphi}
===================================================
In each section below, we provide a characterization for the subdifferential and for the dual norm, and compute or upper bound $\varphi$.
Auxiliary Lemmas
----------------
\[lem:dist-ei-simplex\] For ${\mathbf{\Delta}}_p = \{u\geq {\boldsymbol{0}}_p:~ {\boldsymbol{1}}^{{\sf T}}u = 1\}$ we have ${{\operatorname{dist}}}^2(-e_i, {\mathbf{\Delta}}_p ) = 1+1/\max\{p-1,1/3\}$.
The case of $p=1$ is easy to verify; hence assume $p\geq 2$. Since the projection is unique, we provide a candidate and verify its optimality. In fact, we claim that ${\Pi}(-e_i;{\mathbf{\Delta}}_p) = b = \frac{1}{p-1}({\boldsymbol{1}}_p - e_i)$. By Kolmogorov criteria, for this to be the projection, we need ${\langle}-e_i - b, u-b{\rangle}\leq 0$ for any $u\in {\mathbf{\Delta}}_p$; which can be easily verified. This establishes the claim.
\[lem:dist-weighted-simplex\] For $w\in\mathbb{R}_{++}^p$, $A = \{u\geq 0:~ \langle w,u\rangle = 1\}$, and any $i\in[p]$, we have $${{\operatorname{dist}}}^2( -\frac{1}{w_i}e_i , A ) =
\begin{cases}
\frac{4}{w_i^2}& p=1, \\
\frac{1}{w_i^2} + \frac{1}{{\|w\|}_2^2 - w_i^2}& p > 1,~ 2w_i^2 \leq {\|w\|}_2^2, \\
\frac{4}{{\|w\|}_2^2}& p > 1,~ 2w_i^2 \geq {\|w\|}_2^2, \\
\end{cases}$$ where $e_i$ is the $i$-th standard basis vector.
The case of $p=1$ is easy to verify; hence assume $p\geq 2$. Since the projection is unique, we provide a candidate and verify its optimality. There are two cases (illustrated in ):
- If $2w_i^2 \leq {\|w\|}_2^2$ then we claim that ${\Pi}(-\frac{1}{w_i}e_i;A) = b = \frac{1}{{\|w\|}_2^2 - w_i^2} (w - w_ie_i)$. To prove the claim, consider any $u\in A$ and observe that, $$\langle -\frac{1}{w_i}e_i - b, u-b \rangle
= {\|b\|}_2^2 - \langle b,u\rangle - \frac{u_i}{w_i}
= \frac{1}{{\|w\|}_2^2 - w_i^2} - \frac{1-w_iu_i}{{\|w\|}_2^2 - w_i^2} - \frac{u_i}{w_i}
= \frac{u_i}{w_i} ( \frac{2w_i^2 - {\|w\|}_2^2}{{\|w\|}_2^2 - w_i^2} ) \leq 0$$ which establishes the claim by Kolmogorov criteria.
- If $2w_i^2 \geq {\|w\|}_2^2$ then we claim that ${\Pi}(-\frac{1}{w_i}e_i;A) = b = \frac{2}{{\|w\|}_2^2}w - \frac{1}{w_i}e_i \geq 0$ with ${\|b\|}_2^2 = \frac{1}{w_i^2}$. To prove the claim, consider any $u\in A$ and observe that, $$\begin{aligned}
\langle -\frac{1}{w_i}e_i - b, u-b \rangle
= {\|b\|}_2^2 - \langle b,u\rangle - \frac{u_i}{w_i} + \frac{2}{{\|w\|}_2^2} - \frac{1}{w_i^2} = 0 \end{aligned}$$ which establishes the claim by Kolmogorov criteria. Note that the condition was used to make sure $b\in A$.
With the projection at hand, calculating the distances is straightforward.
(-3,0)–(3,0); (0,-0.5)–(0,3); (2.5,0)–(0,1); (-2.5,0) circle \[radius=.06\]; (-2.5,0)–(0,1) node\[pos=1.1\] [$b$]{};
(-3,0)–(3,0); (0,-0.5)–(0,3); (0,2.5)–(1,0); (-1,0) circle \[radius=.06\]; (-1,0)–($(1,0)!(-1,0)!(0,2.5)$) node\[pos=1.1\] [$b$]{};
\[lem:min-norm-simplex\] For a given $w\in\mathbb{R}_{++}^p$, we have $\min\{{\|u\|}_2^2:~ \langle w,u\rangle=1,~ u\geq 0\} = \frac{1}{{\|w\|}_2^2}$ and ${\mathop{{\operatorname{argmin}}}}\{{\|u\|}_2^2:~ \langle w,u\rangle=1,~ u\geq 0\} = \frac{1}{{\|w\|}_2^2}w$.
Writing down the Lagrange dual of this optimization problem we get the desired result.
\[lem:ext-inf-conv\] Consider two atomic norms and their infimal convolution. The extreme points of the ball for infimal convolution is a subset of the union of extreme points for each norm ball.
Easy from .
Weighted $\ell_1$ and $\ell_\infty$ Norms
-----------------------------------------
Given a positive vector $w\in\mathbb{R}_{++}^p$, one can define a pair of dual norms as $$\sum_{i=1}^p w_i{|\beta_i|} \qquad \text{and} \qquad \max_{i\in[p]} \frac{1}{w_i}{|\beta_i|}$$ which are commonly referred to as the [*weighted $\ell_1$ norm*]{} and the [*weighted $\ell_\infty$ norm*]{}, respectively.
\[lem:varphi-weighted-l1-exact\] Consider the weighted $\ell_1$ norm $f(\beta) = \sum_{i=1}^p w_i{|\beta_i|}$ where $w\in\mathbb{R}_{++}^p$. Then, $$\varphi^2(\beta; f) = 4 {\|w_{{{\operatorname{Supp}}}(\beta)}\|}_2^2$$
Define $S = \{i\in[p]:~ \beta_i\neq 0\}$ and observe that $$\begin{aligned}
\partial f(\beta)
&= \{g:~ \langle g, \beta\rangle = f(\beta),~ \max_{i\in [p]}\frac{1}{w_i}{|\beta_i|}<1\}\\
&= \{g:~ g_i=w_i {{\operatorname{sign}}}(\beta_i) \text{ if } \beta_i\neq 0,~ {|g_i|}\leq w_i \text{ otherwise}\}\end{aligned}$$ where we used the form of dual norm in the first equality. Then, implies $$\begin{aligned}
\varphi^2(\beta; f)
&= \max_{z} \min_{g} \;\bigl\{{\|z-g\|}_2^2 :~ {|z_i|}\leq w_i ~~i\in[p],~ g_i = w_i {{\operatorname{sign}}}(\beta_i)~~i\in S,~ {|g_i|}\leq w_i ~~ i\in S^c \bigr\}\\
&= \max_{z}\; \bigl\{ \sum_{i\in S}(z_i-w_i {{\operatorname{sign}}}(\beta_i))_2^2 + \sum_{i\in S^c} ({|z_i|} -w_i)_+^2:~ {|z_i|}\leq w_i ~~i\in[p]\bigr\}\\
&= 4{\|w_S\|}_2^2\end{aligned}$$ where $(a)_+ \equiv \max\{a,0\}$.
recovers the earlier result $\varphi(\beta; {{\|\cdot\|}}_1) = 2\sqrt{{\|\beta\|}_0}$.
\[lem:subdiff-weighted-linf\] For the weighted $\ell_\infty$ norm, namely $f(\beta) = \max_{i\in[p]} \frac{1}{w_i}{|\beta_i|}$ with $w\in\mathbb{R}_{++}^p$, and $\beta\neq 0$, we have $$\begin{aligned}
\partial f(\beta)
&= \{g:~ g\circ \beta \geq 0,~ g_{T^c}=0,~ \sum_{i\in T} w_i {|g_i|} = 1\} \end{aligned}$$ where $T\equiv \{i\in[p]:~ \frac{1}{w_i}{|\beta_i|} = f(\beta)\}\neq \emptyset$. Moreover, $\min_{g\in\partial f(\beta)}{\|g\|}_2^2 = \frac{1}{{\|w_T\|}_2^2}$.
For $\beta\neq 0$, consider $T$ and observe that $$\begin{aligned}
\partial f(\beta)
&= \bigl\{g:~ \langle g,\beta\rangle = \max_{i\in[p]}\frac{1}{w_i}{|\beta_i|},~ \sum_{i=1}^p w_i{|g_i|} = 1 \bigr\}.\end{aligned}$$ For any $g$ in the above, we have $$\langle g,\beta\rangle
= \sum_{i\in S} g_i\beta_i
= \sum_{i\in S} (w_ig_i)(\frac{1}{w_i} \beta_i)
\leq \sum_{i\in S} (w_i{|g_i|})(\frac{1}{w_i} {|\beta_i|})
\leq (\max_{i\in[p]} \frac{1}{w_i} {|\beta_i|}) \sum_{i\in S}w_i{|g_i|}
= \max_{i\in[p]} \frac{1}{w_i} {|\beta_i|}$$ which implies that the inequalities have to hold with equality, establishing $g_i = 0$ for $i\not\in T$, $g\circ \beta \geq 0$, as well as $\sum_{i\in T}w_i{|g_i|} = 1$. This completes the characterization of the subdifferential (checking that each such $g$ is a subgradient is straightforward).
The last statement follows from .
\[lem:varphi-weighted-linf-exact\] For the weighted $\ell_\infty$ norm, namely $f(\beta) = \max_{i\in[p]} \frac{1}{w_i}{|\beta_i|}$ with $w\in\mathbb{R}_{++}^p$, and $\beta\neq 0$, we have $$\varphi^2(\beta; f) =
\begin{cases}
\max\left\{
\frac{1}{\omega^2}+\frac{1}{{\|w_T\|}_2^2}~,~
\frac{4}{{\|w_T\|}_2^2}
\right\}
&\text {if } {|T|}=1
\\
\max\left\{
\frac{1}{\omega^2}+\frac{1}{{\|w_T\|}_2^2}~,~
\frac{1}{\tau^2}+\frac{1}{{\|w_T\|}_2^2-\tau^2}
\right\}
& \text{if } {|T|}\geq 2
\end{cases}$$ where $T\equiv \{i\in[p]:~ \frac{1}{w_i}{|\beta_i|} = f(\beta)\}\neq \emptyset$, $T_1\equiv \{i\in T:~ 2w_i^2 \leq {\|w_T\|}_2^2\}$, $\omega \equiv \min_{i\not\in T} w_i$, and $\tau \equiv \min_{i\in T_1} w_i$. Note that ${|T|}\geq 2$ if and only if $T_1$ is non-empty. Moreover, if ${|T|}\geq 2$ then $2\tau^2\leq {\|w_T\|}_2^2$ which implies $\varphi^2 \leq \max\{\frac{1}{\omega^2},\frac{1}{\tau^2}\}+\frac{1}{{\|w_T\|}_2^2-\tau^2} \leq \max\{\frac{1}{\omega^2},\frac{1}{\tau^2}\}+\frac{1}{\tau^2} \leq 2\max\{\frac{1}{\omega^2},\frac{1}{\tau^2}\}\leq 2/(\min_{i\in[p]}w_i)^2$.
Consider the characterization of $\partial f(\beta)$, for any $\beta\neq 0$, from . Moreover, note that ${{\operatorname{ext}}}({\mathcal{B}}^\star) = \{\pm \frac{1}{w_i}e_i:~ i\in[p]\}$ where $e_i$ is the $i$-th standard basis vector. There are three cases:
- If $i\not\in T$ and $z = \pm \frac{1}{w_i}e_i$, then $z_T =0$ and ${\|z_{T^c}\|}_2 = \frac{1}{w_i}$. Therefore, $$\begin{aligned}
\min_{g\in\partial f(\beta)} {\|z-g\|}_2^2
= \min_{g\in\partial f(\beta)} {\|g_T\|}_2^2 + {\|z_{T^c}\|}_2^2
= \frac{1}{{\|w_T\|}_2^2} + \frac{1}{w_i^2},\end{aligned}$$ where we used .
- If $i\in T$ and $z = \frac{1}{w_i}e_i$ then $z\in \partial f(\beta)$ which implies ${{\operatorname{dist}}}(z,\partial f(\beta)) = 0$.
- If $i\in T$ and $z = -\frac{1}{w_i}e_i$ then we use to get $${{\operatorname{dist}}}^2(-\frac{1}{w_i}e_i, \partial f(\beta)) =
\begin{cases}
\frac{4}{w_i^2}& {|T|}=1, \\
\frac{1}{w_i^2} + \frac{1}{{\|w_T\|}_2^2 - w_i^2}& {|T|} \geq 2,~ i\in T_1, \\
\frac{4}{{\|w_T\|}_2^2}& {|T|} \geq 2,~ i\in T\backslash T_1. \\
\end{cases}$$ Observe that for any $i\in T$, we have $\frac{1}{w_i^2} + \frac{1}{{\|w_T\|}_2^2 - w_i^2} > \frac{4}{w_i^2}$.
Combining all of the above, to find the maximum over all $z\in{{\operatorname{ext}}}({\mathcal{B}}^\star)$, we get $$\varphi^2(\beta; f) =
\begin{cases}
\max\left\{
\max_{i\not\in T} (\frac{1}{w_i^2}+\frac{1}{{\|w_T\|}_2^2})~,~
\frac{4}{{\|w_T\|}_2^2}
\right\}
&\text {if } {|T|}=1
\\
\max\left\{
\max_{i\not\in T} (\frac{1}{w_i^2}+\frac{1}{{\|w_T\|}_2^2})~,~
\max_{i\in T_1} (\frac{1}{w_i^2}+\frac{1}{{\|w_T\|}_2^2-w_i^2})
\right\}
& \text{if } {|T|}\geq 2
\end{cases}$$ The claim follows by defining $\omega$ and $\tau$.
provides an alternative proof for .
Bounds for the Ordered Weighted $\ell_1$ Norm {#app:OWL}
---------------------------------------------
Given $w_1\geq w_2 \geq \cdots \geq w_p \geq 0$, the ordered weighted $\ell_1$ norm is defined as $$\begin{aligned}
\label{eq:def-OWL}
{\|\beta\|}_{w}= \sum_{i=1}^p w_i \bar\beta_i\end{aligned}$$ where $\bar\beta$ is the sorted absolute value of $\beta$ satisfying $\bar\beta_1\geq \bar\beta_2 \geq \cdots \geq \bar\beta_p \geq 0$. The above is clearly 1-homogeneous. It is also convex due to the assumption on $w$.
The dual norm for ${{\|\cdot\|}}_{w}$ is given by $$\begin{aligned}
\label{eq:OWL-dual}
{\|z\|}_{w}^\star = \max_{i\in [p]} ~\frac{\sum_{j=1}^i \bar z_j}{\sum_{j=1}^i w_j}.\end{aligned}$$
In the following, we present results on these norms, which to the best of our knowledge, are new.
Using the characterization of the norm ball for ${{\|\cdot\|}}_{w}$ in [@zeng2014ordered Theorem 1], it is easy to see that ${{\|\cdot\|}}_{w}$ is a structure norm (all of the extreme points lie on the unit sphere) if and only if $w_i = \sqrt{i}-\sqrt{i-1}$ for $i\in[p]$. Observe that such $w$ satisfies $w_1\geq \cdots \geq w_p>0$ which is required in defining ${{\|\cdot\|}}_{w}$. In the approach of [@obozinski2016unified], for such norm ${{\|\cdot\|}}_{w}$ we have ${\|\beta\|}_{w}^\star = \max\{ \frac{1}{\sqrt{{|A|}}} {\|\beta_A\|}_1:~ A\subseteq [p]\}$.
\[lem:extBst-owl\] Consider $w\in\mathbb{R}^p$ with $w_1\geq w_2\geq \cdots \geq w_p> 0$ and ${\mathcal{B}}_{{{\|\cdot\|}}_{w}}^\star ={\mathcal{B}}_{{{\|\cdot\|}}_{w}^\star} = \{z:~{\|z\|}_{w}^\star \leq 1\}$. Then, ${{\operatorname{ext}}}({\mathcal{B}}_{{{\|\cdot\|}}_{w}^\star}) = \{Qw:~ Q\in \mathcal{P}_\pm \}$. This implies that ${\|w\|}_2{{\|\cdot\|}}_{w}^\star$ is a structure norm in the sense of .
First, the support function for the right-hand side is equal to ${{\|\cdot\|}}_{w}$. Therefore, the convex hull of the right-hand side is ${\mathcal{B}}^\star$. On the other hand, without loss of generality consider $w=I\cdot w$ and assume $w=\alpha x+ (1-\alpha) y$ for some $\alpha\in[0,1]$ and $x,y\in{\mathcal{B}}^\star$. Then, for any $i\in[p]$, $$1
= \frac{\sum_{j=1}^i w_j}{\sum_{j=1}^i w_j}
= \frac{\alpha \sum_{j=1}^i x_j + (1-\alpha) \sum_{j=1}^i y_j}{\sum_{j=1}^i w_j}
\leq \frac{\alpha \sum_{j=1}^i \bar x_j + (1-\alpha) \sum_{j=1}^i \bar y_j }{\sum_{j=1}^i w_j}
\leq 1$$ which implies that $x=y=w$. Therefore, $w$ is an extreme point of the dual norm ball.
Given $\beta$, sort ${|\beta|}$ in descending order to get $\bar\beta$. Moreover, consider $d = {|\{ {|\beta_i|}\neq 0:~ i\in[p]\}|}$. Then, define ${\mathcal{G}}= ({\mathcal{G}}_1, \cdots, {\mathcal{G}}_d)$ as a partition of ${{\operatorname{Supp}}}(\bar\beta)$ into $d$ [*intervals*]{} where for any $i,j\in{{\operatorname{Supp}}}(\bar\beta)$ and any $t\in[d]$: $i,j\in {\mathcal{G}}_t$ if and only if $\bar\beta_i = \bar\beta_j$. Moreover, define ${\mathcal{G}}_0 \equiv [p]\backslash {{\operatorname{Supp}}}(\bar\beta)$.
\[lem:OWL-subdiff\] Given $w_1\geq w_2 \geq \cdots \geq w_p \geq 0$, the ordered weighted $\ell_1$ norm defined by . Then, the subdifferential at $\beta\in\mathbb{R}^p$ is given by $$\begin{aligned}
\label{eq:OWL-subdiff}
\partial {\|\beta\|}_{w}=
\Bigl\{ g:~
&g\circ \beta \geq 0,~
\text{${|g|}$ and ${|\beta|}$ are similarly sorted},~
\sum_{j=1}^i \bar g_j \leq \sum_{j=1}^i w_j ~~ \forall i\in [p],
\nonumber\\
&\sum_{j\in {\mathcal{G}}_t} \bar g_j = \sum_{j\in {\mathcal{G}}_t} w_j ~~ \forall\, t\in[d],~
\sum_{j\in {\mathcal{G}}_0} \bar g_j \leq \sum_{j\in {\mathcal{G}}_0} w_j
\Bigr\}. \end{aligned}$$
Consider from [@watson1992characterization] the characterization of the subdifferential for a norm as $$\partial {\|\beta\|}_{w}= \{g:~ \langle g,\beta\rangle = {\|\beta\|}_{w},~ {\|g\|}_{w}^\star =1 \}.$$ For any $g\in \partial {\|\beta\|}_{w}$, we have $$\begin{aligned}
\label{eq:owl-dummy1}
{\langle}w, \bar\beta{\rangle}= {\|\beta\|}_{w}= {\langle}g, \beta{\rangle}\leq {\langle}\bar{g}, \bar\beta{\rangle}\end{aligned}$$ where the last inequality holds by the rearrangement inequality. Therefore, with the convention $\bar\beta_{p+1}=0$ we have, $$\begin{aligned}
\sum_{i=1}^p w_i \bar\beta_i
&\stackrel{(a)}{=}
\sum_{i=1}^p \left( (\bar\beta_i-\bar\beta_{i+1}) \sum_{j=1}^i w_j \right) \\
&\stackrel{(b)}{\geq}
\sum_{i=1}^p \left( (\bar\beta_i-\bar\beta_{i+1}) \sum_{j=1}^i \bar g_j \right) \\
&\stackrel{(a)}{=}
\sum_{i=1}^p \bar g_i \bar\beta_i\\
&\stackrel{(c)}{\geq}
\sum_{i=1}^p w_i \bar\beta_i\end{aligned}$$ where $(a)$ is a trick we use, $(b)$ is by ${\|g\|}_{w}^\star \leq 1$ and , and $(c)$ is by . Therefore, all of the inequalities we have used must hold with equality: From we get that $g$ and $\beta$ are similarly signed, and, ${|g|}$ and ${|\beta|}$ are similarly sorted. Moreover, equality in $(b)$ implies $$\begin{aligned}
\label{eq:OWL-dummy2}
\sum_{j=1}^i w_j=\sum_{j=1}^i \bar g_j
~~\text{whenever}~~
\bar\beta_i>\bar\beta_{i+1} \end{aligned}$$ with the previous convention $\bar\beta_{p+1}=0$. Recall the definitions $d = {|\{ {|\beta_i|} \neq 0:~ i\in[p]\}|}$ and ${\mathcal{G}}= ({\mathcal{G}}_1, \cdots, {\mathcal{G}}_d)$ for $\beta$, from right before the statement of . Then, we get where the last two conditions have been derived from .
For example, consider $w=e_1$ which gives ${\|\cdot\|}_{w}= {\|\cdot\|}_\infty$ whose subdifferential is given in .
We use the min-max inequality to get $$\begin{aligned}
\varphi^2(\beta)
&= \max_{z\in {\mathcal{B}}^\star}\, \min_{g\in \partial\|\beta\|} ~{\|g-z\|}_2^2 \nonumber \\
&\leq \min_{g\in \partial\|\beta\|} \,\max_{z\in {\mathcal{B}}^\star} ~{\|g-z\|}_2^2 \nonumber \\
&= \min_{g\in \partial\|\beta\|} \,\max_{z\in {\mathcal{B}}^\star} ~{\|g+z\|}_2^2 \nonumber
$$ where we used the symmetry of ${\mathcal{B}}^\star$. We now focus on the inner optimization problem. Fix $g\in \partial {\|\beta\|}_{w}$ and consider $$\max_z \bigl\{ {\|z\|}_2^2 + 2{\langle}z,g{\rangle}:~
{\sum_{j=1}^i \bar z_j} \leq {\sum_{j=1}^i w_j}
~~\forall\,i\in[p] \bigr\}.$$ Observe that 1) the optimal $z$ will have the same sign pattern as $g$, 2) ${|z|}$ and ${|g|}$ are similarly ordered. Furthermore, we claim that the optimal $z$ satisfies $\bar z = w$, hence providing the optimal $z$ completely (one can use to establish this claim). For this, we show that such choice of $z$ maximize each of the two terms in the objective subject to the constraint. Take any $z$ on the boundary of the dual norm ball. First, observe that $$\begin{aligned}
{\langle}z,g{\rangle}= {\langle}\bar z,\bar g{\rangle}= \sum_{i=1}^p \left( (\bar g_i-\bar g_{i+1}) \sum_{j=1}^i \bar z_j \right)
\leq \sum_{i=1}^p \left( (\bar g_i-\bar g_{i+1}) \sum_{j=1}^i w_j \right)
= {\langle}w, \bar g{\rangle}.\end{aligned}$$ Secondly, $$\begin{aligned}
{\|z\|}_2^2 = {\langle}\bar z,\bar z{\rangle}&= \sum_{i=1}^p \left( (\bar z_i-\bar z_{i+1}) \sum_{j=1}^i \bar z_j \right) \nonumber\\
&\leq \sum_{i=1}^p \left( (\bar z_i-\bar z_{i+1}) \sum_{j=1}^i w_j \right) \nonumber\\
&= {\langle}w, \bar z{\rangle}\nonumber\\
&= \sum_{i=1}^p \left( (w_i-w_{i+1}) \sum_{j=1}^i \bar z_j \right)\nonumber\\
&\leq \sum_{i=1}^p \left( (w_i-w_{i+1}) \sum_{j=1}^i w_j \right) \nonumber\\
&= {\langle}w,w{\rangle}= {\|w\|}_2^2\end{aligned}$$ Finally, note that $w$ (and any signed permuted version of it) is feasible in the above optimization program; i.e., $w\in{\mathcal{B}}^\star$. Therefore, the optimal value for the original inner optimization program is given by $${\|g\|}_2^2 + {\|w\|}_2^2 + 2 {\langle}w,\bar g{\rangle}.$$ Next, we are interested in minimizing the above for all $g\in \partial {\|\beta\|}_{w}$ where the subdifferential is characterized in . After a slight change in variable $g$, we would like to solve $$\begin{aligned}
\label{eq:OWL-minmax-min}
\min_g \Bigl\{ {\|g+w\|}_2^2 :~
g_1 \geq \cdots \geq g_k \geq 0 ~,
\sum_{j\in {\mathcal{G}}_t} \bar g_j = \sum_{j\in {\mathcal{G}}_t} w_j ~~ \forall\, t\in[d] ~,
\sum_{j\in {\mathcal{G}}_0} \bar g_j \leq \sum_{j\in {\mathcal{G}}_0} w_j \Bigr\}\end{aligned}$$ where $k = {\|\beta\|}_0$. We upper bound the above by plugging in $$g = \left[
\frac{\sum_{j\in {\mathcal{G}}_1} w_j}{{|{\mathcal{G}}_1|}} {\boldsymbol{1}}_{{|{\mathcal{G}}_1|}}^{{\sf T}}~,~ \cdots ~,~
\frac{\sum_{j\in {\mathcal{G}}_d} w_j}{{|{\mathcal{G}}_d|}} {\boldsymbol{1}}_{{|{\mathcal{G}}_d|}}^{{\sf T}}~,~ -w_{{\mathcal{G}}_0}^{{\sf T}}\right]^{{\sf T}}$$ which gives $$\begin{aligned}
\varphi^2(\beta)
\leq {\|w_{{\mathcal{G}}}\|}_2^2 + 3\sum_{t=1}^d \frac{(\sum_{j\in {\mathcal{G}}_t} w_j)^2}{{|{\mathcal{G}}_t|}} \end{aligned}$$ where we abuse the notation to denote ${\mathcal{G}}= \cup_{t=1}^d {\mathcal{G}}_t = {{\operatorname{Supp}}}(\beta)$, and where $d = {|\{ {|\beta_i|}\neq 0:~ i\in[p]\}|}$, and the partition ${\mathcal{G}}= ({\mathcal{G}}_1, \cdots, {\mathcal{G}}_d)$ is according to equal absolute values in $\beta$. This finishes proof. Moreover, $$\begin{aligned}
\sum_{t=1}^d \frac{(\sum_{j\in {\mathcal{G}}_t} w_j)^2}{{|{\mathcal{G}}_t|}}
= {\|w_{\mathcal{G}}\|}_2^2 - {{\operatorname{dist}}}^2(w; {\mathcal{S}}_{\mathcal{G}}(\beta))\end{aligned}$$ where ${\mathcal{S}}_{\mathcal{G}}(\beta) = \{u:~ \bar\beta_i=\bar\beta_j \implies \bar u_i = \bar u_j\}$.
We use the min-max inequality to get $$\begin{aligned}
\varphi^2(\beta^\star)
&= \max_{z\in {\mathcal{B}}^\star}\, \min_{g\in \partial\|\beta^\star\|} ~{\|g-z\|}_2^2 \nonumber \\
&\leq \min_{g\in \partial\|\beta^\star\|} \,\max_{z\in {\mathcal{B}}^\star} ~{\|g-z\|}_2^2 \nonumber \\
&= \min_{g\in \partial\|\beta^\star\|} \,\max_{z\in {\mathcal{B}}^\star} ~{\|g+z\|}_2^2 \nonumber \\
&\leq \min_{g\in \partial\|\beta^\star\|} \, \bigl\{ {\|g\|}_2^2 + \max_{z\in {\mathcal{B}}^\star} ~{\|z\|}_2^2 + 2\langle g,z \rangle \bigr\} $$ where we used the symmetry of ${\mathcal{B}}^\star$. We now focus on the inner optimization problem.
It is easy to see that vertices of the (scaled) $\ell_1$ norm ball maximize both ${\|z\|}_2^2$ and $\langle g,z \rangle$. Therefore, the optimal value of the original inner problem is given by $${\|g\|}_2^2 + 1 + 2{\|g\|}_\infty .$$ Now, we would like to minimize the above over all $g\in \partial {\|\beta^\star\|}_\infty$ where $$\begin{aligned}
\partial {\|\beta^\star\|}_\infty
&= \bigl\{ g:~ \langle g, \beta^\star\rangle = {\|\beta^\star\|}_\infty,~ {\|g\|}_1\leq 1 \bigr\} \nonumber\\
&= \bigl\{ g:~ g_i = 0 \text{ if } {|\beta^\star_i|}<{\|\beta^\star\|}_\infty,~
{\|g\|}_1=1,~ g \circ \beta \geq 0 \bigr\}.\label{eq:subdiff-linf}\end{aligned}$$ This time, note that a vector with all equal values minimizes both the $\ell_2$ and the $\ell_\infty$ norm subject to $\ell_1$ constraints. Therefore, for $t = {|\{i\in[p]:~ {|\beta^\star_i|} = {\|\beta^\star\|}_\infty\}|}$, the optimal $g$ has $t$ nonzero entries with absolute values equal to $1/t$, which yields $$\begin{aligned}
\varphi^2(\beta^\star)
\leq \frac{1}{t} + 1 + 2 \cdot\frac{1}{t}
= 1 + \frac{3}{t}
\leq 4\end{aligned}$$ and finishes the proof.
Consider and observe that ${{\operatorname{ext}}}({\mathcal{B}}^\star) = \{\pm e_i:~ i\in[p]\}$ where $e_i$ is the $i$-th standard basis vector. Define $S = \{i\in[p]:~ \beta^\star_i = {\|\beta^\star\|}_\infty \} $ and $t={|S|}$.
- Case 1: For $i\not\in S$ and $z=\pm e_i$ we have ${{\operatorname{dist}}}^2(z, \partial {\|\beta^\star\|}_\infty) = \min_{g\in \partial {\|\beta^\star\|}_\infty} 1+{\|g\|}_2^2=1 + \frac{1}{t}$. If $S=[p]$, we ignore this case in the maximum over $z\in {{\operatorname{ext}}}({\mathcal{B}}^\star)$ in .
- Case 2: For $i\in S$ and $z = {{\operatorname{sign}}}(\beta^\star_i)e_i$ we have ${{\operatorname{dist}}}(z, \partial {\|\beta^\star\|}_\infty) =0$.
- Case 3: For $i\in S$ and $z = -{{\operatorname{sign}}}(\beta^\star_i)e_i$, the distance is equal to the distance of $-{|z|}$ to a $t$-dimensional simplex whose square, by , is equal to $1+\frac{1}{t-1}$ when $t\geq 2$ and is equal to $4$ when $t=1$.
Gathering all of the above into the maximum over $z\in {{\operatorname{ext}}}({\mathcal{B}}^\star)$ in yields the desired result.
Doubly-sparse Norms: $(k,1)$
----------------------------
\[lem:all-k-1\] We have
1. \[lem:all-k-1-normk1\] ${\|\beta\|}_{k\square 1} = \max\{\frac{1}{\sqrt{k}}{\|\beta\|}_1 , \sqrt{k}{\|\beta\|}_\infty \}$
2. \[lem:all-k-1-normk1-dual\] ${\|\beta\|}_{k\square 1}^\star = \frac{1}{\sqrt{k}}\sum_{i=1}^k \bar\beta_i = \inf_{u,v} \bigl\{ \frac{1}{\sqrt{k}}{\|u\|}_1 + \sqrt{k}{\|v\|}_\infty:~ \beta = u+v \bigr\}$ which leads to a representation as an ordered weighted $\ell_1$ norm, ${\|\cdot\|}^\star_{k\square 1} = {\|\cdot\|}_w$, with $w = \frac{1}{\sqrt{k}}[{\boldsymbol{1}}_k^{{\sf T}}~,~ {\boldsymbol{0}}_{p-k}^{{\sf T}}]^{{\sf T}}$.
3. \[lem:all-k-1-normk1-ext\] ${{\operatorname{ext}}}({\mathcal{B}}_{k\square 1}) = {\mathcal{S}}_{k,1}\cap \mathbb{S}^{p-1} = \{Q\theta:~ Q\in \mathcal{P}_\pm,~ \theta = \frac{1}{\sqrt{k}}[{\boldsymbol{1}}_k^{{\sf T}}~,~ {\boldsymbol{0}}_{p-k}^{{\sf T}}]^{{\sf T}}\}$.
4. \[lem:all-k-1-normk1-dual-ext\] ${{\operatorname{ext}}}({\mathcal{B}}_{k\square 1}^\star) = \{Q\theta:~\theta\in A,~Q\in\mathcal{P}_\pm\}$ where $A = \{\sqrt{k}e_1, \frac{1}{\sqrt{k}}{\boldsymbol{1}}_p\}$.
The duality of $\frac{1}{\sqrt{k}}\sum_{i=1}^k \bar\beta_i$ and $\max\{\frac{1}{\sqrt{k}}{\|\beta\|}_1 , \sqrt{k}{\|\beta\|}_\infty \}$ is well-known; e.g., see Exercise IV.1.18 in [@bhatia1997matrix]. The representation of $\sum_{i=1}^k \bar\beta_i$ as an infimal convolution can be found in [@bhatia1997matrix Proposition IV.1.5].
Recall the definition of ${\mathcal{S}}_{k,d}$ from which gives $$\begin{aligned}
{\mathcal{S}}_{k,1}
&= \bigl\{\beta:~ {{\operatorname{card}}}(\beta) \leq k \,,~ {|\{\bar{\beta}_1,\ldots,\bar{\beta}_k\}|} \leq 1 \bigr\} \\
&= \bigl\{\beta:~ {{\operatorname{card}}}(\beta) = k \,,~ {|\{\bar{\beta}_1,\ldots,\bar{\beta}_k\}|} = 1 \bigr\} \cup\{0\}\\
&= \bigl\{\eta Q\theta:~ \theta = [{\boldsymbol{1}}_k^{{\sf T}}, {\boldsymbol{0}}_{p-k}^{{\sf T}}],~ Q\in\mathcal{P}_\pm ,~ \eta \in\mathbb{R}\bigr\}.\end{aligned}$$ Therefore, $${\|\beta\|}_{k\square 1}^\star
= \sup\{\langle \theta,\beta\rangle: \theta\in{\mathcal{S}}_{k,1},~{\|\theta\|}_2 =1 \}
= \sup\{\frac{1}{\sqrt{k}}\langle Q\theta,\beta\rangle: \theta= [{\boldsymbol{1}}_k^{{\sf T}}, {\boldsymbol{0}}_{p-k}^{{\sf T}}],~ Q\in\mathcal{P}_\pm \}
= \frac{1}{\sqrt{k}} \sum_{i=1}^k \bar\beta_i.$$ These establish and . To prove , observe that by the representation of ${\mathcal{S}}_{k,1}$ above, and by the definitions in and , we have $${{\operatorname{ext}}}({\mathcal{B}}_{k\square 1}) \subseteq {\mathcal{S}}_{k,1}\cap \mathbb{S}^{p-1} = \{Q\theta:~ Q\in \mathcal{P}_\pm,~ \theta = \frac{1}{\sqrt{k}}[{\boldsymbol{1}}_k^{{\sf T}}~,~ {\boldsymbol{0}}_{p-k}^{{\sf T}}]^{{\sf T}}\}.$$ Then, since each element on the right-hand side has $\ell_2$ norm equal to $1$, no one can be in the convex hull of others. Therefore, we get equality which establishes the claim. Alternatively, assuming , then ${{\|\cdot\|}}_{k\square 1}^\star$ is an ordered weighted $\ell_1$ norm with $w = [\frac{1}{\sqrt{k}}{\boldsymbol{1}}_k^{{\sf T}}~,~ {\boldsymbol{0}}_{p-k}^{{\sf T}}]^{{\sf T}}$. Therefore, can also be seen from . follows from and .
The representation of ${{\|\cdot\|}}_{k\square 1}^\star$ as an ordered weighted $\ell_1$ norm in and the atomic representation for this family of norms in [@zeng2014ordered Theorem 1], provide $${{\operatorname{ext}}}({\mathcal{B}}_{k\square 1}^\star) \subseteq \{Q\theta:~ Q\in\mathcal{P}_\pm ,~ \theta = \frac{\sqrt{k}}{\min\{r,k\}} [{\boldsymbol{1}}_r^{{\sf T}}, {\boldsymbol{0}}_{p-r}^{{\sf T}}]^{{\sf T}}~~~ r\in[p] \}.$$ However, as evident from (), many of the points on the right-hand side are redundant (lie in the convex hull of others).
\[lem:varphi-k-1\] For a given $\beta\neq 0$, define $k^\star = {\|\beta\|}_0$ and $t^\star = {|\{i\in[p]:~ {|\beta_i|} = {\|\beta\|}_\infty\}|}$. Then,
- If ${\|\beta\|}_1 > k {\|\beta\|}_\infty$ then $\varphi^2(\beta; {{\|\cdot\|}}_{k\square 1}) = \max\bigl\{\frac{4k^\star}{k}, 2+\frac{k^\star}{k}+k \bigr\}$.
- If ${\|\beta\|}_1 < k {\|\beta\|}_\infty$ then $\varphi^2(\beta; {{\|\cdot\|}}_{k\square 1}) = \max\bigl\{
k(1+\frac{1}{\max\{t^\star-1,1/3\}}),
2+\frac{k}{t^\star}+\frac{p}{k}
\bigr\}$
- If ${\|\beta\|}_1 = k {\|\beta\|}_\infty$ then $\varphi^2(\beta; {{\|\cdot\|}}_{k\square 1})$ is bounded from above by the [*minimum*]{} of the two above values.
As an example, consider $k=p$ and assume $\beta$ is not a multiple of ${\boldsymbol{1}}_p$, hence $t<p$. Then, using the second item above, we recover the result of .
Recall from () that ${\|\beta\|}_{k\square 1} = \max\{\frac{1}{\sqrt{k}}{\|\beta\|}_1 , \sqrt{k}{\|\beta\|}_\infty \}$. Therefore, $$\begin{aligned}
\partial {\|\beta\|}_{k \square 1} = \begin{cases}
\frac{1}{\sqrt{k}}\partial {\|\beta\|}_1 & {\|\beta\|}_1 > k {\|\beta\|}_\infty, \\
\sqrt{k}\partial {\|\beta\|}_\infty & {\|\beta\|}_1 < k {\|\beta\|}_\infty, \\
{{\operatorname{conv}}}(\frac{1}{\sqrt{k}}\partial {\|\beta\|}_1 \cup \sqrt{k}\partial {\|\beta\|}_\infty) & {\|\beta\|}_1 = k {\|\beta\|}_\infty.
\end{cases}\end{aligned}$$ Consider $S = {{\operatorname{Supp}}}(\beta)$. In the following, we first compute the distance to the subdifferential in each case.
- If ${\|\beta\|}_1 > k {\|\beta\|}_\infty$, we have $k^\star = {\|\beta\|}_0 \geq {\|\beta\|}_1/ {\|\beta\|}_\infty > k$. Fix $z\in {\mathcal{B}}^\star$ and observe that $${{\operatorname{dist}}}^2(z, \frac{1}{\sqrt{k}}\partial {\|\beta\|}_1)
= \sum_{i\in S} (z_i - \frac{1}{\sqrt{k}} {{\operatorname{sign}}}(\beta_i) )^2
+ \sum_{i\not\in S} ({|z_i|} - \frac{1}{\sqrt{k}})_+^2.$$ In maximizing the above over all $z\in {\mathcal{B}}^\star$, we use the sign-invariance property to arrive at $$\varphi^2 = \max_z\, \bigl\{
\sum_{i\in S} ({|z_i|} + \frac{1}{\sqrt{k}} )^2
+ \sum_{i\not\in S} ({|z_i|} - \frac{1}{\sqrt{k}})_+^2
:~ \sum_{i=1}^k \bar z_i = \sqrt{k}
\bigr\}.$$ Denote by $z^\star$ an optimal solution to the above. Using the permutation-invariance property, it is easy to show that if $i\in S$ and $j\in S^c$, then ${|z_i^\star|} \geq {|z_j^\star|}$. This allows for replacing $S$ with $[k^\star]$ as well as for adding a constraint ${|z_1|}\geq {|z_2|} \geq \cdots \geq {|z_p|}$ (or ${|z|}=\bar z$) to the above optimization without changing the optimal solution.
Therefore, as the constraint is insensitive to the lowest $p-k$ values, we can set $\bar z_{k} = \bar z_{k+1} = \cdots = \bar z_{p} = \theta $. Then, we get $$\begin{aligned}
\varphi^2 = \max_{0\leq \theta \leq 1/\sqrt{k}} ~ \max_{h} ~ \Bigl\{
\sum_{i=1}^{k-1} (h_i + \theta + \frac{1}{\sqrt{k}})^2 + (k^\star-k+1)(\theta + \frac{1}{\sqrt{k}})^2 + (p-k^\star)(\theta - \frac{1}{\sqrt{k}})_+^2 \\
\sum_{i=1}^{k-1} h_i = \sqrt{k}-k\theta ,~ h_1,\ldots,h_{k-1} \geq 0
\Bigr\}\end{aligned}$$ where we used the assumption $k^\star>k$ to break $[k^\star]$ into $[k-1]$ and $[k^\star]\backslash [k-1]$. The optimization problem over $h$ is a continuous-convex maximization over a compact convex domain. Hence, by Bauer’s Maximum Principle (e.g., see @schirotzek2007nonsmooth [Proposition 1.7.8]), the maximum is attained by one of the extreme points of the feasible set, which due to symmetry in variables can be taken to be $h_{\rm opt}=[\sqrt{k}-k\theta, {\boldsymbol{0}}_{k-2}]^{{\sf T}}$. Plugging this into the above gives $$\begin{aligned}
\varphi^2 = \max_{0\leq \theta \leq 1/\sqrt{k}} ~ \Bigl\{
(\sqrt{k}-k\theta + \theta + \frac{1}{\sqrt{k}})^2 + (k^\star-1)(\theta + \frac{1}{\sqrt{k}})^2
\Bigr\}.\end{aligned}$$ Again, we are dealing with a convex maximization problem which will attain its maximum at the boundary. Therefore, plugging $\theta=0$ and $\theta=\frac{1}{\sqrt{k}}$ in the objective yields $$\begin{aligned}
\label{eq:varphi-k-1-gtrk}
\varphi^2 = \max\bigl\{ k+2+\frac{k^\star}{k} , \frac{4k^\star}{k}\bigr\}.\end{aligned}$$
- If ${\|\beta\|}_1 < k {\|\beta\|}_\infty$, then $t=t^\star = {|T|} \leq {\|\beta\|}_1/{\|\beta\|}_\infty < k$ where $T=\{i\in[p]:~ {|\beta_i|} = {\|\beta\|}_\infty\}$. Fix $z\in{\mathcal{B}}^\star$ and observe that $${{\operatorname{dist}}}^2(z, \sqrt{k}\partial {\|\beta\|}_\infty)
= \min_h \bigl\{ \sum_{i\in T} (z_i - \sqrt{k} h_i {{\operatorname{sign}}}(\beta_i) )^2
+ \sum_{i\not\in T} z_i^2:~ h\geq {\boldsymbol{0}}_{t},~ {\boldsymbol{1}}^{{\sf T}}h=1 \bigr\}.$$ In maximizing the above over all $z\in {{\operatorname{ext}}}({\mathcal{B}}^\star)$, we use the sign-invariance property to arrive at $$\varphi^2 = \max_z\min_h \bigl\{
k\cdot \sum_{i\in T} ( \frac{1}{\sqrt{k}}{|z_i|} + h_i )^2
+ \sum_{i\not\in T} z_i^2
:~
h\geq {\boldsymbol{0}}_{t},~ {\boldsymbol{1}}^{{\sf T}}h=1,~
z\in{{\operatorname{ext}}}({\mathcal{B}}^\star)
\bigr\}.$$ Denote by $z^\star$ an optimal solution to the above. Using the permutation-invariance property, it is easy to show that if $i\in T$ and $j\in T^c$, then ${|z_i^\star|} \geq {|z_j^\star|}$. This allows for replacing $T$ with $[t]$ as well as for adding a constraint ${|z_1|}\geq {|z_2|} \geq \cdots \geq {|z_p|}$ (or ${|z|}=\bar z$) to the above optimization without changing the optimal solution. Hence, we get $$\begin{aligned}
\label{eq:varphi-k-1-dumm}
\varphi^2 = \max_z \min_h \bigl\{
k\cdot \sum_{i=1}^t ( \frac{1}{\sqrt{k}}z_i + h_i )^2
+ \sum_{i=t+1}^p z_i^2
:~
h\geq {\boldsymbol{0}}_{t},~ {\boldsymbol{1}}^{{\sf T}}h=1,~
z\in\{\sqrt{k}e_1, \frac{1}{\sqrt{k}}{\boldsymbol{1}}_p\}
\bigr\}\end{aligned}$$ where we used . We now divide the maximization in two parts depending on the choice of $z$:
- If $z=\sqrt{k}e_1$, then ${{\operatorname{dist}}}^2(-\frac{1}{\sqrt{k}}z_{1:t}; {\mathbf{\Delta}}_t) = 1+1/\max\{t-1,1/3\}$ leading to a corresponding value for objective in of $k(1+1/\max\{t-1,1/3\})$.
- If $z=\frac{1}{\sqrt{k}}{\boldsymbol{1}}_p$, then $\frac{1}{\sqrt{k}}z_{1:t} = \frac{1}{k}{\boldsymbol{1}}_t$. Since $z_{1:t}$ is a multiple of ${\boldsymbol{1}}_t$, it is easy to see that ${{\operatorname{dist}}}^2(-\frac{1}{\sqrt{k}}z_{1:t}; {\mathbf{\Delta}}_t) = t(\frac{1}{t}+\frac{1}{k})^2$ leading to a corresponding value for objective in of $$kt(\frac{1}{t}+\frac{1}{k})^2 + \frac{p-t}{k} = 2+\frac{k}{t}+\frac{p}{k}.$$
Taking the maximum over the above three cases, we get $$\begin{aligned}
\label{eq:varphi-k-1-lessk}
\varphi^2 = \max\bigl\{
k(1+\frac{1}{\max\{t-1,1/3\}}),
2+\frac{k}{t}+\frac{p}{k}
\bigr\}\end{aligned}$$ for when ${\|\beta\|}_1 < k {\|\beta\|}_\infty$.
- If ${\|\beta\|}_1 = k {\|\beta\|}_\infty$, we proceed with upper bounding $\varphi$ using and . Observe that in this case, $\partial {\|\beta\|}_{k\square 1}$ contains both $\frac{1}{\sqrt{k}}\partial{\|\beta\|}_1$ and $\sqrt{k}\partial{\|\beta\|}_\infty$. Therefore, for any fixed vector, the distance to $\partial {\|\beta\|}_{k\square 1}$ is smaller than the distance to either of the other two subdifferentials. Therefore, $$\begin{aligned}
\varphi^2 \leq \min\{\eqref{eq:varphi-k-1-gtrk} , \eqref{eq:varphi-k-1-lessk} \}\end{aligned}$$ for when ${\|\beta\|}_1 = k {\|\beta\|}_\infty$. For such condition to hold, it is necessary that $t\leq k \leq k^\star$.
\[lem:varphi-bnd-norm-k-1-dual\] $\varphi^2(\beta; {{\|\cdot\|}}^\star_{k\square 1}) \leq 4 \min\{1 ,\frac{{\|\beta\|}_0}{k}\}$.
Recall the representation of ${{\|\cdot\|}}_{k\square 1}^\star$ as an ordered weighted $\ell_1$ norm in () with $w = \frac{1}{\sqrt{k}}[{\boldsymbol{1}}_k^{{\sf T}}~,~ {\boldsymbol{0}}_{p-k}^{{\sf T}}]^{{\sf T}}$. Moreover, from we have $\varphi^2(\beta; {{\|\cdot\|}}_{w}) \leq 4{\|w_{\mathcal{G}}\|}_2^2$ where ${\mathcal{G}}= {{\operatorname{Supp}}}(\bar\beta)$. These establish the result.
[^1]: Technicolor AI Lab. Email: `[email protected]`.
[^2]: Marshall School of Business, University of Southern California. Email: `[email protected]`.
[^3]: Department of Electrical Engineering, University of Washington. Email: `[email protected]`.
[^4]: This is in contrast to the specific constrained loss minimization setups required in IHT.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the set of periods of degree 1 continuous maps from $\sigma$ into itself, where $\sigma$ denotes the space shaped like the letter $\sigma$ (i.e., a segment attached to a circle by one of its endpoints). Since the maps under consideration have degree 1, the rotation theory can be used. We show that, when the interior of the rotation interval contains an integer, then the set of periods (of periodic points of any rotation number) is the set of all integers except maybe $1$ or $2$. We exhibit degree 1 $\sigma$-maps $f$ whose set of periods is a combination of the set of periods of a degree 1 circle map and the set of periods of a $3$-star (that is, a space shaped like the letter $Y$). Moreover, we study the set of periods forced by periodic orbits that do not intersect the circuit of $\sigma$; in particular, when there exists such a periodic orbit whose diameter (in the covering space) is at least $1$, then there exist periodic points of all periods.'
---
<span style="font-variant:small-caps;">Lluís Alsedà</span>
<span style="font-variant:small-caps;">Sylvie Ruette</span>
Introduction
============
In this paper we study the set of periods of continuous maps from the space $\sigma$ to itself, where the space $\sigma$ consists of a circle with a segment attached to it at one of the segment’s endpoints. Our results continue the progression of results which began with Sharkovskii’s Theorem on the characterization of the sets of periods of periodic points of continuous interval maps [@SharOri; @SharTrans] and continued with the study of the periods of maps of the circle [@BGMY; @Block; @Mis], trees [@AJM1; @AJM2; @AJM3; @AJM4; @Bern; @ALMY; @BaldLli] and other graphs [@LLl; @LPR].
A full characterization of the sets of periods for continuous self maps of the graph $\sigma$ having the branching fixed is given in [@LLl]. Our goal is to extend this result to the general case. The most natural approach is to follow the strategy used in the circle case which consists in dividing the problem according to the degree of the map [@BGMY; @Block; @Mis]. The cases considered for the circle are degree different from $\{-1,
0, 1\}$, and separately the cases of degree $0$, $-1$ and $1$. A characterization of the set of periods of the class of continuous maps from the space $\sigma$ to itself with degree different from $\{-1, 0, 1\}$ can be found in [@Mal]. In this paper, we aim at studying the set of periods of continuous $\sigma$-maps of degree 1. Following again the strategy of the circle case, we shall work in the covering space and we shall use rotation theory. This theory for graphs with a single circuit was developed in [@AlsRue2008]; the current paper is thus an application of the theory developed there.
We shall follow three main directions in studying the set of periods of $\sigma$-maps. The first very natural one follows from the trivial observation that the space $\sigma$ contains both a circle and a subset homeomorphic to a $Y$ (also called a $3$-star). It is quite obvious that there exist $\sigma$-maps of degree $1$ whose set of periods is equal to the set of periods of any given degree $1$ circle map, as well as the set of periods of any given $3$-star map. We shall show that there exist $\sigma$-maps $f$ whose set of periods is any combination of both kinds of sets, provided that $0$ is an endpoint of the rotation interval of $f$: the whole rotation interval gives a set of periods as for circle maps whereas the set of periods of a given $3$-star map appears with rotation number $0$.
The second direction is the study of periodic orbits that do not intersect the circuit of the space $\sigma$; this study is necessary because the rotation interval does not capture well the behaviors of such orbits. We shall show that the existence of such a periodic orbit of period $n$ implies all periods less than $n$ for the Sharkovsky ordering; this is quite natural because this ordering rules the sets of periods of interval maps and the branch of $\sigma$ is an interval. Moreover, we shall show that if, in the covering space, there exists a periodic orbit living in the branches and with diameter greater than or equal to $1$, then the set of periods contains necessarily all integers.
The third direction focuses on the rotation number $0$. For degree $1$ circle maps, the strategy is to characterize the set of periods for a given rotation number $p/q$ in the interior of the rotation interval, which comes down to do the same for the rotation number $0$ for another map. Unfortunately, mimicking this strategy fails for $\sigma$-maps because the set of periods of rotation number $0$ can be complicated and we do not know how to describe it. However, we shall characterize the set of periods (of any rotation number) when $0$ in the interior of the rotation interval of a $\sigma$ map: in this case, the set of periods is, either ${\ensuremath{\mathbb{N}}}$, or ${\ensuremath{\mathbb{N}}}\setminus\{1\}$, or ${\ensuremath{\mathbb{N}}}\setminus\{2\}$.
Moreover, we shall stress some difficulties that appear when one tries to follow the same strategy as for degree $1$ circle maps.
In the next section, we state and discuss the main results of the paper, after introducing the necessary notation to formulate them.
Definitions and statements of the main results {#sec:statements}
==============================================
Covering space, periodic (mod 1) points, rotation set {#ss:coveringS}
-----------------------------------------------------
As it has been said, in this paper we want to study the set of periods of the $\sigma$-maps. Given a map , we say that a point $x \in X$ is *periodic of period $n$* if $f^n(x) = x$ and $f^i(x) \ne x$ for all $i=1,2,\dots,n-1$. Moreover, for every $x \in X$, the set $$\operatorname{Orb}(x,f) := {\ensuremath{\{f^{n}(x) \,\colon n \ge 0\}}}$$ is called the *orbit of $x$*. Observe that if $x$ is periodic with period $n$, then we have $\operatorname{Card}(\operatorname{Orb}(x,f)) = n$ (where $\operatorname{Card}(\cdot)$ denotes the cardinality of a finite set). The set of periods of all periodic points of $f$ will be denoted by $\operatorname{Per^{\circ}}(f)$.
Following the strategy of the circle it is advisable to work in the covering space and we shall use the rotation theory developed in [@AlsRue2008]. We also shall consider periodic [[$\kern -0.55em\pmod{1}$]{}]{} points and orbits for liftings instead of the true ones defined above. The results obtained in this setting can be obviously pushed down to the original map and space.
We start by introducing the framework to use the rotation theory developed in [@AlsRue2008].
We consider the universal covering of $\sigma$. More precisely, we take the following realization of the covering space (see Figure \[FigS\]): $$S = {\ensuremath{\mathbb{R}}}\cup B,$$ where $$B := {\ensuremath{\{z \in {\ensuremath{\mathbb{C}}}\,\colon \Re(z) \in {\ensuremath{\mathbb{Z}}}\text{ and }\Im(z) \in [0,1]\}}},$$ and $\Re(z)$ and $\Im(z)$ denote respectively the real and imaginary part of a complex number $z$. The set $B$ is called the *set of branches of $S$*.
(7,2)(-0.5,-0.5) (-0.5,0)[(1,0)[7]{}]{} (0,0)(1,0)[7]{}[(0,1)[1]{}]{} (0,-0.1)[(0,0)\[t\][0]{}]{} (1,-0.1)[(0,0)\[t\][1]{}]{} (2,-0.1)[(0,0)\[t\][2]{}]{} (3,-0.1)[(0,0)\[t\][3]{}]{} (4,-0.1)[(0,0)\[t\][4]{}]{} (5,-0.1)[(0,0)\[t\][5]{}]{} (6,-0.1)[(0,0)\[t\][6]{}]{} (-0.2,0.5)[(0,0)\[r\][$\cdots$]{}]{} (6.3,0.5)[(0,0)\[l\][$\cdots$]{}]{}
Observe that $S \subset \C$ and that $\R$ actually means the copy of the real line embedded in $\C$ as the real axis. Also, the maps $z \mapsto z + n$ for $n \in \Z$ (since $S\subset {\ensuremath{\mathbb{C}}}$, the operation $+$ is just the usual one in ${\ensuremath{\mathbb{C}}}$) are the covering (or deck) transformations. So, they leave $S$ invariant: $S = S + \Z = {\ensuremath{\{z+k \,\colon z \in S \text{ and } k \in \Z\}}}.$ Moreover, the real part function $\Re$ defines a retraction from $S$ to ${\ensuremath{\mathbb{R}}}$. That is, $\Re(z) = z$ for every $z \in {\ensuremath{\mathbb{R}}}$ and, when $z \in S\setminus {\ensuremath{\mathbb{R}}}$, then $\Re(z)$ gives the integer in the base of the segment where $z$ lies.
For every $m \in {\ensuremath{\mathbb{Z}}}$, we set $$\begin{aligned}
B_m &:= {\ensuremath{\{z \in S \,\colon \Re(z) = m \text{ and }\Im(z) \in [0,1]\}}}
= S \cap \Re^{-1}(m), \text{ and}\\
{\mathring{B}}_m &:= B_m \setminus \{m\}.\end{aligned}$$ Each of the sets $B_m$ is called *a branch of $S$*. Clearly, $B = \cup_{m \in {\ensuremath{\mathbb{Z}}}} B_m$, $B_m \cap {\ensuremath{\mathbb{R}}}= \{m\}$ and ${\mathring{B}}_m \cap {\ensuremath{\mathbb{R}}}= \emptyset$. Each branch $B_m$ is endowed with a linear ordering $\le$ as follows. If $x,y \in B_m$, we write $x < y$ if and only if $\Im(x) < \Im(y)$.
In what follows, ${\ensuremath{\mathcal{L}_{d}(S)}}$ will denote the class of continuous maps $F$ from $S$ into itself of degree $d \in {\ensuremath{\mathbb{Z}}}$, that is, $F(z+1)=F(z)+d$ for all $z \in S$. We also set ${\ensuremath{\mathcal{L}(S)}}= \cup_{d \in {\ensuremath{\mathbb{Z}}}} {\ensuremath{\mathcal{L}_{d}(S)}}$. Observe that $\Re \in {\ensuremath{\mathcal{L}_{1}(S)}}$ and thus, if $F\in {\ensuremath{\mathcal{L}_{1}(S)}}$, then $\Re \circ F^n \in {\ensuremath{\mathcal{L}_{1}(S)}}$ for every $n \in \N$.
Let $F \in {\ensuremath{\mathcal{L}(S)}}$ and $z\in S$. The set $${\ensuremath{\{F^{n}(z)+m \,\colon n \ge 0 \text{ and } m\in {\ensuremath{\mathbb{Z}}}\}}}$$ is called the *lifted orbit of $z$*, and denoted by $\operatorname{\mathsf{L}Orb}(z,F)$. The point $z$ is called *periodic [$\kern -0.55em\pmod{1}$]{}* if there exists $n \in {\ensuremath{\mathbb{N}}}$ such that $F^n(z) \in z+{\ensuremath{\mathbb{Z}}}$. The *period [$\kern -0.55em\pmod{1}$]{}* of $z$ is the least positive integer $n$ satisfying this property, that is, $F^n(z) \in z+{\ensuremath{\mathbb{Z}}}$ and $F^i(z) \notin z+{\ensuremath{\mathbb{Z}}}$ for all $1\leq i\leq n-1$. When $z$ is periodic [[$\kern -0.55em\pmod{1}$]{}]{}, then $\operatorname{\mathsf{L}Orb}(z,F)$ is also called a *lifted periodic orbit*. It is not difficult to see that, for all $k \in {\ensuremath{\mathbb{Z}}}$, $\operatorname{Card}\left(\operatorname{\mathsf{L}Orb}(z,F) \cap \Re^{-1}\bigl([k,k+1)\bigr)\right)$ coincides with the period [[$\kern -0.55em\pmod{1}$]{}]{} of $z$. The set of all periods of the periodic [[$\kern -0.55em\pmod{1}$]{}]{} points of $F \in {\ensuremath{\mathcal{L}(S)}}$ will be denoted by $\operatorname{Per}(F)$.
Wen talking about periodic points and periodic [[$\kern -0.55em\pmod{1}$]{}]{} points we shall sometimes write *true period* or *true periodic point* to emphasize that they are not [[$\kern -0.55em\pmod{1}$]{}]{}.
Let $\map{\pi}{S}[\sigma]$ be the standard projection from $S$ to $\sigma$, that is, $\pi{\bigr\rvert_{\Re^{-1}([0,1))}}$ is continuous onto and one-to-one and $\pi(z) = \pi(z + k)$ for all $z \in S$ and all $k \in \Z$. Clearly, for every $F\in {\ensuremath{\mathcal{L}(S)}}$, $\pi_{\star}F := \pi \circ F \circ \pi^{-1}$ is a well defined continuous self map of $\sigma$. Reciprocally, for every continuous map $f$ from $\sigma$ into itself, there exists a lifting $F\in{\ensuremath{\mathcal{L}(S)}}$ such that $\pi_{\star}F=f$, and this lifting is unique up to an integer (that is, if $G$ is another lifting, there exists $k\in{\ensuremath{\mathbb{N}}}$ such that $G=F+k$). Moreover, $\pi(\operatorname{\mathsf{L}Orb}(z,F)) = \operatorname{Orb}(\pi(z),\pi_{\star}F),$ and $z$ is a periodic [[$\kern -0.55em\pmod{1}$]{}]{} point of $F$ of period $n$ if and only if $\pi(z)$ is a true periodic point of $\pi_{\star}F$ of (true) period $n.$ Consequently, $\operatorname{Per}(F) = \operatorname{Per^{\circ}}(\pi_{\star}F)$ and *characterizing the sets of periods [[$\kern -0.55em\pmod{1}$]{}]{} of maps from ${\ensuremath{\mathcal{L}(S)}}$ is equivalent to characterizing the sets of periods of continuous self maps of $\sigma$*.
This paper will deal with maps of degree $1$, for which rotation numbers can be defined. Next we recall the notion of rotation number in our setting and its basic properties.
Let $F\in {\ensuremath{\mathcal{L}_{1}(S)}}$ and $z\in S$. We define the *rotation number of $z$* as $${\rho_{_{F}}}(z) := \lim_{n\to+\infty} \frac{\Re(F^n(z))-\Re(z)}{n}$$ if the limit exists. We also define the following *rotation sets of $F$*: $$\begin{aligned}
\operatorname{Rot}(F) &= {\ensuremath{\{{\rho_{_{F}}}(z) \,\colon z \in S\}}},\\
{\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F) &= {\ensuremath{\{{\rho_{_{F}}}(z) \,\colon z \in \R\}}}.\end{aligned}$$
For every $z \in S,$ $k\in{\ensuremath{\mathbb{Z}}}$ and $n\in{\ensuremath{\mathbb{N}}}$, it follows that ${\rho_{_{F}}}(z+k)={\rho_{_{F}}}(z),$ ${\rho_{_{(F+k)}}}(z)={\rho_{_{F}}}(z)+k$ and ${\rho_{_{F^n}}}(z)=n{\rho_{_{F}}}(z)$ (c.f [@AlsRue2008 Lemma 1.10]). The second property implies that, if $F$, $G$ are two liftings of the same continuous map from $\sigma$ into itself, then their rotation sets differ from an integer ($\exists k\in{\ensuremath{\mathbb{Z}}}$ such that $G=F+k$, and hence $\operatorname{Rot}(G)=\operatorname{Rot}(F)+k$).
Unfortunately, the set $\operatorname{Rot}(F)$ may not be connected as it has been shown in [@AlsRue2008]. However, the set ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)$, which is a subset of $\operatorname{Rot}(F)$, has better properties. Next result is [@AlsRue2008 Theorem 3.1].
\[theo:RotR\] For every $F\in{\ensuremath{\mathcal{L}_{1}(S)}},$ ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)$ is a non empty compact interval. Moreover, if $\alpha\in {\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)$, then there exists a point $x\in{\ensuremath{\mathbb{R}}}$ such that ${\rho_{_{F}}}(x)=\alpha$ and $F^n(x)\in{\ensuremath{\mathbb{R}}}$ for infinitely many $n$. If $p/q\in{\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)$, then there exists a periodic [[$\kern -0.55em\pmod{1}$]{}]{} point $x\in S$ with ${\rho_{_{F}}}(x)=p/q$.
Given $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ and $\alpha \in {\ensuremath{\mathbb{R}}}$, let $\operatorname{Per}(\alpha,F)$ denote the set of periods of all periodic [[$\kern -0.55em\pmod{1}$]{}]{} points of $F$ whose rotation number is $\alpha$.
It is easy to see that every periodic [[$\kern -0.55em\pmod{1}$]{}]{} point has a rational rotation number (see also Lemma \[lem:FF+k\](e)). Therefore, Theorem \[theo:RotR\] implies that, when $\alpha\in{\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)$, $\operatorname{Per}(\alpha,F)$ is non-empty if and only if $\alpha\in{\ensuremath{\mathbb{Q}}}$.
Observe that the class of maps $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ such that $F(\R) \subset
\R$ and $F(B_m) = F(m)$ for every $m \in \Z$ can be identified with the class of liftings of continuous circle maps of degree $1$. Therefore any possible set of periods of a continuous circle map of degree $1$ can be a set of periods of a map in ${\ensuremath{\mathcal{L}_{1}(S)}}$. On the other hand, set $Y_0:=B_0 \cup [-1/3,1/3]$ (this space is called a *$3$-star*) and consider the class of maps $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ such that $F(Y_0) \subset Y_0$, $F(x) \in Y_0 \cup [1/3,x)$ for every $x \in [1/3,1/2)$ and $F(x) \in (Y_0 + 1 )\cup (x,2/3]$ for every $x \in (1/2,2/3]$ (in particular $F(1/2) = 1/2$). This implies that $\operatorname{Per}(F) = \operatorname{Per^{\circ}}(F{\bigr\rvert_{Y_0}})$ and thus, every possible set of periods of a map from a $3$-star into itself can be a set of periods of a map from ${\ensuremath{\mathcal{L}_{1}(S)}}$. Clearly, this includes the sets of periods of interval maps. Moreover, it might happen that this phenomenon occurs for rotation numbers different from 0, that is, there may exist a map from ${\mathcal{X}_{3}}$ with set of periods $A\subset
{\ensuremath{\mathbb{N}}}$, $p \in \Z,$ $q \in \N$ and $\widetilde{S} \subset S$ such that $\operatorname{Per^{\circ}}((F^q -p){\bigr\rvert_{\widetilde{S}}}) = A$ and $\operatorname{Per}(p/q,F) = q \cdot \operatorname{Per^{\circ}}((F^q -p){\bigr\rvert_{\widetilde{S}}}).$ Therefore, a natural conjecture for the structure of the set of periods of maps from ${\ensuremath{\mathcal{L}_{1}(S)}}$ could be that it is the union of the set of periods of a circle map of degree $1$ with some sets of the form $q\cdot \operatorname{Per^{\circ}}(f)$ with $q\in\N$ and $f\in {\mathcal{X}_{3}}$ much in the spirit of the characterization of the set of periods for circle maps of degree one. We shall see that it is unclear that all possibilities can occur.
To explain these ideas in detail, and to state the main results of the paper, we need to recall the characterization of the sets of periods of circle maps of degree $1$ and of star maps. We are going to do this in the next two subsections; we shall also introduce the necessary notations.
Tree maps
---------
A *tree* is a compact uniquely arcwise connected space which is a point or a union of a finite number of segments glued together at some of their endpoints (by a *segment* we mean any space homeomorphic to $[0,1]$). Any continuous map $f$ from a tree into itself is called a *tree map*. The space $S$ is often called an infinite tree by similarity.
Consider a tree $T$ or the space $S$. For every $x$ in $T$ or $S$, the *valence* of $x$ is the number of connected components of $T\setminus\{x\}$. A point of valence different from $2$ is called a *vertex*. A point of valence $1$ is called an *endpoint*. The points of valence greater than or equal to $3$ (that is, vertices that are not endpoints) are called the *branching points*. If $K$ is a subset of $T$ or $S$, then $\chull{K}$ denotes the *convex hull* of $K$, that is, the smallest closed connected set containing $K$ (which is well defined since the trees and the space $S$ are uniquely arcwise connected). An *interval* in $T$ or $S$ is any subset homeomorphic to an interval of ${\ensuremath{\mathbb{R}}}$. For a compact interval $I$, it is equivalent to say that there exist two points $a,b$ such that $I=\chull{a,b}$; in this case, $\{a,b\}=\operatorname{Bd}(I)$ (where $\operatorname{Bd}(\cdot)$ denotes the boundary of a set). When a distance is needed in a tree or $S$, we use a taxicab metric, that is, a distance $d$ such that, if $z\in\chull{x,y}$, then $d(x,y)=d(x,z)+d(z,y)$. In $S$, the distance is simply defined by $$d(x,y) = \begin{cases}
|x-y| & \text{if $x,y\in B_m;\ m\in{\ensuremath{\mathbb{Z}}}$,}\\
|x-\Re(x)| + |\Re(x)-\Re(y)| + |y-\Re(y)| & \text{otherwise}
\end{cases}$$ for every $x,y\in S.$ Consider a compact interval $I$ in an tree $T$ or in $S,$ and a continuous map [\[S\]]{}. We say that $f$ is *monotone* if, either $f(I)$ is reduced to one point, or $f(I)$ is a non degenerate interval and, given any homeomorphisms [\[I\]]{}, [\[f(I)\]]{}, the map is monotone. We say that $f$ is *affine* if $f(I)$ is an interval and there exists a constant $\lambda$ such that $\forall x,y \in I,$ $d(f(x),f(y))=\lambda d(x,y).$
A tree that is a union of $n \ge 2$ segments whose intersection is a unique point $y$ of valence $n$ is called an *$n$-star*, and $y$ is called its *central point*. For a fixed $n$, all $n$-stars are homeomorphic. In what follows, $X_n$ will denote an $n$-star, ${\mathcal{X}_{n}}$ the class of all continuous maps from $X_n$ to itself and ${\mathcal{X}_{n}}^{\circ}$ the class of all maps from ${\mathcal{X}_{n}}$ that leave the unique branching point of $X_n$ fixed.
A crucial notion for periodic orbits of maps in ${\mathcal{X}_{n}}$ is the *type* of an orbit [@Bald]. Let $f \in {\mathcal{X}_{n}}$ and let $P$ be a periodic orbit of $F$. Let $y$ denote the branching point of $X_n$. If $y \in P$, then we say that $P$ has *type 1*. Otherwise, let $\mathrm{Br}$ be the set of branches of $X_n$ that intersect $P$ (by a branch we mean a connected component of $X_n\setminus\{y\}$). For each $b \in \mathrm{Br}$ we denote by $\mathrm{sm}_b$ the point of $P \cap b$ closest to $y$ (that is, $\mathrm{sm}_b \in b$ and $\chull{y,\mathrm{sm}_b} \cap P = \{\mathrm{sm}_b\}$). Then we define a map by letting $\phi(b)$ be the branch of $\mathrm{Br}$ containing $f(\mathrm{sm}_b).$ Since $\mathrm{Br}$ is a finite set, $\phi$ has periodic orbits. Each period of a periodic orbit of $\phi$ is called a *type* of $P$. Clearly the type may not be unique. However, it is clearly unique in the case when $P$ has type $n$.
We shall also speak of the type of a (true) periodic orbit $P$ of a map $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$ such that $\chull{P}$ is homeomorphic to $X_n$ (indeed $X_3$). The definition of type extends straightforwardly to this situation.
We now recall the Sharkovsky total ordering and Baldwin partial orderings, which are needed to state the characterization of the sets of periods of star maps.
The *Sharkovsky ordering* $\leso{\Sho}$ is defined on $\N_{\Sho} = \N \cup \{ {\ifx\empty\empty\else \empty\cdot \fi2^{\infty}}\}$ by: $$\begin{aligned}
& 3 \gtso{\Sho} 5 \gtso{\Sho} 7 \gtso{\Sho} \dots
2 \cdot 3 \gtso{\Sho} 2 \cdot 5 \gtso{\Sho}
2 \cdot 7 \gtso{\Sho} \dots\\
&2^2 \cdot 3 \gtso{\Sho} 2^2 \cdot 5 \gtso{\Sho}2^2 \cdot 7 \gtso{\Sho}
\dots
\gtso{\Sho} \dots\\
& {\ifx\empty\empty\else \empty\cdot \fi2^{\infty}}\gtso{\Sho} \dots
2^n \gtso{\Sho} \dots \gtso{\Sho} 2^4 \gtso{\Sho}
2^3 \gtso{\Sho} 2^2 \gtso{\Sho} 2 \gtso{\Sho} 1.\end{aligned}$$ That is, this ordering starts with all the odd numbers greater than 1, in increasing order, then $2$ times the odd numbers $>1$, then $2^2$ times, $2^3$ times, …$2^n$ times the odd numbers $>1$; finally the last part of the ordering consists of all powers of $2$ in decreasing order; the symbol $2^\infty$ being greater than all powers of $2$ and $1=2^0$ being the smallest element.
For every integer $t\ge 2$, let $\N_t$ denote the set $(\N \cup \{ {\ifxt\empty\else t\cdot \fi2^{\infty}} \}) \setminus\{2,3,\dots,t-1\}$ and ${\N^{\scriptscriptstyle \vee}_{t}}:={\ensuremath{\{mt \,\colon m \in \N\}}} \cup \{1,{\ifxt\empty\else t\cdot \fi2^{\infty}}\}.$ Then the *Baldwin partial ordering* $\leso{t}$ is defined in $\N_t$ as follows. For all $k,m \in \N_t$, we write $k \leso{t} m$ if one of the following cases holds:
(i) $k=1$ or $k=m$,
(ii) $k,m \in {\N^{\scriptscriptstyle \vee}_{t}}\setminus \{1\}$ and $m/t \gtso{\Sho} k/t $,
(iii) $k \in {\N^{\scriptscriptstyle \vee}_{t}}$ and $m \notin {\N^{\scriptscriptstyle \vee}_{t}}$,
(iv) $k,m \notin {\N^{\scriptscriptstyle \vee}_{t}}$ and $k = i m + j t$ with $i,j \in \N$,
where in case (ii) we use the following arithmetic rule for the symbol ${\ifxt\empty\else t\cdot \fi2^{\infty}}$: ${\ifxt\empty\else t\cdot \fi2^{\infty}}/t = {\ifx\empty\empty\else \empty\cdot \fi2^{\infty}}$.
There are two parts in the structure of the orderings $\leso{t}$. The smallest part consists of all elements of ${\N^{\scriptscriptstyle \vee}_{t}}$ ordered as follows. The smallest element is 1. Then all the multiples of $t$ (including ${\ifxt\empty\else t\cdot \fi2^{\infty}}$) come in the ordering induced by the Sharkovsky ordering and the largest element of ${\N^{\scriptscriptstyle \vee}_{t}}$ is $3\cdot t$. Then the ordering $\geso{t}$ divides $\N_t \setminus {\N^{\scriptscriptstyle \vee}_{t}}$ into $t-1$ “branches”. The $l$-th branch ($l \in \{1,2,\dots,t-1\}$) is formed by all positive integers (except $l$) which are congruent to $l$ modulo $t$ in decreasing order. All elements of these branches are larger than all elements of ${\N^{\scriptscriptstyle \vee}_{t}}$.
We note that, by means of the inclusion of the symbol ${\ifxt\empty\else t\cdot \fi2^{\infty}}$, each subset of $\N_t$ has a maximal element with respect to the ordering $\leso{t}$. We also note that the ordering $\leso{2}$ on $\N_2$ coincides with the Sharkovsky ordering on $\N_{\Sho}$ (by identifying the symbol ${\ifx2\empty\else 2\cdot \fi2^{\infty}}$ with ${\ifx\empty\empty\else \empty\cdot \fi2^{\infty}}$).
A non empty set $A \subset \N_t \cap \N$ is called a *tail of the ordering $\leso{t}$* if, for all $m \in A$, we have ${\ensuremath{\{k \in \N \,\colon k \leso{t} m\}}} \subset A$. Moreover, for all $s\in \N_{\Sho}$, $\operatorname{S\mbox{\tiny\textup{sh}}}(s)$ denotes the initial segment of the Sharkovsky ordering starting at $s$, that is, $\operatorname{S\mbox{\tiny\textup{sh}}}(s) = {\ensuremath{\{k \in \N \,\colon k \leso{\Sho} s\}}}$.
The following result, due to Baldwin [@Bald], characterizes the set of periods of star maps.
\[GMT1\] Let $f \in {\mathcal{X}_{n}}$. Then $\operatorname{Per^{\circ}}(f)$ is a finite union of tails of the orderings $\geso{t}$ for all $t \in \{2,\ldots, n\}$ (in particular, $1\in\operatorname{Per^{\circ}}(f)$). Conversely, if a non empty set $A$ can be expressed as a finite union of tails of the orderings $\geso{t}$ with $2 \le t \le n$, then there exists a map $f
\in {\mathcal{X}_{n}}^{\circ}$ such that $\operatorname{Per^{\circ}}(f) = A$.
Note that the case $n=2$ in the above theorem is, indeed, Sharkovsky’s Theorem for interval maps [@SharOri]. Moreover, since every tail of $\geso{t}$ contains $1 \in \operatorname{Per^{\circ}}(f),$ then the order $\geso{t}$ does not contribute to $\operatorname{Per^{\circ}}(f)$ if the tail with respect to $\geso{t}$ in the above lemma is reduced to $\{1\}$.
Circle maps of degree 1
-----------------------
Let ${\ensuremath{\mathbb{S}^1}}$ be the unit circle in the complex plane, that is, ${\ensuremath{\mathbb{S}^1}}=
{\ensuremath{\{z \in \C \,\colon |z| = 1\}}}$, and let ${\mathcal{L}}_1({\ensuremath{\mathbb{R}}})$ denote the class of all liftings of continuous circle maps of degree one. If $F\in
{\mathcal{L}}_1({\ensuremath{\mathbb{R}}})$, $\operatorname{Rot}(F)$ denotes the rotation set of $F$ and, by [@Ito], is a compact non empty interval.
To study the connection between the set of periods and the rotation interval, we need some additional notation. For all $c\le d$, we set $M(c,d):={\ensuremath{\{n \in \N \,\colon c < k/n < d \text{ for some integer $k$}\}}}$. Notice that we do not assume here that $k$ and $n$ are coprime. Obviously, $M(c,d)= \emptyset$ if and only if $c=d$. Given $\rho \in \R$ and $S \subset \N$, we set $$\Lambda(\rho, S) = \begin{cases}
\emptyset & \text{if $\rho \notin \Q$}, \\
{\ensuremath{\{nq \,\colon q\in S\}}} & \text{if $\rho=k/n$ with $k$ and $n$ coprime}.
\end{cases}$$
The next theorem recalls Misiurewicz’s characterization of the sets of periods for degree $1$ circle maps (see [@Mis; @ALM]).
\[S9.5\] Let $F\in {\mathcal{L}}_1({\ensuremath{\mathbb{R}}})$, and let $\operatorname{Rot}(F) = [c,d]$. Then there exist numbers $s_c,s_d \in \N_{\Sho}$ such that $\operatorname{Per}(F)= \Lambda(c,\operatorname{S\mbox{\tiny\textup{sh}}}(s_c)) \cup M(c,d) \cup \Lambda(d,\operatorname{S\mbox{\tiny\textup{sh}}}(s_d))$. Conversely, for all $c,d \in \R$ with $c \le d$ and all $s_c, s_d\in\N_{\Sho}$, there exists a map $F \in {\mathcal{L}}_1({\ensuremath{\mathbb{R}}})$ such that $\operatorname{Rot}(F) = [c,d]$ and $\operatorname{Per}(F) = \Lambda(c,\operatorname{S\mbox{\tiny\textup{sh}}}(s_c)) \cup M(c,d) \cup \Lambda(d,\operatorname{S\mbox{\tiny\textup{sh}}}(s_d))$.
Statement of main results {#ss:main-statements}
-------------------------
In view of what we said at the end of Subsection \[ss:coveringS\], a reasonable conjecture about the set of periods for maps from ${\ensuremath{\mathcal{L}_{1}(S)}}$ could be the following:
\[WishList\] Let $F\in {\ensuremath{\mathcal{L}_{1}(S)}}$ be with ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F) = [c,d]$. Then there exist sets $E_c, E_d \subset {\ensuremath{\mathbb{N}}}$ which are finite unions of of tails of the orderings $\leso{2}$ and $\leso{3}$ such that $$\operatorname{Per}(F) = \Lambda(c,E_c) \cup M(c,d) \cup \Lambda(d,E_d).$$
Conversely, given $c,d \in \R$ with $c \le d,$ and non empty sets $E_c, E_d \subset {\ensuremath{\mathbb{N}}}$ which are finite union of of tails of the orderings $\leso{2}$ and $\leso{3},$ there exists a map $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ such that ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F) = [c,d]$ and $$\operatorname{Per}(F) = \Lambda(c,E_c) \cup M(c,d) \cup \Lambda(d,E_d).$$
As we shall see, some facts seem to indicate that this conjecture is not entirely true (though they do not disprove it). However, we shall use this conjecture as a guideline: on the one hand, we shall prove that it is partly true; on the other hand, we shall stress some difficulties.
We start by discussing the second statement of Conjecture \[WishList\]. This statement holds in two particular cases, stated in Corollary \[ConverseSigmaCircleCase\] and Theorem \[YinSigma\] below. The first one is an easy corollary of Theorem \[S9.5\] and the second one deals with the particular case when $0$ is an endpoint of the rotation interval. Recall that $\leso{2}$ coincide with $\leso{\Sho}$.
\[ConverseSigmaCircleCase\] Given $c,d \in \R$ with $c \le d$ and $s_c, s_d\in\N_{\Sho}$, there exists a map $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ such that ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F) = \operatorname{Rot}(F) = [c,d]$ and $\operatorname{Per}(F) = \Lambda(c,\operatorname{S\mbox{\tiny\textup{sh}}}(s_c)) \cup M(c,d) \cup \Lambda(d,\operatorname{S\mbox{\tiny\textup{sh}}}(s_d))$.
Notice that, when both $c$ and $d$ are irrational, Corollary \[ConverseSigmaCircleCase\] implies the second statement of Conjecture \[WishList\]. Therefore it remains to consider the cases when $c$ and/or $d$ are in ${\ensuremath{\mathbb{Q}}}$ and when the order $\leso{3}$ is needed (or equivalently when one refers to the set of periods of any $3$-star map). The next theorem deals with the case when $c$ (or $d$) is equal to $0$ (or, equivalently, to an integer) and $\leso{3}$ is needed only for this endpoint.
\[YinSigma\] Let $d \ne 0$ be a real number, $s_d\in\N_{\Sho}$ and $f
\in {\mathcal{X}_{3}}$. Then there exists a map $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ such that ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)=\operatorname{Rot}(F)$ is the closed interval with endpoints $0$ and $d$ (i.e., $[c,d]$ or $[d,c]$), $\operatorname{Per}(0,F) = \operatorname{Per^{\circ}}(f)$ and $\operatorname{Per}(F) = \operatorname{Per^{\circ}}(f) \cup M(0,d) \cup \Lambda(d,\operatorname{S\mbox{\tiny\textup{sh}}}(s_d))$.
A natural strategy to prove the second statement of Conjecture \[WishList\] in the general case (i.e. when no endpoint of the rotation interval is an integer) is to construct examples of maps $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$ with a *block structure* over maps $f\in {\mathcal{X}_{3}}$ in such a way that $p/q$ is an endpoint of the rotation interval ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)$ and $\operatorname{Per}(p/q,F) = q\cdot\operatorname{Per^{\circ}}(f)$. The next result shows that this is not possible. Hence, if the second statement of Conjecture \[WishList\] holds, the examples must be built by using some more complicated behavior of the points of the orbit in $\R$ and on the branches than a block structure.
Let $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ and let $P$ be a lifted periodic orbit of $F$ with period $nq$ and rotation number $p/q$. For every $x \in P$ and $i=0,1,\dots,q-1$, we set $$P_i(x):=\{ F^i(x), G(F^i(x)), G^2(F^i(x)), \dots, G^{n-1}(F^i(x)) \},$$ where $G := F^q -p$. By Lemma \[blocksareperiodic\], every $P_i(x)$ is a (true) periodic orbit of $G$ of period $n$.
\[ConverseEndInteger\] Let $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ and let $P$ be a lifted periodic orbit of $F$ with period $nq$ and rotation number $p/q$. Assume that there exists $x \in P$ such that $\chull{P_0(x)}$ is homeomorphic to a 3-star and $\chull{P_1(x)} \subset [n,n+1] \subset \R$ for some $n \in \Z$. Assume also that $P_0(x)$ is a periodic orbit of type 3 of $G:=F^q - p,$ $F^i(m) \in \chull{P_i(x)}$ for $i=0,1,\dots,q-1$ and $G(m) = m$, where $m \in \Z \cap \chull{P_0(x)}$ denotes the branching point of $\chull{P_0(x)}$. Then $\operatorname{Per}(p/q,F) = q\cdot\N$.
Next we study the first statement of Conjecture \[WishList\]. It turns out that there are two completely different types of lifted orbits according to the way that they force the existence of other periods. Namely, the lifted periodic orbits contained in $B$ (viewed at $\sigma$ level, this means that these periodic orbits do not intersect the circuit of $\sigma$) or the “rotational orbits” that visit the ground $\R$ of our space $S$. We start by studying the periods forced by the lifted periodic orbits contained in $B$. We also consider the special case of large orbits (i.e., orbits of large diameter) and show that any orbit of this kind implies periodic [[$\kern -0.55em\pmod{1}$]{}]{} points of all periods. To do this, we have to introduce some notation.
Let $F \in {\ensuremath{\mathcal{L}(S)}}$ and let $P$ be a lifted periodic orbit of $F$. We say that $P$ *lives in the branches* when $P \subset B$. Observe that, since $P$ is a lifted orbit, for every $m \in {\ensuremath{\mathbb{Z}}}$, $B_m \cap P
= (B_0 \cap P) + m$.
The following result holds for any degree. It extends [@LLl Proposition 5.1] (which deals with $\sigma$ maps fixing the branching point of $\sigma$) to all $\sigma$ maps.
\[TheoremSharkovskiiintheBranches\] Let $F \in {\ensuremath{\mathcal{L}(S)}}$ and let $P$ be a lifted periodic orbit of $F$ of period $p$ that lives in the branches. Then $\operatorname{Per}(F) \supset \operatorname{S\mbox{\tiny\textup{sh}}}(p)$. Moreover, for every $d \in {\ensuremath{\mathbb{Z}}}$ and every $p \in \N_{\Sho}$, there exists a map $F_p \in {\ensuremath{\mathcal{L}_{d}(S)}}$ such that $\operatorname{Per}(F_p) = \operatorname{S\mbox{\tiny\textup{sh}}}(p)$.
Let $F \in {\ensuremath{\mathcal{L}(S)}}$ and let $Q$ be a (true) periodic orbit of $F$. We say that $Q$ is a *large orbit* if $\operatorname{diam}(\Re(Q)) \ge 1$, where $\operatorname{diam}(\cdot)$ denotes the diameter of a set.
If $F \in {\ensuremath{\mathcal{L}(S)}}$ and if $Q$ is a true periodic orbit of $F$, then $Q + {\ensuremath{\mathbb{Z}}}$ is a lifted periodic orbit of $F$ of period $\operatorname{Card}(Q)$. Clearly, $Q \subset B$ if and only if $Q + {\ensuremath{\mathbb{Z}}}\subset B$. Therefore we shall also say that $Q$ *lives in the branches* whenever $Q
\subset B$. Moreover, when $F$ is of degree $1$, true periodic orbits correspond to lifted periodic orbits of rotation number $0$. Observe that a periodic orbit $Q$ living in the branches is large if and only if $Q$ intersects two different branches.
In the case of large orbits living in the branches and degree $1$ maps, we obtain the next result, much stronger than Theorem \[TheoremSharkovskiiintheBranches\]
\[LargeOrbitsintheBranches\] Let $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ and let $Q$ be a large orbit of $F$ such that $Q$ lives in the branches. Then $\operatorname{Per}(F) = {\ensuremath{\mathbb{N}}}$.
Large orbits contained in ${\ensuremath{\mathbb{R}}}$ work as in the circle case by using $\Re \circ F$. More precisely, if $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$ has a large orbit contained in $\R$, then so does the map $\Re \circ F$. Thus, by [@AlsRue2010 Theorem 2.2], there exists $n \in \N$ such that $$\left[-\tfrac{1}{n}, \tfrac{1}{n} \right] \subset
\operatorname{Rot}(\Re \circ F).$$ In the proof of [@AlsRue2008 Theorem 4.17], it is shown that, if $0\in\operatorname{Int}{\operatorname{Rot}(\Re \circ F)}$, then $F$ has a positive horseshoe and $\operatorname{Per}(0,F)={\ensuremath{\mathbb{N}}}$. Consequently, $\operatorname{Per}(F) \supset \operatorname{Per}(0, F) = \N$.
The set of periods of maps from ${\ensuremath{\mathcal{L}_{1}(S)}}$ having a large orbit that intersects both $\R$ and the branches remain unknown. Example \[ex:0inintRot-1\] shows that the existence of a large orbit does not ensure that $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$.
Next we study the orbits forced by the existence of lifted periodic orbits that intersect $\R.$ We obtain the following theorem, which is the main result of this paper.
\[theo:0inInterior\] Let $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$. If $\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)) \cap \Z \ne \emptyset$, then $\operatorname{Per}(F)$ is equal to, either ${\ensuremath{\mathbb{N}}}$, or ${\ensuremath{\mathbb{N}}}\setminus\{1\}$, or ${\ensuremath{\mathbb{N}}}\setminus\{2\}$. Moreover, there exist maps $F_0, F_1, F_2 \in {\ensuremath{\mathcal{L}_{1}(S)}}$ with $0 \in \operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F_i))$ for $i=0,1,2$ such that $\operatorname{Per}(F_0) = {\ensuremath{\mathbb{N}}}$, $\operatorname{Per}(F_1) = {\ensuremath{\mathbb{N}}}\setminus\{1\}$ and $\operatorname{Per}(F_2) = {\ensuremath{\mathbb{N}}}\setminus\{2\}$.
The paper is organized as follows. In Section \[sec:covering\], we state some relations about periodic points of different liftings, we recall the notions of covering and positive covering and give some of their properties, which are key tools for finding periodic points. In Section \[sec:Y\], we prove Corollary \[ConverseSigmaCircleCase\] and Theorems \[YinSigma\] and \[ConverseEndInteger\]. In Section \[WeAreInTheBranches\], we prove Theorems \[TheoremSharkovskiiintheBranches\] and \[LargeOrbitsintheBranches\]. Section \[sec:0inIntRotR\], devoted to Theorem \[theo:0inInterior\], starts with the construction of examples, then states some more technical lemmas about the set of periods and finally gives the proof of Theorem \[theo:0inInterior\]. In the last section, we stress some difficulties in the characterization of the set of periods: a first example shows that, in Theorem \[theo:0inInterior\], one cannot replace $\operatorname{Per}(F)$ by $\operatorname{Per}(0,F)$ (i.e., periods [[$\kern -0.55em\pmod{1}$]{}]{} by true periods), which is an obstacle to apply to $\sigma$ maps the same method as for circle maps; two other examples show that orderings $\leso{n}$ with $n>3$ may be needed to characterize $\operatorname{Per}(0,F)$, which might let us think that, in the first statement of Conjecture \[WishList\], considering orderings $\leso{2}$ and $\leso{3}$ may not be sufficient.
Coverings and periodic points {#sec:covering}
=============================
Relations between periodic points of *F* and of *F+k*
-----------------------------------------------------
Next easy lemma summarizes some basic properties of liftings; in particular, periodic [[$\kern -0.55em\pmod{1}$]{}]{} points do not depend on the choice of the lifting of a given $\sigma$-map.
\[lem:FF+k\] Let $F\in{\ensuremath{\mathcal{L}_{d}(S)}}$. The following statements hold for all $k,m\in{\ensuremath{\mathbb{Z}}}$ and all $n\ge 0$:
1. $F^n(x+m)=F^n(x)+md^n$; in particular, if $d=1$ then $F^n(x+m)=F^n(x)+m$,
2. $(F+k)^n(x)=F^n(x)+k(1+d+\cdots+d^{n-1})$; in particular, if $d=1$ then $(F+k)^n(x)=F^n(x)+kn$ and ${\rho_{_{F+k}}}(x)={\rho_{_{F}}}(x)+k$,
3. If $F'\in{\ensuremath{\mathcal{L}_{d'}(S)}}$, then $F'\circ F\in{\ensuremath{\mathcal{L}_{dd'}(S)}}$,
4. A point $x$ is periodic [[$\kern -0.55em\pmod{1}$]{}]{} of period $n$ for $F$ if and only if $x+m$ is periodic [[$\kern -0.55em\pmod{1}$]{}]{} of period $n$ for $F+k$. This implies in particular that $\operatorname{Per}(F)=\operatorname{Per}(F+k)$,
5. if $d=1$ and $F^n(x)=x+m$, then ${\rho_{_{F}}}(x)=m/n$; thus all periodic [[$\kern -0.55em\pmod{1}$]{}]{} points have rational rotation numbers.
Statements (a), (b) and (c) are [@AlsRue2008 Lemma 1.6] (see also [@AlsRue2008 Lemma 1.10(b)]), and (e) is [@AlsRue2008 Remark 1.14(ii)].
We set $G:=F+k$. By (a) and (b), $$\forall x\in S,
\forall i\in {\ensuremath{\mathbb{N}}},
G^i(x+m) = F^i(x) + m d^i + k\sum_{j=0}^{i-1} d^j.$$ Therefore $F^i(x)- x\in{\ensuremath{\mathbb{Z}}}$ if and only if $G^i(x+m)- (x+m)\in{\ensuremath{\mathbb{Z}}}$, which proves (d).
The next lemma is implicitly contained in [@AlsRue2008 Theorem 3.11]. It is a tool to relate the periods and rotation numbers of lifted periodic orbits with the periods of true orbits of appropriate powers of the map.
\[relationF\_Fqmp\] Let $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$, $p \in \Z$ and $q \in \N$ be such that $p,q$ are relatively prime. Then $x$ is a periodic [[$\kern -0.55em\pmod{1}$]{}]{} point of $F$ of period $mq$ and rotation number $p/q$ if and only if $x$ is a (true) periodic point of $F^q -p$ of period $m$.
Set $G:= F^q-p$. Assume first that $x$ is a period [[$\kern -0.55em\pmod{1}$]{}]{} point of $F$ of period $mq$ and rotation number $p/q$. From the definition of periodic [[$\kern -0.55em\pmod{1}$]{}]{} point, we have $F^{mq}(x) = x+k$ for some $k\in \Z$. Then $p/q = {\rho_{_{F}}}(x) = k/(mq)$ by Lemma \[lem:FF+k\](e). Hence $k= mp$.
By Lemma \[lem:FF+k\](b), $G^j(x) = F^{qj}(x) -jp$ for every $j \ge 0$. Consequently, $G^m(x) = F^{qm}(x) - mp = x + k - mp = x$ and $x$ is a true periodic point of $G$ of period a divisor of $m$. Now we have to prove that $G^j(x) \ne x$ for $j=1,2,\dots,m-1$. Assume on the contrary that $G^d(x) = x$ for some $d\in \{1,2,\dots,m-1\}$. From above, we have $x = G^d(x) = F^{qd}(x) - dp$. Hence $F^{qd}(x) - x \in \Z;$ a contradiction with the fact that $x$ is a periodic [[$\kern -0.55em\pmod{1}$]{}]{} point of $F$ of period $mq$. We deduce that $x$ is of period $m$ for $G$.
Assume now that $x$ is a (true) periodic point of $G$ of period $m$. From above, $x = G^m(x) = F^{qm}(x) - mp$. Thus, $F^{qm}(x) = x + mp$, ${\rho_{_{F}}}(x)=\tfrac{p}{q}$ and the period [[$\kern -0.55em\pmod{1}$]{}]{} of $x$ for $F$ is an integer $d$ that divides $qm$. Let $l\in{\ensuremath{\mathbb{N}}}$ and $a\in{\ensuremath{\mathbb{Z}}}$ be such that $d=\tfrac{mq}{l}$ and $F^d(x)=x+a$. To end the proof, we have to show that $d=qm$, that is, $l=1$. Assume on the contrary that $l > 1$. Then, by Lemma \[lem:FF+k\](b), $$x + mp = F^{mq}(x) = F^{ld}(x) = x + la = x + \frac{mq}{d} a.$$ Consequently, $a = d\tfrac{p}{q} \in \Z$. Thus $d$ must be a multiple of $q$ because $p,q$ are coprime. Write $d = bq$. Since $d=\tfrac{mq}{l}$, we obtain $b = \tfrac{m}{l} < m$. But, on the other hand, $F^d(x) = x + a$ can be written as $F^{bq}(x) = x + bp$, which is equivalent to $
x = (F^{bq}-bp)(x) = G^b(x).
$ This contradicts the fact that $x$ is a periodic point of $G$ of period $m$. We deduce that the period [[$\kern -0.55em\pmod{1}$]{}]{} of $x$ for $F$ is $mq$.
The following technical lemma will be useful to relate true periodic orbits of maps from ${\ensuremath{\mathcal{L}(S)}}$ wit lifted periodic orbits.
\[PeriodsAndPeriodsmodiAreFriends\] Let $F \in {\ensuremath{\mathcal{L}(S)}}$, $x \in S$ and $m,k \in {\ensuremath{\mathbb{Z}}}$. Set $G := F+k$ and $\widetilde{x} := x + m$.
1. If $\widetilde{x}$ is a true periodic point of $G$ of period $q$, then $x$ is a periodic [[$\kern -0.55em\pmod{1}$]{}]{} point of $F$ of period $q$. In particular, for $k=m=0$, it states that a true periodic point of $F$ is also a periodic [[$\kern -0.55em\pmod{1}$]{}]{} point of $F$ of the same period.
2. If $x$ is a periodic [[$\kern -0.55em\pmod{1}$]{}]{} point of $F$ of period $q$ and $
\operatorname{diam}(\operatorname{Orb}(\widetilde{x},G)) < 1,
$ then $\widetilde{x}$ is a true periodic point of $G$ of period $q$.
Let $d$ denote the degree of $F$. Suppose that $\widetilde x$ is a periodic point of $G$ of period $q$. Then $\widetilde x$ is periodic [[$\kern -0.55em\pmod{1}$]{}]{} of period $p$ for $G$ with $p$ a divisor of $q$. Let $n\in{\ensuremath{\mathbb{Z}}}$ and $a\in{\ensuremath{\mathbb{N}}}$ be such that $G^p(\widetilde x)=\widetilde x+n$ and $q=ap$. According to Lemma \[lem:FF+k\](a,c), the map $G^p$ is of degree $d^p$ and $$G^q(\widetilde x)=
G^{ap}(\widetilde x)=
\widetilde{x} + n\sum_{i=0}^{a-1} d^{pi}.$$ This equality is possible only if $n=0$. Thus $G^p(\widetilde x)=\widetilde
x$, which implies that $p=q$. Then (a) follows from Lemma \[lem:FF+k\](d).
Let $x$ be a periodic [[$\kern -0.55em\pmod{1}$]{}]{} point of $F$ of period $q$. Then $\widetilde x=x+m$ is periodic [[$\kern -0.55em\pmod{1}$]{}]{} of period $q$ for $G$ by Lemma \[lem:FF+k\](d). If $\operatorname{diam}(\operatorname{Orb}(\widetilde{x},G)) < 1$, the fact that $G^n(\widetilde{x}) - \widetilde{x} \in {\ensuremath{\mathbb{Z}}}$ is equivalent to $G^n(\widetilde{x}) = \widetilde{x}$. This implies that $\widetilde x$ is actually a true periodic point of period $q$ for $G$.
Coverings and periods
---------------------
Let $F \in {\ensuremath{\mathcal{L}(S)}}$ and let $I, J$ be compact non-degenerate subintervals of $S$. We say that $I$ *$F$-covers* $J$ if there exists a subinterval $I' \subset I$ such that $F(I') = J$. If $I_1,\ldots, I_k$ are compact non-degenerate intervals, the *$F$-graph* of $I_1,\ldots, I_k$ is the directed graph whose vertices are $I_1,\ldots, I_k$ and there is an arrow from $I_i$ to $I_j$ in the graph if and only if $I_i$ $F$-covers $I_j$. Then we write $I_i {\nolinebreak[4]\longrightarrow\nolinebreak[4]}I_j$ (or $I_i {\xrightarrow[F]{\hspace*{1.35em}}}I_j$ if the map needs to be specified) to mean that $I_i$ $F$-covers $I_j$. A *path of coverings of length $n$* is a sequence $$J_0{\xrightarrow[F_0]{\hspace*{1.35em}}}J_1{\xrightarrow[F_1]{\hspace*{1.35em}}}\cdots {\xrightarrow[F_{n-1}]{\hspace*{1.35em}}}J_n,$$ where $J_0,\ldots, J_n$ are compact non-degenerate intervals and [\[S\]]{} are continuous maps (generally of the form $F^{n_i}-p_i$) for all $0\le i\le n-1$. Such a path is called a *loop* if $J_n=J_0$. If all the maps $F_i$ are equal to $F$ and $J_0,\ldots, J_n\in
\{I_1,\ldots, I_k\}$, we speak about paths (resp. loops) in the $F$-graph of $I_1,\ldots, I_k$.
Consider two paths of the form $$\begin{gathered}
{\mathcal{A}}=J_0{\xrightarrow[F_0]{\hspace*{1.35em}}}J_1{\xrightarrow[F_1]{\hspace*{1.35em}}}\cdots {\xrightarrow[F_{n-1}]{\hspace*{1.35em}}}J_n,\\
{\mathcal{B}}= J_n{\xrightarrow[F_n]{\hspace*{1.35em}}}J_{n+1}{\xrightarrow[F_{n+1}]{\hspace*{1.35em}}}\cdots
{\xrightarrow[F_{n+m-1}]{\hspace*{1.35em}}}J_{n+m}.\end{gathered}$$ Then ${\mathcal{A}}{\mathcal{B}}$ will denote the concatenation of these two paths, that is, $${\mathcal{A}}{\mathcal{B}}= J_0{\xrightarrow[F_0]{\hspace*{1.35em}}}J_1{\xrightarrow[F_1]{\hspace*{1.35em}}} \cdots
{\xrightarrow[F_{n-1}]{\hspace*{1.35em}}}J_n {\xrightarrow[F_n]{\hspace*{1.35em}}}\cdots
{\xrightarrow[F_{n+m-1}]{\hspace*{1.35em}}}J_{n+m}.$$ If $J_n=J_0$, it is possible to concatenate ${\mathcal{A}}$ with itself and, for every $n\in{\ensuremath{\mathbb{N}}}$, ${\mathcal{A}}^n$ will denote the concatenation of ${\mathcal{A}}$ with itself $n$ times.
When considering an $F$-graph, the intervals are often defined from a finite collection of points.
\[def:basic-int\] Let $P$ be a finite subset of $S$. A *$P$-basic interval* is any set $\chull{a,b}$, where $a,b$ are two distinct points in $P$ such that $\chull{a,b}\cap
\chull{P}=\{a,b\}$. Observe that, if $P$ contains all the branching points ${\ensuremath{\mathbb{Z}}}\cap\chull{P}$, then the $P$-basic intervals are equal to the closure of the connected components of $\chull{P}\setminus P$.
If $\operatorname{Int}(I)$ and $\operatorname{Int}(J)$ contain no branching point, the fact that $F(I) \supset J$ implies $I {\nolinebreak[4]\longrightarrow\nolinebreak[4]}J.$ In what follows, we shall only use coverings with intervals containing no branching point in their interior.
The next result is the key property for finding periodic points with coverings. It is [@ALM Lemma 1.2.7] generalized to intervals in $S$.
\[prop:covering\] Let $I_0,I_1,\ldots, I_n$ be compact subintervals of $S$ with $I_n=I_0$ and, for every $0\le i\le n-1$, let [\[S\]]{} be a continuous map such that $I_i$ $F_i$-covers $I_{i+1}$. Then there exist points $x_i\in I_i$, $i=0,\ldots, n$, such that $F_i(x_i)=x_{i+1}$ for all $0\le i\le n-1$ and $x_n=x_0$. In particular,
- if $F_i=F$ for all $0\le i\le n-1$ (that is, $I_0 {\nolinebreak[4]\longrightarrow\nolinebreak[4]}I_1 {\nolinebreak[4]\longrightarrow\nolinebreak[4]}\cdots {\nolinebreak[4]\longrightarrow\nolinebreak[4]}I_{n-1} {\nolinebreak[4]\longrightarrow\nolinebreak[4]}I_0$ is a loop in the $F$-graph of $I_1,\ldots, I_{n-1}$), then $F^n(x_0)=x_0$;
- if $F_i=F+k_i$ with $k_i\in{\ensuremath{\mathbb{Z}}}$ for all $0\le i\le n-1$, then $F^n(x_0)\in x_0+{\ensuremath{\mathbb{Z}}}$.
The next lemma shows that, under certain hypotheses (that is, in presence of “semi horseshoes”), we have periodic points of all periods. It is a generalization of [@ALM Proposition 1.2.9] and its proof is a variant of the proof of that result. However, we include it for clarity.
\[prop:SemiHorseshoe\] Let $F \in {\ensuremath{\mathcal{L}(S)}}$ and assume that there exist two compact non-degenerate subintervals $K$ and $L$ of $S$ such that $K$ and $L$ do not contain branching points in their interior, $\operatorname{Int}(K) \cap \operatorname{Int}(L) = \emptyset$ and $F(K) \supset L$ and $F(L) \supset K \cup L$. Then, for every $n \in
{\ensuremath{\mathbb{N}}},$ the map $F$ has a periodic orbit of period $n$ contained in $K \cup L$.
By assumption, $K {\nolinebreak[4]\longrightarrow\nolinebreak[4]}L$ and $L {\nolinebreak[4]\longrightarrow\nolinebreak[4]}K, L$. Since $K,L$ contain no branching point in their interior, the set $J := \chull{K \cup L}$ is an interval (which may contain branching points). By continuity of $F$, there exist subintervals $L' \subset L$ and $K' \subset K$ such that $F(L') \supset J$, $F(\operatorname{Bd}(L')) = \operatorname{Bd}(J)$, $F(K') = L'$ and $F(\operatorname{Bd}(K')) = \operatorname{Bd}(L')$. Therefore, for every $n\in{\ensuremath{\mathbb{N}}}$, there is a loop $$\CPath{K'}>{L'}>{L'}>{\dots}>{L'}>{K'}$$ of length $n$ in the $F$-graph of $K', L'$ (if $n = 1$, the loop we take is $L' {\nolinebreak[4]\longrightarrow\nolinebreak[4]}L'$). By Proposition \[prop:covering\], $F$ has a periodic point $x \in K'$ such that $F^i(x) \in L'$ for $i = 1,2,\dots,n-1$ and $F^n(x) = x$ (if $n = 1$, $F(x)=x\in L'$). To prove that $x$ has period $n$, we have to show that $F^i(x) \neq x$ for all $i = 1,2,\dots,n-1$.
Suppose now that $F^i(x) = x$ for some $i \in \{1,2,\dots,n-1\}$ (in particular $n > 1$). Then $x=F^i(x)$ belongs to $K'\cap L'$, and hence $$\label{eq:xinL}
x\in \operatorname{Bd}(L').$$ Consequently, $F(x) = F^{i+1}(x) \in \operatorname{Bd}(J)$. If $i+1 \le n-1$, then $F(x) = F^{i+1}(x)$ also belongs to $L'$ and, hence, it is the unique point in $\operatorname{Bd}(L') \cap \operatorname{Bd}(J)$ and, again, $F^2(x) = F^{i+2}(x) \in \operatorname{Bd}(J)$. Iterating this argument, we see that $F^l(x) = F^{i+l}(x) \in \operatorname{Bd}(J)$ for all $l=0,1,\dots, n-i$. Then $x = F^n(x) = F^{n-i}(x) \in K' \cap \operatorname{Bd}(J)$, which implies that $x$ is the endpoint of $J$ that does not belong to $L'$. But this contradicts . We conclude that the period of $x$ is equal to $n$.
The next lemma is similar to the previous one, except that the coverings are [[$\kern -0.55em\pmod{1}$]{}]{}.
\[lem:SemiHorseshoe-mod1\] Let $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$. Let $I,J$ be two non empty compact intervals in $S$ such that $\operatorname{Int}(I),\operatorname{Int}(J)$ are disjoint and contain no branching point. Suppose that there exist $k_1,k_2,k_3\in{\ensuremath{\mathbb{Z}}}$ such that $$I {\xrightarrow[F-k_1]{\hspace*{1.35em}}} I, \quad
I {\xrightarrow[F-k_2]{\hspace*{1.35em}}} J, \quad
J {\xrightarrow[F-k_3]{\hspace*{1.35em}}} I.$$ Suppose in addition that
- either $I,J$ are disjoint [[$\kern -0.55em\pmod{1}$]{}]{} (that is, $(I+{\ensuremath{\mathbb{Z}}}) \cap (J+{\ensuremath{\mathbb{Z}}})=\emptyset$),
- or $k_3=k_1$.
Then $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$.
We fix $n\in{\ensuremath{\mathbb{N}}}$. For $n = 1$, we consider the loop $I {\xrightarrow[F-k_1]{\hspace*{1.35em}}} I$, and there exists a fixed [[$\kern -0.55em\pmod{1}$]{}]{} point in $I$ by Proposition \[prop:covering\]. For $n\ge 2$, we consider the loop of length $n$ $$J {\xrightarrow[F-k_3]{\hspace*{1.35em}}} I {\xrightarrow[F-k_1]{\hspace*{1.35em}}} I
{\xrightarrow[F-k_1]{\hspace*{1.35em}}} \cdots
{\xrightarrow[F-k_1]{\hspace*{1.35em}}} I {\xrightarrow[F-k_2]{\hspace*{1.35em}}} J.$$ By Proposition \[prop:covering\], $F$ has a periodic [[$\kern -0.55em\pmod{1}$]{}]{} point $x\in J$ such that $F^n(x)=x+k_3+(n-2)k_1+k_2$ and $F^i(x)\in I+k_3+(i-1)k_1$ for all $1\le i\le n-1$. Let $d$ denote the period [[$\kern -0.55em\pmod{1}$]{}]{} of $x$.
If $I,J$ are disjoint [[$\kern -0.55em\pmod{1}$]{}]{}, then $F^i(x)-x\notin {\ensuremath{\mathbb{Z}}}$ for all $1\le i\le n-1$, and thus $d=n$.
Suppose now that $k_3=k_1\neq k_2$. Then $${\rho_{_{F}}}(x) =
\frac{k_3+(n-2)k_1+k_2}{n} =
k_1+\frac{k_2-k_1}{n}.$$ If $d<n$, then $F^d(x)=x+k_3+(d-1)k_1$ and hence $${\rho_{_{F}}}(x)=\frac{k_3+(d-1)k_1}{d}=k_1.$$ But this is impossible because $\frac{k_2-k_1}{n}\neq 0$. We deduce that, if $k_3=k_1\neq k_2$, then $d=n$.
Finally, if $k_1=k_2=k_3$, then Proposition \[prop:SemiHorseshoe\] applies to the map $G:=F-k_1$ and $\operatorname{Per}(G)={\ensuremath{\mathbb{N}}}$. Thus $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$ by Lemma \[lem:FF+k\](d). This concludes the proof.
Positive coverings
------------------
The notion of positive covering for subintervals of ${\ensuremath{\mathbb{R}}}$ was introduced in [@AlsRue2008]. It can be extended to all subintervals on which a retraction can be defined. This is in particular the case of all intervals which have an infinite tree as the ambient space.
If $I \subset S$ is an interval, it can be endowed with two opposite linear orders; we denote them by $<_{_I}$ and $>_{_I}$. When $I\subset {\ensuremath{\mathbb{R}}}$, we choose $<_{_I}$ so that it coincide with the order $<$ in ${\ensuremath{\mathbb{R}}}$; when $I\subset B$, we choose $<_{_I}$ so that $x<_{_I} y\Leftrightarrow \Im(x)<\Im(y)$. In the other cases, $<_{_I}$ is chosen arbitrarily. The notations $\le_{_I}$ and $\ge_{_I}$ are defined consistently.
\[def:positivecover\] Let $F \in {\ensuremath{\mathcal{L}(S)}}$ and let $I,J$ be compact non-degenerate subintervals of $S$, endowed with orders $<_{_I}, <_{_J}$. We say that $(I, <_{_I})$ *positively* (resp. *negatively*) *$F$-covers $(J,<_{_J})$* and we write $(I, <_{_I}) { \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} (J,<_{_J})$ (resp. $(I, <_{_I}) { \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}-\hspace*{.1em}} \nolinebreak[4]} (J,<_{_J})$) if there exist $x,y\in I$ such that $x \le_{_I} y$, $F(x)=\min J$ and $F(y)=\max J$ (resp. $F(x)=\max J$ and $F(y)=\min J$). When there is no ambiguity on the orders (or no need to precise them), we simply write $I { \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} J$ or $I { \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}-\hspace*{.1em}} \nolinebreak[4]} J$.
We remark that the notion of positive or negative covering does not imply (unlike the usual notion of $F$-covering) that there exists a closed subinterval of $I' \subset I$ such that $F(I') = J$. However, it does for the retracted map.
We recall that the retraction [\[I\]]{} is defined as follows: $${r}_{_I}(x) = \begin{cases}
x & \text{if $x \in I$}\\
c_x & \text{if $x \notin I$,}
\end{cases}$$ where $c_x$ is the only point in $I$ such that $\chull{c_x, x} \cap I = \{c_x\}$ (it exists since $S$ is uniquely arcwise connected).
$(I, <_{_I})$ positively (resp. negatively) $F$-covers $(J,<_{_J})$ if and only if there exist $x,y\in I$, $x \le_{_I} y$, such that ${r}_{_J} \circ F(x)=\min J$ and ${r}_{_J} \circ F(y)=\max J$ (resp. ${r}_{_J} \circ F(x)=\max J$ and ${r}_{_J} \circ F(y)=\min J$). Moreover, if $I$ positively or negatively $F$-covers $J$, then there exists a closed subinterval $I' \subset I$ such that ${r}_{_J}(F(I')) = J$ and $F(\operatorname{Bd}(I'))=\operatorname{Bd}(J)$.
If ${\varepsilon},{\varepsilon}'\in\{+,-\}$, the product ${\varepsilon}{\varepsilon}'\in\{+,-\}$ denotes the usual product of signs, and $-{\varepsilon}$ denotes the opposite sign.
A *loop of signed coverings of length $k$* is a sequence $$(I_0, <_{_0}) { \nolinebreak[4] \xrightarrow[F_1]{\hspace*{.25em}{\varepsilon}_1\hspace*{.1em}} \nolinebreak[4]}
(I_1, <_{_1}) { \nolinebreak[4] \xrightarrow[F_2]{\hspace*{.25em}{\varepsilon}_2\hspace*{.1em}} \nolinebreak[4]} \cdots
(I_{k-1}, <_{_{k-1}}) { \nolinebreak[4] \xrightarrow[F_k]{\hspace*{.25em}{\varepsilon}_k\hspace*{.1em}} \nolinebreak[4]} (I_0, <_{_0}),$$ where $(I_0,<_{_0}),(I_1, <_{_1}),\dots,(I_{k-1}, <_{_{k-1}})$ are compact non-degenerate intervals of $S$ endowed with an order, ${\varepsilon}_i\in \{+,-\}$ and [\[S\]]{} are continuous maps (usually of the form $F^{n_i}-p_i$) for all $1\le i\le k$. The *sign* of the loop is defined to be the product ${\varepsilon}_1{\varepsilon}_2\cdots{\varepsilon}_k$. The loop is said *positive* (resp. *negative*) depending on its sign. We shall use the same notations for concatenations of paths of signed coverings as for coverings. It is clear that the sign of the concatenation is the product of the signs of the paths involved.
The next lemma studies the dependence of the sign of a loop of signed coverings on the chosen orderings.
\[lem:make-coverings-positive\] Let $$(I_0, <_{_0}) { \nolinebreak[4] \xrightarrow[F_1]{\hspace*{.25em}{\varepsilon}_1\hspace*{.1em}} \nolinebreak[4]}
(I_1, <_{_1}){ \nolinebreak[4] \xrightarrow[F_2]{\hspace*{.25em}{\varepsilon}_2\hspace*{.1em}} \nolinebreak[4]} \cdots
(I_{k-1}, <_{_{k-1}}){ \nolinebreak[4] \xrightarrow[F_k]{\hspace*{.25em}{\varepsilon}_k\hspace*{.1em}} \nolinebreak[4]}(I_0, <_{_0}),$$ be a loop of signed coverings of sign ${\varepsilon}$.
1. For every $0\le i\le k-1$, let $\widetilde{<_{_i}}\in\{<_{i},>_{i}\}.$ Then, there exist ${\varepsilon}_1',\ldots,{\varepsilon}_k'\in \{+,-\}$ such that $$(I_0, \widetilde{<_{_0}}){ \nolinebreak[4] \xrightarrow[F_1]{\hspace*{.25em}{\varepsilon}_1'\hspace*{.1em}} \nolinebreak[4]}
(I_1, \widetilde{<_{_1}}) { \nolinebreak[4] \xrightarrow[F_2]{\hspace*{.25em}{\varepsilon}_2'\hspace*{.1em}} \nolinebreak[4]} \cdots
(I_{k-1}, \widetilde{<_{_{k-1}}}){ \nolinebreak[4] \xrightarrow[F_k]{\hspace*{.25em}{\varepsilon}_k'\hspace*{.1em}} \nolinebreak[4]}
(I_0, \widetilde{<_{_0}}),$$ and the sign of this loop is equal to ${\varepsilon}$. Consequently, the sign of a loop is independent of the orders.
2. For every $1\le i\le k-1$, there exists $\widetilde{<_{_i}}\in\{<_{_i},>_{_i}\}$ such that $$(I_0, <_{_0}){ \nolinebreak[4] \xrightarrow[F_1]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}
(I_1, \widetilde{<_{_1}}){ \nolinebreak[4] \xrightarrow[F_2]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}
\cdots { \nolinebreak[4] \xrightarrow[F_{k-1}]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}(I_{k-1},
\widetilde{<_{_{k-1}}}){ \nolinebreak[4] \xrightarrow[F_k]{\hspace*{.25em}{\varepsilon}\hspace*{.1em}} \nolinebreak[4]} (I_0, <_{_0}).$$
Consider a sequence of two signed coverings $
(I,<_{_I}){ \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}{\varepsilon}\hspace*{.1em}} \nolinebreak[4]}
(J,<_{_J}){ \nolinebreak[4] \xrightarrow[G]{\hspace*{.25em}{\varepsilon}'\hspace*{.1em}} \nolinebreak[4]}
(K, <_{_K}).
$ If we reverse the order on $J$, it is clear from the definition that we reverse the signs of both coverings. That is, $$\label{eq:reverse-covering}
(I,<_{_I}) { \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}-{\varepsilon}\hspace*{.1em}} \nolinebreak[4]}
(J,>_{_J}) { \nolinebreak[4] \xrightarrow[G]{\hspace*{.25em}-{\varepsilon}'\hspace*{.1em}} \nolinebreak[4]}
(K, <_{_K}).$$ To prove (a), it is sufficient to show that reversing any order gives a new loop of signed coverings with the same sign. If $1 \le i \le k-1$, according to , changing $<_{_i}$ into $>_{_i}$ changes ${\varepsilon}_{i-1}$ and ${\varepsilon}_i$ into $-{\varepsilon}_{i-1}$ and $-{\varepsilon}_i$ respectively. Changing $<_{_0}$ into $>_{_0}$ changes ${\varepsilon}_1$ and ${\varepsilon}_k$ into $-{\varepsilon}_1$ and $-{\varepsilon}_k$ respectively. In both cases, we obtain a new loop of signed coverings with the same sign.
To prove (b), we define inductively $\widetilde{<_{_i}}$ for $i=1,\ldots k-1$.
Let $i \in \{1,\ldots, k-1\}$ and suppose that $\widetilde{<_{_1}},\dots, \widetilde{<_{i-1}}$ have already been chosen such that $$(I_0, <_{_0}) { \nolinebreak[4] \xrightarrow[F_1]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}
(I_1, \widetilde{<_{_1}}) { \nolinebreak[4] \xrightarrow[F_2]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}
\cdots { \nolinebreak[4] \xrightarrow[F_{i-1}]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}
(I_{i-1}, \widetilde{<_{i-1}}) { \nolinebreak[4] \xrightarrow[F_i]{\hspace*{.25em}{\varepsilon}_i'\hspace*{.1em}} \nolinebreak[4]}
(I_i, <_{_i}) { \nolinebreak[4] \xrightarrow[F_{i+1}]{\hspace*{.25em}{\varepsilon}_{i+1}'\hspace*{.1em}} \nolinebreak[4]}
\cdots { \nolinebreak[4] \xrightarrow[F_k]{\hspace*{.25em}{\varepsilon}_k'\hspace*{.1em}} \nolinebreak[4]}
(I_0, <_{_0}),$$ for some ${\varepsilon}_i',\ldots,{\varepsilon}_k'\in\{+,-\}$. If ${\varepsilon}_i'=+$, let $\widetilde{<_{_i}}$ be equal to $<_{_i}$ and ${\varepsilon}_{i+1}'':={\varepsilon}_{i+1}'$. Otherwise, let $\widetilde{<_{_i}}$ be equal to $>_{_i}$ and ${\varepsilon}_{i+1}'':=-{\varepsilon}_{i+1}'$. According to , we obtain $$\begin{gathered}
(I_0, <_{_0}) { \nolinebreak[4] \xrightarrow[F_1]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}
\cdots { \nolinebreak[4] \xrightarrow[F_{i-1}]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}
(I_{i-1}, \widetilde{<_{i-1}}) { \nolinebreak[4] \xrightarrow[F_i]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}
(I_i, \widetilde{<_{_i}}) { \nolinebreak[4] \xrightarrow[F_{i+1}]{\hspace*{.25em}{\varepsilon}_{i+1}''\hspace*{.1em}} \nolinebreak[4]}\\
(I_i, <_{i+1}) { \nolinebreak[4] \xrightarrow[F_{i+2}]{\hspace*{.25em}{\varepsilon}_{i+2}'\hspace*{.1em}} \nolinebreak[4]}
\cdots { \nolinebreak[4] \xrightarrow[F_k]{\hspace*{.25em}{\varepsilon}_k'\hspace*{.1em}} \nolinebreak[4]}
(I_0, <_{_0}).\end{gathered}$$ Then, when all orderings $\widetilde{<_{_1}},\ldots, \widetilde{<_{_{k-1}}}$ are defined, we obtain $$(I_0, <_{_0}) { \nolinebreak[4] \xrightarrow[F_1]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}
(I_1, \widetilde{<_{_1}}) { \nolinebreak[4] \xrightarrow[F_2]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}
\cdots { \nolinebreak[4] \xrightarrow[F_{k-1}]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}
(I_{k-1}, \widetilde{<_{_{k-1}}}) { \nolinebreak[4] \xrightarrow[F_k]{\hspace*{.25em}{\varepsilon}'\hspace*{.1em}} \nolinebreak[4]}
(I_0, <_{_0})$$ for some ${\varepsilon}'\in\{+,-\}$. The sign of this loop is ${\varepsilon}'$, which is equal to ${\varepsilon}$ according to (a).
The next result is the analogous of Proposition \[prop:covering\] for signed coverings.
\[prop:signedcover\] Let $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ and let $(I_0,<_{_0}), (I_1,<_{_1}), \dots, (I_{k-1},<_{_{k-1}})$ be compact non degenerate intervals of $S$ endowed with an order such that $$(I_0,<_{_0}) { \nolinebreak[4] \xrightarrow[F^{n_1}-p_1]{\hspace*{.25em}{\varepsilon}_1\hspace*{.1em}} \nolinebreak[4]}
(I_1,<_{_1}) { \nolinebreak[4] \xrightarrow[F^{n_2}-p_2]{\hspace*{.25em}{\varepsilon}_2\hspace*{.1em}} \nolinebreak[4]}
\cdots { \nolinebreak[4] \xrightarrow[F^{n_{k-1}}-p_{k-1}]{\hspace*{.25em}{\varepsilon}_{k-1}\hspace*{.1em}} \nolinebreak[4]}
(I_{k-1},<_{_{k-1}}) { \nolinebreak[4] \xrightarrow[F^{n_k}-p_k]{\hspace*{.25em}{\varepsilon}_k\hspace*{.1em}} \nolinebreak[4]}
(I_0,<_{_0})$$ is a positive loop of signed coverings, where $n_i\in{\ensuremath{\mathbb{N}}}$ and $p_i\in{\ensuremath{\mathbb{Z}}}$. For every $i\in\{1,2,\ldots, k\}$, set $m_i:=\sum_{j=1}^i n_j$ and $\widehat{p_i}:=\sum_{j=1}^i p_j$. Then there exists $x_0 \in I_0$ such that $F^{m_k}(x_0) = x_0 + \widehat{p_k}$ and $F^{m_i}(x_0) \in I_i + \widehat{p_i}$ for all $1\le i\le k-1$.
According to Lemma \[lem:make-coverings-positive\], for every $1\le i\le k-1$, there exists $\widetilde{<_{_i}} \in \{<_{_i},>_{_i}\}$ such that $$(I_0,<_{_0}) { \nolinebreak[4] \xrightarrow[F^{n_1}-p_1]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}
(I_1,\widetilde{<_{_1}}) { \nolinebreak[4] \xrightarrow[F^{n_2}-p_2]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}
\cdots { \nolinebreak[4] \xrightarrow[F^{n_{k-1}}-p_{k-1}]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}
(I_{k-1},\widetilde{<_{_{k-1}}}) { \nolinebreak[4] \xrightarrow[F^{n_k}-p_k]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}
(I_0,<_{_0}).$$ Thus we can consider a loop in which all coverings are positive. In this case, we have the same situation as [@AlsRue2008 Proposition 2.3] except that [@AlsRue2008 Proposition 2.3] is stated for subintervals of ${\ensuremath{\mathbb{R}}}$. Actually this assumption plays no role (except simplifying the notations), and the proof in our context works exactly the same by using the map $F$ composed with appropriate retractions.
The next result is analogous to Lemma \[lem:SemiHorseshoe-mod1\] (indeed to a particular case of Lemma \[lem:SemiHorseshoe-mod1\]) with the semi horseshoe being made of positive coverings.
\[cory:+horseshoeFF-1\] Let $F\in {\ensuremath{\mathcal{L}_{1}(S)}}$ and let $I \subset S$ be a compact interval such that $(I+n)_{n\in{\ensuremath{\mathbb{Z}}}}$ are pairwise disjoint. If $I { \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} I$ and $I { \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} I+k$ for some $k \in {\ensuremath{\mathbb{Z}}}\setminus\{0\}$, then $\operatorname{Per}(F) = {\ensuremath{\mathbb{N}}}$.
Fix $n\in{\ensuremath{\mathbb{N}}}$. We consider the following loop of positive coverings of length $n$: $$I { \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} I { \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} I \cdots { \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} I
{ \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} I+k.$$ By Proposition \[prop:signedcover\], there exists a point $x\in I$ such that $F^n(x) = x+k$ and $F^i(x) \in I$ for all $1 \le i \le n-1$. In particular, ${\rho_{_{F}}}(x) = k/n \neq 0$. Suppose that $F^i(x) \in x+{\ensuremath{\mathbb{Z}}}$ for some $i \in \{1,2,\dots,n-1\}$. Both $x$ and $F^i(x)$ belong to $I$, and thus $F^i(x) = x$ because $(I+n)_{n\in{\ensuremath{\mathbb{Z}}}}$ are pairwise disjoint. But this implies that ${\rho_{_{F}}}(x) = 0$, which is a contradiction. Therefore the period ${\ensuremath{\kern -0.55em\pmod{1}}}$ of $x$ is equal to $n$. Finally, $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$.
The next lemma is a technical result in the spirit of the previous one. It shows that, when certain signed loops are available, the set of periods contains ${\ensuremath{\mathbb{N}}}\setminus\{2\}.$
\[lem:+-loop\] Let $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$. Let $K, L\subset S$ be two compact intervals in $S$ and let $e\in S$ be such that $(K+{\ensuremath{\mathbb{Z}}}) \cap (L+{\ensuremath{\mathbb{Z}}}) \subset \{e\}+{\ensuremath{\mathbb{Z}}}$ and $F(e) \notin L+{\ensuremath{\mathbb{Z}}}$. Suppose that there exist $k_1,k_2,k_3,k_4\in{\ensuremath{\mathbb{Z}}}$ such that $$L { \nolinebreak[4] \xrightarrow[F-k_1]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} L,\quad L { \nolinebreak[4] \xrightarrow[F-k_2]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} K,\quad
K { \nolinebreak[4] \xrightarrow[F-k_3]{\hspace*{.25em}-\hspace*{.1em}} \nolinebreak[4]} L,\quad K { \nolinebreak[4] \xrightarrow[F-k_4]{\hspace*{.25em}-\hspace*{.1em}} \nolinebreak[4]} K.$$ Then $\operatorname{Per}(F)\supset {\ensuremath{\mathbb{N}}}\setminus\{2\}.$
According to Proposition \[prop:signedcover\] applied to the loop $L{ \nolinebreak[4] \xrightarrow[F-k_1]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} L$, there exists a fixed point [[$\kern -0.55em\pmod{1}$]{}]{} of $F$ in $L$. Hence $1 \in \operatorname{Per}(F)$.
We now fix $n \ge 3$ and we consider the following positive loop of length $n$: $$(L { \nolinebreak[4] \xrightarrow[F-k_2]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} K { \nolinebreak[4] \xrightarrow[F-k_4]{\hspace*{.25em}-\hspace*{.1em}} \nolinebreak[4]} K { \nolinebreak[4] \xrightarrow[F-k_3]{\hspace*{.25em}-\hspace*{.1em}} \nolinebreak[4]} L)
(L { \nolinebreak[4] \xrightarrow[F-k_1]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} L)^{n-3}.$$ By Proposition \[prop:signedcover\], there exists a point $x \in L$ such that $F(x) \in K+{\ensuremath{\mathbb{Z}}}$, $F^2(x) \in K + {\ensuremath{\mathbb{Z}}}$, $F^i(x) \in L + {\ensuremath{\mathbb{Z}}}$ for all $3\le i\le n$ and $F^n(x) -x\in{\ensuremath{\mathbb{Z}}}$. Thus $x$ is a periodic [[$\kern -0.55em\pmod{1}$]{}]{} point for $F$ and its period $p$ divides $n$. It remains to prove that the period [[$\kern -0.55em\pmod{1}$]{}]{} of $x$ is exactly $n$. Suppose on the contrary that $p < n$. Then $1 \le p \le n-2$ because $p$ divides $n \ge 3$. Thus $F^2(x)\in K+{\ensuremath{\mathbb{Z}}}$, $F^{2+p}(x)\in L+{\ensuremath{\mathbb{Z}}}$ and $F^{2+p}(x)-F^2(x)\in{\ensuremath{\mathbb{Z}}}$. By assumption, this is possible only if $F^2(x)\in e+ {\ensuremath{\mathbb{Z}}}$. This leads to a contradiction because $F^3(x)\in L+{\ensuremath{\mathbb{Z}}}$ whereas $F(e)\notin L+{\ensuremath{\mathbb{Z}}}$. This proves that $p = n$, and hence, $\forall n\ge 3$, $n\in \operatorname{Per}(F)$. Consequently, $\operatorname{Per}(F)\supset {\ensuremath{\mathbb{N}}}\setminus\{2\}.$
Sets of periods of 3-star and degree 1 circle maps occur for degree 1 sigma maps {#sec:Y}
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Misiurewicz’s Theorem \[S9.5\] gives a characterization of the sets of periods of circle maps of degree $1$. It is very easy to build a map in ${\ensuremath{\mathcal{L}_{1}(S)}}$ whose set of periods [[$\kern -0.55em\pmod{1}$]{}]{} is equal to the set of periods of a given degree $1$ circle maps. This leads to the following result (see Section \[sec:statements\] for the notations).
Given $c,d \in \R$ with $c \le d$ and $s_c, s_d\in\N_{\Sho}$, there exists a map $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ such that ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F) = \operatorname{Rot}(F)=[c,d]$ and $\operatorname{Per}(F) = \Lambda(c,\operatorname{S\mbox{\tiny\textup{sh}}}(s_c)) \cup M(c,d) \cup
\Lambda(d,\operatorname{S\mbox{\tiny\textup{sh}}}(s_d))$.
By Theorem \[S9.5\], there exists a map $\widetilde{F} \in {\mathcal{L}}_1({\ensuremath{\mathbb{R}}})$ such that $\operatorname{Rot}(\widetilde{F}) = [c,d]$ and $\operatorname{Per}(\widetilde{F}) =
\Lambda(c,\operatorname{S\mbox{\tiny\textup{sh}}}(s_c)) \cup M(c,d) \cup \Lambda(d,\operatorname{S\mbox{\tiny\textup{sh}}}(s_d))$. Then we define $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ by $$F(x) = \begin{cases}
\widetilde{F}(x) & \text{if $x \in \R$},\\
\widetilde{F}(m) & \text{if $x \in B_m$}.
\end{cases}$$ Clearly, $F$ is continuous, $\operatorname{Rot}(F)={\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F) = \operatorname{Rot}(\widetilde{F})$ and every periodic [[$\kern -0.55em\pmod{1}$]{}]{} point of $F$ is contained in $\R$. Hence, $\operatorname{Per}(F) = \operatorname{Per}(\widetilde{F})$. This ends the proof of the corollary.
It is also easy to build a map in ${\ensuremath{\mathcal{L}_{1}(S)}}$ whose set of periods is equal to the set of periods of a given $3$-star map. This construction can be done in such a way that the rotation interval is any interval of the form $[0,d]$ or $[d,0]$. The set of periods [[$\kern -0.55em\pmod{1}$]{}]{} is then a combination of a set of periods of a $3$-star map and a set of periods of a degree $1$ circle map, as stated in the next result.
Let $d \ne 0$ be a real number, $s_d\in\N_{\Sho}$ and $f \in {\mathcal{X}_{3}}$. Then there exists a map $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ such that ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)=\operatorname{Rot}(F)$ is the closed interval with endpoints $0$ and $d$ (i.e., $[c,d]$ or $[d,c]$), $\operatorname{Per}(0,F) = \operatorname{Per^{\circ}}(f)$ and $\operatorname{Per}(F) = \operatorname{Per^{\circ}}(f) \cup M(0,d)
\cup \Lambda(d,\operatorname{S\mbox{\tiny\textup{sh}}}(s_d))$.
We shall only consider the case $d > 0$. The case $d$ negative is analogous.
From Theorem \[S9.5\], it follows that there exists a map $G \in {\mathcal{L}}_1({\ensuremath{\mathbb{R}}})$ such that $\operatorname{Rot}(G) = [0,d]$ and $\operatorname{Per}(G) = \{1\} \cup M(c,d) \cup \Lambda(d,\operatorname{S\mbox{\tiny\textup{sh}}}(s_d))$. Moreover, from the proof of Theorem \[S9.5\] (see [@ALM Theorem 3.10.1]), the map $G$ is constructed in such a way that $G(0) = 0$, there exist $u \le 1/2 \le v$ such that $G{\bigr\rvert_{[0,u]}}$ and $G{\bigr\rvert_{[v,1]}}$ are affine and ${\rho_{_{G}}}(x) = d$ for every $x \in [u,v]$, and ${\rho_{_{G}}}(x) \ne 0$ for every $x \in \R \setminus \bigcup_{n\ge 0}G^{-n}(\Z)$.
To prove the theorem, we shall construct a map $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ such that ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F) = \operatorname{Rot}(F)=\operatorname{Rot}(G) = [0,d],$ $\operatorname{Per}(0,F) = \operatorname{Per^{\circ}}(f)$ and $\operatorname{Per}(F) = \operatorname{Per}(0,F) \cup \operatorname{Per}(G)$.
Let $0 < b < a < 1/2$. For every $m \in \Z$, let $Y^a_m$ (resp. $Y^b_m$) denote the set $[m-a, m+a] \cup B_m$ (resp. $[m-b, m+b] \cup B_m$). Observe that $Y^a_m \cap Y^a_j = \emptyset$ whenever $m \ne j$, $Y^b_m \subset Y^a_m$, and the set $Y^a_m \setminus Y^b_m$ has two connected components: $[m-a,m-b)$ and $(m+b,m+a]$. Moreover, the sets $Y^a_m$ and $Y^b_m$ are homeomorphic to $X_3.$ Let $\beta_0$ denote a homeomorphism from $Y^b_0$ to $X_3$.
Set $Z:=\bigcup_{i=0}^{\infty} G^{-i}(\Z)$. Since $G(m) = m$ for every $m \in \Z$, both sets $Z$ and $\R \setminus Z$ are $G$-invariant and ${\ensuremath{\mathbb{Z}}}\subset Z$. Moreover, ${\rho_{_{G}}}(x) = 0$ for all $x\in Z$. Thus, $Z \cap \left(G([u,v]) + \Z\right) = \emptyset$ and $$\label{eq:tt}
Z \subset ([0,u) \cup (v,1]) + \Z$$ because $d \ne 0.$ Since $G|_{[0,u]}$ and $G|_{[v,1]}$ are affine, this implies that every point in $Z$ has finitely many preimages and, hence, $Z$ is countable. Moreover, since $G$ has degree one (Lemma \[lem:FF+k\](a)), $Z + \Z = Z$. Therefore, there exists a continuous map $\map{\varphi}{S}[\R]$ such that $\varphi(x+1) = \varphi(x) +1$ for all $x\in S$, $\varphi^{-1}(m)=Y_m^a$ for every $m \in \Z,$ $\varphi{\bigr\rvert_{\R}}$ is non-decreasing, $\varphi^{-1}(x)$ is a point for every $x \notin Z$ and $\varphi^{-1}(x)$ is a non-degenerate interval for every $x \in Z\setminus\Z$. The idea is similar to Denjoy’s construction: under the action of $\varphi^{-1}$, every integer $m$ is blown up into the 3-star $Y^a_m$, then the preimages of $m$ under $G$ are blown up too, in order to be able to define a map $\map{F}{S}$ which is a semiconjugacy of $G$.
Now we define our map $F$ as follows:
[**$\mathbf{F{\bigr\rvert_{Y^a_m}}}$: **]{}
we set $F{\bigr\rvert_{Y^b_0}} = \beta^{-1}_0 \circ f \circ \beta_0$, $F(a) = a,$ $F(-a) = -a$ and we define $F{\bigr\rvert_{[-a,-b]}}$ and $F{\bigr\rvert_{[b,a]}}$ affinely in such a way that $F{\bigr\rvert_{Y^a_0}}$ is continuous. Then, for every $m\in{\ensuremath{\mathbb{Z}}}$ and $x\in Y^a_m,$ we set $F(x):=F(x-m)+m.$ In particular, $F(Y^a_m) \subset Y^a_m$ for every $m \in \Z.$
[**$\mathbf{F{\bigr\rvert_{\varphi^{-1}(Z \setminus G^{-1}(\Z))}}}$: **]{}
For every $y \in Z \setminus G^{-1}(\Z)$, the sets $\varphi^{-1}(y)$ and $\varphi^{-1}(G(y))$ are intervals because $y$ and $G(y)$ belong to $Z \setminus \Z$. Moreover, by , the map $G$ is, either increasing, or decreasing at $y$. We define $F{\bigr\rvert_{\varphi^{-1}(y)}}$ to be the unique affine map from $\varphi^{-1}(y)$ onto $\varphi^{-1}(G(y))$ which is increasing (respectively decreasing) when $G$ is increasing (respectively decreasing) at $y$. In particular $F(\operatorname{Bd}(\varphi^{-1}(y))) = \operatorname{Bd}(\varphi^{-1}(G(y))).$
[**$\mathbf{F{\bigr\rvert_{\varphi^{-1}(G^{-1}(\Z)\setminus \Z)}}}$: **]{}
For every $y \in G^{-1}(\Z)\setminus \Z$, it follows that $y\in Z\setminus \Z$ and $G(y) \in \Z$. We define $F{\bigr\rvert_{\varphi^{-1}(y)}}$ to be the unique affine map from $\varphi^{-1}(y)$ onto $[G(y)-a, G(y)+a]$ which is increasing (respectively decreasing) when $G$ is increasing (respectively decreasing) at $y$. In this case we have $F(\operatorname{Bd}(\varphi^{-1}(y))) = \{G(y)-a, G(y)+a\}.$
[**$\mathbf{F{\bigr\rvert_{\varphi^{-1}({\ensuremath{\mathbb{R}}}\setminus Z)}}}$: **]{}
For every $y\in {\ensuremath{\mathbb{R}}}\setminus Z$, $G(y) \notin Z$ and $\varphi^{-1}(y)$ and $\varphi^{-1}(G(y))$ are single points. We set $F(\varphi^{-1}(y))=\varphi^{-1}(G(y))$.
Observe that, by definition, $F$ is continuous in every connected component of $\varphi^{-1}(Z)$. To see that $F$ it is globally continuous, notice that, for every $y \in Z$, $F(z)$ has one-sided limits as $z \in \varphi^{-1}(\R \setminus Z)$ tends to the endpoints of $\varphi^{-1}(y)$, and these limits are equal to the endpoints of $\varphi^{-1}(G(y))$. Consequently, $F$ is continuous. Moreover, from the definition of $F$ and the fact that $\varphi(x+1) = \varphi(x) +1,$ $F$ has degree 1. Hence, $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$. Furthermore, the fact that $F(Y^a_m) \subset Y^a_m$ implies that $\forall m\in{\ensuremath{\mathbb{Z}}}$, $\forall x\in Y^a_m$, ${\rho_{_{F}}}(x)={\rho_{_{F}}}(m)=0$, and hence $\operatorname{Rot}(F)={\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)$.
On the other hand, from the definition of $F$, it follows that $F$ is semiconjugate with $G$ through $\varphi$, that is, $G \circ \varphi = \varphi \circ F$. Hence, $$\label{FGsemiconj}
G^n \circ \varphi = \varphi \circ F^n
\quad
\text{for every $n \in \N.$}$$ From , it follows that ${\rho_{_{F}}}(x) = {\rho_{_{G}}}(\varphi(x))$ for all $x\in S$. Consequently, $\operatorname{Rot}(F)={\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F) = \operatorname{Rot}(G) = [0,d]$ and $$\label{eq:rhoG0}
{\rho_{_{F}}}(x) = 0\quad\text{if and only if}\quad \exists\, i\ge 0, m\in{\ensuremath{\mathbb{Z}}}\text{ such that } F^i(x)\in Y^a_m,$$ i.e., $F^i(x-m)\in Y^a_0$. Thus, $\operatorname{Per}(0,F) = \operatorname{Per^{\circ}}(F{\bigr\rvert_{Y^a_0}})$.
Now we are going to prove that $\operatorname{Per^{\circ}}(F{\bigr\rvert_{Y^a_0}}) = \operatorname{Per^{\circ}}(f)$, which implies $\operatorname{Per}(0,F) = \operatorname{Per^{\circ}}(f).$ By definition, $\operatorname{Per^{\circ}}(F{\bigr\rvert_{Y^b_0}}) = \operatorname{Per^{\circ}}(f) \ni 1$ (recall that a star map always has a fixed point by Theorem \[GMT1\]). So, we only have to prove that all periodic points of $F$ in $[-a,-b] \cup [b,a]$ are fixed points. Recall that we defined $F$ so that $F(a) = a,$ $F(-a) = -a,$ $F(b),F(-b) \in Y^b_0$ and $F{\bigr\rvert_{[-a,-b]}}$ and $F{\bigr\rvert_{[b,a]}}$ are affine. Thus, either $F{\bigr\rvert_{[-a,-b]}}$ is the identity map, or it is expansive; and the same holds for $F{\bigr\rvert_{[b,a]}}.$ Hence, the only periodic points of $F$ in $[-a,-b] \cup [b,a]$ are fixed points.
To end the proof of the theorem, we have to show that $\operatorname{Per}(F) = \operatorname{Per}(0,F) \cup \operatorname{Per}(G)$. Since $G(0) = 0$ and ${\rho_{_{G}}}(x) \ne 0$ for every $x \in \R \setminus Z$, it follows that $
\operatorname{Per}(G) = \{1\} \cup
\left(\bigcup_{\alpha \in (0,d]} \operatorname{Per}(\alpha,G) \right).
$ Consequently, $$\operatorname{Per}(0,F) \cup \operatorname{Per}(G) = \operatorname{Per}(0,F) \cup
\left(\bigcup_{\alpha \in (0,d]} \operatorname{Per}(\alpha,G) \right)$$ because $1 \in \operatorname{Per}(0,F).$ On the other hand, by definition, $
\operatorname{Per}(F) = \operatorname{Per}(0,F) \cup
\left(\bigcup_{\alpha \in (0,d]} \operatorname{Per}(\alpha,F)\right).
$ Therefore, we only have to show that $\operatorname{Per}(\alpha,F) = \operatorname{Per}(\alpha,G)$ for every $\alpha \in (0,d]$.
Fix $\alpha \in (0,d]$ and let $x\in S$ be such that ${\rho_{_{F}}}(x) = \alpha$. Then ${\rho_{_{G}}}(\varphi(x)) = {\rho_{_{F}}}(x)$ by . We are going to prove that $x$ is a periodic [[$\kern -0.55em\pmod{1}$]{}]{} point of $F$ of period $n$ if and only if $\varphi(x)$ is a periodic [[$\kern -0.55em\pmod{1}$]{}]{} point of $G$ of period $n$.
Assume first that $x$ periodic [[$\kern -0.55em\pmod{1}$]{}]{} point of period $n$ for $F$, that is, $F^n(x) = x + k$ for some $k \in \Z$ and $F^j(x) - x \notin \Z$ for all $j = 1,2,\dots, n-1$. From , it follows that $$G^n(\varphi(x)) = \varphi(F^n(x)) = \varphi(x+k) = \varphi(x) + k.$$ Therefore, $\varphi(x)$ is a periodic point [[$\kern -0.55em\pmod{1}$]{}]{} of $G$ with period, either $n$, or a divisor of $n$. To see that $x$ has indeed $G$-period [[$\kern -0.55em\pmod{1}$]{}]{} $n$, suppose by way of contradiction that there exists $j \in \{1,2,\dots,n-1\}$ such that $G^j(\varphi(x)) = \varphi(x) + l$ for some $l \in \Z$. Then $\varphi(F^j(x)) = G^j(\varphi(x)) = \varphi(x+l).$ Note that the fact that $
{\rho_{_{G}}}(\varphi(x+l)) = {\rho_{_{G}}}(\varphi(x)) =
\alpha \ne 0
$ implies that $\varphi(F^j(x)) = \varphi(x+l) \notin Z$ by . Consequently, since $\varphi^{-1}(y)$ is a point for every $y \notin Z$, $F^j(x) = x+l;$ a contradiction.
Now assume that $G^n(\varphi(x)) = \varphi(x) + k$ for some $k \in \Z$ and $G^j(\varphi(x)) - \varphi(x) \notin \Z$ for all $j\in\{ 1,2,\dots, n-1\}$. From , it follows that $
\varphi(F^n(x)) = G^n(\varphi(x)) = \varphi(x+k).
$ As above, ${\rho_{_{G}}}(\varphi(x)) = \alpha \ne 0$ implies that $\varphi(F^n(x)) = \varphi(x+k) \notin Z$ and thus $F^n(x) = x+k$. If there exists $j \in \{1,2,\dots,n-1\}$ such that $F^j(x) \in x + {\ensuremath{\mathbb{Z}}}$, then $G^j(\varphi(x)) = \varphi(F^j(x)) \in \varphi(x) +{\ensuremath{\mathbb{Z}}}$; a contradiction. Thus $x$ is periodic [[$\kern -0.55em\pmod{1}$]{}]{} of period $n$ for $F$.
Theorem \[YinSigma\] gives a map with a non-degenerate rotation interval. It is even easier to obtain a degenerate interval (take $G={\rm Id}$ in the proof), which shows that, for every $f
\in {\mathcal{X}_{3}}$, there exists a map $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ such that $\operatorname{Rot}(F)={\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)=\{0\}$ and $\operatorname{Per}(0,F) = \operatorname{Per}(F)=\operatorname{Per^{\circ}}(f)$.
One may wonder if Theorem \[YinSigma\] can be generalized in order to obtain a map $F\in {\ensuremath{\mathcal{L}_{1}(S)}}$ such that ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)= [c,d]$ and $\operatorname{Per}(c,F)=q\cdot\operatorname{Per^{\circ}}(f)$ for any $f\in {\mathcal{X}_{3}}$ and any rational number $c=p/q$ with $p,q$ relatively prime. As we said in Subsection \[ss:main-statements\], the natural strategy is to use a block structure. The next result shows that this strategy fails.
Let $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ and let $P$ be a lifted periodic orbit of $F$ with period $nq$ and rotation number $p/q$. Assume that there exists $x \in P$ such that $\chull{P_0(x)}$ is homeomorphic to a 3-star and $\chull{P_1(x)} \subset [n,n+1] \subset \R$ for some $n \in \Z$. Assume also that $P_0(x)$ is a periodic orbit of type 3 of $G:=F^q - p,$ $F^i(m) \in \chull{P_i(x)}$ for $i=0,1,\dots,q-1$ and $G(m) = m$, where $m \in \Z \cap \chull{P_0(x)}$ denotes the branching point of $\chull{P_0(x)}$. Then $\operatorname{Per}(p/q,F) = q\cdot\N$.
Recall that, when $P$ and $G$ are as in Theorem \[ConverseEndInteger\], $$P_i(x) := \{ F^i(x), G(F^i(x)), G^2(F^i(x)), \dots,
G^{n-1}(F^i(x)) \}$$ for every $x \in P$ and $i=0,1,\dots,q-1.$ To simplify the notation, in what follows we shall set $P_q(x) := P_0(x) + p$.
Before proving Theorem \[ConverseEndInteger\], we are going to develop the tools needed in its proof.
\[blocksareperiodic\] For all $x\in P$ and all $0\le i\le q-1$, $P_i(x)$ is a true periodic orbit of $G$ of period $n$. In particular, $P_i(x) = {\ensuremath{\{G^s(F^i(x)) \,\colon s \ge 0\}}}.$
Since the point $F^i(x)$ belongs to $P$, it is periodic [[$\kern -0.55em\pmod{1}$]{}]{} of period $nq$ and rotation number $p/q$ for $F$. Then the result follows from Lemma \[relationF\_Fqmp\].
We say that $P$ has an *increasing block structure* whenever, for some $x \in P,$ $$\max \Re(P_i(x)) < \min \Re(P_{i+1}(x))\qquad\forall i\in\{0,1,\dots,q-1\}$$ (when $i=q-1$ this amounts to $\max \Re(P_{q-1}(x))<\min \Re(P_0(x))+p$).
By the next lemma, the fact that a lifted periodic orbit has an increasing block structure is independent on the point $x$ chosen to build the blocks. So, the notion of *increasing block structure* is well defined.
For every $z \in P$ there exist $k \in \Z$ and $j \in \{0,1,\dots,q-1\}$ such that $z \in P_j(x) + k,$ $P_i(z) = P_{i+j}(x)+k$ for all $0\le i \le q-1-j$ and $P_i(z) = P_{i+j-q}(x)+k+p$ for all $q-j \le i \le q.$
By definition, for every $z \in P$ there exist $k_1 \in \Z$ and $j_1 \in \N$ such that $z = F^{j_1}(x) + k_1.$ On the other hand, by Lemma \[lem:FF+k\](b), $
G^n(x) = F^{nq}(x) - np,
$ for every $x \in S$ and $n \ge 0.$
We can write $j_1 = rq + j$ with $r \ge 0$ and $0 \le j < q.$ Hence, by Lemma \[blocksareperiodic\], $$z = F^{rq + j}(x) + k_1
= F^{rq}(F^j(x)) + k_1
= G^{r}(F^j(x)) + k
\in P_j(x) + k,$$ where $k = k_1 + rp.$ This proves the first statement of the lemma.
By Lemma \[blocksareperiodic\], $
P_i(z) = {\ensuremath{\{G^s(F^i(z)) \,\colon s \ge 0\}}}.
$ From above and Lemma \[lem:FF+k\](a), $$G^s(F^i(z)) = G^s(F^i(G^{r}(F^{j}(x)) + k)) = G^{r+s}(F^{i+j}(x)) + k$$ for every $i,s \in \N.$ Consequently, $
P_i(z) = {\ensuremath{\{G^{r+s}(F^{i+j}(x)) \,\colon s \ge 0\}}} + k.
$ If $0\le i \le q-1-j,$ by Lemma \[blocksareperiodic\], $
P_{i+j}(x) = {\ensuremath{\{G^s(F^{i+j}(x)) \,\colon s \ge 0\}}}
= {\ensuremath{\{G^{r+s}(F^{i+j}(x)) \,\colon s \ge 0\}}},
$ which proves the second statement of the lemma. In particular, $P_q(z) = P_0(z) + p = P_j(x) + k + p.$
If $q-j \le i < q$ then, $
G^{r+s}(F^{i+j}(x)) = G^{r+s+1}(F^{i+j-q}(x)) + p
$ with $i+j-q \ge 0.$ Hence, as above, $P_i(z) = P_{i+j-q}(x) + k + p.$
We are going to show that every lifted periodic orbit with period $nq$ and rotation number $p/q$ will have an increasing block structure by changing the lifting and the number $p$, if necessary. To this end, we want to look at the lifted orbit $P$ under the action of $\overline{F} := F + \ell$ with $\ell \in \Z$. By Lemma \[lem:FF+k\](b,d), the $\overline{F}-$rotation number of $P$ is $\tfrac{p}{q} + \ell = \tfrac{p + q\ell}{q}$ while the $\overline{F}-$period is still $nq$. So, by using $\overline{F}$ instead of $F$, we can define $$\overline{P}_i(x) := \{
\overline{F}^i(x),
\overline{G}(\overline{F}^i(x)),
\overline{G}^2(\overline{F}^i(x)), \dots,
\overline{G}^{n-1}(\overline{F}^i(x))
\}$$ for all $i\in\{0,1,\dots,q-1\}$, where $\overline{G} :=\overline{F}^q - (p + q\ell)$. We also set $\overline{P}_q(x) := \overline{P}_0(x) + (p + q\ell).$
\[sepblocks\] The following statements hold:
1. $\overline{G} = G$.
2. For every $i\in\{0,1,\dots,q\}$, $\overline{P}_i(x) = P_i(x) + i\ell$.
3. Assume that $\ell > \max \Re(P_i(x)) - \min \Re(P_{i+1}(x))$ for all $i\in\{0,1,\dots,q-1\}$. Then, the orbit $\overline{P}$ under $\overline{F}$ has an increasing block structure, that is, $\max \Re(\overline{P}_i(x)) < \min \Re(\overline{P}_{i+1}(x))$ for all $i\in\{0,1,\dots,q-1\}$.
For every $i \ge 0$, we have $$\overline{F}^i = (F+\ell)^i = F^i + i\ell$$ by Lemma \[lem:FF+k\](a-b). Hence, $$\overline{G} :=
\overline{F}^q - (p+q\ell) =
F^q + q\ell - (p+q\ell) =
G,$$ and (a) holds.
For all $i,j\ge 0$, we have $$\overline{G}^j(\overline{F}^i(x)) =
G^j(F^i(x) + i\ell) =
G^j(F^i(x)) + i\ell.$$ This gives (b) for $i=0,1,\dots,q-1$. The fact that $\overline{P}_q(x) = P_q(x) + q\ell$ follows from (b) for $i=0$ and from the definition of these two sets.
Suppose that $\ell$ satisfies the assumption of (c). From (b) and the choice of $\ell$, we have $$\min \Re(\overline{P}_{i+1}(x)) - \max \Re(\overline{P}_i(x)) =
\min \Re(P_{i+1}(x)) - \max \Re(P_i(x)) + \ell > 0$$ for every $i \in \{0,1,\dots,q-1\}.$ Hence (c) holds.
It is not difficult to show that, for every $\ell \in \Z$, $\operatorname{Per}(p/q,F) = \operatorname{Per}((p+q\ell)/q,F+\ell).$ Consequently, by changing the lifting and the number $p$, if necessary, we may assume that $P$ has an increasing block structure by Lemma \[sepblocks\]. Moreover, by replacing the point $x$ by $x-m$, we may also assume that the branching point of $\chull{P_0(x)}$ is 0 (that is, $m=0$). To simplify the notation, we shall omit the dependence from $x$ of the blocks $P_i(x)$ in what follows.
Let $I_1,I_2,I_3$ denote the three $P_0 \cup \{0\}$-basic intervals in $\chull{P_0}$ that have an endpoint equal to $0$ and let ${\mathcal{G}}$ be the directed graph with vertices $I_1,I_2,I_3$ such that there is an arrow $I_i {\nolinebreak[4]\longrightarrow\nolinebreak[4]}I_j$ if and only if $\chull{G(\partial
I_i)}\supset I_j$ (notice that arrows in ${\mathcal{G}}$ are $G$-coverings and ${\mathcal{G}}$ is a subgraph of the $G$-graph of $\{I_1,I_2,I_3\}$). Since $P_0$ is a periodic orbit of type 3 of $G$ and $G(0) = 0$, we can label the intervals $I_1,I_2,I_3$ so that $$\label{eq:Gtype3}
I_1 {\xrightarrow[G]{\hspace*{1.35em}}} I_2 {\xrightarrow[G]{\hspace*{1.35em}}} I_3 {\xrightarrow[G]{\hspace*{1.35em}}} I_1\quad
\text{is a loop in }{\mathcal{G}}.$$ Let ${\mathcal{I}}$ be the collection of $P_i\cup\{F^i(0)\}$-basic intervals for all $0\le i\le q$ (recall that $F^i(0)\in\chull{P_i}$ by assumption, and thus the elements of ${\mathcal{I}}$ are intervals in $\bigcup_{i=1}^q \chull{P_i}$). We are going to relate paths in the $F$-graph of ${\mathcal{I}}$ with coverings for $G$. Observe that, if $
\alpha = J_0 {\xrightarrow[F]{\hspace*{1.35em}}}J_1 {\xrightarrow[F]{\hspace*{1.35em}}}\dots{\xrightarrow[F]{\hspace*{1.35em}}}J_q
$ is a path in the $F$-graph of ${\mathcal{I}}$ with $J_0\subset \chull{P_0}$ then, since the blocks $P_i$ have an increasing block structure, $J_i$ is a basic interval of $P_i\cup\{F^i(0)\}$ for all $i\in\{0,1,\dots,q\}$. Moreover, the fact that $\alpha$ is a path for $F$ implies $J_0{\xrightarrow[G]{\hspace*{1.35em}}}J_q-p$. Reciprocally, if $J_0{\xrightarrow[G]{\hspace*{1.35em}}}J_q$ is an arrow in ${\mathcal{G}}$, then $$\label{eq:GFpath}
\exists\, J_1,\ldots J_{q-1}\in {\mathcal{I}},\
J_0 {\xrightarrow[F]{\hspace*{1.35em}}}J_1 {\xrightarrow[F]{\hspace*{1.35em}}}\dots{\xrightarrow[F]{\hspace*{1.35em}}}J_q+p.
$$ Let us prove . We have $F^i(\partial J_0)\subset P_i\cup\{F^i(0)\}$ for all $1\le i\le q$ because $\partial J_0\subset P_0\cup\{0\}$. Then an induction on $i=1,\ldots, q$ shows that, for all $P_i\cup\{F^i(0)\}$-basic intervals $J\subset \chull{F^i(\partial J_0)}$, there exists a path $$\label{eq:Fpath-induction}
J_0{\xrightarrow[F]{\hspace*{1.35em}}}J_1^J {\xrightarrow[F]{\hspace*{1.35em}}}\dots {\xrightarrow[F]{\hspace*{1.35em}}}J_{i-1}^J{\xrightarrow[F]{\hspace*{1.35em}}}J$$ where $J_j^J$ are $P_j\cup\{F^j(0)\}$-basic intervals for all $1\le j\le i-1$. The fact that $J_0{\xrightarrow[G]{\hspace*{1.35em}}}J_q$ is an arrow in ${\mathcal{G}}$ means that $\chull{G(\partial J_0)}\supset J_q$, that is, $\chull{F^q(\partial J_0)}
\supset J_q+p$. Therefore is given by for $i=q$ and $J=J_q+p$.
Combining and , we see that there exist three pairwise different paths $$\begin{aligned}
\alpha_1 &= I_1 {\xrightarrow[F]{\hspace*{1.35em}}}J_1 {\xrightarrow[F]{\hspace*{1.35em}}}\dots{\xrightarrow[F]{\hspace*{1.35em}}}I_2 + p\\
\alpha_2 &= I_2 {\xrightarrow[F]{\hspace*{1.35em}}}J_2 {\xrightarrow[F]{\hspace*{1.35em}}}\dots{\xrightarrow[F]{\hspace*{1.35em}}}I_3 + p\\
\alpha_3 &= I_3 {\xrightarrow[F]{\hspace*{1.35em}}}J_3 {\xrightarrow[F]{\hspace*{1.35em}}}\dots{\xrightarrow[F]{\hspace*{1.35em}}}I_1 + p\end{aligned}$$ in the $F$-graph of ${\mathcal{I}}$, of length $q$.
Now we consider two cases:
[1]{} Two of the intervals $J_i$ coincide.
By relabeling, if necessary, we may assume that $J_1 = J_2$. Denote the interval $J_1 = J_2$ by $L$ and consider the following three loops: $$\begin{aligned}
\overline{\alpha}_1 &=
L {\xrightarrow[F]{\hspace*{1.35em}}}\dots{\xrightarrow[F]{\hspace*{1.35em}}}I_2+p {\xrightarrow[F]{\hspace*{1.35em}}}L + p,\\
\overline{\alpha}_2 &=
L {\xrightarrow[F]{\hspace*{1.35em}}}\dots{\xrightarrow[F]{\hspace*{1.35em}}}I_3+p {\xrightarrow[F]{\hspace*{1.35em}}}J_3 + p,\\
\overline{\alpha}_3 &=
J_3 {\xrightarrow[F]{\hspace*{1.35em}}}\dots{\xrightarrow[F]{\hspace*{1.35em}}}I_1+p {\xrightarrow[F]{\hspace*{1.35em}}}L + p\ .\end{aligned}$$ Then $$G(L) \supset L \cup J_3\quad \text{and}\quad
G(J_3) \supset L.$$ By assumption, $\chull{P_1}$ is included in $[n,n+1]$. Thus $\operatorname{Int}(L)$ and $\operatorname{Int}(J_3)$ do not contain branching points since $L\cup J_3\subset \chull{P_1}$. Then the theorem holds by Proposition \[prop:SemiHorseshoe\] and Lemma \[relationF\_Fqmp\].
[2]{} The intervals $J_i$ are pairwise different.
In this case, we have the following loop: $$J_1 {\xrightarrow[G]{\hspace*{1.35em}}} J_2 {\xrightarrow[G]{\hspace*{1.35em}}} J_3 {\xrightarrow[G]{\hspace*{1.35em}}} J_1.$$ By assumption, $\chull{P_1}$ is an interval in ${\ensuremath{\mathbb{R}}}$. Moreover, $J_1,J_2,J_3$ are included in $\chull{P_1}$ and have pairwise disjoint interiors. Thus, by relabeling if necessary, we can assume that the intervals $J_1, J_2,J_3$ are ordered as: $$\begin{gathered}
\text{either }J_1\le J_2\le J_3,\\
\text{or }
J_1\ge J_2\ge J_3,\end{gathered}$$ with the convention that $J_i\le J_j$ if $\max J_i\le \min J_j$. Both cases being similar, we assume that we are in the first one, that is, $$\max J_1 \le \min J_2 < \max J_2 \le \min J_3.$$ Then,
- since $J_1 {\xrightarrow[G]{\hspace*{1.35em}}} J_2,$ there exists $x_1 \in J_1$ such that $G(x_1) = \min J_2;$
- since $J_2 {\xrightarrow[G]{\hspace*{1.35em}}} J_3,$ there exists $x_2 \in J_2$ such that $G(x_2) = \max J_3$ and
- since $J_3 {\xrightarrow[G]{\hspace*{1.35em}}} J_1,$ there exists $x_3 \in J_3$ such that $G(x_3) = \min J_1.$
Now we set $K = [x_1,x_2]$ and $L= [x_2, x_3]$. By continuity of $G$, $$\begin{aligned}
G(K) &\supset [\min J_2, \max J_3] \supset [x_2, x_3] = L,
\ \text{and}\\
G(L) &\supset [\min J_1, \max J_3] \supset [x_1, x_3] = K \cup L,\end{aligned}$$ and the theorem holds by Proposition \[prop:SemiHorseshoe\] and Lemma \[relationF\_Fqmp\], as in Case 1.
Orbits in the branches {#WeAreInTheBranches}
======================
The aim of this section is to prove Theorems \[TheoremSharkovskiiintheBranches\] and \[LargeOrbitsintheBranches\], which deal with the periods forced by the lifted periodic orbits contained in $B$.
Situations that imply periodic points of all periods
----------------------------------------------------
This subsection is devoted to two technical lemmas that characterize simple situations where $\operatorname{Per}(F) = {\ensuremath{\mathbb{N}}}$ in terms of images of distinguished points. They will also be used in Section \[sec:0inIntRotR\].
Given $F \in {\ensuremath{\mathcal{L}(S)}}$ and $x \in S$ we define the map $F_0$ by $$\label{eq:F0}
F_0(x) := F(x) - \Re(F(x)).$$ To understand the map $F_0$, observe first that $F_0(x) = 0$ whenever $F(x) \in {\ensuremath{\mathbb{R}}}$. Moreover, for every $x \in S$ it follows that $F(x) \in B$ if and only if $\Re(F(x)) \in {\ensuremath{\mathbb{Z}}}$ (more precisely, $F(x) \in B_m$ if and only if $\Re(F(x)) = m$). Thus, $F_0$ is a continuous map from the whole $S$ to $B_0$. From Lemma \[lem:FF+k\](a), we deduce that $F_0(x+k) = F_0(x)$ for all $x \in S$ and all $k \in {\ensuremath{\mathbb{Z}}}$ (that is, $F_0 \in {\ensuremath{\mathcal{L}_{0}(S)}}$).
Recall that, if $x,y$ are in the same branch $B_m$, then $x<y$ means $\Im(x)<\Im(y)$; the other notations related to the order in $B_m$ are defined consistently.
\[twoarrowscrossing\] Let $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$. Let $x,y \in B_0$ and $m\in{\ensuremath{\mathbb{Z}}}$ be such that $F(x)\in B_m$, $x < y \le F_0(x)$ and $F(y) \notin {\mathring{B}}_m$. Assume additionally that $F(0) \notin (x+m, \max B_m]$. Then $\operatorname{Per}(F) = {\ensuremath{\mathbb{N}}}$.
First of all, observe that the assumptions $x < y \le F_0(x)$ and the definition of $F_0$ imply that $F(x) \ge y+m > x+m$. Hence, $F(0) \notin (x+m, \max B_m]$ implies $F(0) \ne F(x)$, and thus, $x \ne 0$.
Consider $K = [x,y]$ and $L = [0,x]$, which are closed non-degenerate intervals in $B_0$. We have $$\begin{aligned}
F(K) & \supset \chull{F(x), F(y)}
\supset \chull{F(x), m} \quad
\text{because $F(x)\in B_m$ and $F(y) \notin {\mathring{B}}_m$,} \\
& \supset (K+m) \cup (L+m) \quad
\text{because $F(x) \ge y+m > x+m \ge m$.}\end{aligned}$$ Moreover, since $F(0) \notin (x+m, \max B_m]$ and $y\le F_0(x)$, $$F(L) \supset \chull{F(0), F(x)} \supset K+m.$$ By Proposition \[prop:SemiHorseshoe\], the map $F-m$ has periodic points of all periods in $K \cup L \subset B_0$. Therefore, $\operatorname{Per}(F) = {\ensuremath{\mathbb{N}}}$ by Lemma \[PeriodsAndPeriodsmodiAreFriends\]. This ends the proof of the lemma.
\[twoarrowscrossingLargeOrbits\] Let $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$. Let $x,y \in B_0$ and $m\in{\ensuremath{\mathbb{Z}}}$ be such that $F(x)\in B_m$, $x < y \le F_0(x)$ and $|\Re(F(x)) - \Re(F(y))| \ge 1$. Then $\operatorname{Per}(F) = {\ensuremath{\mathbb{N}}}$.
\[RelatingTheTwoLemmas\] Lemma \[twoarrowscrossingLargeOrbits\] is a particular case of Lemma \[twoarrowscrossing\] whenever $F(0)$ is not in a wrong place, i.e. $F(0) \notin (x+m, \max B_m]$.
We can assume additionally that $F(0) \in B_m$ and $F(0) > x+m$, otherwise Lemma \[twoarrowscrossing\] gives the conclusion (see Remark \[RelatingTheTwoLemmas\]). We shall also assume that $\Re(F(y)) \le m-1$; the case $\Re(F(y))\ge m+1$ follows in a similar way.
We set $G:=F-m$. Then the three points $x,y, G(x)=F_0(x)$ are in $B_0$ and $G(x) \ge y > x.$ According to Lemma \[lem:FF+k\](d), $\operatorname{Per}(F)=\operatorname{Per}(G)$, and thus we need to show that $\operatorname{Per}(G)={\ensuremath{\mathbb{N}}}$. We consider two cases.
[1]{} $G(0) \ge y$.
The proof of this case is similar to that of Lemma \[twoarrowscrossing\] by taking $K = [x,y]$ and $L = [-1,0]$. Since $\Re(G(y)) \le -1$ we have $$G(K) \supset \chull{G(x), G(y)}
\supset \chull{G(x), -1}
\supset K \cup L.$$ Moreover, since $G(0) \ge y$, we have $G(-1) \in B_{-1}$. Hence, $$G(L) \supset \chull{G(0), G(-1)}
\supset [x,y] \cup [-1,0]
= K \cup L.$$ By Proposition \[prop:SemiHorseshoe\], the map $G$ has periodic points of all periods in $K \cup L$.
[2]{} $x < G(0) < y$.
In this case, we set $K = [x,y ]$ and $L = \chull{-1,x}$, and we endow the interval $L$ with the order $<_L$ such that $-1=\min L$. Observe that $0\neq x$ because $G(0)<y\le G(x)$, and thus $L$ contains the branching point $0$ in its interior.
As in the previous case, $$\begin{aligned}
G(K) & \supset K \cup L,\\
G(L) & \supset \chull{G(-1),G(x)} \supset \chull{-1,G(x)}
\supset K \cup L.\end{aligned}$$
However, observe that the covering is negative in the first case and positive in the second. In other words, we have $K { \nolinebreak[4] \xrightarrow[G]{\hspace*{.25em}-\hspace*{.1em}} \nolinebreak[4]} K,L$ and $L { \nolinebreak[4] \xrightarrow[G]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} K,L$. Moreover, $(K+{\ensuremath{\mathbb{Z}}})\cap (L+{\ensuremath{\mathbb{Z}}})=\{x\}+{\ensuremath{\mathbb{Z}}}$, and $G(x)\notin L+{\ensuremath{\mathbb{Z}}}$. Thus Lemma \[lem:+-loop\] applies and gives $\operatorname{Per}(G)\supset {\ensuremath{\mathbb{N}}}\setminus\{2\}.$ So, we have to prove that $2 \in \operatorname{Per}(G)$. To this end, we shall consider several subcases and several loops.
Since $G(x) \ge y$ and $\Re(G(y)) \le -1$, there exist points $x \le x_1 < x_2 < \alpha < y$ in $B_0$ such that $G(x_1) = y$, $G(x_2) = x_1$ and $G(\alpha) = 0$. Moreover, we can take $x_2$ and $\alpha$ so that $$\begin{aligned}
x_2 &= \max {\ensuremath{\{t \in [x_1, y] \,\colon G(t) = x_1\}}}, \text{ and}\\
\alpha &= \max {\ensuremath{\{t \in [x_2, y] \,\colon G(t) = 0\}}} = \max {\ensuremath{\{t \in
[x_2,y] \,\colon G(t) \in B_0\}}}.\end{aligned}$$ Now we consider two subcases.
[2.1]{} $x < G(0) \le \alpha $.
We look at the interval $[x_2,\alpha]$. Observe that, by Lemma \[lem:FF+k\](a), $$\begin{aligned}
G^2(x_2) &= G(x_1) = y > x_2 \text{ and}\\
G^2(\alpha) &= G(0) \le \alpha.\end{aligned}$$ Hence, $G^2([x_2,\alpha])\supset [x_2+\alpha]$ and, since $G^2$ is continuous and there is no branching point in $[x_2,\alpha]$, there exists a point $z \in (x_2, \alpha]$ such that $G^2(z) = z$. From the definition of $x_2$, it follows that $G([x_2,\alpha]) \cap [x_2, \alpha] = \emptyset$. Therefore, $(G(z) + {\ensuremath{\mathbb{Z}}}) \cap [x_2,\alpha] = \emptyset$ and, consequently, $G(z) - z \notin {\ensuremath{\mathbb{Z}}}$. Thus, $z$ is periodic [[$\kern -0.55em\pmod{1}$]{}]{} point of period $2$.
[2.2]{} $\alpha < G(0) < y$
In this subcase, we need a couple of additional points. Since $G(0) \in {\mathring{B}}_0$, it follows that $G(-1) \in {\mathring{B}}_{-1}$ and, hence, there exists a point $\beta \in (-1,0)$ such that $G(\beta) =0$. Using again that $G(\alpha) = 0$ and $\Re(G(y)) \le -1$, we see that there exists a point $\alpha < u < y$ such that $G(u) = \beta$. Now we look at the interval $[\alpha,u]$. We have $$\begin{aligned}
G^2(\alpha) &= G(0) > \alpha \text{ and}\\
G^2(u) &= G(\beta) = 0 < u.\end{aligned}$$ Hence, there exists a point $z \in (\alpha,u) \subset B_0$ such that $G^2(z) = z$. From the definition of $\alpha$, it follows that $G((\alpha,u)) \cap {\mathring{B}}_0 = \emptyset$. So, as in the previous case, $G(z) - z \notin {\ensuremath{\mathbb{Z}}}$ and $z$ is a periodic [[$\kern -0.55em\pmod{1}$]{}]{} point of period $2$. This ends the proof of the lemma.
Proofs of Theorem \[TheoremSharkovskiiintheBranches\] and Theorem \[LargeOrbitsintheBranches\]
----------------------------------------------------------------------------------------------
The next lemma relates the maps $F$ and $F_0$ in the situation that interests us.
\[F0powern\] Let $F\in{\ensuremath{\mathcal{L}_{d}(S)}}$. Then the following statements hold:
1. Assume that there exists $x \in {\mathring{B}}_0$ and $n \in {\ensuremath{\mathbb{N}}}$ such that $F_0^i(x) \in {\mathring{B}}_0$ for all $0\le i\le n$. Then $F^i(x)
\in \cup_{m\in{\ensuremath{\mathbb{Z}}}} {\mathring{B}}_m$ for all $0\le i\le n$.
2. Assume that there exists $x \in B$ and $n \in {\ensuremath{\mathbb{N}}}$ such that $F^i(x) \in B$ for all $0\le i\le n$. Then $$F^n(x) = F_0^n(x) + \sum_{k=0}^{n-1} d^k \Re(F(F_0^{n-1-k}(x))) \in
F_0^n(x) + {\ensuremath{\mathbb{Z}}}.$$
Observe that if $F(x) \in {\ensuremath{\mathbb{R}}}$ then $F_0(x) = 0 \notin {\mathring{B}}_0$. Thus (a) holds. Statement (b) follows from the iterative use of Lemma \[lem:FF+k\](a) and the definition of $F_0$.
Given a lifted periodic orbit $P$ that lives in the branches (that is, $P\subset B$), we set $$\label{eq:P0}
P_0 := P \cap B_0 = {\ensuremath{\{x - \Re(x) \,\colon x \in P\}}}.$$
\[P0almostperiodic\] From the definitions of $F_0$ and $P_0$, we deduce that $F_0(P_0) \subset P_0$ and the cardinality of $P_0$ coincides with the $F$-period of $P$.
The next lemma summarizes the relation between $P$, $P_0$ and $F_0$. Its proof is omitted since it follows easily from Lemma \[F0powern\] and Remark \[P0almostperiodic\].
\[P0isperiodic\] Let $F \in {\ensuremath{\mathcal{L}(S)}}$ and let $P$ be a lifted periodic orbit of $F$ that lives in the branches. Then $P_0$ is a periodic orbit of $F_0$ and the $F_0$-period of $P_0$ coincides with the $F$-period of $P$.
We are ready to prove Theorems \[TheoremSharkovskiiintheBranches\] and \[LargeOrbitsintheBranches\]. We recall their statement before the proof.
Let $F \in {\ensuremath{\mathcal{L}(S)}}$ and let $P$ be a lifted periodic orbit of $F$ of period $p$ that lives in the branches. Then $\operatorname{Per}(F) \supset \operatorname{S\mbox{\tiny\textup{sh}}}(p)$. Moreover, for every $d \in {\ensuremath{\mathbb{Z}}}$ and every $p \in \N_{\Sho}$, there exists a map $F_p \in {\ensuremath{\mathcal{L}_{d}(S)}}$ such that $\operatorname{Per}(F_p) = \operatorname{S\mbox{\tiny\textup{sh}}}(p)$.
Since $\chull{P_0}$ is a compact interval included in $B_0$, the retraction on $\chull{P_0}$ is the continuous map $\map{{r}_{\chull{P_0}}}{S}[\chull{P_0}]$ defined by: $${r}_{\chull{P_0}}(x) = \begin{cases}
x & \text{if $x \in \chull{P_0}$}\\
\max P_0 & \text{if }x \in B_0 \text{ and } x\ge \max P_0,\\
\min P_0 &\text{otherwise}.
\end{cases}$$ We define $\psi:={r}_{\chull{P_0}}\circ F_0{\bigr\rvert_{\chull{P_0}}}$. Then $\map{\psi}{\chull{P_0}}$ is a continuous interval map such that $\psi{\bigr\rvert_{P_0}} = F_0{\bigr\rvert_{P_0}}$ and $$\label{SpecialCondition}
\psi(z) = F_0(z) \text{ for every }
z \in \chull{P_0}\setminus \psi^{-1}(\{\min P_0,\max P_0\}).$$
By Lemma \[P0isperiodic\], $P_0$ is a periodic orbit of $\psi$ of period $p$. Fix $q\in\operatorname{S\mbox{\tiny\textup{sh}}}(p)$ with $q\neq p$. By Sharkovsky’s theorem on the interval (see [@SharOri; @SharTrans] or Theorem \[GMT1\] for $n=2$), there exists a periodic orbit $Q \subset
\chull{P_0}$ of $\psi$ of period $q$. We have to show that $F$ has a lifted periodic orbit of period $q$.
Notice that $Q \cap P_0 = \emptyset$ and $Q \cap \psi^{-1}(P_0) = \emptyset$ since both are periodic orbits of $\psi$ of different period. Therefore, $Q \subset {\mathring{B}}_0$ and $\psi{\bigr\rvert_{Q}} = F_0{\bigr\rvert_{Q}}$ by . Let $d$ denote the degree of $F$. Then, by Lemma \[F0powern\], $$\label{SpecialConditionpsi}
\forall x\in Q,\ \forall n\in{\ensuremath{\mathbb{N}}},\ F^n(x) =
\psi^n(x) + \sum_{k=0}^{n-1} d^k \Re(F(\psi^{n-1-k}(x)))\in \psi^n(x)+{\ensuremath{\mathbb{Z}}}.$$
To prove that $F$ has a periodic [[$\kern -0.55em\pmod{1}$]{}]{} point of period $q$, we take any $x \in Q$ and we prove that $F^k(x) - x \notin {\ensuremath{\mathbb{Z}}}$ for $k=1,2,\dots,q-1$ and $F^q(x) -x \in {\ensuremath{\mathbb{Z}}}$. This last statement follows trivially from because $\psi^q(x) = x$. Assume that $F^k(x) = x + l$ for some $k \in \{1,2,\dots,q-1\}$ and some $l \in {\ensuremath{\mathbb{Z}}}$. Then, again from , $\psi^k(x) = x + \widetilde{l}$ for some $\widetilde{l} \in {\ensuremath{\mathbb{Z}}}$. Since both $x$ and $\psi^k(x)$ belong to $Q \subset \chull{P_0} \subset B_0$, it follows that $\widetilde{l} = 0$ and, hence, $\psi^k(x) = x$; a contradiction. Consequently, $F^k(x) - x \notin {\ensuremath{\mathbb{Z}}}$ for $k=1,2,\dots,q-1$.
The proof of the second part is easy. Fix $p\in\N_{\Sho}$. By [@Stefan] (see also [@ALM]), there exists a map $f_p \in {\mathcal{C}}^0([0,1])$ such that the set of periods of $f_p$ is precisely $\operatorname{S\mbox{\tiny\textup{sh}}}(p)$. Now we define the map $F_p \in {\ensuremath{\mathcal{L}_{d}(S)}}$ as follows. First we define $F_p$ on $B_0$ by setting $$\forall x\in[0,1],\ F_p(x\iota) := f_p(x)\iota,$$ where $\iota$ denotes the square root of $-1$. Notice that this formula defines $F_p(0)$. Then we define $F_p$ such that it maps the interval $[0,1]$ onto $\chull{F_p(0), F_p(0) + d}$ in an expansive (affine) way. With this we have defined $F_p$ in the set of all $x\in S$ such that $\Re(x) \in
[0,1)$. Finally, we extend $F_p$ to the whole $S$ by the formula $F_p(x) = F_p(x-\lfloor\Re(x)\rfloor) + d\lfloor\Re(x)\rfloor$, where $\lfloor\cdot\rfloor$ denotes the integer part function. Clearly, the map $F_p$ is continuous and has degree $d$. Moreover, each periodic orbit of $f_p$ corresponds to a periodic orbit of $F_p{\bigr\rvert_{B_0}}$. Hence, $\operatorname{Per}(F_p) \supset \operatorname{S\mbox{\tiny\textup{sh}}}(p)$. To end the proof of the theorem we have to show that, indeed, both sets coincide.
To see this, we note that $F_p(B) \subset B$ because $F_p(B_0) \subset B_0$. We claim that $F_p$ has no periodic [[$\kern -0.55em\pmod{1}$]{}]{} points in $S\setminus B = {\ensuremath{\mathbb{R}}}\setminus {\ensuremath{\mathbb{Z}}}$ other that fixed [[$\kern -0.55em\pmod{1}$]{}]{} points. Indeed, when $d = 0,$ $F_p({\ensuremath{\mathbb{R}}}) = F_p(0) \in B_0$ and there are no periodic [[$\kern -0.55em\pmod{1}$]{}]{} points in ${\ensuremath{\mathbb{R}}}\setminus {\ensuremath{\mathbb{Z}}}$. When $d \ne 0$, there exist points $0 \le x_1 < x_2 \le 1$ such that $F_p([0,x_1])
\subset B_0,$ $F_p([x_2,1]) \subset B_d$ and $F_p([x_1,x_2]) = [0,d]$. Therefore, there are no periodic [[$\kern -0.55em\pmod{1}$]{}]{} points in $[0,x_1] \cup [x_2,1]$ other than, perhaps, $0$ and $1$ (which are already contained in B); and the only periodic [[$\kern -0.55em\pmod{1}$]{}]{} points in $(x_1,x_2)$ are fixed [[$\kern -0.55em\pmod{1}$]{}]{} points because $F_p{\bigr\rvert_{(x_2,x_2)}}$ is expansive. This proves the claim. Since $F_p$ has already fixed [[$\kern -0.55em\pmod{1}$]{}]{} points in $B$, there are no new periods of $F_p$ in $S \setminus B$.
Now we are going to show that, if $x \in B$ is a periodic [[$\kern -0.55em\pmod{1}$]{}]{} point of period $q$, then $q \in \operatorname{S\mbox{\tiny\textup{sh}}}(p)$. Clearly, $\widetilde{x} := x - \Re(x) \in B_0$ and $F_p^n(\widetilde{x}) \in B_0$ for every $n \ge 0$. Then, by Lemma \[PeriodsAndPeriodsmodiAreFriends\], $\widetilde{x}$ is a periodic point of $F_p$ of period $q$ whose orbit is contained in $B_0$. Therefore, $q$ is a period of the original map $f_p$ and, thus, $\operatorname{Per}(F_p) = \operatorname{S\mbox{\tiny\textup{sh}}}(p)$. This ends the proof of the theorem.
Let $F \in {\ensuremath{\mathcal{L}_{1}(S)}}$ and let $Q$ be a large orbit of $F$ such that $Q$ lives in the branches. Then $\operatorname{Per}(F) = {\ensuremath{\mathbb{N}}}$.
Let $P = Q + {\ensuremath{\mathbb{Z}}}\subset B$ be the lifted orbit corresponding to $Q$ and set $q:=\operatorname{Card}(Q)$. Recall that $F_0$ and $P_0$ are defined by and . By Lemma \[P0isperiodic\], $P_0$ is a periodic orbit of $F_0$ of period $q$. We are going to show, by a recursive argument, that there exist $x,y \in P_0$ such that $x < y \le F_0(x)$ and $\Re(F(x)) \ne \Re(F(y))$. Then the theorem follows from Lemma \[twoarrowscrossingLargeOrbits\].
We set $A_0 := \{\min P_0\}$ and, for all $i \ge 0$, we define $$A_{i+1} := {\ensuremath{\{z \in P_0 \,\colon z \le \max F_0(A_i)\}}}.$$ It follows from this definition that, if $\max F_0(A_i)\le \max A_i$, then $F_0(A_i)\subset A_i$, which implies that $A_i=P_0$ because $A_i$ is included in $P_0$, which is a periodic orbit of $F_0$. Therefore, either $A_i \varsubsetneq A_{i+1}$ (when $\max F_0(A_i)> \max A_i$), or $A_i=P_0$. Clearly, $A_{i+1}=P_0$ whenever $A_i=P_0$. This implies that $$\label{eq:Ai}
\forall i\ge 0,\ A_i\subset A_{i+1}\quad\text{and}\quad
\forall i\ge q-1,\ A_i = P_0.$$ On the other hand, the function $\Re(F(\cdot))$ is not constant on $P_0$. To prove this, assume that there exists $m\in {\ensuremath{\mathbb{Z}}}$ such that $$\label{eq:FP0m}
\Re(F(P_0)) = \{m\}.$$ Choose $z \in P_0$ and let $s \in {\ensuremath{\mathbb{N}}}$ be such that $z + s \in Q$. Then, since $Q$ is a true periodic orbit of $F$ and $P_0$ is a periodic orbit of $F_0$, both of period $q$, we have $F^q (z+s) = z+s$ and $F_0^q (z) = z$. Lemma \[F0powern\](b) implies that $F^q(z)=F_0^q(z)+qm$ (note that $\forall k, \Re \circ F \circ F_0^{q-1-k} = m$ by ). We then have $$\begin{aligned}
z+s &= F^q (z + s) = F^q (z) + s\quad\text{by Lemma~\ref{lem:FF+k}(a)}\\
&= F_0^q (z) + q m + s\\
&= z + q m + s.\end{aligned}$$ Hence, $m = 0$ and, consequently, $\forall n\ge 0$, $F^n(z+s) = F^n(z) + s = F_0^n(z) + s$, again by Lemma \[F0powern\](b) and . So, $$\begin{aligned}
Q &= {\ensuremath{\{F^n(z+s) \,\colon n=0,1,\dots,q - 1\}}} \\
&= {\ensuremath{\{F_0^n(z) \,\colon n=0,1,\dots,q - 1\}}} + s \\
&= P_0 + s \subset B_s.\end{aligned}$$ This contradicts the fact that $Q$ is a large orbit and, hence, the function $\Re(F(\cdot))$ is not constant on $P_0$. Using this fact and , we see that there exists $1 \le k \le q-1$ such that the function $\Re(F(\cdot))$ is constant on $A_{k-1}$ (and hence its value is $\Re(F(\min P_0))$) but there exists $y \in A_k \setminus A_{k-1}$ such that $\Re(F(y)) \ne \Re(F(\min P_0))$. By definition, $y \le \max F_0(A_{k-1})$. Let $x \in A_{k-1}$ be such that $F_0(x) = \max F_0(A_{k-1})$. Then, since $y \notin A_{k-1}$, we have $x < y \le \max F_0(A_{k-1}) = F_0(x)$. Moreover, $\Re(F(\min P_0)) = \Re(F(x))$ because $x\in A_{k-1}$, and thus we have $\Re(F(y)) \ne \Re(F(x))$. This ends the proof of the theorem.
Periods (mod 1) when 0 is in the interior of the rotation interval {#sec:0inIntRotR}
==================================================================
This section is devoted to prove the next theorem.
Let $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$. If $\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)) \cap \Z \ne \emptyset$, then $\operatorname{Per}(F)$ is equal to, either ${\ensuremath{\mathbb{N}}}$, or ${\ensuremath{\mathbb{N}}}\setminus\{1\}$, or ${\ensuremath{\mathbb{N}}}\setminus\{2\}$. Moreover, there exist maps $F_0, F_1, F_2 \in {\ensuremath{\mathcal{L}_{1}(S)}}$ with $0 \in \operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F_i))$ for $i=0,1,2$ such that $\operatorname{Per}(F_0) = {\ensuremath{\mathbb{N}}}$, $\operatorname{Per}(F_1) = {\ensuremath{\mathbb{N}}}\setminus\{1\}$ and $\operatorname{Per}(F_2) = {\ensuremath{\mathbb{N}}}\setminus\{2\}$.
In the first subsection, we construct the maps $F_0$, $F_1$ and $F_2$ from the statement of Theorem \[theo:0inInterior\]. Then, in Subsection \[subsec:OIntRot\], we prove two lemmas, both giving conditions to obtain $\operatorname{Per}(F)\supset \N \setminus \{1\}$. Finally we prove the first statement of Theorem \[theo:0inInterior\] in the last and biggest subsection.
Construction of examples
------------------------
We give below two examples of maps with $0\in\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F))$ and $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}\setminus\{1\}$ (resp. $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}\setminus\{2\}$). The case $0\in\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F))$ and $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$ is trivially obtained from a lifting of a circle map with this property (just extend the map to $S$ by collapsing $B_0$ to $F(0)$ under the action of $F$); see e.g. [@ALM Section 3.10] for such circle maps.
\[ex:0inintRot-1\] We are going to build a map $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$ such that $0\in\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F))$ and $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}\setminus\{1\}$. Moreover, there is a large orbit of period $n$ for some fixed $n\ge 3$, which shows that the existence of a large orbit is not enough to imply all periods [$\kern -0.55em\pmod{1}$]{}.
We fix an integer $n\ge 3$. Let $a_0,a_1,\ldots, a_n\in [0,1]$ be such that $0=a_0<a_1<a_2< \cdots<a_{n-1}<a_n=1$. We set $A_i=[a_{i-1},a_i]$ for all $1\le i\le n$. We define $F\in {\ensuremath{\mathcal{L}_{1}(S)}}$ such that $F(a_i)=a_{i-1}$ for all $3\le i\le n$, $F(a_2)=\max B_0$, $F(a_1)=0$, $F(\max B_0)=a_2+1$, and $F$ is affine on $B_0$ and $A_i$ for all $1\le i\le n$. The map $F$ and its Markov graph are illustrated in Figure \[fig:0inintRot-1\].
![Above: the map $F$ of Example \[ex:0inintRot-1\]. Below: its Markov graph. The arrow from $B_0$ to the dotted set means that there are arrows $B_0{ \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}0\hspace*{.1em}} \nolinebreak[4]}A_i$ for all $1\le i\le
n$.[]{data-label="fig:0inintRot-1"}](periods-sigma-fig11)
By using the tools from [@AlsRue2008 Subsection 6.1] one can compute from its Markov graph that $\operatorname{Per}(F)=\operatorname{Per}(0,F)=\{n\ge
2\}$ and ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)=\left[-\frac1{n-1},\frac12\right]\ni 0$. The loop $$B_0 { \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}1\hspace*{.1em}} \nolinebreak[4]}A_1{ \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}-1\hspace*{.1em}} \nolinebreak[4]}A_n
{ \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}0\hspace*{.1em}} \nolinebreak[4]}A_{n-1}{ \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}0\hspace*{.1em}} \nolinebreak[4]}\cdots
{ \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}0\hspace*{.1em}} \nolinebreak[4]}A_3{ \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}0\hspace*{.1em}} \nolinebreak[4]}B_0$$ gives a large orbit of period $n$.
\[ex:0inintRot-2\] We are going to build a map $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$ such that $0\in\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F))$ and $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}\setminus\{2\}$.
Let $t_0,t_1,t_2, z_0,z_1\in [0,1]$ be such that $0<t_2<t_1<t_0<z_0<z_1<1$. We set $I_2=[0,t_2]$, $I_1=[t_2,t_1]$, $I_0=[t_1,t_0]$, $C=[t_0,z_0]$, $J_0=[z_0,z_1]$ and $J_1=[z_1,1]$. We define $F\in {\ensuremath{\mathcal{L}_{1}(S)}}$ such that $F(t_0)=t_1$, $F(t_1)=t_2$, $F(t_2)=t_0-1$, $F(z_0)=z_1$, $F(z_1)=\max B_1$, $F(\max B_0)=z_0$, $F(0)=0$ and $F$ is affine on $B_0, I_0,I_1, I_2, J_0, J_1, C$. The map $F$ and its Markov graph are illustrated in Figure \[fig:0inintRot-2\].
![Above: the map $F$ of Example \[ex:0inintRot-2\]. Below: its Markov graph. The arrows from the dotted set mean that there are arrows $I_i{ \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}-1\hspace*{.1em}} \nolinebreak[4]}C,J_0,J_1$ for $i=1,2$.[]{data-label="fig:0inintRot-2"}](periods-sigma-fig16)
By using the tools from [@AlsRue2008 Subsection 6.1] and using the loops $$C { \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}0\hspace*{.1em}} \nolinebreak[4]}
J_0 { \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}1\hspace*{.1em}} \nolinebreak[4]}
B_0 { \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}0\hspace*{.1em}} \nolinebreak[4]} C,
\quad
C { \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}0\hspace*{.1em}} \nolinebreak[4]} C
\quad\text{and}\quad
C { \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}0\hspace*{.1em}} \nolinebreak[4]}
I_0 { \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}0\hspace*{.1em}} \nolinebreak[4]}
I_0 { \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}-1\hspace*{.1em}} \nolinebreak[4]} C,$$ one can compute that $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}\setminus\{2\}$ and ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)=\left[-\frac13,\frac13\right]$.
Situations that imply periodic points of all periods except 1 {#subsec:OIntRot}
-------------------------------------------------------------
The aim of this subsection is to prove Lemmas \[lem:allperiods-1\] and \[lem:F(R)-left\] below, both giving conditions to obtain $\operatorname{Per}(F)\supset \N \setminus \{1\}$. They will be used in the proof of Theorem \[theo:0inInterior\].
There is a common idea in the hypotheses of both lemmas: some points of ${\ensuremath{\mathbb{R}}}$ go to the left whereas others go sufficiently to the right and have an orbit passing through the branches. In Lemma \[lem:allperiods-1\], the assumption is that there is a point $x\in{\ensuremath{\mathbb{R}}}$ such that $F(x)$ is in the branch $B_0$ and $F^2(x)$ is much to the right (or much to the left) of $F(0)$. In Lemma \[lem:F(R)-left\], assumption (a) means that all points in ${\ensuremath{\mathbb{R}}}$ go rather to the left (or at least do not go much to the right) under one iteration, whereas assumption (b) implies that there is one point $x_0$ in ${\ensuremath{\mathbb{R}}}$ whose orbit tends to $+\infty$; because of (a), the orbit of $x_0$ must pass through the branches.
Intuitively, the fact that some points of the real line go to the left whereas others go to the right is clearly related to the fact that there exist points $x_,x'\in{\ensuremath{\mathbb{R}}}$ such that ${\rho_{_{F}}}(x)<0$ and ${\rho_{_{F}}}(x')>0$, and hence $0\in\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F))$.
\[lem:allperiods-1\] Let $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$. Suppose that there exists $y_0\in F({\ensuremath{\mathbb{R}}})\cap B_0$ such that, either $\Re(F(y_0))\ge \lceil \Re(F(0))\rceil +1$, or $\Re(F(y_0))\le \lfloor \Re(F(0))\rfloor -1$. Then $\operatorname{Per}(F)\supset \N \setminus \{1\}$.
\[lem:F(R)-left\] Let $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$. Suppose that
1. $\forall x\in {\ensuremath{\mathbb{R}}}, x<0\Longrightarrow \Re(F(x))<0$,
2. $\exists x_0\in{\ensuremath{\mathbb{R}}}, \rho(x_0)>0$.
Then $\operatorname{Per}(F)\supset{\ensuremath{\mathbb{N}}}\setminus\{1\}$.
We also need two lemmas that, unfortunately, are rather technical. Roughly speaking, the conclusion of Lemma \[lem:PotImp:ToutVaBien\] is that, either we have a “good” point in $F({\ensuremath{\mathbb{R}}})$ and we may hope to apply Lemma \[lem:allperiods-1\], or we are in a “good” situation in view of Lemmas \[twoarrowscrossing\] or \[twoarrowscrossingLargeOrbits\]. Lemma \[lem:super\] summarizes the various conclusions we can obtain in this situation.
\[lem:PotImp:ToutVaBien\] Let $F\in {\ensuremath{\mathcal{L}_{1}(S)}}$, $z \in {\ensuremath{\mathbb{R}}}$ and $u \in \operatorname{Orb}(z,F) \setminus
{\ensuremath{\mathbb{R}}}.$ Then there exists $y \in \operatorname{Orb}(z,F) \setminus {\ensuremath{\mathbb{R}}}$ satisfying $$y-\Re(y) \le u -\Re(u)
\quad\text{and}\quad
\Re(F(y)) - \Re(y) = \Re(F(u)) - \Re(u)$$ and such that
- either $y \in F({\ensuremath{\mathbb{R}}})$,
- or there exists $x\in B_0$ such that $x < y-\Re(y) \le F_0(x)$ and $\Re(F(x)) \ne \Re(F(y)) - \Re(y)$.
\[lem:super\] Let $F\in {\ensuremath{\mathcal{L}_{1}(S)}}$, $z \in {\ensuremath{\mathbb{R}}}$ and $u \in \operatorname{Orb}(z,F) \cap B_0$. Then, there exists $y \in \operatorname{Orb}(z,F) \cap B_0$ such that $y \le u$ and $\Re(F(y)) = \Re(F(u))$, and one of the following situations occurs:
- $y \in F({\ensuremath{\mathbb{R}}})$,
- $\operatorname{Per}(F) = {\ensuremath{\mathbb{N}}}$,
- $y \notin F({\ensuremath{\mathbb{R}}})$ and there exists a point $x\in B_0$ such that $x < y \le F_0(x),$ $F(0) \in (x+m, \max B_m]$ and $F(y) \in (m-1, m+1)\setminus\{m\}\subset \R$, where $m:=\Re(F(x)) \in \Z$.
Next we prove the above four lemmas.
We assume that $\Re(F(y_0)) \ge \lceil \Re(F(0)) \rceil + 1;$ the other case is symmetric. In particular $0 \ne y_0 \in {\mathring{B}}_0$. By the continuity of $F$, there exist $y_1,y_2 \in B_0$, $y_1 < y_2 \le y_0$, such that $F(y_1) = \lceil \Re(F(0)) \rceil$ and $F(y_2) = \lceil \Re(F(0)) \rceil + 1.$ Let $D = [y_1,y_2] \subset B_0.$ We have $F(D) \supset [F(y_1), F(y_2)]$, and hence $D {\nolinebreak[4]\longrightarrow\nolinebreak[4]}[0,1] + \lceil \Re(F(0)) \rceil.$ Let $\widetilde{a} \in {\ensuremath{\mathbb{R}}}$ be such that $F(\widetilde{a}) = \max F({\ensuremath{\mathbb{R}}}) \cap B_0,$ $q = \lfloor \widetilde{a} \rfloor$ and $a = \widetilde{a} - q \in [0,1).$ We have $F(a) \in B_{-q}$ and $F(a) + q \ge y_0$. In the rest of the proof, all the coverings are for the map $F$ and the notation $I {\nolinebreak[4]\longrightarrow\nolinebreak[4]}J\ {\ensuremath{\kern -0.55em\pmod{1}}}$ means that $I {\nolinebreak[4]\longrightarrow\nolinebreak[4]}J+n$ for some $n\in{\ensuremath{\mathbb{Z}}}$.
[1]{} $F(0) \notin B$ (see Figure \[fig:case1.1\]).
This assumption implies that $y_1 \neq 0$, and thus $D \cap {\ensuremath{\mathbb{R}}}= \emptyset$. Set $A_1 = [0,a]$ and $A_2 = [a,1]$. Since $F(a) \in B_{-q}$, the set $F(A_1)$ contains $\chull{F(0),F(a)} \supset \chull{F(0),-q},$ and similarly $F(A_2)$ contains $\chull{-q,F(1)} = \chull{-q, F(0)+1}.$ Thus, if $F(0) \le a-q-1$ then $A_1 {\nolinebreak[4]\longrightarrow\nolinebreak[4]}A_{2}-q-1,$ and if $F(0) \ge a-q-1$ then $A_2 {\nolinebreak[4]\longrightarrow\nolinebreak[4]}A_{1}-q$. Moreover, we have $A_1 {\nolinebreak[4]\longrightarrow\nolinebreak[4]}D-q$ and $A_2 {\nolinebreak[4]\longrightarrow\nolinebreak[4]}D-q$ because $F(a) + q \ge y_0 \ge y_2$ and $F(0), F(1)\notin B$. Therefore we have one of the covering graphs of Figure \[fig:Markov-diagram\].
![The two possible covering graphs in case 1 (arrows are ${\ensuremath{\kern -0.55em\pmod{1}}}$).[]{data-label="fig:Markov-diagram"}](periods-sigma-fig4)
Suppose that we are in the first case, i.e. $A_1 {\nolinebreak[4]\longrightarrow\nolinebreak[4]}A_2\ {\ensuremath{\kern -0.55em\pmod{1}}}$ (see Figure \[fig:case1.1\]). Since $A_2 {\nolinebreak[4]\longrightarrow\nolinebreak[4]}D\ {\ensuremath{\kern -0.55em\pmod{1}}}$, there exists $c \in A_2$ such that $F(c) = y_1\ {\ensuremath{\kern -0.55em\pmod{1}}}$. Moreover $c\notin \{a,1\}$ because $F(a) \ge y_2$ and $F(1) \in {\ensuremath{\mathbb{R}}}$. Similarly, there exist $y_3 \in (y_1,y_2)$ such that $F(y_3) = c\ {\ensuremath{\kern -0.55em\pmod{1}}}$, and $b \in (a,c)$ such that $F(b) = y_3\ {\ensuremath{\kern -0.55em\pmod{1}}}$.
![Positions of the different points in Case 1, where $k = \lceil F(0) \rceil$ (the figure is drawn with $q=0$). []{data-label="fig:case1.1"}](periods-sigma-fig6){width="\textwidth"}
Let $D' = [y_1,y_3] \subset D$ and $A_2'= [b,c] \subset A_2$. Then $D' {\nolinebreak[4]\longrightarrow\nolinebreak[4]}A_1 \cup A_2'\ {\ensuremath{\kern -0.55em\pmod{1}}}$ and $A_2 {\nolinebreak[4]\longrightarrow\nolinebreak[4]}D'\ {\ensuremath{\kern -0.55em\pmod{1}}}.$ That is, we have the covering graph shown on the left picture of Figure \[fig:Markov-diagram\] by replacing $A_2$ and $D$ by $A_2'$ and $D',$ respectively. Moreover, the sets $A_1 + {\ensuremath{\mathbb{Z}}},\ A_2' + {\ensuremath{\mathbb{Z}}}$ and $D' + {\ensuremath{\mathbb{Z}}}$ are disjoint, and $A_1, A_2', D'$ contain no branching point in their interior. Therefore, to show that there exist periodic ${\ensuremath{\kern -0.55em\pmod{1}}}$ points of period $n$, it is enough to show that there exists a non-repetitive loop of length $n$ in the covering graph. Consider the following loops in the covering graph: $$\begin{aligned}
{\mathcal{C}}_2 & := D' {\nolinebreak[4]\longrightarrow\nolinebreak[4]}A_1 {\nolinebreak[4]\longrightarrow\nolinebreak[4]}D',\\
{\mathcal{C}}_2' & := D' {\nolinebreak[4]\longrightarrow\nolinebreak[4]}A_2' {\nolinebreak[4]\longrightarrow\nolinebreak[4]}D',\text{ and}\\
{\mathcal{C}}_3 & := D' {\nolinebreak[4]\longrightarrow\nolinebreak[4]}A_1 {\nolinebreak[4]\longrightarrow\nolinebreak[4]}A_2' {\nolinebreak[4]\longrightarrow\nolinebreak[4]}D',\end{aligned}$$ where the arrows are [[$\kern -0.55em\pmod{1}$]{}]{}. Fix $n\ge 2$. If $n$ is even, we write $n=2m$ and we consider the loop ${\mathcal{C}}_2'({\mathcal{C}}_2)^{m-1}$. If $n$ is odd, we write $n=2m+1$ and we consider the loop ${\mathcal{C}}_3({\mathcal{C}}_2)^{m-1}$. In both cases, we obtain a non-repetitive loop of length $n$. By Proposition \[prop:covering\], there exists a point $x\in D'$ such that $F^n(x)-x\in{\ensuremath{\mathbb{Z}}}$ and $$\begin{gathered}
\forall\, 0\le i\le m-1,\ F^{n-2i}(x)\in D'+{\ensuremath{\mathbb{Z}}},
\quad\forall\, 1\le i\le m-1,\
F^{n-2i+1}(x)\in A_1+{\ensuremath{\mathbb{Z}}},\\
F^{n-2m+1}(x)\in A_2'+{\ensuremath{\mathbb{Z}}}\quad
\text{and, if $n$ is odd, } F(x)\in A_1+{\ensuremath{\mathbb{Z}}}.\end{gathered}$$ Thus $x$ is periodic [[$\kern -0.55em\pmod{1}$]{}]{} for $F$ and its period divides $n$. Since the intervals $A_1,A_2', D'$ are disjoint [[$\kern -0.55em\pmod{1}$]{}]{}, one can show that its period [[$\kern -0.55em\pmod{1}$]{}]{} is exactly $n$. Indeed, consider $1<d<n$. Then $F^{n-2m+1}(x)\in A_2'+{\ensuremath{\mathbb{Z}}}$ and $F^{n-2m+1+d}(x)$ belongs to, either $A_1+{\ensuremath{\mathbb{Z}}}$ , or $D'+{\ensuremath{\mathbb{Z}}}$, and thus the period [[$\kern -0.55em\pmod{1}$]{}]{} of $x$ is not $d$.
The second case (i.e. when $A_2 {\nolinebreak[4]\longrightarrow\nolinebreak[4]}A_1$) is similar: there exist $c\in (0,a)$, $y_3\in (y_1,y_2)$ and $c\in (b,a)$ such that $F(c)=y_1\ {\ensuremath{\kern -0.55em\pmod{1}}}$, $F(y_3)=c\ {\ensuremath{\kern -0.55em\pmod{1}}}$ and $F(b)=y_3\ {\ensuremath{\kern -0.55em\pmod{1}}}$. If we let $A_1'=[c,b]$ and $D'=[y_1,y_3]$, then we have the covering graph shown on the right picture of Figure \[fig:Markov-diagram\] by replacing $A_1$ and $D$ by $A_1'$ and $D',$ respectively. The rest of the proof is the same as before by interchanging the roles of $A_1, A_2$. Therefore, $F$ has periodic ${\ensuremath{\kern -0.55em\pmod{1}}}$ points of period $n$ for all $n\ge 2$.
![ Left side: positions of the different points in Case 2, where $k=\lceil F(0)\rceil$ and $k<-q$ (the figure is drawn with $q=0$). Right side: the covering graph in Case 2 (both when $k\ge -q$ and $k<-q$).[]{data-label="fig:case2"}](periods-sigma-fig5bis){width="\textwidth"}
[2]{} $F(0)\in B$.
Let $k=\Re(F(0))\in {\ensuremath{\mathbb{Z}}}$ (that is, $F(0) \in B_k$ and $F(1) \in B_{k+1}$). Observe that the set $F([0,1])$ contains the points $F(a), F(0), F(1)$, with $F(a)\in B_{-q}$ and $F(a)+q \ge y_0$. When $k \ge -q$, we set $L=[a,1]$. Then, $$F(L) \supset \chull{F(a),F(1)} \supset \chull{y_0-q,k+1} \supset
\chull{y_0-q,-q} \cup \chull{k, k +1} \supset (D-q) \cup (L+k).$$
When $k < -q$, we set $L=[0,a]$ (see Figure \[fig:case2\]). Then, $$F(L) \supset \chull{F(0),F(a)} \supset \chull{k,y_0-q} \supset
\chull{k,k+1} \cup \chull{-q,y_0-q} \supset (D-q)\cup (L+k).$$ Observe that, in both cases, $F(D) \supset [0,1] + k \supset L+k$ and, hence, $F$ has the covering graph on the right side of Figure \[fig:case2\]. Thus, $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$ by Lemma \[lem:SemiHorseshoe-mod1\].
We set $E_0 := \R$ and $E_i := F(E_{i-1})$ for $i \ge 1.$ Since $F(\R) \supset \R$, $E_i$ is a non-decreasing sequence of closed connected subsets of $S$. Thus, $E_i \cap B_0$ is a closed subinterval of $B_0$ containing 0.
The sets $E_i$ are periodic [[$\kern -0.55em\pmod{1}$]{}]{}, i.e. $E_i = E_i + k$ for every $i \in \N$ and $k \in \Z.$ Indeed, $E_0$ is clearly periodic [[$\kern -0.55em\pmod{1}$]{}]{}. If $E_i = E_i + k$ for some $i \in \N$ and every $k \in \Z,$ then $$E_{i} = F(E_i) = F(E_i + k) = F(E_i) + k = E_{i+1} + k.$$
We claim that there exists $n \in \N$ such that $\max \Re(F(E_n \cap B_0)) \ge 1.$ To prove the claim, set $
\R_{<1} := {\ensuremath{\{x\in S \,\colon \Re(x) < 1\}}}
= (-\infty,1) \cup \bigcup_{k\le 0} B_k
$ and assume that $\Re(F(E_i \cap B_0)) < 1$ for every $i \in \N$. By Lemma \[lem:FF+k\](a) and assumption (a), $$F(E_i \cap \R_{<1}) \subset E_{i+1} \cap \R_{<1}$$ for every $i \in \N$. Consequently, $$F^i(E_0 \cap \R_{<1}) \subset E_i \cap \R_{<1} \subset \R_{<1}$$ for every $i \in \N$. Thus, for all $x \in (-\infty,1) = E_0 \cap \R_{<1},$ $\rho(x) \le 0.$ Since ${\rho_{_{F}}}(x+k)={\rho_{_{F}}}(x)$ for every $x \in S$ and $k \in \Z$ we get ${\rho_{_{F}}}(x) \le 0$ for every $x \in \R;$ a contradiction with assumption (b). This proves the claim.
Let $n \in \N$ be the smallest integer such that $\max \Re(F(E_n \cap B_0)) \ge 1.$
Observe that the continuity of $F$ and the assumption (a) imply that $\Re(F(0)) \le 0$ (in particular $\Re(F(E_0 \cap B_0)) < 1$). Hence, $n \ge 1.$ If $n = 1$ then Lemma \[lem:allperiods-1\] applies and we have $\operatorname{Per}(F) \supset \N \setminus \{1\}$.
So, in the rest of the proof we assume $n \ge 2.$ Since $E_n \cap B_0$ is a closed subinterval of $B_0$ containing 0, and $\Re(F(0)) \le 0,$ the continuity of $F$ implies that there exists $y \in E_n \cap {\mathring{B}}_0$ such that $F(y) = 1.$ By the minimality of $n,$ $y \notin E_{n-1}.$
Let $\overline{x} \in E_{n-1}$ be such that $F(\overline{x}) = \max E_n \cap B_0 \ge y.$ If $\overline{x} \in E_{n-2}$ then $E_{n-1} \cap B_0 \supset [0, F(\overline{x})] \ni y;$ a contradiction. Consequently, $\overline{x} \in {\mathring{B}}_{k}$ for some $k \in \Z$ because $\R \subset E_{n-2}.$ Set $x = \overline{x} - k \in E_{n-1} \cap {\mathring{B}}_0.$ If $x \ge y$ then the connectedness of $E_{n-1}$ implies that $y \in E_{n-1};$ a contradiction. Hence, $x < y.$ On the other hand, $F_0(x) = F(\overline{x}) \ge y$ and $F(x) = F(\overline{x}) - k \in B_{-k}.$ In particular $\Re(F(x)) \in \Z.$ The minimality of $n$ and the fact that $x \in E_{n-1} \cap B_0$ implies that $\Re(F(x)) < 1$ and, hence, $\Re(F(x)) \le 0.$ Therefore, $
|\Re(F(x))-\Re(F(y))| = \Re(F(y)) - \Re(F(x)) = 1 - \Re(F(x)) \ge 1.
$ Then the lemma follows from Lemma \[twoarrowscrossingLargeOrbits\].
If $u \in F({\ensuremath{\mathbb{R}}})$ then we are done by taking $y=u$. So, in what follows we assume that $u \notin F({\ensuremath{\mathbb{R}}})$. Then, since $z \in {\ensuremath{\mathbb{R}}}$ and $u \in \operatorname{Orb}(z,F)$ there exists $\overline{z} \in \operatorname{Orb}(z,F) \cap F({\ensuremath{\mathbb{R}}})$ and $l \ge 1$ such that $$\label{eq:Fi(z)}
F^l(\overline{z}) = u\text{ and }
F^i(\overline{z}) \notin F({\ensuremath{\mathbb{R}}})\text{ for } i=1,2,\dots,l.$$ Since $F(\overline{z}) \notin F({\ensuremath{\mathbb{R}}}),$ $\overline{z} \notin {\ensuremath{\mathbb{R}}}$. Also, since $F({\ensuremath{\mathbb{R}}}) \supset {\ensuremath{\mathbb{R}}},$ $F^i(\overline{z}) \in \cup_{j\in{\ensuremath{\mathbb{Z}}}} {\mathring{B}}_j$ for $i=1,2,\dots,l$. Notice that $\overline{z} - \Re(\overline{z}), u - \Re(u) \in {\mathring{B}}_0$ and $0 < \overline{z} - \Re(\overline{z}) < u - \Re(u)$. Otherwise, $\overline{z}-\Re(\overline{z}) \ge u - \Re(u)$ and, since $F({\ensuremath{\mathbb{R}}})$ contains ${\ensuremath{\mathbb{R}}}\cup\{\overline{z}\}$ and is connected, we obtain $F({\ensuremath{\mathbb{R}}}) \supset \chull{0,\overline{z}-\Re(\overline{z})}+{\ensuremath{\mathbb{Z}}}\ni u;$ a contradiction.
If $\Re(F(u)) - \Re(u) = \Re(F(\overline{z})) - \Re(\overline{z})$, then we set $y = \overline{z}$ and the lemma follows.
So, in the rest of the proof, we set $\widetilde{z} := \overline{z} - \Re(\overline{z})
\in {\mathring{B}}_0 \cap F({\ensuremath{\mathbb{R}}})$ and we assume that $$\Re(F(\widetilde{z})) = \Re(F(\overline{z})) - \Re(\overline{z})
\ne \Re(F(u)) - \Re(u).$$ By Lemma \[F0powern\](b) and the fact that $F_0$ has degree 0, $F^i(\overline{z}) - F_0^i(\widetilde{z}) \in \Z$ for $i=0,1,2,\dots,l.$ Consequently, $F_0^i(\widetilde{z}) \in {\mathring{B}}_0$ and $$\label{eq:displ}
F^i(\overline{z}) = F_0^i(\widetilde{z}) + \Re(F^i(\overline{z}))$$ for $i=0,1,2,\dots,l.$ In particular $u = F^l(\overline{z}) = F_0^l(\widetilde{z}) + \Re(u).$ Hence $F_0^l(\widetilde{z}) = u-\Re(u) > \widetilde{z}$ and, hence, $$\label{eq:p}
\text{there exists } p \in \{0,1,2,\dots,l-1\} \text{ such that }
F_0^p(\widetilde{z}) < u - \Re(u) \le F_0^{p+1}(\widetilde{z}).$$ If $\Re(F(F_0^p(\widetilde{z}))) \ne \Re(F(u)) - \Re(u)$, then we set $x = F_0^p(\widetilde{z})$ and $y = u$ and the lemma follows.
Otherwise, we set $l_1 := p < l$ and $u_1 := F^p(\overline{z}) \in \operatorname{Orb}(z,F) \setminus {\ensuremath{\mathbb{R}}}$ and from and Lemma \[lem:FF+k\](a) we obtain $$\begin{aligned}
u - \Re(u) &> F_0^p(\widetilde{z}) = u_1 - \Re(u_1) \text{ and}\\
\Re(F(u)) - \Re(u)
&= \Re(F(F_0^p(\widetilde{z})))
= \Re(F(F_0^p(\widetilde{z}) + \Re(u_1))) - \Re(u_1) \\
&= \Re(F(u_1)) - \Re(u_1).\end{aligned}$$ As in , the first of these inequalities implies that there exists $p_1\in\{0,\ldots, p-1\}$ such that $$F_0^{p_1}(\widetilde{z}) <
u_1-\Re(u_1) \le
F_0^{p_1+1}(\widetilde{z}).$$ If $l_i = p = 0$ then $u_1 = \overline{z},$ $\widetilde{z} = u_1 - \Re(u_1)$ and, hence, $\Re(F(\widetilde{z})) = \Re(F(u_1)) - \Re(u_1).$ This contradicts the fact that $\Re(F(\widetilde{z})) \ne \Re(F(u)) - \Re(u).$ Consequently, $l_1 = p > 0$ and $u_1 \notin F({\ensuremath{\mathbb{R}}})$ according to . As above, this implies that $u_1 - \Re(u_1) > \widetilde{z}.$ So we can replace $u$ by $u_1$ and $l$ by $l_1$ without modifying the current assumptions and we can repeat iteratively the above process to obtain a sequence $0 < l_m < l_{m-1} < \dots < l_1 < l$ with $1 \le m < l$ and $p_m \in \{0,1,2,\dots,l_m-1\}$ such that
- $u_i := F^{l_i}(\overline{z}) \in \operatorname{Orb}(z,F) \setminus {\ensuremath{\mathbb{R}}}$ and $\Re(F(u)) - \Re(u) = \Re(F(u_i)) - \Re(u_i)$ for $i=1,2,\dots,m;$
- $u - \Re(u) > u_1 - \Re(u_1) > u_2 - \Re(u_2) > \dots
> u_m - \Re(u_m) > \widetilde{z};$
- $F_0^{p_m}(\widetilde{z}) < u_m - \Re(u_m) \le
F_0^{p_m+1}(\widetilde{z})$ and $\Re(F(F_0^{p_m}(\widetilde{z}))) \ne \Re(F(u_m)) - \Re(u_m)$.
Notice that such a sequence exists because we are in the case when $\Re(F(\widetilde{z})) \ne \Re(F(u)) - \Re(u).$ Then the lemma follows by taking $x = F_0^{p_m}(\widetilde{z})$ and $y = u_m$.
If $u=0$, then $u\in F({\ensuremath{\mathbb{R}}})$ and we take $y=u$. From now on, we assume that $u\in {\mathring{B}}_0$. By Lemma \[lem:PotImp:ToutVaBien\], we know that there exists $y \in \operatorname{Orb}(z,F) \cap {\mathring{B}}_0$ satisfying $y \le u$ and $\Re(F(y)) = \Re(F(u))$ and such that,
1. either $y \in F({\ensuremath{\mathbb{R}}})$,
2. or there exists $x\in B_0$ such that $x < y \le F_0(x)$ and $m:=\Re(F(x)) \ne \Re(F(y))$.
In case (a), the lemma holds. So, assume that there exists a point $x$ as in case (b). Observe that $m \in \Z$ and $F(y) \notin B_m$ because $F_0(x) \notin \R$. So, by Lemma \[twoarrowscrossing\], the lemma holds unless $F(0) \in (x+m, \max B_m]$.
Assume that $F(0) \in (x+m, \max B_m]$. In view of Lemma \[twoarrowscrossingLargeOrbits\], we have again that $\operatorname{Per}(F) = {\ensuremath{\mathbb{N}}}$ unless $|m - \Re(F(y))| < 1$. Finally, if $|m - \Re(F(y))| <1$, then $F(y)\in (m-1,m+1)\setminus\{m\}$ because $F(y) \notin B_m$. This ends the proof of the lemma.
Proof of Theorem \[theo:0inInterior\]
-------------------------------------
The proof of Theorem \[theo:0inInterior\] is quite long. In the rest of the section, we are going to assume that $\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F))$ contains $0$ (if it contains another integer $m$, we come down to $0$ by considering the map $F-m$). The first step consists in exhibiting a particular configuration of points. Then we shall split the proof into several cases, depending of the positions of these points.
### A particular configuration of points
We proceed along the lines of the proof of [@ALM Lemma 3.9.1]. We first introduce some notation.
Since $0 \in \operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F))$, there exist $a,b \in \operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F))$ such that $a < 0 < b$, and there exist $x_a,x_b \in \R$ such that ${\rho_{_{F}}}(x_a) = a < 0 < b = {\rho_{_{F}}}(x_b).$ We may assume that $x_b < x_a$ (by taking $x_b - k$ instead of $x_b$ with $k \in \Z$ appropriate).
\[rem:boundedcompactnumber\] Since ${\rho_{_{F}}}(x_a)<0$ (resp. ${\rho_{_{F}}}(x_b)>0$), the sequence $\left(\Re(F^n(x_a))\right)_{n\ge 0}$ tends to $-\infty$ (resp. $\left(\Re(F^n(x_b))\right)_{n\ge 0}$ tends to $+\infty$). Thus the orbits of both points have a finite number of elements in each compact subset of $S$.
Now we define $$\begin{aligned}
\overline{M} &:=
{\ensuremath{\{F^k(x_b) \,\colon k \ge 0 \text{ and } \Re(F^l(x_b)) > \Re(F^k(x_b))
\text{ for every } l>k\}}},\text{ and}\\
\underline{M} &:=
{\ensuremath{\{F^k(x_a) \,\colon k \ge 0 \text{ and } \Re(F^l(x_a)) < \Re(F^k(x_a))
\text{ for every } l>k\}}}.\end{aligned}$$ Observe that $\overline{M} \subset \operatorname{Orb}(x_b,F)$ and $\underline{M} \subset \operatorname{Orb}(x_a,F).$ Hence, $\overline{M} \cap \underline{M} = \emptyset$ because $x_a$ and $x_b$ have different rotation numbers.
The next lemma summarizes the properties of $\overline{M}$ and $\underline{M}.$
\[lem:M&M\] The following statements hold for the sets $\overline{M}$ and $\underline{M}.$
1. For every $x \in \R,$ $\operatorname{Card}(\Re^{-1}(x) \cap \overline{M}) \le 1$ and $\operatorname{Card}(\Re^{-1}(x) \cap \underline{M}) \le 1.$
2. Let $L\in{\ensuremath{\mathbb{R}}}$. For every $w\in \operatorname{Orb}(x_b,F)$ there exists a point $\overline{x}\in\overline{M}$ such that $\Re(\overline{x})=\min (\Re(\operatorname{Orb}(w,F))\cap [L,+\infty))$ and for every $w'\in \operatorname{Orb}(x_a,F)$, there exists $\underline{x}\in\underline{M}$ such that $\Re(\underline{x})=\max (\Re(Orb(w',F))\cap (-\infty,L])$.
3. $\min \Re(\overline{M}) = \min \Re(\operatorname{Orb}(x_b,F)) \le x_b,$ and\
$\max \Re(\underline{M}) = \max \Re(\operatorname{Orb}(x_a,F)) \ge x_a.$
4. $\sup \Re(\overline{M}) = +\infty$ and $\inf \Re(\underline{M}) = -\infty.$
5. If $x\in \overline M$, there exists $x'\in \overline M\cap\operatorname{Orb}(x,F)$ such that $\Re(x)<\Re(x')\le \Re(F(x))$. The same holds with reverse inequalities with $x,x'\in \underline M$.
6. For any $x_0\in{\ensuremath{\mathbb{R}}}$ and $x\in \overline M$ with $\Re(x)\le x_0$, there exists $x'\in\overline M$ such that $\Re(x)\le \Re(x')\le x_0<\Re(F(x'))$. If $\Re(x')=\Re(x)$ then $x'=x$. The same holds with reverse inequalities if $x\in \underline M$.
We prove the lemma for the set $\overline{M}.$ The proofs for the set $\underline{M}$ follow similarly.
Let $F^k(x_b),F^l(x_b) \in \overline{M}$ with $k < l.$ From the definition of the set $\overline{M}$, it follows that $\Re(F^l(x_b)) > \Re(F^k(x_b)).$ So, (a) holds.
We have $\lim_{n\to+\infty}\Re(F^n(x_b))=+\infty$ (Remark \[rem:boundedcompactnumber\]) and thus, for every $L \in \R$ and every $w\in\operatorname{Orb}(x_b,F)$, the set $\Re(\operatorname{Orb}(w,F)) \cap [L,+\infty)$ contains infinitely many elements. We can define $\xi:=\min (\Re(\operatorname{Orb}(w,F))\cap [L,+\infty)).$ The set $\Re^{-1}(\xi) \cap \operatorname{Orb}(w,F)$ is finite by Remark \[rem:boundedcompactnumber\]. Thus we can define $i:=\max{\ensuremath{\{n\ge 0 \,\colon \Re(F^n(w))=\xi\}}}$. It follows that, for every $j > i$, $F^j(w) \notin \Re^{-1}(\xi)$ and hence, by the minimality of $\xi$, $\Re(F^j(w)) > \xi = \Re(F^i(w)).$ So $F^i(w) \in \overline{M}.$ This proves (b) with $\overline{x}=F^i(w)$. To prove (c) we repeat the proof of (b) by choosing $w=x_b$ and $L \le \min \Re(\operatorname{Orb}(x_b,F)).$ Then, we obtain $\xi=\min (\Re(\operatorname{Orb}(x_b,F))$ by the definition of $\xi$. Since $\overline{M}\subset \operatorname{Orb}(x_b,F)$ and $\xi\in \Re(\overline{M})$, this implies that $\min \Re(\overline{M})=\min \Re(\operatorname{Orb}(x_b,F))$. Moreover, it is obvious that $ \min \Re(\operatorname{Orb}(x_b,F))\le x_b$, and thus we obtain (c). To prove (d), it is enough to use (b) with $L$ tending to $+\infty$.
Suppose that $x\in\overline{M}$. Consider the set $A={\ensuremath{\{F^i(x) \,\colon i>0\}}}$. Then $\min A >x$ because $x\in \overline M$. Applying (b) with $w=x$ and $L=\min A\in \operatorname{Orb}(x,F)$, we see that there exists $x'\in\overline M$ such that $\Re(x')=\min \Re(A)$. By definition of $A$, we have $\Re(x')\le \Re(F(x))$ and this gives (e).
Let $x_0\in{\ensuremath{\mathbb{R}}}$ and let $x\in\overline M$ be such that $\Re(x)\le x_0$. The set $\Re(\overline M)\cap (-\infty, x_0]$ is non-empty because it contains $\Re(x)$. Thus there exists $x'\in\overline M$ such that $\Re(x')$ is equal to the maximum of this set. Clearly, $\Re(x)\le \Re(x')\le x_0$. Suppose that $\Re(F(x'))\le x_0$ and consider the set $A={\ensuremath{\{F^i(x') \,\colon i>0\}}}$. Then $\min \Re(A)\le x_0$ and there exists $x''\in \overline M$ with $\Re(x'')= \min (\Re(A))$ by (b). By the definitions of $A$ and $\overline M$, we have $\min \Re(A)>x'$. Thus the existence of $x''$ contradicts the definition of $x'$, and hence $\Re(F(x'))>x_0$. If $x'\neq x$, then $x'=F^i(x)$ for some $i>0$, and thus $\Re(x')>\Re(x)$ by definition of $\overline M$. This proves (f).
Lemma \[lem:M&M\](c) states that $\min \Re(\overline{M}) \le x_b < x_a \le \max \Re(\underline{M}).$ Consequently, by Lemma \[lem:M&M\](d), there exist points $z \in \overline{M}$ and $t\in \underline{M}$ such that $\Re(z) < \Re(t)$ and there are no points of $\Re(\overline{M} \cup \underline{M})$ in the interval $(\Re(z),\Re(t))$. By Lemma \[lem:M&M\](b), the inequality $\Re(F(z))<\Re(t)$ (resp. $\Re(F(t))>\Re(z)$) would contradict the definition of $z,t$. Hence $\Re(F(t))\le \Re(z) < \Re(t) \le \Re(F(z))$. Let $z'\in \overline M$ (resp. $t'\in\underline M$) be given by Lemma \[lem:M&M\](e) for $x=z$ (resp. $x=t$). The summary of the properties of $z,t, z', t'$ is then: $$\label{eq:all-inequalities}
\begin{split}
\Re(F(t)) \le \Re(t') & \le \Re(z) < \Re(t) \le \Re(z') \le \Re(F(z))
\text{ and}\\
\Re(F(t')) & < \Re(t') < \Re(z') < \Re(F(z')).
\end{split}$$
We shall keep the notations $z,z',t,t'$ in the whole section. Moreover, without loss of generality, we assume that $\Re(t) \in[0,1).$ The points $z$ and $t$ can have the following respective positions:
(A) $\Re(t)-\Re(z)\ge 1$,
(B) $z,t\in{\ensuremath{\mathbb{R}}}$ and $t-z<1$,
(C) $z \in {\mathring{B}}_0$ and $t \in (0,1),$
(D) $t \in {\mathring{B}}_0$ and $z \in (-1,0).$
In the next three subsections, we shall consider Cases (A), (B) and (C) respectively. Case (D) follows symmetrically from Case (C).
Before dealing with these three cases, we state some lemmas which imply the existence of all periods [[$\kern -0.55em\pmod{1}$]{}]{}, except perhaps 1, when the points $t,t',z,z'$ defined above and $F(0)$ satisfy some simple conditions.
\[lem:F(t)<t-1\] Suppose that $t\in{\ensuremath{\mathbb{R}}}$ and $\Re(F(t))\le t-1$. If either $z'\in{\ensuremath{\mathbb{R}}}$ or $\Re(F(0))\ge 0$, then $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$.
If $z'\in{\ensuremath{\mathbb{R}}}$, we have $z'<\Re(F(z'))$ by . Let $x$ be the point in $z'+{\ensuremath{\mathbb{Z}}}$ such that $t< x<t+1$ (the case $x=t$ is not possible because $x$ and $t$ have different rotation numbers). By Lemma \[lem:FF+k\](a) we also have $x <
\Re(F(x)).$
When $\Re(F(0))\ge 0$ we set $x=1$ and, as above, $x < \Re(F(x)).$ Since $t\in \R$, $0 \le t < 1$. If $t=0$ then, $0 \le \Re(F(t)) \le -1;$ a contradiction. Hence, as in the previous case, $t < x < t+1.$
Thus the interval $I=[t,x]$ is of length less than 1 and we have $I{ \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}[t-1,x]$ and hence $I{ \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} I \cup (I-1)$. Then $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$ by Corollary \[cory:+horseshoeFF-1\].
\[lem:F(0)>t\] Suppose that $z\in B_0$, $t,t'\in{\ensuremath{\mathbb{R}}}$ and $\Re(F(0))\ge t$. Then $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$.
The fact that $z\in B_0$ and imply that $t'\le 0=\Re(z)<t$. Let $t''\in t'+{\ensuremath{\mathbb{Z}}}$ be such that $t''\in (-1,0]$. Necessarily, $t'\le t''$. Using , we obtain $F([t'',0])\supset [t'',t]=[t'',0]\cup [0,t]$ and $F([0,t])\supset [t',t]\supset [t'',t]$. Since $[t'',0]$ and $[0,t]$ contain no branching points in their interior, Proposition \[prop:SemiHorseshoe\] applies to the intervals $[t'',0]$ and $[0,t]$, and $\operatorname{Per}(0,F)={\ensuremath{\mathbb{N}}}$. This clearly implies that $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$.
\[lem:bigF(0)\] Suppose that $z\in B_0$, $t\in{\ensuremath{\mathbb{R}}}$ and $|\Re(F(0))|\ge 1$. Then $\operatorname{Per}(F)\supset{\ensuremath{\mathbb{N}}}\setminus\{1\}$.
The fact that $z\in B_0$ and imply that $\Re(F(t))\le 0=\Re(z)<t$. First we suppose that $\Re(F(0))\ge 1$. Then $F([0,t])\supset [0,1]$ and $F([t,1])\supset [0,1]$. Moreover, the two intervals $[0,t]$ and $[t,1]$ contain no branching point in their interior. Thus Proposition \[prop:SemiHorseshoe\] applies and $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$.
Secondly we suppose that $\Re(F(0))\le -1$. If, for all $x\in (-\infty, 0)$, $\Re(F(x))<0$, then Lemma \[lem:F(R)-left\] applies (with $x_0=x_b$) and $\operatorname{Per}(F)\supset{\ensuremath{\mathbb{N}}}\setminus\{1\}$. Otherwise there exists $x\in (-\infty, 0)$ such that $\Re(F(x))\ge 0$. Let $b$ be the unique point in $x+{\ensuremath{\mathbb{Z}}}\cap [0,1)$. Thus $b\ge x+1$ and $\Re(F(b))\ge 1$. Set $I=[0,b]$. Then $I{ \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]} I\cup (I-1)$ and $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$ by Corollary \[cory:+horseshoeFF-1\].
### Case (A): $\Re(t) - \Re(z) > 1$.
This case is solved in the next lemma.
\[lem:largegap\] Assume that $\Re(t) -\Re(z) \ge 1$. Then $\operatorname{Per}(F) \supset \N\setminus\{1\}$.
We assume that $\Re(F(0)) \ge 0$ and we shall use the point $t$. If $\Re(F(0)) \le 0$, the proof is similar by using the point $z$ instead of $t$.
By and our assumption, $$\Re(F(t)) \le \Re(t') \le \Re(z) \le \Re(t) -1.$$ So, when $t \in B_0$, then $\Re(F(t))<\Re(t)=0$. Therefore, $t \ne 0$ because $\Re(F(0)) \ge 0.$ Consequently, either $t \in {\mathring{B}}_0$, or $t \in (0,1)$.
When $t \in (0,1)$, we have $\Re(t) = t$ and, hence, $\Re(F(t)) \le t -1$. Thus, the lemma follows from Lemma \[lem:F(t)<t-1\].
Assume now that $t \in {\mathring{B}}_0$ (and, hence, $\Re(F(t)) \le \Re(t)-1=-1$). By Lemma \[lem:super\] (applied with $x_a$ and $t$ instead of $z$ and $u$), we know that, either $Per(F)={\ensuremath{\mathbb{N}}}$, and the lemma holds; or there exists $y \in B_0$ satisfying $y \le t$ and $\Re(F(y)) = \Re(F(t))$ and such that
- either $y \in F({\ensuremath{\mathbb{R}}})$,
- or there exists a point $x\in B_0$ such that $x < y \le F_0(x),$ $F(0) \in (x+m, \max B_m]$ and $F(y) \in (m-1, m+1)\setminus\{m\}\subset \R$, where $m:=\Re(F(x)) \in \Z$.
In the second case, $m \ge 0$ because $\Re(F(0)) \ge 0.$ But $\Re(F(y))=\Re(F(t))\le -1.$ Hence, $\Re(F(y)) \le m-1$, and thus the second case is not possible. Consequently, $y \in F({\ensuremath{\mathbb{R}}}).$ Since $\Re(F(y)) \le -1 \le \lfloor \Re(F(0)) \rfloor -1$, we can use Lemma \[lem:allperiods-1\]. Hence, $\operatorname{Per}(F) \supset
\N\setminus\{1\}$ in this case.
### Case (B): $z,t\in{\ensuremath{\mathbb{R}}}$ and $t-z<1$
This case is dealt by the next lemma.
\[lem:sortgapR\] Assume that $t,z \in \R$ and $t -z < 1$. Then $\operatorname{Per}(F) = \N$.
We assume that $\Re(F(0)) \ge 0$ and we shall use the point $t$. If $\Re(F(0)) \le 0$, the proof is similar by using the point $z$ instead of $t$.
Assume first that $z \ge 0$. From , it follows that $$\Re(F(t)) \le z < t < \Re(z')\le \Re(F(z))\quad\text{and}\quad \Re(z') <
\Re(F(z')).$$ Let $I=[z,t]$. There is no branching point in the interior of $I$ since we have assumed $z \ge 0$. If $\Re(F(z))<1$, then $z'\in (0,1)$ and we set $J=[t,z']$ (see the left part of Figure \[fig:caseB1\]). If $\Re(F(z))\ge 1$, we set $J=[t,1]$ (see the right part of Figure \[fig:caseB1\]). In both cases, there is no branching point in $J$, $F(I)\supset I\cup J$ and $F(J)\supset I\cup J$. Then Proposition \[prop:SemiHorseshoe\] applies and $\operatorname{Per}(0,F) = \N$, and hence $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$.
![When $z\ge 0$, the two possible locations of the intervals $I,J$, forming a horseshoe in both cases.[]{data-label="fig:caseB1"}](periods-sigma-fig22 "fig:")$\qquad$ ![When $z\ge 0$, the two possible locations of the intervals $I,J$, forming a horseshoe in both cases.[]{data-label="fig:caseB1"}](periods-sigma-fig23 "fig:")
When $\Re(F(t)) \le t -1$, the lemma follows from Lemma \[lem:F(t)<t-1\]. So, in the rest of the proof we can we assume that $t-1 < \Re(F(t)) \le z < 0$. From , it follows that $$t-1 < \Re(F(t)) \le t' < z < 0, \
\Re(F(t')) < t'\quad
\text{and}\quad
\Re(F(z)) \ge t.$$ This configuration is depicted in Figure \[fig:caseB2\]. Then $$\begin{aligned}
F([t', z]) & \supset [t', t] \supset [t',z] \cup [0,t],
\text{ and}\\
F([0,t]) & \supset [t',0] \supset [t',z].\end{aligned}$$
![When $t-1 < \Re(F(t)) \le z < 0$, the intervals $I=[t',z']$ and $J=[0,t]$ form a horseshoe.[]{data-label="fig:caseB2"}](periods-sigma-fig24)
Since the intervals $(t',z)$ and $(0,t)$ contain no branching points, Proposition \[prop:SemiHorseshoe\] applies and $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$.
### Case (C): $z\in {\mathring{B}}_0$ and $t\in (0,1)$
We want to show that, in this situation, either $\operatorname{Per}(F)\supset{\ensuremath{\mathbb{N}}}\setminus \{1\}$ or $\operatorname{Per}(F)\supset{\ensuremath{\mathbb{N}}}\setminus \{2\}$. This is the most difficult case. To deal with it we need some additional points.
Lemma \[lem:PotImp:ToutVaBien\] applied with $z\in\operatorname{Orb}(x_b) \setminus{\ensuremath{\mathbb{R}}}$ instead of $u\in\operatorname{Orb}(z) \setminus{\ensuremath{\mathbb{R}}}$ gives a point $y$ such that $y_0:=y-\Re(y) \in {\mathring{B}}_0,$ $\Re(F(y_0)) = \Re(F(y)) - \Re(y) = \Re(F(z))$ and, either $$\label{eq:Cx}
\begin{split}
& y \in F({\ensuremath{\mathbb{R}}}),\text{ or}\\
& \text{$\exists$ $x\in B_0$ such that
$x< y_0\le F_0(x)$ and $\Re(F(x))\neq \Re(F(y_0))$.}
\end{split}$$ Observe that, since $F$ has degree one, $F({\ensuremath{\mathbb{R}}})$ is periodic [[$\kern -0.55em\pmod{1}$]{}]{} and, hence, $y \in F({\ensuremath{\mathbb{R}}})$ implies $y_0 \in F({\ensuremath{\mathbb{R}}}).$ Also, $z\in {\mathring{B}}_0$ implies $\Re(F(y_0)) = \Re(F(z)) > \Re(z) = 0$ by .
Let $a\in [0,1)$ be such that $F_0(a)=\max (F({\ensuremath{\mathbb{R}}})\cap B_0)$, and let $q\in{\ensuremath{\mathbb{Z}}}$ be such that $F(a)\in B_q$. In the rest of this subsection, we shall keep the notations $y_0,a, q$ to refer to these objects.
We are going to consider three subcases, depending on the positions of $y_0$ and $t'$:
1. $y_0\not\in F({\ensuremath{\mathbb{R}}})$,
2. $y_0\in F({\ensuremath{\mathbb{R}}})$ and $t'\in B_0$,
3. $y_0\in F({\ensuremath{\mathbb{R}}})$ and $t'\not\in B_0$.
Cases (C1), (C2) and (C3) are respectively proved in Lemmas \[lem:caseC1\], \[lem:caseC2\] and \[lem:caseC3\]. Altogether, they give Case (C).
\[lem:caseC1\] If $y_0\notin F({\ensuremath{\mathbb{R}}})$ then, either $\operatorname{Per}(F)\supset {\ensuremath{\mathbb{N}}}\setminus\{2\}$ or $\operatorname{Per}(F)\supset {\ensuremath{\mathbb{N}}}\setminus\{1\}$.
We first state a part of the proof as a separate lemma because it will be used again in Case (C2).
\[lem:yinB\] Suppose that there exist points $w,x,y\in B_0$ and $m\in{\ensuremath{\mathbb{Z}}}$ such that $|\Re(F(w))|<1$, $\Re(F(w))=\Re(F(y))$, $F(x)\in B_m$, $x<y\le F_0(x)$, $w\in\overline M$ (resp. $w\in\underline M$) and $m\le 0$ (resp. $m\ge 0$). Then $\operatorname{Per}(F)\supset
{\ensuremath{\mathbb{N}}}\setminus\{2\}$.
We prove the lemma in the case $w\in\overline M.$ The other one is symmetric. According to Lemma \[lem:M&M\](e), there is a point $w'\in \overline M$ such that $\Re(w)<\Re(w')\le \Re(F(w))$. Since $w\in B_0$ and $|\Re(F(w))|<1$, the point $w'$ belongs to $(0,1)$. Moreover, $\Re(F(w'))>w'$ because $w'\in \overline M$. Let $I=\chull{w',x}$, endowed with the order for which $\min I=w'$, and $J=[x,y]\subset B_0$ (with the order of $B_0$); see Figure \[fig:C1\].
![Intervals $I$ and $J$, with arrows indicating their order. Though not needed in the proof, it can be noticed that the assumptions imply $\Re(F(w))\in (0,1)$, and hence $F(y)=F(w)\in (0,1)$.[]{data-label="fig:C1"}](periods-sigma-fig25)
Then $I$ positively covers $I+m$ and $J+m$, and $J$ negatively covers $I+m$ and $J+m$. Moreover, $(I+{\ensuremath{\mathbb{Z}}})\cap (J+{\ensuremath{\mathbb{Z}}})=\{x\}+{\ensuremath{\mathbb{Z}}}$, and $F(x)\notin I+{\ensuremath{\mathbb{Z}}}$. Thus Lemma \[lem:+-loop\] applies and gives $\operatorname{Per}(F)\supset {\ensuremath{\mathbb{N}}}\setminus\{2\}$.
Since $y_0\notin F({\ensuremath{\mathbb{R}}})$, there exists $x\in B_0$ such that $x< y_0\le F_0(x)$ and $\Re(F(x))\neq \Re(F(y_0))$ by . Set $m:=\Re(F(x))\in{\ensuremath{\mathbb{Z}}}$ (thus, $F(x) \in B_m$). If $|\Re(F(y_0))-m| \ge 1$, Lemma \[twoarrowscrossingLargeOrbits\] applies and $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$. Since $\Re(F(y_0)) > 0,$ the condition $|\Re(F(y_0))-m| \ge 1$ is satisfied, in particular, when $m \le -1$ or $F(y_0) \in B$. On the other hand, if $\Re(F(0))\ge 1$, then $\operatorname{Per}(F)\supset{\ensuremath{\mathbb{N}}}\setminus\{1\}$ by Lemma \[lem:bigF(0)\]. So, in the rest of the proof we can assume that $F(y_0) \notin B,$ $m \ge 0$ and $\Re(F(0)) < 1.$ If $m \ge 1,$ Lemma \[twoarrowscrossing\] gives $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}.$ Therefore, we are left with the case $m=0$, $F(0)\in (x,\max B_0]$ and $\Re(F(y_0))<m+1=1$. Then $\Re(F(z))=\Re(F(y_0))\in (0,1)$, and finally Lemma \[lem:yinB\], applied to $w=z\in {\mathring{B}}_0$, $x$, $y=y_0$, gives $\operatorname{Per}(F)\supset {\ensuremath{\mathbb{N}}}\setminus\{2\}$.
Now we study Case (C2).
\[lem:caseC2\] Assume that $y_0\in F({\ensuremath{\mathbb{R}}})$ and $t'\in B_0$. Then, either $\operatorname{Per}(F)\supset{\ensuremath{\mathbb{N}}}\setminus\{2\}$, or $\operatorname{Per}(F)\supset{\ensuremath{\mathbb{N}}}\setminus\{1\}$.
Again, we state a part of the proof as a lemma, in order to use it again in Case (C3).
\[lem:3M\] If there exist $z_0, t_1, t_2\in {\ensuremath{\mathbb{R}}}$ such that $0\le t_1\le z_0\le
t_2\le 1$, $z_0\in \overline M$ and $t_1,t_2\in \underline M+{\ensuremath{\mathbb{Z}}}$, then $\operatorname{Per}(0,F)={\ensuremath{\mathbb{N}}}$.
Let $k_1,k_2\in{\ensuremath{\mathbb{Z}}}$ be such that $t_1\in\underline M+k_1$ and $t_2\in
\underline M+k_2$. The points $t_1, t_2$ cannot be equal to $z_0$ because ${\rho_{_{F}}}(z_0)>0$ and ${\rho_{_{F}}}(t_1)={\rho_{_{F}}}(t_2)<0$. According to Lemma \[lem:M&M\](f) (applied with $x_0=z_0-k_2$ and $x=t_2-k_2$), there exists $t_2'\in \underline M+k_2$ such that $\Re(F(t_2'))<z_0\le \Re(t_2')\le t_2$. We choose this point so that $\Re(t_2')$ is minimal. Since $0<z_0<t_2\le 1$ then, either $t_2'$ is in $(0,1)$, or $\Re(t_2')=1=t_2$, in which case $t_2'=t_2$. Thus $t_2'$ is in $(0,1]\subset {\ensuremath{\mathbb{R}}}$. Similarly, there exists $z_0'\in (0,1) \cap \overline{M}$ such that $z_0\le z_0'\le t_2' <\Re(F(z_0'))$. Since $z_0' \in \overline{M}$ and $t_2'\in \underline{M}+k_2,$ $z'_0 < t'_2$ because they have different rotation numbers. By Lemma \[lem:M&M\](e), there exists $t_2'' \in \underline{M}+k_2$ such that $\Re(F(t_2'))\le t_2''< t_2'$. Moreover, $t_2''<z_0$ by the minimality of $\Re(t_2')$. We set $t_1':=\max (t_1, t_2'')$. Then $t_1'\in(\underline M+k_1)\cup (\underline M+k_2)$ and $\max(t_1, \Re(F(t_2'))\le t_1' < z_0$.
![Positions of the points in Lemma \[lem:3M\]; the intervals $I=[t_1',z_0']$ and $J=[z_0,t_2']$ form a horseshoe.[]{data-label="fig:C2"}](periods-sigma-fig26)
Thus $t_1'\in [0,1)$ and $\Re(F(t_1'))\le t_1'$ because $t_1'\in\underline M+{\ensuremath{\mathbb{Z}}}$. Then the points have the following positions (see Figure \[fig:C2\]): $$\max(\Re(F(t_2'), \Re(F(t_1')) \le t_1'< z_0' < t_2' < \Re(F(z_0')).$$ So, Proposition \[prop:SemiHorseshoe\] with $[t_1',z_0']$ and $[z_0', t_2']$ applies. Thus, $\operatorname{Per}(0,F)={\ensuremath{\mathbb{N}}}$.
If $\Re(F(0))\notin (-1,1),$ the result follows from Lemma \[lem:bigF(0)\]. So, we can assume that $\Re(F(0))\in (-1,1).$
We apply Lemma \[lem:super\] with $z=x_a$ and $u=t'\in\operatorname{Orb}(x_a)\cap B_0$, to obtain a point $y\in B_0$ such that $\Re(F(y))=\Re(F(t'))$, and:
(i) either $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$ (and we are over),
(ii) or $y\in F({\ensuremath{\mathbb{R}}})$,
(iii) or there exists $x'\in B_0$ such that $x'<y\le F_0(x')$ and $F(0)\in
B_{m'}$, where $m':=\Re(F(x'))\in{\ensuremath{\mathbb{Z}}}$ and $F(y) \in (m'-1,m'+1)\setminus\{m'\}.$
In the last case, necessarily $m'=0$ because we have assumed $\Re(F(0))\in (-1,1)$. Hence, $\Re(F(t')) = \Re(F(y)) \in(-1,1),$ and we can apply Lemma \[lem:yinB\] with $w=t'$, $x'$, $y$ to obtain $\operatorname{Per}(F)\supset{\ensuremath{\mathbb{N}}}\setminus\{2\}$.
From now on, we suppose that we are in case (ii), that is, $y \in
F({\ensuremath{\mathbb{R}}})$. Since we have assumed that $y_0\in F({\ensuremath{\mathbb{R}}})$, we have $F_0(a)\ge
\max(y,y_0)$ (in $B_0$). Let $J = \chull{y,y_0};$ this interval is included in $B_0$ and thus contains no branching in its interior. If, for every $x \in (-\infty,0)$, $\Re(F(x))<0$, then Lemma \[lem:F(R)-left\] applies (with $x_0=x_b$) and $\operatorname{Per}(F)\supset {\ensuremath{\mathbb{N}}}\setminus\{1\}$. Otherwise, there exists a point $x\in (-\infty,0)$ such that $\Re(F(x))\ge 0$. Let $b$ be the unique point in $(x+{\ensuremath{\mathbb{Z}}})\cap
[0,1)$. Then $b\ge x+1$ and $\Re(F(b))\ge 1$. Since $t,b\in [0,1]$ and $\Re(F(t))\le \Re(z)=0$ by , we have $F([0,1])\supset [0,1]$. Moreover, since $a\in [0,1]$ and $F(a)\in B_q$, we have $F([0,1])\supset [0, F_0(a)]+q$ because, either $F(0)\notin B_q$ or $F(1)\notin B_q$. Thus $F([0,1])\supset J+q$. On the other hand, $F(J)\supset [\Re(F(y)),\Re(F(y_0))]$ and $\Re(F(y))=\Re(F(t'))\le \Re(z)=0$. Thus, if $$\label{eq:Fy0}
\Re(F(y_0))\ge 1,$$ then $F(J)\supset [0,1]$ and we have the situation and the coverings represented in Figure \[fig:C22\]. Then $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$ by Lemma \[lem:SemiHorseshoe-mod1\].
![Left side: points $t,a,b$ are in $[0,1]$ but maybe not in this order; point $y$ may be below $y_0$ in $B_0$. In all cases, we have the coverings on the right.[]{data-label="fig:C22"}](periods-sigma-fig27)
From now on, we assume that does not hold, that is, $\Re(F(y_0))<1.$ This implies that $z'\in (0,1)$ and $\Re(F(y_0)) \ge z'$ by (recall that $\Re(F(y_0))=\Re(F(z))$). If there exists $t_2\in (\underline M+1)\cap [z',1]$, then Lemma \[lem:3M\] applies (with $z_0=z'$, $t_1=t$ and $t_2$) and $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$. So, in the rest of the proof we assume that $$\label{eq:Mz'}
(\underline M+1)\cap [z',1]=\emptyset.$$ Lemma \[lem:M&M\](f), applied with $x_0=z'-1$ and $x=t'\in\underline{M}$, implies that $\Re(F(t'+1))< z'$ (otherwise, there would exist $t''\in \underline{M}$ such that $z'\le \Re(t'')+1\le \Re(t')+1$, which would contradict since $\Re(t')+1\le 1$). Since $F$ has degree one, $\Re(F(y)) = \Re(F(t')) < z'-1$ and, hence, $F(J)\supset [z'-1,z'].$
Now we split the proof of this remaining case into three subcases, depending on the values of $a$ and $q$.
- If $a\le z'$, we have the situation represented in Figure \[fig:C23\].
![Left side: points $t,a$ are in $[0,z']$ but maybe not in this order; point $y$ may be below $y_0$ in $B_0$. In all cases, we have the coverings on the right.[]{data-label="fig:C23"}](periods-sigma-fig28){width="\textwidth"}
We set $I=[0,z']$ and there is no branching point in $(0,z')$ because $z'\in (0,1)$. The interval $I$ contains $t,z'$ and $a$, with $\Re(F(t))\le 0$ and $\Re(F(z'))>z'>0$. Either $F(t)\notin B_q$, or $F(z')\notin B_q$, and thus $F(I)$ contains $[q,F(a)] \subset B_q$. Hence $I {\nolinebreak[4]\longrightarrow\nolinebreak[4]}J+q$. Moreover, $I {\nolinebreak[4]\longrightarrow\nolinebreak[4]}I$ and $J {\nolinebreak[4]\longrightarrow\nolinebreak[4]}I$. Thus $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$ by Lemma \[lem:SemiHorseshoe-mod1\].
- Suppose that $a>z'$ and $q\ge 1$. By Lemma \[lem:M&M\](e), there exists $t''\in \underline M+1$ such that $$\label{eq:t't''}
\Re(F(t'+1))\le \Re(t'')<\Re(t'+1)=1.$$ We have $\Re(F(t''))<\Re(t'')$ because $t''\in \underline M+1$. Moreover, $\Re(t'')<z'$ by . We set $\widetilde{t}= \max (\Re(t''),t) \in (0,z')$; then we have $\Re(F(\widetilde{t})) < \widetilde{t}$ (see Figure \[fig:C24\]).
![Positions of points and covering graph of $I,K$.[]{data-label="fig:C24"}](periods-sigma-fig29){width="\textwidth"}
Let $I=[\widetilde{t},a]\subset {\ensuremath{\mathbb{R}}}$ and $K=\chull{a,y+1}$ endowed with the order such that $\min K=a$. Then $I$ positively covers $I$ and $K+q-1$ (because $F(a)\in B_q$ with $q \ge 1$) and $K$ negatively covers $I$ and $K+q-1$ (because $q\ge 1$ and $\Re(F(y'))=\Re(F(t'))$ and $\Re(F(t'+1))\le \Re(t'') \le \widetilde{t}$ by ). Moreover, $(I+{\ensuremath{\mathbb{Z}}})\cap (K+{\ensuremath{\mathbb{Z}}})=\{a\}+{\ensuremath{\mathbb{Z}}}$, and $F(a)\notin I+{\ensuremath{\mathbb{Z}}}$. Thus Lemma \[lem:+-loop\] applies and gives $\operatorname{Per}(F)\supset {\ensuremath{\mathbb{N}}}\setminus\{2\}$.
- Suppose that $a > z'$ and $q \le 0$. Let $I = \chull{a,b} \subset [0,1)$. If $0\le b\le t$, then $[b,t]$ and $[t,z']$ form a horseshoe; and if $t\le b\le a$, then $[t,b]$ and $I$ form a horseshoe (see Figure \[fig:C25\]). In both cases, Proposition \[prop:SemiHorseshoe\] applies and $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$.
![The two possibilities when $b<z'$. In both cases, there is a horseshoe (either $L,I$ or $L,I'$).[]{data-label="fig:C25"}](periods-sigma-fig30)
It remains to consider the case when $b>a$, which implies that $b> z'$; see Figure \[fig:C26\]. Then $J$ covers $I-1$ (recall that $\Re(F(y))=\Re(F(t'))\le z'-1$) and $I$ covers $I$ and $J+q$. Notice that $I\subset (0,1)$ because $b\ge z'>\Re(z)=0$, which implies that the sets $I+{\ensuremath{\mathbb{Z}}}$ and $J+{\ensuremath{\mathbb{Z}}}$ are disjoint. Then $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$ by Lemma \[lem:SemiHorseshoe-mod1\].
We have covered all the possible cases, and thus Lemma \[lem:caseC2\] is proved.
![When $b\ge a$.[]{data-label="fig:C26"}](periods-sigma-fig31){width="95.00000%"}
Finally, in the next lemma we study Case (C3).
\[lem:caseC3\] Suppose that $y_0\in F({\ensuremath{\mathbb{R}}})$ and $t'\notin B_0$. Then, either $\operatorname{Per}(F) \supset{\ensuremath{\mathbb{N}}}\setminus\{1\}$, or $\operatorname{Per}(F)\supset{\ensuremath{\mathbb{N}}}\setminus\{2\}$.
In order to make the proof easier to read, we first deal with a special configuration of points.
\[lem:t’+1\] Suppose that $y_0\in F({\ensuremath{\mathbb{R}}})$, $t', z'\in{\ensuremath{\mathbb{R}}}$, $\Re(F(0))\le t$ and $t'+1\le z'\le a<1$. Then $\operatorname{Per}(F)\supset {\ensuremath{\mathbb{N}}}\setminus\{2\}$.
Let $I:=[t,z']$, $J:=[t',t]$ and $K:=[0,y_0]$. Notice that these three intervals have disjoint interiors, and $\operatorname{Int}(J)$ contains the branching point $0$ (see Figure \[fig:C31\]).
![The intervals $I,J,K$ and their covering graph in Lemma \[lem:t’+1\].[]{data-label="fig:C31"}](periods-sigma-fig32)
It is clear that $I{ \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}I$, $I{ \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}J$ and $K{ \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}I$. By assumption, $t'<z'-1< a-1<0.$ Thus, all these points belong to $J$. Moreover, either $F(t')\notin B_{q-1}$, or $F(z'-1)\notin B_{q-1}$ (because $\Re(F(t'))<t'$ and $\Re(F(z'))>z'$). Hence $J{ \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}K+q-1$. Now, we are going to show that these coverings imply that $\operatorname{Per}(F)\supset {\ensuremath{\mathbb{N}}}\setminus\{2\}$. We set $${\mathcal{C}}:=I{ \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}I\quad\text{and}\quad
{\mathcal{C}}':=I-q+1{ \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}J-q+1{ \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}K{ \nolinebreak[4] \xrightarrow[F]{\hspace*{.25em}+\hspace*{.1em}} \nolinebreak[4]}I.$$ Proposition \[prop:signedcover\], applied to the loop ${\mathcal{C}}$, shows that there exists a fixed point. We fix $n\ge 3$ and we consider the chain of coverings ${\mathcal{C}}' {\mathcal{C}}^{n-3}$. This gives a loop of length $n$ from $I-q+1$ to $I$. According to Proposition \[prop:signedcover\], there exists a point $x\in I-q+1$ such that $F^n(x)=x+q-1$, $F(x)\in J-q+1$, $F^2(x)\in K$ and $F^i(x)\in I$ for all $3\le i\le
n$. It remains to prove that the period [[$\kern -0.55em\pmod{1}$]{}]{} of $x$ is exactly $n$. Let $p$ be the period ${\ensuremath{\kern -0.55em\pmod{1}}}$ of $x$. If $p< n$, then $p\le n-2$ because $p$ divides $n\ge 3$. Thus $F^2(x)\in K$, $F^{2+p}(x)\in I$ and $F^{2+p}(x)- F^2(x)\in{\ensuremath{\mathbb{Z}}}$. But this is impossible because $I\subset (0,1)$, and hence $(I+{\ensuremath{\mathbb{Z}}})\cap (K+{\ensuremath{\mathbb{Z}}})=\emptyset$. This proves that $p=n$. Therefore, $\operatorname{Per}(F)\supset {\ensuremath{\mathbb{N}}}\setminus\{2\}$.
We can assume that $\Re(F(0))\in (-1,1)$ since, otherwise, Lemma \[lem:bigF(0)\] gives the conclusion. Then, applying Lemma \[lem:allperiods-1\] to $y_0$ (knowing that $\Re(F(y_0))>0$), we see that, either $\operatorname{Per}(F)\supset{\ensuremath{\mathbb{N}}}\setminus\{1\}$, or we are in one of the following cases:
(I) $F(0)\in (-1,0)\cup B_0$ and $F(y_0)\in (0,1)$,
(II) $F(0)\in (0,1)$ and $F(y_0)\in (0,1)$,
(III) $F(0)\in (0,1)$ and $\Re(F(y_0))\in [1,2)$.
Notice that in Cases (I) and (II), we have $z'\in (0, 1)$ because $t<\Re(z')\le \Re(F(y_0))<1$. In addition, we can assume that $\Re(F(t))\ge t-1$, otherwise Lemma \[lem:F(t)<t-1\] gives the result (using $z'\in{\ensuremath{\mathbb{R}}}$ in Cases (I) and (II), and $\Re(F(0))\ge 0$ in Case (III)). Recall that $\Re(F(t))\le \Re(t')\le \Re(z)=0$, $t\in (0,1)$ and $t'\notin B_0$ by assumption. Thus $$-1 < t-1 \le \Re(F(t)) \le \Re(t')) < 0$$ and both points $t'$ and $F(t)$ belong to $(-1, 0)$. Now we consider several cases.
1. If $\Re(F(0))\ge t$, then $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$ by Lemma \[lem:F(0)>t\].
2. Suppose that $a<t$ and $0<F(0)\le t$. If $q\ge 1$, then we are in the situation depicted in Figure \[fig:C3b1\] and we can apply Proposition \[prop:SemiHorseshoe\] to $[a,t]$ and $[t,1]$.
![Case (b) with $q\ge 1$ ($q=1$ in the picture): the intervals $[a,t]$ and $[t,1]$ form a horseshoe.[]{data-label="fig:C3b1"}](periods-sigma-fig33)
Now assume that $q\le 0$, which implies that $a\neq 0$. Let $I=[a,t]$ and $J=[0,y_0]$. Since $F(1)>1$, there exists $d'\in (t,1)$ such that $F(d')>1$. If $\Re(z')\ge 1$, we set $d=d'$; otherwise $z'\in (0,1)$ and we set $d=z'$. In both cases, $t<d< 1$ and $\Re(F(d))\ge d$. Since $\Re(F(t))\le 0<a$, there exists $c\in (t,d)$ such that $F(c)=a$. Let $K=[c,d]$. Then the three intervals $I,J,K$ contain no branching point in their interior and they are disjoint ${\ensuremath{\kern -0.55em\pmod{1}}}$ (that is, the sets $I+{\ensuremath{\mathbb{Z}}}, J+{\ensuremath{\mathbb{Z}}},K+{\ensuremath{\mathbb{Z}}}$ are disjoint). Moreover we have $F(I)\supset J+q$, $F(J)\supset K$ (because $F(0)\le t$ and $\Re(F(y_0))=\Re(F(z))\ge \Re(z')\ge d$) and $F(K)\supset I\cup K$ (see Figure \[fig:C3b2\]).
![Case (b) with $q\le 0$; on the right: covering graph of $I,J,K$.[]{data-label="fig:C3b2"}](periods-sigma-fig34){width="90.00000%"}
We define the loops of coverings $${\mathcal{C}}:=K {\nolinebreak[4]\longrightarrow\nolinebreak[4]}K
\quad \text{and} \quad
{\mathcal{C}}':= K-q {\nolinebreak[4]\longrightarrow\nolinebreak[4]}I-q {\nolinebreak[4]\longrightarrow\nolinebreak[4]}J {\nolinebreak[4]\longrightarrow\nolinebreak[4]}K.$$ The loop ${\mathcal{C}}$ gives a fixed point. For $n\ge 3$, we consider ${\mathcal{C}}' {\mathcal{C}}^{n-3}$, which is a loop of length $n$. According to Proposition \[prop:covering\], there exists a periodic ${\ensuremath{\kern -0.55em\pmod{1}}}$ point $x\in K-q$ such that $F^n(x)=x+q$, $F(x)\in I-q$, $F^2(x)\in J$ and $F^i(x)\in K$ for all $3\le i\le n$. It remains to prove that the period [[$\kern -0.55em\pmod{1}$]{}]{} of $x$ is exactly $n$. Let $p$ be the period ${\ensuremath{\kern -0.55em\pmod{1}}}$ of $x$. If $p< n$, then $p\le n-2$ because $p$ divides $n\ge 3$. Thus $F^2(x)\in J$, $F^{2+p}(x)\in K$ and $F^{2+p}(x)- F^2(x)\in{\ensuremath{\mathbb{Z}}}$. But this is impossible because $(J+{\ensuremath{\mathbb{Z}}})\cap (K+{\ensuremath{\mathbb{Z}}})=\emptyset$. This proves that $p=n$. Therefore, $\operatorname{Per}(F)\supset{\ensuremath{\mathbb{N}}}\setminus\{2\}$.
3. If $0<F(0)<t\le a$ and $\Re(F(y_0))\ge 1$, we set $I=[t,1]$ and $J=[0,y_0]\subset B_0$ (see Figure \[fig:C3c\]). We have $F(I)\supset I$ (because $F(t)<t$ and $F(1)>1$), $F(I)\supset J=q$ (because $a\in I$ and $F(1)\notin B$), $F(J)\supset I$ (because $F(0)<t$ and $\Re(F(y_0))\ge 1$ by assumption). Hence $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$ by Lemma \[lem:SemiHorseshoe-mod1\].
![Case (c); on the right: covering graph of $I,J$.[]{data-label="fig:C3c"}](periods-sigma-fig35){width="90.00000%"}
4. If $z'\in (0,1)$ and $\Re(F(0))\le t\le a\le z'$, we set $I=[t,z']\subset {\ensuremath{\mathbb{R}}}$ and $J=[0,y_0]\subset B_0$ (see Figure \[fig:C3d\]). Then $F(J) \supset I$ (because $\Re(F(0))\le t$ and $\Re(F(y_0))=\Re(F(z))\ge z'$), $F(I)\supset I$ (because $\Re(F(t))\le t$ and $\Re(F(z'))\ge z'$) and $F(I)\supset J+q$ (because $a\in I$ and either $F(t)\notin B_q$ or $F(z')\notin B_q$).
![Case (d): the two arrows starting from $0$ mean that it is only known that $\Re(F(0))\le t$; on the left: covering graphs of $[0,y_0]$ and $J=[t,z']$.[]{data-label="fig:C3d"}](periods-sigma-fig36)
Hence $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$ by Lemma \[lem:SemiHorseshoe-mod1\].
5. Suppose that $z'\in (0,1)$, $\Re(F(0))\le t$ and $a>z'$. If $z'\le t'+1$, we apply Lemma \[lem:3M\] with $t_1=t$, $t_2=t'+1$, $z_0=z'$ and we obtain $\operatorname{Per}(F)={\ensuremath{\mathbb{N}}}$. If $z'\ge t'+1$, we apply Lemma \[lem:t’+1\] and we obtain $\operatorname{Per}(F)\supset{\ensuremath{\mathbb{N}}}\setminus\{2\}$.
Case (III) is covered by items (a), (b) and (c). Case (II) is covered by items (a), (b), (d) and (e), and Case (I) is covered by items (d) and (e). This concludes the proof.
### Conclusion of the proof
Suppose that $m\in\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F))$ with $m\in{\ensuremath{\mathbb{Z}}}$. We may assume that $0\in\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F))$ by considering $F-m$ instead of $F$, which has the same set of periods. Lemmas \[lem:caseC1\], \[lem:caseC2\] and \[lem:caseC3\] give the conclusion in Case (C). In a similar but symmetric way Case (D) holds. This, together with Lemmas \[lem:largegap\] and \[lem:sortgapR\], gives at last Theorem \[theo:0inInterior\].
The set of periods of rotation number 0 — some surprises {#sec:0suprises}
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For a lifting of a circle map $F \in {\mathcal{L}}_1({\ensuremath{\mathbb{R}}})$, the strategy to determine $\operatorname{Per}(F)$ is to characterize $\operatorname{Per}(p/q,F)$ for every rational rotation number $p/q$ (see [@ALM]). The situation is different depending whether $p/q$ belongs to the interior of the rotation interval or to its boundary. Assume that $p,q$ are coprime. If $p/q\in \operatorname{Int}(\operatorname{Rot}(F))$, it is known that $\operatorname{Per}(p/q,F)=q{\ensuremath{\mathbb{N}}}$. If $p/q\in\operatorname{Bd}(\operatorname{Rot}(F))$, there exists $s\in{\ensuremath{\mathbb{N}}}\cup\{{\ifx\empty\empty\else \empty\cdot \fi2^{\infty}}\}$ such that $\operatorname{Per}(p/q,F)=q\cdot\operatorname{S\mbox{\tiny\textup{sh}}}(s)$. In both cases, the strategy is to prove the result for $0$ (i.e. $p/q=0/1$) and then apply it to $G:=F^q-p$ to obtain the result for $\operatorname{Per}(p/q,F)$. When one deals with the set of periods of a map $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$, the first, natural idea is to adopt the same strategy and, first, (try to) characterize $\operatorname{Per}(0,F)$. However, this idea does not work as expected, neither for $\operatorname{Per}(0,F)$, nor for the step relating $\operatorname{Per}(p/q,F)$ to what can occur for $0$. The aim of this section is to show the problems that can arise for the rotation number $0$. Recall that Theorem \[theo:0inInterior\] states that, if $0\in\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F))$, then $\operatorname{Per}(F)$ contains all integers except maybe $1$ or $2$. Notice that this result deals with all periods [[$\kern -0.55em\pmod{1}$]{}]{} and not true periods. The conditions $p/q\in\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F))$ and $0\in\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F^q-p))$ are equivalent; but, whereas it is straightforward to deduce $\operatorname{Per}(p/q,F)$ from $\operatorname{Per}(0, F^q-p)$, there is no easy way to determine $\operatorname{Per}(F)$ when one knows $\operatorname{Per}(F^q-p)$. On the other hand, Theorem \[ConverseEndInteger\] deals with a difficulty arising for rotation numbers $p/q\in \operatorname{Bd}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F))$ when $p/q\notin{\ensuremath{\mathbb{Z}}}$.
In all examples of this section, the map $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$ will satisfy $F({\ensuremath{\mathbb{R}}})=S$, and hence ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)=\operatorname{Rot}(F)$ by [@AlsRue2008 Proposition 3.4].
Per(0, F) when 0 is in the interior of the rotation interval
------------------------------------------------------------
The general rotation theory for a degree 1 map on an infinite tree states that, if $0\in\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F))$, there exists $n$ such that $\operatorname{Per}(0,F)\supset{\ensuremath{\{k\in{\ensuremath{\mathbb{N}}}\,\colon k\ge n\}}}$ [@AlsRue2008 Theorem 3.11]. Unfortunately, the integer $n$ can be arbitrarily large, even for the space $S$, as shown by the next example.
\[ex:Per(0,F)\] **A map such that $0\in\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F))$ and $\operatorname{Per}(0,F)=\{k\in{\ensuremath{\mathbb{N}}}\mid k\ge n\}$.**
We fix $n\ge 3$. Let $b=\max B_0$ and choose $a\in (-1,0)$. We define $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$ such that $F(0)=-1$, $F(b)=
b+1$, $F(a)=b-n-1$ and $F$ is affine on $B_0$, $[-1,a]$ and $[a,0]$. The map $F$ is illustrated in Figure \[fig:Per(0,F)\].
![The map $F$ of Example \[ex:Per(0,F)\] and the covering graph of $B_0$ and $A=[-1,0]$. The Markov graph can be easily deduced from this graph by splitting $A$ into $[-1,a]$ and $[a,0]$.[]{data-label="fig:Per(0,F)"}](periods-sigma-fig12)
Using the Markov graph of $F$ and the tools from [@AlsRue2008 Subsection 6.1], one can compute that ${\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)=\operatorname{Rot}(F)=[-(n-2),1]$ (which contains $0$ in its interior for every $n\ge 3$) and $\operatorname{Per}(0,F)=\{k\in{\ensuremath{\mathbb{N}}}\mid k\ge n\}$.
Sets of periods living in complicated trees can be obtained for rotation number 0
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Although the whole space $S$ is an infinite tree, a periodic orbit of rotation number $0$ is a true periodic orbit, and thus it is compact and lives in a finite subtree of $S$. This makes possible to study $\operatorname{Per}(0,F)$ by using the works on periodic orbits for finite trees [@AGLMM; @AJM4]. In Section \[sec:Y\], we saw that the sets $\operatorname{Per}(0,F)$ can display all possible sets of periods of maps in ${\mathcal{X}_{3}}$. In this subsection, we show that the converse is not true: there exist maps in ${\ensuremath{\mathcal{L}_{1}(S)}}$ with $0\in{\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)$ and such that $\operatorname{Per}(0,F)$ is not the set of periods of a map in ${\mathcal{X}_{3}}$. We are going to exhibit examples in which $\operatorname{Per}(0,F)$ can be deduced from the set of periods of a tree map, where the tree is more complicated than a $3$-star.
Let us introduce some notation. Let $P$ be a true periodic orbit of $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$. We will denote by $T_P\subset S$ the finite tree defined by $$T_P := \chull{\Re(P)} \cup
\bigcup_{i \in \chull{r\circ P} \cap \Z} B_i.$$ Observe that $T_P$ and the closure of $S \setminus T_P$ have at most two points in common: $\min \Re(P) \in {\ensuremath{\mathbb{R}}}$ and $\max \Re(P) \in {\ensuremath{\mathbb{R}}}.$ Moreover, $\min \Re(P)$ and $\max \Re(P)$ are either points of $P$ or branching points.
We also define the map by $F_P := {r}_{T_P} \circ F{\bigr\rvert_{T_P}},$ where ${r}_{T_P}$ is the standard retraction from $T$ to $T_P.$ More precisely, for every $x\in T_P,$ $$F_P(x) = \begin{cases}
F(x) & \text{if $F(x)\in T_P$,}\\
\min \Re(P) & \text{if $\Re(F(x)) < \min \Re(P)$,}\\
\max \Re(P) & \text{if $\Re(F(x)) > \max \Re(P)$.}
\end{cases}$$
Let $x\in T_P$. If $F^n(x)\in T_P$ for all $n\ge 0$, then the orbits of $x$ under $F$ and $F_P$ coincide. In particular, $x$ is $F$-periodic of period $k$ if and only if it is $F_P$-periodic of period $k$. When the orbits of $x$ under $F$ and $F_P$ do not coincide, it follows that $x$ is eventually mapped by $F_P$ either to $\min \Re(P)$ or $\max \Re(P)$. Therefore, these are the only points that may be periodic for $F_P$ but not for $F$. This leads to the next lemma, showing that it is worth studying the set of periods of $F_P$.
\[lem:Fp-F\] There exists $E\subset {\ensuremath{\mathbb{N}}}$ with $\#E\le 2$ such that $\operatorname{Per^{\circ}}(F_P)\setminus E\subset \operatorname{Per}(0,F)$.
Now we briefly define (in a slightly restricted case) the notions of patterns and linear models introduced in [@AGLMM] to study the sets of periods of tree maps. Let $T$ be a (finite) tree, $P$ a finite subset of $T$ with at least two elements and $\varphi$ a cyclic permutation of $P$. The *discrete components* of $P$ are the sets $\overline{C_i}\cap P, i=1, \ldots, n$, where $C_1,\ldots, C_n$ are the connected components of $\chull{P}\setminus P$. If $x,y$ are two distinct elements of the same discrete component, $\chull{x,y}$ is called a *$P$-basic path*. If $T'$ (resp. $P$, $\varphi'$) is also a tree (resp. a finite subset of $T'$ with at least two elements, a cyclic permutation of $P'$), we write $(T,P,\varphi)\sim_{pat}(T',P',\varphi')$ if there exists a bijection [\[P’\]]{} such that $h\circ \varphi=\varphi'\circ h$ and $h$ preserves the discrete components. This gives an equivalence relation; the equivalence class of $(T,P,\varphi)$ is denoted $[T,P,\varphi]$ and is called a *periodic pattern*. If is a tree map, $P$ a periodic orbit of $f$ and $A$ a periodic pattern, we say that $f$ *exhibits $A$* over $P$ if $[T,P,f{\bigr\rvert_{P}}]=A$. The set of periods *forced* by a pattern $A$ is the maximal subset $E_A\subset {\ensuremath{\mathbb{N}}}$ such that every tree map exhibiting the pattern $A$ also has periodic orbits of period $n$ for all $n\in E_A$.
The triple $(T,f,P)$ is called an *$A$-linear model* if
- $f$ exhibits $A$ over $P$,
- $f$ is monotone on all $P$-basic paths,
- for every connected component $I$ of $T\setminus (P\cup V(T))$ (where $V(T)$ denotes the set of vertices of $T$), $f{\bigr\rvert_{\overline{I}}}$ is affine.
Notice that the monotonicity on $P$-basic paths implies that the image of each vertex $v$ is uniquely determined and belongs to $P\cup V(T)$ (consider three $P$-basic paths containing $v$ and their images in order to find $f(v)$ – see also [@AGLMM Proposition 4.2]). Thus an $A$-linear model is Markov with respect to the partition generated by $P\cup V(T)$. The $A$-linear model is the analogous of the “connect-the-dots” map associated to a periodic orbit of an interval map, but the difficulty for tree maps is that the linear model may live in a different tree than the original one — some of the vertices may collapse or explode.
The key results are the following ones. For every periodic pattern $A$, there exists an $A$-linear model (and it is unique up to isomorphism) [@AGLMM Theorem A]. Moreover, if a tree map $f$ exhibits the periodic pattern $A$, then the set of periods of significant periodic points of an $A$-linear model is included in $\operatorname{Per^{\circ}}(f)$ [@AJM4 Corollary B]. A periodic point is called *significant* if its orbit is not equivalent, by iteration of the map, to the orbit of a vertex, see e.g. for the precise definition. Significant periodic points essentially correspond to loops in the Markov graph, therefore the set of periods forced by a periodic pattern $A$ can be computed using the Markov graph of an $A$-linear model.
The characterization of the whole set of periods of a tree map uses the $p$-orderings of Baldwin, where $p$ ranges in a finite set of integers depending on the tree, in particular on the valences of the vertices. When the tree is a $k$-star, one may need the $p$-orderings $\leso{p}$ for $2\le p\le k$.
Let us come back to the map $F_P$ coming from a periodic orbit $P$ of $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$. Although all the vertices of $T_P$ have valence 3, the linear model of $[T_P,P,F_P{\bigr\rvert_{P}}]$ may have vertices of arbitrarily large valence. In Example \[ex:n-star-model\], we show that, for all $k\ge 3$, there exist $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$ and $P$ a periodic orbit of $F$ such that the linear model of $[T_P,P,F_P]$ lives in a $k$-star and the $k$-th partial ordering of Baldwin is needed to express the set of periods of $F_P$. More complicated trees than stars can even be obtained, as shown in Example \[ex:2gluedstars-model\].
\[ex:n-star-model\] Fix an integer $k\ge 3$. Choose $a\in (0,1)$ and $b_0,b_1,\ldots, b_{k-1}\in B_0$ such that $1=b_0>b_1>\cdots>b_{k-1}>0$. We set $x_i=i+b_i\in B_i$ for all $0\le i\le k-1$ and $x_k=a+k-2\in{\ensuremath{\mathbb{R}}}$. In addition, we set $$A_i = [b_{i+1}, b_i] \text{ for all } 0\le i\le k-2,\
A_{k-1} = [0,b_{k-1}], \
L= [0,a] \text{ and }
R=[a,1].$$ We define the map $F\in{\ensuremath{\mathcal{L}_{1}(S)}}$ such that $F(x_i)=x_{i+1}$ for all $0\le
i\le k-1$, $F(x_k)=x_0$, $F(1)=0$, $F$ is affine in restriction to each of the intervals $L$, $R$ and $A_i, 0\le i\le k-1$, and the map is defined on the rest of $S$ using degree 1. Then $P=(x_0,x_1,\ldots,x_k)$ is a true periodic orbit of period $k+1$ for $F$, and $F$ is linear Markov. The map $F$ and its Markov graph are represented in Figure \[fig:Fk\].
The map $F_P$ is defined on $T_P=B_0\cup\cdots\cup B_{k-1}\cup [0, k-1]$. If $F_P(x)\neq F(x)$ then, either $F_P(x)=k-1$, or $F_P(x)=0$. The point $0$ is fixed under $F_P$ and $F_P^{k-1}(k-1)=0$ Thus $\operatorname{Per^{\circ}}(F_P)\setminus\{1\}\subset \operatorname{Per}(0,F)$.
The linear model of $F_P$ is supported by a $k$-star; it is represented in Figure \[fig:model-Fk\]. To prove this fact, the easiest (but not most convincing) way is to see that the map in Figure \[fig:model-Fk\] does exhibit the right pattern, then the uniqueness of the linear model gives the conclusion. We leave to the interested readers the checking that the only way to realize a linear model of $F_P$ is to collapse the $k-2$ vertices of $T_P$. This can be done by looking at all basic paths and their images.
![On the right: the linear model of $[T_P,P,F_P{\bigr\rvert_{P}}]$, the map being affine on each of the intervals $B_0,\ldots, B_k$ (picture is for $k=5$). On the left: its Markov graph.[]{data-label="fig:model-Fk"}](periods-sigma-fig17){width="95.00000%"}
![On the right: the linear model of $[T_P,P,F_P{\bigr\rvert_{P}}]$, the map being affine on each of the intervals $B_0,\ldots, B_k$ (picture is for $k=5$). On the left: its Markov graph.[]{data-label="fig:model-Fk"}](periods-sigma-fig18){width="95.00000%"}
From the linear model, one can show that the pattern $[T_P,P,F_P{\bigr\rvert_{P}}]$ forces all the periods $n$ for $n\le_k k+1$, where $\le_k$ is the $k$-ordering of Baldwin. A direct computation from the Markov graph of $F$ gives $\operatorname{Rot}(F)=[-k+2,0]$ and $$\operatorname{Per}(0,F) = \{k,k+1\} \cup
{\ensuremath{\{ik+j(k+1) \,\colon i,j \ge 1\}}}=
{\ensuremath{\{n\in{\ensuremath{\mathbb{N}}}\,\colon n \le_k k+1\}}} \setminus \{1\}.$$ Therefore, the inclusions ${\ensuremath{\{n\in{\ensuremath{\mathbb{N}}}\,\colon n\le_k k+1\}}}\subset \operatorname{Per^{\circ}}(F)$ and $\operatorname{Per^{\circ}}(F)\setminus\{1\}\subset \operatorname{Per}(0,F)$ are equalities.
\[ex:2gluedstars-model\] Given $p,q\ge 3$, it is possible to build a map $G\in{\ensuremath{\mathcal{L}_{1}(S)}}$ with a true periodic orbit $P$ of period $p+2q-4$ such that the linear model of $G_P$ lives in a tree consisting in a $p$-star glued to a $q$-star. To remain readable, we illustrate the construction for $p=6$ and $q=7$ (hence the period of $P$ is $16$) instead of giving the definition for arbitrary $p,q$. We choose points $x_0\in (0,1)$ and $x_1,\ldots, x_{15}\in B$ as in Figure \[fig:2gluedstars\].
![The map $G$ from Example \[ex:2gluedstars-model\] and its periodic orbit $P=\{x_0,\ldots, x_{15}\}$; $G$ is of degree 1 and affine on each interval of the partition generated by $(P\cup\{0\})+{\ensuremath{\mathbb{Z}}}$.[]{data-label="fig:2gluedstars"}](periods-sigma-fig19){width="95.00000%"}
Then $G$ is defined by $G(x_i)=x_{i+1}$ for all $0\le i\le 15$, $G(x_{15})=x_0$, $G(0)=-5$ and $G$ is of degree $1$ and affine on each interval of the partition generated by these points ${\ensuremath{\kern -0.55em\pmod{1}}}$. We do no draw the Markov graph of $G$, which is rather big, but one may check that $\operatorname{Rot}(G)=[-5,1]$ (in the Markov graph, the endpoints of $\operatorname{Rot}(G)$ are reached by the loops $[0,x_0]{ \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}-5\hspace*{.1em}} \nolinebreak[4]}[0,x_0]$ and, e.g., $[x_{7}+2,x_{12}]{ \nolinebreak[4] \xrightarrow[]{\hspace*{.25em}1\hspace*{.1em}} \nolinebreak[4]}[x_{7}+2,x_{12}]$). The tree $T_P$ is equal to $[-4,6]\cup\bigcup_{-4\le i\le 6}B_i$. The point $-4$ is fixed for $G_P$ and the point $6$ is sent to $-4$ by $G_P^2$. Therefore, as in Example \[ex:n-star-model\], $\operatorname{Per^{\circ}}(G_P)\setminus\{1\}\subset\operatorname{Per}(0,G)$. The linear model of $G_P$ is represented in Figure \[fig:2gluedstars-model\]; the $p-2$ vertices of $T_P$ less than or equal to $0$ collapse into a fixed vertex, and the $q-2$ vertices greater than or equal to $1$ collapse to another fixed vertex. It is possible to compute that the set of (significant) periods of the linear model is $\{1\}\cup\{n\ge 6\}$ and that $\operatorname{Per}(0,F)=\{n\ge 6\}$.
![The linear model of $G_P$ (from Example \[ex:2gluedstars-model\]): the points $x_0,\ldots, x_{15}$ are mapped cyclically, the two vertices are fixed and the map is affine on each interval generated by this partition.[]{data-label="fig:2gluedstars-model"}](periods-sigma-fig20)
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Evgeny Andronov[^1], for the NA61/SHINE Collaboration'
title: 'Recent results from the NA61/SHINE strong interaction physics programme'
---
Introduction {#intro}
============
NA61/SHINE [@NA61] is a fixed target experiment at the Super Proton Synchrotron (SPS) of the European Organization for Nuclear Research (CERN). The layout of the NA61/SHINE detector is sketched in Fig. \[fig1\]. It consists of a large acceptance hadron spectrometer with excellent capabilities in charged particle momentum measurements and identification by a set of five Time Projection Chambers as well as Time-of-Flight detectors. The geometrical layout of the TPCs allows particle detection down to $p_{T}=0$ GeV/c in a broad interval of the forward rapidity semisphere, which is practically impossible at collider experiments. The high resolution, modular forward calorimeter, the Projectile Spectator Detector, measures forward going energy $E_{F}$, which in nucleus-nucleus reactions is primarily a measure of the number of spectator (non-interacted) nucleons and thus related to the centrality of the collision.
![Schematic layout of the NA61/SHINE experiment at the CERN SPS (horizontal cut in the beam plane, not to scale).[]{data-label="fig1"}](./figure_1.png)
The main goal of the strong interaction programme of the experiment is to discover the Critical Point (CP) [@Fodor:2004nz] of strongly interacting matter and study the properties of the onset of deconfinement (OD) [@Gazdzicki:1998vd; @Alt:2007aa]. To achieve this goal a two-dimensional phase diagram scan - energy versus system size - is being performed by NA61/SHINE. Both primary and secondary beams are available to the experiment, allowing measurements of hadron production in collisions of protons and various nuclei (p+p, Be+Be, Ar+Sc, Xe+La) at a range of beam momenta (13[*A*]{} - 158[*A*]{} GeV/c). Figure \[datatak\] shows for which systems and energies data has already been collected (green), is scheduled for recording (red) or is planned (gray). This scan allows to probe different values of temperature $T$ and baryochemical potential $\mu_{B}$ of the strongly interacting matter at the freeze-out stage [@Becattini:2006].
![Data taking status of the strong interaction programme of NA61/SHINE.[]{data-label="datatak"}](Acr3326270866064817651.png){width="5cm"}
Spectra and yields: studying the properties of the onset of deconfinement {#spec}
=========================================================================
One of the main signals of the onset of deconfinement are the kink, horn, and step [@Gazdzicki:1998vd] structures observed in Pb+Pb collisions by the NA49 collaboration. Analysis of spectra and yields by NA61/SHINE allows to check whether these structures are present also in collisions of small and intermediate mass nuclei. Recent measurements of Argon on Scandium collisions are an important step in this program. Figure \[spectra\] shows the spectra of $\pi^{-}$ from strong and electromagnetic processes in Ar+Sc collisions at 150[*A*]{} GeV/c obtained with the $h^{-}$ analysis method [@Lewicki:2016]. The fact that approximately 90$\%$ of negatively charged hadrons produced in the SPS energy range are $\pi^{-}$ mesons is at the core of this method. Correction for contribution from other negatively charged particles is done using Monte Carlo simulations based on the EPOS 1.99 model [@Pierog:2009zt] together with the GEANT-3.2 code for particle transport and detector simulation.
Rapidity spectra (see Fig. \[spectra\], middle) were fitted to obtain $4\pi$ mean multiplicities of $\pi^{-}$ mesons [@Naskret:2016]. As the measurements via the $h^{-}$ analysis method are possible only for $\pi^{-}$ mesons, the multiplicities of $\pi^{+}$ and $\pi^{0}$ mesons were approximated by ${\langle\pi\rangle}_{p+p}=3{\langle\pi^{-}\rangle}_{p+p}+1$ and ${\langle\pi\rangle}_{A+A}=3{\langle\pi^{-}\rangle}_{A+A}$. In order to compare matter created in the collisions of different nuclei the mean pion multiplicity $\langle \pi\rangle$ is divided by the mean number of wounded nucleons $\langle W\rangle$ corresponding to the given class of collision centrality. This quantity was obtained using the Monte Carlo model - Glissando 2.73 [@Broniowski:2007nz]. Figure \[kink\] shows the kink plot, where the $\langle \pi\rangle$ multiplicity, normalized to $\langle W\rangle$, increases faster with $F={\left(\frac{{\left(\sqrt{s_{NN}}-2m_{N}\right)}^{3}}{\sqrt{s_{NN}}}\right)}^{1/4}$ in the SPS energy range for central Pb+Pb collisions than in p+p interactions. This behaviour violates the prediction of the Wounded Nucleon Model [@Bialas:1976ed] $\langle\pi\rangle_{A+A}/\langle W\rangle = \langle\pi\rangle_{p+p}/2$, but is successfully explained by the entropy increase due to formation of quark-gluon plasma in the Statistical Model of the Early Stage (SMES) [@Gazdzicki:2010iv]. The new results obtained for central Ar+Sc collisions follow the Pb+Pb trend for high SPS energies and are close to the p+p results for low SPS energies whereas the new results for Be+Be show the opposite tendency. One should mention that the mean number of wounded nucleons $\langle W\rangle$ is a model-dependent quantity, leading to uncertainties when comparing results obtained for different systems.
![Preliminary results for the mean pion multiplicity ${\langle\pi\rangle}$ divided by the mean number of wounded nucleons $\langle W\rangle$ versus the Fermi energy measure $F\simeq s^{1/4}_{NN}$ for inelastic p+p interactions and central Be+Be, Ar+Sc collisions from NA61/SHINE and for world data on p+p and central A+A collisions [@Golokhvastov:2001; @Abbas:2013].[]{data-label="kink"}](Acr33262708660648-6767.png){width="5cm"}
The NA49 collaboration observed a plateau (step) in the inverse slope parameter ($T$) of transverse mass ($m_{T}$) spectra of kaons for Pb+Pb collisions as expected from the SMES model for constant temperature and pressure in a mixed phase. The recent NA61/SHINE results [@Pulawski:2015tka], presented in Fig. \[step\], show that even in p+p collisions the energy dependence of $T$ for kaons (identified using the $dE/dx$ method) exhibits a rapid change in the SPS energy range.
![Energy dependence of the inverse slope parameter T of transverse mass spectra of K$^-$ and K$^+$ in inelastic p+p interactions measured by the NA61/SHINE experiment (full blue circles) and other experiments (open blue circles) and central Au+Au and Pb+Pb interactions. Blue band represents the systematic uncertainty.[]{data-label="step"}](step.png){width="10cm"}
Moreover, sharp peaks in the energy dependence of the ratios $K^{+}/\pi^{+}$ and $\Lambda/\pi$ were found for Pb+Pb collisions by the NA49 collaboration. Figure \[horn\] shows a comparison of the new measurements by NA61/SHINE for inelastic p+p interactions with the world data. Candidates of charged decays of Lambda hyperons were identified by the standard topological cuts applied to pairs of positively and negatively charged particles detected by the TPCs [@Aduszkiewicz:2015dmr; @Stroebele:2016]. One observes that even in p+p interactions the $K^{+}/\pi^{+}$ ratio exhibits rapid changes with energy whereas new measurements of the $\Lambda/\pi$ ratio were done only for two energies and do not give a clear picture of the energy dependence.
Fluctuation observables: search for the critical point {#fluc}
======================================================
The strategy of looking for the critical point (CP) of strongly interacting matter is based on the expectation that the correlation length $\xi$ diverges at the CP. This divergence may lead to the growth of fluctuations for different observables such as multiplicity, net charge etc. Therefore, one can expect that a scan over freezeout points close to the CP will show non-monotonic behavior of these fluctuation observables.
This search is complicated by the fact that the size of the system created in collisions of two nuclei changes significantly from event to event. Observables can be classified according to their dependence on this volume and its fluctuations: 1) extensive quantity - proportional to the system volume in the Grand Canonical Ensemble or the number of the wounded nucleons in the Wounded Nucleon Model [@Bialas:1976ed] 2) intensive quantity - independent of the system volume 3) strongly intensive quantity [@Gorenstein:2011vq] - independent of the system volume and fluctuations of this volume. Strongly intensive quantities are best suited to study fluctuations in nucleus-nucleus collisions because of the unavoidable event-to-event variations of the volume.
Scaled variance of multiplicity {#scvar}
-------------------------------
The most common way to characterize the strength of fluctuations of the quantity $A$ is to determine the variance of its distribution $Var\left(A\right)$ which is an extensive quantity. In turn, the scaled variance $$\omega[A]=\frac{Var\left(A\right)}{\langle A\rangle}$$ is intensive. Here, $\langle \cdots\rangle$ stands for averaging over all events.
In the Wounded Nucleon Model one can get the following expression for the scaled variance of the distribution of the multiplicity $N$: $$\omega[N]={\omega}^{*}[N]+\langle N\rangle/\langle W\rangle \cdot\omega[W],$$ where ${\omega}^{*}[N]$ is the scaled variance calculated for the fixed value $\langle W\rangle$ (i.e. ${\omega}^{*}[N]=\omega[N]_{p+p}$). This leads to the following ineqality $$\omega[N]_{A+A}\geq \omega[N]_{p+p}
\label{ineq}$$
The recent measurements by the NA61/SHINE collaboration show that this inequality is violated [@Seryakov:2016]. Figure \[scaled\] shows the system size dependence of the scaled variance $\omega[N]$ for negatively charged hadrons for two beam momenta - 19 and 150[*A*]{} GeV/c. One observes that fluctuations of multiplicity in very central Ar+Sc collisions are suppressed compared to p+p interactions, in disagreement with Eq. (\[ineq\]).
![Preliminary results on $\omega[N]$ of the multiplicity distribution of negatively charged hadrons in inelastic p+p interactions and for the 0.2$\%$ of events with the lowest $E_{F}$ in Ar+Sc collisions at 19 and 150[*A*]{} GeV/c. Statistical uncertainties are smaller than the symbol size. The horizontal dashed line divides the plot into WNM exluded (below the line) and WNM allowed (above the line) zones.[]{data-label="scaled"}](W_omega_neg_na61_19GeV.pdf "fig:"){width="6.5cm"} ![Preliminary results on $\omega[N]$ of the multiplicity distribution of negatively charged hadrons in inelastic p+p interactions and for the 0.2$\%$ of events with the lowest $E_{F}$ in Ar+Sc collisions at 19 and 150[*A*]{} GeV/c. Statistical uncertainties are smaller than the symbol size. The horizontal dashed line divides the plot into WNM exluded (below the line) and WNM allowed (above the line) zones.[]{data-label="scaled"}](W_omega_neg_na61_150GeV.pdf "fig:"){width="6.5cm"}
Joint fluctuations of multiplicity and transverse momentum {#joint}
----------------------------------------------------------
In order to suppress contributions from these ’trivial’ fluctuations strongly intensive observables are used: $$\Delta[A,B] = \frac{1}{C_{\Delta}} \biggl[ \langle B \rangle \omega[A] -
\langle A \rangle \omega[B] \biggr]$$ $$\Sigma[A,B] = \frac{1}{C_{\Sigma}} \biggl[ \langle B \rangle \omega[A] +
\langle A \rangle \omega[B] - 2 \bigl( \langle AB \rangle -
\langle A \rangle \langle B \rangle \bigr) \biggr]$$ In case of joint transverse momentum $P_{T}$ and multiplicity $N$ fluctuations we define: $A=P_{T}=\sum_{i=1}^{N}p_{T_{i}}$, $B=N$, $C_{\Delta}=C_{\Sigma}=\langle N\rangle \omega[p_{T}]$.
The NA61/SHINE collaboration performed measurements of these two quantities for inelastic p+p interactions, as well as Be+Be and Ar+Sc collisions for the smallest 5$\%$ of forward energies [@Andronov:2016]. Figure \[deltasigma\] shows $\Delta[P_{T},N]$ (left) and $\Sigma[P_{T},N]$ (right) as a function of $\sqrt{s_{NN}}$ and mean number of wounded nucleons $\langle W\rangle$ (from [@Broniowski:2007nz]). No dip-like or hill-like structures that could be related to the critical point of strongly interacting matter are observed in these results.
![Preliminary results on $\Delta[P_{T},N]$ (left) and $\Sigma[P_{T},N]$ (right) of all charged hadrons in inelastic p+p and central Be+Be and Ar+Sc collisions. Statistical uncertainties are smaller than the symbol size.[]{data-label="deltasigma"}](Delta2d.pdf "fig:"){width="6.5cm"} ![Preliminary results on $\Delta[P_{T},N]$ (left) and $\Sigma[P_{T},N]$ (right) of all charged hadrons in inelastic p+p and central Be+Be and Ar+Sc collisions. Statistical uncertainties are smaller than the symbol size.[]{data-label="deltasigma"}](Sigma2d.pdf "fig:"){width="6.5cm"}
Higher order moments of the net-charge distribution
---------------------------------------------------
The fluctuation quantities analyzed in sections \[scvar\] and \[joint\] include only first and second moments of the studied observables. Higher order moments of fluctuations should depend more sensitively on $\xi$, possibly making the signal of the CP more visible. Another reason to measure higher order moments of the net electric charge distribution is that for conserved quantum numbers there is a possibility to compare with susceptibilities calculated in lattice QCD [@Karsch:2011]. The volume independent combinations of the higher order moments are selected as $$S\sigma = \frac{{\langle Q^{3}\rangle}_{c}}{Var(Q)}, \,\,\,\,\,\,\, \kappa\sigma^{2} = \frac{{\langle Q^{4}\rangle}_{c}}{Var(Q)},$$ where ${\langle Q^{3}\rangle}_{c}$ and ${\langle Q^{4}\rangle}_{c}$ are the third and fourth order cumulants of the net charge distribution.
Figure \[net\] shows preliminary results on fluctuations of net-charge in inelastic p+p interactions [@Mackowiak:2016]. Both the scaled variance $\omega$ and $S\sigma$ depend weakly on collision energy whereas $\kappa\sigma^{2}$ rises significantly. Measured net-charge fluctuations do not agree with independent particle production (Skellam distribution). The difference may come from production of multi-charged particles and quantum statistics [@PBM:2012]. The EPOS 1.99 model provides a reasonably good description of the data.
Summary and conclusion
======================
NA61/SHINE data taking for the system size $-$ energy scan is well advanced: data for p+p, ${}^{7}$Be+${}^{9}$Be and ${}^{40}$Ar+${}^{45}$Sc collisions have already been recorded. Although preliminary results on transverse momentum and multiplicity fluctuations and higher order moments of the net-charge distribution did not yet give evidence of the critical point of strongly interacting matter, a number of intriguing results were observed, such as suppression of the scaled variance of the multiplicity distribution in nucleus-nucleus collisions, as well as rapid changes in the energy dependence of the kaon inverse slope parameter $T$ and of the ratio $\langle K^{+}\rangle /\langle {\pi}^{+}\rangle$ in p+p interactions.
This work was supported by the Hungarian Scientific Research Fund (grants OTKA 68506 and 71989), the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, the Polish Ministry of Science and Higher Education (grants 667/N-CERN/2010/0, NN202484339 and NN202231837), the Polish National Center for Science (grants 2011/03/N/ST2/03691, 2013/11/N/ST2/03879, 2014/13/N/ST2/02565, 2014/14/E/ST2/00018 and 2015/18/M/ST2/00125), the Foundation for Polish Science — MPD program, co-financed by the European Union within the European Regional Development Fund, the Federal Agency of Education of the Ministry of Education and Science of the Russian Federation (SPbSU research grant 11.38.242.2015), the Russian Academy of Science and the Russian Foundation for Basic Research (grants 08-02-00018, 09-02-00664 and 12-02-91503-CERN), the Ministry of Education, Culture, Sports, Science and Technology, Japan, Grant-in-Aid for Scientific Research (grants 18071005, 19034011, 19740162, 20740160 and 20039012), the German Research Foundation (grant GA1480/2-2), the EU-funded Marie Curie Outgoing Fellowship, Grant PIOF-GA-2013-624803, the Bulgarian Nuclear Regulatory Agency and the Joint Institute for Nuclear Research, Dubna (bilateral contract No. 4418-1-15/17), Ministry of Education and Science of the Republic of Serbia (grant OI171002), Swiss Nationalfonds Foundation (grant 200020117913/1), ETH Research Grant TH-0107-3 and the U.S. Department of Energy.
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H. Stroebele in proceedings of SQM2016, Berkeley, USA (2016)
M.I. Gorenstein, M. Gazdzicki Phys. Rev. C [**84**]{} 014904 (2011)
A. Seryakov, in proceedings of CPOD2016, Wroclaw, Poland (2016)
E. Andronov, in proceedings of CPOD2016, Wroclaw, Poland (2016), arXiv:1610.05569 \[nucl-ex\]
F. Karsch [*et al.*]{}, Phys. Lett. B [**695**]{}, 136 (2011)
M. Mackowiak-Pawlowska, in proceedings of CPOD2016, Wroclaw, Poland (2016), arXiv:1610.03838 \[nucl-ex\]
P. Braun-Munzinger [*et al.*]{}, Nucl. Phys. A [**880**]{}, 48 (2012)
[^1]:
| {
"pile_set_name": "ArXiv"
} |
---
address: 'Fermi National Accelerator Laboratory, Batavia, IL 60510, USA\'
author:
- |
MARTIJN MULDERS\
(on behalf of the CDF and DØ collaborations)
title: TOP QUARK MASS MEASUREMENTS AT THE TEVATRON
---
Introduction
============
The recent publication of the improved Run I measurement of the top mass by DØ [@ME] was exciting for two reasons. First of all it demonstrated how much improvement in measurement precision could be achieved using a more advanced analysis technique like the Matrix Element method. Secondly, it was a reminder of how little we yet know about the properties of the top quark and that new experimental information about the top quark can have big implications for electroweak fits in the Standard Model. The current (Run I only) world average value for the top quark mass is $178.0 \pm 4.3 $ GeV$/c^2$. In the coming years the measurements of CDF and DØ combined should lead to a precision of about 2 GeV. Together with expected improvements in the measurement of the W boson mass this will allow to further constrain the Higgs boson mass to a relative precision of approximately 30%, as discussed elsewhere in these proceedings [@hayes].
Since the start of Run II both CDF and DØ have recorded more than 600 pb$^{-1}$ of data, already 5 times the Run I luminosity. The preliminary results presented here are based on fraction of the recorded data ranging from 160 to 230 pb$^{-1}$.
Run II Top mass results
=======================
In $p\bar{p}$ collisions with $\sqrt{s} = 1.96$ TeV at the Tevatron, top quarks can be produced via the strong interaction in $t\bar{t}$ pairs, or as single top quarks through the weak interaction. Single top production is predicted to have a lower cross-section and a more challenging event signature, and has not yet been observed at the time of this conference. For the Top mass measurement therefore only top pair events are used. Each top quark decays immediately to a $W$ boson and a $b$ quark, and the $W$ bosons decay either hadronically or leptonically, giving rise to 3 possible decay channels: di-lepton, lepton+jets and all-jets.
An overview of recent $t\bar{t}$ cross-section results from the CDF and DØ experiments in all three of the above final states is given elsewhere in these proceedings [@nielsen]. In both collaborations several top mass analyses are being developed in the di-lepton and lepton+jets decays channels, mostly based on very similar event selections. No preliminary Run II results in the all-jets channel have been presented so far.
A complete and up-to-date overview of ongoing Run II analyses can be found on the collaborations’ public results web pages [@cdfpub; @d0pub]. A description of all analyses is outside the scope of these proceedings. Below a few of the analyses are briefly described in order to highlight some important aspects of the top mass measurement.
Final states with two leptons plus jets
---------------------------------------
The striking signature due to the presence of two leptons in the final state allows for a relatively pure selection of top events, typically with a signal-to-background ratio of 4/1. The main challenge however is to fully reconstruct the kinematics of the final state, which are underconstrained due to the presence of two neutrinos. Different approaches exist to add an extra constraint to the system, and see for which value of the top mass the observed events are most likely.
In Table \[tab:overview\] several Run II analyses are listed with their preliminary results. Currently the most precise result was obtained by CDF with the neutrino weighting analysis using a loosened lepton identification (one lepton + one isolated track), optimizing the statistical precision by using a higher efficiency (and slightly lower purity) selection. In this method the rapidities of both neutrinos are used as extra constraints, and a weight as function of the top mass is calculated by integrating over all possible rapidity values and comparing the reconstructed missing transverse momentum with the observed momentum imbalance using a Gaussian resolution. For each event the top mass value which leads to the highest weight is plotted and fitted using Monte Carlo Templates, as shown in Figure \[fig:plots\].
data set (pb$^{-1}$) top mass (GeV$/c^2$)
------------------------------------------------------------- ---------------------- -------------------------------------------------
di-lepton channel
CDF neutrino-weighting 200 168.1 $^{+11}_{-9.8}$ (stat) $\pm$ 8.6 (sys)
CDF M$_{\rm reco}$ Template + $t\bar{t}$ $p_z$ 194 176.5 $^{+17.2}_{-16.0}$ (stat) $\pm$ 6.9 (sys)
CDF M$_{\rm reco}$ Template + $\phi$ of $\nu_1$ and $\nu_2$ 194 170.0 $\pm$ 16.6 (stat) $\pm$ 7.4 (sys)
DØ Dalitz and Goldstein 230 155 $^{+14}_{-13}$ (stat) $\pm$ 7 (sys)
lepton+jets channel
CDF Template with b-tagging 162-193 177.2 $^{+4.9}_{-4.7}$ (stat) $\pm$ 6.6 (sys)
CDF Multi-Variate Template 162 179.6 $^{+6.4}_{-6.3}$ (stat) $\pm$ 6.8 (sys)
CDF Dynamic Likelihood 162 177.8 $^{+4.5}_{-5.0}$ (stat) $\pm$ 6.2 (sys)
DØ Ideogram 160 177.5 $\pm$ 5.8 (stat) $\pm$ 7.1 (sys)
DØ Template topological 230 169.9 $\pm$ 5.8 (stat) $^{+7.8}_{-7.1}$ (sys)
DØ Template with b-tagging 230 170.6 $\pm$ 4.2 (stat) $\pm$ 6.0 (sys)
: Overview of preliminary Run II top mass results\[tab:overview\]
Final states with one lepton plus jets
--------------------------------------
While the lepton+jets channel benefits from a higher branching ratio, it suffers from significant backgrounds from $W$+jets and non-$W$ multi-jet events.
Since only one neutrino is present the final state can be fully reconstructed. Some analyses use a constrained kinematic fit to further improve the measurement of lepton and jets beyond detector resolution. The CDF Dynamic Likelihood Method (DLM) follows a different approach, similar to the DØ Matrix Element method [@ME]; transfer functions are derived from Monte Carlo simulation describing the jet energy resolution. These functions are subsequently used in a multi-dimensional integration over phase space calculating the likelihood that the event is compatible with matrix elements describing top pair production and decay.
In order to reconstruct the invariant mass of the top decay products, a choice has to be made to assign jets and lepton to the corresponding top or anti-top quark. In a lepton+jets event 12 ways exist to do this assignment. Some analyses take only one jet assignment per event in consideration. The CDF Dynamic Likelihood Method and the DØIdeogram analysis include all possible jet assignments in the fit.
The CDF and DØ template methods use an overall fit of Monte Carlo templates to the data in order to extract the mass. The CDF Dynamic Likelihood Method and DØ Ideogram analysis derive an event-by-event likelihood to maximize the statistical information extracted from each event. The Ideogram method also includes the hypothesis that the event could be background, weighted according to an estimated event purity.
Both experiments apply b-tagging in some of the top mass analyses. One advantage of b-tagging is to strongly reduce the backgrounds. A second advantage of b-tagging for the top mass measurement in the lepton+jets channel is the reduction of the number of possible jet assignments in the case that one or two jets are b-tagged. The CDF Template analysis combines the 0-tag, 1-tag and double tagged event samples in the fit to optimize the statistical precision. DØ’s first top mass analysis with b-tagging uses events with at least one tag, which applied to a data set of 230 pb$^{-1}$ leads to the most precise preliminary Run II top mass result presented so far. Figure \[fig:plots\] shows the fitted mass for the lowest-$\chi^2$ solution for the b-tagged DØ Template analysis, compared to the Monte Carlo prediction.
An overview of the current preliminary results is shown in Table \[tab:overview\].
Prospects for the Top mass measurement
======================================
In all results reported here the dominant component of the systematic uncertainty is the uncertainties related to the jet energy scale. In the last year a lot of work has been done to improve the calibration of the reconstructed jet energies. CDF reports an improvement of a factor two or more in jet energy scale uncertainties compared to a year ago. Similar improvements are expected in DØ. This will have a direct effect on the systematic uncertainties quoted.
Further improvements in understanding the Jet Energy Scale can come from performing an in-situ calibration of the light-jet energy scale using the jets from the hadronic decay of the $W$ in the same $t\bar{t}$ events used to measure the top mass, and from studies in progress aimed at determining the b-jet energy scale from data.
Other systematics that are being studied are the modeling of initial state and final state gluon radiation in the $t\bar{t}$ Monte Carlo.
Very soon both experiments hope to present preliminary results with updated jet energy scale and an integrated luminosity of more than 300 pb$^{-1}$.
All together the prospects are very good for having new top mass results this year with a precision comparable to or better than the current world average for each of the Tevatron experiments. This will open the door to an exciting new area of top physics to be further explored in the coming years at the Tevatron.
References {#references .unnumbered}
==========
[99]{} DØ Collaboration, Nature 429 (2004) p638.
C. Hays, these proceedings, hep-ex/0505064.
J. Nielsen, these proceedings, hep-ex/0505051.
CDF Collaboration, http://www-cdf.fnal.gov/physics/new/top/top.html
DØ Collaboration, http://www-d0.fnal.gov/Run2Physics/WWW/results/top.htm
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We reformulate the Kazhdan-Lusztig theory for the BGG category $\mathcal{O}$ of Lie algebras of type D via the theory of canonical bases arising from quantum symmetric pairs initiated by Weiqiang Wang and the author. This is further applied to formulate and establish for the first time the Kazhdan-Lusztig theory for the BGG category $\mathcal{O}$ of the ortho-symplectic Lie superalgebra $\mathfrak{osp}(2m|2n)$.'
address: 'Department of Mathematics, University of Maryland, College Park, MD 20742'
author:
- Huanchen Bao
title: 'Kazhdan-Lusztig Theory of super type D and quantum symmetric pairs'
---
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Introduction {#introduction .unnumbered}
============
The Kazhdan-Lusztig theory provides the solution to the problem of determining the irreducible characters in the BGG category $\mc{O}$ of semisimple Lie algebras ([@KL; @BB; @BK]). The theory was originally formulated in terms of the canonical bases (i.e., Kazhdan-Lusztig bases) of Hecke algebras. On the other hand, the classification of finite-dimensional simple Lie superalgebras over complex numbers has been obtained by Kac ([@Kac77]) in 1970’s, while the representation theory of Lie superalgebras turns out to be very difficult. One of the main reasons is that the corresponding Weyl group of a Lie superalgebra is not enough to control the linkage principle in the BGG category $\mc{O}$. Thus the relevant Hecke algebras do not play significant roles in the representation theory of Lie superalgebras as in the representation theory of semisimple Lie algebras.
The Lie superalgebras $\mathfrak{gl}(m|n)$ and $\mathfrak{osp}(m|2n)$, which generalize the classical Lie algebras, are arguably the most important classes of Lie superalgebras. In 2003, Brundan in [@Br03] formulated a Kazhdan-Lusztig type conjecture for the full category $\mc{O}$ of general linear Lie superalgebras. The Jimbo-Schur ([@Jim]) duality plays a crucial role in Brundan’s conjecture, which allows a reformulation of the Kazhdan-Lusztig theory in type A in terms of the canonical bases of the quantum group $\U_q(\mathfrak{sl}_{k})$ of type A. Brundan’s conjecture was proved first by Cheng, Lam and Wang [@CLW15] and later by Brundan, Losev and Webster [@BLW].
Recently in [@BW13], Weiqiang Wang and the author initiated a theory of canonical bases arising from quantum symmetric pairs. We showed that a coideal subalgebra of $\U_q(\mathfrak{sl}_{k})$ centralizes the Hecke algebra of type B (of equal parameters) when acting on $\VV^{\otimes m}$, the tensor product of the natural representation $\VV$ of $\U_q(\mathfrak{sl}_{k})$. We constructed a (new) $\imath$-canonical basis on $\VV^{\otimes m}$, which allows a reformulation of the Kazhdan-Lusztig theory of type B independent of the Hecke algebra. The theory was further applied to formulate and establish for the first time the Kazhdan-Lusztig theory for the BGG category $\mathcal{O}$ of the ortho-symplectic Lie superalgebra $\mathfrak{osp}(2m+1|2n)$. The geometric realization of the coideal subalgebras considered there and the canonical bases on the modified coideal subalgebras have been given in [@BKLW] and [@LW15] using partial flag varieties of type B/C.
On the other hand, the problem of determining the irreducible characters in the BGG category $\mc{O}$ of the ortho-symplectic Lie superalgebra $\mathfrak{osp}(2m|2n)$ is still open since 1970’s.
In this paper, we provide a complete solution to the irreducible character problem in the BGG category $\mc{O}$ of modules of integral and half-integral weights of the ortho-symplectic Lie superalgebra $\mathfrak{osp}(2m|2n)$. We adapt the theory of canonical bases from [@BW13] to quantum symmetric pairs with different parameters. The non-super specialization the work here amounts a reformulation of the classical Kazhdan-Lusztig theory of type C/D. This paper is a sequel of [@BW13].
A naive idea to follow [@BW13] is to find the subalgebra of $\U_q(\mathfrak{sl}_k)$, whose action on the tensor product $\VV^{\otimes m}$ centralizing the action of the Hecke algebra $\mc{H}_{D_m}$ of type D on $\VV^{\otimes m}$. Such (new) subalgebra has been constructed using the geometry of isotropic partial flag varieties of type D in [@FL14]. However the subalgebra is very involved, as expected, due to the complicated structure of isotropic flag varieties of type D, which makes it not suitable for further application to the category $\mc{O}$ of Lie superalgebras.
We realize a natural and simple way to overcome the difficulty is to first consider the Hecke algebra $\mathcal{H}^{1}_{B_m}$ of type B with unequal parameters. Let $\mathcal{H}^{p}_{B_m}$ be the Iwahori-Hecke algebra of type $B_m$ with two parameters $p$ and $q$ over $\mathbb Q(q, p)$, generated by $H^p_0, H_1, H_2, \dots , H_{m-1}$, and subject to certain relations (see ). The Hecke algebra $\mathcal{H}^{1}_{B_m}$ is the specialization of $\mathcal{H}^{p}_{B_m}$ at $p=1$. We observe that $\mathcal{H}^{1}_{B_m}$ naturally contains the Hecke algebra $\mc{H}_{D_m}$ of type D as a subalgebra.
Then we look for the subalgebra of $\U_q(\mathfrak{sl}_k)$, whose action on the tensor product $\VV^{\otimes m}$ centralizing the action of the Hecke algebra $\mathcal{H}^{1}_{B_m}$ on $\VV^{\otimes m}$. The subalgebra is a coideal subalgebra of the quantum group $\U_q(\mathfrak{sl}_{k})$ of type A, denoted by $\U^{\imath}_q(\mathfrak{sl}_{k})$. Since the Hecke algebra $\mathcal{H}^{1}_{B_m}$ contains $\mc{H}_{D_m}$ as a subalgebra, the actions of $\U^{\imath}_q(\mathfrak{sl}_{k})$ and $\mc{H}_{D_m}$ on the tensor space $\VV^{\otimes n}$ clearly commute.
The coideal subalgebra comes in different forms depending on the parity of $k$. The quantum group $\U_q(\mathfrak{sl}_{k})$ and the coideal subalgebra $\U^{\imath}_q(\mathfrak{sl}_{k})$ form a quantum symmetric pair ([@Ko14]).
Ehrig and Stroppel used the same coideal subalgebra ${\U^{\imath}}_q(\mathfrak{sl}_{k})$ of the quantum group $\U_q(\mathfrak{sl}_{k})$ to study the parabolic category $\mc{O}$ of the Lie algebra $\mathfrak{so}(2m)$ in [@ES13] independently and simultaneously from [@BW13]. They established the commutativity between the actions of ${\U^{\imath}}_q(\mathfrak{sl}_{k})$ and $\mathcal{H}_{D_m}$ on the tensor space $\VV^{\otimes m}$. The actions of the Chevalley type generators of ${\U^{\imath}}_q(\mathfrak{sl}_{k})$ on $\VV^{\otimes m}$ have been identified with the actions of translation functors on the category $\mc{O}$ of the Lie algebra $\mathfrak{so}(2m)$. However, neither the establishment of the $\imath$-canonical basis nor the (re)formulation of the (super) Kazhdan-Lusztig theory of type D was available there.
It turns out the action of $\U^{\imath}_q(\mathfrak{sl}_{2r+1})$ (that is $k=2r+1$, being an odd number) on $(\VV^*)^{\otimes n}$, tensor product of the restricted dual $\VV^*$ of the natural representation $\VV$ of $\U_q(\mathfrak{sl}_{2r+1})$, centralizes the action of $\mc{H}^q_{B_n}$ ($p=q$) on $(\VV^*)^{\otimes n}$. It is actually more suitable to consider $\mc{H}^q_{B_n}$ as the Hecke algebra $\mc{H}_{C_n}$ of type C (of equal parameters) due to the connection with the BGG category $\mc{O}$ of the Lie superalgebra $\mathfrak{osp}(2m|2n)$ (in particular, the specialization $\mathfrak{sp}(2n)$ when $m =0$).
In [@BW13] we considered the Hecke algebra $\mathcal{H}^{q}_{B_m}$ with equal parameters $p =q$. It was showed there that certain coideal subalgebra of the quantum group $\U_q(\mathfrak{sl}_{k})$ of type A forms double centralizers with $\mathcal{H}^{q}_{B_m}$ when acting on the tensor space $\VV^{\otimes m}$. In this paper, we consider the Hecke algebra $\mathcal{H}^{1}_{B_m}$ with $p =1$. Thus the (different) centralizing coideal subalgebra $\U^{\imath}_q(\mathfrak{sl}_{k})$ of $\U_q(\mathfrak{sl}_{k})$ consider in this paper is of different parameters than the one considered in [@BW13], where the choice of the parameters in $\U^{\imath}_q(\mathfrak{sl}_{k})$ corresponds to the choice of the parameter $p$ in the two parameters Hecke algebra $\mathcal{H}^{p}_{B_m}$.
The construction of $\imath$-canonical bases developed in [@BW13] applies to the coideal subalgebras with different parameters without difficulty, that is, simple $\U_q(\mathfrak{sl}_{k})$-modules and their tensor products admit $\imath$-canonical bases. In the ongoing work [@BW16], we generalize the construction of $\imath$-canonical bases to more general quantum symmetric pairs (see also Remark \[rem:BW16\]).
Thanks to the (weak) Schur type dualities for the type D (and type C), the classical Kazhdan-Lusztig theory of type D (and type C, respectively) can be reformulated in terms of $\imath$-canonical bases on $\VV^{\otimes m}$ (on $(\VV^*)^{\otimes n}$, respectively). More precisely speaking, the entries of the transition matrix between the $\imath$-canonical basis and the standard basis (i.e., the monomial basis) are exactly the Kazhdan-Lusztig polynomials of type D (and type C, respectively.)
We apply the theory of $\imath$-canonical bases to the BGG category $\mathcal{O}$ of the Lie superalgebra $\mathfrak{osp}(2m|2n)$ in Section \[sec:rep\]. In this section we consider the infinite rank limit of the quantum symmetric pair $(\U_q(\mathfrak{sl}_{\infty}), {\U^{\imath}}_q(\mathfrak{sl}_{\infty}))$. The theory of the super duality developed in [@CLW11] plays the essential role.
For a $0^m1^n$-sequence ${\bf b}$ (which consists of $m$ zeros and $n$ ones), we define a tensor space $\mathbb{T}^{\bf b}$ using $m$ copies of $\VV$ and $n$ copies of $\VV^*$ with the tensor order prescribed by ${\bf b}$ (with 0 corresponds to V). In this approach, $\mathbb{T}^{\bf b}$ (more precisely, its integral form) at $q = 1$ is identified with the Grothendieck group $[\mc{O}^{\bf b}]$ of the BGG category $\mc{O}^{\bf b}$ of $\mathfrak{osp}(2m|2n)$-modules (relative to a Borel subalgebra of type ${\bf b}$). We construct the $\imath$-canonical basis and dual $\imath$-canonical basis on (a suitable completion of) the tensor space $\mathbb{T}^{\bf b}$. The construction of these bases is exactly the same as in [@BW13], while only the precise formulas of these bases are different (which is irrelevant to the construction).
For the Lie superalgebra $\mathfrak{osp}(2m|2n)$, there are generally two types of fundamental systems, hence related Dynkin diagrams, with respect to different choices of the Borel subalgebras ${\bf b}$: with a type D branch (where ${\bf b}$ starts with $0^2$) in the Dynkin diagram; or with a type C branch (where ${\bf b}$ starts with $1$) in the Dynkin diagram. Those two types of fundamental systems are not conjugate under the Weyl group actions, but differ by odd reflections. We prove the Kazhdan-Lusztig theory for the category $\mc{O}^{\bf b}$ of the Lie superalgebra $\mathfrak{osp}(2m|2n)$ with respect to ${\bf b}$ of the first type by induction on $n$, where the base case $n=0$ follows from the classical Kazhdan-Lusztig theory of type D. On the other hand, the proof of the Kazhdan-Lusztig theory for the category $\mc{O}^{\bf b}$ of the Lie superalgebra $\mathfrak{osp}(2m|2n)$ with respect to ${\bf b}$ of the second type follows by induction on $m$, where the base case $m=0$ follows from the classical Kazhdan-Lusztig theory of type C. The induction processes of the two types are actually similar, where we compare category $\mc{O}$’s with respect to adjacent Borel subalgebras (switching adjacent $0$ and $1$ in the sequence ${\bf b}$), as well as compare the parabolic category $\mc{O}$ with the full category $\mc{O}$. We also study certain infinite rank limits of the parabolic category $\mc{O}$.
[**Acknowledgement.**]{} The author would like to thank Weiqiang Wang for helpful discussion and suggestion on this paper. This research is partially supported by the AMS-Simons Travel Grant. A part of this paper was written when the author was visiting the Max-Planck-Institute in Bonn during summer 2015. He would like to thank the institute for its excellent working environment and support.
Preliminaries on quantum groups {#sec:prelim}
===============================
In this preliminary section, we review some basic definitions and constructions on quantum groups from Lusztig’s book [@Lu94]. We also introduce the involution $\inv$ and its quotient $\La_\inv$ which will be used in quantum symmetric pairs.
The involution $\inv$ and the lattice $\Lambda_\inv$ {#subsec:theta}
----------------------------------------------------
Let $q$ be an indeterminate. For $r \in \N$, we define the following index sets: $$\begin{aligned}
\label{eq:I}
\begin{split}
\I_{2r+1} &= \{i \in \Z \mid -r \le i \le r\},
\\
\I_{2r} &= \big\{i \in \Z+\hf \mid -r < i < r \big\}.
\end{split}\end{aligned}$$
Set $k =2r+1$ or $2r$, and we use the shorthand notation $\I =\I_k$ in the remainder of Section \[sec:prelim\], and very often through out this paper. Let $$\Pi (= \Pi_k)= \big \{\alpha_i=\varepsilon_{i-\hf}-\varepsilon_{i+\hf} \mid i \in \I \big \}$$ be the simple system of type $A_{k}$, and let $\Phi$ be the associated root system. Denote by $$\Lambda (= \Lambda_k) = \sum_{{i \in \I} } \big( \Z \varepsilon_{i - \hf} + \Z \varepsilon_{i + \hf} \big)$$ the integral weight lattice, and denote by $(\cdot, \cdot)$ the standard bilinear pairing on $\Lambda$ such that $(\varepsilon_a,
\varepsilon_b) = \delta_{ab}$ for all $a,b$. For any $\mu = \sum_{i}c_i\alpha_i \in \N {\Pi}$, set $\hgt(\mu) = \sum_{i} c_i$. Let $\inv$ be the involution of the weight lattice $\Lambda$ such that $$\inv(\varepsilon_{i-\hf}) = - \varepsilon_{-i+\hf}, \quad \text{ for all } i \in \I.$$ We shall also write $\lambda^{\inv} = \inv(\lambda)$, for $\lambda \in \Lambda$. The involution $\inv$ preserves the bilinear form $(\cdot,\cdot)$ on the weight lattice $\Lambda$ and induces an automorphism on the simple system $\Pi$ such that $ \alpha^{\inv}_i = \alpha_{-i}$ for all $i \in \I$.
Let $\Lambda^{\inv} = \{\mu \in \Lambda \mid \mu^{\inv} +\mu\}$ and $\Lambda_{\inv} = \Lambda/\Lambda^{\inv}$. For $\mu \in \Lambda$, denote by $\ov{\mu}$ the image of $\mu$ under the quotient map. There is a well-defined bilinear pairing $\Z[\alpha_i - \alpha_{-i}]_{i \in \I} \times \BLambda \rightarrow \Z$, such that $(\sum_{i > 0}a_i(\alpha_i-\alpha_{-i}), \ov{\mu}) := \sum_{i > 0}a_i(\alpha_i-\alpha_{-i}, \mu)$ for any $\ov{\mu} \in \BLambda$ with any preimage $\mu\in \Lambda$.
The quantum group
------------------
The quantum group $\U = \U_q(\mf{sl}_{k+1})$ is defined to be the associative $\Q(q)$-algebra generated by $E_{\alpha_i}$, $F_{\alpha_i}$, $K_{\alpha_i}$, $K^{-1}_{\alpha_i}$, $i \in \I$, subject to the following relations (for $i$, $j \in \I$): $$\begin{aligned}
K_{\alpha_i} K_{\alpha_i}^{-1} &= K_{\alpha_i}^{-1} K_{\alpha_i} =1,
\\
K_{\alpha_i} K_{\alpha_j} &= K_{\alpha_j} K_{\alpha_i},
\\
K_{\alpha_i} E_{\alpha_j} K_{\alpha_i}^{-1} &= q^{(\alpha_i, \alpha_j)} E_{\alpha_j}, \\\displaybreak[0]
K_{\alpha_i} F_{\alpha_j} K_{\alpha_i}^{-1} &= q^{-(\alpha_i, \alpha_j)} F_{\alpha_j}, \\\displaybreak[0]
E_{\alpha_i} F_{\alpha_j} -F_{\alpha_j} E_{\alpha_i} &= \delta_{i,j} \frac{K_{\alpha_i}
-K^{-1}_{\alpha_i}}{q-q^{-1}}, \\ \displaybreak[0]
E_{\alpha_i}^2 E_{\alpha_j} +E_{\alpha_j} E_{\alpha_i}^2
&= (q+q^{-1}) E_{\alpha_i} E_{\alpha_j} E_{\alpha_i}, \quad &&\text{if } |i-j|=1, \\
E_{\alpha_i} E_{\alpha_j} &= E_{\alpha_j} E_{\alpha_i}, \,\qquad\qquad &&\text{if } |i-j|>1, \\
F_{\alpha_i}^2 F_{\alpha_j} +F_{\alpha_j} F_{\alpha_i}^2
&= (q+q^{-1}) F_{\alpha_i} F_{\alpha_j} F_{\alpha_i}, \quad\, &&\text{if } |i-j|=1,\\
F_{\alpha_i} F_{\alpha_j} &= F_{\alpha_j} F_{\alpha_i}, \qquad\ \qquad &&\text{if } |i-j|>1.\end{aligned}$$
Let $\U^+$, $\U^0$ and $\U^-$ be the $\Qq$-subalgebra of $\U$ generated by $E_{\alpha_i}$, $K^{\pm 1}_{\alpha_i}$, and $F_{\alpha_i}$ respectively, for $i \in \I$. We introduce the divided power $F^{(a)}_{\alpha_i} = F^a_{\alpha_i}/[a]!$, where $a \ge 0$, $[a] = (q^{a}- q^{-a})/(q-q^{-1})$ and $[a]! = [1][2] \cdots [a]$. Let $\mA =\Z[q,q^{-1}]$. Let $_\mA\U^{+}$ be the $\mA$-subalgebra of $\U^{+}$ generated by $E^{(a)}_{\alpha_i}$ for various $a \ge 0$ and $i \in \I$. Similarly let $_\mA\U^{-}$ be the $\mA$-subalgebra of $\U^{-}$ generated by $E^{(a)}_{\alpha_i}$ for various $a \ge 0$ and $i \in \I$.
\[prop:invol\]
1. There is an involution $\omega$ on the $\Qq$-algebra $\U$ such that $\omega(E_{\alpha_i}) =F_{\alpha_i}$, $\omega(F_{\alpha_i}) =E_{\alpha_i}$, and $\omega(K_{\alpha_i}) = K^{-1}_{\alpha_i}$ for all $i \in \I$.
2. There is an anti-linear ($q \mapsto q^{-1}$) bar involution of the $\Q$-algebra $\U$ such that $\ov{E}_{\alpha_i}= E_{\alpha_i}$, $\ov{F}_{\alpha_i}=F_{\alpha_i}$, and $\ov{K}_{\alpha_i}=K_{\alpha_i}^{-1}$ for all $i \in \I$.
(Sometimes we denote the bar involution on $\U$ by $\psi$.)
Recall that $\U$ is a Hopf algebra with a coproduct $$\begin{aligned}
\label{eq:coprod}
\begin{split}
\Delta: &\U \longrightarrow \U \otimes \U,
\\
\Delta (E_{\alpha_i}) &= 1 \otimes E_{\alpha_i} + E_{\alpha_i} \otimes K^{-1}_{\alpha_i}, \\
\Delta (F_{\alpha_i}) &= F_{\alpha_i} \otimes 1 + K_{\alpha_i} \otimes F_{\alpha_i},\\
\Delta (K_{\alpha_i}) &= K_{\alpha_i} \otimes K_{\alpha_i}.
\end{split}\end{aligned}$$ There is a unique $\Qq$-algebra homomorphism $\epsilon: \U \rightarrow \Qq$, called counit, such that $\epsilon(E_{\alpha_i}) = 0$, $\epsilon(F_{\alpha_i}) = 0$, and $\epsilon(K_{\alpha_i}) =1$.
Braid group actions and canonical bases {#subsec:CB}
---------------------------------------
Let $W := W_{A_{{k}}} = \mf{S}_{{k}+1}$ be the Weyl group of type $A_{{k}}$. Recall [@Lu94] for each $\alpha_i$ and each finite-dimensional $\U$-module $M$, a linear operator $T_{\alpha_i}$ on $M$ is defined by, for $\lambda \in \Lambda$ and $m \in M_\lambda$, $$T_{\alpha_i}(m) = \sum_{a, b, c \ge 0; -a+b-c
=(\lambda, \alpha_i)}(-1)^b q^{b-ac}E^{(a)}_{\alpha_i}F^{(b)}_{\alpha_i}E^{(c)}_{\alpha_i} m.$$ These $T_{\alpha_i}$’s induce automorphisms of $\U$, denoted by $T_{\alpha_i}$ as well, such that $$T_{\alpha_i}(um) = T_{\alpha_i}(u)T_{\alpha_i}(m), \qquad \text{ for all } u \in \U, m \in M.$$ As automorphisms on $\U$ and as $\Qq$-linear isomorphisms on $M$, the $T_{\alpha_i}$’s satisfy the braid group relations ([@Lu94 Theorem 39.4.3]): $$\begin{aligned}
T_{\alpha_i}T_{\alpha_j} &= T_{\alpha_j}T_{\alpha_i}, &\text{ if } |i-j| >1 ,\\
T_{\alpha_i}T_{\alpha_j}T_{\alpha_i} &= T_{\alpha_j}T_{\alpha_i}T_{\alpha_j}, &\text{ if } |i-j| =1,\end{aligned}$$ Hence for each $w \in W$, $T_w$ can be defined independent of the choices of reduced expressions of $w$. (The $T_{\alpha_i}$ here is $T''_{i,+}$ in [@Lu94]).
Denote by $\ell (\cdot)$ the length function of $W$, and let $w_0$ be the longest element of $W$. The following lemma is well-known (cf. [@BW13 Lemma 1.5]).
\[lem:Tw0\] The following identities hold: $$T_{w_0}(K_{\alpha_i}) =K^{-1}_{\alpha_{-i}}, \quad T_{w_0}(E_{\alpha_i}) = -F_{\alpha_{-i}}K_{\alpha_{-i}}, \quad
T_{w_0}(F_{\alpha_{-i}}) = - K^{-1}_{\alpha_i}E_{\alpha_i}, \quad \text{ for } i \in \I.$$
Let $\Lambda^+ = \{\lambda \in \Lambda \mid 2(\alpha_i , \lambda)/(\alpha_i,\alpha_i) \in {\N}, \forall i \in \I\}$ be the set of dominant weights. Note that $\mu \in \Lambda^+$ if and only if $\mu ^{\inv} \in \Lambda^+$, since the bilinear pairing $(\cdot,\cdot)$ on $ \Lambda$ is invariant under $\inv : \Lambda \rightarrow \Lambda$.
Let $M(\lambda)$ be the Verma module of $\U$ with highest weight $\lambda\in \Lambda$ and with a highest weight vector denoted by $\eta$ or $\eta_{\lambda}$. We define a $\U$-module $^\omega M(\lambda)$, which has the same underlying vector space as $M(\lambda)$ but with the action twisted by the involution $\omega$ given in Proposition \[prop:invol\]. When considering $\eta$ as a vector in $^\omega M(\lambda)$, we shall denote it by $\xi$ or $\xi_{-\lambda}$. The Verma module $M(\lambda)$ associated to dominant $\la \in \La^+$ has a unique finite-dimensional simple quotient $\U$-module, denoted by $L(\lambda)$. Similarly we define the $\U$-module $^\omega L(\lambda)$. For $\la \in \Lambda^+$, we let ${}_\mA L(\lambda) ={}_\mA\U^- \eta$ and $^\omega _\mA L(\lambda) ={}_\mA \U^+ \xi$ be the $\mA$-submodules of $L(\lambda)$ and $^\omega L(\lambda) $, respectively.
We call a $\U$-module $M$ equipped with an anti-linear involution $\Abar$ is called [*involutive*]{} if $
\Abar(u m) = \Abar(u) \Abar(m)$, $\forall u \in \U, m \in M$. The $\U$-modules ${^{\omega}L}(\lambda)$ and $L(\lambda)$ are both involutive. Given any two involutive $\U$-modules $M$ and $M'$, Lusztig showed that their tensor product $M \otimes M'$ is also involutive ([@Lu94 §27.31]).
In [@Lu90; @Lu94] and [@Ka], the canonical basis $\bold{B}$ of ${_\mA \U^+} \cong {_\mA \U^-}$ has been constructed. For any element $b \in \B$, when considered as an element in $\U^-$ or $\U^+$, we shall denote it by $b^-$ or $b^+$, respectively. In [@Lu94], subsets $\B(\lambda)$ of $\B$ is also constructed for each $\lambda \in \Lambda^+$, such that $\{b^-\eta_{\lambda} \mid b \in \B(\lambda)\}$ gives the canonical basis of ${}_\mA L(\lambda)$. Similarly $\{b^+ \xi_{-\lambda} \mid b \in \B(\lambda)\}$ gives the canonical basis of ${^{\omega}_\mA L(\lambda)}$.
Quantum Symmetric pairs {#sec:QSP}
=======================
In this section we shall develop the theory of $\imath$-canonical bases for the quantum symmetric pairs : $$(\U_q(\mathfrak{sl}_{2r+1}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}))\quad \text{ and } \quad (\U_q(\mathfrak{sl}_{2r+2}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+2})).$$
The definitions of the quantum symmetric pairs shall be given in the first two sections, separately (see also [@ES13]). The theory of $\imath$-canonical bases are nevertheless uniform in both cases. Therefore after stating their definitions we shall formulate their general theory together. Section $2.1$ - Section $2.3$ are analogous to [@BW13 Part 1] hence we shall omit the proofs almost entirely. Some of the quantum symmetric paris considered here are of different parameters than the ones considered in [@BW13] (See also [@BW16] for more general construction). We refer to [@Ko14] for general theory of quantum symmetric pairs.
The quantum symmetric pair $(\U_q(\mathfrak{sl}_{2r+1}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}))$ {#sec:Ui}
------------------------------------------------------------------------------------------------
We define $$\I^{\imath} _{2r} = (\hf +\N) \cap \I_{2r} = \Big\{\hf, \frac32, \ldots, r-\hf \Big\}.$$ The Dynkin diagram of type $A_{2r}$ together with the involution $\inv$ are depicted as follows:
(-1.5,0) node [$A_{2r}:$]{}; (0.5,0) node\[below\] [$\alpha_{-r+\hf}$]{} – (2.5,0) node\[below\] [$\alpha_{-\hf}$]{} ; (2.5,0) – (3.5,0) node\[below\] [$\alpha_{\hf}$]{}; (3.5,0) – (5.5,0) node\[below\] [$\alpha_{r-\hf}$]{} ; (0.5,0) node (-r) [$\bullet$]{}; (2.5,0) node (-1) [$\bullet$]{}; (3.5,0) node (1) [$\bullet$]{}; (5.5,0) node (r) [$\bullet$]{}; (-r.north east) .. controls (3,1) .. node\[above\] [$\theta$]{} (r.north west) ; (-1.north) .. controls (3,0.5) .. (1.north) ;
The algebra ${\U^{\imath}}_q(\mathfrak{sl}_{2r+1})$ is defined to be the associative algebra over $\Q(q)$ generated by $\be_{\alpha_i}$, $\bff_{\alpha_i}$, $\bk_{\alpha_i}$, $\bk^{-1}_{\alpha_i}$, $i \in \I^{\imath} _{2r} $, subject to the following relations for $i, j \in \I^{\imath} _{2r} $: $$\begin{aligned}
\ibk{i} \ibk{i}^{-1} &= \ibk{i}^{-1} \ibk{i} =1, \displaybreak[0] \notag\\
\ibk{i} \ibk{j} &= \ibk{j} \ibk{i}, \displaybreak[0] \notag\\
\ibk{i} \ibe{j} \ibk{i}^{-1} &= q^{(\alpha_i-\alpha_{-i},\alpha_j)} \ibe{j}, \displaybreak[0] \notag\\
\ibk{i} \ibff{j} \ibk{i}^{-1} &= q^{-(\alpha_i-\alpha_{-i},\alpha_j)}\ibff{j}, \displaybreak[0] \notag\\
\be_{\alpha_i} \ibff{j} -\ibff{i} \ibe{j} &= \delta_{i,j} \frac{\bk_{\alpha_i}
-\bk^{-1}_{\alpha_i}}{q-q^{-1}}, \qquad\; \qquad \text{if } i, j \neq \hf, \displaybreak[0] \notag\\
\ibe{i}^2 \ibe{j} +\ibe{j} \ibe{i}^2 &= (q+q^{-1}) \ibe{i} \ibe{j} \ibe{i}, \qquad \text{if } |i-j|=1, \displaybreak[0] \notag \\
\ibff{i}^2 \ibff{j} +\ibff{j} \ibff{i}^2 &= (q+q^{-1}) \ibff{i} \ibff{j} \ibff{i}, \qquad \text{if } |i-j|=1,\displaybreak[0]\notag \\
\ibe{i} \ibe{j} &= \ibe{j} \ibe{i}, \quad\qquad\qquad\qquad\; \text{if } |i-j|>1, \displaybreak[0]\notag \\
\ibff{i} \ibff{j} &=\ibff{j} \ibff{i}, \quad\qquad\qquad\qquad\; \text{if } |i-j|>1,\displaybreak[0]\notag \\
\ibff{\hf}^2\ibe{\hf} + \ibe{\hf}\ibff{\hf}^2
&= (q+q^{-1}) \Big(\ibff{\hf}\ibe{\hf}\ibff{\hf}-q^2\ibff{\hf}\ibk{\hf}^{-1}-q^{-2}\ibff{\hf}\ibk{\hf} \big),\displaybreak[0] \label{eq:Serre:1}\\
\ibe{\hf}^2\ibff{\hf} + \ibff{\hf}\ibe{\hf}^2
&= (q+q^{-1}) \Big(\ibe{\hf}\ibff{\hf}\ibe{\hf}-q^{-2}\ibk{\hf}\ibe{\hf} -q^2\ibk{\hf}^{-1}\ibe{\hf} \Big).\displaybreak[0] \label{eq:Serre:2}\end{aligned}$$ We introduce the divided powers $\be^{(a)}_{\alpha_i} = \be^a_{\alpha_i} / [a]!$, $\bff^{(a)}_{\alpha_i} = \bff^a_{\alpha_i} / [a]!$.
Note that the last two “Serre" type relations and are different from [@BW13 §6.1].
The algebra ${\U^{\imath}}_q(\mathfrak{sl}_{2r+1})$ has an anti-linear ($q \mapsto q^{-1}$) bar involution such that $\ov{\bk}_{\alpha_i} = \bk^{-1}_{\alpha_i}$, $\ov{\be}_{\alpha_i} = \be_{\alpha_i}$, and $\ov{\bff}_{\alpha_i} = \bff_{\alpha_i}$, for all $i \in \I^{\imath} _{2r} $.
(Sometimes we denote the bar involution on ${\U^{\imath}}_q(\mathfrak{sl}_{2r+1}))$ by ${\psi_{\imath}}$.)
\[int:prop:embedding\] There is an injective $\Qq$-algebra homomorphism $\imath : {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \rightarrow \U_q(\mathfrak{sl}_{2r+1})$ defined by, for all $i \in \I^{\imath} _{2r}$, $$\begin{aligned}
\bk_{\alpha_i} \mapsto K_{\alpha_i}K^{-1}_{\alpha_{-i}}, \quad
\be_{\alpha_i} \mapsto E_{\alpha_i} + F_{\alpha_{-i}}K^{-1}_{\alpha_i}, \quad
\bff_{\alpha_i} \mapsto K^{-1}_{\alpha_{-i}} F_{\alpha_i} + E_{\alpha_{-i}}.\end{aligned}$$
In [@BW13 Proposition 2.2], we consider the embedding that maps $\be_{\alpha_i}$ to $E_{\alpha_i} + q^{-\delta_{i, \hf}} F_{\alpha_{-i}}K^{-1}_{\alpha_i}$ and maps $\bff_{\alpha_i}$ to $q^{\delta_{i, \hf}} K^{-1}_{\alpha_{-i}} F_{\alpha_i} + E_{\alpha_{-i}}$.
Note that $E_{\alpha_i} (K^{-1}_{\alpha_i}F_{\alpha_{-i}})
= q^{2} (K^{-1}_{\alpha_i}F_{\alpha_{-i}}) E_{\alpha_i}$ for $i \in \I^{\imath} _{2r} $. We have for $i \in \I^{\imath} _{2r} $, $$\begin{aligned}
\label{int:eq:beZ} \imath(\be^{(a)}_{\alpha_i})
&= \sum^{a}_{j=0} q^{j(a-j)}{(F_{\alpha_{-i}} K^{-1}_{\alpha_i} )^j \over [j]!}\frac{E^{a-j}_{\alpha_i}}{[a-j]!},
\\
\label{int:eq:bffZ} \imath(\bff^{(a)}_{\alpha_i})
&= \sum^{a}_{j=0} q^{j(a-j)}{(K^{-1}_{\alpha_{-i}}F_{\alpha_{i}})^j \over [j]!}\frac{E^{a-j}_{\alpha_{-i}}}{[a-j]!}.\end{aligned}$$
\[int:prop:coproduct\] The coproduct $\Delta$ on $\U_q(\mathfrak{sl}_{2r+1})$ restricts under the embedding $\imath$ to a $\Qq$-algebra homomorphism $$\Delta : {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \longrightarrow {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \otimes \U_q(\mathfrak{sl}_{2r+1})$$ such that for all $i \in \I^{\imath} _{2r} $, $$\begin{aligned}
\Delta(\bk_{\alpha_i}) &= \bk_{\alpha_i} \otimes K_{\alpha_i} K^{-1}_{\alpha_{-i}},
\\
\Delta({\be_{\alpha_i}}) &= 1 \otimes E_{\alpha_i} + \be_{\alpha_i} \otimes
K^{-1}_{\alpha_i} + \bk^{-1}_{\alpha_i} \otimes F_{\alpha_{-i}} K^{-1}_{\alpha_i} ,
\\
\Delta (\bff_{\alpha_i}) &= \bk_{\alpha_i} \otimes K^{-1}_{\alpha_{-i}} F_{\alpha_i}
+ \bff_{\alpha_i} \otimes K^{-1}_{\alpha_{-i}} + 1 \otimes E_{{\alpha_{-i}}}.\end{aligned}$$ Similarly, the counit $\epsilon$ of $\U_q(\mathfrak{sl}_{2r+1})$ induces a $\Qq$-algebra homomorphism $$\epsilon : {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \rightarrow \Qq$$ such that $\epsilon(\be_{\alpha_i}) =\epsilon(\bff_{\alpha_i})=0$ and $\epsilon(\bk_{\alpha_i}) =1$ for all $i \in \I^{\imath} _{2r} $.
It follows by Proposition \[int:prop:coproduct\] that ${\U^{\imath}}_q(\mathfrak{sl}_{2r+1})$ is a (right) coideal subalgebra of $\U$. The map $\Delta : {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \rightarrow {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \otimes \U_q(\mathfrak{sl}_{2r+1})$ will be called the coproduct of ${\U^{\imath}}_q(\mathfrak{sl}_{2r+1})$ and $ \epsilon : {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \rightarrow \Qq$ will be called the counit of ${\U^{\imath}}_q(\mathfrak{sl}_{2r+1})$. The coproduct $\Delta : {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \rightarrow {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \otimes \U_q(\mathfrak{sl}_{2r+1})$ is coassociative, i.e., $
(1 \otimes \Delta) \Delta = (\Delta \otimes 1)\Delta: {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \rightarrow {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \otimes \U_q(\mathfrak{sl}_{2r+1}) \otimes \U_q(\mathfrak{sl}_{2r+1})$. The counit map $\epsilon$ makes $\Qq$ a (trivial) ${\U^{\imath}}_q(\mathfrak{sl}_{2r+1})$-module. Let $m : \U_q(\mathfrak{sl}_{2r+1}) \otimes \U_q(\mathfrak{sl}_{2r+1}) \rightarrow \U_q(\mathfrak{sl}_{2r+1})$ denote the multiplication map. We have $
m (\epsilon \otimes 1)\Delta = \imath : {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \longrightarrow \U_q(\mathfrak{sl}_{2r+1})
$ by direct computation.
The quantum symmetric pair $(\U_q(\mathfrak{sl}_{2r+2}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}))$ {#sec:Uj}
------------------------------------------------------------------------------------------------
We set $$\label{eq:Ihf}
\I^{\imath}_{2r+1} :=\Z_{>0} \cap \I_{2r+1} =\{1, \ldots, r\}.$$
The Dynkin diagram of type $A_{2r+1}$ together with the involution $\inv$ can be depicted as follows:
(-2,0) node [$A_{2r+1}:$]{}; (0,0) node\[below\] [$\alpha_{-r}$]{} – (2,0) node\[below\] [$\alpha_{-1}$]{} ; (2,0) – (3,0) node\[below\] [$\alpha_{0}$]{} – (4,0) node\[below\] [$\alpha_{1}$]{}; (4,0) – (6,0) node\[below\] [$\alpha_{r}$]{} ; (0,0) node (-r) [$\bullet$]{}; (2,0) node (-1) [$\bullet$]{}; (3,0) node (0) [$\bullet$]{}; (4,0) node (1) [$\bullet$]{}; (6,0) node (r) [$\bullet$]{}; (-r.north east) .. controls (3,1.5) .. node\[above\] [$\theta$]{} (r.north west) ; (-1.north) .. controls (3,1) .. (1.north) ; (0) edge\[<->, loop above\] (0);
The algebra ${\U^{\imath}}_q(\mathfrak{sl}_{2r+2})$ is defined to be the associative algebra over $\Q(q)$ generated by $\be_{\alpha_i}$, $\bff_{\alpha_i}$, $\bk_{\alpha_i}$, $\bk^{-1}_{\alpha_i}$ ($i \in \I^{\imath}_{2r+1}$) , and $\bt$, subject to the following relations for $i$, $j \in \I^{\imath}_{2r+1}$: $$\begin{aligned}
\bk_{\alpha_i} \bk_{\alpha_i}^{-1} &= \bk_{\alpha_i}^{-1} \bk_{\alpha_i} =1,\displaybreak[0]\\
\bk_{\alpha_i} \bk_{\alpha_j} &= \bk_{\alpha_j} \bk_{\alpha_i}, \displaybreak[0]\\
\bk_{\alpha_i} \be_{\alpha_j} \bk_{\alpha_i}^{-1} &= q^{(\alpha_i-\alpha_{-i}, \alpha_j)} \be_{\alpha_j}, \displaybreak[0]\\
\bk_{\alpha_i} \bff_{\alpha_j} \bk_{\alpha_i}^{-1} &= q^{-(\alpha_i-\alpha_{-i}, \alpha_j)}
\bff_{\alpha_j}, \displaybreak[0]\\
\bk_{\alpha_i}\bt\bk^{-1}_{\alpha_i} &= \bt, \displaybreak[0]\\
\be_{\alpha_i} \bff_{\alpha_j} -\bff_{\alpha_j} \be_{\alpha_i} &= \delta_{i,j} \frac{\bk_{\alpha_i}
-\bk^{-1}_{\alpha_i}}{q-q^{-1}}, \displaybreak[0]\\
\be_{\alpha_i}^2 \be_{\alpha_j} +\be_{\alpha_j} \be_{\alpha_i}^2 &= (q+q^{-1}) \be_{\alpha_i} \be_{\alpha_j} \be_{\alpha_i},
\ \ \quad\quad &\text{if }& |i-j|=1, \displaybreak[0]\\
\be_{\alpha_i} \be_{\alpha_j} &= \be_{\alpha_j} \be_{\alpha_i}, \ \qquad\qquad\ \ \ \ \qquad &\text{if }& |i-j|>1, \displaybreak[0]\\
\bff_{\alpha_i}^2 \bff_{\alpha_j} +\bff_{\alpha_j} \bff_{\alpha_i}^2 &= (q+q^{-1}) \bff_{\alpha_i} \bff_{\alpha_j} \bff_{\alpha_i},
\ \ \quad\quad &\text{if }& |i-j|=1,\displaybreak[0]\\
\bff_{\alpha_i} \bff_{\alpha_j} &= \bff_{\alpha_j} \bff_{\alpha_i}, \ \qquad\qquad\ \ \ \ \qquad &\text{if }& |i-j|>1, \displaybreak[0]\\
\be_{\alpha_i}\bt &=\bt\be_{\alpha_i}, \quad\qquad\quad &\text{if }& i > 1, \displaybreak[0]\\
\be_{\alpha_1}^2\bt + \bt\be_{\alpha_1}^2 &= (q+q^{-1}) \be_{\alpha_1}\bt\be_{\alpha_1},\displaybreak[0]\\
\bt^2\be_{\alpha_1} + \be_{\alpha_1}\bt^2 &= (q + q^{-1}) \bt\be_{\alpha_1}\bt + \be_{\alpha_1},\displaybreak[0]\\
\bff_{\alpha_i}\bt &=\bt\bff_{\alpha_i}, \quad\qquad&\text{if }& i > 1, \displaybreak[0]\\
\bff_{\alpha_1}^2\bt + \bt\bff_{\alpha_1}^2 &= (q+q^{-1}) \bff_{\alpha_1}\bt\bff_{\alpha_1},\displaybreak[0]\\
\bt^2\bff_{\alpha_1} + \bff_{\alpha_1}\bt^2 &= (q + q^{-1}) \bt\bff_{\alpha_1}\bt + \bff_{\alpha_1}.\displaybreak[0]\end{aligned}$$ We introduce the divided powers $\be^{(a)}_{\alpha_i} = \be^a_{i} / [a]!$, $\bff^{(a)}_{\alpha_i} = \bff^a_{i} / [a]!$ for $a \ge 0$, $i \in \I^{\imath}_{2r+1}$.
\[rem:Uisame\] The generating relations of the algebra ${\U^{\imath}}_q(\mathfrak{sl}_{2r+2})$ are the same as the generating relations of the algebra considered in [@BW13 §2.1].
\[lem:3inv\] The $\Q$-algebra ${\U^{\imath}}_q(\mathfrak{sl}_{2r+2})$ has an anti-linear ($q \mapsto q^{-1}$) bar involution such that $\ov{\bk}_{\alpha_i} = \bk^{-1}_{\alpha_i}$, $\ov{\be}_{\alpha_i} = \be_{\alpha_i}$, $\ov{\bff}_{\alpha_i} = \bff_{\alpha_i}$, and $\ov{\bt} = \bt$ for all $i \in \I^{\imath}_{2r+1}$.
(Sometimes we denote the bar involution on ${\U^{\imath}}_q(\mathfrak{sl}_{2r+2})$ by $\Bbar$.)
\[prop:embedding\] There is an injective $\Qq$-algebra homomorphism $\imath : {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \rightarrow \U_q(\mathfrak{sl}_{2r+2})$ which sends (for $i \in \I^{\imath}_{2r+1}$) $$\begin{aligned}
\bk_{\alpha_i} \mapsto K_{\alpha_i}K^{-1}_{\alpha_{-i}}, \qquad & \bt \mapsto E_{\alpha_0} +qF_{\alpha_0}K^{-1}_{\alpha_0} \\
\be_{\alpha_i} \mapsto E_{\alpha_i} + K^{-1}_{\alpha_i}F_{\alpha_{-i}},\qquad
& \bff_{\alpha_i} \mapsto F_{\alpha_i} K^{-1}_{\alpha_{-i}}+ E_{\alpha_{-i}}.\end{aligned}$$
The embedding $\imath$ in Proposition \[prop:embedding\] is different from the embedding in [@BW13 Proposition 2.2], although the two subalgebras are (abstractly) isomorphic (see Remark \[rem:Uisame\]). This phenomenon for quantum symmetric pairs was first observed in [@Le Section 5].
Note that $E_{\alpha_i} (K^{-1}_{\alpha_i}F_{\alpha_{-i}}) = q^{2} (K^{-1}_{\alpha_i}F_{\alpha_{-i}}) E_{\alpha_i}$ for all $ 0 \neq i \in \I$. Using the quantum binomial formula [@Lu94 1.3.5], we have, for all $i \in \I^{\imath}_{2r+1}$, $a \in \N$, $$\begin{aligned}
\label{eq:beZ}
\imath(\be^{(a)}_{\alpha_i}) &= \sum^{a}_{j=0} q^{j(a-j)}F^{(j)}_{\alpha_{-i}}K^{-j}_{\alpha_i} E^{(a-j)}_{\alpha_i},
\\
\label{eq:bffZ}
\imath(\bff^{(a)}_{\alpha_i}) &= \sum^{a}_{j=0} q^{j(a-j)}F^{(j)}_{\alpha_{i}}K^{-j}_{\alpha_{-i}} E^{(a-j)}_{\alpha_{-i}}.\end{aligned}$$
\[prop:coproduct\] The coproduct $\Delta$ on $\U_q(\mathfrak{sl}_{2r+2})$ restricts via the embedding $\imath$ to a $\Qq$-algebra homomorphism $$\Delta : {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \longrightarrow {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \otimes \U_q(\mathfrak{sl}_{2r+2})$$ such that, for all $i \in \I^{\imath}_{2r+1}$, $$\begin{aligned}
\Delta(\bk_{\alpha_i}) &= \bk_{\alpha_i} \otimes K_{\alpha_i} K^{-1}_{\alpha_{-i}},
\\
\Delta({\be_{\alpha_i}}) &= 1 \otimes E_{\alpha_i} + \be_{\alpha_i} \otimes K^{-1}_{\alpha_i}
+ \bk^{-1}_{\alpha_i} \otimes K^{-1}_{\alpha_i}F_{\alpha_{-i}},
\\
\Delta (\bff_{\alpha_i}) &= \bk_{\alpha_i} \otimes F_{\alpha_i}K^{-1}_{\alpha_{-i}} + \bff_{\alpha_i}
\otimes K^{-1}_{\alpha_{-i}} + 1 \otimes E_{{\alpha_{-i}}},
\\
\Delta(\bt) &= \bt \otimes K^{-1}_{\alpha_0} + 1 \otimes q F_{\alpha_0}K^{-1}_{\alpha_0}+ 1 \otimes E_{\alpha_0}.\end{aligned}$$ Similarly, the counit $\epsilon$ of $\U_q(\mathfrak{sl}_{2r+2})$ induces a $\Qq$-algebra homomorphism $$\epsilon : {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \rightarrow \Qq$$ such that $\epsilon(\be_{\alpha_i}) =\epsilon(\bff_{\alpha_i})=0$, $\epsilon(\bt) = 0$, and $\epsilon(\bk_{\alpha_i}) =1$ for all $i \in \I^{\imath}_{2r+1}$.
The map $\Delta : {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \mapsto {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \otimes \U_q(\mathfrak{sl}_{2r+2})$ is coassociative, i.e., we have $(1 \otimes \Delta) \Delta = (\Delta \otimes 1)\Delta: {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \longrightarrow {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \otimes \U_q(\mathfrak{sl}_{2r+2}) \otimes \U_q(\mathfrak{sl}_{2r+2}).
$ This $\Delta$ will be called the [ coproduct]{} of ${\U^{\imath}}_q(\mathfrak{sl}_{2r+2})$, and $ \epsilon : {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \rightarrow \Qq$ will be called the [counit]{} of ${\U^{\imath}}_q(\mathfrak{sl}_{2r+2})$. The counit map $\epsilon$ makes $\Qq$ a (trivial) ${\U^{\imath}}_q(\mathfrak{sl}_{2r+2})$-module. Let $m : \U_q(\mathfrak{sl}_{2r+2}) \otimes \U_q(\mathfrak{sl}_{2r+2}) \rightarrow \U_q(\mathfrak{sl}_{2r+2})$ denote the multiplication map. We have $
m (\epsilon \otimes 1)\Delta = \imath : {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \longrightarrow \U_q(\mathfrak{sl}_{2r+2})
$ by direct computation.
The $\imath$-canonical bases
----------------------------
The rest of the section we shall develop the theory of $\imath$-canonical bases for the quantum symmetric pairs $(\U_q(\mathfrak{sl}_{2r+1}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}))$, $(\U_q(\mathfrak{sl}_{2r+2}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}))$ and its applications. The formulation of the theory is uniform for both cases. Hence we shall drop the subscript, and denote both quantum symmetric pairs simply by $(\U, {\U^{\imath}})$, and denote the correspond index sets simply by $\I = \I_k$ and $\I^{\imath} = \I^{\imath}_k$, for $k= 2r+2$ or $2r+1$. In this section we shall assume all modules are finite dimensional.
Let $\widehat{\U}$ be the completion of the $\Qq$-vector space $\U$ with respect to the following descending sequence of subspaces $\U^+ \U^0 \big(\sum_{\hgt(\mu) \geq N}\U_{-\mu}^- \big)$, for $N \ge 1.$ Then we have the obvious embedding of $\U$ into $\widehat{\U}$. We let $\widehat{\U}^-$ be the closure of $\U^-$ in $\widehat{\U}$, and so $\widehat{\U}^- \subseteq \widehat{\U}$. By continuity the $\Q(q)$-algebra structure on $\U$ extends to a $\Q(q)$-algebra structure on $ \widehat{\U}$. The bar involution $\bar{\ }$ on $\U$ extends by continuity to an anti-linear involution on $\widehat{\U}$, also denoted by $\bar{\, }$. The following proposition is the counterpart of [@BW13 §2.3, §2.4 and §4.4].
There is a unique family of elements $\Upsilon_\mu \in {}_\mA\U_{-\mu}^-$ for $\mu \in {\N}{\Pi}$ such that $\Upsilon_0 = 1$, and $\Upsilon = \sum_{\mu}\Upsilon_\mu \in \widehat{\U}^-$ intertwines the bar involution $\psi_{\imath}$ on $\bun$ and the bar involution $\psi$ on $\U$ via the embedding $\imath$; that is, $\Upsilon$ satisfies the following identity (in $\widehat{\U}$): $$\label{eq:star}
\imath(\psi_{\imath}{u}) \Upsilon = \Upsilon\ \psi({\imath(u)}), \quad \text{ for all } u \in \bun.$$ Moreover, $\Upsilon_\mu = 0$ unless $\mu^{\inv} = \mu$. We also have $\Upsilon \cdot \ov{\Upsilon} =1.$
Consider a $\Qq$-valued function $\zeta$ on $\Lambda$ such that $$\begin{aligned}
\zeta (\mu+\alpha_0)&=-q \zeta (\mu) \quad (\text{only for the pair $(\U_q(\mathfrak{sl}_{2r+1}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+1})$}),
\notag \\
\zeta (\mu+\alpha_i) &= -q^{(\alpha_i, \mu+\alpha_i)-(\alpha_{-i},\mu)} \zeta (\mu),
\label{eq:zeta0} \\
\zeta (\mu+\alpha_{-i}) &= -q^{(\alpha_{-i}, \mu+\alpha_{-i}) - (\alpha_{i}, \mu)-1} \zeta (\mu),
\quad \forall \mu \in \Lambda, \; i \in \Ihf.
\notag\end{aligned}$$ Such $\zeta$ clearly exists (but not unique). For any weight $\U$-module $M$, define a $\Qq$-linear map on $ M$ as follows: $$\begin{aligned}
\label{eq:zeta}
\begin{split}
\widetilde{\zeta}&: M \longrightarrow M,
\\
\widetilde{\zeta} (m &) = \zeta (\mu)m, \quad \forall m \in M_{\mu}.
\end{split}\end{aligned}$$
Recall that $w_0$ is the longest element of $W$ and $T_{w_0}$ is the associated braid group element from Section \[subsec:CB\]. The following proposition is the analog of [@BW13 Theorem 2.18].
\[prop:mcT\] For any finite-dimensional $\U$-module $M$, the composition map $$\mc{T} := \Upsilon\circ \widetilde{\zeta} \circ T_{w_0}: M \longrightarrow M$$ is a $\bun$-module isomorphism.
Recall the bar involutions on $\U$ and its modules are denoted by $\Abar$, and the bar involution on $\bun$ is denoted by $\Bbar$. It is also understood that $\Abar(u) =\Abar(\imath(u))$ for $u\in \bun$.
We call a $\bun$-module $M$ equipped with an anti-linear involution $\Bbar$ [*involutive*]{} (or [*$\imath$-involutive*]{}) if $
\Bbar(u m) = \Bbar(u) \Bbar(m)$, $\forall u \in \bun, m \in M$. For any involutive $\U$-module $M$ with anti-linear involution $\Abar$, the anti-linear involution $$\Bbar := \Upsilon \circ \Abar : M \longrightarrow M$$ makes $M$ an $\imath$-involution ${\U^{\imath}}$-module (cf. [@BW13 Proposition 3.10]). In particular, since we know both $L(\la)$ and ${^{\omega}L}(\lambda)$ are involutive $\U$-modules, they are $\imath$-involutive ${\U^{\imath}}$-modules as well. The following theorem is the counterpart of [@BW13 Proposition 4.20].
\[thm:BCB\] Let $\la \in \La^+$. The $\bun$-module ${^{\omega}L}(\lambda)$ admits a unique basis $$\B^\imath(\lambda) := \{T^{\lambda}_{b} \mid b \in \B(\lambda)\}$$ which is $\Bbar$-invariant and of the form $$T^{\lambda}_{b} = b^+ \xi_{-\lambda} +\sum_{b' \prec b}
t^{\lambda}_{b;b'} b'^+ \xi_{-\lambda},
\quad \text{ for }\; t^{\lambda}_{b;b'} \in q\Z[q].$$
\[def:CB\] $\B^\imath(\lambda)$ is called the $\imath$-canonical basis of the $\bun$-module ${^{\omega}L(\lambda)}$.
Recall in [@Lu94 Chapter 27] Lusztig has developed a theory of based $\U$-modules $(M,B)$ (for a general quantum group $\U$ of finite type). The basis $B$ generates a $\Z[q]$-submodule $\mc{M}$ and an $\mA$-submodule ${}_\mA M$ of $M$.
Let $(M,B)$ be a finite-dimensional based $\U$-module.
1. The $\bun$-module $M$ admits a unique basis (called $\imath$-canonical basis) $
B^\imath := \{T_{b} \mid b \in B \}
$ which is $\Bbar$-invariant and of the form $$\label{iCB}
T_{b} = b +\sum_{b' \in B, b' \prec b}
t_{b;b'} b',
\quad \text{ for }\; t_{b;b'} \in q\Z[q].$$
2. $B^\imath$ forms an $\mA$-basis for the $\mA$-lattice ${}_\mA M$, and $B^\imath$ forms a $\Z[q]$-basis for the $\Z[q]$-lattice $\mc{M}$.
Recall that a tensor product of finite-dimensional simple $\U$-modules is a based $\U$-module by [@Lu94 Theorem 27.3.2].
\[cor:iCBontensor\] Let $\la_1, \ldots, \la_r \in \La^+$. The tensor product of finite-dimensional simple $\U$-modules ${^{\omega} L (\la_1)} \otimes \ldots \otimes {^{\omega} L (\la_r)}$ admits a unique $\Bbar$-invariant basis of the form (called $\imath$-canonical basis).
\[rem:BW16\] The construction of the $\imath$-canonical bases in this paper follows straightforwardly from [@BW13]. In the ongoing work [@BW16], we construct the $\imath$-canonical bases for general quantum symmetric pairs. In their preprint [@BK15], Balagovic and Kolb constructed the intertwiners for general quantum symmetric pairs (with some overlap with our [@BW16]), which leads to the universal solutions of the (quantum) reflection equation (a generalization of the Yang-Baxter equation).
Dualities
=========
In this section, we study various dualities between the coideal algebras and the Hecke algebras of type B/C/D. The theory is again uniform in the most cases, except in subsection \[subsec:C\], where we only study the duality between ${\U^{\imath}}_q(\mathfrak{sl}_{2r+1})$ and $\mc{H}_{C_n}$. Hence we shall simplify the notation (except in subsection \[subsec:C\]), and denote both quantum symmetric pairs simply by $(\U, {\U^{\imath}})$, and denote the correspond index sets simply by $\I = \I_k$ and $\I^{\imath} = \I^{\imath}_k$, for $k= 2r+2$ or $2r+1$.
The $({\U^{\imath}}, \mc{H}^1_{B_m})$-duality
---------------------------------------------
We set $I = I_k = \{a \pm \hf \vert a \in \I = \I_k\}$. Let the $\Qq$-vector space $\VV := \sum _{a \in I}\Qq v_{a} $ be the natural representation of $\U$, hence a ${\U^{\imath}}$-module. The action of $\U$ on $\VV$ is given by ($i \in \I$ and $a \in I$) $$E_{\alpha_i} v_{a} = \delta_{i+\hf, a} v_{a-1}, \quad F_{\alpha_i} v_{a} = \delta_{i-\hf,a} = v_{a+1} \quad \text{ and } \quad K_{\alpha_i} v_{a} = q^{\delta_{i-\hf,a} - \delta_{i+\hf},a} v_a.$$ We shall call $\VV$ the natural representation of ${\U^{\imath}}$ as well. For $m \in {\Z_{> 0}}$, $\VV^{\otimes m}$ becomes a natural $\U$-module (hence a ${\U^{\imath}}$-module) via the iteration of the coproduct $\Delta$. Note that $\VV$ is an involutive $\U$-module with $\Abar$ defined as $$\Abar(v_a) :=v_a, \quad \text{ for all } a \in I.$$ Therefore $\VV^{\otimes m}$ is an involutive $\U$-module and hence a $\imath$-involutive ${\U^{\imath}}$-module.
Let $W_{B_m}$ be the Coxeter groups of type $\text{B}_m$ with simple reflections $s_j, 0 \leq j \leq m-1$, where the subgroup generated by $s_i$, $1\leq i \leq m-1$ is isomorphic to $W_{A_{m-1}} \cong \mathfrak{S}_m$. The group $W_{B_m}$ and its subgroup $\mathfrak{S}_m$ act naturally on $I^m$ on the right as follows: for any $f \in I^m$, $ 1 \leq i \leq m$, we have $$\label{eq:rightW}
f \cdot s_j =
\begin{cases}
(\dots, f(j+1), f(j), \dots) , &\text{if } j > 0;\\
(-f(1), f(2), \dots, f(m)),& \text{if } j =0.
\end{cases}$$
Let $\mathcal{H}^p_{B_m}$ be the Iwahori-Hecke algebra of type $B_m$ with two parameters $p$ and $q$ over $\mathbb Q(q, p)$. It is generated by $H^p_0, H_1, H_2, \dots , H_{m-1}$, subject to the following relations ($i, j >0$), $$\label{eq:HeckeB}
\begin{split}
(H^p_0-p^{-1})(H^p_0 +p) &= 0 \quad \text{ and } \quad (H_i -q^{-1})(H_i +q) = 0,
\\
H_i H_{i+1} H_i &= H_{i+1} H_i H_{i+1},
\\
H_i H_j &= H_j H_i, \qquad \qquad\qquad\qquad \quad\qquad \qquad \text{for } |i-j| >1,
\\
H^p_0 H_1 H^p_0 H_1&=H_1H^p_0 H_1H^p_0 \quad \text{ and } \quad H^p_0 H_i = H_i H^p_0 , \qquad \text{for } i >1.
\end{split}$$ The bar involution on $\mathcal
H^p_{B_m}$ is the unique anti-linear ($\overline{q} =q^{-1}$ and $\overline{p} = p^{-1}$) automorphism defined by $\overline{H_i} =H_{i}^{-1}$ and $\overline {H^p_0} = (H^p_0)^{-1}$.
Let $\mathcal{H}^1_{B_m}$ be the degenerate Iwahori-Hecke algebra of type $B_m$ over $\mathbb Q(q)$ with the parameter $p=1$. We shall write the generator $H^p_0$ as $s_0$ in this case. Note that we have $s_0^2 =1$ and $\overline{s_0} = s_0$. For any $f \in I^m$, we can view $f$ as a function from the set $\{1, 2, \dots, m\}$ to $I^m$. Thus we define $
M_f= v_{f(1)} \otimes \cdots \otimes v_{f(m)}.
$ The Weyl group $W_{B_m}$ acts on $I^m$ by as before. Now the degenerate Hecke algebra $\HBm^1$ acts on the $\Qq$-vector space $\VV^{\otimes m}$ as follows ($ a > 0$): $$\begin{aligned}
M_f \cdot H_i &=
\begin{cases}
q^{-1}M_f, & \text{ if } f(i) = f(i+1);\\
M_{f \cdot s_i}, & \text{ if } f(i) < f(i+1);\\
M_{f \cdot s_i} + (q^{-1} - q) M_{f}, & \text{ if } f(i) > f(i+1);
\end{cases}\\
M_f \cdot s_0 &= M_{f \cdot s_0}.\end{aligned}$$ Introduce the $\Qq$-subspaces of $\VV$: $$\begin{aligned}
\VV_{-} &=\bigoplus_{i \in I} \Qq (v_{-i} - v_{i}),
\\
\VV_{+} &= \bigoplus_{i \in I} \Qq (v_{-i} + v_{i} ).\end{aligned}$$
The following lemma follows from direct computation.
\[int:lem:V+-\] $\VV_-$ and $\VV_+$ are ${\U^{\imath}}$-submodules of $\VV$. Moreover, we have $
\VV = \VV_- \oplus \VV_+.
$
Let $s$ be the largest number in $\I$. Now we fix $\zeta$ in such that $\zeta (\varepsilon_{-s}) = 1$. It follows that $$\zeta({\varepsilon_{s -i}}) =
(-q)^{-2s+i}, \qquad \text{ for } s-i \in \I.$$ Let us compute the ${\U^{\imath}}$-homomorphism $\mc{T} = \Upsilon\circ \widetilde{\zeta} \circ T_{w_0}$ (see Proposition \[prop:mcT\]) on the $\U$-module $\VV$; we remind that $w_0$ here is associated to $\U$ instead of $W_{B_m}$ or $W_{A_{m-1}}$.
\[int:lem:T\] The ${\U^{\imath}}$-isomorphism $\mc{T}^{-1}$ on $\VV$ acts as a scalar $- \id$ on the submodule $\VV_-$ and as $ \id$ on the submodule $\VV_+$.
First one computes that the action of $T_{w_0}$ on $\VV$ is given by $$T_{w_0} (v_{-s+i}) =(-q)^{2s-i}v_{s-i},
\qquad \text{ for } 0 \le i \le 2s.$$ Hence $$\label{eq:mcTtos0C}
\widetilde{\zeta} \circ T_{w_0}(v_a) =
v_{ a\cdot s_0}$$
One computes the first few terms of $\Upsilon$. For example we have $\Upsilon_{\alpha_{-\hf}+\alpha_{\hf}} = -\qq F_{\alpha_{-\hf}}F_{\alpha_{\hf}}$ for the quantum symmetric pair $(\U_q(\mathfrak{sl}_{2r+2}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}))$. Therefore using $\mc{T} = \Upsilon\circ \widetilde{\zeta} \circ T_{w_0}$ we have (for $i \in \I^{\imath}$) $$\begin{aligned}
\mc{T}^{-1} v_0 &= v_0\label{int:eq:mcT1} \quad (\text{for the pair $(\U_q(\mathfrak{sl}_{2r+2}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}))$}),\\
\mc{T}^{-1} ( v_{-i} - v_{i}) &= (-1) (v_{-i} - v_{i})\label{int:eq:mcT2},\\
\mc{T}^{-1} (v_{-i}+v_{i}) &= (v_{-i}+v_{i})\label{int:eq:mcT3}.\end{aligned}$$ The lemma now follows from Lemma \[int:lem:T\], since $\mc{T}^{-1}$ is a ${\U^{\imath}}$-isomorphism.
We remind the readers that the intertwiner $\Upsilon$ associated with the quantum symmetric pair $(\U, {\U^{\imath}})$ in this paper are different from the one in [@BW13], since we are considering quantum symmetric pairs with different parameters. This leads to different actions of $\mc{T}$ on the natural representation $\VV$ of $\U$ (c.f. [@BW13 Lemma 5.3]).
We have the following generalization of Schur-Jimbo duality, whose proof follows from [@BW13 Theorem 5.4].
\[int:thm:SchurB\]
1. The action of $\mc{T}^{-1} \otimes \id^{m-1}$ coincides with the action of $s_0 \in \mathcal{H}^1_{B_m}$ on $\VV^{\otimes m}$.
2. The actions of ${\U^{\imath}}$ and $\mathcal{H}^1_{B_m}$ on $\VV^{\otimes m}$ commute with each other, and they form double centralizers.
By variations of the choices of the subalgebras (Proposition \[int:prop:embedding\] and Proposition \[prop:embedding\]), we can obtain the coideal subalgebras ${\U^{\imath}}$ that forms double centralizers with the Hecke algebra $\mathcal{H}^p_{B_m}$ of two parameters, when acting on the tensor product $\VV^{\otimes m}$.
The Hecke algebra $\mathcal{H}_{D_m}$ of type D {#subsec:UiD}
-----------------------------------------------
Let $W_{D_m}$ be the Coxeter group of type $D_m (m \ge 2)$ with simple reflections $s^d_0$ and $s_j$, $1 \le j \le m-1$. The Coxeter group $W_{D_m}$ can be realized as a subgroup of $W_{B_m}$ via the following embedding : $s^d_0 \mapsto s_0 s_1 s_0$ and $s_i \mapsto s_i$ for $i \ge 1$. When $m=1$, we understand $W_{D_m}$ as the trivial group. The Weyl group $W_{D_m}$ acts on the set $I^m$ via the embedding.
Let $\mathcal{H}_{D_m}$ be the Iwahori-Hecke algebra of type $D_m$ over $\Qq$. It is generated by $H_0$, $H_1$, $\dots$, $H_{m-1}$, subject to the following relations: $$\begin{aligned}
(H_i -q^{-1})(H_i +q) &= 0, & \text{for } i \geq 0, &
\\
H_i H_{i+1} H_i &= H_{i+1} H_i H_{i+1}, & \text{for } i> 0,&
\\
H_i H_j &= H_j H_i, & \text{for } |i-j| >1, &
\\
H_0 H_2 H_0 = H_2 H_0 H_2 \quad &\text{ and } \quad H_0H_i=H_iH_0, &\text{for } i \neq 2.\end{aligned}$$ The bar involution on $\mathcal
H_{D_m}$ is the unique anti-linear involution defined by $\overline{H_i} =H_{i}^{-1}$ and $\overline{q} =q^{-1},$ for all $ 0 \le i \le m-1$.
\[lem:HDtoHB\] There is a $\Qq$-algebra embedding $\rho : \mc {H}_{D_m} \rightarrow \mc{H}^1_{B_m}$ such that $$\rho (H_0) = s_0 H_1 s_0 \quad \text{ and } \quad \rho (H_i) = H_i, \quad \text{for } i \ge 1.$$ Moreover, $\rho$ commutes with the bar involutions, that is, $\rho (\overline {h}) = \overline {\rho (h)}$ for $h \in \mc{H}_{D_m}$. (The bar involution on the left hand side is the bar involution on $\mc{H}_{D_m}$, while the bar involution on the right hand side is the bar involution on $\mc{H}^1_{B_m}$.)
It suffices to check the relations involving $H_0$. Note that we have $$\begin{aligned}
(s_0 H_1 s_0 - q^{-1})( s_0 H_1 s_0 +q) &= s_0(H_1 - q^{-1})(H_1 +q ) s_0 = 0, \\
s_0 H_1 s_0 \cdot H_2 \cdot s_0 H_1 s_0 = s_0 H_1 H_2 H_1 s_0 &= s_0 H_2 H_1 H_2 s_0 = H_2 \cdot s_0 H_1 s_0 \cdot H_2,\\
s_0 H_1 s_0 \cdot H_i &= H_i \cdot s_0 H_1 s_0 \quad \text{ for } i \neq 2, 0.\end{aligned}$$ This shows that $\rho$ is an embedding of $\Qq$-algebras. To show that $\rho$ commutes with the bar involutions, it suffices to show that $\rho(\overline{H_0}) = \overline{\rho (H_0)}$. This is clear since (recall $\overline{s_0} = s_0$) $$\rho (\overline{H_0}) = \rho (H_0^{-1}) = (s_0 H_1 s_0 ) ^{-1} = s_0 H^{-1}_1 s_0 = \overline{\rho (H_0)}.$$ The lemma follows.
Via the embedding $\rho : \mc {H}_{D_m} \rightarrow \mc{H}^1_{B_m}$, the Hecke algebra $\mc{H}_{D_m}$ has a natural action on the tensor space $\VV^{\otimes m}$ as follows (Note that $(f(1), f(2), \dots ) \cdot s^d_0 = (-f(2), -f(1), \dots)$): $$\label{int:eq:HBm}
M_f H_a=
\begin{cases}
q^{-1}M_f, & \text{ if } a>0, f(a) = f(a+1);\\
M_{f \cdot s_a}, & \text{ if } a > 0, f(a) < f(a+1);\\
M_{f \cdot s_a} + (q^{-1} - q) M_{f}, & \text{ if } a > 0, f(a) > f(a+1);\\
M_{f \cdot s^d_0}, & \text{ if } a = 0, -f(1) < f(2) ;\\
M_{f \cdot s^d_0} + (q^{-1} -q)M_f, & \text{ if } a=0, -f(1) > f(2);\\
q^{-1} M_f, & \text{ if } a =0, -f(1) = f(2).
\end{cases}$$
The following corollary follows immediately from Theorem \[int:thm:SchurB\].
1. The action of $((\mc{T}^{-1} \otimes \id) \cdot \mc{R}^{-1} \cdot (\mc{T}^{-1} \otimes \id)) \otimes \id^{m-2}$ coincides with the action of $H_0 \in \mathcal{H}_{D_m}$ on $\VV^{\otimes m}$.
2. The actions of ${\U^{\imath}}$ and $\mathcal{H}_{D_m}$ on $\VV^{\otimes m}$ commute with each other.
The commuting relation of the actions of ${\U^{\imath}}$ and $\mathcal{H}_{D_m}$ on $\VV^{\otimes m}$ has also been observed in [@ES13 §7.6] by direct computation without using the Theorem \[int:thm:SchurB\].
An element $f \in I^m$ is called ($D$-)anti-dominant, if $ |f(1) | \le f(2) \le f(3) \cdots \le f(m)$. ($ |f(1) |$ denotes the absolute value of $f(1)$.)
\[thm:samebar\] The bar involution ${\psi_{\imath}}: \VV^{\otimes m} \rightarrow \VV^{\otimes m}$ is compatible with both the bar involution of $\mathcal{H}_{D_m}$ and the bar involution of ${\U^{\imath}}$; that is, for all $v \in \VV^{\otimes m}$, $h \in \mathcal{H}_{D_m}$, and $u \in {\U^{\imath}}$, we have $${\psi_{\imath}}(u v h) = {\psi_{\imath}}(u) \, {\psi_{\imath}}(v) \ov{h} \quad \text{ and } \quad {\psi_{\imath}}(M_f) = M_f \text{ for all $D$-anti-dominant } f .$$ Moreover such bar involution on $\VV^{\otimes m}$ is unique.
The exact same proof as [@BW13 Theorem 5.8] shows that the bar involution ${\psi_{\imath}}= \Upsilon \circ \Abar$ is compatible with both the bar involution of $\mathcal{H}^1_{B_m}$ and the bar involution of ${\U^{\imath}}$. But since the embedding $\rho: \mc{H}_{D_m} \rightarrow \mc{H}^0_{B_m}$ is compatible with bar involutions (Proposition \[lem:HDtoHB\]), we know that ${\psi_{\imath}}$ is compatible with the bar involution of $\mc{H}_{D_m}$. Therefore we only need to show that ${\psi_{\imath}}(M_f) = M_f \text{ for all $D$-anti-dominant } f $.
For any $D$-anti-dominant $f \in I^m$ with $0 \le f(1)$, we have ${\psi_{\imath}}(M_f) = M_f$ by [@BW13 Theorem 5.8]. For any $D$-anti-dominant $f \in I^m$ with $f(1) < 0$, we see that $f \cdot s_0$ is still $D$-anti-dominant and $0 \le -f(1) = f(1) \cdot s_0$. We have $${\psi_{\imath}}(M_f) = {\psi_{\imath}}(M_{f \cdot s_0} s_0) = {\psi_{\imath}}(M_{f \cdot s_0}) \overline{s_0} = M_{f \cdot s_0} s_0 = M_f.$$ Thus ${\psi_{\imath}}(M_f) = M_f$ for all $D$-anti-dominant $f \in I^m$.
The uniqueness of such bar involution on $\VV^{\otimes m}$ follows from a standard argument (cf. [@BW13 Theorem 5.8]). The theorem follows.
It is well-known that via the action defined in , the tensor product $\VV^{\otimes m}$ becomes a direct sum of permutation modules of $\mc{H}_{D_m}$. Therefore the (parabolic) Kazhdan-Lusztig basis of $\mc{H}_{D_m}$ induces a (parabolic) Kazhdan-Lusztig basis (of type D) on $\VV^{\otimes m}$. Recall that $\VV^{\otimes m}$ admits an $\imath$-canonical basis by Corollary \[cor:iCBontensor\]. The following Corollary follows immediately from Theorem \[thm:samebar\].
\[cor:samebar\] The $\imath$-canonical basis on the tensor space $\VV^{\otimes m}$ is the same as the Kazhdan-Lusztig basis of type $D$.
Theorem \[thm:samebar\] and Corollary \[cor:samebar\] make sense in the case $m=1$ as well, where we understand $W_{D_m}$ as the trivial group. More precisely, the $\imath$-canonical basis on $\VV$ is the same as the canonical basis on $\VV$.
Bruhat orderings {#subsec:Bruhat}
----------------
In this subsection we show that the bar involution ${\psi_{\imath}}$ on $\VV^{\otimes m}$ satisfies the type D Burhat ordering, which should be expected in light of Theorem \[thm:samebar\]. We do not need results from this section for any other part of this paper.
Let $X(m) = \hf \Z [\epsilon_1, \epsilon_2, \dots \epsilon_m]$ and set $$\rho = (0 \epsilon_1) - \epsilon_2 - \cdots - (n-1)\epsilon_n.$$ There is a non-degenerate symmetric bilinear form $(\cdot \vert \cdot)$ on $X(m)$ such that $(\epsilon_i \vert \epsilon_j) = \delta_{ij}
$.
There is a natural injective map $I^m \rightarrow X(m)$, defined as $$\begin{aligned}
f &\mapsto \lambda_f , \text{ where } \lambda_f = \sum^m_{i=1} f(i) \epsilon_i - \rho, \quad \text{ for } f \in I^m.
$$
For any $f \in I^m$, the $\U$-weight of $M_f$ is ${\rm wt}(f) = \sum^m_{i=1} \varepsilon_{f(i)} \in \Lambda$. We define the ${\U^{\imath}}$-weight of $M_f$ to be ${\rm{wt}_{\imath}}(f) = \sum^m_{i=1} \overline{\varepsilon_i}$, i.e., the image of ${\rm wt}(f)$ in the quotient $\Lambda_{\inv}$.
Note that we always have $\lambda_f - \lambda_g \in \Z[\epsilon_1, \dots, \epsilon_m]$ for any $f, g \in I^{m}=I^{m}_k$ (for both $k = 2r+1 \text{ or } 2r+2$).
We define the following two partial orderings on $I^m$.
1. For any $f, g \in I^m$, we say $g \preceq_B f$ if $$\rm{wt}_{\imath} (f) = \rm{wt}_{\imath} (g) \quad \text{ and }\quad \lambda_f - \lambda _g = a_0 (-\epsilon_1) + \sum^{m-1}_{i=1} a_i (\epsilon_i - \epsilon_{i+1}), \text{ where }a_i \in \N.$$
2. For any $f, g \in I^m$, we say $g \preceq_D f$ if $$g \preceq_B f \quad \text{ and } \quad g\cdot s_0 \preceq_B f \cdot s_0.$$(Recall $ (f(1), \dots) \cdot s_0 = (-f(1), \dots)$.)
\[prop:bruhatD\] Let $g ,f \in I^m$ such that $g \preceq_D f$.
1. If $m=1$, then $f = g$.
2. If $m \ge 2$, then we have $$\lambda_f - \lambda_g = a_0 (-\epsilon_1 - \epsilon_2 ) + \sum^{m-1}_{i=1} a_i (\epsilon_i - \epsilon_{i+1}), \qquad \text{ where } a_i \in \N.$$
When $m =1$, the proposition follows from direct computation. So let us assume $m \ge 2$. It suffices to consider the case where $$\lambda_f - \lambda_g \in \Z[\epsilon_1, \epsilon_2].$$ Otherwise we can always find $h \in I^m$ (and then replace $g$ by $h$) such that $g \preceq_D h \preceq_D f$ and $
\lambda_h - \lambda_g \in \Z[\epsilon_3, \epsilon_4, \dots, \epsilon_m]$, $\lambda_f -\lambda_h \in \Z[\epsilon_1, \epsilon_2]$. Thus let us simply assume $m=2$.
We know that ${\rm wt}_{\imath}(f) = {\rm wt}_{\imath}(g)$ by our assumption. All elements in $I^{2}$ of the same ${\U^{\imath}}$-weight ${\rm wt}_{\imath}(f)$ has the Hasse diagram with respect to the partial ordering $\preceq_B$ as the following ($a \ge 0$, $b \ge 0$, $a \le b$):
(0,0) node (0101) [(-a, -b)]{}; (-2,-1) node (010) [(-b, -a)]{}; (-2, -2) node (10) [(b, -a)]{}; (-2, -3) node (0) [(-a, b)]{}; (0, -4) node (e) [(a, b)]{}; (2, -1) node (101) [(a, -b)]{}; (2, -2) node (01) [(-b, a)]{}; (2, -3) node (1) [(b, a)]{};
\(1) – node\[below right\] [$\preceq_B$]{} (e); (01) – (1); (101) – (01); (0101) – (101); (0) – (e); (10) – (0); (010) – (10); (0101) – node\[above left\] [$\preceq_B$]{} (010); (010) – (01); (101) – (10); (10) – (1); (01) – (0);
Applying $s_0$ to the vertices, which preserves the ${\U^{\imath}}$-weight ${\rm wt}_{\imath}(f)$, we can rewrite the Hasse diagram with respect to $\preceq_B$ as:
(0,0) node (101) [(a, -b)]{} ; (-2,-1) node (10) [(b, -a)]{} ; (-2, -2) node (010) [(-b, -a)]{}; (-2, -3) node (e) [(a, b)]{} ; (0, -4) node (0) [(-a, b)]{}; (2, -1) node (0101) [(-a, -b)]{} ; (2, -2) node (1) [(b, a)]{} ; (2, -3) node (01) [(-b, a)]{} ;
\(1) – (e); (01) – (1); (101) – (01); (0101) – (101); (0) – (e); (10) – (0); (010) – (10); (0101) – (010); (010) – (01); (101) – (10); (10) – (1); (01) – (0);
Combining the two diagrams, we have the following Hasse diagram with respect to the partial ordering $\preceq_D$:
(0,0) node (0101) [(-a, -b)]{}; (-2,-1) node (010) [(-b, -a)]{}; (-2, -2) node (10) [(b, -a)]{}; (-2, -3) node (0) [(-a, b)]{}; (0, -4) node (e) [(a, b)]{}; (2, -1) node (101) [(a, -b)]{}; (2, -2) node (01) [(-b, a)]{}; (2, -3) node (1) [(b, a)]{};
\(0101) – node\[above left\] [$\preceq_D$]{} (010); (0101) – (1); (010) – (e); (010) – (01); (101) – (10); (10) – (0); (10) – (1); (01) – (0); (101) – (01); (1) – node\[below right\] [$\preceq_D$]{} (e);
The rest of the proposition follows from case by case computation. For example, we have $$\lambda_{(-b,-a)} -\lambda_{(-b, a)} = -2a \epsilon_2 = a (-\epsilon_1 - \epsilon_2) + a (\epsilon_1- \epsilon_2).$$
Note that the set $\{\epsilon_1, \epsilon_1- \epsilon_2, \dots, \epsilon_{m-1}- \epsilon_m \}$, and the set $\{\epsilon_1 + \epsilon_2, \epsilon_1- \epsilon_2, \dots, \epsilon_{m-1}- \epsilon_m \}$ are the sets of simple roots for type B, and type D root systems, respectively. So for $f$, $g \in I^m$, $g \preceq_B f$ means that $\lambda_f -\lambda_g $ is a non-negative integral linear combination of type B simple roots, and $g \preceq_D f$ means that $\lambda_f -\lambda_g $ is a non-negative integral linear combination of type D simple roots, respectively.
Actually if we know $g \preceq_B f$ and $g \cdot s_0 \preceq_B f \cdot s_0$, we have $$\begin{aligned}
\lambda_f - \lambda_g &= a_0 (-\epsilon_1) + \sum^{m-1}_{i=1} a_i (\epsilon_i - \epsilon_{i+1}) \quad & \text{ where } a_i \in \N;\\
\lambda_{f\cdot s_0}- \lambda_{g \cdot s_0} &= (2a_1 -a_0) (-\epsilon_1) + \sum^{m-1}_{i=1} a_i (\epsilon_i - \epsilon_{i+1}) \quad & \text{ where } a_i \in \N, 2a_1 - a_0 \in \N.\end{aligned}$$ But we can write $\lambda_f - \lambda_g$ as $$\lambda_f - \lambda_g = \hf(a_0) (-\epsilon_{1} - \epsilon_2) + \hf(2a_1 - a_0) (\epsilon_1- \epsilon_2) + \sum^{m-1}_{i=2} a_i (\epsilon_i - \epsilon_{i+1}).$$ We already know that $a_0 \ge 0$ and $\hf(2a_1 - a_0) \ge 0 $. So Proposition \[prop:bruhatD\] is essentially showing that $\hf(a_0)$ and $\hf(2a_1 - a_0)$ are actually integers, if we in addition have ${\rm wt}_{\imath} (f) = {\rm wt}_{\imath} (g)$.
In light of the proposition we shall see that the bar involution ${\psi_{\imath}}$ on the tensor space $\VV^{\otimes m}$ actually respects the coarser partial ordering $\preceq_D$.
For any $f \in I^m$, we have $${\psi_{\imath}}(M_f) = \Upsilon \Abar (M_f) = M_f + \sum_{g \preceq_D f} c_{g, f} M_g.$$
Following [@BW13 Lemma 9.4], we have $${\psi_{\imath}}(M_f) = \Upsilon \Abar (M_f) = M_f + \sum_{g \preceq_B f} c_{g, f} M_g.$$ Thanks to compatibility in Theorem \[thm:samebar\], we have $$M_{f\cdot s_0} + \sum_{g' \preceq_B f\cdot s_0} c_{g', f\cdot s_0} M_{g'} = {\psi_{\imath}}(M_f \cdot s_0) = {\psi_{\imath}}(M_f) \cdot \overline{s_0} = M_f \cdot s_0 + \sum_{g \preceq_B f} c_{g,f} M_{g \cdot s_0}.$$ Therefore we have $c_{g', f \cdot s_0} = c_{g, f}$ if $g' = g\cdot s_0$. Thus we have $g \preceq_B f$ and $g\cdot s_0 \preceq_B f \cdot s_0$. By Proposition \[prop:bruhatD\], this implies $g \preceq_D f$. The proposition follows.
We shall NOT use the partial ordering $\preceq_D$, or any variation of the this partial ordering in this paper. It is the partial ordering $\preceq_B$ and its variants that we shall use in this paper, which suffices to establish the $\imath$-canonical bases and for the application to the category $\mathcal{O}$.
The $({\U^{\imath}}_q(\mathfrak{sl}_{2r+1}), \mathcal{H}_{C_n})$-duality {#subsec:C}
------------------------------------------------------------------------
In this subsection, we shall only consider the quantum symmetric pair $(\U_q(\mathfrak{sl}_{2r+1}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}))$. We shall add the subscripts to avoid confusion in this subsection.
Let $\WW :=\VV^*$ be the (restricted) dual module of $\VV$ with basis $\{w_a \mid a \in I_{2r+1}\}$ such that $\langle w_a, v_b \rangle = (-q)^{-a} \delta_{a,b}$. The action of $\U_q(\mathfrak{sl}_{2r+1})$ on $\WW$ is given by the following formulas (for $i \in \I_{2r+1}$, $a \in I_{2r+1}$): $$E_{\alpha_i} w_a = \delta_{i-\hf, a} w_{a+1}, \quad F_{\alpha_i}w_a
= \delta_{i+\hf, a}w_{a-1}, \quad K_{\alpha_i} w_a = q^{-(\alpha_i, \varepsilon_a)}w_a.$$ By restriction through the embedding $\iota$, $\WW$ is naturally a ${\U^{\imath}}_q(\mathfrak{sl}_{2r+1})$-modules. For $n \in \Z_{>0}$, $\WW^{\otimes n}$ is naturally a $\U_q(\mathfrak{sl}_{2r+1})$-module, hence a ${\U^{\imath}}_q(\mathfrak{sl}_{2r+1})$-module, via the iteration of the coproduct $\Delta$. Note that $\WW$ is an involutive $\U_q(\mathfrak{sl}_{2r+1})$-module with $\psi$ defined as $$\psi(w_a) = w_a, \quad \text{ for all }a \in I_{2r+1}.$$ Therefore $\WW^{\otimes n}$ is an involutive $\U_q(\mathfrak{sl}_{2r+1})$-module and hence an $\imath$-involutive ${\U^{\imath}}_q(\mathfrak{sl}_{2r+1})$-module.
Let $\mathcal{H}_{C_n} = \mathcal{H}^{q}_{B_n}$ be the Hecke algebra of type $C$ with equal parameters ($p=q$). For $f \in I_{2r+1}^n$, let $M^*_f = w_{f(1)} \otimes \cdots \otimes w_{f(n)} \in \WW^{\otimes n}$. The Hecke algebra $\mathcal{H}_{C_n}$ acts on $\WW^{\otimes n}$ as follows: $$\label{eq:typeC}
\begin{split}
M^*_f H_a=&
\begin{cases}
q^{-1}M^*_f, & \text{ if } a>0, f(a) = f(a+1);\\
M^*_{f \cdot s_a}, & \text{ if } a > 0, f(a) > f(a+1);\\
M^*_{f \cdot s_a} + (q^{-1} - q) M_{f}, & \text{ if } a > 0, f(a) < f(a+1).
\end{cases}\\
M^*_f H^q_0= &
\begin{cases}
M^*_{f \cdot s_0}, & \text{ if } f(1) < 0;\\
M^*_{f \cdot s_0} + (q^{-1} - q) M_{f}, & \text{ if } f(1) > 0;\\
q^{-1} M^*_{f}, &\text{ if }f(1)=0.
\end{cases}
\end{split}$$
An element $f \in I_{2r+1}^n$ is called ($C$-)anti-dominant, if $ 0 \le f(1) \le f(2) \le f(3) \cdots \le f(n)$.
Let us fix a choice of $\zeta$ in such that $\mc{T}^{-1} : \WW \rightarrow \WW$ maps $w_{-s}$ to $w_{s}$, where $s$ is the largest number in $I_{2r+1}$. The tensor product $\WW^{\otimes n}$ becomes a direct sum of permutation modules of $\mathcal{H}_{C_n}$ via the action defined in . Hence $\WW^{\otimes n}$ admits a Kazhdan-Lusztig basis (of type C). The following theorem is the counterpart of the Theorem \[int:thm:SchurB\] and Theorem \[thm:samebar\], Corollary \[cor:samebar\].
\[thm:KLC\]
1. The action of $\mc{T}^{-1} \otimes \id^{n-1}$ coincides with the action of $H_{0} \in \mathcal{H}_{C_n}$ on $\WW^{\otimes n}$.
2. The actions of ${\U^{\imath}}_{q}(\mathfrak{sl}_{2r+1})$ and $\mathcal{H}_{C_n}$ on $\WW^{\otimes n}$ commute with each other, and they form double centralizers.
3. There exists a unique bar involution $\psi_{\imath} : \WW^{\otimes n} \rightarrow \WW^{\otimes n}$ such that for all $w \in \WW^{\otimes n}$, $h \in \mathcal{H}_{C_n}$, $u \in {\U^{\imath}}_q(\mathfrak{sl}_{2r+1})$, and all $C$-anti-dominant $f \in I_{2r+1}$, we have $${\psi_{\imath}}(u v h) = {\psi_{\imath}}(u) \, {\psi_{\imath}}(v) \ov{h} \quad \text{ and } \quad {\psi_{\imath}}(M_f) = M_f.$$
4. The $\imath$-canonical basis on the tensor space $\WW^{\otimes m}$ is the same as the Kazhdan-Lusztig basis of type $C$.
The actions of ${\U^{\imath}}_q(\mathfrak{sl}_{2r})$ and $\mc{H}_{C_n}$ on $\WW^{n}$, the restricted dual of the natural representation $\VV$ of $\U_q(\mathfrak{sl}_{2r})$, do not commute.
Kazhdan-Lusztig theory of super type D {#sec:rep}
======================================
In this section we shall apply the theory of $\imath$-canonical bases from Section \[sec:QSP\] to study the BGG category $\mc{O}$ of the Lie superalgebra $\mathfrak{osp}(2m|2n)$ with respect to various Borel subalgebras. We shall formulate and establish the Kazhdan-Lusztig theory for the Lie superalgebra $\mathfrak{osp}(2m|2n)$.
We shall first set up various Fock spaces and establish the $\imath$-canonical bases on suitable completions of those Fock spaces. Then we study various versions of the category $\mc{O}$ of the Lie superalgebra $\mathfrak{osp}(2m|2n)$. Finally we can formulate and establish the Kazhdan-Lusztig theory for the BGG category $\mc{O}$ of the Lie superalgebra $\mathfrak{osp}(2m|2n)$.
We emphasize that we shall use the same partial orderings as [@BW13 Definition 8.3], even though §\[subsec:Bruhat\] suggests that we can use some coarser partial orderings (type B vs type D). Most proofs shall be similar to [@BW13], hence shall be omitted and referred to [@BW13].
Infinite-rank constructions and notations
-----------------------------------------
We set $$\begin{aligned}
\label{eq:III}
\I_{odd} = \cup^{\infty}_{r=0} & \I_{2r+1} = \Z,
\qquad
\I^{\imath}_{odd} = \cup^{\infty}_{r=0} \I^{\imath}_{r} = \Z_{>0},
\qquad
I_{odd} = \Z+\hf.\\
\label{eq:II}
\I_{ev} = \cup^{\infty}_{r=0} & \I_{2r+2} = \Z+\hf,
\qquad
\I_{ev}^{\imath} = \cup^{\infty}_{r=0} \I^{\imath}_{r} = \Z_{\ge 0}+\hf,
\qquad
I_{ev} = \Z.\end{aligned}$$
When it is not necessary to distinguish the even or odd cases, we shall abuse the notation and simply write $\I$, $\I^{\imath}$, $I$ (of course, they have to be consistent, i.e., all even or all odd.).
We have the natural inclusions of $\Qq$-algebras: $$\begin{aligned}
\cdots \subset & \U_q(\mathfrak{sl}_{2r+1}) \subset \U_q(\mathfrak{sl}_{2r+3}) \subset \cdots ,\qquad
\cdots \subset & {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}) \subset {\U^{\imath}}_q(\mathfrak{sl}_{2r+3}) \subset \cdots,\\
\cdots \subset & \U_q(\mathfrak{sl}_{2r+2}) \subset \U_q(\mathfrak{sl}_{2r+4}) \subset \cdots ,\qquad
\cdots \subset & {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \subset {\U^{\imath}}_q(\mathfrak{sl}_{2r+4}) \subset \cdots.\end{aligned}$$ Define the following infinite rank $\Qq$-algebras: $$\begin{aligned}
\U_{odd} := \bigcup^{\infty}_{r=0} \U_q(\mathfrak{sl}_{2r+2}) \quad &\text{ and } \quad {\U^{\imath}}_{odd} := \bigcup^{\infty}_{r=0} {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}),\displaybreak[0]\\
\U_{ev} := \bigcup^{\infty}_{r=0} \U_q(\mathfrak{sl}_{2r+1}) \quad &\text{ and } \quad {\U^{\imath}}_{ev} := \bigcup^{\infty}_{r=0} {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}).\end{aligned}$$
We also abuse the notation and simply write the pair ($\U, {\U^{\imath}}$) (with the same subscripts). The embeddings of finite rank algebras induce an embedding of $\Qq$-algebras, denoted also by $\iota : \bun \longrightarrow \U$. Again $\U$ is naturally a Hopf algebra with coproduct $\Delta$, and its restriction under $\iota$, $
\Delta: \bun \rightarrow \bun \otimes \U,
$ makes $\bun$ (or more precisely $\iota(\bun)$) naturally a (right) coideal subalgebra of $\U$. The anti-linear bar involutions on finite rank algebras induce anti-linear bar involution $\psi$ on $\U$ and anti-linear bar involution $\psi_{\imath}$ on ${\U^{\imath}}$, respectively.
Recall $\Pi_{k}$ denotes the simple system of $\U_q(\mathfrak{sl}_{k})$. Let $
\Pi_{odd} := \bigcup^{\infty}_{r=0} \Pi_{2r+1}
$ ($\Pi_{ev} := \bigcup^{\infty}_{r=0} \Pi_{2r+2}$, respectively) be a simple system of $\U_{odd}$ ($\U_{ev}$, respectively). We again shall write $\Pi$ for both $\Pi_{odd}$ and $\Pi_{ev}$. Recall we denote the integral weight lattice of $\U_{k}$ by $\Lambda_{k}$. Then let $$\Lambda_{odd} := \oplus_{i \in I_{odd}} \Z[\varepsilon_i] = \bigcup^{\infty}_{r =0} \Lambda_{2r+1} \quad \text{ and } \quad \Lambda_{ev} := \oplus_{i \in I_{ev}} \Z[\varepsilon_i] = \bigcup^{\infty}_{r =0} \Lambda_{2r+2}$$ be the integral weight lattice of $\U_{odd}$ and $\U_{ev}$, respectively. Thus by abuse of notations, we have (for both cases) $$\Lambda = \oplus_{i \in I} \Z[\varepsilon_i].$$ Following §\[subsec:theta\], we have the quotient lattice $\Lambda_{\inv}$ of the lattice $\Lambda$.
Following [@BW13 §8.1] we can define the intertwiner $\Upsilon$ (which lies in some completion of $\U^{-}$) for the quantum symmetric pair $(\U, {\U^{\imath}})$ such that $$\Upsilon := \sum_{\mu \in \N\Pi} \Upsilon_{\mu}, \quad \Upsilon_\mu \in \U^{-}_{\mu}.$$ We shall see that $\Upsilon$ is a well-defined operator on $\U$-modules that we are concerned.
The Lie superalgebra $\mf{osp}(2m|2n)$ {#subsec:osp}
--------------------------------------
In this subsection, we recall some basics on ortho-symplectic Lie superalgebras and set up notations to be used later on (cf. [@CW12] for more on Lie superalgebras).
Let $\Z_2 = \{\ov{0}, \ov{1}\}$. Let $\C^{2m|2n}$ be a superspace of dimension $(2m|2n)$ with basis $\{e_i \mid 1 \leq i \leq 2m\} \cup \{e_{\ov j} \mid 1 \leq j \leq 2n\}$, where the $\Z_2$-grading is given by the following parity function: $$p(e_i) = \ov 0, \qquad p(e_{\ov j}) = \ov 1 \quad (\forall i,j).$$ Let $B$ be a non-degenerate even supersymmetric bilinear form on $\C^{2m|2n}$. The general linear Lie superalgebra $\mf{gl}(2m|2n)$ is the Lie superalgebra of linear transformations on $\C^{2m|2n}$ (in matrix form with respect to the above basis). For $s \in \Z_2$, we define $$\begin{aligned}
\osp(2m|2n)_s &:= \{ g \in \mf{gl}(2m|2n)_s \mid B(g(x), y) = -(-1)^{s \cdot p(x)}B(x, g(y))\},\\
\osp(2m|2n) &:=\osp(2m|2n)_{\ov 0} \oplus \osp(2m|2n)_{\ov 1}.\end{aligned}$$
We now give a matrix realization of the Lie superalgebra $\osp(2m|2n)$. Take the supersymmetric bilinear form $B$ with the following matrix form, with respect to the basis $(e_1, e_2, \dots, e_{2m}, e_{\ov 1}, e_{\ov 2}, \dots, e_{\ov{2n}})$: $$\mc J_{2m|2n} :=
\begin{pmatrix}
0 & I_m & 0& 0\\
I_m & 0 & 0 & 0\\
0 & 0 & 0 & I^n\\
0 & 0 & -I^n & 0
\end{pmatrix}$$
Let $E_{i,j}$, $1 \leq i,j \leq 2m$, and $E_{\ov k , \ov h}$, $ 1 \leq k,h \leq 2n$, be the $(i,j)$th and $(\ov k, \ov h)$th elementary matrices, respectively. The Cartan subalgebra of $\osp(2m|2n)$ of diagonal matrices is denoted by $\mf h_{m|n}$, which is spanned by $$\begin{aligned}
&H_i := E_{i,i}-E_{m+i,m+i}, \quad 1 \leq i \leq m,\\
&H_{\ov j} :=E_{\ov j, \ov j} - E_{\ov{n+j}, \ov{n+j}}, \quad 1 \leq j \leq n.\end{aligned}$$ We denote by $\{\ep_i, \ep_{\ov j} \mid 1 \leq i \leq m, 1 \leq j \leq n \}$ the basis of $\mf h^*_{m|n}$ such that $$\ep_{a}(H_b) = \delta_{a,b}, \quad \text{ for } a, b \in \{i, \ov j \mid 1 \leq i \leq m, 1 \leq j \leq n\}.$$ We denote the lattice of integral weights of $\osp(2m|2n)$ by $$\label{eq:Xmn}
X_{ev}(m|n) := \sum^{m}_{i=1}\Z\ep_{i} + \sum^n_{j=1}\Z\ep_{\ov j}.$$ Denote the set of half integral weights of $\osp(2m|2n)$ by $$X_{odd}(m|n) := \sum^{m}_{i=1}(\Z+\hf) \ep_{i} + \sum^n_{j=1}(\Z+\hf) \ep_{\ov j}.$$ When it is not necessary to distinguish the integral or half-integral weights we shall abuse the notation, and simply write $X (m |n)$ for both of them.
The supertrace form on $\osp(2m|2n)$ induces a non-degenerate symmetric bilinear form on $\mf h^*_{m|n}$ denoted by $(\cdot | \cdot)$, such that $$(\ep_{i}\vert\ep_{a}) = \delta_{i,a}, \quad (\ep_{\ov j}| \ep_{a})
= -\delta_{\ov j, a}, \quad \text{ for } a \in \{i, \ov j \mid 1 \leq i \leq m, 1 \leq j \leq n\}.$$ We have the following root system of $\osp(2m|2n)$ with respect to $\mf h_{m|n}$ $$\Phi = \Phi_{\ov 0} \cup \Phi_{\ov 1}
= \{\pm\ep_{i}\pm\ep_{j}, \pm\ep_{\ov k}\pm\ep_{\ov l}, \pm2\ep_{\ov q}\} \cup \{\pm\ep_{p}\pm\ep_{\ov q}\},$$ where $1 \leq i < j \leq n$, $1 \leq p \leq n$, $1\leq q \leq m$, $1 \leq k < l\leq m$.
In this paper we shall need to deal with various Borel subalgebras, hence various simple systems of $\Phi$. Let ${\bf b}=(b_1,b_2,\ldots,b_{m+n})$ be a sequence of $m+n$ integers such that $m$ of the $b_i$’s are equal to ${0}$ and $n$ of them are equal to ${1}$. We call such a sequence a [*$0^m1^n$-sequence*]{}. Associated to each $0^m1^n$-sequence ${\bf b} =(b_1, \ldots, b_{m+n})$, we have the following fundamental system $\Pi_{\bf {b}}$, and hence a positive system $\Phi_{\bf b}^+ =\Phi_{{\bf b},\bar{0}}^+ \cup \Phi_{{\bf b},\bar{1}}^+$, of the root system $\Phi$ of $\mathfrak{osp}(2m|2n)$: $$\begin{aligned}
\Pi_{\bf {b}} &= \{-\ep^{b_1}_1 - \ep^{b_2}_2, \ep^{b_i}_i -\ep^{b_{i+1}}_{i+1} \mid 1 \leq i \leq m+n-1\}, \qquad & \text{ for } b_1=0;\\
\Pi_{\bf {b}} &= \{-2\ep^{b_1}_1, \ep^{b_i}_i -\ep^{b_{i+1}}_{i+1} \mid 1 \leq i \leq m+n-1\}, \qquad & \text{ for } b_1=1.\end{aligned}$$ where $\ep^{0}_i = \ep_{x}$ for some $1 \leq x \leq m$, $\ep^1_{j} = \ep_{\ov y}$ for some $1 \leq y \leq n$, such that $\ep_{x} -\ep_{x+1}$ and $\ep_{\ov y} - \ep_{\ov{y+1}}$ are always positive. It is clear that $\Pi_{\bf b}$ is uniquely determined by these restrictions. The Weyl vector associate with the fundamental system $\Pi_{{\bf b}}$ is defined to be $\rho_{\bf b}:= \hf \sum_{\alpha \in \Phi^+_{{\bf b}, \bar{0}}} \alpha -\hf \sum_{\beta \in \Phi^+_{{\bf b}, \bar{1}}} \beta$.
Corresponding to ${\bf b}^{\text{st}} =(0,\ldots, 0,1,\ldots,1)$, we have the following standard Dynkin diagram associated to $\Pi_{{\bf b}^{\text{st}}}$ (for $m\ge 2$):
(-1,1) node\[label=below:$\epsilon_1 - \epsilon_2$\] (1) [$\bigcirc$]{} ; (-1,-1) node\[label=below:$-\epsilon_1 - \epsilon_2$\] (0) [$\bigcirc$]{}; (1,0) node\[label=below:$\epsilon_2 - \epsilon_3$\] (2) [$\bigcirc$]{}; (2.5,0) node (3) [$\cdots$]{}; (4,0) node\[label=below:$\epsilon_m - \epsilon_{\overline{1}}$\] (4) [$\bigotimes$]{}; (6,0) node\[label=below:$\epsilon_{\overline{1}} - \epsilon_{\overline{2}}$\] (5) [$\bigcirc$]{}; (7,0) node (6) [$\cdots$]{}; (8,0) node\[label=below:$\epsilon_{\overline{n-1}} - \epsilon_{\overline{n}}$\] (7) [$\bigcirc$]{};
\(1) – (2); (0) – (2); (2) – (3); (3) – (4); (4) – (5); (5) – (6); (6) – (7);
If we have $m =1$, the corresponding Dynkin diagram becomes (with $n \ge 2$):
(-1,1) node\[label=left:$\epsilon_1 - \epsilon_{\overline{1}}$\] (1) [$\bigotimes$]{} ; (-1,-1) node\[label=below:$-\epsilon_1 - \epsilon_{\overline{1}}$\] (0) [$\bigotimes$]{}; (1,0) node\[label=below:$\epsilon_{\overline{1}} - \epsilon_{\overline{2}}$\] (2) [$\bigcirc$]{}; (2.5,0) node (3) [$\cdots$]{}; (4,0) node\[label=below:$\epsilon_{\overline{n-1}} - \epsilon_{\overline{n}}$\] (4) [$\bigcirc$]{};
\(1) – (2); (0) – (2); (2) – (3); (3) – (4); (1) – (0);
As usual, $\bigotimes$ stands for an isotropic simple odd root, $\bigcirc$ stands for an simple even root. A direct computation shows that $$\label{eq:rhobst}
\rho_{{\bf b}^{\text{st}}} = 0\epsilon_1 - \epsilon_2 -\ldots -(m-1) \epsilon_m
+ (m-1) \epsilon_{\bar{1}} +\ldots + (m-n) \epsilon_{\bar{n}}.$$
More generally, associated to a sequence $\bf b$ which starts with two $0$’s is a Dynkin diagram which always starts on the left with a type $D$ branch:
(-1,1) node\[label=below:$\epsilon_1 - \epsilon_2$\] (1) [$\bigcirc$]{} ; (-1,-1) node\[label=below:$-\epsilon_1 - \epsilon_2$\] (0) [$\bigcirc$]{}; (1,0) node (2) [$\bigodot$]{}; (2.5,0) node (3) [$\cdots$]{}; (4,0) node (4) [$\bigodot$]{}; (6,0) node (5) [$\bigodot$]{}; (7,0) node (6) [$\cdots$]{}; (8,0) node (7) [$\bigodot$]{};
\(1) – (2); (0) – (2); (2) – (3); (3) – (4); (4) – (5); (5) – (6); (6) – (7);
Here $\bigodot$ stands for either $\bigotimes$ or $\bigcirc$ depending on $\bf b$.
On the other hand, corresponding to ${\bf b}^{\text{st}'} =(1,\ldots, 1,0,\ldots,0)$, we have the following another often used Dynkin diagram associated to $\Pi_{{\bf b}^{\text{st}'}}$:
(-0.2,0) node\[label=below:$-2\epsilon_{\overline{1}}$\] (0) [$\bigcirc$]{}; (1,0) node\[label=below:$\epsilon_{\overline{1}} - \epsilon_{\overline{2}}$\] (2) [$\bigcirc$]{}; (2.5,0) node (3) [$\cdots$]{}; (4,0) node\[label=below:$\epsilon_{\overline{n}} - \epsilon_{1}$\] (4) [$\bigotimes$]{}; (6,0) node\[label=below:$\epsilon_1 - \epsilon_2$\] (5) [$\bigcirc$]{}; (7,0) node (6) [$\cdots$]{}; (8,0) node\[label=below:$\epsilon_{m-1} - \epsilon_m$\] (7) [$\bigcirc$]{}; (0.4,0) node [$\Longrightarrow$]{}; (2) – (3); (3) – (4); (4) – (5); (5) – (6); (6) – (7);
A direct computation shows that $$\rho_{{\bf b}^{\text{st}'}} = - \epsilon_{\overline{1}} - 2\epsilon_{\overline{2}} - \cdots - n \epsilon_{\overline{n}} + n\epsilon_{1} + (n-1)\epsilon_{1}+\cdots + (n-m)\epsilon_{m}.$$
The fundamental system $\Pi_{{\bf b}_1}$ with ${\bf b}_1 = (0,1,{\bf b}')$ and the fundamental system $\Pi_{{\bf b}_2}$ with ${\bf b}_2 = (1,0,{\bf b}')$ differ by an odd reflection ([@CW12 Remark 1.31]), even though their corresponding Dynkin diagrams look quite different.
More generally, associated to a sequence $\bf b$ which starts with one $1$ is a Dynkin diagram which always starts on the left with a type $C$ branch:
(-0.2,0) node\[label=below:$-2\epsilon_{\overline{1}}$\] (0) [$\bigcirc$]{}; (1,0) node (2) [$\bigodot$]{}; (2.5,0) node (3) [$\cdots$]{}; (4,0) node (4) [$\bigodot$]{}; (6,0) node (5) [$\bigodot$]{}; (7,0) node (6) [$\cdots$]{}; (8,0) node (7) [$\bigodot$]{}; (0.4,0) node [$\Longrightarrow$]{}; (2) – (3); (3) – (4); (4) – (5); (5) – (6); (6) – (7);
Now we can write the non-degenerate symmetric bilinear form on $\Phi$ as follows: $$(\ep^{b_i}_i | \ep^{b_j}_j) = (-1)^{b_i} \delta_{ij}, \quad \quad \quad 1 \leq i, j \leq m+n.$$ We define $\mathfrak{n}_{\bf b}^\pm$ to be the nilpotent subalgebra spanned by the positive/negative root vectors in $\osp(2m|2n)$. Then we obtain a triangular decomposition of $\osp(2m|2n)$: $$\osp(2m|2n) = \mathfrak{n}_{\bf b}^+ \oplus \mathfrak{h}_{m|n} \oplus \mathfrak{n}_{\bf b}^-,$$ with $\mathfrak{n}_{\bf b}^+ \oplus \mathfrak{h}_{m|n}$ as a Borel subalgebra.
Fix a $0^m1^n$-sequence ${\bf b}$ and hence a positve system $\Phi^+_{\bf b}$. We denote by $Z(\osp(2m|2n))$ the center of the enveloping algebra $U(\osp(2m|2n))$. There exists a standard projection $\phi: U(\osp(2m|2n)) \rightarrow U(\mf h_{m|n})$ which is consistent with the PBW basis associated to the above triangular decomposition ([@CW12 §2.2.3]). For $\lambda \in \mf h^*_{m|n}$, we define the central character $\chi_{\lambda}$ by letting $$\chi_\lambda(z) :=\lambda(\phi(z)),\quad \text{ for }z \in Z(\osp(2m|2n)).$$ Denote the Weyl group of (the even subalgebra of) $\osp(2m|2n)$ by $W_{\osp}$, which is isomorphic to $W_{D_m} \times W_{C_n}$. Then for $\mu$, $\nu \in \mf h^*_{m|n}$, we say $\mu$, $\nu$ are linked and denote it by $\mu \sim \nu$, if there exist mutually orthogonal isotropic odd roots $\alpha_1, \alpha_2, \dots, \alpha_l$, complex numbers $c_1, c_2, \dots, c_l$, and an element $w \in W_{\osp}$ satisfying $$\mu + \rho_{\bf b}= w(\nu +\rho_{\bf b} - \sum_{i=1}^{l}c_i\alpha_i), \quad (\nu + \rho_{\bf b}| \alpha_j)= 0, \quad j=1 \dots, l.$$ It is clear that $\sim$ is an equivalent relation on $\mf h^*_{m|n}$. Versions of the following basic fact went back to Kac, Sergeev, and others.
[@CW12 Theorem 2.30] Let $\lambda$, $\mu \in \mf h^*_{m|n}$. Then $\lambda$ is linked to $\mu$ if and only if $\chi_{\lambda} = \chi_{\mu}$.
The BGG categories {#subsec:cat}
------------------
In this subsection, we shall define various (parabolic) BGG categories for ortho-symplectic Lie superalgebras.
Let ${\bf b}$ be a $0^m1^n$-sequence. The Bernstein-Gelfand-Gelfand (BGG) category $\mathcal{O}_{{\bf b}}$ ($= \mathcal{O}_{{\bf b},ev}$ or $\mathcal{O}_{{\bf b},odd}$, respectively) is the category of $\h_{m|n}$-semisimple $\mathfrak{osp}(2m|2n)$-modules $M$ such that
- $M=\bigoplus_{\mu\in X(m|n)}M_\mu$ and $\dim M_\mu<\infty$; (for $X(m|n) = X_{ev}(m|n)$ or $X(m|n) =X_{odd}(m|n)$, respectively)
- there exist finitely many weights ${}^1\la,{}^2\la,\ldots,{}^k\la\in X(m|n)$ (depending on $M$) such that if $\mu$ is a weight in $M$, then $\mu\in{{}^i\la}-\sum_{\alpha\in{\Pi_{\bf b}}}\N \alpha$, for some $i$.
The morphisms in $\mathcal{O}_{\bf b}$ are all (not necessarily even) homomorphisms of $\mathfrak{osp}(2m|2n)$-modules.
If the $0^m1^n$-sequence ${\bf b}$ starts with $0$, then we are interested in both $ \mathcal{O}_{{\bf b},ev}$ and $\mathcal{O}_{{\bf b},odd}$. If the $0^m1^n$-sequence ${\bf b}$ starts with $1$, then we are interested in only $ \mathcal{O}_{{\bf b},ev}$.
Similar to [@CLW15 Proposition 6.4], all these categories $\mc O_{\bf b}$ are identical for various $\bf b$, since the even subalgebras of the Borel subalgebras $\mathfrak{n}_{\bf b}^+ \oplus \mathfrak{h}_{m|n}$ are identical and the odd parts of these Borels always act locally nilpotently.
Denote by $M_{\bf b}(\lambda)$ the [**b**]{}-Verma modules with highest weight $\lambda$. Denote by $L_{\bf b}(\lambda)$ the unique simple quotient of $M_{\bf b}(\lambda)$. They are both in $\mathcal{O}_{\bf b}$.
It is well known that the Lie superalgebra $\gl(2m|2n)$ has an automorphism $\tau$ given by the formula: $$\tau(E_{ij}):=-(-1)^{p(i)(p(i)+p(j))}E_{ji}.$$ The restriction of $\tau$ on $\osp(2m|2n)$ gives an automorphism of $\osp(2m|2n)$. For an object $M = \oplus_{\mu \in X(m|n)}M_\mu \in \mathcal{O}_{\bf b}$, we let $$M^{\vee}:=\oplus_{\mu \in X(m|n)}M^*_{\mu}$$ be the restrictd dual of $M$. We define the action of $\osp(2m|2n)$ on $M^{\vee}$ by $(g \cdot f)(x) := -f(\tau(g)\cdot x)$, for $ f \in M^\vee, g\in \osp(2m|2n)$, and $ x\in M$. We denote the resulting module by $M^\tau$.
An object $M \in \mathcal{O}_{\bf b}$ is said to have a ${\bf b}$-Verma flag (respectively, dual ${\bf b}$-Verma flag), if $M$ has a filtration $
0=M_0 \subseteq M_1 \subseteq M_2 \subseteq \dots \subseteq M_t = M,
$ such that $M_i/M_{i-1} \cong M_{\bf b}(\gamma_i), 1 \leq i \leq t$ (respectively, $M_i/M_{i-1} \cong M^\tau_{\bf b}(\gamma_i)$) for some $\gamma_i \in X(m|n)$.
Associated to each $\lambda \in X(m|n)$, a ${\bf b}$-tilting module $T_{\bf b}(\lambda)$ is an indecomposable $\osp(2m|2n)$-module in $\mathcal{O}_{\bf b}$ characterized by the following two conditions: $T_{\bf b}(\la)$ has a ${\bf b}$-Verma flag with $M_{\bf b}(\la)$ at the bottom; $\text{Ext}^1_{\CatO_{\bf b}}(M_{\bf b}(\mu),T_{\bf b}(\la))=0$, for all $\mu\in X(m|n)$.
Fock spaces and their completions
---------------------------------
Let $\VV := \sum_{a \in I} \Qq v_a$ be the natural representation of $\U$, where the action of $\U$ on $\VV$ is defined as follows (for $i \in \I$, $a \in I$): $$E_{\alpha_i} v_a = \delta_{i+\hf, a} v_{a-1}, \quad F_{\alpha_i}v_a
= \delta_{i-\hf, a}v_{a+1}, \quad K_{\alpha_i} v_a = q^{(\alpha_i, \varepsilon_a)}v_a.$$ Let $\WW :=\VV^*$ be the restricted dual module of $\VV$ with basis $\{w_a \mid a \in I\}$ such that $\langle w_a, v_b \rangle = (-q)^{-a} \delta_{a,b}$. The action of $\U$ on $\WW$ is given by the following formulas (for $i \in \I$, $a \in I$): $$E_{\alpha_i} w_a = \delta_{i-\hf, a} w_{a+1}, \quad F_{\alpha_i}w_a
= \delta_{i+\hf, a}w_{a-1}, \quad K_{\alpha_i} w_a = q^{-(\alpha_i, \varepsilon_a)}w_a.$$ By restriction through the embedding $\iota$, $\VV$ and $\WW$ are naturally $\bun$-modules.
Fix a ${0^m1^n}$-sequence ${\bf b} =(b_1,b_2,\ldots,b_{m+n})$. We have the following tensor space over $\Q(q)$, called the [*$\bf b$-Fock space*]{} or simply [*Fock space*]{}: $$\label{eq:Fock}
{\mathbb T}^{\bf b} :={\mathbb V}^{b_1}\otimes {\mathbb
V}^{b_2}\otimes\cdots \otimes{\mathbb V}^{b_{m+n}},$$ where we denote $${\mathbb V}^{b_i}:=\begin{cases}
{\mathbb V}, &\text{ if }b_i={0},\\
{\mathbb W}, &\text{ if }b_i={1}.
\end{cases}$$ The tensors here and in similar settings later on are understood to be over the field $\Q(q)$. Note that both algebras $\U$ and $\bun$ act on $\mathbb T^{\bf b}$ via an iterated coproduct.
For $f\in I^{m+n}$, we define $$\label{eq:Mf}
M^{\bf b}_f :=\texttt{v}^{b_1}_{f(1)}\otimes
\texttt{v}^{b_2}_{f(2)}\otimes\cdots\otimes
\texttt{v}^{b_{m+n}}_{f(m+n)},$$ where we use the notation $\texttt{v}^{b_i}:=
\begin{cases}v,\text{ if }b_i={0},
\\
w,\text{ if }b_i={1}.
\end{cases}$ We refer to $\{M^{\bf b}_f \mid f\in
I^{m+n}\}$ as the [*standard monomial basis*]{} of ${\mathbb
T}^{\bf b}$.
Let ${\bf b} =(b_1, \cdots, b_{m+n})$ be an arbitrary $0^m1^n$-sequence. We first define a partial ordering on $I^{m+n}$, which depends on the sequence ${\bf b}$. There is a natural bijection $I^{m+n} \leftrightarrow X(m|n)$ (recall $X(m|n)$ from ), defined as $$\begin{aligned}
&f \mapsto \lambda^{\bf b}_f, \text{ where } \lambda^{\bf b}_f
= \sum_{i=1}^{m+n}(-1)^{b_i}f(i)\ep^{b_i}_i -\rho_{\bf b}, \quad \text{for } f \in I^{m+n},
\label{osp:eq:ftolambda}
\\
& \lambda \mapsto f^{\bf b}_{\lambda},
\text{ where } f(i)=(\lambda + \rho_{\bf b} \vert \ep^{b_i}_i), \quad \quad \quad \text{for } \lambda \in X(m|n).
\label{osp:eq:lambdatof}\end{aligned}$$
Fix a ${\bf b} = (b_1, \dots, b_{m+n})$. For any $f\in I^{m+n}$, let $\varepsilon_f = \sum^{m+n}_{i=1} (-1)^{b_i}\varepsilon_{f(i)} \in \Lambda$. Let $\overline{\varepsilon_f}$ be the image of $\varepsilon_f$ in the quotient $\Lambda_{\inv}$. Define the (${\bf b}$-)Bruhar ordering on the set $I^{m+n}$ (hence on $X(m|n)$) as follows: for $f, g \in I^{m+n}$, we say $g \preceq_{\bf b} f$ if $\overline{\varepsilon_{f}} = \overline{\varepsilon_{g}}$ and $$\lambda^{\bf b}_f - \lambda^{\bf b}_g = a_0 (-\epsilon^{b_1}_1) + \sum^{m+n}_{i=1}a_i (\epsilon^{b_i}_i - \epsilon^{b_{i+1}}_{i+1}), \quad \text{ for } a_i \in \N.$$
This is exactly the same partial ordering used in [@BW13 §8.4].
Let the $B$-completion $\widehat{\mathbb{T}}^{\bf b}$ be the space spanned by elements of the form (possibly infinitely many non-zero $c_{gf}^{\bf b}$) $$\begin{aligned}
M_f+\sum_{g\prec_{\bf b}f}c_{gf}^{\bf b}(q) M_g, \quad \text{ for } c_{gf}^{\bf b}(q) \in\Q(q).\end{aligned}$$
We know from [@BW13 Lemma 9.8] that ${\psi_{\imath}}= \Upsilon \circ \psi : \widehat{\mathbb{T}}^{\bf b} \rightarrow \widehat{\mathbb{T}}^{\bf b}$ is an anti-linear involution such that $${\psi_{\imath}}(M_f) = M_f + \sum_{g \prec_{\bf b} f} r_{gf}(q)M_g, \quad \text{ for } r_{gf}(q) \in \mA.$$
\[thm:iCBb\] The $\Qq$-vector space $\widehat{\mathbb{T}}^{\bf b}$ has unique $\Bbar$-invariant topological bases $$\{T^{\bf b}_f \mid f \in I^{m+n}\} \text{ and } \{L^{\bf b}_f \mid f \in I^{m+n}\}$$ such that $$T^{\bf b}_f = M_f + \sum_{g \preceq_{\bf b} f }t^{\bf b}_{gf}(q)M^{\bf b}_g,
\quad
L^{\bf b}_f = M_f + \sum_{g \preceq_{\bf b} f }\ell^{\bf b}_{gf}(q)M^{\bf b}_g,$$ with $t^{\bf b}_{gf}(q) \in q\Z[q]$, and $\ell^{\bf b}_{gf}(q) \in q^{-1}\Z[q^{-1}]$, for $g \preceq_{\bf b}f $. (We shall write $t^{\bf b}_{ff}(q) = \ell^{\bf b}_{ff}(q) = 1$, $t^{\bf b}_{gf}(q)=\ell^{\bf b}_{gf}(q)=0$ for $ g \not\preceq_{\bf b} f$.)
$\{T^{\bf b}_f \mid f \in I^{m+n}\} \text{ and } \{L^{\bf b}_f \mid f \in I^{m+n}\}$ are call the [*$\imath$-canonical basis*]{} and [*dual $\imath$-canonical basis*]{} of $\widehat{\mathbb{T}}^{\bf b}$, respectively. The polynomials $t^{\bf b}_{gf}(q)$ and $\ell^{\bf b}_{gf}(q)$ are called [*$\imath$-Kazhdan-Lusztig (or $\imath$-KL) polynomials*]{}.
The different ${\psi_{\imath}}$, which coming from different $\Upsilon$, leads to different (dual) $\imath$-canonical basis on the tensor space $\mathbb{T}^{\bf b}$ than the ones in [@BW13 Definition 9.10].
The following theorem is a counterpart of [@BW13 Theorem 9.11].
1. (Positivity)We have $t^{\bf b}_{gf} \in \N[q]$.
2. The sum $T^{\bf b}_f = M_f + \sum_{g \preceq_{\bf b} f }t^{\bf b}_{gf}(q)M^{\bf b}_g$ is finite for all $f \in I^{m+n}$.
Translation functors
--------------------
In [@Br03], Brundan established a $\U$-module isomorphism between the Grothendieck group of the category $\mc O$ of $\gl(m|n)$ and a Fock space (at $q=1$), where some properly defined translation functors acting as Chevalley generators of $\U$ at $q=1$. In [@BW13], the analogue in the setting of $\osp(2m+1|2n)$ has been developed. Here we generalize the construction to the setting of the Lie superalgebra $\osp(2m|2n)$.
Let $V$ be the natural $\mathfrak{osp}(2m|2n)$-module. Notice that $V$ is self-dual. Recalling §\[subsec:osp\], we have the following decomposition of $\mathcal{O}_{\bf b}$ (for fixed ${\bf b}$): $$\mathcal{O}_{\bf b} = \displaystyle\bigoplus _{\chi_\lambda} \mathcal{O}_{{\bf b}, \chi_{\lambda}},$$ where $\chi_\lambda$ runs over all integral or half-integral central characters, i.e. $\lambda$ runs over the equivalent classes $X(m|n) / \sim$ (recall this means $X_{ev}(m|n) / \sim$ or $X_{odd}(m|n) / \sim$, respectively).
We write $\mathcal{O}_{{\bf b}, \gamma} := \oplus_{\chi_\lambda} \mathcal{O}_{{\bf b}, \chi_{\lambda}}$ for all $\lambda$ such that $\overline{\varepsilon_{f^{\bf b}_{\lambda}} }= \gamma \in \Lambda_{\inv}$. For $r \geq 0$, let $S^r V$ be the $r$th supersymmetric power of $V$. For $i \in \I^{\imath}$, $M \in \mathcal{O}_{{\bf b}, \gamma}$, we define the following translation functors in $\mathcal{O}_{\bf b}$: $$\begin{aligned}
\bff^{(r)}_{\alpha_{i}} M &:= \text{pr}_{\gamma - r(\varepsilon_{i-\hf}-\varepsilon_{i+\hf})}(M \otimes S^rV),
\\
\be^{(r)}_{\alpha_{i}} M &:= \text{pr}_{\gamma + r(\varepsilon_{i-\hf}-\varepsilon_{i+\hf})}(M \otimes S^rV),
\label{eq:eft}
\\
\bt M &:=\text{pr}_{\gamma}(M \otimes V), \qquad \text{ (for the case ${\U^{\imath}}_{odd}$)},\end{aligned}$$ where $\text{pr}_{\mu}$ is the natural projection from $\mathcal{O}_{\bf b}$ to $\mathcal{O}_{{\bf b}, \mu}$ for $\mu \in \Lambda_{\inv}$.
Note that the (exact) translation functors naturally induce operators on the Grothendieck group $[\CatO^{\Delta}_{\bf b}]$, denoted by $\bff^{(r)}_{\alpha_{i}}$, $\be^{(r)}_{\alpha_{i}}$, and $\bt$ as well. The following two lemmas are analoges of [@Br03 Lemmas 4.23 and 4.24]. Since they are standard, we shall skip the proofs.
On the category $\mathcal{O}_{\bf b}$, the translation functors $\bff^{(r)}_{\alpha_{i}}$, $\be^{(r)}_{\alpha_{i}}$, and $\bt$ are all exact. They commute with the $\tau$-duality.
\[osp:lem:MOSV\] Let $\nu_1$, $\dots$, $\nu_N$ be the set of weights of $S^rV$ ordered so that $v_i > v_j$ if and only if $ i <j$. Let $\lambda \in X(m|n)$. Then $M_{\bf b}(\lambda) \otimes S^rV$ has a multiplicity-free Verma flag with subquotients isomorphic to $M_{\bf b}(\lambda + \nu_1)$, $\dots$, $M_{\bf b}(\lambda + \nu_N)$ in the order from bottom to top.
Let $\Tb_\mA$ be the $\mA$-lattice spanned by the standard monomial basis of the $\Qq$-vector space $\mathbb{T}^{\bf b}$. We define $\mathbb{T}^{\bf b}_{\Z} =\Z \otimes_\mA \Tb_{\mA}$ where $\mA$ acts on $\Z$ with $q=1$. For any $u$ in the $\mA$-lattice $\Tb_\mA$, we denote by $u(1)$ its image in $\mathbb{T}^{\bf b}_{\Z}$.
Let $\mathcal{O}^{\Delta}_{\bf b}$ be the full subcategory of $\mathcal{O}_{\bf b}$ consisting of all modules possessing a finite ${\bf b}$-Verma flag. Let $\left[\mathcal{O}^{\Delta}_{\bf b}\right]$ be its Grothendieck group. The following lemma is immediate from the bijection $I \leftrightarrow {}X(m|n)$ (with consistent choice of the subscript, i.e., both $ev$ or $odd$).
\[int:lem:OtoT\] The map $$\Psi : \left[\mathcal{O}^{\Delta}_{\bf b}\right] \longrightarrow \mathbb{T}_{\Z}^{\bf b},
\quad \quad \quad [M_{\bf b}(\lambda)] \mapsto M^{\bf b}_{f^{\bf b}_{\lambda}}(1),$$ defines an isomorphism of $\Z$-modules.
Denote by ${_\Z \U} = \Z \otimes_{\mA}\, {_{\mA}\U}$ the specialization of the $\mA$-algebra ${_{\mA}\U}$ at $q=1$. Hence we can view $\mathbb{T}_{\Z}^{\bf b}$ as a ${_\Z \U} $-module. Thanks to , , and , we know $\iota(\bff^{(r)}_{\alpha_{i}})$ and $\iota(\be^{(r)}_{\alpha_{i}})$ lie in ${_{\mA}\U}$, hence their specializations at $q=1$ in ${_\Z \U} $ act on $\mathbb{T}_{\Z}^{\bf b}$. The following proposition is a counterpart of [@BW13 Proposition 11.9].
\[prop:translation\] Under the identification $[\CatO^{\Delta}_{\bf b}]$ and $\mathbb{T}_{\Z}^{\bf b}$ via the isomorphism $\Psi$, the translation functors $\bff^{(r)}_{\alpha_{i}}$, $\be^{(r)}_{\alpha_{i}}$, and $\bt$ act in the same way as the specialization of $\bff^{(r)}_{\alpha_{i}}$, $\be^{(r)}_{\alpha_{i}}$, and $\bt$ in $\bun$.
$\imath$-Kazhdan-Lusztig theory for $\osp(2m|2n)$
-------------------------------------------------
We define $\left[\left[ \mathcal{O}^{\Delta}_{\bf b} \right]\right]$ as the completion of $\left[\mathcal{O}^{\Delta}_{\bf b}\right]$ such that the extension of $\Psi$ $$\Psi: \left[\left[ \mathcal{O}^{\Delta}_{\bf b} \right]\right] \longrightarrow \widehat{\mathbb{T}}_\Z^{\bf b}$$ is an isomorphism of $\Z$-modules.
1. For any $0^m1^n$-sequence ${\bf b}$ starting with $0^2$, the isomorphism $\Psi : \left[ \left[\CatO^{\Delta}_{\bf b}\right]\right] \rightarrow \widehat{\mathbb{T}}_{\Z}^{\bf b}$ satisfies $$\Psi([L_{{\bf b}}(\lambda)]) = L^{{\bf b}}_{f^{{\bf b}}_{\lambda}}(1), \quad \quad \quad \Psi([T_{{\bf b}}(\lambda)])
= T^{{\bf b}}_{f^{{\bf b}}_{\lambda}}(1), \quad \quad \text{ for } \lambda \in {}X(m|n).$$
2. For any $0^m1^n$-sequence ${\bf b}$ starting with $1$, the isomorphism $\Psi : \left[ \left[\CatO^{\Delta}_{{\bf b},ev}\right]\right] \rightarrow \widehat{\mathbb{T}}_{{\Z},ev}^{\bf b}$ satisfies $$\Psi([L_{{\bf b}}(\lambda)]) = L^{{\bf b}}_{f^{{\bf b}}_{\lambda}}(1), \quad \quad \quad \Psi([T_{{\bf b}}(\lambda)])
= T^{{\bf b}}_{f^{{\bf b}}_{\lambda}}(1), \quad \quad \text{ for } \lambda \in {}X_{ev}(m|n).$$
This proof is essentially the same induction as the one in [@BW13 Theorem 11.13] (or its predecessor [@CLW15]). The setting on the Fock spaces is exactly the same as [@BW13], since the only difference is the precise formula of the bar involution ${\psi_{\imath}}$. (Recall we are using the same partial ordering as in [@BW13].) Hence here we will be contented with specifying how each step follows and refer the reader to the proof of [@BW13 Theorem 11.13] (and the references therein) for details.
The inductive procedure case (1), denoted by $ \imath\texttt{KL}(m|n) \, \forall m \ge 2 \Longrightarrow \imath\texttt{KL}(m|n+1)$, is divided into the following steps: [$$\begin{aligned}
\imath\texttt{KL}(m+k|n) \;\; \forall k
& \Longrightarrow \imath\texttt{KL}(m|n|k) \;\; \forall k, \text{ by changing Borels}
\label{ind:oddref} \\
&\Longrightarrow \imath\texttt{KL}(m|n|\underline{k})
\;\; \forall k, \text{ by passing to parabolic}
\label{ind:para} \\
&\Longrightarrow \imath\texttt{KL}(m|n|\underline{\infty}), \text{ by taking $k\mapsto \infty$}
\label{ind:infty} \\
& \Longrightarrow \imath\texttt{KL}(m|n+\underline{\infty}), \text{ by super duality}
\label{ind:SD} \\
& \Longrightarrow \imath\texttt{KL}(m|n+1)\;\; \forall m, \text{ by truncation}.
\label{ind:trunc}\end{aligned}$$ ]{} It is instructive to write down the Fock spaces corresponding to the steps above: [$$\begin{aligned}
\VV^{\otimes (m+k)} \otimes \WW^{\otimes n}\;\; \forall k
&\Longrightarrow \VV^{\otimes m} \otimes \WW^{\otimes n} \otimes
\VV^{\otimes k}\;\; \forall k
\\
&\Longrightarrow \VV^{\otimes m} \otimes \WW^{\otimes n}\otimes
\wedge^k\VV\;\; \forall k
\\
&\Longrightarrow \VV^{\otimes m} \otimes \WW^{\otimes n}\otimes
\wedge^\infty \VV
\\
& \Longrightarrow \VV^{\otimes m} \otimes \WW^{\otimes n}\otimes
\wedge^\infty\WW
\\
& \Longrightarrow
\VV^{\otimes m} \otimes \WW^{\otimes (n+1)}\;\;\; \forall m \ge 2.\end{aligned}$$ ]{} Thanks to Theorem \[thm:samebar\] and Corollary \[cor:samebar\], the base case for the induction, $ \imath\texttt{KL}(m|0)$, is equivalent to the original Kazhdan-Lusztig conjecture [@KL] for $\mf{so}(2m)$. Step follows from [@BW13 Proposition 11.14].
Step follows from [@BW13 §11.2].
Step follows from [@BW13 Proposition 11.4].
Step is based on [@BW13 Proposition 11.12].
Step is based on [@BW13 Propositions 7.7, 11.4 and 9.17].
The inductive procedure for case (2), denoted by $ \imath\texttt{KL}(n|m) \, \forall n \ge 1 \Longrightarrow \imath\texttt{KL}(n|m+1)$, is divided into the following steps: [$$\begin{aligned}
\imath\texttt{KL}(n+k|m) \;\; \forall k
& \Longrightarrow \imath\texttt{KL}(n|m|k) \;\; \forall k, \text{ by changing Borels}
\label{C:oddref} \\
&\Longrightarrow \imath\texttt{KL}(n|m|\underline{k})
\;\; \forall k, \text{ by passing to parabolic}
\label{C:para} \\
&\Longrightarrow \imath\texttt{KL}(n|m|\underline{\infty}), \text{ by taking $k\mapsto \infty$}
\label{C:infty} \\
& \Longrightarrow \imath\texttt{KL}(n|m+\underline{\infty}), \text{ by super duality}
\label{C:SD} \\
& \Longrightarrow \imath\texttt{KL}(n|m+1)\;\; \forall n, \text{ by truncation}.
\label{C:trunc}\end{aligned}$$ ]{} The Fock spaces corresponding to the steps above are the following: [$$\begin{aligned}
\WW^{\otimes (n+k)} \otimes \VV^{\otimes m}\;\; \forall k
&\Longrightarrow \WW^{\otimes n} \otimes \VV^{\otimes m} \otimes
\WW^{\otimes k}\;\; \forall k
\\
&\Longrightarrow \WW^{\otimes n} \otimes \VV^{\otimes m}\otimes
\wedge^k\WW\;\; \forall k
\\
&\Longrightarrow \WW^{\otimes n} \otimes \VV^{\otimes m}\otimes
\wedge^\infty \WW
\\
& \Longrightarrow \WW^{\otimes n} \otimes \VV^{\otimes m}\otimes
\wedge^\infty\VV
\\
& \Longrightarrow
\WW^{\otimes n} \otimes \VV^{\otimes (m+1)}\;\;\; \forall n \ge 1.\end{aligned}$$ ]{} Thanks to Theorem \[thm:KLC\], the base case for the induction, $ \imath\texttt{KL}(n|0)$, is equivalent to the original Kazhdan-Lusztig conjecture [@KL] for $\mf{sp}(2n)$. The rest of the proof is similar to the previous case.
The theorem is proved.
There is a similar Fock space formulation for various parabolic subcategories of $\osp(2m|2n)$-modules.
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Y. Li and W. Wang, [*Positivity vs negativity of canonical bases*]{}, arXiv:1501.00688v3.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We report an angle-resolved photoemission spectroscopy (ARPES) study on IrTe$_2$ which exhibits an interesting lattice distortion below 270 K and becomes triangular lattice superconductors by suppressing the distortion via chemical substitution or intercalation. ARPES results at 300 K show multi-band Fermi surfaces with six-fold symmetry which are basically consistent with band structure calculations. At 20 K in the distorted phase, topology of the inner Fermi surfaces is strongly modified by the lattice distortion. The Fermi surface reconstruction by the distortion depends on the orbital character of the Fermi surfaces, suggesting importance of Ir 5$d$ and/or Te 5$p$ orbital symmetry breaking.'
author:
- 'D. Ootsuki$^1$'
- 'S. Pyon$^2$'
- 'K. Kudo$^2$'
- 'M. Nohara$^2$'
- 'M. Horio$^3$'
- 'T. Yoshida$^3$'
- 'A. Fujimori$^3$'
- 'M. Arita$^4$'
- 'H. Anzai$^4$'
- 'H. Namatame$^4$'
- 'M. Taniguchi$^{4,5}$'
- 'N. L. Saini$^{6,1}$'
- 'T. Mizokawa$^{1}$'
title: 'Electronic structure reconstruction by orbital symmetry breaking in IrTe$_2$'
---
Transition-metal compounds with multi-band Fermi surfaces often exhibit rich and interesting physical properties such as spin-charge-orbital order and superconductivity which originate from the topology of their multi-band Fermi surfaces. For example, the multi-orbital electronic structures of transition-metal oxides and chalcogenides including CuIr$_2$S$_4$ and Ca$_{2-x}$Sr$_x$RuO$_4$ provide various metal-insulator transitions with spin-charge-orbital ordering [@Imada1998; @Nagata1994; @Radaelli2002; @Nakatsuji2000]. Also the multi-band structure of the Fe $3d$ orbitals play important roles in superconductivity and magnetism of Fe pnictides and chalcogenides such as LaFeAsO$_{1-x}$F$_x$ [@Kamihara2008]. Recently, Pyon [*et al.*]{} [@Pyon2012] and Yang [*et al.*]{} [@Yang2012] have discovered interesting interplay between lattice distortion and superconductivity in triangular lattice IrTe$_2$ in which multi-band Fermi surfaces are expected to play significant roles. Since the large spin-orbit interaction of Ir 5$d$ electrons is expected to entangle the spin and orbital degrees of freedom in IrTe$_2$ and the derived superconductors, Yang [*et al.*]{} pointed out that the IrTe$_2$ system provides a new playground to explore and/or realize topological quantum states, which are currently attracting great interest in physics community [@Yang2012].
IrTe$_2$ exhibits a structural phase transition at $\sim$ 270 K from the trigonal (P3m-1) to the monoclinic (C2/m) structure accompanied by anomalies of electrical resistivity and magnetic susceptibility [@Matsumoto1999]. When the lattice distortion is suppressed by chemical substitution of Pt or Pd for Ir or intercalation of Pd, IrTe$_2$ becomes superconductors [@Pyon2012; @Yang2012]. An electron diffraction study by Yang [*et al.*]{} [@Yang2012] observed the superlattice peaks with wave vector of $q$ = (1/5, 0, -1/5) below the structural transition temperature. Such superstructure can be explained by charge density wave (CDW) driven by perfect or partial nesting of multi-band Fermi surfaces. In multi-band Fermi surfaces derived from Ir 5$d$ and Te 5$p$ orbitals, the nesting character can be enhanced by orbitally-induced Peierls mechanism [@Khomskii2005]. In addition, charge modulation of Ir 5$d$ electrons is indicated by an Ir 4$f$ x-ray photoemission study [@Ootsuki2012]. On the other hand, a recent optical study by Fang [*et al.*]{} on single crystal samples shows that there is no gap opening expected for CDW and, instead, band structure is reconstructed over a broad energy scale up to $\sim$ 2 eV [@Fang2012]. Fang [*et al.*]{} conclude that the structural transition of IrTe$_2$ is not of CDW type but of a novel type driven by Te 5$p$ holes [@Fang2012].
In this context, it is very interesting and important to study the geometry of multi-band Fermi surfaces of IrTe$_2$ using angle-resolved photoemission spectroscopy (ARPES). In the present ARPES study, above the transition temperature, the flower-shaped outer Fermi surface and the inner Fermi surfaces like six connected beads, which are predicted by band structure calculations, are partly identified. Across the structural transition, the topology of the inner Fermi surfaces is modified more strongly than that of the outer Fermi surface. Below the transition temperature, the inner Fermi surfaces consist of two straight portions, suggesting Fermi surface nesting. However, clear gap opening expected for CDW is not observed in the ARPES spectra, consistent with the optical study [@Fang2012]. Instead, spectral weight is partially suppressed at specific points of the straight Fermi surfaces.
Single crystal samples of IrTe$_2$ were prepared using a self-flux method [@Fang2012; @Pyon2012b]. The ARPES measurements were carried out at beamline 9A, Hiroshima Synchrotron Radiation Center using a SCIENTA R4000 analyzer with circularly polarized light of photon energy $h\nu$ = 23 eV. The data were collected at 300 K and 20 K with an angular resolutions of $\sim$ 0.3$^{\circ}$ and energy resolution of 18 meV for excitation energy of $h\nu$ = 23 eV. The incident beam is 50$^{\circ}$ off the sample surface. The base pressure of the spectrometer was in the $10^{-9}$ Pa range. The samples were cleaved at $300$ K under the ultrahigh vacuum and cooled across the structural transition, and then warmed to 300 K to check the reproducibility at 300 K. The samples were oriented by [*ex situ*]{} Laue measurements. The spectra were acquired within 8 hours after the cleavage. Binding energies were calibrated using the Fermi edge of gold reference samples.
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The Fermi surface mapping of IrTe$_2$ measured at 300 K above the structural transition temperature are displayed in Figure 1(a). At $h\nu$ = 23 eV, the momentum perpendicular to the Ir plane approximately corresponds to $\pi/c$, where $c$ is the out-of-plane lattice constant, and the center of the hexagonal Brillouin zone is the A point. The direction from the A point to the L (H) point corresponds to the direction of Ir-Ir (Ir-Te) bond. In Figs. 1(a), several Fermi surfaces can be identified as predicted by the band structure calculations [@Yang2012; @Fang2012] although the strong intensity asymmetry due to transition-matrix element effect does not allow perfect identification. In order to extract the shapes of the Fermi surfaces, the second derivative along the cut direction $d^2I(k_x,k_y)/d^2k_x$ is plotted in Fig. 1(b). The flower shape for the outer Fermi surface is more clearly seen which is schematically indicated by the dashed curve. In addition, the inner Fermi surfaces like six connected beads can be identified as indicated by the dotted curves although effect of thermal excitations at 300 K tends to obscure the relatively small Fermi pockets. The inner Fermi surfaces observed around the A point at 300 K are roughly consistent with the prediction of band-structure calculations [@Yang2012; @Fang2012].
Figure 1(c) shows the Fermi surface mapping at 20 K well below the transition temperature. Across the structural transition, in the region where the outer Fermi surface is close to the inner Fermi surfaces, while the outer Fermi surface at 300 K is observed separately from the inner Fermi surfaces, the outer Fermi surface at 20 K disappears due to partial gap opening or overlaps with the inner Fermi surfaces. In addition, in going from 300 K to 20 K, intensity evolves in the six leaves of the outer Fermi surface flower probably due to band folding with $q$ = (1/5, 0, -1/5). In contrast to the limited effect on the outer Fermi surface, the inner Fermi surfaces dramatically change their shapes by the structural transition. The band structure calculations show that the outer Fermi surface is mainly constructed from the Ir 5$d$ $a_{1g}$ \[$\frac{1}{\sqrt{3}}(XY + YZ + ZX)$\] and Te 5$p_z$ orbitals, and that the inner Fermi surfaces mainly have the Ir 5$d$ $e_{g}'$ \[$\frac{1}{\sqrt{3}}(XY+e^{\pm2\pi i/3}YZ+e^{\pm4\pi i/3}ZX)$\] and Te 5$p_{x,y}$ orbital components (Here, the X-, Y-, and Z-axes are along the three Ir-Te bonds of a regular IrTe$_6$ octahedron). This assignment is supported by the good agreement between the ARPES result at 300 K and the band structure calculations [@Yang2012; @Fang2012]. As for the band structure change across the transition, the experimental result that the inner Fermi surfaces are more strongly affected by the transition is consistent with the band structure calculation by Fang [*et al*]{} [@Fang2012]. However, the geometry of inner Fermi surfaces at 20 K deviates from the prediction of the calculation. Interestingly, two straight portions of Fermi surfaces are observed at 20 K. The straight portions are perpendicular to the A-H direction or the direction of Ir-Te bond. Therefore, both of the Te 5$p$ and Ir 5$d$ orbitals would be involved in the structural transition if the straight Fermi surfaces are driven by orbitally-induced Peierls mechanism [@Khomskii2005].
Figures 2(a) and (b) show the broad-range band dispersions along the A-H direction at 300 K and 20 K, respectively. Whereas the broad-range band dispersions at 300 K roughly agree with the predictions of the band-structure calculations, those at 20 K deviate from the predictions [@Yang2012; @Fang2012]. In going from 300 K to 20 K, broad-range band structures up to -3 eV are strongly modified, which is consistent with the optical study [@Fang2012]. The broad-range spectral change is more clearly seen in the energy distribution curves for 300 K and 20 K shown in Figs. 2(c) and (d), respectively. In going from 300 K to 20 K, spectral peaks at 300 K tend to be split into several structures probably due to complicated Jahn-Teller-like effect and band folding effect due to the charge and orbital ordering with the (1/5, 0, -1/5) superstructure. Consequently, in the energy range from -0.2 eV to -3 eV, the spectral peaks at 20 K are much broader than those at 300 K.
On the other hand, spectral peaks from $E_F$ to -0.2 eV are rather sharp at 20 K compared to those at 300 K as shown in Fig. 3. For 300 K, the outer hole band is indicated by the dashed curve in Fig. 3(a) which form the flower-shaped outer Fermi surface of Fig. 1(a). The inner hole-like band indicated by the dotted curves in Fig. 3(a) creates the hole pockets which corresponds to the inner Fermi surfaces of Fig. 1(a). By comparing between Figs. 3(a) and (b), while the outer band at 300 K is observed separately from the inner one, the outer band at 20 K disappears near $E_F$. Across the transition, the outer band is shifted towards the inner one in this momentum region and is probably gapped due to the interaction with the inner band. This is consistent with the partial disappearance of the outer Fermi surface in Fig. 1(c). The inner hole-like band at 300 K is also strongly affected by the structural transition. The band located around $\sim$ -0.15 eV of the A point at 300 K disappears at 20 K probably because it is shifted above $E_F$. Consequently, the hole band indicated by the dotted curve in Fig. 3(b) crosses $E_F$ at 20 K and form the straight portions of the Fermi surfaces of Fig. 1(b). Such band reconstruction cannot be explained by a simple band folding picture, indicating orbital reconstruction by Jahn-Teller-like effect.
The Fermi surface mapping around the A point at 20 K is shown in Fig. 4(a). In general, straight Fermi surfaces with nesting wave vector $q$ are expected to be gapped due to density wave formation with $q$. In IrTe$_2$, instead of gap opening, spectral weight at $E_F$ is partially suppressed at specific points of the straight Fermi surfaces. In cuts 1 and 3 along the A-H direction \[Figs. 4(b) and (d)\], the hole band clearly crosses $E_F$ and the spectral weight at $E_F$ is not suppressed. On the other hand, in cut 2 \[Fig. 4(c)\], the intensity of the hole band is suppressed near $E_F$ as seen in the EDC plot of Fig. 4(e). There are four points where the spectral weight at $E_F$ is suppressed as seen in Fig. 4(a). Such spectral weight suppression at the specific points (cold spots) would be related to the origin of the superstructure of bulk IrTe2 since the wave vectors connecting the two cold spots \[indicated by the arrows in Fig. 4(a)\] are approximately 2/5 of the A-L distance or 1/5 of the L-L’ distance, partly consistent with its period. However, the partial spectral weight suppression would be due to surface effect or transition-matrix element effect, and no decisive conclusion can be obtained at the present stage. Here, it should be noted that the observed Fermi surfaces correspond to one of the Brillouin zone boundaries for possible domains. However , the crystal structure of low temperature phase is highly controversial (refs. 7, 11, and 13), and that, at the present stage, it is difficult to discuss relationship between the observed Fermi surfaces and the band folding due to the superstructure.
In conclusion, above the transition temperature, the observed Fermi surfaces and band dispersions are consistent with the band structure calculations. The flower-shaped outer Fermi surface (hole character) with six-fold symmetry and the inner Fermi surfaces (hole pockets) are observed. Across the structural transition, the geometry of the inner Fermi surfaces is strongly modified. In the distorted phase, the inner Fermi surfaces consist of two straight portions, suggesting that nesting character is enhanced. However, the gap opening expected for CDW is not observed in the ARPES spectra, consistent with the optical study. Also the electronic structure up to $\sim$ -3 eV is reconstructed by the lattice distortion, which is also consistent with the optical study. In addition, the spectral weight at $E_F$ is suppressed at the specific points of the straight Fermi surfaces, which would be related to the origin of the superstructure.
The authors would like to thank valuable discussions with D. I. Khomskii and H. Takagi. This work was partially supported by a Grants-in-Aid for Young Scientists (B) (23740274, 24740238) from the Japan Society of the Promotion of Science (JSPS) and the Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program) from JSPS. The synchrotron radiation experiment was performed with the approval of HSRC (Proposal No.12-A-12).
[99]{} M. Imada, A. Fujimori, Y. Tokura: Rev. Mod. Phys. $\bf70$ (1998) 1039. S. Nagata, T. Hagino, Y. Seki, and T. Bitoh: Physica B $\bf194-196$ (1994) 1077. P. G. Radaelli, Y. Horibe, M. J. Gutmann, H. Ishibashi, C. H. Chen, R. M. Ibberson, Y. Koyama, Y. S. Hor, V. Kirykhin, and S. W. Cheong: Nature $\bf416$ (2002) 155. S. Nakatsuji and Y. Maeno: Phys. Rev. Lett. [**84**]{}, 2666 (2000). Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono: J. Am. Chem. Soc. $\bf130$ (2008) 3296. S. Pyon, K. Kudo, and M. Nohara: J. Phys. Soc. Jpn. $\bf81$ (2012) 053701. J. J. Yang, Y. J. Choi, Y. S. Oh, A. Hogan, Y. Horibe, K. Kim, B. I. Min, and S-W. Cheong: Phys. Rev. Lett. $\bf108$ (2012) 116402. N. Matsumoto, K. Taniguchi, R. Endoh, H. Takano, and S. Nagata: J. Low Temp. Phys. $\bf117$ (1999) 1129. D. I. Khomskii and T. Mizokawa: Phys. Rev. Lett. $\bf94$ (2005) 156402. D. Ootsuki, Y. Wakisaka, S. Pyon, K. Kudo, M. Nohara, M. Arita, H. Anzai, H. Namatame, M. Taniguchi, N. L. Saini, and T. Mizokawa: Phys. Rev. B $\bf86$ (2012) 014519. A. F. Fang, G. Xu, T. Dong, P. Zheng, and N. L. Wang: Scientific Reports $\bf3$ (2013) 1153. S. Pyon, K. Kudo, and M. Nohara: Physica C (2013)(in press) \[DOI:10.1016/j.physc.2013.04.055\]. H. B. Cao, B. C. Chakoumakos, J. -Q. Yan, H. D. Zhou, R. Custelcean, and D. Mandrus: arXiv:1302.5369v1
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper we study the quantum dynamics of an electron/hole in a two-dimensional quantum ring within a spherical space. For this geometry, we consider a harmonic confining potential. Suggesting that the quantum ring is affected by the presence of an Aharonov-Bohm flux and an uniform magnetic field, we solve the Schrödinger equation for this problem and obtain exactly the eigenvalues of energy and corresponding eigenfunctions for this nanometric quantum system. Afterwards, we calculate the magnetization and persistent current are calculated, and discuss influence of curvature of space on these values.'
author:
- 'A. L. Silva Netto'
- 'B. Farias, J. Carvalho'
- 'C. Furtado'
title: A Quantum Ring in a Nanosphere
---
Introduction
============
In recent years, the study of confined quasiparticles in nanostructures with annular geometry has attracted great interest in condensed matter physics. These quantum rings exhibit several interesting physical phenomena, such as the Aharonov-Bohm effect [@9; @10], spin-orbit interaction effects [@spin], persistent currents [@12; @13], quantum Hall effect [@11] and the manifestation of Berry geometric quantum phase [@14]. More recent works have demonstrated that assumption of finite width brings intricacy from an experimental point of view, although even more important results have been also found. So, in Ref. [@lorke] it was shown through use of experiments with quantum rings of very small radii containing few electrons, that there are some electron modes representing different radii of electronic orbits in these nanometric systems. Several results show that magnetic field penetration depth in a conducting region plays an important role for physical properties of finite width quantum rings, including multiple channels which were experimentally observed [@12; @13; @inkson; @margulis; @bogachek]. One-dimensional quantum rings pierced by Aharonov-Bohm solenoid were used to observe the quantum interference effect [@spin; @14; @15; @16]. There are several exactly solvable models known for two-dimensional quantum rings, for example, those ones considered in refs. [@12; @inkson; @margulis; @bogachek]. The theoretical approach developed by Tan and Inkson [@inkson; @12] is of a special interest since it presents good agreement with experimental results, for instance, concerning effects of magnetic field penetration in the conducting region.
Landau levels in negative [@dune; @comtet; @comtet1] and positive [@dune; @greiter] curvature cases have been intensively studied in order to explore quantum Hall effect in these spaces [@bulaphysb; @takuya; @jelal; @iengo; @nair; @hasebe]. Quantum Hall effect in a Lobachevsky plane was considered in ref. [@bulaphysb], where the effect of negative curvature was observed for a Hall conductivity of these systems. The study of quantum Hall effect in a spherical space was carried out for different scenarios in Refs. [@hasebe2; @nair2; @nair3; @nair3; @nair4]. Advances in the development of techniques for low dimensional materials motivate many investigations concerning curvature and topology influence on nanostructures, since now it is possible to obtain several kinds of curved two-dimensional surfaces [@16] and objects of nanometric size with desired shapes [@prinz]. The impacts of curvature and topology for magnetic, spectral and transport properties of nanostructured materials have recently been studied by several authors [@17; @18; @19; @20]. The magnetic moment of two-dimensional electron gas on a negative curvature surface was studied in ref. [@bulaemag]. The effect of a negative curvature for a quantum dot with impurity was investigated in ref. [@geyler]. The zero mode in systems for spin-$1/2$ particle in the presence of an Aharonov-Bohm solenoid in Lobachevsky plane was obtained in ref. [@geyler1]. Recently, Bulaev, Geyler and Margulis [@bulacurva] have studied the Tan-Inkson model [@12; @inkson] in hyperbolic spaces and provided theoretical frameworks, comprising potentials with adjustable parameters, capable of describing nanostructures like quantum dots, antidots, rings and wires in this surface of negative curvature. Recently the effect of topology in quantum rings and dots was investigated in the refs. [@22; @23; @lincoan].
In this work we study a nanosystem in a positive curvature case. This case is interesting, among other reasons, because of the characteristics of the growth techniques for nanometric systems, such as quantum rings. Therefore, we have a motivation for probing how curvature influences physical properties of quantum rings. In our work we study a nanometric system grown over a surface with positive curvature, more specifically, a quantum ring in a spherical space in the presence of an Aharonov-Bohm flux and an uniform magnetic field through that space. We obtain the spectrum of energy and the wave function for Schrödinger equation solved exactly for this system. The magnetization in the zero temperature case is obtained, and the influence of curvature on the magnetization is investigated. The persistent current is obtained using the Byers-Yang relation [@byers], and the influence of curvature on it is discussed. An uniform magnetic field in this case is introduced through the curved space in order to observe what happens in the conducting region. We also compare our results in the appropriated limit with results obtained in Ref. [@bulacurva] for a ring on a negative curvature surface.
This paper is organized as follows. In Section \[sect2\] we investigate the quantum dynamics in a two-dimensional spherical space. In Section \[sec3\], we describe the confinement potential for this positive curvature space. In Section \[sec4\] the quantum dynamics of a charged particle confined in a Tan-Inkson potential is investigated and the eigenvalues and eigenvectors of energy are obtained. In Section \[sec5\] the magnetization for $T=0$ is found and the physical properties is discussed. The persistent current is f calculated in Section \[sec6\]. Finally, in Section \[sec7\] we present the concluding remarks.
Quantum dynamics in a two-dimensional spherical space {#sect2}
======================================================
First of all, we write the Hamiltonian for a free particle in a two-dimensional space $S^{2}$ described by a sphere embedded in the Euclidean three-dimensional space, $ x^{2}+ y^{2} + z^{2}=a^{2}$, where $a$ is the radius of sphere. In this case the metric on the sphere, in terms of angular coordinates $(\theta, \varphi)$ and sphere radius $a$, is given by $$ds^{2}=a^{2}d\theta^{2}+a^{2} \sin^{2}\theta d\varphi^{2},
\label{sphMETRIC}$$ where angular coordinates are restricted to the range $0 < \theta < \pi$ and $ 0 < \varphi < 2\pi$. In this study we use a stereographic projection from the points in a sphere with radius $a$ on a plane. It is worth noting that the stereographic projection is a kind of map preserving angles and circles. In this way, first, angles between curves on original space are mapped into equal angles comprised by respective curves on projected plane; second, image of a circle on the original space is also a circle on the projected space. After this process, the points are at the distance $\rho$ from the origin (that is, from the sphere’s center) on the projection plane. Here the zenith angle is denoted by $\theta$ (which corresponds to the $\psi$ angle in the Figure ($\ref{fig:stereo}$).
![(a) Stereographic projection of a sphere on a plane. (b) Trigonometric relation useful for obtaining the metric for the projected space.[]{data-label="fig:stereo"}](stereoproject-1e2_reduzido.pdf){width="\linewidth"}
In this way, we obtain the following relations: $$\tan\frac{\theta}{2} = \frac{\rho}{2a} \quad ,
\label{relation1}$$ and $$d\theta^{2}=\frac{1}{a^{2}\left[1+\left(\frac{\rho}{2a}\right)^{2}\right]^{2}}d\rho^{2} \quad,
\label{relation2}$$ and after some algebra we find the metric describing our stereographically projected system: $$ds^2=\frac{d\rho^2 +\rho^2 d\varphi^2}{\left[1+\left(\frac{\rho}{2a}\right)^2\right]^2} \ ,
\label{sphere-metric}$$ where $0<\rho<\infty $ and $0< \varphi< 2\pi$. We will consider an uniform magnetic field $B$ on the spherical surface. So, for the projected representation, the equivalent magnetic field will be along the $z$-direction, perpendicular to the projection plane. The vector potential for this field configuration for the geometry described by (\[sphere-metric\]) is given by $$\vec{A_{1}}=\left(0,\frac{B\rho}{2\left[1+\left(\frac{\rho}{2a}\right)^2\right]^2}\right)\label{uniformmag}.$$ Now, we introduce a Aharonov-Bohm magnetic flux ($\Phi_{AB}$) [@Landau:hydrogen3; @sakurai; @bogachek; @Furtado:density; @Dunne] in $z$-direction on the sphere. Therefore, the only non-vanishing component of the magnetic vector potential is the azimuthal one, and the corresponding vector potential is $\vec{A_{2}}$ given by $$\vec{A_{2}}=\left(0,\frac{\Phi_{AB}}{2\pi\rho}\right).\label{abflux}$$ The Hamiltonian in the curved space characterized by the metric $g_{ij}$ in the presence of external magnetic fields is given by $$H_{0}=\frac{1}{2\mu \sqrt{g}} \left(-i\hbar \frac{\partial}{\partial
x^i}-\frac{e}{c}A_{i}\right)\sqrt{g}g^{ij} \left(-i\hbar \frac{\partial}{\partial
x^j}-\frac{e}{c}A_{j}\right). \label{curvedham}$$ Hence, the Hamiltonian (\[curvedham\]) in the space with the metric (\[sphere-metric\]) looks like $$\begin{split}
\hat{H_{0}}\,=\,-\frac{\hbar^{2}}{2\mu a^{2}}\left\{a^{2}\left[1+\left(\frac{\rho}{2a}\right)^{2}\right]^{2}\left[\frac{1}{\rho}\frac{d}{d\rho}\left(\rho\frac{d}{d\rho}\right)+\frac{1}{\rho^{2}}\left(\frac{\partial}{\partial\varphi}+i\frac{\Phi_{AB}}{\Phi_{0}}\right)\right]\right\}\,
\\-\,i\frac{\hbar\omega_{c}}{2}\left[1+\left(\frac{\rho}{2a}\right)^{2}\right]\left(\frac{\partial}{\partial\varphi}+i\frac{\Phi_{AB}}{\Phi_{0}}\right)+\frac{\mu\omega_{c}^{2}\left(-a^{2}\right)}{2}\left(\frac{\rho}{2\left(ia\right)}\right)^{2}\,+\,\frac{\hbar^{2}}{8\mu a^{2}}\,.
\end{split}$$ This operator describes a quantum particle in a two-dimensional spherical space submitted to an uniform magnetic field, in the presence of Aharonov-Bohm solenoid in $z$-direction.
The Tan-Inkson Confinement Potential in a Two-dimensional Spherical Space {#sec3}
=========================================================================
Let us introduce a confinement potential in a two-dimensional spherical space $S^{2}$. We generalize the Tan-Inkson [@inkson] potential for this geometry. This potential is a harmonic confining potential describing different kinds of nanostructures in a spherical space, after a simple change of parameters, is given by $$V(\rho)=\lambda_{1}\rho^2 + \frac{\lambda_{2}}{\rho^2}\left[1+\left(\frac{\rho}{2a}\right)^2\right]^2 - V_{0},\label{confpoten}$$ where $\lambda_{1}$ and $\lambda_{2}$ are the parameters of the potential and $V_{0}$ looks like $$V_{0}=\frac{\lambda_{2}}{2a^2}+2\sqrt{\lambda_{2}\left(\lambda_{1}+\frac{\lambda_{2}}{\left(2a\right)^4}\right)}.
\label{vo}$$ The potential (\[confpoten\]) has a minimum in $\rho_{0}$ equal to $$\rho_{0}=\left(\frac{\lambda_{2}}{\lambda_{1}+\frac{\lambda_{2}}{(2a)^4}}\right)^{1/4}.
\label{minimum}$$ It is worth noting that, in the limit of $\lambda_{2} \to 0$ for the potential presented by (\[confpoten\]), in stereographic coordinates, one recovers the harmonic potential for a quantum dot in a flat space. Besides, if we consider this system in the spherical coordinates characterizing the metric (\[sphere-metric\]), the quantum dot takes the form $V(\theta)=4\lambda_{1}a^{2}\tan^{2}(\theta)$. For the case $\lambda_{2} \to 0$, we arrive at the antidot potential in spherical space. In the limit $a \to \infty$, we obtain the flat Tan-Inkson potential in the form $$V(\rho)=\lambda_{1}\rho^2 + \frac{\lambda_{2}}{\rho^2} - V_{0},\label{confpotenflat}$$ where $V_{0}$ is given by $$V_{0}=2\sqrt{\lambda_{2}\lambda_{1}}.
\label{vopla}$$
![Quantum ring on sphere[]{data-label="fig:dir"}](stereoproject-3_reduzido.pdf){width="40.00000%"}
The quantum Dynamics in a Quantum Ring in Spherical Space {#sec4}
=========================================================
Now we solve the Schrödinger equation for an electron/hole confined by the potential (\[confpoten\]), in the presence of magnetic fields (\[uniformmag\]) and (\[abflux\]). In this case the Hamiltonian of an electron is given by $$\label{totalHamilt}
\begin{split}
\hat{H}\,=\,-\frac{\hbar^{2}}{2\mu a^{2}}\left\{a^{2}\left[1+\left(\frac{\rho}{2a}\right)^{2}\right]^{2}\left[\frac{1}{\rho}\frac{d}{d\rho}\left(\rho\frac{d}{d\rho}\right)+\frac{1}{\rho^{2}}\left(\frac{\partial}{\partial\varphi}+i\frac{\Phi_{AB}}{\Phi_{0}}\right)\right]\right\}\,
\\-\,i\frac{\hbar\omega_{c}}{2}\left[1+\left(\frac{\rho}{2a}\right)^{2}\right]\left(\frac{\partial}{\partial\varphi}+i\frac{\Phi_{AB}}{\Phi_{0}}\right)+\frac{\mu\omega_{c}^{2}\left(-a^{2}\right)}{2}\left(\frac{\rho}{2\left(ia\right)}\right)^{2}\,+\,\frac{\hbar^{2}}{8\mu a^{2}}\,
\\+\,\lambda_{1}\rho^{2}\,+\,\frac{\lambda_{2}}{\rho^{2}}\left[1+\left(\frac{\rho}{2a}\right)^{2}\right]^{2}\,-\,V_{0} \,.
\end{split}$$ We must solve the Schrödinger equation $\hat{H}\Psi(\rho,\varphi)=E\Psi(\rho,\varphi)$. To do it, first we make the following change of the variable $$x\equiv\frac{1}{1\,+\,\left(\frac{\rho}{2a}\right)^{2}}\,.
\label{def-x}$$ Substituting this in (\[totalHamilt\]), we obtain the following equation $$\label{schroequat}
\begin{split}
-\frac{\hbar^{2}}{2\mu a^{2}}\left\{a^{2}\frac{1}{x^{2}}\left[\frac{1}{a^{2}}\left(x^{2}\frac{d}{dx}\left[x(1-x)\right]\frac{d}{dx}\right)-\frac{1}{4 a^{2}}\left(\frac{x}{1-x}\right)\left(\frac{\partial}{\partial\varphi}+i\frac{\Phi_{AB}}{\Phi_{0}}\right)\right]\right\}\Psi(x,\varphi)\,
\\ -\,i\frac{\hbar\omega_{c}}{2}\frac{1}{x}\left(\frac{\partial}{\partial\varphi}+i\frac{\Phi_{AB}}{\Phi_{0}}\right)\Psi(x,\varphi)+\frac{\mu\omega_{c}^{2}a^{2}}{2}\left(\frac{1-x}{x}\right)\Psi(x,\varphi)\,+\,\frac{\hbar^{2}}{8\mu a^{2}}\Psi(x,\varphi) \\
+\lambda_{1}\,4a^{2}\,\left(\frac{1-x}{x}\right)\Psi(x,\varphi) \,+\,\lambda_{2}\,\frac{1}{4a^{2}}\left(\frac{x}{1-x}\right)\frac{1}{x^{2}}\Psi(x,\varphi) \,-\frac{\lambda_{2}}{2a^{2}}\Psi(x,\varphi) - \\-\,\frac{\mu}{4}\omega_{0}^{2}\Psi(x,\varphi) \,\rho_{0}^{2}\, = E\Psi(x,\varphi).
\end{split}$$ From (\[schroequat\]), using the *ansatz* $\Psi\left(\rho,\,\varphi\right)\,=\,\frac{e^{i\,m\varphi}}{\sqrt{2\pi}}\,f_{m}(x)$, we obtain the following Schrödinger equation $$\begin{split}
-\frac{\hbar^{2}}{2\mu a^{2}} \left\{\frac{1}{x^{2}}\left[x^{2}\,\frac{d}{dx}(x(1-x))\frac{d}{dx}\,-\,\frac{1}{4}\left(\frac{x}{1-x}\right)\left(m\,+\,\frac{\Phi_{AB}}{\Phi_{0}}\right)^{2}\right]\right\}\,f_{m}(x)\\
+\,\frac{\hbar\omega_{c}}{2}\left(m\,+\,\frac{\Phi_{AB}}{\Phi_{0}}\right)\,\frac{1}{x}\,f_{m}(x)\\
+\,\left\{\frac{\mu\,\omega_{c}^{2}\,a^{2}}{2}\frac{1}{x}\,-\,\frac{\mu\,\omega_{c}^{2}\,a^{2}}{2}\,+\,\frac{\hbar^{2}}{8\mu a^{2}}\,+\,\frac{\left[\frac{1}{2}\mu\,a^{2}\,\omega_{0}^{2}\,-\,\frac{1}{2}\,\mu\,a^{2}\,\omega_{0}^{2}\left(\frac{\rho_{0}}{2a}\right)^{4}\right]\,x}{1-x}\,+\,\frac{\frac{\mu\,a^{2}\,\omega_{0}^{2}}{2}}{x(1-x)}\right\}\,f_{m}(x)\\
-\left\{\,\frac{\left[\mu\,a^{2}\,\omega_{0}^{2}\,-\,\mu\,a^{2}\,\omega_{0}^{2}\left(\frac{\rho_{0}}{2a}\right)^{4}\right]}{1-x}\,+\,\mu\,a^{2}\,\omega_{0}^{2}\left(\frac{\rho_{0}}{2a}\right)^{4}\,+\,\frac{\mu}{4}\,\omega_{0}^{2}\,\rho_{0}^{2}\right\}\,f_{m}(x)=\,E\,f_{m}(x)\, ,
\end{split}
\label{sch1}$$ where $\omega_{c}=\frac{eB}{m}$ is a cyclotron frequency and $$\omega_{0} = \sqrt{\frac{8}{\mu}\left[\lambda_{1}+\frac{\lambda_{2}}{\left(2a\right)^4}\right]} \quad .
%\label{}$$ Now, after some algebra we obtain $$\begin{split}
\left\{-\frac{d}{dx}(x(1-x))\frac{d}{dx}\,+\,\frac{M^{2}}{4}\frac{1}{1-x}\,+\,\frac{\mu^{2}a^{4}}{\hbar^{2}}\omega_{m}^{2}\frac{1}{x}\,-\,\frac{\mu^{2}a^{4}}{\hbar^{2}}\left[\omega_{c}^{2}\,+\,\omega_{0}^{2}\left(1\,+\,\left(\frac{\rho_{0}}{2a}\right)^{2}\right)^{2}\right]\right\}f_{m}(x)\\
=\,\left\{\frac{2\mu a^{2}}{\hbar^{2}}\,E\,-\,\frac{1}{4}\right\}f_{m}(x)
\label{schequatio1}
\end{split}$$ where $$M\equiv \sqrt{\left(m\,+\,\frac{\Phi_{AB}}{\Phi_{0}}\right)^{2}\,+\,\left(\frac{\mu\omega_{0}\rho_{0}^{2}}{2\hbar}\right)^{2}}\,,$$
$$\omega_{m}\equiv\sqrt{\left(\omega_{c}\,+\,\frac{\hbar}{2\mu a^{2}}\left(m\,+\,\frac{\Phi_{AB}}{\Phi_{0}}\right)\right)^{2}\,+\,\omega_{0}^{2}},$$
$$E'\,=\,\frac{2\mu a^{2}}{\hbar^{2}}\,E.$$
Thus, rearranging terms in (\[schequatio1\]), we write $$\begin{split}
\left\{\frac{d^{2}}{dx^{2}}\,+\,\left(\frac{1}{x}\,-\,\frac{1}{1-x}\right)\frac{d}{dx}\,+\,\left[\frac{-\frac{\mu^{2}a^{4}\omega_{m}^{2}}{\hbar^{2}}}{x}\,+\,\frac{-\frac{M^{2}}{4}}{1-x}\right]\frac{1}{x(1-x)}\right\}f_{m}(x)\\
-\,\left\{\frac{1}{4}-\frac{2\mu a^{2}}{\hbar^{2}}E\,-\,\frac{\mu^{2}a^{4}}{\hbar^{2}}\left[\omega_{c}^{2}\,+\,\omega_{0}^{2}\left(1\,+\,\left(\frac{\rho_{0}}{2a}\right)^{2}\right)^{2}\right]\right\}\frac{1}{x(1-x)}f_{m}(x)=\,0.
\label{sch5}
\end{split}$$ The equation (\[sch5\]) replays the form of following differential equation $$\begin{aligned}
&& \frac{d^{2}\,P(\xi)}{d\xi^{2}}\,+\,\left(\frac{1-\alpha-\alpha'}{\xi}\,-\,\frac{1-\gamma-\gamma'}{1-\xi}\right)\frac{d\,P(\xi)}{d\xi}\,+\,\left(\frac{\alpha\,\alpha'}{\xi}\,-\,\frac{\gamma\,\gamma'}{1-\xi}\,-\,\beta\,\beta'\right)\times\nonumber\\&\times& \frac{1}{\xi\left(\xi-1\right)}P(\xi)\,=\,0\,,
\label{hipergeo1}\end{aligned}$$ where $$\alpha\,+\alpha'\,+\,\beta\,+\,\beta'\,+\,\gamma\,+\gamma'\,=\,1\,.
\label{hipergeo-relat}$$ It is important to note that (\[hipergeo1\]) and (\[hipergeo-relat\]) are similar to Eqs. ($5$) and ($5a$) in Ref. [@rubino]. Further, we can see that (\[hipergeo1\]) has the form of the hypergeometric equation, whose solution reads as $$P(\xi)\,=\,\xi^{\alpha}\,\left(1-\xi\right)^{\gamma}\,F\left(a',b',c';\xi\right)\,,$$ where $$a'\,=\,\alpha\,+\,\beta\,+\,\gamma,\quad b'\,=\,\alpha\,+\,\beta'\,+\,\gamma,\quad c'\,=\,1\,+\,\alpha\,-\,\alpha'\,.$$ Comparing (\[hipergeo1\]) and (\[sch5\]), we see that $$1\,-\,\alpha\,-\,\alpha'\,=1\quad \to \alpha\,=\,-\alpha'.$$ and at the same way $$\gamma\,=\,-\gamma'\quad\mbox{and}\quad \beta'\,=\,1\,-\,\beta\,.$$ Let us define $$\alpha\equiv \frac{\mu a^{2}}{\hbar}\,\omega_{m}\,,
\label{alfa-def}$$ and $$\gamma\equiv \frac{M}{2}\,.
\label{gamma-def}$$ We can also use the relation $$\beta\,\left(1\,-\,\beta\right)\,=\,\frac{1}{4}\,-\,\frac{2\mu a^{2}}{\hbar^{2}}E\,-\,\frac{\mu^{2} a^{4}}{\hbar^{2}}\left[\omega_{c}^{2}\,+\,\omega_{0}^{2}\left(1\,+\,\left(\frac{\rho_{0}}{2a}\right)^{2}\right)^{2}\right]\,.
\label{find-beta}$$ From last relations it is easy to see that we can assume $$P(x)\,=\,x^{\alpha}\,\left(1\,-\,x\right)^{\gamma}\,F\left(\alpha\,+\,\beta\,+\,\gamma,\,\alpha\,+\,\gamma\,+\,1\,-\,\beta,\,1\,+\,2\alpha,\,x\right)\,.
\label{hipergeo2}$$ From (\[find-beta\]) we find the following expression $$\beta\,=\,\frac{1}{2}\,\pm\,\sqrt{\frac{2\mu a^{2}}{\hbar^{2}}E\,+\,\frac{\mu^{2}a^{4}}{\hbar^{2}}\omega_{c}^{2}\,+\,\frac{\mu^{2}a^{4}}{\hbar^{2}}\omega_{0}^{2}\,+\,2\frac{\mu^{2}a^{4}}{\hbar^{2}}\omega_{0}^{2}\left(\frac{\rho_{0}}{2a}\right)^{2}\,+\,\frac{\mu^{2}a^{4}}{\hbar^{2}}\omega_{0}^{2}\left(\frac{\rho_{0}}{2a}\right)^{4}}.
\label{betas}$$ Assuming, because of (\[hipergeo2\]), that $$\alpha\,+\,\beta\,+\,\gamma\,\leqslant\,-n\,
\label{limit-n}$$ and considering (\[alfa-def\]), (\[gamma-def\]) and (\[betas\]), and solving condition (\[limit-n\]) when equality holds, after some algebra and substituting the explicit values for $\alpha,\,\beta,\,\gamma,\,\omega_{m}^{2},\,M^{2}$, basing on previously obtained relations, we obtain the following energy eigenvalues $$\begin{split}
E\,=\,\frac{\hbar^{2}}{2\mu a^{2}}\left[\left(n\,+\,\frac{1}{2}\right)^{2}\,+\,\left(n\,+\,\frac{1}{2}\right)M\,+\,\frac{1}{2}\left(m\,+\,\frac{\Phi_{AB}}{\Phi_
{0}}\right)^{2}\right]\,+\,\hbar\,\omega_{m}\left(n\,+\,\frac{1}{2}\,+\,\frac{M}{2}\right)\\
+\,\hbar\,\omega_{c}\left(m\,+\,\frac{\Phi_{AB}}{\Phi_{0}}\right)\,-\,\frac{\mu\omega_{0}^{2}\rho_{0}^{2}}{4},
\end{split}
\label{stripe-energy}$$ with $n\,\in\,\mathbb{N}:\,0\,\leqslant\,n\,<\,\frac{\mu\omega_{m} a^{2}}{\hbar}\,-\,\frac{M}{2}\,-\,\frac{1}{2}$. In the limit $\lambda_{1} =\lambda_{2}=0$ and $\Phi_{AB}=0$ we find the results obtained by Dunne [@Dunne] for Landau levels in a spherical space. Here we can see that eigenvalues are only discrete, in contrast with the case of hyperbolic space where the Landau levels are studied in Refs. [@comtet; @comtet1; @Dunne] and the eigenvalues can be discrete as well as and continuous. In the limit $a \to \infty$ we recover the results obtained by Tan and Inkson [@12] given by $$\begin{aligned}
\label{eingtanink}
E=\hbar\,\omega_{fm}\left(n\,+\,\frac{1}{2}\,+\,\frac{M}{2}\right) + \hbar\,\omega_{c}\left(m\,+\,\frac{\Phi_{AB}}{\Phi_{0}}\right)\, -V_{0}\end{aligned}$$ where flat definition $V_{0}$ is given by Eq. (\[vopla\]), and $\omega_{fm}= \sqrt{(\omega_{c}^{2}+\omega_{0})^{2}}$.
The Magnetization for Quantum Ring in Spherical space {#sec5}
=====================================================
Considering our system as a canonical ensemble, one can obtain the magnetization [@landau-stat], from a Helmholtz free energy and a magnetic field, as $$\mathcal{M}(B)\,=\,-\left(\frac{\partial F}{\partial B}\right)_{T,N}\,=\,{\sum}_{n,m}\mathcal{M}_{n,m}\,f_{0}\left(E_{n,m}\right)\,
\label{magA}$$ where $N$ represents the number of electrons and $f_{0}$ is the Fermi distribution function. In this case, magnetic moment for each $(n,m)$th state is given by $$\mathcal{M}_{n,m}\,=\,-\frac{\partial E_{n,m}}{\partial B},
\label{mag3}$$ with $$N\,=\,{\sum}_{n,m}\,f_{0}\left(E_{n,m}\right).
\label{magB}$$
Noting that $\omega_{m}$ depends on the magnetic field $B$, $\omega_{c}$, one can write $$\frac{\partial}{\partial B}\,=\,\frac{\partial\omega_{c}}{\partial B}\,\frac{\partial}{\partial\omega_{c}},
\label{derivaB}$$ which implies the following relation $$\mathcal{M}_{n,m}\,=\,-\frac{e}{\mu c}\,\frac{\partial}{\partial\omega_{c}}\,E_{n,m}.
\label{mag-omegC}$$ In this way, we can use the above relation and the energy eigenvalues ($\ref{stripe-energy}$), and obtain $$\mathcal{M}_{n,m}\,=\,\mu_{B}\frac{m_{0}}{\mu}\left[\hbar\left(2n+M+1\right)\frac{\partial}{\partial\omega_{c}}\omega_{m}\,+\,\left(m+\frac{\Phi_{AB}}{\Phi_{0}}\right)\right],
\label{comp-flat}$$ where $m_{0}$ represents the electron rest mass and $\mu_{B}$ is Bohr magneton given by $$\mu_{B}\,=\,\frac{e\,\hbar}{2 m_{0} c}.$$ Taking into account that $$\frac{\partial}{\partial\omega_{c}}\omega_{m}\,=\,\frac{1}{\omega_{m}}\left[\omega_{c}\,+\,\frac{\hbar}{2\mu a^{2}}\left(m+\frac{\Phi_{AB}}{\Phi_{0}}\right)\right],$$ finally we find $$\frac{\mathcal{M}_{n,m}}{\mu_{B}}\,=\,-\frac{m_{0}}{\mu}\,\left[\left(2n\,+\,M\,+\,1\right)\frac{\omega_{c}\,+\,\frac{\hbar\left(m+\frac{\Phi_{AB}}{\Phi_{0}}\right)}{2\mu a^{2}}}{\omega_{m}}\,+\,m+\frac{\Phi_{AB}}{\Phi_{0}}\right].
\label{magneticmoment}$$ The expression (\[magneticmoment\]) is the magnetization for $T=0$ for a two-dimensional quantum ring in a spherical space. Applying limit $a \to \infty $, we recover the flat case [@12; @footnote1] for Gaussian CGS units.
The Persistent current in Quantum Ring in Spherical Space {#sec6}
=========================================================
In this section we investigate the arising of the persistent current in a two-dimensional quantum ring for the spherical geometry. Then, we also obtain persistent currents from ($\ref{stripe-energy}$). It was showed in [@byers-yang] that for known energy eigenvalues, we can obtain the current from the following relation $$I_{n,m}\,=\,-c\,\frac{\partial E_{n,m}}{\partial\Phi_{AB}},\,$$ that is, the Byers-Yang relation. In this way, the persistent currents will be given by $$\begin{split}
I_{n,m}\,=\,-c\left\{\frac{\hbar^{2}}{2\mu a^{2}}\left[\left(n+\frac{1}{2}\right)\frac{\partial M}{\partial\Phi_{AB}}\,+\,\frac{1}{2}\frac{\partial}{\partial\Phi_{AB}}\,\left(m+\frac{\Phi_{AB}}{\Phi_{0}}\right)^{2}\right]\,+\,\hbar\left(n+\frac{1}{2}+\frac{M}{2}\right)\frac{\partial\omega_{m}}{\partial\Phi_{AB}}\right\}\\
\,-\frac{c}{2}\hbar\omega_{m}\frac{\partial M}{\partial\Phi_{AB}}\,-\,c\,\frac{\hbar\omega_{c}}{2}\frac{1}{\Phi_{0}}\,.
\label{persist-1}
\end{split}$$
Taking into account that $$\frac{\partial M}{\partial\Phi_{AB}}\,=\,\frac{1}{2}\,\frac{1}{M}\,2\,\left(m+\frac{\Phi_{AB}}{\Phi_{0}}\right)\,\frac{1}{\Phi_{0}}\,=\,\frac{1}{M}\left(m+\frac{\Phi_{AB}}{\Phi_{0}}\right)\,\frac{1}{\Phi_{0}}\,,
\label{diff-M}$$ and $$\frac{\partial\omega_{m}}{\partial\Phi_{AB}}\,=\,2\,\left(m+\frac{\Phi_{AB}}{\Phi_{0}}\right)\,\frac{1}{\Phi_{0}}\,=\,\frac{\partial\omega_{m}}{\partial\Phi_{AB}}\,=\,\frac{\hbar}{2\mu a^{2}}\,\frac{1}{\Phi_{0}}\,\left[\frac{\omega_{c}\,+\,\frac{\hbar}{2\mu a^{2}}\left(m+\frac{\Phi_{AB}}{\Phi_{0}}\right)}{\omega_{m}}\right]\,,
\label{diff-omegam}$$ and substituting (\[diff-M\]) and (\[diff-omegam\]) in (\[persist-1\]), one finds $$\begin{split}
I_{n,m}\,=\,(-c)\left\{\left[\frac{\hbar^{2}}{2\mu a^{2}}\frac{\left(2n+1\right)}{2}\,+\,\frac{\hbar\omega_{m}}{2}\right]\frac{1}{\Phi_{0}}\frac{1}{M}\left(m+\frac{\Phi_{AB}}{\Phi_{0}}\right)\,+\,\frac{\hbar^{2}}{2\mu a^{2}}\left(m+\frac{\Phi_{AB}}{\Phi_{0}}\right)\frac{1}{\Phi_{0}} \right\}\\
-c\left\{\frac{\hbar}{2}\left(2n+M+1\right)\frac{\hbar}{2\mu a^{2}}\frac{1}{\Phi_{0}}\left[\frac{\omega_{c}\,+\,\frac{\hbar}{2\mu a^{2}}\left(m+\frac{\Phi_{AB}}{\Phi_{0}}\right)}{\omega_{m}}\right]\,+\,\frac{\hbar\omega_{c}}{2}\frac{1}{\Phi_{0}}\right\}\,.
\end{split}$$ Here we used $$\Phi_{0}\,\equiv\,\frac{hc}{e}\,=\,\frac{2\pi\hbar c}{e}\,,$$ where $e$ is the electron/hole charge. After some algebraic manipulations we obtain $$\label{currentspherical}
I_{n,m}\,=\,\frac{c}{\pi\rho_{m}^{2}}\left\{\mathcal{M}_{n,m}\left[1+\left(\frac{\rho_{m}}{2a}\right)^{2}\right]\,+\,\mu_{B}\frac{m_{0}}{\mu}\frac{\omega_{c}}{\omega_{m}}\left(2n+1\right)\right\}\,,$$ where $$\rho_{m}\,\equiv\,\sqrt{\frac{2\hbar M}{\mu\omega_{m}}}\,,$$ is the effective radius of the state with a quantum number $m$.
Concluding remarks {#sec7}
==================
In this paper we have investigated the two-dimensional quantum ring in the presence of the Aharonov-Bohm quantum flux and an uniform magnetic field in the spherical space. We obtained the eigenvalues and eigenfunctions of the Hamiltonian and demonstrated the influence of curvature on these physical quantities. We have found the magnetization and the persistent current for $T=0$ and showed the influence of the curvature on these cases. In the zero curvature limit ($a \to \infty$) we reproduced the previous results obtained by Tan and Inkson [@12]. In the case where $\lambda_{1}=\lambda_{2}=0$ and $\Phi_{AB}=0$ we obtained the Landau levels in spherical space [@Dunne]. Notice that in the expression (\[currentspherical\]) the first contribution is caused by the classical current in a quantum ring of radius $\rho_{m}$ exposed to a magnetic field, the second contribution arises due to the magnetic field penetration in the conduction region of the two-dimensional ring, and this contribution is responsible for breaking the proportionality of the magnetic moment and the persistent current, a similar case was observed by Bulaev [*et al.*]{} for a two-dimensional quantum ring in hyperbolic space [@bulaphysb]. In the limit $\omega_{c} <<\omega_{0}$ the current is proportional to the magnetic moment. We can write the magnetization in the following way: $$\label{magnetizationcurrent}
\mathcal{M}_{n,m}=\left\{ \frac{c\pi\rho_{m}^{2}}{\left[1+\left(\frac{\rho_{m}}{2a}\right)^{2}\right]} I_{n,m}\,-\,\mu_{B}\frac{m_{0}}{\mu}\frac{\omega_{c}}{\omega_{m}}\left(2n+1\right)\right\}\,.$$ It follows from Eq. (\[magnetizationcurrent\]) that the magnetization can be presented as a sum of two terms. The first one arises due to a magnetic dipole moment of a current loop within spherical geometry, and in the limit $a \to \infty$ we recover the classical results [@jackson]. The another contribution corresponds to a diamagnetic shift. This term has a contribution due to the curvature dependence in the term $\frac{\omega_{c}}{\omega_{m}}$. Finally, we claim that with the development on nanotechnology the possibility to investigate this spherical system from the experimental point of view can be realized technically which can allow to obtain this spherical thin shell material . We emphasize the interest to investigating the spherical systems taken place in recent years, see f.e [@jelal; @nair; @hasebe; @hasebe2; @nair2; @nair3].
We can use the geometry of the quantum ring in spherical shell to construct an experimental set-up to investigate the physical properties obtained here for this theoretical model. A nanometric system for a quantum ring on a sphere can be experimentally constructed between two well-determined $\theta$ angles in a quasi-two-dimensional nanostructured hemisphere. Electron or holes may be injected by terminals attached to the ring and the persistent current can be measured in this experimental scheme similar to that described in reference Ref.[@gao] For flat case. Other properties of these systems that can be investigated in a future publication, such as, de Haas- van Alphen effect and a more detailed numerical study of the contribution of the persistent current and magnetization obtained in previous sections to a system with many particles for $T = 0$ and $T \neq 0$.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A review is given of hypothetical faster-than-light tachyons and the development of the author’s $3+3$ model of the neutrino mass states, which includes one tachyonic mass state doublet. Published empirical evidence for the model is summarized, including an interpretation of the mysterious Mont Blanc neutrino burst from SN 1987A as being due to tachyonic neutrinos having $m^2=-0.38 eV^2.$ This possibility requires an 8 MeV antineutrino line from SN 1987A, which a new dark matter model has been found to support. Furthermore, this dark matter model is supported by several data sets: $\gamma-$rays from the galactic center, and the Kamiokande-II neutrino data on the day of SN 1987A. The KATRIN experiment should serve as the unambiguous test of the $3+3$ model and its tachyonic mass state.'
author:
- Robert Ehrlich
title: |
Review of the empirical evidence for superluminal particles\
and the $3+3$ model of the neutrino masses
---
Published as:\
R. Ehrlich. Advances in Astronomy, Vol. 2019, 2820492\
v $>$ c Tachyons
================
Hypothetical faster-than-light particles, now known as tachyons, were first suggested in 1962 by Bilaniuk, Deshpande, and Sudarshan as a way to extend special relativity to the $v>c$ realm. [@Bi1962] Sudarshan and colleagues noted that if a particle was allowed to have a rest mass that was imaginary, or $m^2<0$ one could use the usual formula to compute its real total energy $E=mc^2/\sqrt{1-v^2/c^2},$ as long as the particle was never allowed to have $v<c.$ For those concerned about the meaning of an imaginary rest mass, Ref. [@Bi1962] reminds us that only energy and momentum, by virtue of their direct observability and conservation in interactions, must be real and that the hypothetical imaginary rest mass particles offend only the traditional way of thinking. In this scheme $v=c$ becomes a two-way infinite energy barrier – an upper limit to normal ($m^2>0$) particles and a lower limit to hypothetical tachyons, thus allowing all matter to be divided into three classes with $m^2$ being positive, negative or zero. Moreover, tachyons have the weird property as Fig. 1 shows of speeding up as they lose energy, and approaching infinite speed as E approaches zero. There are, of course, cases of allowed superluminal motion. Thus, Recami and others have considered localized X-shaped solutions to Maxwell’s equations, [@Re1998], quantum tunneling through two successive barriers, [@Re2002] and the apparent separation speed of quasars [@Ba1989]. A nice overview of these and other allowed types of superluminal motion can be found in Recami [@Re2001; @Re2008]. However, in these cases there is no superluminal motion of particles or information with the possibility of a violation of causality, making them outside the scope of this review.
Since the original tachyon paper [@Bi1962] Recami and Mignani, [@Re1974], Recami [@Re1986] and later Cohen and Glashow [@Co2011] and other theorists have suggested various ways to accommodate $v>c$ particles, including the adoption of nonstandard dispersion relations, which can avoid imaginary rest masses, but at the price (in the Cohen-Glashow case) of making the value of a particle’s rest mass dependent the choice of reference frame.
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$v>c$ or “classical" tachyons are not taken seriously by most physicists because of their obnoxious theoretical properties, and the repeated failed attempts to find unambiguous evidence for their existence. These attempts include some well-known mistaken claims, most recently by the OPERA Collaboration in 2011. [@Ad2011] In fairness to OPERA, the initial paper made no discovery claim, and it merely announced their observed $v>c$ anomaly with the intent of promoting further inquiry and debate. As is well-known the group later found several experimental flaws and their corrected neutrino speed was consistent with c. [@Ad2012] In any case, most theoretical (and experimental) physicists have little use for the faster-than-light variety of tachyon, which has been considered a violation of relativity (Lorentz Invariance) and Causality (prohibition against backward-time signalling), although it is also true that some theorists have postulated ways around such difficulties by for example postulating a preferred reference frame or small violations of Lorentz Invariance. [@Re1987; @Ci1999; @Ra2010] Moreover, while most physicists abhor the $v>c$ classical tachyon they have much greater affinity for another variety that is widely used in field theory. [@Se2002] These more reputable tachyons have imaginary mass quanta, but no $v>c$ propagation speed, the field associated with the Higgs particle being the best known example. In particular, the imaginary mass quanta of the Higgs field cause instabilities leading to a spontaneous decay or condensation, but again no $v>c$ propagation. In the rest of this paper the word tachyon refers to the $v>c$ and $m^2<0$ “disreputable" variety.
Given the current state of experimental physics, the only known particle that could be a tachyon is one of the neutrinos, a possibility raised by Chodos, Kostelecky and Hauser in a 1985 paper. [@Ch1985] Since the neutrino’s observed mass is so close to zero, we cannot be certain yet whether $m_\nu^2>0$ or $m_\nu^2<0,$ although it is known that $m_\nu^2\ne 0,$ for at least some neutrinos in order for neutrino oscillations to be possible – a connection that was explored in a 1986 paper by Giannetto et al. [@Gi1986]. Considering the two types of measurements, $v$ or $m^2,$ it is the latter that permits us to put much tighter constraints on whether the neutrino is or is not a tachyon. Thus, if neutrinos in fact had a velocity that was slightly in excess of c by an amount say half the present experimental uncertainty, then their computed $|m^2|$ would need to be orders of magnitude above what would have been readily observed by now in direct mass experiments.
Direct neutrino mass experiments
================================
The most common “direct" (model independent) method of measuring the neutrino (or antineutrino) mass is to look for distortions of the $\beta-$decay spectrum near its endpoint. In these experiments an antineutrino is emitted is in the electron flavor state $\nu_e$ which is a quantum mechanical mixture of states $\nu_j$ having specific masses $m_j$ with weights $U_{ej}, $ i.e., $\nu_e=\sum U_{ej}\nu_j.$ In general, if one can ignore final state distributions, the phase space term describes the spectrum fairly well near the endpoint $E_0,$ and it can be expressed in terms of the effective electron neutrino mass using the square of the Kurie function.
$$K^2(E)=(E_0-E)\sqrt{(E_0-E)^2-m^2_\nu\rm{(eff)}}$$
In Eq.1 the $\nu_e$ effective mass is defined in single $\beta-$decay by this weighted average of the individual $m_j^2$: $$m^2_\nu\rm{(eff)}=\sum |U_{ej}|^2 m_j^2$$
However, if the individual $m_j$ could be distinguished experimentally, one would need to use a weighted sum of spectra for each of the $m_j$ with weights $|U_{ej}|^2$ [@Gi2007]
$$K^2(E)=(E_0-E)\sum |U_{ej}|^2\sqrt{(E_0-E)^2-m_j^2}$$
Note that when $(E_0-E)^2-m_j^2$ is negative it is replaced by zero in Eq. 1 and 3 so as to avoid negative values under the square root. Given the form of Eq. 1 a massless neutrino yields a quadratic result: $K^2(E)=(E_0-E)^2$ near the endpoint, while a neutrino having an effective $\nu_e$ mass $m^2_\nu\rm({eff})>0$ would result in the spectrum ending a distance $m_\nu\rm({eff})$ from the endpoint defined by the decay Q-value. Moreover using Eq. 3 in the case of $m_j^2>0$ neutrinos of distinguishable mass, we would find that the spectrum shows kinks for each mass at a distances $m_j$ from the endpoint defined by the decay Q-value. These direct mass experiments are extraordinarily difficult in light of systematic effects that also distort the spectrun, and the very small number of electrons observed near the spectrum endpoint. As of October 2018 they have only set upper limits on $m_\nu\rm{(eff)}<2eV,$ [@Pa2016] at least according to conventional wisdom. The possibility of observing a $m^2_\nu<0$ neutrino in direct mass experiments is discussed later, but for now we merely note that the results of nearly all such experiments that have in fact found best fit $m_\nu^2<0$ values should not be taken at face value. If they are not due to systematic errors, these results have a simple explanation within the $3+3$ model, as will be discussed later.
Some experiments hope to look for massive (sterile) neutrinos in oscillation experiments, but these experiments do not measure neutrino masses directly, but rather differences in the $m^2$ values of the states making up an oscillating pair, so they would not be sensitive to whether one or both of those states have $m^2>0$ or $m^2<0.$ One could however possibly observe $m_\nu^2<0$ neutrinos from a galactic supernova. Any neutrinos having $v>c$ would of course arrive earlier than those having $m^2>0,$ assuming they all started out approximately simultaneously, and those having higher energy would arrive [*later*]{} than those with lower energy.
SN 1987A and the Mont Blanc burst
=================================
Galactic supernovae are quite rare, occurring an estimated $2\pm 1$ times per century, making SN 1987A a precious treasure that has deserved the very careful attention it has received, with thousands of papers written about it to date. Although many supernovae have now been observed in other galaxies, only SN 1987A was close enough to study the neutrinos it produced during the final collapse of the core of its progenitor star. In fact four neutrino detectors then operating (see Fig. 2 caption) each observed a burst lasting 5-15 seconds, representing a mere 30 neutrinos (or antineutrinos) in total. Three of the bursts occurred within a matter of seconds of each other as expected, but the fourth detector located under Mont Blanc detected its burst of 5 neutrino events almost 5 hours (16,900 sec) earlier than the others. [@Ag1987; @Ag1988] As a result, most physicists with a few notable exceptions, [@Br1992; @Fr2015; @Gi1999] have chosen to dismiss the Mont Blanc burst as having nothing to do with SN 1978A.
Using SN 1987A to find the neutrino mass
----------------------------------------
SN 1987A like all supernovae create huge numbers of neutrinos and antineutrinos having all three flavors, e, $\mu,$ and $\tau,$ but it is the electron flavor that is typically detected. Conventional wisdom has it that SN 1987A was only able to set an upper limit on the mass of the electron neutrino $\nu_e$ of 5.7 eV [@Pa2016] or more depending on how the analysis is done, and that there was no hint of a $m^2<0$ mass state, assuming one ignores the Mont Blanc burst. As in the case of direct mass experiments, these standard analyses assume that the separate active mass states comprising the electron neutrino are so close in mass that one can only hope to observe a single effective mass for $\nu_e.$ This assumption of only a single effective mass being observable is supported by the standard model of three active neutrinos whose $m^2$ values are separated from one another by two very small quantities, the solar and atmospheric mass differences: $\Delta m^2_{sol} =7.53\times 10^{-5} eV^2$ and $\Delta m^2_{atm} =2.44\times 10^{-3} eV^2.$
Origin of the $3+3$ model
-------------------------
In a 2012 paper [@Eh2012] the author analyzed the SN 1987A data without making the assumption that only a single effective mass could be found, and he rediscovered a most peculiar fact that Huzita[@Hu1987] and Cowsik [@Co1988] had pointed out soon after SN 1987A was observed. If one assumes near-simultaneous emissions then all the 25 neutrinos (ignoring the five from Mont Blanc) were consistent with having one of two outlandishly large masses, $m_1=4.0\pm 0.5$ eV and $m_2=21.4\pm 1.2$ eV. [@Eh2012] This result depends on a kinematic relation between the $i^{th}$ neutrino energy $E_i$ and its travel time $t_i$ (relative to a photon) which in the limit $E_i>>m_i$ can be written:
$$\frac{1}{E_i^2} = \left(\frac{2}{Tm_i^2}\right)t_i$$
Here T is the travel time of a photon (around 168 ky), and $m_i$ is the mass of the $i^{th}$ observed neutrino, and $t_i>0$ $(t_i<0)$ means slower (faster) than light. Essentially, Eq. 4 requires that neutrinos having a given mass $m_i$ should lie on or near a straight line in a plot of $1/E^2$ versus time $t_i$ whose slope reveals the value of $m_i^2.$ Thus, the question of whether the neutrinos are consistent with a single mass is left up to the data to answer, which as can be seen from Fig. 2 taken from Ref. [@Eh2012] would seem to favor two separate $m^2>0$ masses and not one effective mass. Regarding the assumption of simultaneous emissions Ref. [@Eh2012] argues that most or all of the observed neutrinos from SN 1987A were emitted within an interval of $\pm0.2s.$ Moreover, given the usual choice of $t=0$ for the first arriving neutrino in each of the unsynchronized detectors (except Mont Blanc) there are probably $\pm1$ s implied horizontal error bars for the points in Fig. 2.
![\[fig3\]A plot of $1/E_\nu^2$ versus observation time for the events seen in the four neutrino detectors operating at the time of SN 1987A: open diamonds (5 Baksan events), open squares (12 Kamiokande-II events), open triangles (8 IMB events). The single dot near t = - 5h shown without error bars represents the 5 Mont Blanc events. The dashed positive sloped lines correspond to masses $m_1=4.0$ eV, and $m_2=21.4$ eV according to Eq. 2. The $3+3$ model called for a third mass $m_3$ that was a tachyon, but it initially explicitly rejected the idea that the 5 Mont Blanc events defined one, as discussed in the text.](2-mass-states.pdf){width="0.8\linewidth"}
The $3+3$ model was proposed in 2013 [@Eh2013] based on the anomalously large values for $m_1$ and $m_2$ that are implied by Fig. 2. Clearly, the only way to accommodate the very small $\Delta m^2_{sol}$ and $\Delta m^2_{atm}$ was to assume that $m_1$ and $m_2$ were each active-sterile doublets – see Fig. 3. This model is in marked contrast to the $(3 + 0)$ conventional model lacking sterile neutrinos which is described by three mixing angles. If any sterile neutrinos are assumed to exist within the conventional framework they must mix very little with the three active neutrino states so as to preserve unitarity of the $U_{ij}$ matrix. Furthermore, for any model (like $3+3$) that has two active-sterile doublets each with large mixing there must of course be a third doublet, since it is well established that there are three active states. In such a model there are a total of 15 mixing angles and 10 phases, making it much more complex to describe oscillation phenomena than the conventional model.
The $m^2<0$ doublet in the $3+3$ model
--------------------------------------
In an earlier paper [@Eh2015] the author had suggested a tachyonic value for the $\nu_e$ effective flavor state mass, i.e., $m^2_\nu\rm{(eff)}=-0.11\pm 0.02eV^2$ and for this to be the case Eq. 2 would require $m_3^2<0.$ The particular choice $m^2_3 \approx -0.2 keV^2$ was made after a remarkable numerical coincidence was discovered, namely that with the pair of doublet splittings $\Delta m_1^2=\Delta m^2_{sol}$ and $\Delta m_2^2=\Delta m^2_{atm},$ one finds identical fractional splittings $\Delta m_1^2/m_1^2=\Delta m_2^2/m_2^2$ for the two doublets. The choice of the 3rd doublet mass then became obvious. Given that short baseline experiments have suggested an oscillation having $\Delta m^2_{sbl}\approx 1 eV^2$ if one chose $m_3^2\approx -0.2keV^2,$ all three doublets would then have identical fractional splittings, i.e., $$\frac{\Delta m^2_1}{m_1^2}=\frac{\Delta m^2_2}{m_2^2}=
\frac{\Delta m^2_3}{m_3^2}$$
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Support for the $3+3$ model
---------------------------
A model as speculative as $3+3$ especially considering its $m_3^2<0$ doublet clearly needs empirical support before it deserves to be taken seriously. Previous papers [@Ch2014; @Eh2016] have in fact provided such support, which is very briefly summarized here. First it was shown that the dark matter radial distribution in the Milky Way Galaxy could be fit using a nearly degenerate gas of neutrinos having a mass very close to 21.4 eV, and that for clusters of galaxies the dark matter distributions could be fit using neutrinos having a 4.0 eV mass [@Ch2014] – these being the $m^2>0$ masses in the $3+3$ model. The $m^2<0$ mass would not be associated with dark matter, but rather with dark energy, as suggested by various authors. [@Ba2003; @Sc2018]. More recently, it has been shown that fits to the $\beta-$spectrum near its endpoint for the three most precise pre-KATRIN tritium $\beta$-decay experiments (by the Mainz, Troitsk and Livermore Collaborations) could be achieved using the three masses in the $3+3$ model, and moreover these fits were significantly better than the fit to a single effective mass, which only gives an upper limit $m_\nu\rm{(eff)}<2 eV$ for the $\bar{\nu}_e$ mass. [@Eh2016]
The most prominent spectral feature in the $3+3$ model is a kink 21.4 eV before the endpoint, which appears in the data from all three experiments – see Figs. 4, 5 and 6 taken from Ref. [@Eh2016]. The evidence for this kink in the Mainz data rests on a single data point in their 1998-99 data that is $5\sigma$ above the $m=0$ curve, for reasons explained in the caption to Fig. 4. The Troitsk spectrum published 1999 (Fig. 5) clearly shows the kink at the up arrow in Fig. 5. [@Lo1999] However, the location of that kink agrees well with the $3+3$ model fit (the solid curve added by the author) only after an adjustment is made to the scale of the energy axis. Such an adjustment to the data (moving the kink from 10 to 20 eV before the endpoint) might seem unwarranted were it not actually called for in the most recent 2012 Troitsk publication – see Fig. 7 in Ref. [@As2012]. In this newer analysis, the Troitsk authors “ did not employ overly short runs and runs in which external parameters have large uncertainties." As a result of this elimination of some runs, they have withdrawn any claim of statistical significance of their unexplained anomaly. Unfortunately in that 2012 reassessment the authors have chosen no longer to display the spectrum, which is why the originally published Troitsk spectrum was used in our Fig. 5. Moreover, despite their withdrawal of a claim of statistical significance, some evidence for a kink clearly must remain in their data, because as shown by the dashed and solid horizontal lines in Fig. 8 in Ref. [@As2012] the amplitude of the kink averaged over all runs is about 3/4 that in the original spectrum.
![The published data for the Mainz Collaboration taken during different years as it appeared in ref. [@Eh2016]. The solid curve was their $m = 0$ fit to the 1998-99 data. The dashed $3+3$ curve has been added after adjusting the background level and the vertical scale so as to fit the 1994 data. The Mainz data from later years do not show the predicted kink (at the location of the up arrow), because they Mainz did not publish their data beyond 20 eV from the spectrum endpoint in those years.](Mainz_data.pdf){width="1.0\linewidth"}
![The published spectrum for the Troitsk Collaboration taken from Ref. [@Lo1999]. The solid curve is this authors fit to the $3+3$ model from Ref.[@Eh2016] after an adjustment to the energy axis, which fits the data much better than the standard quadratic (dashed curve) for the zero mass case. The most recent (2012) Troitsk reassessment of their anomaly no longer regards it as statistically significant as discussed in the text and in more detail in Ref. [@Eh2016].](Troitsk.pdf){width="1.0\linewidth"}
The Livermore Collaboration also has chosen not to display the spectrum itself but instead the residuals from a fit to the data using the standard $m_\nu\rm{(eff)}=0$ expected spectrum. When only residuals are plotted the kink predicted by the $3+3$ model shows up not as a kink but instead as a spurious spectral line broadened by resolution near the endpoint or alternatively a best fit value for $m^2_\nu\rm{(eff)}$ which is negative – see Fig. 6. Thus, as noted earlier, the $3+3$ model can account for the artifactual $m^2_\nu\rm{(eff)}<0$ fitted value found in nearly all direct mass experiments.\
![The published data for the Livermore Collaboration shown as residuals to a fit to the spectrum assuming zero effective mass, taken from Ref. [@Eh2016]. The curve added by the author shows what the residual graph would look like assuming the true spectrum was described by the $3+3$ model masses. (a) and (b) are for two subsamples of the data, and the arrow shows the spectrum endpoint.](Livermore.pdf){width="1.0\linewidth"}
Evidence for the $m^2<0$ mass
-----------------------------
It has been shown in the previous section (and in more detail in Ref. [@Eh2016]) that the $3+3$ model gives better fits than the conventional $m_\nu \rm{(eff)}=0$ parabolic curve to three tritium beta decay experiments. However, given the sparcity of data near the endpoint it has only been the most prominent feature of the model (the kink in the spectrum 21.4 eV before the endpoint) that the data were able to reveal. A recent paper has provided evidence for the tachyonic mass in the model by showing how it explains the mysterious Mont Blanc neutrino burst seen on the date of SN 1987A, [@Eh2017] a possibility first raised by Giani. [@Gi1999] This burst has been a mystery, not only because of its early arrival, but also because the 5 neutrinos comprising it have virtually the same energy (8 MeV) within measurement uncertainties – something no model has previously explained. [@Vi2015]
Using Eq. 4, one can deduce a value for the tachyonic mass for the burst. Thus, with $\Delta t = 16,900$ s and $E_{avg}=8.0$ MeV, Eq. 4 yields $m_{avg}^2 = -0.38$ keV$^2$ – a mass value that is within a factor of two of the originally hypothesized value $m^2 \approx -0.2$ keV$^2.$ [@Eh2013] One should not expect any better agreement because the mass value in the $3+3$ model was based on the estimated $\Delta m^2_{sbl}\approx 1$ eV$^2$ for the large $\Delta m^2$ oscillation claimed in short baseline experiments, which in fact is uncertain by over a factor of two. [@Gi2013]
It should be noted that the author initially ignored the Mont Blanc burst, assuming it could not be evidence for a tachyon. The basis for that initial rejection follows from Eq. 4 and Fig. 2 which show that neutrinos having a specific $m^2<0$ mass should lie on a negatively sloped line and be distributed over the 5 h before $t=0.$ Thus, given the brief (7 sec) time interval of the 5 neutrinos making up the burst the only way they could correspond to a specific mass state would be for them to have almost exactly the same energy. More specifically, the constancy of the energy of the 5 neutrinos needs to be (by Eq. 4) constant to a precision of $\Delta E/E_{avg}=7s/(2\times 16,900)=0.02 \%,$ which is essentially a line in the (anti)neutrino spectrum. For the tachyonic interpretation of the Mont Blanc burst to be remotely plausible there needs to be some model of a core collapse supernova that gives rise to monochromatic 8 MeV neutrinos.
Model for an 8 MeV neutrino line
================================
Neutrino lines are known to exist in the solar spectrum, but until now no existing model of core collapse supernovae has such a feature. In a recent paper [@Eh2017] the author has proposed a new supernova model for such a 8 MeV $\bar{\nu}$ line that invokes dark matter X particles of mass 8.4 MeV. This particular mass for dark matter is based on recently discovered isoscalar gauge bosons of mass $m_{Z'}=16.7\pm 0.6 MeV.$ These Z’ particles have been postulated as carriers of a fifth force which could serve as a mediator between dark matter particles and standard model leptons. [@Fe2016; @Ch2016] Thus, were one to postulate cold dark matter X particles of mass $m_X=m_{Z'}/2=8.4 MeV$ that annihilate they would yield monochromatic $8.4\pm 0.3$ MeV $\nu,\bar{\nu}$ pairs via the reaction $XX\rightarrow Z' \rightarrow \nu\bar{\nu},$ just as required. Given that supernovae in our galaxy are quite rare, how could such a model be tested without waiting or the next one? This $Z'$ mediated reaction model would apply not just to supernovae but also to any place with abundant dark matter, and sufficiently high temperature, such as the the galactic center. Furthermore it is shown in Ref. [@Eh2017] that the dark matter model predicts successfully three observed properties of the of MeV $\gamma-$rays from the galactic center – see Table I and Fig. 7.
![Spectrum, i.e., $E\times\frac{dF}{dE} (cm^{-2}s^{-1})$ versus energy for $\gamma-$rays from the inner galaxy for $E>511$ keV, as measured by 4 instruments: SPI(open circle), COMPTEL (open squares), EGRET (filled circles), and OSSE (filled triangles), as it appeared in ref. [@Eh2017]. All but the 7 OSSE points extracted from ref. [@Ki2001] are from Prantzos et al. [@Pr2010], as are the 3 predicted enhancement curves above the straight line for positrons injected into a neutral medium at initial energies $E_0=5, 10, 50$ MeV displayed as the lower grey curve, the black curve, and the upper grey curve, respectively. The sloped straight line (also from Ref. [@Pr2010] is a power law fit to the spectrum at high and low energies.](OSSE.pdf){width="1.1\linewidth"}
![\[fig3\]Evidence for a neutrino line centered on 7.5 MeV in the Kamiokande-II data taken on the date of SN 1987A from Ref. [@Eh2017]. (b) shows a histogram of the raw data taken in the hours before and after the main burst, with $N_{hit}$ being a proxy for the neutrino energy, $E_\nu,$ and the solid and dashed curves being two versions of the background for the detector. After subtracting the background (obtained from other K-II data) and converting the horizontal scale to neutrino energy one obtains (a) the background-subtracted curve.](K-II-line.pdf){width="1.0\linewidth"}
Additional empirical support for an 8 MeV neutrino line (the basis of the dark matter model) is provided in ref. [@Eh2017] based on Kamiokande-II data taken on the day of SN 1987A in the minutes and hours before and after the main 12-event burst in that detector. These data are consistent with there being a antineutrino spectral line centered near 8 MeV that is broadened by $25\%$ energy resolution – see Fig. 8. The strength of the support for this claim of an 8 MeV line, of course, depends on the reliability of the background, which is discussed at length in ref. [@Eh2017].
Quantity observed value predicted value
------------------ ------------------- -------------------
$m_X$ $10^{+5}_{-2}MeV$ $ 8.4\pm 0.3 MeV$
$\sigma(\theta)$ $2.5^0$ $2.4^0$
$T$ $10^3K$ $10^3K$
: Values of the dark matter particle mass $m_X,$ the temperature of the $\gamma-$ ray source T, and its angular radius $\sigma (\theta).$ The “observed” values are based on either direct observations or fits to the data, and the predictions follow from the $Z'_e/Z'_\nu$ mediated reaction model as discussed in Ref.[@Eh2017].
![\[fig3\]Simulated KATRIN data based on the $3+3$ model masses for the last 10 eV of the spectrum as it appeared in ref. [@Eh2017]. The error bars are based on statistical uncertainties for one hour of data-taking for each of 24 energy bins of 0.5 eV width, with the event count normalized to yield the expected count rate at $E_0-E=20$ eV. The dashed curve shows the all $m_j=0$ case after final state distributions and energy resolution have been included. The insert shows the last 5 eV with an expanded vertical scale.](Katrin.pdf){width="0.8\linewidth"}
The Mont Blanc burst again\[Giani\]
===================================
Even though there were only 5 neutrinos in the anomalous Mont Blanc burst, and most physicists have dismissed this burst as being unrelated to SN 1987A, as the preceding sections have shown it played a very significant role in validating the $3 +3$ model with its $m^2<0$ mass state. Many speculative explanations have been suggested for the 5 hour early Mont Blanc neutrino burst, assuming that it was not just a statistical fluctuation of the background. These include (a) a double bang involving formation of a neutron star and followed by a black hole, [@Br1992] (b) a new core collapse mechanism involving dark matter balls, [@Fr2015] and (c) having the 5 neutrinos in the burst be a tachyon mass state, which was consistent with their equal energies $E\sim 8 MeV$ within the 15% uncertainty [@Gi1999]. The tachyonic explanation of the Mont Blanc burst had until recently been regarded as extremely unlikely, even by this “tachyon enthusiast,” [@Eh2013] given the lack of any known mechanism that would generate the required 8 MeV monochromatic neutrinos from a core collapse supernova. A radical reassessment, however, is now warranted in favor of the tachyonic mass state explanation based on the author’s 2016-2018 publications [@Eh2016; @Eh2017] which showed that:
1. A new dark matter model explains why one would expect an 8 MeV monochromatic component of SN 1987A neutrinos
2. Empirical evidence supports that DM model based on the spectrum of galactic center $\gamma-$rays
3. Strong evidence ($S\sim30\sigma$) supports the existence of an 8 MeV line in the SN 1987A spectrum from the $N\sim 1000$ events in Kamiokande II on the day of SN 1987A
4. The value of the $m^2<0$ mass inferred from the Mont Blanc neutrinos is consistent with that originally postulated in the earlier $3+3$ model.
5. The $3+3$ model gives excellent fits to the three most accurate pre-KATRIN direct mass experiments.
The KATRIN experiment
=====================
The KATRIN experiment [@Dr2013] should prove or refute the existence of a tachyonic mass $m_3^2=-0.38 keV^2$ (and the two other masses in the $3+3$ model) in a short period of data-taking. Specifically, if the model is correct KATRIN should observe three features in the spectrum (associated with each of the three masses in the model). The most prominent of these seen in Figs. 4-6 is the kink 21.4 eV before the endpoint. Fig. 9 shows the two other predicted features: a kink 4.0 eV before the endpoint due to the 4.0 eV mass state, and a linear decline in the last 4 eV, a feature based on the form of Eq. 3 when $m_3^2<0.$ The value used in generating this plot was the original mass in the model, $m_3^2\approx -0.2$ keV$^2$. However, the interactive spreadsheet at Ref. [@slider] allows the reader to see how these features in the spectrum change when one uses alternative masses including $m_3^2=-0.38$ keV$^2.$ Given that KATRIN has the sensitivity to see all three spectral features predicted by the $3+3$ model it should serve as an unambiguous test of the model’s validity, including the tachyonic mass state. However, even if the model should be proven incorrect KATRIN might still be consistent with a tachyonic flavor state, and in particular the earlier noted prediction by the author for the $\nu_e$ effective mass: $m^2_\nu\rm{(eff)}=-0.11\pm 0.02eV^2$[@Eh2015].
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arxiv.org/pdf/1301.6640 J. S. Bagla, H. K. Jassal, and T. Padmanabhan, Phys. Rev. D 67, 063504 (2003). C. Schwartz, Int. J. of Mod. Phys. 33, [10]{}, 1850056 (2018). V. M. Lobashev et al., Phys. Lett. B 460, 227 (1999). V. N. Aseev et al., Phys. of Atomic Nuclei, 75, 4, 464 (2012). C. Giunti, M. Laveder, Y. F. Li, and H. W. Long, Phys. Rev. D 88, 073008 (2013). R. Ehrlich, Astropart. Phys., 99, 21 (2018)\
arXiv.org/pdf/1701.00488. R. L. Kinzer et al., Ap. J., 559:282, È295 (2001). N. Prantzos, et al. Rev. Mod. Phys., [**83**]{}, 1001 (2011). F. Vissani, J. Phys. G. 42 (1) 013001 (2015). J. Feng et al., Phys. Rev. Lett. [**117**]{}, 071803 (2016). C. Chen et al., https://arxiv.org/pdf/1609.07198.pdf K. S. Hirata, et al. (Kamiokande collaboration), Phys. Rev. D [**38**]{}, 448-458 (1988). K. S. Hirata, et al. (Kamiokande collaboration), Phys. Rev. Lett., [**63**]{}, 1 (1989). G. Drexlin, V. Hannen, S. Mertens and C. Weinheimer, Adv. in High Energy Phys., 293986 (2013). See: `mason.gmu.edu/~rehrlich/Kurie_slider.xlsx`
The author declares that there is no conflict of interest regarding the publication of this paper.
| {
"pile_set_name": "ArXiv"
} |
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author:
- |
Enrique Canessa[^1]\
[ ]{}\
[PACS numbers: 05.65.+b, 05.70.-a, 07.05.Mh, 45.80.+r ]{}
title: |
Comment on Phys. Rev. Lett. [**110**]{}, 168702 (2013):\
“Causal Entropic Forces”
---
The recent Letter by Wissner-Gross and Freer [@Wis13] proposes a relationship between intelligence and entropy maximization based on a causal generalization of entropic forces over configuration space paths, which may beautifully induce sophisticated behaviors associated with competitive adaptation on time scales faster than natural evolution. These authors suggest a potentially general thermodynamic model of adaptive behavior as a non-equilibrium process in open systems. On the basis of the force-entropy correlations published by us a decade ago [@Can04], we point out that their main relations have been previously reported within a simpler statistical thermodynamic model where non-interacting moving particles are assumed to form an elastic body.
The claim in [@Wis13] that spontaneous emergence of adaptive behaviour (driven by the systems degrees of freedom $j$ with internal Gaussian forces $f_{j}(t)$) maximizes the overall diversity of accessible future paths is only partially true. There is an alternative approach to understand these complex networks as delineated from a probabilistic perspective within the canonical Gibbs distribution.
In our discretized formalism [@Can04], the probability $p_{i}$ that the system is in the state $i=1, \dots N$, is given by two positive functions satisfying the normalization $\sum_{i=1}^{N} u_{i}w_{i} \equiv 1$. This simple multiplicative form also gives interesting connections between an applied tension and thermodynamics quantities of dynamical systems. Such class of normalized product of positive functions for $p_{i}$ appears formally, [*e.g.*]{}, in the analysis of stochastic processes on graphs according to the Hammersley-Clifford Theorem. As we have shown the product $u_{i}w_{i}$ leads to reveal intrinsic molecular-mechanical properties on classical and non-extensive dynamical systems in relation to a distinct tensile force acting on these systems at constant volume and number of particles with trajectories $x(t) = ({\bf q}(t),{\bf p}(t))$. A new scenario for the entropic $q$-index in Tsallis statistics in terms of the energy of the system was also reported earlier –which has been applied to study, [*e.g.*]{}, brain dynamics.
For completeness the causal entropic forces found in [@Wis13] and [@Can04] and derived from rather alternative thermodynamic analytical models are listed below where the force $f_{i}$ represents variations in the energy states with respect to particle displacements.
-------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------
[**Continous Theory**]{} [@Wis13] [**Discretized Theory**]{} [@Can04]
[*A.D. Wissner-Gross and C.E. Free*]{} [*E. Canessa*]{}
\[5ex\] $F_{j}( {\bf X_{o}},\tau ) = T_{c} \; \frac{ \partial S_{c}({\bf X},\tau) }{ \partial q_{j}(0) }) |_{ {\bf X=X_{o}} }$ $F = k_{B}T \; \frac{\partial}{\partial x}
\left( \frac{\sum_{i=1}^{N} p_{i}^{q}}{q-1} \right)_{T} = T \; \frac{\partial S_{q}}{\partial x}$
\[3ex\] $F_{j}( {\bf X_{o}},\tau ) = -k_{B}T \;\int_{_{{{\bf X}}(t)}} \frac{ \partial \Pr( {\bf x}(t) | {\bf x}(0) ) }{ \partial q_{j}(0) } $ F = - k_{B}T \; \sum_{i=1}^{N} (\frac{\partial p_{i}}{\partial x}) \ln p_{i}$
\ln \Pr( {\bf x}(t) | {\bf x}(0) ) D {\bf x}(t)$
\[3ex\] $\frac{ \partial \Pr( {\bf x}(\epsilon) | {\bf x}(0) ) }{\partial q_{j}(0)} = \frac{2 f_{j}(0)}{k_{B}T} \Pr( {\bf x}(\epsilon) | {\bf x}(0) )$ $\frac{\partial p_{i}}{\partial x} = - ( \frac{\partial \epsilon_{i}}{\partial x} ) \frac{p_{i}}{k_{B}T} \; \rightarrow
\frac{f_{i}}{k_{B}T} \; p_{i}$
\[3ex\] $F_{j}( {\bf X_{o}},\tau ) = - \frac{2 T_{c}}{T_{r}} \int_{_{{{\bf X}}(t)}} f_{j}(0) $F \rightarrow \; - \sum_{i=1}^{N} f_{i} \; p_{i} \ln p_{i}$
\Pr( {\bf x}(t) | {\bf x}(0) ) \ln \Pr( {\bf x}(t) | {\bf x}(0) ) D {\bf x}(t)$
\[3ex\]
-------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------
[99]{}
A.D. Wissner-Gross and C.E. Free, Phys. Rev. Lett. [**110**]{} (2013) 168702 E. Canessa, Physica A [**341**]{} (2004) 165 -also at: arXiv:cond-mat/0403724
[^1]: [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study alternating good-for-games (GFG) automata, i.e., alternating automata where both conjunctive and disjunctive choices can be resolved in an online manner, without knowledge of the suffix of the input word still to be read. We show that they can be exponentially more succinct than both their nondeterministic and universal counterparts. Furthermore, we lift many results from nondeterministic parity GFG automata to alternating ones: a single exponential determinisation procedure, an upper bound to the GFGness problem, a algorithm for the GFGness problem of weak automata, and a reduction from a positive solution to the $G_2$ conjecture to a algorithm for the GFGness problem of parity automata with a fixed index. The $G_2$ conjecture states that a nondeterministic parity automaton $\A$ is GFG if and only if a token game, known as the $G_2$ game, played on $\A$ is won by the first player. So far, it had only been proved for Büchi automata; we provide further evidence for it by proving it for coBüchi automata. We also study the complexity of deciding “half-GFGness”, a property specific to alternating automata that only requires nondeterministic choices to be resolved in an online manner. We show that this problem is strictly more difficult than GFGness check, already for alternating automata on finite words.'
author:
- Udi Boker
- Denis Kuperberg
- Karoliina Lehtinen
- Michał Skrzypczak
bibliography:
- 'gfg.bib'
title: 'On Succinctness and Recognisability of Alternating Good-for-Games Automata'
---
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We numerically investigated the quantum-classical transition in rf-SQUID systems coupled to a dissipative environment. It is found that chaos emerges and the degree of chaos, the maximal Lyapunov exponent $\lambda _{m}$, exhibits non-monotonic behavior as a function of the coupling strength $D$. By measuring the proximity of quantum and classical evolution with the uncertainty of dynamics, we show that the uncertainty is a monotonic function of $\lambda _{m}/D$. In addition, the scaling holds in SQUID systems to a relatively smaller $\hbar _{eff}$, suggesting the universality for this scaling.'
author:
- Ting Mao
- Yang Yu
title: 'Parameter Scaling in the Decoherent Quantum-Classical Transition for chaotic rf-SQUIDs'
---
INTRODUCTION
============
How classical behavior arises in a quantum mechanical system is one of the essential questions in quantum theory, and has long attracted intense interest. The quantum to classical transition (QCT), which has been well understood to be mainly induced by decoherence caused by the coupling with the environment,[@Zurek; @notes] attains some progresses in recent years. It is proposed that the QCT is controlled by relevant parameters including the effective Planck constant $\hbar _{eff}$ (i.e., the relative size of the Planck constant), a measure of the coupling with the environment $D$, and the Lyapunov exponent $\lambda $, for chaotic systems.[@Pattanayak] By computing measures which directly reflect the distance between quantum and classical evolutions, it is shown that the distance is controlled by a composite parameter of the form $%
\zeta =\hbar ^{\alpha }\lambda ^{\beta }D^{\gamma }$. Many efforts on investigating the coefficients $\alpha $, $\beta $, $\gamma $ have been made [@Toscano; @Gammal] in different systems such as the kicked harmonic oscillator and the Duffing oscillator. However, in the previous systems, $%
\lambda $ is generally a constant. Therefore, the direct illustration of the effect of the Lyapunov exponent $\lambda $ on the computed distance is still open.
In this article we try to explore the parameter scaling in QCT by using the system of the superconducting quantum interference device (SQUID). Rf-SQUID system has been demonstrated as a well controllable decoherent quantum system. Macroscopic quantum phenomena such as resonant tunneling[@Rouse] and level quantization[@Silvestrini] and quantum superposition[Friedman]{} have been reported. On the other hand, the strong coupling between the SQUID and the environment can introduce chaos. As early as 1983, the chaotic behavior of the SQUID treated as a semi-classical model had been found.[@Fesser] Recently, a research shows that a three-junction SQUID can be used to study the dynamics of quantum chaos.[@Pozzo] Such works motivate us to study the chaotic behavior of SQUID under decoherence induced by environment, which enables us to directly demonstrate the effect of the Lyapunov exponent on QCT.
This article is organized as follows. In Sec.II we numerically investigate the chaotic dynamics of SQUID with coupling to an external environment, and it is shown that the maximal Lyapunov exponent $\lambda _{m}$, which quantifies the chaotic degree of SQUID, is non-monotonic as a function of $D$, a measure of the coupling. Thus we can say in some regimes of $D$, the chaos of SQUID is suppressed by the decoherence induced by environment[Yamazaki]{}. In Sec.III we use the uncertainty of dynamics as the distance between quantum and classical evolutions, and show that the uncertainty behaves rightly, even in the chaos suppressed region, as a monotonic function of $\lambda _{m}/D$. To the best of our knowledge, this is the first direct demonstration of the scaling relation since it was proposed[Pattanayak]{}.
chaotic dynamics of SQUID
=========================
The rf-SQUID system considered here consists of a large superconducting loop interrupted by a single Josephson junction with a critical current $I_{c}$. Under the driving of a external flux $\phi _{ex}(t)$ with the form of $\phi
_{ex}(0)\cos (\omega _{d}t)$ (where $\phi _{ex}(0)$ and $\omega _{d}$ respectively denote the driving amplitude and driving frequency), the Hamiltonian for the SQUID system can be given as $$\hat{H}_{D}=\frac{\hat{q}^{2}}{2C}+\frac{(\hat{\phi}-\phi _{ex}(t))^{2}}{2L}+%
\frac{I_{c}\phi _{0}}{2\pi }\cos (2\pi \hat{\phi}/\phi _{0}),
\label{Squid Hamiltonian}$$where $C$ is the junction capacitance, $L$ is the rf-SQUID inductance and $%
\phi _{0}=h/2e$ denotes the superconducting flux quantum. The magnetic flux threading the rf-SQUID $\hat{\phi}$ and the total charge on the capacitor $%
\hat{q}$ are the conjugate variables of the system with the imposed commutation relation $[\hat{\phi},\hat{q}]=i\hbar $.
We can rewrite this Hamiltonian into a dimensionless one[@Everitt] as $$\hat{H}_{D}=\frac{\hat{Q}^{2}}{2}+\frac{(\hat{\Phi}-\Phi _{ex}(t))^{2}}{2}+%
\frac{I_{c}}{2\omega _{0}e}\cos (\frac{2e}{\sqrt{\hbar \omega _{0}C}}\hat{%
\Phi}), \label{reduced Squid Hamiltonian}$$in which $\omega _{0}=1/\sqrt{LC}$, $\Phi _{ex}(t)=\sqrt{\frac{\omega _{0}C}{%
\hbar }}\phi _{ex}(t)$, and $\hat{Q}=\sqrt{1/\hbar \omega _{0}C}\hat{q}$, $%
\hat{\Phi}=\sqrt{\omega _{0}C/\hbar }\hat{\phi}$ satisfy the commutation relation $[\hat{\Phi},\hat{Q}]=i$.
Since no chaos can be seen in the dynamics of isolated quantum systems,[Habib]{} to study the chaotic behaviors of the SQUID system, we couple the system to a dissipated environment in the Markovian limit. We adopt the quantum state diffusion (QSD) [@Percival] approach which is widely used in studying open quantum systems [@Brun; @Kapulkin; @Ota] to describe the evolution of this coupled system. The QSD equation for the evolution of the state vector $|\psi \rangle $ reads $$\begin{aligned}
|d\psi \rangle &=&-\frac{i}{\hbar }\hat{H}|\psi \rangle dt+\sum_{j}\Big(%
\langle \hat{L}_{j}^{\dagger }\rangle \hat{L}_{j}-\frac{1}{2}\hat{L}%
_{j}^{\dagger }\hat{L}_{j} \nonumber \label{QSD} \\
&&-\frac{1}{2}\langle \hat{L}_{j}^{\dagger }\rangle \langle \hat{L}%
_{j}\rangle \Big)|\psi \rangle dt+\sum_{j}(\hat{L}_{j}-\langle \hat{L}%
_{j}\rangle )|\psi \rangle d\xi _{j},\end{aligned}$$where $\hat{H}$ is the system Hamiltonian and $\hat{L}_{j}$ are the Lindblad operators representing the coupling with the environment. $d\xi _{j}$ are independent complex differential Gaussian random variables satisfying $%
M(d\xi _{j})=M(d\xi _{i}d\xi _{j})=0$, $M(d\xi _{i}^{\ast }d\xi _{j})=\delta
_{ij}dt$ (where $M$ denotes the ensemble mean). For the SQUID system considered here, we have $\hat{H}$ and $\hat{L}$ for Equation (\[QSD\]) as $\hat{H}=\hat{H}_{D}+\hat{H}_{R}$, $\hat{L}=\sqrt{D}(\hat{\Phi}+i\hat{Q})$, where $\hat{H}_{D}$ is shown in Equation (\[reduced Squid Hamiltonian\]), $%
\hat{H}_{R}=\frac{D}{2}(\hat{\Phi}\hat{Q}+\hat{Q}\hat{\Phi})$ [Brun,Kapulkin]{} is a damping term added to recover the correct equation of motion in the classical limit, and $D$ is the strength of the coupling with the environment mentioned in the beginning.
{width="3.4in"}
Using the powerful QSD library,[@Schack] we numerically solve the Equation (\[QSD\]) and investigate the change in the dynamics of the SQUID system when increasing the strength of dissipation. A typical set of SQUID parameters is selected here, $C=0.1pF$, $L=300pH$, $I_{c}=2.2\mu A$, $\omega
_{d}=1.14\omega _{0}$, $\phi _{ex}(0)=0.2684\phi _{0}$, which insures the action of this system is small enough compared with fixed $\hbar $.[Habib]{} Then we examine 28 different values of $D$ from slightly dissipated ($%
D=0.23$) to heavily dissipated regime ($D=1$) in our calculation, during which we have the same initial state $|\psi (t=0)\rangle =|\sqrt{2}(\langle
\hat{\Phi}\rangle +i\langle \hat{Q}\rangle )=(0.877-0.566i)\rangle $–the coherent state–and same realization of generating the random numbers. The quantum Poincaré sections, which each comprises of $500$ points taken at a fixed phase of the external driving once a driving period, are shown for three representative values of $D$ in Fig.1(a)-1(c). It can be clearly seen in Fig.1(a) that points forms a uniformly stretched Poincaré profile in the phase space which indicates chaosfor $D=0.25$. However, for $D=0.35$ most of points are confined in three relatively small regions as shown in Fig.1(b), which indicates the suppression of chaos. Then the Poincaré profile similar to the one in Fig.1(a) is recovered in Fig.1(c) when $D$ is increased to $0.45$. Some non-monotonic analogous phenomena have been studied in classical chaotic systems,[Yamazaki,Matsumoto]{} and a qualitative explanation has been proposed there. If the chaotic attractors are narrowly and non-uniformly distributed in phase space, the fluctuation induced by dissipation may cause the neighboring trajectory jump over it, which results in the suppression of chaos. While further increasing dissipation intensity, the structure of the chaotic attractor may be modified and thus spread wider than before. Therefore the system becomes chaotic again. Since $(\langle \hat{\Phi}%
(t)\rangle ,\langle \hat{Q}(t)\rangle )$ form classical-like trajectories in our calculation, we expect that the explanation is also valid for the suppression of chaos in quantum region.
{width="3.4in"}
To describe this transition of chaos quantitatively, we calculate the maximal Lyapunov exponent $\lambda_{m}$ for a time series–the expectation value of the magnetic flux $\langle\hat{\Phi}(t)\rangle$–at each value of $%
D $. The calculation is based on the method and programs [@Kantz; @Hegger] which are specifically designed for the analysis of nonlinear time series. With carefully chosen parameters as the delay time $d=3$, the embedding dimension $m=3$ and the scaling length $s=1.4\%$ for the calculation to best meet the requirements in Ref.18, the sufficient convergency of the Lyapunov exponent is guaranteed. The result is shown in Fig.2, in which the graph of $\lambda _{m}$ versus $D$ has a distinctive dip in a approximate region of $D=0.25\sim 0.45,$ indicating the suppression of chaos. We also repeat the whole calculation above in some different realization of random numbers with the SQUID parameters and the initial state fixed, and find the curves are quite analogous to the one in Fig.2.
{width="3.5in"}
effect of maximal lyapunov exponent on QCT
==========================================
With the non-monotonic relationship between maximal Lyapunov exponent $%
\lambda _{m}$ and the strength of the coupling with the environment $D$, we can directly investigate the effect of $\lambda _{m}$ on QCT. To measure the distance between quantum and classical evolution, we use the well known quantity–the uncertainty of dynamics $\Delta =\sqrt{\langle (\hat{\Phi}%
-\langle \hat{\Phi}\rangle )^{2}\rangle }\sqrt{\langle (\hat{Q}-\langle \hat{%
Q}\rangle )^{2}\rangle }$, which is simple for calculation and adequate to describe the QCT. According to the commutation relation $[\hat{\Phi},\hat{Q}%
]=i$, it follows that $\Delta \geq 0.5$. By solving Equation (\[QSD\]) with same calculating parameters as in Sec.II, we get a time series of the uncertainty $\Delta (t)$ at each value of $D$. After averaging each series of $\Delta (t)$ over a reasonably long time ($>$ 100 periods of the external driving), we obtained the curve of the averaged uncertainty $%
\Delta _{a}$ versus $D$ and showed in Fig.3(a), where $D$ has the same sequence of values as in Fig.2. It can be clearly seen that in Fig.3(a) a obvious dip emerges in the very regime where chaos is suppressed by the dissipation, which implies QCT directly depends on the degree of chaos. Motivated by this, we combine $\lambda _{m}$ and $D$ with the form of $%
\lambda _{m}/D$ which is inferred in Ref.2 and look into the relationship between $\Delta _{a}$ and such composite single parameter. Shown Fig.3(b) is an example of $\Delta _{a}$ vs. $\lambda _{m}/D$. One can find that the dip is rubbed out and $\Delta _{a}$ approximately shows a monotonic increasing in $\lambda _{m}/D$ with two distinct regimes of small and large increasing rates.[@Pattanayak] Therefore we demonstrate the scaling between $\lambda _{m}
$ and $D$ holds over a considerable range in $\Delta _{a}$. It is noticed that the points which lie in the dip in Fig.3(a) spread slightly around the curve in Fig.3(b). We conjecture this spread could be mainly attributed to the calculating error [@Kantz] of $\lambda _{m}$ which is induced by the inevitable quantum noise added into the trajectory of $(\langle \hat{\Phi}%
(t)\rangle ,\langle \hat{Q}(t)\rangle )$, especially when chaos is suppressed and the value of $\lambda _{m}$ is comparatively small.
{width="3.6in"}
$\hbar_{eff}$ $I_{c}(\mu A)$ $L(pH)$ $C(pF)$ $\omega_{d}(\omega_{0})$ $\phi_{ex}(0)(\phi_{0})$
--------------- ---------------- --------- --------- -------------------------- --------------------------
1 4.6 100 3.95 0.65 0.081
1.2 3.35 150 2.16 0.71 0.1041
1.4 2.67 200 1.29 0.78 0.1273
1.9 2.46 250 0.36 0.99 0.1851
: $\hbar_{eff}$ and corresponding parameters
{width="3.9in"}
Now we examine this scaling law for the SQUID system with a smaller effective Planck constant $\hbar_{eff}$. To obtain a smaller $\hbar_{eff}$, it is not straightforward for the SQUID system to directly manipulate the value of $\hbar$ .[@Everitt]Instead, we enlarge the action of the SQUID system simply by changing parameters in the Hamiltonian; the larger the action the smaller $\hbar_{eff}$, and vice versa.[@Habib] By deliberately selecting the set of parameters including $I_{c}$, $L$, $C$, $\omega _{d}$ and $\phi _{ex}(0)$, we can enlarge the action and maintain the chaotic dynamics of the system at the same time. The values of these parameters are not difficult to modulate for a realistic SQUID system where $I_{c}$ could become controllable by replacing the single Josephson junction with a small loop (dc SQUID) which contains two identical Josephson junctions,[@Rouse] $C$ and $L$ are both under the upper realistic limit of typical Josephson junctions. We select four sets of parameters for the SQUID systems each of which has a smaller $\hbar _{eff}$ compared with the foregoing system’s. Assuming the smallest $\hbar _{eff}$ is equal to 1 and comparing the actions of the systems which are measured with the system size,[@Habib] we approximately gain the value of other effective Planck constants as follow, 1.2, 1.4, 1.9, 2.6, where 2.6 is the value of the foregoing system’s $\hbar _{eff}$. Then we apply the same calculating procedures to these systems, and the results are shown in Fig.4 and Fig.5 which also include the data of the foregoing system for comparison. Fig.4 shows the averaged uncertainty $\Delta _{a}$ as a function of $D$, $\hbar _{eff}$. For each $\hbar _{eff}$,a distinct dip exists as expected in the region where chaos is suppressed by the dissipation of environment. Fig.5 shows the same data plotted as a function of $\lambda
_{m}/D$, in which, the behavior of $\Delta _{a}$ for each $\hbar _{eff}$ is considerably the same, which demonstrates the scaling between $\lambda _{m}$ and $D$ is still valid for a system with relatively small $\hbar _{eff}$. For clarity, we separately show the curve with $\hbar _{eff}=1$ in the inset of Fig.5. Since a larger action is helpful to undermine the effect of quantum noise, more accurate $\lambda _{m}$ can be gained for the system with smaller $\hbar _{eff}$, which, is reflected in the lack of noticeable spread around the curve in the inset.
We also chose some different random numbers generator to repeat the calculation for SQUID systems with different $\hbar _{eff}$, and succeed in getting same qualitative conclusions as discussed above.
conclusion
==========
In summary, we investigated QCT in chaotic rf-SQUIDs. The suppression of chaos induced by environment dissipation was observed in quantum regime. It is found that the quantum to classical transition in the presence of a dissipated environment is governed by a composite parameter $\lambda _{m}/D$. It could be expected the scaling law between $\lambda _{m}$ and $D$ would holds over a wide range of $\hbar _{eff}$. However, to generalize this scaling to the one involving $\hbar _{eff}$, $\lambda _{m}$ and $D$ and to reveal the coefficients between them are still open questions needed to explore.
ACKNOWLEDGMENTS
================
This work was partially supported by the NSFC (under Contract Nos. 10674062,10725415), the State Key Program for Basic Research of China (under Contract Nos. 2006CB921801), and the Doctoral Funds of the Ministry of Education of the People’s Republic of China (under Contract No. 20060284022 ).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'To study how Andreev reflection (AR) is affected by itinerant antiferromagnetism, we perform $d$-wave AR spectroscopy with superconducting YBa$_2$Cu$_3$O$_{7-\delta}$ on TiAu and on variously-oxidized Nb (NbO$_x$) samples. X-ray photoelectron spectroscopy is also used on the latter to measure their surface oxide composition. Below the Néel temperatures ($T_N$) of both TiAu and NbO$_x$, the conductance spectra show a dip-like structure instead of a zero-bias peak within the superconducting energy gap; for NbO$_x$, higher-oxidized samples show a stronger spectral dip at zero bias. These observations indicate that itinerant antiferromagnetic order suppresses the AR process. Interestingly, the spectral dip persists above $T_N$ for both TiAu and NbO$_x$, implying that spin fluctuations can also suppress AR. Our results suggest that $d$-wave AR spectroscopy may be used to probe the degree of spin ordering in itinerant antiferromagnets.'
author:
- 'C. R. Granstrom'
- 'R.-X. Liang'
- 'Y. Li'
- 'P. Li'
- 'Z. - H. Lu'
- 'E. Svanidze'
- 'E. Morosan'
- 'J. Y.T. Wei'
bibliography:
- './Bibliography.bib'
title: 'Probing itinerant antiferromagnetism with $d$-wave Andreev reflection spectroscopy'
---
There is general interest in the interplay of superconductivity and itinerant magnetism, both on a fundamental level and for technological applications [@Buzdin2005; @Dai2012]. At a normal-metal/superconductor (N/S) interface, Andreev reflection (AR) is the process that converts electrons into Cooper pairs through retro-reflection of holes [@Andreev1964; @Blonder1982]. There has been a considerable amount of theoretical and experimental work on AR in itinerant ferromagnet/superconductor interfaces [@Jong1995; @Upadhyay1998; @Soulen1998; @Ji2001; @parker2002; @Zutic2004; @Buzdin2005; @Nadgorny2011; @Turel2011], where AR has been utilized to probe spin-polarization. In contrast, there has been very little work on AR for itinerant antiferromagnet/superconductor (IAFM/S) interfaces, aside from the theoretical work in Refs. [@Andersen2002; @Andersen2005; @Bobkova2005]. However, there are topics of fundamental and applied importance that motivate study of such interfaces. For example, the interplay between antiferromagnetism and superconductivity is thought to be important in the high critical temperature ($T_c$) superconducting cuprates [@Moriya1990; @Monthoux1992; @Monthoux1994], and in the high-$T_c$ iron-pnictides there is thought to be coexistence of superconducting and spin-density wave (SDW) states [@Hirschfeld2011; @Dai2012]. On the technological side, Josephson junctions involving IAFMs are predicted to exhibit unique properties [@Gorkov2001], and it is desirable to find new probes to characterize IAFMs, e.g. to use for antiferromagnetic spintronics applications [@Gomonay2017; @Baltz2018].
There has been recent theoretical work for AR onto IAFMs, and in particular, AR with $d$-wave superconductors (dSCs) [@Andersen2002; @Andersen2005; @Bobkova2005]. In analogy with AR in N/S junctions, a N/IAFM junction is predicted to exhibit a spin-dependent Q-reflection, whereby the SDW gap $\Delta_{\mathrm{SDW}}$ in the IAFM plays the role of the superconducting gap. In Q-reflection, quasiparticles with energy $E<\Delta_{\mathrm{SDW}}$ and momentum $\mathbf{k}_F$ incident upon the IAFM undergo a spin-dependent retroreflection into states with momentum $\mathbf{k}_F+\mathbf{Q}$, where $\mathbf{Q}$ is the AFM wavevector [@Andersen2005; @Bobkova2005]. If Q-reflection is combined with AR in an dSC/IAFM junction, a variety of low-energy interfacial bound states are predicted to form, showing up as peaks in the differential conductance spectrum.
$d$-wave AR spectroscopy using superconducting YBa$_{2}$Cu$_{3}$O$_{7-\delta}$ (YBCO) tips and films is potentially a powerful technique for probing IAFMs, as it is expected to give spin-sensitive information on electronic states at the sub-nanometer length scale over a wide temperature range ($\sim 0-90$ K) [@Turel2011; @Granstrom2018]. The sensitivity to spin polarization of such $d$-wave AR measurements was demonstrated with nanoscale YBCO point contacts on both Au and CrO$_2$, the latter showing suppression of $d$-wave AR as expected for a half-metallic ferromagnet [@Turel2011], and recently with non-contact tip-sample junctions onto another half-metallic ferromagnet La$_{2/3}$Ca$_{1/3}$MnO$_{3}$ (LCMO), which also showed suppression of $d$-wave AR [@Granstrom2018].
Compound Oxidation state $\rho$ ($\Omega$ cm) $\Theta_{\mathrm{CW}}$ (K) $T_N$ (K)
------------------------------------ ----------------- ----------------------------------- ---------------------------- -----------
Nb$_{2}$O$_{5}$ (NbO$_{2.5}$) +5 $3\times10^{4}$ — —
Nb$_{25}$O$_{62}$ (NbO$_{2.48}$) +5 $3\times10^{-1}$ 0 —
Nb$_{47}$O$_{116}$ (NbO$_{2.468}$) +5 $1.6\times10^{-2}$ 7 —
Nb$_{22}$O$_{54}$ (NbO$_{2.455}$) +5 $1.5\times10^{-2}$ 12 —
Nb$_{12}$O$_{29}$ (NbO$_{2.417}$) +5 $4\times10^{-3}$ 24 12
NbO$_{2}$ +4 $6.3\times10^{3}$ [@Janninck1966] — —
NbO +2 $2.1\times10^{-5}$ [@Hulm1972] — —
In this work, we perform $d$-wave AR spectroscopy with superconducting YBCO to probe itinerant antiferromagnetism in two systems. One system is TiAu, a recently-discovered IAFM with no magnetic constituents [@Svanidze2015]. The other system is the Nb oxides (NbO$_x$), which have tunable antiferromagnetic order [@Cava1991] and potentially itinerant antiferromagnetism [@McQueen2007], both arising from oxygen vacancies. X-ray photoelectron spectroscopy (XPS) is also used on the latter to measure their surface oxide composition. Below the Néel temperatures ($T_N$) of both TiAu and NbO$_x$, the conductance spectra show a dip-like structure instead of a zero-bias peak within the superconducting energy gap; for NbO$_x$, higher-oxidized samples show a stronger spectral dip at zero bias. These observations indicate that itinerant antiferromagnetic order suppresses the AR process. Interestingly, the spectral dip persists above $T_N$ for both TiAu and NbO$_x$, implying that spin fluctuations can also suppress AR. Our results suggest that $d$-wave AR spectroscopy may be used to probe the degree of spin ordering in itinerant antiferromagnets.
TiAu antiferromagnetically orders below 36 K, and XPS data suggests that Ti is close to its non-magnetic $4+$ oxidation state, ruling out the presence of local moments [@Svanidze2015]. Muon spin-relaxation data indicates 100% volume fraction of magnetic order at 0 K and strong spin-fluctuations, but the exact role and strength of the latter are not currently known. Neutron diffraction measurements indicate long-range antiferromagnetic order, with a small itinerant moment of 0.15 $\mu_B$ per Ti atom. The fact that TiAu has no magnetic constituents defies existing theories [@Svanidze2015].
The most stable of NbO$_x$, Nb$_{2}$O$_{5}$, is electrically insulating when pure. However, it tends to have *local* oxygen vacancies [@Grundner1984; @Halbritter1987], creating local moments via Nb$^{4+}$ ions. Additionally, the constituent NbO$_{6}$ octahedra of Nb$_{2}$O$_{5}$ can accommodate extended oxygen vacancies via crystallographic shear [@VanLanduyt1974; @Nico2016], forming several ordered and non-stoichiometric Nb$_{2}$O$_{5-x}$ compounds (table \[tab:NbOx\]). As $x$ increases in Nb$_{2}$O$_{5-x}$, electrical conductivity and antiferromagnetic coupling between local moments increase, until finally Nb$_{12}$O$_{29}$ antiferromagnetically orders with a Néel temperature of 12 K. Of the Nb$_{2}$O$_{5-x}$ compounds, only Nb$_{12}$O$_{29}$ is metallic, and it is generally believed that some of its Nb$^{4+}$ sites have localized magnetic electrons, while other sites have itinerant and non-magnetic electrons [@Cava1991; @Waldron2004; @Ohsawa2011]. However, Nb$_{12}$O$_{29}$ may also be an IAFM, as recent work suggests its antiferromagnetism comes from delocalized electrons [@McQueen2007].
Proslier and Cao et al. recently studied the influence of Nb$_{2}$O$_{5}$ on the superconductivity of Nb using XPS, point-contact tunneling (PCT) with normal-metal tips, and magnetic susceptibility measurements on NbO$_x$ samples [@Proslier2008; @Proslier2008a; @Proslier2009; @Proslier2011; @Cao2014]. Their PCT spectra showed signs of antiferromagnetic exchange-scattering between tunneling electrons and local moments in oxygen-deficient Nb$_{2}$O$_{5}$, as described by the Anderson-Appelbaum theory [@Appelbaum1966; @Anderson1966; @Appelbaum1967]. Magnetic susceptibility vs. temperature measurements on these same samples showed Curie-Weiss-like temperature dependence with positive Curie temperatures, also indicating antiferromagnetic coupling [@Proslier2011; @Cao2014]. The success of these normal-metal tip PCT studies in detecting signatures of antiferromagnetism motivates us to extend them to using superconducting $d$-wave YBCO tips, where the high $T_c$ of 90 K allows spin ordering to be probed over a wide temperature range.
In the present work, $d$-wave AR spectroscopy measurements were made using point-contact and scanning tunneling spectroscopy geometries. These measurements were made using a home-built $^4$He dipper-probe scanning tunneling microscope (STM), designed to allow tip and sample loading within a glovebox filled with dry N$_2$ gas. Measurement electronics are described elsewhere [@Granstrom2018; @Granstrom2016]. XPS measurements were performed with a PHI 5500 analytical chamber using monochromated Al K$_{\alpha}$ radiation of 1486.7 eV at a take-off angle of 75$^{\circ}$ (for details see Ref. [@Adinolfi2016]).
The YBCO single crystals used as tips were grown using the self-flux technique [@Liang2012]. The preparation of the crystals for use as a tip is described elsewhere [@Granstrom2018]. The Nb tips were cut from 0.25mm diameter wire and were passively oxidized by leaving in air for an hour. The Nb foils used as samples were 25 $\mu$m thick and of 99.8% purity. The Nb films used as samples were 30-200 nm thick and were grown on natively-oxidized Si wafers by sputtering. To provide a control sample for AR measurements, 40 nm thick Ag films were grown on mica by sputtering. Natively-oxidized Nb samples were reduced either by sputtering with 0.5-3 keV Ar ions in a background pressure of 0.1 $\mu$Torr, or annealing between 880-930 $^{\circ}$C under a vacuum of 2-4 $\mu$Torr for 25-105 hours. Immediately following each reduction treatment, samples were kept in a dry N$_2$ environment before quickly being mounted either onto the STM inside the glovebox for AR spectroscopy measurements, or into a vacuum chamber for XPS measurements. The (110)-YBCO films were 100 nm thick and grown by pulsed laser deposition on (110)- (LaAlO$_{3}$)$_{0.3}$(Sr$_{2}$TaAlO$_{6}$)$_{0.7}$ (LSAT) substrates, and were similarly transported in an N$_2$ filled container before being loaded onto the STM.
The TiAu polycrystals were grown by arcmelting [@Svanidze2015], and were cut into $\sim$ 5 x 5 x 2 mm$^3$ slabs for use as a sample, or $\sim$ 1 x 1 x 5 mm$^3$ bars for use as a tip. The slabs were polished to a mirror finish, cleaned with Ar ion sputtering, then quickly transferred to an N$_2$ filled glovebox, where they were loaded onto the STM. The bars were broken inside the glovebox to expose a fresh edge, before being mounted onto the STM. Since both TiAu and Nb have chemically reactive surfaces at room temperature in air, various tip/sample combinations were measured.
![\[fig:XPS\] XPS measurements of oxidized Nb samples. (a) and (b): The surface of natively-oxidized samples is predominantly Nb$_{2}$O$_{5}$, as schematically shown in (g). Layer roughness indicates non-uniformity in oxide depth composition [@Darlinski1987]. (c) and (d): Ar-ion sputtering thins the Nb$_{2}$O$_{5}$ surface layer by injecting oxygen into bulk Nb, thickening the lower oxidation states beneath the surface as shown in (h). (e) and (f): Vacuum annealing reduces the surface oxides similar to Ar-sputtering. Intensity is normalized to the 207.55 eV peak of Nb$_{2}$O$_{5}$. Colors of curves match corresponding samples in figure \[fig:YBCO\_PCS\_Nb\_Ag\](a). ](./XPS_NbOx.eps){width="49.00000%"}
![\[fig:YBCO\_PCS\_Nb\_Ag\] (a) Normalized point-contact spectra at 6 K between YBCO tips and NbO$_x$ samples reduced similarly to the NbO$_x$ samples in figure \[fig:XPS\] (colors of data match figure \[fig:XPS\]). The depth of the ZBC dip here and the Nb$_{2}$O$_{5}$ spectral weight in figure \[fig:XPS\] are clearly correlated. (b) Raw data at 6 K on a Ag film and (c) Nb foil. Both spectra exhibit a similar conductance background and peaks near $\pm$ 20 mV, but the Ag data show a ZBC peak while the Nb data show a ZBC dip. (d) and (e): Temperature dependence of normalized data from panels (b) and (c), respectively. The ZBC features weaken at higher temperatures and disappear at YBCO’s $T_{c}$ of 90 K. Insets of panels (b) and (c) show temperature evolution of ZBC for panels (d) and (e), respectively. ](./PCS_Ag_NbOx.eps){width="45.00000%"}
We first demonstrate control over the surface oxide composition of NbO$_x$ with XPS, which is sensitive to the top $\sim$10 nm of our samples’ surfaces [^1]. Figure \[fig:XPS\] shows XPS measurements of the Nb 3d core level. Panels (a) and (b) show that the surface of both natively-oxidized foils and films is predominately Nb$_{2}$O$_{5}$, as expected. The film data show asymmetric Nb peaks, indicating the presence of lower Nb oxidation states beneath the surface Nb$_{2}$O$_{5}$, as shown schematically in panel (g) and reported by others [@Darlinski1987]. Note that it is difficult to distinguish stoichiometric Nb$_{2}$O$_{5}$ from non-stoichiometric Nb$_{2}$O$_{5-x}$ compounds with XPS [@Ohsawa2011], since the predominant oxidation state is +5 in all compounds. Nevertheless, in practice numerous oxygen vacancies exist in Nb$_{2}$O$_{5}$ [@Grundner1984; @Halbritter1987; @Proslier2011], so our large Nb$_{2}$O$_{5}$ peaks suggest that a combination of Nb$_{2}$O$_{5}$, non-stoichiometric Nb$_{2}$O$_{5-x}$ compounds, and locally oxygen-deficient Nb$_{2}$O$_{5}$ are present in our samples.
The XPS spectra in figures \[fig:XPS\](c) and (d) show that sputtering with 3 keV Ar ions shifts spectral weight from Nb$_{2}$O$_{5}$ to lower Nb oxidation states, reducing the film more than the foil. Ar-sputtering oxidized Nb is believed to both preferentially remove oxygen from Nb$_{2}$O$_{5}$ and diffuse oxygen into bulk Nb [@Karulkar1981], thickening the lower oxidation states beneath the surface as shown schematically in panel (h). Panels (e) and (f) show that vacuum annealing has a similar effect as Ar-sputtering: heating oxidized Nb to $\sim900$ $^{\circ}$C in low oxygen partial pressures diffuses oxygen atoms from Nb$_{2}$O$_{5}$ into bulk Nb, while temperatures above 1600 $^{\circ}$C are needed to completely evaporate the oxygen [@Strongin1972]. These XPS data clearly demonstrate that we can significantly thin the native Nb$_{2}$O$_{5}$ surface layer of Nb samples with Ar-sputtering and vacuum annealing.
Next, we show that the reduction treatments in figure \[fig:XPS\](c)-(f) enhance AR. Figure \[fig:YBCO\_PCS\_Nb\_Ag\](a) shows point-contact spectra taken with YBCO tips on Nb samples oxidized similarly to those in figure \[fig:XPS\], where the data colors match between the two figures. The depth of the ZBC dip in figure \[fig:YBCO\_PCS\_Nb\_Ag\](a) and the Nb$_{2}$O$_{5}$ spectral weight in figure \[fig:XPS\] are clearly correlated. Reduced samples occasionally exhibit ZBC peaks (stars), suggesting spatial variation of the surface oxide composition in the reduced samples. The point-contact spectrum for the 0.5 keV Ar-sputtered film (squares) shows a higher ZBC than the 3 keV Ar-sputtered film (triangles), consistent with lower-energy Ar ions more effectively reducing Nb$_{2}$O$_{5}$ [@Karulkar1981]. Estimates of the tip-sample junction sizes using the Wexler formula [@Wexler1966] show that all junctions are ballistic [^2], indicating that the ZBC features in figure \[fig:YBCO\_PCS\_Nb\_Ag\](a) are not due to junction imperfections.
![\[fig:YBCO\_PCS\_Nb\_TiAu\] Normalized point-contact spectra taken between YBCO and both NbO$_x$ and TiAu. For comparison, a spectrum for a YBCO tip onto an Ag film is also shown. Both the Nb and TiAu samples show a strong ZBC dip, in contrast to the ZBC peak in the YBCO-Ag spectrum.](./PCS_NbOx_TiAu.eps){width="40.00000%"}
To confirm that the ZBC dips in figure \[fig:YBCO\_PCS\_Nb\_Ag\](a) are from AR suppression, we compare raw point-contact spectra taken on a Ag film and an Ar-sputtered Nb foil in figures \[fig:YBCO\_PCS\_Nb\_Ag\](b) and (c), respectively. The YBCO-Ag data show a prominent zero-bias conductance (ZBC) peak, flanked by weak peaks near $\pm$ 20 mV. Similar ZBC peaks have been observed in a variety of point-contact studies on YBCO, primarily on crystal faces not normal to a principal crystal axis, and attributed to $d$-wave Andreev resonance [@Deutscher2005]. As for the side peaks, their energies are comparable to the superconducting gap maximum of YBCO [@Wei1998a; @Sharoni2001; @Ngai2007]. A quasi-linear conductance background is also present, as commonly observed for YBCO and other cuprates [@Geerk1988; @Sun1994; @Wei1998]. The YBCO-Nb foil data show similar features, except a ZBC dip is present instead of a ZBC peak. The temperature dependence of the ZBC features in these junctions is shown in their respective insets, which were extracted from the normalized data in panels (d) and (e). As temperature increases, the ZBC features weaken and finally disappear above YBCO’s $T_{c}$ of 90 K, confirming that they are related to AR.
![\[fig:TiAu\_STS\] Tunneling spectra taken between YBCO and TiAu, as well as a reference PtIr tip-(110) YBCO film spectrum, shifted for clarity. The TiAu-YBCO spectra show a ZBC peak of comparable height as the PtIr-YBCO spectrum. ](./STS_TiAu.eps){width="45.00000%"}
We next show that similar point-contact spectra are obtained with natively-oxidized Nb tips on (110)-oriented YBCO films, as well as TiAu tip-YBCO film junctions. Figure \[fig:YBCO\_PCS\_Nb\_TiAu\] shows normalized point-contact spectra taken between YBCO and both TiAu and oxidized Nb samples. Both the Nb oxides (squares and triangles) and TiAu (pentagons) exhibit a strong ZBC dip, in contrast to the ZBC peak in the YBCO-Ag spectrum (circles). The stronger ZBC dip for the Nb tip-YBCO film spectrum (triangles) compared to the YBCO tip-Nb foil spectrum (squares) is consistent with the different preparation procedures used for Nb. That is, when used as a film it was sputtered with 3 keV Ar ions, which XPS showed to significantly thin the native Nb$_2$O$_5$ layer (figure \[fig:XPS\]d). In contrast, when used as a tip the Nb was natively oxidized, so it likely had a relatively thick Nb$_2$O$_5$ layer, similar to the natively-oxidized Nb foil XPS data (figure \[fig:XPS\]a). As figure \[fig:YBCO\_PCS\_Nb\_Ag\](a) showed, Nb samples with thicker Nb$_2$O$_5$ layers exhibit stronger AR suppression, consistent with the above observations.
Having shown that TiAu- and NbO$_x$-YBCO point-contact junctions exhibit AR suppression, we now examine conductance spectra from TiAu-YBCO tunnel junctions, shown in figure \[fig:TiAu\_STS\] along with a reference PtIr-YBCO junction. The PtIr-YBCO spectrum (triangles) exhibits a prominent ZBC peak flanked by side peaks near $\pm$ 20 mV, as expected from $d$-wave Andreev resonance. The TiAu-YBCO data (squares and circles) show a ZBC peak of similar size to the PtIr-YBCO junction. The fact that we observe ZBC peaks for TiAu tunnel junctions indicates that AR suppression is the origin of the ZBC dip in the YBCO-TiAu point-contact junctions.
Finally, we examine conductance spectra from YBCO-NbO$_x$ tunnel junctions, shown in figure \[fig:Nb\_STS\] along with a reference YBCO-Ag junction. The YBCO-Ag spectrum (circles) again exhibits a prominent ZBC peak. The YBCO-NbO$_x$ data (lines and diamonds) show a variety of ZBC features depending on the tip’s $(x,y)$ position on the sample, including a strong ZBC peak, weak ZBC peak, and ZBC peaks split by 6-8 mV. Again, the fact that we observe ZBC peaks for these NbO$_x$ tunnel junctions provides further evidence that AR suppression is the origin of the ZBC dips in YBCO-NbO$_x$ point-contact junctions.
![\[fig:Nb\_STS\] Tunneling spectra taken between YBCO and NbO$_x$, as well as a reference YBCO tip-Ag film spectrum, shifted for clarity. The Nb film was sputtered with 3 keV Ar ions. Compared to the YBCO-Ag film spectra, the YBCO-NbO$_x$ spectra show strong ZBC peaks, weak ZBC peaks, and ZBC peaks split by 6-8 mV.](./STS_NbOx.eps){width="45.00000%"}
We now discuss our point-contact AR data, which show that both TiAu and NbO$_x$ suppress $d$-wave AR. To understand this suppression, we consider the interaction of AR and Q-reflection in $d$-wave superconductor/IAFM (dSC/IAFM) junctions, as described in Refs. [@Andersen2005; @Bobkova2005]. For no interfacial tip/sample barrier, the ZBC peak from $d$-wave Andreev resonance is predicted to be suppressed [@Andersen2005], in which the quasiparticles Q-reflected from the IAFM can re-enter the dSC and disrupt the Andreev resonance. This disruption of AR initially would seem to agree with our observed ZBC dips. However, a finite interfacial barrier is present in all experimental junctions, and such junctions are predicted to show ZBC peaks as well as asymmetric peaks at other energies [@Andersen2005], in disagreement with our data. This disagreement is likely related to discrepancies between the assumptions used in the theoretical work in Ref. [@Andersen2005] and our measurementsnamely, the orientation of the IAFM lattice with respect to the sample normal direction, and the relative size of the magnetic and superconducting order parameters in the IAFM and dSC, respectively. Regardless, our observed sensitivity of $d$-wave AR to itinerant antiferromagnetism in point-contact junctions is qualitatively consistent with the Q-reflection mechanism. Theoretical models that allow for arbitrary orientation of the IAFM lattice with respect to the sample normal direction would help in more accurately modeling our point-contact data. Considering that the energies of the aforementioned asymmetric peaks are predicted to depend on the size of the magnetic order parameter in the IAFM, such theoretical models could potentially be utilized to extract quantitative information about the antiferromagnetic ordering in IAFMs from point-contact measurements like ours.
In understanding our YBCO-NbO$_x$ point-contact and tunneling spectra, one factor to consider is the Nb$_{12}$O$_{29}$ content in the NbO$_x$ samples, since the former is the only Nb oxide that is a candidate IAFM. On the one hand, while our NbO$_x$ samples likely have regions of Nb$_{12}$O$_{29}$, they are certainly not pure Nb$_{12}$O$_{29}$, as shown by our XPS data and noting that a specific annealing procedure is required to isolate this compound in bulk form [@Cava1991]. However, since Nb$_{12}$O$_{29}$ is the most conductive oxide [^3] in table \[tab:NbOx\], our YBCO-NbO$_x$ junctions are likely preferentially probing Nb$_{12}$O$_{29}$ regions. Furthermore, since AR is only known to be sensitive to itinerant moments, our AR data are likely preferentially sensitive to Nb$_{12}$O$_{29}$. To clarify whether our YBCO-NbO$_x$ data are strongly influenced by the presence of other NbO$_{x}$ compounds, it would be useful to perform point-contact and tunneling measurements between YBCO and pure Nb$_{12}$O$_{29}$ crystals.
One interesting feature of both our TiAu- and NbO$_x$-YBCO point-contact data is its temperature dependence. Surprisingly, both types of junctions show AR suppression all the way up to YBCO’s $T_c$ of 90 K (TiAu also shows suppression up to $T_c$, not shown). If the AR suppression was solely due to Q-reflection, one would reasonably expect the AR suppression to subside near the Néel temperature $T_N$ of TiAu (36 K) and Nb$_{12}$O$_{29}$ (12 K). Thus, our observed AR suppression above $T_N$ implies that antiferromagnetic ordering is not necessary to suppress AR. Rather, spin fluctuations may be able to suppress AR. Spin fluctuations are thought to be important in IAFMs [@Hasegawa1974], and indeed, the low $T_N$ and small moment of 0.15 $\mu_B$ in TiAu are consistent with the presence of spin fluctuations [@Svanidze2015]. To investigate the sensitivity of $d$-wave AR to spin fluctuations, it might be useful to extend the models from Refs. [@Andersen2005; @Bobkova2005] to include the effects of disorder in the AFM lattice. For NbO$_x$, if Nb$_{12}$O$_{29}$ is indeed an IAFM, then spin fluctuations would likely also be important. Furthermore, the local moments from oxygen vacancies in Nb$_2$O$_5$ are known to be sources of localized charge and spin [@Proslier2008; @Proslier2008a; @Proslier2009; @Proslier2011] and can antiferromagnetically exchange-scatter tunneling electrons, which has been measured up to 40 K and is expected to persist to higher temperatures [@Proslier2011; @Cao2014]. While it is not experimentally established that AR is affected by local moments, it seems plausible that AR could be upset via spin-flip effects from local moments. More theoretical work is needed to investigate this possibility.
Our tunnel junctions involving TiAu show no AR suppression, which is qualitatively consistent with the Q-reflection work in Refs. [@Andersen2002; @Andersen2005; @Bobkova2005]. Namely, the ZBC peak from $d$-wave Andreev resonance is only predicted to be suppressed in IAFM/dSC junctions with no interfacial barrier, in which the Q-reflected quasiparticles in the IAFM can re-enter the dSC and disrupt the Andreev resonance. For tunnel junctions with large interfacial barriers, the re-entrance of the Q-reflected quasiparticles into the dSC is expected to be negligible, and thus, the IAFM would have little effect on the ZBC peak height, consistent with our data. In contrast to TiAu, tunnel junctions involving NbO$_x$ data show variation in AR suppression, exhibiting ZBC peaks of varying heights as well as ZBC peaks split by 6-8 mV. This spectral variation is consistent with spatial variation of the surface oxide composition. That is, since tunnel junctions are atomic-scale [@Granstrom2018] compared to $\sim$ nm-scale point-contact junctions, tunnel junctions can more easily find regions with less Nb$_{2}$O$_{5-x}$ where AR can apparently occur more robustly. However, it is not obvious in the first place why NbO$_x$ would cause weakened or split ZBC peaks for tunnel junctions. The split peaks are not due to the SC gap of Nb $\Delta_{\mathrm{Nb}}$, since $\Delta_{\mathrm{Nb}}\sim$ 1 meV and thermally broadens to only 2.5 mV (e.g. see Ref. [@Granstrom2016]). In any case, the peak separation is 6-8 mV for both NbO$_x$ tip-YBCO sample and YBCO tip-NbO$_x$ sample combinations, indicating that it is not sensitive to sample preparation procedure. Since the split and weakened peaks are only observed for NbO$_x$ and not TiAu, it would seem they are related to local oxygen vacancies. To check this observation, it would again be useful to perform tunneling measurements between YBCO and pure Nb$_{12}$O$_{29}$. If the split and weakened ZBC peaks were due to local oxygen vacancies, pure Nb$_{12}$O$_{29}$ would exhibit no such peaks.
In summary, we have used $d$-wave AR spectroscopy with YBCO to probe itinerant antiferromagnetism in TiAu and NbO$_x$ samples. XPS was also used on the latter to measure their surface oxide composition. For NbO$_x$, samples with a greater degree of oxidation exhibited greater suppression of $d$-wave AR. For both TiAu and NbO$_x$, low-impedance tip-sample junctions suppressed AR more than high-impedance junctions. Furthermore, AR suppression was observed above the Néel temperature in both compounds, implying that spin fluctuations can suppress $d$-wave AR. Our data demonstrate that $d$-wave AR is suppressed by itinerant antiferromagnetism, and suggest that $d$-wave AR spectroscopy may be utilized as a nanoscale probe to gauge the degree of spin ordering in IAFMs.
We thank Piotr Bartnicki and Yvette De Sereville for laboratory assistance. We are grateful to Tianhan Liu and Peng Xiong from Florida State University, and Zhijie Chen and Kai Liu from UC Davis for supplying the Nb films.
[^1]: $\gtrsim 95$% of photoelectrons come from within 3 inelastic mean free paths of the sample surface. Using the 1486.7 eV energy of Al K$_{\alpha}$ x-rays, the inelastic mean free path of photoelectrons in our samples ranges from 2.5 nm in Nb to 3.9 nm in Nb$_{2}$O$_{5}$, calculated using the expressions from Ref. [@Seah1979].
[^2]: The resistance $R$ of a point-contact can be related to the contact’s radius $a$ with the Wexler formula [@Wexler1966]: $R=4\rho\ell/(3\pi a^{2})+\rho/2a$, where $\rho$ is electrical resistivity and $\ell$ is electronic mean free path. For our $R$ of 0.15-100 k$\Omega$, we use YBCO’s $\ell$ of 10 nm and normal state $\rho$ of 50 $\mu\Omega$ cm [@Wei1998a] to estimate that $a\approx$0.2-5 nm, well below 10 nm.
[^3]: Second to NbO, which is less prevalent in all of our NbO$_x$ samples except the 930 $^{\circ}$C vacuum-annealed foil as shown by the XPS data.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Recent research has documented a significant rise in the volatility (e.g., expected squared change) of individual incomes in the U.S. since the 1970s. Existing measures of this trend abstract from individual heterogeneity, effectively estimating an increase in [*average*]{} volatility. We decompose this increase in average volatility and find that it is far from representative of the experience of most people: there has been no systematic rise in volatility for the vast majority of individuals. The rise in average volatility has been driven almost entirely by a sharp rise in the income volatility of those expected to have the most volatile incomes, identified [*ex-ante*]{} by large income changes in the past. We document that the self-employed and those who self-identify as risk-tolerant are much more likely to have such volatile incomes; these groups have experienced much larger increases in income volatility than the population at large. These results color the policy implications one might draw from the rise in average volatility. While the basic results are apparent from PSID summary statistics, providing a complete characterization of the dynamics of the volatility distribution is a methodological challenge. We resolve these difficulties with a Markovian hierarchical Dirichlet process that builds on work from the non-parametric Bayesian statistics literature.'
author:
- 'Shane T. Jensen and Stephen H. Shore[^1] [^2] [^3] [^4]'
bibliography:
- 'shsrefs30.bib'
title: Changes in the Distribution of Income Volatility
---
Introduction
============
A large literature argues that income volatility – the expectation of squared individual income changes – has increased substantially since the 1970s in the U.S., with further increases since the 1990s.[^5] To the degree that people are risk-averse and income volatility is taken as a proxy for risk, [*ceteris paribus*]{} such rising volatility may carry substantial welfare costs. As a consequence, there has been a great deal of recent interest by politicians and journalists in this finding. [@Gosselin2004; @Scheiber2004; @HouseHearings2007]
To date, research on income volatility trends has ignored individual heterogeneity, effectively estimating an increase in [*average*]{} volatility. We decompose this increase in the average and find that it is far from representative of the experience of most people: there has been no systematic increase in volatility for the vast majority of individuals. The increase has been driven almost entirely by a sharp increase in the income volatility of those with the most volatile incomes. In turn, we find that these individuals with high – and increasing – volatility more likely to be self-employed and more likely to self-identify as risk-tolerant.
Our main finding is apparent in simple summary statistics from the PSID. For example, divide the sample into cohorts, comparing the minority who experienced very large absolute one-year income changes in the past (e.g., four years ago) to those who did not. Since volatility is persistent, those identified [*ex-ante*]{} by large past income changes naturally tend to have more volatile incomes today. The income volatility of this group identified [*ex-ante*]{} as high-volatility has increased since the 1970s while the income volatility of others has remained roughly constant.[^6] This divergence of sample moments identifies our key result.
Obviously, these findings could affect substantially the welfare and policy implications of the rise in average volatility. The individuals whose volatility has increased – who we find are those with the most volatile incomes – may be those with the highest tolerance for risk or the best risk-sharing opportunities. Such risk tolerance is apparent not only from the willingness of these individuals to undertake volatile incomes or self-employment in the first place, but also from their answers to survey questions.
While the basic results can be seen in summary statistics, providing a complete characterization of the dynamics of the volatility distribution is a methodological challenge. We use a standard model for income dynamics that allows income to change in response to permanent and transitory shocks. What is less standard is that we allow the variance of these shocks – our income volatility parameters – to be heterogeneous and time-varying.
We estimate a discrete non-parametric model in which volatility parameters are assumed to take one of L unique values, where the number L and the values themselves are determined by the data. We add structure and get tractability with a variant on the Dirichlet process (DP) prior commonly used in Bayesian statistics. The Markovian hierarchical DP prior model we develop accounts for the grouped nature of the data (by individual) as well as the time-dependency of successive observations within individuals. Implicitly, we place a prior on the probability that an individual’s parameter values will change from one year to the next, on the number of unique parameter values an individual will hold over his lifetime, and on the number of unique parameter values found in the sample.
In Section \[section: data\], we discuss our data and the summary statistics that drive our results. In Section \[section: model\], we present our statistical model including the income process (Section \[section: income process\]), the structure we place on heterogeneity and dynamics in volatility parameters (Section \[section: heterogeneity\]), and our estimation strategy (Section \[section: estimation\]). In Section \[section: results\], we show the results obtained by estimating our model on the data. Increases in the average volatility parameter are due to increases in volatility among those with the most volatile incomes (Section \[subsection: pop evol results\]). We find that the increase in volatility has been greatest among the self-employed and those who self-identify as risk-tolerant (Section \[subsection: whose vol\]), and that these groups are disproportionately likely to have the most volatile incomes (Section \[subsection: who is risky\]). Increases in risk are present throughout the age distribution, education distribution, and income distribution (Section \[subsection: whose vol\]). Section \[section: conclusion\] concludes with a discussion of welfare implications.
Data and summary statistics\[section: data\]
============================================
Data and variable construction\[subsection: data basics\]
---------------------------------------------------------
Data are drawn from the core sample of the Panel Study of Income Dynamics (PSID). The PSID was designed as a nationally representative panel of U.S. households. It tracked families annually from 1968 to 1997 and in odd-numbered years thereafter; this paper uses data through 2005. The PSID includes data on education, income, hours worked, employment status, age, and population weights to capture differential fertility and attrition. In this paper, we limit the analysis to men age 22 to 60; we use annual labor income as the measure of income.[^7] Table \[table: sumstat\] presents summary statistics from these data.
----------------------------- ------------ ------------ -------- ---------------
mean st. dev. min max
year $1986.3$ $10.0$ $1968$ $2005$
age (years) $~40.0~$ $~10.5~$ $22$ $60$
education (years) $~13.1~$ $~2.9~$ $0$ $17$
\# of observations/person $17.2$ $9.0$ $1$ $34$
married (1 if yes, 0 if no) $~0.80~$ $.$ $.$ $.$
black (1 if yes, 0 if no) $0.05$ $.$ $.$ $.$
annual income (2005 \$s) $\$50,553$ $\$57,506$ $0$ $\$3,714,946$
annual income (\$s) $\$29,277$ $\$46,818$ $0$ $\$3,500,000$
family size $3.1$ $1.5$ $1$ $14$
----------------------------- ------------ ------------ -------- ---------------
: Summary Statistics[]{data-label="table: sumstat"}
[This table summarizes data from 52,181 observations on 3,041 male household heads. ]{}
We want to ensure that changes in income are not driven by changes in the top-code (the maximum value for income entered that can be entered in the PSID). The lowest top code for income was \$99,999 in 1982 (\$202,281 in 2005 dollars), after which the top-code rises to \$9,999,999. So that top-codes will be standardized in real terms, this minimum top-code is imposed on all years in real terms, so the top-code is \$99,999 in 1982 and \$202,281 in 2005. Since our income process in Section \[section: income process\] does not model unemployment explicitly, we need to ensure that results for the log of income are not dominated by small changes in the level of income near zero (which will imply huge or infinite changes in the log of income). To address this concern, we replace income values that are very small or zero with a non-trivial lower bound. We choose as this lower-bound the income that would be earned from a half-time job (1,000 hours per year) at the real equivalent of the 2005 federal minimum wage (\$5.15 per hour). This imposes a bottom-code of \$5,150 in 2005 and \$2,546 in 1982. Note that the difference in log income between the top- and bottom-code is constant over time, so that differences over time in the prevalence of predictably extreme income changes cannot be driven by changes in the possible range of income changes. The vast majority of the values below this bound are exactly zero. This bound allows us to exploit transitions into and out of the labor force. At the same time, the bound prevents economically unimportant changes that are small in levels but large and negative in logs from dominating the results. Results are robust to other values for this lower bound, such as the income from full-time work (2,000 hours per year) at the 2005 minimum wage (in real terms).[^8]
------------------------------------------------------------- -------------------------------------------------------------- -- --------- ---------- -----------
Real Income
Level Level One-Year Five-Year
$\begin{array}{c} \rm{One-Year} \\ \rm{Change} \end{array}$ $\begin{array}{c} \rm{Five-Year} \\ \rm{Change} \end{array}$
Mean \$50,553 (\$48,867) 0 0.0017 0.0043
St. Dev. \$57,506 (\$34,943) 0.7307 0.4870 0.6863
Observations 52,181 52,181 43,261 34,972
Minimum \$0 (\$5,150) -2.9325 -3.6877 -3.8361
5$^{\rm{th}}$ Percentile \$668 (\$5,150) -1.6283 -0.7323 -1.3046
25$^{\rm{th}}$ Percentile \$26,174 -0.2964 -0.1089 -0.2126
50$^{\rm{th}}$ Percentile \$42,887 0.1246 0.0134 0.0653
75$^{\rm{th}}$ Percentile \$62,012 0.4601 0.1442 0.3072
95$^{\rm{th}}$ Percentile \$113,500 0.9757 0.6673 0.9764
Maximum \$3,714,946 (\$202,381) 2.6435 3.5862 4.0678
------------------------------------------------------------- -------------------------------------------------------------- -- --------- ---------- -----------
: Distribution of Income, Excess Log Income, and Income Changes for Men[]{data-label="table: ex distribution"}
[Table \[section: data\] describes the distribution of labor income for men in the PSID over the period from 1968 to 2005. See Section \[section: data\] for a detailed description of the income variable and the top- and bottom-coding procedure. Column 1 shows the distribution of real annual income for men (in 2005 dollars). The numbers in parentheses are the values with top- and bottom-coding restrictions. Column 2 shows the distribution of excess log income, the residual from the regression of log labor income (with top- and bottom-code adjustments) on the covariates enumerated in Section \[section: data\]. Column 3 presents the distribution of one-year changes in excess log income. Column 4 repeats the results for column 3, but presents five-year changes instead of one-year changes. ]{}
In this paper, we model the evolution of excess log income. This is taken as the residual from a regression to predict the natural log of labor income (top- and bottom-coded as described). The regression is weighted by the PSID-provided sample weights, with the weights normalized so that the average weight in each year is the same. We use as regressors: a cubic in age for each level of educational attainment (none, elementary, junior high, some high school, high school, some college, college, graduate school); the presence and number of infants, young children, and older children in the household; the total number of family members in the household, and dummy variables for each calendar year. Including calendar year dummy variables eliminates the need to convert nominal income to real income explicitly. While this step is standard in the income process literature, it is not necessary to obtain our results. The results to follow are qualitatively the same and quantitatively similar when we use log income in lieu of excess log income.
Table \[table: ex distribution\] presents data on the distribution of real annual income in column 1 (imposing top- and bottom-code restrictions in parentheses). While the mean real income is nearly identical with and without top- and bottom-code restrictions (\$50,553 versus \$48,867), these restrictions on extreme values reduce the standard deviation of real income from \$57,506 to \$34,943. Column 2 shows the distribution of excess log income. Since excess log income is the residual from a regression, its mean is zero. The inter-quartile range of excess log income is $-0.30$ to $0.46$. Column 3 presents the distribution of one-year changes in excess log income. Naturally, the mean of one-year changes is close to zero. The inter-quartile range of one-year changes is $-0.11$ to $0.14$; excess income does not change more than $11$ to $14$ percent from year to year for most individuals. However, there are extreme changes in income, so the standard deviation of changes to log income ($0.49$) is far great than the inter-quartile range. This implies either that changes to income have fat tails (so that everyone faces a small probability of an extreme income change), or alternatively that there is heterogeneity in volatility (so that a few people face a non-trivial probability of an extreme income change). Unless a model is identified from parametric assumptions, these are observationally equivalent in a cross-section of income changes. However, heterogeneity and fat tails have different implications for the time-series of volatility, and we exploit these in the paper.
Column 4 repeats the results from column 3, but presents five-year excess log income changes instead of one-year changes. These long-term changes have only slightly higher standard deviations than the one-year change, $0.69$ vs. $0.49$, suggesting some mean-reversion in income. @AbowdCard89 show that while one-year income changes are highly negatively correlated at one-year lags, there is no evidence of autocorrelated income changes at lags greater than two years.
Volatility summary statistics\[subsection: moments basics\]
-----------------------------------------------------------
----------- -------- -------- ------------------- -- --------- -------- -------------------
Mean Median 95$^{\rm{th }}\%$ Mean Median 95$^{\rm{th }}\%$
Average 0.1091 0.0099 0.8264 0.3561 0.0314 2.0042
% Change
1970-2003
Slope 0.0015 0.0000 0.0205 0.0106 0.0002 0.0775
(t-stat) (4.11) (0.52) (8.76) (11.96) (1.26) (11.18)
1970 . . . 0.1555 0.0210 0.7709
1971 . . . 0.1823 0.0229 0.8004
1972 0.0665 0.0059 0.4003 0.2142 0.0277 1.1276
1973 0.0786 0.0048 0.4423 0.2296 0.0269 1.1500
1974 0.0792 0.0054 0.5090 0.2324 0.0264 1.1059
1975 0.0986 0.0129 0.6243 0.2496 0.0380 1.2286
1976 0.0997 0.0179 0.6749 0.3124 0.0498 1.6006
1977 0.0933 0.0095 0.7058 0.2983 0.0316 1.8058
1978 0.0706 0.0062 0.5958 0.2751 0.0296 1.3344
1979 0.0838 0.0061 0.6415 0.2931 0.0269 1.6711
1980 0.1388 0.0115 0.9270 0.2811 0.0292 1.4495
1981 0.1159 0.0123 0.8844 0.2932 0.0296 1.5200
1982 0.1004 0.0150 0.7256 0.2514 0.0305 1.2840
1983 0.0859 0.0150 0.6630 0.2912 0.0330 1.5820
1984 0.1220 0.0126 0.8786 0.3185 0.0331 1.8609
1985 0.1109 0.0118 0.7869 0.3283 0.0370 1.7499
1986 0.1002 0.0110 0.6905 0.3089 0.0358 1.5483
1987 0.1089 0.0093 0.7739 0.3015 0.0295 1.6058
1988 0.1224 0.0087 0.7969 0.3121 0.0300 1.6476
1989 0.1161 0.0077 0.8171 0.3278 0.0276 1.8996
1990 0.1174 0.0091 0.7770 0.2998 0.0261 1.5937
1991 0.1312 0.0121 0.9905 0.3523 0.0309 1.8485
1992 0.1013 0.0111 0.9119 0.3168 0.0295 1.7572
1993 0.1272 0.0112 1.0935 0.4166 0.0333 2.3561
1994 0.1083 0.0104 0.9270 0.4479 0.0347 2.6530
1995 0.1346 0.0077 1.1290 0.4914 0.0333 3.3055
1996 . . . 0.4768 0.0264 3.1923
1997 0.0898 0.0074 0.8660 0.4671 0.0282 2.9644
1999 0.1142 0.0080 0.9632 0.4539 0.0317 2.7189
2001 0.1190 0.0073 1.1174 0.4463 0.0271 2.9567
2003 0.1487 0.0182 1.2951 0.6348 0.0574 3.9098
----------- -------- -------- ------------------- -- --------- -------- -------------------
: Income Volatility Sample Moments[]{data-label="table: momentsyby"}
[[The year $t$ permanent variance is the product of two-year changes in excess log income (from $t-2$ to $t$) and the six-year changes that span them (from $t-4$ to $t+2$). The year $t$ squared change is from $t-2$ to $t$. The first row shows full sample moments. The second row shows the percent change over the sample, calculated as the coefficient of a weighted OLS regression of year-specific sample moments on a time trend, multiplied by the number of years (2005-1968) and divided by the full sample moment. The coefficient and t-statistic are shown below. ]{} ]{}
Table \[table: momentsyby\] shows the evolution of volatility sample moments over time. The first three columns show the variance of permanent income changes.[^9] The final three columns present two-year squared changes in excess log income, a raw measure of income volatility.[^10] Note that while the mean size of an income change (columns 1 and 4, Table \[table: momentsyby\]) has increased over time, the median (columns 2 and 5) has not. This divergence can be explained by an increase in the magnitude of large unlikely income changes (columns 3 and 6). While not framed in this way, these features of the data have been identified in previous research, including @Dynanetal2007.
Table \[table: persistmomyby\] and Figure \[fig:volpersist\] show the evolution of volatility sample moments separately for those who are [*ex-ante*]{} likely or unlikely to have volatile incomes. The left panel of Table \[table: persistmomyby\] presents the sample mean of the permanent variance; the right panel presents the mean two-year squared excess log income change. For each year, the sample is split into two groups (below median or above 95$^{\rm{th}}$ percentile) based on the absolute magnitude of permanent (left panel) or squared (right panel) changes four years prior. Unsurprisingly, individuals with large past income changes tend to have larger subsequent income changes. The tendency to have large income changes is persistent, which indicates that some individuals have *ex-ante* more volatile incomes than others.
If (as we argue) volatility is increasing for high-volatility individuals but not for low-volatility individuals, then the gap in the sample variance between those with and without large past income changes should be increasing over time. This divergence over time in volatility between past low- and high-volatility cohorts is clear in both Table \[table: persistmomyby\] and Figure \[fig:volpersist\]. The magnitude of income changes has been increasing more for those with large past income changes (who are more likely to be inherently high-volatility) than for those without such large past income changes (who are not). This is particularly apparent for the permanent variance; for the transitory variance, the finding is obscured slightly by the jump in volatility for everyone in the early- to mid-nineties (when the PSID changed to an automated data collection system which may have led to increased measurement error in income). This divergence illustrates the key stylized fact developed in this paper: the increase in income volatility can be attributed to an increase in volatility among those with the most volatile incomes, identified [*e*x-ante]{} by large past income changes.
Statistical model\[section: model\]
===================================
Income process\[section: income process\]
-----------------------------------------
Here, we present a standard process for excess log income for individual $i$ at time $t$ [following @CarrollSamwick97; @MeghirPistaferri2004 and many others]:
$$\begin{aligned}
y_{i,t} &=&p_{i,t}+\xi_{i,t}+\se_{i,t} \label{eq: income process} \\
p_{i,t} &=&p_{i,0}+\sum\limits_{k=1}^{t-\sOmega }\somegaik
+\sum\limits_{k=t-\sOmega +1}^{t}\stheta _{\omega,t-k}\somegaik\rm{.}
\notag \\
\xi _{i,t} &=&\sum\limits_{k=t-\sepsilon +1}^{t}\stheta _{\varepsilon,t-k}\svarepsilonik
\notag\end{aligned}$$
Excess log income ($y_{i,t}$) is the sum of permanent income ($p_{i,t}$), transitory income ($\xi _{i,t}$), and measurement error ($\se_{i,t}$). The permanent shock, transitory shock, and measurement error are assumed to be normally distributed with mean zero as well as independent of one another, over time and across individuals. Permanent income is initial income ($p_{i,0}$) plus the weighted sum of past permanent shocks ($\somega_{i,k}, 0 < k \le t$) with variance $\ssigmasqit\equiv E\left[
\somegait^{2}\right] $. Transitory income is the weighted sum of recent transitory shocks ($\svarepsilon_{i,k}$) with variance $\stausqit\equiv E\left[ \svarepsilonit^{2}\right] $. We refer to $\ssigsqit \equiv (\stausqit,\ssigmasqit)$ jointly as the volatility parameters. These will be allowed to differ between individuals to accommodate heterogeneity, and to evolve over time. This accommodates not just an evolving distribution of volatility parameters, but also systematic changes over the life-cycle in volatility paramters, as suggested by @ShinSolon2008. Subcripts for $i$ and $t$ indicate that volatility parameters may differ across individuals and over time, as discussed in Section \[section: heterogeneity\]. Noise variancerefers to the variance of measurement error, $\sgammasq\equiv E\left[
\se_{i,t}^{2}\right] $. This measurement error could be subsumed into transitory income; it is kept separate only to accommodate our estimation strategy.
Here, permanent shocks come into effect over $\sOmega $ periods, and transitory shocks fade completely after $\sepsilon$ periods.[^11] As an example of our notation, $\stheta_{\omega,2}$ denotes the weight placed on a permanent shock from two periods ago, $\somegaitm2$, in current excess log income; $\stheta_{\varepsilon,2}$ denotes the weight placed on a transitory shock from two periods ago, $\svarepsilonitm2$, in current excess log income. While we use the word shock for parsimony, these innovations to income may be predictable to the individual, even if they look like shocks in the data. Without loss of generality, we impose the constraint that the weights placed on transitory shocks sum to one ($\sum_{k}\stheta_{\varepsilon,k}=1$).
Heterogeneity and dynamics\[section: heterogeneity\]
----------------------------------------------------
We characterize the dynamics of volatility parameters, $\svolit$, using a discrete non-parametric approach. In a discrete non-parametric model, the variable of interest – here, the pair $\svolit \equiv (\stausqit,\ssigmasqit) $ – can take one of $\sN$ possible values, $\svolNset$ (where $\sN$ and $\svolNset$ for any given sample are determined by the data). The probability that $\svolit$ takes a given value is a function of a) the distribution of values in the population, $\sPiNunboundset$, where $\sPin$ is the proportion of the population whose parameter values are equal to $\svoln$, b) the distribution of values for each individual $i$, $\sPiNiunboundset$, where $\sPini$ is the proportion of individual $i$’s observations with parameter values are equal to $\svoln$,, and c) the number of consecutive years $Q_{i,t}$ with the most recent value.[^12] In other words, $\svolit$ has a given probability of changing from one year to the next; when it changes, it changes to a value drawn from the individual’s distribution, $\sPiNiunboundset$, which in turn consists of values drawn from the population distribution, $\sPiNunboundset$.
We add structure and get tractability by adding a prior commonly used in Bayesian analysis of such discrete non-parametric problems: the Dirichlet process (DP) prior. In a standard DP model, there is a “tuning parameter”, $\Theta$, which implicitly places a prior on the total number of unique parameter values in the sample, $\sN$.[^13] $\Theta$ is defined more formally in Section \[section: estimation\]. We set $\Theta = 1$, though our inference is not sensitive to this choice. In a hierarchical DP (HDP) model [recently developed by @TehJorBea06], the usual DP model is extended so by adding a second tuning parameter, $\Theta_{i}$, which implicitly places a prior on the total number of unique parameter values for any given individual, $\sNi$; we set $\Theta_{i}=1$.
We extend this approach further to address panel data by including a Markovian structure on the hierarchical DP, giving us a Markovian hierarchical DP (MHDP) model. In our Markovian approach, the prior probability that the parameter is unchanged from the previous period depends on the number of consecutive years with that value, $Q_{i,t}$. We add a third tuning parameter, $\theta$, to place a prior on the probability of changing the parameter value, $p \left( \svolit =\svolitm1 \ | i,t \right)=Q_{i,t} / (\theta+ Q_{i,t})$; we set $\theta=1$. In the MHDP model, our prior parameters can then be characterized with the triple $\bTheta\equiv\{\Theta,\Theta_{i},\theta\}=\{1,1,1\}$.
Given our research question, a key advantage of this set-up is that it does not restrict the shape (or the evolution of the shape) of the cross-sectional volatility distribution. We view our discrete non-parametric model and the structure placed on it by our MHDP prior as providing a sensible middle ground between tractability and flexibility.
Estimation\[section: estimation\]
---------------------------------
We estimate the income process from Section \[section: income process\] on annual data from the PSID (detailed in Section \[section: data\]) for excess log income. When data are missing, mostly because no data was collected by the PSID in even-numbered years following 1997, we impute bootstrapped guesses of income.[^14] These bootstrapped values add no additional information; they merely accommodate our estimation strategy in a setting with missing data in a way that is intended to minimize the possible impact on our results. Here, we outline an approach for combining the prior from Section \[section: heterogeneity\] with data on excess log income, $\by$, to form a posterior on the distribution of volatility parameters, $\bheteroparams$.[^15] Further details and an algorithm for implementation are provided in the appendix.
Consider the problem of estimating $\svolit$, the volatility parameters for person $i$ in year $t$, if all other parameters $\bvolnit$ (and $\bthetaphi$) were known. The decision tree for estimation is shown in Figure \[fig:hierarchy\] and described here, both with references to relevant equations in the appendix.
1. $\svolit$ can remain unchanged from last year ($\svolit=\svolitm1$, eq: \[eq: prevchoice1\]) or can change ($\svolit \neq \svolitm1$, eq: \[eq: prevchoice2\]). If $\svolit$ changes;
2. $\svolit$ can change to a value from the set of *other values for that individual* ($\svolit \in \bvolint$ and $\svolit \ne \svolitm1$, eq: \[eq: personchoice1\]) or can take on a value new to the individual ($\svolit \notin \bvolint$, eq: \[eq: personchoice2\]). If $\svolit$ takes on a value new to the individual;
3. $\svolit$ can be a value held by *other individuals* ($\svolit\in\bvolnit$ and $\svolit \notin \bvolint$, eq: \[eq: popchoice1\]) or can be a new value not shared with other individuals ($\svolit\notin\bvolnit$, eq: \[eq: popchoice2\]).
The probability that $\svolit$ takes a given value is a function of a) the likelihood of generating estimated shocks $(\somegait,\svarepsilonit)$ given $\svolit$ and b) the prior probability of $\svolit$.
The prior probability that the parameter remains unchanged in Level 1 ($\svolit=\svolitm1$) is proportional to $ Q_{i,t}$; the prior probability that the parameter changes is proportional to $\theta$. If the parameter changes in Level 1 ($\svolit \neq \svolitm1$), the prior probability that $\svolit$ changes to a value held by that individual in another year in Level 2 is proportional to the number of times that value occurs in other years for that individual; the prior probability that $\svolit$ changes to a new value not seen for that individual in another year is proportional to $\Theta_{i}$. If the parameter changes to a new value not seen for that individual in another year in Level 2, the prior probability that $\svolit$ changes to one of the other population values in Level 3 is proportional to the number of times that value occurs within the population; the prior probability that $\svolit$ changes to a new value not seen elsewhere in the population is proportional to $\Theta$.
A detailed outline of this estimation algorithm is given in the appendix. The appendix shows this compound prior algebraically, and also shows how it is combined with the data to produce a posterior for $\svolit$. We proceed iteratively through all $t$ within an individual and all $i$ across individuals. This entire scheme for choosing volatility values $\bheteroparams$ is nested within a larger Gibbs sampling algorithm [@GemGem84]. This Markov Chain Monte Carlo (MCMC) approach simultaneously estimates the other parameters of our model, namely shocks ($\bshockparams$) and income coefficients ($\bhomoparams$).
(6,7) (1,6)[(1,-1)[2]{}]{} (1,6)[(-1,-1)[2]{}]{} (1,6) (1.1,6)[1]{} (1.7,6.1) (-2.3,3.5)
(3,4)[(1,-1)[2]{}]{} (3,4)[(-1,-1)[2]{}]{} (3,4) (3.1,4)[2]{} (3.7,3.5) (-0.2,1.3)
(5,2)[(1,-1)[2]{}]{} (5,2)[(-1,-1)[2]{}]{} (5,2) (5.1,2)[3]{} (5.7,1.3) (1.8,-0.8) (6.5,-0.8)
(-12,-3) (-13,-4.5)[(1,0)[19]{}]{}
Results\[section: results\]
===========================
Distribution of Variance Parameters
-----------------------------------------------------------------------------
$\begin{array}{c} $\begin{array}{c}
\rm{Permanent} \\ \rm{Transitory} \\
\rm{Variance}\end{array}$ \rm{Variance}\end{array}$
--------------------- --------------------------- ---------------------------
Mean 0.0713 0.2771
St. Dev. 0.4685 1.0471
N 67,725 67,725
1$^{\rm{st}}$ $\%$ 0.0200 0.0499
5$^{\rm{th}}$ $\%$ 0.0250 0.0506
10$^{\rm{th}}$ $\%$ 0.0301 0.0510
25$^{\rm{th}}$ $\%$ 0.0313 0.0518
50$^{\rm{th}}$ $\%$ 0.0321 0.0530
75$^{\rm{th}}$ $\%$ 0.0331 0.0572
90$^{\rm{th}}$ $\%$ 0.0356 0.2452
95$^{\rm{th}}$ $\%$ 0.0498 1.2187
99$^{\rm{th}}$ $\%$ 0.8909 5.5030
-----------------------------------------------------------------------------
: Basic Model Results[]{data-label="table:basic results"}
[Distribution of posterior means of $\bheteroparams$]{}
Shocks’ Rate of Entry/Exit
lag $\stheta_{\omega,k}$ $\stheta_{\varepsilon,k}$
------- ---------------------- ---------------------------
$k=0$ 0.381 0.784
(0.088) (0.029)
$k=1$ 0.865 0.180
(0.072) (0.025)
$k=2$ 0.951 0.037
(0.064) (0.017)
: Basic Model Results[]{data-label="table:basic results"}
\
[$\stheta_{\omega,k}$: impact of permanent shock\
from $k$ periods ago\
$\stheta_{\varepsilon,k}$: impact of transitory shock\
from $k$ periods ago\
Standard errors in parentheses.\
]{}
[The left panel presents the posterior mean estimates of the volatility parameters, $\bheteroparams$. The distributions presented here consider all years and all individuals together. The right panel of this table present $\bthetaphi$, the mapping of shocks to income changes. ]{}
------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------
Permanent Variance Transitory Variance
![Distribution of Permanent and Transitory Variance[]{data-label="fig:unconditionaldist"}](fig2_permhist.pdf "fig:"){width="2.8in"} ![Distribution of Permanent and Transitory Variance[]{data-label="fig:unconditionaldist"}](fig2_tranhist.pdf "fig:"){width="2.8in"}
------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------
[This figure presents the distribution of $\btausq$ and $\bsigmasq$. These are the distribution of posterior means estimated from the data, as presented numerically in Table \[table:basic results\]. These posteriors of the permanent variance and transitory variance are calculated for each individual in each year, as described in Section \[section: estimation\]. The distributions presented here show all years and individuals together. Values are truncated at the 95$^{\rm{th}}$ percentile for the permanent variance and at the 90th percentile for the transitory variance. Mean and median of the truncated part of each distribution is given.\
(1,0)[430]{}]{}
Here, we present the model parameters estimated using the methods from Section \[section: estimation\]. The chief object of interest is the evolution of the cross-sectional distribution of volatility parameters, $\bvolt$, over time. These are shown in Section \[subsection: pop evol results\]. We begin with more basic results. In subsection \[subsection: basic results\], we present estimates of the homogeneous parameters $\bthetaphi$ that map shocks to income changes and the unconditional distribution of volatility parameters, $\bheteroparams$. In Section \[subsection: alternative explanations\], we rule out alternative explanations. In Sections \[subsection: who is risky\] and \[subsection: whose vol\], we map these volatility parameter estimates to individuals’ demographic or risk attributes.
Basic results\[subsection: basic results\]
------------------------------------------
Table \[table:basic results\] presents the basic parameter estimates obtained from fitting our model to the PSID income data described in Section \[section: estimation\]. The left panel shows the distribution of risk in the population, $\btausq$ and $\bsigmasq$. Formally, we present the distribution of posterior means of permanent and transitory variance parameters. The right panel show the mapping from shocks to income changes, $\bthetaphi$, which we constrained to be constant over time and across individuals.
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Permanent Shock Transitory Shock
![Impulse Response Function for Permanent and Transitory Shocks[]{data-label="fig:irf"}](fig3_irf_omega.pdf "fig:"){width="2.8in"} ![Impulse Response Function for Permanent and Transitory Shocks[]{data-label="fig:irf"}](fig3_irf_epsilon.pdf "fig:"){width="2.8in"}
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[This figure presents an estimated impulse response function for a permanent (left panel) and transitory (right panel) shock. (1,0)[430]{}]{}
Note the extreme skew and fat tails (kurtosis) in the distribution of volatility parameters, $\bheteroparams$, shown in the left panel of Table \[table:basic results\]). While medians are modest, means far exceed medians. At the median, transitory shocks have a standard deviation of approximately 23% annually; permanent shocks have a standard deviation of just under 18% annually. However, the highest volatility observations imply shocks with standard deviations well above 100% annually. Figure \[fig:unconditionaldist\] plots these skewed and fat-tailed distributions by truncating the right tail.
As shown in the right panel of Table \[table:basic results\], permanent shocks enter in quickly ($\stheta _{\omega,k}$ are close to one) while transitory shocks damp out quickly ($\stheta _{\varepsilon,k}$ fall to zero). The impact of a shock on the evolution of income is presented in Figure \[fig:irf\]. These present impulse response functions for a permanent (left panel) and transitory (right panel) shock. Shocks were calibrated as a one standard-deviation shock for an individual with volatility parameters at the estimated means (pulled from Table \[table:basic results\]).
Evolution of the volatility distribution\[subsection: pop evol results\]
------------------------------------------------------------------------
---------- -------- -------- ------------------- -- -------- -------- -------------------
Mean Median 95$^{\rm{th }}\%$ Mean Median 95$^{\rm{th }}\%$
Average 0.0713 0.0321 0.0498 0.2771 0.0530 1.2186
% Change 73$\%$ 0$\%$ 71$\%$ 99$\%$ 1$\%$ 154$\%$
Slope 0.0014 0.0000 0.0010 0.0074 0.0000 0.0508
(t-stat) (6.84) (3.78) (6.31) (7.02) (9.37) (6.25)
1970 0.0573 0.0321 0.0424 0.1568 0.0526 0.4498
1971 0.0502 0.0321 0.0406 0.1901 0.0526 0.6419
1972 0.0411 0.0320 0.0374 0.1909 0.0527 0.7775
1973 0.0550 0.0321 0.0389 0.2027 0.0528 0.7997
1974 0.0481 0.0322 0.0437 0.1848 0.0528 0.5520
1975 0.0547 0.0321 0.0397 0.1923 0.0530 0.7597
1976 0.0663 0.0321 0.0464 0.2746 0.0529 1.3527
1977 0.0540 0.0321 0.0409 0.2424 0.0529 1.1020
1978 0.0557 0.0321 0.0411 0.1865 0.0529 0.6785
1979 0.0738 0.0321 0.0432 0.2226 0.0528 1.0134
1980 0.0748 0.0321 0.0452 0.2012 0.0529 0.7139
1981 0.0651 0.0321 0.0504 0.1986 0.0529 0.7762
1982 0.0594 0.0321 0.0502 0.2055 0.0529 0.8885
1983 0.0744 0.0321 0.0457 0.2550 0.0531 1.2691
1984 0.0660 0.0321 0.0503 0.2307 0.0531 0.9686
1985 0.0593 0.0321 0.0477 0.2260 0.0530 1.0063
1986 0.0672 0.0321 0.0441 0.2557 0.0529 1.1042
1987 0.0679 0.0321 0.0477 0.2448 0.0530 1.1468
1988 0.0714 0.0321 0.0467 0.2286 0.0531 0.9494
1989 0.0629 0.0321 0.0490 0.2462 0.0529 1.3182
1990 0.0801 0.0321 0.0607 0.2387 0.0530 0.9812
1991 0.0726 0.0321 0.0600 0.2708 0.0530 1.2466
1992 0.0633 0.0321 0.0539 0.2431 0.0531 1.0536
1993 0.0887 0.0321 0.0701 0.4290 0.0532 2.6502
1994 0.0916 0.0321 0.0628 0.4229 0.0532 2.3884
1995 0.0764 0.0321 0.0583 0.4080 0.0532 2.2152
1996 0.0609 0.0321 0.0541 0.4167 0.0531 2.4093
1997 0.0721 0.0321 0.0499 0.3916 0.0531 2.3408
1999 0.0769 0.0321 0.0519 0.3059 0.0532 1.5679
2001 0.0975 0.0322 0.0719 0.2616 0.0531 1.0974
2003 0.1026 0.0322 0.0967 0.4771 0.0534 2.4896
2005 0.1294 0.0324 0.0592 0.4379 0.0538 2.2246
---------- -------- -------- ------------------- -- -------- -------- -------------------
: Year-by-Year Income Volatility Parameters[]{data-label="table: volyby"}
[[The construction of posterior means for $\ssigmasq$ and $\stausq$ for each individual in each year is detailed in the text. The first row shows the full sample distribution, so that the second column shows the median value of the posterior mean of $\ssigmasq$ over all individual-years. The second row shows the percent change over the sample, calculated as the coefficient of a weighted OLS regression of year-specific sample moments on a time trend, multiplied by the number of years (2005-1968) and divided by the full sample value. The coefficient and t-statistic are shown below. ]{} ]{}
Here, we show how the distribution of posterior means of variance parameters has evolved over time. This evolution is shown in Tables \[table: volyby\] and also in Figure \[fig:voldistevol\]. Table \[table: volyby\] shows the year-by-year distribution of volatility parameters ($\bvolt$) posterior means. This table mirrors Table \[table: momentsyby\], with volatility parameter ($\svolit$) posterior means replacing reduced form moments. The first three columns show results for the permanent variance parameter, $\ssigmasq$; the final three columns show results for the transitory variance parameter, $\stausq$. The first and fourth columns present means of the permanent and transitory variance parameter posterior means, the second and fifth columns present medians of parameter posterior means, and the third and sixth columns present 95$^{\rm{th}}$ percentiles. All use weights from the PSID. The first row shows whole-sample results. The second row shows the percent change in the mean, median, or 95$^{\rm{th}}$ percentile over the sample.[^16] The coefficient and t-statistic from this regression are shown just below. Year-by-year values are then shown.
Table \[table: volyby\] shows that the mean of permanent and transitory parameters have increased substantially over the sample (by 73 and 99 percent, respectively) while the medians have not (0 and 1 percent increases, respectively). This divergence can be explained by an increase in the magnitude of permanent and transitory variance parameters at the right tail, among individuals with the highest parameters (the 95$^{\rm{th}}$ percentile values increasing 71 percent and 154 percent, respectively). Colloquially, the kind of people whose incomes had always moved around a lot are moving around even more than they used to; the median person’s income does not move more than it used to. This pattern can be seen graphically in Figure \[fig:voldistevol\], which shows the year-by-year evolution of many quantiles of the distribution of permanent and transitory variance posterior means. In the bottom panels of Figure \[fig:voldistevol\], we plot the 1st, 5$^{\rm{th}}$, 10th, 25$^{\rm{th}}$, 50th, and 75$^{\rm{th}}$ percentile values of the posterior mean of the permanent ($\ssigmasq$, left) and transitory ($\stausq$, right) variance parameters by year. These are very stable and show no clear upward trend. The size of this increase is extremely small economically. Looking at all but the riskytail of the distributions, the distributions look very stable.
In the middle and upper panels of Figure \[fig:voldistevol\], we show the evolution of the risky tail of the distribution of posterior means. In this case, variance parameters increase strongly and significantly. This increase in the right tail of the distribution explains the increase in the mean completely.
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Permanent Income Changes Transitory Income Changes
Mean and Median Mean and Median
![Evolution of Percentiles of Volatility Distribution[]{data-label="fig:voldistevol"}](fig4_permdistevolmeanmed.pdf "fig:"){height="1.65in"} ![Evolution of Percentiles of Volatility Distribution[]{data-label="fig:voldistevol"}](fig4_trandistevolmeanmed.pdf "fig:"){height="1.65in"}
$99^{\rm{th}}$ Percentile $99^{\rm{th}}$ Percentile
![Evolution of Percentiles of Volatility Distribution[]{data-label="fig:voldistevol"}](fig5_permdistevolhigh2.pdf "fig:"){height="1.65in"} ![Evolution of Percentiles of Volatility Distribution[]{data-label="fig:voldistevol"}](fig5_trandistevolhigh2.pdf "fig:"){height="1.65in"}
$90^{\rm{th}}$ and $95^{\rm{th}}$ Percentiles $90^{\rm{th}}$ and 95$^{\rm{th}}$ Percentiles
![Evolution of Percentiles of Volatility Distribution[]{data-label="fig:voldistevol"}](fig5_permdistevolhigh1.pdf "fig:"){height="1.65in"} ![Evolution of Percentiles of Volatility Distribution[]{data-label="fig:voldistevol"}](fig5_trandistevolhigh1.pdf "fig:"){height="1.65in"}
$\le 75^{\rm{th}}$ Percentiles $\le 75^{\rm{th}}$ Percentiles
![Evolution of Percentiles of Volatility Distribution[]{data-label="fig:voldistevol"}](fig5_permdistevollow.pdf "fig:"){height="1.65in"} ![Evolution of Percentiles of Volatility Distribution[]{data-label="fig:voldistevol"}](fig5_trandistevollow.pdf "fig:"){height="1.65in"}
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[These figures show the evolution of various percentiles of the posterior mean of the permanent (left) and transitory (right) variance for various percentiles of the distribution of variance parameters.\
(1,0)[430]{}]{}
Heterogeneity or fat tails?\[subsection: alternative explanations\]
-------------------------------------------------------------------
So far, we have shown that the increases in income volatility can be attributed solely to increases in the right tail of the volatility distribution. To obtain this result, our model assumes that the distribution of shocks is normal conditional on the volatility parameters. When the unconditional distribution of shocks is fat-tailed (has high kurtosis), this is automatically attributed to heterogeneity in volatility parameters. An alternative hypothesis is that there is little or no heterogeneity in volatility parameters, but that shocks are conditionally fat-tailed.
When looking at the cross-section of income changes, heterogeneity in volatility parameters (with conditionally normal shocks) and conditionally fat-tailed shocks (without no heterogeneity in volatility parameters) are observationally equivalent; they both imply a fat-tailed unconditional distribution of income changes. By examining serial dependence, it is possible to reject the hypothesis that everyone has the same volatility parameter. If shocks are conditionally fat-tailed but everyone has the same volatility parameters, then those with large past income changes should be no more likely than others to experience large subsequent income changes. If individuals differ in their volatility parameters and those volatilities are persistent, then individuals with large past income changes will be more likely than others to have large subsequent income changes.
This possibility is investigated in Table \[table: persistmomyby\] and shown graphically in Figure \[fig:volpersist\]. These compare the sample variance of income changes for individuals with and without large past income changes. In each year, a cohort without large income changes is formed as the set of individuals whose measure of variance, either permanent variance or squared income change, was below median four years ago; a cohort with large income changes is formed as the set of individuals whose measure of variance was above the 95$^{\rm{th}}$ percentile four years ago. This four-year period is chosen so that income shocks are far enough apart to be uncorrelated. [@AbowdCard89]
Note that individuals with large past income changes tend to have larger subsequent income changes. The tendency to have large income changes is persistent, which indicates that some individuals have *ex-ante* more volatile incomes than others.
The divergence over time in volatility between past low- and high-volatility cohorts is clear in both Figure \[fig:volpersist\] and Table \[table: persistmomyby\]. The magnitude of income changes has been increasing more for those with large past income changes (who are more likely to be inherently high-volatility) than for those without such large past income changes (who are not). This increase in volatility falls primarily on those who could be expected to have volatile incomes to begin with. This shows that the increase in volatility among the volatile we find in the model cannot be attributed to increasingly fat-tailed shocks for everyone.
------------------------- --------------- ----------------
Dependent Permanent Transitory
Variable Variance Variance
self-employed? 1 or 0 0.6001 0.7794
(24.07)\*\*\* (32.22)\*\*\*
$[0.1085]$ $[0.1533]$
risk-tolerant? 1 or 0 0.1303 0.0950
(5.91)\*\*\* (4.31)\*\*\*
$[0.0180]$ $[0.0131]$
age 0.0104 0.0082
(7.82)\*\*\* (6.20)\*\*\*
$[0.0014]$ $[0.0011]$
years of education -0.0041 -0.0123
(-0.89) (-2.67)\*\*\*
$[-0.0006]$ $[-0.0017]$
income$>$median? 1 or 0 -0.2277 -0.2922
(-9.84)\*\*\* (-12.65)\*\*\*
$[-0.0308]$ $[-0.0398]$
have children? 1 or 0 -0.0498 -0.0686
(-1.48) (-2.04)\*\*
$[-0.0068]$ $[-0.0094]$
number of children 0.0120 0.0068
(0.90) (0.51)
$[0.0016]$ $[0.0009]$
married? 1 or 0 -0.1009 -0.1815
(-3.00)\*\*\* (-5.56)\*\*\*
$[-0.0143]$ $[-0.0270]$
$R^{2}$ 0.0469 0.0751
observations 31,898 31,898
------------------------- --------------- ----------------
: Determinants of High Income Volatility (Probit)[]{data-label="table: probitreg"}
[Results from a probit regression to predict an indicator variable for whether posterior mean variance (permanent or transitory volatility) estimate is is above the 90$^{\rm{th}}$ percentile for that year. “Risk tolerant” is set to 1 if the PSID risk tolerance variable exceeds 0.3. Above-median income indicates that four-year lagged income is above-median for that (lagged) year. \*, \*\*, and \*\*\* indicate significance at the 10$\%$, 5$\%$, and 1$\%$ levels, respectively. z-statistics are in parentheses. Marginal effects are in square brackets.]{}
Whose incomes are volatile?\[subsection: who is risky\]
-------------------------------------------------------
In this paper, we have identified increasing volatility for men in the U.S. since 1968 as being driven solely by the right (volatile) tail of the volatility distribution. Here, we examine the attributes of men with highly volatile incomes.
Table \[table: probitreg\] presents the results from a probit regression to predict whether a person-year estimate of the (posterior mean) volatility parameter is above the 90$^{\rm{th}}$ percentile for that year. Note from the first row that self-employed individuals are much more likely to have highly volatile incomes. The second row shows that “risk tolerant” individuals are also much more likely to have highly volatile incomes. Risk tolerance is identified from answers to hypothetical questions about lotteries, designed to elicit the individual’s coefficient of relative risk-aversion; risk-tolerant individuals are defined as those with an estimated coefficient of relative risk-aversion below 1/0.3. High income individuals (those with incomes above median four years before the observation in question) are less likely to have volatile incomes. Individuals with more years of education are also less likely to have volatile incomes. Older individuals are more likely to have volatile incomes, a result driven by the large number of high-volatility individuals between ages 50 and 60. Unsurprisingly, men who are married and/or who have children are less likely to have volatile incomes.
Whose incomes are increasingly volatile?\[subsection: whose vol\]
-----------------------------------------------------------------
Section \[subsection: who is risky\] identified attributes of individuals with volatile incomes. In particular, the self-employed and those whose answers to survey questions suggest they are risk-tolerant are more likely to have volatile incomes. Here, we examine the increase in volatility over time among these groups.
Permanent Variance
--------------------- --------------- --------------- -- --------------- --------------- -- -------------- --------------
self- not self- $>$ med. $\le$ med. risk not risk
sample employed employed income income tolerant tolerant
change per year 0.0048 0.0011 0.0018 0.0009 0.0035 0.0012
$\%$ change ’68-’05 194$\%$ 58$\%$ 135$\%$ 36$\%$ 172$\%$ 76$\%$
(6.17)\*\*\* (4.58)\*\*\* (5.99)\*\*\* (2.75)\*\*\* (4.61)\*\*\* (4.50)\*\*\*
N 6,068 41,766 10,336 23,876 23,958 18,029
--------------------- --------------- --------------- -- --------------- --------------- -- -------------- --------------
: Volatility Trends by Self-Employment, Income, and Risk Tolerance[]{data-label="table: voltrendsample"}
Transitory Variance
--------------------- --------------- --------------- -- --------------- --------------- -- -------------- --------------
self- not self- $>$ med. $\le$ med. risk not risk
sample employed employed income income tolerant tolerant
change per year 0.0262 0.0061 0.0040 0.0116 0.0100 0.0076
$\%$ change ’68-’05 176$\%$ 101$\%$ 125$\%$ 101$\%$ 117$\%$ 114$\%$
(11.27)\*\*\* (13.80)\*\*\* (10.84)\*\*\* (13.22)\*\*\* (7.81)\*\*\* (9.45)\*\*\*
N 6,068 41,766 23,876 23,958 10,336 18,029
--------------------- --------------- --------------- -- --------------- --------------- -- -------------- --------------
: Volatility Trends by Self-Employment, Income, and Risk Tolerance[]{data-label="table: voltrendsample"}
[Results from a weighted OLS regression to predict the posterior mean variance (volatility) estimate with a linear time trend. The “change” row shows the coefficient on calendar time; the “percent change” row shows the expected percent change over the sample implied by this coefficient. This is (100 percent) times (2005 minus 1968) times (the coefficient on calendar time) divided by (the average posterior mean in the sample). The top panel presents results for the permanent variance; the bottom panel presents results for the transitory variance. Each column presents results for a different sub-sample. “Risk tolerant” means that the PSID risk tolerance variable exceeds 0.3. Above-median income indicates that four-year lagged income is above-median for that (lagged) year. t-statistics are in parentheses. ]{}
Permanent Variance
------------------------------------ -------------- -------------- -- --------------- -------------- -------------
less than at least more than high less than
sample 40 yrs old 40 yrs old high school school high school
mean change/year 0.0006 0.0018 0.0024 0.0005 0.0004
$\%$ change ’68-’05 44$\%$ 76$\%$ 120$\%$ 28$\%$ 22$\%$
(3.66)\*\*\* (4.55)\*\*\* (6.08)\*\*\* (1.71)\* (1.17)
median change/year 0.0000 0.0000 0.0000 0.0000 0.0000
$\%$ change ’68-’05 0$\%$ 0$\%$ 1$\%$ 0$\%$ 0$\%$
(0.79) (4.29)\*\*\* (5.20)\*\*\* (-1.36) (0.20)
95$^{\rm{th}}$ $\%$tile chnge/year 0.0007 0.0010 0.0008 0.0007 0.0012
$\%$ change ’68-’05 53$\%$ 67$\%$ 63$\%$ 55$\%$ 72$\%$
(8.35)\*\*\* (6.47)\*\*\* (10.32)\*\*\* (6.50)\*\*\* (2.31)\*\*
N 23,928 23,906 23,455 15,516 8,863
------------------------------------ -------------- -------------- -- --------------- -------------- -------------
: Volatility Trends by Age and Education[]{data-label="table: voltrendageedy"}
Transitory Variance
------------------------------------ -------------- --------------- -- --------------- --------------- --------------
less than at least more than high less than
sample 40 yrs old 40 yrs old high school school high school
mean change/year 0.0057 0.0096 0.0093 0.0065 0.0066
$\%$ change ’68-’05 86$\%$ 123$\%$ 120$\%$ 102$\%$ 95$\%$
(9.36)\*\*\* (13.27)\*\*\* (12.14)\*\*\* (8.76)\*\*\* (6.91)\*\*\*
median change/year 0.0000 0.0000 0.0000 0.0000 0.0000
$\%$ change ’68-’05 1$\%$ 2$\%$ 2$\%$ 2$\%$ 3$\%$
(6.87)\*\*\* (18.73)\*\*\* (11.18)\*\*\* (13.69)\*\*\* (7.60)\*\*\*
95$^{\rm{th}}$ $\%$tile chnge/year 0.0378 0.0649 0.0598 0.0483 0.0467
$\%$ change ’68-’05 124$\%$ 211$\%$ 183$\%$ 188$\%$ 135$\%$
(7.87)\*\*\* (17.10)\*\*\* (12.15)\*\*\* (11.04)\*\*\* (5.78)\*\*\*
N 23,928 23,906 23,455 15,516 8,863
------------------------------------ -------------- --------------- -- --------------- --------------- --------------
: Volatility Trends by Age and Education[]{data-label="table: voltrendageedy"}
[Results from a weighted OLS regression to predict the posterior mean variance (volatility) estimate with a linear time trend. The “change” row shows the coefficient on calendar time; the “percent change” row shows the expected percent change over the sample implied by this coefficient. This is (100 percent) times (2005 minus 1968) times (the coefficient on calendar time) divided by (the average posterior mean in the sample). The top panel presents results for the permanent variance; the bottom panel presents results for the transitory variance. Each column presents results for a different sub-sample. t-statistics are in parentheses. ]{}
Table \[table: voltrendsample\] predicts the posterior mean variance (volatility) estimates described earlier with a linear time trend. The “change” row shows the coefficient on calendar time; the “percent change” row shows the expected percent change over the sample implied by this coefficient. The top panel presents results for the permanent variance; the bottom panel presents results for the transitory variance. Each column presents results for a different sub-sample. By comparing the first two columns, note that that volatility has increased dramatically more for self-employed people than for others. These individuals have much higher average levels of volatility, but their percentage change in volatility is still higher than for other individuals. Self-employed individuals account for a substantial proportion of the overall increase in income volatility. Similarly, the increase in permanent volatility (the variance of permanent shocks) is much greater for those who self-identify as risk tolerant (those whose estimated coefficient of relative risk aversion less than $1/0.3$) than those who do not. Transitory volatility does not show major differences in trend for risk tolerant and not risk tolerant individuals.
Table \[table: voltrendsample\] shows that the increase in volatility is apparent throughout the income distribution. While increases in the average variance of transitory shocks are similar (in proportional terms) for those with above- and below-median income, the variance of permanent shocks has increased more for those with above-median income than for those with below-median income. While below-median individuals are over-represented among those with the highest volatilities (Section \[subsection: whose vol\]), low income individuals are not driving the increase in volatility among those with the most volatile incomes.
Table \[table: voltrendageedy\] presents results by age and educational attainment. Note that while magnitudes vary, the increase in volatility at the right tail is present for those below and above 40, and across the education distribution.
Conclusion\[section: conclusion\]
=================================
Increases in the size of income changes in the PSID can be attributed almost entirely to the right tailof the volatility distribution. Taking volatility as a proxy for risk, those who would have had risky incomes in the past now face even more risk. Everyone else has had no substantial change.
Without knowing more, the welfare implications of this finding are unclear. Depending on what kind of people have volatile incomes, an increase in volatility at the volatile end of the distribution could be more or less bad than an increase in volatility for everyone. Consider the possibility (which we refute in Section \[subsection: who is risky\]) that risk tolerance is independent of income volatility or expected income. In this case, increasing volatility at the volatile end of the distribution decreases welfare more than increasing risk throughout the distribution. When individuals have decreasing absolute risk aversion, high levels of income risk (proxied here by volatility) make people more vulnerable to additional risk. [@Gollier2001] If there is a compensating differential for risk so that volatile incomes are also higher on average, then this effect will be mitigated or reversed.
This paper shows that those with the most volatile incomes are also the most risk-tolerant. In this case, the increase in risk has hit those best able to handle it. To the degree that income volatility is chosen (e.g., by choosing an occupation), we would expect those with the highest tolerance for risk or the best risk-sharing opportunities to take on the most volatile incomes. If it is these individuals whose volatility has increased, it could blunt substantially any welfare costs associated with increased income volatility. Since the increase in volatile has fallen disproportionately on the self-employed, it could also reflect an increase in profitable (but volatile) business opportunities. In this case, there could even be welfare gains associated with increased income volatility.
Appendix A: Estimation
======================
We estimate the joint posterior distribution of all unknown parameters conditional on our observed data as: $$p(\bhomoparams,\bshockparams,\bheteroparams|\by)\propto
p(\by|\bhomoparams,\bshockparams)\cdot
p(\bshockparams|\bheteroparams)\cdot p(\bheteroparams) \label{eq: fullpost}$$Following Bayes rule, the distribution of parameters given the data – $p(\bhomoparams,\bshockparams,\bheteroparams|\by)$ – is proportional to the product of the distribution of the data given those parameters – $p(\by|\bhomoparams,\bshockparams)$ – and the probability of those parameters – $p(\bshockparams|\bheteroparams)\cdot
p(\bheteroparams)$. We will estimate the posterior distribution of our unknown parameters by Markov Chain Monte Carlo (MCMC) simulation, specifically the Gibbs sampler. [@GemGem84] The Gibbs sampler estimates the full posterior distribution in equation (\[eq: fullpost\]) by iteratively sampling a value for each unknown parameter conditional on the current values of the other unknown parameters. In other words, we iterate over the following steps.
1. Sample new values of $(\bhomoparams)$ from $p(\bhomoparams|\by,\bshockparams,\bheteroparams) $
2. Sample new values of $(\bshockparams)$, the shock parameters for each person and year, from $p(\bshockparams|\by,\bhomoparams,\bheteroparams)$
3. Sample new values of $(\bheteroparams)$, the volatility parameters for each person and year, from $p(\bheteroparams|\by,\bhomoparams,\bshockparams)$.
These sampling steps form a Markov chain that is iterated until the set of all parameters has converged to their joint posterior distribution. This algorithm is programmed in Python and run on a grid cluster of computers. One run of this model (with 10,000 iterations) takes several weeks, though multiple runs can be done simultaneously. Each of the runs was started from a randomly sampled set of initial parameter values. These multiple runs were used to evaluate convergence of the algorithm to a reasonable set of samples from the posterior distribution of all parameters. The first 5000 iterations of each chain was discarded as the pre-convergence burn-in period, and our inference was based upon the remaining sampled values.
Step 1: Sampling income process parameters ($\protect\bhomoparams$) {#homoestimation}
-------------------------------------------------------------------
In this step, we take realized shocks ($\bomega,\bvarepsilon$) as well as excess log income data ($\by$) as given, to estimate the rate at which shocks pass through to income ($\bthetaphi$). Reorganizing equation (\[eq: income process\]) and setting limits of $\sOmega=\sepsilon =3$ [a conservative choice according to @AbowdCard89], we get the following dynamic linear model, $$y_{i,t}=
\sum\limits_{k=0}^{t-3}\somega _{i,k}
+\sum\limits_{k=t-2}^{t}\stheta _{\omega,t-k}\somega _{i,k}
+\sum\limits_{k=t-2}^{t}\stheta _{\varepsilon,t-k}\svarepsilon _{i,k}
\label{eq: dynamiclinearmodel}$$ For each individual $i$, the dynamic linear model for their excess log income ($\by_{i}$) is a combination of the homogeneous parameters $(\bthetaphi)$ and realized shocks ($\bshockparams$). In our Gibbs sampling model implementation, we take advantage of the fact that sampling new values of the homogeneous parameters conditional on fixed values of the realized shocks is relatively simple, and vice versa.
If we are given values of the realized shocks ($\bomega _{i},\bvarepsilon_{i}$), we can calculate the scalar $y_{i,t}^{\star }$ and the $1\times 6$ (since $\sOmega+\sepsilon$=6) vector $X_{i,t}$ ,
$$y_{i,t}^{\star }\equiv
y_{i,t}-\sum\limits_{k=0}^{t-3}\somega _{i,k}
\qquad \qquad
X_{i,t} \equiv
(\somega _{i,t-2},\somega _{i,t-1},\somegait,
\svarepsilon_{i,t-2},\svarepsilon _{i,t-1},\svarepsilonit)$$
Let $\by^{\star }$ be the $N(T-3)\times 1$ vector of all $y_{i,t}^{\star }$ across individuals $i$ and time $t$, and let $\bX$ be the $N(T-3)\times 6$ matrix whose rows are all $X_{i,t}$ across individuals $i$ and time $t$. We can then write equation (\[eq: dynamiclinearmodel\]) as a simple linear regression model, $$\by^{\star }=
\bX \cdot \bbeta + \be
\qquad \mathrm{where} \quad
\be \sim \mathrm{Normal}(\bzero,\sgammasq \cdot \bI)$$ where $\bbeta=(\stheta _{\omega,2},\stheta _{\omega,1},\stheta _{\omega,0},
\stheta _{\varepsilon,2},\stheta _{\varepsilon,1},\stheta _{\varepsilon,0})$ are the homogeneous parameters of interest. Note that this is the stage at which we use measurement error () as distinct from transitory shocks.
We use non-informative prior distributions for both $\sgammasq$ and $\bbeta$, which leads to the following posterior distributions (the Bayesian analog of a least-squares estimate): $$\begin{aligned}
\sgammasq &\sim &\mathrm{Inv-Gamma}\left( \frac{TN}{2}\,,\,\frac{(\by^{\star }-\bX\hat{\beta})^{\prime }(\by^{\star }-\bX\hat{\bbeta})}{2}\right) \notag \\
\bbeta &\sim &\mathrm{Normal}\left( \hat{\bbeta}\,,\,\gamma
^{2}\cdot (\bX^{\prime }\bX)^{-1}\right) \label{eq: regressionpost}\end{aligned}$$where $\hat{\bbeta}=(\bX^{\prime }\bX)^{-1}\bX^{\prime }\by^{\star }$ as in a least-squares regression. We sample new values of $\sgammasq$ and $\bthetaphi$ from the distributions in (\[eq: regressionpost\]), but with the additional constraint that $\sum_{k}\stheta _{\varepsilon,k}=1$.
Step 2: Sampling realized shocks ($\protect\bshockparams$) {#shockestimation}
----------------------------------------------------------
In this step, we take excess log income data ($\by$), the homogeneous parameters ($\bthetaphi$), and the volatility parameters $(\bheteroparams)$ as given. We use these to sample realized shocks ($\bshockparams$).
If we are now given values of the homogeneous parameters $(\bthetaphi)$, then the only unmeasured variables in our dynamic linear model (\[eq: dynamiclinearmodel\]) are the realized shocks ($\bomega_{i},\bvarepsilon _{i}$). We use maximum likelihood estimates from a Kalman filter [@Kal60] to sample new values of the realized shocks ($\bomega _{i},\bvarepsilon _{i}$), as outlined in @CarKoh94. Given the homogeneous parameters $(\bhomoparams)$ and the collection of volatility parameters $(\bheteroparams)$, each individual’s income process is independent, so run the Kalman filter and sampling procedure for the realized shocks ($\bomega _{i},\bvarepsilon _{i}$) for each individual $i$ separately.
Step 3: Sampling volatility parameters ($\protect\bheteroparams$) {#heteroestimation}
-----------------------------------------------------------------
In this step, we take sampled realized shocks ($\bshockparams$) as given and use these to sample estimates of volatility parameters ($\bheteroparams$). In order to sample a full set of volatility parameters $\bheteroparams$ from the distribution $p(\bheteroparams|\by,\bhomoparams,\bshockparams)$, it is easiest to proceed sequentially by sampling (one-by-one), the volatility parameters $\svolit$ for individual $i$ and year $t$ from the distribution $p(\bheteroparams|\by,\bhomoparams,\bshockparams,\bvolnit)$. Note that under this scheme, information about ($\svolit$) comes from our sampled permanent and transitory shocks ($\sshockit$) as well as our current estimates of the volatility parameters, $\bvolnit$, from other years within the individual as well as other individuals. We link these other volatility values $\bvolnit$ to our shock parameters $(\sshockit)$ through the posterior distribution, $$p(\svolit|\sshockit,\bvolnit)
\propto p(\sshockit|\svolit)
\cdot p(\svolit|\bvolnit)
\label{eq: mhdpposterior}$$The first term of equation (\[eq: mhdpposterior\]) comes from the likelihood of our realized shocks $(\sshockit)$ from our dynamic linear model, $$p(\sshockit|\svolit)\propto
\left( \ssigmasqit\stausqit\right) ^{-\frac{1}{2}}\exp \left(-
\frac{1}{2}\frac{\somegait^{2}}{\ssigmasqit}-\frac{1}{2}\frac{\svarepsilonit^{2}}{\stausqit}
\right)
\label{eq: mhdplikelihood}$$The second term of equation (\[eq: mhdpposterior\]) is our Markovian hierarchical Dirichlet process (MHDP) prior, $p(\svolit|\bvolnit)$, described in Sections \[section: heterogeneity\] and \[section: estimation\]. Sampling new values $\svolit$ from the posterior distribution (\[eq: mhdpposterior\]) is a multi-step process that acknowledges the structure of our population. First, we sample a volatility parameter proposal value ($\ssigsqstar\equiv\{\ssigmasqstar,\stausqstar\}$) from a continuous distribution $f(\cdot )$. For our implementation, we used an inverse-Gamma distribution, which is commonly used for variance parameters. We will set $\svolit=\ssigsqstar$ only if we cannot find a suitable $\svolit \in \bvolnit $ i.e. among our currently existing values in the population.
### Level 1: Is volatility unchanged from last year?
We first consider the posterior probability that $\svolit=\svolitm1$, $$\begin{aligned}
p(\svolit = \svolitm1) &
\propto & Q_{i,t} \cdot p( \sshockit | \svolitm1) \label{eq: prevchoice1} \\
p(\svolit \ne \svolitm1) &
\propto & \theta \ \ \ \cdot p( \sshockit | \ssigsqstar)
\label{eq: prevchoice2}\end{aligned}$$ Recall that $Q_{i,t}$ is the number of consecutive years with parameter values $\svolitm1$; $\theta$ is the prior tuning parameter for Level 1. We compare the posterior probability that volatility values are unchanged from last year ($\svolitm1$ in equation (\[eq: prevchoice1\])) to the posterior probability that volatility values are equal to the proposal value ($\ssigsqstar$ in equation (\[eq: prevchoice2\])). We sample a possible value for $\svolit$ from this posterior distribution, either $\svolitm1$ or $\ssigsqstar$, where choice is made stochastically by flipping a weighted coin with weights equal to the probabilities in equations (\[eq: prevchoice1\]) and (\[eq: prevchoice2\]). If this weighted coin flip selects $\svolitm1$, then we set $\svolit=\svolitm1$. If the coin flip selects $\ssigsqstar$, we do not set $\svolit=\svolitm1$ and instead proceed to Level 2 to find $\svolit$.
### Level 2: Is volatility the same as in another year?
Given that we did not choose to set $\svolit \ne \svolitm1$, we consider the posterior probability that $\svolit \in \bvolint$. If there are $\sNi$ unique values $\svolni \in \bvolint$, the posterior probability that $\svolit=\svolni$ is, $$\begin{aligned}
p(\svolit = \svolni) & \propto &
n_l \cdot p( \sshockit | \svolni) \quad l = 1, \ldots, \sNi \label{eq: personchoice1} \\
p(\svolit \notin \,\,
\bvolint) & \propto & \Theta_{i} \cdot p( \sshockit | \ssigsqstar) \label{eq: personchoice2}\end{aligned}$$ $n_l$ is the number of occurrences of value $\svolni$ within the set of possible values $\bvolnit$; $\Theta_{i}$ is the prior tuning parameter for Level 2. We sample one of these $\sNi +1$ choices by flipping a weighted coin with weights proportional to the probabilities above. If this weighted coin flip selects $\svolni \in \bvolint$, then we set $\svolit=\svolni$. If the coin flip selects $\ssigsqstar$, we do not set $\svolit=\svolni$ for any $\svolit \notin \bvolint$ but instead proceed to Level 3 to find $\svolit$.
### Level 3: Is volatility the same as another person’s?
Given that $\svolit \notin \bvolint$, we consider the posterior probability that $\svolit \in \pmb{\ssigma^{2}_{-i}}$, where $\pmb{\ssigma^{2}_{-i}}$ are the volatility values that currently exist in the population outside of individual $i$. If there are $\sN$ unique values $\svoln \in\pmb{\ssigma^{2}_{-i}}$, the posterior probability that $\svolit=\svoln$ is,
$$\begin{aligned}
p(\svolit = \svoln) & \propto & n_l
\cdot p( \sshockit | \svoln) \qquad
l = 1, \ldots, \sN \label{eq: popchoice1} \\
p(\svolit \notin \,\,
\bvolnit ) & \propto & \Theta \cdot p( \sshockit |
\ssigsqstar) \label{eq: popchoice2}\end{aligned}$$
$n_l$ is the number of occurrences of $\svoln$ within the set of current volatility values over all people other than person $i$; $\Theta$ is the prior tuning parameter for Level 3.We sample one of these $\sN +1$ values by flipping a weighted coin with weights proportional to the probabilities above. If this weighted coin flip selects $\svoln \in \bvolnit$, then we set $\svolit=\svoln$. If the coin flip selects $\ssigsqstar$, we set $\svolit=\ssigsqstar$. $\ssigsqstar$ represent new volatility values that have not yet been seen in the population.
The three steps outlined above result in a sampled volatility value $\svolit$ for person $i$ and year $t$, conditional on the other volatility values $\bvolnit$. We can repeat this procedure for all other years and individuals to update our full set of volatility values $\bheteroparams$.
[^1]: Jensen: Wharton School, Department of Statistics, [email protected]. Shore: Johns Hopkins University, Department of Economics, [email protected]. Please contact Shore at: 458 Mergenthaler Hall, 3400 N. Charles Street, Baltimore, MD, 21218; 410-516-5564.
[^2]: JEL Classification: D31 - Personal Income, Wealth, and Their Distributions; C11 - Bayesian Analysis; C14 - Semiparametric and Nonparametric Methods.
[^3]: keywords: Markovian hierarchical Dirichlet process, income risk, income volatility, heterogeneity
[^4]: We thank Christopher Carroll, Jon Faust, Robert Moffitt, and Dylan Small, for helpful comments, as well as seminar participants at the University of Pennsylvania Population Studies Center, the Wharton School, the 2008 Society of Labor Economists Annual Meeting, the 2008 Seminar on Bayesian Inference in Econometrics and Statistics, and the 2008 North American Annual Meeting of the Econometric Society, and the 2008 Annual Meeting of the Society for Economic Dynamics.
[^5]: [@Dahletal2007 is a noteable exception. @Dynanetal2007 provide an excellent survey of research on this subject in their Table 2, including @GottschalkMoffitt94 [@GottschalkMoffitt95; @DalyDuncan97; @DynarskiGruber97; @CameronTracy98; @Haider2001; @Hyslop2001; @GottschalkMoffitt2002; @Batchelder2003; @Hacker2006; @Cominetal2006; @GottschalkMoffitt2006; @Hertz2006; @Winship2007; @BollingerZiliak2007; @BaniaLeete2007; @Dahletal2007]. See also @ShinSolon2008.]{}
[^6]: [Our finding is consistent with @Dynanetal2007 who find that increasing income volatility has been driven by the increasing magnitude of extreme income changes, by the increasingly fat tails of the unconditional distribution of income changes. The fat tails of the unconditional distribution of income changes has also been documented in @GewekeKeane2000. In its reduced form, our paper shows that these increasingly fat tails are borne largely by individuals who are *ex-ante* likely to have volatile incomes. The increasingly fat tails of the unconditional distribution are not attributable – or at least not solely attributable – to increasingly fat tails of the *expected* distribution for everyone.]{}
[^7]: [Labor income in 1968 is labeled v74 for husbands and has a constant definition through 1993. From 1994, we use the sum of labor income (HDEARN94 in 1994) and the labor part of business income (HDBUSY94), with a constant definition through 2005. Note that data is collected on household “heads" and “wives" (where the husband is always the “head" in any couple). We use data for male heads so that men who are not household heads (as would be the case if they lived with their parents) are excluded.]{}
[^8]: The Winsorizing strategy employed here is obviously second-best to a strategy of modeling a zero income explicitly. Unfortunately, such a model is not feasible given the complexity added by evolving and heterogeneous volatility parameters. The other alternative would be simply to drop observations with low incomes, though we view this approach is much more problematic in our context; it would explicitly rule out the extreme income changes that are the subject of this paper.
[^9]:
[^10]:
[^11]: In @CarrollSamwick97, $\stheta_{\omega,k}=\stheta_{\varepsilon,k}=0$ is assumed for $k>0$, though the authors acknowledge that this assumption is unrealistic and design an estimation strategy that is robust to this restriction but do not estimate $\stheta_{k}$. In @MeghirPistaferri2004 and @Blundelletal2008, $\stheta_{\omega,k}=0$ is assumed for $k>0$ but $\stheta_{\varepsilon,k}=0$ is not.
[^12]:
[^13]:
[^14]:
[^15]:
[^16]:
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
**Tanwi Mallick**\
Mathematics and Computer Science Division\
Argonne National Laboratory, Lemont, IL\
[email protected]\
**Prasanna Balaprakash**\
Mathematics and Computer Science Division\
Argonne National Laboratory, Lemont, IL\
[email protected]\
**Eric Rask**\
Energy Systems Division\
Argonne National Laboratory, Lemont, IL\
[email protected]\
**Jane Macfarlane**\
Sustainable Energy Systems Group\
Lawrence Berkeley National Laboratory, Berkeley, CA\
[email protected]\
bibliography:
- 'trb\_template.bib'
title: 'Graph-Partitioning-Based Diffusion Convolution Recurrent Neural Network for Large-Scale Traffic Forecasting'
---
Abstract
========
Traffic forecasting approaches are critical to developing adaptive strategies for mobility. Traffic patterns have complex spatial and temporal dependencies that make accurate forecasting on large highway networks a challenging task. Recently, diffusion convolutional recurrent neural networks (DCRNNs) have achieved state-of-the-art results in traffic forecasting by capturing the spatiotemporal dynamics of the traffic. Despite the promising results, adopting DCRNN for large highway networks still remains elusive because of computational and memory bottlenecks. We present an approach to apply DCRNN for a large highway network. We use a graph-partitioning approach to decompose a large highway network into smaller networks and train them simultaneously on a cluster with graphics processing units (GPU). For the first time, we forecast the traffic of the entire California highway network with 11,160 traffic sensor locations simultaneously. We show that our approach can be trained within 3 hours of wall-clock time using 64 GPUs to forecast speed with high accuracy. Further improvements in the accuracy are attained by including overlapping sensor locations from nearby partitions and finding high-performing hyperparameter configurations for the DCRNN using DeepHyper, a hyperparameter tuning package. We demonstrate that a single DCRNN model can be used to train and forecast the speed and flow simultaneously and the results preserve fundamental traffic flow dynamics. We expect our approach for modeling a large highway network in short wall-clock time as a potential core capability in advanced highway traffic monitoring systems, where forecasts can be used to adjust traffic management strategies proactively given anticipated future conditions.
Introduction
============
In the United States alone, the estimated loss in economic value due to traffic congestion reaches into the tens or hundreds of billions of dollars, impacting not only the productivity lost due to additional travel time but also the additional inefficiencies and energy required for vehicle operation. To address these issues, Intelligent Transportation Systems (ITS) [@bishop2005intelligent] seek to better manage and mitigate congestion and other traffic-related issues via a range of data-informed strategies and highway traffic monitoring systems. Near-term traffic forecasting is a foundational component of these strategies; and accurate forecasting across a range of normal, elevated, and extreme levels of congestion is critical for improved traffic control, routing optimization, probability of incident prediction, and identification of other approaches for handling emerging patterns of congestion [@teklu2007genetic; @tang2005traffic]. Furthermore, these predictions and the related machine learning configurations and weights associated with a highly accurate model can be used to delve more deeply into the dynamics of a particular transportation network in order to identify additional areas of improvement above and beyond those enabled by improved prediction and control [@fadlullah2017state; @abdulhai2003reinforcement; @lv2014traffic]. These forecasting methodologies are also expected to enable new and additional forms of intelligent transportation system strategies as they become integrated into larger optimization and control approaches and highway traffic monitoring systems [@pang1999adaptive; @decorla1997total]. For example, the benefits of highly dynamic route guidance and alternative transit mode pricing in real time would be greatly aided by improved traffic forecasting.
Traffic forecasting is a challenging problem: The key traffic metrics such as flow[^1] and speed[^2] exhibit complex spatial and temporal correlations that are difficult to model with classical forecasting approaches [@williams2003modeling; @chan2012neural; @karlaftis2011statistical; @castro2009online]. From the spatial perspective, locations that are close geographically in the Euclidean sense (for example, two locations located in opposite directions of the same highway) may not exhibit a similar traffic pattern, whereas locations in the highway network that are far apart (for example, two locations separated by a mile in the same direction of the same highway) can show strong correlations. Many traditional predictive modeling approaches cannot handle these types of correlation. From the temporal perspective, because of different traffic conditions across different locations (e.g., diverse peak hour patterns, varying traffic flow and volume, highway capacity, incidents, and interdependencies), the time series data becomes nonlinear and non-stationary, rendering many statistical time series modeling approaches ineffective.
Recently, deep learning (DL) approaches have emerged as high-performing methods for traffic forecasting. In particular, Li et al. [@li2017diffusion] developed a diffusion convolution recurrent neural network (DCRNN) that models complex spatial dependencies using a diffusion process[^3] on a graph and temporal dependencies using a sequence to sequence recurrent neural network. The authors reported forecasting performances for 15, 30, and 60 minutes on two data sets: a Los Angeles data set with 207 locations collected over 4 months and a Bay Area data set with 325 locations collected over 6 months. They showed improvement on the state-of-the-art baselines methods such as historical average [@williams2003modeling], an autoregressive integrated moving average model with a Kalman filter [@xu2017real], a vector autoregressive model [@hamilton1995time], a linear support vector regression, a feed-forward neural network [@raeesi2014traffic], and an encoder-decoder framework using long short-term memory [@sutskever2014sequence]. Despite these results, modeling large highway networks with DCRNN remains challenging due to the computational and memory bottlenecks.
We focus on developing and applying DCRNN to a large highway network with thousands of traffic sensor locations. Our study is motivated by the fact that the highway network of a state such as California is $\approx$30 times larger than the Los Angeles or Bay Area dataset. Training a DCRNN with $\approx$30 times more data poses two main challenges. First, the training data size for thousands of locations is too large to fit in a single computer’s memory. Second, the time required for training a DCRNN on a large data set can be prohibitive, rendering the method ineffective for large highway networks. Two common approaches to overcome this issue in deep learning literature are distributed data-parallel training or model-parallel training [@dean2012large]. In data-parallel training, different computing nodes train the same copy of the model on different subsets of the data and synchronize the information from these models. The number of trainable parameters is the same as for single-instance training because the whole highway network graph is considered together. Speedup is achieved only by the reduced amount of training data per compute node. In model-parallel training, the model is split across different computing nodes, and each node estimates a different part of the model parameters. It is used mostly when the model is too large to fit in a single node’s memory. Implementation, fault tolerance, and better cluster utilization are easier with data-parallel training than with model-parallel training. Therefore, data-parallel training is arguably the preferred approach for distributed systems [@hegde2016parallel]. On the other hand, in traditional high-performance computing (HPC) domains, a common approach for scaling is domain decomposition, wherein the problem is divided into a number of subproblems that are then distributed over different compute nodes. While domain decomposition approaches are not applicable in scaling typical DL training such as image and text classification, for the traffic forecasting problem with DCRNN it is well suited. The reason is that traffic flow in one part of the highway network does not affect another part when the parts are separated by a large driving distance.
In this paper, we develop a graph-partitioning-based DCRNN for traffic forecasting on a large highway network. The main contributions of our work are as follows.
1. We demonstrate the efficacy of the graph-partitioning-based DCRNN approach to model the traffic on the entire California highway network with 11,160 sensor locations. We show that our approach can be trained within 3 hours of wall-clock time to forecast speed with high accuracy.
2. We develop two improvement strategies for the graph-partitioning-based DCRNN. The first is an overlapping sensor location approach that includes data from partitions that are geographically close to a given partition. The second is an adoption of DeepHyper, a scalable hyperparameter search, for finding high-performing hyperparameter configurations of DCRNN to improve forecast accuracy of multiple sensor locations.
3. We adopt and train a single DCRNN model to forecast both flow and speed simultaneously as opposed to the previous DCRNN implementation that predict either speed or flow.
Methodology
===========
In this section, we describe the DCRNN approach for traffic modeling, followed by graph partitioning for DCRNN, the overlapping node method, and the hyperparameter search approach.
Diffusion convolution recurrent neural network {#sec_dcrnn}
----------------------------------------------
Formally, the problem of traffic forecasting can be modeled as spatial temporal time series forecasting defined on a weighted directed graph $G = (V, \epsilon, A)$, where $V$ is a set of $N$ nodes that represent sensor locations, $\epsilon$ is the set of edges connecting the sensor locations, and $A\in R^{N\times N}$ is the weighted adjacency matrix that represents the connectivity between the nodes in terms of highway network distance. Given the graph $G$ and the time series data $X_{t-T'+1}$ to $X_t$, the goal of the traffic forecasting problem is to learn a function $\text{h(.)}$ that maps historical data $T'$ at given $t$ to future $T$ time steps: $X_{t-T'+1}, ...,X_t; G \xrightarrow{\text{h(.)}} X_{t+1},... , X_{t+T}$
In DCRNN, the temporal dependency of the historical data has been captured by the encoder-decoder architecture [@cho2014learning; @sutskever2014sequence] of recurrent neural networks. The encoder steps through the input historical time series data and encodes the entire sequence into a fixed length vector. The decoder predicts the output of the next $T$ time steps while reading from the vector. Along with the encoder-decoder architecture of RNN, a diffusion convolution process has been used to capture the spatial dependencies. The diffusion process [@teng2016scalable] can be described by a random walk on $G$ with a state transition matrix $D^{-1}A$. The traffic flow from one node to the neighbor nodes can be represented as a weighted combination of infinite random walks on the graph. The diffusion kernel is used in the convolution operation to map the features of the node to the result of the diffusion process beginning at that node. A filter learns the features for graph-structured data during training as a result of the diffusion convolution operation over a graph signal.
During the training phase, historical time series data and the graph are fed into the encoder, and the final stage of the encoder is used to initialize the decoder. The decoder predicts the output of the next $T$ time steps, and the layers of DCRNN are trained by using backpropagation through time. During the test, the ground truth observations are replaced by previously predicted output. The discrepancy between the input distributions of training and testing can cause performance degradation. In order to resolve this issue, scheduled sampling [@bengio2015scheduled] has been used, where the model is fed a ground truth observation with probability of $\epsilon_i$ or the prediction by the model with probability $1 - \epsilon_i$ at the $i$th iteration. The model is trained with MAE loss function, defined as $\text{MAE} = \frac{1}{s}\sum_{i=1}^s | y_i - \hat{y}_i |$, where $y_i$ is the observed value and $\hat{y}_i$ corresponds to the forecasted values for the $i^{th}$ training data.
Graph-partitioning-based DCRNN
------------------------------
To scale DCRNN, we adopt a divide-and-conquer approach for solving a large problem by solving subproblems defined on smaller subdomains. The overall idea of scaling is shown in Figure \[fig\_archi\]. Here, the graph has been divided into multiple subgraphs shown as partition 1 to partition M. Each of the partitions is then trained on M compute nodes simultaneously. Simultaneous training of subgraphs on multiple GPUs speeds up the overall training time in comparison with single-node training. The speedup with graph partitioning can be expressed as $S_m = t_1/t_m$, and the efficiency can be expressed as $E_m = t_1/(m*t_m)$. Here, $t_1$ is the time to execute an algorithm on a single node, and $t_m$ is the time to execute the same algorithm on $m$ nodes. $E = 1$ in a perfectly parallel algorithm.
{width="\linewidth"}
We use Metis [@metis], a graph-partitioning package, to decompose the large network graph into smaller subgraphs. First, to reduce the size of the input graph, Metis coarsens the graph iteratively by collapsing the connected nodes into supernodes. The process of coarsening helps reduce edge-cut. Then, the coarsened graph is partitioned by using either multilevel $k$-way partitioning [@karypis1998multilevelk] or multilevel recursive bisection algorithms [@karypis1998fast]. The next step is to map the partitions into the original graph by backtracking through the coarsened graph. In order to reduce the edge-cut, the nodes are swapped between partitions by using the Kernighan-Lin algorithm [@hendrickson1995multi] during uncoarsening. The method produces roughly $k$ equally sized partitions. Metis’s multilevel $k$-way partitioning algorithm provides additional capabilities such as minimizing the resulting subdomain connectivity graph, enforcing contiguous partitions, and minimizing alternative objectives. Therefore, we use the $k-$way partitioning algorithm in our work. Metis is extremely fast and provides high-quality partitions in a few seconds. For example, to perform 64 partition on a graph of 11, 160 nodes, metis takes only 0.030 seconds.
Various graph clustering and community detection methods [@liu2015empirical] have been developed, such as spectral clustering, Louvain, SlashBurn [@koutra2015summarizing], and $k-$core-based clustering [@giatsidis2011evaluating]. Compared with all these methods, Metis is a fast graph-partitioning algorithm [@liu2015empirical] that is capable of partitioning a million-node graph in a few tightly connected clusters. It generates roughly equally sized partitions. Our approach is agnostic to the graph-partitioning method adopted.
Overlapping nodes {#sec_overlap}
-----------------
An issue that affects the prediction accuracy in DCRNN due to graph partitioning is that nodes that are spatially correlated will end up in different partitions. While the graph-partitioning methods try to minimize this effect, the nodes at the boundary of the partitions will not have nearby spatially correlated nodes. To address this issue, we develop an overlapping nodes approach, wherein for each partition, we find and include spatially correlated nodes from other partitions. Consequently, the nodes that are near the boundary of the partition will appear in more than one partition. A naive approach for finding these nodes consists of computing nearest neighbors for each node in the partition based on the driving distance and excluding the nodes already included in the partition. The disadvantage of this approach is that it can include, for a given node, several spatially correlated nodes that are close to each other. This can lead to an increase in the number of nodes per partition, and consequently higher training time and memory requirement. Therefore, we down sample the spatially correlated nodes from other partitions as follows: given two spatially correlated overlapping nodes from a different partition, we select only one and remove the other if they are within $D'$ driving distance miles, where $D'$ is a parameter.
Hyperparameter tuning {#sec_hpc}
---------------------
The forecasting accuracy of the DCRNN depends on a number of hyperparameters such as batch size, filter type (i.e., random walk, Laplacian), maximum diffusion steps, number of RNN layers, number of RNN units per layers, a threshold `max_grad_norm` to clip the gradient norm to avoid exploring gradient problem of RNN [@pascanu2013difficulty], initial learning rate, and learning rate decay. Li et al. [@li2017diffusion] used a tree-structured Parzen estimator [@bergstra2011algorithms] for tuning the hyperparameters of the DCRNN; the obtained values are used as the default configuration. However, our dataset has a lot more variability because we consider all the districts of California. Therefore, finding the appropriate hyperparameter values is critical in our setting.
We use DeepHyper [@balaprakash2018deephyper], a scalable hyperparameter search (HPS) package for neural networks, to search for high performing hyperparameters values for DCRNN. DeepHyper adopts an asynchronous model-based search (AMBS) method, which relies on fitting a surrogate model that tries to learn the relationship between the hyperparameter configurations and their corresponding model validation errors. The surrogate model is then used to prune the search space and identify promising regions of the search space. The surrogate model is iteratively refined in the promising regions of the hyperparameter search space by obtaining new outputs at inputs that are predicted by the model to be high performing.
Given that we use a graph partition approach, finding the best hyperparameter configuration for each partition, although feasible, will be computationally expensive. Therefore, we select an arbitrary partition, run a hyperparameter search on it, and use the same best hyperparameter configuration for all the partitions.
Multi-output forecasting with a single model {#sec_multioutput}
--------------------------------------------
In the previous study, DCRNN was used to forecast only speed based on historical speed data. In this paper, we customize the input and output layers of the DCRNN for multi-output forecasting and demonstrate that a single DCRNN model can be trained and used for forecasting speed and flow simultaneously. The three key modifications for multi-output forecasting are as follows: 1) normalization of speed and flow: to bring speed and flow to the same scale, normalization has been done separately on the two features using the standard scalar transformation. The normalized values of speed are given by: $z_{sp} = \frac{x_{sp} - \mu_{sp}}{\sigma_{sp}}$, where $\mu_{sp}$ is the mean and $\sigma_{sp}$ is the standard deviation of the speed values $x_{sp}$. The same method is applied for normalizing the flow values ($z_{fl} = \frac{x_{fl} - \mu_{fl}}{\sigma_{fl}}$, where $\mu_{fl}$ and $\sigma_{fl}$ are the standard deviation of the flow values $x_{fl}$). We apply an inverse transformation to the normalized speed and flow forecasting values to transform them to the original scale (for computing error on the test data). 2) multiple output layers in the DCRNN: in the previous study of DCRNN, the convolution filter $f_{\theta}$ learns the graph-structured data from input graph signal $X$. This filter is parameterized by $\theta_{P,Q}$ to take P-dimensional input (such as speed and flow) and predict Q-dimensional output (such as speed and flow). Though multiple output prediction is reported as a capability of DCRNN, but its implementation had the format to take only 1-dimensional input and predict same as output. We changed the input/output format in our implementation with which $P-$dimensional input can be given to predict $Q-$dimensional output. 3) loss function: for multioutput training, we use a loss function of the form $\text{MAE} = \text{MAE}_{sp} + \text{MAE}_{fl} = \frac{1}{s}\sum_{i=1}^s | y_{sp_{i}} - \hat{y}_{sp_{i}} | + \frac{1}{s}\sum_{i=1}^s | y_{fl_{i}} - \hat{y}_{fl_{i}}|$, where $y_{sp_{i}}$ and $y_{fl_{i}}$ are observed speed and flow values and $\hat{y}_{sp_{i}}$ and $\hat{y}_{fl_{i}}$ are corresponding forecast values, respectively, for the $i^{th}$ training data, and $s$ is the total number of training points.
California highway network {#sec_data_cal}
==========================
For modeling the California highway network, we used data from PeMS [@pems]. It provides access to real-time and historical performance data from over 39,000 individual sensors. The individual sensors placed on the different highways are aggregated across several lanes and are fed into vehicle detector stations. The PeMS dataset contains raw detector data for over 18,000 vehicle detector stations. These include a variety of sensors such as inductive loops, side-fire radar, and magnetometers. The sensors may be located on High-occupancy Vehicle lanes, mainlines, on ramps, and off ramps. The dataset covers 9 districts of California—D3 (North Central) with 1,212 stations, D4 (Bay Area) with 3,880 stations, D5 (Central coast) with 382 stations, D6 (South Central) with 624 stations, D7 (Los Angeles) with 4,864 stations, D8 (San Bernardino) with 2,115 stations, D10 (Central) with 1,195 stations, D11 (San Diego) with 1,502 stations, and D12 (Orange County) with 2,539 stations. A total of 18,313 stations are listed by site. Detectors capture samples every 30 seconds. PeMS then aggregates that data to the granularity of 5 minutes, an hour, and a day. The data includes timestamp, station ID, district, freeway, direction of travel, total flow, and average speed(mph). The time series data is available from 2001 to 2019.
PeMS details the station IDs, district, freeway, direction of travel, and absolute postmile markers. This list does not contain the latitude and longitude for the stations IDs, which is essential to defining the connectivity matrix used by the DCRNN. In the PeMS database, the latitude and longitude are associated with postmile markers of every freeway given the direction. We downloaded the entire time series data of the California highway network and find the latitude and longitude for sensor IDs by matching the absolute postmile markers of every freeway. Linear interpolation is used to find the exact latitude and longitude if the absolute postmile markers do not match exactly.
The official PeMs website shows that 69.59% of the $\approx$18K stations are in good working condition. The remaining 30.41% do not capture time series data throughout the year. These are excluded from our dataset. Our final dataset has 11,160 stations for the year 2018 with the granularity of 5 minutes. We observed that flow and speed values are missing for multiple time periods in the time series data. We calculate the missing data by taking the average of the past one week data of that particular timestamp. Holidays are handled separately from normal working days.
Experimental results
====================
We represent the highway network of 11,160 detector stations as a weighted directed graph. The speed and flow data of each node of the graph is collected over one year ranging from January 1, 2018, to December 31, 2018, from PeMS [@pems]. From the one-year data, we used the first 70% of the data (36 weeks approx.) for training and the next 10% (5 weeks approx.) and 20% (10 weeks approx.) of the data for validation and testing, respectively. Given 60 minutes of time series data on the nodes in the graph, we forecast for the next 60 minutes. We prepared the dataset in a way to look back ($T'$ as mentioned in \[sec\_dcrnn\]) for 60 minutes or 12 time steps (granularity of the data is 5 minutes as mentioned in Section \[sec\_data\_cal\]) to predict ($T$) next 60 minutes or 12 time steps. The look back ($T'$) window slides by 5 minutes or 1 time steps and repeat until the whole data is consumed. The forecasting performance of the models were evaluated on the test data using MAE =$\frac{1}{u}\sum_{i=1}^u | y_i - \hat{y_i}|$, where $y_1$, . . . , $y_u$ represent the observed values, $\hat{y_1} \ldots \hat{y_u}$ represent the corresponding predicted values, and $u$ denotes the number of prediction samples.
The adjacency matrix for DCRNN requires the highway network distance between the nodes. We used the Open Source Routing Machine (OSRM) [@osrm] running locally for the area of interest to compute the highway network distance. Given the latitude and longitude of two nodes, OSRM gives the shortest driving distance between them using OpenStreetMap data [@osm]. To speed up the highway network distance computation, first we find 30 nearest neighbors for each node using the Euclidean distance and then limit the OSRM queries only to the nearest neighbors. As in the original DCRNN work, we compute the pairwise highway network distances between nodes to build the adjacency matrix using a thresholded Gaussian kernel [@shuman2012emerging]: $A_{ij} = \exp (- \frac{ dist(v_i, v_j)^2}{\sigma^2})\;\; if \;\; dist(v_i, v_j)^2 \leq thresh,$ otherwise $0$, where $A_{ij}$ represents the edge weight between node $v_i$ and node $v_j$; $dist(v_i, v_j )$ denotes the highway network distance from node $v_i$ to node $v_j$; $\sigma$ is the standard deviation of distances; and $thresh$ is the threshold, which introduces the sparsity in the adjacency matrix.
For the experimental evaluation, we used Cooley, a GPU-based cluster at the Argonne Leadership Computing Facility. It has 126 compute nodes, where each node consists of two 2.4 GHz Intel Haswell E5-2620 v3 processors (6 cores per CPU, 12 cores total), one NVIDIA Tesla K80 (two GPUs per node), 384 GB RAM per node, and 24 GB GPU RAM per node (12 GB per GPU). The compute nodes are interconnected via an InfiniBand fabric. We used Python 3.6.0, TensorFlow 1.3.1, and Metis 5.1.0. We customized the DCRNN code of [@li2017diffusion], which is available on Github [@li2018dcrnn_traffic]. Given $p$ partitions of the highway network, we trained partition-specific DCRNNs simultaneously on Cooley GPU nodes. We used two MPI ranks per node, where each rank ran a partition-specific DCRNN using one GPU. The input data for different partitions (time series, and adjacency matrix of the graph) were prepared offline and loaded into the partition-specific DCRNN before the training started.
We used a bidirectional graph random walk [@lovasz1993random] to model the stochastic nature of highway traffic. Random walk on a directed graph is random process that gives a path composed of successive random steps on the graph. The default hyperparameter configuration for the DCRNN is: batch size: 64, filter type: random walk, number of diffusion steps: 2 , RNN layers: 2, and RNN units per layer: 16 , a threshold for gradient clipping: 5, initial learning rate: 0.01, and learning rate decay of 0.1. We trained our model by minimizing MAE using the Adam optimizer [@kingma2014adam].
Impact of number of graph partitions on accuracy and training time
------------------------------------------------------------------
![Distribution of MAE for different number of partitions[]{data-label="fig_accuracy"}](images/strong_scaling.png){width="1\linewidth"}
Here, we experiment with different number of graph partitions and show that partitions with larger number of nodes require longer training time and partitions with fewer nodes can reduce the forecasting accuracy.
We used Metis to obtain 2, 4, 8, 16, 32, 64, and 128 partitions of the California highway network graph. The average number of nodes in each case is 5,580, 2,790, 1395, 697, 348, 174, and 87, respectively. Partition of size 1 (the whole network) and 2 were not presented because the training data was too large to fit in the memory of a single K80 node of Cooley. Given $p$ partitions, we used $p/2$ nodes (or $p$ GPUs) on Cooley to run the partition-specific DCRNNs simultaneously. We consider the training time as the maximum time taken by any partition-specific DCRNN training (excluding the data loading time).
Figure \[fig\_accuracy\] shows the distribution of MAE of all nodes obtained using box-and-whisker plots. Each box represents distribution of MAE of 11,160 nodes. The ends of each box are 25% (bottom) and 75% (top) quantiles of the distribution, the median of the distribution is shown as the horizontal line in the middle of the box, the two vertical lines on the two sides of the whisker represent 5% and 95% of the distribution, and the diamonds mark the outliers of the distribution. From the results we can observe that medians, 75% quantiles, and the maximum MAE values show a trend in which an increase in the number of partitions decreases the MAE. From 4 to 64 partitions, the median of MAE decreases from 2.11 to 2.02. The increase in accuracy can be attributed to the effectiveness of the graph partitioning of Metis that separates nodes that were not temporally and spatially correlated. For smaller number of partitions, presence of such nodes increases MAE. For 128 partitions (with only 87 nodes per partition), the observed MAE values are higher than that of 64 partitions. This is because the graph partition results in significant number of spatially correlated nodes ending up in different partitions. This can be assumed as a tipping point for graph partitioning, which relates to the size and spread of the actual network.
![Training time for DCRNNs with different number of partitions[]{data-label="fig_strong_scaling_time"}](images/strong_scaling_time.png){width="1\linewidth"}
Figure \[fig\_strong\_scaling\_time\] shows the training time required for different numbers of partitions. We can observe that the time decreases significantly with an increase in the number of partitions. We can also observe that our approach reduces the training time from 2,820 minutes on 4 partitions(= 4 GPUs) to 178.67 minutes on 64 partitions (= 64 GPUs), resulting in a 15.78x speedup. Until 64 partitions, we observe almost a liner speedup, where doubling the number of partitions (and GPUs) results in $\approx$2X speedup. However, the speedup gains drop significantly with 128 nodes. This can be attributed to the reduction in the workload per GPU, where there is not enough workload for the GPU given that there are only 87 nodes per partition.
Since the best forecasting accuracy and speedup were obtained by using 64 partitions, we used it as a default number of partitions in rest of the experiments.
Impact of training data size
----------------------------
Here, we assess the impact of training data size and show that it has a significant impact on the predictive accuracy.
\[!ht\] ![Impact of training data size on MAE of the graph-partition-based DCRNN with 64 partitions[]{data-label="fig_reduction"}](images/data_reduction.png "fig:"){width="1\linewidth"}
From the full 36 weeks of training data, we selected the last 1, 2, 4, 12, and 20 weeks of data for training the DCRNN. The last weeks of data were chosen to minimize the impact of highway and sensor upgrades. Figure \[fig\_reduction\] shows the distribution of MAE of all nodes obtained using box-and-whisker plots. From the plots it can be observed that the medians, the 75% quantiles, and the maximum MAE values show that increasing the training data size decreases the MAE. These results show that DCRNN, similar to other state of the art neural networks [@cai2018cascade; @al2019character], can leverage large amount of data to improve accuracy. Therefore, we use the entire 36 weeks of training data in rest of the experiments.
Impact of overlapping nodes and hyperparameter tuning
-----------------------------------------------------
Here, we demonstrate that the graph-partitioning-based DCRNN achieves high forecasting accuracy using overlapping nodes and hyperparameter search.
We trained the graph-partitioning-based DCRNN with 64 partitions for the California highway network on 32 nodes of Cooley (two DCRNNs per node; 64 GPUs). We refer this variant to `DCRNN_64_naive`. It took a total training time of 178 minutes. After training, we forecast the speed for 60 minutes on the test data and calculated the MAE for each node. The results are summarized in the first row of Table \[tab\_HPS\_results\]. We observe that MAE values of 1,716, 6,729, 2,266, and 449 nodes are less than 1, between 1 and 3, between 3 and 5, and greater than 5, respectively.
Next, we trained the graph-partitioning-based DCRNN with 64 partitions with overlapping nodes as described in Section \[sec\_overlap\]. We down sampled nodes with different distance threshold ($D'$) values: 0.5 mile, 1 mile, 1.5 miles, 2 miles, and 3 miles. The result showed no significant improvement beyond the 1 mile of threshold; therefore, we used 1 mile as distance threshold for our experiments. In a given partition, while calculating the MAE for each node, we did not consider the overlapping nodes as they originally belong to a different partition, where their MAE values will be computed. We refer this variant to `DCRNN_64_overlap`. The results are shown in the row 2 of Table \[tab\_HPS\_results\]. We observe that `DCRNN_64_overlap` completely outperforms `DCRNN_64_naive`. With reference to the latter, the number of nodes with MAE values less than 1 has increased from 1,716 to 1,837; on the other hand, the number of nodes with MAE values between 1 and 3, 3 and 5, and greater than 5 reduced from 6,729 to 6,687, 2,266 to 2,204, and 449 to 432, respectively. We observe that the training time increased from 178.67 minutes to 221.04 minutes, which can be attributed to the increase in the number of nodes per partition.
Finally, we ran hyperparameter search with DeepHyper for `DCRNN_64_naive` and\
`DCRNN_64_overlap`. We used 5 months of data (from May 2018 to October 2018) from partition 1. We used 32 nodes of Cooley with a 12 hours of wall-clock time as stopping criterion. DeepHyper sampled 518 and 478 hyperparameter configurations for naive and overlapping approaches, respectively. The best hyperparameter configurations are selected from each and used to train and infer the forecasting accuracy. We refer these two variants as `DCRNN_64_naive_hps` and `DCRNN_64_overlap_hps`. The results are shown in the rows 3 and 4 of the Table \[tab\_HPS\_results\]. We observe that `DCRNN_64_naive_hps` outperforms `DCRNN_64_naive`, where hyperparameter tuning improved the accuracy of several nodes. The number of nodes with MAE values less 1 and between 1 and 3, have increased from 1,716 to 1,920 and 6,729 to 6,897, respectively. The number of nodes with MAE values between 3 and 5, and greater than 5 got reduced from 2,266 to 1,980, and 449 to 363, respectively. We did not see a significant improvement with `DCRNN_64_overlap_hps`. The number of node in the MAE bins are similar to `DCRNN_64_overlap`. Moreover, hyperparameter tuning resulted in an increase in the number of trainable parameters, which led to training time increase from 221.04 min to 461.57 mins.
We did not notice a significant difference in the time required for forecasting on the test data. An exception is `DCRNN_64_overlap_hps`, where the large number of trainable parameters increases the forecasting time by 1 minute (5.83 mins).
To summarize, we can improve the graph-partitioning-based DCRNNs either by using overlapping nodes from other partitions or by tuning the hyperparameters of DCRNN. Combining both did not show any benefit in our study.
[ll|r|r|r|r|r|r|r|]{} & & & & & & & &\
& `DCRNN_64_naive` & 1,716 & 6,729 & 2,266 & 449 & 14,608 & 178.67 & 4.38\
&
--------------------
`DCRNN_64_overlap`
--------------------
: Results of graph-partitioning-based DCRNN with overlapping region and hyperparameter search[]{data-label="tab_HPS_results"}
& 1,837 & 6,687 & 2,204 & 432 & 14,608 & 221.04 & 4.88\
&
----------------------
`DCRNN_64_naive_hps`
----------------------
: Results of graph-partitioning-based DCRNN with overlapping region and hyperparameter search[]{data-label="tab_HPS_results"}
& 1,920 & 6,897 & 1,980 & 363 & 19,808 & 287.05 & 4.92\
&
------------------------
`DCRNN_64_overlap_hps`
------------------------
: Results of graph-partitioning-based DCRNN with overlapping region and hyperparameter search[]{data-label="tab_HPS_results"}
& 1,897 & 6,940 & 1,972 & 351 & 38,048 & 461.57 & 5.83\
Multioutput forecasting
-----------------------
Here, we show that a single DCRNN model can be used to predict the speed and flow simultaneously and the forecasting results preserve the fundamental properties of traffic flow.
Figure \[fig\_mult\_output\] shows the distribution of MAE of all nodes using box-and-whisker plots. The first and second box plots show the speed forecast from the DCRNN models that are trained to forecast only speed and to forecast speed and flow simultaneously. Similarly, the third and forth box plots are for flow forecasts. The median of MAE from speed only model (first box plot) is 2.02, which got reduced to 1.98 when multioutput model (second box plot) is used. Similarly, the median of MAE from flow only model (third box plot) is 21.20, which got reduced to 20.64 when multioutput model (fourth box plot) is used. We adopted a statistical test to check if the observed MAE values between the two models are significant. We used the paired t-test and found that the multioutput model obtains MAE values that are significantly better than the speed only or flow only model ($p-$values of $9.20 \times 10^{-4}$ for speed and $5.77 \times 10^{-5}$ for flow). The superior performance of multioutput forecasting can be attributed to the multitask learning [@sener2018multi]. The key advantage is that it leverages the commonalities and differences across speed and flow learning tasks. This results in improved learning efficiency and consequently forecasting accuracy when compared to training the models separately.
![Box plot distribution of MAE for speed and flow forecasting. From left to right the box plots show the results of: speed forecasting from speed only model, speed forecasting from multioutput model, flow forecasting from flow only model, and flow forecasting from multioutput model[]{data-label="fig_mult_output"}](images/multiple_output.png){width="1\linewidth"}
In Figure \[fig\_flow\_diag\], we show speed and flow forecasting forecasting results of a congested node (ID: 717322 located on the highway 60-E in Los Angeles area) in a scatter plot. We can observe that the speed and flow forecast values closely follow the fundamental flow diagram with three distinct phases of congestion, bounded, and free flow. This forecasting pattern of DCRNN shows that the model has learned and preserved the properties of traffic flow.
![Closeness of the predicted flow and speed with fundamental traffic flow diagram[]{data-label="fig_flow_diag"}](images/flow_diag.png){width=".8\linewidth"}
Related work
============
Modeling the flow and speed patterns of traffic in a highway network has been studied for decades. Capturing the spatiotemporal dependencies of the highway network is a crucial task for traffic forecasting. The methods for traffic forecasting are broadly classified into two main categories: knowledge-driven and data-driven approaches. In transportation and operational research, knowledge-driven methods usually apply queuing theory [@cascetta2013transportation; @romero2018queuing; @lartey2014predicting; @yang2014application] and Petri nets [@ricci2008petri] simulate user behaviors of the traffic. Usually, those approaches estimate the traffic flow of one intersection at a time. Traffic prediction for the full highway system of an entire state has not been attempted to date using knowledge-driven approaches.
Data-driven approaches have received notable attention in recent years. Traditional methods include statistical techniques such as autoregressive statistics for time series [@williams2003modeling] and Kalman filtering techniques [@kumar2017traffic]. These models are mostly used to forecast at a single sensor location and are based on a stationary assumption about the time series data. Therefore, they often fail to capture nonlinear temporal dependencies and cannot predict overall traffic in a large-scale network [@li2017diffusion]. Recently, statistical models have been challenged by machine learning methods on traffic forecasting. More complex data modeling can be achieved by these models, such as artificial neural networks (ANNs) [@chan2012neural; @karlaftis2011statistical], and support vector machines (SVMs) [@castro2009online; @ahn2016highway].
However, SVMs are computationally expensive for large networks, and ANNs cannot capture the spatial dependencies of the traffic network. Furthermore, the shallow architecture of ANNs make the network less efficient compared with a deep learning architecture. Recently,deep learning models such as deep belief networks [@huang2014deep] and stacked autoencoders [@lv2015traffic] have been used to capture effective features for traffic forecasting. Recurrent neural networks (RNNs) and their variants, long short-term memory (LSTM) networks [@ma2015long] and gated recurrent units [@fu2016using], show effective forecasting [@cui2018deep; @yu2017deep] because of their ability to capture the temporal dependencies. RNN-based methods can capture contextual dependency in the temporal domain, but spatial dynamics are often missed. To capture the spatial dynamics, researchers have used convolutional neural networks (CNNs). Ma et al. [@ma2017learning] proposed an image-based traffic speed prediction method using CNNs, whereas Yu et al. [@yu2017spatiotemporal] proposed spatiotemporal recurrent convolutional networks for traffic forecasting. Spatial dynamics have been captured by deep CNNs, and temporal dynamics have been learned by LSTM networks. In both, the highway network has been represented as an image, and the speed of each link is mapped by using color in the images. The model has been tested on 278 links of the Beijing transportation network. Zhang et al. [@zhang2016dnn; @zhang2017deep] also represented the flow of crowds in a traffic network using grid-based Euclidean space. The temporal closeness, period, and trend of the traffic were modeled by using a residual neural network framework. They evaluated the model on Beijing and New York City crowd flows. They used two datasets: (1) trajectory of taxicab GPS data of four time intervals and (2) trajectory of NYC bike data of one time interval. Trip data included trip duration, starting and ending sensor IDs, and start and end times. The key limitation of these approaches is that they do not capture non-Euclidean spatial connectivity. Du et al. [@du2018hybrid] proposed a model with one-dimensional CNNs and GRUs with the attention mechanism to forecast traffic flow on UK traffic data. The contribution of this method is multimodal learning by multiple features (flow, speed, events, weather, and so on) fusion on single time series data of one year (34,876 timestamps in 15-minute intervals). The proposed approach is limited to a narrow spacial dimension, however. Recently, CNNs have been generalized from a 2D grid-based convolution to a graph-based convolution in non-Euclidean space. Yu et al.[@yu2017spatio] modeled the sensor network as a undirected graph and proposed a deep learning framework, called spatiotemporal graph convolutional networks, for speed forecasting. They applied graph convolution and gated temporal convolution through spatiotemporal convolutional blocks. The experiments were done on two datasets, BJER4 and PeMSD7, collected by the Beijing Municipal Traffic Commission and California Department of Transportation, respectively. The maximum size of their data set was 1,026 sensors of California district 7. However, these spectral-based convolution methods require the graph to be undirected. Hence, moving from a spectral-based to a vertex-based method, Atwood and Towsley [@atwood2016diffusion] first proposed convolution as a diffusion process across the node of the graph. Later, Hechtlinger et al. [@hechtlinger2017generalization] developed convolution to graphs by convolving every node and its closest neighbors selected by a random walk. However, none of these methods capture the temporal dependencies Li et al. [@li2017diffusion] first represented diffusion-convolutional recurrent neural network (DCRNN) to capture the spatiotemporal dynamics of the highway network. Our approach differs from these works in many respects. From the problem perspective, none have addressed a problem size of 11,160 sensor locations covering the fully monitored California highway system. From the solutions perspective, graph-partitioning-based approach for large-scale traffic forecasting, adoption of multinode GPUs, and multioutput forecasting were never investigated before.
Conclusion and future work
==========================
We described a traffic forecasting approach for a large highway network comprising the entire state of California with 11,160 sensor locations. We developed a graph-partitioning approach to partition the large highway network into a number of small networks, and trained them simultaneously on a moderately sized GPU cluster. We studied the impact of the number of partitions on the training time and accuracy. We showed that 64 partitions gave the best forecasting accuracy and GPU resource usage efficiency with a training time of 178 minutes. We demonstrated that our approach leverages a large training data to improve forecasting accuracy. We developed overlapping nodes approach to include spatially correlated nodes from different partitions and showed significant improvement in accuracy. We tuned the hyperparameters of the graph-partitioning-based DCRNN using DeepHyper and showed improvement in forecasting accuracy. We adapted and trained a single DCRNN model to forecast speed and flow and showed that the accuracy is better than models that predict either speed or flow and that the forecasts preserve the fundamental traffic flow dynamics. The DCRNN model once trained can be run on traditional hardware such as CPUs for forecasting without the need for multiple GPUs and could be readily integrated into a traffic management center. Once integrated into a traffic management center, the scale and accuracy of the forecasting techniques discussed in this work would likely lead to more proactive decision making as well as better decisions themselves given the capability to make large-scale and accurate forecasts regarding future traffic states.
Our current and future work includes 1) Extending the approach for large scale traffic forecasting with mobile device data. Our goal will be to determine if mobile device data can act as a proxy for inductive loop data, which could either be used as a substitute for poorly working loops or extending the scope of the monitoring to areas where loops would be prohibitively expensive. 2) Combining DCRNN with large scale simulation to integrate realistic speed and flow forecasts into active traffic management decision algorithms; and 3) Developing models for route and policy scenario evaluation in adaptive traffic routing and management studies.
Acknowledgments {#acknowledgments .unnumbered}
===============
This material is based in part upon work supported by the U.S. Department of Energy, Office of Science, and under contract DE-AC02-06CH11357. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility under contract DE-AC02-06CH11357. This report and the work described were sponsored by the U.S. Department of Energy (DOE) Vehicle Technologies Office (VTO) under the Big Data Solutions for Mobility Program, an initiative of the Energy Efficient Mobility Systems (EEMS) Program. The following DOE Office of Energy Efficiency and Renewable Energy (EERE) managers played important roles in establishing the project concept, advancing implementation, and providing ongoing guidance: David Anderson and Prasad Gupte.
Government license {#government-license .unnumbered}
==================
The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (“Argonne”). Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan. http://energy.gov/downloads/doe-public-access-plan.
[^1]: Flow (volume) is a quantity representing an estimate of the number of vehicles that passed over each detector on the highway in a given time period
[^2]: Speed is the estimated rate of motion at which a detector records drivers operating their vehicles
[^3]: In physics, diffusion is a process of movement of particle from a region of higher concentration to a region of lower concentration. The diffusion process can be represented as a weighted combination of infinite random walks on a graph.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Disordered 2D chiral fermions provide an effective description of several materials including graphene and topological insulators. While previous analysis considered delta correlated disorder and no ultraviolet cut-offs, we consider here the effect of short range correlated disorder and the presence of a momentum cut-off, providing a more realistic description of condensed matter models. We show that the density of states is anomalous with a critical exponent function of the disorder and that conductivity is universal only when the ultraviolet cut-off is removed, as consequence of the supersymmetric cancellation of the anomalies.'
author:
- Vieri Mastropietro
title: 'Universality, exponents and anomaly cancellation in disordered Dirac fermions'
---
Introduction
============
It is known that several materials exhibit fermionic excitations with linear dispersion relation close to the Fermi level, which can be effectively described in terms of $2+1$dimensional Dirac fermions. Early examples include systems displaying integer quantum Hall effect [@LFSG] and [*d-wave superconductors*]{} [@NTW; @ASZ] and more recently [*graphene*]{} [@V3; @H0; @Mi; @CG; @Z1; @FCO] and [*topological insulators*]{} [@N1; @Mo]. In particular, in the case of graphene at half filling it has been observed [@N2] that the [*optical*]{} conductivity (for frequencies greater than the temperature) is essentially constant in a wide range of frequencies and very close to the [*universal*]{} value $(\pi/2)(e^2/h)$, which also happens to be the value found for the system of non-interacting $2d$ Dirac fermions [@LFSG], a remarkable result in view of the fact that interactions are not particularly weak. In transport measurements an [*universal*]{} value for the conductivity is also found, of order of the conductivity quantum $e^2/h$ [@N3]; again a surprising result in view of the presence of disorder which is surely relevant in such experiments.
It is of course important to understand if and under which conditions such universality can be understood theoretically. In presence of weak short range interactions, after first perturbative computations claiming non vanishing corrections, it was finally rigorously proved [@GMP] that the optical conductivity is [*exactly equal*]{} to its non interacting value. Note that the emerging description is in terms of a Nambu-Jona Lasinio model and the natural cut-off provided by honeycomb lattice ensures the correct symmetries and allows the proof of the complete cancellation of the interaction corrections. On the other hand, in the case of [*long range*]{} Coulomb interaction it has been predicted that the optical conductivity is still equal to the non interacting value [@H0], the argument this time being based on the divergence of the Fermi velocity. However, the Fermi velocity divergence found in the Coulomb case at very low frequencies is clearly rather unphysical, and simply signals ultimate inadequacy of the usual model of instantaneous Coulomb interaction. With the increase of the Fermi velocity the retardation effects eventually become important, so that the retarded current-current interaction must be added to the Coulomb density-density interaction; the emerging model is in this case $QED_{4,3}$ (with an ultraviolet cut-off ) in which the fermionic velocity is different from the light velocity. Such system have been analyzed before in [@GGV] , [@GMPgauge] and it was found that the flow of the Fermi velocity stops at the velocity of light $c$, and, maybe most importantly, that the coupling constant (i. e. the charge) in the theory is [*exactly*]{} marginal (anomalous critical exponents are found); as a consequence of that, the optical conductivity is [*not*]{} equal to its non interacting value but corrections are found [@HM], which are however quite small and still universal at lowest order (they depend only only from the fine structure constant).
When we turn to the analysis of the effect of disorder on the conductivity, the natural emerging description is in terms of [*disordered Dirac fermions*]{}, which were extensively analyzed along the years. In the case of [*chiral preserving disorder*]{} it was found that the density of states is vanishing with a critical exponent (non trivial function of the disorder strength) but the conductivity is [*universal*]{} and not depending from the disorder amplitude, see [@LFSG; @NTW]. Such results, obtained using the replica trick, were confirmed and extended by a Supersymmetric analysis of such models [@ASZ; @Mu1; @Mu2] leading to a functional integral in Bosonic and Grassmann variables and a [*local*]{} quartic interaction. It is rather natural to relate such results to the universal conductivity found by transport measurements in graphene [@Mi], despite the understanding of why the dominant disorder in graphene should preserve chirality is an open issue which may be related to how the sample is produced. However, even assuming that disorder preserves chirality, several questions still remain to be understood. The results in [@LFSG; @NTW; @ASZ; @Mu1; @Mu2] on Dirac fermions with disorder where found assuming [*delta correlated*]{} disorder and an unbounded fermionic dispersion relation (no ultraviolet cut-offs). Such features makes an exact analysis possible (even non perturbative, see [@Mu2] and references therein) but produce [*ultraviolet divergences*]{} similar to the one present in local Quantum Field Theory in $d=1+1$ (for instance in the Thirring model), which could lead to some discrepancy with respect to lattice models (see [@Z0; @Z] and the discussion in [@Zirn]) which are of course free from ultraviolet divergences. As the dispersion relation (in graphene or in the other condensed matter applications) is approximately conical (“relativistic”) only in a small region around the Fermi level, it is natural to consider the presence of a momentum cut-off; moreover, a [*short-range*]{} correlated disorder is a much more realistic description for condensed matter systems, see [*e.g.*]{} [@FCO; @N1; @Mo]. Both such features make disordered Dirac fermions [*free*]{} from ultraviolet divergences, and it is therefore natural to ask if the results with no cut-off and $\d$-correlated disorder are sufficiently robust to persists under the above more realistic conditions. Our main results are the following:
1. In the case of short range disorder, if the momentum cut-off is removed the density of states vanishes with a critical exponent and the conductivity is universal; that is, the system has the same qualitative behavior than the case of $\d$-correlated disorder.
2. If the momentum cut-off is not removed, the density of states is still anomalous but the conductivity has in general disorder-dependent corrections.
Therefore, the vanishing of the density of states with an anomalous exponent is a robust property for chiral disordered fermions, but the [*exact*]{} vanishing of the disorder corrections to the conductivity does not survive in general to the presence of a momentum cut-off. From a Renormalization Group point of view this is rather natural. In presence of chiral disorder the theory is [*marginal*]{} with a line of fixed points; therefore corrections are expected, as in the case of the optical conductivity in presence of e.m. interaction. From this perspective, it is the [*absence*]{} of corrections the more surprising feature of disordered Dirac fermions with no cut-off; as it will be clear from the subsequent analysis, it is a direct consequence of the validity of the Adler-Bardeen theorem and the exact cancellation of the chiral anomaly due to the supersymmetry, which is valid only when the momentum cut-off is removed. The presence of corrections to the conductivity in presence of an ultraviolet cut-off may have of course implications for the physics of graphene, in which a natural ultraviolet cut-off is provided by the honeycomb lattice.
The presence of momentum cut-off and of non local disorder prevents the use of any [*exact*]{} methods, like the ones adopted in [@LFSG; @NTW; @ASZ; @Mu1; @Mu2], and one has therefore to rely on functional integral methods, which are more lengthy but of more general applicability. In particular we will use multiscale methods based on Wilsonian Renormalization Group (RG), in the more advanced form used in constructive Quantum Field Theory, see e.g. [@GJ]. Such form is exact, in the sense that the irrelevant terms (in the technical RG sense) are fully taken into account, while in most non exact RG implementations the irrelevant terms are simply neglected; as non local disorder or finite cut-offs are irrelevant in the infrared regime, non exact RG cannot distinguish between local and non local disorder, or the presence or absence of an ultraviolet cut-off.
Using the supersymmetric formalism we can rewrite disordered Dirac fermions in terms of functional integrals. The fermionic sector strongly reminds the [*non local Thirring model*]{} which was constructed using a multiscale analysis respectively in [@Le; @M2] for the ultraviolet problem and in [@M3] for the infrared part; therefore restricting to the fermionic sector a full non-perturbative construction of the model can be achieved, in the sense of a proof of the well definiteness of the functional integrals removing cut-offs; this would be parallel to [@DZ], in which the restriction to the bosonic sector of an hyperbolic sigma model coming by a disordered electron system was constructed.
The plan of the paper is the following. In §2 we define the model and we explain its supersymmetric representation. In §3 we analyze the critical theory at $E=0$, we derive Ward Identities and we show the validity of the Adler-Bardeen theorem and the supersymmetric cancellation of the anomalies in the limit of removed ultraviolet momentum cut-off; also, the relation with universality will be explained. In §4 we consider the non critical theory $E\not=0$ and we discuss the infrared problem. Finally, in §5 the main conclusions are discussed.
Disordered Chiral fermions and Supersymmetric representation
============================================================
The Dirac equation with vector disorder
---------------------------------------
The (regularized) first quantized Hamiltonian describing chiral disordered Dirac fermions is H=\_[i=1]{}\^2 \_i (i\_[i]{}+g A\_i())\[11\] with $\xx=(x_1,x_2)\in \L_a$, $\L_a$ is a square lattice with step $a$ with antiperiodic boundary conditions, $\s_i$ are Pauli matrices $$\s_1=\begin{pmatrix} 0 & 1 \\ 1\0 & 0 \end{pmatrix}\quad\quad \s_2=
\begin{pmatrix} 0 & -i\\ i\0 & 0 \end{pmatrix}\quad\quad \s_0=
\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}$$ and $\tilde\partial_i$ is a regularized (smeared) derivative \_i f\_=\_\^[-1]{}(-)\_i f\_ \_[i]{}f()=[12 a]{}(f(+a[**e**]{}\_i)-f(-a [**e**]{}\_i)) with $\chi(\xx)$ is a cut-off function defined as the Fourier transform of $\hat\chi(\kk)$, with $\hat\chi(\kk)$ a smooth function which is $\hat\chi(\kk)=0$ for $|\kk|\ge 2^{N+1}$ and $\hat\chi(\kk)=1$ for $|\kk|\le 2^{N}$; $A_i(\xx)$ is a Gaussian random field with short range (but non local) correlation (A\_[i]{}()A\_j())=\_[i,j]{}v(-) and |v(-)|C e\^[-|-|]{}\[alal\] and we will set $\k=1$ for definiteness.
One is mainly interested in the average of the two-point function, from which the density of states can be computed \[<|[1i H- E]{}|0>\]\[s1\] and in the average of the product of two functions \[<|[1i H- E]{}|0>\_[+-]{} <0|[1i H+ E]{}|>\_[+-]{}\]\[s2\] which is related to the conductivity. In the absence of disorder <|[1i H- E]{}|0> =[1L\^2]{}\_e\^[i]{}()
-E & [1a]{}\[ia k\_1+a k\_2\]\
[1a]{}\[ia k\_1-a k\_2\] & -E
\^[-1]{}\[pro\] In the following we will assume that $2^{N}<<{\pi\over a}$ in order to avoid the [*fermion doubling*]{} problem. Indeed at $E=0$ the denominator in the r.h.s. of is vanishing, in the $L\to\io$ limit, not only at $\kk=(0,0)$ but also at $\kk=(0,\pi/a),(\pi/a,0),(\pi/a,\pi/a)$ modulo $2\pi/a$. The condition $2^{N}<<{\pi\over a}$ ensures that the only remaining pole is the one at $\kk=0$, so preventing the fermionic species multiplication but at the same time preserving the chiral symmetry. The role of the lattice cut-off is just to make the functional integrals appearing below well defined and it will removed first.
Supersymmetric formalism
------------------------
It is well known, see for instance [@Mu1], that the average of the two-point function and the average of the product of two functions can be represented in terms of a supersymmetric functional integral in the chiral basis. One introduces a finite set of Grassmann variables $\psi^+_{\o,\xx},\psi^-_{\o,\xx}$ with $\o=\pm$ and defines, if $\e=\pm$, the Grassmann integration by d\^\_[ø,]{} \^\_[ø,]{}=1d\^\_[ø,]{}=0 where $d\psi^\e_{\o,\xx}$ is another set of Grassmann variables. Therefore if $
\DD\psi=\prod_{\xx,\o=\pm}d\psi^+_{\o,\xx}d\psi^-_{\o,\xx}$ we can write |\_[ø,-ø’]{}=[e\^[-(\^+, A \^-) ]{} \^-\_[,ø]{}\^+\_[,ø’]{}e\^[-(\^+, A \^-) ]{}]{} \[aaxx\]where $A=\s_1( i H- E)$. Note that the denominator of is equal to ${\rm Det } A$, and that $\int \DD\phi e^{-(\phi^+, A \phi^-)}={1\over {\rm Det }A}$, for any $n\times n$ invertible matrix $A$ with $Re A>0$ and $\phi^+,\phi^-$ complex numbers with $(\phi^+)^*=\phi^-$. Therefore one obtains the following representation of the average of the two point function G\_[ł,E,N;ø,ø’]{}() =\[ e\^[-(\^+, A \^-)-(\^+, A \^-)]{} \]\[88\]and integrating over the disorder, calling $\Psi^+=(\psi^+_+,\psi^+_-,\phi^+_+,\phi^+_-)$ and $\Psi^-=(\psi^-_+,\psi^-_-,\phi^-_+,\phi^-_-)$, $\phi^+=(\phi^-)^*$ G\_[ł,E,N;ø,ø’]{}()=P(d) P(d) e\^ \^-\_[ø,]{}\^+\_[ø’,0]{}\[66\] where, if $\l=2g^2$ =- łddv(-)\_[,’=,]{}\_[ø=]{}\^+\_[,ø,]{}\^-\_[,ø,]{} \^+\_[’,-ø,]{}\^-\_[’,-ø,]{}\[67\] with $\int d\xx=a^2\sum_\xx$ and $P(d\psi)$ and $P(d\phi)$ are the fermionic and bosonic integration with propagator $\d_{\a,\a'} g(\xx)$ with g() =[1L\^2]{}\_e\^[i]{}()
[1a]{} \[ia k\_1-a k\_2\]& -E\
-E & [1a]{}\[ia k\_1+a k\_2\]
\^[-1]{}\[pro\] and we have used that the normalization of the bosonic and fermionic integration are one inverse to the other. The fermionic sector of the above functional integral coincides with a massless Thirring model with a [*non local*]{} current-current interaction.
In the same way we can rewrite the averaged product of two functions as &&K\_[ł,E,N]{}()= {\_a e\^[-(\_a\^+,A \^-\_a)-(\_a\^+,A \^-\_a)- ]{} \^-\_[a,ø,]{}\^+\_[a,ø,0]{}\
&& \_b e\^[-(\_b\^+, B\_b\^-) -(\_b\^+, B\^-\_b)]{} \^-\_[b,ø,]{}\^+\_[b,ø,0]{}}\[89\] with $A=i H- E$, $B=i H+ E$, and averaging over the disorder K\_[ł,E,N]{}()=P(d\_a) P(d\_b)\^ \^-\_[a,ø,]{}\^+\_[a,ø,0]{} \^-\_[b,ø,]{}\^+\_[b,ø,0]{}\]\[24\] where = -łddv(-)\_[,=,,’=a,b]{}\_[ø=]{}\^+\_[,,ø,]{}\_[,,ø,]{}\^+\_[’,’,-ø,]{}\_[’,’,-ø,]{}
The critical theory
===================
The averaged two point function
-------------------------------
We define the [*generating function*]{} as e\^[\_N(J,)]{}= P(d) e\^[()+d]{}\[al\] where $\r_{\o,\a,\xx}=\Psi^+_{\o,\a,\xx}\Psi^-_{\o,\a,\xx}$ and we define, for $\a=(\psi,\phi)$, $\o=\pm$, the truncated correlations &&<\^-\_[,ø,]{}\^+\_[,ø,]{}>\_[T,E,N]{} =[\^2\^+\_[,ø,]{}\^-\_[,ø,]{}]{}\_N(J,)|\_[0]{} \[zaz\]\
&&<\_[’,ø’,]{}\^-\_[,ø]{}\^+\_[,ø]{}>\_[T,E,N]{} =[\^3J\_[’,ø’,]{}\^+\_[,ø,]{}\^-\_[,ø,]{}]{}\_N(J,)|\_[0]{}\
where $<AB>_T=<AB>-<A><B>$ and $<\Psi^-_{\psi,\o,\xx}\Psi^+_{\psi,\o',0}>_{T,E,N}\equiv <\psi^-_{\o,\xx}\psi^+_{\o',0}>_{T,E,N}
\equiv G_{\l,E,N;\o,\o'}(\xx)$ defined by .
Using a smooth decomposition of the unity, we write the propagator as sum of single scale propagators at $E=0$ g()=\_[j=h\_L]{}\^N g\^[(j)]{}() where $h_L\sim -\log L$ and $g^{(j)}(\xx)$, the single scale propagator, is similar to $g(\xx)$ with $\hat\chi(\kk)$ replaced by $f_j(\kk)$ , with $f_j(\kk)$ non vanishing in $2^{j-1}\le |\kk|\le 2^{j+1}$. The presence of a minimal scale $h_L$ comes from the fact that antiperiodic boundary conditions are assumed, and therefore the momenta are of the form $\kk={2\pi\over L}({\bf n}+{1\over 2})$, so that $|k_i|\ge {\pi\over L}$. $L$ plays the role of an [*infrared cut-off*]{} while $2^N$ is the [*ultraviolet*]{} cut-off. Note that &&|g\^[(j)]{}|\_[L\_1]{}=d|g\^[(j)]{}()|C 2\^[-j]{}\
&&|g\^[(k)]{}|\_[L\_]{}=\_|g\^[(j)]{}()| C 2\^[j]{} \[fon111\] We use now the following basic property of gaussian integrations, bosonic or fermionic, called [*addition property*]{} and we get , calling the exponent in the r.h.s. of simply $V(\Psi,\h,J)$ P(d)e\^[V(,,J) ]{}=P(d\^[(N-1)]{}) P(d\^[(N)]{}) e\^[V(,,J)]{} =P(d\^[(N-1)]{})e\^[V\^[(N-1)]{}(\^[(N-1)]{},J)]{} \[all\] where $P(d\Psi^{(\le N-1)})$ and $P(d\Psi^{(N)})$ are gaussian integrations with propagator respectively $g^{(\le N-1)}(\xx)$ and $g^{(\le N)}(\xx)$ and V\^[(N-1)]{}(,,J)=\_[n=1]{}\^ \^T(V;n)with $\EE^T$ are the [*truncated expectations*]{} with respect to $P(d\Psi^{(N)})$ \^T(V();n)=[\^n\^n]{}P(d\^[(N)]{}) e\^[V(\^[(N)]{}+)]{}|\_[=0]{} When expressed in terms of Feynman graphs, the truncated expectation are written in terms of [*connected*]{} diagrams only. The multiscale analysis continues integrating the fields $\Psi^{(N-1)},..,\Psi^{(h+1)}$ obtaining e\^[\_N(J,)]{} =P(d\^[(h)]{})e\^[V\^[(h)]{}(\^[(h)]{},,J)]{}with $V^{(h)}$, called effective potential, being a sum of integral of monomials with $n\ge 0$ $\Psi,\h$ and $m\ge 0$ $J$ fields multiplied by kernels $W_{n,m}^{(h)}$; moreover $P(d\Psi^{(\le h)})$ is the integration with propagator $g^{(\le h)}(\xx)= \sum_{k \le h}
g^{(k)}(\xx)$.
The range of the disorder ($\k=1$ in ) provides a natural [*momentum scale*]{} separating the scales $j$ in [*ultraviolet scales*]{}, between $0$ and $N$, and [*infrared scales*]{}, namely between $h_L$ and $-1$. Let us consider first the integration of the ultraviolet scales. The [*scaling dimension*]{} in the case of $\d$-correlated disorder is the same in the ultraviolet and infrared region and equal to $D=2-n/2-m$, that is greater or equal to zero in the case $n,m=(2,0), (4,0), (2,1)$. Therefore there are in general ultraviolet divergences and this requires that the ultraviolet $N\to\io$ limit can be taken only choosing properly the bare parameters to $N$-dependent and possibly singular value. For instance, in the case of the Thirring model with a local $\d$-like interaction, the $N\to\io$ limit can be taken only choosing the bare wave function renormalization vanishing as $2^{-\h N}$ with $\h>0$.
In the case of short-ranged correlated disorder the situation is different; the non locality of the disorder induces an [*improvement*]{} in the scaling dimension, and indeed no ultraviolet divergences are present; the kernels of the effective potential are bounded uniformly in the ultraviolet cut-off $N$. Consider for instance $W^{(h)}_{2,0}$, $h\ge 0$, with scaling dimension $D=1$. We can decompose $W^{(h)}_{2,0}$, using general properties of truncated expectations (or the fact that they are expressed in terms of connected diagrams), as in Fig. 1. Note that the first and third contributions are vanishing by parity considerations (remember that $E=0$ here); regarding the second, we can use the following bound &&|d\_1 d\_2 d\_3 v(\_1-\_2) g\^[\[h,N\]]{}(\_1-\_3) W\_[2,1]{}\^[(h)]{}(\_2;\_3,0)|\
&&|g\^[\[h,N\]]{}|\_[L\_1]{} |v|\_[L\_]{} d\_2 d\_3 |W\_[2,1]{}\^[(h)]{}(\_2;\_3,0)|C 2\^[-h]{}, where we have inductively bounded $|W_{2,1}^{(h)}|_{L_1}$ by a constant, as its dimension is $D=0$. Note the crucial role played by the non locality of the disorder; in the case of $\d$-correlated disorder one needs to integrate over the wiggly lines instead than over the propagator (as $|v|_{L_\io} $ is unbounded) so that in the above bound one gets $|g^{[h,N]}|_{L_\io} |v|_{L_1}$ instead of $|g^{[h,N]}|_{L_1} |v|_{L_\io}$ and the resulting bound would be diverging as $N\to\io$ as $2^N$. Similar considerations could be done for $W_{0, 2}^{(h)}$ which can be decomposed as in the r.h.s. Fig. 2; the second term can again be bounded by && |v|\_[L\^]{} |W\^[(k)]{}\_[2,2]{}|\_[L\^1]{}\_[hi’jiN]{}|g\^[(j)]{}|\_[L\^1]{} |g\^[(i)]{}|\_[L\^1]{}|g\^[(i’)]{}|\_[L\^]{}\
&& C\_1 ł2\^[-2h]{}\_[hiN]{} (i-h)2\^[-i+h]{}C\_2 ł2\^[-2h]{}\[aza\] This argument again cannot be repeated for the first term in Fig. 2, but the local part vanishes since the local part of the bubble graph is zero by symmetry \_\_[N]{}()[k\_0\^2- k\^2+2 i k\_0 k(k\_0\^2+ k\^2)\^2]{}= 0. \[xxx\] A similar analysis can be repeated for the other terms to show that the scaling dimension is always negative. The conclusion is that the effective potential is uniformly bounded in $N$ and that there are [*no ultraviolet divergences*]{} even when the momentum cut-off is removed, that is for $N\to\io$.
Ward Identities and cancellation of the anomalies
-------------------------------------------------
A crucial role is played by [*Ward Identities*]{}, which can be obtained by performing in with $E=0$ the [*chiral*]{} local phase transformation $
\Psi^\pm_{\o,\a,\xx}\to e^{\pm i a_{\o,\a,\xx}} \Psi^\pm_{\o,\a,\xx}
$ and performing a derivative with respect to $a_{\o,\a,\xx}$ and the external fields; due to the presence of cut-offs the Jacobian is equal to $1$ but, with respect to the formal Ward Identities obtained neglecting cut-offs, one has an extra term; indeed it is found &&D\_ø()<\_[,ø,]{}\^-\_[’,ø’]{}\^+\_[’,ø’,+]{}>\_[T,0,N]{}= \_[,’]{}\_[ø,ø’]{}\[<\^-\_[’,ø’,]{}\^+\_[’,ø’,]{}>\_[T,0,N]{}\
&& -<\^-\_[’,ø’,+]{}\^+\_[’,ø’,+]{}>\_[T,0,N]{}\] +<\_[,ø,]{}\^-\_[’,ø’,]{}\^+\_[’,ø’,+]{}>\_[T,0,N]{}\[pp\] where $D_\o(\kk)=i k_1-\o k_2$, with $\o=\pm $, $\r_{\a,\o,\xx}=\Psi^+_{\a,\o,\xx}\Psi^-_{\a,\o,\xx}$, \_[,ø,]{}=dC\_[N]{}(,)\^+\_[,ø,]{}\^-\_[,ø,+]{} with C\_[N]{}(,)=\[(+ )\^[-1]{} - 1\] D\_[ø]{}(+) - \[ (()\^[-1]{} - 1\] D\_[ø]{}()\[fonddd1\]. The last term in is due the the presence of the momentum cut-off which breaks the [*local*]{} chiral invariance. Remarkably, such term it is not vanishing even removing the ultraviolet cut-off, but the following identity holds &&\_[T,0,N]{}=\
&& \_ł[14]{} D\_[-ø]{}()\_[”=,]{}<\_[”,-ø,]{}\^-\_[’,ø’,]{}\^+\_[’,ø’,+]{}>\_[T, 0,N]{}+R\_[N,]{}(,)\[aaaa7\] with \_=-1,\_=1\[aaadd\] and $R_{N,\a}(\kk,\pp)$ is in absolute value smaller than ${
2^{-N}\over |\kk||\kk-\pp|}$, that is vanishing for $N\to\io$.
If we restrict to the fermionic sector, in the limit $N\to\io$ the first term in the r.h.s. of (32) would be the chiral anomaly and the second term is vanishing. The fact that the chiral anomaly is [*linear*]{} in the coupling is a property called Adler-Bardeen theorem. It is important to stress the presence of the correction term $R_N$ in the l.h.s. of (32), which is vanishing [*only*]{} in the $N\to\io$ limit. The derivation of (32) is based on a multiscale integration also for the correction term in , and the main difference is that the source term $(J_\xx,\Psi^+_{\xx}\Psi_\xx)$ is replaced by $\int d\xx \chi_{\a,\o}\d\r_{\a,\o,\xx}$ where $\chi$ is a source term. After the integration of the fields $\Psi^{(N)},\Psi^{(N-1)},..,\Psi^{(h+1)}$ the effective potential can be again written as sum of monomials with $n$ $\Psi$ fields, and $m$ $\chi$ fields with kernels $\tilde W^{(h)}_{n,m}$. The analysis of $\tilde W^{(h)}_{2,1}$ is very similar to the analysis of $W^{(h)}_{2,1}$ in the previous section. An important difference with respect to the bound comes from the fact that $C_{N}(\kk,\pp)
g^{(i)}(\kk)g^{(j)}(\kk+\pp)$ vanishes unless either $i$ or $j$ equals the cut-offs scales $N$. Therefore, the second term in Fig. 3, which contributes to $R_{N,\a}$ in (32), can be bounded as with the difference that one of the scales of the propagator attached to the back dot have scale $N$; therefore one obtains the bound &&|ł| |v|\_[L\^]{} | W\^[(h)]{}\_[4,1]{}|\_[L\^1]{}\_[hi’iN]{}|g\^[(N)]{}|\_[L\^1]{} |g\^[(i)]{}|\_[L\^1]{}|g\^[(i’)]{}|\_[L\^]{}\[xaxa\]\
&&C\_1ł\^2 2\^[-2h]{}(N-h)2\^[-N+h]{}C\_2ł\^2 2\^[-2h]{}2\^[-(N-h)/2]{}leading to the vanishing of this contribution for $N\to\io$. On the other hand the non-connected contributions, that is the first term in Fig. 3, is now non vanishing and contribute to the first term in (32) ; the bubble is now given by \_\_[C\_[N]{}(,)D\_[-ø]{}()]{} g\_ø()g\_ø(+)=\_+O(2\^[-N]{}) Remarkably, the anomalies cancel out in the WI for the total density due to supersymmetry (that is, due to ) &&D\_ø()\_[=,]{}<\_[,ø,]{}\^-\_[’,ø’,]{}\^+\_[’,ø’,+]{}>\_[T,0,N]{}=\_[ø,ø’]{} \[<\^-\_[’,ø’,]{}\^+\_[’,ø’,]{}>\_[T,0,N]{}\
&& -<\^-\_[’,ø’,+]{}\^+\_[’, ø’,+]{}>\_[T,0,N]{}\] +\_[=,]{} R\_[N,]{}(,)\[pp1\] We can write the Schwinger-Dyson equation \_[T,0,N]{}=g\_ø()+łg\_[ø]{}() dv() \_[’=,]{}<\_[’,-ø]{}()\^+\_[,ø,+]{}\^-\_[,ø,]{}>\_[T,0,N]{} \[sd\]and inserting we obtain \_[T,0,N]{}=g\_ø()+łg\_ø() dv() \_[’=,]{} [R\_[N,’]{}(,)D\_ø()]{}\[sd1\] It can be shown, by an analysis similar to the one for $R_{N,\a}$ , that that $\int d\pp \hat v \sum_{\b=\phi,\psi} {R_{N,\a}\over D_\o}$ is smaller than $2^{-N}$, that is vanishing when the ultraviolet cut-off is removed. Therefore at $E=0$ the averaged 2-point function is equal to the free one up to corrections which are vanishing [*only*]{} when the ultraviolet cut-off is removed $N\to\io$; on the other hand for any finite cut-off non vanishing corrections are expected.
It is indeed useful to compare the present result to the analogous computation for the Thirring model with non local interaction, that is neglecting the bosons; in such a case the Schwinger-Dyson equation is still given by but in the r.h.s. $\sum_{\a=\phi,\psi}
\hat\r_{\a,-\o,\a}(\pp)$ should be replaced by $\hat\r_{\psi,-\o}(\pp)$; by using the WI , (32) one would get an extra term in function of the chiral anomaly. As a result, one would find that the asymptotic behavior of $<\hat\Psi^-_{\psi,\o,\kk}\hat\Psi^+_{\psi,\o,\kk}>_{T,0,N}$ is [*different*]{} with respect to the non interacting case; for small $\kk$ $<\hat\Psi^-_{\psi,\o,\kk}
\hat\Psi^+_{\psi,\o,\kk}>_{T,0,N}$ would behave as $|\kk|^{-1+\h}$ with $\h>0$. In the present case, instead, the cancellation of the anomalies due to the supersymmetry has the effect that the asymptotic behavior of the two point function is equal to the free one, up to a small correction vanishing when the cut-off is removed.
The average of the product
--------------------------
Starting from for $E=0$, and using the notation (with $P(d\Psi)$ replaced by $P(d\Psi_a) P(d\Psi_b)$) we can write K\_[ł,0]{}()= <\^-\_[a,ø,]{}\^+\_[a,ø,0]{}>\_[T,0,N]{} <\^-\_[b,ø,]{}\^+\_[b,ø,0]{}>\_[T,0,N]{}+<\^-\_[a,ø,]{}\^+\_[a,ø,0]{} \^-\_[b,ø,]{}\^+\_[b,ø,0]{}>\_[T,0,N]{}\[ma1\] The computation of $<\psi^-_{a,\o,\xx}\psi^+_{a,\o,\yy}>_{T,0,N}$ can be done exactly as in the previous case. The Schwinger-Dyson equation is given by \_[T,0,N]{}=g\_ø()+łg\_ø() dv() \_[’=,]{}\_[’=a,b]{}<\_[’,’,-ø]{}()\^+\_[,ø,+]{}\^-\_[,ø,]{}>\_[T,0,N]{} and using the WI &&D\_ø()<\_[,,ø,]{}\^-\_[’,ø’,]{}\^+\_[’,ø’,+]{}>\_[T,0,N]{}= \_[,]{}\_[,’]{}\_[ø,ø’]{}\[<\^-\_[’,ø’,]{}\^+\_[’,ø’,]{}>\_[T,0,N]{}\
&& -<\^-\_[’,ø’,+]{}\^+\_[’,ø’,+]{}>\] +<\_[,,ø,]{}\^-\_[’,ø’,]{}\^+\_[’,ø’,+]{}>\_[T,0,N]{}\[pp3\] where $\r_{\a,\b,\o,\xx}=\Psi^+_{\a,\b,\o,\xx}\Psi^-_{\a,\b,\o,\xx}$ and again && <\_[,,ø,]{}\^-\_[’,ø’,]{}\^+\_[’,ø’,+]{}>\_[T,0,N]{}=\
&& \_ł[14]{} D\_[-ø]{}()\_[”=,”=a,b]{} <\_[”,”,-ø,]{}\^-\_[’,ø’,]{}\^+\_[’,ø’,+]{}>\_[T,0,N]{}+R\^[(2)]{}\_[N,,]{}(,)\[aaa1\]we get \_[T,0,N]{}=g\_ø()+łg\_ø() dv() \_[’=,,’=a,b]{} [R\^[(2)]{}\_[N,’,’]{}(,)D\_ø()]{}\[ma2\] with $\int d\pp \hat v(\pp) {R^{(2)}_{N,\a',\b'}\over D_\o}$ is $O(2^{-N})$. In a similar way one analyze the connected part of ; we write the Schwinger-Dyson equation for the four point function \_[T,0,N]{}= dv() \_[’=,’=a,b]{} <\_[’,’,-ø,]{} \^-\_[a,ø,\_1]{}\^+\_[a,ø,\_2]{}\^-\_[b,ø,\_3]{}\^+\_[b,ø,\_4-]{}>\_[T,0,N]{}The WI for the four point function is &&D\_ø() <\_[’,’,-ø,]{} \^-\_[a,ø,\_1]{}\^+\_[a,ø,\_2]{}\^-\_[b,ø,\_3]{}\^+\_[b,ø,\_4-]{}>\_[T,0,N]{} +\
&&<\_[’,’,-ø,]{} \^-\_[a,ø,\_1]{}\^+\_[a,ø,\_2]{}\^-\_[b,ø,\_3]{}\^+\_[b,ø,\_4-]{}>\_[T,0,N]{} =0 with && <\_[’,’,-ø,]{}\^-\_[a,ø,\_1]{}\^+\_[a,ø,\_2]{}\^-\_[b,ø,\_3]{}\^+\_[b,ø,\_4-]{}>\_[T,0,N]{}=\
&& \_[’]{}ł[14]{} D\_[ø]{}()\_[”,”]{}<\_[”,”,ø,]{}\^-\_[a,ø,\_1]{}\^+\_[a,ø,\_2]{}\^-\_[b,ø,\_3]{}\^+\_[b,ø,\_4-]{}>\_[T,0,N]{}+R\^[(4)]{}\_[N,’,’]{}(\_1,\_2,\_3,\_4,)\[aaa\]and using that $\sum_{\a'=\phi,\psi} \e_{\a'}=0$ we finally obtain \_[T,0,N]{}= \_[’,’]{}dv() [R\^[(4)]{}\_[N,’,’]{}D\_[ø]{}()]{}\[ma3\] and again the r.h.s. is vanishing as $O(\l 2^{-N}))$. Therefore, by ,, the interacting average of the product $K_{\l,0,N}(\xx)$ differs from its non interacting value $K_{0,0,N}(\xx)$ by terms which are order $O(\l 2^{-N})$ for large $N$; exact universality for such quantity (and therefore for the conductivity) is achieved only in the limit of removed ultraviolet cut-off.
The non critical theory and the infinite volume limit
=====================================================
We have to discuss finally the removal of the infrared cut-off and the case $E\not =0$. Again we perform a multiscale decomposition of $\Psi$ in the $E\not=0$ case and the integration of the ultraviolet (positive) scales is done as in the previous section (the fact that $E\not=0$ plays no role in the ultraviolet regime). We consider now the integration of the negative infrared scales, in the $L\to\io$ limit. In this case there is no improvement with respect to the scaling dimension, and one has to define a [*renormalized*]{} multiscale integration in the following way. Assume that we have integrated the fields $\Psi^{(N)},..,\Psi^{(h)}$, $h\le 0$ obtaining e\^[(J,)]{} =P\_[Z\_h,E\_h]{}(d\^[(h)]{})e\^[V\^[(h)]{}(,,J)]{}where $P_{Z_h,E_h}(d\Psi^{(\le h)})$ is the gaussian integration with propagator, $\a=\psi,\phi$ g\_(,) =[1L\^2]{} \_\_h() e\^[i(-)]{} [1Z\^[()]{}\_h]{}
D\_+() & E\^[()]{}\_h\
E\^[()]{}\_h & D\_-()
\^[-1]{} and $\hat\chi_h(\kk)=\sum_{j=-\io}^h f_j(\kk)$ and again $V^{(h)}$ being a sum of integral of monomials with $n\ge 0$ $\Psi,\h$ and $m\ge 0$ $J$ fields multiplied by kernels $\hat W_{n,m}^{(h)}$. We decompose the kernels as W\_[n,m]{}\^[(h)]{}()=W\_[n,m;a]{}\^[(h)]{}( )+W\_[n,m;b]{}\^[(h)]{}( )+W\_[n,m;c]{}\^[(h)]{}( ) where $\hat W_{n,m;a}^{(h)}$ and $\hat W_{n,m;b}^{(h)}$ are respectively the zero-th and first order contribution in $E$ to $\hat W_{n,m}^{(h)}$ and $\hat W_{n,m;c}^{(h)}$ is the rest. We define a [*localization operator*]{} on the terms with positive scaling dimension $D=2-n/2-m$ in the following way &&W\^[(h)]{}\_[4,0]{}(\_1,\_2,\_3,\_4)=W\^[(h)]{}\_[4,0;a]{}([**0**]{}, [**0**]{},[**0**]{},[**0**]{})\
&&W\^[(h)]{}\_[2,1]{}(\_1,\_2,\_3)=W\^[(h)]{}\_[2,1;a]{}([**0**]{},[**0**]{},[**0**]{})\
&&W\^[(h)]{}\_[2,0;ø,ø]{}()= W\^[(h)]{}\_[2,0:ø,ø;a]{}([**0**]{})+ W\^[(h)]{}\_[2,0;ø,ø;a]{}([**0**]{})\
&&W\^[(h)]{}\_[2,0;ø,-ø]{}()=W\^[(h)]{}\_[2,0:ø,-ø;a]{}([**0**]{})+ W\^[(h)]{}\_[2,0:ø,-ø;b]{}([**0**]{}) and we write e\^[(J,)]{} =P\_[Z\_h,E\_h]{}(d\^[(h)]{})e\^[V\^[(h)]{}(,,J)+(1-)V\^[(h)]{}(,,J) ]{}\[fg\]The action of $(1-\LL)$ on the kernels improve their scaling dimension. For instance (1-)W\^[(h)]{}\_[4,0]{}= \[W\^[(h)]{}\_[4,0;a]{}()- W\^[(h)]{}\_[4,0;a]{}()\]+W\^[(h)]{}\_[4,0;b]{}()\[41\] and the first term in the r.h.s. has negative dimension while regarding the other term one has simply to use that the bound for $\hat W^{(h)}_{4,0;a}$ as an extra ${E_h^{(\a)}\over 2^h}$.
We use now the following symmetries of the propagator at $E=0$ (g\^[(k)]{}\_ø)\^\*(k\_1,k\_2)=g\_[ø]{}\^[(h)]{}(-k\_1,k\_2)g\^[(h)]{}\_ø(k\_1,k\_2)=-iøg\^[(h)]{}\_[ø]{}(k\_2,-k\_1)\[bb1\] and that at $E=0$ there is global phase invariance $
\Psi^\pm_{\a,\o}\to e^{\pm i \a_{\a,\o}}\Psi^\pm_{\a,\o}
$. Therefore
1. The local part of the terms with four fields with the same $\o$ is vanishing; indeed if $n$ is the order there are $n-2$ $(\o)$-propagators and $n$ $(-\o)$-propagators; then by $\hat W^{(h)}_{4,0;a}(\underline k_1,\underline k_2)=(i\o)^{-2} W^{(h)}_{4,0}(-\underline k_2,\underline k_1)$ so that $\hat W^{(h)}_{4,0;a}
(\underline 0,\underline 0)=-\hat W^{(h)}_{4,0;a}(\underline 0,\underline 0)=0$; moreover by global phase invariance there is an even number of fields with the same $(\a,\o)$.
2. The quartic terms are real. Indeed by $(\hat W^{(h)}_{4,0;a})^*(\underline k_1,\underline k_2)=
\hat W^{(h)}_{4,0;a}(-\underline k_1,\underline k_2)$, so that the local part is real
3. The local part of the terms with two external line and the same $\o$ is vanishing by the parity of the propagator.
4. Finally $\partial_1 \hat W^{(h)}_{2,0;a}(0)=i\o \partial_2 \hat W^{(h)}_{2,0:a}(0)$
The only quadratic terms in $\LL\VV^{(h)}$ are the one multiplying $\partial W^{(h)}_{2,0;\e,\e;a}({\bf 0})$ and $ W^{(h)}_{2,0:\e,-\e}({\bf 0})$ producing respectively a renormalization of $Z^{(\a)}_h$ and $E^{(\a)}_h$. Therefore we can move the quadratic terms in the gaussian integration so obtaining P\_[Z\_[h-1,]{}E\_[h-1]{}]{}(d\^[(h)]{})e\^[V\^[(h)]{}(,,J)+(1-)V\^[(h)]{}(,,J) ]{}\[fga\]and &&V\^[(h)]{}(,0,0)= ł\_[1,h]{}\_[ø]{}d\^+\_[,ø]{}\^-\_[,ø]{} \^+\_[,-ø]{}\^-\_[,-ø]{}+\
&& ł\_[2,h]{}\_[ø]{}d\^+\_[,ø]{}\^-\_[,ø]{} \^+\_[,-ø]{}\^-\_[,-ø]{}+ ł\_[3,h]{}\_[ø]{}d\^+\_[,ø]{}\^-\_[,ø]{} \^+\_[,-ø]{}\^-\_[,-ø]{}One can write as P\_[Z\_[h-1,]{}E\_[h-1]{}]{}(d\^[(h-1)]{})P\_[Z\_[h-1,]{}E\_[h-1]{}]{}(d\^[(h)]{}) e\^[V\^[(h)]{}(,,J)+(1-)V\^[(h)]{}(,,J) ]{}\[fgaa\]and the procedure can be iterated up to a scale $h^*_\a$ (that is $h^*_\psi$ for the fermionic fields and $h^*_\phi$ for the bosonic ones) such that $E_{h^*_\a}=2^{h^*_\a}$; one can see that $g^{(-\io, h^*)}$ obey exactly to the same bounds as the single scale propagator $g^{(h)}$ with $h>h^*_\a$. The outcome of this procedure is a sequence of $V^{(h)}(\Psi,\h,J)$ with kernels $W^{(h)}_{n,m}$, expressed in terms of the effective coupling constants $\l_{i,k}$, $k=h,h+1,..0$; the kernels are finite uniformly in $h$ provided that the running coupling constants stay bounded. On the other hand the running coupling constants are the same in the critical theory at $E=0$. Therefore in order to study their flow can consider the theory with $E=0$ and infrared cut-off $2^h$, replacing $\hat\chi(\kk)$ with $\hat\chi_{h,N}(\kk)=\sum_{j=h}^{N} f_j(\kk)$ with $h\le 0$. The Schwinger-Dyson equations for the two and four point function coincide with the ones derived in the previous sections up to negligible corrections due to the presence of the infrared cut-off $2^h$. Therefore fixing the value of the external momenta at the scale of the infrared cut-off we get ł\_[h-1,i]{}=ł\_[0,i]{}+O(ł\_0\^2) Z\_h\^[()]{}=1+O(ł\_0\^2)\[x1\] and $\l_0=\l\hat v(0)+O(\l^2)$. This means that the effective couplings $\l_{h,i}$ converge to a line of fixed points (the beta function is asymptotically vanishing) and the critical exponent for the wave function renormalization is zero (contrary to what happens in the fermionic theory in which is positive).
The flow equation for the energy is given by =1+ał\_[1,h]{}+O(ł\_h\^2)=1+ał\_[2,h]{}+O(ł\_h\^2) with $a={1\over 2\pi}>0$ and by symmetry the contributions with different $\a$ do not mix, by the global phase symmetry valid at $E=0$. Therefore E\^[()]{}\_h= E 2\^[-\_h]{} with $\h_\a=a\hat v(0)\l+O(\l^2)$; this implies $2^{h^*_\a}=E^{1\over 1+\h_\a }$. For $h\ge h^*=\max (h^*_\phi, h^*_\psi)$ this makes clear why the second term in has the correct scaling dimension; indeed $
E^{(\a)}_h 2^{-h}$ can be bounded by $2^{(1+\h_\a)(h^*-h)}$ which is sufficient to make the dimension negative. For $h\le h^*$ the theory becomes purely fermionic or bosonic. Therefore = \_[h=h\^\*\_]{}\^(1+łF\_h) with $|F_h(\kk)|\le \l$ and $E^{(\psi)}_h=E$, $Z_h=1$ f or $h\ge 0$; we have used that the contributions from the scales $h \le h^*$ are summable. The density of states (with imaginary energy) is therefore bounded by \_[h=h\^\*\_]{}\^0 |E\^[()]{}\_h|+E\_[h=0]{}\^e\^[-2\^h]{} E\^[11+]{} where $C$ is a suitable constant and $\e$ is an extra ultraviolet cut-off ; that is the density of states vanishes with a critical exponent.
Conclusions
===========
We have analyzed for the first time chiral Dirac fermions in presence of a momentum cut-off and short range disorder, extending previous results in which only delta correlated disorder without ultraviolet cut-off was considered. The model provides a more realistic description in view of applications to condensed matter models, and is free from any ultraviolet divergence. We have shown that the density of states is anomalous with a critical exponent function of the disorder and that the conductivity is exactly universal [*only*]{} when the ultraviolet cut-off is removed; this may have implications for the physics of graphene in which a natural ultraviolet cut-off is provided by the honeycomb lattice.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We present the `COLIBRI` code for computing the evolution of stars along the TP-AGB phase. Compared to purely synthetic TP-AGB codes, `COLIBRI` relaxes a significant part of their analytic formalism in favour of a detailed physics applied to a complete envelope model, in which the stellar structure equations are integrated from the atmosphere down to the bottom of the hydrogen-burning shell. This allows to predict self-consistently: (i) the effective temperature, and more generally the [*convective envelope and atmosphere structures*]{}, correctly coupled to the changes in the surface chemical abundances and gas opacities; (ii) the conditions under which [*sphericity effects*]{} may significantly affect the atmospheres of giant stars; (iii) the [*core mass-luminosity relation and its possible break-down due to the occurrence of hot bottom burning*]{} (HBB) in the most massive AGB stars, by taking properly into account the nuclear energy generation in the H-burning shell and in the deepest layers of the convective envelope; (iv) the [*HBB nucleosynthesis*]{} via the solution of a complete nuclear network (including the pp chains, and the CNO, NeNa, MgAl cycles) coupled to a diffusive description of mixing, suitable to follow also the synthesis of $^{7}$Li via the Cameron-Fowler beryllium transport mechanism; (v) the [*intershell abundances*]{} left by each thermal pulse via the solution of a complete nuclear network applied to a simple model of the pulse-driven convective zone; (vi) the [*onset and quenching of the third dredge-up*]{}, with a temperature criterion that is applied, at each thermal pulse, to the result of envelope integrations at the stage of the post-flash luminosity peak.
At the same time `COLIBRI` pioneers new techniques in the treatment of the physics of stellar interiors, not yet adopted in full TP-AGB models. It is the first evolutionary code ever to use accurate [*on-the-fly*]{} computation of the [*equation of state*]{} for roughly 800 atoms, ions, molecules, and of the Rosseland mean [*opacities*]{} throughout the atmosphere and the deep envelope. This ensures a complete consistency, step by step, of both EoS and opacity with the evolution of the chemical abundances caused by the third dredge-up and HBB. Another distinguishing aspect of `COLIBRI` is its high computational speed, that allows to generate complete grids of TP-AGB models in just a few hours. This feature is absolutely necessary for calibrating the many uncertain parameters and processes that characterize the TP-AGB phase.
We illustrate the many unique features of `COLIBRI` by means of detailed evolutionary tracks computed for several choices of model parameters, including initial star masses, chemical abundances, nuclear reaction rates, efficiency of the third dredge-up, overshooting at the base of the pulse-driven convection zone, etc. Future papers in this series will deal with the calibration of all these and other parameters using observational data of AGB stars in the Galaxy and in nearby systems, a step that is of paramount importance for producing reliable stellar population synthesis models of galaxies up to high redshift.
author:
- |
Paola Marigo$^{1}$[^1], Alessandro Bressan$^{2}$, Ambra Nanni$^{2}$, Léo Girardi$^{3}$, and Maria Letizia Pumo$^{1,3}$\
$^{1}$Department of Physics and Astronomy G. Galilei, University of Padova, Vicolo dell’Osservatorio 3, I-35122 Padova, Italy\
$^{2}$Astrophysics Sector, SISSA, Via Bonomea 265, I-34136 Trieste, Italy\
$^{3}$Astronomical Observatory of Padova – INAF, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy
date: 'Accepted 2013 xxx. Received 2013 January xxx; in original form 2013 February xxx'
---
\[firstpage\]
stars: evolution – stars: AGB and post-AGB – stars: carbon – stars: mass-loss – stars: abundances – Physical Data and Processes: equation of state.
Context and motivation
======================
The modelling of the Thermally Pulsing Asymptotic Giant Branch (TP-AGB) stellar evolutionary phase plays a critical role in many astrophysical issues, from the chemical composition of meteorites belonging to the pre-solar nebula [e.g. @Zinner_etal05], up to the cosmological context of galaxy evolution in the high-redshift Universe [e.g. @Maraston_etal06]. Indeed, luminous TP-AGB stars are potentially the dominant contribution to a galaxy’s flux, particularly at the red wavelengths and high redshifts that are much of the focus of modern extragalactic astronomy. In spite of its importance, the TP-AGB phase is still affected by large uncertainties which uncomfortably propagate into the field of current population synthesis models of galaxies that, for this reason, are strongly debated [e.g. @Conroy_etal09; @Kriek_etal10; @Zibetti_etal13].
As a matter of fact, the evolution along TP-AGB phase is determined in a crucial way by processes which are challenging to model from first principles: turbulent convection, stellar winds, and long-period variability. Also, these processes do not take place in a steady and smooth way during the TP-AGB evolution, but greatly vary in both character and efficiency over the single thermal pulse cycles (TPC) – the $10^2$ to $10^5$-yr long periods that go from one He-shell flash, through quiescent H-shell burning, up to the next He-flash. Moreover, the rich nucleosynthesis in the intershell convective region followed by recurrent dredge-up episodes, and the nuclear burning at the base of the convective envelope (hot-bottom burning, HBB) of the most massive TP-AGB stars ($M \ga 4\, M_{\odot}$), can dramatically change the surface abundances, and hence the envelope structure, over a timescale much shorter than a single TPC.
The result is that the modelling of the TP-AGB phase is quite difficult, time consuming, and affected by large uncertainties. Efforts to follow this phase with “full models”, which solve the time-dependent equations of stellar structure with the aid of classical 1D stellar evolution codes, are becoming increasingly successful thanks to the speeding-up of modern processors, and to the particular care devoted to the nucleosynthesis [e.g. @Ventura_etal02; @Cristallo_etal09; @Karakas_10]. However, full TP-AGB models still meet three fundamental difficulties.\
(1) They are affected by quite subtle and nasty numerical uncertainties, that can greatly affect the predicted efficiency of convective dredge-up episodes even within the same set of models [@FrostLattanzio_96; @Mowlavi_99a].\
(2) Full TP-AGB models need to resort to parametrized descriptions of crucial processes (mass loss, convection, overshoot), with theoretical formulations and “efficiency parameters” that may largely vary from study to study, so that to date no universally accepted set of prescriptions exists. This intrigued situation is well exemplified by fact that, for instance, the so-called carbon-star mystery, pointed out by @Iben_81 in the far past, is now claimed to have been solved by full TP-AGB models [@Stancliffe_etal05; @WeissFerguson_09; @Cristallo_etal11]. However, it is somewhat disturbing to recognize that the same observable, i.e. the carbon star luminosity function of carbon stars in the Large Magellanic Cloud, seems to be recovered by different full TP-AGB models in which the third dredge-up takes place with very different characteristics (in this respect, see Sect. \[ssec\_3dup\] and Fig. \[fig\_3dup3z02\]).\
(3) The range of parameters to be covered, and prescriptions to be tested, in order to obtain grids of TP-AGB models that reproduce the wide variety of observational data for AGB stars in resolved galaxies, is simply too large.
In this tricky context, a valuable contribution may be provided by the so-called “synthetic models", in which the evolution from one thermal pulse to the next is described with analytical relations that synthesize the results of full models. Being very agile and hence suitable to explore wide ranges of parameters and prescriptions, synthetic models can help to constrain the physical domain towards which full models should converge in order to reproduce observations of TP-AGB stars (e.g. carbon star luminosity functions (CSLF), C/M ratios, H-R diagrams, etc.). For instance, following the work of @GroenewegendeJong_93, based on synthetic models and focussed on the CSLF in the Large Magellanic Cloud, it became clear that the third dredge-up should not only be much more efficient, but also start earlier, at fainter luminosities, than usually predicted by full TP-AGB models up to that time.
On the other hand, synthetic models are often criticised because they lack the accurate physics involved in the evolution of these stars. Moreover, they are completely subordinate to the relations fitting the results of full AGB model calculations, which severely limits their capability of exploring new evolutionary effects. A notable example is the effective temperature, for which various formulas have been proposed in the past in the usual form $T_{\rm eff}={\rm func}(L,M,Z)$, involving luminosity, stellar mass and metallicity. Unfortunately, their validity is extremely narrow as they can apply only to oxygen-rich stars (with surface C/O$<1$), hence being unable to account for the Hayashi limits of carbon stars. Moreover, these relations reflect the specific set of input physics adopted in the underlying full models, e.g. mixing-length parameter, gas opacities, equation of state, etc.
If this criticism reasonably applies to the purely analytic TP-AGB models that rely on a mere compilation of fitting formulas [e.g. @Hurley_etal00; @Izzard_etal04; @Izzard_etal06; @Cordier_etal07], it is not as well suited to the class of hybrid models [e.g. @Marigo_etal96; @Marigo_etal98; @Marigo_etal99; @Marigo_07; @MarigoGirardi_07], in which the analytic formalism is complemented with numerical integrations of the stellar structure equations, carried out from the atmosphere down to the bottom of the convective envelope. In the latter case both the HBB nucleosynthesis and the basic changes in envelope structure – including effective temperature and radius – can be followed with the same richness of detail as in full models, but still in a much quicker and more versatile way.
It is not by accident that the crucial role of the surface C/O ratio and C-rich opacities in determining the evolution of TP-AGB stars was established just with the aid of these “envelope-based models” [@Marigo_02; @Marigo_07; @Marigo_etal03; @MarigoGirardi_07]. Although the same effect could have been assessed with the aid of full models, the latter were fighting with so many numerical and physical difficulties related to the occurrence of the third dredge-up, that the key aspect of the C-rich opacities was ignored, and likely forgotten, for long time in the field of AGB stellar evolution. Since @Marigo_02, molecular opacities for C-rich mixtures have been progressively adopted in full TP-AGB models [e.g. @Kamath_etal12; @VenturaMarigo_10; @VenturaMarigo_09; @WeissFerguson_09; @Cristallo_etal07].
This example tells clearly that progresses in the description of the TP-AGB phase do not rely only on full models, but they can come also from other complementary approaches.
With this work we go a few steps ahead in the development of our “envelope-based TP-AGB models”. We describe a code, called `COLIBRI`, that implements a number of improvements which, effectively, make our models to perform much more like “almost-full” models than “improved synthetic” ones. Among the most relevant points we mention: i) a spherically-symmetric deep envelope model extending from the atmosphere down to the bottom of the quiescent H-burning shell, so that the classical core-mass luminosity relation (CMLR) is naturally predicted and not taken as an input prescription; ii) the first ever on-the-fly accurate calculation of molecular chemistry and Rosseland mean opacities, fully consistent with the changing surface abundances, iii) a detailed HBB nucleosynthesis coupled with a diffusive description of convection, iv) a model for the pulse-driven convection zone to predict the chemical composition of the dredged-up material, and v) improved prescriptions to determine the onset and quenching of the third dredge-up.
Of course, in the development of the `COLIBRI` code full TP-AGB models still play a paramount role: they are taken as a reference to check the accuracy of some basic predictions, and they are used to derive quantitative information, via fitting relations, on those aspects that the `COLIBRI` code cannot, by construction, address by itself like, for example, the evolution of the intershell convection zone during thermal pulses.
In any case, all these aspects are treated fulfilling two extremely important conditions: a robust numerical stability which allows to follow the TP-AGB evolution until the complete ejection of the envelope, and a high computational speed which is kept comparable to the levels that made the success of the very first synthetic TP-AGB models. In this way the `COLIBRI` code is a tool perfectly suitable to perform a multi-parametric, but still accurate, calibration of the TP-AGB phase, our final goal.
The plan of the paper is as follows. Section \[sec\_outline\] presents an outline of the `COLIBRI` code. Section \[sec\_physmod\] describes in detail all input physics and the solution methods adopted to integrate the deep envelope model, and to predict the nucleosynthesis in the pulse-driven convective zone and during HBB. Section \[sec\_synthmod\] summarises the analytic ingredients of `COLIBRI`. Accuracy tests of `COLIBRI` predictions against full stellar models are discussed in Sect. \[sec\_tests\]. The present sets of TP-AGB evolutionary tracks are introduced in Sect. \[sec\_tracks\], while the whole Sect. \[sec\_results\] is dedicated to illustrate several examples of possible `COLIBRI` calculations. Finally, Sect. \[sec\_finalsum\] closes the paper giving a résumé of `COLIBRI`’s features, and briefly mentioning current and planned applications.
Overview of the COLIBRI code {#sec_outline}
============================
The `COLIBRI` code computes the TP-AGB evolution from the first thermal pulse up to the complete ejection of the stellar mantle by stellar winds. While maintaining a few basic features of our original TP-AGB model developed and revised over the years [@Marigo_etal96; @Marigo_etal98; @Marigo_98; @Marigo_etal99; @MarigoGirardi_07], we have introduced substantial improvements that notably enhance the predictive power of our TP-AGB calculations. The main variables of the TP-AGB model, which are also frequently cited in the text, are operatively defined in Table \[tab\_mod\].
`COLIBRI` consists of three main components, that we conveniently refer to as 1) the [*physics module*]{}, 2) the [*synthetic module*]{}, and 3) the [*parameter box*]{}.
The [*physics module*]{} involves all detailed input physics (equation of state, opacities, nuclear reactions rates) and differential equations necessary to numerically integrate a stationary [*deep envelope model*]{}, extending from the atmosphere down to the bottom of the H-burning shell (see Sect. \[sec\_physmod\]). At each time step, the run of mass $M_r$, temperature $T_r$, pressure $P_r$, and luminosity $L_r$ is determined across the deep envelope during the quiescent interpulse periods. By adopting proper boundary conditions at the bottom of the convective envelope, we obtain the effective temperature, and the luminosity provided by the hydrogen burning shell. In this way we are able to follow consistently the occurrence of HBB in the most massive AGB stars, being responsible for the break-down of the CMLR (see Sect. \[ssec\_lum\]), as well as a significant nucleosynthesis (see Sect. \[ssec\_HBBnuc\]).
The [*synthetic module*]{} contains the analytic formalism of the code, which includes both fitting formulas that synthesize the results of full AGB models (e.g. the core mass-interpulse period relation, the core mass-intershell mass relation, the efficiency of the third dredge-up as a function of stellar mass and metallicity, etc.), and other auxiliary relations (e.g. mass-loss prescription, period-mass-radius relations for variable AGB stars, etc.). It is outlined in Sect. \[sec\_synthmod\].
The [*parameters box*]{} collects all free parameters that we think need to be calibrated (e.g. minimum base temperature for the occurrence of the third dredge-up, efficiency of mass loss, dependence on mass and metallicity, overshoot at the base of the convective envelope) in order to reproduce basic observables. Since a fine calibration of the TP-AGB phase is not the primary purpose of this paper, the results presented here are obtained with a particular set of parameters, as specified in Sect. \[ssec\_tpagbev\].
These three components clearly represent a sequence of decreasing accuracy, and increasing uncertainty. While for most ingredients of the physics module we rely on detailed and well-established prescriptions, in the synthetic module we have to resort to the results of various sets of full TP-AGB models in the literature that share a general agreement, but present also unavodaible differences due to specific model details. The parameter box, instead, hides a big deal of our ignorance about basic physical processes in AGB stars. The coupling of these components, with very different degrees of accuracy, is inescapable at this point. The situation resembles the one that persists in practically all full stellar evolutionary codes to date, in which rough descriptions for convective processes – such as the mixing length theory and overshooting – are routinely adopted, and anyhow being able to produce very useful results. Although we all know that “fake physics” is being used to some extent in all these codes, it is also a matter of fact that, at some stages, these approximations have opened the way for advancing the theory of stellar evolution on other fronts. Our wish is that the same strategy can turn out to be useful also for the TP-AGB phase.
The physics module {#sec_physmod}
==================
Equation of state {#ssec_eos}
-----------------
The equation of state (EoS) for temperatures in the interval from $5\times 10^4$ K to $10^8$ K is that of a fully-ionized gas, in the way described by @Girardi_etal00.
For temperatures in the range from $5\times 10^4$ K to $10^3$ K all relevant thermodynamic quantities and their partial derivatives (mass density, electron density, mean molecular weight, entropy, specific heats, etc.) are computed [*on-the-fly*]{} with the `ÆSOPUS` code [@MarigoAringer_09]. We briefly recall that `ÆSOPUS` solves the EoS for atoms and molecules in the gas phase, under the assumption of an ideal gas in both thermodynamic equilibrium and instantaneous chemical equilibrium. We consider the ionisation stages from I to V for all elements from C to Ni (up to VI for O and Ne), and from I to III for heavier atoms from Cu to U. Saha equations for ionisation and dissociation are solved for $\approx 800$ species, including $\approx 300 $ atoms (neutral and ionised) from H to U, and $\approx 500$ molecules.
An example of the EoS calculations across the outermost layers of a TP-AGB model is given in Fig. \[fig\_molec\], that also illustrates the dramatic change in the equilibrium molecular chemistry as the surface C/O ratio passes from ${\rm C/O} < 1$, typical of M stars, to ${\rm C/O} > 1$, characteristic of C stars.
Gas opacities {#ssec_opac}
-------------
Rosseland mean gas opacities, in the whole temperature range $8.0 \le \log T \le 3.2$, are computed [*on-the-fly*]{}, i.e. contemporary with the atmospheric and envelope integrations that constitute the kernel of our TP-AGB code.
We remark that this is the [*first time ever*]{} that accurate opacities are computed on-the-fly, just starting from the monochromatic absorption coefficients of the opacity sources, without interpolation in pre-exiting tables of Rosseland mean opacities.
This choice is motivated by the demand of accurately describing the tight coupling of the opacity sources (mainly in the molecular regime) with the frequent and significant changes in the envelope chemical composition that characterise the TP-AGB phase. In this way we avoid the loss in accuracy that one must otherwise pay when performing multi-dimensional interpolation.
To this aim we have constructed a routine which, for any given set of chemical abundances of $92$ elements from H to U, and a specified pair of state variables (e.g. gas pressure $P_{\rm g}$ and temperature $T$), makes direct calls to one of two opacity codes, depending on the temperature:
- The [*Opacity Project*]{}[^2] (OP) [OP; @Seaton05; @Badnell_etal05] for $4.2 < \log T \le 8.0$;
- The `ÆSOPUS`[^3] code [@MarigoAringer_09] for $3.2 \le \log T \le 4.2$.
The OP data provides the monochromatic opacities for several atoms (H, He, C, N, O, Na, Mg, Al, Si, S, Ar, Ca, Cr, Mn, Fe, Ni) over a wide range of values of temperature $T$ and electron density $N_{\rm e}$. We have employed the routines [*mixv.f*]{} and [*opfit.f*]{} to calculate the Rosseland mean opacities on a pre-determined grid of OP$(T,\,N_{\rm e})$ meshes and then to interpolate to any specified values of $T$ and $\rho$. Since the original OP version assumes a fixed mixture of elements (i.e. scaled-solar chemical composition), we have suitably modified the OP routines to compute the Rosseland mean for any chemical composition involving the $16$ species for which the OP monochromatic opacities are available. This is an important improvement compared to the common practice in which the chemical parameters (besides the H or He abundances) are limited to few metal abundances. For instance, the widely-used OPAL web tool [@RogersIglesias_96] allows the on-line computation and provides the interpolating routines of Rosseland mean opacity tables with a fixed partition of metals, but for the abundances of two species (e.g. C and O), which are enhanced according to a specified grid of values. We notice that in this case, the possible depletion of a metal, due for instance to nuclear burning, cannot be considered. At variance, the OP utility gives us an important flexibility in this respect.
Suitably converted into an internal routine of our `COLIBRI` code, for each pair of $P_{\rm g}$ and $T$, `ÆSOPUS` calculates the monochromatic true absorption and scattering cross sections due to a number of continuum and discrete processes, i.e. bound-free absorption due to photoionisation, free-free absorption, Rayleigh and Thomson scattering, collision-induced absorption, atomic bound-bound absorption and molecular absorption. We note that the monochromatic cross sections for atoms (C, N, O, Na, Mg, Al, Si, S, Ar, Ca, Cr, Mn, Fe, Ni) are taken from the OP database, thus assuring a complete consistency with the high-temperature opacities. Then, after summing up all contributions, the Rosseland mean (RM) opacity is computed.
The incorporation of `ÆSOPUS` in the `COLIBRI` code allows us to follow accurately the changes in molecular opacities driven by any variation of the envelope composition, especially by the C/O ratio which plays the key role in determining the molecular chemistry [see e.g. @MarigoAringer_09]. The complex behaviour of the RM opacities as a function of the C/O ratio is exemplified with the aid of Fig. \[fig\_opac\]. It turns out that while the C/O ratio increases from $0.1$ to $0.9$ the opacity bump peaking at ($\log(T)\simeq 3.25-3.35$) – mostly due to H$_2$O – becomes more and more depressed because of the smaller availability of O atoms. Then, passing from C/O $=0.9$ up to C/O $=0.95$ the H$_2$O feature actually disappears and $\kappa_{\rm R}$ drastically drops by more almost two orders of magnitude. In fact, at this C/O value the chemistry enters in a transition region where most of both O and C atoms are trapped in the very stable CO molecule at the expense of the other molecular species, belonging to both the O- and C-bearing groups. At C/O $=1$ the RM opacity reaches its minimum throughout the temperature range, $3.2 \la \log(T) \la 3.4$, while a sudden upturn is expected as soon as C/O slightly exceeds unity, as displayed by the curve for C/O $=1.05$ of Fig. \[fig\_opac\] (right panel). This fact reflects the drastic change in the molecular equilibria from the O- to the C-dominated regime. Then, at increasing C/O the opacity curves move upward following a more gradual trend, which is related to the strengthening of the C-bearing molecular absorption bands.
Note, however, that the C-rich opacity does not rise linearly with C/O, but less and less steeply as the C/O ratio increases. This is mainly due to the underlying equilibrium chemistry of the most efficient absorbers, in particular of the CN and HCN molecules, whose abundances are conditioned not only by the carbon excess (C-O), but also by the availability of the N atoms (having a fixed abundance in the case under consideration). As we will see in Sect. \[ssec\_hayashi\], the non-linear dependence of the opacity on the C/O ratio impacts on the maximum extension of the Hayashi lines for C stars towards lower effective temperatures.
Nuclear reactions {#ssec_nrat}
-----------------
Our nuclear network consists of the p-p chains, the CNO tri-cycle, and the Ne-Na, Mg-Al chains, and the most important $\alpha$-capture reactions, including explicitly $N_{\rm el} = 25$ chemical species: $^1$H, $^2$H, $^3$He, $^4$He, $^7$Li, $^7$Be, $^{12}$C, $^{13}$C, $^{14}$N, $^{15}$N, $^{16}$O, $^{17}$O, $^{18}$O, $^{19}$F, $^{20}$Ne, $^{21}$Ne, $^{22}$Ne, $^{23}$Na, $^{24}$Mg, $^{25}$Mg, $^{26}$Mg, $^{26}$Al$^m$, $^{26}$Al$^g$, $^{27}$Al, $^{28}$Si. The latter nucleus acts as the “exit element”, which terminates the network. In total we consider $42$ reaction rates, listed in Tab. \[tab\_rates\]. For all of them we adopt analytic relations, with fitting coefficients taken from the JINA reaclib database [@Cyburt_etal10]. The alternative of using detailed tables of reaction rates as a function of the temperature can be easily implemented in `COLIBRI`, and may be done in future studies dedicated to nucleosynthesis calculations.
----------------------------------------------------------------------- ----------------------
[$\rm\,{}^{}\kern-0.8pt{p}\,({p}\,,{\beta^+\,\nu}) @Cyburt_etal10
\,{}^{}\kern-0.8pt{D}\,$]{}
[$\rm\,{}^{}\kern-0.8pt{p}\,({D}\,,{\gamma}) @Descouvemont_etal04
\,{}^{3}\kern-0.8pt{He}\,$]{}
[$\rm\,{}^{3}\kern-0.8pt{He}\,({^{3}He}\,,{\gamma}) @Angulo_99
\,{}^{}\kern-0.8pt{2\,p + ^{4}\kern-0.8pt{He}}\,$]{}
[$\rm\,{}^{4}\kern-0.8pt{He}\,({^{3}He}\,,{\gamma}) @Descouvemont_etal04
\,{}^{7}\kern-0.8pt{Be}\,$]{}
[$\rm\,{}^{7}\kern-0.8pt{Be}\,({e^-}\,,{\gamma}) @CaughlanFowler_88
\,{}^{7}\kern-0.8pt{Li}\,$]{}
[$\rm\,{}^{7}\kern-0.8pt{Li}\,({p}\,,{\gamma}) @Descouvemont_etal04
\,{}^{}\kern-0.8pt{^{4}\kern-2.0pt{He} + ^{4}\kern-2.0pt{He}}\,$]{}
[$\rm\,{}^{7}\kern-0.8pt{Be}\,({p}\,,{\gamma}) @Angulo_99
\,{}^{8}\kern-0.8pt{B}\,$]{}
[$\rm\,{}^{12}\kern-0.8pt{C}\,({p}\,,{\gamma}) @Angulo_99
\,{}^{13}\kern-0.8pt{N}\,$]{}
[$\rm\,{}^{13}\kern-0.8pt{C}\,({p}\,,{\gamma}) @Angulo_99
\,{}^{14}\kern-0.8pt{N}\,$]{}
[$\rm\,{}^{14}\kern-0.8pt{N}\,({p}\,,{\gamma}) @Imbriani_etal05
\,{}^{15}\kern-0.8pt{O}\,$]{}
[$\rm\,{}^{15}\kern-0.8pt{N}\,({p}\,,{\gamma}) @Angulo_99
\,{}^{}\kern-0.8pt{^4He + ^{12}\kern-2.0pt{C}}\,$]{}
[$\rm\,{}^{15}\kern-0.8pt{N}\,({p}\,,{\gamma}) @Angulo_99
\,{}^{16}\kern-0.8pt{O}\,$]{}
[$\rm\,{}^{16}\kern-0.8pt{O}\,({p}\,,{\gamma}) @Angulo_99
\,{}^{17}\kern-0.8pt{F}\,$]{}
[$\rm\,{}^{17}\kern-0.8pt{O}\,({p}\,,{\gamma}) @Chafa_etal07
\,{}^{}\kern-0.8pt{\,^4He + ^{14}\kern-2.0pt{N}}\,$]{}
[$\rm\,{}^{17}\kern-0.8pt{O}\,({p}\,,{\gamma}) @Chafa_etal07
\,{}^{18}\kern-0.8pt{F}\,$]{}
[$\rm\,{}^{18}\kern-0.8pt{O}\,({p}\,,{\gamma}) @Angulo_99
\,{}^{}\kern-0.8pt{\,^4He + ^{15}\kern-2.0pt{N}}\,$]{}
[$\rm\,{}^{18}\kern-0.8pt{O}\,({p}\,,{\gamma}) @Angulo_99
\,{}^{19}\kern-0.8pt{F}\,$]{}
[$\rm\,{}^{19}\kern-0.8pt{F}\,({p}\,,{\gamma}) @Angulo_99
\,{}^{}\kern-0.8pt{\,^4He + ^{16}\kern-2.0pt{O}}\,$]{}
[$\rm\,{}^{19}\kern-0.8pt{F}\,({p}\,,{\gamma}) @Angulo_99
\,{}^{20}\kern-0.8pt{Ne}\,$]{}
[$\rm\,{}^{20}\kern-0.8pt{Ne}\,({p}\,,{\gamma}) @Angulo_99
\,{}^{21}\kern-0.8pt{Na}\,$]{}
[$\rm\,{}^{21}\kern-0.8pt{Ne}\,({p}\,,{\gamma}) @Iliadis_etal01
\,{}^{22}\kern-0.8pt{Na}\,$]{}
[$\rm\,{}^{22}\kern-0.8pt{Ne}\,({p}\,,{\gamma}) @Hale_etal02
\,{}^{23}\kern-0.8pt{Na}\,$]{}
[$\rm\,{}^{23}\kern-0.8pt{Na}\,({p}\,,{\gamma}) @Hale_etal04
\,{}^{}\kern-0.8pt{\,^4He + ^{20}\kern-2.0pt{Ne}}\,$]{}
[$\rm\,{}^{23}\kern-0.8pt{Na}\,({p}\,,{\gamma}) @Hale_etal04
\,{}^{24}\kern-0.8pt{Mg}\,$]{}
[$\rm\,{}^{24}\kern-0.8pt{Mg}\,({p}\,,{\gamma}) @Iliadis_etal01
\,{}^{25}\kern-0.8pt{Al}\,$]{}
[$\rm\,{}^{25}\kern-0.8pt{Mg}\,({p}\,,{\gamma}) @Iliadis_etal01
\,{}^{26}\kern-0.8pt{Al^g}\,$]{}
[$\rm\,{}^{25}\kern-0.8pt{Mg}\,({p}\,,{\gamma}) @Iliadis_etal01
\,{}^{26}\kern-0.8pt{Al^m}\,$]{}
[$\rm\,{}^{26}\kern-0.8pt{Mg}\,({p}\,,{\gamma}) @Iliadis_etal01
\,{}^{27}\kern-0.8pt{Al}\,$]{}
[$\rm\,{}^{26}\kern-0.8pt{Al^g}\,({p}\,,{\gamma}) @Iliadis_etal01
\,{}^{27}\kern-0.8pt{Si}\,$]{}
[$\rm\,{}^{27}\kern-0.8pt{Al}\,({p}\,,{\gamma}) @Iliadis_etal01
\,{}^{}\kern-0.8pt{\,^4He + ^{24}\kern-2.0pt{Mg}}\,$]{}
[$\rm\,{}^{27}\kern-0.8pt{Al}\,({p}\,,{\gamma}) @Iliadis_etal01
\,{}^{28}\kern-0.8pt{Si}\,$]{}
[$\rm\,{}^{4}\kern-0.8pt{He}\,({2\,^{4}He}\,,{\gamma}) @Fynbo_etal05
\,{}^{12}\kern-0.8pt{C}\,$]{}
[$\rm\,{}^{12}\kern-0.8pt{C}\,({^{4}He}\,,{\gamma}) @Buchmann_96
\,{}^{16}\kern-0.8pt{O}\,$]{}
[$\rm\,{}^{14}\kern-0.8pt{N}\,({^{4}He}\,,{\gamma}) @Gorres_etal00
\,{}^{18}\kern-0.8pt{F}\,$]{}
[$\rm\,{}^{15}\kern-0.8pt{N}\,({^{4}He}\,,{\gamma}) @Wilmes_etal02
\,{}^{19}\kern-0.8pt{F}\,$]{}
[$\rm\,{}^{16}\kern-0.8pt{O}\,({^{4}He}\,,{\gamma}) @Angulo_99
\,{}^{20}\kern-0.8pt{Ne}\,$]{}
[$\rm\,{}^{18}\kern-0.8pt{O}\,({^{4}He}\,,{\gamma}) @Dababneh_etal03
\,{}^{22}\kern-0.8pt{Ne}\,$]{}
[$\rm\,{}^{20}\kern-0.8pt{Ne}\,({^{4}He}\,,{\gamma}) @Angulo_99
\,{}^{24}\kern-0.8pt{Mg}\,$]{}
[$\rm\,{}^{22}\kern-0.8pt{Ne}\,({^{4}He}\,,{\gamma}) @Angulo_99
\,{}^{26}\kern-0.8pt{Mg}\,$]{}
[$\rm\,{}^{24}\kern-0.8pt{Mg}\,({^{4}He}\,,{\gamma}) @CaughlanFowler_88
\,{}^{28}\kern-0.8pt{Si}\,$]{}
[$\rm\,{}^{13}\kern-0.8pt{C}\,({^{4}He}\,,{n}) @Angulo_99
\,{}^{16}\kern-0.8pt{O}\,$]{}
[$\rm\,{}^{17}\kern-0.8pt{O}\,({^{4}He}\,,{n}) @Angulo_99
\,{}^{20}\kern-0.8pt{Ne}\,$]{}
[$\rm\,{}^{18}\kern-0.8pt{O}\,({^{4}He}\,,{n}) @Angulo_99
\,{}^{21}\kern-0.8pt{Ne}\,$]{}
[$\rm\,{}^{21}\kern-0.8pt{Ne}\,({^{4}He}\,,{n}) @Angulo_99
\,{}^{24}\kern-0.8pt{Mg}\,$]{}
[$\rm\,{}^{22}\kern-0.8pt{Ne}\,({^{4}He}\,,{n}) @Angulo_99
\,{}^{25}\kern-0.8pt{Mg}\,$]{}
[$\rm\,{}^{25}\kern-0.8pt{Mg}\,({^{4}He}\,,{n}) @Angulo_99
\,{}^{28}\kern-0.8pt{Si}\,$]{}
----------------------------------------------------------------------- ----------------------
: Nuclear reaction rates adopted in this work.
\[tab\_rates\]
The atmosphere model {#ssec_atmo}
--------------------
For given chemical composition of the gas, an atmosphere model is generally specified by three stellar parameters, e.g: total mass $M$, luminosity $L$, and radius $R$. The effective temperature derives from the Stefan-Boltzmann law $L=4\pi R^2 \sigma T_{\rm eff}^4$. In our TP-AGB code the atmospheric structure can be obtained by choosing among two different options, namely: i) static plane-parallel atmosphere, and ii) static spherically symmetric atmosphere.
### Plane-parallel atmospheres {#sssec_ppatmo}
The plane-parallel grey atmosphere model is described by a temperature stratification given by a modified Eddington approximation for radiative transport: $$T^{4} = \frac{3}{4} T_{\rm eff}^{4} \left [ \tau +q\left ( \tau \right ) \right ]
\label{eq_ttau}$$ where $\tau(r)$ is the optical depth defined by the differential equation $$d\tau = - \kappa \rho dr
\label{eq_tau}$$ with the boundary condition $\tau(+\infty)=0$. Here $\kappa$ is the opacity which is usually described by the Rosseland mean, and $\rho$ is the mass density. The quantity $q(\tau)$ in the right-hand side of Eq. (\[eq\_ttau\]) is the Hopf function.
Under the plane-parallel assumption the variations across the atmospheres of mass, radius, and luminosity can be neglected so that we have $$\nonumber
M_r \approx M,\; \; \; \; \; \; r \approx R,\; \; \; \; \; \; L_r \approx L.$$
Let us denote with $\tilde{\tau}$ the optical depth of the photosphere (approximately $2/3$), and $r_{\tilde{\tau}}$ its radial coordinate. In the plane-parallel approximation, it defines the radius of the star, i.e. $R=r_{\tilde{\tau}}$, and the corresponding temperature $T_{\tilde{\tau}}$ coincides with the effective temperature $T_{\rm eff}$, defined by the Stefan-Boltzmann law $T_{\textup{eff}} = \left (L/4\pi \sigma R^2 \right )^{1/4}$.
Combining the equations of mass continuity, hydrostatic equilibrium and Eq. (\[eq\_tau\]), we obtain the atmospheric equation for the total pressure $$\frac{d\tau }{d P} = \frac{\kappa R^2}{G M}
\label{eq_dtaudp}$$ where $P= P_{\rm gas} + P_{\rm rad}$ includes the contributions from gas and radiation and obeys the boundary condition that $P_{\rm gas}=0$ for $\tau=0$. The integration of Eq. (\[eq\_dtaudp\]) is accomplished by a standard extrapolation-interpolation procedure, from $\tau=0$ to $\tau=\tilde{\tau}$. The solution is obtained through iteration on the total pressure $P$. Starting from the top of the atmosphere, with $P=P_{\rm rad}$ and $\tau=0$, we integrate Eq. (\[eq\_dtaudp\]) inward with a sequence of extrapolation-interpolation steps. The adopted scheme is a combination of a third-order Adams-Bashforth predictor followed by a fourth-order Adams-Moulton corrector [chapter XVI of “Numerical Recipes”; @Press_etal88]. In brief, for a given increment $\Delta P$, to proceed from the mesh-point $j$ to mesh-point $j+1$, we first extrapolate the optical depth $\tau_{j+1}^{\rm extr}$ with the predictor part, using the known value $\tau_j$. Then, we use the corrector to interpolate the derivative at $j+1$, and hence to obtain the value $\tau_{j+1}^{\rm int}$. The integration step is considered successful if the extrapolated $\tau_{j+1}^{\rm extr}$ and interpolated $\tau_{j+1}^{\rm int}$ values agree to within a given tolerance, normally set to $10^{-4}$ for the logarithmic optical depth. Otherwise, the integration step is repeated halving the pressure step-width $\Delta P$.
### Spherically-symmetric atmospheres
We have implemented the spherical-symmetry geometry following the formalism described in @Lucy76, but with the addition that the mass above the atmosphere is not neglected compared to that of the entire star. Introducing the variable $z=r/R$, the temperature stratification accounts for the geometrical dilution of the radiation field and is given by: $$T^{4} = \frac{3}{4} T_{\rm eff}^{4}
\left [\tilde\tau + \frac{4}{3} W \right]\,,
\label{eq_spheatm}$$ where $$W = \frac{1}{2} \left ( 1 - \frac{\sqrt{z^2 - 1}}{z} \right )$$ is the dilution factor; $\tilde\tau$ is the optical depth defined by the differential equation: $$\frac{d\tilde\tau}{dz} = -\frac{\kappa \rho R}{z^2}\,.$$
In this case, the radial extension of the atmosphere is not neglected, and $r=R_0$ refers to the maximum outer radius of the atmosphere, where by definition $\tau(R_0)=0$ and $P_{\rm gas}(R_0)=0$. Since in principle these two boundary conditions are met for $r \rightarrow + \infty$, we define the outer boundary $R_0$ of the atmosphere the radial coordinate of the point at which $P_{\rm gas} = 10^{-4}$ dyne cm$^{-2}$. The parameter $$\delta R = \frac{R_0-R}{R}
\label{eq_dr}$$ quantifies the geometrical extension of the atmosphere.
In an extended atmosphere an effective temperature cannot be uniquely defined; therefore we refer to it as the photospheric temperature obeying the relation $$T_{\textup{eff}} = T(\tilde{\tau}) = \left ( \frac{L}{4\pi \sigma R^2} \right )^{1/4} \,\,\,\,\,\,\,{\rm and} \,\,\,\,\,\,\,\tilde{\tau}=2/3
\label{eq_tef}$$ which is formally analogous to that of a compact atmosphere star.
In summary, together with the auxiliary relation Eq. (\[eq\_spheatm\]), our extended atmosphere model requires the integration of three differential equations for the unknowns optical depth $\tau$, non-dimensional radial coordinate $z=r/R$, and mass coordinate $m$, which are conveniently expressed in the form $d \tau /d\log P $, $d z /d\log P $, and $d \log m /d\log P $, where the total pressure $P$ is the independent variable.
For any given atmosphere model specified by a choice of $L$, $M$, $T_{\rm eff}$ (hence with $R$ known from Eq. \[eq\_tef\]), and chemical composition, we proceed as follows. We make an initial guess of the ratio $R_0/R$. Then the differential equations, reduced to a finite-difference form, are solved starting from the provisional outermost point at $r=R_0$, with the boundary conditions $$\tau(R_0)=0,\;\;\;\; m(R_0)=M,\;\;\;\; P(R_0)=P_{\rm rad},$$ and proceeding inward by using the same extrapolation-interpolation method already described in Sect. \[sssec\_ppatmo\], but this time extended to the three differential equations in the unknowns $\tau$, $r$, and $m$. Integration is stopped when the photosphere at $\tau = \tilde{\tau}$ is reached. In general the temperature at the photospheric layer, $T_{\tilde{\tau}}$, will differ from $T_{\rm eff}$ given by Eq. (\[eq\_tef\]), so we adopt a new value for $R_0/R$ and integrate another atmospheric structure. The procedure is repeated until the $\left | \log(T_{\tilde{\tau}}) - \log(T_{\textup{eff}})\right| < \varepsilon$, where the tolerance $ \varepsilon$ is normally set to $10^{-4}$.
The quiescent interpulse phases
-------------------------------
### The deep envelope model {#ssec_envmod}
In synthetic AGB models $L$, $T_{\rm eff}$, and the temperature at the base of the convective envelope, $T_{\rm bce}$, are usually obtained with the aid of formulas that fit the results of full models calculations [e.g. @Hurley_etal00; @Izzard_etal04; @Izzard_etal06; @Cordier_etal07]. In `COLIBRI` the approach is completely different: during the quiescent interpulse periods the four stellar structure equations (i.e. mass continuity, hydrostatic equilibrium, energy transport, and energy balance) are integrated from the photosphere down to the bottom of the quiescent H-burning shell, a region which we globally refer to as [*deep envelope*]{}.
The energy balance equation reads $$\frac{\partial l}{\partial m } = \varepsilon _{\rm nuc} + \varepsilon _{\rm g}
- \varepsilon _{\rm \nu}\,,
\label{eq_energ}$$ where the right-hand side member accounts for the energy contributions/losses from nuclear, gravitational, and neutrino sources, with rates (per unit time and unit mass) $\varepsilon _{\rm nuc}$, $\varepsilon _{\rm g}$, and $\varepsilon _{\rm \nu}$, respectively.
The efficiency of nuclear energy generation is computed as $\varepsilon _{\rm nuc} = \varepsilon _{\rm pp} + \varepsilon _{\rm CNO}$, that is including the contributions of the p-p chains and CNO cycles. The corresponding nuclear reaction rates are listed in Table \[tab\_rates\].
In our deep envelope model we assume $\varepsilon _{\rm \nu}=0$, which is a safe approximation since thermal neutrinos mainly come from the degenerate core.
The gravitational energy generation, given by $$\varepsilon _{\textup{g}} = - T \, \frac{\partial S}{\partial t}\,,$$ where $S$ is the gas entropy and $t$ denotes the time variable, is computed in the [*stationary*]{} wave approximation [@Weigert_66; @Iben_77]: $$\frac{\partial S}{\partial t} = \frac{\mathrm{d} M_{\textup{c}}}{\mathrm{d} t} \: \frac{\partial S}{\partial m}$$ where $T$ is the local temperature, $\partial S/\partial m$ is the local derivative of entropy with respect to mass, and $\mathrm{d} M_{\textup{c}}/{\mathrm{d} t}$ denotes the rate at which the mass coordinate of the centre of the hydrogen-burning shell advances outward.
The rate of displacement of the H-burning shell actually measures the growth rate of the core mass and it is computed with $$\frac{\mathrm{d} M_{\textup{c}}}{\mathrm{d} t} = \frac{q}{X_{\textup{env}}} L_{\textup{H}}$$ where $L_{\textup{H}}$ is the total luminosity produced by the [*radiative*]{} portion of the hydrogen burning shell, $X_{\textup{env}}$ corresponds to the hydrogen abundance (in mass fraction) in the convective envelope, and $q = 1.05\times 10^{-11} + 0.017\times 10^{-11}\,\log(Z)\;
[M_{\odot}\,L_{\odot}^{-1}\,{\rm yr}^{-1}]$ [@Wagenhuber_96].
#### Method of solution. {#sssec_envsol .unnumbered}
Since we deal with a set of four stellar structure equations, we need to set up four boundary conditions to close the system.
The first pair of boundary conditions applies to the surface, and corresponds to the photospheric values of radius and temperature, $r(\tilde{\tau})$, and $T(\tilde{\tau})$, provided by the atmosphere model (either in the plane-parallel or spherically-symmetric assumption as described in Sect. \[ssec\_atmo\]): $$\label{c1}
T(\tilde{\tau}) = T_{\rm eff}\,,$$ $$\label{c2}
r(\tilde{\tau}) = R\,.$$
The second pair of boundary conditions applies to the interior. Moving inward across the [*deep envelope*]{}, the bottom of the H-burning shell corresponds to the radiative layer where the hydrogen abundance first goes to zero ($X=0$). We choose the mass coordinate of the corresponding mesh, $m(X=0)$, to identify a key parameter of the AGB evolution, the core mass $M_{\rm c}$.
The third boundary condition is therefore: $$\label{c3}
m({X=0}) = M_{\rm c}\,.$$ The fourth inner boundary condition is given by the temperature $T_{\rm c}$ at the bottom of the H-burning shell: $$\label{c4}
T({X=0}) = T_{\rm c}\, .$$ Full stellar AGB models calculated with `PARSEC` show that $T_{\rm c}=T(M_ {\rm c}, Z_{\rm i})$ is a well-behaving, increasing function of the core mass, with some moderate dependence on metallicity. After the first sub-luminous thermal pulses, in the full-amplitude regime $T_{\rm c}$ is found to vary within a narrow range (i.e. $\log(T_{\rm c})\approx 7.9-8.0$), reflecting the thermostatic property of the shell-hydrogen burning (mainly via the CNO cycle), occurring at a well-defined temperature. This fact makes the boundary condition Eq. (\[c4\]) a robust choice, only little dependent on technical and model details.
In summary, Eqs. (\[c1\]), (\[c2\]), (\[c3\]), and (\[c4\]) provide the four boundary constraints necessary to determine the entire structure of the [*deep envelope*]{}. The total pressure $P$ is chosen as the independent variable, and the four differential equations of the stellar structure are suitably expressed in the form $d\log m /d\log P $, $d\log r /d\log P $, $d\log l /d\log P $, and $d\log T /d\log P$. Inward numerical integrations are carried out using an Adams-Bashforth-Moulton extrapolation-interpolation scheme, that combines a third-order predictor with a fourth-order corrector. The procedure is formally the same as that described in Sect. \[sssec\_ppatmo\], but applied to the four equations in the unknowns $m$, $r$, $l$, $T$. The integration accuracy is usually set to $10^{-4}$ for all logarithmic variables.
We adopt a very fine mass resolution, the width of the innermost shells (where the structural gradients become extremely steep) typically amounting to $10^{-7} - 10^{-8} M_{\odot}$. The chemical composition is assumed homogeneous throughout the convective envelope (possible deviations for specific elements are discussed in Sect. \[ssec\_HBBnuc\]). Once in the deep interior the radiative temperature gradient falls below the adiabatic one and the energy transport becomes radiative, a chemical profile is built with abundances that change with mass in direct proportion to the rate of energy generation by the hydrogen-burning reactions, until hydrogen vanishes The procedure is the same as that described by @Iben_77.
The integration method just illustrated is adopted to obtain the atmosphere-envelope structure at the quiescent stage just preceding each thermal pulse. In particular, this yields the quiescent pre-flash luminosity maximum, $L_{\rm Q}$. To follow the subsequent structural variations, driven by the occurrence of thermal pulses, we proceed as follows. Let us denote with $$\phi\equiv t/\tau_{\rm ip}
\label{eq_phi}$$ the pulse-cycle phase, where $\tau_{\rm ip}$ is the interpulse period and $t$ is the current time, counted from the stage of quiescent pre-flash luminosity maximum, such that $\phi =0$ at $t=0$, and $\phi =1$ at $t=\tau_{\rm ip}$ (and $L=L_{\rm Q}$). According to @WoodZarro_81 and @WagenGroen_98 the star luminosity as a function of the pulse-cycle phase, $L(\phi)$, when normalized to $L_{\rm Q}$, has a very well-known and almost universal form (${\rm f}(\phi) = L(\phi)/L_{\rm Q}$), independent of $Z_{\rm i}$ [@WagenGroen_98 see their equation 15]. Therefore, once we determine $L_{\rm Q}$ at $\phi =1$ by solving the complete set of stellar equations, then the structure of the envelope over the next thermal TPC (for each value of the phase $0\le \phi< 1$) is obtained iteratively in a similar fashion, but this time adopting $L=L(\phi)={\rm f}(\phi)\:L_{\rm Q}$, and fulfilling three out of four boundary conditions. While the first pair, Eqs. (\[c1\]) and (\[c2\]), is the same for any value of $\phi$, the third boundary condition depends on phase of the pulse cycle.
Following the thorough analysis by @WagenGroen_98, in the initial phases of a TPC, for $0\le \phi \la 0.1$, (that include the so-called “‘rapid dip’’, “rapid peak” and part of the “slow dip”, i.e. from A to D in their figure 1), the H-shell is extinguished, while the He-shell is on. During these very short-lived stages, immediately after the onset of a TP, we adopt $M(R_{\rm c}) = M_{\rm c}$ (Eq. \[c5\]) as the third boundary condition for the envelope integrations. More details can be found in Sect. \[ssec\_TP\]. At later stages, for $0.1 < \phi \le 1$ (i.e. from D to A’), when the helium burning drops and the quiescent H-shell recovers becoming the dominant energy source, the third boundary condition is again given by $m({X=0}) = M_{\rm c}$ (Eq. \[c3\]).
It is worth remarking that the integration of the [*deep envelope*]{} allows us to predict the integrated luminosity provided by the quiescent H-burning shell, both in the relatively simple case of low-mass TP-AGB stars (in which the H-burning shell is completely radiative and thermally decoupled from the convective envelope), and in the more complex case of intermediate-mass TP-AGB stars experiencing HBB (in which the bottom of the convective envelope lies inside the H-burning shell, providing an extra-luminosity $\Delta L_{\rm HBB}$ contribution above the classical CMLR). Section \[sec\_tests\] is devoted to compare and test our results against those from various sets of full AGB models in the literature.
Another important implication is that our method assures a correct treatment of HBB, i.e. a full consistency between energetics and associated nucleosynthesis. In other words, the rates of variation of the surface chemical abundances caused by HBB (i.e via the CNO, NeNa, and MgAl cycles) are precisely those that correspond to the luminosity contribution $\Delta L_{\rm HBB}$. Despite being a basic requirement [@MarigoGirardi_01], the strict coupling between the consumption of the nuclear fuel and the chemical composition changes, are in general not fulfilled by analytical approximations of HBB, often adopted in synthetic TP-AGB models.
### Nucleosynthesis in convective envelope layers {#ssec_HBBnuc}
Besides being an important energy source for AGB stars with $M_{\rm i} > 3-4\, M_{\odot}$, HBB significantly alters the chemical composition of their envelopes through the nuclear reactions (pp chains, and CNO, NeNa, MgAl cycles) taking place in the innermost convective layers [e.g. @Boothroyd_etal95; @ForestiniCharbonnel_97; @Marigo_01; @Karakas_10; @Ventura_etal11].
In `COLIBRI` the HBB nucleosynthesis is treated in detail. Once the structure of the convective envelope is determined, as explained in Sect. \[ssec\_envmod\], nucleosynthesis occurring in the convective envelope is treated in detail, by coupling nuclear burning to a diffusive description of convection. In a one-dimensional, spherically-symmetric system the conservation equation for an arbitrary chemical species $i$, locally defined at the Lagrangian coordinate $m_r$, reads $$\begin{aligned}
\label{eq_difmix}
\frac{\partial Y_i}{ \partial t}\Bigl\lvert_{m_r} & = & \frac{1}{\rho r^2} \frac{\partial}{\partial r}
\left( r^2 \rho D \frac{\partial Y_i }{\partial r} \right)\\
\nonumber
& & \pm \sum_{j} Y_j \lambda_k(j) \pm \sum_{j\ge k} Y_j Y_k r_{j k}\,,\end{aligned}$$ where $Y_i = X_i/A_i$ (in units of mole/mass) is the ratio between the abundance (in mass fraction) of the nucleus $i$ and its atomic weight $A_i$. The term on the left-hand side gives the local rate of change of abundance of element $i$ at the coordinate $m_r$, which is due to two different processes, namely: mixing and nucleosynthesis.
On the right-hand side of Eq. (\[eq\_difmix\]) the first term is the mixing contribution, that is the local abundance variation produced by the convective motions in the gas. In our approach convection is treated as a diffusion process, with the diffusive coefficient approximated as $$\label{eq_difco}
D = \frac{1}{3} v_{\rm conv} l_{\rm conv}\,,$$ where $v_{\rm conv}$ and $l_{\rm conv}$ denote the velocity and the mean-free path of the convective eddies, respectively. Both quantities are computed in the framework of the standard mixing length theory [@mlt_58]. The mixing length $l_{\rm conv}$ is assumed linearly proportional to the pressure scale height, $H_{\rm p}$, with the proportionality coefficient $\alpha_{\rm MLT}=1.74$, as derived from a recent calibration of the solar model [@Bressan_etal12]. The convective velocity is obtained from the only real root of the “cubic equation” [equation $14.82$, Vol. I of “Principles of Stellar Structure”; @CoxGiuli_68], under the condition that the total energy flux is specified.
The second and third terms on the right-hand side of Eq. (\[eq\_difmix\]) describe the abundance change due to nuclear reactions involving the species $i$, being related to single-body decays (with rates $\lambda$) and two-body reactions (with rate $r$), respectively. As usual, the negative (positive) sign is used to denote destruction (production) of the species $i$.
#### Method of solution. {#method-of-solution. .unnumbered}
The convective envelope is divided into a number $N_{\rm mesh}$ of concentric shells, so as to ensure smooth enough variations of the physical variables (radius, temperature, density, etc.) between consecutive mesh points. For instance, in the deepest zones, where nuclear burning takes place the temperature difference of consecutive shells is chosen $\delta\log(T) = 0.01-0.02$ dex.
We deal with a system of coupled, non-linear, partial differential equations, given by Eq. (\[eq\_difmix\]), for each chemical species at all mesh points. The equations are first converted to finite central-difference equations and the quadratic terms, $Y_j Y_k$, are linearized according to @ArnettTruran_69. To estimate the diffusion coefficient between two shells, $D_{k \pm 1/2}$, we adopt the prescription proposed by @Meynet_etal04: $$D_{k \pm 1/2} =\frac{D_{k \pm 1} D_k}{f D_{k \pm 1} + (1-f) D_k}$$ with $f=0.5$, which appears to be more physically sound than adopting a simple arithmetic mean.
Following the scheme proposed by @Sackmann_etal74, we set up a matrix equation $A=Y\,b$ in the unknown abundances $Y_{i,k}^{n+1}$ at the time $n+1$, where $i=1,\dots,N_{\rm el}$ denotes the element, and $k=1,\dots, N_{\rm mesh}$ refers to the mesh-point. $A$ is the $(N,N)$ matrix of the coefficients with $N=N_{\rm el}\times N_{\rm mesh}$. Since we assume that each species is coupled to all others at the same mesh point and to its own abundance at adjacent mesh-points, the matrix $A$ has a band-diagonal structure with $k_l=N_{\rm el}$ sub-diagonals and $k_u=N_{\rm el}$ super-diagonals (hence the band width is $k_l+k_u+1$). This property is taken into consideration to reduce the computing-time requirement of the adopted numerical algorithm. The $(N,1)$ matrix $b$ contains the known terms, which depend on the chemical abundances across the envelope, $Y_{i,k}^{n}$, at the previous time $n$.
Finally, the system is solved by means of a fully implicit method that, when applied to diffusion problems, proves to yield robust results in terms of numerical stability and accuracy [see the thorough analysis in @Meynet_etal04]. Compared to explicit and “Crank-Nicholson” methods the great advantages of the implicit technique are that i) we are not forced to stick to the “Courant condition”, that imposes short integration time steps to assure stability, ii) in most cases it does not yield unphysical solution (e.g. negative abundances), and iii) the conservation of the mass, i.e. the normalization condition of the abundances, at each mesh-point is reasonably fulfilled, typically not exceeding $\simeq 10^{-5}$.
Fortran routines taken from the LAPACK[^4] software package are employed to get the numerical solution of the matrix equation, which is accomplished through three main steps, namely: 1) LU decomposition[^5] of the matrix $A$, which is conveniently stored in a compact form so as to get rid of most of the useless null terms outside the main diagonal band; 2) solution of the system of linear equations by partial pivoting, and 3) iterative improvement of the solution. The latter step attempts to refine the solution by reducing the backward errors (mainly due to round-off and truncation errors) as much as possible.
### Time integration
To follow the time evolution along the TP-AGB phase we proceed as follows. Each interpulse period is divided into a suitable number, $N_{\phi}$, of phase intervals, $\Delta\phi_j=(t_{\rm j+1}-t_j))/\tau_{\rm ip}=
\Delta t_j/\tau_{\rm ip}$, so as to assure a good sampling of the complex luminosity variations driven by the pulse (see Eq. (\[eq\_phi\]) and Sect. \[ssec\_envmod\]). This defines a first guess of the time step. A subsequent adjustment may be done by imposing the condition that the time step does not exceed a given limit, i.e. $\zeta (M-M_{\rm c})/\dot M$, where $(M-M_{\rm c})/\dot M$ is a measure of the time-scale required to expel the envelope at the current mass loss rate $\dot M$. The coefficient $\zeta$ is normally set to $10^{-3}$. This condition determines a sizable reduction of the time step in the last evolutionary stages, when the super-wind regime of mass loss is attained.
Once $\Delta t_j$ is fixed, the increment of the core mass and the decrease of the total mass are predicted with the explicit Eulerian method: $$\begin{aligned}
M_{\rm c, j+1} & = & M_{\rm c, j}
+ (q L_{\textup{H}}/ X_{\textup{env}})_j \, \Delta t_j \\
M_{j+1} & = & M_{j} - \dot M_j \, \Delta t_j \end{aligned}$$ At this point all other variables (e.g. $T_{\rm eff}$, $L$, $T_{\rm bce}$, and chemical abundances in case of HBB, etc.) at the time $t_{\rm j+1}$ are obtained from envelope integrations with the new values $M_{\rm c,j+1}$ and $M_{j+1}$.
With the current set of prescriptions, typical values of $N_{\phi}$ over one TPC range from few to several hundreds, depending on stellar parameters and evolutionary status.
The thermal-pulse phases {#ssec_TP}
------------------------
In addition to the quiescent interpulse phases (see Sect. \[ssec\_envmod\]), we carry out envelope integrations to test whether appropriate thermodynamic conditions exist for the occurrence of the third dredge-up. This approach replaces the use of the parameter $M_{\rm c}^{\rm min}$, i.e. the minimum core mass for the third dredge-up (see Sect. \[ssec\_tbdred\]), used in previous models [@Marigo_etal96; @Marigo_etal98; @MarigoGirardi_07]. Also, we set up a nuclear network to follow the synthesis of C, O, Ne, Na, and Mg in the flash-driven convective zone, which determines the chemical composition of the dredged-up material. All details are given in Sect. \[ssec\_pdcz\].
### Onset and quenching of the third dredge-up {#ssec_tbdred}
We follow the method first proposed by @Wood_81 and later adopted by @Marigo_etal99 to predict [*if*]{} and [*when*]{} the third dredge-up may take place during the TP-AGB evolution of a star of given current mass and chemical composition. We refer to the quoted papers for all details, and recall here the basic scheme.
The technique makes use of suitable envelope integrations at the stage of post-flash luminosity maximum, $L_{\rm P}$, when the envelope is close to hydrostatic and thermal equilibrium [@Wood_81]. TP-AGB models show that $L_{\rm P}$ is essentially controlled by the core mass of the star, in analogy with the existence of the CMLR relation during the quiescent interpulse periods for low-mass AGB stars. Following @Wood_81 and @BoothroydSackmann_88b, at the post-flash luminosity peak the nuclearly processed material involved in the He-shell flash is pushed out and cooled down to its minimum temperature over the flash-cycle, $T_{\rm N}^{\rm min}$, approaching a limiting characteristic value, as the thermal pulses reach the full-amplitude regime. This latter typically lies in the range $\log(T_{\rm N}^{\rm min})\approx 6.5-6.7$ [@BoothroydSackmann_88b; @Karakas_etal02], being little dependent on chemical composition and core mass. At the same time the envelope convection reaches its maximum inward penetration (in mass fraction) and the maximum base temperature, $T_{\rm bce}^{\rm max}$.
Hence it is reasonable to assume that the third dredge-up takes place if, at the stage of post-flash luminosity maximum, the condition $T_{\rm bce}^{\rm max} \ge T_{\rm N}^{\rm min}$ is satisfied.
Operatively, let us denote with $T_{\rm dup}$ the parameter representing the minimum temperature that the envelope base must exceed to activate the third dredge-up, that is: $$\label{eq_tbdred}
T_{\rm bce}^{\rm max} \ge T_{\rm dup}\,.$$ In order to check it, at each thermal pulse, we integrate our envelope model described in Sect. \[ssec\_envmod\]. These numerical integrations are computed under particular conditions[^6], namely: i) we set $\varepsilon _{\rm nuc} = \varepsilon _{\rm pp} + \varepsilon _{\rm CNO} = 0$, since at this stage the H-burning shell is extinguished; ii) the two inner boundary conditions Eqs. (\[c3\]) and (\[c4\]) are replaced with $$\label{c5}
M(R_{\rm c}) = M_{\rm c}\,.$$ This condition means that the mass of the degenerate core is equal to the mass contained inside the radius of a [*warm*]{} white dwarf, $R_{\rm c}= \delta \times R_{\rm WD}$. In the latter expression $R_{\rm WD}$ is the radius of a zero-temperature white dwarf (WD) with mass $M=M_{\rm c}$, while the coefficient $\delta> 1$ accounts for the fact that the nearly isothermal degenerate core is warm, i.e. it has a non-zero temperature. To compute $R_{\rm c}$ we follow the same prescriptions as in @Marigo_etal99, and adopt the $M_{\rm c} - L_{\rm P}$ relation of @WagenGroen_98.
Then, for given stellar mass, core mass, surface chemical composition, and peak-luminosity $L_{\rm P}$, envelope integrations are performed iterating on the effective temperature, $T_{\rm eff}$, until when $M(R_{\rm c})=M_{\rm c}$. At this point, the structure of the envelope is entirely and uniquely determined.
Since the typical values of $T_{\rm N}^{\rm min}$ may vary between different sets of models (reflecting its dependence on the adopted input physics and on the description of convection), we take $T_{\rm dup}$ as a free parameter. An advantage is that with the condition given by Eq. (\[eq\_tbdred\]) we can also test the eventual quenching of the third dredge-up due, for instance, to a drastic reduction of the envelope mass, without the need for another external assumption (see Sect. \[ssec\_3dup\]). For the present set of TP-AGB models we have adopted the temperature parameter $\log(T_{\rm dup})=6.40$.
### Pulse-driven nucleosynthesis {#ssec_pdcz}
We have developed a simplified model to predict the intershell chemical composition produced by the flash-driven nucleosynthesis, using an approach similar in some aspects to those proposed by @IbenTruran_78, @Mowlavi_99a [@Mowlavi_99b], and @DenissenkovHerwig_03.
The assumed scheme for the pulse-driven convection zone (PDCZ) is sketched with the aid of a Kippenhahn diagram in Fig. \[fig\_pdcz\], showing the time evolution of the PDCZ borders from its appearance to its final quenching. Several relevant variables are defined in Table \[tab\_mod\].
[ll]{} $Z_{\rm i}$ & initial (zero-age-main-sequence) metallicity (mass fraction)\
$Y_{\rm i}$ & initial (zero-age-main-sequence) helium abundance (mass fraction)\
$X_{\rm i}$ & initial (zero-age-main-sequence) hydrogen abundance (mass fraction)\
$Z$ & current metallicity (mass fraction)\
$M_{\rm c}$ & current core mass $\equiv$ mass of the H-exhausted core\
$M_{\rm c, 1}$ & core mass at the first thermal pulse\
$M_{\rm c, nodup}= M_{\rm c, 1}+\displaystyle{\int_{0}^{t}\frac{d M_{\rm c}}{dt'} dt'}$ & core mass in absence of the third dredge-up, where $t=0$ is the time of the first TP.\
$M_{\rm i}$ & initial stellar mass at the zero-age main sequence\
$M_{1}$ & stellar mass at the first thermal pulse\
$M$ & current stellar mass\
$T_{\rm bce}$ & temperature at the base of the convective envelope\
$\tau_{\rm ip}$ & interpulse period\
$\phi\equiv t/\tau_{\rm ip}$ $(0 \le \phi \le 1)$ & pulse-cycle phase; the time $t=0$ refers to the quiescent pre-flash luminosity maximum.\
\
$\Delta M_{\rm c, tpc}$ & core mass growth over one interpulse period\
$\Delta M_{\rm c} = M_{\rm c} - M_{\rm c, 1}$ & cumulative core mass growth since the $1^{\rm st}$ TP\
$\Delta M_{\rm c, nodup} = M_{\rm c,nodup} - M_{\rm c, 1}$ & cumulative core mass growth in absence of the third dredge-up\
\
$M_{\rm Pt}$ & mass coordinate of the top of the current PDCZ at its maximum extension\
$M_{\rm Pt}^{\prime}$ &mass coordinate of the top of the previous PDCZ at its maximum extension\
$M_{\rm He}$ & mass coordinate of the He-exhausted core\
$M_{\rm Pb}$ &mass coordinate of the bottom of the current PDCZ at its maximum extension\
$f_{\rm ov}$ & parameter to mimic overshoot applied to the bottom of the PDCZ\
$\Delta M_{\rm pdcz}$ &PDCZ mass at its maximum extension\
$\tau_{\rm pdcz}$ & total duration of the PDCZ\
$\tau_{\rm q}$ & quenching time since maximum extension\
$T_{\rm pdcz}^{\rm max}$ & maximum temperature reached in a TP at the inner border of the PDCZ\
$\rho_{\rm pdcz}^{\rm max}$ & maximum density reached in a TP at the inner border of the PDCZ\
\
$M_{\rm c}^{\rm min}$ & minimum core mass for the occurrence of the third dredge-up\
$M_{\rm c}^{\rm 3dup}$ & actual core mass at the first episode of the third dredge-up\
$T_{\rm N}^{\rm min}$ & minimum temperature reached by the pulse at the stage of post-flash luminosity maximum\
$T_{\rm dup}$ & minimum temperature at the base of the convective envelope for the occurrence of the third dredge-up\
$\Delta M_{\rm dup}$ & dredged-up mass at a given thermal pulse\
$\Delta M_{\rm overlap}=M_{\rm Pt}^{\prime}-M_{\rm He}$ & overlap mass between two consecutive PDCZs\
$\lambda = \displaystyle{\frac{\Delta M_{\rm dup}}{\Delta M_{\rm c, tpc}}}$ & efficiency of the third dredge-up\
$r=\displaystyle{\frac{\Delta M_{\rm overlap}}{\Delta M_{\rm pdcz}}}
$ & degree of overlap between two consecutive PDCZ\
\[tab\_mod\]
At the onset of each TP the quantities $\Delta M_{\rm pdcz}$, $\tau_{\rm pdcz}$, $\tau_{\rm q}$, $T_{\rm pdcz}^{\rm max}$, $\rho_{\rm pdcz}^{\rm max}$ are preliminarily computed with the aid of analytic relations as a function of the core mass and metallicity, that can be obtained as fits to full AGB models (see Sect. \[sec\_synthmod\] for more details). For the present work we use mainly the results by @IbenTruran_78, @Wagenhuber_96, @Karakas_etal02, @Straniero_etal03.
A nuclear network is set up which includes the triple-$\alpha$ reaction and the most important $\alpha$-captures listed in Table \[tab\_rates\]. Among them we consider the main reactions which may be important as neutron sources: [$\rm\,{}^{13}\kern-0.8pt{C}\,({^{4}He}\,,{n})
\,{}^{16}\kern-0.8pt{O}\,$]{}, [$\rm\,{}^{17}\kern-0.8pt{O}\,({^{4}He}\,,{n})
\,{}^{20}\kern-0.8pt{Ne}\,$]{}, [$\rm\,{}^{18}\kern-0.8pt{O}\,({^{4}He}\,,{n})
\,{}^{21}\kern-0.8pt{Ne}\,$]{}, [$\rm\,{}^{21}\kern-0.8pt{Ne}\,({^{4}He}\,,{n})
\,{}^{24}\kern-0.8pt{Mg}\,$]{}, [$\rm\,{}^{22}\kern-0.8pt{Ne}\,({^{4}He}\,,{n})
\,{}^{25}\kern-0.8pt{Mg}\,$]{}, and [$\rm\,{}^{25}\kern-0.8pt{Mg}\,({^{4}He}\,,{n})
\,{}^{28}\kern-0.8pt{Si}\,$]{}.
At time $t=0$, just before the development of a TP, the chemical composition of the region over which the flash-driven convection will extend, is assumed to be stratified over three zones:
a) $M_{\rm Pt}- M_{\rm Pt}^{\prime}$ containing the ashes, with abundances $\{X_{\rm Hb}\}$, left by the quiescent radiative H-shell over the previous interpulse period;
b) $M_{\rm Pt}^{\prime} - M_{\rm He}$ containing the nuclear products of the PDCZ developed during the [*previous*]{} TP;
c) $M_{\rm He}- M_{\rm Pb}$ containing the products of radiative He burning.
For simplicity each of the three zones is assigned an average chemical composition, though a chemical profile exists in the a) and c) regions where nuclear burning has occurred in radiative conditions.
Denoting with $X^{\rm s}$ the homogeneous surface abundances, the composition of the hydrogen free layer left by the H-burning shell is estimated following the indications by @Mowlavi_99a [@Mowlavi_99b], which can be summarised as follows:
- all hydrogen is burnt into helium: $X_{\rm Hb}({\rm H})=0$;
- all available CNO isotopes are converted into $^{14}$N: $X_{\rm Hb}(^{14}{\rm N})=14 \times \sum_{i=12}^{i=18}
X_{i}^{\rm s}/A_i$ (where $A_i$ is the mass number);
- all $^{22}$Ne is burnt into $^{23}$Na by the NeNa chain: $X_{\rm Hb}(^{22}{\rm Ne})=0$;
- the abundance of $^{23}{\rm Na}$ is computed with:\
$X_{\rm Hb}(^{23}{\rm Na})=
f_{\rm Na}[23/22\times X^{\rm s}(^{22}{\rm Ne})+ X^{\rm s}(^{23}{\rm Na})]$.
The factor $f_{\rm Na}$ accounts for the possible destruction of $^{23}$Na by proton captures at $T > 6\times10^7$ K. Its value typically ranges from $f_{\rm Na}=1$ (no destruction) down to $f_{\rm Na}=0.2$ [see figure A.3 in @Mowlavi_99a]. For the present set of calculations we have adopted $f_{\rm Na}=1$. The effects of the Mg-Al chain on the resulting $X_{\rm Hb}$ abundances is not considered in this work, and it will be implemented in a future study.
During each TP we follow the progressive development of pulse convection and related nucleosynthesis, over the duration $\tau_{\rm pdcz}$. The process is divided into two consecutive phases:
I. from the onset of the PDCZ at time $t=0$ up to maximum extension at time $t=\tau_{\rm pdcz}-\tau_{\rm q}$;
II. from maximum PDCZ extension to final pulse quenching at time $t=\tau_{\rm pdcz}$, with duration $\tau_{\rm q}$.
The PDCZ is resolved both in time and in space. The entire duration $\tau_{\rm pdcz}$ is subdivided in typically $\simeq 100$ time steps, while at each time a suitable grid of mass meshes is set up across the current PDCZ, with a maximum mass resolution of $\simeq 10^{-4}\,M_{\odot}$. The evolution of $T_{\rm pdcz}^{\rm max}$ and $T_{\rm rho}^{\rm max}$ over $\tau_{\rm pdcz}$, and the temperature and density stratifications across the PDCZ mass are described on the basis of detailed calculations of thermal pulses [@Wagenhuber_96; @WagenGroen_98 and private communications]. Illustrative examples are discussed later, in Sect. \[ssec\_xpdcz\].
During the phase I the evolution of the PDCZ is followed by cycling over the sequence of steps: nucleosynthesis $\rightarrow$ homogenization $\rightarrow$ expansion/recession$\rightarrow$ homogenization. At each time step, starting from the current PDCZ bottom (with mass coordinate $m_{\rm Pb}$) up to the current PDCZ top border (with mass coordinate $m_{\rm Pt}$) the nuclear network is solved locally in each mesh point.
A homogeneous chemical composition is assigned to the PDCZ by mass-averaging the mesh abundances. Then, the PDCZ is made expand i.e. inner/upper borders of the PDCZ are shifted inward/outward, and elements of new material, stratified according to the initial composition, are engulfed. Eventually, a new PDCZ composition is obtained by averaging the abundances with weights proportional to the masses of the corresponding meshes.
The entire process, i.e. convective burning followed by expansion and homogenization, is iterated until the maximum extension is reached, i.e. $m_ {\rm Pb}= M_{\rm Pb}$ and $m_{\rm Pt} = M_{\rm Pt}$, and the mass contained in the PDCZ is equal to $\Delta M_{\rm pdcz}$. At this point $t=\tau_{\rm pdcz}-\tau_{\rm q}$.
The quenching phase II is described by a similar scheme, except that now the PDCZ convection retreats and the inner/upper borders are shifted outward/inward until $t=\tau_{\rm pdcz}$. The nuclear network is integrated over the pulse quenching phase and a final homogeneous chemical composition is obtained. This sets the chemical mixture of the material that may be brought up to the surface by the subsequent third dredge-up phase.
Despite its simplicity the PDCZ model yields results that nicely agree with those of full TP-AGB computations. A detailed discussion of the predictions and their main dependencies is given in Sect. \[ssec\_xpdcz\].
The synthetic module {#sec_synthmod}
====================
Most analytical ingredients of the `COLIBRI` code are formulas accurately fitting the results of full AGB models covering wide ranges of initial stellar mass and metallicity. The formulas are taken either from the extensive compilations by @Wagenhuber_96 [@WagenGroen_98; @Karakas_etal02; @Izzard_etal04; @Izzard_etal06], and other sources [@Straniero_etal03], or they are directly derived from AGB model data sets by using standard $\chi^2$-minimization techniques. New fits can be found in Appendix \[app\_fit\].
Importantly, all these analytic relations include a metallicity dependence, and take into account the peculiar behaviour of the first sub-luminous pulses while approaching the full-amplitude regime.
Among the most important prescriptions we mention the flash-driven luminosity variations as a function of the pulse-cycle phase [@WagenGroen_98], the core mass-interpulse period relation [@WagenGroen_98], the maximum mass of the PDCZ and its duration, the maximum temperature attained at the bottom of the PDCZ during a TP [@KarakasLattanzio_07][^7], the efficiency $\lambda$ of the third dredge-up [@Karakas_etal02].
Due to their particular relevance, below we will discuss in more detail a few analytic relations adopted in the present version of `COLIBRI`.
The third dredge-up: the need for a parametric description {#ssec_3dup}
----------------------------------------------------------
It is common practice describing the third dredge-up by means of two characteristic quantities, namely:
- $M_{\rm c}^{\rm min}$: the minimum core mass for the onset of the third dredge-up;
- $\lambda= \displaystyle\frac{\Delta M_{\rm dup}}{\Delta M_{{\rm c, tpc}}}$: the efficiency of the third dredge-up, defined as the fraction of the core-mass growth over the interpulse period that is dredged-up to the surface at the next TP.
Compared to earlier computations, recent full TP-AGB evolutionary models have allowed a wide exploration of the third dredge-up characteristics as a function of stellar mass and metallicity [e.g. @Karakas_etal02; @Herwig_00; @Herwig_04a; @Herwig_04b; @WeissFerguson_09; @Cristallo_etal11]. A few general trends can be extracted from these calculations.
The efficiency $\lambda$ is expected to increase with stellar mass $M$, such that TP-AGB stars with initial masses $M > 3\,M_{\odot}$ are predicted to reach $\lambda \simeq 1$, which implies no, or very little, core mass growth. Lower metallicities favour an earlier onset of the third dredge-up and a larger efficiency, resulting in an easier formation of low-mass carbon stars. Full TP-AGB models exist which are found to reproduce, or at least to be reasonably consistent with, basic observables, such as the luminosity functions of carbon stars in the Magellanic Clouds [e.g. @Stancliffe_etal05; @WeissFerguson_09; @Cristallo_etal11].
Together with these improvements, present TP-AGB models also document that the third dredge-up is plagued by severe theoretical uncertainties. They are due mainly to our still deficient knowledge of convection and mixing, as well to a nasty sensitivity of the depth of the third dredge-up to technical and numerical details (see @FrostLattanzio_96, and @Mowlavi_99b for thorough analyses).
As a consequence we still lack a robust assessment for $M_{\rm c}^{\rm min}$ and $\lambda$, and these parameters are found to vary considerably from author to author even for the same combination ($M_{\rm i}$,$Z_{\rm i}$) of initial stellar mass and metallicity. The theoretical dispersion is exemplified in Fig. \[fig\_3dup3z02\]. The dynamical ranges of the parameters covered by the various sets of computations are large, amounting to almost a factor of 3 for the maximum $\lambda$ attained in a ($M_{\rm i}=3.0 \, M_{\odot}$, $Z_{\rm i}=0.02$) model, and more than $\simeq 0.1\, M_{\odot}$ for $M_{\rm c}^{\rm min}$ for the ($M_{\rm i}=2.0 \, M_{\odot}$, $Z_{\rm i}=0.008$) case. It is clear that these variations propagate dramatically in terms of the predicted stellar properties: significant differences are expected in the luminosities spanned during the C star phase, the final masses, the chemical yields, etc. The situation appears even more unclear considering, for instance, that two independent sets of calculations, i.e. @Stancliffe_etal05 and @WeissFerguson_09, with largely different predictions for $M_{\rm c}^{\rm min}$ (see the right-hand side panel of Fig. \[fig\_3dup3z02\]) are found by the authors to recover the same observable, i.e. the carbon star luminosity function in the LMC. This uncomfortable convergence of the results is likely due to the combination of other critical parameters (e.g. efficiency $\lambda$, and mass loss). In fact, it is differences in details of the chosen input physics, such as the treatment of convective boundaries and the inclusion or not of overshoot, that produces most of the variations seen in full models, such as those shown in Fig. \[fig\_3dup3z02\].
All these reasons amply justify the approach of taking $\lambda$ and $M_{\rm c}^{\rm min}$ (or, in alternative, $\lambda$ and the temperature parameter $T_{\rm dup}$; see Sect. \[ssec\_tbdred\]), as free parameters, and to calibrate them with the largest possible set of observations to reduce the likely degeneracy between different factors.
Properties of the pulse-driven convection zone
----------------------------------------------
In Fig. \[fig\_pdczfit\] we show three key quantities of the PDCZ as a function of the core mass (starred symbols), as predicted by @Karakas_etal02 [@KarakasLattanzio_07] for five values of the initial metallicity $(Z_{\rm i}=0.0001,\,Z_{\rm i}=0.004,\,Z_{\rm i}=0.008,\,Z_{\rm i}=0.012,\,{\rm and}\,Z_{\rm i}=0.02)$. Superimposed we plot the results obtained with the analytic relations (grey triangles) for the same stellar parameters ($M_{\rm i}$, $M_{\rm c}$, and $Z_{\rm i}$) as in the original full computations, The fitting relations behave well all over the core-mass range covered by the full models. The formulas and their coefficients are given in Appendix \[app\_fit\].
For comparison we draw two more relations taken from literature, namely @IbenTruran_78 [black line] and @Straniero_etal03 [magenta solid line]. We have extrapolated the @IbenTruran_78 relations over the whole $M_{\rm c}$ range, but one should consider that they were originally derived from the high core mass $(0.96 \la M_{\rm c}/M_{\odot} \la 1.33)$ AGB models of @Iben_77. We see that for $M_{\rm c} \ga 0.85\, M_{\odot}$ the @IbenTruran_78 relations for $\Delta M_{\rm pdcz}$ and $\tau_{\rm pdcz}$ are in general agreement with the average trend predicted by the recent AGB computations of @KarakasLattanzio_07. The earlier results of @Iben_77 for $T_{\rm pdcz}^{\rm max}$ are systematically lower by up to $0.6-0.8$ dex.
The other relation proposed by @Straniero_etal03, on the basis of their full AGB calculations, appears to be consistent with the @KarakasLattanzio_07 results inside its validity range, (i.e. $0.6 \la M_{\rm c}/M_{\odot} \la 0.7$). However, we notice that it does not allow to describe the initial rise of the temperature typical of the first pulses.
Tests: COLIBRI vs full stellar models {#sec_tests}
=====================================
Effective temperature and convective-base temperature {#ssec_teftest}
-----------------------------------------------------
As a first test we compare the effective temperatures obtained with `COLIBRI` from envelope integrations (the method is outlined in Sect. \[ssec\_envmod\]), against the predictions of full stellar models computed with `PARSEC` [@Bressan_etal12]. A detailed discussion is given in Appendix \[sec\_dtests\].
Figure \[fig\_dtefzvar\] quantifies the comparison in relation to the quiescent stage just preceding the occurrence of the $1^{\rm st}$ thermal pulse for several values of stellar masses and metallicities. We see that the differences are in most cases quite low, amounting to few tens of degrees, well below the typical observational errors for $T_{\rm eff}$ of AGB stars, about equal to $\pm (100-200)$ K.
The results shown in the two panels of Fig. \[fig\_dtefzvar\] differ in the chemical distributions of metals assumed in `COLIBRI`. They are usually expressed in terms of the ratios $X_i/Z$, where $X_i$ denotes the fractional mass of a given metal $i$. While in one case (top panel) both EoS and opacities are computed with the `ÆSOPUS` and [*Opacity Project*]{} codes adopting, for each model, the actual set of surface abundances predicted by `PARSEC` at the $1^{\rm st}$ TP, in the other case (bottom panel) the mixtures are assumed to be all scaled-solar for any metallicity, i.e. $X_i/Z=X_{i,\odot}/Z_{\odot}$ for each metal $i$.
In principle, the former case is the correct one as it couples consistently EoS and opacities with the current metal abundances, that may have varied with respect to the values at the zero-age main sequence, following the $1^{\rm st}$ and second dredge-up processes. On the other hand, the latter case, which is also adopted in the `PARSEC` models and, more generally, by most full stellar codes, neglects the variation of the elemental ratios, e.g. the lowering of the C/O, due to mixing episodes prior to the TP-AGB phase.
It follows the accuracy degree of `COLIBRI` against `PARSEC ` is best represented by the temperature differences in the bottom panel of Fig. \[fig\_dtefzvar\], since the same metal ratios, $X_i/Z=X_{i,\odot}/Z_{\odot}$, are assumed in both sets of computations. In fact, passing from the top to the bottom panel of Fig. \[fig\_dtefzvar\] it is evident that the agreement between the `COLIBRI` and `PARSEC` predictions improves, particularly for models of larger masses which are most affected by the second dredge-up. A more detailed discussion of this aspect and other related effects can be found in Appendix \[ssec\_teff\].
The temperature at the base of the convective envelope, $T_{\rm bce}$, provides an additional test for our envelope-integration method, and it is particularly relevant for massive AGB models ($M > 4\,M_{\odot}$) as it measures the efficiency of HBB. As analysed in Appendix \[ssec\_tbot\], the results are affected by several technical details not dealing with the envelope integration method, such as differences in the operative definition of the convective border, inclusion or not of convective overshooting, assumed metal partitions, adopted equation of state, high-temperature opacities, etc. All these aspects, together with the fact that the base of the convective envelope may fall inside a region characterized by an extremely steep temperature gradient, concur to somewhat amplify the differences in $T_{\rm bce}$.
Figure \[fig\_dtbot\] shows the temperature differences between `COLIBRI` and `PARSEC ` predictions for initial masses $M_{\rm i} \ge 2.6\, M_{\odot}$ and various metallicities. Two cases are considered in the `COLIBRI` definition of the innermost stable mesh-point of the convective envelope, namely: the strict application of the Schwarzschild criterion (top panel), and the inclusion of convective overshoot by the same amount as adopted in `PARSEC` (bottom panel). In both cases the differences remain fairly small, i.e. $|\log(T_{\rm bce}^{\rm full}) - \log(T_{\rm eff}^{\rm env})| < 0.05$ dex.
In conclusion our tests indicate that:
- the agreement in effective temperatures between our envelope integrations and full stellar modelling is extremely good, with differences $|T_{\rm eff}^{\rm env} - T_{\rm eff}^{\rm full} |< 40$ K and in many cases practically negligible;
- the differences $T_{\rm eff}^{\rm env} - T_{\rm eff}^{\rm full}$ are always negative and tend to systematically decrease at lower metallicity, suggesting that they are likely related to the elemental abundances and the way they are treated in the EoS and opacity computations. Indeed cooler $T_{\rm eff}^{\rm env}$ compared to $T_{\rm eff}^{\rm full}$ are partly explained by the differences in the assumed $X_i/Z$ used in the EoS and opacities, i.e. actual chemical abundances in `COLIBRI` against frozen scaled-solar ratios adopted by `PARSEC`.
- A very good agreement is found also for $T_{\rm bce}$ (within $0.05$ dex), which strongly supports the ability of our envelope-integration method to account correctly for the occurrence of HBB in more massive AGB models.
Quiescent luminosity on the TP-AGB {#ssec_lum}
----------------------------------
Thanks to the extension of the [*deep envelope*]{} model to include the H-burning shell, we can predict the luminosity during the quiescent stages without adopting any auxiliary CMLR, as usually done in synthetic TP-AGB models .
Figure \[fig\_cmlr\] shows the pre-flash luminosity as a function of the core mass for two sets of TP-AGB models with initial metallicity $Z_{\rm i}=0.008$ and a few values of the initial stellar mass, computed by @Karakas_etal02, and with the `COLIBRI` envelope-integration technique adopting the same stellar parameters (e.g. total mass, core mass, dredged-up mass, mixing-length parameter, and initial metallicity). Considering that the two sets of calculations differ both in technical details (e.g. solution method of the stellar structure equations, zone-meshing, etc.) and in the input physics (e.g. EoS, opacities, nuclear reaction rates, etc.) the overall agreement is quite striking. We derive two main implications: i) in absence of HBB, i.e. for TP-AGB models with smaller cores ($M_{\rm c} \la 0.75\,M_{\odot}$) and less massive envelopes ($M_{\rm env} \la 2.5\,M_{\odot}$), the CMLR is a robust prediction of the theory (essentially reflecting the thermostatic character of the H-burning shell), ii) in our [*deep envelope*]{} integrations the treatment of the H-burning energetics is reliable.
In fact, in the range $0.5\, M_{\odot} \la M_{\rm c} \la 0.7\, M_{\odot}$ our predictions for the pre-flash luminosity maximum recover the @Karakas_etal02 results remarkably well, and more generally the classical CMLRs [e.g. @BoothroydSackmann_88a red line]. The brightening of the tracks beyond the CMLR, as shown by @Karakas_etal02 models with $M_{\rm c} \la 0.75\, M_{\odot}$ and $M \la 3.5\, M_{\odot}$, is driven by the occurrence of a deep third dredge-up. This effect is discussed in Sect. \[sssec\_lum3dup\].
At larger core masses, $M_{\rm c} \ga 0.75\,M_{\odot}$ (see the models with initial masses $M_{\rm i} = 4,\,5,\,6\, M_{\odot}$ in Fig. \[fig\_cmlr\]), HBB is expected to produce the break-down of the CMLR: similarly to the tracks by @Karakas_etal02, the `COLIBRI` sequences with $M \ge 4\,M_{\odot}$ exhibit a steep luminosity increase at almost constant core mass ($\lambda \simeq 1$ in these models). After reaching a maximum, the luminosity starts to decline quickly from pulse to pulse until the CMLR is recovered again. The luminosity peak and the subsequent decrease are controlled by the onset of the super-wind phase, which determines a rapid reduction of the envelope mass, hence the weakening and eventual extinction of HBB. We note that the `COLIBRI` tracks with HBB reach higher luminosity maxima than the @Karakas_etal02 models with the same initial masses, a circumstance that confirms the sensitivity of the HBB process on the adopted input physics and details of the convection treatment [@Ventura_etal05].
### The effect of deep third dredge-up {#sssec_lum3dup}
Full AGB calculation indicate that the occurrence of deep dredge-up events make the models brighter than expected by the CMLR [@Herwig_etal98; @Mowlavi_99b; @Karakas_etal02], due to the intervening non-linear relation between the core mass and the core radius.
To account for this effect we have analysed a large number of full TP-AGB models from @Karakas_etal02. These models are characterised by a large range of dredge-up efficiencies, from $\lambda \approx 0$ to $\lambda \approx 1$, depending on stellar mass and metallicity.
We find that, in presence of dredge-up, the quiescent pre-flash luminosity $L_{\rm Q}$ of a TP-AGB model with a core mass $M_{\rm c}$ is well recovered with our envelope-integration method by applying the boundary condition for the core temperature (Eq. \[c4\]) in the form $T_{\rm c}=T(M_ {\rm c}^{\rm fict})$, where we introduce a fictitious core mass $$M_ {\rm c}^{\rm fict} = M_{\rm c} + \xi\,(M_{\rm c,nodup} - M_{\rm c, 1})\,,
\label{eq_mcfic}$$ with the multiplicative factor $\xi \simeq 0.3-0.4$.
The variable $M_{\rm c, nodup}$ has been already used in past synthetic TP-AGB models [e.g. @Hurley_etal00; @Izzard_etal04; @Izzard_etal06]. It was introduced to account for effects due to an increase in core degeneracy during the quiescent interpulse growth, so that stars with the same core mass, but different dredge-up histories, may have different quiescent luminosities. Since in `COLIBRI` the integrations of stellar structure are performed down to the bottom of the H-burning shell, for the electron-degenerate core beneath it we need to resort to a parametrized description. The variable $M_{\rm c, nodup}$ is a suitable choice for the case under consideration.
The results are illustrated in Fig. \[fig\_cmlr3dup\], where the `COLIBRI` tracks computed with Eq. (\[eq\_mcfic\]) setting $\xi = 0.3$ (right-hand side panel) are compared to the original sequences @Karakas_etal02 (left-hand side panel). Despite the simple formulation of the corrective term in Eq. (\[eq\_mcfic\]), the agreement is quite satisfactory.
It is also instructive to look at the middle panel of Fig. \[fig\_cmlr3dup\] showing the `COLIBRI` predictions for $\xi=0$, i.e. without the effect of the third dredge-up. In this case all the tracks comply with the classical CMLR by @BoothroydSackmann_88a, and reproduce quite well the dimming of the quiescent luminosity at decreasing metallicity. As a matter of fact, the TP-AGB models from which @BoothroydSackmann_88a derived their analytic CMLR were characterised by rather shallow, in most cases absent, convective dredge-up events, and were mostly limited to the first few thermal pulses. This fact explains why the over-luminosity effect due to the third dredge-up does not show up in the @BoothroydSackmann_88a models.
It follows that the very nice accordance between the CMLR of @BoothroydSackmann_88a and the `COLIBRI` predictions for $\xi=0$ adds a further confirmation on the validity of our envelope-integration method in terms of the H-burning energetics (see also Sect. \[ssec\_lum\]).
Computational agility {#ssec_cpu}
---------------------
A key feature of the `COLIBRI` code is the computational agility, that is kept to competitive levels despite the several numerical operations performed at each time step, i.e. iterative solution of the atmosphere and envelope structures, integration of nuclear networks, [*on-the-fly*]{} computation of the EoS and Rosseland mean opacities across all meshes.
Figure \[fig\_cpu\] compares the performance of the `COLIBRI` and the `PARSEC` codes, in terms of the typical CPU time required to compute one thermal pulse cycle, i.e. the time interval between two consecutive pre-flash luminosity maxima. The two histograms correspond to the distributions of $N^{\rm tot}_{\rm tpc}= 507$ thermal pulse cycles followed over a wide range of initial stellar masses $(0.6 M_{\odot}\, \la M_{\rm i} \la 6 M_{\odot})$, and metallicities $(0.0005 \le Z_{\rm i} \le 0.07)$.
The difference in CPU time[^8] requirements is noticeable. The `COLIBRI` distribution shows a broad peak at $ \tau_{\rm tpc}\sim 30-40$ s, and a low tail extending down to $3-4$ s. The median of the distribution is $\tilde{\tau}_{\rm tpc}\simeq 14$ s. Bins at longer $\tau_{\rm tpc}$ are populated by TP cycles referring to i) the last TP-AGB stages in which the high mass-loss rates impose the reduction of the evolutionary time steps, and ii) more massive AGB stars experiencing both the third dredge-up and HBB, with consequent intensive computing of EoS and opacities to follow the continuous changes in the envelope chemical abundances.
The `PARSEC` distribution is located over much longer time scales, with $\tau_{\rm tpc}$ ranging from $\approx 10$ min to $\approx 200$ min. The median of the `PARSEC` distribution is $\tilde{\tau}_{\rm tpc} \simeq 29$ min.
In any case, the gain in terms of CPU time with `COLIBRI` is sizable: the integrated CPU time to compute $N^{\rm tot}_{\rm tpc}= 507$ thermal pulse cycles is roughly $4$ hours for `COLIBRI` and $\simeq 21$ days for `PARSEC`.
While we acknowledge that the continuing increase in computing speed of modern computers enables present-day full evolution codes to compute extended grids of TP-AGB tracks, we should also realize that performing a multi-parametric fine calibration of the uncertain processes/assumptions is extremely more demanding in terms of computational agility and numerical stability, characteristics that do not ordinarily apply to the full approach.
Processes and assumptions that are known to dramatically affect the TP-AGB evolutionary phase are, for instance, mass loss, third dredge-up, nucleosynthesis, convection efficiency, overshooting, initial chemical abundances, etc. For each of them, we could single out more than one characteristic parameter, depending on the theoretical picture one aims investigating at. A dozen parameters may represent a reasonable estimate of the number of factors one should take into consideration for an extensive analysis.
To get an order of magnitude of the time requirements, let us consider our specific working case. At present we are dealing with $14$ metallicity sets (limited to the scaled-solar compositions, other sets are planned), from very low to super-solar $Z$. From the `PARSEC` database of models, we extract the initial conditions at the first TP for $65-70$ values of the initial stellar mass (on average), from $\simeq 0.5 \, M_{\odot}$ to $\simeq 5-6 \, M_{\odot}$. The fine grid in mass is important to allow for the construction of accurate and detailed stellar isochrones.
The total number of TP-AGB tracks to be calculated is $951$. With the set of parameters adopted in this exploratory work, all the TP-AGB tracks followed by `COLIBRI` cover $14293$ thermal pulse cycles, for a true CPU time of $7854\,{\rm s} \simeq 5.2$ days.
With the conservative assumption that the `PARSEC` code takes a computing time $\approx 100$ longer (probably more), the whole TP-AGB tracks would be ready after $\simeq 520$ days, that is $\approx 1.5$ yr. These are likely optimistic estimates, considering that the current `PARSEC` distribution of CPU times is biased towards shorter values since, in general, each evolutionary track includes the first few TPs, that usually involve a lighter computational effort compared to the later, well-developed TPs. Moreover, the PARSEC tracks are calculated at constant mass, while the inclusion of a mass-loss prescription would certainly impose a further reduction of the time steps, hence an increase of the CPU time.
It is also worth noting that the computing time request is expected to increase with the stellar mass, given that the pace at which TPs take place correlates with the core mass, while HBB gets stronger. In a recent study @Siess_10 reported that $\approx 6$ months of CPU time were required by his full evolution code to follow the whole Super-AGB phase of just one model with strong HBB. Of course, this may not be the same for other full codes, but a trend of increasing computational cost with the stellar mass is of general validity.
In any case, we emphasize here that what makes the computational effort particularly challenging for full models is the calibration process. In fact, promptly producing extended and dense (in mass and metallicity) sets of TP-AGB tracks is a necessary requisite to build accurate stellar isochrones spanning the whole relevant ranges of ages and metallicities. In turn, the stellar isochrones are the building blocks of population synthesis simulations of galaxies including AGB stars, which can be readily put in direct comparison with observations. Possible discrepancies between predictions and observed data will bring the work-flow back to the theoretical side, and new sets of TP-AGB tracks with a different set of input assumptions should be put in execution. This calibration cycle may be repeated several times before a satisfactory match between models and observations is attained.
Even before starting the calibration loop, in this preliminary and exploratory phase, we have already computed ten complete grids, for a total of $9510$ TP-AGB tracks, each time changing a technical/physical parameter (e.g. an efficiency mass-loss parameter, the mass meshing, the time-step regulation, or a subset of nuclear reaction rates). It seems realistic that many more iterations, maybe hundreds, are necessary for an adequate global calibration. As a consequence, numerical stability and computational agility are essential conditions, both fully met by our `COLIBRI` code.
Evolutionary tracks {#sec_tracks}
===================
We consider $14$ sets of stellar tracks covering a wide range of the initial metallicity, namely for $Z_{\rm i}=$0.0001, 0.0005, 0.001, 0.004, 0.006, 0.008, 0.01, 0.014, 0.017, 0.02, 0.03, 0.04, 0.05, and 0.06 with initial scaled-solar abundances of metals. The reference solar mixture is that recently revised by @Caffau_etal11, corresponding to a Sun’s metallicity $Z\!\simeq\!0.0152$.
Up to the onset of the TP-AGB
-----------------------------
The evolution prior to the TP-AGB phase, from the pre-main sequence to the occurrence of the first TPs, is computed at constant mass with the `PARSEC` code, as described in the paper by @Bressan_etal12 to which we refer for all details. We recall here only a few relevant points. For each value of $Z_{\rm i}$, the initial helium abundance is determined by the $Y_{\rm i}=0.2845+1.78\,Z_{\rm i}$ enrichment law. The energy transport in the convective regions is described according to the mixing-length theory of @mlt_58. The mixing length parameter $\alpha_{\rm MLT}$ is fixed by means of the solar model calibration, and turns out to be $\alpha_{\rm MLT}=1.74$. The PARSEC tracks include overshoot applied to the borders of both convective cores and envelopes, with overshooting scales that vary with the stellar mass as described in @Bressan_etal12. Envelope overshoot is discussed also in Sects. \[ssec\_teftest\] and \[ssec\_tbot\], in relation to the accuracy checks performed on `COLIBRI` results.
For each `PARSEC` set of stellar tracks of given ($Z_{\rm i},Y_{\rm i}$) combination, we extract the initial conditions at the $1^{\rm st}$ TP for all the values of the initial stellar mass in the grid, ranging from $\simeq 0.5\, M_{\odot} $ to $M_{\rm up}$, the latter being the maximum mass for a star to develop an electron-degenerate C-O core. We deal typically with $60-70$ low- and intermediate-mass tracks for each initial chemical composition.
The core mass at $1^{\rm st}$ thermal pulse, $M_{\rm c,1}$, fixes a lower limit to the mass of the remnant white dwarf, and it is closely connected to the initial-final mass relation.
Figure \[fig\_mc1\] shows the `PARSEC` predictions for $M_{\rm c,1}$, as a function of the stellar mass for several choices of the initial metallicity. The stellar mass, $M_1$, is the value at the onset of the TP-AGB phase, so that, in principle, one should correct for the amount of mass lost by low-mass stars $(M_{\rm i} \la 2\, M_{\odot})$ during the red giant branch (RGB) phase in order to translate the $M_{\rm c,1}$ relation as a function of the initial stellar mass.
Two are the main features common to all the curves, namely: i) the almost constancy of $M_{\rm c,1}$ for stellar masses lower than $1.6-2.0\, M_{\odot}$ (depending on $Z_{\rm i}$), which simply reflects the fact that these stars develop He-cores of very similar mass due to the electron degeneracy after the main sequence; ii) the change of slope at stellar masses in the range $2.5-3.5\, M_{\odot}$ (depending on $Z_{\rm i}$) and the subsequent flattening of the $M_{\rm c,1}$ relations. This is the fingerprint of the occurrence of the second dredge-up during the Early-AGB of intermediate-mass stars, that causes a significant reduction of their core masses.
In Table \[tab\_mc1\] of Appendix \[app\_fit\] we present the fitting coefficients that we derive following the parametrization proposed by @WagenGroen_98, for several metallicities. In each panel of Fig. \[fig\_mc1\] the fitting curves are over-imposed to the `PARSEC` data for $M_{\rm c,1}$. We note, however, that our TP-AGB calculations use the true $M_{\rm c,1}$ values, and not those derived from the formulas.
TP-AGB evolution {#ssec_tpagbev}
----------------
For each stellar model with initial parameters $(M_{\rm i},Z_{\rm i})$ the characteristic quantities at the $1^{\rm st}$ thermal pulse (core mass, luminosity, effective temperature, envelope chemical composition), obtained from the [*PARSEC*]{} database, are fed as initial conditions to the `COLIBRI` code, which computes the TP-AGB evolution until when almost the entire envelope is lost by stellar winds. Operatively the `COLIBRI` calculations are stopped when the mass of the residual envelope falls below a limit of $0.002 M_{\odot} -0.005 M_{\odot}$. At this stage all evolutionary tracks are already evolving off the AGB towards higher effective temperatures, with a luminosity that depends mainly on the mass of the C-O core, and the phase of the pulse cycle at which the last event of mass ejection took place (see Fig \[fig\_hr\]).
For the present work we adopt a specific set of prescriptions for the mass loss and the third dredge-up, which we briefly outline below. These models will serve as a reference case for our ongoing TP-AGB calibration, and therefore the current parameters may be somewhat changed in future calculations. Anyhow, from various preliminary tests made with the present models, we expect that they already yield a fairly good description of the TP-AGB phase.
#### Mass loss. {#mass-loss. .unnumbered}
It has been included under the hypothesis that it is driven by two main mechanisms, dominating at different stages. Initially, before radiation pressure on dust grains becomes the main agent of stellar winds, mass loss is described with the semi-empirical relation by @SchroderCuntz_05, which essentially assumes that the stellar wind originates from magneto-acoustic waves operating below the stellar chromosphere. The corresponding mass-loss rates are indicated with $\dot M_{\rm pre-dust}$.
Later on the AGB the star enters the dust-driven wind regime, which is treated with an approach similar to that developed by @Bedijn_88, and recently adopted by @Girardi_etal10, to which the reader is referred for all details. Briefly, assuming that the wind mechanism is the combined effect of two processes, i.e., radial pulsation and radiation pressure on the dust grains in the outermost atmospheric layers, we adopt a formalism for the mass-loss rate as a function of basic stellar parameters, mass $M$ and radius $R$, expressed in the form $\dot M \propto e^{M^{a} R^{b}}$. The free parameters $a$ and $b$ have been calibrated on a sample of Galactic long-period variables with measured mass-loss rates, pulsation periods, stellar masses, radii, and effective temperatures. More details about the fit procedure will be given elsewhere. We denote the corresponding mass-loss rates with $\dot M_{\rm dust}$.
The key feature of this formalism is that it predicts an exponential increase of the mass-loss rates as the evolution proceeds along the TP-AGB, until typical super-wind values, around $10^{-5}-10^{-4}\, M_{\odot} {\rm yr}^{-1}$, are eventually reached. The super-wind mass loss is described in the same fashion as in @VassiliadisWood_93, and corresponds to a radiation-driven wind, $\dot M_{\rm sw}=L/c\, v_{\rm exp}$, where $c$ is the speed of light and $v_{\rm exp}$ is the terminal velocity of the wind.
At any time during the TP-AGB calculations the actual mass-loss rate is taken as $$\dot M =
{\rm max}[\dot M_{\rm pre-dust}, {\rm min}(\dot M_{\rm dust},
\dot M_{\rm sw})].$$
#### The third dredge-up. {#the-third-dredge-up. .unnumbered}
The onset of the third dredge-up is predicted according to the scheme described in Sect. \[ssec\_tbdred\]. The minimum temperature parameter is set to $\log(T_{\rm dup})=6.4$. This rather low value favours an early occurrence of the third dredge-up episodes. The efficiency $\lambda$ of the third dredge-up is computed with the analytic fits provided by @Karakas_etal02, as a function of current stellar mass and metallicity.
Figure \[fig\_hr\] illustrates a few selected evolutionary tracks of low- and intermediate-mass stars, zooming in their brightest portions in the H-R diagram, that include the whole TP-AGB computed with `COLIBRI` and some earlier evolution calculated with `PARSEC`. The transition from `PARSEC` to `COLIBRI` is not even distinguishable in most cases, except for the higher mass models with HBB ($M_{\rm i}=5.0 M_{\odot},\,Z_{\rm i}=0.001$ and $M_{\rm i}=5.8 M_{\odot},\,Z_{\rm i}=0.01$) for which `COLIBRI` predicts somewhat cooler effective temperature at the $1^{\rm st}$ TP compared to `PARSEC`. This difference has been discussed in Sect. \[ssec\_teftest\], and can be partly explained in terms of the small differences in molecular opacities adopted by the two codes (see Fig. \[fig\_dtefzvar\]).
We also note in Fig. \[fig\_hr\] that low-mass models ($M_{\rm i}=0.6 M_{\odot},\,1.0 M_{\odot}$) are characterised by quite narrow TP-AGB tracks since at given metallicity, as long as the surface C/O$<1$, the effective temperature is mostly determined by stellar mass and luminosity. Differently, models with larger masses ($M_{\rm i}=2.0 M_{\odot},\,3.0 M_{\odot}$), which are expected to undergo the transition to carbon stars, exhibit a pronounced displacement towards lower effective temperatures, mainly driven by the increase in molecular opacities. Finally, models with the highest masses ($M_{\rm i}=5.0 M_{\odot},\,5.8 M_{\odot}$) present TP-AGB tracks with the typical bell-shape modulated by the occurrence of HBB, and with the peak in luminosity reached when the envelope mass starts being drastically reduced by stellar winds. These considerations apply in general to both metallicity cases here considered ($Z_{\rm i}=0.001$ and $Z_{\rm i}=0.01$), with some systematic differences, i.e. lower effective temperatures are expected at higher metallicities, again due to surface opacity effects. A more detailed analysis of this aspect is given in Sect. \[ssec\_hayashi\].
Finally, we note that in our TP-AGB calculations no particular convergence problem was met all the way to the complete ejection of the envelope, whereas other studies, based on full TP-AGB calculations, report the divergence of the models in the late stages of evolution [e.g., @WoodFaulkner_86; @WagenhuberWeiss_94; @Lau_etal12]. In the latter paper the authors suggest that the cause of the instability in the most massive TP-AGB models may be related to a local opacity maximum of Fe at the base of the convective envelope. At present we cannot identify the reason for the different behaviour of `COLIBRI`, this delicate point will deserve a closer look in follow-up studies.
Overview and analysis of the COLIBRI predictions {#sec_results}
================================================
In the following we will discuss some relevant predictions of the `COLIBRI` code, with the aim of understanding a few key dependencies of the various physical processes at work and their complex interplay, as well as giving a general overview of the `COLIBRI` predictive capability.
Molecular concentrations at the photosphere
-------------------------------------------
The [*on-the-fly*]{} use of the `ÆSOPUS` code during the TP-AGB calculations enables us to predict, for the first time, the evolution of the abundances of $\simeq 500$ molecular species in the outermost layers of the envelope. In Fig. \[fig\_molchem\] (left-hand side panel) we show the results at the photosphere of a $M_{\rm i}=2\,M_{\odot}$, $Z_{\rm i}=0.008$ model. We see clearly how the occurrence of thermal pulses produces large variations of the photospheric temperature and density (top panel), which in turn cause similar “pulses” in the concentrations of the molecules.
As amply discussed in Sects. \[ssec\_eos\] and \[ssec\_opac\] the other critical factor determining the molecular chemistry is the surface C/O ratio. The model under consideration experiences several third dredge-up episodes, that make the C/O ratio increase above unity (top panel). At the stage C/O$\approx 1$ we note an abrupt change in the molecular equilibria: while the abundances of the O-bearing molecules drop (middle panel), the C-bearing molecules suddenly start dominating the atmospheric chemistry (bottom panel). The abundance variations due to the increase of the C/O ratio are indeed remarkable, and they may span many orders of magnitudes! In this respect we also acknowledge the numerical stability of `ÆSOPUS` code, which is able to handle molecular species down to trace concentrations (e.g. SO$_2$ drops down to $\simeq 10^{-30}$ in the last TPs). At variance with the other molecules, the concentration of the carbon monoxide (CO) remains almost unperturbed by the evolution of the C/O ratio (except for a modest increment following the increase of C), due to its extremely large bond energy.
To better appreciate the role of the C/O ratio as the main driving factor of molecular chemistry, Fig. \[fig\_molchem\] (right-hand side panel) zooms in the evolution of just six molecules, among the most abundant ones, during the TP-AGB phase of a $M_{\rm i}=5\,M_{\odot}$, $Z_{\rm i}=0.001$ star. This model is predicted to suffer significant changes in its envelope chemical composition due to both the third dredge-up and HBB, which produce a complex evolution of the C/O ratio. We expect that the surface C/O follows a sawtooth trend crossing the critical region around unity several times, even during the single TPs. This may happen under particular conditions such that one dredge-up episodes brings the C/O$>1$ and later, during the interpulse period, HBB is able to burn C into N, hence lowering C/O below unity again.
In particular we note that the during the last TPs HBB is extinguished while the third dredge-up keeps on taking place, so that a significant increase of the C/O ratio is predicted in the last stages, as already noted by [@Frost_etal98]. Correspondingly, the molecular species exhibit quite drastic variations: the C-bearing molecules (CN, HCN, C$_2$) follow the steep increase of the C/O ratio, whereas those of the O-bearing molecules (SiO, H$_2$O, CO$_2$) show a specular behaviour. Eventually the abundances of all molecules drop when the atmosphere starts warming up as the star evolves off the AGB.
Extended atmospheres in the H-R diagram
---------------------------------------
Figure \[fig\_atmo\] displays the gas pressure – temperature stratifications of a few static atmosphere models (with the same stellar mass and luminosity), under the assumption of either plane parallel or spherically-symmetric geometry (see Sect. \[ssec\_atmo\]). Computations were carried out for three values of the effective temperature and two choices of the C/O ratio.
It is interesting to note that, at least for the models under consideration, at given $T_{\rm eff}$ and C/O, the photospheric pressure is almost insensitive to the geometry, while the separation between the thin and the extended atmospheres grows wider and wider at lower pressures. On the contrary a major effect is produced by the C/O ratio: at fixed $T_{\rm eff}$, the photospheric pressure is lower for C-rich than for O-rich models. This will have a sizable impact on the inner envelope structure of AGB stars with different C/O ratios, since the photosphere sets two of the four boundary conditions for the envelope integrations described in Sect. \[ssec\_envmod\].
By comparing the atmospheric structures for the two geometry options in Fig. \[fig\_atmo\], it is clear the relevance of the dilution of the radiation field in the extended atmospheres of AGB stars. For instance, for C/O$=2.0$ the plane-parallel model with $T_{\rm eff} \ge 3.4$ remains too cool and does not enter the condensation region of SiC and graphite, while the corresponding spherical model does it successfully. On the other hand, almost all models with C/O$=0.5$ stay outside the condensation area even at the lowest $T_{\rm eff}=3.3$. Indeed, a detailed analysis on the nucleation and growth of dust grains in the outer envelopes of AGB stars requires abandoning the static approximation in favour of an expanding envelope model. This important issue is beyond the scope of the present work, and is addressed in a forthcoming paper [@Nanni_etal13].
Figure \[fig\_sphat\] illustrates the areas in the H-R diagram where AGB and post-AGB stars (cooler than $\sim 3 \times 10^4$ K) are expected to have extended atmospheres, i.e. the radial extension of the atmosphere being a non-negligible fraction of the photospheric radius. The geometrical thickness $\delta R$ is defined according to Eq. (\[eq\_dr\]).
First of all we note that, at given stellar mass, $\delta R$ increases at higher $L$ and lower $T_{\rm eff}$. Giants with lower masses have thicker atmospheres (higher $\Delta R$), since smaller $M/L$ values tend to reduce the effective gravitational acceleration, $g_{\rm eff}= (1-\Gamma) g$, by increasing the Eddington factor $$\Gamma = \frac{\kappa }{4 \pi G c}\frac{L}{M}
\label{eq_gamma}$$ where $g=GM/R^2$ is the gravitational acceleration, $\kappa$ is the flux-averaged opacity, while the other constants have their usual meanings. As shown in the top panels of Fig. \[fig\_sphat\], these conditions are preferably met by evolved M-type stars of low mass, a circumstance already discussed by e.g. @Schmid_etal81, and @Laskarides_etal90.
At higher $L$ and increasing $T_{\rm eff}$ atmospheres may even become gravitationally unbound, as the Eddington factor rises above unity due to the increasing opacity $\kappa$ in the outermost layers. In fact, for temperatures $\log(T)\ga 3.8$ K, the Rosseland mean opacity is expected to grow steeply due to the increasing contributions of the hydrogen bound-free and free-free absorptions [see e.g. @MarigoAringer_09]. It follows that this condition may apply, for instance, to post-AGB stars with high mass ($\ga\! 1 M_{\odot}$) (evolved from more massive AGB stars with HBB) on their way towards the hotter regions of the H-R diagram (see the dotted area top-right panel of Fig. \[fig\_sphat\]).
Hayashi lines on the TP-AGB {#ssec_hayashi}
---------------------------
Figure \[fig\_hrco\] displays several sequences of AGB Hayashi lines, with the aim of illustrating their basic dependencies on stellar mass, envelope mass, metallicity and C/O ratio. To this aim we consider two choices of the stellar mass, $1.0\,M_{\odot}$ and $2.0\,M_{\odot}$, and two values of the initial metallicity $Z_{\rm i}=0.0005,\,{\rm and}\,0.017$.
The surface C/O ratio is made vary from $0.1$ to $10$ in steps of $\Delta($C/O$)=0.2$, by increasing the C abundance, while keeping O constant (to mimic the effect of the third dredge-up). Therefore the actual metallicity $Z$ increases as C/O increases.
For each value of C/O, the core mass $M_{\rm c}$ is made increase from $0.5\,M_{\odot}$ in steps of $\Delta M_{\rm c}=0.1\,M_{\odot}$, until either the luminosity reaches $\log(L/L_{\odot})=4.6$, or the envelope mass falls below $10\%$ of the total stellar mass, i.e. $(M-M_{\rm c})/M<0.1$. While the former condition is first met by the $2.0\,M_{\odot}$ sequences, the latter applies to the $1.0\,M_{\odot}$ tracks, that are terminated when $M_{\rm c} =0.94\, M_{\odot}$.
The effective temperature and the luminosity are determined by complete integrations of envelope models (see Sect. \[ssec\_envmod\]), with gas opacities calculated on-the-fly consistently with the current chemical composition (and C/O ratio).
We remark that these calculations are simply grids of envelope integrations and are meant to yield an overall picture of the Hayashi lines of C stars and their critical dependencies, but they cannot, by construction, be strictly representative of the TP-AGB evolution. For instance, the over-luminosity effect due to a deep third dredge-up is not taken into account and the Hayashi lines in Fig. \[fig\_hrco\] are those corresponding to a standard CMLR (for $\lambda=0$). As a consequence, at a given stellar mass, luminosity, and C/O ratio the “actual” effective temperature of an evolving C star model should be somewhat lower than that predicted in Fig. \[fig\_hrco\]. This said, the following discussion is nevertheless instructive since the general trends remain valid.
Examining Fig. \[fig\_hrco\] several features can be noticed. As long as C/O$<1$ the Hayashi lines have a steep slope and span a limited $T_{\rm eff}$ range, which becomes narrower at decreasing metallicity. This interval defines the expected location of M and S stars. The value C/O$=1$ corresponds to the warmest Hayashi line, due to a deep minimum in the molecular opacities .
As soon as C/O overcomes unity we expect a sudden jump of the Hayashi lines to lower effective temperatures, the amplitude of the temperature jump being more pronounced at increasing metallicity. The cooling rate, expressed by the derivative $|d(\log T_{\rm eff})/d({\rm C/O})|$, progressively decreases at increasing C/O ratio, so that larger and larger C/O ratios are required to reach lower effective temperatures. This is evident by looking at the thickening of the iso-C/O curves in Fig. \[fig\_hrco\] (dashed lines), which become gradually closer one to the next.
It means that, above some critical C/O ratio, the atmospheric structure becomes less and less sensitive to a further increase of the carbon abundance. This kind of “saturation” effect shows up at lower C/O ratio for decreasing metallicity, as can be better appreciated in Fig. \[fig\_slope\]. We notice that at higher $Z_{\rm i}$ the cooling rate is large for C/O values slightly above $1$, then it decreases until it flattens out to a nearly constant, small value. This trend is found also at lower $Z_{\rm i}$, but with smoother features: the initial drop of $T_{\rm eff}$ becomes less pronounced and $|d(log T_{\rm eff})/d({\rm C/O})|$ levels off at lower C/O ratios. Note, for instance, the extremely low cooling rate at $Z_{\rm i}=0.0005$ all over the C/O ratio under consideration ($1 \le {\rm C/O} \le 10$).
The core mass at the onset of the third dredge-up {#ssec_mcmin}
-------------------------------------------------
As already mentioned in Sect. \[ssec\_tbdred\], we can determine the minimum core mass for the occurrence of the third dredge-up $M_{\rm c}^{\rm min}$, checking if and when the $T_{\rm bce}$ exceeds a critical value $T_{\rm dup}$ at the stage of post-flash luminosity peak. The quantity $T_{\rm dup}$ is assumed as a free parameter.
In Figure \[fig\_mc3\] the left-hand side panels display the $M_{\rm c}^{\rm min}$ predictions for $\log(T_{\rm dup})= 6.2, 6.4, 6.5, 6.6, 6.7, 6.8$ and three values of the initial metallicity, $Z_{\rm i}=0.02, \,Z_{\rm i}=0.008, \,{\rm and}\, Z_{\rm i}=0.004$. The numerical method described in Sect. \[ssec\_tbdred\] has been applied for stellar masses ranging from $1\,M_{\odot}$ to $3\,M_{\odot}$ in steps of $0.05\,M_{\odot}$. In practice, once set the minimum temperature $T_{\rm dup}$, for each initial stellar mass and chemical composition, $M_{\rm c}^{\rm min}$ is the value of the core mass for which $T_{\rm bce}=T_{\rm dup}$ is satisfied. The solution is found iteratively with envelope integrations adopting the Brent root-finding algorithm [chapter IX of “Numerical Recipes”; @Press_etal88]. In each case $M_{\rm c}^{\rm min}$ is taken as the maximum between the value obtained by the envelope-integration method and the core mass at the first thermal pulse, $M_{\rm c,1}$. We do not show the results for $M> 3M_{\odot}$, since for the higher masses the temperature criterion is always satisfied since the onset of the TP-AGB, regardless of the value $T_{\rm dup}$. We see that all the curves share the same trend. Starting from lower masses towards the higher ones, $M_{\rm c}^{\rm min}$ slightly decreases, reaches a minimum and then steeply increases. It is interesting to note that the minimum in $M_{\rm c}^{\rm min}$ corresponds exactly to the critical maximum mass, $M_{\rm HeF}$, for a star to develop a degenerate He-core and experience the He-flash at the tip of the RGB. This reflects the same correspondence between $M_{\rm HeF}$ and the minimum of $M_{\rm c,1}$ (see Fig. \[fig\_mc1\]), already pointed out long ago by @Lattanzio86.
For a given initial metallicity, at decreasing $T_{\rm dup}$, the sequences move downward and reach lower stellar masses, that is $M_{\rm c}^{\rm min}$ decreases and the third dredge-up is expected to take place in stars of lower and lower masses. We note that for $T_{\rm dup}\le 6.4$ the minimum core mass $M_{\rm c}^{\rm min}$ coincides with $M_{\rm c,1}$.
The values of the core mass, $M_{\rm c}^{\rm 3dup}$, when the third dredge-up effectively occurs for the first time during the TP-AGB evolution, are shown in the right-hand side panels of Figure \[fig\_mc3\]. We note that, in general, $M_{\rm c}^{\rm 3dup} \ge M_{\rm c}^{\rm min}$, as expected. The `COLIBRI` results for $M_{\rm c}^{\rm 3dup}$, corresponding to $\log(T_{\rm dup})= 6.4$, show a similar trend with the stellar mass compared to full TP-AGB models calculations. At the same initial metallicity and stellar mass our predictions for $\log(T_{\rm dup})= 6.4$ are lower than @Karakas_etal02, but somewhat larger than @WeissFerguson_09.
Clearly significant differences exist between the two sets of full calculations, which supports the need to accurately calibrate $M_{\rm c}^{\rm min}$ with the aid of observations of M and C giants of different ages and metallicities. This calibration is presently underway and will be presented in subsequent papers.
Intershell abundances {#ssec_xpdcz}
---------------------
The standard chemical composition of the intershell region, left after the development of a thermal pulse, amounts to roughly $20\%-25\%$ of $^{12}$C, $1\%-2\%$ of $^{16}$O, $1\%-2\%$ of $^{22}$Ne, with $^{4}$He essentially comprising all the rest [@Schoenberner_79; @BoothroydSackmann_88b; @Mowlavi_99a], almost regardless of metallicity and core mass.
These standard values are presently debated. @Izzard_etal04 find a lower value for $^{16}$O, typically amounting to $\approx\,0.5\%$, while the inclusion of convective diffusive overshooting applied to all convective boundaries of the PDCZ determines a substantial increase of the $^{12}$C and $^{16}$O abundances at the expense of $^{4}$He [@Herwig_etal97]. @Herwig_00 shows that with his calibrated parametric scheme for overshoot, the $^{12}$C and $^{16}$O intershell abundances reach typical values of $0.45$, and $0.25$, respectively.
We will now discuss our predictions obtained from the semi-analytic scheme detailed in Sect. \[ssec\_pdcz\]. Figure \[fig\_pdcztp\] exemplifies the evolution of the main characteristics of the PDCZ during a thermal pulse, in two models with different core masses. Let us first analyse the results for the model with $M_{\rm c}=0.576\,M_{\odot}$ (left-hand side panels).
We see that while the density at the bottom of the PDCZ is continuously dropping, the corresponding temperature first rises up to the maximum value, $T_{\rm pdcz}^{\rm max}$, and then decreases (top panels).
Before reaching the maximum temperature, the chemical composition of the PDCZ may vary mainly due to its growth in mass, as the ashes of the H-burning shell are reached by the expanding convection. As a consequence the abundances of $^{4}$He, $^{14}$N, and $^{23}$Na are expected to increase. The sharp rise of $^{14}$N is evident in the bottom left-hand side panel of Fig. \[fig\_pdcztp\]. The increase of $^{14}$N is only temporary: as soon as the PDCZ heats up nitrogen is completely destroyed by the chain $^{14}{\rm N}(^4{\rm He},\gamma)^{18}{\rm F}(\beta^+ \nu)^{18}{\rm O}$.
In the short phase around the temperature maximum the PDCZ reaches its widest mass extension. At this point the main $\alpha$-capture reactions are turned on, leading to the production of primary carbon via the [$\rm\,{}^{4}\kern-0.8pt{He}\,({2\,^{4}He}\,,{\gamma})
\,{}^{12}\kern-0.8pt{C}\,$]{} reaction, together with some synthesis of $^{16}$O from [$\rm\,{}^{12}\kern-0.8pt{C}\,({^{4}He}\,,{\gamma})
\,{}^{16}\kern-0.8pt{O}\,$]{}, and $^{22}$Ne from [$\rm\,{}^{18}\kern-0.8pt{O}\,({^{4}He}\,,{\gamma})
\,{}^{22}\kern-0.8pt{Ne}\,$]{}. Correspondingly the $^{4}$He abundance decreases.
Finally, when the PDCZ cools and the convection recedes the chemical composition barely changes, so that the entire intershell with mass $\Delta M_{\rm pdcz}$ is assigned the final mixture at $\phi=1$.
Basically the same analysis holds for the model with the higher core mass (right-hand side panels), but for a few differences that are explained mainly by the higher $T_{\rm pdcz}^{\rm max}$, the shorter duration $\tau_{\rm PDCZ}$ of the PDCZ, and by the previous dredge-up history. As we discuss later, the intershell abundances do depend on the indirect interaction of one pulse with the preceding one, which can be quantified by the so-called “degree of overlap”, denoted with $r$. We also note that in the model with higher $M_{\rm c}$ a higher $T_{\rm pdcz}^{\rm max}$ is attained so that [$\rm\,{}^{22}\kern-0.8pt{Ne}\,({^{4}He}\,,{n})
\,{}^{25}\kern-0.8pt{Mg}\,$]{} is also activated. This reaction is recognized as a source of neutrons for the s-process nucleosynthesis in more massive AGB stars [e.g. @Busso_etal99; @Pumo_etal09].
Figure \[fig\_xpdcz2\] shows the evolution of the final PDCZ abundances left after each thermal pulse (bottom panels), during the entire TP-AGB evolution of the ($M_{\rm i}=2.6,\,Z_{\rm i}=0.017$) model. The left- and right-hand side panels compare the results in the cases the third dredge-up takes place (left-hand side panel; $\lambda > 0$) or does not (left-hand side panel; $\lambda = 0$). In the $\lambda > 0$ case, the efficiency of the third dredge-up is described following the analytic relations presented by @Karakas_etal02 which fit the results of their full AGB models (see also Sect. \[ssec\_tpagbev\]), the $\lambda = 0$ case is simply treated setting the efficiency to zero at each thermal pulse. This is equivalent to assume a high value for $T_{\rm dup}$. Several remarks can be made.
#### The “standard” intershell abundances. {#the-standard-intershell-abundances. .unnumbered}
Our intershell abundances of $^{4}$He, $^{12}$C, and $^{16}$O recover nicely the “standard” values obtained by the class of full AGB models [e.g. @Schoenberner_79; @BoothroydSackmann_88b; @Izzard_etal04; @KarakasLattanzio_07] in which the borders of the PDCZ are determined by the classical Schwarzschild criterion applied to the temperature gradients. We find typical values of $\approx 20\,\%$ for $^{12}$C and $\approx 0.5\%-1\,\%$ for $^{16}$O (Figs. \[fig\_xpdcz2\], \[fig\_over\], \[fig\_rate\], \[fig\_xpdcz\]). More specifically, our predictions for $^{16}$O are in closer agreement with the lower abundances reported by @Izzard_etal04, than the higher values of $1\% - 2 \%$ defining the “standard” intershell composition [@BoothroydSackmann_88b]. This difference will be discussed below, being likely related to the efficiency of the third dredge-up.
#### Dependence on the degree of overlap. {#dependence-on-the-degree-of-overlap. .unnumbered}
The degree of overlap $r$ is defined as the fraction of the matter contained in a given PDCZ that is incorporated into the PDCZ produced at the next thermal pulse. The reader may refer to Table \[tab\_mod\] for the operative definition of $r$ in terms of mass coordinates, and to Fig. \[fig\_pdcz\] for a graphical representation. The parameter $r$ was originally introduced and discussed in early studies [e.g. @Ulrich_73; @Iben_75; @TruranIben_77; @Iben_77] to highlight the importance of the overlap between successive pulses to the slow-neutron capture nucleosynthesis of heavy elements, especially in relation to the synthesis of $^{22}$Ne and its role in the release of neutrons via the [$\rm\,{}^{22}\kern-0.8pt{Ne}\,({^{4}He}\,,{n})
\,{}^{25}\kern-0.8pt{Mg}\,$]{} reaction.
We also find a significant dependence of the intershell abundances on the degree of overlap $r$ between consecutive PDCZs. This can be better appreciated by looking at the bottom panels of Fig. \[fig\_xpdcz2\]. We see that the degree of overlap tends in general to decrease from pulse to pulse, but the occurrence of the third dredge-up ($\lambda > 0$) makes $r$ to drop more steeply, eventually reaching zero in the last TPs. The smooth decline of $r$, expected for $\lambda=0$, is mostly due to the inverse correlation between $\Delta M_{\rm pdcz}$ and $M_{\rm c}$ (see top panel of Fig. \[fig\_pdczfit\]), so that less massive PDCZs are produced in later TPs. On the other hand, every time a dredge-up event takes place the upper border, $M_{\rm Pt}$, of the PDCZ is shifted inward in mass coordinate, by an amount which is larger at increasing $\lambda$. This circumstance causes a further reduction of $r$. We find that for $\lambda \ga 0.7$ the overlap $r \simeq 0$, implying that the PDCZs are almost decoupled one from the next.
This effect is clearly shown in Fig. \[fig\_moverlap\], where the mass difference, $M_{\rm Pt}-M_{\rm c}$, becomes more and more negative when the third dredge-up is active ($\lambda >0$), at variance with the nearly constancy, or even small increase, expected if the process does not take place ($\lambda =0$). Consequently, the decrease of the overlap mass between two consecutive PDCZs, $\Delta M_{\rm overlap}$, is steeper at increasing $\lambda$. We note that in the TP-AGB model with $\lambda>0$ the overlap mass gradually reduces to almost zero, and then it grows again in the very last thermal pulses when the third dredge-up does not take place anymore.
The consequences on the PDCZ nucleosynthesis are exemplified in Fig. \[fig\_xpdcz2\]. While in absence of dredge-up events $(\lambda=0)$ all intershell abundances tend to flatten out at nearly constant values, when the third dredge-up takes place this pattern is modified. In particular, as the third dredge-up starts to occur we expect that the intershell abundance of $^{16}$O somewhat declines levelling off in the last TPs, while those of $^{22}$Ne and $^{23}$Na increase, steadily. These findings for $^{22}$Ne and $^{23}$Na are in full agreement with @Mowlavi_99a [@Mowlavi_99b], to which the reader should also refer for a very detailed analysis.
The increase of $^{22}$Ne, that reaches up to $\simeq 2\% - 3 \%$ in the cases under consideration, is directly related to the increase of primary $^{12}$C in the envelope composition caused by the third dredge-up. The more abundant the surface $^{12}$C is, the larger amount of $^{14}$N is synthesized during the interpulse period by the CNO-cycle operating in the radiative H-burning shell. In turn, the more abundant $^{14}$N is, the more $^{22}$Ne will be produced by the chain of reactions $^{14}{\rm N}(^4{\rm He},\gamma)^{18}{\rm F}(\beta^+ \nu)^{18}{\rm O}(^4{\rm He}, \gamma)^{22}{\rm Ne}$ occurring inside the PDCZ.
The increase of $^{23}$Na is related to the larger envelope abundance of $^{22}$Ne, that we expect as a consequence of the third dredge-up. We recall that the $^{23}$Na in the PDCZ is not synthesized in situ during the pulse, but is it inherited as part of the material processed by the radiative H-burning shell, where the conversion $^{22}{\rm Ne}(p, \gamma)^{23}{\rm Na}$ took place.
The trends of the $^{12}$C and $^{16}$O intershell abundances, mainly synthesized as primary products during the TPs, are also affected by the third dredge-up, hence by the degree of overlap between consecutive PDCZs.
To better investigate this aspect, we have performed a few test calculations, assuming each time a different value of the overlap parameter $r$, which is kept constant along a pre-determined sequence of thermal pulses. Given a selected value of $\widehat{r}$, at each thermal pulse the mass coordinate of the top of the previous PDCZ, $M_{\rm pdcz}^{\prime}$, is artificially varied such that the condition $\Delta M_{\rm overlap}=M_{\rm Pt}^{\prime}-M_{\rm He}
= \widehat{r}\, \Delta M_{\rm pdcz}$ is fulfilled (see Fig. \[fig\_pdcz\]). This is equivalent to suitably adjusting the maximum depth of the third dredge-up, hence its efficiency $\lambda$.
The results are presented in Fig. \[fig\_over\], together with a TP-AGB sequence computed without the third dredge-up (black triangles). From the intersections with the bunch of lines we can read out the corresponding values of the degree of overlap $r$, which is found to decrease slowly from roughly $0.8$ to $0.4$.
As for the PDCZ abundances, we notice that, after the first pulses, the curves tracing the evolution of the intershell abundances at constant $r$, run almost parallel at increasing core mass. For instance, at $M_{\rm c}=0.65\, M_{\odot}$, passing from $r=0.8$ to $r=0.0$ the $^{12}$C ($^{16}$O) concentration decreases from $\simeq 41\,\%$ ($\simeq 2.7\, \%$) to $\simeq 17\,\%$ ($\simeq 0.5\, \%$). The relative change with $r$ appears larger for $^{16}$O ($approx$ a factor of six) than for $^{12}$C ($approx$ a factor of two).
From these results we suggest that the lower $^{16}$O intershell abundances ($< 1\%$) reported by @Izzard_etal06, compared to the standard values ($1\%-2\%$) found by @BoothroydSackmann_88b reflect the larger efficiency of the third dredge-up (i.e. higher $\lambda$ hence lower $r$) found in the more recent works compared to the past.
#### Dependence on the nuclear reaction rates. {#dependence-on-the-nuclear-reaction-rates. .unnumbered}
We have investigated the robustness of the “standard” intershell composition against reasonable changes in two key nuclear reaction rates, namely $^4{\rm He}(2\,^4{\rm He}, \gamma)^{12}{\rm C}$ and $^{12}{\rm C}(^4{\rm He}, \gamma)^{16}{\rm O}$. A few versions for the latter rate are compared in Fig. \[fig\_rate\_oxy\]. The results for the PDCZ abundances of $^{12}{\rm C}$ and $^{16}{\rm O}$ are shown in Fig. \[fig\_rate\]. There is an almost perfect overlap of the $^{12}$C predictions obtained with the @CaughlanFowler_88 and the @Fynbo_etal05 rates. This is not surprising since the two versions are quite similar (with a relative difference always below $1\%$) in the temperature range of interest for the pulse nucleosynthesis, i.e. $2\times 10^8 {\rm K} \la T \la 4 \times 10^8 {\rm K}$.
The results for $^{16}$O exhibit a somewhat larger dependence, but still modest, on the assumed $^{12}{\rm C}(^4{\rm He}, \gamma)^{16}{\rm O}$. A comparison of four rates for this reaction is displayed in Fig. \[fig\_rate\_oxy\]. In the relevant temperature range the largest difference reaches roughly a factor of $2$ between the @CaughlanFowler_88 and the @Fynbo_etal05 rates, while the variation in the intershell abundance of $^{16}$O remain quite small, $\approx 10\%$. The rather low sensitivity of the $^{16}$O abundance PDCZ to significant changes of the $^{12}{\rm C}(^4{\rm He}, \gamma)^{16}{\rm O}$ rate was already noticed by @BoothroydSackmann_88b and is essentially explained by the fact that the proper temperature conditions are kept for too short a time to allow a sizable conversion of $^{12}{\rm C}$ into $^{16}$O.
#### Dependence on stellar mass and metallicity. {#dependence-on-stellar-mass-and-metallicity. .unnumbered}
Figure \[fig\_xpdcz\] shows the evolution of the final PDCZ abundances left after each thermal pulse, during the entire TP-AGB evolution of models with a few values of initial stellar masses and two choices of the initial metallicity $Z_{\rm i}=0.017$ and $Z_{\rm i}=0.001$. As already mentioned, our predictions are essentially consistent with the recent results from full stellar models without overshooting applied to the PDCZ [@Mowlavi_99a; @Karakas_etal02; @Izzard_etal04]. In particular the $^{12}$C abundance evolves towards an asymptotic value of $\simeq 20\%$, independent of mass and metallicity, while in most cases the $^{16}$O abundance settles down around a value of $\simeq 0.005-0.008$, in any case lower than $2\%$ reported by @BoothroydSackmann_88b.
The abundance of $^{22}$Ne is nearly always larger than that of $^{16}$O, reaching up to $\approx 2\% -3 \% $ in models with $Z_{\rm i}=0.017$, while lower values up to $\approx 1\% -2 \% $ are attained for $Z_{\rm i}=0.001$. However we note that, relative to its value at the first TP, the PDCZ concentration of $^{22}$Ne shows a larger increase in lower metallicity models, while at larger metallicity the increment is by one order of magnitude at most. This result is likely related to the fact that at lower metallicity we expect a more efficient enrichment of primary $^{12}$C, hence of the total CNO abundance, in the envelope caused by the third dredge-up. In this way the synthesis of $^{22}$Ne is favoured, as it is the end product of a chain of $\alpha$-capture reactions that just start with $^{14}$N, the most abundant product of the CNO cycle (after $^{4}$He) operating in the H-burning shell.
A similar trend characterizes the evolution of the $^{23}$Na intershell abundance, which depends on the proton capture reactions occurring in the H-burning shell during the quiescent interpulse periods. High-metallicity models show, in general, higher values of $^{23}$Na ingested in the PDCZ, up to $\simeq 10^{-3}$, but the relative increase over the TP-AGB evolution is larger in low-metallicity models.
#### Dependence on overshoot. {#dependence-on-overshoot. .unnumbered}
The scheme depicted in Fig. \[fig\_pdcz\] for the PDCZ can be easily modified to account for overshoot applied to the base of the convective pulse. As a test case, we have considered the results obtained by @Herwig_00, who applied an exponential diffusive overshoot at the convective boundaries of the PDCZ. One major consequence is a depletion of helium and enhancement of carbon and oxygen in the intershell abundance distribution. Typical abundances (by mass) are $0.4 -0.5$ for $^{12}$C, $0.15-0.20$ for $^{16}$O, and $0.30-0.40$ for $^{4}$He, obtained by @Herwig_00 with his calibrated overshoot parameter.
We underline that in our model the physical structure of the PDCZ is described via analytic fits to the results of full TP-AGB model (see Sect. \[ssec\_pdcz\]), so that we cannot perform physical tests of stability against convection at the borders of the convective intershell. Nonetheless, we can simulate the effect of different prescriptions with the aid of a simple parametric approach. To mimic the effect of overshoot applied to the PDCZ boundaries, we shift inward the mass coordinate of its bottom, adopting the parametrization: $$M_{\rm Pb}^{\rm oversh} = M_{\rm He} - f_{\rm ov} (M_{\rm He}-M_{\rm Pb})\,,$$ where $f_{\rm ov}\ge 1$ is an adjustable factor. For $f_{\rm ov}=1$ we recover the typical intershell chemical composition that is predicted by full TP-AGB models when using the Schwarzschild criterion, while the effect of convective overshoot is simulated adopting $f_{\rm ov} >1$. As mentioned by @Herwig_00, there is no noticeable effect of overshoot at the top of the PDCZ, so that we keep the mass coordinate $M_{\rm Pt}$ unchanged.
The mass difference ($M_{\rm He}-M_{\rm Pb}$), derived from full TP-AGB calculations as a function of the core mass, is plotted in Fig. \[fig\_herwig\]. We find that @Herwig_00 results are reasonably well reproduced with $f_{\rm ov} \simeq 7$, in terms of both PDCZ mass and abundances (see his figures 7d and 11 for the $(M_{\rm i}=3 M_{\odot}, Z_{\rm i}=0.02)$ model).
Without pretending to investigate in more detail complex aspects of the PDCZ nucleosynthesis (e.g. the formation of the $^{13}$C pocket is not considered here), we underline that this simple parametric approach may be useful to explore the impact of the overshoot option on the formation and evolution of carbon stars, by comparing population synthesis simulations including overshoot with observations, an important test which is still to be done to our knowledge. An example of test calculation is discussed in the next Sect. \[ssec\_hbbres\], and illustrated in Fig. \[fig\_hbbnuc\].
Hot-bottom burning nucleosynthesis {#ssec_hbbres}
----------------------------------
Figure \[fig\_tauli\] demonstrates the importance of including [*a time-dependent convective diffusion algorithm*]{} to treat the synthesis of lithium in intermediate-mass AGB stars with HBB. As thoroughly discussed by @SackmannBoothroyd_92, such an approach is necessary when the usual [*instantaneous mixing*]{} [^9] approximation is no longer valid. This is the case for nuclei, like $^7$Li and $^{7}$Be, whose lifetimes may become shorter or comparable to the convective timescale in some parts of the convective envelope. The circumstance $\tau_{\rm conv} \approx \tau_{\rm nuc}$ occurs in the inner regions for $^{7}$Li, and in the external layers for $^{7}$Be (see Fig. \[fig\_tauli\], left panel). As a consequence, the abundances of these species may vary considerably across the convective envelope, at variance with the concentrations of other nuclei (e.g. $^{3}$He, C, N, O) made homogeneous by the rapid convective mixing.
In particular, the convective envelopes of intermediate-mass AGB stars present the suitable thermodynamic conditions to put the [*Cameron-Fowler beryllium transport mechanism*]{} [@CameronFowler_71] at work: $^7$Li is efficiently produced and sustained in the outermost layers by electron captures on $^{7}$Be nuclei until either the reservoir of $^{3}$He (involved in the reaction [$\rm\,{}^{4}\kern-0.8pt{He}\,({^{3}He}\,,{\gamma})
\,{}^{7}\kern-0.8pt{Be}\,$]{}) is exhausted, or HBB is extinguished due to envelope ejection by stellar winds.
The model displayed in Fig. \[fig\_tauli\] shows the envelope structure of a TP-AGB star with $M_{\rm i} = 5.4,\, Z_{\rm i}=0.008$, that may be considered as representative of the most luminous M giants in the Large Magellanic Cloud. The structure is taken at the maximum surface Li enrichment corresponding to $\epsilon(^7{\rm Li}) \simeq 4.6$, and $M_{\rm bol} \simeq -6.5$, in nice agreement with the luminosities and the highest measured values of Li in the LMC super-rich Lithium stars [@SmithLambert_89; @SmithLambert_90; @Smith_etal95]. Note the mirror behaviours of $^{7}$Be and $^{7}$Li abundances: towards the surface $^{7}$Li is efficiently produced by electron captures on $^{7}$Be nuclei.
Figure \[fig\_hbbz\] compares the evolution of luminosity and surface $^{7}$Li abundance in TP-AGB stars with the same initial mass of $5 M_{\odot}$ but different metallicities. A few points are worth noting. Since at decreasing $Z$ higher temperatures at the base of the envelope are reached, the brightening of stars with HBB along the TP-AGB becomes steeper at lower metallicity, so that the classical Paczy[ń]{}ski limit[^10] [@Paczynski_70], at $M_{\rm bol} ~\simeq -7.1$, may be even exceeded, like the $M_{\rm i}=5 M_{\odot}, \,Z_{\rm i}=0.0005$ model does. In fact, because of the break-down of the CMLR in stars with HBB, the Paczy[ń]{}ski limit is no longer a true upper bound to the AGB luminosity [@BloeckerSchoenberner_91; @BoothroydSackmann_92], so that AGB stars brighter than $M_{\rm bol} ~\simeq -7.1$ could be effectively be observed with a core mass $M_{\rm c} < 1.4\, M_{\odot}$.
At the same time, the synthesis of lithium is more efficient at lower metallicity due to the larger amounts of $^{7}$Be produced in the innermost layers of the envelope by the [$\rm\,{}^{4}\kern-0.8pt{He}\,({^{3}He}\,,{\gamma})
\,{}^{7}\kern-0.8pt{Be}\,$]{} reaction. But for very high metallicities, e.g. $Z_{\rm i}=0.04$, at which the Li production remains quite modest (left-hand side panels of Fig. \[fig\_hbbz\]), in the other cases under consideration a maximum value around $\log[n(^{7}{\rm Li})/n({\rm H})]+12 \simeq 4-4.5$ is reached, that is only moderately dependent on $Z_{\rm i}$. This limiting value is in full agreement with earlier computations by @SackmannBoothroyd_92, and it is the result of the high temperature sensitivity of $\tau(^{7}{\rm Be})$ from one side, and of similar temperature conditions for the maximum Li synthesis in envelope models, on the other side.
Finally, we note that there should be a limited range of metallicity for which we expect AGB stars to contribute to the lithium enrichment of the interstellar medium. Comparing the trends of the $^{7}$Li abundance and the current stellar mass (bottom panels of Fig. \[fig\_hbbz\]), we see that for only models with $Z_{\rm i}=0.02$ significant mass loss takes place when the surface $^{7}$Li is high, while at higher and lower metallicities, the ejecta are practically $^{7}$Li free. In fact at $Z_{\rm i}=0.04$ the $^{7}$Li synthesis is just a small and short-lived event, whereas at $Z_{\rm i}=0.008$ and $Z_{\rm i}=0.0005$ the $^{7}$Li production is quite efficient but confined to the earliest stages of the AGB evolution, so that when the super-wind regime of mass loss is attained, practically whole $^{7}$Li has been destroyed, following the progressive exhaustion of the $^{3}$He reservoir. These conclusions are drawn for a particular set of stellar models, while a more general analysis should be extended also to other values of the stellar mass, which will be done a future investigation.
Figure \[fig\_hbbnuc\] exemplifies the results of the nucleosynthesis calculations made by `COLIBRI` over the entire TP-AGB evolution of a $M_{\rm i}=5.0, \,Z_{\rm i}=0.001$ model, corresponding to a low-metallicity star experiencing strong HBB.
The nucleosynthesis of the CNO, NeNa and MgAl cycles at low metallicities is of particular interest, in relation to the possible role of primordial AGB (and Super-AGB) stars as polluters of the gas out of which the old stars, presently observed in Galactic Globular Clusters (GGCs), may have formed [@VenturaDantona_08; @Pumo_etal08]. In this so-called self-enrichment scenario the HBB nucleosynthesis in metal-poor AGB (and Super-AGB) stars could have left its signatures in the prominent chemical anti-correlations (C-N, O-Na, Mg-Al) currently detected in GGC stars [@Carretta_etal09].
Indeed, our `COLIBRI` code may be fruitfully employed to investigate the several debated issues about the AGB chemical yields in the low $Z$ regime. An example is given in Fig. \[fig\_hbbnuc\], where we compare the results of four sets of computations obtained with exactly the same set of parameters, but varying a few key assumptions that should sample the spread in the predictions of current TP-AGB models. The effects on the predicted evolution of several light elements is remarkable. The results of our reference model, computed with the default set of input prescriptions, are shown in panel a) of Fig. \[fig\_hbbnuc\].
In the first test case (panel b), we have changed the rates of three nuclear reactions, namely [$\rm\,{}^{22}\kern-0.8pt{Ne}\,({p}\,,{\gamma})
\,{}^{23}\kern-0.8pt{Na}\,$]{}, [$\rm\,{}^{23}\kern-0.8pt{Na}\,({p}\,,{\gamma})
\,{}^{}\kern-0.8pt{\,^4He + ^{20}\kern-2.0pt{Ne}}\,$]{}, and [$\rm\,{}^{23}\kern-0.8pt{Na}\,({p}\,,{\gamma})
\,{}^{24}\kern-0.8pt{Mg}\,$]{}, replacing those quoted in Table \[tab\_rates\] with the theory rates labeled “ths8” in the JINA REACLIB database, that were calculated with the NON-SMOKER code$^{\rm WEB}$ version 5.0w developed by T. Rauscher[^11] and presented in @Cyburt_etal10. At the typical temperatures $T_{\rm bce}\ga 10^8$ K, the “ths8” rates are higher than the default ones. In particular, the “ths8” destruction rate [$\rm\,{}^{22}\kern-0.8pt{Ne}\,({p}\,,{\gamma})
\,{}^{23}\kern-0.8pt{Na}\,$]{} can be larger by up to 3 orders of magnitude! The large impact is evident by comparing the abundance trends of $^{22}$Ne, $^{23}$Na, and $^{24}$Mg in panels a) and b).
In the second test case (panel c), we assume that no third dredge-up takes place, i.e. $\lambda = 0$ at each TP, a condition found, for instance, in the recent models of super-AGB stars by @Siess_10, where the absence of extra-mixing at the edge of the convective boundaries prevents the development of dredge-up episodes. The evolution of the elemental abundances in the envelope is simply regulated by the CNO, NeNa, and MgAl cycles. A very significant depletion of $^{16}$O is responsible for the transition to C/O$>1$. At the same time we see that, compared to the other models, the lack of carbon enrichment in the envelope favours the attainment of higher base temperatures $T_{\rm bce}$.
In the third test case (panel d), we mimic the effect of convective overshoot at the bottom of the PDCZ following the scheme described in Sect. \[ssec\_xpdcz\]. As a consequence, the intershell abundance distribution becomes carbon- and oxygen-enhanced compared to the classical composition, resembling the findings by @Herwig_00 (see Fig. \[fig\_herwig\]). The differences with respect to the standard model shown in panel a) are sizable. The enrichment of $^{16}$O due to the third dredge-up prevails over the its destruction by HBB, producing a continuously increasing surface abundance of $^{16}$O. The C/O ratio remains lower than one for most of the TP-AGB evolution. Moreover, we note that the large increase of the metallicity due to the very efficient third dredge-up contributes to reach lower temperature $T_{\rm bce}$.
Closing remarks {#sec_finalsum}
===============
Summary of `COLIBRI`’s features
-------------------------------
In this paper we have presented the main improvements and novelties characterizing the `COLIBRI` code for the computation of the TP-AGB phase. They are briefly recalled below.
Compared to purely synthetic TP-AGB codes, `COLIBRI` relaxes a significant part of their analytic formalism in favour of a detailed physics which, applied to a complete envelope model, allows to predict self-consistently:
- the [*effective temperature*]{}, and more generally the convective envelope and atmosphere structures, suitably coupled to the changes in the surface chemical abundances and gas opacities;
- the [*CMLR and its possible break-down due to the occurrence of HBB*]{} in the most massive AGB stars, by taking properly into account the nuclear energy generation in the H-burning shell and in the deepest layers of the convective envelope;
- the [*HBB nucleosynthesis*]{} via the solution of a complete nuclear network coupled to a diffusive description of mixing, in which the current stratifications of temperature and density are derived from integrations of complete envelope models;
- the [*intershell abundances*]{} left by each thermal pulse via the solution of a complete nuclear network applied to a simple model of the pulse-driven convective zone;
- the [*onset and quenching of the third dredge-up*]{}, with a temperature criterion that is tested, at each thermal pulse, with the aid of envelope integrations at the stage of the post-flash luminosity peak.
At the same time `COLIBRI` pioneers new techniques in the treatment of the physics of stellar interiors, not yet adopted in full TP-AGB models. Compared to present-day full stellar evolutionary codes, the prerogatives of `COLIBRI` are related to 1) the computation of the equation of state and opacities, and 2) computation requirements, as below summarized.
- `COLIBRI` is able to perform the first ever [*on-the-fly*]{} accurate computation of the [*equation of state*]{} for roughly 800 atoms, ions, molecules, and of the Rosseland mean [*opacities*]{} throughout the atmosphere and the deep envelope. This has been accomplished by incorporating the `ÆSOPUS` code [@MarigoAringer_09] and the [*Opacity Project* ]{} software package [@Seaton05] as internal routines of the `COLIBRI` code. Avoiding the preliminary preparation of static tables and their subsequent interpolations, the new approach assures a complete consistency, step by step, of both EoS and opacity with the evolution of the chemical abundances caused by the third dredge-up and HBB. For the first time we show the evolution of the photospheric molecular concentrations during the TP-AGB phase, and their modulation driven not only by changes in the chemical compositions but also by the periodic occurrence of the TPs.
- [*Flexibility and optimized computation requirements*]{}. `COLIBRI` is competitive in terms of low computing-time requests. Tests made with a standard 2.2 GHz CPU processor have shown that `COLIBRI`, on average, computes one complete pulse-cycle in ${0.5-1.0}$ min against the ${60-90}$ min taken by full evolution codes, e.g. `PARSEC` [@Bressan_etal12], with a gain factor of ${\approx\,100}$. This characteristic makes `COLIBRI` an agile tool suitable to carry out extensive calculations of the TP-AGB evolutionary tracks covering large and dense grids of stellar masses and metallicities.
Figures \[fig\_stru\] and \[fig\_chem\] collect a representative sample of the most significant quantities that can be predicted by `COLIBRI` throughout the entire TP-AGB evolution of a star with given initial mass and chemical composition. The quantity of available information is indeed large, including both structural and chemical properties. We plan to keep the same level of richness also in the stellar isochrones we are going to construct from the `COLIBRI` tracks.
Ongoing and planned applications {#ssec_future}
--------------------------------
It should be mentioned that the present set of TP-AGB models is a preliminary release, since we are currently working to a global TP-AGB calibration as a function of stellar mass and metallicity, aimed at reproducing a large number of AGB observables at the same time (star counts, luminosity functions, C/M ratios, distributions of colors, pulsation periods, etc.) in different star clusters and galaxies. Since the calibration is still ongoing the current parameters (e.g. efficiency of the third dredge-up and mass loss) of the TP-AGB model may be changed in future calculations.
Anyhow, various tests indicate that the present version of the `COLIBRI` models already yields a fairly good description of the TP-AGB phase. Compared to our previously calibrated sets [@MarigoGirardi_07; @Marigo_etal08; @Girardi_etal10] the new TP-AGB models yield somewhat shorter, but still comparable, TP-AGB lifetimes, and they successfully recover various observational constraints dealing with e.g. the Galactic initial–final mass relation (Kalirai et al., in prep.), spectro-interferometric determinations of AGB stellar parameters [@Klotz_etal13], the correlation between mass-loss rates and pulsation periods, and the trends of the effective temperature with the C/O ratio observed in Galactic M, S and C stars.
Further important support comes from the results of our new model for the condensation and growth of dust grains in the outflows of AGB stars [@Nanni_etal13], which has been applied to the `COLIBRI` TP-AGB tracks. The results are extremely encouraging as they are found to nicely reproduce other independent sets of key observations, i.e. the correlation between expansion velocities and mass-loss rates/pulsation periods of Galactic AGB stars.
Acknowledgments {#acknowledgments .unnumbered}
===============
It is a pleasure to thank Julianne Dalcanton and Luciana Bianchi for their strong encouragement to this work, Phil Rosenfield and Marco Gullieuszik for their contribution to test the preliminary versions of the new TP-AGB tracks. Warm thanks go to Anita and Alessio for having inspired the name of the code. We acknowledge financial support from contract ASI-INAF n. I/009/10/0, and from [*Progetto di Ateneo 2012*]{}, University of Padova, ID: CPDA125588/12.
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Fitting relations {#app_fit}
=================
Properties of the pulse-driven convection zone
----------------------------------------------
Here we present relations for characteristic quantities of the PDCZ, based on full TP-AGB calculations by @Wagenhuber_96 [@Karakas_etal02; @KarakasLattanzio_07]. All masses are expressed in solar units, $\tau_{\rm pdcz}$ is given in years, and $T_{\rm pdcz}^{\rm max}$ in Kelvin degrees, $Z_{\rm i}$ denotes the initial metallicity. $$\begin{aligned}
\label{eq_taupdcz}
\log (\tau_{\rm pdcz}) & = & a_1 + a_2 Z_{\rm i}+ (a_3+a_4 Z_{\rm i}) M_{\rm c}\\
\nonumber
& & +10^{\displaystyle(a_5 + a_6 M_{\rm c} + a_7 \Delta M_{\rm c, nodup})}\end{aligned}$$
$$\begin{aligned}
\label{eq_tpdcz}
\log (T_{\rm pdcz}^{\rm max}) & = & (b_1 +b_2 \log(Z_{\rm i})) + (b_3+b_4 \log(Z_{\rm i})) M_{\rm c}\\
\nonumber
& & - 10^{\displaystyle( b_5 + b_6 \Delta M_{\rm c, nodup})} \end{aligned}$$
$$\begin{aligned}
\label{eq_rhopdcz}
\log(\rho_{\rm pdcz}^{\rm max}) = {\rm max}(3.7, c_1 + c_2 Z_{\rm i} +c_3 M_{\rm c})\end{aligned}$$
$$\begin{aligned}
\label{eq_mpdcz}
\log (\Delta M_{\rm pdcz}) & = & d_1 +d_2 M_{\rm c} + d_3 M_{\rm c}^2+d_4 \log(Z_{\rm i})\\
\nonumber
& & -10^{\displaystyle(d_5 + d_6 M_{\rm c} + d_7 \Delta M_{\rm c, nodup})} \\
\nonumber
& & + d_8 M_{\rm c} \log(Z_{\rm i})\end{aligned}$$
$$\begin{aligned}
\label{eq_tqti}
x_q =\tau_{\rm q}/\tau_{\rm pdcz}& = &(e_1+e_2 Z_{\rm i})M_{\rm c}+e_3 Z_{\rm i} +e_4 \\
\nonumber
& & -10^{\displaystyle(e_5 M_{\rm c,1} + e_6 \Delta M_{\rm c, nodup})}\end{aligned}$$
The core mass at the $1^{\rm st}$ thermal pulse
-----------------------------------------------
We follow the parametrization proposed by @WagenGroen_98, where $M$ denotes the stellar mass at the onset of the TP-AGB phase. All masses are expressed in solar units. Coefficients are obtained by fitting the predictions from the `PARSEC` sets of stellar models [@Bressan_etal12]. $$\begin{aligned}
\label{eq_mc1}
M_{\rm c,1} & = & [-p_{1} (M - p_{2})^2 + p_{3}]f\\
\nonumber
& & + (p_{4}M + p_{5})(1-f)\,, \\
\nonumber
f & = & \left(1 + {\rm exp}^{\frac{M - p_{6}}{p_{7}}} \right)^{-1} \end{aligned}$$
[lllllllll]{}\
& & & & & & &\
4.675 & -18.56 & 3.793 & 22.65 & -2.451 & 2.216 & 116.7 &\
\
& & & & & & &\
8.037 & -0.06876 & 0.5697 & 0.07701 & -0.8459 & -22.18 & &\
\
& & & & & & &\
4.96 & - 2.4 & - 1.25 & & & & &\
\
& & & & & & &\
-1.134 & 0.2884 & -1.898 & -0.08295 & -2.171 & 1.429 & -21.55 & 0.09189\
\
& & & & & & &\
0.8220 & 0.9602 & 5.481 & - 0.4321 & -0.8632 & -26.23 & &\
\[tab\_fpdcz\]
-------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
0.0005 9.616573E-02 1.300268E+00 5.567979E-01 9.204736E-02 5.204188E-01 1.947073E+00 1.607459E-01
0.001 1.173875E-01 1.188889E+00 5.505528E-01 9.301397E-02 5.100448E-01 1.954574E+00 1.670251E-01
0.004 1.074609E-01 1.150773E+00 5.389349E-01 9.559346E-02 4.645270E-01 2.170495E+00 1.949511E-01
0.006 9.772655E-02 1.148381E+00 5.347831E-01 9.128342E-02 4.641443E-01 2.254396E+00 2.278098E-01
0.008 9.020493E-02 1.156664E+00 5.318839E-01 8.671702E-02 4.719326E-01 2.319841E+00 2.560683E-01
0.01 7.480933E-02 1.193024E+00 5.300704E-01 9.499056E-02 4.257837E-01 2.365426E+00 2.470678E-01
0.014 7.496712E-02 1.189756E+00 5.286927E-01 9.300582E-02 4.175395E-01 2.375119E+00 2.651535E-01
0.017 6.956924E-02 1.227015E+00 5.275279E-01 8.479260E-02 4.427424E-01 2.477161E+00 2.505828E-01
0.02 6.530806E-02 1.243030E+00 5.269612E-01 8.581963E-02 4.315992E-01 2.459101E+00 2.572425E-01
0.03 5.160226E-02 1.249103E+00 5.268402E-01 7.668322E-02 4.601484E-01 2.516399E+00 2.637952E-01
0.04 4.661234E-02 1.274814E+00 5.324125E-01 7.903245E-02 4.494590E-01 2.481670E+00 2.438550E-01
0.05 5.827199E-02 1.337793E+00 5.441922E-01 8.204387E-02 4.402451E-01 2.389034E+00 2.424820E-01
-------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
\[tab\_mc1\]
Accuracy tests {#sec_dtests}
==============
Effective temperature {#ssec_teff}
---------------------
A fundamental check is to compare our determination of the effective temperatures, based on envelope integrations ($T_{\rm eff}^{\rm env}$; the method is detailed in Sect. \[ssec\_envmod\]), against the results of full stellar models ($T_{\rm eff}^{\rm full}$).
In Fig. \[fig\_dtefz01\] we show the results for the set of stellar evolutionary tracks with initial chemical composition ($Z_{\rm i} =0.01$, $Y_{\rm i}=0.267$), computed with `PARSEC` [@Bressan_etal12]. In the top panel we compare directly the effective temperatures, $T_{\rm eff}^{\rm full}$ and $T_{\rm eff}^{\rm env}$, relative to the quiescent pre-flash luminosity maximum at the $1^{\rm st }$ thermal pulse. We can already see that the agreement is very good for all stellar masses here considered. We also note that $T_{\rm eff}^{\rm env}$ is systematically lower than $T_{\rm eff}^{\rm full}$ by a small amount, which appears to increase somewhat with the stellar mass.
Considering that part of the differences is likely due to unavoidable numerical effects impossible to be disentangled, we have also investigated other possible physical causes that may explain some systematic trends. In particular we have considered the effects due to different descriptions of the EoS and the opacities in the `PARSEC` and `COLIBRI` codes.
In the bottom panel of Fig. \[fig\_dtefz01\] we zoom in the difference $T_{\rm eff}^{\rm env} - T_{\rm eff}^{\rm full}$ (in $K$ degrees), as a function of the stellar mass. The three sequences are obtained with three combinations of the EoS and low-$T$ opacities used in the `COLIBRI` code. The lowest sequence (black empty triangles), showing the largest deviations from `PARSEC`, corresponds to the $T_{\rm eff}^{\rm env}$ predictions with the optimal configuration of all input physics in `COLIBRI`. Specifically, envelope integrations have been carried out with both the EoS and the Rosseland mean opacities computed with `ÆSOPUS` on-the-fly according to the actual chemical mixture of all elements.
This implies that the molecular chemistry is accurately solved, exactly complying with the true surface C/O ratio that characterizes each stellar model at the onset of the TP-AGB phase. In fact, the surface C/O ratio may have decreased, compared to its initial value at the main sequence (${\rm C/O} < {\rm C/O}_{\rm initial} = ({\rm C/O})_{\odot}\simeq 0.55$ for the scaled-solar case under consideration), as a consequence of the first dredge-up and, in stars with $M > 4\,M_{\odot}$, because of the second dredge-up.
In contrast, in `PARSEC` the opacities are derived through interpolations on pre-computed opacity tables as a function of temperature, density, hydrogen abundance, and current metallicity $Z$, while keeping the distribution of metals fixed to the initial configuration, $X_i/Z=X_{i,\odot}/Z_{\odot}$. In particular this means that that no change in the C/O ratio is considered, i.e. ${\rm C/O}=({\rm C/O})_{\odot}$ is assumed in all opacity tables.
To test the effect produced on the effective temperatures by low-$T$ opacities with a fixed chemical partition, we have performed a second run of envelope integrations setting the metals partition in the `ÆSOPUS` chemistry routine frozen to the scaled-solar one ($X_i/Z=X_{i,\odot}/Z_{i,\odot}$), as in `PARSEC`. The differences $(T_{\rm eff}^{\rm env} - T_{\rm eff}^{\rm full})$ are now smaller, as one can see in Fig. \[fig\_dtefz01\] comparing the sequence of magenta crosses with that of black triangles. In this case the temperature differences are mostly comprised within $25$ K, and in all cases lower than $40$ K. The fact the assumed solar C/O ratio is higher than the actual values at the $1^{\rm st}$ TP, implies that a smaller excess of oxygen atoms, (O-C), is available to form the H$_2$O molecule, the most efficient opacity source at the atmospheric temperatures under consideration. The effect seems to be somewhat larger at increasing stellar mass.
Finally, we have explored possible additional EoS effects. At this stage we cannot obtain a quantitative comparison with respect to `PARSEC`, in which the EoS is solved with the `FreeEOS` code[^12], since these latter is not implemented in our `COLIBRI` code. Anyway, to obtain an order-of-magnitude estimate, we have carried out a third run of envelope integrations, switching the EoS option from the `ÆSOPUS` routine to an older and simpler EoS description based on @Kippenhahn_etal65. We see that now the deviations $T_{\rm eff}^{\rm env} - T_{\rm eff}^{\rm full}$ reduce further, keeping of the order of $\approx 20\,K$ or lower. Therefore we may conclude that the EoS treatment may also explain part of the differences $T_{\rm eff}^{\rm env} - T_{\rm eff}^{\rm full}$, by an amount that is comparable to that driven by the opacities.
Temperature at the base of the convective envelope {#ssec_tbot}
--------------------------------------------------
The quantity $T_{\rm bce}$ provides an additional performance test of our envelope-integration method, and it is particularly relevant for massive AGB models ($M > 4\,M_{\odot}$) as it measures the efficiency of hot-bottom burning.
In full stellar models calculated with `PARSEC` convective overshoot is applied to the formal Schwarzschild border of the envelope, with an efficiency parameter[^13] $\Lambda_{\rm e}=0.05$ for $M<M01$ and $\Lambda_{\rm e}=0.7$ for $M>M02$. The transition masses, with approximate values $M01\approx 1.0-1.5\,M_{\odot}$ and $M01\approx 1.5-2.0\,M_{\odot}$, are operatively defined in Bressan et al. (2012) and depend on chemical composition.
We apply the same scheme to our envelope integrations and then compare the predictions for $T_{\rm bce}$ as a function of stellar mass and metallicity. Results are shown in Fig. \[fig\_dtbotz01\]. We have verified that variations in the EoS and opacities, as those discussed in Sect. \[ssec\_teff\], produce almost negligible changes in $T_{\rm bce}$ for the models under considerations, so that we do not show the corresponding results.
The effect of convective overshoot on $T_{\rm bce}$ is illustrated in Fig. \[fig\_dtbotz01\] for the set with initial chemical composition $Z_{\rm i}=0.01, Y_i=0.267$. As a general rule models with $\Lambda_{\rm e} > 0$ tend to have higher $T_{\rm bce}$ since the base of the convective envelope penetrates more deeply inward. For masses $M<M01$ the differences in $T_{\rm bce}^{\rm env}$ remain small among models with or without overshoot, with $[\log T_{\rm bce}^{\rm env}
(\Lambda_{\rm e}=0.05)- \log T_{\rm bce}^{\rm env}(\Lambda_{\rm e}=0)]
\la 0.006$, reflecting the little overshoot efficiency adopted in since this mass range. In all cases $\log(T_{\rm bce}^{\rm full}) - \log(T_{\rm bce}^{\rm env})$ keep positive, i.e. the envelope-integration method yields somewhat higher temperatures than full stellar models.
Larger differences in $T_{\rm bce}^{\rm env}$ arise instead for masses $M>M02$, depending on whether we assume or not convective overshoot. We see that passing from $\Lambda_{\rm e}=0.7$ to $\Lambda_{\rm e}=0$ in our envelope integrations the differences $\log(T_{\rm bce}^{\rm full}) - \log(T_{\rm bce}^{\rm env})$ tend to become negative, i.e. the envelope-integration method yields lower temperatures than full stellar models. A systematic decrease of $[\log T_{\rm bce}^{\rm env}
(\Lambda_{\rm e}=0.7)- \log T_{\rm bce}^{\rm env}(\Lambda_{\rm e}=0)]
\simeq 0.03-0.05$ is predicted for these models.
In general the deviations from the full stellar models are larger than those for the effective temperatures, with $|\log(T_{\rm bce}^{\rm full}) - \log(T_{\rm eff}^{\rm env})|$ reaching up to a few hundredths of a dex. Part of the reason likely resides in the operative definition of the convective border and the adopted mass meshing across the envelope. In our `COLIBRI` code the classical Schwarzschild border is determined by the equality between the radiative and adiabatic temperature gradients, $\nabla_{\rm rad}=\nabla_{\rm ad}$, and all physical quantities are derived from interpolation between the last convective mesh and the first radiative one during the inward envelope integration. In `PARSEC` the Schwarzschild border is assumed to coincide with the last convective mesh, without interpolation in temperature gradients.
Limiting to the `COLIBRI` models with $\Lambda_{\rm e} > 0$, we note that larger deviations from $T^{\rm full}_{\rm bce}$ are found at larger stellar masses ($M > 4\, M_{\odot}$) where HBB starts to be operative. Part of these differences are likely related to the arrangement of the mesh points across the envelope; in fact the base of the convective envelope locates inside an extremely thin (in mass) region characterised by very steep gradients of all thermodynamic quantities ($T,P,\rho$, etc.), As a consequence, even small differences in mass resolution in this region may produce somewhat appreciable differences in the thermodynamic profile of the innermost layers of envelope.
We conclude that our envelope-integration method yields a description of the deepest envelope layers which is in satisfactory agreement with full stellar models, but unavoidable differences exist mainly due numerical and technical details. The size of such deviations are in any case lower than the current differences between various sets of AGB models, the latter reflecting the uncertainties of a still ill-defined theory of convection in stars.
[^1]: E-mail [email protected]
[^2]: We have used the OPCD$\_$3.3 open-source package available at the WEB page <http://cdsweb.u-strasbg.fr/topbase/op.html>
[^3]: The `ÆSOPUS` tool is accessible via the web interface at <http://stev.oapd.inaf.it/aesopus>
[^4]: LAPACK is a freely-available copyrighted library of Fortran 90 with subroutines for solving the most commonly occurring problems in numerical linear algebra. It can be obtained via <http://www.netlib.org/lapack/>
[^5]: In linear algebra LU decomposition factorizes a matrix as the product of a lower (L) triangular matrix and an upper (U) triangular matrix.
[^6]: The absence of nuclear energy sources in the envelope implies that the system of the stellar structure can be reduced from four to three equations (following @Wood_81 the local luminosity is reasonably constant across the envelope, $l = L$), so that we need to specify three boundary conditions, i.e. two at the photosphere Eqs. (\[c1\])-(\[c2\]), and one at the core border Eq. (\[c5\]).
[^7]: AGB models by @KarakasLattanzio_07 are available for download at <http://www.mso.anu.edu.au/~akarakas/model_data/>
[^8]: In our discussion we refer to the CPU time taken by a typical $2.2$-GHz processor.
[^9]: The instantaneous mixing approximation is based on the assumption $\tau_{\rm conv} \ll \tau_{\rm nuc}$, that is the convective timescale, $\tau_{\rm conv}$, is much shorter than the nuclear lifetime $\tau_{\rm nuc}$, such that any element produced by nucleosynthesis is immediately homogenized all over the convective region. This brings a big simplification in nucleosynthesis calculations: nuclear reactions rates are mass-averaged throughout the convective region, which can be then treated as a single radiative zone.
[^10]: In the old-fashion terminology the Paczy[ń]{}ski limit, also known as “AGB limit”, corresponds to the maximum luminosity that an AGB star, complying with the CMLR, may reach when its core mass has grown up to the Chandrasekhar limit, $M_{\rm c} \simeq 1.4\, M_{\odot}$. Its physical meaning has been dismissed since the prediction of the break-down of the CMLR by hot-bottom burning in massive AGB stars.
[^11]: Online code NON-SMOKER$^{\rm WEB}$, version 5.0w and higher available at <http://nucastro.org/websmoker.html>
[^12]: `FreeEOS` is a software package developed by A.W. Irwin, and freely available under the GPL licence at <http://freeeos.sourceforge.net/>
[^13]: The radial extension of the overshooting region is given by $\Lambda_{\rm e}\times H_P$, where $H_P$ is the local pressure scale height at the Schwarzschild border.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We implemented Simon’s quantum period finding circuit for functions $\F_2^n \rightarrow \F_2^n$ with period $\vec s \in \F_2^n$ up to $n=7$ on the 14-qubit quantum device IBM Q 16 Melbourne. Our experiments show that with a certain probability $\tau(n)$ we measure erroneous vectors that are not orthogonal to $\vec s$. While Simon’s algorithm for extracting $\vec s$ runs in polynomial time in the error-free case $\tau(n)=0$, we show that the problem of extracting $\vec s \in \F_2^n$ in the general setting $0 \leq \tau(n) \leq \frac 1 2$ is as hard as solving LPN (Learning Parity with Noise) with parameters $n$ and $\tau(n)$. Hence, in the error-prone case we may not hope to find periods in time polynomial in $n$. However, we also demonstrate theoretically and experimentally that erroneous quantum measurements are still useful to find periods faster than with purely classical algorithms, even for large errors $\tau(n)$ close to $\frac 1 2$.'
author:
- 'Alexander May[^1] [](https://orcid.org/0000-0001-5965-5675)'
- 'Lars Schlieper$^\star$ [](https://orcid.org/0000-0002-4870-1012)'
- Jonathan Schwinger
bibliography:
- 'Bib/abbrev3.bib'
- 'Bib/crypto.bib'
- 'Bib/IBM.bib'
title: |
Practical Period Finding on IBM Q –\
Quantum Speedups in the Presence of Errors
---
Introduction
============
The discovery of Shor’s quantum algorithm [@FOCS:Shor94] for factoring and computing discrete logarithms in 1994 had a dramatic impact on public-key cryptography, initiating the fast growing field of post-quantum cryptography that studies problems supposed to be hard even on quantum computers, such as e.g. Learning Parity with Noise (LPN) [@FOCS:Alekhnovich03] and Learning with Errors (LWE) [@STOC:Regev05].
For some decades, the common belief was that the impact of quantum algorithms on symmetric crypto is way less dramatic, since the effect of Grover search can be easily handled by doubling the key size. However, starting with the initial work of Kuwakado, Morii [@KuwakadoM12] and followed by Kaplan, Leurent, Leverrier and Naya-Plasencia [@C:KLLN16] it was shown that (among others) the well-known Even-Mansour construction can be broken with quantum CPA-attacks [@C:BonZha13] in polynomial time using Simon’s quantum period finding algorithm [@FOCS:Simon94]. This is especially interesting, because Even and Mansour [@AC:EveMan91] proved that in the ideal cipher model any classical attack on their construction with $n$-bit keys requires $\Omega(2^{\frac n 2})$ steps.
These results triggered a whole line of work that studies the impact of Simon’s algorithm and its variants for symmetric key cryptography, including e.g. [@SantoliSchaffner16; @AC:LeaMay17; @EC:AlaRus17; @SAC:Bonnetain17; @RSA:HosSas18; @AC:BonNay18; @DBLP:asiacrypt19]. In a nutshell, Simon’s quantum circuit produces for a periodic function $f:\F_2^n \rightarrow \F_2^n$ with period $\vec s \in \F_2^n$, i.e. $f(\vec x)=f(\vec z)$ iff $\vec z \in \{\vec x, \vec x+ \vec s\}$, via quantum measurements uniformly distributed vectors $\vec y$ that are orthogonal to $\vec s$. It is not hard to see that from a basis of $\vec y$’s that spans the subspace orthogonal to $\vec s$, the period $\vec s$ can be computed via elementary linear algebra in time polynomial in $n$. Thus, Simon’s algorithm finds the period with a linear number of quantum measurements (and therefore calls to $f$), and some polynomial time classical post-processing. On any purely classical computer however, finding the period of $f$ is equivalent to collision finding and thus requires $\Omega(2^{\frac n 2})$ operations.
#### Our contributions.
We implemented Simon’s algorithm on IBM’s freely available <span style="font-variant:small-caps;">Q 16 Melbourne</span> [@IBMQ16], called in the following, that realizes $14$-qubit quantum circuits. Since Simon’s quantum circuit requires for $n$-bit periodic functions $2n$ qubits, we were able to implement functions up to $n=7$ bits. Due to its limited size, is not capable of performing any error correction [@calderbank1997quantum] on the circuits.\
[**Implementation.**]{} Our experiments show that with some (significant) error probability $\tau$, we measure on vectors $\vec y$ that are [*not orthogonal*]{} to $\vec s$. The error probability $\tau$ depends on many factors, such as the number of $1$- and $2$-qubit gates that we use to realize Simon’s circuit, ’s topology that allows only limited $2$-qubit applications, and even the individual qubits that we use. We optimize our Simon implementation to achieve minimal error $\tau$. Since increasing $n$ requires an increasing amount of gates, we discovered experimentally that $\tau(n)$ increases as a function of $n$. For the function $f$ that we implemented, we found $\tau$-values ranging between $\tau(2)=0.1$ and $\tau(7)=0.15$.
Although produces faults for Simon’s quantum circuit, we still observe qualitatively the desired quantum effect: Vectors $\vec y$ orthogonal to $\vec s$ appear with significant larger probabilities than vectors not orthogonal to $\vec s$. Moreover, experimentally our distribution among those vectors that are orthogonal (respectively not orthogonal) to $\vec s$ is close to uniform. Notice that intuitively it should be hard to distinguish orthogonal vectors from not orthogonal ones.\
[**Hardness.**]{} Based on our experiments, we obtain a (simplified) error model that any quantum measurement yields with probability $1-\tau$ a uniformly chosen vector $\vec y$ orthogonal to $\vec s$, and with probability $\tau$ a uniformly chosen vector $\vec y$ not orthogonal to $\vec s$. We call [*Learning Simon with Noise*]{} (LSN) the problem of recovering $\vec s \in \F_2^n$ from quantum measurements. We show that solving LSN with parameters $n, \tau$ is polynomial time equivalent to solving the famous [*Learning Parity with Noise*]{} (LPN) problem with the same parameters $n, \tau$. The core of the reduction shows that LSN samples coming from quantum measurements of Simon’s circuit can be turned into perfectly distributed LPN samples, and vice versa.
Hence, quantum measurements of Simon’s circuit realize a [*physical LPN oracle*]{}. To the best of our knowledge, this is the first known physical realization of such an oracle. Moreover, from our hardness result we obtain a quite surprising link between symmetric and public key cryptography: Handling errors (i.e. not orthogonal vectors) in Simon’s algorithm, the most important quantum algorithm in symmetric crypto, is as hard as LPN, one of the major problems in post-quantum public key crypto.
From a cryptanalyst’s perspective, this result may at first sound quite negative, since we believe that we cannot solve (and thus by the -to- reduction also ) in time polynomial in $(n,\tau)$ — not even on a quantum computer. On the positive side, the -to- reduction accurately tells us how harmful errors $\tau$ from quantum computers are in practice, and how they affect the time complexity for quantum-assisted period finding.\
[**Error Handling.**]{} We may use the LSN-to-LPN reduction to handle errors from $\IBMQ$ via LPN-solving algorithms. In theory, the best algorithm for solving LPN with constant $\tau$ is the BKW-algorithm of Blum, Kalai and Wasserman [@STOC:BluKalWas00] with time complexity $2^{\bigO\big(\frac{n}{\log(\frac n {\tau})}\big)}$. This already improves on the classical time $2^{\frac n 2}$ for period finding. However, the BKW-algorithm has a huge sample and memory complexity, which hinder its practical implementation.
At the moment, the largest LPN instances with errors in ’s range $\tau \in [0.1, 0.15]$ are solved with variants of the low-memory algorithms <span style="font-variant:small-caps;">Pooled Gauss</span> and <span style="font-variant:small-caps;">Well-Pooled Gauss</span> of Esser, Kübler, May [@C:EssKubMay17]. We show that <span style="font-variant:small-caps;">Pooled Gauss</span> solves LSN for $\tau \leq 0.292$ faster than classical period finding algorithms. <span style="font-variant:small-caps;">Well-Pooled Gauss</span> even improves on any classical period finding algorithm for all errors $\tau < \frac 1 2$.
<span style="font-variant:small-caps;">Well-Pooled Gauss</span> is able to handle errors in time $2^{cn}$, where $c<\frac 1 2$ is constant for constant $\tau$. For $\tau=0$, we obtain polynomial time as predicted by Simon’s analysis. However, for $0 <\tau < \frac 1 2$ we achieve exponential run time, but still improve over the purely classical computation. This indicates that we achieve [*quantum supremacy*]{} for the period finding problem on sufficiently large computers, even in the presence of errors: Our quantum oracle helps us in speeding up computation! But as opposed to the exponential speedup from the (overly optimistic) error-free Simon setting $\tau =0$, we obtain in the (realistic) general error-prone Simon setting $0 <\tau < \frac 1 2$ only a [*polynomial speedup*]{} with a polynomial of degree $\frac{1}{2c} >1$.
Concerning quantum supremacy, assume that one could build a quantum device with $486$ qubits performing Simon’s circuit on a $243$-bit periodic function with error $\tau = \frac 1 8$. Then the error handling would translate into an -instance with $(n,\tau)=(243, \frac 1 8)$. Such an LPN instance was solved in [@C:EssKubMay17] on 64 threads in only $15$ days, whereas classically we would need $2^{121}$ steps for period finding.\
The paper is organized as follows. In \[sec:simon\] we recall Simon’s original quantum circuit, and already introduce our error model that we experimentally verify in \[sec:ibmq\] on . In \[sec:reduction\] we show the polynomial time equivalence of and . In \[sec:correct\] we theoretically show that quantum measurements with error $\tau$ in combination with LPN-solvers outperform classical period finding for any $\tau < \frac 1 2$. Eventually, in \[sec:cexperiments\] we experimentally extract periods out of erroneous measurements.
Simon’s Algorithm in the Presence of Errors {#sec:simon}
===========================================
#### Notation.
All $\log$s in this paper are base $2.$ Let $\vec x \in \F_2^n$ denote a binary vector with coordinates $\vec x = (x_{n-1}, \ldots, x_0)$. Let $\vec 0 \in \F_2^n$ be the vector with all-zero coordinates. We denote by ${\cal U}$ the uniform distribution over $\F_2$, and by ${\cal U}_n$ the uniform distribution over $\F_2^n$. If a random variable $X$ is chosen from distribution ${\cal U}$, we write $X \sim {\cal U}$. We denote by $\textrm{Ber}_{\tau}$ the Bernoulli distribution for $\F_2$, i.e. a $0,1$-valued $X \sim \textrm{Ber}_{\tau}$ satisfies $\Pr[X=1] = \tau$.
Two vectors $\vec x, \vec y$ are [*orthogonal*]{} if their inner product $\langle \vec x, \vec y \rangle:= \sum_{i=0}^{n-1} x_i y_i \bmod~2$ is $0$, otherwise they are called [*not orthogonal*]{}. Let $\vec s \in \F_2^n$. Then we denote the subspace of all vectors orthogonal to $\vec s$ as $$\vec s^{\perp} = \left\{ \vec x \in \F_2^n \; | \; \langle \vec x, \vec s \rangle = 0 \right\}.$$ Let $Y=\{\vec y_1, \ldots, \vec y_k\} \subseteq \F_2^n$. Then we define $Y^{\perp} = \{\vec x \mid \langle \vec x, \vec y_i\rangle = 0 \textrm{ for all } i \}$.
For a Boolean function $f: \F_2^n \rightarrow \F_2^n$ we denote its [*universal (quantum) embedding*]{} by $$U_f: \F_2^{2n} \rightarrow \F_2^{2n} \textrm{ with } (\vec x, \vec y) \mapsto (\vec x, f(\vec x)+\vec y).$$ Notice that $U_f(U_f(\vec x,\vec y)) = (\vec x, \vec y)$.
Let $\ket{x} \in \mathbb{C}^2$ with $x \in \F_2$ be a qubit. We denote by $H$ the [*Hadamard function*]{} $$x \mapsto \frac 1 {\sqrt 2} (\ket{0} + (-1)^x \ket{1}).$$ We briefly write $H_n$ for the $n$-fold tensor product $H \otimes \ldots \otimes H$. Let $\ket{x}\ket{y}\in \mathbb{C}^4$ be a $2$-qubit system. The $\cnot$ (controlled **not**) function is the universal embedding of the identity function, i.e. $\ket{x}\ket{y} \mapsto \ket{x}\ket{x+y}$. We call the first qubit $\ket{x}$ [*control bit*]{}, since we perform a **not** on $\ket{y}$ iff $x=1$.
A [*Simon function*]{} is a periodic $(2:1)$-Boolean function defined as follows.
\[def:simon\] Let $f:\F_2^n\to \F_2^n$. We call $f$ a [*Simon function*]{} if there exists some period $\vec s \in \F_2^n \setminus\Zero$ such that for all $\vec x, \vec y \in \F_2^n$ we have $$f( \vec x) = f( \vec y) \Leftrightarrow \vec y = \vec x + \vec s.$$ In [*Simon’s problem*]{} we have to find $\vec s$ given oracle access to $f$.
In order to solve Simon’s problem classically, we have to find some collision $\vec x \not= \vec y$ satisfying $f(\vec x) = f(\vec y)$. It is well-known that this requires $\Omega( 2^{\frac n 2})$ function evaluations.
Simon’s quantum algorithm [@FOCS:Simon94], called <span style="font-variant:small-caps;">Simon</span> (see \[alg:simon\]), solves Simon’s problem with only $\bigO(n)$ function evaluations on a quantum circuit. It is known that on input $\ket{0^{n}} \otimes \ket{0^{n}}$ a measurement of the first $n$ qubits of the quantum circuit $Q^{\Simon}_{f}$ depicted in \[circuit:simon\] yields some $\vec y \in \F_2^n$ that is orthogonal to $\vec s$. Moreover, $\vec y \in \F_2^n$ is uniformly distributed in the subspace ${\vec s}^{\perp}$, t.i. we obtain each $\vec y \in {\vec s}^{\perp}$ with probability $\frac 1 {2^{n-1}}$. <span style="font-variant:small-caps;">Simon</span> repeats to measure $Q^{\Simon}_{f}$ until it has collected $n-1$ linearly independent vectors $\vec y_1, \ldots, \vec y_{n-1}$, from which $\vec s$ can be computed via linear algebra in polynomial time. It is not hard to see that a collection of $n-1$ linearly independent vectors requires only $\bigO(n)$ function evaluations.
![Quantum circuit $Q^{\Simon}_{f}$[]{data-label="circuit:simon"}](circuit_Simon.pdf)
Set $Y=\emptyset$. Compute the unique $\vec s \in Y^{\perp} \setminus \{ \vec 0 \}$.
At this point we should stress that <span style="font-variant:small-caps;">Simon</span> only works for [*error-free*]{} quantum computations. Hence we have to ensure that each $\vec y$ is indeed in ${\vec s}^{\perp}$. Assume that we obtain in line \[line:choicey\] of algorithm <span style="font-variant:small-caps;">Simon</span> at least a single $\vec y$ with $\langle \vec y, \vec s \rangle=1$. Then the output of <span style="font-variant:small-caps;">Simon</span> is always false! Thus, <span style="font-variant:small-caps;">Simon</span> is not robust with respect to computational errors on the quantum device.
More precisely, if we obtain in line \[line:choicey\] erroneous $\vec y \notin {\vec s}^{\perp}$ with probability $\tau$, $0 < \tau \leq \frac 1 2$, then <span style="font-variant:small-caps;">Simon</span> outputs the correct $\vec s$ only with exponentially small probability $(1-\tau)^n$. This motivates our following quite simple error model.
\[def:error\_model\] Let $\tau \in \mathbb{R}$ with $0 \leq \tau \leq \frac 1 2$. Upon measuring the first $n$ qubits of $Q^{\Simon}_{f}$, our quantum device outputs with probability $1-\tau$ some uniformly random $\vec y \in {\vec s}^{\perp}$, and with probability $\tau$ some uniformly random $\vec y \in \F_2^n \setminus {\vec s}^{\perp}$. That is, the output distribution is $$\Pr[Q^{\Simon}_{f} \textrm{ outputs } \vec y] =
\begin{cases}
\frac{1-\tau}{2^{n-1}} & \textrm{if } \vec y \in {\vec s}^{\perp} \\
\frac{\tau}{2^{n-1}} & \textrm{else}
\end{cases}\;.$$ We call $\tau$ the [*error rate*]{} of our quantum device.
In the subsequent \[sec:ibmq\] we show that the realization of quantum circuits approximately follows our error model of \[def:error\_model\].
Notice that intuitively there is no efficient way to tell whether $\vec y \in {\vec s}^{\perp}$. This intuition is stated more precisely in \[sec:reduction\], where we show that computing $\vec s$ from the distribution in \[def:error\_model\] is as hard as solving the Learning Parity with Noise (LPN) problem.
Quantum Period Finding on {#sec:ibmq}
==========================
We ran our experiments on the Melbourne, which (despite its name) realizes $14$-qubit circuits. Let us number ’s qubits as $0, \ldots, 13$. Our implementation goal was to realize quantum period finding for Simon functions $f_{\vec s}: \F_2^n \rightarrow \F_2^n$ with error rate as small as possible. To this end we used the following optimization criteria.
#### Gate count.
realizes several $1$-qubit gates such as Hadamard and rotations, but only the $2$-qubit gate $\cnot$. On , the application of any gates introduces some error, where especially the $2$-qubit $\cnot$ introduces approximately as much error as ten $1$-qubit gates (see \[sec:appendix\], \[table:calibration\]). Therefore, we introduce a circuit norm that defines a weighted gate count, which we minimize in the following.
\[def:cnnorm\] Let $Q$ be a quantum circuit with $g_1$ many 1-qubit gates and $g_2$ many 2-qubit gates. Then we define $Q$’s [*circuit-norm*]{} as $\CN(Q):=g_1 + 10 g_2.$
#### Topology.
can only process 2-qubit gates on qubits that are adjacent in its topology graph, see \[fig:IBM\_topo\]. Let $G=(V,E)$ be the directed topology graph, where node $i$ denotes qubit $i$. Moreover, let $\bar G = (V, \bar E)$ be the undirected version of $G$, i.e. we have $\{u,v\} \in \bar E$ iff $(u,v) \in E$ or $(v,u) \in E$.
![Topology graph $G(V,E)$ of .[]{data-label="fig:IBM_topo"}](Topologie_IBM_Q_16.png){width="\textwidth"}
If $(u,v) \in E$ then we can directly implement $\cnot(u, v)$, where $u$ serves as the control bit. If we wish to implement $\cnot(v, u)$ instead, we may use the identity of \[fig:CNOT\] at the cost of an additional $4$ Hadamard gates. Hence, we call qubits $u, v$ [*adjacent*]{} iff $\{u, v\} \in \bar E$.
![Control bit change[]{data-label="fig:CNOT"}](circuit_CNOT.pdf)
Let us assume that we want to realize $\cnot(1, 3)$ in our algorithm. Since $\{1,3\} \notin \bar E$ we cannot directly realize this operation. But we may first swap the contents of qubits $2$ and $3$ by realizing a $\mathbf{swap}$ gate via 3 $\cnot$s as depicted in \[fig:SWAP\]. Since $(2,3) \in E$, we realize the first and third $\cnot$ directly, whereas the second $\cnot$ is realized as in \[fig:CNOT\]. Thus, with a total of 3 $\cnot$ and 4 Hadamards we swap the content of qubit $3$ into $2$. Since $(1,2) \in E$, we may now apply $\cnot(1, 2)$.
![Realisation of **swap** via 3 $\cnot$s and 4 Hadamards.[]{data-label="fig:SWAP"}](circuit_SWAP.pdf)
#### Function choice.
Let $\vec s \in \F_2^n \setminus \{\vec 0\}$, and let $i \in [0,n-1]$ with $s_i = 1$. We define $$f_{\vec s}: \F_2^n \rightarrow \F_2^n, \quad \vec x \mapsto \vec x + x_i \cdot \vec s.$$ Let us first show that $f_{\vec s}$ is indeed a Simon function as given in \[def:simon\]. We have for all $\vec x \in \F_2^n$ that $$f_{\vec s}(\vec x + \vec s) = \vec x + \vec s + (\vec x + \vec s)_i \cdot \vec s = \vec x + \vec s + (x_i + 1) \cdot \vec s = \vec x + x_i \cdot \vec s = f_{\vec s}(\vec x).$$ Thus, $f$ has period $\vec s$. It remains to show that $f_{\vec s}$ is $(2:1)$, i.e. that $f_{\vec s}(\vec x) = f_{\vec s}(\vec y)$ implies that $\vec y = \vec x$ or $\vec y = \vec x + \vec s$. From $f_{\vec s}(\vec x) = f_{\vec s}(\vec y)$ we conclude $$\vec x + x_i \cdot \vec s = \vec y + y_i \cdot \vec s.$$ In the case $x_i=y_i$ this implies $\vec x = \vec y$, whereas in the case $x_i \not= y_i$ this implies $\vec y = \vec x + \vec s$.
#### Instantiation and Discussion of Function Choice.
Throughout the paper, we instantiate our function $f_{\vec s}$ with the period $\vec s = (s_{n-1}, \ldots, s_0) = 0^{n-2}11$ and $x_i=x_0$. We may realize $f_{\vec s}$ with $n$ $\cnot$-gates for copying $\vec x$, and an additional $2$ $\cnot$-gates for the controlled addition of $\vec s$ via control bit $0$. See \[fig:dida\_circ\] for an implementation of $f_{\vec s}$ with $n=3$.
![Simon circuit $Q_1$ with our realization of $f_{\vec s}$ and $\CN(Q_1)=56$. The first 3 $\cnot$s copy $\vec x$, the remaining two $\cnot$s add $\vec s=110$.[]{data-label="fig:dida_circ"}](circuit_Didactic.pdf)
Our function choice has the advantage that it can be implemented with only $n+2$ $\cnot$ gates (if we are able to avoid $\mathbf{swap}$s) and $2n$ Hadamards. Thus we obtain a small circuit norm $CN=10(n+2) + 2n$, which in turn implies a relatively small error on . We perform further circuit norm minimization in \[sec:minimize\].
We would like to point out that as a downside of its simplicity, for our class of functions $f_\vec s$ it is classically [*not hard*]{} to find the period $\vec s$. Since $f(1^{n}) + 1^{n} = \vec s$, a single classical $f$-query directly reveals $\vec s$. However, we want to stress that our quantum algorithm does not exploit this property of $f$ in any manner, but instead works for any Simon function. The only reason that we use our simple form $f$ is that forces us to have a low circuit norm for producing a tolerable error.
Minimizing the gate count of $f_{\vec s}$ {#sec:minimize}
-----------------------------------------
We may implement $f_s$ on directly as the circuit $Q_1$ from \[fig:dida\_circ\]. Since $Q_1$ uses $6$ Hadamard- and $5$ $\cnot$-gates, we have circuit norm $\CN(Q_1) = 56$, but only when ignoring ’s topology. As already discussed, only allows $\cnot$s between adjacent qubits in the topology graph $G=(V,E)$ of \[fig:IBM\_topo\].
Thus, compiles $Q_1$ to $Q_2$ as depicted in \[fig:swap\_circ\]. Let us check that $Q_2$ realizes the same circuit as $Q_1$, but only acts on adjacent qubits. Let $U_{f_{\vec s}}: \F_2^6 \rightarrow \F_2^6$ be the universal quantum embedding of $f_{\vec s}$ with $(\vec x, \vec y) \mapsto (\vec x, f(\vec x) + \vec y) = \vec x + x_0 \vec s + \vec y)$. In $U_{f_{\vec s}}$ we first add each $x_i$ to $y_i$ via $\cnot$s, see \[fig:dida\_circ\]. Thus, we have to make sure that each $x_i$ is adjacent to its $y_i$. Second, we add $\vec s = 011$ via $\cnot$s controlled by $x_0$. Thus, we have to ensure that $x_0$ is adjacent to $y_0$ and $y_1$.
We denote by $i: j$ that qubit $i$ contains the value $j$. This allows us to define the starting [*configuration*]{} as $$0: x_0 \quad 1:x_1 \quad 2: x_2 \quad 3: y_0 \quad 4:y_1 \quad 5:y_2.$$ Step 1 of $Q_2$ (see \[fig:dida\_circ\]) performs $\swap(2,3)$ and thus results in configuration $$0:x_0 \quad 1:x_1 \quad 2:y_0 \quad 3:x_2 \quad 4:y_1 \quad 5:y_2.$$ Step 2 of $C_2$ performs $\swap(1, 2)$ as well as $\swap(4, 3)$. This results in configuration $$0:x_0 \quad 1:y_0 \quad 2:x_1 \quad 3:y_1 \quad 4:x_2 \quad 5:y_2.$$ Eventually, Step 3 of $C_2$ performs $\swap(0, 1)$ and $\swap(2, 3)$ resulting in $$0:y_0 \quad 1:x_0 \quad 2:y_1 \quad 3:x_1 \quad 4:x_2 \quad 5:y_2.$$ Since $(1,0), (2,3), (5,4) \in E$, in Step 4 we now compute $\cnot(1,0)$, $\cnot(3,2)$ and $\cnot(4,5)$ by changing for the second and third operation the control bit (see \[fig:CNOT\]). This realizes the computation of $\vec x + \vec y$. For realizing the addition of $x_i \cdot \vec s = x_0 \cdot 011$, in Step 5 we compute $\cnot(1, 0)$ and $\cnot(1, 2)$ using $(1,0), (1,2) \in E$.
![ compiles $Q_1$ to $Q_2$ with $\CN(Q_2)=234$.[]{data-label="fig:swap_circ"}](circuit_Trans_SWAP.pdf){width="\textwidth"}
In total $Q_2$ consumes $34$ $1$-bit gates and $20$ $2$-bit gates and thus has $\CN(Q_2) = 234$, as compared to $\CN(Q_1)=56$. In the following, our goal is the construction of a quantum circuit that implements $Q_1$’s functionality with minimal circuit norm on .
In \[fig:circ:opt\_step\] we start with circuit $Q_3$, for which our optimization eventually results in circuit $Q_4$ (\[fig:circ:opt\_H\]) that can be realized on with gate count only $\CN(Q_4)=33$.
![Circuit $Q_3$.[]{data-label="fig:circ:opt_step"}](circuit_Opt_step.pdf)
From the discussion before, it should not be hard to see that $Q_3$ realizes $Q_{f_{\vec s}}^{\Simon}$, but yet it has to be optimized for . First of all observe that $\cnot$ is self-inverse, and thus we can eliminate the two $\cnot(2,3)$ gates. Afterwards, we can safely remove qubit 3. The resulting situation for qubits $0, 1, 2$ is depicted in \[fig:circ:Optimierung\].
![Optimization of $Q_3$.[]{data-label="fig:circ:Optimierung"}](circuit_Optimierung.pdf){width="\textwidth"}
From \[fig:circ:Optimierung\] we see that the change of control bits from $\cnot(0,1)$, $\cnot(2,1)$ to $\cnot(1,0)$, $\cnot(1,2)$ leads to some cancellation of self-inverse Hadamard gates. Moreover, the second Hadamard of qubit $1$ can be eliminated, since it does not influence the measurement. We end up with circuit $Q_4$ with an optimized gate count of $\CN(Q_4) = 33$.
![Optimized circuit $Q_4$ on with $\CN(Q_4)=33$.[]{data-label="fig:circ:opt_H"}](circuit_Opt.pdf)
Since $(1,0), (1,2),(6,8) \in E$, all three $\cnot$s of $Q_4$ can directly by realized on .
Notice that a configuration with optimal circuit norm is in general not unique. For our example, the following configuration yields the same circuit norm as the configuration of $Q_4$:
$$3:y_0 \quad 4:x_0 \quad 5:y_1 \quad 6:x_1 \quad 8:y_2 \quad 9:x_2.$$
We optimized our implementation by choosing among all configurations with minimal circuit norm the one using ’s qubits of smallest error rate (see \[fig:IBM\_topo\]). The choice of our configurations is given in Table \[tab:config\], a complete list of optimized circuits can be found in \[sec:appendix\], \[fig:collection\_circuit\].
\[1\][>p[\#1]{}]{}
Experiments on IBM Q 16 {#sec:qexperiments}
-----------------------
For each dimension $n=2, \ldots , 7$ we took $8192$ measurements on of our optimized circuits from the previous section. The resulting relative frequencies are depicted in \[fig:simon\_experiment\]. For each $n$, let $S(n)$ denote the set of erroneous measurements in $\F_2^n \setminus \sorth$. Then we compute the error rate $\tau(n)$ as $\tau(n) = \frac{|S(n)|}{8192}$. In \[fig:simon\_experiment\] we draw horizontal lines $\frac{1-\tau(n)}{2^{n-1}}$, respectively $\frac{\tau(n)}{2^{n-1}}$, for the probability distributions of our error model for orthogonal, respectively not orthogonal, vectors.
[0.49]{} ![ measurements of our optimized circuits (see \[sec:appendix\], \[fig:collection\_circuit\]).[]{data-label="fig:simon_experiment"}](n=2_Label.pdf "fig:")
[0.49]{} ![ measurements of our optimized circuits (see \[sec:appendix\], \[fig:collection\_circuit\]).[]{data-label="fig:simon_experiment"}](n=3_Label.pdf "fig:")
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[0.49]{} ![ measurements of our optimized circuits (see \[sec:appendix\], \[fig:collection\_circuit\]).[]{data-label="fig:simon_experiment"}](n=4_Label.pdf "fig:")
[0.49]{} ![ measurements of our optimized circuits (see \[sec:appendix\], \[fig:collection\_circuit\]).[]{data-label="fig:simon_experiment"}](n=5_Label.pdf "fig:")
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[0.49]{} ![ measurements of our optimized circuits (see \[sec:appendix\], \[fig:collection\_circuit\]).[]{data-label="fig:simon_experiment"}](n=6_Label.pdf "fig:")
[0.49]{} ![ measurements of our optimized circuits (see \[sec:appendix\], \[fig:collection\_circuit\]).[]{data-label="fig:simon_experiment"}](n=7_Label.pdf "fig:")
We observe the following:
- [**Vectors in $\vec s^{\perp}$ are more frequent.**]{} We see that in principle $\IBMQ$ works well for period finding. E.g. for $n=3$, we have $\{\vec s\}^{\perp}= \{011\}^{\perp} = \{000, 011,100, 111\}$, and we measure one of these vectors with probability $1- \tau \approx 90 \%$. But this in turn implies that we also measure erroneous vectors in $\F_2^n \setminus \{\vec s\}^{\perp}$ with probability $\tau \approx 10 \%$.
- [**Triangular structure.**]{} In \[fig:simon\_experiment\] we ordered our measurements lexicographically. It seems that the first vectors in lexicographic order are measured with a larger probability than the last vectors. This is not overly surprising, since we also ordered our qubits by quality – starting with lowest error rate for the least significant bit $x_0$ up to highest error rate for the most significant bit $x_{n-1}$ (nevertheless e.g. for $n=3$ it seems that the qubit for $x_2$ performed worse than the one for $x_3$). We can mitigate the effect of different qubit quality by permuting the qubits in our starting configuration such that we retain the same circuit norm. However, we deliberately ordered the qubits in descending quality to make the quality effect visible. The triangular structure would vanish for qubits of similar quality.
- [**Increasing $\tau(n)$.**]{} The error rate $\tau(n)$ is a function increasing in $n$. This is what we expected, since the circuit norm increases with $n$ and for larger $n$ we also had to include lower quality qubits.
- [**Hamming weight.**]{} Usually, measurements with small Hamming weight appear with larger frequencies than large Hamming weight measurements. This is a physical effect that is mainly due to the readout error of the measurements in (see \[sec:appendix\], \[table:calibration\]) and its significant bias towards $0$.
All in all, our error model is an oversimplified model that for ease of exposition ignores facts like error quality of different qubits and issues with Hamming weight. It is not surprising that a single parameter like the error rate $\tau$ cannot all too precisely capture complex physical effects and complex probability distributions. Nevertheless, we show in \[sec:cexperiments\] that our simple error model is accurate enough to predict the run times for extracting the secret vector $\vec s$ from quantum measures with error rate $\tau$.
is Polynomial Time Equivalent to {#sec:reduction}
==================================
In the previous section, we checked experimentally on our error model (\[def:error\_model\]). Recall that our model states that with probability $\tau$ we measure in the quantum circuit $Q^{\Simon}_{f_{\vec s}}$ some uniformly distributed $\vec y \in \F_2^n \setminus \sorth$. The question is now whether such erroneous $\vec y$ can easily be handled.
In this section, we answer this question in the negative. Namely, we show that handling these errors is as hard as the well-studied , which is supposed to be hard even on quantum computers.
\[def:LPN\] Let $\vec s \in \F_2^n \setminus \Zero$ be chosen uniformly at random, and let $\tau\in[0,\frac{1}{2})$. In the [*Learning Parity with Noise*]{} problem, denoted $\LPN_{n,\tau}$, one obtains access to an oracle ${\cal O}_{\LPN}(\vec s)$ that provides samples $(\vec a,\langle \vec a,\vec s \rangle + \epsilon)$, where $\vec a \sim {\cal U}_n$ and $\epsilon \sim \mathrm{Ber}_{\tau}$. The goal is to compute $\vec s$.
\[def:LPN\] explicitly excludes $\vec s = \vec 0$ in . Notice that the case $\vec s = \vec 0$ implies that the oracle has distribution $U_n \times \textrm{Ber}_{\tau}$. However, in the case $\vec s \not= \vec 0$, we have $\Pr_{\vec a}[\langle \vec a , \vec s \rangle=0] = \frac 1 2$, which implies $\Pr_{\vec a}[\langle \vec a , \vec s \rangle + \epsilon=0] = \frac 1 2$. Therefore the samples for $\vec s \not= \vec 0$ have distribution $U_n \times U$. This allows us to easily distinguish both cases by a majority test, whenever $\tau$ is polynomially bounded away from $\frac 1 2$. Hence, $\vec s= \vec 0$ is not a hard case for and may wlog excluded.
Let us now define the related [*Learning Simon with Noise*]{} problem that reflects our error model.
\[def:LSN\] Let $\vec s \in \F_2^n \setminus \Zero$ be chosen uniformly at random, and let $\tau\in[0,\frac{1}{2})$. In the [*Learning Simon with Noise*]{} problem, denoted $\LSN_{n,\tau}$, one obtains access to an oracle ${\cal O}_{\LSN}(\vec s)$ that provides samples $\vec y$, where $\vec y \in \F_2^n$ is distributed as in \[def:error\_model\], i.e. $$\Pr[\vec y] =
\begin{cases}
\frac{1-\tau}{2^{n-1}} & \textrm{, if } y \in \vec s^{\perp} \\
\frac{\tau}{2^{n-1}} & \textrm{, else}
\end{cases}\; \textrm{ and therefore } \Pr[ \langle \vec y , \vec s \rangle = 0 ] = 1-\tau.$$ The goal is to compute $\vec s$.
In the following we prove that $\LSN_{n,\tau}$ is polynomial time equivalent to $\LPN_{n,\tau}$ by showing that we can perfectly mutually simulate ${\cal O}_{\LPN}(\vec s)$ and ${\cal O}_{\LSN}(\vec s)$. The purpose of excluding $\vec s \neq \vec 0$ from $\LPN_{n,\tau}$ is to guarantee in the reduction non-trivial periods $\vec s \neq \vec 0$ in $\LSN_{n,\tau}$.
\[theo:equivalence\] Let ${\cal A}$ be an algorithm that solves $\LPN_{n,\tau}$ (respectively $\LSN_{n,\tau}$) using $m$ oracle queries in time $T$ with success probability $\epsilon_{\cal A}$. Then there exists an algorithm ${\cal B}$ that solves $\LSN_{n,\tau}$ (respectively $\LPN_{n,\tau}$) using $m$ oracle queries in time $T$ with success probability $\epsilon_{\cal B} \geq \frac {\epsilon_{\cal A}} 2$.
Assume that we want to solve $\LSN$ via an algorithm ${\cal A}_{\LPN}$ with success probability $\epsilon_{\cal A}$ as in \[alg:LPNtoLSN\].
Choose $\vec z \sim {\cal U}_n$. \[line:z\] $\vec s \leftarrow {\cal A}_{\LPN}(n, \tau, (\vec y_1 + b_1\vec z, b_1), \ldots, (\vec y_m + b_m\vec z, b_m))$
We show in the following that \[alg:LPNtoLSN\] perfectly simulates the oracle ${\cal O}_{\LPN}(\vec s)$ via ${\cal O}_{\LSN}(\vec s)$ if the vector $\vec z \sim {\cal U}_n$ chosen in \[line:z\] satisfies $\langle \vec z, \vec s \rangle =1$. Since $\vec s \not= \vec 0$, we have $\Pr_{\vec z}[\langle \vec z, \vec s \rangle =1] = \frac 1 2$. Therefore Algorithm \[alg:LPNtoLSN\] succeeds with probability $$\epsilon_{\cal B} \geq \Pr_{\vec z}[\langle \vec z, \vec s \rangle = 1 \cap {\cal A} \textrm{ outputs } \vec s] = \frac {\epsilon_{\cal A}} 2.$$
Let us now show correctness of \[alg:LPNtoLSN\]. We first show that the constructed samples $(\vec y + b \vec z, b)$ have the correct distribution. Let $\epsilon = \langle \vec y + b\vec z, \vec s \rangle + b$. Since $\langle \vec z, \vec s \rangle =1$, we have $$\Pr_{\vec y}[\epsilon =1] = \Pr_{\vec y}[\langle \vec y + b \vec z, \vec s \rangle + b = 1] = \Pr_{\vec y}[\langle \vec y, \vec s \rangle + b \langle \vec z, \vec s \rangle + b = 1] = \Pr_{\vec y}[\langle \vec y , \vec s \rangle = 1] = \tau.$$ It remains to show that $\vec y + b \vec z$ is uniformly distributed. To this end, we show that $$p_0 = \Pr_{\vec y, b}[\vec y + b \vec z \mid \langle \vec y, \vec s \rangle = 0] = \frac 1 {2^n}.$$ Analogous, it follows that $p_1=\Pr_{\vec y, b}[\vec y + b \vec z \mid \langle \vec y, \vec s \rangle = 1] = \frac 1 {2^n}$. From both statements we obtain $$\Pr_{\vec y, b}[\vec y + b \vec z] = \Pr_{\vec y}[\langle \vec y, \vec s \rangle = 0] \cdot p_0 + \Pr_{\vec y}[\langle \vec y, \vec s \rangle = 1] \cdot p_1 = \frac{1-\tau}{2^n} + \frac{\tau}{2^n} = \frac 1 {2^n},$$ as desired. It remains to show that $$\begin{aligned}
p_0 & = \Pr_{\vec y, b}[\vec y + b \vec z \mid \langle \vec y, \vec s \rangle = 0] \\
& = \Pr_b[b=0] \cdot \Pr_{\vec y}[\vec y \mid \langle \vec y, \vec s \rangle = 0] + \Pr_b[b=1] \cdot \Pr_{\vec a}[\vec y + \vec z \mid \langle \vec y, \vec s \rangle = 0] \\
& = \frac 1 2 \left( \frac{1-\tau}{2^{n-1}} + \frac{\tau}{2^{n-1}} \right) = \frac{1}{2^n}.\end{aligned}$$ This completes the analysis of Algorithm \[alg:LPNtoLSN\].
Choose $\vec z \sim {\cal U}_n$. $\vec s \leftarrow {\cal A}_{\LSN}(n, \tau, \vec a_1 + b_1\vec z, \ldots, \vec a_m + b_m\vec z)$
For Algorithm \[alg:LSNtoLPN\] we conclude the success probability analogous to the reasoning for Algorithm \[alg:LPNtoLSN\], i.e. we succeed when $\langle \vec z , \vec s \rangle =1$ and ${\cal A}_{\LSN}$ succeeds. So let us assume in the following correctness analysis that we are in the case $\langle \vec z , \vec s \rangle =1$. This implies for the constructed samples $\vec a + b \vec z$ that $$\langle \vec a + b \vec z, \vec s \rangle = 0 \Leftrightarrow \langle \vec a , \vec s \rangle + b \langle \vec z , \vec s \rangle = 0 \Leftrightarrow \langle \vec a, \vec s \rangle = b.$$ Let $\epsilon= \langle \vec a, \vec s \rangle + b$. It follows that $$\Pr_{\vec a, b}[\langle \vec a + b \vec z, \vec s \rangle = 0] = \Pr_{\vec a, b}[ \langle \vec a, \vec s \rangle = b] = \Pr_{\vec a, b}[\epsilon=0] = 1- \tau.$$ We also have to show that we obtain a uniform distribution among all $\vec a + b \vec z \in \vec s^{\perp}$. This follows from $$\begin{aligned}
& \ \Pr_{\vec a, b}[\vec a + b \vec z \mid \langle \vec a + b \vec z, \vec s \rangle = 0 ] = \Pr_{\vec a, b}[\vec a + b \vec z \mid \langle \vec a , \vec s \rangle = b ] \\
= & \ \Pr_{\vec a}[\langle \vec a, \vec s \rangle=0] \cdot \Pr_{\vec a, b}[\vec a + b \vec z \mid \langle \vec a , \vec s \rangle = b = 0 ] \ + \\
& \ \Pr_{\vec a}[\langle \vec a, \vec s \rangle=1] \cdot \Pr_{\vec a, b}[\vec a + b \vec z \mid \langle \vec a , \vec s \rangle = b = 1 ] \\
= & \ \frac 1 2 \cdot \Pr_{\vec a}[\vec a \mid \langle \vec a , \vec s \rangle = 0 ] + \frac 1 2 \cdot \Pr_{\vec a}[\vec a + \vec z \mid \langle \vec a , \vec s \rangle = 1 ] \\
= & \ \frac 1 2 \cdot \frac{1}{2^{n-1}} +\frac 1 2 \cdot \frac{1}{2^{n-1}} = \frac{1}{2^{n-1}}.\end{aligned}$$
Analogous, we can show that we obtain a uniform distribution among all $\vec a + b \vec z \in\F_2^n\setminus\sorth$. This proves that we perfectly simulate $\LSN$-samples via ${\cal O}_{\LPN}$, and thus shows correctness of \[alg:LSNtoLPN\].
\[theo:equivalence\] shows that under the $\LPN$ assumption we cannot expect to solve $\LSN$ — i.e. to handle error-prone quantum measurements in Simon’s algorithm — in polynomial time. However, it does not exclude that quantum measurements are still useful in the sense that they help us to solve period finding faster than on classical computers. In the following section, we show that our quantum output indeed leads to speedups even for large error rates $\tau$.
Theoretical Error Handling for Simon’s Algorithm {#sec:correct}
================================================
Recall that period finding for $n$-bit Simon functions classically requires time $\Omega(2^{\frac n 2})$. So despite the hardness results of \[sec:reduction\] we may still hope that even error-prone quantum measurements lead to period finding speedups. Indeed, it is well-known that for any fixed $\tau < \frac 1 2 $ the BKW algorithm [@STOC:BluKalWas00] solves $\LPN_{n, \tau}$ — and thus by \[theo:equivalence\] also $\LSN_{n, \tau}$ — in time $2^{\bigO\big(\frac{n}{\log n}\big)}$. This implies that asymptotically the combination of quantum measurements together with a suitable LPN-solver already outperforms classical period finding.
However, the BKW algorithm has sample and memory consumption $2^{\Theta\big(\frac{n}{\log n}\big)}$, which makes it quite impractical in practice. Therefore, we want to focus on $\LPN$-algorithms that consume only a small amount of samples and memory. We start with the analysis of the <span style="font-variant:small-caps;">Pooled Gauss</span> algorithm that was introduced at Crypto ’17 by Esser, Kübler and May [@C:EssKubMay17]. <span style="font-variant:small-caps;">Pooled Gauss</span> solves $\LPN_{n, \tau}$ in time $\tilde{\Theta}\left( 2^{\log\left(\frac{1}{1-\tau}\right)\cdot n} \right)$ using $\tilde{\Theta}\left( n^2 \right)$ samples and $\tilde{\Theta}\left( n^3 \right)$ memory.
The following theorem shows that period finding with error-prone quantum samples in combination with <span style="font-variant:small-caps;">Pooled Gauss</span> is superior to purely classical period finding whenever the error $\tau$ is bounded by $\tau \leq 0.293$.
\[theo:pgauss\] In our error model (\[def:error\_model\]), <span style="font-variant:small-caps;">Pooled Gauss</span> finds the period $\vec s \in \F_2^n$ of a Simon function $f_{\vec s}$ using $\tilde{\Theta}\left( n^2 \right)$ many $\LSN_{n,\tau}$-samples, coming from practical measurements of Simon’s circuit $Q^{\Simon}_{f_{\vec s}}$ with error rate $\tau$, in time $\tilde{\Theta}\left( 2^{\log\left(\frac{1}{1-\tau}\right)\cdot n} \right)$. This improves over classical period finding for error rates $$\tau < 1-\frac{1}{\sqrt 2} \approx 0.293.$$
We use \[alg:LPNtoLSN\], where any $\bigO_{\LPN}(\vec s)$-call is provided by a measurement of $Q^{\Simon}_{f_{\vec s}}$. In our error model, this gives us an $\LSN_{n,\tau}$-instance which is transformed by \[alg:LPNtoLSN\] into an $\LPN_{n,\tau}$-instance. We use <span style="font-variant:small-caps;">Pooled Gauss</span> as the LPN-solver ${\cal A}_{\LPN}$ inside \[alg:LPNtoLSN\]. This immediately implies time complexity $\tilde{\Theta}\left( 2^{\log\left(\frac{1}{1-\tau}\right)\cdot n} \right)$.
It remains to show outperformance of the classical algorithm, i.e. $\log\left(\frac{1}{1-\tau}\right) < \frac 1 2$. Notice that our condition $\tau < 1-\frac{1}{\sqrt 2}$ implies that $\frac{1}{1-\tau} < \sqrt{2}$ and therefore $$\log\left(\frac{1}{1-\tau}\right) < \log(\sqrt{2}) = \frac 1 2.$$
\[theo:pgauss\] already shows the usefulness of a quite limited quantum oracle that only allows us polynomially many measurements, whenever its error rate $\tau$ is small enough.
If we allow for more quantum measurements, the <span style="font-variant:small-caps;">Well-Pooled Gauss</span> algorithm of Esser, Kübler and May [@C:EssKubMay17] solves $\LPN_{n,\tau}$ in improved time and query complexity $\tilde{\Theta}( 2^{f(\tau) n})$, where $f(\tau) = 1-\frac{1}{1+\log(\frac{1}{1-\tau})}$, using $\tilde{\Theta}\left( n^3 \right)$ memory. The following theorem shows that <span style="font-variant:small-caps;">Well-Pooled Gauss</span> in combination with error-prone quantum measurements improves on classical period finding for [*any*]{} error rate $\tau$.
\[theo:wpgauss\] In our error model (\[def:error\_model\]), <span style="font-variant:small-caps;">Well Pooled Gauss</span> finds the period $\vec s \in \F_2^n$ of a Simon function $f_{\vec s}$ using $\tilde{\Theta}( 2^{f(\tau) n})$ many $\LSN_{n,\tau}$-samples, coming from practical measurements of Simon’s circuit $Q^{\Simon}_{f_{\vec s}}$ with error rate $\tau$, in time $\tilde{\Theta}( 2^{f(\tau) n})$, where $$f(\tau) = 1-\frac{1}{1+\log(\frac{1}{1-\tau})}.$$ This improves over classical period finding for [*all error rates*]{} $\tau < \frac 1 2$.
As in the proof of \[theo:wpgauss\] we use \[alg:LPNtoLSN\], where measurements of $Q^{\Simon}_{f_{\vec s}}$ provide the $\bigO_{\LPN}(\vec s)$-calls and <span style="font-variant:small-caps;">Well Pooled Gauss</span> is the LPN-solver ${\cal A}_{\LPN}$. Correctness and the claimed complexities follow immediately.
It remains to show outperformance of any classical period finding algorithm. Notice that $\tau < \frac 1 2$ implies $\frac{1}{1-\tau} < 2$ and therefore $\log(\frac{1}{1-\tau}) < 1$. This in turn implies $$f(\tau) = 1-\frac{1}{1+\log(\frac{1}{1-\tau})} < 1-\frac{1}{2} = \frac 1 2.$$
The results of \[theo:pgauss\] and \[theo:wpgauss\] show that quantum measurements of $Q^{\Simon}_{f_{\vec s}}$ always help us (asymptotically) even for large error rates $\tau$, provided that our error model is sufficiently accurate. In the following section, we show that our simple error model is in practical experiments sufficiently precise to predict run times.
Practical Error Handling for Simon’s Algorithm {#sec:cexperiments}
==============================================
In this section, we want to handle errors in quantum measurements of Simon’s circuit $Q_{f_{\vec s}}^{\Simon}$ in practice. By the result of \[sec:reduction\] we may first transform our quantum measurements into LPN samples, and then use one of the LPN-algorithm from \[sec:correct\]. Since the error rates from our measurements in \[sec:qexperiments\] are below error rate $\tau \leq 0.15$, according to \[theo:pgauss\] for sufficiently large $n$ LPN-solver <span style="font-variant:small-caps;">Pooled Gauss</span> already outperforms classical period finding.
The goal of this section is to check the accuracy of our error model with respect to the prediction of run times in the presence of errors. Therefore, we do not implement the error handling detour via reduction to LPN, but we tackle the LSN problem directly. To this end, we simply adapt the <span style="font-variant:small-caps;">Pooled Gauss</span> algorithm to the LSN setting. This is done in \[alg:SPG\], called <span style="font-variant:small-caps;">Pooled Simon</span>.
In practice, the input pool $P$ consists of $Q_{f_{\vec s}}^{\Simon}$-measurements. It is not hard to see that <span style="font-variant:small-caps;">Pooled Simon</span> succeeds iff $y_i \in \vec s^{\perp}$ for all $\vec y_i \in Y$. Thus, <span style="font-variant:small-caps;">Pooled Simon</span> works similar to the original Simon algorithm (\[alg:simon\]), but repeats until it finds some error free set $Y$ of measurements. If we would take fresh quantum measurements in each iteration of the **repeat**-loop, then we succeed in a single iteration with probability $(1-\tau)^n$. This implies an expected run time of $(\frac{1}{1-\tau})^n$ iterations for <span style="font-variant:small-caps;">Pooled Simon</span>. It was shown in [@C:EssKubMay17] that this analysis also holds if we choose $Y \subseteq P$ for sufficiently large pools $P$.
In order to check the accuracy of our error model, we first ran <span style="font-variant:small-caps;">Pooled Simon</span> for every $n=2, \ldots, 7$ with our pools $P$ of $2^{13}$ quantum measurements from \[sec:qexperiments\]. Second, we also ran <span style="font-variant:small-caps;">Pooled Simon</span> with pools $P$ of $2^{13}$ randomly chosen, perfectly distributed LSN samples. As the run time cost we took the number of $f_{\vec s}$ evaluations, i.e. the number of iterations plus one for evaluating $f_{\vec s}(\vec 0)$, averaged over $1000$ runs of <span style="font-variant:small-caps;">Pooled Simon</span>. In \[tab:pool\_runtime\] we give the resulting run times $T_\text{QM}$ for our quantum measurements and $T_\text{LSN}$ for LSN samples.
[|\*[4]{}[c|]{}]{} $(n,\tau(n))$& $T_\text{QM}$ & $T_\text{LSN}$ & $\frac{T_\text{QM} - T_\text{LSN}}{n^2} \vphantom{\frac{\frac{8}{8}}{\frac{8}{8}}}$\
$\ (2,0.098) \ $ & $\ \mathbf{1.238} \ $ & $\ \mathbf{1.216} \ $ & $\ 5.5 \cdot 10^{-3} \ \vphantom{\frac{1^1}{1^1}}$\
$(3,0.099)$ & $ \mathbf{1.443} $ & $ \mathbf{1.398} $ & $5.0 \cdot 10^{-3}\vphantom{\frac{1^1}{1^1}}$\
$(4,0.109)$ & $ \mathbf{1.727} $ & $ \mathbf{1.644} $ & $5.2 \cdot 10^{-3}\vphantom{\frac{1^1}{1^1}}$\
$(5,0.126)$ & $ \mathbf{2.160} $ & $ \mathbf{2.066} $ & $3.8 \cdot 10^{-3}\vphantom{\frac{1^1}{1^1}}$\
$(6,0.118)$ & $ \mathbf{2.366} $ & $ \mathbf{2.245} $ & $3.4 \cdot 10^{-3}\vphantom{\frac{1^1}{1^1}}$\
$(7,0.147)$ & $ \mathbf{3.440} $ & $ \mathbf{3.251} $ & $3.9 \cdot 10^{-3}\vphantom{\frac{1^1}{1^1}}$\
From \[tab:pool\_runtime\] we see that $T_\text{LSN}$ quite accurately predicts $T_{QM}$, but that in general it slightly underestimates $T_{QM}$. This in turn implies that for <span style="font-variant:small-caps;">Pooled Simon</span> it might be a bit harder to handle errors in quantum measurement than solving $\LSN$ (or equivalently $\LPN$). This seems reasonable because <span style="font-variant:small-caps;">Pooled Simon</span> should profit from the $\LSN$ samples’ uniformity. However, this does not exclude other algorithms that might be tailored to and profit from the specific distribution of quantum measurements.
But of course, we should not over-interpret our very small run times in very small dimension. Assume e.g. that $T_\text{QM}$ and $T_\text{LSN}$ differ only by an additive term $\bigO(n^2)$. Then the term $\frac{T_\text{QM} - T_\text{LSN}}{n^2}$ should be upper bounded by a constant, which seems to hold quite well for the limited data in \[tab:pool\_runtime\]. In this case, our error model would be (asymptotically) highly accurate. Only experiments on quantum devices with more qubits can tell us more.
#### Acknowledgement.
We acknowledge use of the IBM Q for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Q team.
Appendix {#sec:appendix}
========
Average measurement Range of measurements
----------------------- ------------------------- ----------------------------------------------------------------------
Gate error (Hadamard) $\ 6.28\cdot10^{-3} \ $ $\ (1.67\cdot10^{-3},14.43\cdot10^{-3})\vphantom{\frac{1^1}{1^1}}\ $
Gate error ($\cnot$) $7.83\cdot10^{-2}$ $(3.15\cdot10^{-2},13.47\cdot10^{-2})\vphantom{\frac{1^1}{1^1}}$
Readout error $6.48\cdot10^{-2}$ $(2.58\cdot10^{-2},17.83\cdot10^{-2})\vphantom{\frac{1^1}{1^1}}$
T1 ($\mu$s) $50.54$ $(23.65,91.36)\vphantom{\frac{1^1}{1^1}}$
T2 ($\mu$s) $69.14$ $(25.21,119.98)\vphantom{\frac{1^1}{1^1}}$
: Calibration facts for our measurements[]{data-label="table:calibration"}
[0.49]{} ![Optimized circuits for $n=2, \ldots, 7$ with $\vec s = 0^{n-2}11$. We omit qubit $7$ with input $y_0$, which is not required after optimization.[]{data-label="fig:collection_circuit"}](circuit_Opt_n=2.pdf "fig:"){width="83.00000%"}
[0.49]{} ![Optimized circuits for $n=2, \ldots, 7$ with $\vec s = 0^{n-2}11$. We omit qubit $7$ with input $y_0$, which is not required after optimization.[]{data-label="fig:collection_circuit"}](circuit_Opt_n=3.pdf "fig:"){width="83.00000%"}
\
[0.49]{} ![Optimized circuits for $n=2, \ldots, 7$ with $\vec s = 0^{n-2}11$. We omit qubit $7$ with input $y_0$, which is not required after optimization.[]{data-label="fig:collection_circuit"}](circuit_Opt_n=4.pdf "fig:"){width="83.00000%"}
[0.49]{} ![Optimized circuits for $n=2, \ldots, 7$ with $\vec s = 0^{n-2}11$. We omit qubit $7$ with input $y_0$, which is not required after optimization.[]{data-label="fig:collection_circuit"}](circuit_Opt_n=5.pdf "fig:"){width="83.00000%"}
\
[0.49]{} ![Optimized circuits for $n=2, \ldots, 7$ with $\vec s = 0^{n-2}11$. We omit qubit $7$ with input $y_0$, which is not required after optimization.[]{data-label="fig:collection_circuit"}](circuit_Opt_n=6.pdf "fig:"){width="83.00000%"}
[0.49]{} ![Optimized circuits for $n=2, \ldots, 7$ with $\vec s = 0^{n-2}11$. We omit qubit $7$ with input $y_0$, which is not required after optimization.[]{data-label="fig:collection_circuit"}](circuit_Opt_n=7.pdf "fig:"){width="83.00000%"}
[^1]: Funded by DFG under Germany’s Excellence Strategy - EXC 2092 CASA - 390781972.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Using numerical integrations, we find that the orbital eccentricity of Saturn’s moon Iapetus undergoes prominent multi-Myr oscillations. We identify the responsible resonant argument to be $\varpi-\varpi_{g5}+\Omega-\Omega_{eq}$, with the terms being the longitudes of pericenter of Iapetus and planetary secular mode $g_5$, Iapetus’s longitude of the node and Saturn’s equinox. We find that this argument currently (on a $10^7$ yr timescale) appears to librate with a very large amplitude. On longer timescales, the behavior of this resonant angle is strongly dependent on the resonant interaction between Saturn’s spin axis and the planetary mode $f_8$, with long-term secular resonance being possible if Saturn’s equinox is librating relative to the node of the $f_8$ eigenmode. We present analytical estimates of the dependence of the resonant argument on the orbital elements of Iapetus. We find that this Iapetus-$g_5$ secular resonance could have been established only after the passage of Iapetus through the 5:1 mean-motion resonance with Titan, possibly in the last Gyr. Using numerical simulations, we show that the capture into the secular resonace appears to be a low-probability event. While the Iapetus-$g_5$ secular resonance can potentially help us put new constraints on the past dynamics of the Saturnian system, uncertainties in both the spin axis dynamics of Saturn and the tidal evolution rate of Titan make it impossible to make any firm conclusions about the resonance’s longevity and origin.'
author:
- |
Matija [Ć]{}uk,$^{1}$[^1] Luke Dones,$^{2}$ David Nesvorn[ý]{}$^{2}$ and Kevin J. Walsh$^{2}$\
$^{1}$SETI Institute, 189 North Bernardo Avenue, Suite 200, Mountain View, CA 94043, USA\
$^{2}$Southwest Research Institute, 1050 Walnut Street, Suite 400, Boulder, CO 80302, USA
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: Secular Resonance Between Iapetus and the Giant Planets
---
\[firstpage\]
planets and satellites: dynamical evolution and stability – planets and satellites: individual: Iapetus – celestial mechanics
Introduction
============
Iapetus is the third-largest moon of Saturn, as well as the major moon that is the most distant from the planet. Iapetus is notable for its albedo dichotomy [@bur95; @por05], oblate shape [@tho07; @cas11], and equatorial ridge [@lev11; @dom12; @sti18], but here we will restrict ourselves to studying its orbital motion. Like other regular satellites, Iapetus has a relatively low orbital eccentricity ($e_I=0.028$), but it also has a substantial orbital inclination ($i_I=8^{\circ}$ with respect to its Laplace plane[^2]), the origin of which has been a long-standing problem [@war81; @nes14]. As the solar perturbations on Iapetus’s orbit are comparable to those arising from Saturn’s oblateness and the inner moons (chiefly Titan), the Laplace plane of Iapetus is significantly tilted to Saturn’s equator($i_L=14^{\circ}$). As Iapetus’s orbit precesses around its Laplace plane, the instantaneous inclination of Iapetus to Saturn’s equator varies approximately over a $5^{\circ}-21^{\circ}$ range over Iapetus’s nodal precession period of about 3400 yr.
Iapetus’s inclination contradicts the established opinion that Iapetus and other regular satellites formed from a flat disk surrounding Saturn. Any disk consisting of gas and/or small particles that is inclined to the local Laplace plane would be subject to differential nodal precession at different distances. Through collisions and other dissipative mechanisms, the disk would soon settle into the local Laplace plane. A satellite that forms from such a disk should have no inclination at all. Therefore, if Iapetus formed in orbit around Saturn (as suggested by its prograde, low-eccentricity orbit), some dynamical process had to impart inclination to Iapetus after its formation. @war81 suggested that Iapetus’s inclination could have been generated through rapid gas disk dissipation. If the circumplanetary disk could disappear in a time comparable to or shorter than the 3400-year nodal precession period of Iapetus, the resulting change in the Laplace plane could induce a substantial free inclination. However, it is not clear that the circumplanetary disk would disappear on such a short timescale [@mar11].
Another potential source of Iapetus’s inclination would be close encounters between Saturn and ice giants during planetary migration [@tho99; @tsi05]. If these encounters were to operate as a classic random-walk process, they would excite a distant satellite’s eccentricity more than its inclination [@pah15]. However, @nes14 found that in a significant number of planetary flybys they simulated, the inclination of Iapetus was excited by several degrees while its eccentricity stayed well below 0.01. This behavior was associated with distant encounters ($r>0.1$ AU), and the inclination excitation was apparently driven by secular torques from highly-inclined passing ice-giants, which had little effect on the eccentricity. Such distant encounters between Saturn and the ice giants were also found to be capable of capturing the existing irregular satellites of Saturn [@nes07a; @nes14a].
Recently, there has been some reconsideration of the dynamical history of the Saturnian system, prompted by observations of much faster than expected tidal evolution [@lai12; @lai17]. While in the classical picture [e.g. @md99] Iapetus does not take part in any resonances with other satellites, faster tidal evolution would make Titan and Iapetus cross their mutual 5:1 mean-motion resonance in the past. This crossing should have happened about 500 Myr ago if we assume a uniform tidal quality factor $Q=1500-2000$ for all satellites [@cuk13], or could have happened at a very different epoch if the tidal evolution of Saturn’s moons is driven by resonant modes inside the planet [@ful16]. Since this paper deals with the relatively recent past (a few hundred Myr), we will mostly assume that Titan’s orbital evolution is driven by Saturn’s constant tidal quality factor $Q=1500$ and tidal Love number $k_2$ (tidal evolution of Iapetus is negligible in this model).
Current Dynamics of Iapetus with a Fixed-Obliquity Saturn
=========================================================
We start our study by importing position and velocity vectors for Iapetus, Titan and the four giant planets (with the epoch of January 1, 2000) from the Jet Propulsion Laboratory’s HORIZONS ephemeris system[^3]. We use these vectors as initial conditions in simulations using numerical integrators derived from [simpl]{}, which was previously employed by @cuk16. Briefly, [simpl]{} is a mixed-variable, symplectic integrator based on an algorithm of @cha02 that simultaneously integrates the orbits of the planets and satellites of one of the planets. The basic version of [simpl]{} includes all mutual perturbations (except the satellites’ effects on planets), as well as the parent planet’s oblateness, tidal torques on satellites and additional migration forces (to account for ring or disk torques, when necessary). One important limitation of [simpl]{} is that the planet’s spin axis is stationary and not affected by any of the torques that would act on it in the real system (this includes both precession-inducing gravitational torques and tidal dissipation with the planet). In the case of Saturn, this approximation is justified when studying the relatively fast dynamics of the inner satellites [@cuk16], as their orbital precession periods are on the order of years and decades, while the precession period of Saturn’s spin axis is longer than 1 Myr [@fre17]. Even when dealing with Titan and Iapetus, precession periods are still shorter than $10^4$ yr, seemingly making Saturn’s pole precession irrelevant. However, when studying longer-period dynamics, precession and other motions of Saturn’s spin axis will need to be taken into account, as detailed below.
![Top: Eccentricity of Iapetus during a 10 Myr integration of Iapetus’s orbit using [psimpl]{}. Bottom: Evolution of the resonant argument $\varpi-\varpi_J+\Omega-\Omega_{eq}$ in the same simulation.[]{data-label="psim2"}](psim2.eps){width="\columnwidth"}
Our first and simplest modification of [simpl]{} so we can study the dynamics of Iapetus’s orbit over Myr timescales is the introduction of uniform precession of Saturn’s spin axis around the invariable plane. The version of [simpl]{} modified in this manner is designated [psimpl]{}, with a “[p]{}” signifying precession. Figure \[psim2\] (top panel) shows the evolution of Iapetus’s eccentricity over 10 Myr integrated using [psimpl]{}, assuming Saturn’s axial precession period to be 1.96 Myr. In this integration we included the full orbital dynamics of the four giant planets, as well as Titan and Iapetus. We ignored Hyperion and included the satellites interior to Titan into Saturn’s $J_2$ obliquity term. A periodic variation with a period of about 4 Myr is clearly present in Fig. \[psim2\], with the variation comparable to the average eccentricity of Iapetus. This variation is clearly caused by a very slow-changing resonant (or near-resonant) argument, and its very long period compared to the 3400-year apsidal precession period of Iapetus (which is the conjugate of angular momentum and must be present in a eccentricity-affecting term) suggests a near-canceling of two similar precession terms. @cuk16 found a somewhat similar resonance involving the sum of apsidal and nodal precessions of the inner moons. The near-identical precession rates (with opposite signs) of the apsidal and nodal precession for Tethys and Dione produce very slow-changing secular terms, with a rate of change more than two orders of magnitude slower than the basic precession frequencies [@cuk16]. This inspired us to investigate terms including the angle $\varpi+\Omega$, which has a period of about $3 \times 10^5$ yr. This term is close to secular resonance with the $g_5$ mode of planetary eccentricities (i.e. the “slow” or “aligned” mode of Jupiter and Saturn). In order to satisfy the D’Alembert rules for the arguments of the disturbing function [@md99], an additional very slowly evolving node-type angle is necessary; we opted for the longitude of Saturn’s equinox (with respect to the invariable plane), as it determines the orientation of Iapetus’s Laplace plane. The evolution of the resulting resonant argument $\varpi-\varpi_J+\Omega-\Omega_{eq}$ is plotted in the bottom panel of Fig. \[psim2\], where $\varpi$ and $\varpi_J$ are the longitudes of pericenter of Iapetus and Jupiter[^4], while $\Omega$ and $\Omega_{eq}$ are the longitudes of Iapetus’s ascending node and Saturn’s vernal equinox. Fig. \[psim2\] clearly indicates that a term with this argument is responsible for the variations in Iapetus’s eccentricity, and that the resonant argument appears to librate with a large amplitude over the next 10 Myr.
While the secular resonances usually evolve on precession timescales, Iapetus-$g_5$ secular resonance described here has a more slowly evolving argument involving $\varpi+\Omega$, i.e. the sum of the apsidal and nodal precession rates of the same body, which are usually opposite and approximately equal for regular satellites. The only other currently known examples of a similar resonant argument among regular satellites are the Pallene-Mimas secular resonance found by @cal10, and the past Tethys-Dione secular resonance proposed by @cuk16, and in both cases the secular resonance is caused by proximity to a mean-motion resonance (MMR). Resonances including combinations of $\varpi+\Omega$ are also found among asteroids, where they are referred to as the $z_1$ and $z_2$ secular resonaces [@mil92; @mil94].
Among regular satellites with orbital precession dominated by the planet’s oblateness, the angles $\varpi + \Omega$ precess at rates that decrease monotonically with distance from the planet, and an additional perturbation (such as a nearby MMR) is needed to make these angles for two moons enter a resonance. The unique dynamics of Iapetus, which is at the transition between oblateness-dominated and solar perturbation-dominated orbits, allows for the observed secular resonance in the absence of any MMRs. Iapetus has about the slowest orbital precession that is possible for a Saturnian satellite, which then places the rate of change of its $\varpi+\Omega$ angle right in the parameter space occupied by planetary secular frequencies (in this case $g_5$). We will address the relevant terms affecting the precession of the angle $\varpi+\Omega$ in more detail in Section 4.
![Top: Eccentricity of Iapetus during a 100 Myr integration of Iapetus’s orbit using [psimpl]{}. Bottom: Evolution of the resonant argument $\varpi_I-\varpi_J+\Omega_I-\Omega_{eq}$ in the same simulation. Here we used a constant precession of Saturn’s spin axis around the invariable plane with a 1.96 Myr period, as in Fig. \[psim2\].[]{data-label="psim3"}](psim3.eps){width="\columnwidth"}
![Top: Eccentricity of Iapetus during a 100 Myr integration of Iapetus’s orbit using [psimpl]{}. Bottom: Evolution of the resonant argument $\varpi_I-\varpi_J+\Omega_I-\Omega_{eq}$ in the same simulation. Here we used a constant precession of Saturn’s spin axis around the invariable plane with a 1.91 Myr period, equal to the period of the secular mode $f_8$ [@md99].[]{data-label="psim4"}](psim4.eps){width="\columnwidth"}
In Fig. \[psim2\], the resonant argument of the Iapetus-$g_5$ secular resonance librates for 10 Myr under the above stated assumptions of Saturn’s pole precession. To study longer-term stability of this resonance, we extended the simulation to 100 Myr, and the results are plotted in Fig. \[psim3\]. About halfway through the integration in Fig. \[psim3\], the Iapetus-$g_5$ secular resonance breaks and the argument enters circulation (bottom), while the eccentricity now oscillates with only about half of the previous amplitude. Before deciding that the secular resonance is ephemeral, we need to consider the limitations of our model. Apart from the assumption of constant-rate, constant-obliquity precession of Saturn’s pole built into [psimpl]{}, we also had to select a precession rate for Saturn. The rate we chose (-0.66 arcsec yr$^{-1}$, with a 1.96 Myr period) is based on the observational results of @fre17 for the current precession rate of Saturn’s pole (-0.45 arcsec yr$^{-1}$), which had to be converted into the long-term average rate. @war04 [using the moon precession models of @vie92] find that the current precession rate of Saturn’s pole should be about 68% percent of the long-term rate due to the 700-year cycle of Titan’s orbital precession. Therefore we used that value to adjust the results of @fre17, obtaining the rate of -0.66 arcsec yr$^{-1}$. Given the approximate way we combined the results of these authors, it is very likely that the evolution plotted in Fig. \[psim3\] does not reflect the real dynamics of the system.
Another way to estimate the long-term precession rate of Saturn’s pole is to assume that it is locked in a spin-orbit secular resonance with the node associated with the $f_8$ secular mode of the Solar System [@war04; @ham04]. The precession rate of the secular mode $f_8$ is -0.69 arcsec yr$^{-1}$, equivalent to a period of 1.91 Myr [@md99; @las11; @vok15; @zee17]. Figure \[psim4\] shows the 100 Myr evolution of Iapetus’s eccentricity and Iapetus-$g_5$ secular resonant argument using [psimpl]{} and assuming the $f_8$ precession rate for Saturn’s pole. In this case, libration in the Iapetus-$g_5$ resonance is preserved over 100 Myr, with the current periodic oscillations in eccentricity persisting throughout the simulation. This demonstrates the sensitivity of the Iapetus-$g_5$ secular resonance to the precessional dynamics of Saturn’s spin axis, and shows the need for a more sophisticated model of Saturn’s precessional motion, which we will address in the next section.
Current Dynamics of Iapetus with a Variable Obliquity of Saturn
===============================================================
In order to model the full spin dynamics of Saturn, we needed to modify [simpl]{} further to include the realistic response of Saturn’s spin axis to solar, satellite and planetary torques. Since Saturn’s precessional period is much slower than any of the periods studied here, we are justified in using an azimuthally symmetric, oblate model of Saturn, despite known azimuthal asymmetries [@elm17]. Similarly, the large distances between interacting bodies involved here (Titan is the closest perturber) justify restricting ourselves to the $J_2$ moment of Saturn (which also includes Rhea and interior satellites). We decided to use the same approach as @cuk16b did for Earth in their integrations of the Earth-Moon system. In every timestep, Saturn’s spin axis suffered a “kick” [cf. @vok15]: $$d{\bf {\hat n}} = {\Sigma \ 3 m_i J_2 ( {\bf r}_i \times {\bf{\hat n}} ) ({\bf r}_i \cdotp {\bf {\hat n}}) dt \over \alpha R^2 \omega_R r_i^5}$$ where ${\bf{\hat n}}$ is the spin axis unit vector, $m_i$ is the mass of the perturber (in units of AU$^3$ yr$^{-2}$), $J_2$ is the usual oblateness moment (including effective oblateness due to satellites interior to Titan), ${\bf r}_i$ is the radius-vector of the perturber w.r.t. Saturn, $dt$ is the timestep, and $\alpha$, $R$ and $\omega_R$ are Saturn’s dimensionless moment of inertia, radius and spin rate, respectively. While the orbits of Titan and Iapetus were affected by the oblateness of Saturn (effectively the reverse of the above torque, but calculated independently), we ignored the back-reaction of Saturn’s spin on heliocentric orbits. We interwove this kick with the other perturbations in the usual “leapfrog” manner. While this is the simplest possible implementation of Saturn’s spin dynamics in a fully numerical integrator, we find that there are no discernible errors over the 100s of Myr we studied (which are only hundreds of Saturn’s precession periods). We term the version of [simpl]{} with a freely precessing planet [ssimpl]{}, with the extra “[s]{}” standing for “spin”.
![(Top) Evolution of Saturn’s obliquity as a function of the spin-orbit resonant argument $\Omega_{eq}-\Omega_N$ and over the next 100 Myr, obtained using [ssimpl]{} and six different values for Saturn’s moment of inertia; see Table \[table\] for details. The square symbol plots the initial conditions (i.e., the current state). (Bottom) The same integrations, now with the ratio of the moment of inertia $\alpha$ and the cosine of obliquity (as a measure of precession rate) plotted on the $y$-axis. Both $\Omega_{eq}$ and $\Omega_N$ were determined with respect to the invariable plane of the Solar System.[]{data-label="eye"}](eye.eps){width="\columnwidth"}
In [ssimpl]{}, as it fully integrates the precessional dynamics, the only adjustable parameter is Saturn’s principal moment of inertia $\alpha$, which then determines Saturn’s angular momentum. In reality, $\alpha$ is convolved with the differential rotation of Saturn to produce angular momentum, but here we will use a constant rotation rate of 5211.3 rad yr$^{-1}$, which corresponds to a period of 10.569 h. The value of $\alpha$ is not known directly, and the observations of Saturn’s pole precession are the most promising way of measuring it. Therefore we integrated Saturn’s pole precession (and the associated dynamics of Iapetus) for six different values of $\alpha$, which we refer to as cases A-F (Table \[table\]). Figure \[eye\] shows some of the solutions (A, B, F) circulating and some (C-E) librating, meaning that the pole of Saturn is in secular resonance with Neptune’s longitude of the node.[^5]
Case $\alpha$ prec. rate obs. rate
------ ---------- ------------ -----------
A 0.2 0.799 \[0.49\]
B 0.215 0.744 \[0.46\]
C 0.23 0.6955 0.427
D 0.235 0.681 0.42
E 0.24 0.667 0.41
F 0.245 0.6535 \[0.40\]
: Parameters for the six different integrations plotted in Figs. \[eye\] and \[prec\]. The units for precession rates are arcsec yr$^{-1}$, and all the values are negative. The third column lists the long-term precession rates (with respect to the invariable plane) fitted over $\simeq 1$ Myr, while the fourth column lists the average precession rates (with respect to the ecliptic) for the 1975-2015 period. The values in square brackets are not fits to integrations but estimates (assuming 61.5% of the third column).[]{data-label="table"}
![Evolution of the Iapetus-$g_5$ secular resonant argument over the next 100 Myr, obtained using [ssimpl]{} and six different values for Saturn’s moment of inertia. See Table \[table\] for details.[]{data-label="prec"}](prec_middle.eps){width="\columnwidth"}
Figure \[prec\] plots the evolution of the Iapetus-$g_5$ secular argument over each of the six 100 Myr simulations. While Iapetus is initially in resonance, it remains in the resonance for the whole of 100 Myr only in one of the six cases: case D, in which Saturn’s pole is librating in the spin-orbit resonance, with the present obliquity being close to the maximum one. However, it is not certain that the difference between the different solutions is systematic, and not stochastic, and to answer that question we would need to run many more computationally intensive simulations. However, if we could identify the correct solution for Saturn’s pole precession from observations, we could greatly constrain the problem and we should be able to predict the future of the Iapetus-$g_5$ secular resonance with more confidence.
![(Top panel) The precession of Saturn’s longitude of equinox (with respect to the ecliptic) in the 1975-2015 period in our simulation C. A short-term average precession rate of $-0.427$ arcsec yr$^{-1}$ is plotted with a dashed line. (Bottom panel) Residuals between the Saturn’s longitude of the equinox and the linear fit using the rate of $-0.427$ arcsec yr$^{-1}$. The most prominent periodic feature is associated with the $2 \lambda_S - 2 \Omega_{eq}$ term.[]{data-label="obs"}](obs.eps){width="\columnwidth"}
Determining where the current system is among the six simulations plotted in Fig. \[eye\] is non-trivial. @fre17 have measured the rate of Saturn’s pole precession sinve the Voyager encounter to be $-0.451 \pm 0.014$ arcsec yr$^{-1}$. This rate cannot be compared directly to the long-term precession rate, which we fit to our simulations over $\simeq 1$ Myr and list in the third column of Table \[table\] for every simulation (1 Myr is longer than most of the periodic terms but shorter than the libration in the spin-orbit resonance). In order to be able to compare the theory and observation more directly, we also computed current observable precession rates for Saturn’s pole (for the years 1975-2015, and relative to the J2000 ecliptic, rather than the invariable plane we use in our long-term fits). The short-term fits are listed for cases C, D, and E (which have a librating pole of Saturn) in Table \[table\]. We find that the current precession rate is between $-0.41$ and $- 0.427$ arcsec yr$^{-1}$ for the three librating cases, which can be compared to the value reported by @fre17. Formally, our case C is within 2 $\sigma$ of the observed value, and @fre17 state that their formal errors may underestimate the true uncertainties. Figure \[obs\] plots the short-term variation in the longitude of Saturn’s equinox for 1975-2015 in simulation C (top panel), and the same results with the average rate of $-0.41$ arcsec yr$^{-1}$ subtracted (bottom panel). A strong periodic feature proportional to $\sin(2 \lambda_S - 2 \Omega_{eq})$ (where $\lambda_S$ is Saturn’s mean longitude), with an amplitude of $\simeq 0.4$ arcsec yr$^{-1}$, can be seen in the bottom panel. We speculate that this periodic term has lowered the precession rate in @fre17 Fit \#1, which is based on Cassini data for 2004-2010, as well as for pole position variations in their Figure 12. In any case, it is clear that a linear fit is not sufficient to fit Saturn’s pole precession to observations with high accuracy.
![The residuals between the longitude of Saturn’s equinox and the long-term average precession rate -0.6955 arcsec yr$^{-1}$ in our simulation C. The solid line plots the longitude of the equinox measured relative to the ecliptic J2000, while the dashed line plots one with respect to the invariable plane (the latter definition was used to calculate the long-term average precession rate). The periodic terms due to Titan (700 yr period) and Iapetus (3500 yr period) are visible, and the slight upward trend in the dashed line is due to a 50,000 yr $\Omega_{eq}-\Omega_S$ term.[]{data-label="obs2"}](obs2.eps){width="\columnwidth"}
A related question is why our simulations suggest that the current precession rate is about 61.5% of the long term one, while @war04 [based on the model of @vie92] obtained 68%. Figure \[obs2\] plots the evolution of Saturn’s longitude of equinox in simulation C over the next 8000 yr, once the long-term average rate of 0.6955 arcsec yr$^{-1}$ has been removed. First of all, the precession rate is different when measured relative to the ecliptic (as observers do) and the invariable plane of the Solar System (which is used in theoretical calculations). Apart from Titan’s main 700 yr nutation period, we can also see a smaller periodic term associated with Iapetus (with a period of about 3500 yr) and a secular trend associated with the $\Omega_{eq}-\Omega_S$ periodic term which has a period of about 50,000 yr. While @vie92 definitely included the dynamics of Iapetus into their model, @war04 took only the dominant effect of Titan into account when determining the current/mean precession ratio of 68%. We conclude that our numerical simulations may be consistent with past analytical estimates once all periodic terms are included. We also find that the maximum allowed value for Saturn’s moment of inertia from the empirical model of @hel09, $\alpha=0.226$, would still put Saturn’s pole in libration within the spin-orbit resonance, due to the substantial resonance width.
The results of this section indicate that the current state of knowledge does not allow us to predict the long-term stability of the Iapetus-$g_5$ secular resonance, or even the future behavior of Saturn’s spin pole. However, it is clear that data can be consistent with Saturn’s pole librating in the resonance with secular eigenmode $f_8$, and @war04 and @ham04 have made a strong case on theoretical grounds that this resonance is present. The phase of Saturn’s pole precession close to the libration center is also indicative of the resonance. It is tempting to use the Iapetus-$g_5$ secular resonance as the constraint in Saturn’s pole precession, i.e. argue that simulation D is closest to the real solution as it preserves the Iapetus-$g_5$ secular resonance. However, we cannot make such pronouncements based on six simulations, and also it is not impossible that the Iapetus-$g_5$ secular resonance is short-lived or intermittent, especially if it is less than 1 Gyr old (Section 5). We conclude that the way forward will be to compare observations of Saturn’s pole position to a full numerical model of its precession, which is outside the scope of this paper.
Location of the Secular Resonance
=================================
A secular argument which includes the angle $\varpi+\Omega$ evolves very slowly, due to near-cancellation of the precession rates of the pericenter and the node. In a first-order approximation, these two precessional rates are indeed the same (with an opposite sign), so we have to look to higher order terms to identify the sources of secular frequencies.
The first effect to consider is orbital precession due to perturbations from Saturn’s oblateness and all the satellites interior to Iapetus (including Titan). While the leading term oblateness-driven precession is symmetric for the pericenter and the node, the symmetry is broken for eccentric and inclined orbits [@dan92]: $${\dot \Omega}_2 = - {3 J_2 n \over 2 (a/R)^2 (1-e^2)^2} \cos{i}$$ $${\dot \omega}_2 = {3 J_2 n \over 2 (a/R)^2 (1-e^2)^2} ({5 \over 2} \cos^2{i} -{1 \over 2})$$ So, assuming small-inclination orbits, the sum of Iapetus’s apsidal and nodal precession is ${\dot \varpi}+ {\dot \Omega} = {\dot \omega}+ 2 {\dot \Omega}$: $${\dot \varpi}_2+ {\dot \Omega}_2 = {3 J_2 n \over 4 (a/R)^2 (1-e^2)^2} (5 \cos^2{i} - 1 - 4 \cos{i})$$ We introduce $s=\sin{i}$, and assume that both $e$ and $i$ are small quantities (so $\cos{i}=1-s^2/2$, and we ignore $O(e^2 s^2)$, $O(s^4)$): $${\dot \varpi}_2+ {\dot \Omega}_2 = {3 J_2 n \over 4 (a/R)^2} (5 - 5 s^2 -1 - 4 + 2 s^2) = - {9 J_2 n \over 4 (a/R)^2} s^2
\label{j2}$$ To the lowest order, the sum of apsidal and nodal precession due to planetary oblateness does not depend on eccentricity and is negative (i.e. retrograde) for orbits with non-zero inclination. For Iapetus, this term amounts to about -5.0 arcsec yr$^{-1}$; note that 3/4 of the $J_2$ in Eq. \[j2\] comes from Titan, and the rest mostly from Saturn’s oblateness. While $J_2^2$ term is the dominant non-zero part of the sum of apsidal and nodal precession for the inner moons [@cuk16], it can be ignored for Iapetus, due to its small size and dependence on distance as $(a/R)^{-4}$ [@md99].
Next terms we need to consider are Titan’s perturbations not included in the $J_2$ term. Since the orbits of Titan and Iapetus are relatively well-separated ($a/a_T \simeq 3$), we will restrict ourselves to terms arising from $J_4$ perturbations of Titan (here $J_4=(3/8) (m_T/M)(a_T/R)^4)$). The precession of a satellite’s orbit due a $J_4$ moment is given in @bro59: $$\dot{\omega}_4 = B_4 [21 - 9 \eta^2 + (-270 + 126 \eta^2) \vartheta^2 + (385 - 189 \eta^2) \vartheta^4]$$ $$\dot{\Omega}_4 = 4 B_4 [(5 - 3 \eta^2) \vartheta (3 - 7 \vartheta^2)]$$ Where $$B_4={15 J_4 n \over 128 (a/R)^4 (1-e^2)^4}$$ where $\vartheta=\cos{i}$ and $\eta=\sqrt{1-e^2}$. Since we are interested in the small $e$ and $i$ case, we switch from $\eta$ and $\vartheta$ to $e$ and $s$: $$\dot{\omega}_4 \simeq B_4 [64 + 72 e^2 - 248 s^2]$$ $$\dot{\Omega}_4 \simeq 4 B_4 [-8 -12 e^2 +18 s^2]$$ The net contribution to the rate of change of the secular resonance argument is then: $$\dot{\varpi}_4+\dot{\Omega}_4 = {15 J_4 n \over 16 (a/R)^4} (-3 e^2 - 13 s^2)$$ which equates to about -1.8 arcsec yr$^{-1}$ for Iapetus.
The next term we need to examine is one due to solar secular perturbations, commonly referred to Kozai-Lidov interaction [@lid62; @koz62]. The expression for the precession of a satellite’s orbit due to solar quadrupole-order perturbations averaged over mean motions (assuming no coupling between mean-motion and secular terms) is [@inn97; @cuk04]: $$\dot{\omega}_K = K_2 [2 (1-e^2) + 5 \sin^2{\omega}(e^2 -\sin^2{i}]$$ $$\dot{\Omega}_K = - K_2 [1 + 4 e^2 -5 e^2 \cos^2{\omega}] \cos{i}$$ where $$K_2 = {3 n_S^2 \over 4 (1-e_S^2) \sqrt{1-e^2} n}$$ where $e_S$ and $n_S$ are Saturn’s eccentricity and mean motion, respectively. Since Iapetus has very weak oscillations in $e$ and $i$ as $\omega$ precesses, we can average over $\omega$, so $\cos^2{\omega}=\sin^2{\omega}=1/2$. Therefore, assuming small $s$, we get: $$\begin{aligned}
\dot{\omega}_K = K_2 (2 + {1 \over 2} e^2 - {5 \over 2} s^2) \\
\dot{\Omega}_K = - K_2 (1 + {3 \over 2} e^2 - {1 \over 2} s^2)\end{aligned}$$ So, finally, the Kozai contribution to the change of secular resonance argument is: $$\dot{\varpi}_K+\dot{\Omega}_K \simeq {3 n_S^2 \over 8 n} (- 5 e^2 - 3 s^2)$$ which for Iapetus amounts to -8.0 arcsec yr$^{-1}$.
The three secular contributions derived above amount to about -14.8 arcsec yr$^{-1}$, which would make the secular resonance argument $\dot{\varpi} +\dot{\Omega}$ precess in the retrograde direction with a period $< 10^5$ yr, while numerical integrations show this angle in resonance with $\dot{\varpi}_J+\dot{\Omega}_{eq}$ which has the prograde precession rate of 3.6 arcsec yr$^{-1}$. Before we lose faith in secular theory, we must remember that the Kozai-Lidov precession assumes no coupling between short-period and secular terms, which fails spectacularly when applied to apsidal precession of the Moon, as discovered by Clairaut [@bau97]. In reality, there is notable coupling between the apsidal precession of the satellite and the mean motion of the Sun, leading to the so-called “evection” term. While the averaged evection term is much more important for secular behavior of the Moon and irregular satellites which are much more strongly perturbed by the Sun, is is relevant to the Iapetus-$g_5$ secular resonance as it affects the pericenter much more strongly than the node. The leading Clairaut terms for the precession driven by evection, as well as the analogous coupling between the solar mean-motion and the nodal precession, are [@cuk04]: $$\dot{\varpi}_C + \dot{\Omega}_C = \Bigl( {225 \over 32 }(1-s^2)+ {9 \over 32} \Bigr) {n_S^3 \over n^2} + {4071 \over 128}{n^4_S \over n^3}$$ Here we kept only the inclination dependence of the largest apsidal term, and ignored the $e$ and $i$ dependence for the other two. This gives us a positive precession contribution of 17.7 arcsec yr$^{-1}$ to the rate of change of the secular resonance argument.
For completeness, since we are accounting for non-zero average contributions of periodic terms, we must include the average effect of the octupole apsidal secular interaction between Titan and Iapetus. If we average the precessional effects of the $\varpi_T-\varpi$ term [@lee03] the same way @cuk04 did for solar evection, we get: $$\dot{\varpi}_3= {225 \over 512} {n^2 \over (\dot{\varpi}_T-\dot{\varpi})} \Bigl( {m_T \over M}\Bigr)^2 \Bigl({a_T \over a}\Bigr)^6 {e_T^2 \over e^2}
\label{octupole}$$ where $(\dot{\varpi}_T-\dot{\varpi})$ has a period of about 900 yr, and the net contribution of this term is 0.7 arcsec yr$^{-1}$. This brings the total $\dot{\varpi} + \dot{\Omega}$ rate from all five contributing terms (effective $J_2$ and $J_4$, Kozai-Lidov, Clairaut and averaged-octupole) to a prograde rate of 3.6 arcsec yr$^{-1}$, which is within 0.1 arcsec yr$^{-1}$ of $\dot{\varpi}_J+\dot{\Omega}_{eq}$ and therefore satisfactorily explains the current secular resonance of Iapetus.
It is interesting that the current very slow resonance exist only because several larger terms mostly cancel each other out. Since the major positive contribution to $\dot{\varpi} + \dot{\Omega}$ is from the Clairaut terms which (to the first order) do not depend on $e$ and $i$ like the retrograde terms, the fastest $\dot{\varpi} + \dot{\Omega}$ possible at Iapetus’s distance from the Sun would be 17.4 arcsec yr$^{-1}$ for circular orbits in the Laplace plane. So a resonance with the $g_6$ planetary eigenmode is not possible in the same way as the observed one with the $g_5$ eigenmode. The only other secular resonance we found for Iapetus has the argument $\varpi+\Omega-\varpi_T-\Omega_T$ which requires an almost zero $\dot{\varpi} + \dot{\Omega}$ for Iapetus, and happens at somewhat higher $e$ and/or $i$.
![The location in the $e-i$ plane where $\dot{\varpi} + \dot{\Omega}=3.6$ arcsec yr$^{-1}$ according to Eq. \[total\]. The open square plots the approximate current average orbital elements of Iapetus. The behavior of the curve close to $e=0$ and $i=0$ is poorly constrained, as at those points the definitions of $\varpi$ and $\Omega$ become unreliable.[]{data-label="loc"}](location.eps){width="\columnwidth"}
Now we can plot where the resonance is in the $e-i$ phase space for Iapetus, assuming the current semimajor axes for Iapetus, Titan and Saturn (this level of approximation ignores the eccentricity of Saturn). Using all of the precession contributions we derived, we get: $$\dot{\varpi} + \dot{\Omega} = P_0 - P_i \sin^2{i} - P_e e^2 + P_3 e^{-2}
\label{total}$$ where $$P_0= {117 \over 16} {n_S^3 \over n^2} + {4071 \over 128}{n^4_S \over n^3}$$ $$P_e = {15 \over 8} \Bigl({{ 3 J_4 n \over 2 (a/R)^4 } + {n^2_S \over n}}\Bigr)$$ $$P_i= {3 \over 8}\Bigl( { 6 J_2 n \over (a/R)^2} + {65 J_4 n \over 2 (a/R)^4 } + { 3 n^2_S \over n} + { 75 n^3_S \over 4 n^2}\Bigr)$$ and $P_3$ is defined by Eq. \[octupole\]. In Fig. \[loc\] we plot the locations in the $e-i$ plane where $\dot{\varpi} + \dot{\Omega}=3.6$ arcsec yr$^{-1}$ according to Eq. \[total\]. There is a continuous line of locations in $e-i$ phase space for which the resonance is possible, along which eccentricity decreases while the inclination increases.
It is tempting to observe Fig. \[loc\] and envision past evolution (or diffusion) of Iapetus’s orbit along the secular resonance. The orbit of Iapetus is more inclined and less eccentric than would be expected (on average) from excitation by planetary fly-bys [@nes14], and some of the combinations of $e$ and $i$ along the resonant location would be a more likely outcomes of encounters of Saturn with the Ice Giants. Diffusion of orbits along the secular resonance is known to be a major effect in the dynamics of asteroids and meteoroids [@nes07]. However, the resonant perturbations must affect $e$ and $i$ equally and with the same sign, as the coefficients of $\varpi$ and $\Omega$ in the resonant argument are the same. Therefore secular resonant perturbations (chaotic or not) can only move Iapetus’s orbit along a diagonal line in $e-i$ space along which both $e$ and $i$ are increasing or decreasing at the same time. However, this line is close to perpendicular to the line plotting the locations of the secular resonance, greatly reducing the potential for evolution or diffusion through the resonance. So it is likely that the secular resonance with the $g_5$ mode was established only once Iapetus reached its current orbit, and could not be responsible for its unusually high inclination and much lower eccentricity.
The Origin of the Secular Resonance
===================================
Iapetus’s semimajor axis is practically fixed, as the tidal acceleration of Iapetus due to tides on Saturn raised by Iapetus is negligible [@lai12]. However, Titan does migrate appreciably due to tides, which changes the Titan/Iapetus mean motion ratio and affects the secular dynamics of Iapetus. The most notable dynamical event in the history of the Titan-Iapetus pair was likely their mutual 5:1 mean-motion resonance crossing [@cuk13; @pol17; @pol18]. If we ignore the satellite tides within Titan, this crossing should have happened about 0.5 Gyr ago, assuming $Q/k_2 \simeq 5000$ for Saturn [@lai12]. Prior work has found that this resonance can excite the eccentricity of Iapetus from zero to the present value, while the inclination of Iapetus could not have been changed substantially. While in the majority of cases Iapetus survives this resonance crossing when we assume $e < 0.01$ for Titan, Iapetus is almost always lost if Titan had its present eccentricity of $e_T=0.029$ during MMR crossing [@cuk13; @pol17; @pol18]. This prompted @cuk16 to propose that the eccentricity of Titan was recently ($\simeq$100 Myr) excited, as a side-effect of a massive instability among the inner moons.
![Evolution of Iapetus’s orbit using [psimpl]{} and assuming that Iapetus was in the $g_5$ secular resonance (with present $e$ and $i$) before the Titan-Iapetus 5:1 MMR. We used an axial precession rate of $-0.069$ arcsec yr$^{-1}$ (same as in Fig. \[psim4\]) and a tidal $Q/k_2=5000$ for Saturn. The secular resonance moves to higher eccentricities due to proximity of the 5:1 MMR, making Iapetus more eccentric in the process. Iapetus enters the 5:1 MMR with $e \simeq 0.1$, which usually leads to instability. Here and in Figs. \[sres42\]-\[sres52\], the “stair step" texture of Titan’s semimajor axis plot is an artifact of a low-precision output.[]{data-label="res6"}](res6.eps){width="\columnwidth"}
The first question to ask is whether the Iapetus-$g_5$ secular resonance can be ancient, pre-dating the Titan-Iapetus 5:1 MMR crossing. Figure \[res6\] shows a simulation (using [psimpl]{}, see caption for details) in which we integrated the 5:1 resonance crossing with Iapetus initially in the secular resonance. In this and other simulations we consistently get eccentricity growth for Iapetus as it approaches the 5:1 MMR. Apparently, Titan’s near-resonant perturbations on Iapetus lead to a positive precession of the secular resonance argument, most likely by affecting ${\dot\varpi}$ [cf. @cuk16 who found similar secular-MMR interference for the Dione-Rhea 5:3 resonance]. This additional positive rate of change of the secular resonance argument must be balanced by the increase in eccentricity in order to preserve the resonance (as the $e^2$ term in Eq. \[total\] has a negative coefficient). The secular resonance is broken when $e$ reaches about 0.1. Eccentricity is then constant until the 5:1 MMR is encountered, at which point the orbit of Iapetus becomes chaotic and is eventually destabilized. Since this is a systematic result, we conclude that Iapetus was unlikely to be in the secular resonance with the $g_5$ mode before the 1:5 MMR with Titan and that the resonance must have been established more recently.
![A simulation of the Titan-Iapetus 5:1 MMR using [ssimpl]{}, with tidal evolution accelerated 10x. The bottom panel plots the secular resonant argument $\varpi_I-\varpi_J+\Omega_I-\Omega_{eq}$. In this run, Iapetus is captured into the secular resonance soon after crossing the 5:1 MMR. []{data-label="sres42"}](sres42b.eps){width="\columnwidth"}
![A simulation of the Titan-Iapetus 5:1 MMR using [ssimpl]{}, with tidal evolution accelerated 10x. The bottom panel plots the secular resonant argument $\varpi_I-\varpi_J+\Omega_I-\Omega_{eq}$. In this run, Iapetus crosses the secular resonance at about 33 Myr without being captured.[]{data-label="sres34"}](sres34b.eps){width="\columnwidth"}
An intriguing possibility is an evolution that is in a way reversed from that shown in Fig. \[res6\]: Iapetus encountering the $g_5$ secular resonance as it exits the 5:1 MMR, subsequently evolving to lower eccentricities. One problem in modeling this process is the high degree of stochasticity. The outcome of the 5:1 MMR resonance is unpredictable even with low initial eccentricities for Titan and Iapetus. Sometimes Iapetus is outright destabilized, while in other cases it “jumps” through the resonance instantaneously with no significant changes to the orbit, and we find that the probability for each outcome is about 20%. Sometimes the final eccentricity is too low and Iapetus never encounters the secular resonance. A very common case is when the inclination of Iapetus increases or decreases slightly, which shifts the location of the secular resonance to substantially lower or higher eccentricities, respectively (Fig. \[loc\]). Often we end up with an outcome where there is no $g_5$ secular resonance at all for $e > 0$, or it is shifted to high eccentricities and therefore will be “missed" by Iapetus if its eccentricity is comparable to the present one. While this does not invalidate the idea that the resonance was crossed, it does make this process hard to model numerically. Additionally, the chaotic phase of the 5:1 MMR can last anything from a few to hundreds of Myr, making these integrations very computationally expensive given the uncertain outcome. Therefore, in order to complete a large number of simulations in reasonable time, we used a sped-up tidal evolution with $Q/k_2=500$ for Saturn. Figures \[sres42\] and \[sres34\] show two such integrations (made using [ssimpl]{} with $\alpha=0.235$, case D in Fig. \[prec\]). In Fig. \[sres42\] Iapetus is captured into the secular resonance, while in Fig. \[sres34\] it jumps through the resonance. Resonance capture happens only in about 10-20% of our outcomes (which assume $e=0$ and the current inclination for Iapetus before the 5:1 MMR), making it somewhat unlikely, but not prohibitively so. We find that resonance “jumps" and captures were about equally likely (with Iapetus not encountering the secular resonance in the rest of the cases).
In order to verify this rate of resonant capture, we also performed a number of simulations using the realistic tidal properties of Saturn ($Q/k_2=5000$), but starting Titan immediately outside the 5:1 MMR with Iapetus, with Iapetus having its current or a somewhat higher eccentricity. While not as comprehensive as simulations which include the 5:1 resonance crossing, these runs should reflect the range of outcomes that are possible if Iapetus exits the MMR with $e$ and $i$ close to current values. This time we find that “jumps” through the resonance are an order of magnitude more common than captures. Apparently, the direction of resonance encounter we have here does not lead to capture under adiabatic conditions, and slower resonance encounters are “worse” for capture than faster ones. It is only because of the extremely slow libration period within the resonance that capture is possible in our simulations at all (i.e. even the realistic tidal evolution simulations are barely adiabatic). Therefore, while we cannot exclude the secular resonance capture in the aftermath of the Titan-Iapetus 5:1 MMR crossing, this is not a likely outcome.
![A simulation of the dynamics of Iapetus starting after the Titan-Iapetus 5:1 MMR using [ssimpl]{}, with tidal $Q/k_2=5000$ for Saturn [@lai12]. The bottom panel plots the secular resonant argument $\varpi_I-\varpi_J+\Omega_I-\Omega_{eq}$. In this run, Iapetus is temporarily captured into a three-body resonance with partial argument $5\lambda - \lambda_T -\lambda_S$ between 50 and 100 Myr. During the three-body resonance, Iapetus enters the secular resonance and remains in it for most of the rest of the simulation.[]{data-label="sres52"}](sres52b.eps){width="\columnwidth"}
Realistic-rate simulations of the 5:1 MMR aftermath uncovered some additional dynamical effects. Hundreds of Myr after Titan has crossed the main 5:1 MMR with Iapetus, new mean-motion-type resonances are observed in our simulation. These resonances seem to happen at Titan’s semimajor axes that are an integer number of Saturn’s mean motions away from the 5:1 resonance with Iapetus, meaning that they are three-body resonances involving Titan, Iapetus and the Sun. Fig. \[sres52\] shows Iapetus encountering such a resonance, becoming temporarily captured in the three-body resonance (50-100 Myr). While Iapetus is in the three-body resonance, its eccentricity grows and the secular resonance is encountered. About 100 Myr into the simulation, Iapetus leaves the three-body resonance and remains in the secular resonance for the remainder of the simulation (with one short break). We are certain that the mean-motion part of the three-body resonance is $5\lambda - \lambda_T -\lambda_S$, but we were unable to find a complete librating argument, possibly indicating that the resonance is chaotic and what looks like temporary capture is constant shifting between different sub-resonances. In any case, we find that such three-body resonances are the major cause of both capture and escape from the secular resonance in realistic-rate simulations, greatly complicating the dynamics. We conclude that a capture into the Iapetus-$g_5$ secular resonance following the Titan-Iapetus 5:1 MMR crossing is a possible but not a very likely outcome, and that additional work is needed to further examine the dynamics of this phase of the system’s evolution.
![A simulation of the dynamics of Iapetus starting after the Titan-Iapetus 5:1 MMR using [ssimpl]{}, with tidal $Q=100$ and $k_2=0.37$ for Saturn, implying orbital evolution dominated by Saturn’s normal modes [@ful16]. The bottom panel plots the secular resonant argument $\varpi_I-\varpi_J+\Omega_I-\Omega_{eq}$. As in Fig. \[sres52\], we used the current inclination of Iapetus. In this run, the starting eccentricity of Iapetus is just right for it to be captured into the secular resonance. The capture is apparently non-adiabatic, and there is no noticeable evolution of Iapetus’s orbit along the resonance.[]{data-label="fres72"}](fres72.eps){width="\columnwidth"}
Since the capture into the reasonance appears non-adiabatic, and simulations with examples shown in Figs. \[sres42\]-\[sres52\] indicate that faster migration of Titan improves the chances of resonance capture, we may want to reconsider some of our assumptions about Saturn’s tidal response. While the $Q/k_2=5000$ for Titan we used so far is based on results of @lai12, theoretical predictions [@ful16] and some recent observational results [@lai17] suggest that the orbital evolution of Rhea and (possibly) Titan may be much faster than would be expected if Saturn’s tidal $Q$ was the same for all satellites. Therefore we perfomed some additional integrations using $Q=100$ and $k_2=0.37$ for Saturn (V. Lainey, pers. comm.), with an example shown in Fig. \[fres72\]. These integrations are similar to that shown in Fig. \[sres52\] by starting with present $e$ and $i$ just after the Titan-Iapetus MMR. The dynamics of the 5:1 resonance crossing in the $Q \simeq 100$ regime is beyond the scope of this paper and is addressed by @pol17 and @pol18. We find that the evolution is non-adiabatic as the evolution is fast relative to resonant librations, and capture is likely for a narrow range of initial eccentricities around $e=0.03$, but impossible for any other $e$. If Titan’s orbit does indeed evolve this fast, then the Iapetus-$g_5$ secular resonance is an accidental side-effect of the stochastic 5:1 Titan-Iapetus MMR crossing. While this mechanism of resonance capture is promising and appears more straightforward than the one shown in Fig. \[sres52\], more work on the Titan-Iapetus 5:1 MMR is needed to properly evaluate the probabilities of either scenario. Note that a sustained rapid orbital evolution of Titan (equivalent to Saturn’s tidal $Q \simeq 100$) would make the secular resonance only about 50 Myr old, which would naturally explain its existence despite its apparent dynamical fragility.
Summary
=======
This paper represent a first exploration of a previously unknown orbital resonance between Iapetus and the planetary system. Our conclusions can be summarized as follows:
1\. Iapetus is currently in a secular resonance with an argument $\varpi-\Omega+\varpi_J-\Omega_{eq}$ librating around 180$^{\circ}$. The libration period is several Myr and the libration is likely to persists for several tens of Myr.
2\. Longer-term stability of this resonance is tied to the precession of Saturn’s spin axis, and more definite predictions need to wait for better determinations of Saturn’s precession rate. Most allowable solutions for Saturn’s pole precession lead to eventual breaking of the Iapetus-$g_5$ secular resonance, but some solutions preserve the secular resonance for at least 100 Myr.
3\. We use analytical considerations to establish that the current occurence of the Iapetus-$g_5$ secular resonance is enabled by near-canceling of the precession term $\dot{\varpi} +\dot{\Omega}$, arising from several different secular and averaged periodic terms in the disturbing function.
4\. The Iapetus-$g_5$ secular resonance was almost certainly established more recently than the proposed 5:1 MMR crossing between Titan and Iapetus (500-50 Myr ago, depending on the Titan’s unknown tidal evolution rate). While we find cases when the secular resonance was established in the aftermath of this MMR (with the more rapidly evolving Titan offering promissing results), we yet have to find a high-probability mechanism for establishing the secular resonance. More work is clearly needed to fully understand the rich dynamical history of Iapetus.
Acknowledgements {#acknowledgements .unnumbered}
================
Work by M[Ć]{} on this project was supported by NASA Outer Planets Research Program award NNX14AO38G. The authors thank Valery Lainey and William Polycarpe for very useful discussions. We thank the reviewer Doug Hamilton for alerting us to very relevant work by Callegari and Yokoyama.
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\[lastpage\]
[^1]: E-mail: [email protected] (MC)
[^2]: The instantaneous Laplace plane can be defined for every perturbed orbit as plane normal to the vector around which the orbit normal is precessing.
[^3]: https://ssd.jpl.nasa.gov/?horizons accessed on January 24, 2013
[^4]: Here and elsewhere in this paper we used $\varpi_J$ as a directly-observable proxy for the orientation of the $g_5$ eccentricity vector, as Jupiter’s eccentricity is dominated by the $g_5$ mode.
[^5]: Here and throughout we used $\Omega_N$ as a directly-observable proxy for the phase of the secular eigenmode $f_8$. The presence of other modes in Neptune’s inclination vector leads to some smearing in the $x$-direction of the curves plotted in Fig. \[eye\].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'To derive an eigenvalue problem for the associated Askey–Wilson polynomials, we consider an auxiliary function in two variables which is related to the associated Askey–Wilson polynomials introduced by Ismail and Rahman. The Askey–Wilson operator, applied in each variable separately, maps this function to the ordinary Askey–Wilson polynomials with different sets of parameters. A third Askey–Wilson operator is found with the help of a computer algebra program which links the two, and an eigenvalue problem is stated.'
address:
- 'Mathematics and Computer Science, Colorado College, Tutt Science Center, 14 E. Cache la Poudre St., Colorado Springs, CO 80903, U.S.A.'
- 'Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Vienna, Austria'
- 'School of Mathematical and Statistics Sciences & Mathematical. Computational and Modeling Sciences Center, Arizona State University, Tempe, AZ 85287-1804, U.S.A.'
author:
- Andrea Bruder
- 'Christian Krattenthaler$^\dagger$'
- 'Sergei K. Suslov'
title: |
An eigenvalue problem for\
the associated Askey–Wilson polynomials
---
[^1]
Introduction
============
Throughout this paper, we use the standard notation for the $q$-shifted factorials: $$\begin{aligned}
2
\left( a;q\right) _{n}&:=\prod\limits_{j=0}^{n-1}\left( 1-aq^{j}\right) ,&%
\qquad \left( a_{1},a_{2},\dots,a_{r};q\right)
_{n}&:=\prod_{k=1}^{r}\left( a_{k};q\right) _{n}, %\label{in1}
\\
\left( a;q\right) _{\infty }&:=\lim_{n\rightarrow \infty }\left( a;q\right)
_{n},&\qquad \left( a_{1},a_{2},\dots,a_{r};q\right) _{\infty
}&:=\prod_{k=1}^{r}\left( a_{k};q\right) _{\infty }, %\label{in2}\end{aligned}$$provided $\left\vert q\right\vert <1.$ The basic hypergeometric series is defined by (cf. [@Ga:Ra]) $$_{r}\varphi _{s}\left(
\begin{array}{c}
a_{1},a_{2},\dots,a_{r} \\
b_{1},\dots,b_{s}%
\end{array}%
;\,q\,,\,z\right) :=\sum_{n=0}^{\infty }\frac{\left(
a_{1},a_{2},\dots,a_{r};q\right) _{n}}{\left(
q,b_{1},b_{2},\dots,b_{s};q\right) _{n}}\,((-1)^{n}q^{n(n-1)/2})^{1+s-r}%
\,z^{n}.$$If $0<|q|<1,$ the series converges absolutely for all $z$ if $r\leq s,$ and for $|z|<1$ if $r=s+1.$
The Askey–Wilson polynomials are the most general extension of the classical orthogonal polynomials [@An:As], [@An:As:Ro], [@As:Wi], [Koe:Sw]{}, [@Ni:Su:Uv], [@Sz]. They are most conveniently given in terms of a $_4\varphi_3$-series, $$\begin{aligned}
p_{n}(x)& =p_{n}(x;a,b,c,d)=p_{n}(x;a,b,c,d|q) \notag \\
& =a^{-n}\,(ab,ac,ad;q)_{n}\;{}_{4}\varphi _{3}\!\left( \!\!%
\begin{array}{c}
q^{-n},\ abcdq^{n-1},\ az,\ a/z\smallskip \\[0.1cm]
ab,\ ac,\ ad%
\end{array}%
\!\!;q,\,q\!\right) , \notag\end{aligned}$$where $x=\left( z+z^{-1}\right) /2,$ and $\left\vert z\right\vert <1.$ In this normalization, the Askey–Wilson polynomials are symmetric in all four parameters due to Sears’ transformation [@As:Wi].
The Askey–Wilson polynomials satisfy the 3-term recurrence relation$$2x\,\,p_{n}(x;a,b,c,d)=A_{n}\,p_{n+1}(x;a,b,c,d)\,+\,B_{n}%
\,p_{n}(x;a,b,c,d)+\,C_{n}\,p_{n-1}(x;a,b,c,d), \label{in5}$$where$$\begin{aligned}
A_{n}& =\frac{a^{-1}(1-abq^{n})(1-acq^{n})(1-adq^{n})(1-abcdq^{n-1})}{%
(1-abcdq^{2n-1})(1-abcdq^{2n}-q^{2n})}, \label{in6} \\
C_{n}& =\frac{a(1-bcq^{n-1})(1-bdq^{n-1})(1-cdq^{n-1})(1-q^{n})}{%
(1-abcdq^{2n-1})(1-abcdq^{2n})}, \label{in7} \\
B_{n}& =a+a^{-1}-A_{n}-C_{n}. \label{in8}\end{aligned}$$The weight function with respect to which the polynomials $p_{n}(x)$ are orthogonal was found by Askey and Wilson in [@As:Wi]. The Askey–Wilson divided difference operator is defined by$$\begin{aligned}
L(x)u&:=L\left( s;a,b,c,d\right) u\left( s\right) \notag \\
&\hphantom{:} =\frac{\sigma \left( -s\right) \nabla x\left( s\right) u\left(
s+1\right) +\sigma \left( s\right) \Delta x\left( s\right) u\left(
s-1\right) -\left[ \sigma \left( s\right) \Delta x\left( s\right) +\sigma
\left( -s\right) \nabla x\left( s\right) \right] u\left( s\right) }{\Delta
x\left( s\right) \nabla x\left( s\right) \nabla x_{1}\left( s\right) },
\label{in9}\end{aligned}$$where $\sigma \left( s\right) =q^{-2s}\left( q^{s}-a\right) \left(
q^{s}-b\right) \left( q^{s}-c\right) \left( q^{s}-d\right) $ and, by definition, $$\begin{aligned}
x(s)& =\frac{1}{2}\left( q^{s}+q^{-s}\right) \text{\ }, & \qquad x_{1}(s)&
=x\left( s+\frac{1}{2}\right) , \\
\Delta f(s)& =f(s+1)-f(s), & \qquad \nabla f(s)& =f(s)-f(s-1).\end{aligned}$$(We follow the notation in [@At:Su:DHF] and [@At:Su1].) We will make use of an analogue of the power series expansion method, where a function is expanded in terms of generalized powers. For a positive integer $m,$ the generalized powers are defined by$$\lbrack
x(s)-x(z)]^{(m)}=\prod_{n=0}^{m-1}[x_{n}(s)-x_{n}(z-k)],\qquad
x_{n}(z)=x\left( z+\frac{n}{2}\right) \label{in10}$$(see [@Su4 Exercises 2.9–2.11, 2.25] and [@Su2] for more details).
The Associated Askey–Wilson Polynomials
=======================================
The associated Askey–Wilson polynomials, $p_{n}^{\alpha }(x)=p_{n}^{\alpha
}(x;a,b,c,d)=p_{n}^{\alpha }(x;a,b,c,d|q),$ were introduced by Ismail and Rahman in [@Is:Rah]. They are solutions of the 3-term recurrence relation$$2x\,\,p_{n}^{\alpha }(x;a,b,c,d)=A_{n+\alpha }\,\,p_{n+1}^{\alpha
}(x;a,b,c,d)\,+\,B_{n+\alpha }\,\,p_{n}^{\alpha }(x;a,b,c,d)+\,C_{n+\alpha
}\,\,p_{n-1}^{\alpha }(x;a,b,c,d), \label{aaw1}$$where $0<\alpha <1,$ with initial values $\,p_{-1}^{\alpha }(x)=0,$ $%
p_{0}^{\alpha }(x)=1$, and $A_{n+\alpha },$ $B_{n+\alpha },$ $C_{n+\alpha }$ are given as in (\[in6\])–(\[in8\]) with $n$ replaced by $n+\alpha .$ The two linearly independent solutions to (\[in5\]) found in [Is:Rah]{} are$$\begin{gathered}
R_{n+\alpha } =\frac{(abq^{n+\alpha },acq^{n+\alpha },adq^{n+\alpha
},bcdq^{n+\alpha }/z;q)_{\infty }}{(bcq^{n+\alpha },bdq^{n+\alpha
},cdq^{n+\alpha },azdq^{n+\alpha };q)_{\infty }}\left( \frac{a}{z}\right)
^{n+\alpha }\smallskip \smallskip \medskip \label{aaw2} \\
\times \,_{8}W_{7}(bcd/qz;b/z,c/z,d/z,abcdq^{n+\alpha
-1},q^{-\alpha -n};q,qz/a)\end{gathered}$$and$$\begin{gathered}
S_{n+\alpha } =\frac{(abcdq^{2n+2\alpha },bzq^{n+\alpha +1},czq^{n+\alpha
+1},dzq^{n+\alpha +1},bcdzq^{n+\alpha +1};q)_{\infty }}{(bcq^{n+\alpha
},bdq^{n+\alpha },cdq^{n+\alpha },q^{n+\alpha +1},bcdzq^{2n+2\alpha
+1};q)_{\infty }}(az)^{n+\alpha }\smallskip \smallskip \medskip \label{aaw3}
\\
\times \,_{8}W_{7}(bcdzq^{2n+2\alpha };bcq^{n+\alpha
},bdq^{n+\alpha },cdq^{n+\alpha },q^{n+\alpha +1},zq/a;q,az).\end{gathered}$$The weight function for the associated Askey–Wilson polynomials and an explicit polynomial representation were found by Ismail and Rahman in [Is:Rah]{}. The latter is given by$$\begin{aligned}
p_{n}^{\alpha }(x)& =p_{n}^{\alpha }(x;a,b,c,d|q) \notag \\
& =\sum_{k=0}^{n}\frac{(q^{-n},abcdq^{2\alpha +n-1},abcdq^{2\alpha
-1},ae^{i\theta },ae^{-i\theta };q)_{k}}{(q,abq^{\alpha },acq^{\alpha
},adq^{\alpha },abcdq^{\alpha -1};q)_{k}}\ q^{k} \notag \\
& \qquad \times ~_{10}W_{9}(abcdq^{2\alpha +k-1};q^{\alpha },bcq^{\alpha
-1},bdq^{\alpha -1},cdq^{\alpha -1},q^{k+1},abcdq^{2\alpha
+n+k-1},q^{k-n};q,a^{2}). \label{aaw4}\end{aligned}$$There is another useful representation of the associated Askey–Wilson polynomials in terms of a double series due to Rahman, $$\begin{aligned}
p_{n}^{\alpha }(x)& =p_{n}^{\alpha }(x;a,b,c,d|q)\smallskip \notag \\
& =\frac{(abcdq^{2\alpha -1},q^{\alpha +1};q)_{n}}{(q,abcdq^{\alpha
-1};q)_{n}}q^{-\alpha n}\sum_{k=0}^{n}\frac{(q^{-n},abcdq^{2\alpha
+n-1};q)_{k}}{(q^{\alpha +1},abq^{\alpha };q)_{k}}\smallskip \notag \\
& \qquad \times \frac{(aq^{\alpha }e^{i\theta },aq^{\alpha }e^{-i\theta
};q)_{k}}{(acq^{\alpha },acq^{\alpha };q)_{k}}\sum_{j=0}^{k}\frac{(q^{\alpha
},abq^{\alpha -1},acq^{\alpha -1},adq^{\alpha -1};q)_{j}}{(q,abcdq^{2\alpha
-2},aq^{\alpha }e^{i\theta },aq^{\alpha }e^{-i\theta };q)_{j}}q^{j},
\label{aaw5}\end{aligned}$$where $x=\cos \theta $ (see [@Ga:Ra Exercises 8.26–8.27] and [Rahman96]{}, [@Rah2000]). This formula will be the starting point for our investigation.
An Overview of the Main Result
==============================
To construct an eigenvalue problem for the associated Askey–Wilson polynomials, let us consider an auxiliary function $u_{n}^{\alpha }(x,y)$ in two variables, which for $x=y$ coincides with the associated Askey–Wilson polynomials (up to a factor). We observe that the Askey–Wilson operator $%
L_{0}(x)$ (in one variable $x$) maps $u_{n}^{\alpha }(x,y)$ to the $n$-th degree ordinary Askey–Wilson polynomial (up to some factors). A similar result is obtained for the operator $L_{1}(y)$ applied to $u_{n}^{\alpha
}(x,y)$ with respect to the second independent variable $y.$ We will find an operator $L_{2}(x),$ which maps certain multiples of $\left(
L_{1}(y)+\lambda \right) u_{n}^{\alpha }(x,y)$ to $(L_{0}(x)+\lambda
)u_{n}^{\alpha }(x,y).$ As a result, we obtain an eigenvalue problem of the form$$\begin{gathered}
\frac{(aq^{s},aq^{-s};q)_{\infty }}{(aq^{\alpha +s-1},aq^{\alpha
-s-1};q)_{\infty }}(L_{2}(x)+\lambda )\frac{(aq^{\alpha +s},aq^{\alpha
-s};q)_{\infty }}{(aq^{s},aq^{-s};q)_{\infty }}\left( L_{1}(y)+\mu _{\alpha
}\right) u_{n}^{\alpha }(x,y) \label{ep1} \\
=\frac{4q^{9/2}}{(1-q)^{2}\gamma }(L_{0}(x)+\lambda _{\alpha
+n})u_{n}^{\alpha }(x,y)\end{gathered}$$related to the associated Askey–Wilson polynomials of Ismail and Rahman (see Theorem \[thm:1\] below for an exact statement). We shall use the normalization $$p_{n}(x;a,b,c,d)={}_{4}\varphi _{3}\!\left( \!\!%
\begin{array}{c}
q^{-n},\ abcdq^{n-1},\ aq^{s},\ aq^{-s}\smallskip \\[0.1cm]
ab,\ ac,\ ad%
\end{array}%
\!\!;q,\,q\!\right) \label{AskeyWilsonPlinomials}$$for the ordinary Askey–Wilson polynomials throughout this paper.
\[lem:1\] Let $u_{n}^{\alpha }(x,y)$ be the function in the two variables $x$ and $y$ defined by $$\begin{aligned}
u_{n}^{\alpha }(x,y)& :=\frac{(aq^{s},aq^{-s},aq^{\alpha +z},aq^{\alpha
-z};q)_{\infty }}{(aq^{\alpha +s},aq^{\alpha -s},aq^{z},aq^{-z};q)_{\infty }}
\notag \\
& \qquad \times \sum_{m=0}^{n}\frac{(q^{-n},\gamma q^{2\alpha
+n-1},aq^{\alpha +s},aq^{\alpha -s};q)_{m}}{(q^{\alpha +1},abq^{\alpha
},acq^{\alpha },adq^{\alpha };q)_{m}}q^{m} \notag \\
& \qquad \qquad \times \sum_{k=0}^{m}\frac{(q^{\alpha },abq^{\alpha
-1},acq^{\alpha -1},adq^{\alpha -1};q)_{k}}{(q,\gamma q^{2\alpha
-2},aq^{\alpha +z},aq^{\alpha -z};q)_{k}}q^{k}, \label{AssAWFuncs}\end{aligned}$$with $x(s)=(q^{s}+q^{-s})/2$ and $y(z)=(q^{z}+q^{-z})/2.$ Then $%
u_{n}^{\alpha }(x,y)$ satisfies an equation of the form $$(L_{0}(x)+\lambda _{\alpha +n})u_{n}^{\alpha }(x,y)=f_{n}^{\alpha }(x,y),
\label{AWOpLemma1}$$where $L_{0}(x)=L\left( s;a,b,c,d\right) $ is the Askey–Wilson divided difference operator in the variable $x$ given by . Here,$$\begin{aligned}
f_{n}^{\alpha }(x,y)& =-\frac{4q^{3/2-\alpha }}{(1-q)^{2}}\frac{%
(aq^{s},aq^{-s},aq^{\alpha +z},aq^{\alpha -z};q)_{\infty }}{(aq^{\alpha
+s-1},aq^{\alpha -s-1},aq^{z},aq^{-z};q)_{\infty }} \notag \\
& \qquad \quad \quad
\times (q^{\alpha },abq^{\alpha -1},acq^{\alpha -1},adq^{\alpha
-1};q)_{1}\,\, \notag \\
& \qquad \qquad \times p_{n}(x;aq^{\alpha -1},bcdq^{\alpha
-1},q^{1+z},q^{1-z}), \notag\end{aligned}$$and$$\lambda _{\alpha +n}=\frac{4q^{3/2}}{\left( 1-q\right) ^{2}}\left(
1-q^{-\alpha -n}\right) \left( 1-\gamma q^{\alpha +n-1}\right) ,\qquad
\gamma =abcd.$$
Note that $f_{n}^{\alpha }(x,y)$ contains the $n$-th degree ordinary Askey–Wilson polynomial of the form (\[AskeyWilsonPlinomials\]) in the variable $x.$ Our function $u_{n}^{\alpha }(x,y)$ is the Askey–Wilson polynomial when $\alpha =0$ and a constant multiple of the associated Askey–Wilson polynomial if $x=y.$
\[lem:2\] The function $u_{n}^{\alpha }(x,y)$ satisfies another equation, namely$$(L_{1}(y)+\mu _{\alpha })u_{n}^{\alpha }(x,y)=g_{n}^{\alpha }(x,y),$$where $L_{1}(y):=L\left( y;q/a,q/b,q/c,q/d\right) $ is the Askey–Wilson divided difference operator in $y.$ Here,$$\begin{aligned}
g_{n}^{\alpha }(x,y)& =-\frac{4q^{9/2-\alpha }}{(1-q)^{2}\gamma }\frac{%
(aq^{s},aq^{-s},aq^{\alpha +z+1},aq^{\alpha -z+1};q)_{\infty }}{(aq^{\alpha
+s},aq^{\alpha -s},aq^{z},aq^{-z};q)_{\infty }} \notag \\
& \qquad\quad \quad
\times (q^{\alpha },abq^{\alpha -1},acq^{\alpha -1},adq^{\alpha
-1};q)_{1}\,\, \notag \\
& \qquad \qquad \times p_{n}(x;aq^{\alpha },bcdq^{\alpha -2},q^{1+z},q^{1-z})
\notag\end{aligned}$$and$$\mu _{\alpha }=\frac{4q^{3/2}}{\left( 1-q\right) ^{2}}\left( 1-q^{\alpha
}\right) \left( 1-q^{3-\alpha }/\gamma \right) .$$
Note that $g_{n}^{\alpha }(x,y)$ contains another $n$-th degree Askey–Wilson polynomial (\[AskeyWilsonPlinomials\]) in the same variable $%
x.$
\[lem:3\] The difference differentiation formula $$(L\left( x\right) +\lambda )p_{n}(x;a,b,c,d)=\lambda p_{n}(x;a/q,bq,c,d)
\label{DiffDiffFormulaAW}$$holds for the Askey–Wilson polynomials given by . Here, $L\left( x\right) =L\left( s;a,a/q,c,d\right) $ is the Askey–Wilson divided difference operator and$$\lambda =\frac{4q^{3/2}}{\left( 1-q\right) ^{2}}\left( 1-ac/q\right) \left(
1-ad/q\right) .$$
Lemmas \[lem:1\]–\[lem:3\] allow us to establish the eigenvalue problem (\[ep1\]) for the associated Askey–Wilson functions (\[AssAWFuncs\]), see the next section.
Main Result {#sec:5}
===========
With the help of Lemmas \[lem:1\]–\[lem:3\], we now identify an operator $L_{2}(x)$ linking $(L_{0}(x)+\lambda _{\alpha +n})u_{n}^{\alpha }(x,y)$ and $%
(L_{1}(y)+\lambda _{-\alpha })u_{n}^{\alpha }(x,y)$ in such a way that an eigenvalue problem is formulated.
\[thm:1\] Let $L_{2}(x)=L(s;aq^{\alpha },aq^{\alpha -1},q^{1+z},q^{1-z})$ be the Askey–Wilson divided difference operator defined by with $$\sigma (s)=q^{-2s}\left( q^{s}-aq^{\alpha }\right) \left( q^{s}-aq^{\alpha
-1}\right) \left( q^{s}-q^{1+z}\right) \left( q^{s}-q^{1-z}\right)$$and$$\lambda =\frac{4q^{3/2}}{(1-q)^{2}}\left( 1-aq^{\alpha -z}\right) \left(
1-aq^{\alpha +z}\right) .$$Then an eigenvalue problem for the associated Askey–Wilson functions $%
u_{n}^{\alpha }(x,y)$ can be stated as$$\begin{gathered}
\frac{\gamma }{q^{3}}\frac{\left( aq^{s},aq^{-s};q\right) _{\infty }}{%
\left( aq^{\alpha +s-1},aq^{\alpha -s-1};q\right) _{\infty }}\left(
L_{2}(x)+\lambda \right) \frac{\left( aq^{\alpha +s},aq^{\alpha -s};q\right)
_{\infty }}{\left( aq^{s},aq^{-s};q\right) _{\infty }}\left( L_{1}(y)+\mu
_{\alpha }\right) u_{n}^{\alpha }(x,y) \label{MainTh3} \\
=\frac{4q^{3/2}}{(1-q)^{2}}\left( L_{0}(x)+\lambda _{\alpha
+n}\right) u_{n}^{\alpha }(x,y),\end{gathered}$$where $L_{0},$ $L_{1},$ $\lambda _{\alpha +n},$ $\mu _{\alpha }$ and $%
u_{n}^{\alpha }(x,y)$ are defined as in Lemmas *\[lem:1\]–\[lem:3\]*.
Computational details are left to the reader. The explicit form of the difference operator in two variables on the left-hand side of the last equation has also been calculated, but it is too long to be displayed here.
Proofs
======
Proof of Lemma \[lem:1\]
------------------------
Let $\lambda _{\nu }$ be an arbitrary number. We are looking for solutions of a generalization of the equation (\[AWOpLemma1\]), namely,$$(L_{0}(x)+\lambda _{\nu })u_{n}^{\alpha }(x,y)=f_{n}^{\alpha }(x,y),$$in terms of generalized powers (see for the definition) $$u_{n}^{\alpha }(x,y)=\sum_{m=0}^{n}c_{m}v_{m}[x(s)-x(\xi )]^{(\alpha +m)},$$where $$v_{m}=v_{m}(y)=\frac{(aq^{\alpha +z},aq^{\alpha -z};q)_{\infty }}{%
(aq^{z},aq^{-z};q)_{\infty }}\sum_{k=0}^{n}\frac{(q^{\alpha },abq^{\alpha
-1},acq^{\alpha -1},adq^{\alpha -1};q)_{k}}{(q,\gamma q^{2\alpha
-2},aq^{\alpha +z},aq^{\alpha -z};q)_{k}}q^{k},$$and $\gamma =abcd.$ (This is an analogue of the power series expansion; see [@At:Su:DHF], [@Su4 Exercises 2.9–2.11], and [@Su2] for properties of the generalized powers.)
Apply the Askey–Wilson operator to $u_{n}^{\alpha }(x,y)$ to obtain $$(L_{0}(x)+\lambda _{\nu })u_{n}^{\alpha }(x,y)=\lambda _{\nu
}\sum_{m=0}^{n}c_{m}v_{m}[x(s)-x(\xi )]^{(\alpha +m)}
+\sum_{m=0}^{n}c_{m}v_{m}\ L_{0}(x)[x(s)-x(\xi )]^{(\alpha +m)},$$since $v_{m}$ is independent of $x.$ By [@At:Su:DHF], we have $$\begin{aligned}
L_{0}(x)[x(s)-x(\xi )]^{(\alpha +m)}
& =\gamma (\alpha +m)\gamma (\alpha +m-1)\sigma (\xi -\alpha
-m+1)[x(s)-x(\xi -1)]^{(\alpha +m-2)} \notag \\
& \qquad +\gamma (\alpha +m)\tau _{\alpha +m-1}(\xi -\alpha -m+1)[x(s)-x(\xi
-1)]^{(\alpha +m-1)} \notag \\
& \qquad -\lambda _{\alpha +m}[x(s)-x(\xi )]^{(\alpha +m)}.
\notag\end{aligned}$$We use the same notations as in [@At:Su:DHF], [@Su4 Exercise 2.25], or [@Su2]. Choose $a_{0}:=\xi -\alpha -m+1$ to be a root of the equation $\sigma (a_{0})=0.$ Then $\xi =a_{0}+\alpha +m-1$, and one obtains $$\begin{aligned}
& (L_{0}(x)+\lambda _{\nu })u_{n}^{\alpha
}(x,y)=\sum_{m=0}^{n}c_{m}v_{m}\gamma (\alpha +m)\tau _{\alpha
+m-1}(a_{0})[x(s)-x(a_{0}+\alpha +m-2)]^{(\alpha +m-1)} \notag \\
& \qquad +\sum_{m=0}^{n}c_{m}v_{m}(\lambda _{\nu }-\lambda _{\alpha
+m})[x(s)-x(a_{0}+\alpha +m-1)]^{(\alpha +m)} \notag \\
& =c_{0}v_{0}\gamma (\alpha )\tau _{\alpha -1}(a_{0})[x(s)-x(a_{0}+\alpha
-2)]^{(\alpha -1)} \notag \\
& \qquad +\sum_{m=1}^{n}c_{m}v_{m}\gamma (\alpha +m)\tau _{\alpha
+m-1}(a_{0})[x(s)-x(a_{0}+\alpha +m-2)]^{(\alpha +m-1)} \notag \\
& \qquad \qquad +\sum_{m=0}^{n}c_{m}v_{m}(\lambda _{\nu }-\lambda _{\alpha
+m})[x(s)-x(a_{0}+\alpha +m-1)]^{(\alpha +m)}. \notag\end{aligned}$$Letting $m=k+1,$ we get $$\begin{gathered}
(L_{0}(x)+\lambda _{\nu })u_{n}^{\alpha }(x,y)=c_{0}v_{0}\gamma (\alpha
)\tau _{\alpha -1}(a_{0})[x(s)-x(a_{0}+\alpha -2)]^{(\alpha -1)}
\label{L_0-eqn} \\
\kern4cm
+\sum_{k=0}^{n-1}c_{k+1}v_{k+1}\gamma (\alpha +k+1)\tau _{\alpha
+k}(a_{0})[x(s)-x(a_{0}+\alpha +k-1)]^{(\alpha +k)} \\
+\sum_{k=0}^{n}c_{k}v_{k}(\lambda _{\nu }-\lambda _{\alpha
+k})[x(s)-x(a_{0}+\alpha +k-1)]^{(\alpha +k)}.\end{gathered}$$Note that for$$v_{k}=\sum_{l=0}^{k}e_{l},\qquad e_{l}:=\frac{(aq^{\alpha +z},aq^{\alpha
-z};q)_{\infty }}{(aq^{z},aq^{-z};q)_{\infty }}\frac{(q^{\alpha
},abq^{\alpha -1},acq^{\alpha -1},adq^{\alpha -1};q)_{l}}{(q,\gamma
q^{2\alpha -2},aq^{\alpha +z},aq^{\alpha -z};q)_{l}}q^{l}$$one has $$v_{k+1}=v_{k}+e_{k+1}\text{ \ \ \ \ and \ \ \ \ }v_{0}=e_{0.}$$After choosing $\lambda _{\nu }=\lambda _{\alpha +n},$ equation (\[L\_0-eqn\]) becomes $$\begin{aligned}
(L_{0}(x)+\lambda _{\alpha +n})u_{n}^{\alpha
}(x,y)&=\sum_{k=-1}^{n-1}c_{k+1}e_{k+1}\gamma (\alpha +k+1)\tau _{\alpha
+k}(a_{0})[x(s)-x(a_{0}+\alpha +k-1)]^{(\alpha +k)} \\
& \kern1cm
+\sum_{k=0}^{n-1}c_{k+1}v_{k}\gamma (\alpha +k+1)\tau _{\alpha
+k}(a_{0})[x(s)-x(a_{0}+\alpha +k-1)]^{(\alpha +k)} \\
& \kern1cm +\sum_{k=0}^{n-1}c_{k}v_{k}(\lambda _{\alpha +n}-\lambda
_{\alpha +k})[x(s)-x(a_{0}+\alpha +k-1)]^{(\alpha +k)}.\end{aligned}$$The latter two sums vanish if $$c_{k+1}\gamma (\alpha +k+1)\tau _{\alpha +k}(a_{0})=c_{k}(\lambda _{\alpha
+n}-\lambda _{\alpha +k}).$$Therefore, $$\begin{gathered}
(L_{0}(x)+\lambda _{\alpha +n})u_{n}^{\alpha
}(x,y)=\sum_{k=-1}^{n-1}c_{k+1}e_{k+1}\gamma (\alpha +k+1)\tau _{\alpha
+k}(a_{0})[x(s)-x(a_{0}+\alpha +k-1)]^{(\alpha +k)} \\
=\sum_{m=0}^{n}c_{m}e_{m}\gamma (\alpha +m)\tau _{\alpha
+m-1}(a_{0})[x(s)-x(a_{0}+\alpha +m-2)]^{(\alpha +m-1)}=:f_{n}^{\alpha
}(x,y).\end{gathered}$$Finally, we show that the function $f_{n}^{\alpha }(x,y)$ is, up to a factor, the $n$-th ordinary Askey–Wilson polynomial. The generalized powers have the property (see [@Su4]) $$\lbrack x(s)-x(z)]^{(n+1)}=[x(s)-x(z)][x(s)-x(z-1)]^{(n)},$$which leads to $$f_{n}^{\alpha }(x,y)=\sum_{m=0}^{n}c_{m}e_{m}\gamma (\alpha +m)\tau _{\alpha
+m-1}(a_{0})\frac{[x(s)-x(a_{0}+\alpha +m-1)]^{(\alpha +m)}}{%
[x(s)-x(a_{0}+\alpha +m-1)]}.$$Moreover, $$\begin{aligned}
c_{m}[x(s)-x(a_{0}& +\alpha +m-1)]^{(\alpha +m)} \\
& =c_{0}\frac{(q^{-n},\gamma q^{2\alpha +n-1};q)_{m}}{(q^{\alpha
+1},abq^{\alpha },acq^{\alpha },adq^{\alpha };q)_{m}}q^{m}\
[x(s)-x(a_{0}+\alpha +m-1)]^{(\alpha +m)} \\
& =c_{0}\,\varphi _{m}(x)\,[x(s)-x(a_{0}+\alpha -1)]^{(\alpha )},\end{aligned}$$where, by definition, $$\varphi _{m}(x):=\frac{(aq^{s},aq^{-s};q)_{\infty }}{(aq^{\alpha
+s},aq^{\alpha -s};q)_{\infty }}\frac{(q^{-n},\gamma q^{2\alpha
+n-1},aq^{\alpha +s},aq^{\alpha -s};q)_{m}}{(q^{\alpha +1},abq^{\alpha
},acq^{\alpha },adq^{\alpha };q)_{m}}q^{m}.$$Therefore, $$\begin{aligned}
f_{n}^{\alpha }(x,y)& =\frac{(aq^{s},aq^{-s};q)_{\infty }}{(aq^{\alpha
+s},aq^{\alpha -s};q)_{\infty }}\frac{(q^{-n},\gamma q^{2\alpha
+n-1},aq^{\alpha +s},aq^{\alpha -s};q)_{m}}{(q^{\alpha +1},abq^{\alpha
},acq^{\alpha },adq^{\alpha };q)_{m}}q^{m} \notag \\
& \qquad \times \frac{(aq^{\alpha +z},aq^{\alpha -z};q)_{\infty }}{%
(aq^{z},aq^{-z};q)_{\infty }}\frac{(q^{\alpha },abq^{\alpha -1},acq^{\alpha
-1},adq^{\alpha -1};q)_{m}}{(q,\gamma q^{2\alpha -2},aq^{\alpha
+z},aq^{\alpha -z};q)_{m}}q^{m} \notag \\
& \qquad \qquad \times \frac{\gamma (\alpha +m)\tau _{\alpha +m-1}(a_{0})}{%
\,[x(s)-x(a_{0}+\alpha +m-1)]}.
\label{FLemma1}\end{aligned}$$Recall that $a=q^{a_{0}}$ and $$\begin{aligned}
\gamma (\alpha +m)& =q^{-\frac{\alpha +m-1}{2}}\,\frac{1-q^{\alpha +m}}{1-q},
\\
x(s)-x(a_{0}+\alpha +m-1)& =-\frac{1}{2a}q^{-\alpha -m+1}(1-aq^{\alpha
-s+m-1})(1-aq^{\alpha +s+m-1}),\text{ \ \ \ \ } \\
\tau _{\alpha +m-1}(a_{0})& =\frac{2}{a(1-q)}q^{-2(\alpha +m-1)+\frac{\alpha
+m}{2}}(1-abq^{\alpha +m-1})(1-acq^{\alpha +m-1})(1-adq^{\alpha +m-1}),\end{aligned}$$which allows us to simplify the last term of (\[FLemma1\]) to $$q^{m}\frac{\gamma (\alpha +m)\tau _{\alpha +m-1}(a_{0})}{\,[x(s)-x(a_{0}+%
\alpha +m-1)]}=-4q^{\frac{3}{2}-\alpha }\frac{1-q^{\alpha +m}}{1-q}\frac{%
(1-abq^{\alpha +m-1})(1-acq^{\alpha +m-1})(1-adq^{\alpha +m-1})}{%
(1-aq^{\alpha -s+m-1})(1-aq^{\alpha +s+m-1})}.$$Thus $f_{n}^{\alpha }(x,y)$ becomes $$\begin{aligned}
f_{n}^{\alpha }(x,y)& =\frac{-4q^{\frac{3}{2}-\alpha }}{(1-q)^{2}}\frac{%
(aq^{s},aq^{-s},aq^{\alpha +z},aq^{\alpha -z};q)_{\infty }}{(aq^{\alpha
+s-1},aq^{\alpha -s-1},aq^{z},aq^{-z};q)_{\infty }}\ (q^{\alpha
},abq^{\alpha -1},acq^{\alpha -1},adq^{\alpha -1};q)_{1} \\
& \qquad \times \sum_{m=0}^{n}\frac{(q^{-n},\gamma q^{2\alpha
+n-1},aq^{\alpha +s-1},aq^{\alpha -s-1};q)_{m}}{(q,\gamma q^{2\alpha
-2},aq^{\alpha +z},aq^{\alpha -z};q)_{m}}q^{m} \notag \\
& =\frac{-4q^{\frac{3}{2}-\alpha }}{(1-q)^{2}}\frac{(aq^{s},aq^{-s},aq^{%
\alpha +z},aq^{\alpha -z};q)_{\infty }}{(aq^{\alpha +s-1},aq^{\alpha
-s-1},aq^{z},aq^{-z};q)_{\infty }}\medskip \ (q^{\alpha },abq^{\alpha
-1},acq^{\alpha -1},adq^{\alpha -1};q)_{1}\, \notag \\
& \qquad \times p_{n}(x;aq^{\alpha -1},bcdq^{\alpha -1},q^{1+z},q^{1-z}),
\notag\end{aligned}$$which completes the proof of the lemma.
Proof of Lemma \[lem:2\]
------------------------
Consider the equation $$(L_{1}(y)+\lambda _{\nu })u_{n}^{\alpha }(x,y)=g_{n}^{\alpha }(x,y),$$and rewrite $u_{n}^{\alpha }(x,y)$ in the form $$u_{n}^{\alpha }(x,y)=\sum_{m=0}^{n}c_{m}^{\alpha }(aq^{\alpha +s},aq^{\alpha
-s};q)_{m}\frac{(aq^{s},aq^{-s};q)_{\infty }}{(aq^{\alpha +s},aq^{\alpha
-s};q)_{\infty }}v_{m}^{\alpha }(y),$$where $$c_{m}^{\alpha }=\frac{(q^{-n},\gamma q^{2\alpha +n-1};q)_{m}}{(q^{\alpha
+1},abq^{\alpha },acq^{\alpha },adq^{\alpha };q)_{m}}q^{m},\text{ \ \ \ \ }%
\gamma =abcd,$$and $$v_{m}^{\alpha }(y)=\frac{(aq^{\alpha +z},aq^{\alpha -z};q)_{\infty }}{%
(aq^{z},aq^{-z};q)_{\infty }}\sum_{k=0}^{m}\frac{(q^{\alpha },abq^{\alpha
-1},acq^{\alpha -1},adq^{\alpha -1};q)_{k}}{(q,\gamma q^{2\alpha
-2},aq^{\alpha +z},aq^{\alpha -z};q)_{k}}q^{k}.$$Apply the Askey–Wilson operator $L_{1}(y):=L\left( y;q/a,q/b,q/c,q/d\right)
$ to $u_{n}^{\alpha }(x,y)$ to obtain $$(L_{1}(y)+\lambda _{\nu })u_{n}^{\alpha }(x,y)=\sum_{m=0}^{n}c_{m}^{\alpha
}(aq^{\alpha +s},aq^{\alpha -s};q)_{m}\frac{(aq^{s},aq^{-s};q)_{\infty }}{%
(aq^{\alpha +s},aq^{\alpha -s};q)_{\infty }}\ \left( L_{1}(y)+\lambda _{\nu
}\right) v_{m}^{\alpha }(y).$$
Let $$v_{m}^{\alpha }(y):=\sum_{k=0}^{m}\frac{c_{k}}{[x(s)-x(\xi )]^{(\alpha +k)}}$$in analogy with [@At:Su:DHF]. Then $$(L_{1}(y)+\lambda _{\nu })v_{m}^{\alpha }(y)=\lambda _{\nu }\sum_{k=0}^{m}%
\frac{c_{k}}{[x(s)-x(\xi )]^{(\alpha +k)}}+\sum_{k=0}^{m}c_{k}\
L_{1}(y)\left( \frac{1}{[x(s)-x(\xi )]^{(\alpha +k)}}\right) .$$By [@At:Su:DHF], we have $$\begin{gathered}
L_{1}(y)\left( \frac{1}{[x(s)-x(\xi )]^{(\alpha +k)}}\right) =\frac{\gamma
(\alpha +k)\gamma (\alpha +k+1)\sigma (\xi +1)}{[x(z)-x(\xi +1)]^{(\alpha
+k+2)}} \notag \\
-\frac{\gamma (\alpha +k)\tau _{-\alpha -k-1}(\xi +1)}{[x(z)-x(\xi
)]^{(\alpha +k+1)}}-\frac{\lambda _{-\alpha -k}}{[x(z)-x(\xi )]^{(\alpha +k)}%
} \notag\end{gathered}$$(see also [@Su4 Exercise 2.25]). Upon choosing $a_{0}:=\xi +1$ to be a root of the equation $\sigma (a_{0})=0,$ we obtain $$\begin{aligned}
(L_{1}(y)+\lambda _{\nu })v_{m}^{\alpha }(y)& =\lambda _{\nu }\sum_{k=0}^{m}%
\frac{c_{k}}{[x(s)-x(a_{0})]^{(\alpha +k)}} \notag \\
& \qquad -\sum_{k=0}^{m}c_{k}\left( \frac{\gamma (\alpha +k)\tau _{-\alpha
-k-1}(a_{0})}{[x(z)-x(a_{0}-1)]^{(\alpha +k+1)}}+\frac{\lambda _{-\alpha -k}%
}{[x(z)-x(a_{0}-1)]^{(\alpha +k)}}\right) \notag \\
& =\sum_{k=0}^{m}\frac{c_{k}\left( \lambda _{\nu }-\lambda _{-\alpha
-k}\right) }{[x(z)-x(a_{0}-1)]^{(\alpha +k)}}-\sum_{k=0}^{m}\frac{%
c_{k}\,\gamma (\alpha +k)\tau _{-\alpha -k-1}(a_{0})}{[x(z)-x(a_{0}-1)]^{(%
\alpha +k+1)}} \notag \\
& =\frac{c_{0}\left( \lambda _{\nu }-\lambda _{-\alpha }\right) }{%
[x(z)-x(a_{0}-1)]^{(\alpha )}}+\sum_{k=1}^{m}\frac{c_{k}\left( \lambda _{\nu
}-\lambda _{-\alpha -k}\right) }{[x(z)-x(a_{0}-1)]^{(\alpha +k)}} \notag \\
& \qquad -\frac{c_{m}\,\gamma (\alpha +m)\tau _{-\alpha -m-1}(a_{0})}{%
[x(z)-x(a_{0}-1)]^{(\alpha +m+1)}}-\sum_{k=0}^{m-1}\frac{c_{k}\,\gamma
(\alpha +k)\tau _{-\alpha -k-1}(a_{0})}{[x(z)-x(a_{0}-1)]^{(\alpha +k+1)}}.
\notag\end{aligned}$$Now choose $\lambda _{\nu }=\lambda _{-\alpha }$ and let $k=l+1.$ Then we obtain$$\begin{gathered}
(L_{1}(y)+\lambda _{\nu })v_{m}^{\alpha }(y)=-\frac{c_{m}\,\gamma
(\alpha +m)\tau _{-\alpha -m-1}(a_{0})}{[x(z)-x(a_{0}-1)]^{(\alpha +m+1)}} \\
+\sum_{l=0}^{m-1}\frac{c_{l+1}\left( \lambda _{-\alpha }-\lambda
_{-\alpha -l-1}\right) }{[x(z)-x(a_{0}-1)]^{(\alpha +l+1)}}-\sum_{l=0}^{m-1}%
\frac{c_{l}\,\gamma (\alpha +l)\tau _{-\alpha -l-1}(a_{0})}{%
[x(z)-x(a_{0}-1)]^{(\alpha +l+1)}}.\end{gathered}$$The latter two sums vanish if $$c_{l+1}\left( \lambda _{-\alpha }-\lambda _{-\alpha -l-1}\right)
=c_{l}\,\gamma (\alpha +l)\tau _{-\alpha -l-1}(a_{0}).$$In that case, we have $$(L_{1}(y)+\lambda _{\nu })v_{m}^{\alpha }(y)=-\frac{c_{m}\,\gamma (\alpha
+m)\tau _{-\alpha -m-1}(a_{0})}{[x(z)-x(a_{0}-1)]^{(\alpha +m+1)}}=-\frac{%
c_{m+1}\left( \lambda _{-\alpha }-\lambda _{-\alpha -m-1}\right) }{%
[x(z)-x(a_{0}-1)]^{(\alpha +m+1)}}=:h_{m}^{\alpha }(y).$$Here, $$\begin{aligned}
\frac{c_{m+1}}{[x(z)-x(a_{0}-1)]^{(\alpha +m+1)}}& =\frac{c_{0}}{%
[x(z)-x(a_{0}-1)]^{(\alpha )}}\varphi _{m+1}(z), \\
\varphi _{m+1}(z)& =\frac{(q^{\alpha },abq^{\alpha -1},acq^{\alpha
-1},adq^{\alpha -1};q)_{m+1}}{(q,\gamma q^{2\alpha -2},aq^{\alpha
+z},aq^{\alpha -z};q)_{m+1}}q^{m+1}, \\
\frac{c_{0}}{[x(z)-x(a_{0}-1)]^{(\alpha )}}& =\frac{(aq^{\alpha
+z},aq^{\alpha -z};q)_{\infty }}{(aq^{z},aq^{-z};q)_{\infty }}, \\
\lambda _{-\alpha }-\lambda _{-\alpha -m-1}& =\frac{4}{(1-q)^{2}\gamma }q^{%
\frac{7}{2}-\alpha -m}(1-q^{m+1})(1-\gamma q^{2\alpha +m-2})\end{aligned}$$and $$h_{m}^{\alpha }(y)=-\frac{4q^{\frac{9}{2}-\alpha }}{(1-q)^{2}\gamma }\frac{%
(aq^{\alpha +z},aq^{\alpha -z};q)_{\infty }}{(aq^{z},aq^{-z};q)_{\infty }}%
\frac{(q^{\alpha },abq^{\alpha -1},acq^{\alpha -1},adq^{\alpha -1};q)_{m+1}}{%
(q,\gamma q^{2\alpha -2};q)_{m}(aq^{\alpha +z},aq^{\alpha -z};q)_{m+1}}.$$Therefore, $$\begin{aligned}
(L_{1}(y)& +\lambda _{\nu })u_{n}^{\alpha }(x,y) \notag \\
& =\sum_{m=0}^{n}c_{m}^{\alpha }(aq^{\alpha +s},aq^{\alpha -s};q)_{m}\frac{%
(aq^{s},aq^{-s};q)_{\infty }}{(aq^{\alpha +s},aq^{\alpha -s};q)_{\infty }}%
L_{1}(y)v_{m}^{\alpha }(y) \notag \\
& =-\sum_{m=0}^{n}c_{m}^{\alpha }(aq^{\alpha +s},aq^{\alpha -s};q)_{m}\frac{%
(aq^{s},aq^{-s};q)_{\infty }}{(aq^{\alpha +s},aq^{\alpha -s};q)_{\infty }}
\notag \\
& \qquad \times \frac{4q^{\frac{9}{2}-\alpha }}{(1-q)^{2}\gamma }
\frac{%
(aq^{\alpha +z},aq^{\alpha -z};q)_{\infty }}{(aq^{z},aq^{-z};q)_{\infty }}%
\frac{(q^{\alpha },abq^{\alpha -1},acq^{\alpha -1},adq^{\alpha -1};q)_{m+1}}{%
(q,\gamma q^{2\alpha -2};q)_{m}(aq^{\alpha +z},aq^{\alpha -z};q)_{m+1}}
\notag \\
& =-\frac{4q^{\frac{9}{2}-\alpha }}{(1-q)^{2}\gamma }\frac{%
(aq^{s},aq^{-s},aq^{\alpha +z+1},aq^{\alpha -z+1};q)_{\infty }}{(aq^{\alpha
+s},aq^{\alpha -s},aq^{z},aq^{-z};q)_{\infty }} \notag \\
& \qquad \times \sum_{m=0}^{n}\frac{(q^{-n},\gamma q^{2\alpha
+n-1},aq^{s},aq^{-s};q)_{m}}{(q^{\alpha +1},abq^{\alpha },acq^{\alpha
},adq^{\alpha };q)_{m}}q^{m} \notag \\
& \qquad \times \frac{(1-q^{\alpha })(1-abq^{\alpha -1})(1-acq^{\alpha
-1})(1-adq^{\alpha -1})}{(q,\gamma q^{2\alpha -2};q)_{m}(aq^{\alpha
+z+1},aq^{\alpha -z+1};q)_{m}} \notag \\
& =-\frac{4q^{\frac{9}{2}-\alpha }}{(1-q)^{2}\gamma }\frac{%
(aq^{s},aq^{-s},aq^{\alpha +z+1},aq^{\alpha -z+1};q)_{\infty }}{(aq^{\alpha
+s},aq^{\alpha -s},aq^{z},aq^{-z};q)_{\infty }} \notag \\
& \qquad \times \left( q^{\alpha },abq^{\alpha -1},acq^{\alpha
-1},adq^{\alpha -1};q\right) _{1} \notag \\
& \qquad \times \text{ }_{4}\varphi _{3}\!\left( \!\!%
\begin{array}{c}
q^{-n},\gamma q^{2\alpha +n-1},aq^{\alpha +s},aq^{\alpha -s} \\[0.1cm]
\gamma q^{2\alpha -2},aq^{\alpha +z+1},aq^{\alpha -z+1}%
\end{array}%
\!\!;q,\,q\!\right) \notag \\
& =-\frac{4q^{\frac{9}{2}-\alpha }}{(1-q)^{2}\gamma }\frac{%
(aq^{s},aq^{-s},aq^{\alpha +z+1},aq^{\alpha -z+1};q)_{\infty }}{(aq^{\alpha
+s},aq^{\alpha -s},aq^{z},aq^{-z};q)_{\infty }} \notag \\
& \qquad \times \left( q^{\alpha },abq^{\alpha -1},acq^{\alpha
-1},adq^{\alpha -1};q\right) _{1}\times \text{ }p_{n}(x;aq^{\alpha
},bcdq^{\alpha -2},q^{1+z},q^{1-z}). \notag\end{aligned}$$This completes the proof of the lemma.
Proof of Lemma \[lem:3\]
------------------------
The structure of the Askey–Wilson operator in (\[in9\]) and the basic hypergeometric series representation (\[AskeyWilsonPlinomials\]) suggest to look for a $4$-term relation of the form$$\begin{gathered}
K_{1}\ {}_{4}\varphi _{3}\!\left( \!\!%
\begin{array}{c}
A,B,C,D \\[0.1cm]
F,G,H%
\end{array}%
\!\!;q,\,q\!\right) +K_{2}\ {}_{4}\varphi _{3}\!\left( \!\!%
\begin{array}{c}
A,B,Cq,D/q \\[0.1cm]
F,G,H%
\end{array}%
\!\!;q,\,q\!\right) \\
+K_{3}\ {}_{4}\varphi _{3}\!\left( \!\!%
\begin{array}{c}
A,B,C/q,Dq \\[0.1cm]
F,G,H%
\end{array}%
\!\!;q,\,q\!\right) +K_{4}\ {}_{4}\varphi _{3}\!\left( \!\!%
\begin{array}{c}
A,B,C/q,D/q \\[0.1cm]
F,G/q,H/q%
\end{array}%
\!\!;q,\,q\!\right) =0,
\label{4TermRecurrenceAWPols}\end{gathered}$$for some undetermined coefficients $K_{1},$ $K_{2},$ $K_{3}$ and $K_{4}$ (up to a common factor). Doing a term-wise comparison, we may hope to find $K_{1},$ $K_{2},$ $K_{3},$ $K_{4}$ which satisfy$$\begin{aligned}
K_{1}(1-C)& (1-D)(1-Cq^{k-1})(1-Dq^{k-1})(1-G/q)(1-H/q) \notag \\
& +K_{2}(1-Cq^{k})(1-Cq^{k-1})(1-D/q)(1-D)(1-G/q)(1-H/q) \notag \\
& +K_{3}(1-Dq^{k})(1-Dq^{k-1})(1-C/q)(1-C)(1-G/q)(1-H/q) \notag \\
& +K_{4}(1-C/q)(1-C)(1-D/q)(1-D)(1-Gq^{k-1})(1-Hq^{k-1})=0. \notag\end{aligned}$$If we are successful, then the above equation does indeed imply the contiguous relation . In the equation, we compare coefficients of powers of $q^{k}$. This yields a system of 3 linear equations in the 4 unknowns $K_{1},$ $K_{2},$ $K_{3},$ $K_{4}$. With the help of *Mathematica*, we obtain the solution $$\begin{aligned}
K_{1}& =\frac{(C-q)(D-q)(-GH-CDq+CGq+DGq+CHq+DHq-GHq-CDq^{2})}{%
(G-q)(H-q)(Cq-D)(Dq-C)}, \\
K_{2}& =\frac{(C-1)(D-G)(D-H)(C-q)q}{(D-C)(G-q)(H-q)(Cq-D)}, \\
K_{3}& =\frac{(D-1)(C-G)(C-H)(D-q)q}{(C-D)(G-q)(H-q)(Dq-C)},\end{aligned}$$where the free parameter $K_{4}$ was chosen to be $1$ (see Appendix A for the *Mathematica* code). The required $4$-term contiguous relation is then given by $$\begin{aligned}
& \frac{(C-q)(D-q)(-GH-CDq+CGq+DGq+CHq+DHq-GHq-CDq^{2})}{%
(G-q)(H-q)(Cq-D)(Dq-C)}\text{ } \notag \\
& \qquad \quad \quad \quad \times \text{ }_{4}\varphi _{3}\!\left( \!\!%
\begin{array}{c}
A,B,C,D \\[0.1cm]
F,G,H%
\end{array}%
\!\!;q,\,q\!\right) +{}_{4}\varphi _{3}\!\left( \!\!%
\begin{array}{c}
A,B,C/q,D/q \\[0.1cm]
F,G/q,H/q%
\end{array}%
\!\!;q,\,q\!\right) \notag \\
& \qquad \quad +\frac{(C-1)(D-G)(D-H)(C-q)q}{(D-C)(G-q)(H-q)(Cq-D)}\text{ }%
_{4}\varphi _{3}\!\left( \!\!%
\begin{array}{c}
A,B,Cq,D/q \\[0.1cm]
F,G,H%
\end{array}%
\!\!;q,\,q\!\right) \notag \\
& \qquad \quad +\frac{(D-1)(C-G)(C-H)(D-q)q}{(C-D)(G-q)(H-q)(Dq-C)}%
\text{ }_{4}\varphi _{3}\!\left( \!\!%
\begin{array}{c}
A,B,C/q,Dq \\[0.1cm]
F,G,H%
\end{array}%
\!\!;q,\,q\!\right) =0.
\label{4TermsRecurrenceAWSolved}\end{aligned}$$(This $4$-term contiguous relation for the $_{4}\varphi _{3}$-functions can be extended to an arbitrary $_{r}\psi _{s}$-function, see Appendix A for more details.)
When $qABCD=FGH$, in view of the structure of the Askey–Wilson operator in (\[in9\]), equation (\[4TermsRecurrenceAWSolved\]) should become$$\begin{aligned}
& (L(x)+\lambda )\ _{4}\varphi _{3}\!\left( \!\!%
\begin{array}{c}
A,B,Cq,D/q \\[0.1cm]
F,G,H%
\end{array}%
\!\!;q,\,q\!\right) \\
& =\frac{\sigma (-s)}{\Delta x(s)\nabla x_{1}(s)}\ {}_{4}\varphi
_{3}\!\left( \!\!%
\begin{array}{c}
A,B,Cq,D/q \\[0.1cm]
F,G,H%
\end{array}%
\!\!;q,\,q\!\right) +\frac{\sigma (s)}{\nabla x(s)\nabla x_{1}(s)}{}\
_{4}\varphi _{3}\!\left( \!\!%
\begin{array}{c}
A,B,C/q,Dq \\[0.1cm]
F,G,H%
\end{array}%
\!\!;q,\,q\!\right) \\
& \qquad +\frac{\lambda \Delta x(s)\nabla x(s)\nabla x_{1}(s)-\sigma
(s)\Delta x(s)-\sigma (-s)\nabla x(s)}{\Delta x(s)\nabla x(s)\nabla x_{1}(s)%
}{}\ _{4}\varphi _{3}\!\left( \!\!%
\begin{array}{c}
A,B,C,D \\[0.1cm]
F,G,H%
\end{array}%
\!\!;q,\,q\!\right) .\end{aligned}$$Equating coefficients, one obtains $$\begin{gathered}
(1-C/q)(1-D/q)(D-C)(GH+CDq-CGq-DGq-CHq-DHq+GHq+CDq^{2}) \\
=\frac{2qa^{3}}{1-q}\left( \sigma (-s)\nabla x(s)+\sigma (s)\Delta
x(s)-\lambda \Delta x(s)\nabla x(s)\nabla x_{1}(s)\right)\end{gathered}$$and$$\begin{aligned}
(D-C)(D-C/q)(-C+D/q)&=\frac{2aq^{1/2}}{1-q}\nabla x_{1}(s)\frac{2a}{1-q}%
\Delta x(s)\frac{2a}{1-q}\nabla x(s), \\
(C-1)(G-D)(H-D)(q-C)&=-qa^{2}\sigma (-s), \\
(D-C)(-C+D/q)&=\frac{2aq^{1/2}}{1-q}\nabla x_{1}(s)\frac{2a}{1-q}\Delta
x(s), \\
(D-1)(G-C)(H-C)(q-D)&=-qa^{2}\sigma (s), \\
(D-C)(D-C/q)&=\frac{2aq^{1/2}}{1-q}\nabla x_{1}(s)\frac{2a}{1-q}\nabla x(s),
\\
(G-q)(H-q)&=q^{2}\frac{(1-q)^{2}}{4q^{3/2}}\lambda .\end{aligned}$$This gives the required formula (\[DiffDiffFormulaAW\]) for the Askey–Wilson operator with $$\sigma (s)=q^{-2s}\left( q^{s}-a\right) \left( q^{s}-a/q\right) \left(
q^{s}-c\right) \left( q^{s}-d\right) ,\quad \lambda =\frac{4q^{3/2}}{%
(1-q)^{2}}\left( 1-ac/q\right) \left( 1-ad/q\right) .$$The proof of the lemma is complete.
4-Term Contiguous Relations
===========================
In order to derive the contiguous relation , one can use the following *Mathematica* program:[^2]
In\[1\]:= X1 = K1\*(1 - C) (1 - D) (1 - C\*K/q) (1 - D\*K/q) (1 - G/q) (1 - H/q) + K2\*(1 - C\*K) (1 - C\*K/q) (1 - D/q) (1 - D) (1 - G/q) (1 - H/q) + K3\*(1 - D\*K) (1 - D\*K/q) (1 - C/q) (1 - C) (1 - G/q) (1 - H/q) + K4\*(1 - C/q) (1 - C) (1 - D/q) (1 - D) (1 - G\*K/q) (1 - H\*K/q) ; X1 = Table\[Coefficient\[X1, K, i\] == 0, [i, 0, 2]{}\]; X1 = Solve\[X1, [K1, K2, K3, K4]{}\]; X1 = K1 -> Factor\[K1/.X1\[\[1\]\]\], K2 -> Factor\[K2/.X1\[\[1\]\]\], K3 -> Factor\[K3/.X1\[\[1\]\]\], K4 -> Factor\[K4/.X1\[\[1\]\]\]Out\[1\]= K1 (K4 (C - q) (D - q) (G H + C D q - C G q - D G q - C H q - D H q + G H q + C D q2)) / ((G - q) (H - q) (-D + C q) (C - D q)), K2 - ((-1 + C) (D - G) (D - H) K4 (C - q) q) / ((C - D) (G - q) (H - q) (-D + C q)), K3 - ((-1 + D) (C - G) (C - H) K4 (D - q) q) / ((C - D) (G - q) (H - q) (C - D q)), K4 K4
It is evident from the proof of that, actually, an extension for bilateral series (see [@Ga:Ra equation (5.1.1)] for the definition) with an arbitrary number of parameters holds, namely: $$\begin{aligned}
\notag
&\frac{\left( c-q\right) \left( d-q\right) \left(
-gh-cdq+cgq+dgq+chq+dhq-ghq-cdq^{2}\right) }{\left( g-q\right) \left(
h-q\right) \left( cq-d\right) \left( dq-c\right) } \\
&\quad \quad \quad \qquad \times \ _{r}\psi _{s}\!\left( \!\!%
\begin{array}{c}
a_{1},\dots,a_{i},\ c,d \\[0.1cm]
b_{0},\dots,b_{k},\ g,h%
\end{array}%
\!\!;q,\,t\!\right) \notag \\
&\quad \quad
+\frac{\left( c-1\right) \left( d-g\right) \left( d-h\right) \left(
c-q\right) q}{\left( d-c\right) \left( g-q\right) \left( h-q\right) \left(
cq-d\right) }\ \ _{r}\psi _{s}\!\left( \!\!%
\begin{array}{c}
a_{1},\dots,a_{i},\ cq,d/q \\[0.1cm]
b_{0},\dots,b_{k},\ g,h%
\end{array}%
\!\!;q,\,t\!\right) \notag \\
&\quad \quad
+\frac{\left( d-1\right) \left( c-g\right) \left( c-h\right) \left(
d-q\right) q}{\left( c-d\right) \left( g-q\right) \left( h-q\right) \left(
dq-c\right) }\ \ _{r}\psi _{s}\!\left( \!\!%
\begin{array}{c}
a_{1},\dots,a_{i},\ c/q,dq \\[0.1cm]
b_{0},\dots,b_{k},\ g,h%
\end{array}%
\!\!;q,\,t\!\right) \notag \\
&\qquad +\ _{r}\psi _{s}\!\left( \!\!%
\begin{array}{c}
a_{1},\dots,a_{i},\ c/q,d/q \\[0.1cm]
b_{0},\dots,b_{k},\ g/q,h/q%
\end{array}%
\!\!;q,\,t\!\right) =0.
\label{4Polynomials}\end{aligned}$$Furthermore, in the same way, the following variation can be obtained:[^3] $$\begin{aligned}
\notag
&\frac{\left( g-1\right) \left( h-1\right) \left(
-gh-cdq+cgq+dgq+chq+dhq-ghq-cdq^{2}\right) }{\left( c-1\right) \left(
d-1\right) \left( gq-h\right) \left( hq-g\right) } \\
&\quad \quad \quad \qquad \times \ _{r}\psi _{s}\!\left( \!\!%
\begin{array}{c}
a_{1},\dots,a_{i},\ c,d \\[0.1cm]
b_{0},\dots,b_{k},\ g,h%
\end{array}%
\!\!;q,\,t\!\right) \notag \\
&\quad \quad
+\frac{\left( c-g\right) \left( d-g\right) \left( h-1\right) \left(
h-q\right) }{\left( c-1\right) \left( d-1\right) \left( h-g\right) \left(
gq-h\right) }\ \ _{r}\psi _{s}\!\left( \!\!%
\begin{array}{c}
a_{1},\dots,a_{i},\ c,d \\[0.1cm]
b_{0},\dots,b_{k},\ gq,h/q%
\end{array}%
\!\!;q,\,t\!\right) \notag \\
&\quad \quad
+\frac{\left( c-h\right) \left( d-h\right) \left( g-1\right) \left(
g-q\right) }{\left( c-1\right) \left( d-1\right) \left( g-h\right) \left(
hq-g\right) }\ \ _{r}\psi _{s}\!\left( \!\!%
\begin{array}{c}
a_{1},\dots,a_{i},\ c,d \\[0.1cm]
b_{0},\dots,b_{k},\ g/q,hq%
\end{array}%
\!\!;q,\,t\!\right) \notag \\
&\qquad +\ _{r}\psi _{s}\!\left( \!\!%
\begin{array}{c}
a_{1},\dots,a_{i},\ cq,dq \\[0.1cm]
b_{0},\dots,b_{k},\ gq,hq%
\end{array}%
\!\!;q,\,t\!\right) =0.
\label{4RationalF}\end{aligned}$$
An Inverse of the Askey–Wilson Operator
=======================================
The Askey–Wilson divided difference operator on the left-hand side of equation (\[DiffDiffFormulaAW\]) can be inverted by the method of Ref. [@As:Rah:Sus]. The end result is$$\frac{\left( q,q^{2};q\right) _{\infty }}{2\pi }\int_{-1}^{1}L\left(
x,y\right) \ p_{n}\left( x;a,b,c,d\right) \rho \left( x;a,b,c,d\right) \
dx=p_{n}\left( x;aq,b/q,c,d\right) ,$$where $\rho \left( x;a,b,c,d\right) $ is the weight function of the Askey–Wilson polynomials (\[AskeyWilsonPlinomials\]) and the kernel is given by$$\begin{gathered}
L\left( x,y\right) =\left( ac,ad,qce^{i\varphi },qde^{-i\varphi
};q\right) _{1}\ \frac{\left( be^{i\theta },be^{-i\theta },qde^{i\theta
},qde^{-i\theta },qae^{i\varphi },qae^{-i\varphi },qce^{i\varphi
},qce^{-i\varphi };q\right) _{\infty }}{\left( qe^{i\theta +i\varphi
},qe^{i\theta -i\varphi },qe^{i\varphi -i\theta },qe^{-i\theta -i\varphi
};q\right) _{\infty }} \notag \\
\times \ _{8}\varphi _{7}\!\left( \!\!%
\begin{array}{c}
qde^{-i\varphi },q\sqrt{qde^{-i\varphi }},-q\sqrt{qde^{-i\varphi }}%
,qe^{i\theta -i\varphi },qe^{-i\theta -i\varphi },qd/c,q \\[0.1cm]
\sqrt{qde^{-i\varphi }},\sqrt{qde^{-i\varphi }},qde^{-i\theta },qde^{i\theta
},q^{2},qce^{-i\varphi },qde^{-i\varphi }%
\end{array}%
\!\!;q,\,ce^{i\varphi }\right) .\end{gathered}$$Here, $x=\cos \theta $ and $y=\cos \varphi .$ Computational details are left to the reader.
**Acknowledgment.** We thank Mizan Rahman for valuable discussions and encouragement.
[99]{} G. E. Andrews, R. A. Askey: *Classical orthogonal polynomials*; in *“Polynômes orthogonaux et applications”*, Lecture Notes in Math. **1171**, Springer–Verlag, 1985, pp. 36–62.
G. E. Andrews, R. A. Askey, R. Roy: *Special functions*; Cambridge University Press, Cambridge, 1999.
R. A. Askey, *Orthogonal polynomials and special functions*; CBMS–NSF Regional Conferences Series in Applied Mathematics, SIAM, Philadelphia, Pennsylvania, 1975.
R. A. Askey, T. H. Koornwinder, M. Rahman: *An integral of products of ultraspherical functions and a $q$-extension*; J. London Math. Soc. **33** \#2 (1986), 133–148.
R. A. Askey, M. Rahman, S. K. Suslov : *On a general* $q$*-Fourier transformation with nonsymmetric kernels*, Journal of Computational and Applied Mathematics **68** (1996), 25–55.
R. A. Askey, J. A. Wilson: *Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials*; Memoirs Amer. Math. Soc. **319**, 1985.
N. M. Atakishiyev, S. K. Suslov: *Difference Hypergeometric Functions*; in *Progress in Approximation Theory*, A. A. Gonchar and E. B. Saff, eds., Springer Verlag (1992), 1–35.
N. M. Atakishiyev, S. K. Suslov: *On the Askey–Wilson polynomials*; Constr. Approx. **8** (1992), 363–369.
G. Gasper, M. Rahman: *Basic hypergeometric series*; Second Edition, Encyclopedia of Mathematics and Its Applications, vol. 96, Cambridge University Press, Cambridge, 2004.
M. E. H. Ismail, M. Rahman: *The associated Askey-Wilson polynomials*, Trans. Amer. Math. Soc. **328** (1991) 201–237.
R. Koekoek, R. F. Swarttouw: *The Askey scheme of hypergeometric orthogonal polynomials and its $q$-analogues*; Report 94–05, Delft University of Technology, 1994.
A. F. Nikiforov, S. K. Suslov, V. B. Uvarov: *Classical orthogonal polynomials of a discrete variable*; Nauka, Moscow, 1985 \[in Russian\]; English translation, Springer–Verlag, Berlin, 1991.
M. Rahman: *Askey–Wilson functions of the first and second kind: Series and integral representations of* $C_{n}^{2}(x;\beta
|q)+D_{n}^{2}(x;\beta |q)$; J. Math. Anal. Appl. **164** (1992), 263–284.
M. Rahman: *Askey–Wilson functions of the first and second kind: Series and integral representations of*; J. Comput. Appl. Math. Anal. **68** (1996), 287–296.
M. Rahman: *The Associated Classical Orthogonal Polynomials*; in *Special Functions 2000*, J. Bustoz, M. E. H. Ismail and S. K. Suslov, eds., Kluwer Academic Publishers, Dordrecht, Boston, London, 2001.
S. K. Suslov: *The theory of difference analogues of special functions of hypergeometric type*; Russian Math. Surveys **44** (1989), 227–278.
S. K. Suslov: *An introduction to basic Fourier series*; Kluwer Series Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht, Boston, London, 2003.
G. Szegő: *Orthogonal polynomials*; Fourth Edition, Amer. Math. Soc. Colloq. Publ., Vol. 23, Providence, R. I., 1975.
[^1]: $^\dagger$Research partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and S9607-N13, the latter in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory"
[^2]: A corresponding *Mathematica* notebook is available on the article’s website`http://www.mat.univie.ac.at/0.5exkratt/artikel/AssAWPols.html`.
[^3]: Again, a corresponding [*Mathematica*]{} notebook is available on the article’s website.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- Xueshi Guo
- 'Casper R. Breum'
- Johannes Borregaard
- Shuro Izumi
- 'Mikkel V. Larsen'
- '[Tobias Gehring]{}'
- Matthias Christandl
- 'Jonas S. Neergaard-Nielsen'
- 'Ulrik L. Andersen'
title: Distributed quantum sensing in a continuous variable entangled network
---
Quantum noise associated with quantum states of light and matter ultimately limits the precision by which measurements can be carried out [@Giovannetti2006; @Escher2011; @Giovannetti2011]. However, by carefully designing the coherence of this quantum noise to exhibit properties such as entanglement and squeezing, it is possible to measure various physical parameters with significantly improved sensitivity compared to classical sensing schemes [@CavesPRD1981]. Numerous realizations of quantum sensing utilizing non-classical states of light [@Yonezawa2012; @Berni2015; @Slussarenko2017] and matter [@Muessel2014] have been reported, while only a few applications have been explored. Examples are quantum-enhanced gravitational wave detection [@LIGO], detection of magnetic fields [@Wolfgramm2010; @Li2018; @Jones2009] and sensing of the viscous-elasticity parameter of yeast cells [@Taylor2013]. All these implementations are, however, restricted to the sensing of a single parameter at a single location.
Spatially distributed sensing of parameters at multiple locations in a network is relevant for applications from local beam tracking [@Qi2018] to global scale clock synchronization [@Komar2014]. The development of quantum networks enables new strategies for enhanced performance in such scenarios. Theoretical works [@Humphreys2013; @Knott2016; @Baumgratz2016; @Pezze2017; @Eldredge2018; @Proctor2018; @Ge2018; @Zhuang2018] have shown that entanglement can improve sensing capabilities in a network using either twin-photons or Greenberger-Horne-Zeilinger (GHZ) states combined with photon number resolving detectors [@Proctor2018; @Ge2018] or using CV entanglement for the detection of distributed phase space displacements [@Zhuang2018]. In this Letter, we experimentally demonstrate an entangled CV network for sensing the average of multiple phase shifts inspired by the theoretical proposal of Ref. [@Zhuang2018]. We focus on the task of estimating small variations around a known phase in contrast to *ab initio* phase estimation. For the first time in any system, we demonstrate deterministic distributed sensing in a network of four nodes with a sensitivity beyond that achievable with a separable approach using similar quantum states.
![ **Distributed phase sensing scheme**. The task is to estimate the average value of $M$ spatially distributed small phase shifts $\phi_1,\ldots,\phi_M$. (**a**) Without a network, the average phase shift must be estimated by probing each sample individually. This can be done with homodyne detection of the phase quadrature (HD$_1$,$\ldots$,HD$_M$), and the sensitivity can be increased by using squeezed probes generated by $M$ independent squeezers $S_1,\ldots, S_M$. (**b**) If the $M$ sites are connected by an optical beam splitter network (BSN), a single squeezed probe can be distributed among the sites. This enables entanglement-enhanced sensing of the average phase shift. (**c,d**) The entangled approach of panel (b) shows a gain in sensitivity compared to the separable approach in panel (a) for the same number of photons, $N$, hitting each sample and with optimized probe states. This gain, $G=\sigma^\mathrm{opt}_{s}/\sigma^\mathrm{opt}_{e}$, is here plotted as a function of the number of samples $M$ with $N$ fixed at 10 (c) and as a function of the average number of photons with $M$ fixed at 4 (d) for different values of $\eta$, the efficiency of the channel between pure resource state and phase sample. []{data-label="fig_theory"}](fig1_new.pdf){width="\linewidth"}
We start by introducing a theoretical analysis of the networked sensing scheme assuming the existence of an external phase reference. Consider a network of $M$ nodes with optical inputs that undergo individual phase shifts, $\phi_j\ (j=1, \dots, M)$. The goal is to estimate the averaged phase shift, $\phi_\mathrm{avg}=\sum_{j=1}^M\phi_j/M$, among all nodes with as high precision as possible. Two different sensing setups are considered: A separable system where the nodes are interrogated with independent quantum states (Figure \[fig\_theory\]a) and an entangled system where they are interrogated with a joint quantum state (Figure \[fig\_theory\]b). We assume the squeezers give out pure single-mode Gaussian quantum states described by the state vectors $\hat D(\alpha)\hat S(r)|0\rangle$, where $\hat D$ and $\hat S$ are the displacement and squeezing operators, respectively, $\alpha$ is the displacement amplitude and $r$ is the squeezing factor. We assume that each probe state undergoes loss in a channel with transmission $\eta$. We furthermore restrict the estimator to be the joint phase quadrature, $\hat P_\mathrm{avg}=\sum_{j=1}^M \hat p_j/M$ (where $\hat{p}_j$ are the phase quadratures of the individual modes), practically corresponding to the averaged outcome of $M$ individual homodyne detectors. These states and detectors are of particular interest due to their experimental feasibility, inherent deterministic nature, high efficiency, and robustness to noise.
Using the separable approach, $M$ identical Gaussian probe states are prepared and individually detected, while in the entangled approach, a single squeezed Gaussian state is distributed evenly to the $M$ nodes via a beam splitter array and likewise measured individually with homodyne detectors at the nodes. If one wanted to estimate different linear combinations of the phase shifts than the simple average, other beam splitter divisions would be required [@Eldredge2018; @Proctor2018]. The sensitivity of the measurement can be defined as the standard deviation of the measurement which, by error propagation, is [@Giovannetti2011] $$\label{Eq_sensitivity_def}
\sigma= \frac{\sqrt{\langle\Delta \hat P^2_\mathrm{avg} \rangle}}{|\partial \langle \hat P_\mathrm{avg}\rangle /\partial\phi_\mathrm{avg}|},$$ where $\langle \Delta \hat P^2_\mathrm{avg} \rangle=\langle\hat P^2_\mathrm{avg}\rangle-\langle\hat P_\mathrm{avg}\rangle^2$ is the variance of the estimator. We are only interested in the sensitivity for small phase shifts, since one can always use an initial rough phase estimation to adjust the homodyne detector (the local oscillator phase) to the maximum sensitivity setting [@Berni2015]. For small phase shifts, we obtain the sensitivities for the separable ($\sigma_s$) and entangled ($\sigma_e$) approaches (see Supplementary Material Sec. I): $$\begin{aligned}
\label{eq:sens_sep}
\sigma_s &= \frac{\sqrt{e^{-2 r_s} + 1/\eta - 1}}{2 \alpha_s \sqrt{M}} ,\\
\sigma_e &= \frac{\sqrt{e^{-2 r_e} + 1/\eta - 1}}{2 \alpha_e} .\end{aligned}$$ We now constrain the average number of photons, $N$, hitting each sample. The photons can be separated into those originating from coherent displacement and those originating from squeezing: $N=N_{s,\mathrm{coh}} + N_{s,\mathrm{sqz}} = \eta(\alpha_s^2 + \sinh^2 r_s)$ for the separable case and $N=N_{e,\mathrm{coh}} + N_{e,\mathrm{sqz}} = \eta(\alpha_e^2 + \sinh^2 r_e)/M$ for the entangled case. The ratio between photon numbers, parametrized as $\mu_{s(e)} = N_{s(e),\mathrm{sqz}} / N$ can be tuned to give the optimal sensitivities $$\begin{aligned}
\sigma_s^\mathrm{opt} &= \frac{1}{2\sqrt{M}N} \sqrt{\frac{N(1-\eta) + \frac{\eta}{2} \big(1 + \sqrt{1+4N(1-\eta)} \big)}{1 + \eta/N}} , \\
\sigma_e^\mathrm{opt} &= \frac{1}{2MN}\sqrt{\frac{MN(1-\eta) + \frac{\eta}{2} \big(1 + \sqrt{1+4MN(1-\eta)} \big)}{1 + \eta/(MN)}} .\end{aligned}$$
For perfect efficiency ($\eta=1$), it is clear that the sensitivity of the entangled system yields Heisenberg scaling both in the number of nodes $(1/M)$ and the number of photons per mode ($1/N$) whereas the separable system only achieves the latter and a classical $1/\sqrt{M}$-scaling with the number of modes. The gain in sensitivity of the entangled network relative to the separable network (denoted $G=\sigma^\mathrm{opt}_{s}/\sigma^\mathrm{opt}_{e}$) is thus $G=\sqrt{M}$.
For non-ideal efficiency, the Heisenberg scaling ceases to exist in accordance with previous work on single parameter estimation [@KnyshPRA2011]. In fact, for $\eta\rightarrow 0$, both sensitivities approach $1/2\sqrt{MN}$. Still, it is important to note that the entangled network exhibits superior behavior for any value of $\eta$, $M$ and $N$ for optimized $\mu_s,\mu_e$. Some examples for the sensitivity gain are illustrated in Figures \[fig\_theory\]c and d. From Fig \[fig\_theory\]d where a network of $M=4$ nodes is considered, it is clear that the highest gain in sensitivity is attained at a finite photon number. We also note that for large photon numbers, the gain tends to unity for non-zero loss meaning no enhanced sensitivity when using the entangled approach. However, there is still a practical advantage for the entangled approach: Only one squeezed state is needed compared to the M squeezed states with similar squeezing levels for the separable approach (see Supplementary Material Sec. I).
{width="0.8\linewidth"}
Next, we demonstrate experimentally the superiority of using an entangled network for distributed sensing. The entangled network is realized by dividing equally a displaced single mode squeezed state into four spatial modes by means of three balanced beam splitters (Fig. \[fig\_setup\]a, see Supplementary Material Sec. III for more details). These modes are then sent to the four nodes of the network at which they each undergo a phase shift $\phi_j$ and are finally measured with high-efficiency homodyne detectors (HD) that are set to measure the phase quadrature, $\hat{p}_j$. The external phase reference is set by the local oscillator which co-propagates with the probes through the setup but in a different polarization mode. This ensures that the relative phases between the probes and the local oscillator can be controlled. The resulting photo-currents from the four detectors are further processed and subsequently combined to produce the averaged phase shift. For demonstration purpose, we set all $\phi_j$ to the same value $\phi_j=\phi_{avg}$, but in principle they could be different.
We choose to define our quantum states within a narrow spectral mode at the 3 MHz sideband frequency. There are no fundamental restrictions in the scheme on the optical modes employed. In any practical setting, they would be chosen based on the nature of both the squeezing source and the samples being probed. Here, the 3 MHz sideband is chosen to maximize the squeezing from our source, an optical parametric oscillator (OPO) operating below threshold: At higher frequencies, the squeezing reduces due to the limited bandwidth of the OPO, while at lower frequencies, it is degraded by technical noise. A displaced squeezed state is obtained by injecting into the OPO a coherent state produced by phase modulating the injected beam at 3 MHz. The maximum squeezing measured through the joint measurement of 4 HDs is $\sim$5 dB at 3 MHz. More details on the probe generation are in Supplementary Material Sec. III.
An experimental run is shown in Fig. \[fig\_setup\]b. In this particular run, a displaced squeezed state with an average photon number of $N=2.48\pm0.12$ in each mode is prepared of which $N_{e,\mathrm{sqz}}=0.30\pm0.01$ photons are from the squeezing operation and $N_{e,\mathrm{coh}}=2.19\pm0.11$ are from the phase modulation as this distribution is near-optimal for the entangled case. We then impose 12 different $\phi_\mathrm{avg}$ values by phase shifts at each node while recording the Fourier transformed homodyne detector outputs; the spectra around the 3 MHz sideband for six of the $\phi_\mathrm{avg}$ values are shown in Fig. \[fig\_setup\]b (see Supplementary Material Sec. V for more details). These outputs yield poor estimates of the individual phase shifts (because the squeezing in each mode is only $\sim$0.8 dB) but the averaged phase shift obtained by summing the photo-currents produces an entanglement-enhanced estimate with significantly lower noise. The spectra for the averaged photo-currents are shown in Figure \[fig\_setup\]c. For comparison, we also simulate the separable approach by directing the entire displaced squeezed state (with properly optimized parameters) to a single node. We then perform the phase estimation at that node and scale the obtained sensitivity by $\sqrt{4}$ to get the projected performance for an average over four identical sites. An example is shown in Fig \[fig\_setup\]b for $N=2.63\pm0.11$, with $N_{s,\mathrm{sqz}}=0.31\pm0.01$ and $N_{s,\mathrm{coh}}=2.32\pm0.10$.
{width=".8\linewidth"}
We quantify the performance of the sensing network by estimating the sensitivities of the two approaches based on the averaged homodyne measurement outcomes, $P_\mathrm{avg}$. By extracting the rate of change with respect to a phase rotation, $|\partial\langle \hat P_\mathrm{avg}\rangle / \partial\phi_\mathrm{avg}|$, as well as the variance, $\langle \Delta \hat{P}_\mathrm{avg}^2\rangle$, of $P_\mathrm{avg}$, we deduce the sensitivity using Eq. (\[Eq\_sensitivity\_def\]). For the experimental runs described above, we obtain sensitivities of $\sigma_{e} = 0.099 \pm 0.003$ and $\sigma_{s} = 0.118 \pm 0.002$ for the entangled and separable approach, respectively. This corresponds to single shot resolvable distributed phase shifts (that is, phase shifts for which the signal-to-noise ratio is unity) of $5.66^\circ \pm 0.18^\circ$ for the entangled case and $6.76^\circ \pm 0.11^\circ$ for the separable case with $\sim 2.5$ photons. Using a coherent state in replacement of the squeezed state, the minimal resolvable phase for $2.5$ photons is $9.06^\circ \pm 0.07^\circ$ corresponding to the standard quantum limit. Note that these angles are larger than our small phase shift approximation (which requires $\phi_{avg}$ to be much smaller than $\sim7^{\circ}$ for the conditions in this experimental run, see Supplementary Sec. I). In practice this means that it is necessary to probe the sample more than once to resolve the small phases implemented in the experiment. Sampling the phases $K$ times will result in $\sqrt{K}$ times smaller resolvable phase shift angles. The entangled strategy will still benefit from the enhanced sensitivity per probe.
We find the sensitivities for different total average photon numbers both for the entangled and separable network, and plot the results in Figure \[fig\_result\]a. For every selection of the total photon number, we adjust $\mu$ to a near-optimal value for optimized sensitivity (Figure \[fig\_result\]b,c). It is clear in Figure \[fig\_result\]a that both realizations beat the standard quantum limit (reachable by coherent states of light), and most importantly, we see that the entangled network outperforms the separable network. The ultimate sensitivity of our entangled approach is not reached in our implementation. However, homodyne detection will not even in principle saturate this bound and non-Gaussian measurements are in general needed (see Supplementary Material Sec. II).
Our results experimentally demonstrate how mode entanglement, here in the form of squeezing of a collective quadrature of a multi-mode light field, can enhance the sensitivity in a distributed sensing scenario. Importantly, we show this enhancement in an experimentally feasible setting where the sensitivity of standard coherent probes are enhanced through quadrature squeezing. This approach allows for easily tunable probe powers in order to adapt the setup to the specific application. [ Furthermore, because the entanglement is generated from a simple beam-splitter network, it is straight-forward to scale to more modes where the sensitivity gain may be even larger, cf. Fig. \[fig\_theory\]c. The main limitation will be the channel efficiency which will eventually limit the gain.]{} Consequently, we believe that techniques demonstrated here have direct applications in a number of areas. Specifically, beam tracking relevant for molecular tracking [@Taylor2013; @Qi2018] could directly benefit from these techniques. Such applications impose limits on the allowed probe power to prevent photon damage and heating of the systems. Mode-entanglement can thus be used to increase sensitivity without increasing the probe power. Using squeezed coherent light for quantum non-demolition (QND) measurement has also been exploited for generation of spin squeezing in atomic ensembles [@Hammerer2010] and optical magnetometry [@Wolfgramm2010]. While this is usually considered for single ensembles, the generalization to multiple ensembles can provide enhanced sensitivity and new primitives for quantum information processing. Combining several ensembles for magnetometry and utilizing mode-entanglement would further reduce the shot-noise and increase sensitivity of a collective optical measurement. Performing a collective optical QND measurement of several atomic ensembles can prepare a distributed spin-squeezed state for quantum network applications. In particular, squeezing of multiple optical lattice clocks could be used for collective enhancement of clock stability [@Komar2014; @Eugene2016]. In Ref. [@Eugene2016], this was obtained by letting a single probe interact with all ensembles in a sequential manner. However, utilizing mode-entanglement, this can be performed in a parallel fashion with no quantum signal being transmitted between the ensembles.
**DATA AVAILABILITY**\
Experimental data and analysis code is available on request.
**Acknowledgments**\
M.C. and J.B. acknowledge support from VILLUM FONDEN via the QMATH Centre of Excellence (Grant no. 10059), the European Research Council (ERC Grant Agreements no 337603), and from the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union’s Horizon 2020 Programme (QuantAlgo project) via the Innovation Fund Denmark. X.G., C.B., S.I., M.L., T.G., J.N. and U.A. acknowledge support from Center for Macroscopic Quantum States (bigQ DNRF142). X.G., S.I. and J.N. acknowledge support from VILLUM FONDEN via the Young Investigator Programme (Grant no. 10119).
**AUTHOR CONTRIBUTIONS**\
J.B., U.A., J.N., T.G. and X.G. conceived the experiment. X.G., C.B. and M.L. performed the experiment and analyzed the data. J.B., X.G., S.I., M.C. and J.N. worked on the theoretical analysis. X.G. wrote the paper with contributions from J.B., C.B., S.I., J.N. and U.A. J.N. and U.A. supervised the project.
**ADDITIONAL INFORMATION**\
**Competing interests:** The authors declare that there are no competing interests.
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**Supplemental Materials for Distributed quantum sensing in a continuous variable entangled network**
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We introduce $(k,l)$-regular maps, which generalize two previously studied classes of maps: affinely $k$-regular maps and totally skew embeddings. We exhibit some explicit examples and obtain bounds on the least dimension of a Euclidean space into which a manifold can be embedded by a $(k,l)$-regular map. The problem can be regarded as an extension of embedding theory to embeddings with certain non-degeneracy conditions imposed, and is related to approximation theory.'
author:
- |
Gordana Stojanovic\
[*Brown University and Penn State University*]{}\
title: Embeddings With Multiple Regularity
---
Introduction
============
Two lines in a Euclidean space are called *skew* if they are not parallel and do not intersect. A submanifold $M^n$ of ${{\mathbf R}}^N$ is said to be *totally skew* if arbitrary tangent lines to $M$ at any two distinct points are skew. Equivalently, one can define an immersion $f:M^n\to{{\mathbf R}}^N$ to be *totally skew* if for all $x, y \in M^n$ the tangent spaces $df(T_x M)$ and $df(T_y M)$ as affine subspaces of ${{\mathbf R}}^N$ have an affine span of maximal possible dimension, that of $2n+1$.
Totally skew embeddings have been introduced and studied in [@G-T]. Other closely related classes of embeddings into affine and projective spaces defined in terms of mutual positions of tangent spaces at distinct points are skew embeddings and T-embeddings and they have also received a considerable amount of attention, see [@Gh1; @Gh2; @G-S; @G-T; @S-S; @S-T; @Ta; @T-T].
Another, seemingly less closely related, class of embeddings are so-called $k$-regular maps, and their affine version, introduced by Borsuk in [@Bo]. A continuous map $f:X\to{{\mathbf R}}^N$ is called *k-regular* (respectively *affinely $k-1$-regular*) if the images under $f$ of $k$ distinct points are linearly (respectively affinely) independent.[^1] The study of $k$-regular maps was motivated by the theory of Chebyshev approximation. It was conducted by non-algebro-topological methods in [@Bo; @BRS], while Handel [@C-H; @H1; @H2; @H3; @H4; @H-S] introduced cohomological methods using configuration spaces to obtain various existence and non-existence results. Vassiliev [@V] independently studied $k$-regular maps under the name ‘$k$-interpolating spaces of functions’, using topological methods similar to those of Handel. He was interested in the interpolating properties of a finite dimensional space of continuous functions on a topological space. Namely, he calls a finite dimensional space $L$ of continuous functions on a topological space $M$, *$k$-interpolating* if every real-valued function on $M$ can be interpolated at arbitrary $k$ points of $M$ with an appropriate function from $L$. The connection with $k$-regular maps is as follows: the functions $f_1,\ldots,f_N$ span a $k$-interpolating space of functions if and only if the map $f=(f_1,\ldots,f_N)$ is $k$-regular. In other words, $f$ is $k$-regular if and only if we can prescribe values at any distinct $k$ points of $M$ for functions in the span of coordinate functions of $f$.
One of the main questions that arises in the study of all mentioned maps is to find the lowest possible dimension of the target Euclidean space which allows them. For example, for a given manifold $M^n$, what is the smallest dimension $N=N(M^n)$ such that $M^n$ admits a totally skew embedding in ${{\mathbf R}}^N$? As is, this question has been answered for very few manifolds. Results are available only for line, circle and plane: $N({{\mathbf R}}^1)=3, N(S^1)=4$, $N({{\mathbf R}}^2)=6$, see [@G-T]. Ghomi and Tabachnikov actually give totally skew embeddings of line, circle and plane in the Euclidean space of minimal possible dimension and these are the only known explicit examples of optimal totally skew embeddings. According to the same authors [@G-T], dimension $n$ submanifolds of ${{\mathbf R}}^N$ are generically totally skew when $N\geq 4n+1$. This abundance of totally skew embeddings contrasted with the scarcity of available examples points to another object of investigation: finding more of them.
The same question can be asked for $k$-regular maps, and one result, that both Handel [@H4] and Vassiliev [@V] reached, is for instance, that when $k$ is even, $N(S^1)=k+1$, and when $k$ is odd, $N(S^1)=k$. While the result for odd $k$ is almost immediate, to achieve the result for $k$ even, they both used nonelementary topological methods, in particular, characteristic classes.
We introduce a class of regular maps, so called $(k,l)$-regular maps, which generalize both totally skew embeddings and affinely $k$-regular maps, and ask the same question of determining minimal dimensional target Euclidean space. This problem can be regarded as an extension of investigations that led to the birth of embedding theory - an extension to the embeddings with certain prescribed non-degeneracy conditions. The interpretation of $(k,l)$-regular maps in the language of the approximation theory is as follows: it turns out that a smooth map $f=(f_1,\ldots,f_N):M^n\to {{\mathbf R}}^N$ on a smooth manifold $M^n$ is $(k,l)$-regular if and only if for every function in the span of $1,f_1,\ldots,f_N$ we can prescribe not only values at any distinct points $x_1,\ldots,x_k,y_1,\ldots,y_l$ but directional derivatives as well in any direction at the last $l$ points. Thus, the existence of $(k,l)$-regular maps is equivalent to the possibility of interpolating functions on $M^n$ through any $k+l$ points and up to the first order derivatives in arbitrary directions at the last $l$ points. Finally, let us mention that there is an obvious connection with recent work of Arnold and his school, see [@A].
In this paper, we generalize existing estimates for totally skew and affinely $k$-regular maps to our class, provide explicit examples in the case of line, circle and plane and determine the minimal target spaces for curves. We only employ non-algebro-topological methods, which leaves plenty of room for further investigations in the topology of these embeddings.
[**Acknowledgments**]{}. I would like to thank my advisor Sergei Tabachnikov for suggesting this problem as well as for his constant support and guidance. I would also like to thank Mohammad Ghomi and Bruce Solomon for interesting discussions and Anatole Katok for continual support during my stay at Penn State University, where this work has been carried out.
Definition of $(k,l)$-regular maps
==================================
We will start with the definition of affine independence. There are many different (albeit equivalent) ways to define affine independence, but we settle with the following one.
Affine subspaces $V_1,\ldots,V_k\subset{{\mathbf R}}^N$ are said to be *affinely independent* if their affine span has the maximal possible dimension that the affine span of affine spaces of respective dimensions can have in any given affine ambient space.
For example, the affine span of two lines may have dimension 1, 2 or 3 depending on their position. The maximal of these is three dimensional, and so any two lines are affinely independent if their affine span is three dimensional. Thus no two lines in ${{\mathbf R}}^2$ are affinely independent. From now on, when we say that some span is maximal possible we will mean maximal possible regardless of the ambient space. In this terminology, no two lines in ${{\mathbf R}}^2$ will have maximal possible affine span.
One can calculate that if $V_1,\ldots,V_k\subset{{\mathbf R}}^N$ have dimensions $n_1,\ldots, n_k$ respectively than they are affinely independent if and only if their affine span has the dimension $$(n_1+1)+\ldots+(n_k+1)-1.$$
Let $M^n$ be an $n$-dimensional manifold. Let $k$ and $l$ be non-negative integers, not both equal to 0. We will call a smooth map *$(k,l)$-regular* if for every set of distinct points $x_1,\ldots,x_k,y_1,\ldots,y_l$ of $M$ and of $l$ tangent lines $L_i\subset T_{y_{i}} M, i=1,\ldots,l$ the set of points and lines $$f(x_1),\ldots,f(x_k),df(L_1),\ldots,df(L_l)$$ is affinely independent.
[When $l=0$, the notion of $(k,l)$-regularity coincides with the notion of affine $k-1$-regularity as defined in [@H-S]. On the other hand, a smooth map $f$ is (0,2)-regular if and only if it is totally skew. Thus, the notion of generalizes both totally skew and (affinely) $k$-regular maps.]{}
We will often say that a manifold is $(k,l)$-regular with the understanding that it is a submanifold of some Euclidean space, and the inclusion map is .
The $(0,l)$-regular maps we will also call $l$-totally-skew.
\[obvious\]
The following properties are more or less obvious:
- If a map is $(k,l)$-regular then it is also $(k',l')$-regular, for all $k'\leq k$, $l'\leq l$.
- Any (2,0)-regular map is one-to-one.
- Any 1-totally-skew map is an immersion.
- Any 2-totally-skew map is nothing else but a totally skew embedding.
- Every restriction of a $(k,l)$-regular map is $(k,l)$-regular.
One may be tempted to adopt a more general definition of regularity: instead of requiring affine independence of a collection of points and lines, one may consider arbitrary dimensional vector subspaces of tangent spaces at certain number of distinct points of $M$ and require the affine independence of their images under $df$. However, it turns out that this doesn’t add any generality to the definition, as stated in the following lemma, which we leave without proof.
\[reduction\] Let $M^n$ be an $n$-dimensional manifold and fix some positive integers $n_1,\ldots,n_l$ not greater than n. Then, the map $f:M^n\to{{\mathbf R}}^N$ is $(k,l)$-regular if and only if for arbitrary distinct points and arbitrary vector spaces $V_i\subset T_{y_{i}}M,~ dim(V_i)=n_i,~i=1,\ldots,l$ the affine spaces $$f(x_1),\ldots,f(x_k),df(V_1),\ldots,df(V_l)$$ are affinely independent.
To fix notation we will denote $N_{k,l}=N_{k,l}(M^n)$ to be the least possible integer $N$, such that $M^n$ admits a $(k,l)$-regular map into ${{\mathbf R}}^N$.
[A simple dimension counting argument shows that any $n$-dimensional $(k,l)$-regular manifold requires at least $k+(n+1)l-1$ dimensions of ambient space, immediately yielding that $$\label{count}
N_{k,l}(M^n)\geq k+(n+1)l-1.$$ In particular, $$\label{count1}
N_{k,l}({{\mathbf R}})\geq k+2l-1.$$]{}
[ Since manifolds are locally Euclidean, it follows that $$N_{k,l}(M^n)\geq N_{k,l}({{\mathbf R}}^n).$$]{}
The following observation will be useful in the proofs.
\[notreg\] [An embedding $f:M^n\to{{\mathbf R}}^N$ is *not* $k,l$-regular if and only if there exists an affine subspace that touches $M$ at $l$ points and intersects it in additional $k$ points.]{}
[*Proof.*]{} It follows from Lemma \[reduction\] where $V_j$’s are taken to be full tangent spaces. [$\Box$]{}.
Henceforth we shall assume that $f$ is always an embedding, simply because in most interesting cases (e.g. whenever $k+l>1$), $(k,l)$-regular maps actually are embeddings.
Examples and bounds
===================
The following proposition, together with the last item in Remark \[obvious\] and the existence of $(k,l)$-regular embeddings of real line (Thm \[curves\]) implies the existence of $(k,l)$-regular embeddings of any manifold into some Euclidean space, and as a consequence, $N_{k,l}$ is well defined.
\[tensor\] Let $M$ and $N$ be $(k,l)$-regular submanifolds in euclidean spaces $U$ and $V$ respectively. Then, the embedding $$f:M \times N \to (U\otimes V) \oplus (U\otimes {{\mathbf R}})\oplus ({{\mathbf R}}\otimes V),~ f(x,y)=(x \otimes y, x\otimes1,1\otimes y)$$ is $(k,l)$-regular.
The next theorem improves on the estimate (\[count\]) and provides an upper bound as well. The lower bound is obtained by extending the argument in [@BRS], while the upper bound is obtained similarly as in the case of totally skew embeddings [@G-T].
\[main\] For any manifold $M^n$, $$\Biggl[\frac{k}{2}\Biggr]n+\biggl[\frac{k-1}{2}\biggr]+(n+1)l \leq N_{k,l}(M^n)\leq (n+1)k+(2n+1)l-1.$$Moreover, generically any submanifold $M^n$ in $R^{(n+1)k+(2n+1)l-1}$ is $(k,l)$-regular.
Setting $k=0$ and $l=2$ gives us the theorem for totally skew embeddings obtained in [@G-T]. On the other hand, the upper bound for the opposite case, $k=2$ and $l=0$, says simply that $M^n$ always embeds in $R^{2n+1}$, a well known fact. The theorem also generalizes the lower bound for affinely $k$-regular maps given in [@BRS].
When the manifold is closed, we have a better (by 1) lower bound in the case of $l$-totally-skew embeddings.
\[closed\] Let $M^n$ be a closed manifold. Then, $$N_{0,l}(M^n)\geq (n+1)l.$$
Finally, we give examples of $(k,l)$-regular embeddings of real line, circle and plane. In the case of curves, these embeddings happen to be optimal, deciding $N_{k,l}$ when $n=1$.
\[examples\]
1. The map $\gamma:{{\mathbf R}}\to {{\mathbf R}}^{k+2l-1}$ given by $$\gamma(t)=(t,t^2,\ldots,t^{k+2l-1})$$ is $(k,l)$-regular.
2. The map $\gamma:S^1\to{{\mathbf R}}^{2k'+2l}$, given by $$\gamma({\alpha})=( \cos {\alpha}, \sin {\alpha},\cos 2{\alpha},\sin 2{\alpha},\ldots,\cos (k'+l){\alpha},\sin (k'+l){\alpha})$$ is $(k,l)$-regular for $k=2k'+1$ (and hence for $k=2k'$ as well).
3. The map $\gamma:{{\mathbf R}}^2 \cong {{\mathbf C}}\to {{\mathbf C}}^{k+2l-1} \cong {{\mathbf R}}^{2(k+2l-1)}$ given by $$\gamma(z)=(z,z^2,\ldots,z^{k+2l-1})$$ is $(k,l)$-regular.
\[curves\] One has:
1. $N_{k,l}({{\mathbf R}})= k+2l-1$.
2. $N_{k,l}(S^1)= \left \{ \begin{array}{ll}
k+2l, & \textrm {$k$ is even}\\
k+2l-1, & \textrm{$k$ is odd}.
\end{array} \right.$
Proofs
======
Let $i$ be the embedding of ${{\mathbf R}}^N$ into ${{\mathbf R}}^{N+1}$ as the height 1 hyperplane, that is let $$i:{{\mathbf R}}^N\to {{\mathbf R}}^{N+1},~i:x\mapsto (x,1).$$ Let $AG_n(N)$ denote the (affine Grassmanian) manifold of $n$-dimensional affine subspaces of ${{\mathbf R}}^N$, and $G_{n+1}(N+1)$ be the (Grassmanian) manifold of the ($n+1$)-dimensional subspaces of ${{\mathbf R}}^{N+1}$. Then, $i$ induces the canonical embedding $$AG_n(N)\to G_{n+1}(N+1),$$ given by assigning to each point $p\in{{\mathbf R}}^N$ the line $\ell (p)\subset{{\mathbf R}}^{n+1}$ which passes through the origin and $(p,1)$. We will also call this embedding $i$ without any fear of confusion. The following fact, which is immediate consequence of the definition, we state as a lemma.
\[linear\] The affine span of the affine subspaces $V_1,\ldots,V_k\subset{{\mathbf R}}^N$ is maximal possible if and only if the linear span of the corresponding vector subspaces $i(V_1),\ldots,i(V_k)\subset{{\mathbf R}}^{N+1}$ maximal possible. The dimension of this linear span is one larger than the dimension of the affine span of $V_1,\ldots,V_k$.
\[operative\] [We will use notation $\tilde x=(x,1)$ for points $x\in M$, but for vectors $u$ tangent to $M$. As a consequence of the previous lemma, it follows that $f:M\to {{\mathbf R}}^N$ is $(k,l)$-regular if and only if for distinct points $x_1,\ldots, x_k,y_1,\ldots, y_l\in M$ and non-zero vectors $u_j \in T_{y_j}M,j=1,\ldots,l$, the vectors $\tilde{x_1},\ldots,\tilde{x_k},\tilde{y_1},\ldots,\tilde{y_l},\tilde{u_1},\ldots,\tilde{u_l}\in {{\mathbf R}}^{N+1}$ are linearly independent.]{}
[**Proof of Proposition \[tensor\].**]{}First note that a tangent vector to $f(x,y)$ is of the form $(u\otimes y+x\otimes v,u \otimes 1, 1 \otimes v)$ where $u\in T_x M$ and $v\in T_y N$. Fix arbitrary distinct points $(x_i,y_i)\in M\times N, i =1,\ldots,k+l$ and arbitrary non-zero vectors $(u_j,v_j)\in T_{(x_j,y_j)}M\times N, j=k+1,\ldots,k+l.$ Because of remark \[operative\], it suffices to show, after the identification $$(U \otimes V)\oplus(U\otimes{{\mathbf R}})\oplus(V\otimes {{\mathbf R}})\oplus {{\mathbf R}}\cong (U\oplus {{\mathbf R}})\otimes (V \oplus {{\mathbf R}})$$ that the vectors in the set $$\mathcal S=\{\tilde{x_i}\otimes \tilde{y_i}| i=1,\ldots,k+l\} \cup \{\tilde u_j \otimes \tilde y_j + \tilde x_j \otimes \tilde v_j| j=k+1,\ldots,k+l\}$$ are linearly independent.
Since $M$ is $(k,l)$-regular, we know that the vectors in $\mathcal M=\{\tilde {x_i}, \tilde{u_j}|u_j\neq0\}$ are linearly independent and similarly for $\mathcal N=\{\tilde {y_i}, \tilde {v_j}|v_j\neq0\}.$ But now we observe that every nontrivial linear combination of vectors in $\mathcal S$ would also be a nontrivial linear combination of vectors in $$\{e\otimes f| e\in \mathcal M,f \in \mathcal N\},$$ and that is not possible. [$\Box$]{}
[**Proof of Theorem \[main\].**]{} First we prove the lower bound.We will make use of the following theorem, affine version of which is proved in [@BRS]. For one line proof of the even $k$ case of the theorem stated here, see [@C-H].
[**(Boltyanski-Ryzhkov-Shashkin)**]{}\[thm\] If a $k$-regular map of ${{\mathbf R}}^n$ into $R^N$ exists, then $N \geq [\frac{k}{2}]n+[\frac{k+1}{2}]$.
Since the inequality holds trivially when $k+l=1$, and $f$ is an embedding when $k+l>1$, we can assume without loss of generality that $M^n$ is a submanifold of $R^N$.
Without loss of generality, assume $l>0$; otherwise, there is nothing to prove. Now fix some arbitrary distinct points $y_1,\ldots,y_l\in M^n$, and let $W$ be the affine span of $T_{y_1}M ,\ldots, T_{y_l}M$. The inclusion $M\to {{\mathbf R}}^N$ induces a map $M^n- \{y_1,\ldots,y_l\} \to {{\mathbf R}}^{N+1} /i(W)\cong {{\mathbf R}}^{N+1-(n+1)l}$ . Since $M^n$ is $(k,l)$-regular, the induced map is $k$-regular. Because of Theorem \[thm\] we have that $$N+1-(n+1)l \geq \Biggl[\frac{k}{2}\Biggr]n + \Biggl[\frac{k+1}{2}\Biggr],$$ which proves the lower bound.
To prove the upper bound, the strategy is to first embed the manifold $M$ in a large Euclidean space $R^N$ as a $(k,l)$-regular submanifold. This can be done, for instance, using Proposition \[tensor\]. Then, one reduces the dimension of the target space by succesive projections to hyperplanes all the while preserving $(k,l)$-regularity.
To do that, we project to any hyperplane, centrally from a point $A$ outside of it. The projected manifold will still be $(k,l)$-regular if we choose $A$ away from $U$, the union of affine spans of all $k+l$-tuples of points of $M$ and $l$ tangent spaces at the last $l$ points. Since $U$ consists of points $$\sum_{i=1}^k a_i x_i+ \sum_{j=1}^l b_j y_j+\sum_{j=1}^l v_j$$ where $$a_i,b_j\in{{\mathbf R}},x_i,y_j\in M^n, v_j\in T_{y_j}M^n,
\sum_{i=1}^k a_i+\sum_{j=1}^l b_j=1$$ one can calculate using local charts that dim $U=k(n+1)+l(2n+1)-1$. Thus, for $N>k(n+1)+l(2n+1)-1$ one can find $A$ not in $U$ and reduce $N$ by 1. This proves the upper bound.
To prove that the upper bound is generically true, that is, that every embedding $M^n \to {{\mathbf R}}^{k(n+1)+l(2n+1)-1}$ can become $(k,l)$-regular after an arbitrarily small perturbation, one uses Thom’s transversality theorem. The proof is exactly the same as in [@G-T], so we omit it. [$\Box$]{}
[**Proof of Theorem \[closed\].**]{}Thm \[closed\] follows immediately from the next proposition and remark \[notreg\].
If $M^n$ is a closed submanifold of ${{\mathbf R}}^{(n+1)l-1}$, then there exists a hyperplane tangent to $M$ at $l$ distinct points.
[*[Proof.]{}*]{}Without loss of generality, we may assume that $M=M^n$ does not belong to an affine hyperplane in ${{\mathbf R}}^{(n+1)l-1}$; if it does, we are done. Let us denote by $\mathrm{C}(M)$ the convex hull of $M$.
\[convex\] There exists a point $x\in\partial\mathrm{C}(M)$ that is not a convex combination of $l-1$ or fewer points of $M$.
*Proof.* The set of all convex combinations of all $l-1$-tuples of points of $M$, $$S = \{x= a_1 y_1+\ldots+a_{l-1} y_{l-1}|\sum_{i=1}^{l-1} a_i=1,~ a_i\geq 0, y_i\in\ M, i=1,\ldots,l-1\},$$ is a set of dimension $(n+1)l-2-n$, containing $M$. On the other hand, since $M$ does not belong to any hyperplane, neither does $\mathrm{C}(M)$. Therefore, $\mathrm{C}(M)$ is a convex set of dimension $(n+1)l-1$, its boundary $\partial\mathrm{C}(M)$ has dimension $(n+1)l-2$. Thus, there must exist a point $x$ in $\partial\mathrm{C}(M)$, but not in $S$, as claimed.
Consider now the support hyperplane $H$ at $x$. It is a hyperplane through $x$, so that $\mathrm{C}(M)$ is contained in a closed half-space bounded by $H$. Since $M$ is closed, $\mathrm{C}(M)$ is compact, and therefore $x \in \mathrm{C}(M)$. By the theorem of Carathéodory, see for example [@Ba], every point in $\mathrm{C}(M)$ is a convex combination of *at most* $(n+1)l$ points, so $$x=\ a_1 y_1+\ldots+a_{(n+1)l} y_{(n+1)l},~~~\mathrm{where}~~~ \sum_{i=1}^{(n+1)l} a_i=1,~ a_i\geq 0$$ and $y_i\in\ M$ for $i=1,\ldots,(n+1)l$. However, all the points $y_i$ with non-zero coefficients $a_i$ in the above convex combination must belong to $H$, because otherwise $x$ wouldn’t be in $H$. So, it is exactly at those points where $H$, our support hyperplane, touches $M$. And by claim \[convex\] there must be at least $l$ of them. Which is exactly what we wanted to prove. [$\Box$]{}
[**[Proof of Proposition \[examples\]]{}**]{} [*[1.The open case.]{}*]{}
We will argue by contradiction. Suppose that the curve $$\gamma(t)=(t,t^2,\ldots,t^{k+2l-1}),~ t\in{{\mathbf R}}$$ is not $(k,l)$-regular. Then there exists a hyperplane $H$, $$a_0+\sum_{i=1}^{k+2l-1}a_i x_i=0,$$ tangent to the curve $\gamma(t)$ at $l$ distinct points and intersecting it in at least $k$ additional points. But this means that the polynomial $$f(t)=a_0+\sum_{i=1}^{k+2l-1}a_i t^i,$$ of degree $k+2l-1$, has $l$ double and $k$ simple roots. Contradiction.
[*[2.The closed case.]{}*]{}
Again, suppose that the curve $\gamma:S^1\to{{\mathbf R}}^{2k'+2l}$, $$\gamma({\alpha})=(\cos {\alpha}, \sin {\alpha}, \cos 2{\alpha}, \sin 2{\alpha},\ldots, \cos (k'+l){\alpha}, \sin (k'+l){\alpha})$$ is not $(k,l)$-regular for $k=2k'+1$. Then, just as in the open case we obtain a function $$f({\alpha})= a_0+\sum_{i=1}^{k'+l}a_i \cos i{\alpha}+\sum_{i=1}^{k'+l}b_i \sin i{\alpha}.$$ having $2k'+2l+1$ roots on the interval $[0,2\pi)$ when counted with multiplicites. However, $f({\alpha})$ is a trigonometric polynomial of degree $k'+l$ , and it is a well known fact (see, for example, [@Ch]) that it can have at most $2(k'+l)$ zeros. Thus, our curve is $(k,l)$-regular, just as we claimed.
[3. [*The plane.*]{}]{}
First, we observe that this map is $(k,l)$-regular in the complex sense, that is, that for any given distinct points $z_1,\ldots,z_k,w_1,\ldots w_l\in {{\mathbf C}}$, the points $\gamma(z_1),\dots,\gamma(z_k),\gamma(w_1),\ldots,\gamma(w_l)$ and complex tangent lines at last $l$ points are affinely independent over ${{\mathbf C}}$. In other words, there do not exist complex numbers $a_1,\ldots,a_k,b_1,\ldots,b_l,\xi_1,\ldots,\xi_l$, not all equal to zero, so that the following realtions hold: $$\sum_{i=1}^k a_i \gamma(z_i) + \sum_{j=1}^l b_j \gamma(w_j)+ \sum_{j=1}^l\xi_j \gamma'(w_j)=0,~
\sum_{i=1}^k a_i + \sum_{j=1}^l b_j=0$$ The exact same proof as in the case of real open curves goes through when we replace real coordinates and coefficients with complex ones. Now, we show $(k,l)$-regularity in the real sense. Suppose therefore, towards a contradiction, that there are distinct points and non-zero vectors $\xi_j \in T_{w_j}{{\mathbf R}}^2,j=1,\ldots,l$ so that $(\gamma(z_1),1),\ldots,(\gamma (z_k),1)$, $(\gamma(w_1),1),\ldots,(\gamma(w_l),1)$, $(d\gamma(\xi_1),0),\ldots,(d\gamma(\xi_l),0)$ are linearly dependent. Thus, for some real numbers $a_1,\ldots,a_k,b_1,\ldots,b_l$, not all equal to zero, we have: $$\sum_{i=1}^k a_i \gamma(z_i) + \sum_{j=1}^l b_j \gamma(w_j)+ \sum_{j=1}^l d\gamma_{w_j}(\xi_j)=0,~
\sum_{i=1}^k a_i + \sum_{j=1}^l b_j=0$$ But $d\gamma_{w}(\xi)=\xi \gamma'(w)$, where on the right side of the equation we view $\xi$ and $w$ as complex numbers and $\gamma$ as a map from ${{\mathbf C}}$ to ${{\mathbf C}}^{k+2l-1}$, and this contradicts $(k,l)$-regularity in the complex sense, which we have already established. [$\Box$]{}
[**[Proof of Theorem \[curves\]]{}**]{}We will prove that $N_{k,l}(S^1)=k+2l$ for even k. The other two claims follow immediately from (\[count1\]) and Proposition \[examples\].
\[curvature\] Let $\gamma \subset R^n$ be a smooth curve with non-vanishing curvature vector $v$ at point $x$, and $H$ be a hyperplane, tangent to $\gamma$ at $x$ and transverse to $v$. Then, near $x$, the curve $\gamma$ doesn’t cross $H$.
Proof. Let $\gamma(t)$ be parametrized by arc length with $\gamma(0)=x$. Let . Then $$\gamma(t)=x + tu + \frac{t^2}{2} v +...$$ We can assume $H$ to be the zero level hyperplane of a linear function $l$. Then, the nonvanishing curvature implies $l(v)\neq 0$, say $l(v)>0$. One has: $$l(\gamma(t))=l(x) + tl(u) + \frac{t^2}{2}l(v) + O(t^3)= \frac{t^2}{2}l(v) + O(t^3) > 0$$ for $t$ small enough, which proves the lemma. [$\Box$]{}
Now we prove the lower bound by contradiction. Assume that $S^1\subset{{\mathbf R}}^{k+2l-1}$ as a $(k,l)$-regular submanifold. In order to arrive at a contradiction, we want to find a hyperplane in ${{\mathbf R}}^{k+2l-1}$ that intersects $\gamma$ at $k$ points and touches at $l$ points, all of them distinct. Choose $l$ points $y_j$ where the curvature doesn’t vanish. They must exist unless the curve $\gamma$ is straight. Let $W$ be the affine span of the tangent spaces (to $\gamma$) at these points. Let $v_j$ be the curvature vectors at $y_j$, and let $V_j$ be the affine span of $v_j$ and $W$. Note that dim $V_j \leq 2l$.
Assume that $k \geq 2$ (case $k=0$ holds because of Theorem $\ref {closed}$). Note that $\gamma$ doesn’t lie in $\cup_{j=1}^l V_j$, because then there would exist an open piece of this curve contained in one of the $V_j$’s, which is impossible since $N_{k,l}({{\mathbf R}}) \geq k+2l-1$. Choose generic $k-1$ points $x_i \in \gamma$ that belong to neither of $V_j$ and let $U$ be the affine span of $W$ and these points. Then $U$ is a hyperplane that is transverse to $\gamma$ at points $x_i$ (by their general position) and, by Lemma $\ref{curvature}$, $\gamma$ doesn’t cross $U$ at points $y_i$. Thus we have an odd number of crossings of $\gamma$ and $U$, and since $\gamma$ is closed, there must be another one, say $x_k \in U \cap \gamma$. Therefore $\gamma$ is not $(k,l)$-regular. [$\Box$]{}
[99]{}
5 mm
V. Arnold. On the number of flattening points on space curves. [*Amer. Math. Soc. Transl.*]{} (2) [**171**]{} (1996), 11–22.
A. Barvinok. [*A course in convexity*]{}. AMS, Providence, 2002.
K. Borsuk. On the $k$-independent subsets of the Euclidean space and of the Hilbert Space. [*Bull. Acad. Pol. Sci. Cl.III*]{} [**5**]{}, (1957),351-356.
V. G. Boltyansky, S. S. Ryzhkov and Yu. A. Shashkin. On $k$-regular embeddings and their applications to the theory of approximation of functions.[*Uspekhi Mat. Nauk*]{} [**15**]{} (1960), no. 6 (96), 125–132; [*Amer. Math. Soc. Transl.*]{} (2) [**28**]{} (1963), 211–219.
E. W. Cheney. [*Introduction to Approximation Theory*]{}, $2^{nd}$ Ed., Chelsea Publishing Company, New York, 1996.
F. R. Cohen and D. Handel. $k$-regular embeddings of the plane, [*Proc. Amer. Math. Soc.*]{} [**72**]{} (1978), 201–204.
M. Ghomi. Tangent bundle embeddings of manifolds in Euclidean space. Preprint.
M. Ghomi. Nonexistence of skew loops on ellipsoids. [*Proc. Amer. Math. Soc.*]{}, in print.
M. Ghomi, B. Solomon. Skew loops and quadratic surfaces. [*Comment. Math. Helv.*]{} [**77**]{} (2002), 767–782.
M. Ghomi, S. Tabachnikov. Totally skew embeddings of manifolds. Preprint, ArXiv: math.DG/0302288.
D. Handel. Obstructions to 3-regular embeddings. [*Houston J. Math*]{} [**5**]{} (1979), 339–343.
D. Handel. Approximation theory in the space of sections of a vector bundle. [*Trans. Amer. Math. Soc.*]{} [**256**]{} (1979), 383–394.
D. Handel. Some existence and non-existence theorems for $k$-regular maps. [*Fund. Math.*]{} [**109**]{} (1980), 229–233.
D. Handel. $2k$-regular maps on smooth manifolds. [*Proc. Amer. Math. Soc.*]{} [**124**]{} (1996), 1609–1613.
D. Handel, J. Segal. On $k$-regular embeddings of spaces in Euclidean space. [*Fund. Math.*]{} [**106**]{} (1980), 231–237.
J.-P. Sha, B. Solomon. No skew branes on non-degenerate hyperquadrics. Preprint, ArXiv: math.DG/0412197.
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S. Tabachnikov. On skew loops, skew branes and quadratic hypersurfaces. [*Moscow Math. J.*]{} [**3**]{} (2003), 681–690.
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[*E-mail address:* ]{}[email protected]; [email protected]
[^1]: The definitions of both types of $k$-regularity that we adopt here are those of Handel, see for example [@H-S]. The definitions used by other authors are in essence the same, but the role of $k$ differs from one author to another. We adapt all the quoted results to our definition.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Motivated by the possibility to load multi-color fermionic atoms in optical lattices, we study the entropy dependence of the properties of the one-dimensional antiferromagnetic $SU(N)$ Heisenberg model, the effective model of the $SU(N)$ Hubbard model with one particle per site (filling $1/N$). Using continuous-time world line Monte Carlo simulations for $N=2$ to $5$, we show that characteristic short-range correlations develop at low temperature as a precursor of the ground state algebraic correlations. We also calculate the entropy as a function of temperature, and we show that the first sign of short-range order appears at an entropy per particle that increases with $N$ and already reaches $0.8k_B$ at $N=4$, in the range of experimentally accessible values.'
author:
- Laura Messio
- Frédéric Mila
bibliography:
- 'SUN\_chain.bib'
title: 'Entropy dependence of correlations in one-dimensional SU(N) antiferromagnets'
---
Lattice $SU(N)$ models play an ever increasing role in the investigation of strongly correlated systems, both in condensed matter and in cold atoms. The first systematic use of these models took place in the context of the large-$N$ generalization of the $SU(2)$ Heisenberg model, in which conjugate (or self-conjugate) representations are put on the two sublattices of the square lattice so that a $SU(N)$ singlet can be formed on two sites[@Affleck_largeN; @Sachdev_fermions; @Auerbach_largeN]. Over the years, another class of $SU(N)$ models with the same representation at each site has appeared as the relevant description of the low temperature properties in several contexts. In particular, the $SU(3)$ model corresponds to the spin-1 Heisenberg model with equal bilinear and biquadratic interactions[@laeuchli_2006; @toth_2010; @toth_2012], while the $SU(4)$ model is equivalent to the symmetric version of the Kugel-Khomskii model of Mott insulators with orbital degeneracy[@kugel1982; @li1998]. These models have however attracted renewed attention recently as the appropriate low energy theory of ultracold gases of alkaline-earth-metal atoms in optical lattices in the Mott insulating phase with one atom per site, the parameter $N$ corresponding to the number of internal degrees of freedom of the atoms[@Nature_SUN].
A peculiar characteristic of these $SU(N)$ models is that one needs $N$ sites to form a singlet. This is often reflected in their ground state properties. In one dimension, the $SU(N)$ model has been solved with Bethe ansatz for arbitrary $N$[@Sutherland_SUN], and the dispersion of the elementary fractional excitations has a periodicity $2\pi/N$. On a ladder, the $SU(4)$ model has a plaquette ground state[@vdb_2004]. In two-dimensions, the $SU(3)$ model on both the square and triangular lattices has long-range color order with 3-site periodicity along the lines[@laeuchli_2006; @toth_2010], while on the kagome lattice it is spontaneaously trimerized[@corboz_kagome]. The $SU(4)$ model on the checkerboard lattice also has a plaquette ground state[@corboz_kagome]. Even on the square lattice, where the $SU(4)$ model undergoes spontaneous dimerization[@corboz_2011] with possibly algebraic correlations[@vishwanath_2009], neighboring dimers involved pairs of different colors, so that the 4 colors are indeed present with equal weight on all plaquettes. The general properties for arbitrary $N$ are not known however. An adaptation of the previous large-$N$ studies has been proposed for $m$ atoms per site[@Hermele_largeN_SUN]. If $m=O(N)$, the ground state has been proposed to be a chiral spin liquid for large $N$. The wealth of ground states predicted for different $N$ on various lattices calls for an experimental investigation. Ultra-cold fermionic atoms can a priori lead to very accurate realizations of these models. However, the temperature is a limiting factor. It can be lowered with respect to the initial temperature if the optical lattice is adiabatically switched on[@entropy_cold_atoms], but it cannot be made arbitrarily small. In fact, with adiabatic switching, one can control the entropy rather than the temperature, and in current state-of-the-art experimental setups, the lower limit for fermions with $N=2$ is equal to $0.77 k_B$ per particle[@jordens_2010]. If contact is to be made with experiments on cold atoms, it is thus crucial to know the properties of a given model as a function of entropy. For the SU(2) Heisenberg model on the cubic lattice, Néel ordering takes place at an entropy 0.338 $k_B$, i.e. about half the value that can be achieved today[@jordens_2010].
The first hint that increasing the number of colors might help in beating this experimental limit has been obtained in the context of a high temperature investigation of the $N$-flavour Hubbard model by Hazzard et al[@hazzard_2012], who have shown that the effective temperature reached after introducing the optical lattice decreases with $N$ under fairly general conditions. However, to the best of our knowledge, no attempt has been made so far to determine how the temperature or the entropy below which signatures of the ordering will show up depends on $N$.
In this Letter, we address this issue in the context of the one-dimensional (1D) antiferromagnetic $SU(N)$ Heisenberg model on the basis of extensive Quantum Monte Carlo (QMC) simulations. As we shall see, the ground state algebraic correlations lead to characteristic anomalies in the structure factor upon lowering the temperature. These anomalies only become visible at quite low temperature, but remarkably enough, the corresponding entropy per particle increases with $N$, leading to observable qualitative effects with current experimental setups for $N\ge 4$.
[*The $SU(N)$ Heisenberg model.—*]{} A good starting point to discuss $N$-color fermionic atoms loaded in an optical lattice is the $SU(N)$ Hubbard model defined by the Hamiltonian: $$\widehat H=t\sum_{\langle i,j\rangle\alpha}(\widehat c^\dag_{\alpha i}\widehat c_{\alpha j}+h.c.)+U\sum_{i,\alpha<\beta}\widehat n_{\alpha i}\widehat n_{\beta i},
\label{eq:Ham_Hubbard}$$ where $\widehat c^\dag_{i,\alpha}$ and $\widehat c_{i,\alpha}$ are creation and annihilation operators of a fermion of color $\alpha=1\dots N$ on site $i$ and the sum is over the first-neighbors of a periodic chain of length $L$. $\widehat n_{\alpha i}$ is the number of fermions of color $\alpha$ on site $i$. At filling $1/N$, i.e. with one fermion per site, the ground state is a Mott insulator, and to second order in $t/U$, the low-energy effective Hamiltonian is the $SU(N)$ Heisenberg model with the fundamental $SU(N)$ representation at each site, and with coupling constant $J=2t^2/U$. Setting the energy unit by $J=1$, this Hamiltonian can be written (up to an additive constant): $$\widehat H= \sum_{\langle ij\rangle} \widehat P_{ij}.
\label{eq:Ham}$$ where $\widehat P_{ij}$ permutes the colors on sites $i$ and $j$. If we denote by $\widehat S^{\alpha\beta}_i$ the operator that replaces color $\beta$ by $\alpha$ on site $i$, this permutation operator can be written as: $$\widehat P_{ij}=\sum_{\alpha,\beta} \widehat S^{\alpha\beta}_i\widehat S^{\beta\alpha}_j
\label{eq:Pij}$$
This effective Hamiltonian is an accurate description of the system provided the temperature is much smaller than the Mott gap. In terms of entropy, the criterion is actually quite simple. The high temperature limit of the entropy per site of the $SU(N)$ Hubbard model at $1/N$-filling can be shown to be equal to $k_B (N \ln N - (N-1) \ln (N-1))$, while that of the $SU(N)$ Heisenberg model is equal to $k_B \ln N$. So we expect the description in terms of the Heisenberg model to be accurate when the entropy is below $k_B \ln N$. For $SU(2)$, this is a severe restriction for experiments since $\ln 2\simeq 0.693...$, but already for $SU(3)$, this is less of a problem since $\ln 3 \simeq 1.099$. Of course, this is not the whole story since what really matters is the entropy below which specific correlations develop, but this is an additional motivation to consider $SU(N)$ models with $N>2$.
[*Exact results.—*]{} A number of exact results that have been obtained over the years on the 1D $SU(N)$ Heisenberg model will prove to be useful. The model has been solved with Bethe ansatz by Sutherland[@Sutherland_SUN]. He showed that, in the thermodynamic limit, the energy per site is given by $$E_0(N)=2\sum_{k=2}^\infty\frac{(-1)^k\zeta(k)}{N^k}-1,$$ where $\zeta$ is the Riemann’s zeta function. Some values are given in Tab. \[tab:energies\_SUN\]. In addition, he showed that there are $N-1$ branches of elementary excitations which all have the same velocity $v=2\pi/N$ at small $k$. Affleck has argued that the central charge $c$ should be equal to $N-1$[@affleck_1988], and Lee has shown that[@Lee_SUN], at low temperature $T$, the entropy is given by: $$S(T)=\frac{k_BN(N-1)}{6}T+O(T^2),
\label{eq:entropy_slope}$$ a direct consequence of $c=N-1$ and $v=2\pi/N$ since the linear coefficient is equal to $\pi c/3v$.
[*The QMC algorithm.—*]{} Quantum Monte-Carlo is the most efficient method to study the finite temperature properties of interacting systems provided one can find a basis where there is no minus sign problem, i.e. a basis in which all off-diagonal matrix elements of the Hamiltonian are non-positive. For the $SU(2)$ antiferromagnetic Heisenberg model on bipartite lattices, this is easily achieved by a spin-rotation by $\pi$ on one sublattice. For $SU(N)$ with $N>2$, there is no such general solution, but in 1D one can get rid of the minus sign on a chain with open boundary conditions, as already noticed for the SU(4) model[@frischmuth_1999]. Let us start from the natural basis consisting of the $N^{L}$ product states $\otimes_i|\alpha_i\rangle=|\alpha_0,\dots,\alpha_{L-1}\rangle$, where $\alpha_i$ is the color at site $i$. In this basis, all off-diagonal elements of the $SU(N)$ model of Eq.\[eq:Ham\] are either zero or positive. However, a generalization of the Jordan-Wigner transformation allows to change all these signs on an open chain. This transformation is defined by: $$|\alpha_0,\dots,\alpha_{L-1}\rangle \to (-1)^{r(\alpha_0,\dots,\alpha_{L-1})}|\alpha_0,\dots,\alpha_{L-1}\rangle,$$ where $r(\alpha_0,\dots,\alpha_{L-1})$ is the number of permutations between different color particules on neighboring sites needed to obtain a state such that the $\alpha_i$ are ordered ($\alpha_i\leq\alpha_j$ for $i<j$). This basis change is equivalent to a Hamiltonian transformation, the new Hamiltonian being given by: $$\widehat H=\sum_{\langle ij\rangle}\sum_{\alpha} \left( \widehat S^{\alpha\alpha}_i\widehat S^{\alpha\alpha}_j- \sum_{\beta\neq\alpha} \widehat S^{\alpha\beta}_i\widehat S^{\beta\alpha}_j\right).
\label{eq:Ham2}$$ On a periodic chain, the equivalence with the Hamiltonian of Eq. \[eq:Ham\] is not exact, but the difference disappears in the thermodynamic limit. So in the following we will simulate the Hamiltonian of Eq.\[eq:Ham2\].
To do so, we have developed a continuous time world-line algorithm with cluster updates[@QMC] adapted to the model of Eq. \[eq:Ham2\] with $N$ colors. The partition function $Z$ is expressed as a path integral over the configurations $\phi:\tau\to\phi(\tau)$, where $\tau$ is the imaginary time going from 0 to $\beta=\frac1{k_BT}$ and $\phi(\tau)$ is a basis state. The functions $\phi$ that contribute to the integral can be represented by $\phi(0)$ and by a set of world line crossings $\{(i,j,\tau)\}$ that exchange the colors of two sites $i$ and $j$ at time $\tau$. A local configuration $c$ on a link $ij$ at time $\tau$ is represented by $$c=\left(\begin{matrix}
\alpha_i(\tau^+)\,\alpha_j(\tau^+) \\ \alpha_i(\tau^-)\,\alpha_j(\tau^-)
\end{matrix}\right)$$ Cluster algorithms are well documented for 2-color models. Here we generalize the approach to $SU(N)$ by choosing randomly two different colors $p$ and $q$ out of $N$ and by constructing clusters on which only these two colors are encountered. The steps to construct the clusters are the following. We first randomly place elementary graphs in the configuration using a Poisson distribution. These graphs are drawn in the first column of Tab.\[tab:cluster\] and the Poisson time constant is given in the last column. They are accepted only if $\Delta_G(c)=1$ (if a color which is neither $p$ nor $q$ appears in the local configuration, the graph is rejected). Then we assign graphs to the world-line crossings between $p$ and $q$ colors using the last two columns of Tab.\[tab:cluster\]. At the places where no graph has been attributed, we follow the path with the same color. Finally we follow each constructed cluster and exchange $p$ and $q$ on it with a probablity $1/2$ (Swendsen-Wang algorithm). This constitutes a Monte Carlo step.
$$\begin{array}{!{\vrule width 1pt}c!{\vrule width 1pt}c|c|c!{\vrule width 1pt}c!{\vrule width 1pt}}
\noalign{\hrule height 1pt}
G& \Delta_G\left(\begin{matrix}
p\, p \\ p\, p
\end{matrix}\right)&
\Delta_G\left(\begin{matrix}
p\,q \\ p\,q
\end{matrix}\right)&
\Delta_G\left(\begin{matrix}
q\,p \\ p\,q
\end{matrix}\right)&
W_G\\
\noalign{\hrule height 1pt}
% \includegraphics[width=.025\textwidth]{graphe1} &1 &1 &- &1-d\tau (1+\epsilon)\\
\includegraphics[width=.025\textwidth]{graphe2} &1 &- &1 &\epsilon\\
\includegraphics[width=.025\textwidth]{graphe4} &- &1 &1 &1-\epsilon\\
\includegraphics[width=.025\textwidth]{graphe3} &0 &1 &0 &2\epsilon\\
% \noalign{\hrule height 1pt}
% &1-d\tau&1&d\tau&\\
\noalign{\hrule height 1pt}
\end{array}$$
Using this algorithm, we have calculated the energy per site $E$, which is given by: $$E
=\left\langle \frac{\widehat H}{L}\right\rangle
\simeq\frac{k_BT}{L\,n}\sum_\phi\left(\sum_{\langle ij\rangle}\int d\tau
\delta_{\alpha_i(\tau),\alpha_j(\tau)}-n(\phi)\right),$$ where $n$ is the number of Monte Carlo steps and $n(\phi)$ the number of world-line crossings in the configuration $\phi$, the diagonal correlations defined by $$C(j)
=\left\langle \sum_\alpha \widehat S^{\alpha\alpha}_0 \widehat S^{\alpha\alpha}_j\right\rangle -\frac1{N}
% \simeq\frac{T}{n}\sum_\phi\int d\tau \delta_{\alpha,\alpha_0(\tau)}\delta_{\alpha,\alpha_x(\tau)}-\frac1N ,$$ and the associated structure factor defined by $$\tilde C(k)=\frac1{2\pi} \frac{N}{N-1}\sum_jC(j)e^{ikj}.$$ This structure factor is normalized in such a way that $\frac{2\pi}{L}\sum_k \tilde C(k)=1$.
![ Evolution of the energy per site $E$ and of the entropy per site $S$ as a function of the temperature $T$ for different $N$ on a $L=60$ chain. The inset shows the slope of the entropy at $T=0$, given in Eq. \[eq:entropy\_slope\]. The curvature being positive at $T=0$, the curves go higher than the tangent (dashed line). \[fig:fdeTS\_chain\]](QMC_1D_ES){width="48.00000%"}
[*The results.—*]{} We have studied chains of length $L=60$ for $T$ from $0.01$ to $20$ with a number of colors $N=2,3,4$ and $5$, and a number of Monte Carlo steps $n$ at least equal to $10^6$. The correlation time measured by the binning method indicates that around $N$ steps are needed to obtain uncorrelated configurations, whatever the temperature, and that the precision on the energy per site $E$ is better than $10^{-4}$. This could be confirmed by the comparison of the limit of the energy when $T\to0$ with the exact finite $L$ value for $SU(3)$.[@Martins_SUN] Moreover, the energy of the ground state differs from that of the thermodynamic limit by less than $8.10^{-4}$. So, for our purpose, the finite size effects can be considered to be negligible (see Tab. \[tab:energies\_SUN\]). The entropy per site $S$ has been deduced from the energy $E$ by an integration from high temperature: $$S(T)=S(\infty)-\int_T^\infty d\tau \frac{k_B}{\tau}\frac{dE}{d\tau}$$ where $S(\infty)=k_B\ln(N)$. $E$ and $S$ are plotted in Fig. \[fig:fdeTS\_chain\] for different $N$ as a function of $T$. Since the entropy is the result of a numerical integration, it is important to check its accuracy, especially at low temperature since by construction it has to be correct at high temperature. Now, we know that, at low temperature, the entropy must be linear with a slope equal to $k_BN(N-1)/6$ (see Eq.\[eq:entropy\_slope\]). This is confirmed by the inset of Fig. 1b), in which one clearly sees that the entropies times $6/k_BN(N-1)$ lie on top of eachother at low temperature.
Now, the stabilization of the energy at low $T$ occurs at a temperature that decreases when $N$ increases. Thus, one could naively think that it will be more difficult to observe the development of the ground state correlations when $N$ increases. However, this is not true if one considers the entropy. Indeed, the entropy grows much faster at low temperature when increasing $N$. So, the temperature corresponding to a given entropy decreases very fast when $N$ increases.
$$\begin{array}{!{\vrule width 1pt}c!{\vrule width 1pt}c|c|c!{\vrule width 1pt}}
\noalign{\hrule height 1pt}
N & BA(L=\infty) & BA(L=60) & QMC(L=60) \\
\noalign{\hrule height 1pt}
2& -0.386294 & &-0.38675(2)\\
3& -0.703212 &-0.7038228 &-0.70384(2)\\
4& -0.8251193 & &-0.82577(2)\\
5& -0.884730 & &-0.88541(2)\\
\noalign{\hrule height 1pt}
\end{array}$$
![Structures factor $\tilde C(k)$ at low temperature ($k_BT=0.01$) for different $N$ on a $L=60$ chain, with $n=10^7$ Monte-Carlo steps. Small peaks are clearly visible at $k=\pi$ for $N=4$ and $k=4\pi/5$ and $6\pi/5$ for $N=5$. The data for $N=4$ are in perfect agreement with those of Ref.. \[fig:correlations\_T001\]](corr_T001_severalN.pdf){width=".5\textwidth"}
We now look at the diagonal correlations $\tilde C(k)$. They have been calculated for different temperatures, but, in view of the implications for ultracold fermionic gas, we represent them as a function of the entropy per site $S$. Since the system is 1D, there is no long range order, hence no Bragg peaks. Nevertheless, short-range correlation develop at low entropy. They translate into finite height peaks in $\tilde C(k)$ at finite temperature, and singularities at zero temperature. The number and the position of these peaks depend on the number of colors $N$. From the Bethe ansatz solution, singularities are expected to occur at $k=2p\pi/N$ with $p=1,...,N-1$. The results of Fig. \[fig:correlations\_chain\] agree with this prediction: there is a single peak at $\pi$ for $SU(2)$, while $N-1$ peaks are indeed present for $SU(N)$ at sufficiently small entropy. Note however that all peaks do not have the same amplitude for $N\geq4$. For $N=4$ and $5$, two types of peaks not related by the symmetry $k\rightarrow
2\pi-k$ are present. The peaks at $2\pi/N$ and $2(N-1)\pi/N$ are much more prominent, and they start to be visible at much larger entropy.
At the maximal entropy, the structure factor $\tilde C(k)$ is flat (see Fig. \[fig:correlations\_chain\]). At large but finite entropy, it presents a broad maximum at $k=\pi$ for all $N$. This reflects the simple fact that colors tend to be different on neighboring sites. More specific correlations appear upon lowering the entropy. For $SU(2)$, the peak at $k=\pi$ just gets more pronounced. To observe the development of the singularity typical of the $SU(2)$ ground state algebraic correlations will however require to reach rather low entropy. This should be contrasted with the $N>2$ cases, where a qualitative change in the structure factor occurs upon reducing the entropy: the broad peak at $k=\pi$ is replaced by peaks at $2\pi/N$ and $2(N-1)\pi/N$. One can in principle read off the corresponding entropy from Fig. \[fig:correlations\_chain\]. To come up with a quantitative estimate, we note that, upon reducing the entropy, the curvature of the structure factor at $k=\pi$ changes sign from positive at high temperature to negative when the peaks at $2\pi/N$ and $2(N-1)\pi/N$ appear . This occurs at $S_c/k_B=0.58$, $0.87$ and $1.08$ for $N=3$, $4$ and $5$ respectively. This characteristic entropy $S_c$ increases more or less linearly with $N$ as $S_c \simeq 0.2 Nk_B$, and for $N=4$ and $5$, it lies in the experimentally accessible range. This is mostly a consequence of the temperature dependence of the entropy, which grows much faster with $N$ at low temperature. The characteristic temperature at which deviations from the broad peak at $k=\pi$ occur depends only weakly on $N$. Finally, secondary peaks appear at lower temperature (see Fig. \[fig:correlations\_T001\])
[*Conclusions.—*]{} We have shown that the entropy at which the periodicity characteristic of the zero temperature algebraic order of $SU(N)$ chains is revealed increases significantly with $N$. For $N=4$, this entropy is already larger than the entropy per particle recently achieved in the $N=2$ case in the center of the Mott insulating cloud (0.77 $k_B$)[@jordens_2010]. Whether a similar entropy can be achieved for $N>3$ remains to be seen. As shown by Hazzard et al[@hazzard_2012], if the initial temperature is fixed, the initial entropy in a 3D trap increases with $N$ as $N^{1/3}$, implying that one might have to go to values of $N$ larger than 4 to reach a final entropy low enough to observe characteristic correlations. However, evaporative cooling might allow to reach initial entropies that are less dependent on $N$. In a recent experiment on $^{173}$Yb, the initial entropy reported by Sugawa et al[@sugawa_2011] for this $N=6$ case is not much higher than in $N=2$ experiments[@jordens_2010]. It is our hope that the present results will encourage the experimental investigation of the $1/N$-filled Mott phase of $N$-color ultracold fermionic atoms.
We thank Daniel Greif for useful discussions. LM acknowledges the hospitality of EPFL, where most of this project has been performed. This work has been supported by the Swiss National Fund and by MaNEP.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We give necessary and sufficient conditions for the set of Neumark projections of a countable set of phase space observable to constitute a resolution of the identity, and we give a criteria for a phase space observable to be informationally complete. The results will be applied to the phase space observables arising from an irreducible representation of the Heisenberg group.'
address:
- 'Gianni Cassinelli, Dipartimento di Fisica, Università di Genova, I.N.F.N., Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy'
- 'Ernesto De Vito, Dipartimento di Matematica, Università di Modena, via Campi 213/B, 41100 Modena, Italy and I.N.F.N., Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy'
- 'Pekka Lahti, Department of Physics, University of Turku, 20014 Turku, Finland'
- 'Alberto Levrero, Dipartimento di Fisica, Università di Genova, I.N.F.N., Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy'
author:
- 'G. Cassinelli'
- 'E. De Vito'
- 'P. Lahti'
- 'A. Levrero'
title: Phase space observables and isotypic spaces
---
Introduction {#intro}
============
Phase space observables have turned out to be highly useful in various fields of quantum physics, including quantum communication and information theory, quantum tomography, quantum optics, and quantum measurement theory. Also many conceptual problems, like the problem of joint measurability of noncommutative quantities, or the problem of classical limit of quantum mechanics have greatly advanced by this tool. The monographs [@Davies; @Helstrom; @Holevo; @Busch; @Schroeck; @Hakioglu; @Perinova] exhibit various aspects of these developments.
Any positive trace one operator $T$ (a state) defines a phase space observable $Q_T$ according to the rule $$Q_T(E)= \frac 1{2\pi}\int_E e^{i(qP+pQ)}Te^{-i(qP+pQ)}dq\,dp,$$ where $E$ is a Borel subset of the (two dimensional) phase space. It is well known that all the phase space observables generated by [*pure states*]{} have the same minimal Neumark dilation to a canonical projection measure on $L^2(\runo^2)$. On the other hand, the corresponding Neumark projections depend on the pure state in question. If $T$ is a pure state $|u\rangle\langle u|$ defined by a unit vector $u$, we let $P_u$ denote the Neumark projection associated with $Q_{|u\rangle\langle u|}$ . If two unit vectors $u$ and $v$ are orthogonal then also $P_{u}P_{v}=0$. One could then pose the problem of determining a set of orthonormal vectors $\{u_i\}$ such that the associated Neumark projections $\{P_{u_i}\}$ of the phase space observables $Q_{|u_i\rangle\langle u_i|}$ constitute a resolution of the identity. In [@Lahti] it was shown that the set of number eigenvectors possesses this property. This was proved by a direct method using the properties of the Laguerre polynomials.
It turns out that this problem has a group theoretical background. This follows from the work of A. Borel [@Borel] on the group representations that are square integrable modulo the centre. Using the results of Borel this problem can be traced back to the study of the isotypic spaces of the representations induced by a central character of the Heisenberg group $H^1$. (We recall that a representation $(\pi,\hi)$ is called isotypic if it is the direct sum of copies of the same irreducible representation). More precisely, the phase space observables arise from an irreducible representation of $H^1$ that is square integrable modulo the centre. This is actually a general result: any irreducible representation $\pi$ of a group $G$ that is square integrable modulo the centre gives rise to covariant “phase space observables” with the above properties. We prove that a necessary and sufficient condition for the set of Neumark projections $\{P_{u_i}\}$ to be a resolution of the identity is that the representation of $G$ induced by the central character of $\pi$ be isotypic. This phenomenon occurs in particular for the Heisenberg group, which is behind the phase space observables.
Phase space observables $Q_T$ that are generated by states $T$ such that ${\rm tr\,}[Te^{i(qP+pQ)}]\ne 0$ for almost all $(q,p)\in\runo^2$, are known to have another important property. They are informationally complete, namely, if $W_1$ and $W_2$ are two states for which ${\rm tr\,}[W_1Q_T(E)]={\rm tr\,}[W_2Q_T(E)]$ for all $E$, then $W_1=W_2$, see, eg. [@AliPru; @giape]. We show that, under suitable conditions, this property holds in general for “phase space observables" associated with any irreducible representations $\pi$ of $G$ square integrable modulo centre.
We hope that these results could bring further light on some of the many applications of the phase space observables in quantum mechanics.
Preliminaries and notations {#s1}
===========================
In this paper we use freely the basic concepts and results of harmonic analysis, referring to [@Folland95] as our standard source. Let $G$ be a Hausdorff, locally compact, second countable topological group, and let $Z$ be its centre. $Z$ is a closed, abelian, normal subgroup of $G$. We denote by $X=G/Z$ the quotient space. It is a Hausdorff, locally compact, second countable topological group, and it is also a locally compact $G$-space with respect to the natural action by left multiplication. Let $p:G\to X$ be the canonical projection and $s:X\to G$ a Borel section for $p$, fixed throughout the paper.
Assume further that $G$ is unimodular so that its left Haar measures are also right Haar measures. As an abelian subgroup $Z$ is also unimodular. We denote by $\mu$ and $\mu_0$ two (arbitrarily fixed) Haar measures of $G$ and $Z$, respectively. Then there is a unique $G$-invariant positive Borel measure $\alpha$ on $X$ such that for each compactly supported continuous function $f\in C_c(G)$ $$\label{Weil}
\int_G\, f(g)\,d\mu(g) =\int_X\left(\int_Zf(s(x)h)\,d\mu_0(h)\right)\,
d\alpha(x).$$ Moreover, $f\in L^1(\mu)$ if and only if the function $(x,h)\mapsto f(s(x)h)$ is in $L^1(\alpha\otimes\mu_0)$ and in this case (\[Weil\]) holds for $f$. The measure $\alpha$ is also a Haar measure for $X$ (regarded as a group), both right and left.
We denote by $(\pi,\hi)$ a continuous unitary irreducible representation of $G$ acting on a complex separable Hilbert space $\hi$. Let $h\in Z, g\in G$. Then $\pi(h)\pi(g)=\pi(hg)=\pi(gh)=\pi(g)\pi(h)$, so that $\pi(h)$ commutes with all $\pi(g), g\in G$. By Schur’s lemma, $$\pi(h) = \chi(h)\, I,$$ where $I$ is the identity operator on $\hi$ and $\chi$ is a $\tuno$-valued character of $Z$, $\tuno$ denoting the group of complex numbers of modulus one. We call $\chi$ the central character of $\pi$.
In the following we describe explicitly the imprimitivity system for $G$, based on $X$, induced by the irreducible unitary representation $\chi$ of $Z$. There are several equivalent realizations of this object, and we choose those which are most appropriate for our purposes.
Let $\hic$ denote the space of ($\mu$-equivalence classes of) measurable functions $f:G\to\cuno$ for which
$f(gh) = \chi(h^{-1})f(g)$ for all $h\in Z$,
$f\circ s\in L^2(X,\alpha)$. The definition of the space $\hic$ does not depend on the section $s$. Indeed, if $s'$ is another Borel section for $p$, then for any $x\in X$, $s'(x)=s(x)h$ for some $h\in Z$, so that $$|f(s'(x))|^2=|f(s(x)h)|^2=|\chi(h^{-1})f(s(x))|^2 = |f(s(x))|^2.$$ The space $\hic$ is a complex separable Hilbert space with respect to the scalar product $$\ip{f_1}{f_2}_{\hic} := \int_X \overline{f_1(s(x))}f_2(s(x))\,d\alpha(x),$$ which is independent of $s$. A description of the structure of $\hic$ is given by the following property. Let $K(G)^\chi$ denote the set of continuous functions $f:G\to\cuno$ with the properties
$f(gh) = \chi(h^{-1})f(g)$ for all $g\in G, h\in Z$,
$p({\rm supp}\, f)$ is compact in $X$. If ${\varphi}\in C_c(G)$, then the function $f_{\varphi}$, defined by $$f_{\varphi}(g) := \int_Z\chi(h){\varphi}(gh)\,d\mu_0(h),$$ is in $K(G)^\chi$. Moreover, any function $f\in K(G)^\chi$ is of the form $f=f_{\varphi}$ for some ${\varphi}\in C_c(G)$ (see, e.g., [@Folland95], Proposition 6.1, p. 152). Obviously, $K(G)^\chi\subset\hic$ and $K(G)^\chi$ is dense in $\hic$.
The Hilbert space $\hic$ carries a continuous unitary representation $l$ of $G$ explicitly given by $$(l(a)f)(g) = f(a^{-1}g),\ \ \ g\in G.$$ It is a realization of the representation of $G$ induced by the representation $\chi$ of $Z$.
Let ${\mathcal}B(X)$ be the $\sigma$-algebra of the Borel subsets of $X$. We define a projection measure on $(X,{\mathcal}B(X))$ by $$(P(E)f)(g):= \chi_E(p(g))f(g),$$ where $E\in{\mathcal}B(X)$ and $f\in\hic$. Clearly, ${\mathcal}B(X)\ni
E\mapsto P(E)\in B(\hic)$ is a projection measure and $(l,P)$ is an imprimitivity system for $G$, based on $X$, and acting on $\hic$. Indeed, $$l(a)P(E)l(a)^{-1} = P(a.E),\ \ \ a\in G, E\in{{\mathcal}B}(X).$$ It is a realization of the imprimitivity system canonically induced by $\chi$ and it is irreducible since $\chi$ is irreducible.
Representations that are square integrable modulo the centre {#s3}
============================================================
Let $(\pi,\hi)$ be a continuous unitary representation of $G$ in a complex separable Hilbert space $\hi$. Given $u,v\in\hi$, we denote by $\cuv$ the function on $G$ defined through the formula $$\cuv(g) := \ip{\pi(g)u}{v}.$$ This function is called a [*coefficient*]{} of $\pi$ and it is continuous and bounded, $$|\cuv(g)| = |\ip{\pi(g)u}{v}|\leq \no{\pi(g)u}\,\no{v} \leq
\no{u}\,\no{v}, \ \ \ g\in G,$$ and it has the property $\cuv(gh) = \chi(h)^{-1}\cuv(g)$ for all $h\in Z$.
[Let $(\pi,\hi)$ be a continuous unitary irreducible representation of $G$. We say that $\pi$ is [*square integrable modulo the centre*]{} of $G$, when, for all $u,v\in\hi$, $\cuv\circ s\in L^2(X,\alpha)$. ]{}
This definition is independent of the choice of the function $s$. Indeed, if $s'$ is another section for $p$, then $s'(x)=s(x)h$, $h\in Z$, for all $x\in X$, so that $
\pi(s'(x)) = \pi(s(x)h)=\chi(h)\pi(s(x)),
$ and thus $
|\ip{\pi(s'(x))u}{v}|^2=|\ip{\pi(s(x))u}{v}|^2.
$
We shall list next the basic properties of the square integrable representations modulo the centre. These results are due to A. Borel [@Borel], and they generalize the classical results of R. Godement [@Godement] for square integrable representations.
Let $\pi$ be a unitary irreducible representation of $G$ with central character $\chi$. Then the following three statements are equivalent: $a)$ $\pi$ is square integrable modulo $Z$; $b)$ there exist $u,v\in\hi\setminus\{0\}$ such that $\cuv\circ s \in L^2(X,\alpha)$; $c)$ $\pi$ is equivalent to a subrepresentation of $(l,\hic)$.
If any (hence all) of the preceding conditions is satisfied, then $\cuv\in\hic$ for all $u,v\in\hi$.
If $(\pi,\hi)$ is square integrable modulo $Z$, there exists a number $d_\pi>0$, called [*the formal degree*]{} of $\pi$, such that $$\ip{\cuv}{c_{u',v'}}_{\hic} =
\frac 1{d_\pi}\ip{u'}{u}_\hi\,\ip{v}{v'}_\hi.$$ The formal degree depends on the normalisation of the Haar measure $\mu$ so that, possibily redefining $\mu$, one can always assume that $d_\pi=1$ so that $$\label{ort1}
\ip{\cuv}{c_{u',v'}}_{\hic} = \ip{u'}{u}_\hi\,\ip{v}{v'}_\hi.$$
If $(\pi,\hi)$ and $(\pi',\hi')$ are two representations of $G$ which are square integrable modulo $Z$, whose central characters $\chi$ and $\chi'$ coincide, and which are not equivalent, then $$\label{ort2}
\ip{\cuv}{c'_{u',v'}}_{\hic} = 0,$$ where $c'_{u',v'}$ are coefficients of $(\pi',\hi')$.
Canonical POM associated with a square integrable representation modulo the centre {#s4}
==================================================================================
Let $(\pi,\hi)$ be a fixed representation with central character $\chi$ and square integrable modulo the centre. Fix $u\in\hi\setminus\{0\}$, and define $
W_u:\hi\to\hic
$ by $$W_uv := \cuv,\ \ \ v\in\hi.$$ $W_u$ is a linear map and it is a multiple of an isometry. Indeed, if $v,w\in\hi$, then $$\label{ort3}
\ip{W_uv}{W_uw}_{\hic} =
\no{u}^2_\hi\,\ip{v}{w}_\hi.$$ The range of $W_u$ is a closed subspace of $\hic$, and $1/\no{u}_\hi\, W_u$ is a unitary operator from $\hi$ to the range of $W_u$. The operator $W_u$ intertwines the action of $\pi$ on $\hi$ with the action of $l$ on $\hic$. In fact, for any $a\in G$, (W\_u((a)v))(g) &=& c\_[u,(a)v]{}(g)= \_\
&=& \_= c\_[u,v]{}(a\^[-1]{}g)\
&=& (W\_uv)(a\^[-1]{}g) = (l(a)(W\_uv))(g), showing that $$W_u\,\pi(a) = l(a)\, W_u$$ for all $a\in G$. Hence ${\rm ran}\,W_u$ is invariant with respect to $l$ and the unitary operator $1/\no{u}_\hi\, W_u$ defines an isomorphism of the unitary irreducible representations $(\pi,\hi)$ and $(l|_{\,{\rm ran}\,W_u}, {\rm ran}\,W_u)$ of $G$, $$(\pi,\hi) \simeq (l|_{\,{\rm ran}\,W_u}, {\rm ran}\,W_u).$$ We are in a position to associate to any state $T$ a natural positive operator measure (POM) on $(X,{\mathcal}B(X))$, with values in the positive operators on $\hi$. Given a state $T$, for all $E\in{\mathcal}B(X)$ we define $$\label{ronza}
Q_T(E) = \int_E \pi(s(x)) T \pi(s(x))^{-1} \,d\alpha(x),$$ where the integral is in the weak sense. The definition is well posed. Indeed, let $T=\sum_i \lambda_i |e_i\rangle\langle e_i|$ be the spectral decomposition of $T$ and fix a trace class operator $\W$ with the decomposition $\W=\sum_k w_k |u_k\rangle\langle v_k| $, where $w_k\geq 0$ and $(u_k), (v_k)\subset\hi$ are orthonormal sequences. Since $\pi$ is square integrable modulo $Z$, the function $$\phi_{ik}(x)=\overline{c_{e_i,v_k}(s(x))}c_{e_i,u_k}(s(x))
= \ip{v_k}{\pi(s(x))e_i}\ip{e_i}{\pi(s(x))^{-1}u_k}$$ is $\alpha$-integrable on $X$ and, using the Hölder inequality and the orthogonality relations (\[ort1\]), \_E |\_[ik]{}(x)|d(x) & & ( \_E |c\_[e\_i,v\_k]{}(s(x))|\^2d(x) )\^[12]{}\
& & ( \_E |c\_[e\_i,u\_k]{}(x)|\^2d(x) )\^[12]{}\
&& \_\_\
& & \^2\_\_\_= 1. Since $\sum_{i,k}\lambda_i w_k =
\nor{T}_1\nor{\W}_1=\nor{\W}_1$, the series $\sum_{i,k}\lambda_i w_k \phi_{ik}$ converges $\alpha$-almost everywhere to an integrable function $\phi$ and $$\int_E\phi(x)\,d\alpha(x)=\sum_{i,k}\lambda_i w_k \int_E\phi_{ik}\,d\alpha(x).$$ On the other hand, for $\alpha$-almost all $x\in X$, $\phi(x)=\tr{\W\pi(s(x))T\pi(s(x))^{-1}}$. Hence $\int_E |\tr{\W\pi(s(x))T\pi(s(x))^{-1}}|\,d\alpha(x) \leq
\nor{\W}_1$ and the linear form $$\W\mapsto \int_E \tr{\W\pi(s(x))T\pi(s(x))^{-1}}\,d\alpha(x)$$ is continuous on the Banach space of the trace class operators. Therefore it defines a bounded operator $Q_T(E)$ such that & = & \_E d(x)\
& = & \_[i,k]{}\_i w\_k \_E d(x)\
& = & \_[i,k]{}\_i w\_k \_E c\_[e\_i,u\_k]{}(s(x)) d(x). By choosing $\W=|u\rangle\langle v|$ we see that $Q_T(E)$ has the expression (\[ronza\]).
The mapping $E\mapsto Q_T(E)$ defines a POM on $X$. Indeed, $Q_T(E)$ is a positive operator and, given $u,v\in\hi$, the map $E\mapsto\ip{u}{Q_T(E)v}_\hi$ is a complex measure on $(X,{\mathcal}B(X))$, due to the $\sigma$-additivity of the integral.
Moreover, $Q_T(X)=I$. Indeed, for all $u,v\in\hi$, \_ &=&\_i \_i \_X c\_[e\_i,v]{}(s(x)) d(x)\
&=& \_i\_i\
&=& \_i \_i \^2\_\_= \_.
The operator measure $E\mapsto Q_T(E)$ is covariant under the representation $(\pi,\hi)$, that is, for all $E\in{\mathcal}B(X), a\in G$, $$\pi(a)Q_T(E)\pi(a)^{-1} = Q_T(a. E).$$ Indeed, (a)Q\_T(E)(a)\^[-1]{} & = &\_E (a)(s(x)) T (s(x))\^[-1]{} (a)\^[-1]{} d(x)\
& = &\_E (as(x)) T (as(x))\^[-1]{} d(x)\
& = &\_E (s(a.x)) T (s(a.x))\^[-1]{} d(x)\
& = &\_[a.E]{} (s(x)) T (s(x))\^[-1]{} d(x)\
& = &Q\_T(a. E) where we used the fact that $as(x)=s(a. x)h$, for some $h\in Z$.
The minimal Neumark dilation of $Q_u$
=====================================
In this section we consider the operator measure $Q_{|u\rangle\langle u|}$ associated with a pure state $|u\rangle\langle u|$ and we show that the canonical projection measure $P$ defined in Section \[s1\] is the minimal Neumark dilation of $Q_{|u\rangle\langle u|}$ for any $u$.
Given a unit vector $u\in \hi$, we denote simply by $Q_u$ the POM $Q_{|u\rangle\langle u|}$. Then for any $E\in{{\mathcal}B}(X)$ and for all $v,w\in\hi$, \_ &=& \_X \_E(x) (W\_uw)(s(x)) d(x)\
& = & \_E c\_[u,w]{}(s(x)) d(x)\
&=&\_, which shows that $P$ is a Neumark dilation of $Q_u$.
Furthermore, $P$ is minimal in the sense that $\hic$ is the smallest closed space containing all the vectors of the form $P(E)f$, as $E$ varies in ${\mathcal}B(X)$ and $f$ varies in ${\rm ran}\, W_u$, $$\hic = \overline{{\rm span}}\,\{P(E)f\,|\, E\in{\mathcal}B(X), f\in\ {\rm ran}\, W_u\}.$$ We go on to prove this fact. Due to the irreducibility of $\pi$, all the vectors of $\hi$ are cyclic for $\pi$ itself. Hence for any $v\in\hi$, $v\neq 0$, $$\hi = \overline{{\rm span}}\,\{\pi(a)v\,|\, a\in G\}.$$ Therefore, $${\rm ran}\, W_u = \overline{{\rm span}}\,\{ W_u(\pi(a)v)\,|\, a\in G\}
= \overline{{\rm span}}\,\{ (l(a)W_u)(v)\,|\, a\in G\},$$ so that && {P(E)f| EB(X), f [ran]{} W\_u}\
&=& { P(E)(l(a)W\_u)(v)| EB(X), aG}\
&=& { l(a)P(a\^[-1]{}. E)W\_u(v)| EB(X), aG}. Now $W_uv$ is a nonzero element of $\hic$ and $(l,P)$ is an irreducible imprimitivity system for $G$, acting in $\hic$, so that $$\overline{{\rm span}}\,\{ l(a)P(a^{-1}. E)W_u(v)\,|\, E\in{\mathcal}B(X), a\in G\}=\hic,$$ which completes the proof of the statement.
As a final remark we notice that the Neumark projection $P_u:\hic\to\hic$ onto the range of $W_u$ is explicitly given by $P_u=W_uW_u^*$.
A decomposition of the space $\hic$ {#s5}
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In this section we describe a decomposition of the space $\hic$ associated with the representation $(\pi,\hi)$ of $G$. We denote M()\_0 & := & \_[u]{}[ran]{}W\_u\
& = & [span]{}{| u,v}\
M() & := & . $M(\pi)$ is the smallest closed subspace of $\hic$ that contains all the ranges of the maps $W_u$. If $\pi$ and $\pi'$ are equivalent representations, then $$M(\pi)=M(\pi').$$ In other words, $M(\pi)$ depends only on the equivalence class of $\pi$. On the other hand, if $\pi$ and $\pi'$ are not equivalent, but they have the same central character $\chi$, then the orthogonality condition (\[ort2\]) imply that $$M(\pi)\perp M(\pi').$$ We proceed to study the structure of the subspace $M(\pi)$.
$M(\pi)$ is invariant under the action of $l$. This is clear since $M(\pi)_0$ is invariant with respect to $l$, hence, for any $a\in G$, $l(a)M(\pi)=\overline{l(a)M(\pi)_0}=\overline{M(\pi)_0}=M(\pi)$.
Let $(e_n)_{n\geq 1}$ be a basis of $\hi$. Then $(\wpn)_{n,p\geq 1}$ is a basis of $M(\pi)$. To show this, observe that $\ip{\wpn}{W_{e_q}e_m}_{\hic}= \ip{e_q}{e_p}_{\hi}\ip{e_n}{e_m}_{\hi} $, so that $(\wpn)_{n,p\geq 1}$ is an orthonormal set in $M(\pi)$. Given $u,v\in\hi$, one has that $\sum_{n,p}|\ip{u}{e_n}\ip{e_p}{v}|^2=\nor{u}^2\nor{v}^2$. Hence the series $\sum_{n,p}\ip{u}{e_n}\ip{e_p}{v}\wpn$ converges in $M(\pi)$. Since, for all $g\in G$, $\sum_{n,p}\ip{u}{e_n}\ip{e_p}{v}\wpn(g)$ converges to $W_uv(g)$, the set $(\wpn)_{n,p\geq 1}$ generates $M(\pi)_0$, hence $M(\pi)$.
The space $M(\pi)$ is isotypic, in fact it can be decomposed as $$\mpi=\oplus_{p\geq 1}\,{\rm ran\,}W_{e_p}$$ and, for any $p$ the representation $(l|_{{\rm ran\,}W_{e_p}},{\rm
ran\,}W_{e_p})$ is unitarily equivalent to $(\pi,\hi)$.
The Hilbert sum of the subspaces $\mpi$, as $\pi$ runs through the (inequivalent) irreducible representations of $G$ with the same central character $\chi$ that are square integrable modulo the centre, does not exhaust $\hic$, in general. This Hilbert sum is the [ *discrete part*]{} of $\hic$. In fact, let $V$ be a closed subspace of $\hic$ which is invariant and irreducible under $l$, and denote by $\sigma$ the restriction of $l$ to $V$. Then $\sigma$ is a square integrable representation of $G$ modulo the centre, with the same central character $\chi$, and one has the following result.
\[prop\] The subspace $V$ is contained in $M(\sigma)$.
Let $f\in V$ and denote by $S:\hic\to V$ the orthogonal projection onto $V$. For all $g\in\hic$ and $a\in G$ we have $$\ip{\sigma(a)Sg}{f}_{\hic} = \ip{Sl(a)g}{f}_{\hic} = \ip{l(a)g}{f}_{\hic}.$$ Since $Sg$ and $f$ are in $V$ and $(\sigma,V)$ is square integrable modulo $Z$, we have $$\left( a\mapsto \ip{l(a)g}{f}_{\hic} \right)\in M(\sigma).$$ Explicitly, $$\ip{l(a)g}{f}_{\hic} = \int_X \overline{g(a^{-1}s(x))}
f(s(x))\,d\alpha(x).$$ For any $\phi\in C_c(G)$ the function $f_\phi$ defined in section \[s1\] is in $K(G)^\chi\subset\hic$ and we get $$\ip{l(a)f_\phi}{f}_{\hic}= \int_X d\alpha(x) f(s(x)) \int_Z d\mu_0(h)
\overline{\chi(h)}\overline{\phi(a^{-1}s(x)h)}.$$ We claim that $$\left(
x,h\mapsto f(s(x)) \overline{\chi(h)}\overline{\phi(a^{-1}s(x)h)}
\right)\in L^1(\alpha\otimes\mu_0).$$ Indeed $$\int_Z |f(s(x)) \chi(h)\phi(a^{-1}s(x)h)| \,d\mu_0(h)
= |f(s(x))|\int_Z |\phi(a^{-1}s(x)h)|\,d\mu_0(h)$$ and the function $$x\mapsto \int_Z |\phi(a^{-1}s(x)h)|\,d\mu_0(h)$$ is in $C_c(X)$ (see, for instance, [@Folland95]). Hence its product with $|f(s(x))|$ is in $L^1(\alpha)$ and the claim follows by Tonelli’s theorem. Now we can apply Equation (\[Weil\]) to the function $$f(s(x)) \overline{\chi(h)}\overline{\phi(a^{-1}s(x)h)}=
f(s(x)h) \overline{\phi(a^{-1}s(x)h)}$$ to conclude that \_ & = & \_G f(g) d(g)\
& = & (f\*)(a), where $\tilde\phi(a):=\overline{\phi(a^{-1})}$, and $*$ denotes the convolution. In particular $f*\tilde\phi\in M(\sigma)$. If we let $\phi$ run over a sequence of functions on $G$ which is an approximate identity, see for example [@Folland95], one can prove that $f*\tilde\phi \to f$ in $\hic$ (see the below remark) and, since $M(\sigma)$ is closed, $f\in M(\sigma)$. This shows that $V\subseteq M(\sigma)$.
[The proof of the above proposition uses the fact that $f*\tilde\phi \to f$ in $\hic$ when $\phi$ runs over a sequence of functions on $G$ which is an approximate identity. To show this technical result one can mimic the standard proof in $L^2(G)$, taking into account that there exists a Borel measure $\nu$ on $G$ having density with respect to $\mu$ such that the induced representation $(l,\hic)$ can be realized on a suitable subspace of $L^2(G,\nu)$ (compare Ex. 6, Sect XXII.3 of [@dieu]).]{}
To summarize, $$\hic=\oplus_\pi M(\pi) \oplus R,$$ where the first direct sum ranges over the inequivalent irreducible representations of $G$ with central character $\chi$ that are square integrable modulo the centre and the orthogonal complement $R$ is the continuous part of the decomposition.
We can now state the main result of the paper.
Let $(\pi,\hi)$ be a square integrable representation of $G$ modulo the centre. Let $\{e_i\}$ be a basis of $\hi$. Then the set of orthogonal projections $\{ W_{e_i}W_{e_i}^*\}$ is a resolution of the identity in $\hic$ if and only if $(l,\hic)$ is an isotypic representation.
From items 2 and 3 above it follows that the set $\{ W_{e_i}W_{e_i}^*\}$ is a resolution of the identity of $\mpi$ and $(l,\mpi)$ is an isotypic representation. Hence, $\{ W_{e_i}W_{e_i}^*\}$ is a resolution of the identity in $\hic$ if and only if $\mpi=\hic$ and, in this case, $(l,\hic)$ is isotypic. Conversely, assume that $(l,\hic)$ is an isotypic representation. Let $(\sigma,V)$ be an irreducible subrepresentation of $(l,\hic)$, then $\sigma$ is square integrable modulo the centre and, by Proposition \[prop\], $V\subset
M(\sigma)$. Since $(l,\hic)$ is isotypic and $\pi$ is equivalent to a subrepresentation of $(l,\hic)$, $\sigma$ is equivalent to $\pi$, so that $M(\sigma)=\mpi$. Since $\hic$ is direct sum of copies of $(\sigma,V)$, it follows that $\hic=\mpi$.
The informational completeness
==============================
An interesting property of the phase space observables is related to the notion of informational completeness. We say that the operator measure $Q_T$, associated with the state $T$, is informationally complete if the set of operators $\{Q_T(E)\,|\, E\in{\mathcal}B(X)\}$ separates the set of states, [@giape; @Prugo]. An extensive study of the conditions assuring the informational completeness is given in [@jmp]. In this section, we prove some results suited to our case. First of all,
Let $T$ be a state in $\hi$ and $Q_{T}$ the corresponding POM generated by the representation $\pi$. Then the following conditions are equivalent:
$Q_{T}$ is informationally complete;
if $\W $ is a trace class operator and $\tr{\W \pi(g)T\pi(g^{-1})}=0$ for all $g\in G$, then $\W =0$.
It is known, see for example [@giape], that $Q_T$ is informationally complete if and only if it separates the set of trace class operators. Let $\W$ be a trace class operator, then $\tr{Q_{T}(E)\W}=0$ for any $E\in{\mathcal}B(X)$ if and only if $\tr{\W\pi(s(x))T\pi(s(x)^{-1})}=0$ for $\alpha$-almost all $x\in X$. Observing that $\pi(s(x))T\pi(s(x)^{-1})=\pi(g)T\pi(g^{-1})$ for all $g\in G$ such that $p(g)=x$, this last condition is equivalent to $\tr{\W\pi(g)T\pi(g^{-1})}=0$ for $\mu$-almost all $g\in G$. Since the map $g\mapsto \tr{\W\pi(g)T\pi(g^{-1})}$ is continuous, the lemma is proved.
Let $G_1$ be the commutator subgroup of $G$, [*i.e.*]{} the subgroup of $G$ generated by the elements of the form $ghg^{-1}h^{-1}$, where $g,h\in G$, and assume that there is subspace $\ki$ of $\hi$ such that for all $g\in G_1$ and $v\in\ki$, $\pi(g)v=c(g)v$ where $c$ is a character of $G_1$. Then the following result is obtained, compare with Th. 15 of [@jmp].
\[prop2\] If $T$ is a state such that $T\hi\subset\ki$ and $\tr{T\pi(g)}\neq 0$ for $\mu$-almost all $g\in G$, then $Q_{T}$ is informationally complete.
Let $\W$ be a trace class operator, and consider the decompositions of $T$ and $B$ as given in Section \[s4\], [*i.e.*]{} $T=\sum_i \lambda_i |e_i\rangle\langle
e_i|$ and $\W=\sum_k w_k |u_k\rangle\langle v_k| $ . Since $T\hi\subset\ki$, it follows that $\pi(g)e_i=c(g)e_i$ for all $g\in G_1$. Given $g\in G$, using the orthogonality relations (\[ort1\]), one has & & = \_[i,k]{} \_i w\_k \_\_\
& &= \_[i,k]{} \_i w\_k \_\
&&= \_[i,k]{} \_i w\_k\_X c\_[e\_i,u\_k]{}(s(x))d(x)\
& & = \_[i,k]{} \_i w\_k\_X \_\_d(x)\
& & = \_[i,k]{} \_i w\_k\_X c(s(x)\^[-1]{}g\^[-1]{}s(x)g) \_\_d(x)\
& & =\_X c(s(x)\^[-1]{}g\^[-1]{}s(x)g) d(x), since $\sum_{i,k} \lambda_iw_k \scal{v_k}{\pi(s(x))e_i}\scal{\pi(s(x))e_i}{u_k}$ converges in $L^1(X,\alpha)$ to $\tr{T\pi(s(x)^{-1})\W\pi(s(x))}$, as shown in Section \[s4\], and $c$ is bounded. Hence $$\tr{T\pi(g)}\tr{\W\pi({g^{-1}})} = \int_X c(s(x)^{-1}g^{-1}s(x)g)
\tr{T\pi(s(x)^{-1})\W\pi(s(x))}\,d\alpha(x)$$ and, if $\tr{\W\pi(g)T\pi(g^{-1})}=0$ for all $g\in G$, then $\tr{\W\pi({g^{-1}})}=0$ for $\mu$-almost all $g\in G$. On the other hand, if $\{e_n\}$ is a basis of $\hi$, &=& \_[n,p]{}\
&=& \_[n,p]{} (W\_[e\_p]{}e\_n)(g), where the double series converges in $\hic$. Since the set $\{W_{e_p}e_n\}_{n,p}$ is orthonormal in $\hic$, the condition $\tr{\W\pi({g^{-1}})}=0$ for $\mu$-almost all $g\in G$ implies $\ip{e_n}{\W e_p}=0$ for all $n,p$, [*i.e.*]{} $\W=0$ and this proves that $Q_{T}$ is informationally complete.
[ The condition that $G_1$ is represented by a character is automatically fulfilled (on the whole $\hi$) if $G_1$ is contained in the centre of $G$, whence $\pi|_{G_1}=\chi|_{G_1}$.]{}
[ Suppose $G$ is a Lie group and let $\hi^{\omega}$ be the dense subspace of $\hi$ of analytic vectors for $\pi$. If $T$ has range in $\hi^{\omega}$, then the function $G\ni g\mapsto \tr{T\pi(g)}$ is analytic. This guarantees that $\tr{T\pi(g)}\neq 0$ for $\mu$-almost all $g\in G$.]{}
An example {#s6}
==========
To discuss an example it is convenient to work with another realization of the induced representation $(l,\hic)$.
Let $J$ be the unitary operator from $\hic$ onto $\lta$ given by $$(Jf)(x):=f(s(x)),\ \ \ x\in X.$$ $J$ intertwines the imprimitivity system $(l,P)$ with $(\tilde l,\tilde P)$, where (l(a) f)(x) & = & (s(x)\^[-1]{}as(a\^[-1]{}. x)) f(a\^[-1]{}. x), aG,\
((E) f)(x) & = & \_E(x) f(x), EB(X), with $f\in\lta$.
Given $u\in\hi$, if we compose $W_u:\hi\to\hic$ of Section \[s4\] with $J$ we obtain an operator $\twu:\hi\to\lta$ explicitly given by $$(\twu v)(x) = \cuv(s(x)) = \ip{\pi(s(x))u}{v}_\hi.$$ If $u$ is a unit vector, $\twu$ intertwines the operator measure $Q_u$, defined in Section \[s4\], with the projection measure $\tp$, which is the minimal Neumark dilation of $Q_u$.
We denote by $\tmpi$ the image of $\mpi$ under the map $J$. The analysis of $\mpi$, made in Section \[s5\], can easily be translated into an analysis of $\tmpi$.
$\tmpi$ is a closed subspace of $\lta$, invariant under $\tilde l$.
Let $(e_n)_{n\geq 1}$ be a basis of $\hi$. Then $\tmpi = \oplus_{p\geq 1}\ {\rm ran}\, \widetilde{W}_{e_p}$.
For each $n\geq 1$, $(\tilde l_{{\rm ran}\, \widetilde{W}_{e_n}},{\rm
ran}\, \widetilde{W}_{e_n})$ is equivalent to the irreducible unitary representation $(\pi,\hi)$ of $G$.
For each $n,p\geq 1$, we define $$f_{n,p}(x):= \widetilde{W}_{e_n}e_p.$$ For each $n\geq 1$, $(f_{n,p})_{p\geq 1}$ is a basis of ${\rm ran}\, \widetilde{W}_{e_n}$, and $(f_{n,p})_{n,p\geq 1}$ is a basis of $\tmpi$.
The Heisenberg group
--------------------
We denote by $H^1$ the Heisenberg group. It is $\runo^3$ as a set and we denote its elements by $(t,q,p)$. The product rule is given by $$(t_1,q_1,p_1)(t_2,q_2,p_2)
=
(t_1+t_2+\frac{p_1q_2-q_1p_2}2,q_1+q_2,p_1+p_2).$$ $H^1$ is a connected, simply connected, unimodular Lie group. Its centre is $Z=\{(t,0,0)\,|\, t\in\runo\}$, and the quotient space $X=H^1/Z$ can be identified with $\runo^2$ (with respect to all relevant structures). For the sake of convenience we choose the Haar measures $\mu, \mu_0$, and $\alpha$ on $G$, $Z$, and $X$, respectively, as $\frac 1{2\pi}dtdqdp$, $dt$, and $\frac 1{2\pi}dqdp$. The canonical projection $p:G\to X$ is the coordinate projection $p((t,q,p))=(q,p)$, and we choose the natural, smooth section $s((q,p))=(0,q,p), q,p\in\runo$. With these choices the integral formula of Section \[s1\], which links together the measures $\mu, \mu_0$, and $\alpha$ reads $$\int_{\runo^3} f(t,q,p)\,\frac {dtdqdp}{2\pi} =
\int_{\runo^2}\left( \int_{\runo}f((0,q,p)(t,0,0))\,dt
\right)\,\frac{dqdp}{2\pi},$$ for all $f\in C_c(\runo^3)$, and is simply a consequence of Fubini’s theorem.
Let $\hi$ be a complex separable infinite dimensional Hilbert space, and let $(e_n)_{n\geq 1}$ be an orthonormal basis of $\hi$. There is a natural action of $H^1$ on $\hi$. Let $a,a^*$ denote the ladder operators associated with the basis $(e_n)_{n\geq 1}$, and define Q&=& 1[2]{}(a+a\^\*)\
P&=& 1[i]{}(a-a\^\*) on their natural domains. Then $$(t,p,q)\mapsto e^{i(t+qP+pQ)}$$ is a [*unitary irreducible*]{} representation of $H^1$ on $\hi$. It is the [*only*]{} unitary irreducible representation of $H^1$ whose central character is $t\mapsto e^{it}$, see for instance [@Taylor86]. It is [*unitarily equivalent*]{} to the representation of $H^1$ which acts on $L^2(\runo)$ as $$(\pi(t,q,p)\phi)(x) = e^{i(t+px+qp/2)}\phi(x+q),\ \ \ \phi\in L^2(\runo).$$ We show that $(\pi,L^2(\runo))$ is a representation of $H^1$ that is square integrable modulo the centre $Z$. According to item 1 of section \[s3\], it suffice to show that $c_{\phi,\phi}\circ s\in L^2(\runo^2)$ for some $\phi\in L^2(\runo)$. Explicitly $$c_{\phi,\phi}(s(q,p))= \ip{\pi(s(q,p))\phi}{\phi}=
e^{-i\frac{pq}2}\int e^{-ipx} \overline{\phi(x+q)}\phi(x)\, dx.$$ Choose $\phi\in C_c(\runo)$, then, for any $q\in\runo$ $$\left(x\mapsto \overline{\phi(x+q)}\phi(x) \right)\in\lyr\cap\ltr.$$ Properties of the Fourier transform tell us that $$\left(p\mapsto \int_{\runo}e^{-ipx}\overline{\phi(x+q)}\phi(x)\,dx \right)\in\ltr,\ \ \ q\in\runo.$$ Thus we have, by the Plancherel theorem, $$\int_{\runo} \left|e^{-i\frac{pq}2}\int_{\runo}e^{-ipx}\overline{\phi(x+q)}\phi(x)\,dx\right|^2 dp
=
2\pi \int_{\runo} \left|\overline{\phi(x+q)}\phi(x)\right|^2dx,$$ and, by the Fubini theorem, $$\int_{\runo} \left(2\pi \int_{\runo} |\overline{\phi(x+q)}\phi(x)|^2dx \right) dq= 2\pi\no{\phi}^4_{\ltr}.$$ Tonelli’s theorem tells us that the function $c_{\phi,\phi}\circ s$ is in $L^2(\runo^2)$. Moreover, recalling that $d\alpha=\frac{dq\,dp}{2\pi}$, $$\no{c_{\phi,\phi}\circ s}_{L^2(\runo^2,\alpha)}= \no{\phi}^2_{\ltr}.$$ This shows that $\pi$ is square integrable modulo the centre and that its formal degree is $1$. Since $\pi$ is the only irreducible representation of $H^1$ with the central character $e^{it}$ and it is square integrable modulo the centre we conclude that $$\tmpi = L^2(\runo^2,\alpha),$$ namely, that $(\tilde{l},L^2(\runo^2,\alpha))$ is an isotypic representation. To exhibit this representation, let us observe that the map $\twu:\hi\to L^2(\runo^2,\alpha)$ is given by $$(\twu v)(x,y) = \ip{e^{i(xQ+yP)}u}{v}_\hi.$$ The functions $f_{n,p}$, $p\geq 1$, which constitute a basis of ${\rm ran}\,\widetilde{W}_{e_n}$, are $$f_{n,p}(x,y) = \sqrt{2\pi} \ip{e^{i(xQ+yP)}e_n}{e_p}_\hi.$$ The operator measure $Q_u$ is given by $$\ip{v}{Q_u(E)w}=
\frac 1{2\pi} \int_E\,\ip{v}{\pi(0,q,p)u}_\hi\ip{u}{\pi(0,q,p)^{-1}w}_\hi dqdp,$$ which can be written as $$Q_u(E)= \frac 1\pi\int_E\, D_z|u\rangle\langle u|D_z^{-1}d\lambda(z),$$ where $z=\frac{-q+ip}{\sqrt 2}$, $D_z = e^{it+za^*-\overline{z}a}$, and $\lambda$ is the Lebesgue measure on $\mathbb C$. The action of $\tilde{l}$ on $L^2(\runo^2,\alpha)$ can directly be computed and we get $$(\tilde{l}(t,q,p)\tf)(x,y) = e^{i(t+\frac{xp-yq}2)}\tf(x-q,y-p).$$
As a final remark we note that the commutator group of the Heisenberg group is contained in its center so that if $T$ is a state such that $\tr{T\pi(g)}\neq 0$ for almost all $g\in H^1$, then by Proposition \[prop2\] the operator measure $Q_T$ is informationally complete. This holds, in particular, if the range of $T$ is contained in the subspace of $\hi$ of analytic vectors.
[References]{}
E.B. Davies, [*Quantum Theory of Open Systems*]{}, Academic Press, New York, 1976.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We report the discovery of two, new, rare, wide, double-degenerate binaries that each contain a magnetic and a non-magnetic star. The components of SDSSJ092646.88+132134.5 + J092647.00+132138.4 and SDSSJ150746.48+521002.1 + J150746.80+520958.0 have angular separations of only 4.6 arcsec (a$\sim$650AU) and 5.1 arcsec (a$\sim$750AU), respectively. They also appear to share common proper motions. Follow-up optical spectroscopy reveals each system to consist of a DA and a H-rich high-field magnetic white dwarf (HFMWD). Our measurements of the effective temperatures and the surface gravities of the DA components reveal both to have larger masses than are typical of field white dwarfs. By assuming that these degenerates have evolved essentially as single stars, due to their wide orbital separations, we use them to place limits on the total ages of our stellar systems. These argue that in each case the HFMWD is probably associated with an early type progenitor ($M_{\rm init}$$>$2M$_{\odot}$). We find that the cooling time of SDSSJ150746.80+520958.0 (DAH) is somewhat lower than might be expected had it followed the evolutionary path of a typical single star. This mild discord is in the same sense as that observed for two of the small number of other HFMWDs for which progenitor mass estimates have been made, REJ0317-853 and EG59. The mass of the other DAH, SDSSJ092646.88+132134.5, appears to be smaller than expected on the basis of single star evolution. If this object was/is a member of a hierarchical triple system it may have experienced greater mass loss during an earlier phase of its life as a result of it having a close companion. The large uncertainties on our estimates of the parameters of the HFMWDs suggest a larger sample of these objects is required to firmly identify any trends in their inferred cooling times and progenitor masses. This should shed further light on their formation and the impact magnetic fields have on the late stages of stellar evolution. To serve as a starting point, we highlight two further candidate young, wide magnetic + non-magnetic double-degenerate systems within SDSS, CBS229 and SDSSJ074853.07+302543.5 + J074852.95+302543.4, which should be subjected to detailed (resolved) spectroscopic followed-up studies.'
author:
- |
P. D. Dobbie$^{1}$[^1], R. Baxter$^{2}$, B. Külebi$^{3}$, Q. A. Parker$^{1,2}$, D. Koester$^{4}$, S. Jordan$^{5}$, N. Lodieu$^{6,7}$, F. Euchner$^{8}$\
$^{1}$Australian Astronomical Observatory, PO Box 296, Epping, NSW, 1710, Australia\
$^{2}$Dept. of Physics & Astronomy, Macquarie University, NSW, 2109, Australia\
$^{3}$Institut de Ciències de l$^{\prime}$Espai (CSIC-IEEC), Facultat de Ciències, Campus UAB, Torre C5-parell, 2$^{\rm a}$ planta, 08193 Bellaterra, Spain\
$^{4}$Institut für Theoretische Physik und Astrophysik, Christian-Albrechts-Universität, Kiel, Germany\
$^{5}$Astronomisches Rechen-Institut, Zentrum für Astronomieder Universität Heidelberg, Mönchhofstr. 12–14, D-69120 Heidelberg, Germany\
$^{6}$Instituto de Astrofísica de Canarias, Vía Láctea s/n, E-38200 La Laguna, Tenerife, Spain\
$^{7}$Departmento de Astrofísica, Universidad de La Laguna, E-38205 La Laguna, Tenerife, Spain\
$^{8}$Swiss Seismological Service, ETH Zurich, Schafmattstrasse 30, HPP P3, 8093 Zurich, Switzerland
bibliography:
- 'mnemonic.bib'
- 'therefs.bib'
date: 'Accepted 2011 November 29. Received 2011 November 28; in original form 2011 September 4'
title: 'Two new young, wide, magnetic + non-magnetic double-degenerate binary systems.[^2]'
---
\[firstpage\]
stars: white dwarfs; stars: binaries:general; stars: magnetic field
Introduction
============
[lcccccccc]{} & Name & $u$ & $g$ & $r$ & $i$ & $z$ & $\mu_{\alpha}$cos $\delta$ & $\mu_{\delta}$\
& & &\
SDSSJ092646.88+132134.5 & DAH1 &18.46$\pm$0.02 & 18.34$\pm$0.02 & 18.39$\pm$0.01 & 18.50$\pm$0.02 & 18.60$\pm$0.03 & -8.6$\pm$6.9 & -77.2$\pm$9.6\
SDSSJ092647.00+132138.4 & DA1 &18.74$\pm$0.03 & 18.40$\pm$0.03 & 18.46$\pm$0.05 & 18.60$\pm$0.04 & 18.79$\pm$0.03 &-11.6$\pm$6.9 & -65.3$\pm$9.6\
\
SDSSJ150746.48+521002.1 & DA2 &17.14$\pm$0.02 & 16.91$\pm$0.03 & 17.29$\pm$0.01 & 17.55$\pm$0.02 & 17.84$\pm$0.02 &-30.3$\pm$4.9 & +12.7$\pm$5.8\
SDSSJ150746.80+520958.0 & DAH2 &17.98$\pm$0.03 & 17.76$\pm$0.03 & 18.06$\pm$0.01 & 18.33$\pm$0.02 & 18.55$\pm$0.03 &-31.0$\pm$4.9 & +13.1$\pm$5.8\
\[phot\]
A non-negligible proportion of white dwarfs appear to possess substantial magnetic fields, with strengths typically $>$1MG. A number of studies have determined that they represent between $\sim$5-15 per cent of the white dwarf population yet their origins remain quite unclear [@angel81; @liebert03; @kawka07; @kulebi09]. These are often referred to as the high field magnetic white dwarfs [HFMWDs, e.g. @wickram05]. While the mass distribution of field white dwarfs is found to be strongly peaked around 0.6M$_{\odot}$ [e.g. @liebert05a; @koester09], the mass distribution of the HFMWDs is flatter and skewed towards higher masses, $M$$\sim$0.9M$_{\odot}$ [e.g. @liebert03]. Three of the ten ultramassive ($M$$>$1.1M$_{\odot}$) white dwarfs identified in the extreme ultraviolet surveys appear to be HFMWDs [@vennes99].
At present, there are two principle theories regarding their formation. In the “fossil field” hypothesis the HFMWDs are the descendents of the Ap + Bp stars, a magnetic, chemically peculiar subset of objects with spectral types ranging from late-B to early-F [@angel81]. This is in accord with the similar magnetic fluxes of the HFMWDs and the Ap + Bp stars and with the predicted long decay times of the fields in these objects. Moreover, the higher average mass of the HFMWDs is explained naturally here as a result of the form of the stellar initial-final mass relation, a positive correlation between the main sequence masses of stars and their white dwarf remnant masses [e.g. @weidemann00]. However, in light of more recent results, the proportion of late-B to early-F stars that can be classified as Ap + Bp may be too low by a factor of 2-3 to be consistent with the larger revised estimates of the percentage of HWMWDs in the general white dwarf population [e.g. @kawka03]. To alleviate this apparent shortfall in progenitors, it is required that $\sim$40 per cent of stars with M$>$4.5M$_{\odot}$ also evolve to become HFMWDs [e.g. @wickram05].
Alternatively, [@tout08] have proposed that the magnetic fields of HFMWDs are generated by differential rotation within the common envelope gas which engulfs a primordial close binary system when the primary star expands to giant dimensions and overfills its Roche Lobe. An isolated HFMWD is predicted to form if the cores of the components merge before this envelope is dispersed. However, if the gas is removed prior to this, the outcome is instead expected to be a magnetic-cataclysmic variable. This hypothesis can account for the puzzling lack of detached HFMWD + late-type star binary systems that has emerged from the Sloan Digital Sky Survey [SDSS; e.g. @liebert05c]. It might also explain the reported discrepancies in the cooling times of a small number of the HFMWDs where it has been possible to test them against evolutionary models for single stars [e.g. RE J0317-835 and EG59, @barstow95; @ferrario97; @claver01; @casewell09].
{width="15cm"}
The identification of new HFMWDs where there is the opportunity to set constraints on their masses, cooling times and the ages of their parent populations can further address the questions regarding their origins. Traditionally, white dwarf members of open clusters are used since their masses can be constrained from their observed fluxes using a mass-radius relation [e.g. @casewell09]. Subsequently, their cooling times can be derived and compared to those of the non-magnetic degenerate members for which progenitor masses can be estimated straightforwardly. Unfortunately, less than a handful of HFMWDs have been found in open clusters to date. These have tended to be rather faint due to their substantial distances [e.g. NGC6819-8, $V$$\approx$23.0, @kalirai08] and thus not be particularly amenable to detailed follow-up study.
An alternative approach focuses on field HFMWDs in nearby, wide, double-degenerate systems where the components are sufficiently far apart to have evolved essentially as separate entities yet the system age and distance can be determined from the non-magnetic companion star [e.g. @girven10]. However, only two of these spatially resolved binaries have been identified to date. REJ0317-853 [@barstow95], which resides at d$\sim$30pc [@kulebi10], was discovered in the course of the extreme-ultraviolet all sky surveys undertaken with the [*Röntgensatellit*]{} [e.g. @pye95] and the [*Extreme Ultraviolet Explorer*]{} [e.g. @bowyer96] satellites. It consists of a common proper motion pairing of a $\sim$0.85M$_{\odot}$ DA and an ultramassive HFMWD with a field strength of $B$$\sim$450MG [@ferrario97; @burleigh99] that are separated on the sky by 7 arcsec. The more recently discovered common proper motion system PG1258+593 + SDSSJ130033.48+590407.0 [@girven10] contains a pair of near equal mass ($M$$\approx$0.54M$_{\odot}$) hydrogen rich white dwarfs that are separated on sky by 16 arcsec. The HFMWD component, which has a field strength of $B$$\sim$6MG, is the cooler of the pair. In addition, three unresolved double-degenerate systems containing a magnetic and a non-magnetic object are also currently listed in the refereed literature. LB11146 [@liebert93] is known to be a physically close system [@nelan07] while the orbital separations of the white dwarfs in REJ1439+75 [@vennes99] and G62-46 [@bergeron93] might also be relatively small.
With several large area charged-coupled device (CCD) imaging surveys such as SDSS [@york00], VST ATLAS (http://www.astro.dur.ac.uk/Cosmology/vstatlas/) and SkyMapper [@keller07] having recently been completed or soon to be undertaken, the prospects for unearthing more young, wide, magnetic + non-magnetic white dwarf binaries are excellent. Here we report the discovery and confirmation of two further examples of such systems, SDSSJ092646.88+132134.5 + J092647.00+132138.4 (hereafter, System 1) and SDSSJ150746.48+521002.1 + J150746.80+520958.0 (hereafter, System 2). In the following sections we briefly describe our broader survey for wide double-degenerate binaries and demonstrate that the components of these two new pairings share common proper motions. We perform a spectroscopic analysis of the two white dwarfs in each system to assess their masses and cooling times. Subsequently, we examine the objects in the context of canonical single star evolutionary theory and place constraints on the nature of the progenitors of the HFMWDs. Additionally, we search for evidence that these white dwarfs have more exotic formation histories. We finish by highlighting two further candidate wide, magnetic + non-magnetic double-degenerate binaries which maybe suitable for this type of analysis. These can serve as a starting point for the construction of an enlarged sample of these systems that will be neccessary to better understand the formation of HFMWDs.
Discovery of young, wide, magnetic + non-magnetic white dwarf binaries.
=======================================================================
Imaging search for wide, double-degenerate systems. {#survey}
---------------------------------------------------
We have been conducting a survey for young, wide, double-degenerates using imaging and spectroscopic data from the SDSS. The full details of this study will be described in a forthcoming paper (Baxter et al. in prep) but we provide a brief outline of our approach here so that this result can be placed into context. We selected from DR7 (the 7th SDSS data release) all point sources flagged as photometrically clean with $r$$\le$20.0, $u$-$g$$\le$0.5, $g$-$r$$\le$0.0 and $r$-$i$$<$0.0 (corresponding to white dwarfs with $T_{\rm eff}$$\simgreat$9000K [e.g. @eisenstein06]) which have another object satisfying these colour-magnitude criteria within 30 arcsec. The resulting candidate systems were visually inspected using the SDSS finder chart tool to weed out a number of spurious pairings (e.g. blue point-like detections within resolved galaxies). This procedure led to the identification of 52 candidate systems, including the previously known double-degenerate binaries, PG0922+162 [@finley97] and HS2240+1234 [@jordan98]. The associated SDSS spectroscopy, which is available for 21 objects in 19 of these pairs, reveals 17 DAs, 1 DB and 3 DCs but no quasars or subdwarfs suggesting that contamination levels in this sample are low. We have obtained long-slit spectroscopic follow-up data for 13 additional candidate binaries to date and all have turned out to be comprised of white dwarfs, confirming the low level of contamination.
System 1 and System 2 are two of these 13 pairings which both appear to harbour a HFMWD. Their components are separated on the sky by only 4.35 arcsec and 5.05 arcsec, respectively. Their $u$, $g$, $r$, $i$ and $z$ magnitudes are listed in Table \[phot\]. To ascertain the likelihood of chance alignments we have used Equation 1 [@struve52], under the assumption of a random on-sky distribution of objects, where $N$ is the number of sources satisfying the photometric selection criteria in area $A$ (square degrees) and $\rho$ is the maximum projected separation (degrees).
$$\begin{aligned}
n(\le\rho)= N (N-1) \pi \rho^{2} / 2 A\end{aligned}$$
We estimate $n$(4 arcsec$<$$\rho$$\le$6 arcsec)$\sim$0.3 for $N$=36231 and $A$=11663 square degrees. In the course of our survey we have unearthed a total of 10 pairs of objects with separations in this range, suggesting that the probability of any one being a mere chance alignment is P$\sim$0.03.
Astrometric follow-up and proper motions
----------------------------------------
To probe the reality of the putative associations between the two sources in each of these pairs, we have examined their relative proper motions. We note that [@girven10] used proper motions to confirm the association of PG1258+593 + SDSSJ130033.48+590407.0 in their recent exploration of the bottom end of the IFMR. In principle, measurements of this nature can be obtained from one of a number of online databases featuring imaging of widely separated epochs e.g. the United States Naval Observatory B1.0 catalogue [USNO-B1.0; @monet03], SDSS [e.g. @munn04], the SuperCOSMOS Sky Survey [SSS; @hambly01a] and the extended Position and Proper Motion catalogue [PPMXL; @roeser10]. However, these resources rely heavily on comparatively low spatial resolution photographic plate exposures. Examination of the values reported in these catalogues indicates that the small separations of the components of our candidate binaries have compromised their accuracy. For example, the astrometry reported in the PPMXL catalog suggests that the relative positions of the objects in these two pairs should have changed by several arc seconds in the $\sim$50 years between the imaging of the first Palomar Sky Survey and SDSS. However, visual inspection of these data reveals no discernable difference. Therefore, we have opted to perform our own astrometric measurements of System 1 and System 2. For the first system we have adopted the $g$ band data from SDSS (2006/01/06) and a $B$ band acquisition frame (2010/02/06) from our spectroscopic follow-up program on the European Southern Observatory’s (ESO) Very Large Telescope (VLT) as first and second epoch images respectively. We have used [[SExtractor]{} @bertin96] to determine the positions of $\sim$10 unblended, stellar-like objects lying in close proximity to the candidate white dwarfs. As no suitable second CCD image of System 2 is available to us we have resorted to using data from the first Palomar Sky Survey (Plate 2376 observed on 1954/06/28) and the SDSS $r$ band imaging (2002/06/09) as our first and second epoch images of this system, respectively. Here we have determined the positions of unsaturated stellar-like objects lying within a few arcminutes of the candidate white dwarfs. In detecting these stars and determining their centroids in the photographic data, no smoothing filter was applied to the image (to minimise blending of the stellar profiles) and only those pixels substantially above the background ($\simgreat$5$\sigma$) were included in the calculations.
![VLT + FORS2 spectroscopy of SDSSJ092646.88+132134.5 (upper) and SDSSJ092647.00+132138.4 (lower) with SDSS $u$, $g$, $r$, $i$ and $z$ fluxes (open grey circles) and synthetic spectra (red line) overplotted. The spectrum of SDSSJ092647.00+132138.4 is clearly that of an H-rich white dwarf, while that of SDSSJ092646.88+132134.5 displays several weaker features which are consistent with magnetically broadened and shifted Balmer lines. []{data-label="specs1"}](WD0926.ps){width="\linewidth"}
For each pair of objects we have cross matched the lists of reference star positions using the software. Subsequently, we have employed routines in the library to construct six co-efficient linear transforms between the two images of the putative systems, where $>$3$\sigma$ outliers were iteratively clipped from the fits. The proper motions, in pixels, were determined by taking the differences between the observed and calculated locations of candidates in the 2nd epoch imaging. These were then converted into milli-arcseconds per year in right ascension and declination using the world co-ordinate systems of the 1st epoch datasets and dividing by the time baseline between the two observations, $\sim$4.08yr for System 1 and $\sim$47.95yr for System 2 (Table \[phot\]). The uncertainties on these measurements were estimated from the dispersion observed in the (assumed near-zero) proper motions of stars of comparable brightness surrounding each system. The relative proper motion vector point diagrams for the objects (solid triangles) are shown in Figure \[VPD\].
![WHT + ISIS spectroscopy of SDSSJ150746.48+521002.1 (lower) and SDSSJ150746.80+520958.0 (upper) with SDSS $u$, $g$, $r$, $i$ and $z$ fluxes overplotted (open grey circles). The spectrum of SDSSJ150746.48+521002.1 is clearly that of an H-rich white dwarf, while that of SDSSJ150746.80+520958.0 displays several weaker features which are consistent with magnetically broadened and shifted Balmer lines. []{data-label="specs2"}](WD1507.ps){width="\linewidth"}
[clcrccc]{} SDSS & $T_{\rm eff}$$^{*}$ & log $g$$^{*}$ & M$_{r}$ & $r$-M$_{r}$ & M(M$_{\odot}$) & $\tau_{c}$ (Myr)\
& $10482^{+47}_{-47}$ & $8.54^{+0.03}_{-0.03}$ &$12.57^{+0.17\ddagger}_{-0.17}$ & 5.89$\pm$0.18$^{\ddagger}$ & 0.79$\pm$0.06$^{\ddagger}$ & 833$^{+123\ddagger}_{-123}$\
\
& $17622^{+99}_{-94}$ & $8.13^{+0.02}_{-0.02}$ & $11.38^{+0.11}_{-0.11}$ & 5.91$\pm$0.11 & 0.70$\pm$0.04 & 147$^{+24}_{-21}$\
\[temps\]
$^{*}$ Formal fit errors.\
$^{\ddagger}$ Adjusted to account for spectroscopic overestimate of mass.
Inspection of these plots reveals that our proper motion measurements of the components of the pairs are significant at $>$3$\sigma$ and consistent with each other within their 1$\sigma$ uncertainties. For each case we have estimated the probability that this similarity could have occurred merely by chance. We have selected all spectroscopically confirmed white dwarfs within 10$^{\circ}$ of these systems from the SDSS DR4 white dwarf catalogue [@eisenstein06] which meet the photometric selection criteria outlined in Section \[survey\]. We have cross-correlated these with the SSS database to obtain their proper motions. After cleaning these samples for objects with poorly constrained astrometry (e.g. $\chi^{2}$$>$3 or flagged as bad and/or proper motion uncertainties $>$7.5mas yr$^{-1}$), we are left with 59 and 159 white dwarfs in the general directions of System 1 and System 2, respectively. An examination of this astrometry reveals that none and seven sources in these samples have proper motions that could be deemed as consistent (within 2$\sigma$ of the mean proper motion of the components of the putative system) with System 1 and System 2, respectively. Therefore we estimate the probabilities that the proper motions of these objects are similar by chance are less than 0.02 and 0.05 respectively. Combined with the likelihoods of chance projected angular proximity we find the probabilities of these two systems merely being optical doubles are less than P$\sim$0.0006 and P$\sim$0.0015 giving a potent argument in favour of their association.
Spectroscopic analysis
======================
Follow-up optical spectroscopy of the binary components. {#followup}
--------------------------------------------------------
We have obtained optical spectroscopy of System 1 in visitor mode with the ESO Very Large Telescope and the Focal Reducer and low dispersion Spectrograph (FORS2). A full description of the FORS2 instrument may be found on the ESO webpages[^3]. These observations (1$\times$360s and 1$\times$500s exposures) were conducted on the night of 2010 February 6 when the seeing was good but there was some cirrus scattered across the sky. All data were acquired using the 2$\times$2 binning mode of the $E2V$ CCD, the 600B+24 grism and a 1.3 arcsec slit which gives a notional resolution of $\lambda$/$\Delta$$\lambda$$\sim$600. The components of System 2 were observed with the William Herschel Telescope (WHT) and the double-armed Intermediate dispersion Spectrograph and Imaging System (ISIS) on the night of 2008 July 24. These observations were conducted when the sky was photometric, with seeing $\sim$0.6-0.9 arcsec. The spectrograph was configured with a 1.0 arcsec slit and with the R300B ($\lambda$/$\delta\lambda$$\approx$2000) and the R1200R ($\lambda$/$\delta\lambda$$\approx$10000) gratings on the blue and red arms, respectively. Three 1800s exposures covering the two wavelength ranges, $\lambda$$\approx$3600-5200Å and 6200-7000Å, were obtained simultaneously.
The CCD frames were debiased and flat fielded using the IRAF procedure CCDPROC. Cosmic ray hits were removed using the routine LACOS SPEC (van Dokkum 2001). Subsequently, the spectra were extracted using the APEXTRACT package and wavelength calibrated by comparison with a CuAr+CuNe arc spectrum taken immediately before and after the science exposures (ISIS) or with a He+HgCd arc spectrum obtained within a few hours of the science frames (FORS2). The removal of remaining instrument signature from the science data was undertaken using observations of the bright DC white dwarfs WD1918+386 (ISIS) and LHS2333 (FORS2). The spectra of the components of System 1 and System 2 are shown in Figures \[specs1\] and \[specs2\].
Effective temperatures and surface gravities {#tandg}
--------------------------------------------
The optical energy distributions of SDSSJ092647.00+132138.4 (hereafter, DA1) and SDSSJ150746.48+521002.1 (hereafter, DA2) each display broadened H-Balmer line series and these objects are unmistakably hydrogen rich white dwarfs. In contrast, no prominent features at the notional wavelengths of the lines of either [HI]{} or [HeI]{} are observed in the spectra of SDSSJ092646.88$+$132134.5 (hereafter, DAH1) and SDSSJ150746.80$+$520958.0 (hereafter, DAH2). However, neither dataset is completely smooth like the spectrum of a DC white dwarf and instead they display a number of shallow depressions across the observed wavelength range reminiscent of a H-rich HFMWD such as SDSSJ172045.37+561214.9 [@gaensicke02].
To determine the effective temperatures and the surface gravities of DA1 and DA2 we have compared the observed Balmer lines, H-8 to H-$\beta$, to a grid of synthetic profiles [e.g. @bergeron92]. These are based on recent versions of the plane-parallel, hydrostatic, local thermodynamic equilibrium (LTE) atmosphere and spectral synthesis codes [ATM]{} and [SYN]{} [e.g. @koester10], which include an updated physical treatment of the Stark broadening of [HI]{} lines [@tremblay09]. The fitting procedure has been performed with the spectral analysis package [XSPEC]{} [@shafer91]. [XSPEC]{} folds a model through the instrument response before comparing the result to the data by means of a $\chi^{2}-$statistic. The best model representation of the data is found by incrementing free grid parameters in small steps, linearly interpolating between points in the grid, until the value of $\chi^{2}$ is minimised. Formal errors in the $T_{\rm eff}$s and log $g$s are calculated by stepping the parameter in question away from its optimum value and redetermining minimum $\chi^{2}$ until the difference between this and the true minimum $\chi^{2}$ corresponds to $1\sigma$ for a given number of free model parameters [e.g. see @lampton76]. It is important to be aware that these errors do not take into account shortcomings in the models or systematic issues affecting the data (e.g. flaws in the flat fielding) so undoubtedly underestimate the true uncertainties in the parameters. Therefore in our subsequent analysis we assume more realistic levels of uncertainty of 2.3 per cent and 0.07dex in effective temperature and surface gravity, respectively [e.g. see @napiwotzki99]. The results of our line fitting are shown in Table \[temps\]. Synthetic colours for H-rich white dwarfs at these effective temperatures and surface gravities [e.g. @holberg06] appear to be broadly consistent with the SDSS photometry for DA1 and DA2 (Table \[phot\]).
SDSS $T_{\rm eff}$ $B_{\rm dip}$(MG) $z_{\rm off}({\rm R}_{\rm WD})$ inclination (${}^\circ$) $M$(M$_{\odot}$) $\tau_{c}$ (Myr)
----------------------- ---------------- ------------------- --------------------------------- -------------------------- ------------------ ---------------------
J092646.88$+$132134.5 9500$\pm$500 210$\pm$25.1 -0.09$\pm$0.01 21.8$\pm$7.9 0.62$\pm$0.10 726$^{+140}_{-107}$
J150746.80$+$520958.0 18000$\pm$1000 65.2$\pm$0.3 -0.39$\pm$0.03 36.4$\pm$4.1 0.99$\pm$0.05 321$^{+47}_{-40}$
\[magnetic\]
The energy distributions of the DAH components DAH1 and DAH2 have been compared to a grid of model spectra for HFMWDs. These were calculated with a radiative transfer code for magnetised, high gravity atmospheres. The code calculates both theoretical flux (Stokes $I$) and polarization (Stokes $V$) spectra, for a given temperature and pressure structure ($T_{\rm eff}$, $\log g$) and a specific magnetic field vector with respect to the line of sight and the normal on the surface of the star [see @jordan92; @jordan93]. However, as no polarization information is available from our datasets, our analysis has been limited to the flux spectra (Stokes parameter $I$). For computational efficiency, we have made use of our existing three-dimensional grid of synthetic spectra. This grid spans the effective temperature range $7000\,{\rm K}\le T_{\rm eff} \le 50000\,{\rm K}$ in 14 steps, treats magnetic field strengths between $1\,{\rm MG}\le B\le 1.2\,{\rm GG}$ in 1200 steps and has 17 different values of the field direction $\psi$ relative to the line of sight, as the independent variables (9 entries, equally spaced in $\cos \psi$). All spectra were calculated for a surface gravity of log $g$=8 [see @kulebi09]. Limb darkening is accounted for by a simple linear scaling law [see @euchner02].
The magnetic field geometry of the DAHs has been determined using a modified version of the code developed by @euchner02. This code calculates the total flux distribution for an arbitrary magnetic field topology by adding up appropriately weighted synthetic spectra for a large number of surface elements and then evaluates the goodness of fit. For this work we have used a simple model for the magnetic geometry, namely a dipole with an offset on its polar axis. This is the most parameter efficient way of generating non-dipolar geometries and is representative of dipole plus quadrupole configurations. Moreover, these models are ubiquitous in the diagnosis of single phase HFMWD spectra [@kulebi09]. The resulting magnetic parameters of our fits are summarized in Table \[magnetic\]. As explained in @euchner02, these values might not be unique and it is possible to fit similar spectra with different models. However this analysis is fully satisfactory in the context of this work since we are interested in a good spectral model of a given object for determining the temperature. We find that the magnetic field geometries of these two HFMWDs are quite different. While the field of DAH1 does not deviate significantly from dipolarity, in the case of DAH2 the offset parameter is quite large and the magnetic field distribution used in the construction of the fitted spectrum departs substantially from a simple dipole with field strengths of between 45-317MG.
After obtaining a satisfactory representation of the magnetic field structures, the effective temperature of each star has been assessed from the line strengths and the continuum [e.g. @gaensicke02]. These are also shown in Table \[magnetic\]. While in some cases the magnetic analysis can be hindered by the lack of a realistic and easily applicable theory for the simultaneous impact of Stark and Zeeman effect on the spectral lines, this is not a problem for the field strengths relevant to this work ($B>50$MG).
White dwarf masses and cooling times
------------------------------------
We have used our measurements of the effective temperatures and the surface gravities of DA1 and DA2, in conjunction with the evolutionary models of [@fontaine01], to estimate their masses and cooling times. For consistency with our previous work and a large number of other recent studies [e.g. @dobbie09a; @williams09; @kalirai08; @liebert05a] we have adopted the calculations which include a mixed CO core and thick H surface layer structure. It is important to note that DA1 has an effective temperature of $T_{\rm eff}$$<$12000K. Spectroscopic mass determinations are known to be systematically larger in this regime than at higher effective temperatures where they agree well with those derived from gravitational redshifts [e.g. @bergeron95; @reid96]. This trend is most likely due to shortcomings in the treatment of convection within the model atmosphere calculations [@koester10]. Based on the studies of [@tremblay11], [@koester09] and [@kepler07], we have estimated the size of this effect at $T_{\rm eff}$=10500K to be $\Delta$M=+0.16$\pm$0.04M$_{\odot}$. The properties reported in Table \[temps\] reflect a downward adjustment of this size to the estimated mass of DA1. We have used the grids of synthetic photometry of [@bergeron95], which have been updated by [@holberg06], to derive the absolute $r$ magnitudes of DA1 and DA2 and to determine the distance moduli of their host binary systems (Table \[temps\]). We have neglected foreground extinction in these directions as dust maps suggest A$_{V}$$<$0.1 integrated along these sight lines through the Galaxy [@schlegel98].
To evaluate the masses of DAH1 and DAH2, we first assumed that they reside at the same distance as their non-magnetic companions. We then determined the radius of each HFMWD through scaling it’s distance by the square root of the estimated flux ratio, (f/F)$^{0.5}$, where f is the observed flux at the Earth’s surface and F is the flux at the surface of the white dwarf. The surface flux in the SDSS filters [$r$, $i$ and $z$; e.g. @fukugita96] for each object was derived from the non-magnetic, pure-H model white dwarf atmospheres of [@holberg06], where cubic splines were used to interpolate between points in this grid. Next we used the mass-radius relations for non-magnetic DA white dwarfs predicted by the evolutionary models of [@fontaine01] to obtain estimates of the masses of DAH1 and DAH2 of $M$=0.62$\pm$0.10M$_{\odot}$ and $M$=0.99$\pm$0.05M$_{\odot}$, respectively. Finally, we used the evolutionary models to determine their cooling times to be $\tau$$_{\rm cool}$=726$_{-107}^{+140}$Myr and $\tau$$_{\rm cool}$=321$_{-40}^{+47}$ Myr, respectively. The uncertainties in all derived parameters here were determined using a Monte-Carlo like approach in which we created 25000 realisations of each binary system under the assumption that the adopted errors on the effective temperatures (2.3 per cent), the surface gravities (0.07 dex) and the observed magnitudes (Table \[phot\]) of the component stars are normally distributed. We also noted that the SDSS magnitudes (Table \[phot\]) have an absolute precision of 2 per cent [@adelman_mccarthy08].
The progenitors of the HFMWDs {#disc1}
=============================
In wide double-degenerate systems
---------------------------------
[ccccccc]{} Component & & &\
SDSS & $M$$_{\rm init}$ (M$_{\odot}$) & System age (Myr) & $M$$_{\rm init}$ (M$_{\odot}$) & System age (Myr) & $M$$_{\rm init}$ (M$_{\odot}$) & System age (Myr)\
&3.06$^{+0.66}_{-0.58}$ & 599$^{+357}_{-181}$ & 2.76$^{+0.50}_{-0.47}$ & 753$^{+390}_{-217}$ & 2.76$^{+0.37}_{-0.36}$ & 754$^{+259}_{-167}$\
\
& 5.30$^{+0.86}_{-0.75}$ & 426$^{+67}_{-44}$ & 5.50$^{+0.62}_{-0.59}$ & 416$^{+55}_{-38}$ &5.07$^{+0.41}_{-0.42}$ & 437$^{+52}_{-37}$\
\[progcool1\]
From their observed positions on the sky and the distance moduli in Table \[temps\], we determine that the components of System 1 and System 2 have minimum separations of a$\sim$650AU and a$\sim$750AU, respectively. Even assuming that the original orbital separations of these binaries were substantially smaller [e.g. @valls88], the Roche lobes of their components have likely always been much larger than the dimensions of an asymptotic giant branch (AGB) star [r$\simless$4-5AU; @ibenlivio93]. Thus this work doubles the number of known wide, magnetic + non-magnetic double-degenerate binaries, where through their large orbital separations the components could be expected to have evolved essentially as single stars.
Intruigingly, in three of these four pairings now known, the non-magnetic component has a substantially greater mass than the value observed at the prominent peak in the field white dwarf mass distribution [e.g. $M$$\sim$0.6M$_{\odot}$; @bergeron92; @kepler07]. The intermediate temperatures ($T_{\rm eff}$=10500-16000K) of these three non-magnetic white dwarfs, coupled with their comparatively high masses link their HFMWD companions to an early-type stellar population. For example, DA2 has a mass of $M$$\approx$0.66-0.74M$_{\odot}$ and a corresponding cooling time of $\tau$$\approx$126-171Myr. Assuming it has evolved essentially as a single object from a star with an initial mass of $M_{\rm init}$$\approx$2.3-3.7M$_{\odot}$ [e.g. @williams09; @kalirai08; @dobbie06a], allowing for a stellar lifetime as predicted by the solar metalicity model grid of [@girardi00], the total age of the binary is likely to be $\tau$$\simless$1150Myr. Thus the formation of DA2 appears to be associated with a star of initial mass $M_{\rm init}$$>$2.2M$_{\odot}$. Following a similar line of reasoning, from the mass and cooling time of DA1 we infer the age of the host system to be $\tau$$\simless$1300Myr which corresponds to the lifetimes of stars with initial masses, $M_{\rm init}$$>$2.1M$_{\odot}$.
Due to our adopted colour selection criteria, our search for wide double-degenerate binaries is sensitive only to relatively recently formed white dwarfs. For example, a 0.6M$_{\odot}$ white dwarf cools to $T_{\rm eff}$$\sim$9000K [$g$-$r$$\approx$0.0; @holberg06] in only 800Myr. Assuming single star evolution, a massive white dwarf companion to a degenerate formed from a sufficiently long lived progenitor ($M_{\rm init}$$\simless$1.6M$_{\odot}$) will generally always have cooled below our photometric colour limits before the latter has formed. Thus the detection of systems where the components have quite different evolutionary timescales is somewhat disfavoured. If HFMWDs are frequently associated with relatively short main sequence lifetimes, we might expect in a “blue” colour-selected survey a low probability of finding them paired with white dwarfs which have “average” masses ($M$$\sim$0.6M$_{\odot}$). Interestingly, the DA in the fourth pairing which lies within the SDSS DR7 footprint but which failed our survey selection criteria on the grounds of one component being too red (the DAH), has a comparatively low mass, $M$=0.54M$_{\odot}$. The large difference of $\sim$1.6Gyr between the cooling times of PG1258+593 and SDSSJ130033.48+590407.0 means that, within the measurement uncertainties, this HFMWD could still be the progeny an early-type star with $M_{\rm init}$$\approx$2-3M$_{\odot}$ [@girven10].
![SDSS $z$ band image of two candidate spatially resolved magnetic + non-magnetic double-degenerate systems, CBS229 [@gianninas09] and SDSSJ074853.07+302543.5. Images are approximately 1’$\times$1’ with N at the top and E to the left.[]{data-label="newcands"}](DAH+D.ps){width="8.25cm"}
In closer or less well characterised systems
--------------------------------------------
[ccccccc]{} Component & & &\
SDSS & $M$$_{\rm init}$ (M$_{\odot}$) & System age (Myr) & $M$$_{\rm init}$ (M$_{\odot}$) & System age (Myr) & $M$$_{\rm init}$ (M$_{\odot}$) & System age (Myr)\
& 3.73$^{+0.78}_{-0.69}$ & 1092$^{+202}_{-94}$ & 3.61$^{+0.63}_{-0.65}$ & 1115$^{+178}_{-58}$ & 3.48$^{+0.47}_{-0.51}$ & 1146$^{+134}_{-42}$\
\
& 2.49$^{+0.92}_{-0.85}$ & 1531$^{+1334}_{-372}$ & 2.08$^{+0.96}_{-0.92}$ & 2072$^{+4464}_{-781}$ & 2.18$^{+0.77}_{-0.77}$ & 1892$^{+2215}_{-549}$\
\[progcool2\]
Three further hot ($T_{\rm eff}$$\simgreat$9000K) magnetic + non-magnetic binaries have been identified, and spectroscopically confirmed, to date (the DAH component of G62-46 has only $T_{\rm eff}$$\sim$6000K). The components of at least one of these systems, LB11146 (PG0945+245), are separated by only a$\sim$0.6AU [@nelan07]. As this is less than the radius of an AGB star, it is likely they have interacted during prior phases of their evolution. Detailed analysis of this binary has revealed it to consist of a $T_{\rm eff}$$\sim$14500K, $M$$\approx$0.9M$_{\odot}$ DA and a similarly hot, massive magnetic white dwarf with a field strength $B$$\simgreat$300MG [@liebert93]. 2REJ1440+750 (EUVEJ1439+75.0) was shown, through a detailed spectroscopic (and imaging) analysis, to consist of a $M$$\sim$0.9M$_{\odot}$ DA and a $M$$\sim$1.0M$_{\odot}$ DAH ($B$$\sim$14-16MG), each having $T_{\rm eff}$$>$20000K, and with a projected orbital separation of a$\simless$250AU [@vennes99]. CBS229 was identified as an unresolved DA+DAH binary during the course of a spectroscopic survey of bright white dwarfs ($V$$\le$17.5) drawn from the catalogue of [@mccook99]. [@gianninas09] have performed a preliminary analysis of a composite spectrum of this pair and find that the non-magnetic component has $T_{\rm eff}$$\approx$15000K and log $g$$\approx$8.5, corresponding to a mass of $M$$\approx$0.9M$_{\odot}$. The shape of their deblended spectrum of the DAH suggests that the two objects have similar effective temperatures. Our examination of the SDSS imaging reveals that the components are in fact resolved into two photocenters with a projected separation of $\sim$1.3 arcsec (see Figure \[newcands\], left). From the [@gianninas09] parameters for the DA and its magnitude from the SDSS $z$ imaging (the band in which the objects are most clearly resolved), we provisionally estimate a distance to this binary of d$\sim$140pc and a projected orbital separation of a$\sim$180AU. Thus CBS229 appears to be a wide magnetic + non-magnetic double-degenerate system which escaped detection by our survey. This is probably due to the $u$ band magnitude measurement for the NE component which appears to be anomalous.
Although it is possible that past mass exchange within at least the first of these binaries has influenced the characteristics of their white dwarfs, as is observed in three out of four of the confirmed wide systems, the non-magnetic components in each of LB11146, EUVEJ1439+75.0 and, provisionally, CBS229 also appear to have substantially greater masses than are typical of field degenerates. This is consistent with their magnetic white dwarf companions being related to an early-type stellar population.
Cooling times of the HFMWDs and canonical stellar evolution.
============================================================
The DAH SDSSJ150746.80+520958.0
-------------------------------
DA2 has a mass and a cooling time which are comparable to several degenerate members of the Hyades [e.g. WD0352+098, WD0421+162, @claver01]. This suggests that the host binary system is likely to have a total age which is similar to this cluster [$\tau$=625$\pm$50Myr, @perryman98]. We have made detailed estimates of this age using our determinations of the progenitor masses of DA2 and DAH2 (obtained from three recent, independent, derivations of the IFMR), the stellar lifetimes as predicted by the solar metalicity models of [@girardi00] and by assuming standard single star evolution (Table \[progcool1\]). We find that our estimates are only formally consistent within their quoted (1$\sigma$) error bounds when the oldest, least well constrained of the three approximations to the IFMR [@dobbie06a] is adopted. When either of the two other IFMRs is assumed, the age derived from the HFMWD is lower than that obtained from the DA.
This discord is not statistically significant alone but it is notable for being in the same sense as seen for REJ0317-853 [@ferrario97] and the Praesepe cluster HFMWD, EG59 [@claver01]. It has been proposed that the age paradox of the REJ0317-853 + LB9208 system is due to the former component having formed through the merging of the white dwarf progeny of two stars of more modest initial mass (than is assumed in the case of single star evolution). This could also explain why this HFMWD is observed to rotate with a relatively short period of only 725sec [@ferrario97]. More recently, [@kulebi10] have argued that the cooling time of this HFMWD appears at odds with that of LB9208 only because the mass of the former has been slightly underestimated ($\sim$5 per cent) due to evolutionary models neglecting the effects of the magnetic field on the structure of the white dwarf. However, our revised calculations suggest that this earlier conclusion is erroneous and that these structural effects are unable to fully account for the age discprepancy. In [@kulebi10], the tables of [@holberg06] were extrapolated to estimate the new cooling ages but effectively considered the wrong radii hence incorrect luminosities for the cooling. Here we instead consider a simple approach using Mestel’s equation [see @shapiro83] for a half carbon half oxygen white dwarf, which is a good approximation especially before the onset of crystallization (Equation \[mestel\]),
$$\tau_{\rm cool} \approx 1.1\times10^7 \ \left(\frac{M}{M_{\odot}} \right)^{5/7}
\left(\frac{L}{L_{\odot}} \right)^{-5/7} \ {\rm years}
\label{mestel}$$
where $\tau_{\rm cool}$ is the cooling age, $M$ is the mass of the star in solar masses and $L$ is its luminosity in solar luminosity. Given that luminosity is constant, any difference in mass ($\Delta M$) causes an underestimation of cooling age, linearly (Equation \[linear\]).
$$\frac{\Delta \tau_{\rm cool}}{\tau_{\rm cool}} = \frac{5}{7}\frac{\Delta M}{M}
\label{linear}$$
Hence if the mass of REJ0317-853 is underestimated by $\sim$ 5 per cent, the underestimation of the cooling age would be only $\sim$ 3.5 per cent, which is far smaller than the observed $\sim$ 30 per cent.
It should be noted that our calculation is based on the assumption that the radius is inflated by 8-10 per cent. It is possible that the stellar interior contains a greater level of magnetic energy which might have an even stronger influence on the structure, especially if this approaches 10 per cent of the gravitational binding energy as suggested by [@ostriker68]. However, this supposition was not based on quantitative considerations of the stellar structure. Recently, [@reisenegger09] investigated the magnetic structure of non-barotropic stars and put an upper limit on the internal magnetic energy which can be supported. This value is limited by the entropy of the star. For white dwarfs it is sufficient to consider only the ions hence entropy is directly related to the core temperature. A 1.1$M_{\odot}$ white dwarf with an effective temperature of 18000K can support a magnetic energy at most $\sim$ 2.4 per cent of its gravitational binding energy. In this case the radius is 16 per cent larger, which corresponds to that of a 1.0$M_{\odot}$ non-magnetic white dwarf. Thus the most extreme mass underestimation expected for DAH2 is 10 per cent which translates to a cooling age increase of only $\sim$7 per cent, compared to an observed discrepancy of perhaps $\sim$50 per cent.
----------- ----------------- -------------- ---------------------- -------------- ----------------------- ----------------- ----------------- ---------------- -- --
SDSS M$_{\rm WD}$ R$_{\rm WD}$ H$\alpha$ shift $v$ rv$_{\rm WD}$ U V W
0.695$\pm$0.043 11.9$\pm$0.6 19.1$^{+2.8}_{-4.1}$ 37.1$\pm$4.0 -18.0$^{+4.9}_{-5.7}$ $-$20.6$\pm$3.0 $-$20.8$\pm$2.8 $-$6.4$\pm$4.6
\[rvfit\]
----------- ----------------- -------------- ---------------------- -------------- ----------------------- ----------------- ----------------- ---------------- -- --
The DAH SDSSJ092646.88+132134.5
-------------------------------
We have similarly estimated the progenitor masses of DA1 and DAH1 and calculated the total age of this other new binary, under the assumption of standard single star evolution. These two sets of estimates, which are shown in Table \[progcool2\], are not formally consistent within their 1$\sigma$ errors for any of our three adopted IFMRs. However, as in the case of our other binary system, the discrepancy is not overwhelming, statistically. The variance suggested here is in the opposite sense to that observed for the three HFMWDs discussed previously, with DAH1 appearing too old for its mass or, alternatively, too low in mass for its cooling time. It could be expected, on the basis of single star evolution, that since the white dwarfs in this system have similar cooling times, they should have comparable masses, having descended from stars with similar initial masses and lifetimes. DAH1 may have endured greater mass loss than assumed for a single star. This could be a consequence of having a close companion. Studies of stellar multiplicity have revealed that at least 10 per cent of stars are members of triple or higher order systems [@raghavan10; @abt76]. For reasons of dynamical stability, these systems are frequently hierarchically structured with triples often consisting of a body in a relatively wide orbit around a much closer pairing [@harrington72]. DA1 and DAH1 perhaps trace what was the wider orbit of a putative triple system, with the latter object possibly having been (or perhaps still being) part of a tighter pairing. A $M$$_{\rm init}$$\sim$3.5M$_{\odot}$ star that experiences Roche Lobe overflow around the time of central helium ignition can lead to the formation of a CO white dwarf with a mass towards the lower end of the range estimated for this HFMWD [e.g. @iben85]. However, we note that an observed lack of HFMWDs with close detached companions [@liebert05c] has been one of the main arguments that these stars are formed through close binary interaction [@tout08]. Whether or not this mechanism was required for the manufacture of either of the two HFMWDs identified in this work, the parameters of their DA companions argue that they are associated with an early type stellar population.
The sizeable uncertainties associated with the parameters of the HFMWDs highlight the need to expand substantially the sample of those which are members of either nearby star clusters or wide binary systems so that we can begin to firmly identify any trends in their cooling times and inferred progenitor masses, relative to non-magnetic white dwarfs. For now, a relatively straightforward but useful exercise would involve photometrically monitoring these two new systems to search for short period variability that may reveal evidence which more closely links their evolution and that of the REJ0317-853 + LB9208 system.
The space velocity of SDSSJ150746.48+521002.1
=============================================
![The results of our fitting of a synthetic profile (black line) to the central portions of the observed H-$\alpha$ Balmer line of SDSSJ150746.48+521002.1 (grey lines). The flux$_{\lambda}$ units are arbitrary.[]{data-label="halp"}](H-ALP.ps){width="8cm"}
We have exploited the higher resolution spectroscopy we have in hand for DA2 to determine the radial velocity of this system from the observed shift of the H$\alpha$ line core. We have removed the effects of telluric water vapour from our red arm ISIS data using a template absorption spectrum. The observed H$\alpha$ line and the profile from a non-LTE synthetic spectrum corresponding to $T_{\rm eff}$=17500K and log $g$=8.15, generated using [v200; @hubeny88; @hubeny95] and (v49; Hubeny, I. and Lanz, T. 2001, http://nova.astro.umd.edu/), have both been normalised using a custom written IDL routine. The model has then been compared to the data, allowing a velocity parameter to vary freely[^4] and using a Levenberg-Marquardt algorithm to minimise a $\chi$$^{2}$ goodness-of-fit statistic. The result of this process is displayed in Figure \[halp\] and the line velocity shift, after correction to the heliocentric rest frame, is shown in Table \[rvfit\]. This measurement is not sensitive to the details of the model we adopt for effective temperatures and surface gravities within plausible limits. The uncertainty we quote has been estimated using the bootstrapping method of statistical resampling [@efron82]. Subsequently, the gravitational redshift ($v$ in kms$^{-1}$) component of this velocity shift was derived using Equation \[gre\], where $M$ and $R$ are the mass and radius of the white dwarf in solar units respectively.
$$\label{gre}
v = 0.635 M / R$$
As described above, the mass and radius of DA2 were determined using the evolutionary tracks of [@fontaine01]. The radial velocity was then derived from the difference between the measured shift of the line and the calculated gravitational redshift (Table \[rvfit\]). Finally, adopting the mean of our new (relative)[^5] proper motion measurements for the components of this binary ($\mu_{\alpha}\cos\delta$=30.7$\pm$3.5 mas yr$^{-1}$, $\mu_{\delta}$=12.9$\pm$4.1 mas yr$^{-1}$), we have followed the prescription outlined by [@johnson87] to calculate the heliocentric space velocity of DA2 to be U=$-$20.6$\pm$3.0kms$^{-1}$, V=$-$20.8$\pm$2.8 kms$^{-1}$ and W=$-$6.4$\pm$4.6 kms$^{-1}$. This is coincident with the space velocity of the oldest component of the Pleiades moving group, B3, identified by [@asiain99] in their Hipparcos kinematic analysis of the B, A and F-type stars in the vicinity of the Sun. The sub-populations of this supercluster are estimated to span the range of ages, $\tau$$\approx$60-600Myr [@eggen92]. While this result is not definitive proof of an association between System 2 and the Pleiades moving group, it is at least in accord with our conclusions above, from the cooling time and the inferred main sequence lifetime of the DA component, that this is a relatively young binary.
Summary and future work
=======================
Within a broader photometric, astrometric and spectroscopic survey for wide double-degenerate systems (Baxter et al. in prep) we have discovered two new binaries, each containing a hydrogen rich HFMWD and a non-magnetic (DA) component. We have used synthetic spectra generated from offset dipole magnetic models to estimate the field strengths for DAH1 and DAH2 to be $B_{\rm dip}$$\sim$210MG ($z_{\rm off}$$\sim$-0.09R$_{\rm WD}$) and $B_{\rm dip}$$\sim$65MG ($z_{\rm off}$$\sim$-0.39R$_{\rm WD}$), respectively. Our measurements of the effective temperatures and surface gravities of their non-magnetic companions allow us to infer the masses of these DAHs to be $M$=0.62$\pm$0.10M$_{\odot}$ and $M$=0.99$\pm$0.05M$_{\odot}$, respectively. If we assume that the two components in each of these systems have evolved essentially as single stars we find mild discord in their cooling times, with DAH2 appearing slightly too hot and young relative to expectations, while DAH1 appears to be “too old” for its mass or, alternatively, too low in mass for its cooling time. The former object may represent the third known HFMWD which is apparently “too young”, perhaps hinting at a trend, though the study of more such systems are required to firm up this possibility. The latter white dwarf may have been part of a hierarchical triple system and suffered greater mass loss than expected of a single star during its earlier evolution.
In three of the four of these wide systems which are now known, the non-magnetic components have larger masses than are typical of field white dwarfs. The characteristics (ie. masses, cooling times and kinematics) of the DAs in our two new binaries argue that their HFMWD companions are members of relatively young systems and are therefore associated with early type stars ($M_{\rm init}$$\simgreat$2M$_{\odot}$). The non-magnetic components in three additional but spatially unresolved, young, magnetic + non-magnetic binaries known prior to this work (at least one of which is a physically close system where the components may have previously exchanged mass), also have atypically large masses. This is consistent with the HFMWDs in these systems also being related to an early type stellar population. To re-inforce these findings and to clearly delineate any trends in the cooling times of HFMWDs which could shed light on their formation and the impact of magnetic fields on stellar evolution, an enlarged sample of these objects that are located either in wide double-degenerate binaries or in nearby open clusters, will be required. As a starting point we have flagged the previously known DA+DAH system CBS229 as a probable wide binary. Additionally, we note that the SDSS spectrum of one component of the close ($\sim$1.6 arcsec) pair of relatively bright ($g$$\sim$17.6 mag.), blue point sources, SDSSJ074853.07+302543.5 and SDSSJ074852.95+302543.4 (Figure \[newcands\]), displays a Zeeman split, pressure broadened, Balmer line series and thus also represents a promising candidate wide double-degenerate system containing a DAH.
Acknowledgments {#acknowledgments .unnumbered}
===============
The WHT is operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington. BK acknowledges support by the MICINN grant AYA08-1839/ESP, by the ESF EUROCORES Program EuroGENESIS (MICINN grant EUI2009-04170), by the 2009SGR315 of the Generalitat de Catalunya and EU-FEDER funds. NL acknowledges funding from Spanish ministry of science and innovation through the national program AYA2010-19136.
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: Based on observations made with ESO Telescopes at the La Silla or Paranal Observatories under programme ID 084.D-1097
[^3]: http://www.eso.org/instruments/fors2/
[^4]: An additional flux scaling parameter was allowed to vary by up to 1 per cent
[^5]: The difference between relative and absolute values are comparable to or smaller than the quoted errors on the proper motion
| {
"pile_set_name": "ArXiv"
} |
1.0 cm
\
.1cm [*$^{(a)}$Instituto de Física, Universidade de São Paulo,\
C. Postal 66318, 05314-970 São Paulo, SP, Brazil*]{}\
.3cm [*$^{(b)}$S. N. Bose National Centre for Basic Sciences,\
Block JD, Sector III, Salt Lake, Kolkata$-$700098, India*]{}\
.1cm [E-mails: [email protected]; [email protected]]{}
1.0 cm
[**Abstract:**]{} We derive the complete set of off-shell nilpotent ($s^2_{(a)b} = 0$) and absolutely anticommuting ($s_b s_{ab} + s_{ab} s_b = 0$) Becchi-Rouet-Stora-Tyutin (BRST) ($s_b$) as well as anti-BRST symmetry transformations ($s_{ab}$) corresponding to the combined Yang-Mills and non-Yang-Mills symmetries of the $(2 + 1)$-dimensional Jackiw-Pi model within the framework of augmented superfield formalism. The absolute anticommutativity of the (anti-)BRST symmetries is ensured by the existence of [*two*]{} sets of Curci-Ferrari (CF) type of conditions which emerge naturally in this formalism. The presence of CF conditions enables us to derive the coupled but equivalent Lagrangian densities. We also capture the (anti-)BRST invariance of the coupled Lagrangian densities in the superfield formalism. The derivation of the (anti-)BRST transformations of the auxiliary field $\rho$ is one of the key findings which can neither be generated by the nilpotent (anti-)BRST charges nor by the requirements of the nilpotency and/or absolute anticommutativity of the (anti-)BRST transformations. Finally, we provide a bird’s-eye view on the role of auxiliary field for various massive models and point out few striking similarities and some glaring differences among them.\
[ PACS numbers:]{} 11.15.-q, 03.70.+k, 11.10Kk, 12.90.+b\
[*Keywords*]{}: Jackiw-Pi model; augmented superfield formalism; Curci-Ferrari conditions; (anti-)BRST symmetry transformations; nilpotency and absolute anticommutativity\
Introduction
============
The co-existence of mass and gauge invariance [*together*]{} is still one of the main issues connected with the gauge theories, in spite of the astonishing success of the standard model of particle physics which is based on (non-)Abelian 1-form gauge theories. However, it is worthwhile to mention that, in the case of sufficiently strong vector couplings, the gauge invariance does not entail the masslessness of gauge particles [@Schwinger:1962tn; @Schwinger:1962tp]. Thus, it is needless to say that the mass generation in gauge theories is a crucial issue which has attracted a great deal of interest [@Deser:1981wh; @Deser:1982vy].
In the recent past, many models for the mass generation have been studied in the diverse dimensions of spacetime. In this context, mention can be made of about 4D topologically massive (non-)Abelian gauge theories, with $(B \wedge F)$ term, where 1-form gauge field acquires a mass in a natural fashion [@Freedman:1980us; @Allen:1990gb; @Harikumar:2001eb]. One of the key features associated with such models is that the 1-form gauge field gets a mass without taking any recourse to the Higgs mechanism. We have thoroughly investigated these models within the framework of Becchi-Rouet-Stora-Tyutin (BRST) as well as superfield formalism [@Gupta:2008he; @Gupta:2010xh; @Gupta:2009up; @Kumar:2011zi; @Krishna:2010dc; @Malik:2011pm]. It is interesting to point out that the main issues connected with the 4D Abelian topologically massive models are that they suffer from the problems connected with renormalizability when straightforwardly generalized to the non-Abelian case [@Henneaux:1997mf]. However, this issue can be circumvented by the introduction of extra field (see, e.g. [@Lahiri:1996dm; @Lahiri:1999uc]).
At this juncture, it is worth mentioning about the lower dimensional non-Abelian massive models, such as $(2 + 1)$-dimensional Jackiw-Pi (JP) model [@Jackiw:1997jga], which are free from the above mentioned issues. The silent features of JP model are as follows. First, it is parity conserving model due to the introduction of a 1-form vector field having odd parity. Second, mass and gauge invariance are respected together. Third, it is endowed with the two independent sets of local continuous symmetries, namely; the usual Yang-Mills (YM) symmetries and non-Yang-Mills (NYM) symmetries. Finally, it is free from the problems connected with the 4D topologically massive models. These features make JP model attractive and worth studying in detail.
The JP model has been explored in many different prospects such as constraint analysis and Hamiltonian formalism [@Dayi:1997in], establishment of Slavnov-Taylor identities and BRST symmetries [@DelCima:2011bx]. Furthermore, this model is also shown to be ultraviolet finite and renormalizable [@DelCima:2012bm]. We have applied superfield formalism and derived the full set of off-shell nilpotent and absolutely anticommuting BRST as well as anti-BRST symmetry transformations corresponding to the both YM and NYM symmetries of JP model [@Gupta:2011cta; @Gupta:2012ur]. Within the superfield formalism, we have been able to derive the [*proper*]{} (anti-)BRST transformations for the auxiliary field $\rho$ which can neither be deduced by the conventional means of nilpotency and/or absolute anticommutativity of (anti-)BRST symmetries nor generated by the conserved (anti-)BRST charges. At this stage, we would like to point out that the derivation of the proper anti-BRST symmetries have utmost importance because they play a fundamental role in the BRST formalism (see, e.g. [@Curci:1976ar; @Ojima:1980da; @Hwang:1989mn] for details). In fact, both the symmetries (i.e. BRST and anti-BRST) have been formulated in an independent way [@Hwang:1983sm].
Recently, the (anti-)BRST symmetries for perturbative quantum gravity in curved as well as complex spacetime, in linear as well as in non-linear gauges have been found [@mir1; @mir2] and a superspace formulation of higher derivative theories [@mir3], Chern-Simons and Yang-Mills theories on deformed superspace [@mir4; @mir5] within BV formalism have also been established. Moreover, the study of massless and massive fields with totally symmetric arbitrary spin in AdS space has been carried out in the framework of BRST formalism [@mets].
The main motivations behind our present investigation are as follows. First, the derivation of off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations corresponding to the combined YM and NYM symmetries of JP model. As, in our recent works (cf. [@Gupta:2011cta; @Gupta:2012ur]), we have already established the corresponding proper (anti-) BRST symmetry transformations, individually, for both the YM and NYM cases, within the framework of superfield formalism. Second, to establish the Curci-Ferrari (CF) conditions in the case of combined symmetries. These CF conditions are hallmark of any non-Abelian 1-form gauge theories [@Curci:1976ar] and have a close connection with gerbes [@Bonora:2007hw], within the framework of BRST formalism. Third, to procure appropriate coupled Lagrangian densities which respect the (anti-)BRST symmetries derived from augmented superfield approach. Finally, to point out the role of auxiliary field $\rho$, which is very special to this model (cf. [@Dayi:1997in; @Gupta:2011cta] for details).
This paper is organized in the following manner. In Section 2, we recapitulate the underlying symmetries of 3D JP model. We derive the off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetries corresponding to the combined YM and NYM symmetries of JP model, within the framework of superfield formalism, in Section 3. Section 4 contains the derivation of coupled Lagrangian densities that respect the preceding (anti-) BRST symmetries. The conservation of (anti-)BRST charges is shown in Section 5. We also discuss about the novel observations of our present study in this section. Section 6 is devoted for the discussions of ghost symmetries and BRST algebra. In Section 7, we provide a bird’s-eye view on the role of auxiliary field in the context of various massive models. Finally, in Section 8, we make some concluding remarks.
In Appendix A, we show the nilpotency and absolute anticommutativity of the (anti-) BRST charges within the framework of augmented superfield formalism. We also capture (anti-)BRST invariance of coupled Lagrangian densities in the superfield framework.
[*Conventions and notation:*]{} We adopt here the conventions and notation such that the 3D flat Minkowski metric $\eta_{\mu\nu} =$ diag $(+ 1, - 1, -1)$ and the 3D totally antisymmetric Levi-Civita tensor $\varepsilon_{\mu\nu\eta}$ satisfies $\varepsilon_{\mu\nu\eta}\,\varepsilon^{\mu\nu\eta}= - 3!,\;
\varepsilon_{\mu\nu\eta}\,\varepsilon^{\mu\nu\kappa}$ $= - 2! \delta^\kappa_\eta$, etc. with $\varepsilon_{012} = - \varepsilon^{012} = +1$. The Greek indices $\mu, \nu, \eta, ... = 0, 1, 2$ correspond to the 3D spacetime directions and Latin indices $i, j, k,... = 1,2$ correspond to the space directions only. The dot and cross product between two non-null vectors $P$ and $Q$ in the $SU(N)$ Lie algebraic space are defined as $P \cdot Q = P^a Q^a,\; P \times Q = f^{abc}\, P^a Q^b T^c$. The $SU(N)$ generators $T^a$ (with $a, b, c,... = N^2 - 1$) follow the commutation relation $[T^a,\, T^b] = i f^{abc}\, T^c$ where the structure constants $f^{abc}$ are chosen to be totally antisymmetric in $a,b,c$ for the semi-simple $SU(N)$ Lie algebra [@Wein].
Preliminaries: Jackiw-Pi model
==============================
We start off with the massive, non-Abelian, gauge invariant Jackiw-Pi model in $(2+ 1)$-dimensions of spacetime. The Lagrangian density of this model is given by [@Jackiw:1997jga; @Gupta:2011cta] $$\begin{aligned}
{\cal L}_0 &=& - \frac{1}{4}\, F_{\mu\nu}\cdot F^{\mu\nu}
- \frac{1}{4}\, \big(G_{\mu\nu} + g F_{\mu\nu} \times \rho\big) \cdot \big(G^{\mu\nu} + g F^{\mu\nu} \times \rho\big) \nonumber\\
&+& \frac{m}{2}\, \varepsilon^{\mu\nu\eta}\,F_{\mu\nu}\cdot \phi_\eta, \label{2.1}\end{aligned}$$ where the 2-form $F^{(2)} = d A^{(1)} + i g \big(A^{(1)} \wedge A^{(1)} \big)
= \frac{1}{2!}\,\big(dx^\mu \wedge dx^\nu \big)F_{\mu\nu}\cdot T$ defines the curvature tensor $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - g(A_\mu \times A_\nu)$ for the non-Abelian 1-form \[$A^{(1)} = dx_\mu A^\mu \cdot T$\] gauge field $A_\mu = A_\mu \cdot T$ where $d = dx^\mu \partial_\mu$ is the exterior derivative (with $d^2 = 0$). Similarly, another 2-form $G^{(2)} = d\phi^{(1)} + i g \big(A^{(1)} \wedge \phi^{(1)}\big) + ig \big(\phi^{(1)} \wedge A^{(1)}\big)
= \frac{1}{2!}\,\big(dx^\mu \wedge dx^\nu\big)\,G_{\mu\nu} \cdot T$ defines the curvature tensor $G_{\mu\nu} = D_\mu \phi_\nu - D_\nu \phi_\mu$ corresponding[^1] to 1-form $[\phi^{(1)} = dx^\mu\phi_\mu \cdot T]$ vector field $\phi_\mu = \phi_\mu \cdot T$. In the above, the vector fields $A_\mu$ and $\phi_\mu$ have opposite parity thus the JP model becomes parity invariant, $\rho$ is an auxiliary field, $g$ is the coupling constant and $m$ defines the mass parameter.
Local gauge symmetries: YM and NYM
----------------------------------
The above Lagrangian density respects two sets of local and continuous gauge symmetry transformations, namely; YM gauge transformations $(\delta_1)$ and NYM gauge transformations $(\delta_2)$. These symmetry transformations are [@Gupta:2011cta; @Gupta:2012ur] $$\begin{aligned}
&&\delta_1 A_\mu = D_\mu \Lambda, \quad \delta_1 \phi_\mu = - g(\phi_\mu \times \Lambda),
\quad \delta_1 \rho = - g(\rho \times \Lambda),\nonumber\\
&& \delta_1 F_{\mu\nu} = - g(F_{\mu\nu} \times \Lambda),
\quad \delta_1 G_{\mu\nu} = - g(G_{\mu\nu} \times \Lambda), \label{2.2}\end{aligned}$$ $$\begin{aligned}
\delta_2 A_\mu = 0, \quad \delta_2 \phi_\mu = D_\mu \Sigma, \quad \delta_2 \rho = + \Sigma, \quad
\delta_2 F_{\mu\nu} = 0, \quad \delta_2 G_{\mu\nu} = - g(F_{\mu\nu} \times \Sigma), \label{2.3}\end{aligned}$$ where $\Lambda \equiv \Lambda \cdot T $ and $\Sigma \equiv \Sigma \cdot T$ are the $SU(N)$ valued local gauge parameters corresponding to the YM and NYM gauge transformations, respectively. Under the above local and infinitesimal gauge transformations the Lagrangian density (\[2.1\]) transforms as $$\begin{aligned}
&&\delta_1 {\cal L}_0 = 0, \qquad \delta_2 {\cal L}_0 = \partial_\mu \Big[\frac{m}{2}\, \varepsilon^{\mu\nu\eta}\, F_{\nu\eta}\cdot \Sigma\Big]. \end{aligned}$$ As a consequence, the action integral $S = \int d^3x {\cal L}_0$ remains invariant under both the gauge transformations ($\delta_1$ and $\delta_2$) for the physically well-defined fields which vanish off rapidly at infinity. We would like to point out that in order to maintain the NYM symmetry, we have to have the auxiliary field $\rho$ in the theory (cf. Section 7 for details).
Combined gauge symmetry
-----------------------
In the above, we have seen that both the YM and NYM transformations are the symmetries of the theory. Thus, the combination of the above symmetries \[i.e. $(\delta = \delta_1 + \delta_2$)\] would also be the symmetry of theory. Under the combined gauge transformation $\delta$, namely; $$\begin{aligned}
&&\delta A_\mu = D_\mu \Lambda, \qquad \delta \phi_\mu = D_\mu \Sigma - g(\phi_\mu \times \Lambda), \qquad
\delta \rho = \Sigma - g(\rho \times \Lambda),\nonumber\\
&& \delta F_{\mu\nu} = - g(F_{\mu\nu} \times \Lambda), \qquad \delta G_{\mu\nu} = - g(G_{\mu\nu} \times \Lambda)
-g (F_{\mu\nu} \times \Sigma), \label{2.5}\end{aligned}$$ the Lagrangian density (\[2.1\]) remains quasi-invariant. To be more specific, the Lagrangian density transforms to a total spacetime derivative $$\begin{aligned}
\delta {\cal L}_0 = \partial_\mu \Big[\frac{m}{2}\, \varepsilon^{\mu\nu\eta}\, F_{\nu\eta}\cdot \Sigma\Big]. \end{aligned}$$ Thus, the action integral remains invariant (i.e. $\delta S = \delta \int d^3x {\cal L}_0 = 0$) under the combined symmetry ($\delta$), too.
(Augmented) superfield approach
===============================
We apply Bonora-Tonin’s (BT) superfield approach to the BRST formalism [@Bonora:1980pt; @Bonora:1980ar], to derive the off-shell nilpotent and absolutely anticommuting (anti-) BRST symmetry transformations for the 1-form gauge field $A_\mu$ and corresponding (anti-)ghost fields $(\bar C)C$.
(Anti-)BRST symmetries: Gauge and (anti-)ghost fields
-----------------------------------------------------
For this purpose, we generalize 1-form connection $A^{(1)}$ (and corresponding 2-form curvature $F^{(2)}$) and exterior derivative $d$ onto the $(3,2)$-dimensional supermanifold, as $$\begin{aligned}
d \to \tilde d &=& dZ^M\partial_M = dx^\mu\,\partial_\mu + d\theta\, \partial_\theta + d\bar\theta \,\partial_{\bar\theta},
\qquad \tilde d^2 = 0,\nonumber\\
A^{(1)} \to \tilde{\cal A}^{(1)} &=& dZ^M \tilde {\cal A}_M = dx^\mu\,\tilde {\cal A}_\mu(x, \theta, \bar\theta)
+ d\theta\, {\tilde {\bar{ \cal F}}} (x, \theta, \bar\theta)
+ d \bar\theta\, {\tilde{{\cal F}}} (x, \theta, \bar\theta), \nonumber\\
F^{(2)} \to \tilde {\cal F}^{(2)} &=& \frac{1}{2!}\,(dx^M \wedge dx^N)\,\tilde {\cal F}_{MN} = \tilde d \tilde {\cal A}^{(1)}
+ i g \big(\tilde {\cal A}^{(1)} \wedge \tilde {\cal A}^{(1)} \big) ,\end{aligned}$$ where $Z^M = (x^\mu, \theta, \bar \theta)$ are superspace coordinates characterizing the $(3, 2)$-dimensional supermanifold. In the above expression, $\theta$ and $\bar \theta$ are the Grassmannian variables (with $\theta^2 = \bar \theta^2 = 0,
\theta \bar \theta + \bar \theta \theta = 0$) and $\partial_\theta, \partial_{\bar\theta}$ are corresponding Grassmannian derivatives. We also generalize 3D gauge field \[$A_\mu (x)$\] and (anti-)ghost fields $[(\bar C)C(x)]$ of the theory to their corresponding superfields onto the $(3,2)$-dimensional supermanifold.
Now, these superfields can be expanded along the Grassmannian directions, in terms of the basic fields ($A_\mu, C, \bar C$) and secondary fields ($R_\mu$, $\bar R_\mu$, $S_\mu$, $B_1$, $B_2$, $\bar B_1$, $\bar B_2$, $s$, $\bar s$), in the following manner, $$\begin{aligned}
\tilde {\cal A}_{\mu} (x, \theta, \bar\theta) &=& A_\mu (x) + \theta\, \bar R_\mu (x) + \bar \theta\, R_\mu (x)
+ i \,\theta \,\bar\theta \, S_\mu (x), \nonumber\\
\tilde {\cal F} (x, \theta, \bar\theta) &=& C (x) + i\,\theta\, \bar B_1 (x) + i\,\bar \theta\, B_1 (x)
+ i \,\theta\, \bar\theta \, s (x), \nonumber\\
{\tilde {\bar {\cal F}}} (x, \theta, \bar\theta) &=& \bar C (x) + i\,\theta\, \bar B_2 (x) + i\,\bar \theta\;
B_2 (x) + i \,\theta \,\bar\theta \; \bar s (x). \label{3.2}\end{aligned}$$ Here $ \tilde {\cal A}_{\mu} (x, \theta, \bar\theta), \tilde {\cal F} (x, \theta, \bar\theta), {\tilde{\bar {\cal F}}} (x, \theta, \bar\theta) $ are superfields corresponding to the basic fields $A_\mu (x),$ $C (x)$ and $\bar C (x)$, respectively. Now these secondary fields, in the above expression, can be determined in terms of the basic and auxiliary fields of the underlying theory through the application of horizontality condition (HC) (cf. [@Bonora:1980pt; @Bonora:1980ar] for details). This HC can be mathematically expressed in the following fashion $$\begin{aligned}
d\,A^{(1)} + i \,g \big(A^{(1)} \wedge A^{(1)}\big)= \tilde d \,\tilde{\cal A}^{(1)}
+ i \,g\big(\tilde{\cal A}^{(1)} \wedge \tilde{\cal A}^{(1)}\big) \Longleftrightarrow F^{(2)} = \tilde {\cal F}^{(2)}. \label{3.3}\end{aligned}$$ Exploiting the above HC, we obtain the following relationships among the basic, auxiliary and secondary fields of the theory $$\begin{aligned}
&& R_\mu = D_\mu C, \quad \bar R_\mu = D_\mu \bar C, \quad B_1 = - \frac{i}{2}\,g\,(C \times C),\quad
\bar B_2 = - \frac{i}{2}\, g\,(\bar C \times \bar C), \nonumber\\
&& B + \bar B = -\,i\, g\,(C \times \bar C),\quad
s = -\,g\,(\bar B \times C), \quad \bar s = +\, g\,(B \times \bar C), \nonumber\\
&& S_\mu = D_\mu B \,+ \,i\, g \,(D_\mu C \times \bar C) \equiv - \,D_\mu \bar B - \,i\, g\,(D_\mu \bar C \times C), \label{3.4}\end{aligned}$$ where we have chosen $\bar B_1 = \bar B$ and $B_2 = B$.
Substituting the relationships (\[3.4\]) into the super-expansion of superfields in (\[3.2\]), we procure following explicit expansions $$\begin{aligned}
\tilde {\cal A}^{(h)}_{\mu} (x, \theta, \bar\theta) &=& A_\mu (x) + \theta D_\mu \bar C (x)
+ \bar \theta D_\mu C (x)+ \theta \,\bar\theta \, \big[i D_\mu B - g(D_\mu C \times \bar C)\big](x) \nonumber\\
&\equiv& A_\mu (x) + \theta \big(s_{ab}\, A_\mu (x)\big) + \bar \theta \big(s_b \,A_\mu (x)\big)
+ \theta \bar\theta \big(s_b\, s_{ab} \,A_\mu (x)\big), \nonumber\\
\tilde {\cal F}^{(h)} (x, \theta, \bar\theta) &=& C (x) + \theta \big(i \bar B (x)\big) + \bar \theta \Bigl
[\frac{g}{2}\, (C \times C) \Bigr](x) + \theta \bar\theta \big[-i g\big(\bar B \times C)\big](x) \nonumber\\
&\equiv& C (x) + \theta \big(s_{ab} C (x)\big) + \bar \theta \big(s_b C (x)\big)
+ \theta \bar\theta \big(s_b s_{ab} \,C (x)\big),\nonumber\\
{\tilde {\bar {\cal F}}}^{(h)} (x, \theta, \bar\theta) &=& \bar C (x) +
\theta \Bigl [\frac{g}{2} \,(\bar C \times \bar C)\Bigr](x) + \bar \theta \big(i B (x)\big)
+ \theta \bar\theta \big[i g(B \times \bar C)\big](x) \nonumber\\
&\equiv& \bar C (x) + \theta \big(s_{ab} \bar C (x)\big) + \bar \theta\, \big(s_b \bar C (x)\big)
+ \theta \bar\theta \big(s_b \,s_{ab}\, \bar C (x)\big). \label{3.5}\end{aligned}$$ In the above, the superscript $(h)$ on the superfields denotes the super-expansion of the superfields obtained after the application of HC (\[3.3\]). Thus, from the above expressions, we can easily identify the (anti-)BRST symmetry transformations corresponding to the gauge field $A_\mu$ and (anti-)ghost fields $(\bar C)C$. These transformations are explicitly listed below $$\begin{aligned}
&& s_b A_\mu = D_\mu C, \qquad s_b C = \frac{g}{2}\, \big(C \times C\big),
\qquad s_b \bar B = -\, g\,\big(\bar B \times C\big),\nonumber\\
&& s_b \bar C = i\, B,\qquad s_b B = 0, \label{3.6}\end{aligned}$$ $$\begin{aligned}
&&s_{ab} A_\mu = D_\mu \bar C, \qquad s_{ab} \bar C = \frac {g}{2} \,\big(\bar C \times \bar C\big),
\qquad s_{ab} B = - g\,\big(B \times \bar C\big), \nonumber\\
&& s_{ab} C = i \,\bar B,\qquad s_{ab} \bar B = 0. \label{3.7}\end{aligned}$$ We point out that, the (anti-)BRST symmetry transformations for the Nakanishi-Lautrup auxiliary fields $B$ and $\bar B$ have been derived with the help of absolute anticommutativity and nilpotency properties of the above (anti-)BRST symmetries.
(Anti-)BRST symmetries for $\phi_\mu$, $\beta$ and $\bar \beta$
---------------------------------------------------------------
In the previous subsection, we applied BT superfield approach to derive the off-shell nilpotent and absolutely anti-commuting (anti-)BRST symmetry transformations for the gauge field $(A_\mu)$ and corresponding (anti-)ghost fields $(\bar C)C$. Now, in order to derive the proper (anti-)BRST symmetries for the vector field $(\phi_\mu)$, corresponding (anti-)ghost fields $[(\bar \beta) \beta] $ and auxiliary field $(\rho)$, we have to go beyond the BT approach. For this purpose, we have exploited the power and strength of augmented superfield approach.
To derive the (anti-)BRST symmetries for the vector field $(\phi_\mu)$ and corresponding (anti-) ghost fields $[(\bar \beta) \beta]$, we invoke the following HC $$\begin{aligned}
\tilde{\cal G}^{(2)} + \tilde {\mathscr{F}}^{(2)} \equiv G^{(2)} + {\mathscr{F}}^{(2)}, \label{hc}\end{aligned}$$ where $G^{(2)}$, ${\mathscr{F}}^{(2)}$ are define in the following fashion $$\begin{aligned}
G^{(2)} &=&d\phi^{(1)} + i g\big(A^{(1)} \wedge \phi^{(1)}\big) + ig \big(\phi^{(1)} \wedge A^{(1)}\big)
= \frac{1}{2!}\,\big(dx^\mu \wedge dx^\nu\big)\,G_{\mu\nu},\nonumber\\
\mathscr{F}^{(2)} &=& -ig\big(F^{(2)} \wedge \rho^{(0)}\big) + ig\big(\rho^{(0)} \wedge F^{(2)}\big)
= \frac{g}{2!}\,\big(dx^\mu \wedge dx^\nu\big)(F_{\mu\nu} \times \rho),\end{aligned}$$ and $\tilde{\cal G}^{(2)}$, $\tilde {\mathscr{F}}^{(2)}$ are the generalizations of $G^{(2)}$, ${\mathscr{F}}^{(2)}$ onto the superspace, respectively, which can be explicitly represented in the following manner $$\begin{aligned}
\tilde{\cal G}^{(2)} &=& \tilde d \tilde \Phi^{(1)}
+ i\, g\, \big(\tilde {\cal A}^{(1)}_{(h)} \wedge \tilde \Phi^{(1)}\big)
+ i\, g\, \big(\tilde \Phi^{(1)} \wedge \tilde {\cal A}^{(1)}_{(h)}\big), \nonumber\\
\tilde {\mathscr{F}}^{(2)} &=& - i\, g\, \big(\tilde {\cal F}^{(2)}_{(h)} \wedge \tilde \rho^{(0)} \big)
+ i\, g\, \big(\tilde \rho^{(0)} \wedge \tilde {\cal F}^{(2)}_{(h)}\big).\end{aligned}$$ In the above expression, the quantities $\tilde {\cal A}^{(1)}_{(h)}, \tilde \Phi^{(1)}$ and $\tilde \rho^{(0)}$ are given as $$\begin{aligned}
\tilde {\cal A}^{(1)}_{(h)} (x, \theta, \bar \theta) &=& dx^\mu \,\tilde {\cal A}^{(h)}_\mu(x, \theta, \bar \theta)
+ d\theta \,{\tilde {\bar {\cal F}}}^{(h)}(x, \theta, \bar \theta)
+ d \bar \theta\, \tilde {\cal F}^{(h)}(x, \theta, \bar \theta), \nonumber\\
\tilde \Phi^{(1)} (x, \theta, \bar \theta) &=& dx^\mu\, \tilde\Phi_\mu(x, \theta, \bar\theta)
+ d \theta \; \tilde {\bar \beta}(x, \theta, \bar\theta)
+ d \bar \theta \; \tilde\beta(x, \theta, \bar\theta),\nonumber\\
\tilde \rho^{(0)} (x, \theta, \bar \theta) &=& \tilde \rho(x, \theta, \bar\theta), \end{aligned}$$ where the sub/super script $(h)$ denotes the quantities obtained after the application of HC. The superfields in the above expression, corresponding to the basic fields $\phi_\mu, \beta, \bar\beta$ and $\rho$ of the theory, can be expanded in terms of the secondary fields, as follows $$\begin{aligned}
\tilde \Phi_\mu(x, \theta, \bar\theta) &=& \phi_\mu(x) + \theta\, \bar P_\mu(x) + \bar \theta\, P_\mu(x)
+ i\,\theta\,\bar\theta\, Q_\mu(x),\nonumber\\
\tilde \beta(x, \theta, \bar\theta) &=& \beta(x) + i\, \theta\, \bar R_1(x) + i\, \bar \theta\, R_1(x)
+ i\,\theta\,\bar\theta\, s_1(x),\nonumber\\
\tilde{\bar \beta}(x, \theta, \bar\theta) &=& \bar \beta(x) + i\, \theta\, \bar R_2(x) + i\, \bar \theta\, R_2(x)
+ i\,\theta\,\bar\theta\, s_2(x),\nonumber\\
\tilde \rho (x, \theta, \bar\theta) &=& \rho(x) + \theta\, \bar b(x) + \bar \theta\, b(x)
+ i\,\theta\,\bar\theta\, q(x), \label{4.5}\end{aligned}$$ where $P_\mu, \bar P_\mu, b, \bar b, s_1, s_2$ are fermionic secondary fields and $R_1, \bar R_1, R_2, \bar R_2, Q_\mu, q$ are bosonic in nature.
Exploiting the above HC (\[hc\]) which demands that the coefficients of wedge products $(dx^\mu \wedge d \theta), \, (dx^\mu \wedge d \bar\theta),\, (d \theta \wedge d \theta),\,
(d \bar \theta \wedge d \bar \theta),\, (d \theta \wedge d \bar \theta)$ set equal to zero. We get following expressions: $$\begin{aligned}
&& \tilde {\cal D}_\mu \tilde {\bar \beta} - \partial_\theta \tilde \Phi_\mu
- g\,\Big(\tilde \Phi_\mu \times {\tilde {\bar {\cal F}}}^{(h)}\Big) = 0, \qquad
\partial_\theta \tilde {\bar \beta} - g\,\Big({\tilde {\bar {\cal F}}}^{(h)} \times \tilde {\bar \beta}\Big) = 0,\nonumber\\
&&\tilde {\cal D}_\mu \tilde \beta - \partial_{\bar \theta} \tilde \Phi_\mu
- g\,\Big(\tilde \Phi_\mu \times {\tilde {\cal F}}^{(h)}\Big) = 0,\qquad
\partial_{\bar \theta} \tilde \beta - g\,\Big({\tilde {\cal F}}^{(h)} \times \tilde \beta\Big) = 0,\nonumber\\
&& \partial_\theta \tilde \beta + \partial_{\bar \theta} \tilde {\bar \beta}
- g\,\Big({\tilde {\bar {\cal F}}}^{(h)} \times \tilde \beta\Big)
- g\,\Big(\tilde{\cal F}^{(h)} \times \tilde {\bar \beta}\Big) =0, \label{4.6}\end{aligned}$$ where $\tilde {\cal D}_\mu \bullet = \partial_\mu \bullet - g\big(\tilde {\cal A}^{(h)}_\mu \times \bullet \big)$. Using the expansion (\[4.5\]) in (\[4.6\]), we get following relationships amongst the basic and secondary fields of the theory, namely; $$\begin{aligned}
R_1 &=& - i\,g (C \times \beta), \quad \bar R_2 = - i\,g (\bar C \times \bar \beta), \quad
s_1 = - g (\bar B \times \beta) + g (C \times \bar R),\nonumber\\
s_2 &=& g\,(B \times \bar \beta) - g\,(\bar C \times R),
\quad R + \bar R + i\, g(C \times \bar \beta) + i \,g (\bar C \times \beta)= 0, \nonumber\\
P_\mu &=& D_\mu \beta - g (\phi_\mu \times C), \quad
D_\mu \bar R_2 + i\, g (D_\mu \bar C \times \bar \beta) + i\, g (D_\mu \bar \beta \times \bar C) = 0, \nonumber\\
\bar P_\mu &=& D_\mu \bar \beta - g (\phi_\mu \times \bar C),
\quad D_\mu R_1 + i \,g (D_\mu C \times \beta) + i \,g (D_\mu \beta \times C) = 0,\nonumber\\
Q_\mu &=& D_\mu R + g (B \times \phi_\mu) + i\, g(D_\mu C \times \bar \beta)
+ i \,g[D_\mu \beta \times \bar C - g(\phi_\mu \times C)\times \bar C]\nonumber\\
&\equiv& - D_\mu \bar R - g (\bar B \times \phi_\mu) - i\, g(D_\mu \bar C \times \beta)
- i \,g[D_\mu \bar \beta \times C - g(\phi_\mu \times \bar C)\times C], \nonumber\\ \label{3.14}\end{aligned}$$ where we have chosen $\bar R_1 = \bar R$, $R_2 = R$. Substituting, these values of secondary fields in (\[4.5\]), we have following form of superfield expansions $$\begin{aligned}
{\tilde \Phi_\mu}^{(h)} (x, \theta, \bar \theta) &=& \phi_\mu(x) + \theta \big[D_\mu \bar \beta - g\,(\phi_\mu \times \bar C)\big](x)
+ \bar \theta \big[D_\mu \beta - g\,(\phi_\mu \times C)\big](x)\nonumber\\
&+& \theta\, \bar \theta\,\big[i \,D_\mu R - i\, g\, (\phi_\mu \times B) - g\,(D_\mu C \times \bar \beta) - g\,(D_\mu \beta \times \bar C) \nonumber\\
&+& g^2\,(\phi_\mu \times C)\times \bar C \big](x) \nonumber\\
&\equiv& \phi(x) + \theta \,\big(s_b \,\phi(x)\big) + \bar \theta \,\big(s_{ab} \,\phi (x)\big)
+ \theta\,\bar\theta \,\big(s_b \,s_{ab}\,\phi(x)\big),\nonumber\\
{\tilde \beta}^{(h)} (x, \theta, \bar \theta) &=& \beta (x) + \theta \,\big[i\, \bar R (x)\big]
+ \bar \theta\, \big[g\, (C \times \beta)\big](x) \nonumber\\
&+& \theta\, \bar \theta\,\big[- i\, g\,(\bar B \times \beta) - i\, g\, (\bar R \times C) \big](x) \nonumber\\
&\equiv& \beta(x) + \theta \,\big(s_b \,\beta(x)\big) + \bar \theta \,\big(s_{ab} \,\beta (x)\big)
+ \theta\,\bar\theta \,\big(s_b \,s_{ab}\,\beta(x)\big),\nonumber\\
{\tilde {\bar \beta}}^{(h)} (x, \theta, \bar \theta) &=& \bar \beta (x) + \theta\, \big[g \,(\bar C \times \bar \beta)\big](x)
+ \bar \theta \,\big[i\, R(x)\big] \nonumber\\
&+& \theta \,\bar \theta\,\big[i \,g \,(B \times \bar \beta) + i\, g \,(R \times \bar C)\big](x) \nonumber\\
&\equiv& \bar \beta(x) + \theta \,\big(s_b \,\bar \beta(x)\big) + \bar \theta \,\big(s_{ab} \,\bar \beta (x)\big)
+ \theta\,\bar\theta \,\big(s_b \,s_{ab}\,\bar \beta(x)\big), \label{3.15}\end{aligned}$$ here $(h)$ on the superscript of superfields represents the respective quantities obtained after the application of HC (\[hc\]). Therefore, (anti-)BRST symmetry transformations for vector field $(\phi_\mu)$ and (anti-)ghost fields $[(\bar \beta)\beta]$ are obvious from the above super-expansions.
(Anti-)BRST symmetries for auxiliary field $\rho$
-------------------------------------------------
In order to derive the proper (anti-)BRST symmetry transformations for the auxiliary field $\rho$, we look for a quantity which remains invariant (or should transform covariantly) under the combined gauge transformations (\[2.5\]). Such gauge invariant quantity will serve a purpose of ‘physical quantity’ (in some sense) which could be generalized onto the $(3,2)$-dimensional supermanifold. Furthermore, being a ‘physical quantity’ it should remain unaffected by the presence of Grassmannian variables when the former is generalized onto the supermanifold. Thus, keeping above in mind, we note that under the combined gauge transformations (\[2.5\]), the quantity $(D_\mu \rho - \phi_\mu)$ transforms covariantly (as the quantities $F_{\mu\nu}$ and $G_{\mu\nu} + g (F_{\mu\nu} \times \rho)$ do). This can be explicitly checked as follows $$\begin{aligned}
\delta(D_\mu \rho - \phi_\mu) = - g \,(D_\mu \rho - \phi_\mu)\times \Lambda.\end{aligned}$$ Therefore, the above quantity serves our purpose and it can also be expressed in the language of differential forms as follows $$\begin{aligned}
d \rho^{(0)} + i\, g\, \big(A^{(1)} \wedge \rho^{(0)}\big) - i\, g\, \big(\rho^{(0)} \wedge A^{(1)}\big) - \phi^{(1)}
&=& dx^\mu \big(D_\mu \rho - \phi_\mu \big), \end{aligned}$$ which is clearly a 1-form object. Now, we generalize this 1-form object onto the $(3, 2)$-dimensional supermanifold and demand that it should remain unaffected by the presence of Grassmannian variables. This, in turn, produces the following HC $$\begin{aligned}
d \rho^{(0)} + i\, g \big(A^{(1)} \wedge \rho^{(0)}\big) - i\, g \big(\rho^{(0)} \wedge A^{(1)}\big) - \phi^{(1)} &\equiv&
\tilde d \tilde \rho^{(0)} + i\, g \big(\tilde {\cal A}^{(1)}_{(h)} \wedge \tilde \rho^{(0)}\big) \nonumber\\
&-& i \,g \big(\tilde \rho^{(0)} \wedge \tilde {\cal A}^{(1)}_{(h)}\big) - \tilde \Phi^{(1)}_{(h)}. \qquad\label{gir}\end{aligned}$$ This HC can also be derived from the integrability of (\[hc\]) (see e.g., [@ThierryMieg:1982un] for details on the topic). The r.h.s. of the above HC can be simplified as $$\begin{aligned}
&& \tilde d \tilde \rho^{(0)} + i g \big(\tilde {\cal A}^{(1)}_{(h)} \wedge \tilde \rho^{(0)}\big)
- i g \big(\tilde \rho^{(0)} \wedge \tilde {\cal A}^{(1)}_{(h)} \big) - \tilde \Phi^{(1)}_{(h)} = \nonumber\\
&& dx^\mu\Big[{\tilde {\cal D}_\mu} \tilde \rho - \tilde \Phi_\mu^{(h)}\Big]
+ d\theta \Big[\partial_\theta \tilde \rho - \tilde{\bar \beta}^{(h)}
- g \Big({\tilde {\bar {\cal F}}}^{(h)} \times \tilde \rho\Big)\Big]
+ d\bar \theta \Big[\partial_{\bar \theta} \tilde \rho - \tilde \beta^{(h)}
- g \Big(\tilde {\cal F}^{(h)} \times \tilde \rho\Big)\Big]. \qquad \;\end{aligned}$$ Exploiting (\[gir\]), and set the coefficients of $d\theta, d\bar \theta$ equal to zero, we have the following relationships, namely; $$\begin{aligned}
\partial_\theta \tilde \rho - \tilde{\bar \beta}^{(h)} - g\,\Big({\tilde {\bar {\cal F}}}^{(h)} \times \tilde \rho\Big) =0,\qquad
\partial_{\bar \theta} \tilde \rho - \tilde \beta^{(h)} - g\,\Big(\tilde {\cal F}^{(h)} \times \tilde \rho\Big) =0.\end{aligned}$$ Plugging the values of superfield expansions from (\[3.5\]), (\[4.5\]) and (\[3.15\]) into the above expressions, we get the following relationships amongst the basic and secondary fields $$\begin{aligned}
b &=& \beta - g(\rho \times C), \qquad \bar b = \bar \beta - g(\rho \times \bar C),\nonumber\\
q &=& R + g(B \times \rho) + i g (\bar C \times \beta) - i g^2(\rho \times C)\times \bar C\nonumber\\
&\equiv& - \bar R - i g (C \times \bar \beta) - g (\bar B \times \rho) + i g^2 (\rho \times \bar C)\times C.\end{aligned}$$ We point out that, however, there also exist other relationships but they are same as quoted in equation (\[3.14\]).
Finally, substituting these values of secondary fields into (\[4.5\]), we obtain the following superfield expansion for the super-auxiliary field $\tilde \rho (x, \theta, \bar\theta)$ $$\begin{aligned}
\tilde \rho^{(h)} (x, \theta, \bar \theta) &=& \rho (x) + \theta \,\big[\bar \beta - g\,(\rho \times \bar C)\big](x)
+ \bar \theta \,\big[\beta - g\,(\rho \times C)\big](x) \nonumber\\
&+ & i \theta\, \bar \theta\, \big[R + g\,(B \times \rho) + i\, g\, (\bar C \times \beta)
- i \,g^2\,(\rho \times C)\times \bar C\big](x), \nonumber\\
&\equiv& \rho(x) + \theta \,\big(s_b \,\rho(x)\big) + \bar \theta \,\big(s_{ab} \,\rho (x)\big)
+ \theta\,\bar\theta \,\big(s_b \,s_{ab}\,\rho(x)\big), \label{3.22}\end{aligned}$$ where $(h)$ as the superscript on the generic superfield denotes the corresponding superfield expansion obtained after the application of HC (\[gir\]). The (anti-) BRST symmetry transformations for the auxiliary field $\rho$ can be easily deduced from the above expansion. Thus, we have derived the proper (anti-)BRST symmetry transformations for the vector field $(\phi_\mu)$, corresponding (anti-)ghost fields $[(\bar \beta)\beta]$ and auxiliary field $(\rho)$ within the framework of augmented superfield formalism. Moreover, the (anti-)BRST symmetry transformations for the Nakanishi-Lautrup auxiliary fields $R$ and $\bar R$ have been derived with the help of anticommutativity and nilpotency properties of the (anti-)BRST symmetries. These symmetry transformations are listed below $$\begin{aligned}
&& s_b \phi_\mu = D_\mu \beta - g \big(\phi_\mu \times C\big), \quad s_b \beta = g \big(C \times \beta\big),
\quad s_b \rho = \beta -\, g \big(\rho \times C\big), \nonumber\\
&& s_b \bar \beta = i\, R, \quad s_b R = 0, \quad s_b \bar R = - g \big(\bar R \times C\big)
- g \big(\bar B \times \beta\big), \label{3.23}\end{aligned}$$ $$\begin{aligned}
&&s_{ab} \phi_\mu = D_\mu \bar \beta - g \big(\phi_\mu \times \bar C\big),
\quad s_{ab} \bar \beta = g \big(\bar C \times \bar \beta\big),
\quad s_{ab} \rho = \bar \beta - g \big(\rho \times \bar C\big),\nonumber\\
&&s_{ab} \beta = i \,\bar R, \quad s_{ab} \bar R = 0, \quad
\quad s_{ab} R = - g \big(R \times \bar C\big) - g \big(B \times \bar \beta\big). \label{3.24}\end{aligned}$$ These (anti-)BRST symmetry transformations as well as the transformations listed in (\[3.6\]) and (\[3.7\]) are off-shell nilpotent $(s_{(a)b}^2 \Psi = 0)$ and absolutely anticommuting $[(s_b s_{ab} + s_{ab} s_b) \Psi = 0]$ in nature. Here $\Psi$ represents any generic field of the theory. These properties (i.e. nilpotency and anticommutativity) are two key ingredients of the BRST formalism. The anticommutativity property for the vector fields ($\phi_\mu$ and $A_\mu$) and auxiliary field ($\rho$) is satisfied only on the constrained surface parametrized by the CF conditions (cf. (\[3.26\]) below). For instance, one can check that $$\begin{aligned}
&&\{s_b,\, s_{ab}\} A_\mu = iD_\mu \big[B + \bar B + i (C \times \bar C)],\nonumber\\
&& \{s_b, \,s_{ab}\} \phi_\mu = iD_\mu\big[R + \bar R + i g(C \times \bar \beta) + i g (\bar C \times \beta) \big]
+ i g \big[B + \bar B + i g (C \times \bar C)\big] \times \phi_\mu, \nonumber\\
&& \{s_b, \,s_{ab}\} \rho = i\big[R + \bar R + i g(C \times \bar \beta) + i g (\bar C \times \beta) \big]
+ i g \big[B + \bar B + i g (C \times \bar C)\big] \times \rho, \end{aligned}$$ whereas, for all the [*rest*]{} of the fields (of our present 3D JP model), the absolute anticommutativity property (i.e. $ \{s_b, s_{ab} \} \Psi = 0$) is valid [*without*]{} invoking the CF type conditions.
Before, we wrap up this section, some crucial points are in order. First and foremost, a very careful look at (\[3.4\]) and (\[3.14\]) reveals, respectively, the existence of two sets of Curci-Ferrari (CF) type conditions, namely; $$\begin{aligned}
&(i)& B + \bar B + i g (C \times \bar C) = 0, \nonumber\\
&(ii)& R + \bar R + i \,g\,\big(C \times \bar \beta\big) + i\, g\,\big(\bar C \times \beta\big) = 0. \label{3.26}\end{aligned}$$ These conditions are key signatures of any $p$-form gauge theory when the latter is discussed within the framework of BRST formalism. In our case, the above mentioned CF conditions emerge very naturally within the framework of (augmented) superfield formalism. In fact, CF conditions $(i)$ and $(ii)$ emerge from the HC (\[3.3\]) and (\[hc\]), respectively, when we set the coefficients of $(d\theta \wedge d \bar \theta)$ equal to zero. Second, the absolute anticommutativity of (anti-)BRST symmetries is ensured by these CF type conditions. Third, these CF type conditions are (anti-)BRST invariant. Finally, these CF type conditions play a crucial role in the derivation of the coupled (but equivalent) Lagrangian densities. We have discussed this aspect, in detail, in our next section.
Coupled Lagrangian densities
============================
In this section, we construct the coupled (but equivalent) Lagrangian densities which respect nilpotent as well as anticommuting (anti-)BRST symmetry transformations derived in the previous section (cf. Section 3). In order to proceed further, a few important points are in order. First, the mass dimensions (in natural units $c = \hbar =1$) of the various fields in our present 3D theory are: $[A_\mu] = [\phi_\mu] = [C] = [\bar C] = [\beta] = [\beta]
= [M]^{\frac{1}{2}}, \; [B] = [\bar B] = [R] = [\bar R] = [M]^{\frac{3}{2}},\;
[\rho] = [M]^{-\frac{1}{2}}, \;$ and the coupling constant $g$ has the mass dimension $[g] = [M]^{\frac{1}{2}}$. Second, the fermionic (anti-)ghost fields $(\bar C) C$ and $(\bar \beta) \beta$ carry ghost numbers $(\mp1)$, respectively whereas rest of the (bosonic) fields carry ghost number equal to zero. Third, the nilpotent (anti-)BRST transformations increase the mass dimension by one unit when they operate on any generic field of the theory. In other words, we can say that the (anti-)BRST transformations carry mass dimension equal one (in natural units). Fourth, the (anti-)BRST transformations (decrease)increase the ghost number by one unit when they act on any field of the theory. This means that (anti-)BRST transformations carry ghost number $(\mp 1)$, respectively. These points are very important in constructing the (anti-)BRST invariant coupled Lagrangian densities.
Exploiting the basic tenets of the BRST formalism, the most appropriate (anti-)BRST invariant Lagrangian densities that can be written in terms of nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations are as follows [@ThierryMieg:1982un] $$\begin{aligned}
{\cal L}_b &=& {\cal L}_0 + s_b\,s_{ab}\bigg[\frac{i}{2} A_\mu \cdot A^\mu + C \cdot \bar C
+ \frac{i}{2} \phi_\mu \cdot \phi^\mu + \frac{1}{2}\, \beta \cdot \bar \beta\bigg],\nonumber\\
&&\nonumber\\
{\cal L}_{\bar b} &=& {\cal L}_0 - s_{ab}\,s_b\bigg[\frac{i}{2} A_\mu \cdot A^\mu + C \cdot \bar C
+ \frac{i}{2} \phi_\mu \cdot \phi^\mu + \frac{1}{2}\,\beta \cdot \bar \beta\bigg],\end{aligned}$$ where ${\cal L}_0$ is our starting gauge invariant Lagrangian density (\[2.1\]). We would like to emphasize that each term in the square brackets is Lorentz scalar and chosen in such a way that they have ghost number zero and mass dimension one (in natural units). Moreover, the (constant) factors in front of each term are picked for the algebraic convenience. Utilizing the off-shell nilpotent (anti-) BRST transformations from (\[3.6\]), (\[3.7\]), (\[3.23\]) and (\[3.24\]), we obtain the following explicit Lagrangian densities, namely; $$\begin{aligned}
{\cal L}_b &=& - \frac{1}{4}\, F_{\mu\nu}\cdot F^{\mu\nu}
- \frac{1}{4}\, \big(G_{\mu\nu} + g F_{\mu\nu} \times \rho\big) \cdot \big(G^{\mu\nu} + g F^{\mu\nu} \times \rho\big)
+ \frac{m}{2}\, \varepsilon^{\mu\nu\eta}\,F_{\mu\nu}\cdot \phi_\eta\nonumber\\
&+& \frac{1}{2}\,\big[B \cdot B + \bar B \cdot \bar B\big] + B\cdot \big(\partial^\mu A_\mu\big)
+ \frac{1}{2}\,\big[R + i g (C \times \bar \beta)\big]\cdot \big[R + i g (C \times \bar \beta)\big]\nonumber\\
&+& \big[R + i g (C \times \bar \beta)\big]\cdot \big(D^\mu \phi_\mu\big)
- i \partial_\mu \bar C \cdot D^\mu C - i D_\mu \bar \beta \cdot D^\mu \beta,\nonumber\\
&&\nonumber\\
{\cal L}_{\bar b} &=& - \frac{1}{4}\, F_{\mu\nu}\cdot F^{\mu\nu}
- \frac{1}{4}\, \big(G_{\mu\nu} + g F_{\mu\nu} \times \rho\big) \cdot \big(G^{\mu\nu} + g F^{\mu\nu} \times \rho\big)
+ \frac{m}{2}\, \varepsilon^{\mu\nu\eta}\,F_{\mu\nu}\cdot \phi_\eta\nonumber\\
&+& \frac{1}{2}\,\big[B \cdot B + \bar B \cdot \bar B\big] - \bar B\cdot \big(\partial^\mu A_\mu\big)
+ \frac{1}{2}\,\big[\bar R + i g (\bar C \times \beta)\big]\cdot \big[\bar R + i g (\bar C \times \beta)\big]\nonumber\\
&-& \big[\bar R + i g (\bar C \times \beta)\big]\cdot \big(D^\mu \phi_\mu\big)
- i D_\mu \bar C \cdot \partial^\mu C - i D_\mu \bar \beta \cdot D^\mu \beta, \label{4.2}\end{aligned}$$ where $B, \bar B$ and $R, \bar R$ are the Nakanishi-Lautrup type auxiliary fields. These Lagrangian densities are coupled because these Nakanishi-Lautrup auxiliary fields $B, \bar B$ and $R, \bar R$ are related through the CF conditions (\[3.26\]).
It can be checked that the (anti-)BRST transformations ($s_{(a)b}$) leave the above Lagrangian densities quasi-invariant. To be more specific, under the operations of nilpotent (anti-)BRST transformations, the Lagrangian densities $({\cal L}_{\bar b}) {\cal L}_b$ transform to a total spacetime derivative, in the following fashion, respectively $$\begin{aligned}
s_{ab} {\cal L}_{\bar b} &=& \partial_\mu \bigg[\frac{m}{2}\, \varepsilon^{\mu\nu\eta} F_{\nu\eta} \cdot \bar \beta
- \bar B \cdot (D^\mu \bar C) - \bar R\cdot D^\mu \bar \beta - i g \big(\bar C \times \beta\big)\cdot D^\mu \bar \beta\bigg], \nonumber\\
&&\nonumber\\
s_b {\cal L}_b &=& \partial_\mu \bigg[\frac{m}{2}\, \varepsilon^{\mu\nu\eta} F_{\nu\eta} \cdot \beta + B \cdot (D^\mu C)
+ R\cdot D^\mu \beta + i g \big(C \times \bar \beta\big)\cdot D^\mu \beta\bigg]. \quad\end{aligned}$$ Thus, the action integral corresponding to the above Lagrangian densities remain invariant under ($s_{(a)b}$). Furthermore, it is interesting to note that the following variations are true: $$\begin{aligned}
s_{ab} {\cal L}_b &=& \partial_\mu \Big[\frac{m}{2}\, \varepsilon^{\mu\nu\eta} F_{\nu\eta} \cdot \bar \beta
+ B \cdot \partial^\mu \bar C + \big(R + i g C \times \bar \beta \big)\cdot D^\mu \bar \beta \Big]\nonumber\\
&-& \Big[D_\mu\big(B + \bar B + i g C \times \bar C\big)\Big]\cdot \partial^\mu \bar C
- \Big[D_\mu\big(R + \bar R + ig C \times \bar \beta + i g \bar C \times \beta \big)\Big] \cdot D^\mu\bar \beta\nonumber\\
&-& g \Big[R + ig \big(C \times \bar \beta\big)
+ D^\mu \phi_\mu \Big]\cdot \Big[\big(B + \bar B + i g C \times \bar C\big)\times \bar \beta\Big], \nonumber\\
&&\nonumber\\
s_b {\cal L}_{\bar b} &=& \partial_\mu \Big[\frac{m}{2}\, \varepsilon^{\mu\nu\eta} F_{\nu\eta} \cdot \beta
- \bar B \cdot \partial^\mu C - \big(\bar R + i g \bar C \times \beta \big)\cdot D^\mu \beta \Big]\nonumber\\
&+& \Big[D_\mu\big(B + \bar B + i g C \times \bar C\big)\Big]\cdot \partial^\mu C
+ \Big[D_\mu\big(R + \bar R + ig C \times \bar \beta + i g \bar C \times \beta \big)\Big] \cdot D^\mu \beta \nonumber\\
&-& g \big[\bar R + ig \big(\bar C \times \beta \big)
- D^\mu \phi_\mu\Big]\cdot \Big[\big(B + \bar B + i g C \times \bar C\big)\times \beta\Big].\end{aligned}$$ Therefore, it is evident from the above variations that the Lagrangian densities ${\cal L}_b$ and ${\cal L}_{\bar b}$ also respect the anti-BRST ($s_{ab}$) and BRST ($s_b$) transformations, respectively only on the constrained hypersurface defined by the CF conditions (\[3.26\]). As a result, both the Lagrangian densities are equivalent and they respect BRST as well as anti-BRST symmetries on the constrained hypersurface spanned by CF conditions \[cf. (\[3.26\])\].
Conserved charges: Novel observations
=====================================
In our previous section, we have seen that the coupled Lagrangian densities (and corresponding actions) respect the off-shell nilpotent and continuous (anti-)BRST symmetry transformations. As a consequence, according to Noether’s theorem, the invariance of the actions under the continuous (anti-) BRST transformations lead to the following conserved (anti-)BRST currents ($J^\mu_{(a)b}$), namely; $$\begin{aligned}
J^\mu_{ab} &=& - (D_\nu \bar C) \cdot \Big[F^{\mu\nu} - g \big(G^{\mu\nu} + g F^{\mu\nu} \times \rho\big)\times \rho
- m \,\varepsilon^{\mu\nu\eta} \phi_\eta\Big] - \bar B \cdot (D^\mu \bar C) \nonumber\\
&-& \frac{i}{2}\,g \big(\bar C \times \bar C\big) \cdot \partial^\mu C
- (D_\nu \bar \beta)\cdot \Big(G^{\mu\nu} + g F^{\mu\nu} \times \rho \Big)
+ g (\phi_\nu \times \bar C) \cdot \Big(G^{\mu\nu} + g F^{\mu\nu} \times \rho \Big) \nonumber\\
&+& g (\phi^\mu \times \bar C)\cdot \big(\bar R + i g \bar C\times \beta\big)
- \bar R\cdot D^\mu \bar \beta - i g \big(\bar C \times \bar \beta \big)\cdot D^\mu \beta
- \frac{m}{2}\,\varepsilon^{\mu\nu\eta} F_{\nu\eta} \cdot \bar \beta,\nonumber\\
&&\nonumber\\
J^\mu_b &=& - (D_\nu C) \cdot \Big[F^{\mu\nu} - g \big(G^{\mu\nu} + g F^{\mu\nu} \times \rho\big)\times \rho
- m \,\varepsilon^{\mu\nu\eta} \phi_\eta\Big] + B \cdot (D^\mu C)\nonumber\\
&+& \frac{i}{2}\,g \big(C \times C\big) \cdot \partial^\mu \bar C - (D_\nu \beta)\cdot \Big(G^{\mu\nu} + g F^{\mu\nu} \times \rho \Big)
+ g (\phi_\nu \times C) \cdot \Big(G^{\mu\nu} + g F^{\mu\nu} \times \rho \Big)\nonumber\\
&-& g (\phi^\mu \times C)\cdot \big(R + i g C\times \bar \beta\big)
+ R\cdot D^\mu \beta + i g \big(C \times \beta \big)\cdot D^\mu \bar \beta
- \frac{m}{2}\,\varepsilon^{\mu\nu\eta} F_{\nu\eta} \cdot \beta.\end{aligned}$$ One can check that the conservation (i.e. $\partial_\mu J^\mu_b = 0$) of BRST current ($J^\mu_b$) can be proven by exploiting the Euler-Lagrange (E-L) equations of motion that are derived from the Lagrangian density ${\cal L}_b$. These E-L equations are as listed below: $$\begin{aligned}
&& D_\mu F^{\mu\nu} - g\,D_\mu\big[\big(G^{\mu\nu} + g\, F^{\mu\nu} \times \rho \big)\times \rho\big]
+ g \,\big(G^{\mu\nu} + g\, F^{\mu\nu} \times \rho \big) \times \phi_\mu - m \,\varepsilon^{\mu\nu\eta}\,(D_\mu \phi_\eta)
\nonumber\\
&& - \; \partial^\nu B - i g \big(\partial^\nu \bar C \times C\big)
+ g \big(R + i g C \times \bar \beta\big)\times \phi^\nu
+ i\, g \big(\bar \beta \times D^\nu \beta\big) - i\,g \big(\beta \times D^\nu \bar \beta\big) = 0, \nonumber\\
&& D_\mu \big[G^{\mu\nu} + g (F^{\mu\nu} \times \rho) \big] - D^\nu \big[R + i g(C \times \bar \beta)\big]
- \frac{m}{2}\, \varepsilon^{\mu\nu\kappa}\, F_{\mu\kappa} = 0, \nonumber\\
&& \big[G^{\mu\nu} + g (F^{\mu\nu} \times \rho) \big] \times F_{\mu\nu} = 0, \qquad R + ig (C \times \bar \beta) + D_\mu \phi^\mu = 0,
\qquad B = - \big(\partial_\mu A^\mu\big), \nonumber\\
&& \partial_\mu (D^\mu C) = 0, \qquad D_\mu (\partial^\mu \bar C) = 0, \qquad D_\mu (D^\mu \beta) = 0.
\qquad D_\mu (D^\mu \bar \beta) = 0, \label{5.2}\end{aligned}$$ $$\begin{aligned}
&& D_\mu F^{\mu\nu} - g\,D_\mu\big[\big(G^{\mu\nu} + g\, F^{\mu\nu} \times \rho \big)\times \rho\big]
+ g \,\big(G^{\mu\nu} + g\, F^{\mu\nu} \times \rho \big) \times \phi_\mu - m \,\varepsilon^{\mu\nu\eta}\,(D_\mu \phi_\eta)
\nonumber\\
&& +\; \partial^\nu \bar B + i g \big(\partial^\nu C \times \bar C\big)
- g \big(\bar R + i g \bar C \times \beta\big)\times \phi^\nu
+ i\, g \big(\bar \beta \times D^\nu \beta\big) - i\,g \big(\beta \times D^\nu \bar \beta\big) = 0, \nonumber\\
&& D_\mu \big[G^{\mu\nu} + g (F^{\mu\nu} \times \rho) \big] + D^\nu \big[\bar R + i g(\bar C \times \beta)\big]
- \frac{m}{2}\, \varepsilon^{\mu\nu\kappa}\, F_{\mu\kappa} = 0, \nonumber\\
&& \big[G^{\mu\nu} + g (F^{\mu\nu} \times \rho) \big] \times F_{\mu\nu} = 0, \qquad \bar R + ig (\bar C \times \beta) - D_\mu \phi^\mu = 0,
\qquad \bar B = \big(\partial_\mu A^\mu\big), \nonumber\\
&& D_\mu (\partial^\mu C) = 0, \qquad \partial_\mu (D^\mu \bar C) = 0, \qquad D_\mu (D^\mu \beta) = 0,
\qquad D_\mu (D^\mu \bar \beta) = 0, \label{5.3}\end{aligned}$$ which emerge from the Lagrangian density ${\cal L}_{\bar b}$.
Exploiting the above E-L equations of motion (cf. (\[5.2\]) and (\[5.3\])), the conserved currents $J^\mu_{(a)b}$ can be written in simpler forms as: $$\begin{aligned}
J^\mu_{ab} &=& - \partial_\nu \Big(\big[F^{\mu\nu} - g \big(G^{\mu\nu} + g F^{\mu\nu} \times \rho\big)\times \rho
- m \,\varepsilon^{\mu\nu\eta} \phi_\eta\big]\cdot \bar C + \big[G^{\mu\nu} + g F^{\mu\nu} \times \rho\big]\cdot \bar \beta\Big)\nonumber\\
&+& (\partial^\mu \bar B) \cdot \bar C - \bar B \cdot (D^\mu \bar C)
+ \frac{i}{2}\,g \big(\bar C \times \bar C\big) \cdot \partial^\mu C
- (\bar R + i g \bar C \times \beta \big)\cdot (D^\mu \bar \beta) \nonumber\\
&+& D^\mu\big(\bar R + i g \bar C \times \beta\big) \cdot \bar \beta,\nonumber\\
&&\nonumber\\
J^\mu_b &=& - \partial_\nu \Big(\big[F^{\mu\nu} - g \big(G^{\mu\nu} + g F^{\mu\nu} \times \rho\big)\times \rho
- m \,\varepsilon^{\mu\nu\eta} \phi_\eta\big]\cdot C + \big[G^{\mu\nu} + g F^{\mu\nu} \times \rho\big]\cdot \beta\Big)\nonumber\\
&+& B \cdot (D^\mu C) - (\partial^\mu B) \cdot C - \frac{i}{2}\,g \big(C \times C\big) \cdot \partial^\mu \bar C
+ (R + i g C \times \bar \beta \big)\cdot (D^\mu \beta)\nonumber\\
&-& D^\mu\big(R + i g C \times \bar \beta\big)\cdot \beta. \end{aligned}$$ Now, the proof of conservation laws ($\partial_\mu J^\mu_{(a)b} = 0$) is quite straightforward. The temporal components (i.e. $\int d^2x J^0_{(a)b} = Q_{(a)b}$) of the above conserved currents ($J^\mu_{(a)b}$) lead to the following conserved (i.e. ${\dot Q}_{(a)b} =0$) (anti-)BRST charges ($Q_{(a)b}$), namely; $$\begin{aligned}
Q_{ab} &=& - \int d^2x \bigg[\bar B \cdot (D^0 \bar C) - (\partial^0 \bar B) \cdot \bar C
- \frac{i}{2}\,g \big(\bar C \times \bar C\big) \cdot \partial^0 C
+ (\bar R + i g \bar C \times \beta \big)\cdot (D^0 \bar \beta)\nonumber\\
&-& D^0\big(\bar R + i g \bar C \times \beta\big)\cdot \bar \beta \bigg], \nonumber\\
&&\nonumber\\
Q_b &=& \int d^2x \bigg[ B \cdot (D^0 C) - (\partial^0 B) \cdot C
- \frac{i}{2}\,g \big(C \times C\big) \cdot \partial^0 \bar C
+ (R + i g C \times \bar \beta \big)\cdot (D^0 \beta) \nonumber\\
&-& D^0\big(R + i g C \times \bar \beta\big)\cdot \beta \bigg].\label{5.5}\end{aligned}$$ It turns out that the conserved, nilpotent ($Q^2_{(a)b} = 0$, see below) and anticommuting ($Q_b\,Q_{ab} + Q_{ab}\, Q_b = 0$, see below) (anti-)BRST charges are the generators of the (anti-)BRST symmetry transformations, respectively. For the sake of brevity, these transformations can be obtained by exploiting the following symmetry properties: $$\begin{aligned}
s_b \Psi = - i \big[\Psi, \; Q_b\big]_{\pm}, \qquad s_{ab} \Psi = - i \big[\Psi,\; Q_{ab} \big]_{\pm},\qquad
\Psi = A_\mu, \phi_\mu, C, \bar C, \beta, \bar \beta\end{aligned}$$ The $(\pm)$ signs as the subscript on the square brackets represent (anti)commutators corresponding to the generic field $\Psi$ being (fermionic)bosonic in nature (see, e.g. [@Gupta:2009bu] for details). The (anti-)BRST transformations of the Nakanishi-Lautrup auxiliary fields $B, \bar B, R, \bar R$ have been derived from the basic requirements (i.e. nilpotency and/or absolute anticommutativity properties) of the (anti-)BRST symmetry transformations.
It is worthwhile to mention that, even though, the (anti-)BRST charges $(Q_{(a)b})$ are conserved, nilpotent as well as anticommuting in nature (see below), they are unable to generate the proper (anti-)BRST transformations (i.e. $s_b \rho = \beta - g (\rho \times C)$ and $s_{ab} \rho = \bar \beta - g (\rho \times \bar C)$) of the auxiliary field $\rho$. Furthermore, the nilpotency and absolute anticommutativity properties of the (anti-)BRST transformations also fail to produce the transformations of $\rho$. This is one of the novel observations of our present endeavor. Although, we have derived these transformations by exploiting the power and strength of the augmented superfield formalism which produces the off-shell nilpotent ($s^2_{(a)b} = 0$) as well as absolutely anticommuting ($s_b \,s_{ab} + s_{ab}\, s_b = 0$) (anti-)BRST symmetry transformations for [*all*]{} the basic and auxiliary fields of the theory.
The nilpotency ($Q^2_{(a)b} =0$) of the (anti-)BRST charges reflects the fermionic nature whereas the anticommutativity ($Q_b\,Q_{ab} + Q_{ab}\, Q_b = 0$) shows that the (anti-)BRST charges are linearly independent of each other. These properties can be verified in the following straightforward manner: $$\begin{aligned}
s_b Q_b &=& - i \{Q_b,\; Q_b\} = 0 \Rightarrow Q^2_b = 0,\nonumber\\
s_{ab} Q_{ab} &=& - i \{Q_{ab},\; Q_{ab}\} = 0 \Rightarrow Q^2_{ab} = 0,\nonumber\\
s_b Q_{ab} &=& - i \{Q_{ab},\; Q_b\} = 0 \Rightarrow Q_{ab}\,Q_{b} + Q_{b}\, Q_{ab} = 0,\nonumber\\
s_{ab} Q_b &=& - i \{Q_b,\; Q_{ab}\} = 0 \Rightarrow Q_b\,Q_{ab} + Q_{ab}\, Q_b = 0.\end{aligned}$$ We point out that in proving the anticommutativity property ($Q_b\,Q_{ab} + Q_{ab}\, Q_b = 0$) of the (anti-)BRST charges we have used the CF conditions (\[3.26\]). For the sake of brevity, one can check $$\begin{aligned}
s_b Q_{ab} &=& - i \int d^2x\,\Big[\bar B \cdot \partial^0\Big(B + \bar B + i g C \times \bar C\Big)\Big] \nonumber\\
&+& \int d^2x\,\Big[g \Big(\big(B + \bar B + i g C \times \bar C\big) \times \beta \Big) \cdot D^0\beta
- g D^0\Big(\big(B + \bar B + i g C \times \bar C\big) \times \beta \Big) \cdot \bar \beta\Big] \nonumber\\
&-& i\int d^2x \Big[\big(R + \bar R + i g C \times \bar \beta
+ i g \bar C \times \beta\big) \cdot D^0\big(R + i g C \times \bar \beta\big)\Big],\nonumber\\\end{aligned}$$ $$\begin{aligned}
s_{ab} Q_b &=& i \int d^2x\,\Big[B \cdot \partial^0\Big(B + \bar B + i g C \times \bar C\Big)\Big] \nonumber\\
&-& \int d^2x\,\Big[g \Big(\big(B + \bar B + i g C \times \bar C\big) \times \bar \beta \Big) \cdot D^0 \beta
- g D^0\Big(\big(B + \bar B + i g C \times \bar C\big) \times \bar \beta \Big) \cdot \beta\Big] \nonumber\\
&+& i\int d^2x \Big[\big(R + \bar R + i g C \times \bar \beta
+ i g \bar C \times \beta\big) \cdot D^0\big(\bar R + i g \bar C \times \beta\big)\Big].\end{aligned}$$ It is clear from the above expressions that $s_b Q_{ab} = 0$ and $s_{ab} Q_b = 0$ if and only if CF conditions (\[3.26\]) are satisfied. As a consequence, the (anti-)BRST charges are anticommuting only on the constrained hypersurface defined by the CF conditions (\[3.26\]).
Ghost scale symmetry and BRST algebra
=====================================
The Lagrangian densities (\[4.2\]), in addition to the (anti-)BRST symmetry transformations, also respect the continuous ghost scale symmetry $(s_g)$. These symmetry transformations are given as follows $$\begin{aligned}
&&C \to e^{+\Omega}\,C, \qquad \bar C \to e^{-\Omega}\,\bar C, \qquad \beta \to e^{+\Omega}\,\beta,
\qquad \bar \beta \to e^{-\Omega}\,\bar \beta, \nonumber\\
&& \big(A_\mu, \phi_\mu, \rho, B, \bar B, R, \bar R\big) \to e^0 \big(A_\mu, \phi_\mu, \rho, B, \bar B, R, \bar R\big)\end{aligned}$$ where $\Omega$ is the global scale parameter. The numbers ($\pm1, 0$) in the exponential of the above transformations stand for ghost numbers of the corresponding fields. For instance, the ghost fields $(C, \beta)$ carry ghost number $(+1)$ and anti-ghost fields $(\bar C, \bar \beta)$ have ghost number ($-1$). The rest (bosonic) fields have ghost number zero. The infinitesimal version of the above continuous transformation is given by $$\begin{aligned}
&& s_g C = + \Omega \,C, \qquad s_g \bar C = - \Omega \,\bar C, \qquad s_g \beta = + \Omega \,\beta,
\qquad s_g \bar \beta = - \Omega \bar \beta, \nonumber\\
&& s_g\big(A_\mu, \phi_\mu, \rho, B, \bar B, R, \bar R\big) =0. \label{6.2}\end{aligned}$$ It is straightforward to check that under the above continuous ghost scale symmetry transformations (\[6.2\]) both the Lagrangian densities remain invariant (i.e. $s_g{\cal L}_b = s_g {\cal L}_{\bar b} = 0$). As a consequence, the existence of ghost scale symmetry leads to the following Noether’s conserved current ($J^\mu_g$) and charge ($Q_g$): $$\begin{aligned}
J^\mu_g &=& i \Big[\bar C \cdot (D^\mu C) - (\partial^\mu \bar C) \cdot C
+ \bar \beta \cdot (D^\mu \beta) - (D^\mu \bar \beta) \cdot \beta \Big], \nonumber\\
Q_g &=& i \int d^2x \,\Big[\bar C \cdot (D^0 C) - (\partial^0 \bar C) \cdot C
+ \bar \beta \cdot (D^0 \beta) - (D^0 \bar \beta) \cdot \beta \Big].\end{aligned}$$ The conservation law $(\partial_\mu J^\mu_g = 0)$ can be proven by exploiting the E-L equations of motion (\[5.2\]). The ghost charge $Q_g$ also turns out to be the generator of the ghost scale symmetry transformations (\[6.2\]). For instance, one can check that $s_g C = - i [C, \, \Omega\,Q_g] = +\Omega\, C.$
The above ghost charge $Q_g$ together with the nilpotent (anti-)BRST charges $Q_{(a)b}$ obey a standard BRST algebra. In operator form, this algebra can be given as follow $$\begin{aligned}
&& Q^2_b = 0, \qquad Q^2_{ab} = 0, \qquad \big\{Q_b,\; Q_{ab}\big\} = Q_b\,Q_{ab} + Q_{ab}\, Q_b = 0,\nonumber\\
&& i \big[Q_g, \,Q_b\big] = + Q_b, \qquad i \big[Q_g, Q_{ab}\big] = - Q_{ab}, \qquad Q^2_g \ne 0. \label{6.4}\end{aligned}$$
Let us consider a state $|\psi \rangle _n$, in the quantum Hilbert space of states, such that the ghost number of the state is defined in the following manner $$\begin{aligned}
i Q_g |\psi \rangle _n = n |\psi \rangle _n ,\end{aligned}$$ where $n$ is the ghost number of the state $|\psi \rangle _n$. Now, it is easy to check, with the help of above algebra (\[6.4\]), that following relationships holds $$\begin{aligned}
&& i Q_g Q_b |\psi \rangle _n = (n + 1) Q_b |\psi \rangle _n , \nonumber\\
&& i Q_g Q_{ab} |\psi \rangle _n = (n - 1) Q_{ab} |\psi \rangle _n , \end{aligned}$$ which shows that the BRST charge $Q_b$ increases the ghost number by one unit when it operates on a quantum state whereas the anti-BRST charge $Q_{ab}$ decreases it by one unit. In other words, we can say that the (anti-)BRST charge carry the ghost numbers $(\mp 1)$, respectively. A careful look at the expressions of the (anti-)BRST and ghost charges, where the ghost numbers of the fields are concerned, also reveal the same observations.
Role of auxiliary field: A bird’s-eye view
==========================================
In this section we provide a brief synopsis about few striking similarities and some glaring differences among the 3D non-Abelian JP model, 4D topologically massive non-Abelian 2-form gauge theory [@Malik:2010gu; @Kumar:2010kd] and the 4D modified gauge invariant Proca theory in the realm of well-known St[ü]{}ckelberg formalism (see, e.g. [@Ruegg:2003ps] for details).
Jackiw-Pi model
---------------
It is interesting to note that, if we make the following substitution $$\begin{aligned}
\phi_\mu \;\longrightarrow \; \phi_\mu + D_\mu \rho, \label{7.1}\end{aligned}$$ in our starting Lagrangian density (\[2.1\]), the 2-form $G_{\mu\nu}$ and mass term re-defined as $$\begin{aligned}
G_{\mu\nu}& \longrightarrow & G_{\mu\nu} - g \big(F_{\mu\nu} \times \rho \big),\nonumber\\
\frac{m}{2}\,\varepsilon^{\mu\nu\eta}\,F_{\mu\nu} \cdot \phi_\eta & \longrightarrow &
\frac{m}{2}\,\varepsilon^{\mu\nu\eta}\,F_{\mu\nu} \cdot \phi_\eta
+ \partial_\eta \Big[\frac{m}{2}\, \varepsilon^{\mu\nu\eta}\,F_{\mu\nu} \cdot \rho \Big]
- \frac{m}{2}\,\varepsilon^{\mu\nu\eta}\,\big(D_\eta F_{\mu\nu}\big) \cdot \rho. \qquad\end{aligned}$$ In the above, the term $\frac{m}{2}\,\varepsilon^{\mu\nu\eta}\,\big(D_\eta F_{\mu\nu}\big) \cdot \rho$ is zero due to the validity of the well-known Bianchi identity $(D_\mu F_{\nu\eta} + D_\nu F_{\eta\mu} + D_\eta F_{\mu\nu} = 0)$. Therefore, the mass term remains invariant, modulo a total spacetime derivative, under the re-definition (\[7.1\]). As a consequence, the modified Lagrangian density, modulo a total spacetime derivative, is given by $$\begin{aligned}
\tilde{\cal L}_0 = - \frac{1}{4}\, F_{\mu\nu}\cdot F^{\mu\nu}
- \frac{1}{4}\, G_{\mu\nu}\cdot G^{\mu\nu} + \frac{m}{2}\, \varepsilon^{\mu\nu\eta}\,F_{\mu\nu}\cdot \phi_\eta. \label{7.3}\end{aligned}$$ It is clear that the auxiliary field $\rho$ is completely eliminated from the above Lagrangian density. We point out that, even though, Lagrangian density (\[7.3\]) respects the YM gauge transformations (\[2.2\]) but it fails to respect the NYM gauge transformations (\[2.3\]). The similar observation can also be seen in the case of 4D topologically massive non-Abelain 2-form gauge theory as well as in the 4D modified gauge invariant version of Proca theory.
4D massive non-Abelian 2-form gauge theory
------------------------------------------
The Lagrangian density for the 4D massive non-Abelian 2-form gauge theory is given by (see, for details [@Kumar:2011zi; @Malik:2010gu; @Kumar:2010kd]) $$\begin{aligned}
{\cal L} = - \frac{1}{4}\, F_{\mu\nu}\cdot F^{\mu\nu}
+ \frac{1}{12}\, H_{\mu\nu\eta}\cdot H^{\mu\nu\eta} + \frac{m}{4}\, \varepsilon^{\mu\nu\eta\kappa}\,B_{\mu\nu}\cdot F_{\eta\kappa}, \label{7.4}\end{aligned}$$ where 3-form $H_{\mu\nu\eta} = D_\mu B_{\nu\eta} + D_\nu B_{\eta\mu} + D_\eta B_{\mu\nu} + g(F_{\mu\nu} \times K_\eta)
+ g(F_{\nu\eta} \times K_\mu)+ g (F_{\eta\mu} \times K_\nu)$ is the field strength tensor corresponding to the 2-form gauge field $B_{\mu\nu}$ and the 2-form field strength tensor $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - g (A_\mu \times A_\nu)$ corresponds to the 1-form gauge field $A_\mu$. The coupling constant is represented by $g$ and $D_\mu$ is the covariant derivative. The auxiliary field $K_\mu$ is the compensating field. This Lagrangian density respects the two types of gauge transformations – the scalar gauge transformation $(\tilde \delta_1)$ and vector gauge transformation $(\tilde \delta_2)$, namely; [@Kumar:2011zi; @Malik:2010gu; @Kumar:2010kd] $$\begin{aligned}
&&\tilde \delta_1 A_\mu = D_\mu \Omega, \qquad \tilde \delta_1 B_{\mu\nu} = - g(B_{\mu\nu} \times \Omega),
\qquad \tilde \delta_1 K_\mu = - g(K_\mu \times \Omega), \nonumber\\
&& \tilde \delta_2 A_\mu =0, \qquad \tilde \delta_2 B_{\mu\nu} = - (D_\mu \Lambda_\nu - D_\nu \Lambda_\mu),
\qquad \tilde \delta_2 K_\mu = - \Lambda_\mu, \label{7.5}\end{aligned}$$ where $\Omega (x)$ and $\Lambda_\mu (x)$ are the local scalar and vector gauge parameters, respectively. We note that if we re-define the $B_{\mu\nu}$ field as $$\begin{aligned}
B_{\mu\nu} \;\longrightarrow \; B_{\mu\nu} + (D_\mu K_\nu - D_\nu K_\mu),\end{aligned}$$ the 3-form field strength tensor $H_{\mu\nu\eta}$ and the mass term modify as follows $$\begin{aligned}
H_{\mu\nu\eta} & \longrightarrow& \tilde H_{\mu\nu\eta}\;=\; D_\mu B_{\nu\eta} + D_\nu B_{\eta\mu} + D_\eta B_{\mu\nu}, \nonumber\\
\frac{m}{4}\,\varepsilon^{\mu\nu\eta\kappa}\,B_{\mu\nu} \cdot F_{\eta\kappa} &\longrightarrow&
\frac{m}{4}\,\varepsilon^{\mu\nu\eta\kappa}\,B_{\mu\nu} \cdot F_{\eta\kappa}
+ \partial_\mu \Big[\frac{m}{2}\, \varepsilon^{\mu\nu\eta\kappa}\, K_\nu \cdot F_{\eta\kappa} \Big] \nonumber\\
&& \qquad \qquad \qquad \qquad -\; \frac{m}{2}\,\varepsilon^{\mu\nu\eta\kappa}\,K_\nu \cdot \big(D_\mu F_{\eta\kappa}\big). \end{aligned}$$ and the compensating auxiliary vector field $K_\mu$ disappears from the Lagrangian density (\[7.4\]). Furthermore, the mass term $\displaystyle \frac{m}{4}\,\varepsilon^{\mu\nu\eta\kappa}\,B_{\mu\nu} \cdot F_{\eta\kappa}$ remains intact modulo a total spacetime derivative. Thus, the modified Lagrangian density can be given in the following manner (modulo a total spacetime derivative) $$\begin{aligned}
\tilde {\cal L} = - \frac{1}{4}\, F_{\mu\nu}\cdot F^{\mu\nu}
+ \frac{1}{12}\, {\tilde H}_{\mu\nu\eta}\cdot \tilde{H}^{\mu\nu\eta}
+ \frac{m}{4}\, \varepsilon^{\mu\nu\eta\kappa}\,B_{\mu\nu}\cdot F_{\eta\kappa}. \label{58}\end{aligned}$$ Clearly, the above Lagrangian density is no longer invariant under the vector gauge transformation even though it respects the scalar gauge transformations \[cf. (\[7.5\])\].
It is clear form the above discussions that both the above models (i.e. JP model and 4D massive non-Abelian 2-form gauge theory) are very similar to each other in the sense that under the re-definitions of the fields $\phi_\mu$ and $B_{\mu\nu}$ the auxiliary fields $\rho$ and $K_\mu$ are eliminated from their respective models. As a result, the modified Lagrangian densities (\[7.3\]) and (\[58\]) do not respect the symmetry transformations $(\delta_2)$ and $(\tilde \delta_2)$, respectively. Thus, the auxiliary fields $\rho$ and $K_\mu$ are required in their respective models so that these models respect both the gauge symmetry transformations \[cf. (\[2.2\]), (\[2.3\]) and (\[7.5\])\].
Modified version of Abelian Proca theory
----------------------------------------
The above key observations can also be seen in the case of modified gauge invariant Abelian Proca theory. The gauge invariant Lagrangian density of this model is as follows [@Ruegg:2003ps] $$\begin{aligned}
{\cal L}_s = - \frac{1}{4}\, F_{\mu\nu}\,F^{\mu\nu} + \frac{m^2}{2}\, A_\mu \,A^\mu
+ \frac{1}{2}\,\partial_\mu \phi\,\partial^\mu \phi + m A_\mu\, \partial^\mu \phi, \label{7.8}\end{aligned}$$ where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ is the field strength tensor corresponding to $A_\mu$, $\phi$ is the St[ü]{}ckelberg field and $m$ represents the mass of the photon field $A_\mu$. Under the following local gauge transformations $$\begin{aligned}
\delta_{(gt)}A_\mu = \partial_\mu \chi(x), \qquad \delta_{(gt)} \phi = - m\,\chi(x), \label{7.9}\end{aligned}$$ the Lagrangian density (\[7.8\]) remains invariant. Here $\chi(x)$ is the local gauge transformation parameter. It can be checked that under the following re-definition $$\begin{aligned}
A_\mu \;\longrightarrow\; A_\mu - \frac{1}{m}\,\partial_\mu \phi,\end{aligned}$$ the St[ü]{}ckelberg field $\phi$ completely disappears from the Lagrangian density (\[7.8\]). As a consequence, the resulting Lagrangian density does not respect the above gauge transformations (\[7.9\]).
The above observation is very similar to the JP model and the massive non-Abelian 2-form gauge theory. As a consequence, the field $\rho$ (in JP model) and $K_\mu$ (in massive non-Abelian 2-form theory) are like the St[ü]{}ckelberg field. However, the key difference is that these St[ü]{}ckelberg like fields (i.e. $\rho$ and $K_\mu$) are auxiliary fields in their respective models whereas, in the modified gauge invariant Proca theory, the St[ü]{}ckelberg field $\phi$ is dynamical in nature.
Conclusions
===========
In our present investigation, we have derived the off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations corresponding to the combined YM and NYM symmetries of the JP model. For this purpose, we have utilized the power and strength of augmented superfield approach. The derivation of proper (anti-)BRST symmetries for the auxiliary field $\rho$ is one of the main findings of our present endeavor. These (anti-)BRST symmetry transformations corresponding to the auxiliary field $\rho$ can neither be generated from the conserved (anti-)BRST charges nor deduced by the requirement of nilpotency and/or absolute anticommutativity of the (anti-)BRST symmetry transformations.
One of the main features of the superfield formalism is the derivation of CF conditions which, in turn, ensure the absolutely anticommutativity of (anti-) BRST symmetry transformations. The CF conditions, a hallmark of any non-Abelian 1-form gauge theories [@Curci:1976ar], appear naturally within the framework of superfield formalism and also have connections with gerbes [@Bonora:2007hw]. In our present case of combined YM and NYM symmetries of JP model, there exist [*two*]{} CF conditions (cf. (\[3.26\])). This is in contrast to the YM symmetries case where there exist only [*one*]{} CF condition [@Gupta:2011cta] and in NYM symmetries case, [*no*]{} CF condition was observed [@Gupta:2012ur]. Moreover, these CF conditions have played a central role in the derivation of coupled Lagrangian densities (cf. Section 4).
Furthermore, we have obtained a set of coupled Lagrangian densities which respect the above mentioned (anti-)BRST symmetry transformations. The ghost sector of these coupled Lagrangian densities is also endowed with another continuous symmetry - the ghost symmetry. We have exploited this symmetry to derive the conserved ghost charge. Moreover, we have pointed out the standard BRST algebra obeyed by all the conserved charges of the underlying theory.
At the end, we have provided a bird’s-eye view on the role of auxiliary field in the context of various massive models. For this purpose, we have taken three different cases of 3D JP model, 4D massive non-Abelian 2-form gauge theory and the 4D modified version of Abelian Proca theory. We have shown that the field $\rho$ (in JP model) and $K_\mu$ (in massive non-Abelian 2-form theory) are like St[ü]{}ckelberg field ($\phi$) of Abelian Proca model. However, $\rho$ and $K_\mu$ are auxiliary fields whereas $\phi$ is dynamical, in their respective models. Finally, we capture the (anti-)BRST invariance of the coupled Lagrangian densities (cf. (\[4.2\])), nilpotency and absolute anticommutativity of (anti-)BRST charges (cf. (\[5.5\])) within the framework of superfield approach.
Acknowledgments {#acknowledgments .unnumbered}
===============
The research work of SG is supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) grant 151112/2014-2.
(Anti-)BRST invariance, nilpotency and anticommutativity: Superfield approach
=============================================================================
It is interesting to point out that the super expansions (\[3.5\]), (\[3.15\]) and (\[3.22\]) can be expressed in terms of the translations of the corresponding superfields along the Grassmannian directions of the $(3,2)$-dimensional supermanifold, as $$\begin{aligned}
&& s_b \Psi (x) = \frac {\partial}{\partial \bar \theta} \, \tilde \Psi^{(h)}(x,\theta,\bar\theta)\Big|_{\theta = 0}, \qquad \quad
s_{ab} \Psi (x) = \frac {\partial}{\partial \theta} \, \tilde \Psi^{(h)}(x,\theta,\bar\theta)\Big|_{\bar \theta = 0},\nonumber\\
&&s_b\, s_{ab} \Psi (x) = \frac {\partial}{\partial \bar \theta} \, \frac {\partial}{\partial \theta}
\, \tilde \Psi^{(h)} (x,\theta,\bar\theta), \label{b1}\end{aligned}$$ where $\Psi (x)$ is any generic field of the underlying 3D theory and $\Psi^{(h)} (x,\theta,\bar\theta)$ is the corresponding superfield obtained after the application of HC. The above expression captures the off-shell nilpotency of the (anti-)BRST symmetries because of the properties of Grassmannian derivatives, i.e. $\partial_\theta^2 = \partial_{\bar \theta}^2 = 0$. Moreover, the anticommutativity property of the (anti-)BRST symmetry transformations is also clear from the expansions (\[3.5\]), (\[3.15\]) and (\[3.22\]), in the following manner $$\begin{aligned}
\Big(\frac {\partial}{\partial \theta} \; \frac {\partial}{\partial \bar \theta} +
\frac {\partial}{\partial \bar \theta} \; \frac {\partial}{\partial \theta} \Big) \; \tilde \Psi^{(h)}
(x, \theta, \bar \theta ) = 0. \label{b2}\end{aligned}$$ Thus, the expressions (\[b1\]) and (\[b2\]) provide the geometrical interpretations for the (anti-) BRST symmetry transformations in terms of the translational generators $(\partial_\theta, \partial_{\bar\theta})$ along the Grassmannian directions of the $(3, 2)$-dimensional supermanifold.
Furthermore, the nilpotency of the (anti-)BRST charges can also be realized, within the framework of superfield formalism, in the following manner $$\begin{aligned}
Q_b &=& \frac {\partial}{\partial \bar \theta} \, \int d^2 x \,
\Big[B(x) \cdot \tilde {\cal A}_0^{(h)} (x, \theta, \bar\theta) + i \, {\dot{\tilde {\bar {\cal F}}}}^{(h)}(x, \theta, \bar\theta)
\cdot \tilde {\cal F}^{(h)} (x, \theta, \bar\theta)\nonumber\\
&+& \Big(R(x)+ ig \tilde {\cal F}^{(h)}(x, \theta, \bar\theta) \times {\tilde {\bar \beta}}^{(h)}(x, \theta, \bar\theta) \Big)
\cdot \tilde\Phi^{(h)}_0 (x, \theta, \bar\theta) \nonumber\\
&+& i {\tilde {\cal D}}_0{\tilde {\bar \beta}}^{(h)}
(x, \theta, \bar\theta) \cdot \tilde \beta^{(h)} (x, \theta, \bar\theta)\Big] \bigg|_{\theta = 0} \nonumber\\
&\equiv& \int d^2 x \int d \bar\theta \;
\Big[B(x) \cdot \tilde {\cal A}_0^{(h)} (x, \theta, \bar\theta) + i \; {\dot{\tilde {\bar {\cal F}}}}^{(h)}(x, \theta, \bar\theta)
\cdot \tilde {\cal F}^{(h)} (x, \theta, \bar\theta ) \nonumber\\
&+& \Big(R(x)+ ig \tilde {\cal F}^{(h)}(x, \theta, \bar\theta) \times {\tilde {\bar \beta}}^{(h)}(x, \theta, \bar\theta) \Big)
\cdot \tilde\Phi^{(h)}_0 (x, \theta, \bar\theta) \nonumber\\
&+& i {\tilde {\cal D}}_0{\tilde {\bar \beta}}^{(h)}
(x, \theta, \bar\theta) \cdot \tilde \beta^{(h)} (x, \theta, \bar\theta)\Big] \bigg|_{\theta = 0}. \end{aligned}$$ This, in turn, implies $$\begin{aligned}
\frac {\partial}{\partial \bar\theta} \; Q_b \bigg|_{\theta = 0} = 0
\quad \Longrightarrow \quad Q_b^2 = 0,\end{aligned}$$ because of the nilpotency property of the Grassmannian derivative (i.e. $\partial_{\bar \theta}^2 = 0$). It is interesting to point out that the nilpotency of above BRST charge $(Q_b)$, when written in ordinary 3D spacetime, $$\begin{aligned}
Q_b &=& \int d^2 x\, s_b \ \Big[B(x) \cdot A_0 (x) + i \; \dot{\bar C} (x) \cdot C(x)
+ \Big(R(x) + i g C(x) \times \bar \beta(x)\Big)\cdot \phi_0(x) \nonumber\\
&+& i D_0 \bar \beta(x) \cdot \beta(x) \Big],\end{aligned}$$ is straightforward and encoded in the nilpotency property $(s_b^2 = 0)$ of the BRST transformations $(s_b)$. In other words, $s_b Q_b = -i \{Q_b, Q_b\} = 0$ is true due to above mentioned reason. Moreover, using the CF-conditions, there is yet another way to express the above BRST charge where nilpotency is quite clear, as can be seen from the following expression $$\begin{aligned}
Q_b &=& i \frac {\partial} {\partial \bar \theta} \, \frac {\partial} {\partial \theta} \int
d^2 x \Big[ \tilde {\cal A}_0^{(h)} (x, \theta, \bar \theta) \cdot \tilde {\cal F}^{(h)}
(x, \theta, \bar \theta) + \tilde \Phi_0^{(h)} (x, \theta, \bar \theta) \cdot \tilde \beta^{(h)}
(x, \theta, \bar \theta)\Big] \nonumber\\
&\equiv& i \int d^2 x \, s_b\, s_{ab} \, \Big[\, A_0 (x) \cdot C(x) + \phi_0(x) \cdot \beta(x)\Big].\end{aligned}$$ This is true only on the constrained surface spanned by CF conditions. Similarly, we can express the anti-BRST charge $(Q_{ab})$ in the following two different ways: $$\begin{aligned}
Q_{ab} &= & - \frac {\partial}{\partial \theta} \, \int d^2 x \,
\Big[\bar B(x) \cdot \tilde {\cal A}_0^{(h)} (x, \theta, \bar\theta) + i \, \dot{\tilde{{\cal F}}}^{(h)}(x, \theta, \bar\theta)
\cdot {\tilde {\bar {\cal F}}}^{(h)} (x, \theta, \bar\theta)\nonumber\\
&-& \Big(\bar R(x)+ ig {\tilde {\bar {\cal F}}}^{(h)}(x, \theta, \bar\theta) \times {\tilde {\beta}}^{(h)}(x, \theta, \bar\theta) \Big)
\cdot \tilde\Phi^{(h)}_0(x, \theta, \bar\theta)\nonumber\\
&-& i {\tilde {\cal D}}_0{\tilde {\beta}}^{(h)}
(x, \theta, \bar\theta) \cdot {\tilde {\bar \beta}}^{(h)} (x, \theta, \bar\theta)\Big] \bigg|_{\theta = 0} \nonumber\\
& \equiv & i \frac {\partial}{\partial \theta} \frac {\partial}{\partial \bar \theta}
\int d^2 x \Big[ \tilde {\cal A}_0^{(h)} (x, \theta, \bar\theta) \cdot
{\tilde {\bar {\cal F}}}^{(h)} (x, \theta, \bar\theta)
+ \tilde \Phi_0^{(h)} (x, \theta, \bar \theta) \cdot {\tilde {\bar \beta}}^{(h)} (x, \theta, \bar \theta)\Big]. \label{A7}\end{aligned}$$ In the above, the second expression is valid on the constrained hypersurface parametrized by the CF conditions. The nilpotency of anti-BRST charge (i.e. $Q_{ab}^2 = 0$) is assured by the nilpotency $(\partial_\theta^2 = 0)$ of the Grassmannian derivative $\partial_\theta$, as described below $$\begin{aligned}
\frac {\partial}{\partial \theta} \; Q_{ab} \bigg|_{\bar \theta = 0} = 0
\quad \Longrightarrow \quad Q_{ab}^2 = 0.\end{aligned}$$ In 3D ordinary space, the above expression (\[A7\]) can be written in the following fashion $$\begin{aligned}
Q_{ab} &=& - \int d^2 x \; s_{ab}\; \Big [ \bar B(x) \cdot A_0(x) + i \; \dot C (x) \cdot \bar C (x)
+ \Big(\bar R(x) + i g \bar C(x) \times \beta(x)\Big)\cdot \phi_0(x) \nonumber\\
&+& i D_0 \beta(x) \cdot \bar \beta(x)\Big] \nonumber\\
&\equiv& - i\, \int s_{ab} s_b \,\Big[\, A_0 (x) \cdot \bar C(x) + \phi_0(x) \cdot \bar \beta(x)\Big].\end{aligned}$$ Here, the nilpotency of the anti-BRST charge lies in the equation $s_{ab} Q_{ab} = - i \{Q_{ab}, Q_{ab} \} = 0$ because of the fact that $s_{ab}^2 = 0$.
Furthermore, in order to prove the (anti-)BRST invariance of the coupled Lagrangian densities, within the framework of superfield formalism, we first generalize our starting Lagrangian density (${\cal L}_0$) onto the $(3,2)$-dimensional supermanifold, as follows $$\begin{aligned}
{\cal L}_0 \to \tilde {\cal L}_0 &=& - \frac{1}{4}\, \tilde {\cal F}^{\mu\nu (h)}
\cdot \tilde {\cal F}_{\mu\nu}^{(h)} - \;\frac{1}{4}\, \Big[\tilde {\cal G}^{\mu\nu (h)}
+ g \, \tilde {\cal F}^{\mu\nu (h)} \times \tilde \rho^{(h)} \Big]
\cdot \Big[\tilde {\cal G}_{\mu\nu}^{(h)} + g \, \tilde {\cal F}_{\mu\nu}^{(h)} \times \tilde \rho^{(h)} \Big] \nonumber\\
&+& \frac {m}{2}\,\varepsilon^{\mu\nu\eta} \, \tilde {\cal F}_{\mu\nu}^{(h)} \cdot \tilde \Phi_\eta^{(h)}.\end{aligned}$$ This Lagrangian density ($\tilde {\cal L}_0$) is free from the Grassmannian variables (cf. Section 3, for details). Therefore, the followings are true $$\begin{aligned}
\frac {\partial}{\partial \bar \theta} \; \tilde {\cal L}_0\bigg|_{\theta = 0} = 0, \qquad
\frac {\partial}{\partial \theta} \; \tilde {\cal L}_0\bigg|_{\bar \theta = 0} = 0, \end{aligned}$$ which captures the (anti-)BRST invariance of the starting Lagrangian density ${\cal L}_0$. Similarly, we can also generalize the coupled Lagrangian densities (\[4.2\]) onto the $(3,2)$-dimensional supermanifold in the following manner $$\begin{aligned}
{\cal L}_{\bar b} \longrightarrow \tilde {\cal L}_{\bar b} & =& \tilde {\cal L}_0 - \frac {\partial}
{\partial \theta}\, \frac {\partial} {\partial \bar \theta}\, \Big[ \frac {i} {2} \, \tilde {\cal A}_\mu^{(h)}
\cdot \tilde {\cal A}^{\mu (h)} + \tilde {\cal F}^{(h)} \cdot {\tilde {\bar {\cal F}}}^{(h)}
+ \frac {i} {2} \, \tilde \Phi_\mu^{(h)} \cdot \tilde \Phi^{\mu (h)}
+ \frac{1}{2}\, \tilde \beta^{(h)} \cdot {\tilde {\bar \beta}}^{(h)} \Big],\nonumber\\
&&\nonumber\\
{\cal L}_{b} \longrightarrow \tilde {\cal L}_{b} & = & \tilde {\cal L}_0 + \frac {\partial}
{\partial \bar \theta}\, \frac {\partial} {\partial \theta}\, \Big [ \frac {i} {2} \, \tilde {\cal A}_\mu^{(h)}
\cdot \tilde {\cal A}^{\mu (h)} + \tilde {\cal F}^{(h)} \cdot {\tilde {\bar {\cal F}}}^{(h)}
+ \frac {i}{2} \, \tilde \Phi_\mu^{(h)} \cdot \tilde \Phi^{\mu (h)}
+ \frac{1}{2}\, \tilde \beta^{(h)} \cdot {\tilde {\bar \beta}}^{(h)} \Big].\nonumber\\\end{aligned}$$ Now, the (anti-)BRST invariance of the above coupled Lagrangian densities is straightforward because of the fact $(\partial_\theta^2 = \partial_{\bar \theta}^2 = 0)$. Thus, we have $$\begin{aligned}
\frac {\partial}{\partial \theta}\; \tilde{\cal L}_{\bar b}\bigg|_{\bar \theta = 0} = 0,\qquad
\frac {\partial}{\partial \bar \theta}\; \tilde{\cal L}_b \bigg|_{\theta = 0} = 0,\end{aligned}$$ which imply the (anti-)BRST invariance of the coupled Lagrangian densities within the framework of superfield formalism.
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[^1]: The covariant derivative is defined as $D_\mu * = \partial_\mu * - g (A_\mu \times * )$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This paper presents a prescription for distilling the information contained in the cosmic microwave background radiation from multiple sky maps that also contain both instrument noise and foreground contaminants. The prescription is well-suited for cosmological parameter estimation and accounts for uncertainties in the cosmic microwave background extraction scheme. The technique is computationally viable at low resolution and may be considered a natural and significant generalization of the “Internal Linear Combination” approach to foreground removal. An important potential application is the analysis of the multi-frequency temperature and polarization data from the forthcoming Planck satellite.'
author:
- Steven Gratton
bibliography:
- 'concept.bib'
title: Prescription for Cosmic Information Extraction from Multiple Sky Maps
---
[\[sec:intro\]]{}Introduction
=============================
The detection and subsequent investigation of the cosmic microwave background ([[CMB]{}]{}) radiation over the past four decades has been essential to our current understanding of the universe. Initially providing evidence for an early hot dense radiation-dominated phase to the universe [@1965ApJ...142..419P; @1965ApJ...142..414D], the [[CMB]{}]{}’s spatial distribution is now studied in precise detail (see [@Hinshaw:2008kr; @Gold:2008kp; @Nolta:2008ih; @Dunkley:2008ie; @Wright:2008ib; @Hill:2008hx; @Komatsu:2008hk] for the latest analysis of the [[WMAP]{}]{} satellite data by the [[WMAP]{}]{} science team) for clues both about the state of the universe at the beginning of the radiation era and the universe’s composition, structure and subsequent evolution. As cosmologists we have been exceedingly fortunate that, from our vantage point of the earth, the galaxy (away from its plane) is both sufficiently transparent and sufficiently lacking in emission for us to be able to readily measure the intensity of the cosmic signal shining through. The cosmic signal is also moderately polarized and this carries important additional cosmological information, in particular about the presence of gravity waves in the universe (which could be generated by say a high energy inflationary phase before the radiation era) and about the reionization of the universe. Unfortunately, with the signal being weaker and the polarization of the galactic emission being less understood than its intensity, both the detection and analysis of the polarization part of the signal are much more challenging. In addition, there is also extragalactic contamination, such as that from point sources. This paper presents a technique to extract the cosmological information out of multiple sky maps, including polarization ones, in the presence of both instrument noise and foregrounds. The technique rests upon the assumption that the foregrounds do not have the same frequency response as the [[CMB]{}]{} and requires that the sky maps probe linearly independent parts of the spectrum of the sky signal.
The prescription presented here relates to and builds upon a number of approaches already in the literature. The COBE team considered three approaches to separating galactic emission from the cosmic signal, one involving modelling known emissions using non-[[CMB]{}]{} data, another involving fitting the maps to functions of given spectral index, and a final one involving linearly combining their multifrequency maps to cancel the dominant galactic emission [@1992ApJ...396L...7B]. The [[WMAP]{}]{} team primarily use a template subtraction procedure to mitigate the effects of foreground contaminants in power spectrum estimation, and study the foregrounds themselves via maximum entropy and most recently Markov Chain Monte Carlo parametric methods. However, initially for visualization purposes but later also for analysis they additionally introduced the “Internal Linear Combination” ([[ILC]{}]{}) scheme, forming a linear combination of their sky maps and choosing the weights to minimize the variance between the maps whilst being constrained to preserve unit response to the [[CMB]{}]{}[@Bennett:2003ca; @Hinshaw:2006ia; @Gold:2008kp]. An harmonic mode-by-mode equivalent of [[ILC]{}]{} was presented in [@1996MNRAS.281.1297T] and applied to the [[WMAP]{}]{} data in [@Tegmark:2003ve]. An alternative harmonic-based generalization of the [[ILC]{}]{} technique was recently presented in [@Kim:2008zh]. The “Independent Component Analysis” ([[ICA]{}]{}) signal processing technique (see [@hyvarinen]), which attempts to use non-gaussianity of the one-pixel distribution to separate data into independent signals, has been tested for cosmological uses on the COBE, BEAST and [[WMAP]{}]{}data [@Maino:2001vz; @Maino:2003as; @Donzelli:2005is; @Maino:2006pq]. A possible weakness of [[ICA]{}]{} to [[CMB]{}]{} extraction is that the [[CMB]{}]{} is believed to be very close to gaussian and so can only emerge as what is left behind after the non-gaussian foregrounds have been removed. The related “Correlated Component Analysis” idea uses pixel-pixel cross correlations instead of non-quadratic single-pixel statistics to separate signals and has also been applied to the [[WMAP]{}]{}data [@Bonaldi:2006qw; @Bonaldi:2007mf]. A modification of [[ICA]{}]{} that forces it to take into account the black body nature of the [[CMB]{}]{} (a key feature of the approach described here) was recently presented in [@Vio:2008kw]. The “spectral matching” approach of [@Delabrouille:2002kz; @Patanchon:2004kj] shares many similarities to that presented here, and a recent paper [@Cardoso:2008qt] presents an “additive component”-based separation technique. [@Eriksen:2005dr] suggests fitting model parameters at low resolution and then using these to solve for high resolution maps; see also the very recent work [@Stompor:2008sf] for more on parametric component separation. A Gibbs sampling based approach to component separation and [[CMB]{}]{} power spectrum estimation was presented in [@Eriksen:2007mx] and applied to the [[WMAP]{}]{} data in [@Eriksen:2007mp]. The [[WMAP]{}]{} team also test this approach in [@Dunkley:2008ie].
Unlike many of the above papers, this work focusses on likelihood estimation for cosmological models as opposed to [[CMB]{}]{} sky map production. This requires a quantification of the uncertainties related to the [[CMB]{}]{} extraction (as also stressed by Ref. [@Eriksen:2006pn]). Of course this is a somewhat ill-defined problem seeing as one does not know precisely what the foregrounds are. By putting relatively weak priors on the foregrounds though one might hope that such errors are being estimated if anything conservatively. Our prescription allows one to naturally incorporate non-[[CMB]{}]{} datasets and the information they contain about the foregrounds into the analysis. With the immiment launch of the Planck satellite [@unknown:2006uk] and the significant new information on the polarization of the [[CMB]{}]{} that it should deliver, this work is also notable in treating all Stokes parameters of the [[CMB]{}]{} in a unified manner. Numerical testing of the scheme and application to existing [[WMAP]{}]{} data are underway.
[\[sec:priors\]]{}Data and Priors
=================================
Our starting point shall be a collection of $n$ sky maps. Any that are usefully described in terms of a temperature will be assumed to be in thermodynamic temperature units. Note from the start that these maps do not all have to be “[[CMB]{}]{}” maps; other data sets (e.g. radio surveys, starlight polarization maps, point source maps, the [[WMAP]{}]{}“spurious signal” maps) can be included in the analysis in a unified manner and might be useful if they have physical correlations with contaminants in the [[CMB]{}]{} channels.
We assume the sky map is discretized into elements, typically pixels or spherical harmonic coefficients up to some [$l_\text{max}$]{}, but perhaps say wavelet coefficients. For each element $i$ we have the associated measurements for each sky map. We can stack these measurements into a vector ${\ensuremath{\mathbf{X}}}(i)$. These vectors can be further stacked into a big vector [$\mathbf{X}$]{} (the entire data set). We’ll assume we can estimate or calculate the inverse noise covariance matrix [$\mathbf{N}^{-1}$]{} for [$\mathbf{X}$]{}.
Next, let us assume a linear relation between [$\mathbf{X}$]{} and some underlying “signals” [$\mathbf{S}$]{}, i.e.[$\mathbf{X}$]{}=[$\mathbf{A}$]{}[$\mathbf{S}$]{}+ [$\mathbf{M}$]{}, where [$\mathbf{M}$]{} is the noise realization. Some of the signals will of course be the [[CMB]{}]{}, and the others will be the foregrounds. These “effective” foreground signals don’t necessarily have to be thought of as physical processes, just as unwanted contaminants. We shall not assume that these foregrounds are uncorrelated with each other, but they will, however, almost by definition be taken to independent of the [[CMB]{}]{}. To extract cosmological information, we shall work towards a probability distribution for that part of [$\mathbf{S}$]{} associated with the [[CMB]{}]{} sky.
A key assumption shall be that the [[CMB]{}]{} is a blackbody. As a consequence, the “foreground” signals will implicitly be assumed to be linearly independent of the [[CMB]{}]{} in frequency space.
We’ll typically assume that the “mixing matrix” [$\mathbf{A}$]{} is block-diagonal in pixel space, meaning that measurements in a given direction only depend on the signals in that direction (and we are assuming that any beam convolution effects have already been accounted for). In this case, we can write: [$\mathbf{X}$]{}(i) =[$\mathbf{A}$]{}(ii) [$\mathbf{S}$]{}(i) + [$\mathbf{M}$]{}(i). [\[eq:locallink\]]{} (Here and onwards, a matrix followed by element indices denote the submatrix appropriate to the indices of the original matrix.) ${\ensuremath{\mathbf{S}}}(i)$ encodes the strength of the signals, and ${\ensuremath{\mathbf{A}}}(ii)$ controls how much each signal affects each instrument channel. The degeneracy between the amplitude of a column of ${\ensuremath{\mathbf{A}}}(ii)$ and the normalization of a given foreground signal is in principle handled by assumed priors, and the discrete symmetry of column interchange of ${\ensuremath{\mathbf{A}}}(ii)$ leads to an uniform multiplicative overcounting that can be ignored.
The blackbody nature of the [[CMB]{}]{} is realized as a requirement on the form of the ${\ensuremath{\mathbf{A}}}(ii)$’s. If ${\ensuremath{\mathbf{S}}}(i)^\text{T}$ takes the form $({\ensuremath{\mathbf{S}_{\textsc{cmb}}}}(i)^{\text{T}},{\ensuremath{\mathbf{F}}}(i)^{\text{T}})$, with [$\mathbf{S}_{\textsc{cmb}}$]{} for the [[CMB]{}]{} component(s) and [$\mathbf{F}$]{} for the foregrounds, then ${\ensuremath{\mathbf{A}}}(ii)$ must take the form [$\mathbf{A}$]{}(ii)=(
[c|\*[2]{}[c]{}]{} [$\mathbf{e}$]{}&?&?\
&?&?\
), [\[eq:aform\]]{} where [$\mathbf{e}$]{} is a “tall” matrix whose width equals the number of components of ${\ensuremath{\mathbf{S}_{\textsc{cmb}}}}(i)$ and whose elements are ones or zeroes (since temperature-type maps are assumed to be in thermodynamic temperature units). If a channel is polarization-sensitive, then its corresponding part of [$\mathbf{e}$]{} will be a “small” identity matrix (with [$\mathbf{i}$]{}denoting identity matrices of various sizes from now on). If a channel is independent of the [[CMB]{}]{} its corresponding part of [$\mathbf{e}$]{} will be a “small” (row) matrix of zeroes (all “zero” matrices will be denoted [$\mathbf{0}$]{}). Such a channel is still useful in the analysis because it potentially responds to some of the foreground signals and thus carries information about them that can be used in extracting the [[CMB]{}]{}. The part of [$\mathbf{a}$]{} marked ? encodes the spectral indices of the foreground signals.
Because some physical foregrounds (e.g. galactic synchrotron radiation) are known to have a spectral index that varies across the sky, one might expect that our ${\ensuremath{\mathbf{A}}}(ii)$ should indeed be a function of the element $i$. However, an alternative procedure is possible if we have many frequency channels; we can take ${\ensuremath{\mathbf{A}}}(ii)={\ensuremath{\mathbf{a}}}$ for all $i$, and describe a single physical foreground with varying spectral index as two or more correlated effective signal components which have linearly independent corresponding columns of ${\ensuremath{\mathbf{a}}}$.
From now on we will take [$\mathbf{a}$]{} to be a square $n$-by-$n$ matrix, assuming there to be as many signal components as we have frequency channels. As we’ll be marginalizing out the foreground signals, it seems reasonable to allow for as many of them as we might need to describe the data on top of the noise.
We are aiming towards using Bayes’ Theorem to get a probability for a [[CMB]{}]{}model (or perhaps a [[CMB]{}]{} sky map) given the data, marginalized over foregrounds. Let us set down the relevant conditional probabilities that we shall need. Firstly, p([$\mathbf{S}$]{}| , ) = p( [$\mathbf{S}_{\textsc{cmb}}$]{}| ) . p([$\mathbf{F}$]{}| ) by the independence assumption, where “f.g.” is short for foreground. Here ${\ensuremath{\mathbf{S}_{\textsc{cmb}}}}$ is the [[CMB]{}]{} signal (temperature and perhaps polarization) and [$\mathbf{F}$]{} are the foreground signals, as introduced above. Typically the is describable in terms of the [[CMB]{}]{} power spectra, but in principle we could consider non-gaussian corrections too. In the gaussian case, p([$\mathbf{S}_{\textsc{cmb}}$]{}| ) = e\^[-[$\mathbf{S}_{\textsc{cmb}}^{\text{T}}$]{}[$\mathbf{C}^{-1}$]{}[$\mathbf{S}_{\textsc{cmb}}$]{}/2]{}. For isotropic models in harmonic space, [$\mathbf{C}$]{}is diagonal in $l$ and conventionally expressed in terms of [$C_l$]{}’s (i.e. $ \<a^P_{lm}
a^{Q}_{l'm'}\>=C_l^{PQ} \delta_{l,l'} \delta_{m,-m'}$ where $P$ and $Q$ denote Stokes parameter type.).
Next, the probability of the data given the signals and the mixing matrix is given by the probability of obtaining the noise realization ${\ensuremath{\mathbf{M}}}={\ensuremath{\mathbf{X}}}-{\ensuremath{\mathbf{A}}}{\ensuremath{\mathbf{S}}}$: p([$\mathbf{X}$]{}| [$\mathbf{S}$]{}, [$\mathbf{A}$]{})= e\^[-([$\mathbf{X}$]{}-[$\mathbf{A}$]{}[$\mathbf{S}$]{})\^ [$\mathbf{N}$]{}\^[-1]{} ${\ensuremath{\mathbf{X}}}-{\ensuremath{\mathbf{A}}}{\ensuremath{\mathbf{S}}}$/2]{}. Here we have used maximum entropy to assign a gaussian distribution to the noise (see [@Jaynes]) and have decided not to add in and marginalize over any non-gaussian corrections we could imagine (which would complicate the analysis). In some cases we might wish to consider “inverse noise matrices” ${\ensuremath{\mathbf{N}}}^{-1}$ that are not actually the inverse of anything due to having had say monopole and dipole components, point sources, regions of strong galactic emission or instrumental effects projected out of them. (Such projections ensure that a model is not penalized if the data and model “disagree” along such directions.) In this case the determinant factor will not exist but seeing as it is independent of the model it can be safely ignored. Note that in many applications where the sky map data is actually derived with a maximum-likelihood approach from a timestream it is in fact the inverse noise covariance matrix and not the noise covariance matrix itself that naturally emerges.
Now we need some priors. For the [$C_l$]{}’s, we might employ a Jeffreys’-type prior $l$ by $l$, or alternatively assume they come from a known cosmological model which itself has priors on its parameters. In a middle-ground position we might incorporate a positive correlation between neighbouring [$C_l$]{}’s, expecting them to be given by integrals over related transfer functions times the same underlying three-dimensional primordial power spectrum [@Efstathiou:2003wr]. As simple test cases we could assume that the [[CMB]{}]{} is gaussian white noise on the sky with some unknown amplitude or that the [[CMB]{}]{} is scale-invariant with unknown amplitude. We shall see that the “white noise” assumption leads directly to the [[ILC]{}]{} approach as used by the [[WMAP]{}]{} team.
We’ve already effectively imposed a prior that the mixing matrix [$\mathbf{A}$]{}is block-diagonal in pixel space and have argued that it can be taken to be isotropic across the sky if we have enough sky maps to work with. (If we do relax the isotropy assumption on [$\mathbf{a}$]{} then we should parametrize a slow variation of [$\mathbf{a}$]{} across the sky with a suitable probability distribution over the parameters; see the recent work of [@Kim:2008zh] for an application of this type of approach to [[ILC]{}]{} map making.) One might be concerned whether it is most natural to put a measure on [$\mathbf{a}$]{} or on its inverse, since the former is perhaps more natural when thinking about the spectral response of physical foregrounds whereas the latter is heuristically at least what is needed to go from the data to the [[CMB]{}]{}. In fact we shall develop a measure that is invariant under inversion. A matrix measure naturally involves some power of the determinant of the matrix, and from ${\ensuremath{\mathbf{a}^{-1}}}{\ensuremath{\mathbf{a}}}={\ensuremath{\mathbf{i}}}$ one can see that $\delta{\ensuremath{\mathbf{a}^{-1}}}=-{\ensuremath{\mathbf{a}^{-1}}}\delta{\ensuremath{\mathbf{a}}}{\ensuremath{\mathbf{a}^{-1}}}$ and hence that $\partial ({\ensuremath{\mathbf{a}^{-1}}})
/\partial ({\ensuremath{\mathbf{a}}})=|{\ensuremath{\mathbf{a}^{-1}}}|^{2n}$.[^1] So a measure of $|{\ensuremath{\mathbf{a}}}|^m d {\ensuremath{\mathbf{a}}}$ transforms into a measure $|{\ensuremath{\mathbf{a}^{-1}}}|^{-m-2n} d{\ensuremath{\mathbf{a}^{-1}}}$ for some power $m$. Full form symmetry is achieved if we choose $m=-m-2n$, or $m=-n$. This choice yields the Haar measure on $GL(n,R)$, the group of symmetries of a real $n$-dimensional vector space, and so seems very natural. This measure is also scale-invariant as well as inversion-invariant. From this base we now consider our constraints on the mixing matrix from our taking the [[CMB]{}]{} to be black-body and to be independent from the foregrounds. Start from the matrix equation ${\ensuremath{\mathbf{a}^{-1}}}{\ensuremath{\mathbf{a}}}= {\ensuremath{\mathbf{i}}}$. If we write [$\mathbf{a}^{-1}$]{} as [$\mathbf{a}^{-1}$]{}=(
[\*[1]{}[c]{}]{} [$\mathbf{w}^\text{T}$]{}…\
[$\mathbf{u}^\text{T}$]{}…\
), [\[eq:ainv\]]{} then we must have
[\[eq:amconstraints\]]{} [$\mathbf{w}^\text{T}$]{}[$\mathbf{e}$]{}&=&[$\mathbf{i}$]{},\
[$\mathbf{u}^\text{T}$]{}[$\mathbf{e}$]{}&=&[$\mathbf{0}$]{},
where [$\mathbf{w}$]{} and [$\mathbf{u}$]{} are appropriate “tall” matrices. From this we can see that delta functions required in our measure on [$\mathbf{a}$]{} to force the first columns of ${\ensuremath{\mathbf{a}}}$ to equal [$\mathbf{e}$]{} correspond to the term $|{\ensuremath{\mathbf{a}^{-1}}}|^{-1}
\delta({\ensuremath{\mathbf{w}^\text{T}}}{\ensuremath{\mathbf{e}}}-{\ensuremath{\mathbf{i}}})\delta({\ensuremath{\mathbf{u}^\text{T}}}{\ensuremath{\mathbf{e}}})$ in a measure for ${\ensuremath{\mathbf{a}^{-1}}}$. With the form of [$\mathbf{e}$]{} as discussed above, Eqs. [(\[eq:amconstraints\])]{} render $|{\ensuremath{\mathbf{a}^{-1}}}|$ independent of ${\ensuremath{\mathbf{w}}}$.[^2] Hence the components [$\mathbf{w}$]{}needed to extract the [[CMB]{}]{} only actually appear in the delta function $\delta({\ensuremath{\mathbf{w}^\text{T}}}{\ensuremath{\mathbf{e}}}-{\ensuremath{\mathbf{i}}})$ and not in any prefactor in a base measure on ${\ensuremath{\mathbf{a}^{-1}}}$.[^3]
The final prior is for the foregrounds. We shall take them to be gaussian, with auto- and cross- $l$ by $l$ correlations. The gaussian assumption is not ideal; indeed, the [[WMAP]{}]{}K- and KQ- sequence of masks are actually constructed by excluding pixels whose temperatures fall into the skewed high tail of the one-point temperature distribution function. However, by taking the inverse covariance to zero, a gaussian model does at least provide an analytically-controlled method of approaching a flat prior on the foregrounds. A flat prior might be particularly appropriate for polarization, given our limited physical understanding of galactic polarization, and so is what a maximum entropy argument would yield.[^4]
[\[sec:withnoise\]]{}Probabilities for CMB Power Spectra with Noise
===================================================================
We now move on to our main interest, the calculation of probabilities for cosmological models, expressed in terms of their predicted [[CMB]{}]{}power spectra, in presence of general (anisotropic and correlated) instrument noise and foreground contaminants.
We shall take a limit towards a flat prior on the foregrounds, starting from a gaussian prior on them. As we shall see, this limit conveniently decouples the foreground-related parts of the inverse mixing matrix from the rest of the problem. We’ll call the “grand” covariance matrix, of [[CMB]{}]{} and foregrounds combined, [$\mathbf{G}$]{}, and under our stated assumptions this fully specifies our [[CMB]{}]{} and foreground models.
We have: p([$\mathbf{X}$]{}|[$\mathbf{a}^{-1}$]{},[$\mathbf{G}$]{}) &=& d[$\mathbf{S}$]{}p([$\mathbf{X}$]{},[$\mathbf{S}$]{}|[$\mathbf{a}^{-1}$]{},[$\mathbf{G}$]{})\
&=& d[$\mathbf{S}$]{}p([$\mathbf{X}$]{}|[$\mathbf{S}$]{},[$\mathbf{a}^{-1}$]{},[$\mathbf{G}$]{}) p([$\mathbf{S}$]{}| [$\mathbf{a}^{-1}$]{},[$\mathbf{G}$]{})\
&& d[$\mathbf{S}$]{} e\^[-[$\mathbf{S}$]{}\^T [$\mathbf{G}^{-1}$]{}[$\mathbf{S}$]{}/2]{} and to do the integral over [$\mathbf{S}$]{} it is useful to change variables to ${\ensuremath{\mathbf{U}}}\equiv {\ensuremath{\mathbf{A}}}{\ensuremath{\mathbf{S}}}$, generating a $|{\ensuremath{\mathbf{A}}}|$ determinant factor. With the mixing matrix being uniform across the sky, [$\mathbf{A}$]{} decomposes into [$\mathbf{a}$]{}’s down the diagonal and so $|{\ensuremath{\mathbf{A}}}|=|{\ensuremath{\mathbf{a}}}|^{\ensuremath{{N_\text{elem}}}}$. Note that the right-hand-side has been written so as to not explicitly depend on [$\mathbf{G}$]{}, only on [$\mathbf{G}^{-1}$]{}, to make the limiting process clearer. We can perform the gaussian integral over [$\mathbf{S}$]{} and use the result in Bayes’ theorem, in conjunction with our priors discussed in [Sec. \[sec:priors\]]{}, to find: p([$\mathbf{a}^{-1}$]{}, [$\mathbf{G}$]{}| [$\mathbf{X}$]{}) && p([$\mathbf{X}$]{}| [$\mathbf{a}^{-1}$]{},[$\mathbf{G}$]{}) p([$\mathbf{a}^{-1}$]{},[$\mathbf{G}$]{})\
&& p([$\mathbf{G}$]{}) . [\[eq:postforamandg\]]{} To do this integral we needed ${\ensuremath{\mathbf{N}^{-1}}}+{\ensuremath{\mathbf{A}^{-\text{T}}}}{\ensuremath{\mathbf{G}^{-1}}}{\ensuremath{\mathbf{A}^{-1}}}$ to be invertible. We also see that as long as the number of elements is much larger than the number of frequency channels, the $|{\ensuremath{\mathbf{a}^{-1}}}|^{n+1}$ term from our prior on [$\mathbf{a}^{-1}$]{} becomes insignificant and could be ignored relative to the $|{\ensuremath{\mathbf{a}^{-1}}}|^{{\ensuremath{{N_\text{elem}}}}}$ term from the likelihood. We keep both however, defining $N\equiv {\ensuremath{{N_\text{elem}}}}-n-1$.
It is convenient to take the flat-foreground-prior limit ${\ensuremath{\mathbf{F}^{-1}}}{\rightarrow}0$ at this point. First note that $|{\ensuremath{\mathbf{G}^{-1}}}|=|{\ensuremath{\mathbf{C}^{-1}}}||{\ensuremath{\mathbf{F}^{-1}}}|$ since [[CMB]{}]{} and foregrounds are assumed independent. By taking the limit in the same manner for the different [[CMB]{}]{} models that we are considering, we can safely ignore the vanishing $|{\ensuremath{\mathbf{F}^{-1}}}|$ factor (alternatively we can say that we are only interested in likelihood ratios between models, in which case the factor vanishes).
Next, we consider ${\ensuremath{\mathbf{A}^{-\text{T}}}}{\ensuremath{\mathbf{G}^{-1}}}{\ensuremath{\mathbf{A}^{-1}}}$. Written out more explicitly, this takes the form $
\begin{array}{ccc}
{\ensuremath{\mathbf{a}^{-\text{T}}}}{\ensuremath{\mathbf{G}^{-1}}}(1,1) {\ensuremath{\mathbf{a}^{-1}}}& {\ensuremath{\mathbf{a}^{-\text{T}}}}{\ensuremath{\mathbf{G}^{-1}}}(1,2) {\ensuremath{\mathbf{a}^{-1}}}& \\
{\ensuremath{\mathbf{a}^{-\text{T}}}}{\ensuremath{\mathbf{G}^{-1}}}(2,1) {\ensuremath{\mathbf{a}^{-1}}}& {\ensuremath{\mathbf{a}^{-\text{T}}}}{\ensuremath{\mathbf{G}^{-1}}}(2,2) {\ensuremath{\mathbf{a}^{-1}}}& \\
&&\ddots
\end{array}
$ where we have decomposed ${\ensuremath{\mathbf{G}^{-1}}}$ into element-labelled blocks, ${\ensuremath{\mathbf{G}^{-1}}}(ij)$ corresponding to the block of [$\mathbf{G}^{-1}$]{}relating to elements $i$ and $j$. As ${\ensuremath{\mathbf{F}^{-1}}}{\rightarrow}0$, ${\ensuremath{\mathbf{a}^{-\text{T}}}}{\ensuremath{\mathbf{G}^{-1}}}(ij) {\ensuremath{\mathbf{a}^{-1}}}{\rightarrow}{\ensuremath{\mathbf{w}}}{\ensuremath{\mathbf{C}^{-1}}}(ij) {\ensuremath{\mathbf{w}^\text{T}}}$. So, introducing the $n{\ensuremath{{N_\text{elem}}}}$-by-${\ensuremath{{n_\textsc{cmb}}}}{\ensuremath{{N_\text{elem}}}}$ matrix ${\ensuremath{\mathbf{Q}}}$: [$\mathbf{Q}$]{}$
\begin{array}{cccc}
{\ensuremath{\mathbf{w}}}&{\ensuremath{\mathbf{0}}}&{\ensuremath{\mathbf{0}}}& \cdots \\
{\ensuremath{\mathbf{0}}}&{\ensuremath{\mathbf{w}}}& {\ensuremath{\mathbf{0}}}& \\
\vdots &\vdots &\vdots&
\end{array}
$, we have [$\mathbf{A}^{-\text{T}}$]{}[$\mathbf{G}^{-1}$]{}[$\mathbf{A}^{-1}$]{}[$\mathbf{Q}$]{}[$\mathbf{C}^{-1}$]{}[$\mathbf{Q}^\text{T}$]{}. Thus the probability factors into a term depending on ${\ensuremath{\mathbf{w}}}$ and ${\ensuremath{\mathbf{C}}}$ alone and a term depending on ${\ensuremath{\mathbf{u}}}$ alone (the $\delta ({\ensuremath{\mathbf{u}^\text{T}}}{\ensuremath{\mathbf{e}}})/|{\ensuremath{\mathbf{a}^{-1}}}|^{-N}$ part). When we integrate over ${\ensuremath{\mathbf{a}}}$, or equivalently over ${\ensuremath{\mathbf{w}}}$ and ${\ensuremath{\mathbf{u}}}$, to obtain the marginalized probability for the [[CMB]{}]{} model, the ${\ensuremath{\mathbf{u}}}$ integral thus factors off to give a multiplicative constant that can be ignored.
Hence we obtain an effective probability for a [[CMB]{}]{} model of p([$\mathbf{C}$]{}| [$\mathbf{X}$]{}) d [$\mathbf{w}$]{}([$\mathbf{w}^\text{T}$]{}[$\mathbf{e}$]{}-[$\mathbf{i}$]{}) . [\[eq:effcmbprob\]]{} The integral over [$\mathbf{w}$]{} will be performed using the saddle-point method developed in [Appendix \[sec:saddle\]]{}.
Before integrating though, there may exist opportunities to significantly simplify [(\[eq:effcmbprob\])]{}, depending on the invertibility properties of the matrices remaining. If so, the size of subsequent matrix operations is substantially reduced, easing the computational burden. For the rest of this section we consider the simplest case of both ${\ensuremath{\mathbf{N}^{-1}}}$ and ${\ensuremath{\mathbf{C}^{-1}}}$ being invertible; more complicated cases are deferred to subsections below. Following the discussion in [Appendix \[sec:woodbury\]]{}, we are able to apply formulae [(\[eq:wood\])]{} and [(\[eq:wooddet\])]{} to simplify [(\[eq:effcmbprob\])]{} to p([$\mathbf{C}$]{}| [$\mathbf{X}$]{}) p([$\mathbf{C}$]{}) d [$\mathbf{w}$]{}([$\mathbf{w}^\text{T}$]{}[$\mathbf{e}$]{}-[$\mathbf{i}$]{}) . [\[eq:simplecmbprob\]]{} Note how the size of the matrix that needs to be inverted and have its determinant found has reduced from $n{\ensuremath{{N_\text{elem}}}}$-by-$n{\ensuremath{{N_\text{elem}}}}$ to ${\ensuremath{{n_\textsc{cmb}}}}{\ensuremath{{N_\text{elem}}}}$-by-${\ensuremath{{n_\textsc{cmb}}}}{\ensuremath{{N_\text{elem}}}}$, where [${n_\textsc{cmb}}$]{} is the number of [[CMB]{}]{} Stokes parameters under consideration, saving $O((n/{\ensuremath{{n_\textsc{cmb}}}})^3)$ in required computation.
Comparing with Eq. [(\[eq:theint\])]{}, we have S= $ \ln |{\ensuremath{\mathbf{C}}}+{\ensuremath{\mathbf{Q}^\text{T}}}{\ensuremath{\mathbf{N}}}{\ensuremath{\mathbf{Q}}}| + {\ensuremath{\mathbf{X}^\text{T}}}{\ensuremath{\mathbf{Q}}}({\ensuremath{\mathbf{C}}}+{\ensuremath{\mathbf{Q}^\text{T}}}{\ensuremath{\mathbf{N}}}{\ensuremath{\mathbf{Q}}})^{-1} {\ensuremath{\mathbf{Q}^\text{T}}}{\ensuremath{\mathbf{X}}}$, [\[eq:newexp\]]{} and by varying with respect to ${\ensuremath{\mathbf{w}}}$ the appropriate first and second derivatives for $S$ can be read off for use in [(\[eq:update\])]{}: \_[$\mathbf{w}$]{}S &=& [$\mathbf{w}^\text{T}$]{}{ [$\mathbf{N}$]{}(ij) [$\mathbf{w}$]{}( [$\mathbf{D}^{-1}$]{} (ji) - [$\mathbf{D}^{-1}$]{}(jm) [$\mathbf{X}$]{}(m) [$\mathbf{X}^\text{T}$]{}(n) [$\mathbf{D}^{-1}$]{}(ni) ) + [$\mathbf{X}$]{}(i) [$\mathbf{X}^\text{T}$]{}(j) [$\mathbf{w}$]{}[$\mathbf{D}^{-1}$]{}(ji) }\
\^2\_[$\mathbf{w}$]{}S &=& [$\mathbf{w}^\text{T}$]{}[$\mathbf{N}$]{}(ij) [$\mathbf{w}$]{}( [$\mathbf{D}^{-1}$]{}(ji) - [$\mathbf{D}^{-1}$]{}(jm) [$\mathbf{X}$]{}(m) [$\mathbf{X}^\text{T}$]{}(n) [$\mathbf{D}^{-1}$]{}(ni) ) /2\
&+&[$\delta\mathbf{w}^\text{T}$]{}[$\mathbf{N}$]{}(ij) [$\mathbf{w}$]{}[$\mathbf{D}^{-1}$]{}(jk) [$\delta\mathbf{w}^\text{T}$]{}[$\mathbf{N}$]{}(ki) [$\mathbf{w}$]{}\
&+& [$\delta\mathbf{w}^\text{T}$]{}[$\mathbf{N}$]{}(ij) [$\mathbf{w}$]{}[$\mathbf{D}^{-1}$]{}(jk) [$\mathbf{w}^\text{T}$]{}[$\mathbf{N}$]{}(ki) [$\delta\mathbf{w}$]{}\
&+& [$\delta\mathbf{w}^\text{T}$]{}[$\mathbf{N}$]{}(ij) [$\mathbf{X}$]{}(i)[$\mathbf{X}^\text{T}$]{}(j)[$\mathbf{w}$]{}[$\mathbf{D}^{-1}$]{}(ji) /2\
&-& [$\delta\mathbf{w}^\text{T}$]{}[$\mathbf{X}$]{}(i) [$\mathbf{X}^\text{T}$]{}(j) [$\mathbf{w}$]{}[$\mathbf{D}^{-1}$]{}(jk) ([$\delta\mathbf{w}^\text{T}$]{}[$\mathbf{N}$]{}(km) [$\mathbf{w}$]{}+[$\mathbf{w}^\text{T}$]{}[$\mathbf{N}$]{}(mn) [$\delta\mathbf{w}$]{}) [$\mathbf{D}^{-1}$]{}(ni)\
&+&[$\delta\mathbf{w}^\text{T}$]{}[$\mathbf{N}$]{}(ij) [$\mathbf{w}$]{}[$\mathbf{D}^{-1}$]{}(jk) [$\delta\mathbf{w}^\text{T}$]{}[$\mathbf{N}$]{}(km)[$\mathbf{w}$]{} [$\mathbf{D}^{-1}$]{}(mn) [$\mathbf{w}^\text{T}$]{}[$\mathbf{X}$]{}(n)[$\mathbf{X}^\text{T}$]{}(p)[$\mathbf{w}$]{}[$\mathbf{D}^{-1}$]{}(pi)\
&+&[$\delta\mathbf{w}^\text{T}$]{}[$\mathbf{N}$]{}(ij) [$\mathbf{w}$]{}[$\mathbf{D}^{-1}$]{}(jk) [$\mathbf{w}^\text{T}$]{}[$\mathbf{N}$]{}(km) [$\delta\mathbf{w}$]{}[$\mathbf{D}^{-1}$]{}(mn) [$\mathbf{w}^\text{T}$]{}[$\mathbf{X}$]{}(n) [$\mathbf{X}^\text{T}$]{}(p) [$\mathbf{w}$]{}[$\mathbf{D}^{-1}$]{}(pi) /2\
&+&[$\delta\mathbf{w}^\text{T}$]{}[$\mathbf{D}^{-1}$]{}(ij) [$\delta\mathbf{w}$]{}[$\mathbf{N}$]{}(jk) [$\mathbf{w}$]{}[$\mathbf{D}^{-1}$]{}(km) [$\mathbf{w}^\text{T}$]{}[$\mathbf{X}$]{}(m) [$\mathbf{X}^\text{T}$]{}(n) [$\mathbf{w}$]{}[$\mathbf{D}^{-1}$]{}(np)[$\mathbf{w}^\text{T}$]{}[$\mathbf{N}$]{}(pi) /2 where we have defined ${\ensuremath{\mathbf{D}}}$ to equal ${\ensuremath{\mathbf{C}}}+{\ensuremath{\mathbf{Q}^\text{T}}}{\ensuremath{\mathbf{N}}}{\ensuremath{\mathbf{Q}}}$ and assumed implied summation over repeated element indices. (To obtain full equivalence with Eq. [(\[eq:update\])]{} we need to map from the matrix [$\mathbf{w}$]{} here to the vector $x$ there. This can be done explicitly by “vectorization” of [$\mathbf{w}$]{} if required, as discussed in [Appendix \[sec:veckron\]]{}.) The numerical calculation time is dominated by the required $O(({\ensuremath{{n_\textsc{cmb}}}}{\ensuremath{{N_\text{elem}}}})^3)$ inversion of [$\mathbf{D}$]{}. So as long as the initial guess for [$\mathbf{w}$]{} is reasonable[^5], one should be able to converge to the saddle-point expression for [$\mathbf{w}$]{} in a few $O(({\ensuremath{{n_\textsc{cmb}}}}{\ensuremath{{N_\text{elem}}}})^3)$ steps. Having converged to a saddle point solution ${\ensuremath{\mathbf{w}}}{_\text{s.p.}}$, we evaluate the exponent [(\[eq:newexp\])]{} there, calculate a prefactor[^6] involving second derivatives of $S$ at the saddle point as discussed in [Appendix \[sec:saddle\]]{} and multiply by any prior in order to finally obtain the posterior probability for the [[CMB]{}]{} model in question.
Noise Matrices with Projections
-------------------------------
Often one wishes to project additional degrees of freedom out of a formally invertible inverse noise matrix in order to render our posterior probabilities insensitive to certain complications with the data that would otherwise be unaccounted for. For example, one might project out monopole and dipole contributions of the maps from an experiment like [[WMAP]{}]{}, the monopole because the experiment is basically differential and the dipole because of the earth’s motion relative to the [[CMB]{}]{}. Or one might project out regions of strong emission from the galaxy because one does not expect the relatively simple foreground model used to be accurate there. The [[WMAP]{}]{} team also project out a “transmission loss imbalance” mode from their maps. In some cases one can sometimes just reduce the dimensions of the matrices involved; e.g. one might just use the $l>1$ modes in harmonic space to forget about the monopole and dipole, or for a galactic cut working in pixel space one might simply only consider data from pixels outside of the cut. However, when say both a monopole and dipole projection and a galactic cut are made, no such simple approach is possible, and one must proceed as follows.
Each of the [${n_\text{proj}}$]{} modes to be projected out are expressed as a column vector, and these column vectors are arranged into an $n{\ensuremath{{N_\text{elem}}}}$-by-[${n_\text{proj}}$]{} matrix [$\mathbf{P}$]{}. Then, ${\ensuremath{\mathbf{N}^{-1}}}$ is replaced by: [$\mathbf{N}^{-1}$]{} |\_[[$\mathbf{P}$]{}]{} [$\mathbf{N}^{-1}$]{}-[$\mathbf{N}^{-1}$]{}[$\mathbf{P}$]{}${\ensuremath{\mathbf{P}^\text{T}}}{\ensuremath{\mathbf{N}^{-1}}}{\ensuremath{\mathbf{P}}}$\^[-1]{} [$\mathbf{P}^\text{T}$]{}[$\mathbf{N}^{-1}$]{}. (Note that the normalization of the modes cancels out in the formation of $\overline{{\ensuremath{\mathbf{N}^{-1}}}}|_{{\ensuremath{\mathbf{P}}}}$.) $\overline{{\ensuremath{\mathbf{N}^{-1}}}}|_{{\ensuremath{\mathbf{P}}}} $ has been constructed so that when it is multiplied into any linear combination of the modes to be projected out it returns zero. Hence any discrepancy between the data and a potential signal along these modes is not penalized in the likelihood.
Eq. [(\[eq:effcmbprob\])]{} becomes p([$\mathbf{C}$]{}| [$\mathbf{X}$]{}) d [$\mathbf{w}$]{}([$\mathbf{w}^\text{T}$]{}[$\mathbf{e}$]{}-[$\mathbf{i}$]{}) , [\[eq:projeffcmbprob\]]{} which, after some work involving [(\[eq:wood\])]{} and [(\[eq:wooddet\])]{} and keeping [$\mathbf{N}$]{} to refer to the assumed-invertible formal noise matrix, reduces to: p([$\mathbf{C}$]{}| [$\mathbf{X}$]{}) p([$\mathbf{C}$]{}) d [$\mathbf{w}$]{}([$\mathbf{w}^\text{T}$]{}[$\mathbf{e}$]{}-[$\mathbf{i}$]{}) . [\[eq:projcmbprob\]]{} We can now expand the delta function as a gaussian and integrate as above. Additional terms in the first and second derivates of the exponent come from the projection, further complicating the formulae, but the numerical time of the calculation is still dominated by the inversion of $ {\ensuremath{\mathbf{C}}}+{\ensuremath{\mathbf{Q}^\text{T}}}{\ensuremath{\mathbf{N}}}{\ensuremath{\mathbf{Q}}}$.
There is some subtlety about which projections can be handled in this manner. For example, with temperature maps alone, one cannot independently project out all measurements associated with a given element; the second determinant under the square root turns out to be proportional to $|{\ensuremath{\mathbf{w}}}{\ensuremath{\mathbf{w}^\text{T}}}|^{{\ensuremath{{N_\text{elem}}}}}$ and is thus singular. (This is happening perhaps because as the foregrounds are being marginalized out and the dimensions effectively reducing from $n
{\ensuremath{{N_\text{elem}}}}$ to $ {\ensuremath{{N_\text{elem}}}}$ the previously orthogonal projections might be becoming linearly dependent.) Potentially including polariation again, one can however project out any response to the [[CMB]{}]{} in the $i$’th element say with ${\ensuremath{\mathbf{P}}}(i)$ equal to ${\ensuremath{\mathbf{e}}}$ and all other entries in [$\mathbf{P}$]{} zero. With the help of the delta function ${\ensuremath{\mathbf{Q}^\text{T}}}{\ensuremath{\mathbf{P}}}(i)$ becomes equal to the identity matrix [$\mathbf{i}$]{}. Finally, one obtains just the result one would have gotten by forgetting about the element in question in the first place and starting with a reduced problem with only ${\ensuremath{{N_\text{elem}}}}-1$ elements rather than [${N_\text{elem}}$]{} elements, as can be seen using rules for the determinants of blocked matrices and their inverses.[^7] By combining many such projections into a wide projection matrix one can form a mask, or by considering an appropriately weighted vector sum of such projections one can project out spatially coherent modes such as the monopole or dipole from pixellized data.
The general case
----------------
Finally we mention the case when the inverse noise matrix is completely general. Now, we cannot use the Woodbury formula to simplify the exponential in [(\[eq:effcmbprob\])]{}. Rather, we have to use the effective exponent from [(\[eq:effcmbprob\])]{} as is and directly proceed with the replacement of the delta function with a gaussian. It is in fact very appealing to work like this directly with inverse noise matrices and inverse-noise-weighted maps, as these are what naturally emerge out of a maximum-likelihood analysis of timestream data. However, the practical cost is that one now has to invert full $n {\ensuremath{{N_\text{elem}}}}$-by-$n {\ensuremath{{N_\text{elem}}}}$ matrices for each [[CMB]{}]{} model considered.
[\[sec:nonoise\]]{}The no-noise limit
=====================================
It is interesting to consider what happens if the noise is negligible. Starting from [(\[eq:simplecmbprob\])]{}, we can take the limit ${\ensuremath{\mathbf{N}}}{\rightarrow}0$ to obtain: p([$\mathbf{C}$]{}| [$\mathbf{X}$]{})\_ d[$\mathbf{w}$]{}([$\mathbf{w}^\text{T}$]{}[$\mathbf{e}$]{}-[$\mathbf{i}$]{}) e\^[-S’]{}, with exponent $S'= {\ensuremath{\text{vec}({\ensuremath{\mathbf{w}}})^\text{T}}}{\ensuremath{\widehat{\mathbf{R}}}}\, {\ensuremath{\text{vec}({\ensuremath{\mathbf{w}}})}}/2$ (with “vec” indicating vectorization as discussed in [Appendix \[sec:veckron\]]{}, and using Eq. [(\[eq:veckronid\])]{}) where we have defined [$\widehat{\mathbf{R}}$]{}([$\mathbf{C}$]{}) = \_[ij]{} [$\mathbf{C}^{-1}$]{}(ij) [$\mathbf{X}$]{}(i) [$\mathbf{X}^\text{T}$]{}(j). Here $\otimes$ denotes the Kronecker product, discussed in [Appendix \[sec:veckron\]]{}. Notice that $S'$ is now quadratic in the components of [$\mathbf{w}$]{} and hence the saddle point integration over [$\mathbf{w}$]{} is exact. Re-expressing the ${\ensuremath{\mathbf{w}^\text{T}}}{\ensuremath{\mathbf{e}}}={\ensuremath{\mathbf{i}}}$ constraint as ([$\mathbf{i}$]{}[$\mathbf{e}^\text{T}$]{}) [$\text{vec}({\ensuremath{\mathbf{w}}})$]{}=[$\text{vec}({\ensuremath{\mathbf{i}}})$]{}, Eq. [(\[eq:quadraticsoln\])]{} gives the saddle point for [$\text{vec}({\ensuremath{\mathbf{w}}})$]{} at: [$\text{vec}({\ensuremath{\mathbf{w}}})$]{}[\_]{}=[$\widehat{\mathbf{R}}^{-1}$]{}([$\mathbf{i}$]{}[$\mathbf{e}$]{}) $({\ensuremath{\mathbf{i}}}\otimes {\ensuremath{\mathbf{e}^\text{T}}}) {\ensuremath{\widehat{\mathbf{R}}^{-1}}}({\ensuremath{\mathbf{i}}}\otimes {\ensuremath{\mathbf{e}}}) $\^[-1]{} [$\text{vec}({\ensuremath{\mathbf{i}}})$]{} [\[eq:nonoisew\]]{} (“s.p.” for saddle point). Using [(\[eq:quadraticaction\])]{}, the posterior for the [[CMB]{}]{} model is p([$\mathbf{C}$]{}| [$\mathbf{X}$]{}) e\^[- [$\text{vec}({\ensuremath{\mathbf{i}}})^\text{T}$]{} $({\ensuremath{\mathbf{i}}}\otimes {\ensuremath{\mathbf{e}^\text{T}}}) {\ensuremath{\widehat{\mathbf{R}}^{-1}}}({\ensuremath{\mathbf{i}}}\otimes {\ensuremath{\mathbf{e}}}) $\^[-1]{} [$\text{vec}({\ensuremath{\mathbf{i}}})$]{}/2]{}. [\[eq:nonoiselike\]]{}
Eq [(\[eq:nonoisew\])]{} bears a marked resemblance to the formula for the weights in the [[ILC]{}]{} approach and this will be expanded upon in the following section.
[\[sec:ilc\]]{}Deriving sky maps and relations with the ILC procedure
=====================================================================
So far in this paper we have concentrated on deriving likelihood functions for [[CMB]{}]{} models rather than on producing [[CMB]{}]{} sky maps. However, our procedure can of course be used to generate sky maps and associated noise covariances including contributions from foreground removal.
Now $p({\ensuremath{\mathbf{S}}}|{\ensuremath{\mathbf{X}}})=\int d{\ensuremath{\mathbf{A}}}\,d{\ensuremath{\mathbf{G}}}\, p({\ensuremath{\mathbf{S}}},{\ensuremath{\mathbf{A}}},{\ensuremath{\mathbf{G}}}|{\ensuremath{\mathbf{X}}}) \propto \int
d{\ensuremath{\mathbf{A}}}\,d{\ensuremath{\mathbf{G}}}\, p({\ensuremath{\mathbf{X}}}|{\ensuremath{\mathbf{S}}},{\ensuremath{\mathbf{A}}},{\ensuremath{\mathbf{G}}}) \, p({\ensuremath{\mathbf{S}}},{\ensuremath{\mathbf{A}}},{\ensuremath{\mathbf{G}}})$ using Bayes’ Theorem. Using the same conditional probabilities and priors as above, again introducing ${\ensuremath{\mathbf{U}}}= {\ensuremath{\mathbf{A}}}{\ensuremath{\mathbf{S}}}$, and using Eq. [(\[eq:spmean\])]{}, we obtain: [$\mathbf{S}_{\textsc{cmb}}$]{}(i) && d[$\mathbf{C}$]{}p([$\mathbf{C}$]{}|[$\mathbf{X}$]{}) [$\mathbf{w}^\text{T}$]{}[\_]{}([$\mathbf{C}$]{}) [$\mathbf{X}$]{}(i) [\[eq:generalc\]]{} along with a somewhat messier expression for $\< {\ensuremath{\mathbf{S}_{\textsc{cmb}}}}(i) {\ensuremath{\mathbf{S}_{\textsc{cmb}}^{\text{T}}}}(j) \> $ using Eq. [(\[eq:spcov\])]{}. In the no-noise limit, these expressions become exact and ${\ensuremath{\mathbf{w}^\text{T}}}{_\text{s.p.}}({\ensuremath{\mathbf{C}}})$ and $p({\ensuremath{\mathbf{C}}}|{\ensuremath{\mathbf{X}}})$ are obtainable from Eqs. [(\[eq:nonoisew\])]{} and [(\[eq:nonoiselike\])]{} respectively.
To establish a direct link with [[ILC]{}]{} let us try to construct the [[CMB]{}]{} temperature (and possibly polarization) maps, ${\ensuremath{\mathbf{S}_{\textsc{cmb}}}}$. Now, imagine we are working in pixel space in the no-noise limit and we have a prior on the [[CMB]{}]{} that it is sky-uncorrelated, i.e. it is gaussian white noise. Then ${\ensuremath{\mathbf{C}}}(i,j)={\ensuremath{\mathbf{c}}}\, \delta_{ij}$ say and [$\widehat{\mathbf{R}}$]{} reduces to ${\ensuremath{\mathbf{c}^{-1}}}\otimes {\ensuremath{\widehat{\mathbf{x}}}}$, defining ${\ensuremath{\widehat{\mathbf{x}}}}\equiv \sum_i
{\ensuremath{\mathbf{X}}}(i) {\ensuremath{\mathbf{X}^\text{T}}}(i) $. Then ${\ensuremath{\widehat{\mathbf{R}}^{-1}}}$ is just ${\ensuremath{\mathbf{c}}}\otimes {\ensuremath{\widehat{\mathbf{x}}^{-1}}}$ using [(\[eq:kroninv\])]{}. Eq. [(\[eq:nonoisew\])]{} simplifies to become independent of [$\mathbf{c}$]{}, and so, no matter what our prior on [$\mathbf{c}$]{} actually is, we have [$\mathbf{S}_{\textsc{cmb}}$]{}(i)= [$\mathbf{w}^\text{T}$]{}[\_]{}[$\mathbf{X}$]{}(i) [\[eq:ilcmap\]]{} with [$\mathbf{w}$]{}[\_]{}= [$\widehat{\mathbf{x}}^{-1}$]{}[$\mathbf{e}$]{}$ {\ensuremath{\mathbf{e}^\text{T}}}{\ensuremath{\widehat{\mathbf{x}}^{-1}}}{\ensuremath{\mathbf{e}}}$\^[-1]{}. [\[eq:ilcw\]]{} For temperature alone, in which case [$\mathbf{e}$]{} is just $(1,\ldots,1)^\text{T}$, this is exactly the [[ILC]{}]{} result. So Eqs. [(\[eq:ilcmap\])]{} and [(\[eq:ilcw\])]{} are the natural generalization of [[ILC]{}]{} when treating temperature and polarization in a unified manner. Typically the [[ILC]{}]{} procedure is described as choosing the linear combination of channels that has minimum variance whilst still retaining unit response to the [[CMB]{}]{}. Our Bayesian derivation here (albeit with flat priors on the foregrounds and a very strong, and obviously incorrect, prior on the [[CMB]{}]{}) here gives an alternative, more insightful, perspective. For example, we see that one might not need to correct for “cosmic covariance” [@Hinshaw:2006ia; @Chiang:2007rp]; the [[ILC]{}]{}coefficients already give the mean maps. We might replace the “white noise [[CMB]{}]{}” prior with one based on a fiducial model up to an overall amplitude and thus derive “improved” [[ILC]{}]{} coefficients that correctly take into account spatial correlations in the [[CMB]{}]{}. A very simple case appropriate for low-$l$ would be to take the [[CMB]{}]{} to be scale-invariant, $C_l \propto 1/(l(l+1))$, and work in harmonic space, calculating [$\widehat{\mathbf{R}}$]{} with the appropriate weights.
As mentioned below Eq. [(\[eq:generalc\])]{}, we can calculate a measure of the foreground-induced pixel-pixel covariance of our maps. For clarity we shall do this for temperature alone in the no-noise limit. Then ${\ensuremath{\mathbf{S}_{\textsc{cmb}}}}(i)$ is simply $t(i)$, the temperature in the $i$’th pixel. With the “white noise” [[CMB]{}]{} prior, [$\mathbf{c}$]{} is given by the single number [$\Delta T^2$]{}. With a Jeffreys’ prior on [$\Delta T^2$]{} one finds t(i)t(j)= ([$\mathbf{X}$]{}(i)-t(i) [$\mathbf{e}$]{})\^ ([$\mathbf{X}$]{}(j)-t(j) [$\mathbf{e}$]{}). This is a very reasonable result; uncertainties are given by a quadratic form on differences away from a blackbody, with the quadratic form determined by the covariance of the sample.
One can attempt generalizing the [[ILC]{}]{} procedure in other ways. For example, staying with temperature alone, one could keep the “white noise” [[CMB]{}]{} prior, but try to add in instrument noise at some level, taking it say to be sky-uncorrelated and isotropic, described by the same matrix [$\mathbf{n}$]{} at each pixel. Then the joint likelihood for [$\mathbf{w}$]{} and [$\Delta T^2$]{}, with a flat prior on [$\Delta T^2$]{}, turns out to be: p([$\Delta T^2$]{},[$\mathbf{w}$]{}|[$\mathbf{X}$]{}) e\^[-]{}, [\[eq:jointnoise\]]{} and the joint maximum likelihood point for [$\Delta T^2$]{} and [$\mathbf{w}$]{} together occurs at the just same value for [$\mathbf{w}$]{}, namely ${\ensuremath{\widehat{\mathbf{x}}^{-1}}}{\ensuremath{\mathbf{e}}}/({\ensuremath{\mathbf{e}^\text{T}}}{\ensuremath{\widehat{\mathbf{x}}^{-1}}}{\ensuremath{\mathbf{e}}})$, as in the noise-free case, with unchanged variance for [$\mathbf{w}$]{}. This is at odds with the idea of subtracting the noise contribution [$\mathbf{n}$]{}from the total variance [$\widehat{\mathbf{x}}$]{} (which might seem plausible on the grounds of focussing on the signal) in deriving the [[ILC]{}]{} coefficients.
[\[sec:extensions\]]{}Possible Extensions, Practical Considerations and Conclusion
==================================================================================
While the scheme presented above treats foregrounds very generally, there is scope for further extension. One possibility is to allow for a slow variation of the mixing matrix across the sky. In fact it would be technically easiest to perform a low-order spherical harmonic expansion of the inverse pixel-space mixing matrix; then ${\ensuremath{\mathbf{w}^\text{T}}}({\ensuremath{\mathbf{x}}})
{\ensuremath{\mathbf{e}}}={\ensuremath{\mathbf{i}}}$ would reduce to ${\ensuremath{\mathbf{w}^\text{T}}}_{00} {\ensuremath{\mathbf{e}}}={\ensuremath{\mathbf{i}}}$ and ${\ensuremath{\mathbf{w}^\text{T}}}_{lm}
{\ensuremath{\mathbf{e}}}={\ensuremath{\mathbf{0}}}$ for $l>0$. Ref. [@Kim:2008zh] applies an analagous approach to map making, allowing the [[ILC]{}]{} coefficients to vary across the sky. Notice that this is distinct from the harmonic approach of [@1996MNRAS.281.1297T; @Tegmark:2003ve] and is rather more physical when looked at from a component separation approach; the $l$-by-$l$ analysis there effectively imagines the data from given direction to come from a convolution of signals around that direction, whereas [@Kim:2008zh] and the suggestion here consider the data in a given direction to come from signals in that direction alone.
Another natural variation would be to relax the flat prior on the foregrounds and rather try to marginalize over power spectra for the foregrounds. Now the separation achieved above between [[CMB]{}]{} and foregrounds would not occur and one would have to think about possible priors on the foreground part of the inverse mixing matrix (luckily we have seen that the base prior on [$\mathbf{a}$]{} rapidly becomes irrelevant as the number of elements increases). The underlying model would be close to that used in the Gibbs sampling approach of [@Eriksen:2007mx], and in the context of map making one would be left with a scheme very similar to that of [@Delabrouille:2002kz] (but with the advantages of working with [$\mathbf{a}^{-1}$]{} rather than [$\mathbf{a}$]{}). Now one of the distinguishing features of the foreground signals is that they are non-gaussian, and one could hope to exploit this if one could develop a plausible non-gaussian correlated probability distribution for the foregrounds. As mentioned earlier, one might also consider non-gaussian corrections to the [[CMB]{}]{} itself, but in this case at least we suspect them to be small.
A conceptually simple change would be to relax the assumption that the mixing matrix [$\mathbf{a}$]{} is square, if say one is confident in the number of physical foreground emission processes operating. However, many of the manipulations of our approach depend technically on the invertibility of [$\mathbf{a}$]{}, so this change would be difficult to implement and would only be trustworthy if any neglected foreground components (e.g. coming from non-modelled spectral index variations) were below the level of the detector noise.
Let us now move on to discuss certain practical considerations in implementing a scheme such as that described here. An obvious difficulty with a straightforward implementation is the usual one confronting [[CMB]{}]{}analysis, namely the need to numerically invert large matrices corresponding to the large size of the data sets involved. Although the cosmic signal is naturally band limited, the cutoff is too high for a full resolution analysis over a good fraction of the sky to be practical.[^8] With simplifying assumptions however, some progress may be possible.[^9] In any case, a full resolution analysis is probably not even necessary. Instead, one might perform a split analysis, treating large and small angular scales in a different manner (see [@Efstathiou:2003dj] for a discussion in the absence of foregrounds). While the large angular scales (where some of the most interesting new results might manifest themselves and where particularly B-mode polarization might have some signal relative to detector noise for Planck) might be dealt with accurately by an application of the method of this paper, an heuristic approximation to the method (or a different scheme altogether) might be all that is needed for high $l$. Of course the problem will not precisely factorize into a large scale part and a small scale part and so the coupling will have to be taken into account or shown to be negligible. In addition, there are issues with the production and interpretation of low resolution maps and their (inverse) noise matrices (see the description in [@Jarosik:2006ib] of the procedure used by the [[WMAP]{}]{} team), with difficulties due to aliasing and the finite size of the pixels themselves. In this work we have assumed that the signal has been deconvolved from the beam; the appropriate noise matrices must be obtained by transforming those appropriate to the common case of solving only for the beam-smoothed signal. Any beam uncertainties should be incorporated in the noise matrices also.
Often the time-ordered data is discretized into a map on the sphere using some pixelization scheme as opposed to going directly to harmonic space. Even if all signals are strictly band-limited, with non-zero spherical harmonic coefficients only for $l\leq {\ensuremath{l_\text{max}}}$ say, fully describing such signals in pixel space requires more than $({\ensuremath{l_\text{max}}}+1)^2$ pixels. One then has to be careful about the validity of certain matrix operations that need to be performed. For example, the inverse transformation from an unconstrained map in pixel space to harmonic space is not well-defined. Similarly, the pixel-space signal covariance matrix is non-invertible. The harmonic-space inverse noise matrix can be obtained by transforming the pixel-space inverse noise matrix though. An important issue arises if a galaxy cut projection is to be made: the mask itself needs be band-limited in order for pixel-space and harmonic-space approaches to be equivalent.
While all of these complications need to be investigated and accounted for if necessary, it is not likely that they will lead to a fundamental problem in the scheme defined here for cosmic information extraction in the presence of foregrounds, just as they have not prevented standard cosmological analysis of [[CMB]{}]{} data.
In conclusion, this paper has described a prescription for the analysis of multiple sky maps for cosmological information in the presence of both detector noise and foregrounds. The prescription takes into account uncertainties in the foregrounds by modelling the foregrounds in a very general way and then marginalizing over the foregrounds. The noise can be very general, potentially correlated over the sky and even correlated between frequency channels, and include projections. Work is underway both to subject the prescription to extensive Monte Carlo testing and to apply it to the [[WMAP]{}]{} data. Assuming its performance is good the scheme should be ideal for the analysis of the temperature and polarization data from the forthcoming Planck satellite.
I thank George Efstathiou for directing my attention to the problem of the foreground contamination of the [[CMB]{}]{}, and thank him, Mark Ashdown, Anthony Challinor, Antony Lewis and Francesco Paci for helpful comments and discussions. I am supported by STFC.
[\[sec:saddle\]]{}Constrained Saddle Point Integrals
====================================================
This appendix explains the technique we use to evaluate constrained integrals in a saddle-point approximation.
Imagine we are performing an $n$-dimensional integral over variables $x^i$ of some function $e^{-S(x)}$, with $m<n$ linearly-independent linear constraints of the form $c^{~k}_i x^i=d^k$ (summed over $i$) on the $x$’s applied, i.e.I=d\^n x (c\^[ 1]{}\_i x\^i-d\^1) …(c\^[ m]{}\_i x\^i-d\^m) e\^[-S(x)]{}. [\[eq:theint\]]{}
First, we approximate the delta functions by a narrow gaussian, described with some covariance matrix $D$ which will eventually be taken to zero: (c\^ x -d ) (c\^[ 1]{}\_i x\^i-d\^1) …(c\^[ m]{}\_i x\^i-d\^m) \~ e\^[-(c\^[ m]{}\_i x\^i-d\^m) [D\^[-1]{}]{}\_[mn]{} (c\^[ n]{}\_i x\^i-d\^n)/2]{}. [\[eq:deltaapp\]]{} Next, we find the saddle point of the entire integrand and approximate the integrand as a gaussian around the saddle point. The saddle point is where the first derivative . c D\^[-1]{} (c\^ x-d) |\_i +S\_[,i]{} [\[eq:firstderiv\]]{} of minus the exponent is zero, and the matrix $M_\text{total}$ of second derivatives of minus the exponent has components . c D\^[-1]{} c\^ |\_[ij]{}+S\_[,ij]{}. [\[eq:secondderiv\]]{} We need to evaluate both exponent and prefactor terms to obtain the full saddle-point approximation to [(\[eq:theint\])]{}. At the saddle point, it turns out that the “delta-function” part of the exponent tends to zero as $D{\rightarrow}0$, as can be seen as follows. Multiplying [(\[eq:firstderiv\])]{} by $c$ on the left at the saddle point (“s.p.” in formulae below) and rearranging yields c\^ x-d=- D (c\^ c)\^[-1]{} . S\_[,i]{}|\_ and so the “delta-function” part is . S\^\_[,i]{}|\_ (c\^ c)\^[-1]{} D (c\^ c)\^[-1]{}. S\_[,i]{}|\_/2. If there exists a sensible solution as $D{\rightarrow}0$, then $\left.
S_{,i} \right|_\text{s.p.}$ will only have a weak dependence on $D$ and so in the limit the above term will vanish.
The prefactor comes from the gaussian integral over deviations away from the saddle point, which is proportional to the reciprocal square root of $M_\text{total}$ evaluated at the saddle point. Splitting $M_\text{total}$ into the piece $c D^{-1} c^\text{T}$ from the delta function and the piece $M$ with components $\left. S_{,ij}\right|_\text{s.p.}$, we have $|M_\text{total}|=|M| |D^{-1}| |D+c~\text{T} M^{-1}
c|$. Combining with the $\sqrt{|D|}$ term in the denominator of [(\[eq:deltaapp\])]{} and taking the limit, we obtain an overall prefactor proportional to $1/\sqrt{|M| |c M^{-1} c^\text{T}|}$.
Hence we obtain: I e\^[-S]{}, [\[eq:spanswer\]]{} with all quantities evaluated at the saddle point. Note that the result depends only on the original integrand and its second derivatives but evaluated at the saddle point of the combined integrand. Additionally, d\^n x (c\^x -d) x e\^[-S(x)]{} && x[\_]{}I, [\[eq:spmean\]]{}\
d\^n x (c\^x -d) x x\^ e\^[-S(x)]{} &&$M^{-1}-M^{-1} c
\left(c^\text{T} M^{-1} c \right)^{-1} c^\text{T} M^{-1}
+x{_\text{s.p.}}x^T{_\text{s.p.}}\right) I.
{\label{eq:spcov}}
\ea
We still have to actually find the saddle point and typically this can be done
numerically with a Newton-Raphson-type approach as follows. With the first
and second derivatives from formulae {(\ref{eq:firstderiv})}
and {(\ref{eq:secondderiv})} above,
the Newton-Raphson update is of the form:
\ba
\delta x =-M_\text{total}^{-1} \(c D^{-1}(c^\text{T} x-d) +\nabla S(x)$ .With the saddle point of the limit being the limit of the saddle point we can simply start from a point on the constraint surface and take the limit $D{\rightarrow}0$ straight away to obtain the projected update formula: \_ x = -( M\^[-1]{}-M\^[-1]{} c (c\^ M\^[-1]{} c )\^[-1]{} c\^ M\^[-1]{}) S(x) [\[eq:update\]]{} (“proj” for projected). Iterating this will lead us to the saddle point, which can then be used in [(\[eq:spanswer\])]{} to obtain the approximate value for the integral.
In the special case that $S$ is quadratic in $x$, the saddle point can be solved for analytically and is at $x = M^{-1} c \(
D+c^\text{T} M^{-1} c \)^{-1} d$. So taking $D {\rightarrow}0$, we have x\_= M\^[-1]{} c $ c^\text{T} M^{-1} c $\^[-1]{} d [\[eq:quadraticsoln\]]{} and hence S[\_]{}= d\^ $ c^\text{T} M^{-1} c $\^[-1]{} d /2 . [\[eq:quadraticaction\]]{} Furthermore, Eqs. [(\[eq:spanswer\])]{}, [(\[eq:spmean\])]{} and [(\[eq:spcov\])]{} become exact.
[\[sec:veckron\]]{}Vectorization and Kronecker Products
=======================================================
We here summarize vectorization of matrices and the Kronecker product of two matrices. See Appendix A of [@Hamimeche:2008ai] for further identities and a recent discussion of vectorization in the context of [[CMB]{}]{} analysis.
The vectorization $\text{vec} (A)$ of an $m$-by-$n$ matrix $A$ is the $mn$-by-1 column vector formed by stacking the columns of $A$ on top of each other. Explicitly, $\text{vec} (A)^\text{T}=
(A_{11},\ldots,A_{m1},A_{12},\ldots,A_{m2},\ldots,A_{1n},\ldots,A_{mn})$.
The Kronecker product of an $m$-by-$n$ matrix $A$ and a $p$-by-$q$ matrix $B$ is an $mp$-by-$nq$ matrix, denoted $A \otimes B$, formed by appropriately stacking copies of the $B$ matrix that have been multiplied by the elements of A: A B =
A\_[11]{} B & A\_[12]{} B & & A\_[1n]{} B\
A\_[21]{} B & A\_[22]{} B && A\_[2n]{} B\
&&&\
A\_[m1]{} B & A\_[m2]{} && A\_[mb]{} B\
. When the matrices are of compatible sizes such that the relevant products exist, (AB)(CD)=(AC) (BD). Hence, (A B)\^[-1]{} = A\^[-1]{} B\^[-1]{}. [\[eq:kroninv\]]{} Also, (AB)\^=A\^ B\^.
A useful identity involving both vectorization and the Kronecker product is (A\^ BCD) = (A)\^ (D\^ B) (C). [\[eq:veckronid\]]{}
[\[sec:woodbury\]]{}Summary of Matrix Identities Used
=====================================================
This paper makes extensive use of Sherman-Morrison/Woodbury-type formulae (see e.g. [@numrec]) for matrix inverses and their determinants. Here we briefly show how these results may be derived in order to understand how and when they may be applied. We start by considering two related decompositions of the same blocked matrix:
a & -u\
v\^ & c\^[-1]{}
&=&
1 & -u c\
0 & 1
a+u c v\^ & 0\
0 & c\^[-1]{}
1 & 0\
c v\^ & 1
\
&=&
1 & 0\
v\^ a\^[-1]{} & 1
a & 0\
0 & c\^[-1]{} + v\^ a\^[-1]{}u
1 & - a\^[-1]{} u\
0& 1
. Taking the determinant of both decompositions and rearranging yields | a + u c v\^ | = | a | | c | | c\^[-1]{} + v\^ a\^[-1]{}u |. [\[eq:wooddet\]]{} Inverting both decompositions and equating top-left corners yields ( a + u c v\^ )\^[-1]{} = a\^[-1]{}-a\^[-1]{} u ( c\^[-1]{}+ v\^ a\^[-1]{}u )\^[-1]{}v\^ a\^[-1]{}. [\[eq:wood\]]{}
[^1]: The latter is most easily seen by considering the two linear transformations on $\delta{\ensuremath{\mathbf{a}}}$ that aggregate to give $\delta{\ensuremath{\mathbf{a}^{-1}}}$, i.e. bracketing the latter as $-({\ensuremath{\mathbf{a}^{-1}}}\delta {\ensuremath{\mathbf{a}^{-1}}}) {\ensuremath{\mathbf{a}^{-1}}}$, using the chain rule for Jacobians and performing row and column interchange operations on each resulting $n^2$-by-$n^2$ matrix.
[^2]: This can be seen by using Eqs. [(\[eq:amconstraints\])]{} to substitute for the variables appearing in the first [${n_\textsc{cmb}}$]{} columns of [$\mathbf{a}^{-1}$]{}, then appropriately adding in the other columns to make the first ones equal $({\ensuremath{\mathbf{i}}},{\ensuremath{\mathbf{0}}})^\text{T}$ in the determinant calculation.
[^3]: If we do have useful information on the foreground spectra we would presumably first encode this in [$\mathbf{a}$]{} and then consider transforming to [$\mathbf{a}^{-1}$]{}. If this information is actually coming from other non-[[CMB]{}]{} sky maps, it might perhaps be easier to perform a combined analysis including these maps as discussed above but with a larger [$\mathbf{a}$]{}.
[^4]: As with for the mixing matrix, we could take any data sets that do give us extra information into account by directly including them in the analysis.
[^5]: A suitable starting place should be obtainable from the “noise-free” solution evaluated for a reasonable fiducial model, as derived in [Sec. \[sec:nonoise\]]{}.
[^6]: This prefactor coming from the integral over [$\mathbf{w}$]{} is helping to encode the uncertainties from the foreground separation into the likelihood; this term would be missed in any approach that attempts to use a standard [[CMB]{}]{}likelihood formula applied to some best-fit linearly combined map and effective noise matrix.
[^7]: By only using certain columns of ${\ensuremath{\mathbf{e}}}$ in ${\ensuremath{\mathbf{P}}}(i)$ one can in fact project out individual Stokes components of the [[CMB]{}]{}, paving the way for independent temperature and polarization masks.
[^8]: Of course one might be able to apply the scheme directly to data from high resolution ground based experiments that only focus on small patches of the sky.
[^9]: For example, one might work in harmonic space and assume that the detector noise, along with the signal, is diagonal in this basis. Then many of the matrix operations simplify. Still implementing a general cut using the projection technique, the computational burden might be reduced by the ratio of the cube of the number of modes projected out to the cube of the number of elements. A limited number of further corrections to the noise matrix might also be efficiently implementable using Woodbury formula techniques. Alternatively one might consider perturbative corrections to the noise model away from the (masked) isotropic case.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we propose a framework capable of generating face images that fall into the same distribution as that of a given one-shot example. We leverage a pre-trained StyleGAN model that already learned the generic face distribution. Given the one-shot target, we develop an iterative optimization scheme that rapidly adapts the weights of the model to shift the output’s high-level distribution to the target’s. To generate images of the same distribution, we introduce a style-mixing technique that transfers the low-level statistics from the target to faces randomly generated with the model. With that, we are able to generate an unlimited number of faces that inherit from the distribution of both generic human faces and the one-shot example. The newly generated faces can serve as augmented training data for other downstream tasks. Such setting is appealing as it requires labeling very few, or even one example, in the target domain, which is often the case of real-world face manipulations that result from a variety of unknown and unique distributions, each with extremely low prevalence. We show the effectiveness of our one-shot approach for detecting face manipulations and compare it with other few-shot domain adaptation methods qualitatively and quantitatively.'
author:
- |
Chao YangSer-Nam Lim\
Facebook AI
bibliography:
- 'egbib.bib'
title: 'One-Shot Domain Adaptation For Face Generation'
---
| {
"pile_set_name": "ArXiv"
} |