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+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+NICOLA DE NITTI AND FLORIAN SCHWEIGER
+Abstract. This work is concerned with fractional Gaussian fields, i.e. Gaussian fields whose covari-
+ance operator is given by the inverse fractional Laplacian (−∆)−s (where, in particular, we include
+the case s > 1). We define a lattice discretization of these fields and show that their scaling limits –
+with respect to the optimal Besov space topology – are the original continuous fields. As a byproduct,
+in dimension d < 2s, we prove the convergence in distribution of the maximum of the fields. A key
+tool in the proof is a sharp error estimate for the natural finite difference scheme for (−∆)s under
+minimal regularity assumptions, which is also of independent interest.
+Contents
+1.
+Introduction
+1
+1.1.
+Fractional Gaussian Fields
+1
+1.2.
+Finite difference schemes for fractional operators
+3
+1.3.
+Main results
+4
+1.4.
+Future work
+6
+2.
+Fractional polyharmonic Gaussian fields
+7
+2.1.
+The (continuous) fractional Gaussian field
+7
+2.2.
+The (discrete) fractional Gaussian field
+9
+3.
+Rigorous estimates for the finite difference scheme
+10
+4.
+Proofs of the scaling limits
+13
+4.1.
+Scaling limit in the space of distributions
+13
+4.2.
+Scaling limit in Besov, Sobolev and Hölder spaces
+15
+Appendix A.
+Technical lemmas
+21
+A.1.
+Discretization and restriction
+21
+A.2.
+Discrete inequalities
+21
+Appendix B.
+Fractional Gaussian Fields via eigenfunctions
+23
+Acknowledgments
+25
+References
+25
+1. Introduction
+1.1. Fractional Gaussian Fields. Fractional Gaussian fields form a natural one-parameter family
+of Gaussian interface models.
+For a fixed parameter s ≥ 0, the s-fractional Gaussian field is the
+Gaussian field whose covariance operator is (−∆)−s, the inverse of the fractional Laplacian of order
+s. We emphasize right away that we do not assume s ∈ [0, 1], and in fact our main interest is in the
+polyharmonic case s > 1. A purely formal and non-rigorous way to define the s-fractional Gaussian
+field on some domain Ω ⊂ Rd is to set
+P(dϕ) = 1
+Z exp
+�
+−1
+2
+�
+Ω
+ϕ(x)((−∆)sϕ)(x) dx
+�
+dϕ.
+(1.1)
+This cannot be taken as a rigorous definition, as dϕ refers to the Lebesgue measure on the infinite-
+dimensional space RΩ, which does not exist.
+1
+arXiv:2301.13781v1  [math.PR]  31 Jan 2023
+
+2
+N. DE NITTI AND F. SCHWEIGER
+(a) s = 0 (white noise)
+(b) s = 0.5
+(c) s = 1 (Gaussian free field)
+(d) s = 1.5
+(e) s = 2 (membrane model)
+(f) s = 2.5
+(g) s = 3
+(h) s = 3.5
+(i) s = 4
+Figure 1.1. Surface plots of discrete fractional polyharmonic Gaussian fields on Ω :=
+(0, 1)2 ∩
+1
+100Z2 with zero boundary conditions. These discrete random functions are
+linearly interpolated. See also [15, Fig. 1.1] for further numerical experiments.
+There are also other issues with (1.1): namely, one needs to decide how to define (−∆)s for functions
+ϕ: Ω → R and (closely related to that issue) one needs to decide on boundary values of ϕ. For these
+questions, we have a clear answer, though. We take 0 boundary values (i.e., we take ϕ to be extended
+by 0 to the whole Rd), and we let (−∆)s be the fractional Laplacian on the full space Rd, which is
+defined by using the Fourier transform1. That is, for any ξ ∈ Rd,
+F [(−∆)su] (ξ) = |ξ|2sF[u](ξ)
+with
+F[u](ξ) =
+�
+Rd e−iξ·xu(x) dx.
+These choices are natural from a probabilistic point of view, as we will explain in Remark 2.2, and
+they can be implemented to provide a rigorous meaning to (1.1), for example, as a probability measure
+on the space of tempered distributions. This is discussed in detail in the excellent survey [15] and in
+Section 2.1 we recall the points which are important for us.
+1An equivalent hypersingular integral formulation of the polyharmonic fractional operator of order s ∈ (0, m), for
+any m ∈ N, is given by
+(−∆)su(x) := Cd,m,s
+�
+Rd
+�m
+j=−m(−1)j
+� 2m
+m − j
+�
+u(x + jy)
+|y|d+2s
+dy,
+where
+Cd,m,s :=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+22sΓ(n/2 + s)
+πn/2Γ(−s)
+�
+�
+m
+�
+j=1
+(−1)j
+�
+2m
+m − j
+�
+j2s
+�
+�
+−1
+if s ∈ (0, m)\N,
+22sΓ(n/2 + s)s!
+2πn/2
+�
+�
+m
+�
+j=2
+(−1)j
+�
+2m
+m − j
+�
+j2s ln j
+�
+�
+−1
+if s ∈ (0, m) ∩ N.
+We refer to [1] and references therein for further information on the theory of higher-order fractional Laplacians.
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+3
+The main goal of the present work is to define a discrete version ϕh of the s-fractional Gaussian field
+ϕ on a lattice Ωh = Ω ∩ hZd and to study its properties. It is not immediately obvious how one should
+define this discrete version, but we will argue that our definition is quite natural. Certainly, one would
+expect that in the limit h → 0 the discrete s-fractional Gaussian field converges in law, with respect
+to a suitable topology, to the (continuous) s-fractional Gaussian field. Our main result is that, with
+our definition of a s-fractional Gaussian field, this convergence holds in a rather strong sense, namely
+in law with respect to the topology of Besov spaces for the optimal range of parameters.
+Similar problems have been studied before for specific values of s. If s = 1, the field is the Gaussian
+free field and the convergence of the discrete Gaussian free field to its continuous variant is folklore
+(see [20, Section 4] for related results). The proof relies on the fact that covariances of the discrete
+Gaussian free field can be represented using simple random walk, which in the scaling limit becomes
+Brownian motion. The case 0 ≤ s < 1 is addressed in [15, Section 12]2, and the proof of the scaling
+limit follows a similar strategy as for the case s = 1, just with the 2s-stable Lévy process taking the
+place of Brownian motion.
+These results for s ≤ 1 all rely on some form of a random walk representation. For s > 1 and for our
+choice of boundary values, there is no such random walk representation and so proofs become much
+more difficult.3 The only existing result in this regime is for s = 2, where the s-fractional Gaussian field
+is the so-called membrane model. In [7], it was proven that this field is the scaling limit of its discrete
+version. The main ingredient in the proof were estimates for finite difference schemes for (−∆)2 from
+[22], and estimates for its Green’s function from [16].
+Thus, previous work was restricted to s ∈ [0, 1] ∪ {2}, while our results cover the entire range
+s ∈ [0, ∞). Even in the case s ∈ [0, 1] ∪ {2}, our results improve upon the previous work. Namely, the
+convergence in [15, Section 12] is with respect to the topology of distributions and the convergence in
+[7] is with respect to the topology of some negative Sobolev space (for non-optimal parameters). As
+an easy corollary of our result with respect to the Besov-space topology, one obtains convergence with
+respect to the Sobolev-topology and also with respect to the Hölder topology (both with the optimal
+range of parameters).
+Our method of proof uses estimates for finite difference schemes like [7], but of a different flavor. In-
+stead of the estimate from [22] used in [7] (that needs Ck-regularity of the function to be approximated
+by the scheme), we establish an estimate that needs only minimal regularity assumptions (essentially
+just Hs+ε-regularity for some ε > 0). This is the main technical result of the paper and we will discuss
+it and its context next.
+1.2. Finite difference schemes for fractional operators. There is a close relation between discrete
+versions of Gaussian fields and finite difference approximations of the corresponding operator. Indeed,
+if we want to define a lattice version of (1.1) that is suitably close to (1.1) itself, then we need a lattice
+approximation of (−∆)s; the better this approximation, the closer the resulting lattice field will be to
+its continuous version.
+Before discussing finite difference schemes, let us mention that there has been work on finite element
+approaches to the fractional Laplacian (at least for s ≤ 1). We cannot cover the whole body of relevant
+literature here, but we refer to the very recent survey [3].
+Let us now turn to finite difference schemes. The subject of finite difference schemes for the fractional
+Laplacian (−∆)s has been studied before and various schemes have been proposed. However, the main
+focus has been on the case s < 1 and often also d = 1. We refer to the survey [13] and the references
+therein for an overview.
+The main challenge when constructing a finite difference scheme for the
+2There, a definition of the discrete FGF that is slightly different from ours is used; the proof, however, should apply
+to all reasonable discretizations including ours.
+3If one uses another definition of the fractional Gaussian field in terms of spectral powers of the ordinary Laplacian
+(the so-called eigenfunction FGF from [15, Section 9]), one retains a random walk representations and it is comparably
+easy to establish a scaling limit. In fact, in [2], this is done not in the lattice case, but in the more complicated case of
+a Sierpinski gasket.
+
+4
+N. DE NITTI AND F. SCHWEIGER
+fractional Laplacian is that it is given by convolution with a singular integral kernel, and a naive
+discretization of this kernel might not capture its behavior near the singularity.
+Our preferred way to construct a finite difference scheme arises naturally when working in Fourier
+space. Let us consider first the usual Laplacian, i.e. the case s = 1. Its symbol is |ξ|2 and its standard
+finite difference approximation (the 2d + 1-point stencil in dimension d) has symbol
+(1.2)
+Mh(ξ)2 :=
+d
+�
+j=1
+4
+h2 sin2
+�ξjh
+2
+�
+.
+So, for the fractional Laplacian (−∆)s with symbol |ξ|2s, a natural way to define a finite difference
+scheme is to take the finite difference operator with symbol Mh(ξ)2s. That is, we define
+Fh [(−∆h)suh] (ξ) = Mh(ξ)2sFh[uh](ξ)
+with
+Fh[uh](ξ) = hd �
+x∈hZd
+e−iξ·xuh(x).
+We can also use Mh(ξ)2s as a continuous Fourier multiplier and thereby understand (−∆h)s also as a
+continuous operator. This is consistent with the previous definitions, as pointed out in Lemma A.1.
+This scheme for s ≤ 1 (but general d) has already been studied in [12] (and the special case d = 1
+already in [13, Section 4.2] and, in more detail, in [6]) and has many desirable properties. First of all,
+for s ∈ N, we recover the standard schemes for polyharmonic Laplacians. We also have the property
+that (−∆h)s(−∆h)s′ = (−∆h)s+s′. Moreover, while for other schemes the accuracy often degenerates
+as s ↗ 1, our scheme has accuracy h2 uniformly in s (as follows from Theorem 1.3). In Remark 3.2
+below, we comment on how this scheme might work in practice.
+Now that we have chosen our scheme, let us discuss rigorous estimates for its approximation quality.
+In the literature on finite difference schemes, it is common to derive pointwise estimates on the error
+under a strong regularity assumption (Ck or Ck,α for a large enough k). In fact, for d ∈ {1, 2} and
+s ≤ 1, there are two such results in the literature: in [6], pointwise estimates for the approximation
+error for functions in Hölder spaces (at least C0,2s+ε) are shown and, in [12], such pointwise estimates
+are shown under the assumption that the 2s + ε-th derivative has integrable Fourier transform (which,
+roughly speaking, again corresponds to C0,2s+ε).
+However, as already mentioned in the previous subsection, our interest is more in estimates under
+low-regularity assumptions, i.e. in a Sobolev scale. To the best of our knowledge, such estimates are
+new (even in the case d = 1, s < 1). For the case of the Laplacian or Bilaplacian, though, such results
+are classical (see the textbook [14] or a recent refinement for the Bilaplacian in [19]), and our result is
+inspired by the latter. However, the proof is quite different. Namely, the proof of [19, Theorem 2.3]
+relied on the Bramble-Hilbert lemma and thereby used that s ∈ N. In our general setting, we use a
+different approach, based on the Poisson summation formula and a lengthy estimate of various error
+terms in Fourier space.
+1.3. Main results. Let us now state our main results more precisely. We consider the discrete FGF
+ϕh and the continuous FGF ϕ, as introduced informally in Section 1.1. As the rigorous definitions are
+quite technical, we postpone them to Sections 2.2 and 2.1, respectively.
+We claim that the scaling limit of ϕh in an appropriate sense is ϕ. However, ϕh is defined only
+on hZd, so we need to interpolate it to a function on Rd first. For that purpose, we fix a compactly
+supported function Θ ∈ S(Rd) with
+�
+Rd Θ(x) dx = 1 and define Θh(x) =
+1
+hd Θ
+� x
+h
+�
+.
+Using Θh, we can define the interpolated field
+Ihϕh(x) :=
+�
+y∈hZd
+hdϕh(y)Θh(x − y)
+as a random element of S′(Rd).
+Some of the results below also hold if Θ is just a tempered distribution (and, in fact, in [7] only the
+choice Θ = δ0 was used). However, if we hope to find a scaling limit in some Banach space of optimal
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+5
+regularity, we need to consider more regular Θ (as otherwise Ihϕh might not even be an element of
+the Banach space in question); so, to avoid unneccessarily complicated notations, we directly assume
+that Θ is a measurable function.
+As a first result, we claim that, for any Θ chosen as above, the interpolated fields Ihϕh converge in
+the sense of distributions. Note that our definition of ϕh is made in such a way that we do not need
+to rescale it with some power of h to obtain a scaling limit. Indeed, we have the following result.
+Theorem 1.1 (Scaling limit in the space of distributions). Let Ω ⊂ Rd be a bounded domain with
+Lipschitz boundary, and let s ≥ 0.
+Let Θ be a compactly supported function with integral 1. Then Ihϕh converges in law with respect to
+the topology of S′(Rd) to ϕ. That is, for any f ∈ S(Rd), the random variable (Ihϕh, f)L2
+h(hZd)converges
+in law to (ϕ, f)L2(Rd).
+In Theorem 1.1, we established a scaling limit in the space of distributions. However, as we discuss
+in detail in Section 2, the continuous FGF is defined not just as a distribution-valued random variable,
+but actually has a certain Besov-, Sobolev- and Hölder-regularity. Hence, it is natural that we can
+take the scaling limit of the ϕh also in these spaces. In order to do so, however, we need some further
+assumptions on the regularization Θ (otherwise the interpolated field Ihϕh might not even be an
+element of the space). The result now is the following.
+Theorem 1.2 (Scaling limits in Sobolev and Hölder Spaces). Let Ω ⊂ Rd be a bounded domain with
+Lipschitz boundary and let s ≥ 0. Let Θ be a compactly supported function with integral 1, and suppose
+that there is some k with k > s + d
+2 such that
+(1.3)
+|FΘ(ξ)| ≤ C
+(�d
+j=1 sin2(ξj))k/2
+|ξ|k
+for all ξ.
+Let s′ < s − d
+2, p, q ∈ [1, ∞]. Then the interpolated fields Ihϕh converge in law with respect to the
+topology of ˆBs′
+p,q(Rd) to ϕ.
+Moreover, for any fixed bounded domain ˆΩ with Lipschitz boundary such that Ω ⋐ ˆΩ, Ihϕh are
+supported in ˆΩ for h sufficiently small. For any s′ < s − d
+2, the interpolated fields Ihϕh converge in
+law with respect to the topology of ˙Hs′(ˆΩ) to ϕ. In addition, if H := s − d
+2 > 0, m = ⌈H⌉ − 1, and
+0 < α < H − m, then Ihϕh converges in law with respect to the topology of Cm,α(Rd) to ϕ.
+Here, ˆBs′
+p,q(Rd) is, up to a minor technicality that we again discuss in Section 2, equal to the standard
+Besov space Bs′
+p,q(Rd).
+Several remarks are in order. First of all, a convenient example of a function satisfying (1.3) for
+some k ∈ N is given by the centered B-spline of order k (see, e.g., [14, Section 1.9.4] for a summary of
+their properties).
+Next, the restriction to s′ < s − d
+2 cannot be avoided as the continuous FGF is not in ˆBs−d/2
+p,q
+for
+any p, q. Thus, the range of allowed s′ is optimal.
+Regarding the convergence in Sobolev spaces, we cannot expect convergence with respect to the
+topology of ˙Hs′(Ω) for the simple reason that, because of the mollification, Ihϕh need not have zero
+boundary values outside of Ω.
+A fundamental step in the proof of the results above is establishing the following error estimate for
+a fractional Poisson equation (which is of interest in itself). Our goal is to compare the solutions of
+(−∆)su = f and of (−∆h)suh = f and we will estimate the error u − uh in the (discrete) energy norm
+∥ · ∥ ˙Hs
+h. As we work under minimal regularity assumptions on u, the precise result is somewhat more
+technical. Namely, in general u and f might not be continuous functions and so it is not clear how to
+restrict them to the lattice. We circumvent this by introducing two additional mollifiers.
+The result then takes the following shape.
+Theorem 1.3 (Error estimate on the discrete approximation). Let Ω ⊂ Rd be an open bounded set
+with Lipschitz boundary. Let Θ: Rd → R and θ: Rd → R be mollifiers that are compactly supported,
+
+6
+N. DE NITTI AND F. SCHWEIGER
+symmetric around 0, and have integral 1. Furthermore, let us assume that there exist k, l ≥ 0 such
+that
+|FΘ(ξ)| ≤ C
+(�d
+j=1 sin2(ξj))k/2
+|ξ|k
+,
+|Fθ(ξ)| ≤ C 1
+|ξ|l ,
+and define Θh(x) =
+1
+hd Θ
+� x
+h
+�
+, θh(x) =
+1
+hd θ
+� x
+h
+�
+.
+Let 0 < s < t and let u ∈ Ht(Rd) be the solution of
+�
+(−∆)su(x) = f(x),
+x ∈ Ω,
+u(x) = 0,
+x ∈ Rd \ Ω,
+for some f ∈ Ht−2s(Rd), and uh : hZd → R be the solution of
+�
+(−∆h)suh(x) = Θh ∗ f(x),
+x ∈ hZd ∩ Ω,
+uh(x) = 0,
+x ∈ hZd \ Ω.
+Then, if h ≤ 1, k ≥ s, k > d
+2 + 2s − t, l > d
+2 − t, and t − s ≤ 2, we have the estimate
+∥θh ∗ u − uh∥ ˙Hs
+h(hZd) ≤ Cht−s∥u∥ ˙Ht(Rd),
+where C > 0 does not depend on h.
+Here (as in the rest of the paper) C denotes some generic constant that might change from line to
+line, but is always independent of h.
+Let us give some explanations regarding the linear constraints on the parameters in this result. The
+most important constraint is t − s ≤ 2. It arises from the fact that the proposed finite difference
+scheme is of second order (see [13, Section 5.2]), so that the accurary of our scheme saturates at h2.
+The condition k > d
+2 + 2s − t is needed in order for Θh ∗ f to be continuous (so that it has a well-
+defined restriction to the lattice). Similarly, the condition l > d
+2 − t is needed in order for θh ∗ u to be
+continuous.
+As mentioned in Section 1.2, this is the first rigorous estimate for a finite difference scheme for
+(−∆)s under low regularity assumptions. For finite elements, a result that is similar in spirit can
+be found in [3, Theorem 2.6]. There an estimate for piecewise linear finite elements is shown that is
+similar to our result (albeit with the additional restriction t − s ≤ 1
+2 instead of t − s ≤ 2). The method
+of proof is very different.
+1.4. Future work. The most well-studied discrete Gaussian interface model is certainly the discrete
+Gaussian free field (corresponding to s = 1 in our notation). In recent years there has been a lot
+of activity to extend results known for the discrete Gaussian free field to other discrete (Gaussian or
+non-Gaussian) interface models, and this work is a first step to include the discrete FGFs in the latter
+class.
+Let us highlight one such question, namely regarding the maximum of the field. In case of the
+discrete Gaussian free field, this is well-understood. In the subcritical dimension (d = 1), the field
+is nothing but a random walk bridge, so it is easy to see that the rescaled maximum converges to a
+non-degenerate random variable. In supercritical dimensions (d ≥ 3), correlations decay so rapidly
+that the maximum behaves as if the field values were independent [5]. The most interesting case is the
+critical case, d = 2, where the field is log-correlated and obtains the typical second-order correction
+[4].
+These results have already been extended to the case of the membrane model (corresponding to
+s = 2). The subcritical case (d ≤ 3) was studied in [7] using results from [16], the supercritical case
+in [5], and finally the critical case in [19]. An important tool in the latter proof was an estimate for
+finite difference schemes very similar to the one in Theorem 1.3.
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+7
+For general s, it is very likely that similar results hold true. In fact, in the subcritical case d < 2s,
+convergence of the rescaled maximum is a straightforward corollary of Theorem 1.2.
+Corollary 1.4 (Convergence of the maximum for d < 2s). Let d < 2s, Ω ⊂ Rd be a fixed bounded
+domain with Lipschitz boundary, and consider the family ϕh of s-FGF on Ωh = Ω∩hZd as h → 0. Then
+the random variables maxx∈Ωh ϕh(x) converge in distribution to a non-degenerate random variable.
+While this corollary covers the subcritical case, the critical case (d = 2s) and the supercritical case
+(d > 2s) remain open, and we hope to address them in the future.
+In particular, a study of the
+critical case would be very interesting, as most existing examples of discrete log-correlated fields in the
+literature are in d = 2 or some other even dimension while the 3
+2-discrete FGF, for instance, would be
+a natural example of a log-correlated field in odd dimensions.
+2. Fractional polyharmonic Gaussian fields
+In this section, we give precise definitions for the continuous and discrete FGF. For the continuous
+FGF, we follow [15]. The major difference is that we only require Lipschitz continuity of the boundary
+of our domain Ω (and hence our results cover in particular the important special case Ω = (0, 1)d).
+Because of this, several functional-analytic statements require extra attention and we give precise
+references for the results we use.
+2.1. The (continuous) fractional Gaussian field. We first fix our conventions for the Fourier
+transform, and then use it to define some relevant function spaces.
+For a function u: Rd → R, we let F[u](ξ): Rd → R, defined by
+F[u](ξ) =
+�
+Rd e−iξ·xu(x) dx,
+be its continuous Fourier transform. Then, we have the Fourier inversion formula,
+u(x) = F−1[F[u]](x) =
+1
+(2π)d
+�
+Rd eiξ·xF[u](ξ) dξ,
+and Plancherel’s theorem,
+�
+Rd |u(x)|2 dx =
+1
+(2π)d
+�
+Rd |F[u](ξ)|2 dξ.
+We can also define the Sobolev norms
+∥u∥2
+˙Hs(Rd) =
+�
+Rd |ξ|2s|F[u](ξ)|2 dξ,
+∥u∥2
+Hs(Rd) =
+�
+Rd(1 + |ξ|2)s|F[u](ξ)|2 dξ,
+where |ξ|2 is the Fourier multiplier of the Laplacian −∆.
+Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary and let S′(Rd) be the space of tempered
+distributions on Rd. In what follows, we collect some results on fractional Sobolev spaces on Ω. If Ω
+has a smooth boundary, all of them are well-known (and [23, Chapter 4] is a comprehensive reference).
+If Ω has merely Lipschitz boundary, the situation is slightly more complicated and we rely on the
+reference [24]. A brief version of some of these results is also contained in [15, Section 4.1], but some
+of them are not made explicit there.
+For s ≥ 0, let ˙˜Hs(Ω) be the closure of C∞
+c (Ω) with respect to the norm ∥· ∥ ˙Hs(Rd)
+4 and let ˙H−s(Ω)
+be its dual space. The space ˙H−s(Ω) can alternatively be described as follows. Let
+∥u∥ ˙H−s(Ω) =
+inf
+v∈ ˙H−s(Rd)
+u=v in Ω
+∥v∥ ˙H−s(Rd)
+4In [15] this space is denoted Hs
+0(Ω). However, more commonly ˙Hs
+0(Ω) is defined as the closure of C∞
+c (Ω) with respect
+to the norm ∥ · ∥ ˙Hs(Ω), while our space ˙˜Hs(Ω) is equal to the Lions-Magenes space (which is also denoted by ˙Hs
+00(Ω)).
+The two spaces are different whenever s ∈ N + 1
+2 . Our notation is based on the one in [23, Chapter 4].
+
+8
+N. DE NITTI AND F. SCHWEIGER
+for u ∈ S(Rd). Then if S(Ω) is the quotient of S(Rd) under the equivalence relation that identifies
+functions when they agree in Ω, we have that ˙H−s(Ω) is the closure of S(Ω) with respect to the norm
+∥ · ∥ ˙H−s(Ω) (this follows from [24, Theorem 3.5 (i)] upon observing that our spaces ˙˜Hs(Ω), for s ≥ 0,
+and ˙H−s(Ω), for s < 0, are equal to Triebel’s ¯Bs
+2,2(Ω) by [24, Proposition 3.1]).
+From the Lax-Milgram lemma (and the fact that C∞
+c (Ω) is dense in ˙˜Hs(Ω)) we also obtain that
+(−∆)s is an isometry from ˙˜Hs(Ω) to ˙H−s(Ω).
+For convenience (and with a slight abuse of notation) we define
+˙Hs(Ω) =
+� ˙˜Hs(Ω)
+if s ≥ 0,
+˙Hs(Ω)
+if s < 0.
+This scale of Hilbert spaces has various desirable properties. For any s < t, the embedding from ˙Ht(Ω)
+to ˙Hs(Ω) is compact (cf. [24, Theorem 2.7]). Even more importantly, the spaces form an interpolation
+scale with respect to complex (or equivalently real) interpolation (cf. [24, Theorem 3.5 (iv)]).
+In [15, Section 4.2], the continuous FGF is defined as a probability measure P on S′(Rd). More
+precisely, it is defined such that when ϕ is distributed according to P, then for every Schwartz function
+f ∈ S(Rd) we have that (ϕ, f) is a centered Gaussian with variance ∥f∥2
+˙H−s(Ω). By [15, Theorem 2.3
+and Proposition 2.4], this property defines P as a probability measure on S′(Rd) uniquely.
+Let us remark that one can one alternatively define the FGF as a random sum of eigenfunctions of
+(−∆)s. We give details on this in Appendix B.
+The regularity of ϕ is best measured in Besov spaces. For s′ ∈ R, p, q ∈ [1, ∞], we let ∥ · ∥Bs′
+p,q(Rd)
+be the usual Besov norm (defined, e.g., via Littlewood-Paley decomposition or via wavelets; see,
+for example, [23, Chapter 2]) and let ˆBs′
+p,q(Rd) be the closure of C∞
+c (Rd) with respect to the norm
+∥ · ∥Bs′
+p,q(Rd)
+5.
+Then we have the following regularity results for ϕ.
+Proposition 2.1 (Regularity of the FGF). Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary
+and let s ≥ 0. For any s′ < s − d
+2 and p, q ∈ [1, ∞], the FGF on Ω is P-almost surely an element of
+ˆBs′
+p,q(Rd).
+In particular, for any s′ < s − d
+2 the FGF on Ω is P-almost surely an element of ˙Hs′(Ω).
+Moreover, if H := s − d
+2 > 0, then the FGF on Ω is also P-almost surely an element of Cm,α
+loc (Rd)
+for m = ⌈H⌉ − 1 and any 0 < α < H − m.
+Proof. The Besov regularity could be shown using the tightness criterion in Lemma 4.1 below applied
+to the constant sequence ϕ(m) = ϕ. However, according to Theorem 1.2 we have the much stronger
+statement that the FGF is the limit (with respect to the ˆBs′
+p,q(Rd)-topology) of the discrete fractional
+Gaussian fields, suitably interpolated; so we do not give details for the proof of the Besov regularity
+here.
+It is well-known that ˆBs′
+2,2(Rd) = Hs′(Rd) and ˆBs′
+∞,∞(Rd) �→ Cs′(Rd), where Cs′(Rd) is the Hölder-
+Zygmund space, which embeds into the classical Hölder space C⌊s′′⌋,s′′−⌊s′′⌋ for any 0 < s′′ < s′ (see
+[24, Section 2.1]). These results together with the fact that the FGF is supported in Ω easily imply
+the Sobolev and Hölder regularity results in the proposition.
+Let us remark that the Sobolev regularity alternatively follows from the fact the random series defin-
+ing ˜ϕ converges in ˙Hs′(Ω) almost surely, while the Hölder regularity also follows from [15, Proposition
+6.2 and Theorem 8.3]6.
+□
+5Note that the Besov space Bs′
+p,q(Rd) commonly defined as the set of all tempered distributions for which ∥·∥Bs′
+p,q(Rd)
+is finite. Clearly ˆBs′
+p,q(Rd) ⊂ Bs′
+p,q(Rd), and the inclusion is strict if p = ∞ or q = ∞.
+6N.B. There is a typo in the statement of [15, Proposition 6.2]: it should read H − k instead of H − ⌈H⌉.
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+9
+2.2. The (discrete) fractional Gaussian field. Our definition of the discrete FGF follows the one
+of the continuous FGF as closely as possibly. Let us again begin by fixing our conventions for discrete
+Fourier transforms and discrete function spaces.
+For a function uh : hZd → R, we let Fh[uh](ξ): Rd → R, defined by
+Fh[uh](ξ) = hd �
+x∈hZd
+e−iξ·xuh(x),
+be its discrete Fourier transform (note that this function is 2π
+h -periodic). Then we have the discrete
+Fourier inversion formula,
+uh(x) = F−1
+h [Fh[uh]](x) =
+1
+(2π)d
+�
+(− π
+h , π
+h)
+d eiξ·xFh[uh](ξ) dξ,
+and Plancherel’s theorem,
+hd �
+x∈hZd
+|uh(x)|2 =
+1
+(2π)d
+�
+(− π
+h , π
+h)
+d |Fh[uh](ξ)|2.
+We can define the discrete Sobolev norms
+∥uh∥2
+˙Hs
+h(hZd) =
+�
+(− π
+h , π
+h)
+d Mh(ξ)2s|Fh[uh](ξ)|2 dξ,
+∥uh∥2
+Hs
+h(hZd) =
+�
+(− π
+h , π
+h)
+d(1 + Mh(ξ)2)s|Fh[uh](ξ)|2 dξ,
+where Mh(ξ)2 := �d
+j=1
+4
+h2 sin2 �
+ξjh
+2
+�
+is the discrete Fourier multiplier of the discrete Laplacian.
+Let Ω be as before and let Ωh = Ω ∩ hZd. Similarly as in the continuous setting, we define the
+space ˙˜Hs(Ωh) as the space of functions hZd → R that vanish outside of Ωh (equipped with the norm
+induced by ∥ · ∥ ˙Hs
+h(hZd)). We let ˙H−s(Ωh) be its dual space, and define
+˙Hs(Ω) =
+� ˙˜Hs
+h(Ωh)
+if s ≥ 0,
+˙Hs
+h(Ωh)
+if s < 0.
+We define the discrete FGF as a probability measure on ˙˜Hs(Ωh). More precisely, we consider the
+measure
+Ph( dϕh) = 1
+Zh
+exp
+�
+−1
+2∥ϕh∥2
+˙Hs
+h(hZd)
+� �
+x∈Ωh
+dϕh(x)
+�
+x∈hZd\Ωh
+δ0( dϕh(x))
+= 1
+Zh
+exp
+�
+−1
+2
+�
+x∈Ωh
+hdϕh(x)(−∆h)sϕh(x)
+� �
+x∈Ωh
+dϕh(x)
+�
+x∈hZd\Ωh
+δ0( dϕh(x)).
+This is a well-defined Gaussian measure with mean 0, and variance
+Eh(ϕh, fh)2
+L2
+h(hZ)d = ∥fh∥2
+˙H−s
+h
+(Ωh)
+(2.1)
+for any fh : hZd → R.
+Remark 2.2 (Boundary values). Let us comment on our choice of boundary values. The main advantage
+of our definition is the fact that it is consistent with projections. Namely, let Ω ⊂ ˜Ω be open sets
+and consider the discrete FGF ˜ϕh on ˜Ωh. Then, the restriction of ˜ϕh to Ωh is equal in distribution to
+the sum of the (−∆h)s-harmonic extension of ˜ϕh from hZd \ Ωh to Ωh and of an independent discrete
+FGF ϕh on Ωh 7. In particular, even if we had started with a field without boundary values (i.e., with
+7If s = 1, this reduces to the familiar domain Markov property for the discrete Gaussian free field: the field in a
+subdomain is equal in distribution to the harmonic extension of its boundary values plus an independent zero-boundary
+field.
+
+10
+N. DE NITTI AND F. SCHWEIGER
+˜Ω = Rd), then looking at the field on a subset naturally leads to consider fields with zero boundary
+values outside that subset.
+3. Rigorous estimates for the finite difference scheme
+In this section, we present the proof of Theorem 1.3. As mentioned in Section 1.2, the proof of
+the analogous statement for s = 2 in [19, Theorem 2.3] was based on the Bramble-Hilbert lemma to
+estimate various error terms. Thus it relied on the fact that (−∆h)2 (and hence the finite difference
+scheme) is local in that case.
+In the generic case s ̸∈ N, however, (−∆h)s is not local, and so this proof strategy can no longer
+be applied. Instead, we use the fact that both (−∆)s and (−∆h)s are defined via Fourier multipliers
+and directly estimate all relevant error terms in Fourier space. However, this requires extra care as we
+need to switch from discrete Fourier space to continuous Fourier space at some point. In fact, we need
+a way to compare Fh and F. Fortunately, the following Poisson-type summation formula enables us
+to do so easily.
+Lemma 3.1 (Poisson-type summation formula). Suppose that g: Rd → R is a Schwartz function.
+Then we have the identity
+Fh[g](ξ) =
+�
+ζ∈ 2π
+h Zd
+F[g](ξ + ζ).
+Proof. By the Poisson summation formula (see, e.g., [18, Chapter 4.4]), for any Schwartz function f
+we have
+hd �
+x∈hZd
+f(x) =
+�
+ζ∈ 2π
+h Zd
+F[f](ζ).
+Applying this to f(x) = e−iξ·xg(x), we find
+hd �
+x∈hZd
+e−iξ·xg(x) =
+�
+ζ∈ 2π
+h Zd
+F[e−iξ·g](ζ) =
+�
+ζ∈ 2π
+h Zd
+F[g](ξ + ζ),
+which implies the claim.
+□
+Using Lemma 3.1, we can now turn to the proof of our estimate on finite difference schemes.
+Proof of Theorem 1.3. By density, we can assume that u is smooth; then, in particular, u is a Schwartz
+function and F[u] is a Schwartz function as well. Therefore, all integrals and sums below will be well-
+defined.
+Step 1: Representation of the error. From the definitions we have
+(3.1)
+∥u − uh∥ ˙Hs
+h(hZd) = ∥(−∆h)s(θh ∗ u − uh)∥ ˙H−s
+h
+(Ωh) =
+inf
+v : hZd→R
+v=(−∆h)s(θh∗u−uh) in Ωh
+∥v∥ ˙H−s
+h
+(hZd).
+Using Lemma A.1, we can also rewrite, for x ∈ Ωh,
+(−∆h)s(θh ∗ u − uh)(x)
+= (−∆h)s(θh ∗ u)(x) − Θh ∗ f(x)
+= (−∆h)s(θh ∗ u)(x) − Θh ∗ (−∆)su(x)
+=
+1
+(2π)d
+�
+(− π
+h , π
+h)
+d eiξ·xMh(ξ)2sFh[θh ∗ u](ξ) dξ −
+1
+(2π)d
+�
+Rd eiξ·x|ξ|2sF[Θh](ξ)F[u](ξ) dξ
+= I1 + I2 + I3 + I4 + I5,
+where
+I1(x) :=
+1
+(2π)d
+�
+(− π
+h , π
+h)
+d eiξ·xMh(ξ)2s (Fh[θh ∗ u](ξ) − F[θ ∗ u](ξ)) dξ,
+I2(x) :=
+1
+(2π)d
+�
+(− π
+h , π
+h)
+d eiξ·xMh(ξ)2s (F[θh ∗ u](ξ) − F[u](ξ)) dξ,
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+11
+I3(x) :=
+1
+(2π)d
+�
+(− π
+h , π
+h)
+d eiξ·x (1 − F[Θh](ξ)) Mh(ξ)2sF[u](ξ) dξ,
+I4(x) :=
+1
+(2π)d
+�
+(− π
+h , π
+h)
+d eiξ·x �
+Mh(ξ)2s − |ξ|2s�
+F[Θh](ξ)F[u](ξ) dξ,
+I5(x) := −
+1
+(2π)d
+�
+Rd\(− π
+h , π
+h)
+d eiξ·x|ξ|2sF[Θh](ξ)F[u](ξ) dξ.
+To conclude the proof, we need to show that
+∥Ij∥ ˙H−s
+h
+(hZd) ≤ Cht−s∥u∥ ˙Ht(Rd)
+holds for each j ∈ {1, 2, 3, 4, 5}.
+Step 2: Estimate of I2. Directly from the definition, we see that
+Fh[I2](ξ) = Mh(ξ)2s (F[θh ∗ u](ξ) − F[u](ξ)) = Mh(ξ)2s (F[θh](ξ) − 1) F[u](ξ).
+As F[θh](ξ) = F[θ](hξ), our assumption on θ implies that F[θh](ξ) ≤
+C
+hl|ξ|l ≤ C for ξ ∈
+�
+− π
+h, π
+h
+�d.
+Therefore
+∥I2∥2
+˙H−s
+h
+(hZd) =
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s|Fh[I2](ξ)|2 dξ
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)2s |F[θh](ξ) − 1|2 |F[u](ξ)|2 dξ
+≤ C
+�
+(− π
+h , π
+h)
+d |ξ|2s|F[u](ξ)|2 dξ
+≤ C
+�
+(− π
+h , π
+h)
+d |ξ|2(t−s)h2(t−s)|F[u](ξ)|2 dξ
+≤ Ch2(t−s)∥u∥2
+˙Ht(Rd).
+Step 3: Estimate of I3. The estimate of I3 is quite similar: again, we have that
+Fh[I3](ξ) = (1 − F[Θh](ξ)) Mh(ξ)2sF[u](ξ).
+The assumptions on Θ imply that FΘ(0) = 1 and ∇FΘ(0) = 1. Furthermore, FΘ is a Schwartz
+function. So, by Taylor’s theorem, there is a constant C such that |1 − FΘ(ξ)| ≤ C|ξ|2. This implies
+|1 − F[Θh](ξ)| ≤ Ch2|ξ|2. We can now estimate
+∥I3∥2
+˙H−s
+h
+(hZd) =
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s|Fh[I3](ξ)|2 dξ
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)2s(1 − F[Θh](ξ))2|F[u](ξ)|2 dξ
+≤ C
+�
+(− π
+h , π
+h)
+d |ξ|2sh4|ξ|4|F[u](ξ)|2 dξ
+≤ C
+�
+(− π
+h , π
+h)
+d |ξ|2(t−s)h2(t−s)|F[u](ξ)|2 dξ
+≤ Ch2(t−s)∥u∥2
+˙Ht(Rd).
+Step 4:
+Estimate of I4.
+The argument for I4 is very similar to that for I3:
+we use that
+��Mh(ξ)2s − |ξ|2s�� ≤ Ch2|ξ|2 and proceed as for I3.
+Step 5: Estimate of I1. From the definition and Lemma 3.1, we have that
+Fh[I1](ξ) = Mh(ξ)2s (Fh[θh ∗ u](ξ) − F[θh ∗ u](ξ))
+
+12
+N. DE NITTI AND F. SCHWEIGER
+= Mh(ξ)2s
+�
+ζ∈ 2π
+h Zd\{0}
+F[θh ∗ u](ξ + ζ)
+= Mh(ξ)2s
+�
+ζ∈ 2π
+h Zd\{0}
+F[θh](ξ + ζ)F[u](ξ + ζ).
+The Cauchy-Schwartz inequality then yields
+∥I1∥2
+˙H−s
+h
+(hZd) =
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s|Fh[I1](ξ)|2 dξ
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)2s
+������
+�
+ζ∈ 2π
+h Zd\{0}
+F[θh](ξ + ζ)F[u](ξ + ζ)
+������
+2
+dξ
+≤
+�
+(− π
+h , π
+h)
+d Mh(ξ)2s
+�
+�
+�
+ζ∈ 2π
+h Zd\{0}
+|ξ + ζ|2t|F[u](ξ + ζ)|2
+�
+�
+�
+�
+�
+ζ∈ 2π
+h Zd\{0}
+F[θh](ξ + ζ)
+|ξ + ζ|2t
+�
+� dξ.
+We know that F[θh](ξ + ζ) ≤
+C
+hl|ξ+ζ|l . As 2(t + l) > d, we can bound
+sup
+ξ∈(− π
+h , π
+h)
+d
+�
+ζ∈ 2π
+h Zd\{0}
+F[θh](ξ + ζ)
+|ξ + ζ|2t
+≤
+sup
+ξ∈(− π
+h , π
+h)
+d
+�
+ζ∈ 2π
+h Zd\{0}
+1
+h2l|ξ + ζ|2(t+l)
+≤ C
+�
+ζ∈ 2π
+h Zd\{0}
+1
+h2l|ζ|2(t+l)
+≤ Ch2t
+and deduce
+∥I1∥2
+˙H−s
+h
+(hZd) ≤ C 1
+h2s h2t
+�
+(− π
+h , π
+h)
+d
+�
+ζ∈ 2π
+h Zd\{0}
+|ξ + ζ|2t|F[u](ξ + ζ)|2 dξ
+≤ Ch2(t−s)
+�
+Rd |ξ2t|F[u](ξ)|2 dξ
+≤ Ch2(t−s)∥u∥2
+˙Ht(Rd).
+Step 6: Estimate of I5. We see that
+F[I5](ξ) = |ξ|2sF[Θh](ξ)F[u](ξ)χRd\(− π
+h , π
+h)
+d(ξ),
+where χA is the indicator function of the set A. Lemma 3.1 then implies that, for ξ ∈
+�
+− π
+h, π
+h
+�d,
+Fh[I5](ξ) =
+�
+ζ∈ 2π
+h Zd
+F[I5](ξ + ζ)
+=
+�
+ζ∈ 2π
+h Zd
+|ξ + ζ|2sF[Θh](ξ + ζ)F[u](ξ)χRd\(− π
+h , π
+h)
+d(ξ + ζ)
+=
+�
+ζ∈ 2π
+h Zd\{0}
+|ξ + ζ|2sF[Θh](ξ + ζ)F[u](ξ + ζ)
+and therefore (recalling that Mh(ξ) is 2π
+h -periodic)
+∥I5∥2
+˙H−s
+h
+(hZd)
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s|Fh[I5](ξ)|2 dξ
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+13
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s
+������
+�
+ζ∈ 2π
+h Zd\{0}
+|ξ + ζ|2sF[Θh](ξ + ζ)F[u](ξ + ζ)
+������
+2
+dξ
+=
+�
+(− π
+h , π
+h)
+d
+������
+�
+ζ∈ 2π
+h Zd\{0}
+|ξ + ζ|2s
+Mh(ξ + ζ)s F[Θh](ξ + ζ)F[u](ξ + ζ)
+������
+2
+dξ
+≤
+�
+(− π
+h , π
+h)
+d
+�
+�
+�
+ζ∈ 2π
+h Zd\{0}
+|ξ + ζ|2t|F[u](ξ + ζ)|2
+�
+�
+�
+�
+�
+ζ∈ 2π
+h Zd\{0}
+|F[Θh](ξ + ζ)|2
+Mh(ξ + ζ)2s|ξ + ζ|2(t−2s)
+�
+� dξ.
+Note that F[Θh](ξ) = F[Θ](hξ) and so |F[Θh](ξ)| ≤ C
+(�d
+j=1 sin2(hξj))k/2
+hk|ξ|k
+; moreover,
+(�d
+j=1 sin2(hξj))1/2
+Mh(ξ)
+≤
+Ch. Since k ≥ s, Mh(ξ + ζ)2s is controlled by the sin-terms from |F[Θh](ξ + ζ)|2 and so we can bound
+sup
+ξ∈(− π
+h , π
+h)
+d
+�
+ζ∈ 2π
+h Zd\{0}
+|F[Θh](ξ + ζ)|2
+Mh(ξ + ζ)2s|ξ + ζ|2t−4s ≤ C
+sup
+ξ∈(− π
+h , π
+h)
+d
+�
+ζ∈ 2π
+h Zd\{0}
+h2s
+h2k|ξ + ζ|2(t+k−2s)
+≤ C
+�
+ζ∈ 2π
+h Zd\{0}
+1
+h2(k−s)|ζ|2(t+k−2s)
+≤ Ch2(t−s),
+where we used the fact that 2(t + k − 2s) > d. Hence,
+∥I5∥2
+˙H−s
+h
+(hZd) ≤ Ch2(t−s)
+�
+(− π
+h , π
+h)
+d
+�
+ζ∈ 2π
+h Zd\{0}
+|ξ + ζ|2t|F[u](ξ + ζ)|2 dξ
+≤ Ch2(t−s)
+�
+Rd |ξ|2t|F[u](ξ)|2 dξ
+≤ Ch2(t−s)∥u∥2
+˙Ht(Rd).
+This completes the proof.
+□
+Remark 3.2 (Usage of the finite difference scheme). So far we have not said much regarding the
+practical applications of the finite difference scheme in Theorem 1.3. It would go beyond the scope of
+this work to report on some practical experiments, but let us make a few comments.
+In order to use the scheme to approximate a solution of (−∆)su = f, a first challenge is to compute
+the entries of ((−∆h)s)x,y∈Ωh. Even using the translation-invariance of (−∆h)s, we need to compute
+O
+� 1
+h2
+�
+entries, where each is given as a singular integral. This is quite costly, but avoids introduction
+of additional error. In fact, the pictures in Figure 1.1 were computed using this method.
+If one is willing to accept an additional error term, then a more efficient way to compute an
+approximation to the entries of ((−∆h)s)x,y∈Ωh was suggested in [12]: choose a parameter h′ ≤ h, and
+approximate the integral over
+�
+− π
+h, π
+h
+�d appearing in the definition of (−∆h)s by a Riemann sum on
+a lattice of width h′
+h . The advantage is that this Riemann sum can be computed very efficiently using
+the fast Fourier transform. Moreover, in [12, Section 4.2], it is suggested that this should lead to an
+additional error of order O
+�
+h′d+2s
+h2s
+�
+. In other words, if we choose h′ ≤ h(2s+2)/(2s+d), the error should
+be of order h2 and thus not bigger than the error in Theorem 1.3. While the error estimate in [12,
+Section 4.2] is not rigorous, it should be possible to give a full proof.
+4. Proofs of the scaling limits
+4.1. Scaling limit in the space of distributions. With Theorem 1.3 in hand, we are ready to
+prove that ϕ is indeed the scaling limit of the ϕh. First, we study the scaling limit in the space of
+distributions, Theorem 1.1.
+
+14
+N. DE NITTI AND F. SCHWEIGER
+Proof of Theorem 1.1. Step 1: Characterization of the convergence. Let us consider some f ∈ S(Rd).
+Both (Ihϕh, f)L2(Rd) and (ϕ, f)L2(Rd) are centered Gaussian random variables and so it suffices to
+prove that their variances converge. We have that
+Eh(Ihϕh, f)2 = Eh
+�
+�
+�
+Rd
+�
+y∈hZd
+hdϕh(y)Θh(x − y)f(x) dx
+�
+�
+2
+= Eh
+�
+� �
+y∈hZd
+hdϕh(y)
+�
+Rd Θh(x − y)f(x) dx
+�
+�
+2
+= Eh(ϕh, Θh ∗ f)2
+L2
+h(hZd)
+= ∥Θh ∗ f∥2
+˙H−s
+h
+(Ωh)
+(4.1)
+and so we only need to prove that
+(4.2)
+lim
+h→∞ ∥Θh ∗ f∥2
+˙H−s
+h
+(Ωh) = ∥f∥2
+˙H−s(Ω).
+Step 2: Representation of the error. For each h > 0, let uh : hZd → R be the solution of
+�
+(−∆h)suh(x) = Θh ∗ f(x),
+x ∈ hZd ∩ Ω,
+uh(x) = 0,
+x ∈ hZd \ Ω
+and let u ∈ Hs(Rd) be the solution of
+�
+(−∆)su(x) = f(x),
+x ∈ Ω,
+u(x) = 0,
+x ∈ Rd \ Ω.
+Moreover, let ˜Θ, ˜θ be functions satisfying the assumptions of Theorem 1.3 with k := max
+� d
+2 + s, s
+�
+and l := max
+� d
+2 − s, 0
+�
+; for example, let us take ˜Θ to be a B-spline of order ⌈k⌉ and ˜θ any smooth
+mollifier. Then, let us define ˜Θh and ˜θh as before and let ˜uh : hZd → R be the solution of
+�
+(−∆h)s˜uh(x) = ˜Θh ∗ f(x),
+x ∈ hZd ∩ Ω,
+˜uh(x) = 0,
+x ∈ hZd \ Ω.
+Then, we can write
+∥Θh ∗ f∥2
+˙H−s
+h
+(Ωh) − ∥f∥2
+˙H−s(Ω) = (Θh ∗ f, uh)L2
+h(hZd) − (f, u)L2(Rd)
+= J1 + J2 + J3 + J4 + J5,
+(4.3)
+where
+J1 := (Θh ∗ f, uh)L2
+h(hZd) − (Θh ∗ f, ˜uh)L2
+h(hZd),
+J2 := (Θh ∗ f, ˜uh)L2
+h(hZd) − (Θh ∗ f, ˜θh ∗ u)L2
+h(hZd),
+J3 := (Θh ∗ f, ˜θh ∗ u)L2
+h(hZd) − (f, ˜θh ∗ u)L2
+h(hZd),
+J4 := (f, ˜θh ∗ u)L2
+h(hZd) − (f, ˜θh ∗ u)L2(Rd),
+J5 := (f, ˜θh ∗ u)L2(Rd) − (f, u)L2(Rd).
+We need to show that Ji → 0 as h → 0. This implies (4.2), as required. The most important term is
+J2, for which we shall need to use Theorem 1.3; the other terms will be straightforward to control.
+Step 3: Estimate of J2. Let t > s be a constant to be chosen later. Our choices of k, l ensure that
+the assumptions of Theorem 1.3 are all satisfied. Theorem 1.3 and the discrete Poincaré inequality
+(see Lemma A.2) then imply that
+J2 = (Θh ∗ f, ˜uh − ˜θh ∗ u)L2
+h(hZd)
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+15
+≤ ∥Θh ∗ f∥L2
+h(hZd)∥˜uh − ˜θh ∗ u∥L2
+h(hZd)
+≤ C∥Θh ∗ f∥L2
+h(hZd)∥˜uh − ˜θh ∗ u∥ ˙Hs
+h(hZd)
+≤ C∥f∥L∞(Rd)ht−s∥u∥ ˙Ht(Rd).
+For t−s small enough (depending on s and Ω), Lemma A.4 implies that we have Ht-regularity-estimates
+on Ω and hence in particular ∥u∥ ˙Ht(Rd) < ∞. Thus J2 → 0 as h → 0.
+Step 3: Estimate of J1, J3, J4, J5. For J1, using again the discrete Poincaré inequality, we estimate
+J1 = (Θh ∗ f, uh − ˜uh)L2
+h(hZd)
+≤ ∥Θh ∗ f∥L2
+h(hZ)d∥uh − ˜uh∥L2
+h(hZd)
+≤ C∥f∥L∞(Rd)∥uh − ˜uh∥ ˙Hs
+h(Ωh)
+≤ C∥f∥L∞(Rd)∥Θh ∗ f − ˜Θh ∗ f∥ ˙H−s
+h
+(Ωh)
+≤ C∥f∥L∞(Rd)∥Θh ∗ f − ˜Θh ∗ f∥L2
+h(Ωh)
+≤ Ch∥f∥L∞(Rd)∥∇f∥L∞(Rd);
+where the right-hand side tends to 0 as h → 0. The same argument also applies to J3.
+For J5, it suffices to observe that ˜θh ∗ u tends to u in L2(Rd). Finally, for J4, we use the fact that
+˜θh ∗ u is continuous (and thus f · (˜θh ∗ u) is continuous), and so
+lim
+h→0(f, ˜θh ∗ u)L2
+h(hZd) = (f, u)L2(Rd)
+as a Riemann sum.
+□
+4.2. Scaling limit in Besov, Sobolev and Hölder spaces. We now turn to the proof of the scaling
+limit in Besov spaces (which then implies the result in Sobolev and Hölder spaces as well). As we have
+already established convergence of the fields in the space of distributions, the main challenge is to
+prove tightness in Besov spaces. To this end, we use a very convenient criterion from [9]. As in our
+case we do not need to worry about boundary issues, we do not the full generality of that criterion.
+Let us state the version that we will use.
+Lemma 4.1 (Tightness criterion). Let r ∈ N and let ˆΩ ⊂ Rd be an open bounded set. Then there
+exist functions f, (gj)2d−1
+j=1
+∈ Cr
+c (Rd) such that, for any multi-index m ∈ Nd with |m| < r and any
+j ∈ {1, . . . , 2d − 1}, we have
+(4.4)
+�
+Rd xmgj(x) dx = 0
+and such that the following statement holds. Let (φn)n∈N be a family of random linear forms on Cr
+c (Rd)
+with support in ˆΩ. Let t, t′ ∈ R with t < t′, |t|, |t′| < r and let p ∈ [1, ∞), q ∈ [1, ∞]. Let us suppose
+that there exists a constant C such that
+(4.5)
+lim sup
+n→∞
+sup
+x∈Rd (E |⟨φn, f(· − x)⟩|p)1/p < ∞
+and
+(4.6)
+lim sup
+n→∞
+sup
+k∈N
+sup
+x∈Rd
+max
+1≤j≤2d−1 (E |⟨φn, gj(2a(· − x))⟩|p)1/p ≤
+C
+2a(d+t′) .
+Then the family (φn)n∈N is tight in ˆBt
+p,q(Rd). If t < t′ − d
+p, it is also tight in ˆBt
+∞,q(Rd).
+Proof. This is essentially [9, Theorem 2.30]. There a local version of the theorem is given. The global
+version presented here is obtained by choosing U = Rd, ˆΩ ⊂ K1 ⊂ K2 ⊂ . . . such that already K1 is far
+larger than ˆΩ, k1 = k2 = . . . = 0 and observing that, for functions with uniformly compact support,
+the local and global Besov spaces agree.
+
+16
+N. DE NITTI AND F. SCHWEIGER
+We also used lim supn→∞ instead of supn∈N in (4.5) and (4.6), but this clearly does not make a
+difference. Finally, the assertion that
+�
+Rd xmgj(x) dx = 0 is stated in [9, Equation (2.2)].
+□
+Proof of Theorem 1.2. Step 1: Simplifications. It suffices to prove tightness of Ihϕh in the corre-
+sponding spaces; the convergence then follows easily from Theorem 1.1 by the same argument as in
+[7, Proof of Theorem 3.11]. In order to prove tightness, we will apply Lemma 4.1. We fix some open
+bounded set ˆΩ ⋑ Ω and note that for h small enough Ihϕh is supported in ˆΩ. Let us fix some r ∈ N
+with r >
+��s − d
+2
+�� and let f, (gj) be as in the lemma. We claim that, for any p′ < ∞,
+lim sup
+h→0
+sup
+x∈Rd Eh
+���(Ihϕh, f(· − x))Lp′(Rd)
+���
+p′
+< ∞,
+(4.7)
+lim sup
+h→0
+sup
+a∈N
+sup
+x∈Rd
+max
+1≤j≤2d−1 Eh
+���(Ihϕh, gj(2a(· − x)))Lp′(Rd)
+���
+p′
+≤
+C
+2a(d+2s) .
+(4.8)
+Once we have verified this, Lemma 4.1 (with t′ = s − d
+2) directly implies tightness in ˆBs′
+p,q(Rd) for any
+p ∈ [1, ∞), q ∈ [1, ∞], and choosing p′ sufficiently large such that s′ < t′ − d
+p′ we cover the case p = ∞
+as well. Once we know tightness in Besov spaces, the tightness in Sobolev- and Hölder spaces follows
+directly from Besov embedding.
+Regarding (4.5) and (4.6), we can make some immediate simplifications. First of all, it suffices to
+check the two estimates for p′ ∈ 2N (the result for other p′ then follows from Jensen’s inequality).
+In addition, as ϕh is a Gaussian random variable, all even functions of linear functionals of ϕh are
+controlled by its second moment. This means that we only need to consider p′ = 2. That is, we
+actually only need to verify that
+lim sup
+h→0
+sup
+x∈Rd Eh
+��(Ihϕh, f(· − x))L2(Rd)
+��2 < ∞,
+(4.9)
+lim sup
+h→0
+sup
+k∈N
+sup
+x∈Rd
+max
+1≤j≤2d−1 Eh
+��(Ihϕh, gj(2a(· − x)))L2(Rd)
+��2 ≤
+C
+22a(d+t′) =
+C
+2a(d+2s) .
+(4.10)
+The estimate (4.10) is the crucial one. So we give its proof in detail, and then explain how to prove
+(4.9) as well.
+Step 2: Proof of (4.10). Let us fix some h ≤ 1, a ∈ N, x ∈ Rd, and abbreviate ˜g(a)
+j
+(y) := gj(−2ay).
+A computation similar to the one in (4.1) shows that
+Eh
+��(Ihϕh, gj(2a(· − x)))L2(Rd)
+��2 = Eh
+������
+�
+y∈Rd
+�
+z∈hZd
+hdϕh(z)Θh(y − z)gj(2a(y − x)) dy
+������
+2
+= Eh
+�
+� �
+z∈hZd
+hdϕh(z)
+�
+y∈Rd Θh(y − z) ˜f (a)
+j
+(x − y) dy
+�
+�
+2
+= Eh
+�
+ϕh, (Θh ∗ ˜g(a)
+j
+)(x − ·)
+�2
+L2
+h(Ωh)
+= ∥(Θh ∗ ˜g(a)
+j
+)(x − ·)∥2
+˙H−s
+h
+(Ωh)
+≤ ∥(Θh ∗ ˜g(a)
+j
+)(x − ·)∥2
+˙H−s
+h
+(hZd).
+(4.11)
+We estimate the right-hand side of (4.11) by arguing in Fourier space (similarly as in the proof of
+Theorem 1.3). Namely, using Lemma 3.1 and the fact that the Fourier transform of a convolution is
+the product of the Fourier transforms, we compute
+Eh
+��(Ihϕh, gj(2a(· − x)))L2(Rd)
+��2
+≤
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s|Fh[(Θh ∗ ˜g(a)
+j
+)(x − ·)](ξ)|2 dξ
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+17
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s
+������
+�
+ζ∈ 2π
+h Zd
+F[(Θh ∗ ˜g(a)
+j
+)(x − ·)](ξ + ζ)
+������
+2
+dξ
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s
+������
+�
+ζ∈ 2π
+h Zd
+F[(Θh(x − ·)](ξ + ζ)F[˜g(a)
+j
+(x − ·)](ξ + ζ)
+������
+2
+dξ.
+Next, we fix some t with d
+2 < t < k − s and use the Cauchy-Schwarz inequality (as in the proof of
+Theorem 1.3) to rewrite this as
+Eh
+��(Ihϕh, gj(2a(· − x)))L2(Rd)
+��2
+≤
+�
+(− π
+h , π
+h)
+d Mh(ξ)−2s
+�
+� �
+ζ∈ 2π
+h Zd
+� 1
+h + |ξ + ζ|
+�2t
+|F[Θh(x − ·)](ξ + ζ)|2|F[˜g(a)
+j
+(x − ·)](ξ + ζ)|2
+�
+�
+×
+�
+� �
+ζ∈ 2π
+h Zd
+1
+� 1
+h + |ξ + ζ|
+�2t
+�
+� dξ.
+(4.12)
+Observing that
+sup
+ξ∈(− π
+h , π
+h)
+d
+�
+ζ∈ 2π
+h Zd
+1
+� 1
+h + |ξ + ζ|
+�2t ≤ C
+�
+ζ∈ 2π
+h Zd
+h2t
+(1 + h|ζ|)2t ≤ Ch2t
+as well as the fact that Mh(ξ) is 2π
+h -periodic, we can rewrite (4.12) as
+Eh
+��(Ihϕh, gj(2a(· − x)))L2(Rd)
+���2
+≤ Ch2t
+�
+(− π
+h , π
+h)
+d Mh(ξ + ζ)−2s
+�
+ζ∈ 2π
+h Zd
+� 1
+h + |ξ + ζ|
+�2t
+|F[Θh(x − ·)](ξ + ζ)|2|F[˜g(a)
+j
+(x − ·)](ξ + ζ)|2 dξ
+= Ch2t
+�
+Rd Mh(ξ)−2s
+� 1
+h + |ξ|
+�2t
+|F[Θh(x − ·)](ξ)|2|F[˜g(a)
+j
+(x − ·)](ξ)|2 dξ.
+(4.13)
+After all these manipulations, we have rewritten the term to be estimated as an integral involving the
+absolute values of the Fourier transforms of Θh, ˜g(a)
+j
+. To complete the proof, we use our assumptions
+on Θh, ˜g(a)
+j
+to bound these Fourier transforms.
+Regarding Θh, we know that F[Θh](ξ) = F[Θ](hξ); so assumption (1.3) implies that |F[Θh](ξ)| ≤
+C
+(�d
+j=1 sin2(hξj))k/2
+hk|ξ|k
+and
+(4.14)
+|F[Θh(x·)](ξ)| ≤ C
+(�d
+j=1 sin2(hξj))k/2
+hk|ξ|k
+.
+Regarding ˜g(a)
+j
+, we first note that F[˜g(a)
+j
+](ξ) = F[gj(−2a·)](ξ) =
+1
+2ad F[gj]
+�
+− ξ
+2a
+�
+. As gj ∈ Cr
+c (Rd), we
+know that F[gj] is smooth and decays at least like
+1
+|ξ|r as ξ → ∞. On the other hand, the moments of
+gj up to order r − 1 vanish by (4.4) and so ∇mF[gj](0) = 0 for any m ≤ r − 1. By Taylor’s theorem,
+this implies |F[gj](ξ)| ≤ C|ξ|r. Altogether, we conclude that |F[gj](ξ)| ≤ C
+|ξ|r
+(1+|ξ|)2r and thus also
+|F[˜g(a)
+j
+](ξ)| ≤ C 1
+2ad
+��� ξ
+2a
+���
+r
+�
+1 +
+��� ξ
+2a
+���
+�2r =
+C|ξ|r
+2a(d+r)(1 + 2−a|ξ|)2r
+
+18
+N. DE NITTI AND F. SCHWEIGER
+and
+(4.15)
+|F[˜g(a)
+j
+(x − ·)](ξ)| ≤
+C|ξ|r
+2a(d+r)(1 + 2−a|ξ|)2r .
+Returning to (4.13), we obtain that
+Eh
+��(Ihϕh, gj(2a(· − x)))L2(Rd)
+��2
+≤ Ch2t
+�
+Rd
+h2s
+�d
+j=1 sin2(hξj))s
+(1 + h|ξ|)2t
+h2t
+�����
+(�d
+j=1 sin2(hξj))k/2
+hk|ξ|k
+�����
+2 ����
+|ξ|r
+2a(d+r)(1 + 2−a|ξ|)2r
+����
+2
+dξ
+≤ C h2(s−k)
+22a(d+r)
+�
+Rd
+(1 + h|ξ|)2t(�d
+j=1 sin2(hξj))k−s
+|ξ|2(k−r)(1 + 2−a|ξ|)4r
+dξ.
+(4.16)
+In particular, the integrand decays like
+1
+|ξ|2(k+r−t) and so our assumptions t < k − s and r >
+��s − d
+2
+�� ≥
+d
+2 − s ensure its integrability.
+We distinguish the two cases whether h < 2−a or h ≥ 2−a. In the former case, we can bound the
+integral on the right-hand side of (4.16) as
+�
+Rd
+(1 + h|ξ|)2t(�d
+j=1 sin2(hξj))k−s
+|ξ|2(k−r)(1 + 2−a|ξ|)4r
+dξ
+≤ C
+�
+|ξ|≤2a
+1 · (h|ξ|)2(k−s)
+|ξ|2(k−r) · 1
+dξ + C
+�
+2a<|ξ|≤1/h
+1 · (h|ξ|)2(k−s)
+|ξ|2(k−r) · (2−a|ξ|)4r + C
+�
+|ξ|>1/h
+(h|ξ|)2t · 1
+|ξ|2(k−r)(2−a|ξ|)4r dξ
+≤ C2adh2(k−s)22a(r−s) + C 1
+hd 24arh2(k−s)h2(r+s) + C 1
+hd 24arh2th2(k+r−t)
+≤ C2a(d+2r−2s)h2(k−s) �
+1 + 2a(2r+2s−d)h2r+2s−d + 2a(2r+2s−d)h2r+2s−d�
+≤ C2a(d+2r−2s)h2(k−s),
+where in the last step we used that 2ah < 1 and 2r + 2s − d > 0. In case h ≥ 2−a, we can similarly
+estimate the integral by
+�
+Rd
+(1 + h|ξ|)2t(�d
+j=1 sin2(hξj))k−s
+|ξ|2(k−r)(1 + 2−a|ξ|)4r
+dξ
+≤ C
+�
+|ξ|≤1/h
+1 · (h|ξ|)2(k−s)
+|ξ|2(k−r) · 1
+dξ + C
+�
+1/h<|ξ|≤2a
+(h|ξ|)2t · 1
+|ξ|2(k−r) · 1 + C
+�
+|ξ|>2a
+(h|ξ|)2t · 1
+|ξ|2(k−r)(2−a|ξ|)4r dξ
+≤ C 1
+hd h2(k−s)h2(s−r) + C2adh2t22a(−k+r+t) + C2ad24arh2t22a(−k−r+t)
+≤ C2a(d+2r−2s)h2(k−s) �
+2a(−d−2r+2s)h−d−2r+2s + 22a(−k+s+t)h2(−k+s+t + 22a(−k+s+t)h2(−k+s+t)�
+≤ C2a(d+2r−2s)h2(k−s).
+Thus, in any case, the integral on the right-hand side of (4.16) is bounded by C2a(d+2r−2s)h2(k−s).
+Using this, (4.16) implies (4.10).
+Step 3: Proof of (4.9). The proof of (4.9) is similar. One difference is that we no longer need to
+prove decay of the term in question, only boundedness. On the other hand, the function f does not
+satisfy a moment bound like (4.4) and so we have less control over the behavior of F[f] near 0. To
+deal with the latter problem, we will use the Poincaré inequality on a suitable bounded domain ˜˜Ωh
+right in the beginning of the argument to replace the term
+1
+Mh(ξ)2s with
+1
+(1+Mh(ξ)2)s and thereby make
+sure there is no singularity at 0.
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+19
+In more detail, let us fix some bounded domain ˜Ω ⋑ ˆΩ such that supp(f(x − ·)) ⊂ ˜Ω whenever
+x ∈ ˆΩ. Then (4.9) vanishes for x ̸∈ ˜Ω, and we can restrict attention to x ∈ ˜Ω. Let us fix yet another
+bounded domain ˜˜Ω such that supp(f(x − ·)) ⊂ ˜˜Ω when x ∈ ˜Ω, and let ˜˜Ωh = ˜˜Ω ∩ hZd.
+We also abbreviate ˜f(y) = f(−y). Arguing as for (4.10) we find that, for x ∈ ˜Ω and h ≤ 1,
+Eh
+��(Ihϕh, f(2a(· − x)))L2(Rd)
+��2 ≤ ∥(Θh ∗ ˜f (a))(x − ·)∥2
+˙H−s
+h
+(Ωh) ≤ ∥(Θh ∗ ˜f (a))(x − ·)∥2
+˙H−s
+h
+(˜˜Ωh).
+Since supp(f(x−·))∗Ωh ⊂ ˜˜Ω, we use Poincaré’s inequality on ˜˜Ωh to deduce that the ∥·∥ ˙H−s
+h
+(˜˜Ωh)-norm
+is bounded by a multiple of the ∥ · ∥H−s
+h
+(˜˜Ωh)-norm. Using this, we can now continue with a calculation
+similar as the one for (4.10) to obtain
+Eh
+��(Ihϕh, f(2a(· − x)))L2(Rd)
+��2
+≤ ∥(Θh ∗ ˜f)(x − ·)∥2
+H−s
+h
+(˜˜Ωh)
+≤ ∥(Θh ∗ ˜f)(x − ·)∥2
+H−s
+h
+(hZd)
+=
+�
+(− π
+h , π
+h)
+d(1 + Mh(ξ)2)−s
+������
+�
+ζ∈ 2π
+h Zd
+F[(Θh(x − ·)](ξ + ζ)F[ ˜f(x − ·)](ξ + ζ)
+������
+2
+dξ
+≤ Ch2t
+�
+Rd(1 + Mh(ξ)2)−s
+� 1
+h + |ξ|
+�2t
+|F[(Θh(x − ·)](ξ)|2|F[ ˜f(x − ·)](ξ)|2 dξ.
+Using the bound (4.14) for F[Θh] as well as the estimate
+|F[ ˜f(x − ·)](ξ)| ≤
+C
+(1 + |ξ|)r
+(the analogue of (4.15)), we obtain that
+Eh
+��(Ihϕh, f(2a(· − x)))L2(Rd)
+��2 ≤ Ch2(s−k)
+�
+Rd
+(1 + h|ξ|)2t(�d
+j=1 sin2(hξj))k
+(h2 + (�d
+j=1 sin2(hξj))2)s|ξ|2k(1 + |ξ|)2r dξ
+and (splitting the integral into three integrals over |ξ| ≤ 1, 1 < |ξ| ≤ 1/h, and 1/h < |ξ|) we see as
+before that the right-hand side is indeed bounded by a constant.
+□
+Finally, let us give the argument for convergence of the maximum of the subcritical discrete FGF.
+Some technicalities arise because Theorem 1.2 applies to Ihϕh while we are interested in ϕh itself. So
+we need to argue that the regularity of Ihϕh implies that ϕh is necessarily close to Ihϕh.
+Proof of Corollary 1.4. Step 1: Consequences of Theorem 1.2. Let k = d > s + d
+2 and take Θ to be a
+product of one-dimensional B-splines of order k, i.e.
+F[Θ](ξ) =
+d
+�
+j=1
+�sin(ξ)
+ξ
+�k
+.
+This Θ is a compactly supported non-negative mollifier that satisfies the assumptions of Theorem 1.2.
+Moreover, let us fix some α with 0 < α < min
+�
+s − d
+2, 1
+�
+. Theorem 1.2 implies that Ihϕh converges to
+ϕ in law with respect to the topology of C0,α(Rd). This directly implies that the maximum of Ihϕh
+converges in distribution to the maximum of ϕ. Therefore, it suffices to prove that the maximum of
+ϕh is close enough to the maximum of Ihϕh in the sense that
+(4.17)
+max
+x∈Ωh ϕh(x) − max
+y∈Rd Ihϕh(y) → 0
+in probability as h → 0.
+
+20
+N. DE NITTI AND F. SCHWEIGER
+Step 2: Regularity of ϕh. In order to prove (4.17), we need to quantify the regularity of ϕh. The
+idea here is that if ϕh oscillates a lot, then also Ihϕh oscillates a lot and hence have large C0,α-norm,
+which is unlikely. In making this rigorous, we use our choice of Θ, which simplifies some calculations.
+The function Θh has support precisely
+�
+− hk
+2 , hk
+2
+�d and is piecewise a polynomial of degree at most
+k − 1 in each variable.
+Let us take x ∈ hZd and consider an arbitrary fh : hZd → R. Then Ihfh ↾x+(0,h/2)2 is a polynomial of
+degree at most k−1 in each variable, which depends precisely on the values of fh in x+
+�
+− hk
+2 , − h(k+1)
+2
+�
+∩
+hZd.
+The space of polynomials of degree at most k − 1 in each variable is an R-vector space of dimension
+exactly kd. The same holds true for the space of functions from x +
+�
+− hk
+2 , − h(k+1)
+2
+�
+∩ hZd to R.
+This means that Ihfh induces a linear map between two finite-dimensional vector spaces of the same
+dimension. By standard properties of B-splines, this map is surjective and hence, in fact, bijective.
+As all norms on a finite-dimensional R-vector space are equivalent, we conclude that
+max
+y∈x+(− hk
+2 ,− h(k+1)
+2
+)∩hZd fh(y) ≤ C
+max
+z∈x+(0,h/2)d Ihfh(z)
+and (quotienting out constant functions) also
+max
+y,y′∈x+(− hk
+2 ,− h(k+1)
+2
+)∩hZd |fh(y) − fh(y′)|
+≤ C
+max
+z,z′∈x+(0,h/2)d Ihfh(z) − Ihfh(z′) ≤ Chα∥Ihfh∥C0,α(x+(0,h/2)d).
+The constant in the latter estimate is independent of x. This means that we actually obtain
+(4.18)
+max
+y,y′∈hZd
+|y−y′|y∞≤kh
+|fh(y) − fh(y′)| ≤ Chα∥Ihfh∥C0,α(Rd).
+We know that Ihϕh converges in C0,α(Rd) and so it is, in particular, tight in that space. This means
+that, if we define
+EM =
+�
+[Ihϕh]C0,α(Rd)
+�
+,
+then limM→∞ limh→0 P(EM) = 1. On the other hand, (4.18) implies that, on the event EM, we have
+(4.19)
+max
+y,y′∈hZd
+|y−y′|y∞≤kh
+|ϕh(y) − ϕh(y′)| ≤ CMhα.
+This is the desired regularity estimate for ϕh.
+Step 3: Completion of the proof. Our specific choice of Θ has the property that �
+x∈hZd hdΘh(y −
+x) = 1 for any y ∈ Rd. This means that Ihϕh(y) is a convex combination of the ϕh(x) with |x− y|∞ <
+hk
+2 , and so we have
+max
+x∈Ωh ϕh(x) ≥ max
+y∈Rd Ihϕh(y),
+which implies the lower bound in (4.17). For the upper bound we need to use (4.19). As Ihϕh(y) is
+a convex combination of the ϕh(x) with |x − y|∞ < hk
+2 , (4.19) implies that on the event EM we have
+|ϕh(x) − Ihϕ(x)| ≤ CMhα for any x ∈ hZd. Therefore, on the event EM we have
+max
+x∈Ωh ϕh(x) ≤ max
+x∈Ωh Ihϕh(x) + CMhα ≤ max
+y∈Rd Ihϕh(y) + CMhα.
+Putting these considerations together, we conclude that
+lim
+M→∞ lim
+h→0 P
+�
+max
+y∈Rd Ihϕh(y) ≤ max
+x∈Ωh ϕh(x) ≤ max
+y∈Rd Ihϕh(y) + CMhα
+�
+,
+which yields (4.17).
+□
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+21
+Appendix A. Technical lemmas
+In this appendix, we provide the proof of several technical results that have been used throughout
+the paper.
+A.1. Discretization and restriction. We start by proving that the applications of restricting to
+hZd and applying (−∆h)s commute.
+Lemma A.1 (Discretization and restriction). Let u: Rd → R be a Schwartz function. Then, restricting
+to hZd and applying (−∆h)s commute: i.e.,
+((−∆h)su)↾hZd= (−∆h)s (u↾hZd) .
+This allows us to be rather careless about when we restrict functions to hZd. In fact, we will omit
+writing ↾hZd when (because of Lemma A.1) there is no ambiguity.
+Proof. The crucial fact here is that Mh(ξ) is 2π
+h -periodic. Using this, we compute that, for x ∈ hZd,
+((−∆h)su) (x) =
+�
+Rd Mh(ξ)2sF[u](ξ) dξ
+=
+�
+ζ∈ 2π
+h Zd
+�
+(− π
+h , π
+h)
+d Mh(ξ + ζ)2sF[u](ξ + ζ) dξ
+=
+�
+(− π
+h , π
+h)
+d Mh(ξ)2s
+�
+ζ∈ 2π
+h Zd
+F[u](ξ + ζ) dξ.
+Using Lemma 3.1, we can rewrite this as
+((−∆h)su) (x) =
+�
+(− π
+h , π
+h)
+d Mh(ξ)2sFh[u](ξ) dξ
+= (−∆h)s (u↾hZd) ,
+which is what we wanted to show.
+□
+A.2. Discrete inequalities. Let us state the discrete Poincaré inequality that we used in the proof.
+Lemma A.2. Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary and let s > 0. Then, there
+exists a constant C such that, for any uh : hZd → R that vanishes outside of Ωh, we have
+∥uh∥L2(Ωh) ≤ C∥uh∥ ˙Hs
+h(Ω).
+Proof. We shall present a discrete version of the proof in [8, Theorem 3.7]. The key idea is to use
+Plancherel’s theorem and split low and high frequencies as follows:
+∥uh∥2
+L2(Ω) =
+�
+Bϵ(0)
+|Fhuh(ξ)|2 dξ +
+�
+(−π/h,π/h)d\Bϵ(0)
+|Fhuh(ξ)|2 dξ
+=: I1 + I2,
+where ϵ > 0 is to be fixed later on.
+Step 1. Low-frequencies. For the low-frequency part, I1, Hölder’s inequality yields
+|Fhuh(ξ)| ≤ ∥uh∥L1(Ωh) ≤ |Ωh|1/2hd/2∥uh∥L2(Ωh),
+where we used the notation
+∥uh∥Lp
+h(Ωh) :=
+� �
+x∈Ωh
+hd|uh|p
+� 1
+p
+,
+p ∈ [1, +∞).
+Therefore, we have
+I1 ≤ ϵdB1(0)|Ωh|hd∥uh∥2
+L2
+h(Ωh).
+
+22
+N. DE NITTI AND F. SCHWEIGER
+Step 2. High-frequencies. For the high-frequency part, I2, we compute
+�
+(−π/h,π/h)d\Bϵ(0)
+|Fhuh(ξ)|2 dξ =
+�
+(−π/h,π/h)d\Bϵ(0)
+Mh(ξ)2s|Fhuh(ξ)|2
+Mh(ξ)2s
+dξ
+≤ ϵ−2s∥(−∆h)s/2uh∥2
+L2
+h(Rd).
+Step 3. Conclusion. Choosing 0 < ϵ < (|Ωh|hd|B1(0)|)−1/d, we conclude
+∥uh∥L2(Ωh) ≤
+ϵ−s
+�
+1 − ϵd|Ωh|hd|B1(0)|
+∥uh∥Hs
+h(Ω).
+By considering a square of side L ≥ diam(Ω) (containing Ωh), we deduce that |Ωh| ≤ C
+hd (note that is
+holds even if h ≫ diam(Ω)). This means that Ωh|hd|B1(0)| ≤ C, and so we can make a choice of ε > 0
+independent of h. This concludes the proof.
+□
+Remark A.3 (Generalized Poincaré inequality). Let s ≥ t ≥ 0. Arguing as in [8, Theorem 1.5], Lemma
+A.2 also implies that, for u ∈ ˜Hs(Ω), there exists a constant c = c(d, Ω, s) > 0 such that
+∥(−∆h)t/2uh∥L2(Ωh) ≤ c∥(−∆h)s/2uh∥L2(Ωh)).
+Indeed,
+∥(−∆h)t/2uh∥L2(Ω) = ∥uh∥ ˙Ht
+h(Ω) ≤ ∥uh∥Ht
+h(Ω) ≤ ∥uh∥Hs
+h(Ω) ≤ 2
+s+1
+2 (∥uh∥L2(Ωh) + ∥uh∥ ˙Hs
+h(Ω))
+≤ 2
+s+1
+2 (c∥(−∆h)s/2uh∥L2(Ωh) + ∥(−∆h)s/2uh∥L2(Ωh))
+= c∥(−∆h)s/2uh∥L2(Ωh).
+We also used the fact that solutions of the Dirichlet problem for (−∆)s have a little bit of additional
+regularity.
+Lemma A.4. Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary and let s ≥ 0. Then, there
+exists κ0 > 0 with the following property. If 0 ≤ κ ≤ κ0, then, for each f ∈ H−s+κ(Ω), there exists
+a unique u ∈ ˙Hs+κ(Ω) such that (−∆)su = f in the sense of distributions; moreover, we have the
+estimate
+∥u∥ ˙Hs+κ(Ω) ≤ Cκ∥f∥ ˙H−s+κ(Ω)
+for a constant Cκ depending only on κ.
+Let us remark that according to [3, Theorem 2.3] one can take any κ0 < 1
+2 here. The argument,
+however, is rather complicated; so we prefer to present an easy perturbative argument that gives
+existence of some κ0 > 0 (which is enough for our purposes).
+Proof. We adapt the argument used in [19, Theorem 3.3] for the biharmonic operator to the fractional
+case.
+We first show that the claimed estimate holds for κ = 0. To do so, we test the equation with u and
+deduce
+∥(−∆)s/2u∥2
+L2(Ω) = (u, (−∆)su)L2(Ω) = (u, f)L2(Ω) ≤ ∥u∥ ˙Hs(Ω)∥f∥ ˙H−s(Ω).
+Using Poincaré’s inequality, we see that indeed
+∥u∥ ˙Hs(Ω) ≤ Cκ∥f∥ ˙H−s(Ω)
+To show that we also can take some κ > 0, we use a stability result for analytic families
+of operators on Banach spaces:
+The spaces
+˙Hs(Ω) form an interpolation family with respect to
+complex interpolation; thus, by [21, Proposition 4.1], the set of those α for which the operator
+(−∆)s :
+˙Hα(Ω) →
+˙Hα−2s(Ω) has a bounded inverse is open. We have seen that this set contains
+s, so the existence of κ0 as in the statement of the theorem follows.
+□
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+23
+Appendix B. Fractional Gaussian Fields via eigenfunctions
+In this appendix, we shall present an alternate description of the continuous FGF. As remarked in
+Section 2.1, (−∆)s is an isometry from ˙Hs(Ω) to ˙H−s(Ω). Its inverse, restricted to L2(Ω), is a positive-
+definitive compact operator on L2(Ω); so, by the spectral theorem, there exists an orthonormal basis
+(v1, v2, . . .) of L2(Ω) consisting of eigenfunctions of (−∆)s with associated eigenvalues 0 < λ1 ≤ λ2 ≤
+. . .. Let Xj be a collection of independent standard Gaussians, and let ˜ϕ be the random variable
+˜ϕ = �∞
+j=1
+Xj
+√
+λj vj.
+According to Lemma B.1 below, this sum converges almost surely in ˙Hs′(Ω) ⊂ ˙Hs′(Rd) for any
+s′ < s − d
+2. Therefore, ˜ϕ is a well-defined random variable on ˙Hs′(Ω) ⊂ ˙Hs′(Rd). Every element of
+˙Hs′(Rd) induces an element of S′(Rd) and so we can think of ˜ϕ as a random element of S′(Rd). Again,
+according to Lemma B.1, for any f ∈ S(Rd) we have that ( ˜ϕ, f) is a centered Gaussian with variance
+∥f∥ ˙H−s(Ω). This means that ˜ϕ has the law P on S′(Rd) and so we can identify ϕ and ˜ϕ.
+Let us present the aforementioned lemma.
+Lemma B.1. Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary. Let s ≥ 0 and s′ < s − d
+2 be
+arbitrary.
+(i) The series
+˜ϕ :=
+∞
+�
+j=1
+Xj
+�
+λj
+vj
+converges almost surely in ˙Hs′(Ω).
+(ii) For any f ∈ S(Rd), we have
+E( ˜ϕ, f)2 = ∥f∥2
+˙H−s(Ω).
+For the proof, we need a sharp estimate on the eigenfunction expansion of a function.
+Lemma B.2. Let Ω ⊂ Rd be a bounded domain with Lipschitz boundary. Let s ≥ 0 and s′ ≤ s be
+arbitrary. Then, for any f ∈ ˙Hs′(Ω), we have
+(B.1)
+∥f∥2
+˙Hs′(Ω) ≤ C
+∞
+�
+j=1
+λs′/s
+j
+(f, vj)2
+L2(Rd).
+Note that we make no claim about the ˙Hs′-regularity for s′ > s.
+Proof. For s′ ∈ {−s, 0, s} the estimate (B.1) follows directly from the definition. We next claim that
+(B.1) holds whenever −s ≤ s′ ≤ s. To see this, we adapt the argument in [17, Corollary 1]. Namely
+take first 0 < s′ < s. Consider (−∆)s restricted to functions in
+˙Hs(Ω) and let ((−∆)s)s′/s
+N
+be its
+(spectral) s′
+s -th power. Explicitly,
+((−∆)s)s′/s
+N
+f =
+∞
+�
+j=1
+λs′/s
+j
+(f, vj)L2(Rd)vj
+Note that, if we define ˙Hs′(Ω) to be the space of functions in L2(Ω) such that this quantity is finite,
+then the domain of ((−∆)s)s′/s
+N
+is exactly ˙Hs′(Ω). According to the theory of interpolation of fractional
+powers of self-adjoint operators (see, e.g., [23, Section 1.18.10]), the Hilbert spaces
+˙Hs′(Ω) form an
+interpolation scale. However, we know that the same holds true for the Hilbert spaces ˙Hs′(Ω), and
+moreover ˙Hs′(Ω) = ˙Hs′(Ω) (with equivalent norms) for s′ ∈ {0, s}, and so we have actually have this
+equality for any s′ with 0 ≤ s′ ≤ s. So, for 0 ≤ s′ ≤ s, there is some C > 0 such that
+1
+C
+∞
+�
+j=1
+λs′/s
+j
+(f, vj)2
+L2(Rd) ≤ ∥f∥2
+˙Hs′(Ω) ≤ C
+∞
+�
+j=1
+λs′/s
+j
+(f, vj)2
+L2(Rd)
+
+24
+N. DE NITTI AND F. SCHWEIGER
+By duality, the same holds true for −s ≤ s′ ≤ 0. Putting these considerations together, we obtain a
+statement even stronger than (B.1).
+It remains to study the case that s′ < −s. We proceed inductively. Let us suppose that we know
+that (B.1) holds for s′ ≥ −(2k − 1)s, for some k ∈ N, and let us consider some s′ with −(2k + 1)s ≤
+s′ ≤ −(2k − 1)s. We have that
+∥f∥2
+˙Hs′(Ω) =
+inf
+g∈ ˙Hs′(Rd)
+f=g in Ω
+∥g∥2
+˙H−s(Rd).
+Let u be such that
+�
+(−∆)su(x) = f(x),
+x ∈ Ω,
+u(x) = 0,
+x ∈ Rd \ Ω.
+We can choose g = (−∆)su and obtain, using the induction hypothesis, that
+∥f∥2
+˙Hs′(Ω) ≤ ∥(−∆)su∥2
+˙Hs′(Rd)
+≤ ∥u∥2
+˙Hs′+2s(Rd)
+≤ C
+∞
+�
+j=1
+λ(s′+2s)/s
+j
+(u, vj)2
+L2(Rd)
+= C
+∞
+�
+j=1
+λ(s′+2s)/s
+j
+�
+u, (−∆)svj
+λj
+�2
+L2(Rd)
+= C
+∞
+�
+j=1
+λs′/s
+j
+((−∆)su, vj)2
+L2(Rd)
+= C
+∞
+�
+j=1
+λs′/s
+j
+(f, vj)2
+L2(Rd) .
+This completes the induction step.
+□
+Proof of Lemma B.1. Claim (i). By the Hilbert-space-valued version of Kolmogorov’s two series the-
+orem (see e.g. [11, Corollary on p. 386]), the series
+∞
+�
+j=1
+Xj
+�
+λj
+vj
+converges almost surely in ˙Hs′(Ω) if
+∞
+�
+j=1
+�����
+1
+�
+λj
+uj
+�����
+2
+˙Hs′(Ω)
+< ∞.
+From Lemma B.2, we know in particular that
+∥vj∥2
+˙Hs′(Ω) ≤ λs′/s
+j
+.
+Moreover, by Weyl’s law for the operator (−∆)s restricted to ˙Hs(Ω) (as follows, e.g., from the main
+result of [10]), we have that
+λj ≍ j2s/d.
+Therefore,
+∞
+�
+j=1
+�����
+1
+�
+λj
+vj
+�����
+2
+˙Hs′(Ω)
+≤
+∞
+�
+j=1
+λs′/s−1
+j
+
+SCALING LIMITS FOR FRACTIONAL POLYHARMONIC GAUSSIAN FIELDS
+25
+≤
+∞
+�
+j=1
+(cj)2s/d·(s′/s−1) ≤ C
+∞
+�
+j=1
+j2(s′−s)/d,
+and this sum is indeed convergent if s′ − s < − d
+2.
+Claim (ii). Let f ∈ S(Rd). The functions vj are by definition orthonormal in L2(Ω), and the Xj
+are independent. So we can calculate that
+E( ˜ϕ, f)2 = E
+�
+�
+∞
+�
+j=1
+Xj
+�
+λj
+(vj, f)
+�
+�
+2
+=
+∞
+�
+j=1
+1
+λj
+(vj, f)2 =
+�
+�f,
+∞
+�
+j=1
+1
+λj
+(vj, f)vj
+�
+�
+=
+�
+f, (−∆)−sf
+�
+= ∥f∥2
+H−s(Ω).
+□
+Acknowledgments
+N. De Nitti is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le
+loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). He has been
+supported by the Alexander von Humboldt Foundation and by the TRR-154 project of the Deutsche
+Forschungsgemeinschaft (DFG, German Research Foundation).
+F. Schweiger is supported by the Foreign Postdoctoral Fellowship Program of the Israel Academy
+of Sciences and Humanities, and partially by ISF grant No. 421/20.
+We thank O. Zeitouni and E. Zuazua for helpful comments on the topic of this work.
+References
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+[12] Z. Hao, Z. Zhang, and R. Du. Fractional centered difference scheme for high-dimensional integral fractional Lapla-
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+Mathematics. Springer, London, 2014. For linear partial differential equations with generalized solutions.
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+[19] F. Schweiger. The maximum of the four-dimensional membrane model. Ann. Probab., 48(2):714–741, 2020.
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+(N. De Nitti) Friedrich-Alexander-Universität Erlangen-Nürnberg, Department of Data Science, Chair
+for Dynamics, Control and Numerics (Alexander von Humboldt Professorship), Cauerstr.
+11, 91058
+Erlangen, Germany.
+Email address: [email protected]
+(F. Schweiger) Weizmann Institute of Science, Department of Mathematics, Rehovot 7610001, Israel.
+Email address: [email protected]
+

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+arXiv:2301.00029v1  [math-ph]  31 Dec 2022
+Generalized conformal maps as classical symmetries of
+Yang-Mills fields
+Edward B. Baker III∗
+January 3, 2023
+We show that a class of previously defined maps, called causal and
+self-dual morphisms, form classical symmetries of Yang-Mills fields
+in four complex dimensions. These maps generalize conformal trans-
+formations, and admit a nonlocal pullback connection that preserves
+the equations of the theory. First it is shown that self-dual mor-
+phisms form symmetries of the anti-self-dual Yang-Mills equations
+under this pullback. Then a supersymmetric generalization of causal
+morphisms is defined which preserves solutions of the field equations
+for N=3 supersymmetric Yang-Mills theory. As a special case, this
+implies that a modified definition of causal morphisms form sym-
+metries for the ordinary Yang-Mills field equations.
+1
+Introduction
+Hidden symmetries have played an important role in the study of Yang-Mills (YM) theory.
+As an example, the anti-self dual Yang Mills (ASDYM) equations have an infinite class
+of hidden symmetries which bear some resemblance to the infinite-dimensional conformal
+group in two dimensions [1][2][3]. In addition, an extended conformal symmetry called
+dual superconformal invariance has been uncovered in the study of N=4 supersymmetric
+Yang-Mills (SYM) theory [4][5], leading to an infinite dimensional Yangian symmetry [6].
+This and other advances have led to powerful tools for the study of N=4 SYM.
+Many of these results can be understood best with the use of twistor and ambitwistor
+methods, which have been used extensively in the study of Yang-Mills fields. For example,
+the Penrose-Ward correspondence reformulates the ASDYM equations in Twistor space,
+∗[email protected]
+1
+
+which leads to the ADHM construction of instantons [7][8][9]. This construction was gen-
+eralized to a geometric formulation of the Yang-Mills field equations in ambitwistor space,
+with a natural interpretation in superspace [10][11][12][13]. More recently, twistor and am-
+bitwistor methods have been used in string theory to understand Yang-Mills scattering
+amplitudes and their properties [14][15].
+In this paper we investigate a previously defined class of generalized maps [16] in
+the context of Yang-Mills theory. These maps are motivated by twistor and ambitwistor
+theory, and are called self-dual and causal morphisms, respectively. Under certain as-
+sumptions, one can define a non-local pullback connection under these transformations
+that preserves integrability on certain subspaces. In the case of self-dual morphisms, the
+maps preserve integrability on self-dual planes which imply that they are symmetries of
+the ASDYM equations. A supersymmetric generalization of causal morphisms is then
+developed which preserves integrability on super null lines, implying that these maps are
+symmetries of the N=3 SYM field equations. This fact is used to show that a modified
+version of causal morphisms are symmetries of the YM field equations as a special case.
+It is likely that some of these symmetries are related to known hidden symmetries for
+the different cases, but characterizing these relationships will be left as a topic of future
+investigation.
+2
+Self-dual morphisms as symmetries of ASDYM
+Self-dual morphisms were introduced in a previous paper, where they were defined using
+maps on null surfaces [16]. Here we provide a self contained summary of these results,
+using different but equivalent definitions. To begin, define the twistor correspondence
+space F = C4 × CP1 with the usual double fibration [17][18]
+C4
+π1
+←− F
+π2
+−→ PT ,
+(1)
+where PT = CP3 is the projective twistor space of C4. Now define a self-dual embedding
+as a totally null holomorphic embedding χ : C2 → C4, which means that vectors at a
+point t ∈ C2 are mapped under χ⋆ to vectors of the form vα ˙α = λα˜λ ˙α for ˜λ ˙α fixed. We
+will call the image of such a map a self-dual surface. If ˜λ is independent of t then this
+surface maps to a self-dual plane (or α-plane) Z. Furthermore, for any point on a self-dual
+embedding there is a tangent α-plane passing through χ(t) that is characterized by ˜λ(t).
+For brevity, we say that χ is tangent to ˜λ at t. Now define the self-dual prolongation
+jsχ : C2 → F by jsχ = (χ, ˜λ), where dependence on t is suppressed. The prolongation
+satisfies a contact condition, that χ is tangent to ˜λ for all t ∈ C2. Conversely, given a
+surface ψ : C2 → F, we say that it satisfies the contact condition if ψ = jsχ for some
+self-dual embedding χ. A map f : F → F is said to preserve the contact condition if
+2
+
+f ◦ jsχ satisfies the contact condition for any χ.1 We then define
+Definition 1. A self-dual morphism is a holomorphic map f : F → F which preserves
+the contact condition.
+Defined in this way, a self-dual morphism naturally induces maps on self-dual embed-
+dings and self-dual planes
+Definition 2. Given a self-dual morphism f and a self-dual embedding χ, define the
+contraction map f⌟χ := π1 ◦ f ◦ jsχ. Furthermore, for a self-dual plane Z tangent to ˜λ,
+define f⌟Z : Z → C4 by f⌟Z(x) = π1 ◦ f(x, ˜λ) where x ∈ Z.
+This map on surfaces was the starting point for the definitions in the previous paper,
+and the two definitions are equivalent.
+Now consider a GL(n, C) connection with vector potential A satisfying the ASDYM
+equations on MC = C4. Given a self-dual morphism f, there is a natural definition for
+a pullback connection f ∗A. To see this, first restrict to a self-dual plane Z, which can
+be parameterized linearly by coordinates on C2. The contraction map f⌟Z then gives
+a self-dual embedding, and the pullback connection (f⌟Z)∗A is integrable on Z because
+the curvature of A vanishes on the self-dual planes tangent to f⌟Z as a consequence
+of ASDYM. By varying Z this allows us to define the bundle of parallel sections on
+the twistor space, and to use the Penrose-Ward procedure to define a connection on
+the pullback bundle, which defines the pullback connection f ∗A and gives a solution of
+ASDYM. This requires that the bundle of parallel sections is trivial for points x ∈ C4,
+which will be shown with an explicit construction of f ∗A.
+For the construction, consider two points x1, x2 ∈ Z and their images yi = f⌟Z(xi).
+Define a Wilson line for the pullback connection by
+W ∗
+Z(x1, x2) = Wf⌟Z(y1, y2) = P exp
+��
+γ
+Aµdxµ
+�
+.
+(2)
+Here the path of integration is any path γ confined to the image of f⌟Z starting at y1
+and ending at y2. The Wilson line is independent of path due to the integrability of the
+connection on self-dual surfaces. We can then define the patching matrix used in the
+Penrose-Ward correspondence by
+G = W ∗
+Z(q, p) = ˜HH−1
+(3)
+where
+H = W ∗
+Z(p, x),
+˜H = W ∗
+Z(q, x).
+(4)
+1These constructions are all assumed to be local and defined in some neighborhood, but are written
+globally for ease of notation.
+3
+
+Here p and q are the points of intersection between Z and self-dual planes P and Q with
+twistor coordinates ˆP = (0, 0, 1, 0) and ˆQ = (0, 0, 0, 1), as defined in the usual patching
+construction. The patching matrix descends to the twistor space, and the splitting formula
+guarantees that the bundle is trivial, so the bundle satisfies the conditions of the Penrose-
+Ward correspondence. We therefore can recover a self-dual pullback connection f ∗A. This
+all assumes that the integrals are non-singular, which depends on the details of the maps,
+connections and domains under consideration.
+To gain intuition for this construction, it is useful to derive an explicit formula for f ∗A.
+To this end, restrict to a self-dual plane Z and use the formula ˜λ ˙αf ∗Aα ˙α = H−1˜λ ˙α∂α ˙αH
+from the Penrose-Ward correspondence, in addition to equation (4), to find
+v · f ∗A|x= (f⌟Z)∗v · A|f⌟Z(x),
+x ∈ Z, v ∈ TxZ.
+(5)
+By varying the self-dual plane through a fixed point x this gives the value for null vectors
+v ∈ TxC4, and the value on other vectors can be recovered from linearity of the connection.
+Linearity and self-duality follow from the Penrose-Ward construction, but it is instructive
+to derive these results directly from equation (5), which can be expanded as
+(f⌟Z)∗v · A|f⌟Z(x)= vα ˙α�∂yβ ˙β
+∂xα ˙α Aβ ˙β
+����
+f⌟Z(x), x ∈ Z, v ∈ TxZ,
+(6)
+where yβ ˙β = f⌟Z(x)β ˙β. This equation is degree one in both λ and ˜λ and globally holo-
+morphic on CP 1 × CP 1, and so is linear by Liouville’s theorem. Self-duality is equivalent
+to the connection being integrable on self-dual planes, which follows from using equa-
+tion (5) to show that [v1 · D∗, v2 · D⋆] = 0, where v1,2 ∈ TxZ are linearly independent
+vectors tangent to a self-dual plane Z. These arguments generalize well to the N = 3
+supersymmetric case.
+3
+Super causal morphisms and N=3 SYM
+In this section we will define a super causal morphism, which is an extension of the pre-
+viously defined causal morphisms to superspace. The supersymmetric generalization is
+useful because of an interpretation of the N=3 SYM field equations as an integrability
+condition on supersymmetric null lines [10][12]. This interpretation allows for a gener-
+alization of the arguments of the previous section to the N = 3 SYM field equations.
+Furthermore, solutions of the usual YM field equations are special cases of the supersym-
+metric solutions, and this will be used to show that a modified version of causal morphisms
+are also symmetries of the ordinary YM field equations.
+The definition of super causal morphisms follows closely to the previous definitions.
+To begin, consider the superspace C4|4N with coordinates zA = (xα ˙α, θiα, ˜θ ˙α
+j ) and super-
+symmetry generators qiα =
+∂
+∂θiα + i˜θ ˙α
+i
+∂
+∂xα ˙α and ˜qi
+˙α =
+∂
+∂˜θ ˙α
+i + iθiα
+∂
+∂xα ˙α. The lightlike lines
+4
+
+through z ∈ C4|4N and tangent to (λ, ˜λ) are generated by fermionic translation operators
+Ti = λαqαi and ˜T i = ˜λ ˙α˜qi
+˙α which satisfy the algebra
+{Ti, Tj} = { ˜T i, ˜T j} = 0,
+{Ti, ˜T j} = 2iδj
+i D,
+(7)
+where D = λα˜λ ˙α∂α ˙α. Define the correspondence space (F 6|4N, π1, C4|4N) as the bundle
+over superspace whose fiber at a point z is the set of super null lines that intersect z.
+These fibers are isomorphic to CP 1 ×CP 1, corresponding to the projective spinors λ and
+˜λ that generate bosonic translations along a given null line.
+The super null lines described above have one complex dimension and 2N fermionic
+dimensions, which can be parameterized by coordinates σ = (s, ξi, ˜ξj) ∈ C1|2N.
+The
+supersymmetry generators on this line are ∂s, qi = ∂ξi + i˜ξi∂s and ˜qi = ∂˜ξi + iξi∂s. More
+generally, we can also consider a super null curve, which is a morphism χ : C1|2N → C4|4N
+whose pushforward χ∗ takes the form
+(∂s, qi, ˜qk) → (∂, Mj
+i tj, ˜
+Mk
+l ˜tl),
+(8)
+at every point σ, where the operators (∂, tj, ˜tl) are defined for a super null line that is
+tangent to χ at χ(σ). As with the self-dual case, we can then define the prolongation
+jcχ : C1|2N → F 6|4N by jcχ = (χ, λ, ˜λ), and given a curve ψ : C1|2N → F 6|4N we say that
+it satisfies the contact condition if ψ = jcχ for some nonsingular null curve χ. As before,
+a map f : F 6|4N → F 6|4N preserves the contact condition if f ◦ jcχ satisfies the contact
+condition for any χ. Furthermore, for a super null line L tangent to (λ, ˜λ), we also define
+f⌟L(z) = π1 ◦ f(z, λ, ˜λ) ∀z ∈ L as before.
+In the supersymmetric case there is an extra consideration necessary to ensure inte-
+grability of the pullback connection. To see this, consider a morphism f : F 6|4N → F 6|4N
+which preserves the contact condition. Given a super null line L, we want to demand
+that the supersymmetry relations (7) are preserved under (f⌟L)∗. By construction, this
+pushforward takes the form of equation (8), where the coordinates on L are also chosen
+to satisfy the supersymmetry relations. To preserve these relations, we must demand
+Mi
+j ˜
+Mj
+k = δi
+k.
+(9)
+This condition must be satisfied for every super null line L, which is the condition that
+must be satisfied for integrability.
+We can now define
+Definition 3. A super causal morphism is a holomorphic map f : F 6|4N → F 6|4N which
+preserves the contact condition, and preserves the supersymmetry relations (7) under
+(f⌟L)⋆ for tangent vectors to any super null line L.
+Now consider an N=3 supersymmetric Yang-Mills field satisfying the field equations
+on C4|12. This field can be defined by a superconnection characterized by a one form Φ
+with components ΦA = (ωiα, ˜ωi
+˙α, Aα ˙α), which defines covariant derivative operators
+Qiα = qiα + ωiα, ˜Qi
+˙α = qi
+˙α + ˜ωi
+˙α, Dα ˙α = ∂α ˙α + Aα ˙α.
+(10)
+5
+
+The field equations are equivalent to integrability on super null lines [12], which for a
+given line L tangent to vα ˙α = λα˜λ ˙α are given by equations (7) for the translation operators
+Ti = λαQiα, ˜T i = ˜λ ˙α ˜Qi
+˙α and D = λα˜λ ˙αDα ˙α.
+Now, given a super causal morphism f, we can define the pullback connection f ∗Φ
+similarly to the self-dual case.
+To do so, consider a super null line L, and the super
+null curve f⌟L it generates. The bundle of parallel sections on these super null lines
+then generates a pullback connection in the usual manner. To calculate this pullback
+connection, we can directly generalize equation (5) to
+v · f ∗Φ|z= (f⌟L)∗v · Φ|f⌟L(z),
+z ∈ L, v ∈ TxL.
+(11)
+As in the previous section, linearity of the pullback connection follows from a variant
+of Liouville’s theorem. Integrability on lines follows from writing (7) for the pullback
+translation operators defined on L, and then using (11) and the assumption that Φ is
+integrable on lines.
+This implies that super causal morphisms are symmetries of the
+N = 3 SYM field equations.
+4
+Reduction to Yang-Mills field equations
+The geometric interpretation of the YM field equations using field extensions can be
+naturally understood by viewing these equations as a special case of the N=3 SYM field
+equations for a Yang-Mills super multiplet with the scalar and spinor fields set to zero
+[10][12][13][19]. In a similar spirit, it is possible to use a modified definition of causal
+morphisms to generate an N=3 super causal morphism which preserves the property that
+the scalar and spinor fields equal zero, thus forming a symmetry of the YM field equations.
+A causal morphism can be defined as an N = 0 super causal morphism, or a map
+f : G → G that preserves the contact condition, where G = F 4|0 is the usual ambitwistor
+correspondence space. Given such a function, we can construct an extended morphism
+ˆf : F 6|4N → F 6|4N given by
+ˆf(g, θiα, ˜θ ˙α
+j ) = (f(g), [V −1]α
+βθiβ, [ ˜V −1] ˙α
+˙β ˜θ
+˙β
+j ), g ∈ G,
+(12)
+where V , ˜V are invertible matrix functions of g. Now restrict to a super null line L tangent
+to a null vector vα ˙α = λα˜λ ˙α. Along L, the supersymmetry relations (7) are preserved if
+and only if
+vβ ˙βV α
+β ˜V ˙α
+˙β = ((f⌟L0)∗v)α ˙α,
+(13)
+where L0 is the bosonic projection of L. The existence of holomorphically varying matrix
+functions V and ˜V satisfying this condition is the extra modification necessary to extend
+a causal morphism f to ˆf, and will be assumed.
+The above construction yields a symmetry of the N = 3 SYM field equations, but we
+must also show that solutions of the YM field equations, with scalar and spinor fields set
+6
+
+to zero, are preserved by these extended causal morphisms. To do so, we will use two
+results proved by Harnad et. al. [13]. In that paper, theorem 3.3 characterizes the form
+of the superconnection induced from a solution of the YM field equations when embedded
+as a gauge fixed N=3 SYM connection, which is
+ωiα = ˜θ ˙α
+i hα ˙α(xβ ˙β, τ β ˙β),
+˜ωi
+α = θiα˜hα ˙α(xβ ˙β, τ β ˙β),
+(14)
+where τ β ˙β = �
+i θiβ ˜θ
+˙β
+i , Aα ˙α = Aα ˙α(xβ ˙β, τ β ˙β), and the gauge condition is θiαωiα+˜θi
+˙α˜ω ˙α
+i = 0.
+Conversely, corollary 4.3 shows that a gauge-fixed connection that is integrable on super
+null lines and takes the above form corresponds to an N = 3 extended solution of the YM
+field equations.
+Based on these considerations, showing that the extended causal morphisms preserve
+the form of equation (14) and the gauge condition implies that they preserve solutions
+of the YM field equations. To show this, restrict to a super null line L and use (11) to
+compute the pullback connection of (14), which gives
+λα ˆf ∗ωiα(z) = ˜θ
+˙β
+i λα �
+V β
+α hβ ˙β(x′, τ ′)
+�
+,
+˜λ ˙α ˆf ∗˜ωi
+˙α(z) = θiβ˜λ ˙α �
+˜V
+˙β
+˙α ˜hβ ˙β(x′, τ ′)
+�
+,
+(15)
+where x′, τ ′ are evaluated at ˆf⌟L(z).
+Now consider two points z1, z2 ∈ C4|4N with
+x1 = x2 and τ1 = τ2, lying on two parallel lines. Under ˆf⌟L, τ transforms as τ α ˙α →
+[V −1]α
+β[ ˜V −1] ˙α
+˙βτ β ˙β, so the quantities in parentheses are the same for these two points and
+parallel lines, but the line can be varied, so this is true for any (λ, ˜λ). We therefore see
+that the connection has the form of equation (14), as desired. Furthermore, the gauge
+condition can be written τ α ˙α(hα ˙α + ˜hα ˙α) = 0. For τ α ˙α proportional to a null vector this
+condition is preserved, but this must be true for any null line, so by linearity the gauge
+condition is preserved. This implies that the pullback connection corresponds to an N = 3
+extended solution of the YM field equations.
+5
+Discussion
+We have shown that self-dual, N = 3 super causal and causal morphisms yield symmetries
+of the ASDYM, N = 3 SYM and YM field equations, respectively. To further understand
+these symmetries, it will be necessary to classify their solutions and to investigate their
+action on concrete examples of YM fields. Some partial results were found in the previous
+paper [16], where examples of self-dual morphisms were constructed from holomorphic
+endomorphisms of twistor space. A method was also developed to construct causal mor-
+phisms from these self-dual morphisms. Although these constructions provide examples
+of solutions, it will be important to find a more complete classification. In particular, it is
+likely that there are more general examples than those constructed from endomorphisms
+7
+
+of the twistor space, or super ambitwistor space, which could be analogous to holomorphic
+functions that preserve the real line in two dimensions. This preliminary interpretation
+is based on the CR ambitwistor space used in [20], but will require further investigation
+to make precise. It will also be important to understand how these maps are related to
+other well known hidden symmetries for these equations.
+There are many additional avenues of further research. Here the action of these maps
+was only considered for classical Yang-Mills fields, but the ultimate goal is to further
+understand the quantum theory.
+Furthermore, it will be interesting to consider how
+gravitational fields transform under these maps. In this vein, one could define a causal
+manifold with coordinate transformations that are morphisms of these types, in analogy
+to the definition of Riemann surfaces for holomorphic functions. Due to the nonlocal
+nature of these maps, the theory could lead to interesting new mathematics.
+References
+[1] L. Dolan, A new symmetry group of real self-dual yang-mills theory,
+Physics Letters B 113 (1982) 387.
+[2] L.-L. Chau, G. Mo-Lin, A. Sinha and W. Yong-Shi, Hidden-symmetry algebra for
+the self-dual yang-mills equation, Physics Letters B 121 (1983) 391.
+[3] A.D. Popov, Self-dual yang-mills: Symmetries and moduli space,
+Reviews in Mathematical Physics 11 (1999) 1091.
+[4] L. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and
+grassmannians, Journal of High Energy Physics 2009 (2009) 045.
+[5] N. Arkani-Hamed, F. Cachazo and C. Cheung, The grassmannian origin of dual
+superconformal invariance, Journal of High Energy Physics 2010 (2010) .
+[6] J. Drummond, J. Henn and J. Plefka, Yangian symmetry of scattering amplitudes
+in n = 4 super yang-mills theory, Journal of High Energy Physics 2009 (2009) 046.
+[7] R.S. Ward, On self-dual gauge fields, Physics Letters A 61 (1977) 81.
+[8] M. Atiyah, N. Hitchin, V. Drinfeld and Y. Manin, Construction of instantons,
+Physics Letters A 65 (1978) 185 .
+[9] M. Atiyah, A. nazionale dei Lincei and S. normale superiore (Italy), Geometry of
+Yang-Mills fields, Lezioni fermiane, Scuola normale superiore (1979).
+[10] E. Witten, An interpretation of classical yang-mills theory,
+Physics Letters B 77 (1978) 394 .
+8
+
+[11] J. Isenberg, P.B. Yasskin and P.S. Green, Non-self-dual gauge fields,
+Physics Letters B 78 (1978) 462 .
+[12] J. Harnad, J. Hurtubise, M. Legare and S. Shnider, Constraint equations and field
+equations in supersymmetric n = 3 yang-mills theory,
+Nuclear Physics B 256 (1985) 609.
+[13] J. Harnad, J. Hurtubise and S. Shnider, Supersymmetric yang-mills equations and
+supertwistors, Annals of Physics 193 (1989) 40.
+[14] E. Witten, Perturbative gauge theory as a string theory in twistor space,
+Communications in Mathematical Physics 252 (2004) 189.
+[15] M. Atiyah, M. Dunajski and L.J. Mason, Twistor theory at fifty: from contour
+integrals to twistor strings, Proceedings of the Royal Society A: Mathematical,
+Physical and Engineering Sciences 473 (2017) 20170530.
+[16] E.B. Baker, Causal and self-dual morphisms in four complex dimensions, 2022.
+10.48550/ARXIV.2203.07952.
+[17] M. Dunajski, Solitons, Instantons, and Twistors, Oxford Graduate Texts in
+Mathematics, OUP Oxford (2010).
+[18] R. Ward and R. Wells, Twistor Geometry and Field Theory, Cambridge
+Monographs on Mathematical Physics, Cambridge University Press (1991).
+[19] M. Eastwood, Supersymmetry, twistors, and the yang-mills equations, Transactions
+of the American Mathematical Society 301 (1987) 615.
+[20] L. Mason and D. Skinner, An ambitwistor yang–mills lagrangian,
+Physics Letters B 636 (2006) 60.
+9
+

2NAyT4oBgHgl3EQfPvY8/content/tmp_files/load_file.txt

@@ -0,0 +1,210 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf,len=209
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='00029v1  [math-ph]  31 Dec 2022  Generalized conformal maps as classical symmetries of  Yang-Mills fields  Edward B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Baker III∗  January 3, 2023  We show that a class of previously defined maps, called causal and  self-dual morphisms, form classical symmetries of Yang-Mills fields  in four complex dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' These maps generalize conformal trans-  formations, and admit a nonlocal pullback connection that preserves  the equations of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' First it is shown that self-dual mor-  phisms form symmetries of the anti-self-dual Yang-Mills equations  under this pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Then a supersymmetric generalization of causal  morphisms is defined which preserves solutions of the field equations  for N=3 supersymmetric Yang-Mills theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' As a special case, this  implies that a modified definition of causal morphisms form sym-  metries for the ordinary Yang-Mills field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  1  Introduction  Hidden symmetries have played an important role in the study of Yang-Mills (YM) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  As an example, the anti-self dual Yang Mills (ASDYM) equations have an infinite class  of hidden symmetries which bear some resemblance to the infinite-dimensional conformal  group in two dimensions [1][2][3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' In addition, an extended conformal symmetry called  dual superconformal invariance has been uncovered in the study of N=4 supersymmetric  Yang-Mills (SYM) theory [4][5], leading to an infinite dimensional Yangian symmetry [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  This and other advances have led to powerful tools for the study of N=4 SYM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Many of these results can be understood best with the use of twistor and ambitwistor  methods, which have been used extensively in the study of Yang-Mills fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' For example,  the Penrose-Ward correspondence reformulates the ASDYM equations in Twistor space,  ∗edwardbaker86@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='com  1  which leads to the ADHM construction of instantons [7][8][9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This construction was gen-  eralized to a geometric formulation of the Yang-Mills field equations in ambitwistor space,  with a natural interpretation in superspace [10][11][12][13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' More recently, twistor and am-  bitwistor methods have been used in string theory to understand Yang-Mills scattering  amplitudes and their properties [14][15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  In this paper we investigate a previously defined class of generalized maps [16] in  the context of Yang-Mills theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' These maps are motivated by twistor and ambitwistor  theory, and are called self-dual and causal morphisms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Under certain as-  sumptions, one can define a non-local pullback connection under these transformations  that preserves integrability on certain subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' In the case of self-dual morphisms, the  maps preserve integrability on self-dual planes which imply that they are symmetries of  the ASDYM equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' A supersymmetric generalization of causal morphisms is then  developed which preserves integrability on super null lines, implying that these maps are  symmetries of the N=3 SYM field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This fact is used to show that a modified  version of causal morphisms are symmetries of the YM field equations as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  It is likely that some of these symmetries are related to known hidden symmetries for  the different cases, but characterizing these relationships will be left as a topic of future  investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  2  Self-dual morphisms as symmetries of ASDYM  Self-dual morphisms were introduced in a previous paper, where they were defined using  maps on null surfaces [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Here we provide a self contained summary of these results,  using different but equivalent definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To begin, define the twistor correspondence  space F = C4 × CP1 with the usual double fibration [17][18]  C4  π1  ←− F  π2  −→ PT ,  (1)  where PT = CP3 is the projective twistor space of C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Now define a self-dual embedding  as a totally null holomorphic embedding χ : C2 → C4, which means that vectors at a  point t ∈ C2 are mapped under χ⋆ to vectors of the form vα ˙α = λα˜λ ˙α for ˜λ ˙α fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' We  will call the image of such a map a self-dual surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' If ˜λ is independent of t then this  surface maps to a self-dual plane (or α-plane) Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Furthermore, for any point on a self-dual  embedding there is a tangent α-plane passing through χ(t) that is characterized by ˜λ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  For brevity, we say that χ is tangent to ˜λ at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Now define the self-dual prolongation  jsχ : C2 → F by jsχ = (χ, ˜λ), where dependence on t is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The prolongation  satisfies a contact condition, that χ is tangent to ˜λ for all t ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Conversely, given a  surface ψ : C2 → F, we say that it satisfies the contact condition if ψ = jsχ for some  self-dual embedding χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' A map f : F → F is said to preserve the contact condition if  2  f ◦ jsχ satisfies the contact condition for any χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='1 We then define  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' A self-dual morphism is a holomorphic map f : F → F which preserves  the contact condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Defined in this way, a self-dual morphism naturally induces maps on self-dual embed-  dings and self-dual planes  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Given a self-dual morphism f and a self-dual embedding χ, define the  contraction map f⌟χ := π1 ◦ f ◦ jsχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Furthermore, for a self-dual plane Z tangent to ˜λ,  define f⌟Z : Z → C4 by f⌟Z(x) = π1 ◦ f(x, ˜λ) where x ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  This map on surfaces was the starting point for the definitions in the previous paper,  and the two definitions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Now consider a GL(n, C) connection with vector potential A satisfying the ASDYM  equations on MC = C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Given a self-dual morphism f, there is a natural definition for  a pullback connection f ∗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To see this, first restrict to a self-dual plane Z, which can  be parameterized linearly by coordinates on C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The contraction map f⌟Z then gives  a self-dual embedding, and the pullback connection (f⌟Z)∗A is integrable on Z because  the curvature of A vanishes on the self-dual planes tangent to f⌟Z as a consequence  of ASDYM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' By varying Z this allows us to define the bundle of parallel sections on  the twistor space, and to use the Penrose-Ward procedure to define a connection on  the pullback bundle, which defines the pullback connection f ∗A and gives a solution of  ASDYM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This requires that the bundle of parallel sections is trivial for points x ∈ C4,  which will be shown with an explicit construction of f ∗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  For the construction, consider two points x1, x2 ∈ Z and their images yi = f⌟Z(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Define a Wilson line for the pullback connection by  W ∗  Z(x1, x2) = Wf⌟Z(y1, y2) = P exp  ��  γ  Aµdxµ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  (2)  Here the path of integration is any path γ confined to the image of f⌟Z starting at y1  and ending at y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The Wilson line is independent of path due to the integrability of the  connection on self-dual surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' We can then define the patching matrix used in the  Penrose-Ward correspondence by  G = W ∗  Z(q, p) = ˜HH−1  (3)  where  H = W ∗  Z(p, x),  ˜H = W ∗  Z(q, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  (4)  1These constructions are all assumed to be local and defined in some neighborhood, but are written  globally for ease of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  3  Here p and q are the points of intersection between Z and self-dual planes P and Q with  twistor coordinates ˆP = (0, 0, 1, 0) and ˆQ = (0, 0, 0, 1), as defined in the usual patching  construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The patching matrix descends to the twistor space, and the splitting formula  guarantees that the bundle is trivial, so the bundle satisfies the conditions of the Penrose-  Ward correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' We therefore can recover a self-dual pullback connection f ∗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This  all assumes that the integrals are non-singular, which depends on the details of the maps,  connections and domains under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  To gain intuition for this construction, it is useful to derive an explicit formula for f ∗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  To this end, restrict to a self-dual plane Z and use the formula ˜λ ˙αf ∗Aα ˙α = H−1˜λ ˙α∂α ˙αH  from the Penrose-Ward correspondence, in addition to equation (4), to find  v · f ∗A|x= (f⌟Z)∗v · A|f⌟Z(x),  x ∈ Z, v ∈ TxZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  (5)  By varying the self-dual plane through a fixed point x this gives the value for null vectors  v ∈ TxC4, and the value on other vectors can be recovered from linearity of the connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Linearity and self-duality follow from the Penrose-Ward construction, but it is instructive  to derive these results directly from equation (5), which can be expanded as  (f⌟Z)∗v · A|f⌟Z(x)= vα ˙α�∂yβ ˙β  ∂xα ˙α Aβ ˙β  ����  f⌟Z(x), x ∈ Z, v ∈ TxZ,  (6)  where yβ ˙β = f⌟Z(x)β ˙β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This equation is degree one in both λ and ˜λ and globally holo-  morphic on CP 1 × CP 1, and so is linear by Liouville’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Self-duality is equivalent  to the connection being integrable on self-dual planes, which follows from using equa-  tion (5) to show that [v1 · D∗, v2 · D⋆] = 0, where v1,2 ∈ TxZ are linearly independent  vectors tangent to a self-dual plane Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' These arguments generalize well to the N = 3  supersymmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  3  Super causal morphisms and N=3 SYM  In this section we will define a super causal morphism, which is an extension of the pre-  viously defined causal morphisms to superspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The supersymmetric generalization is  useful because of an interpretation of the N=3 SYM field equations as an integrability  condition on supersymmetric null lines [10][12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This interpretation allows for a gener-  alization of the arguments of the previous section to the N = 3 SYM field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Furthermore, solutions of the usual YM field equations are special cases of the supersym-  metric solutions, and this will be used to show that a modified version of causal morphisms  are also symmetries of the ordinary YM field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  The definition of super causal morphisms follows closely to the previous definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  To begin, consider the superspace C4|4N with coordinates zA = (xα ˙α, θiα, ˜θ ˙α  j ) and super-  symmetry generators qiα =  ∂  ∂θiα + i˜θ ˙α  i  ∂  ∂xα ˙α and ˜qi  ˙α =  ∂  ∂˜θ ˙α  i + iθiα  ∂  ∂xα ˙α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The lightlike lines  4  through z ∈ C4|4N and tangent to (λ, ˜λ) are generated by fermionic translation operators  Ti = λαqαi and ˜T i = ˜λ ˙α˜qi  ˙α which satisfy the algebra  {Ti, Tj} = { ˜T i, ˜T j} = 0,  {Ti, ˜T j} = 2iδj  i D,  (7)  where D = λα˜λ ˙α∂α ˙α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Define the correspondence space (F 6|4N, π1, C4|4N) as the bundle  over superspace whose fiber at a point z is the set of super null lines that intersect z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  These fibers are isomorphic to CP 1 ×CP 1, corresponding to the projective spinors λ and  ˜λ that generate bosonic translations along a given null line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  The super null lines described above have one complex dimension and 2N fermionic  dimensions, which can be parameterized by coordinates σ = (s, ξi, ˜ξj) ∈ C1|2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  The  supersymmetry generators on this line are ∂s, qi = ∂ξi + i˜ξi∂s and ˜qi = ∂˜ξi + iξi∂s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' More  generally, we can also consider a super null curve, which is a morphism χ : C1|2N → C4|4N  whose pushforward χ∗ takes the form  (∂s, qi, ˜qk) → (∂, Mj  i tj, ˜  Mk  l ˜tl),  (8)  at every point σ, where the operators (∂, tj, ˜tl) are defined for a super null line that is  tangent to χ at χ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' As with the self-dual case, we can then define the prolongation  jcχ : C1|2N → F 6|4N by jcχ = (χ, λ, ˜λ), and given a curve ψ : C1|2N → F 6|4N we say that  it satisfies the contact condition if ψ = jcχ for some nonsingular null curve χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' As before,  a map f : F 6|4N → F 6|4N preserves the contact condition if f ◦ jcχ satisfies the contact  condition for any χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Furthermore, for a super null line L tangent to (λ, ˜λ), we also define  f⌟L(z) = π1 ◦ f(z, λ, ˜λ) ∀z ∈ L as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  In the supersymmetric case there is an extra consideration necessary to ensure inte-  grability of the pullback connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To see this, consider a morphism f : F 6|4N → F 6|4N  which preserves the contact condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Given a super null line L, we want to demand  that the supersymmetry relations (7) are preserved under (f⌟L)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' By construction, this  pushforward takes the form of equation (8), where the coordinates on L are also chosen  to satisfy the supersymmetry relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To preserve these relations, we must demand  Mi  j ˜  Mj  k = δi  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  (9)  This condition must be satisfied for every super null line L, which is the condition that  must be satisfied for integrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  We can now define  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' A super causal morphism is a holomorphic map f : F 6|4N → F 6|4N which  preserves the contact condition, and preserves the supersymmetry relations (7) under  (f⌟L)⋆ for tangent vectors to any super null line L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Now consider an N=3 supersymmetric Yang-Mills field satisfying the field equations  on C4|12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This field can be defined by a superconnection characterized by a one form Φ  with components ΦA = (ωiα, ˜ωi  ˙α, Aα ˙α), which defines covariant derivative operators  Qiα = qiα + ωiα, ˜Qi  ˙α = qi  ˙α + ˜ωi  ˙α, Dα ˙α = ∂α ˙α + Aα ˙α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  (10)  5  The field equations are equivalent to integrability on super null lines [12], which for a  given line L tangent to vα ˙α = λα˜λ ˙α are given by equations (7) for the translation operators  Ti = λαQiα, ˜T i = ˜λ ˙α ˜Qi  ˙α and D = λα˜λ ˙αDα ˙α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Now, given a super causal morphism f, we can define the pullback connection f ∗Φ  similarly to the self-dual case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  To do so, consider a super null line L, and the super  null curve f⌟L it generates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The bundle of parallel sections on these super null lines  then generates a pullback connection in the usual manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To calculate this pullback  connection, we can directly generalize equation (5) to  v · f ∗Φ|z= (f⌟L)∗v · Φ|f⌟L(z),  z ∈ L, v ∈ TxL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  (11)  As in the previous section, linearity of the pullback connection follows from a variant  of Liouville’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Integrability on lines follows from writing (7) for the pullback  translation operators defined on L, and then using (11) and the assumption that Φ is  integrable on lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  This implies that super causal morphisms are symmetries of the  N = 3 SYM field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  4  Reduction to Yang-Mills field equations  The geometric interpretation of the YM field equations using field extensions can be  naturally understood by viewing these equations as a special case of the N=3 SYM field  equations for a Yang-Mills super multiplet with the scalar and spinor fields set to zero  [10][12][13][19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' In a similar spirit, it is possible to use a modified definition of causal  morphisms to generate an N=3 super causal morphism which preserves the property that  the scalar and spinor fields equal zero, thus forming a symmetry of the YM field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  A causal morphism can be defined as an N = 0 super causal morphism, or a map  f : G → G that preserves the contact condition, where G = F 4|0 is the usual ambitwistor  correspondence space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Given such a function, we can construct an extended morphism  ˆf : F 6|4N → F 6|4N given by  ˆf(g, θiα, ˜θ ˙α  j ) = (f(g), [V −1]α  βθiβ, [ ˜V −1] ˙α  ˙β ˜θ  ˙β  j ), g ∈ G,  (12)  where V , ˜V are invertible matrix functions of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Now restrict to a super null line L tangent  to a null vector vα ˙α = λα˜λ ˙α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Along L, the supersymmetry relations (7) are preserved if  and only if  vβ ˙βV α  β ˜V ˙α  ˙β = ((f⌟L0)∗v)α ˙α,  (13)  where L0 is the bosonic projection of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' The existence of holomorphically varying matrix  functions V and ˜V satisfying this condition is the extra modification necessary to extend  a causal morphism f to ˆf, and will be assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  The above construction yields a symmetry of the N = 3 SYM field equations, but we  must also show that solutions of the YM field equations, with scalar and spinor fields set  6  to zero, are preserved by these extended causal morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To do so, we will use two  results proved by Harnad et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' In that paper, theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='3 characterizes the form  of the superconnection induced from a solution of the YM field equations when embedded  as a gauge fixed N=3 SYM connection, which is  ωiα = ˜θ ˙α  i hα ˙α(xβ ˙β, τ β ˙β),  ˜ωi  α = θiα˜hα ˙α(xβ ˙β, τ β ˙β),  (14)  where τ β ˙β = �  i θiβ ˜θ  ˙β  i , Aα ˙α = Aα ˙α(xβ ˙β, τ β ˙β), and the gauge condition is θiαωiα+˜θi  ˙α˜ω ˙α  i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Conversely, corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='3 shows that a gauge-fixed connection that is integrable on super  null lines and takes the above form corresponds to an N = 3 extended solution of the YM  field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Based on these considerations, showing that the extended causal morphisms preserve  the form of equation (14) and the gauge condition implies that they preserve solutions  of the YM field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To show this, restrict to a super null line L and use (11) to  compute the pullback connection of (14), which gives  λα ˆf ∗ωiα(z) = ˜θ  ˙β  i λα �  V β  α hβ ˙β(x′, τ ′)  �  ,  ˜λ ˙α ˆf ∗˜ωi  ˙α(z) = θiβ˜λ ˙α �  ˜V  ˙β  ˙α ˜hβ ˙β(x′, τ ′)  �  ,  (15)  where x′, τ ′ are evaluated at ˆf⌟L(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Now consider two points z1, z2 ∈ C4|4N with  x1 = x2 and τ1 = τ2, lying on two parallel lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Under ˆf⌟L, τ transforms as τ α ˙α →  [V −1]α  β[ ˜V −1] ˙α  ˙βτ β ˙β, so the quantities in parentheses are the same for these two points and  parallel lines, but the line can be varied, so this is true for any (λ, ˜λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' We therefore see  that the connection has the form of equation (14), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Furthermore, the gauge  condition can be written τ α ˙α(hα ˙α + ˜hα ˙α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' For τ α ˙α proportional to a null vector this  condition is preserved, but this must be true for any null line, so by linearity the gauge  condition is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This implies that the pullback connection corresponds to an N = 3  extended solution of the YM field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  5  Discussion  We have shown that self-dual, N = 3 super causal and causal morphisms yield symmetries  of the ASDYM, N = 3 SYM and YM field equations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' To further understand  these symmetries, it will be necessary to classify their solutions and to investigate their  action on concrete examples of YM fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Some partial results were found in the previous  paper [16], where examples of self-dual morphisms were constructed from holomorphic  endomorphisms of twistor space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' A method was also developed to construct causal mor-  phisms from these self-dual morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Although these constructions provide examples  of solutions, it will be important to find a more complete classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' In particular, it is  likely that there are more general examples than those constructed from endomorphisms  7  of the twistor space, or super ambitwistor space, which could be analogous to holomorphic  functions that preserve the real line in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' This preliminary interpretation  is based on the CR ambitwistor space used in [20], but will require further investigation  to make precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' It will also be important to understand how these maps are related to  other well known hidden symmetries for these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  There are many additional avenues of further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' Here the action of these maps  was only considered for classical Yang-Mills fields, but the ultimate goal is to further  understand the quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content='  Furthermore, it will be interesting to consider how  gravitational fields transform under these maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
+page_content=' In this vein, one could define a causal  manifold with coordinate transformations that are morphisms of these types, in analogy  to the definition of Riemann surfaces for holomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfPvY8/content/2301.00029v1.pdf'}
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+1
+On the Mutual Information of Multi-RIS
+Assisted MIMO: From Operator-Valued Free
+Probability Aspect
+Zhong Zheng, Member, IEEE, Siqiang Wang, Zesong Fei, Senior Member, IEEE,
+Zhi Sun, Senior Member, IEEE, Jinhong Yuan, Fellow, IEEE
+Abstract
+The reconfigurable intelligent surface (RIS) is useful to effectively improve the coverage and data rate
+of end-to-end communications. In contrast to the well-studied coverage-extension use case, in this paper,
+multiple RIS panels are introduced, aiming to enhance the data rate of multi-input multi-output (MIMO)
+channels in presence of insufficient scattering. Specifically, via the operator-valued free probability theory,
+the asymptotic mutual information of the large-dimensional RIS-assisted MIMO channel is obtained
+under the Rician fading with Weichselberger’s correlation structure, in presence of both the direct and
+the reflected links. Although the mutual information of Rician MIMO channels scales linearly as the
+number of antennas and the signal-to-noise ratio (SNR) in decibels, numerical results show that it requires
+sufficiently large SNR, proportional to the Rician factor, in order to obtain the theoretically guaranteed
+linear improvement. This paper shows that the proposed multi-RIS deployment is especially effective to
+improve the mutual information of MIMO channels under the large Rician factor conditions. When the
+reflected links have similar arriving and departing angles across the RIS panels, a small number of RIS
+panels are sufficient to harness the spatial degree of freedom of the multi-RIS assisted MIMO channels.
+Z. Zheng, S. Wang, and Z. Fei are with the School of Information and Electronics, Beijing Institute of Technology, Beijing,
+China. Z. Sun is with the Department of Electronic Engineering, Tsinghua University, Beijing, China. J. Yuan is with the School
+of Electrical Engineering and Telecommunications, UNSW Sydney, Sydney, Australia.
+DRAFT
+arXiv:2301.12144v1  [cs.IT]  28 Jan 2023
+
+2
+Index Terms
+Reconfigurable intelligent surface, MIMO, Rician channel, mutual information, operator-valued free
+probability.
+I. INTRODUCTION
+In both the current and forthcoming generations of mobile communication systems, multi-input multi-
+output (MIMO) is one of the mainstream physical-layer techniques to improve the spectral efficiency
+and the reliability of the wireless communications [1]. In the favorable environments with rich scattering,
+MIMO is able to increase the achievable data rate linearly with the number of antennas [2]. However,
+when the wireless systems operate in higher frequencies with larger bandwidth, such as the millimeter
+wave and terahertz bands, the radio signals are easily attenuated due to absorption and blockage. In this
+case, the MIMO channels typically have only a few dominating propagation paths and/or limited angular
+spread, which causes rank deficiency in the channel matrix that significantly degrades the MIMO channel
+capacity [3].
+Recently, reconfigurable intelligent surface (RIS) has attracted substantial attentions and is foreseen
+to be an important component in the future communication systems [4]. A typical RIS consists of
+a large number of low-power integrated electronic circuits, which can be programmed to modify the
+electromagnetic properties of the incoming radio waves in the desired frequency band [5], such as the
+phase and amplitude of the reflected signals from each programmable circuit. Therefore, by deploying
+some RIS panels in the environment, the signal’s radiation pattern within the operating spectrum bands
+of the communication systems can be reconfigured to increase the number of independent paths with
+diversified angular spreads, thus increasing the rank of the MIMO channels. As an example, Fig. 1
+illustrates the transmissions between a base station (BS) and a user equipment (UE) in an urban canyon.
+In this scenario, without RIS deployment, the signals have to propagate through a scattering-limited area,
+where the direct propagation link F0 dominates the end-to-end channel, while other scattered/reflected
+components are severally attenuated by the building materials. In comparison, RIS panels are able to
+actively and effectively reflect the signals to increase the number of independent specular components,
+DRAFT
+
+3
+• • • 
+UE
+BS
+RIS K
+RIS 1
+F0
+F1
+FK
+G1
+GK
+Fig. 1.
+Multi-RIS assisted MIMO communications.
+resulting in a total number of K +1 propagation links, including the direct link F0 and K reflected links
+that consist of channels {Fk}1≤k≤K between BS and RIS panels and channels {Gk}1≤k≤K between RIS
+panels and UE.
+There exist a number of studies focusing on the performance evaluation of the end-to-end communi-
+cations assisted by a single or multiple RIS panels. When both the transmitter and receiver are equipped
+with a single antenna and a single RIS is deployed, the signal-to-noise-ratio (SNR) of such RIS-assisted
+single-input single-output (SISO) channel is proportional to the squared amplitude of the end-to-end
+effective channel. There are two typical theoretic frameworks to analyze the statistical properties of
+the SNR: One is based on the Meijer’s G- and Fox’s H-function systems [6], which result in exact
+but rather complicated expressions. The other is to match the moments of the effective channel with
+classical random variables [7]–[10]. In particular, when the direct link is blocked, the SNR distribution and
+the corresponding outage probability of the RIS-assisted communications are approximated by Gamma
+random variables, when the component channels are independently Rayleigh-faded [7], independently
+Rician-faded [8], correlated Rician-faded [9], and independently Nakagami-faded [10], respectively. When
+both the direct and the reflected links exist, the Gamma-approximated SNR distribution and the finite
+block-length rate of the RIS-assisted channel are obtained in [11], when all component channels are
+DRAFT
+
+4
+Rayleigh-faded.
+When multiple RIS panels are deployed in the network, the RIS panels either work in exhaustive mode
+to jointly assist the end-to-end communications [12], or work in opportunistic mode, where only the RIS
+with maximum channel gains is selected [13]. In the former case, the end-to-end SNR is approximated
+by a Gaussian random variable due to the central limit theorem. The SNR and the average symbol error
+probability are derived for the phase shift keying signaling. In the latter case, the SNR of each reflected
+link corresponding to one RIS panel is approximated as a Gaussian random variable and the order-statistics
+is then obtained for the optimally selected RIS-assisting channel. In [14], a comprehensive performance
+comparison of those two operation modes is provided, under the scenario of multi-RIS assisted SISO
+channels assuming Nakagami-faded direct and reflected links.
+In the case of MIMO systems, the available performance analysis of RIS-assisted communications is
+rather limited due to the challenge of understanding the statistical distribution of matrix-valued prop-
+agation channels, and results only exist for the single-RIS deployment. In [15], the authors consider
+the RIS-assisted MIMO communications, where the direct link is blocked and the reflected link is the
+concatenated Rayleigh-faded MIMO channels with single-sided correlation. The exact outage probability
+of such channel is derived by using the Mellin transform [16]. The result is expressed as the integration
+of product of multiple Meijer’s G-functions, which is difficult to solve in practice. When the reflected
+link is the concatenated millimeter wave MIMO channels assuming Saleh-Valenzuela model [17], an
+upper bound of the ergodic achievable rate is derived in [18] using majorization theory and Jensen’s
+inequality. In the RIS-assisted uplink multiple access channel, the asymptotic ergodic sum rate of the
+multi-user MIMO system is derived in [19] by using the replica method, assuming that the reflected links
+are Rician-faded MIMO channels with Kronecker’s correlation. In the same channel model as [19], the
+finite-SNR diversity-multiplexing tradeoff (DMT) of the RIS-assisted MIMO channel is analyzed in [20]
+by the martingale method. When both the direct and the reflected links exist, the asymptotic achievable
+rate of the single-RIS assisted MIMO channel is derived in [21] via replica method, assuming that all
+the component channels are Rician fading with Kronecker’s correlation and all the channel dimensions
+grow to infinity.
+DRAFT
+
+5
+Although the RIS-assisted MIMO communications have been investigated in [15], [18]–[21], the results
+therein are obtained for the single-RIS deployment under certain MIMO channel configurations. In
+contrast, this paper aims to provide the theoretic framework that analyzes the general multi-RIS assisted
+MIMO communications under arbitrary Rician fading with Weichselberger’s correlation structure [22].
+Such a model fits a wider range of realistic MIMO channels compared to the conventional Kronecker’s
+correlation structure. Based on the above system settings, we first embed the component MIMO channel
+matrices into a large block matrix. Then, an operator-valued probability space over the algebra of the
+constructed block matrices is defined, where the operator-valued Cauchy transform is defined and is shown
+to be closely related to the classic Cauchy transform of the channel Gram matrix. The operator-valued
+Cauchy transform is then derived by leveraging the freeness over the defined probability space and the
+additive free convolution machinery. Based on the obtained Cauchy transform, the probability distribution
+of the eigenvalue of the channel Gram matrix as well as the mutual information of the multi-RIS assisted
+MIMO channel can be calculated, which avoids time-consuming Monte Carlo simulations. Numerical
+results show that in presence of strong line-of-sight conditions, although the mutual information could
+scale linearly as the number of antennas and the SNRs (in decibels), the SNR has to be sufficiently large
+in order to exhibit such linear scaling law. On the other hand, deploying additional RIS panels could
+effectively improve the channel’s mutual information and thus, alleviate the SNR requirement.
+The rest of this article is organized as follows. The signal model, the channel model, and the mutual
+information of the MIMO channel under consideration are introduced in Section II. In Section III, the
+operator-valued probability space is introduced and the main result of the Cauchy transform of the channel
+Gram matrix is given. Numerical simulation results on the spectral distribution and the mutual information
+of the MIMO channels are in Section IV. Section V concludes the main findings of this article.
+Notations. Throughout the paper, vectors and matrices are represented by lower-case and upper-case
+bold-face letters, respectively. The complex column vector with length n is denoted as Cn. We use
+CN(0, A) to denote the zero-mean complex Gaussian vector with covariance matrix A and In is an
+n × n identity matrix. The superscript (·)† denotes the matrix conjugate-transpose operation and (·)T
+is matrix transpose. We denote Tr(A) as the trace of n × n matrix A. The notation E[·] denotes the
+DRAFT
+
+6
+expectation, and det(·) denotes the matrix determinant.
+II. SYSTEM MODEL
+A. Signal Model
+Consider a MIMO communication channel between a transmitter equipped with T antennas and a
+receiver equipped with R antennas. The transmissions are assisted by K RIS panels, which reflect the
+impinging signals via their reflecting elements and each RIS panel is equipped with Lk reflecting elements,
+1 ≤ k ≤ K. For notational simplicity, we define R = L0 and use these two symbols interchangeably.
+Denote the transmitted signal as x ∈ CT and the additive noise at the receiver as n ∈ CR. The received
+signal y ∈ CR is expressed as
+y =
+�
+F0 +
+K
+�
+k=1
+√ρkGkFk
+�
+x + n,
+(1)
+where the R × T matrix F0 denotes the direct channel between transmitter and receiver, the Lk × T
+matrix Fk denotes the channels between the transmitter and the k-th RIS panel, and 0 < ρk ≤ 1 denotes
+the relative channel gain of the k-th reflected channel via the k-th RIS, compared to the direct channel.
+The R × Lk matrix Gk denotes phase-shifted reflected channel between the k-th RIS and the receiver,
+modeled as
+Gk = RkΘk,
+(2)
+where the R×Lk matrix Rk denotes the channel coefficients, and the diagonal matrix Θk = diag
+�
+eiφk,1, . . . , eiφk,Lk�
+contains the phase-shifts of the reflecting elements, where 0 ≤ φk,l ≤ 2π denotes the phase-shift of the
+l-th element of the k-th RIS.
+We adopt the following assumptions on the signal and the channels:
+(A1) The signal x is Gaussian distributed with uniform power allocation, i.e., x ∼ CN(0T , PIT ), where
+P is the average power of the signals from each transmit antenna;
+(A2) The noise n is assumed to be a white Gaussian random vector with i.i.d. zero-mean entries, i.e.,
+n ∼ CN(0R, σ2IR), where σ2 denotes the variance of the noise;
+DRAFT
+
+7
+(A3) The channel coefficients {Fk}0≤k≤K and {Rk}1≤k≤K are block-faded, which keeps constant within
+the coherence time, while changing randomly and independently in the next coherence time. The
+phase shifts {Θk}1≤k≤K are assumed to be fixed.
+Note that without the direct link F0, channel models similar to (1) have been also studied in [3],
+[23], [24] for the keyhole channel, the Rayleigh-product channel, and the double-scattering channel,
+respectively, by using different theoretic techniques, which cannot be applied here.
+B. Channel Model
+In order to characterize the directivity and the spatial correlation of the channels between antenna
+arrays, we adopt the non-central Weichselberger’s MIMO model for each component links [22], such
+that
+Fk = Fk + �Fk = Fk + Uk(Mk ⊙ Xk)V†
+k,
+0 ≤ k ≤ K,
+(3)
+Gk = Gk + �Gk = Gk +
+1
+√rk
+Wk(Nk ⊙ Yk)S†
+k,
+1 ≤ k ≤ K,
+(4)
+where Fk and Gk are the fixed specular components of Fk and Gk, respectively. The random scattering
+components are captured by �Fk and �Gk, where Uk, Vk, Wk, and Sk are deterministic unitary matrices.
+The deterministic matrices Mk and Nk represent the variance profiles of �Fk and �Gk, respectively, each
+having non-negative real elements. The Lk × T random matrix Xk and the R × Lk random matrix Yk
+are i.i.d. complex Gaussian distributed with entries having zero mean and variance 1/T, i.e., [Xk]i,j ∼
+CN(0, 1/T) and [Yk]i,j ∼ CN(0, 1/T). We denote rk as the ratio between Lk and T, i.e., rk = Lk/T,
+1 ≤ k ≤ K. The operator ⊙ denotes the element-wise matrix multiplication. Note that the phase-shift
+matrix Θk, 1 ≤ k ≤ K, is also unitary and can be absorbed into the deterministic matrices Gk and
+Sk. The ratio between the power of fixed specular component and the random scattering component is
+defined as the Rician factor of the MIMO channel, i.e.,
+κ(F)
+k
+= ||Fk||2
+F
+E[||�F||2
+F]
+, and κ(G)
+k
+= ||Gk||2
+F
+E[|| �G||2
+F]
+,
+(5)
+where || · ||F denotes the Frobenius norm of a matrix.
+DRAFT
+
+8
+For the correlated MIMO channel Gk, the one-sided correlation function ηk(�C) = E[ �G†
+k �C �Gk] param-
+eterized by an Hermitian matrix �C is given by [22, Thm. 1] as
+ηk(�C) = E[ �G†
+k �C �Gk] = 1
+Lk
+SkΠk(�C)S†
+k,
+1 ≤ k ≤ K,
+(6)
+where the Lk × Lk diagonal matrix Πk(�C) contains the diagonal entries
+�
+Πk(�C)
+�
+i,i =
+R
+�
+j=1
+([Nk]j,i)2 �
+W†
+k �CWk
+�
+j,j ,
+1 ≤ i ≤ Lk.
+(7)
+The other one-sided correlation function �ηk(Ck) = E[ �GkCk �G†
+k] parameterized by Ck is given by
+�ηk(Ck) = E[ �GkCk �G†
+k] = 1
+Lk
+Wk �Πk(Ck)W†
+k,
+1 ≤ k ≤ K,
+(8)
+where the R × R diagonal matrix �Πk(Ck) contains the diagonal entries
+�
+�Πk(Ck)
+�
+i,i =
+Lk
+�
+j=1
+([Nk]i,j)2 �
+S†
+kCkSk
+�
+j,j ,
+1 ≤ i ≤ R.
+(9)
+Similarly, for 0 ≤ k ≤ K, the two parameterized one-sided correlation functions of the matrix �Fk are
+given by:
+ζk(Dk) = E[�F†
+kDk�Fk] = 1
+T VkΣk(Dk)V†
+k,
+(10)
+�ζk( �D) = E[�Fk �D�F†
+k] = 1
+T Uk �Σk( �D)U†
+k,
+(11)
+where the T × T diagonal matrix Σk(Dk) and the Lk × Lk diagonal matrix �Σk( �D) respectively contain
+the diagonal entries
+[Σk(Dk)]i,i =
+Lk
+�
+j=1
+([Mk]j,i)2 �
+U†
+kDkUk
+�
+j,j ,
+1 ≤ i ≤ T,
+(12)
+�
+�Σk( �D)
+�
+i,i =
+T
+�
+j=1
+([Mk]i,j)2 �
+V†
+k �DVk
+�
+j,j ,
+1 ≤ i ≤ Lk.
+(13)
+In addition, since the channels {Fk}0≤k≤K and {Gk}1≤k≤K are spatially separated, channels corre-
+spond to different links are assumed to be independent.
+DRAFT
+
+9
+C. Mutual Information of Multi-RIS MIMO Channel
+Due to the assumptions (A1)-(A3), the channel (1) is a Gaussian MIMO channel and its mutual
+information is given by the well-known Telatar’s formula [2] as
+I(γ) = log det
+�
+IR + γHH†�
+,
+(14)
+where γ = P/σ2 is the average SNR, and the end-to-end channel H is given by
+H = F0 +
+K
+�
+k=1
+√ρkGkFk.
+(15)
+The channel H can be factorized as the product of two matrices G and F as
+H = GF =
+�
+IR
+√ρ1G1
+. . .
+√ρKGK
+�
+�
+���������
+F0
+F1
+...
+FK
+�
+���������
+.
+(16)
+Denoting L = �K
+k=0 Lk, G =
+�
+IR
+√ρ1G1
+. . .
+√ρKGK
+�
+is a R × L block matrix and F =
+�
+FT
+0 , . . . , FT
+K
+�T is a L × T block matrices.
+Letting B = HH† = GFF†G†, the mutual information (14) can be rewritten as
+I(γ) = R VB(γ) = R
+ˆ ∞
+0
+log(1 + γt)fB(t)dt,
+(17)
+where VB(x) is the Shannon transform of the matrix B [25], and fB(t) is the probability density function
+(PDF) of the eigenvalue of B. Applying the relation between the Shannon transform and the corresponding
+Cauchy transform [25], the mutual information (17) can be rewritten as
+I(γ) = R
+ˆ γ
+0
+�1
+t + 1
+t2 GB
+�
+−1
+t
+��
+dt,
+(18)
+where GB(z) is the Cauchy transform of B and is defined as
+GB(z) =
+ˆ ∞
+0
+1
+z − tfB(t)dt = 1
+RTr ◦ E
+�
+(zI − B)−1�
+= τR
+�
+(zI − B)−1�
+.
+(19)
+Here, τR(X) is the composite function 1
+RTr◦E[X]. Note that the PDF fB(t) has an one-to-one mapping
+with the Cauchy transform GB(z) via the inverse transform
+fB(t) = − 1
+π lim
+ϵ→0 ℑ(GB(t + iϵ)),
+(20)
+DRAFT
+
+10
+where ℑ(·) denotes the imaginary part of the complex number. Therefore, the problem of finding the
+mutual information I(γ) and the PDF fB(t) are amount to finding the Cauchy transform GB(z) of
+product of matrices B = GFF†G†. In the next section, we will resort to a linearization trick and the
+operator-valued free probability theory to derive the expression of GB(z).
+III. ASYMPTOTIC EIGENVALUE DISTRIBUTION VIA OPERATOR-VALUED FREE PROBABILITY
+THEORY
+In the classic free probability theory, it is common to combine the Cauchy transform and the free
+multiplicative convolution to obtain the limiting spectral distribution of product of random matrices. For
+example, in [26], the limiting spectral distribution of the concatenated MIMO channels of the form
+� K
+�
+k=1
+Hk
+� � K
+�
+k=1
+Hk
+�†
+(21)
+is derived, where Hk is Nk × Nk−1 random matrix and has i.i.d. zero-mean entries, unitarily invariant,
+and independent of each other. Such assumptions, in the language of free probability theory, is equivalent
+to requiring freeness among families of random variables as specified below.
+Let A be a unital algebra and B ⊂ A be a unital subalgebra. For H ∈ A, a linear map EB[H] : A → B
+is a B-valued conditional expectation, if EB[b] = b for all b ∈ B, and EB[b1Hb2] = b1EB[H]b2 for all
+H ∈ A and b1, b2 ∈ B. Then, a B-valued probability space is denoted as (A, EB, B), consisting of B ⊂ A
+and the linear functional EB. In addition, let A1, . . . , AK be the subalgebras of A with B ⊂ Ak for all
+1 ≤ k ≤ K. We also let {Hk ∈ Ak, 1 ≤ k ≤ K} denote a family of operator-valued random variables,
+which are free with amalgamation over B according to the following definition.
+Definition 1. Let n be an arbitrary integer. The families of random variables {H1, . . . , HK} are free
+with amalgamation over B, if for every family of index {k1, . . . , kn} ⊂ {1, . . . , K} with k1 ̸= k2, . . . ,
+kn−1 ̸= kn, and every family of polynomials {P1, . . . , Pn} satisfying EB[Pj(Hkj)] = 0, j ∈ {1, . . . , n},
+we have EB
+��n
+j=1 Pj(Hkj)
+�
+= 0.
+In order to observe the freeness among families of random matrices {Hk, H†
+k}1≤k≤K in (21), we
+can construct the random variable Hk as Hk = HkH†
+k. Let C denote the algebra of complex random
+DRAFT
+
+11
+variables. We define Ak = MNk(C) as the algebra of Nk × Nk complex Hermitian matrices, B = C and
+the linear functional EB as EB =
+1
+Nk Tr ◦ E. Then, as specified in Definition 1, the asymptotic freeness
+among {Hk}1≤k≤K over C has been established in some of the classic free probability theory, such as
+in [28], which further enables one to apply free multiplicative convolution [26] to obtain the limiting
+spectral distribution of the concatenated MIMO channels.
+However, in the considered problem with B = GFF†G†, both G and F are non-central and with
+non-trivial spatial correlations, and thus, are not free over C in the classic free probability aspect. More
+precisely, GG† and FF† are not free with respect to the linear functional τR = 1
+RTr ◦ E. Yet, as will be
+shown in the remaining of this section, via a linearization trick, the random matrix B of interest can be
+embedded into a larger block matrix, which can be then separated as the sum of a deterministic matrix
+and a random matrix. Instead of invoking the classic freeness over C, we are able to elevate them as the
+operator-valued variables, which are shown to be asymptotically free in the operator-valued probability
+space. The limiting spectral distribution of their sum can be then obtained by using the operator-valued
+free additive convolution.
+A. Linearization Trick and Operator-Valued Probability Space
+Let n denote 2L + R + T and M = Mn(C) denote the algebra of n × n complex random matrices.
+Although the original formulation of B is in the form of product of two random matrices that are not free
+with respect to τR, we could instead construct a block matrix L ∈ M, whose operator-valued Cauchy
+transform can be properly defined and is directly related to the conventional Cauchy transform of B.
+By using the Anderson’s linearization trick as described in [30, Prop. 3.4], we can construct a block
+matrix L ∈ M as follows
+L =
+�
+��
+L(1,1)
+L(1,2)
+L(2,1)
+L(2,2)
+�
+�� =
+�
+���������
+0R×R
+0R×L
+0R×T
+G
+0L×R
+0L×L
+F
+−IL
+0T×R
+F†
+−IT
+0T×L
+G†
+−IL
+0L×T
+0L×L
+�
+���������
+,
+(22)
+DRAFT
+
+12
+where the matrix blocks
+�
+L(i,j)�
+correspond to the partitions shown on the right-hand-side (RHS) of (22).
+In addition, let us consider the sub-algebra D ⊂ M as the n×n block diagonal matrix. For each K ∈ D,
+it is defined as
+K = blkdiag
+�
+�C, D, �D, C
+�
+,
+(23)
+where �C is a R × R sub-matrix and �D is a T × T sub-matrix. The L × L block diagonal matrices C
+and D are defined as C = blkdiag {C0, . . . , CK} and D = blkdiag {D0, . . . , DK}, respectively, where
+Ck and Dk are Lk × Lk sub-matrices. In (23), all the involved sub-matrices �C, �D, {Ck}0≤k≤K, and
+{Dk}0≤k≤K are Hermitian matrices.
+For X ∈ M, we define X �C, X �D, {XCk}0≤k≤K, and {XDk}0≤k≤K as the sub-blocks of X, corre-
+sponding to the same diagonal sub-blocks �C, �D, {Ck}0≤k≤K, and {Dk}0≤k≤K in the matrix K. Then,
+we define the linear functional τD : M → D as
+τD(X) = id ◦ ED [X] ,
+(24)
+where id denotes the identity operator on a Hilbert space and the expectation ED [X] is defined as
+ED [X] =
+�
+���������
+E[X �C]
+E[XD]
+E[X �D]
+E[XC]
+�
+���������
+,
+(25)
+and E[XC] = blkdiag {E[XC0], . . . , E[XCK]}, E[XD] = blkdiag {E[XD0], . . . , E[XDK]}. Then, we
+can define an operator-valued probability space (M, τD, D). For the M-valued random variable L ∈
+(M, τD, D), its D-valued Cauchy transform is defined as
+GD
+L (Λ(z)) = id ◦ ED
+�
+(Λ(z) − L)−1�
+= τD
+�
+(Λ(z) − L)−1�
+,
+(26)
+where Λ(z) ∈ M denotes the n × n diagonal matrix as
+Λ(z) =
+�
+��
+zIR
+0R×(2L+T)
+0(2L+T)×R
+0(2L+T)×(2L+T)
+�
+�� .
+(27)
+DRAFT
+
+13
+By substituting (22) and (27) into (26) and invoking Lemma 2 in the Appendix A, we obtain
+GD
+L (Λ(z)) = id ◦ ED
+�
+��
+�
+zIR + L(1,2) �
+L(2,2)�−1 L(2,1)�−1
+−L(1,2) �
+zL(2,2) + L(2,1)L(1,2)�−1
+−
+�
+zL(2,2) + L(2,1)L(1,2)�−1 L(2,1)
+−
+�
+L(2,2) + z−1L(2,1)L(1,2)�−1
+�
+�� . (28)
+In particular, the upper-left block of (28) can be explicitly written as
+�
+zIR + L(1,2) �
+L(2,2)�−1
+L(2,1)
+�−1
+=
+�
+zIR − GFF†G†�−1
+.
+(29)
+Therefore, the Cauchy transform of B over C is related to the D-valued Cauchy transform of L as
+GB(z) = 1
+RTr
+��
+GD
+L (Λ(z))
+�(1,1)�
+,
+(30)
+where {·}(1,1) denotes the upper-left R × R matrix block.
+B. Operator-Valued Free Additive Convolution
+Let us introduce the following notations:
+G =
+�
+IR
+√ρ1G1
+. . .
+√ρKGK
+�
+,
+(31)
+�G =
+�
+0R
+√ρ1 �G1
+. . .
+√ρK �GK
+�
+,
+(32)
+F =
+�
+F
+T
+0
+. . .
+F
+T
+K
+�T
+,
+(33)
+�F =
+�
+�FT
+0
+. . .
+�FT
+K
+�T
+.
+(34)
+The linearization matrix L can be further expressed as
+L = L + �L,
+(35)
+where L and �L contain the deterministic and the random parts of L, respectively, and are given as follows:
+L =
+�
+���������
+G
+F
+−IL
+F
+†
+−IT
+G
+†
+−IL
+�
+���������
+,
+(36)
+DRAFT
+
+14
+�L =
+�
+���������
+�G
+�F
+�F†
+�G†
+�
+���������
+,
+(37)
+where we omit the all-zero matrix blocks.
+The advantage of working with L as well as its D-valued Cauchy transform GD
+L is that the elements
+of �L are monomials of �G, �G†, �F, and �F†, which are decoupled from each other. This is in contrast
+to the Cauchy transform of B over C, where the random variables are mixed together. Then, following
+similar steps as in [32], �L is shown to be an operator-valued semicircular variable and is free from the
+deterministic matrix L over D. Thus, the limiting spectral distribution of L can be determined by the
+operator-valued free additive convolution of L and �L, over the sub-algebra D, which are summarized in
+the following propositions.
+Proposition 1. The random variable �L is semicircular and is free from L over D.
+Proof: The proof of Proposition 1 is given in Appendix B.
+Due to Proposition 1, the operator-valued Cauchy transform of L in (30) can be calculated as the free
+additive convolution between �L and L, by using a subordination formula [30] as follows:
+GD
+L (Λ(z)) = GD
+L
+�
+Λ(z) − RD
+�L
+�
+GD
+L (Λ(z))
+��
+= ED
+��
+Λ(z) − RD
+�L
+�
+GD
+L (Λ(z))
+�
+− L
+�−1�
+,
+(38)
+where RD
+�L (·) denotes the operator-valued R-transform of L over D. Then, GB(z) can be determined by
+the following proposition.
+Proposition 2. The Cauchy transform of B, with z ∈ C+, is given by
+GB(z) = 1
+RTr
+��
+�Ψ(z) − GΞ(z)−1G
+†�−1�
+,
+(39)
+where
+Ξ(z) = Ψ(z) −
+�
+�Φ(z) − FΦ(z)−1F
+†�−1
+.
+(40)
+DRAFT
+
+15
+The matrix-valued function �Ψ(z), Ψ(z), �Φ(z), Φ(z) satisfy the following fixed-point equations
+�Ψ(z) = zIR −
+K
+�
+k=1
+�ηk(GCk(z)),
+(41)
+Ψ(z) = blkdiag
+�
+0R, −η1(G �C(z)), . . . , −ηK(G �C(z))
+�
+,
+(42)
+�Φ(z) = blkdiag
+�
+−�ζ0(G �D(z)), −�ζ1(G �D(z)), . . . , −�ζK(G �D(z))
+�
+,
+(43)
+Φ(z) = IT −
+K
+�
+k=0
+ζk(GDk(z)),
+(44)
+where blkdiag {A1, . . . , An} constructs a block diagonal matrix with square matrices A1, . . . , An being
+the diagonal blocks, and G �C(z), GCk(z), G �D(z), GDk(z) are given by
+G �C(z) =
+�
+�Ψ(z) − GΞ(z)−1G
+†�−1
+,
+(45)
+GCk(z) =
+��
+Ψ(z) − G
+† �Ψ(z)−1G −
+�
+�Φ(z) − FΦ(z)−1F
+†�−1�−1�
+k+1
+,
+1 ≤ k ≤ K,
+(46)
+G �D(z) =
+�
+Φ(z) − F
+†
+�
+�Φ(z) −
+�
+Ψ(z) − G
+† �Ψ(z)−1G
+�−1�−1
+F
+�−1
+,
+(47)
+GDk(z) =
+��
+�Φ(z) − FΦ(z)−1F
+† −
+�
+Ψ(z) − G
+† �Ψ(z)−1G
+�−1�−1�
+k+1
+,
+0 ≤ k ≤ K.
+(48)
+The notation {A}k+1 with n × n matrix A denotes the (k + 1)-th diagonal matrix block containing
+entries from �k−1
+i=0 Li + 1 to �k
+i=0 Li rows and columns of A.
+Proof: The proof of Proposition 2 is given in Appendix C.
+As indicated by Proposition 2, the Cauchy transform GB(z) as well as the matrix-valued functions
+�Ψ(z), Ψ(z), �Φ(z), Φ(z) can be determined by solving the fixed-point equations. The numerical value
+of GB(z) can be obtained by iterating the set of equations (41)-(44) and (45)-(48).
+IV. NUMERICAL RESULTS
+In this section, numerical simulations are conducted to study the spectral distribution of the RIS-assisted
+MIMO channel as well as its mutual information. In particular, we examine the impacts of the number of
+RIS panels, the number of antennas at the transceivers, and the Rician factor of the propagation channels
+DRAFT
+
+16
+on the mutual information. The mutual information I(γ) and the eigenvalue PDF fB(t) are calculated
+by (17) and (20), respectively, where the involved Cauchy transform GB(z) is given in Proposition 2. In
+each simulation case, the MIMO system without RIS deployment is included for comparison, i.e., K = 0,
+where the eigenvalue PDF and the Cauchy transform can be calculated by using existing result from [32,
+Thm. 2]. Each simulation curve is obtained by averaging over 106 independent channel realizations.
+In the simulations, the antenna elements of the transceivers and the reflecting elements of the RIS
+panels are arranged as the uniform planar arrays (UPAs). Denote T = T (H) × T (V ), R = R(H) × R(V ),
+and Lk = L(H)
+k
+× L(V )
+k
+, where the numbers with the superscripts H and V represent the numbers of
+elements aligned in the horizontal and vertical dimensions, respectively. The specular component of each
+channel is the line-of-sight propagation component between two uniform planar arrays (UPAs), i.e.,
+Fk = a
+�
+ϕ(F)
+k
+, ν(F)
+k
+, L(H)
+k
+, L(V )
+k
+�
+a† �
+θ(F)
+k
+, φ(F)
+k
+, T (H), T (V )�
+,
+0 ≤ k ≤ K,
+(49)
+Gk = a
+�
+ϕ(G)
+k
+, ν(G)
+k
+, R(H), R(V )�
+a† �
+θ(G)
+k
+, φ(G)
+k
+, L(H)
+k
+, L(V )
+k
+�
+,
+1 ≤ k ≤ K,
+(50)
+where θ(i)
+k
+and φ(i)
+k
+are the azimuth and elevation angles of the k-th departing UPA, while ϕ(i)
+k
+and ν(i)
+k
+are the azimuth and elevation angles of the k-th arriving UPA, i ∈ {F, G}. The function a(·) denotes
+the steering vector of an M × N UPA and is defined as
+a(α, β, M, N) =
+�
+1, . . . , eiπ(n sin(α) sin(β)+m cos(β)), . . . , eiπ((N−1) sin(α) sin(β)+(M−1) cos(β))�T
+,
+(51)
+where 0 ≤ m ≤ M − 1 and 0 ≤ n ≤ N − 1.
+Fig. 2 shows the empirical and asymptotic eigenvalue PDF of the RIS-assisted MIMO channels HH†,
+assuming the number of RIS panels is K = 0, 1, 2, and 4, respectively. In all the cases, the numbers of
+transmit and receive antennas are set to T = R = 64, and the number of reflecting elements in each RIS
+panel is set to 144. The channel statistics, such as Fk, Uk, Vk, Mk in (3), and Gk, Wk, Sk, Nk in
+(4) are randomly generated but fix for the Monte Carlo simulations. The numerical results show that the
+asymptotic PDF calculated by (20) provides an excellent approximation to the simulated PDF for all the
+considered parameter configurations. By increasing the number of deployed RIS panels, it is possible to
+increase the maximum eigenvalue, therefore, improve amplitude of the eigen-channels.
+DRAFT
+
+17
+0
+1
+2
+3
+4
+5
+0
+0.5
+1
+1.5
+PDF
+(a) K = 0
+0
+1
+2
+3
+4
+5
+6
+7
+8
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+PDF
+(b) K = 1
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+PDF
+(c) K = 2
+0
+2
+4
+6
+8
+10
+12
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+PDF
+(d) K = 4
+Fig. 2.
+Comparisons of empirical and asymptotic eigenvalue PDFs of the RIS-assisted MIMO channels HH† with different
+numbers of RIS panels. The numbers of transmit and receive antennas are set to T = R = 64, and the number of reflecting
+elements of each RIS panel is set to 144.
+In Fig. 3, we investigate the impacts of the SNR, the number of antennas, and the Rician factors
+on the mutual information of the RIS-assisted MIMO channels. Equal number of antennas is set at the
+transmitter and the receiver, where T = R = 4 in Fig. 3 (a) and T = R = 8 in Fig. 3 (b), respectively.
+The MIMO communication is assisted by K = 6 RIS panels, and each RIS panel is composed of
+16 reflecting elements. Compared to the direct link F0, the relative channel gains [ρ1, . . . , ρ6] in (15)
+corresponding to the reflected links are configured as [0.9, 0.8, 0.7, 0.5, 0.3, 0.1]. All the Rician factors
+are set equal as κ = κ(F)
+k
+= κ(G)
+k
+, where κ is set to 1, 10, or 100. In presence of non-degenerate random
+DRAFT
+
+18
+scattering components �Fk in (3) and �Gk in (4), the RIS-assisted MIMO channels are full-rank, and the
+mutual information at large SNR linearly increases as min{T, R}/10 log10(e) nats/s/Hz for every 1 dB
+SNR improvement, depicted as the dashed lines in Fig. 3. However, as the Rician factor becomes large,
+although the mutual information has the same scaling law, it requires larger SNR levels to exhibit the
+linear improvement. This is illustrated in the insets of Fig. 3. When the Rician factors are κ = 1, 10,
+and 100, the asymptotic mutual information has at least 5% deviation from the high-SNR scaling law at
+SNRs 20.8 dB, 27.5 dB, and 34.2 dB when T = R = 4, and at SNRs 23.7 dB, 29.8 dB, and 36.1 dB
+when T = R = 8, respectively. This is due to the fact that as the Rician factor increases, the random
+scattering components to maintain the rank of the channel have less contributions to the overall MIMO
+channels.
+To further investigate the impacts of the Rician factor on the mutual information of the MIMO channels,
+we plot Fig. 4 to show the mutual information as a function of κ, with the numbers of RIS panels K
+set to 0, 1, 2, and 4, respectively. The number of transmit and receive antennas are set to T = 16 and
+R = 8, while the performance of the MIMO system is evaluated at SNR γ = 10 dB. It is observed that
+when κ is less than 1, the mutual information can be improved as κ increases, while it monotonically
+decreases for κ > 1 in all the considered cases. When the number of RIS panels is larger, the mutual
+information degradation is less prominent as each RIS provides independent reflected link, which increases
+the richness of the MIMO channels.
+In Fig. 5, the impact of the numbers of RIS panels is investigated in more details, when the mutual
+information is evaluated for different transmit antennas T = 8, 16, 32, and 64. The number of receive
+antennas is fixed to R = 10, and each RIS panel has 8 reflecting elements. In this simulation setting, we
+consider the urban canyon communication scenario as depicted in Fig. 1, where the specular components
+of {Fk} channels and of {Gk} channels have relatively small angular variations. That is, in (49)
+and (50), we assume that the departing angles
+�
+θ(F)
+k
+, φ(F)
+k
+�
+0≤k≤K of the transmitter UPA and the
+arriving angles
+�
+ϕ(G)
+k
+, ν(G)
+k
+�
+1≤k≤K of the receiver UPA are uniformly and randomly generated in some
+fixed intervals having length 0.05π. The departing angles
+�
+θ(G)
+k
+, φ(G)
+k
+�
+1≤k≤K and the arriving angles
+DRAFT
+
+19
+0
+10
+20
+30
+40
+5
+10
+15
+20
+25
+30
+35
+40
+Mutual information (nats/s/Hz)
+Asymptotic MI
+High-SNR MI
+Simulation
+20
+25
+30
+35
+15
+16
+17
+18
+ = 1, 10, 100
+(a) T = R = 4
+0
+10
+20
+30
+40
+50
+0
+10
+20
+30
+40
+50
+60
+70
+80
+Mutual information (nats/s/Hz)
+Asymptotic MI
+High-SNR MI
+Simulation
+22
+24
+26
+28
+30
+32
+34
+36
+38
+26
+28
+30
+32
+34
+ = 1, 10, 100
+(b) T = R = 8
+Fig. 3.
+Mutual information of RIS-assisted MIMO channels at varying SNR γ, when the number of antennas at the transceivers
+is T = R = 4 in (a) and T = R = 8 in (b), respectively. In each case, the Rician factor of the component channels is set equal
+to κ = 1, 10, or 100. There are K = 6 deployed RIS panels, each of which has 16 reflecting elements. Insets show the 5%
+deviations of the asymptotic mutual information from the high-SNR scaling law.
+DRAFT
+
+20
+0
+2
+4
+6
+8
+10
+12
+14
+4
+6
+8
+10
+12
+14
+16
+Mutual information (nat/s/Hz)
+Fig. 4. Mutual information of RIS-assisted MIMO channel for varying Rician factor κ. The number of antennas at the transmitter
+and the receiver are T = 16 and R = 8, respectively, and each RIS panel has 8 reflecting elements. The SNR of the end-to-end
+channel is set to γ = 10 dB.
+�
+ϕ(F)
+k
+, ν(F)
+k
+�
+1≤k≤K of the RIS panels are randomly generated in some fixed intervals having length
+0.1π. As K increases, Fig. 5 shows that the mutual information first improves at a larger rate between
+0 ≤ K ≤ 5, and then becomes slower thereafter. This is due to the fact that the richness of the channels
+can be improved more efficiently when the number of reflected links is small. Since the angular ranges
+are restricted, the added RIS panels have similar reflected links that cannot provide additional richness.
+Therefore, it is less effective to deploy more RIS panels to improve the mutual information. Finally, as
+shown in Figs. 3-5, the mutual information calculated by (17) via the Cauchy transform (39) achieves a
+good agreement with the simulation in all the considered simulation cases, and thus, can be applied to
+evaluate the performance of the RIS-assisted MIMO channels.
+V. CONCLUSIONS
+This paper studies the information-theoretic data rate of the RIS-assisted MIMO systems, where
+multiple RIS panels are deployed to improve the scattering-limited MIMO channels. By using the
+DRAFT
+
+21
+0
+5
+10
+15
+6
+7
+8
+9
+10
+11
+12
+13
+Mutual information (nat/s/Hz)
+Fig. 5.
+Mutual information of RIS-assisted MIMO channel for varying numbers of RIS panels K. The number of receive
+antennas is R = 8, the number of elements in each RIS panel is 8, and the SNR of the channel is γ = 10 dB.
+operator-valued free probability theory, the Cauchy transform of the MIMO matrix is obtained using
+the general Rician MIMO model with Weichselberger’s correlation structure. Based on this result, the
+asymptotic eigenvalue distribution of the channel matrix as well as the mutual information of the MIMO
+channel are calculated, which closely match the corresponding simulation results for practical system
+configurations. Numerical results show that the additional reflected links created by the RIS panels can
+increase the range of eigenvalues of the channel matrix, which can be leveraged to improve the amplitude
+of the eigen-channels. In the MIMO communications, the negative impact of a large Rician factor on the
+mutual information can be partly alleviated by deploying more RIS panels. However, the performance
+improvement of the multi-RIS deployment slows down when the added reflected links have similar
+arriving and departing angles.
+DRAFT
+
+22
+APPENDIX A
+SOME USEFUL MATRIX INVERSION IDENTITIES
+For the sake of completeness, the following matrix inversion identities are summarized in Lemmas 1-3,
+which are repeatedly applied throughout this paper. For notational simplicity, in this appendix, we use
+italic bold symbols to define matrices, which are different from those used in the main sections.
+Lemma 1. (Woodbury matrix inversion identity [31, Eq. (0.7.4.1)].) Let A denote a m × m invertible
+matrix, D denote a k × k matrix, B and C denote m × k and k × m matrices, respectively. Then the
+following identity holds
+(A + BDC)−1 = A−1 − A−1B
+�
+D−1 + CA−1B
+�−1 CA−1.
+(52)
+Lemma 2. (2 × 2 block matrix inversion identity [31, Eq. (0.7.3.1)].) Let A, B, C, and D be defined
+as in Lemma 1, the inversion identity of the following 2 × 2 block matrix holds
+�
+��
+A
+B
+C
+D
+�
+��
+−1
+=
+�
+��
+A−1 + A−1B(D − CA−1B)−1CA−1
+−A−1B(D − CA−1B)−1
+−(D − CA−1B)−1CA−1
+(D − CA−1B)−1
+�
+��
+=
+�
+��
+(A − BD−1C)−1
+−A−1B(D − CA−1B)−1
+−(D − CA−1B)−1CA−1
+(D − CA−1B)−1
+�
+�� ,
+(53)
+where the second equality holds when D is also invertible.
+Lemma 3. (3 × 3 block matrix inversion identity.) Let the matrices E, F , G, H, J, K, L, M, and N
+be the conformable partitions of the following 3 × 3 block matrix X
+X =
+�
+������
+E
+F
+G
+H
+J
+K
+L
+M
+N
+�
+������
+.
+DRAFT
+
+23
+When E is invertible, the inversion of X is given by
+X−1 =
+�
+������
+E−1 + E−1(F A−1H + US−1V )E−1
+−E−1(F − US−1C)A−1
+−E−1US−1
+−A−1(H − BS−1V )E−1
+A−1 + A−1BS−1CA−1
+−A−1BS−1
+−S−1V E−1
+−S−1CA−1
+S−1
+�
+������
+,
+where
+A = J − HE−1F ,
+B = K − HE−1G,
+C = M − LE−1F ,
+D = N − LE−1G,
+(54)
+U = G − F A−1B,
+V = L − CA−1H,
+(55)
+S = D − CA−1B.
+(56)
+Proof: Apply Lemma 2 to the inversion of X that is partitioned into a 2 × 2 block matrix as
+X−1 =
+�
+������
+E
+F
+G
+H
+J
+K
+L
+M
+N
+�
+������
+−1
+=
+�
+��
+P
+Q
+R
+Z−1
+�
+�� ,
+(57)
+where the matrix blocks P , Q, and R are given by
+P = E−1 + E−1
+�
+F
+G
+�
+Z−1
+�
+��
+H
+L
+�
+�� E−1,
+(58)
+Q = −E−1
+�
+F
+G
+�
+Z−1,
+(59)
+R = −Z−1
+�
+��
+H
+L
+�
+�� E−1,
+(60)
+and the matrix Z is a 2 × 2 block matrix such that
+Z =
+�
+��
+J
+K
+M
+N
+�
+�� −
+�
+��
+H
+L
+�
+�� E−1
+�
+F
+G
+�
+=
+�
+��
+A
+B
+C
+D
+�
+�� ,
+(61)
+where A, B, C, and D are given in (54).
+Applying again Lemma 2 to the inversion of Z, we obtain
+Z−1 =
+�
+��
+A−1 + A−1BS−1CA−1
+−A−1BS−1
+−S−1CA−1
+S−1
+�
+�� ,
+(62)
+DRAFT
+
+24
+where S is given in (56). Substituting (62) into (58), P can be rewritten as
+P = E−1 + E−1
+�
+F A−1 − US−1CA−1
+US−1
+�
+�
+��
+H
+L
+�
+�� E−1
+= E−1 + E−1 �
+F A−1H + US−1V
+�
+E−1,
+(63)
+where U and V are defined in (55). Similarly, Q and R can be obtained as
+Q =
+�
+−E−1 �
+F − (G − F A−1B)S−1C
+�
+A−1
+−E−1 �
+G − F A−1B
+�
+S−1
+�
+=
+�
+−E−1 �
+F − US−1C
+�
+A−1
+−E−1US−1
+�
+,
+(64)
+R =
+�
+��
+−A−1HE−1 + A−1BS−1(L − CA−1H)E−1
+−S−1(L − CA−1H)E−1
+�
+�� =
+�
+��
+−A−1(H − BS−1V )E−1
+−S−1V E−1
+�
+�� .
+(65)
+Finally, substituting (62)-(65) into (57) completes the proof of Lemma 3.
+APPENDIX B
+PROOF OF PROPOSITION 1
+A random variable �L ∈ M is said to be D-valued semicircular if the free cumulant
+κD
+m(�Lb1, �Lb2, . . . , �Lbm−1, �L) = 0,
+(66)
+for all n ̸= 2, and all b1, . . . , bn−1 ∈ D. The free cumulant κD
+m is a mapping from Mm to D and we
+refer the reader to [29] for detailed explanations on this topic. The proof is followed by expanding �L
+into a sum of n × n matrices, such that
+�L =
+K
+�
+k=0
+�L(F)
+k
++
+K
+�
+k=1
+�L(G)
+k
+,
+(67)
+DRAFT
+
+25
+where the matrices �L(F)
+k
+and �L(G)
+k
+are given by
+�L(F)
+k
+=
+�
+���������
+0R×L
+�Fk
+�F†
+k
+0L×R
+�
+���������
+,
+(68)
+�L(G)
+k
+=
+�
+���������
+�Gk
+0L×T
+0T×L
+�G†
+k
+�
+���������
+,
+(69)
+where �Fk and �Gk are L × T and R × L matrices, respectively, and are given by
+�Fk =
+�
+0T×L0
+. . .
+�F†
+k
+. . .
+0T×LK
+�†
+,
+0 ≤ k ≤ K,
+(70)
+�Gk =
+�
+0R×R
+0R×L1
+. . .
+√ρk �Gk
+. . .
+0R×LK
+�
+,
+1 ≤ k ≤ K.
+(71)
+Recalling the definitions of �Fk and �Gk in (3) and (4), we have
+�L(F)
+k
+= A(F)
+k
+�X kA(F)†
+k
+,
+(72)
+�L(G)
+k
+= A(G)
+k
+�YkA(G)†
+k
+,
+(73)
+where the matrix �X k has the same structure as the block matrix �L(F)
+k
+in (68) while replacing �Fk in (70)
+with �Xk = Mk ⊙ Xk, and �Yk has the same structure as the block matrix �L(G)
+k
+in (69) while replacing
+�Gk in (71) with �Yk =
+1
+√rk Nk ⊙ Yk. The n × n matrices A(F)
+k
+and A(G)
+k
+are given by
+A(F)
+k
+=
+�
+��
+�Uk
+0(R+L)×(T+L)
+0(T+L)×(R+L)
+�Vk
+�
+�� ,
+(74)
+A(G)
+k
+=
+�
+��
+�
+Wk
+0(R+L)×(T+L)
+0(T+L)×(R+L)
+�Sk
+�
+�� ,
+(75)
+DRAFT
+
+26
+where �Uk, �Vk, �
+Wk, �Sk are deterministic diagonal block matrices and are given by
+�Uk = blkdiag(0R, 0L0, . . . , Uk, . . . , 0LK),
+(76)
+�Vk = blkdiag(Vk, 0L),
+(77)
+�
+Wk = blkdiag(Wk, 0L),
+(78)
+�Sk = blkdiag(0T , 0L0, . . . , Sk, . . . , 0LK).
+(79)
+Since { �X k}0≤k≤K, {�Yk}1≤k≤K are Wigner matrices and independent from each other, they are semi-
+circular and free over the sub-algebra Dn ⊂ M of n × n diagonal matrices. Then, following the same
+arguments as in [32, Appendix B], {�L(F)
+k
+}0≤k≤K and {�L(G)
+k
+}1≤k≤K are semicircular and free over sub-
+algebra of block diagonal matrices D. Therefore, the sum of �L(F)
+k
+and �L(G)
+k
+is also semicircular over D
+and is free from any deterministic matrix from M.
+APPENDIX C
+PROOF OF PROPOSITION 2
+Since �L is an operator-valued semicircular variable over D and �L are free from L over D, the
+limiting spectral distribution of L is a free additive convolution of the limiting spectral distributions of
+�L and L. Specifically, the operator-valued Cauchy transform GD
+L can be calculated via the subordination
+formula (38). Recall that the R-transform RD
+�L (·) is the free cumulant generating function of �L with the
+following formal power series expansion:
+RD
+�L (K) = κD
+1 ( �K) + κD
+2 (�LK, �L) + κD
+3 (�LK, �LK, �L) + · · · ,
+(80)
+where κD
+i denotes the i-th free cumulant of �L over D. In addition, since �L is semicircular over D, all
+its cumulants in (80) except κD
+2 are zero. Therefore, the R-transform RD
+�L (K) reduces to the covariance
+DRAFT
+
+27
+function of �L over D parameterized by K, i.e.,
+RD
+�L(K) = ED
+�
+�LK�L
+�
+=
+�
+���������
+�K
+k=1 �ηk(Ck)
+�ζ( �D)
+�K
+k=0 ζk(Dk)
+η(�C)
+�
+���������
+,
+(81)
+where �ζ( �D) = blkdiag
+�
+�ζ0( �D), . . . , �ζK( �D)
+�
+and η(�C) = blkdiag
+�
+0R, η1(�C), . . . , ηK(�C)
+�
+.
+Since GD
+L (Λ(z)) ∈ D, by same matrix partitioning as in (23), GD
+L (Λ(z)) is partitioned into
+GD
+L (Λ(z)) = blkdiag
+�
+G �C(z), GD(z), G �D(z), GC(z)
+�
+,
+(82)
+where GD(z) = blkdiag {GD0(z), . . . , GDK(z)} and GC(z) = blkdiag {0R, GC1(z), . . . , GCK(z)}. Note
+that the upper-left block
+�
+GD
+L (Λ(z))
+�(1,1) = G �C(z), which is then used to compute GB(z) = 1
+RTr(G �C(z)).
+By replacing K in (81) with GD
+L (Λ(z)) in (82), and substituting L and RD
+�L with (36) and (81),
+respectively, we obtain GD
+L (Λ(z)) as
+GD
+L (Λ(z)) =
+�
+���������
+G �C(z)
+GD(z)
+G �D(z)
+GC(z)
+�
+���������
+= ED
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�Ψ(z)
+0
+0
+−G
+0
+�Φ(z)
+−F
+IL
+0
+−F
+†
+Φ(z)
+0
+−G
+†
+IL
+0
+Ψ(z)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−1
+,
+(83)
+where �Ψ(z), Ψ(z), �Φ(z), and Φ(z) are given in (41)-(44). By invoking Lemma 2 to the RHS of (83)
+and taking expectation over D, the matrix-valued function G �C(z) = A−1
+1 , and GD(z), G �D(z), GC(z) are
+DRAFT
+
+28
+the diagonal blocks of the matrix A−1
+2 , where A1 and A2 are given by
+A1 = �Ψ(z) −
+�
+0
+0
+G
+�
+�
+������
+�Φ(z)
+−F
+IL
+−F
+†
+Φ(z)
+0
+IL
+0
+Ψ(z)
+�
+������
+−1 �
+������
+0
+0
+G
+†
+�
+������
+,
+(84)
+A2 =
+�
+������
+�Φ(z)
+−F
+IL
+−F
+†
+Φ(z)
+0
+IL
+0
+Ψ(z)
+�
+������
+−
+�
+������
+0
+0
+G
+†
+�
+������
+�Ψ(z)−1
+�
+0
+0
+G
+�
+.
+(85)
+Applying Lemma 3, the RHS of (84) can be further derived as
+A1 = �Ψ(z) − GS−1G
+†,
+(86)
+where S = Ξ(z) and is calculated in (56) as
+S = Ξ(z) = Ψ(z) − �Φ(z)−1 − �Φ(z)−1F
+�
+Φ(z) − F
+† �Φ(z)−1F
+�−1
+F
+† �Φ(z)−1
+= Ψ(z) −
+�
+�Φ(z) − FΦ(z)−1F
+†�−1
+.
+(87)
+The second equality of (87) is obtained by applying Lemma 1. Then, (45) is established by combining
+(86) and (87).
+The inverse of A2 can be explicitly calculated via Lemma 3, where E = �Φ(z), F = H† = −F,
+G = L = IL, J = Φ(z), K = M = 0, and N = Ψ(z) − G
+† �Ψ(z)−1G. We further let T =
+Φ(z)−F
+† �Φ(z)−1F and �T = �Φ(z)−FΦ(z)−1F
+†. Then, the matrix-valued functions GD(z), G �D(z), and
+GC(z), being the diagonal blocks of A−1
+2 , are given by
+GD(z) = �Φ(z)−1 + �Φ(z)−1FT −1F
+† �Φ(z)−1 + �T −1 �
+N − �T −1�−1 �T −1,
+(88)
+G �D(z) = T −1 + T −1F
+† �Φ(z)−1 �
+N − �T −1�−1 �Φ(z)−1F T −1,
+(89)
+GC(z) =
+�
+N −
+�
+�Φ(z)−1 + �Φ(z)−1FT −1F
+† �Φ(z)−1��−1
+.
+(90)
+Finally, applying Lemma 1 to (88)-(90), we obtain GD(z), G �D(z), and GC(z) as in (46)-(48).
+DRAFT
+
+29
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+DRAFT
+

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+Resonant triad interactions of gravity waves
+in cylindrical basins
+Matthew Durey1 and Paul A. Milewski2
+1School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, UK
+2Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
+Abstract
+We present the results of a theoretical investigation into the existence, evolution and ex-
+citation of resonant triads of nonlinear free-surface gravity waves confined to a cylinder of
+finite depth. It is well known that resonant triads are impossible for gravity waves in laterally
+unbounded domains; we demonstrate, however, that horizontal confinement of the fluid may
+induce resonant triads for particular fluid depths. For any three correlated wave modes arising
+in a cylinder of arbitrary cross-section, we prove necessary and sufficient conditions for the
+existence of a depth at which nonlinear resonance may arise, and show that the resultant crit-
+ical depth is unique. We enumerate the low-frequency triads for circular cylinders, including
+a new class of resonances between standing and counter-propagating waves, and also briefly
+discuss annular and rectangular cylinders. Upon deriving the triad amplitude equations for a
+finite-depth cylinder of arbitrary cross-section, we deduce that the triad evolution is always
+periodic, and determine parameters controlling the efficiency of energy exchange. In order to
+excite a particular triad, we explore the influence of external forcing; in this case, the triad
+evolution may be periodic, quasi-periodic, or chaotic. Finally, our results have potential im-
+plications on resonant water waves in man-made and natural basins, such as industrial-scale
+fluid tanks, harbours and bays.
+1
+Introduction
+Nonlinear resonance is a mechanism by which energy is continuously transferred between a small
+number of linear wave modes. This phenomenon, first observed in Wilton’s analysis of gravity-
+capillary wave trains [63], has been the subject of frequent investigation over the past century
+[36, 37, 38, 54, 52, 19, 15]; indeed, nonlinear resonance has since observed for wave trains in a
+growing number of dispersive wave systems, including gravity waves [48, 21, 33, 2], acoustic-gravity
+waves [27, 26], flexural-gravity waves [58], two-layer flows [1, 25, 53], and atmospheric flows [51, 50].
+Whilst the aforementioned studies typically consider nonlinear resonance for laterally unbounded
+domains, the purpose of this study is to demonstrate that energy exchange between free-surface
+gravity waves may be induced and accentuated by horizontal confinement.
+We focus our study on the collective resonance of three linear wave modes, henceforth referred to
+as a triad [5]. In laterally unbounded domains, the monotonic and concave form of the dispersion
+curve precludes the existence of resonant triads for gravity wave trains at finite depth [48, 21],
+with resonant quartets instead being the smallest possible collective resonant interaction [2, 33, 4].
+However, confinement of the fluid to a vertical cylinder results in linear wave modes that differ in
+1
+arXiv:2301.02163v1  [physics.flu-dyn]  5 Jan 2023
+
+form to sinusoidal plane waves (except for a rectangular cylinder), so the preclusion of resonant
+triads no longer applies. Indeed, our study demonstrates that, under certain conditions, resonant
+triads may arise in cylinders of arbitrary cross-section for specific values of the fluid depth. As
+resonant triads evolve over a much faster time scale than that of resonant quartets, the exchange
+of energy in gravity waves is thus more efficient under the influence of lateral confinement [40],
+with potential implications on resonant sloshing in man-made and natural basins [8].
+Prior investigations of confined resonant free-surface gravity waves have predominantly focused
+on the so-called 1:2 resonance, which arises when two of the three linear wave modes comprising
+a triad coincide. For axisymmetric standing waves in a circular cylinder, Mack [34] determined a
+condition for the existence of critical depth-to-radius ratios at which a 1:2 resonance may arise, a
+result later generalised to cylinders of arbitrary cross-section [44]. Miles [42, 43] then characterised
+the weakly nonlinear evolution of such internal resonances, demonstrating that a 1:2 resonance is
+impossible in a rectangular cylinder [42]. Although Miles’ seminal results provide an informative
+view of the weakly nonlinear dynamics, the influence of fully nonlinear effects was later assessed
+by Bryant [8] and Yang et al. [64]. For the case of a circular cylinder of finite depth, Bryant [8]
+and Yang et al. [64] both characterised new steadily propagating nonlinear waves arising in the
+vicinity of a 1:2 resonance, and Yang et al. [64] also computed nonlinear near-resonant axisymmetric
+standing waves. Finally, broader mathematical properties of water waves exhibiting O(2) symmetry
+(of which a circular cylinder is one example) were analysed by Bridges & Dias [6] and Chossat &
+Dias [12].
+Given the restrictive set of critical depths at which a 1:2 resonance may arise [8, 64], it is
+natural to explore the possibility of nonlinear resonance in cylinders whose depth departs from the
+depths that trigger a 1:2 resonance. To the best of our knowledge, the first and only such study
+was the seminal experimental investigation performed by Michel [40], who focused on resonant
+triads arising for free-surface gravity waves confined to a finite-depth circular cylinder. Notably,
+the cylinder depth in Michel’s experiment was judiciously chosen so as to isolate a specific triad.
+Michel utilised bandlimited random horizontal vibration so as to excite two members of the triad,
+whose nonlinear interaction led to the growth of the third mode. Significantly, the energy of the
+third mode was, on average, the product of the energies of the remaining two modes, thereby
+satisfying the quadratic energy exchange typical of resonant triads.
+In order to exemplify the mechanism of nonlinear resonance, Michel [40] also calculated the
+response of a child mode due to the nonlinear interaction between two parent modes (where all
+three wave modes comprise the triad). Notably, Michel’s calculation is restricted to the early stages
+of growth and to particular relative phases of the wave modes. In addition Michel considered a
+fluid of infinite depth for all but the resonance conditions, for which finite-depth corrections were
+included. In contrast, we consider general resonances in arbitrary cylinders of finite depth and
+derive equations for the triad evolution over long time-scales. We also believe some nonlinear
+contributions to the interactions were omitted from Michel’s calculation, resulting in quantitative
+differences (see §4.3).
+The goal of our study is to unify the existence and evolution of 1:2 and triadic resonances
+into a single mathematical framework, effectively characterising all triad interactions of this type.
+Based on existing theory, it is unclear how the existence of resonant triads depends on the form
+of the cylinder cross-section, and which combinations of wave modes are permissible for judicious
+choice of the fluid depth. Furthermore, the range of depths that may excite a particular triad
+is uncertain, with 1:2 resonances only excited in a very narrow window about each critical depth
+[34, 44]. Once a particular triad is excited, one anticipates that the triad evolution will be governed
+by the canonical triad equations [5, 15]; however, quantifying the triad evolution and relative energy
+2
+
+exchange requires computation of the triad coupling coefficients. Finally, it is unclear how best to
+excite triads in arbitrary cylinders, both with and without external forcing.
+We here present a relatively comprehensive characterisation of the existence, evolution and
+excitation of resonant triads for gravity waves confined to a cylinder of arbitrary cross-section
+and finite depth. In order to reduce the problem to its key components, we first truncate the
+Euler equations, recasting the fluid evolution in terms of a finite-depth Benney-Luke equation
+(§2), incorporating only the nonlinear interactions necessary for resonant triads. In §3, we prove
+necessary and sufficient conditions for there to exist a finite depth at which three linear wave
+modes may form a resonant triad. In particular, we prove that resonant triads are impossible
+for rectangular cylinders, yet there is an abundance of resonant triads for circular cylinders. We
+then use multiple-scales analysis to determine the long-time evolution of a triad in a cylinder of
+arbitrary cross-section (§4), from which we characterise the relative coupling of different triads.
+Finally, we explore the excitation of resonant triads (§5), and discuss the potential extension of
+our theoretical developments to the cases of applied forcing and two-layer flows (§6).
+2
+Formulation
+We consider the irrotational flow of an inviscid, incompressible liquid that is bounded above by
+a free surface, confined laterally by the vertical walls of a cylinder whose horizontal cross-section,
+D, is enclosed by the curve ∂D, and bounded below by a rigid horizontal plane lying a distance H
+below the undisturbed free surface; see figure 1. We consider the fluid evolution in dimensionless
+variables, taking the cylinder’s typical horizontal extent, a, as the unit of length, and
+�
+ag−1 as
+the unit of time, where g is the acceleration due to gravity. It follows that the dimensionless
+free-surface elevation, η(x, t), and velocity potential, φ(x, z, t), evolve according to the equations
+∆φ + φzz = 0
+for x ∈ D,
+−h < z < ϵη,
+(1a)
+φt + η + ϵ
+2
+�
+|∇φ|2 + φ2
+z
+�
+= 0
+for x ∈ D,
+z = ϵη,
+(1b)
+ηt + ϵ∇φ · ∇η = φz
+for x ∈ D,
+z = ϵη,
+(1c)
+n · ∇φ = 0
+for x ∈ ∂D,
+−h < z < ϵη,
+(1d)
+φz = 0
+for x ∈ D,
+z = −h,
+(1e)
+corresponding to the continuity equation, dynamic and kinematic boundary conditions, and no-
+flux through the vertical walls and horizontal base, respectively. In equation (1), the dimensionless
+parameter ϵ is proportional to the typical wave slope, h = H/a is the ratio of the fluid depth to
+the typical horizontal extent, n is a unit vector normal to the boundary ∂D, and the operators ∇
+and ∆ denote the horizontal gradient and Laplacian, respectively. Moreover, conservation of mass
+implies that the free surface satisfies
+��
+D η dA = 0 for all time. Finally, in dimensional variables,
+ax is the two-dimensional horizontal coordinate, az is the upward-pointing vertical coordinate,
+�
+ag−1t denotes time, ϵaη is the free-surface displacement, and ϵa√agφ is the velocity potential.
+We aim to develop a broad framework for understanding resonant triads in a cylinder of finite
+depth; however, care must be taken when modelling fluid-boundary interactions and determining
+the class of permissible cylinder cross-sections.
+From a modelling perspective, we employ an
+assumption generally implicit to the water-wave problem in bounded domains; specifically, we
+neglect the meniscus and dissipation arising near the vertical walls [41], thus determining that
+the free surface intersects the boundary normally, i.e. n · ∇η = 0 for x ∈ ∂D [46]. In order to
+3
+
+Figure 1: Schematic diagram of the cylindrical tank (with cross-section D and boundary ∂D)
+partially filled with liquid. The undisturbed free surface (dashed lines) lies on z = 0, a distance H
+above the rigid bottom plane (grey). The disturbed free surface is sketched in dash-dotted lines.
+maximise the generality of our investigation, we allow the cylinder cross-section, D, to be fairly
+arbitrary; however, the mathematical developments presented herein require D to be bounded
+with a piecewise-smooth boundary, thereby allowing us to utilise the spectral theorem for compact
+self-adjoint operators [29] and the divergence theorem. As most cylinders of practical interest
+consist of a piecewise-smooth boundary, this mathematical restriction fails to limit the breadth of
+our study.
+2.1
+Derivation of the Benney-Luke equation
+As our study is focused on the weakly nonlinear evolution of small-amplitude waves, we proceed
+to simplify (1) in the case 0 < ϵ ≪ 1 and h = O(1). We begin by expanding the dynamic and
+kinematic boundary conditions (equations (1b)–(1c)) about z = 0 in powers of ϵ, which, upon
+eliminating η, gives rise to the equation [2, 47]
+φtt + φz = ϵ
+�
+∂t(φtzφt) + φzzφt − ∂t(|∇φ|2) − φzφzt
+�
++ O(ϵ2)
+for
+x ∈ D,
+z = 0.
+(2)
+To reduce the fluid evolution to the dynamics arising on the linearised free surface, z = 0, we
+define the Dirichlet-to-Neumann operator, L , so that L φ|z=0 = φz|z=0. Here φ satisfies Laplace’s
+equation (1a) over the linearised domain −h < z < 0, with ∂zφ = 0 on z = −h (see equation (1e))
+and n·∇φ = 0 for x ∈ ∂D (see equation (1d)). Notably, the Dirichlet-to-Neumann operator may be
+defined in terms of its spectral representation, as detailed in §2.2. By denoting u(x, t) = φ(x, 0, t),
+we finally obtain the finite-depth Benney-Luke equation [2, 3, 47]
+utt + L u + ϵ
+�
+ut
+�
+L 2 + ∆
+�
+u + ∂
+∂t
+�
+(L u)2 + |∇u|2��
+= O(ϵ2)
+for
+x ∈ D,
+(3)
+where we have simplified the nonlinear terms in equation (2) using φzz = −∆φ and utt = −L u +
+O(ϵ).
+4
+
+H
+DThe remainder of our investigation will be focused on the evolution of resonant triads governed
+by the Benney-Luke equation (3). As resonant triads arising in confined geometries are governed
+primarily by quadratic nonlinearities, it is sufficient to neglect terms of size O(ϵ2) in equation
+(3); however, higher-order corrections to the Benney-Luke equation may be derived by following
+a similar expansion procedure [2, 47, 4].
+Although our investigation is mainly focused on the
+evolution of the velocity potential, u, one may recover the leading-order free-surface elevation from
+the dynamic boundary condition (1b), namely η = −ut + O(ϵ).
+2.2
+Spectral representation of the Dirichlet-to-Neumann operator
+The Dirichlet-to-Neumann operator, L , may be understood in terms of the discrete set of orthog-
+onal eigenfunctions of the horizontal Laplacian operator [29]. Specifically, we consider the set of
+real-valued eigenfunctions, Φn(x), satisfying
+−∆Φn = k2
+nΦn for n = 0, 1, . . . ,
+where the corresponding eigenvalues, k2
+n, are ordered so that 0 = k0 < k1 ≤ k2 ≤ . . .. Moreover,
+each eigenfunction satisfies the boundary condition n · ∇Φn = 0 on ∂D, as motivated by the no-
+flux condition (1d). Finally, the orthogonal eigenfunctions are normalised so that ⟨Φm, Φn⟩ = δmn,
+where
+⟨f, g⟩ = 1
+S
+��
+D
+fg dA
+defines an inner product for real functions f and g, S is the area of D, and δmn is the Kronecker
+delta. Notably, Φ0(x) = 1 is the constant eigenfunction, with corresponding eigenvalue k0 = 0.
+To determine the Dirichlet-to-Neumann operator for sufficiently smooth φ, we first substitute
+the series expansion φ(x, z) = �∞
+n=0 φn(z)Φn(x) into Laplace’s equation (1a), where we have
+temporally omitted the time dependence. We then solve the resulting equation for φn(z) over the
+linearised domain −h < z < 0, in conjunction with the no-flux condition on z = −h (see equation
+(1e)). It follows that ∂zφn(0) =
+ˆ
+Lnφn(0), where
+ˆ
+Ln = kn tanh(knh)
+(4)
+is the spectral multiplier of the Dirichlet-to-Neumann operator, L .
+By expressing the time-
+dependent free-surface velocity potential, u = φ|z=0, in terms of the basis expansion u(x, t) =
+�∞
+n=0 un(t)Φn(x), it follows that the Dirichlet-to-Neumann map has the spectral representation
+L u = �∞
+n=0
+ˆ
+LnunΦn.
+3
+The existence of resonant triads
+Resonant triads arise due to the exchange of energy between linear wave modes, an effect induced
+by nonlinear wave interactions. In order to define resonant triads mathematically, it is necessary
+to first determine the angular frequency associated with each linear wave mode. In the limit ϵ → 0,
+the Benney-Luke equation (3) reduces to the linear equation utt + L u = 0. By seeking a solution
+to the linearised Benney-Luke equation of the form u(x, t) = Φn(x)e−iωnt, we conclude that the
+angular frequency, ωn, satisfies ω2
+n =
+ˆ
+Ln, or the more familiar [32]
+ω2
+n = kn tanh(knh).
+(5)
+5
+
+As we will see, a crucial aspect of the following analysis is that the angular frequency depends on
+the fluid depth, i.e. ωn(h). Finally, we note that the angular frequency is larger for more oscillatory
+eigenfunctions (i.e. for larger values of kn); by analogy to the evolution of plane gravity waves, we
+refer to kn as a ‘wavenumber’ henceforth.
+We proceed by considering three linear wave modes, enumerated n1, n2 and n3, where we denote
+Ωj = ωnj,
+Kj = knj,
+and
+Ψj(x) = Φnj(x)
+for j = 1, 2, 3.
+Notably, we exclude the wavenumber k0 = 0 from consideration as the corresponding eigenmode,
+Φ0, simply reflects the invariance of the Benney-Luke equation (3) under the mapping u �→ u +
+constant; henceforth, we consider only wavenumbers Kj > 0. The three linear wave modes form a
+resonant triad if there is a critical fluid depth, hc, satisfying
+Ω1(hc) ± Ω2(hc) ± Ω3(hc) = 0,
+(6)
+where all four sign combinations are permissible (we consider Ωj > 0 without loss of generality).
+To simplify notation in the following arguments, we restrict our attention to the particular case
+Ω1(hc) + Ω2(hc) = Ω3(hc),
+(7)
+where the other three sign combinations in equation (6) may be recovered by suitable re-indexing
+of the Ωj terms. However, as we will see in §4, an additional constraint necessary for triads to
+exist is the eigenmode correlation condition,
+��
+D
+Ψ1Ψ2Ψ3 dA ̸= 0,
+(8)
+which implies that the product of any two eigenmodes is non-orthogonal to the remaining eigen-
+mode.
+3.1
+The existence of a critical depth
+We proceed to determine necessary and sufficient conditions on the wavenumbers, Kj, for there
+to exist a depth, hc, at which a resonant triad forms, where such a critical depth is unique. We
+summarise our results in terms of the following theorem.
+Theorem 1. There exists a positive and finite value of h such that Ω1 + Ω2 = Ω3 if and only if
+K1 + K2 < K3 <
+��
+K1 +
+�
+K2
+�2.
+(9)
+When this pair of inequalities is satisfied, the corresponding value of h is unique.
+We briefly sketch the proof of Theorem 1, with full details presented in appendix A.
+We
+first demonstrate that no solutions to Ω1 + Ω2 = Ω3 are possible when the bounds in equation
+(9) are violated, i.e. when K1 + K2 ≥ K3 or when √K1 + √K2 ≤ √K3. We then consider the
+case where the inequalities (9) are satisfied and determine the existence of positive roots to the
+function F(h) = (Ω1(h) + Ω2(h))/Ω3(h) − 1. In this case, we demonstrate that limh→0 F(h) < 0
+and limh→∞ F(h) > 0, from which we conclude that F(h) has at least one root (by continuity of
+F). Finally, we deduce that this root is unique by proving that F(h) is a strictly monotonically
+increasing function of h when the inequalities (9) are satisfied.
+6
+
+Two important conclusions may be deduced from Theorem 1. First, it follows from equation (9)
+that the wavenumber, K3, corresponding to the largest angular frequency, Ω3, is larger than both
+the other two wavenumbers (K1 and K2), but it cannot be arbitrarily large (as supplied by the
+upper bound). For a given pair of eigenmodes (say Ψ1 and Ψ2), we conclude that there are likely
+to be only finitely many eigenmodes that can resonate with this pair (indeed, that number might
+fairly small, or even zero). Second, when modes 1 and 2 coincide (a 1:2 resonance), one deduces
+that Ω1 = Ω2 and K1 = K2; as such, the existence bounds (9) simplify to 2K1 < K3 < 4K1, or
+2 < K3/K1 < 4 [34, 44].
+3.2
+Determining the critical depth
+Although Theorem 1 determines necessary and sufficient conditions on the wavenumbers, Kj, for
+there to be a critical depth, hc, at which a resonant triad exists, the critical depth remains to be
+determined. In general, the critical depth must be computed numerically (being the unique root
+of the nonlinear function F(h)); however, we demonstrate that useful quantitative and qualitative
+information may be obtained via asymptotic analysis. For the remainder of this section, we consider
+the rescaled wavenumbers, ξ1 = K1/K3 and ξ2 = K2/K3, and the rescaled depth, ζ = K3h; it
+remains to determine the root, ζc, of
+F(ζ) =
+�
+ξ1 tanh(ξ1ζ)
+tanh(ζ)
++
+�
+ξ2 tanh(ξ2ζ)
+tanh(ζ)
+− 1
+(10)
+when ξ1, ξ2 > 0 satisfy
+ξ1 + ξ2 < 1 <
+�
+ξ1 +
+�
+ξ2.
+(11)
+In figure 2(a), we present contours of the critical rescaled depth, ζc, in the (ξ1, ξ2)-plane, restricted
+to the region demarcated by equation (11). Consistent with the limits limζ→0 F(ζ) = ξ1 + ξ2 − 1
+and limζ→∞ F(ζ) = √ξ1+√ξ2−1, we observe that the root, ζc, tends to zero at the line ξ1+ξ2 = 1,
+and approaches infinity at the curve √ξ1 + √ξ2 = 1. Furthermore, the uniqueness of the root of
+F for given (ξ1, ξ2) is reflected in the observation that the contours of ζc do not cross. Finally, we
+note that the contours are symmetric about the line ξ1 = ξ2, which is a direct consequence of the
+invariance of F(ζ) under the mapping ξ1 ↔ ξ2 (see equation (10)).
+Although we are primarily interested in the physically relevant case for which the cylinder’s
+depth-to-width ratio, h, is of size O(1), an informative analytic result may be obtained by consid-
+ering F(ζ) in the limit ζ ≪ 1 (or K3h ≪ 1). By utilising the Taylor expansion
+�
+tanh(x) ∼ √x
+�
+1 − x2
+6 + 19
+360x4 + O
+�
+x6��
+,
+we obtain
+�
+ξ1 tanh(ξ1ζ)+
+�
+ξ2 tanh(ξ2ζ)−
+�
+tanh(ζ) ∼
+�
+ζ
+��
+ξ1 +ξ2 −1
+�
+− ζ2
+6
+�
+ξ3
+1 +ξ3
+2 −1
+�
++O(ζ4)
+�
+(12)
+for 0 < ζ ≪ 1. Whilst deriving equation (12), we have utilised the bound ξ1, ξ2 < 1 (see equation
+(11)), which additionally ensures that 0 < ξjζ ≪ 1 for j = 1, 2. We note that the left-hand side of
+equation (12) is equal to F(ζ) tanh(ζ), so ζc satisfies
+ξ1 + ξ2 − 1 − ζ2
+c
+6
+�
+ξ3
+1 + ξ3
+2 − 1
+�
+= O(ζ4
+c ),
+(13)
+7
+
+Figure 2: Contours of the rescaled critical depth, ζc = hcK3, as a function of the rescaled wavenum-
+bers, ξ1 = K1/K3 and ξ2 = K2/K3. (a) The contours computed numerically from equation (10).
+The black lines indicate the limiting cases of ζc → 0 (at ξ1+ξ2 = 1) and ζc → ∞ (at √ξ1+√ξ2 = 1).
+(b) The contours are overlaid by the leading-order approximation (equation (17); circles) and the
+higher-order correction (equation (18); diamonds) for ζc equal to 0.5, 1, 1.5 and 2.
+provided that 0 < ζc ≪ 1. By neglecting terms of size O(ζ4
+c ) in equation (13), one may then easily
+solve for ζc in terms of ξ1 and ξ2.
+Alternatively, a more succinct expression for ζc may be found by first noting that
+ξ3
+1 + ξ3
+2 = (ξ1 + ξ2)3 − 3ξ1ξ2(ξ1 + ξ2) = 1 − 3ξ1ξ2 + O(ζ2
+c ),
+(14)
+where we have utilised the leading-order approximation ξ1 + ξ2 = 1 + O(ζ2
+c ) (see equation (13)) to
+determine the second equality. Upon substituting equation (14) into equation (13), we find that
+ξ1, ξ2 and ζc are now related by the notably simpler expression
+ξ1 + ξ2 − 1 + ζ2
+c
+2 ξ1ξ2 = O(ζ4
+c ).
+(15)
+By neglecting terms of O(ζ4
+c ), the leading-order approximation for the rescaled critical depth, ζc,
+is given by
+ζc ∼
+�
+2(1 − ξ1 − ξ2)
+ξ1ξ2
+,
+(16)
+an expression valid when 0 < ζc ≪ 1 and ξ1 + ξ2 < 1 (see equation (11)). Alternatively, one may
+deduce from equation (15) that the contours of ζc satisfy the approximate form
+ξ2 ∼
+1 − ξ1
+1 + 1
+2ζ2
+c ξ1
+,
+(17)
+where the term in the denominator is responsible for the increased ‘bending’ of the contours as
+ζc becomes progressively larger (see figure 2). We note that the additional simplification afforded
+8
+
+(a)
+(b)
+hcK3 = 0.5
+ Leading-order approx.
+hcK3 = 1
+Higher-order correction
+hcK3 = 1.5
+0.8
+0.8
+hcK3 = 2
+hcK3 = 3
+hcK3 = 5
+0.6
+0.6
+K2
+hc
+hcK3 = 10
+个
+K3
+0.4
+0.4
+3
+0.2
+0.2
+8
+0
+0
+0.2
+0.4
+0.6
+0.8
+0.2
+0.4
+0.6
+0.8
+0
+1
+0
+1
+Ki/K3
+Ki/K3by equation (14) allows for a far more tractable representation of the contours relative to solving
+equation (13) directly for ξ2 given ξ1 and ζc.
+Despite being derived under the assumption 0 < ζc ≪ 1, we see in figure 2(b) that the contours
+given by equation (17) agree favorably with the numerical solution even up to ζc ≈ 1. However, it
+is readily verified from equation (16) that the asymptotic approximation of each contour crosses
+the boundary curve √ξ1 + √ξ2 = 1 at ζc = 4 (for which ξ1 = ξ2 = 1
+4), thereby demonstrating
+that the reduced asymptotic form has limited applicability (even in a qualitative sense) for slightly
+larger values of ζc. One may further improve the quantitative (and, to an extent, qualitative)
+agreement between the asymptotic analysis and numerical computation by including terms of size
+O(ζ4) in equation (12); indeed, an analogous calculation gives rise to the following higher-order
+correction to equation (15):
+ξ1 + ξ2 − 1 + ζ2
+c
+2 ξ1ξ2 + ζ4
+c
+72ξ1ξ2
+�
+ξ1ξ2 − 1
+�
+= O(ζ6
+c ).
+(18)
+Although one may then solve for ζc given ξ1 and ξ2 (or, alternatively, determine the contours of
+ζc) by truncating terms of O(ζ6
+c ) in equation (18), the resulting algebraic expressions yield little
+qualitative information. However, one may, in principle, use this reduced form as a reasonable
+initial guess for a numerical root-finding algorithm for determining the root of F(ζ), provided that
+ζc is not too large.
+3.3
+Example cavities
+Our investigation into the emergence of resonant triads has been focused, thus far, on finite-depth
+cylinders with arbitrary horizontal cross-section. However, it is convenient to understand how
+the results of Theorem 1 influence the formation (or not) of resonant triads for some specific
+cross-sections, namely rectangular, circular, and annular cylinders.
+3.3.1
+Rectangular cylinder
+It is well known that resonant triads are impossible for plane gravity waves evolving across an
+unbounded horizontal domain of finite depth [48, 21]. 1 We now utilise Theorem 1 to demonstrate
+a similar result: resonant triads are impossible for gravity waves evolving within a rectangular
+cylinder of finite depth. Our result generalises the special case of a 1:2 resonance, for which the
+impossibility of internal resonance in a rectangular cylinder was demonstrated by Miles [42].
+To proceed, we consider a rectangular cylinder with side lengths Lx and Ly. By orientating the
+Cartesian coordinate system, x = (x, y), so that the cylinder cross-section is defined by the region
+0 < x < Lx and 0 < y < Ly, the eigenmodes are of the form
+Φmn(x, y) =
+1
+Nmn
+cos(pmx) cos(qny),
+where Nmn > 0 is a normalisation constant.
+Notably, the wavenumbers pm = mπ/Lx and
+qn = nπ/Ly are chosen so that the no-flux condition is satisfied (see equation (1d)). For a triad de-
+termined by the non-negative integers mj and nj (for j = 1, 2, 3), the corresponding wavenumbers,
+Pj = pmj and Qj = qnj, must satisfy P1 + P2 = P3 and Q1 + Q2 = Q3 (under suitable reordering
+1Weak interactions are possible, however, in the shallow-water limit, Kjh → 0, for which tanh(Kjh) in the
+dispersion relation (5) is replaced by its leading-order approximation, Kjh [48, 7, 42].
+9
+
+of the subscripts) in order for the eigenmode correlation condition (8) to be satisfied. By defining
+the wave vector kj = (Pj, Qj), the conditions on Pj and Qj simplify to the single requirement
+k1 + k2 = k3, where the triangle inequality supplies that |k3| ≤ |k1| + |k2|. As the eigenvalues,
+K2
+j , of the negative Laplacian operator are related to the wave vectors via Kj = |kj|, we deduce
+that K3 ≤ K1 + K2. Owing to the violation of the left-hand bound in equation (9), we conclude
+that resonant triads cannot exist in a rectangular cylinder of finite depth.
+3.3.2
+Circular cylinder
+We consider a circular cylinder of unit radius in dimensionless variables (i.e. the dimensional radius
+is equal to a; see §2). For polar coordinates x = (r, θ), it is well known that the corresponding
+(complex-valued) eigenmodes may be expressed in the form
+Φmn(r, θ) =
+1
+Nmn
+Jm(kmnr)eimθ,
+where
+Nmn =
+��Jm(kmn)
+��
+�
+1 − m2
+k2
+mn
+(19)
+is the normalisation factor and m is the azimuthal wavenumber (an integer). Furthermore, the
+no-flux condition (1d) determines that the radial wavenumbers, denoted kmn, satisfy J′
+m(kmn) =
+0, where 0 < km1 < km2 < . . . (we exclude k00 = 0 from consideration; see §3).
+Notably,
+the eigenvalues of the negative Laplacian operator are precisely the squared wavenumbers, k2
+mn;
+consequently, the antinodes of each Bessel function play a pivotal role in determining the existence
+of resonant triads.
+Akin to the rectangular cylinder, we find that the eigenmode correlation condition imparts
+an important restriction on the combination of eigenmodes that may resonate.
+For given mj
+and nj (for j = 1, 2, 3), we denote Kj = kmjnj, Ψj = Φmjnj and Nj = Nmjnj. Although the
+correlation condition given in equation (8) is defined for real eigenmodes, a similar condition holds
+for complex-valued eigenmodes, namely
+��
+D Ψ1Ψ2Ψ∗
+3 dA ̸= 0. By considering the quantity
+��
+D
+Ψ1Ψ2Ψ∗
+3 dA =
+1
+N1N2N3
+� � 1
+0
+rJm1(K1r)Jm2(K2r)Jm3(K3r) dr
+�� � 2π
+0
+ei(m1+m2−m3)θ dθ
+�
+,
+we deduce from the azimuthal integral that a necessary condition for the correlation integral to
+be nonzero is m1 + m2 = m3 [40]. This condition thus restricts the permissible combinations of
+angular wavenumbers in a manner similar to the restriction on the permissible planar wavenumbers
+for the case of a rectangular cylinder. Unlike rectangular cylinders, however, we demonstrate that
+resonant triads are possible in a circular cylinder.
+Despite the apparent restriction of the Bessel antinodes, Kj, and summation condition on the
+azimuthal wavenumbers, mj, Theorem 1 determines that a vast array of resonant triads may be
+excited for judicious choices of the fluid depth. In table 1, we list a small number of resonant
+triads and their corresponding critical depth, hc, subject to the restrictions |mj| ≤ 3 and nj ≤ 3;
+for larger values of |mj| and nj, the corresponding wave field becomes increasingly oscillatory, to
+the extent that the effects of surface tension and dissipation may become appreciable. Moreover,
+even marginally relaxing the upper bounds on |mj| and nj vastly increases the number of resonant
+triads; indeed, the restriction |mj| ≤ 4 and nj ≤ 4 introduces 70 additional resonant triads relative
+to table 1. As the upper bounds for |mj| and nj are further increased, the typical difference between
+the various critical depths decreases. Finally, the correlation condition,
+��
+D Ψ1Ψ2Ψ∗
+3 dA ̸= 0, is
+satisfied for each triad; however, the integral is very close to zero in some cases (e.g. triad 18),
+corresponding to an elongation of the triad evolution time-scale (see §4.1).
+10
+
+No.
+m1
+m2
+m3
+n1
+n2
+n3
+K1
+K2
+K3
+hc
+��
+D Ψ1Ψ2Ψ∗
+3 dA
+1
+-3
+3
+0
+1
+1
+3
+4.201
+4.201
+10.173
+0.14591
+0.19061
+2
+-2
+2
+0
+1
+1
+2
+3.054
+3.054
+7.016
+0.17030
+0.46429
+3
+-2
+2
+0
+1
+1
+3
+3.054
+3.054
+10.173
+0.39129
+-0.03050
+4
+-2
+2
+0
+1
+2
+3
+3.054
+6.706
+10.173
+0.06331
+0.68257
+5
+-1
+1
+0
+1
+1
+1
+1.841
+1.841
+3.832
+0.15227
+1.28795
+6
+-1
+1
+0
+1
+1
+2
+1.841
+1.841
+7.016
+1.00970
+-0.02032
+7
+-1
+1
+0
+1
+2
+3
+1.841
+5.331
+10.173
+0.30197
+-0.00603
+8
+0
+0
+0
+1
+1
+3
+3.832
+3.832
+10.173
+0.19814
+0.03327
+9
+-2
+3
+1
+1
+1
+3
+3.054
+4.201
+8.536
+0.15767
+0.30704
+10
+-1
+2
+1
+1
+1
+2
+1.841
+3.054
+5.331
+0.17266
+0.85581
+11
+-1
+2
+1
+1
+1
+3
+1.841
+3.054
+8.536
+0.60375
+-0.02595
+12
+-1
+2
+1
+2
+1
+3
+5.331
+3.054
+8.536
+0.04664
+0.99088
+13
+0
+1
+1
+1
+1
+3
+3.832
+1.841
+8.536
+0.38516
+0.00542
+14
+-1
+3
+2
+1
+1
+2
+1.841
+4.201
+6.706
+0.16313
+0.64211
+15
+-1
+3
+2
+1
+1
+3
+1.841
+4.201
+9.969
+0.48152
+-0.02717
+16
+-1
+3
+2
+1
+2
+3
+1.841
+8.015
+9.969
+0.03928
+1.08903
+17
+-1
+3
+2
+2
+1
+3
+5.331
+4.201
+9.969
+0.06286
+0.66930
+18
+0
+2
+2
+1
+1
+3
+3.832
+3.054
+9.969
+0.26387
+-0.00087
+19
+1
+1
+2
+1
+1
+2
+1.841
+1.841
+6.706
+0.83138
+0.02801
+20
+1
+1
+2
+1
+2
+3
+1.841
+5.331
+9.969
+0.28691
+0.01818
+21
+0
+3
+3
+1
+1
+3
+3.832
+4.201
+11.346
+0.21395
+-0.00640
+22
+0
+3
+3
+2
+1
+3
+7.016
+4.201
+11.346
+0.02782
+1.00669
+23
+1
+2
+3
+1
+1
+2
+1.841
+3.054
+8.015
+0.50595
+0.03712
+24
+1
+2
+3
+1
+2
+3
+1.841
+6.706
+11.346
+0.23678
+0.02590
+25
+1
+2
+3
+2
+1
+3
+5.331
+3.054
+11.346
+0.19839
+0.01522
+Table 1: Combinations of the angular wavenumbers, mj, and radial mode indices, nj, that form
+a resonant triad (m1 + m2 = m3 and Ω1 + Ω2 = Ω3) at critical depth, hc, in a circular cylinder
+of unit radius. For each triad, the corresponding wavenumbers, Kj = kmjnj, satisfy (9), and the
+correlation condition,
+��
+D Ψ1Ψ2Ψ∗
+3 dA ̸= 0, is met. The list is restricted to resonant triads arising
+for |mj|, nj ≤ 3, and we consider m1 ≤ m2 and m3 ≥ 0 without loss of generality. We have omitted
+resonances that give rise to the same critical depth, but with the roles of modes 1 and 2 swapped.
+The triad numbers (left column) and shaded rows are referenced in the text.
+11
+
+At this juncture, it is informative to assess how the triads listed in table 1 relate to the
+resonances explored in prior investigations. First, triad 7 in table 1 (dark grey row) was explored
+by Michel [40] for a circular cylinder of radius 9.45 cm and an approximate fluid depth of 3 cm; it
+follows that the depth-to-radius ratio in Michel’s experiment was approximately 0.317, close to the
+value of 0.30197 reported in table 1. Furthermore, table 1 (grey rows) incorporates two well-known
+examples of a 1:2 resonance, for which modes 1 and 2 coincide: (i) the critical depth hc = 0.83138
+(triad 19) corresponds to the second-harmonic resonance with the fundamental mode [42, 43, 8, 64];
+(ii) the critical depth hc = 0.19814 (triad 8) corresponds to a standing wave composed of two
+resonant axisymmetric modes [34, 64]. Finally, triads 1, 2, 3, 5 and 6 (table 1, light grey rows)
+form an interesting class of resonant triad, for which an axisymmetric mode (m3 = 0) interacts
+with two identical counter-propagating non-axisymmetric modes (m1 = −m2 ̸= 0 and n1 = n2).
+In fact, our investigation in §5.1 demonstrates that the axisymmetric mode is the so-called pump
+mode, and may thus excite the non-axisymmetric modes, even when the initial energy in each
+non-axisymmetric mode is negligible. We draw an analogy between this novel class of resonant
+triad and the excitation of beach edge waves [18] in §6.
+We conclude our exploration of resonant triads arising in a circular cylinder by remarking
+that the fluid depth may, in some cases, be judiciously chosen so as to excite multiple triads. In
+general, the condition on the angular frequencies, Ω1 + Ω2 = Ω3 (see equation (7)), cannot be
+satisfied for two distinct triads at the same fluid depth; however, nonlinear resonance may persist
+for both triads provided that each condition on the angular frequencies is approximately satisfied
+[5, 39, 15], at the cost of weak detuning (see §4.3.1 for further details). Specifically, if triads 1 and
+2 have critical depths hc,1 and hc,2, respectively, then there is potential excitement of both triads
+when the fluid depth, h, satisfies |h−hc,j| = O(ϵ) for j = 1, 2 (where 0 < ϵ ≪ 1 is the typical wave
+slope; see §2), giving rise to the approximation Ω1 + Ω2 − Ω3 = O(ϵ) for each triad. For example,
+if 0 < hc,2 − hc,1 ≪ 1, then it may be sufficient to excite both triads at an intermediate depth,
+hc,1 ≤ h ≤ hc,2. We note, however, that the excitation of multiple triads at a single fluid depth is
+not possible when the depth discrepancy, |h− hc,j|, becomes too large (relative to the typical wave
+slope) for any of the triads under consideration.
+To demonstrate the potential for the simultaneous excitation of two triads within a circular
+cylinder of finite depth, we consider two scenarios: (i) the excitation of two triads that share a
+common wave mode; and (ii) the excitation of two triads that do not share any common wave
+modes. Heuristically, case (ii) is more common than case (i) owing to the number of similar fluid
+depths in table 1; however, case (i) will likely generate a far richer set of dynamics owing to
+the nonlinear interaction between the two triads [35, 15, 13, 11]. As an example of case (i), we
+consider triads 21 and 24 in table 1, with nearby critical depths hc,1 = 0.21395 and hc,2 = 0.23678,
+respectively. As mode (m3, n3) = (3, 3) is common to both triads, inter-triad resonance may arise
+at an intermediate depth, e.g. h = 0.225. Finally, an example of case (ii) arises for triads 8 and 25
+in table 1, with nearby critical depths hc,1 = 0.19814 and hc,2 = 0.19839. Neither of these triads
+share a common wave mode, so one would not expect the inter-triad energy exchange discussed
+in case (i). Nevertheless, one might anticipate a signature of these two triads to be visible in
+the surface evolution for an intermediate depth, e.g. h = 0.19825. The theoretical and numerical
+exploration of coupled triads in a circular cylinder will be the focus of future investigation.
+3.3.3
+Annular cylinder
+A natural variation upon a circular cylinder is an annulus of inner radius r0 ∈ (0, 1) and outer
+radius 1. By varying r0, the annulus approaches a circular cylinder as r0 → 0+, and a quasi-one-
+12
+
+0
+0.5
+1
+0
+0.5
+1
+0
+0.5
+1
+0
+2
+4
+6
+0
+0.5
+1
+0
+0.5
+1
+Figure 3: The existence and predominant characteristics of a triad in an annular cylinder with inner
+radius r0 and outer radius 1. The triad bifurcates from the critical depth hc = 0.17266 as r0 → 0
+(the limiting case of a circular cylinder), with corresponding wavenumbers presented in table 1
+(see triad 10). (a) The critical depth, hc (blue curve), with hc → ∞ as r0 → rc, where rc ≈ 0.57
+(black line). (b) The corresponding wavenumbers, Kj, all of which remain finite for r0 < rc (black
+line). (c) The normalised wavenumbers, K1/K3 and K2/K3, parametrised by increasing r0 (blue
+arrow), with the limiting case r0 → 0 denoted by the white dot. The wavenumbers leave the triad
+existence region (see Theorem 1) via the left-hand boundary (black curve) as r0 → r−
+c .
+dimensional periodic ring as r0 → 1−. Notably, resonant triads are impossible for a one-dimensional
+periodic ring, as can be shown by modifying the arguments presented for the case of a rectangular
+cylinder (see §3.3.1). Thus, one might anticipate that the existence of triads in an annular cylinder
+depends critically on the inner radius, r0. Rather than enumerating some possible triads for given
+values of r0, we instead track the corresponding critical depth, hc, for the triads identified for a
+circular cylinder (see table 1) as r0 is progressively increased from zero. Of particular interest is
+determining whether a given triad exists for all r0 < 1, or whether there is some critical inner
+radius, rc, beyond which the triad ceases to exist, with either hc → 0 or hc → ∞ as r0 → r−
+c .
+The (complex-valued) eigenmodes in an annular domain are cylinder functions of the form
+Φmn(r, θ) =
+1
+Nmn
+�
+Jm(kmnr) cos(γmnπ) + Ym(kmnr) sin(γmnπ)
+�
+eimθ,
+(20)
+where Nmn > 0 is a normalisation constant, Ym is the Bessel function of the second kind with
+order m (an integer), and γmn ∈ [0, 1] determines the weighting between the two Bessel functions.
+As shown in appendix B, the no-flux condition (see equation (1d)) on the inner and outer walls
+determines that the wavenumbers, kmn(r0), satisfy the equation
+J′
+m(kmnr0)Y′
+m(kmn) − J′
+m(kmn)Y′
+m(kmnr0) = 0.
+(21)
+A formula for the corresponding value of γmn is determined in appendix B.
+Once again, the
+wavenumbers, kmn, are ordered so that 0 < km1 < km2 < . . . (excluding k00 = 0) and satisfy
+−∆Φmn = k2
+mnΦmn. Three correlated wave modes may form a resonant triad (for a judicious
+choice of the fluid depth) provided that the corresponding wavenumbers, Kj, which depend on the
+channel width, 1 − r0, satisfy the bounds given in Theorem 1.
+Bifurcating from the limiting case of a circular cylinder, we track the critical depth (when such
+a depth exists) of different triads as r0 is progressively increased. The predominant behaviour is
+13
+
+characterised by the example presented in figure 3, for which we consider the triad whose critical
+depth is hc = 0.17266 as r0 → 0+ (see triad 10 in table 1). Given that hc is fairly small in this limit,
+one might anticipate that the triad ceases to exist with hc → 0; somewhat surprisingly, however,
+the opposite scenario arises, with hc → ∞ as r0 → r−
+c (rc ≈ 0.57 in this example). It follows,
+therefore, that the triad may persist for narrow channels only when the fluid is sufficiently deep.
+We note, however, that there exist (at least) two relatively rare transitions for increasing r0, which
+we briefly describe as follows: (i) the triad ceases to exist when hc → 0 as r0 → r−
+c , which may
+arise when bifurcating from a sufficiently shallow circular cylinder (e.g. triad 22 in table 1); and (ii)
+the triad continues to exist for all r0 < 1, with hc → 0 and Kj → ∞ as r0 → 1, yet the normalised
+depth, hcK3, remains finite, and the normalised wavenumbers, K1/K3 and K2/K3, remain within
+the triad existence region (e.g. triad 8 in table 1). Owing to the appreciable influence of viscous
+effects for relatively shallow fluids, the physical relevance of these latter two scenarios is somewhat
+nebulous, however.
+4
+The evolution of resonant triads
+Having established the existence of resonant triads, we now determine the long-time triad evolution,
+utilising the method of multiple scales. Ostensibly, the calculations necessary for determining the
+triad equations are a variation upon the pioneering work of McGoldrick [36, 37, 38] in the absence
+of surface tension. However, the confinement of the fluid to a cylinder imposes some additional
+considerations, the salient details of which we outline below. Finally, we note that an alternative
+approach to multiple scales is Whitham’s technique of averaging the system’s Lagrangian [59,
+60, 61, 62], which has the advantage of streamlining some algebraic calculations [54, 42, 43];
+nevertheless, multiple-scales analysis is sufficient for our purposes and allows for the possible
+inclusion of higher-order corrections in the asymptotic expansion [38].
+In a manner similar to §3, we consider three linear wave modes (with real-valued eigenfunctions),
+enumerated n1, n2 and n3, where we denote
+Ωj = ωnj,
+Kj = knj,
+Lj =
+ˆ
+Lnj,
+and
+Ψj(x) = Φnj(x)
+for j = 1, 2, 3.
+In contrast to §3, however, we now allow each (nonzero) angular frequency to be either negative
+or positive: the resonance condition on the angular frequencies is henceforth defined
+Ω1 + Ω2 + Ω3 = 0.
+(22)
+The modified requirement on the angular frequencies (equation (22)) is not restrictive on the
+possible triad combinations; one may recover equation (7) by mapping Ω3 �→ −Ω3, for example.
+The decision behind the summation condition on the angular frequencies is motivated by the
+cyclical symmetry of equation (22), a property that will be inherited by the resultant amplitude
+equations [54]. As a consequence, one need only derive the amplitude equation for one of the wave
+modes; the amplitude equations for the remaining two wave modes follow by cyclic permutation
+of the subscripts (1, 2, 3).
+Before embarking on the multiple-scales analysis presented in §4.1, we remark upon two caveats.
+First, we note that equation (22) corresponds to an exact resonance, for which the fluid depth, h,
+is chosen to be precisely equal to the critical depth, hc. In practice, however, there may be a small
+discrepancy between h and hc, resulting in a the sum of the angular frequencies being slightly
+offset from zero. When the frequency detuning is sufficiently weak, e.g. Ω1 + Ω2 + Ω3 = O(ϵ), one
+14
+
+may modify the following asymptotic analysis to derive a similar set of amplitude equations (see
+§4.3.1). Second, our analysis in §4.1 is not valid when two of the wave modes coincide. This case
+corresponds to a 1:2 resonance, for which the corresponding evolution equations were derived by
+Miles [42] using Whitham modulation theory (as summarised in §4.3.2).
+4.1
+Multiple-scales analysis
+In order to determine the evolution of each of the three dominant wave modes involved in an exact
+resonance, we utilise the method of multiple scales [28, 55]. Specifically, we seek a perturbation
+solution to the Benney-Luke equation (3) of the form u ∼ u0+ϵu1+O(ϵ2). The leading-order terms
+in equation (3) determine that u0 satisfies ∂ttu0 + L u0 = 0; we choose to consider a leading-order
+solution comprised only of the three triad modes (all other modes are assumed to be smaller in
+magnitude and appear at higher order), giving rise to the leading-order form
+u0(x, t, τ) =
+3
+�
+j=1
+�
+Aj(τ)Ψj(x)e−iΩjt + c.c.
+�
+.
+(23)
+In equation (23), we have introduced the slow time-scale τ = ϵt, which governs the evolution of
+each complex amplitude, Aj. As ϵ and t are both independent variables, we treat τ and t as
+independent time-scales, giving rise to the transformation of derivatives ∂t �→ ∂t + ϵ∂τ. Finally,
+c.c. denotes the complex conjugate of the preceding term, a contribution necessary for real u0.
+So as to determine coupled evolution equations for each complex amplitude, Aj, we consider
+terms of O(ϵ) in the Benney-Luke equation (3). By substituting the leading-order solution, u0,
+into the nonlinear terms and applying the triad condition for the angular frequencies (equation
+(22)), we obtain the following problem for u1:
+∂ttu1 + L u1 = −
+�
+3
+�
+j=1
+fj(x, τ)e−iΩjt + c.c.
+�
++ nonresonant terms.
+(24)
+As we will see below, each of the functions fj(x, τ) appearing on the right-hand side of equation
+(24) will play a fundamental role when determining the amplitude equations; specifically,
+f1 = −2iΩ1
+dA1
+dτ Ψ1 + iA∗
+2A∗
+3
+��
+Ω2
+�
+L2
+3 − K2
+3
+�
++ Ω3
+�
+L2
+2 − K2
+2
+�
+− 2Ω1L2L3
+�
+Ψ2Ψ3 − 2Ω1∇Ψ2 · ∇Ψ3
+�
+,
+where f2 and f3 follow upon cyclic permutation of the subscripts (1, 2, 3). Finally, we note that
+the ‘nonresonant terms’ in equation (24) are of the general form p(x, τ)eiςt, where we assume that
+the angular frequency, ς, is not equal (or close) to any of the angular frequencies, ±ωn, associated
+with linear wave modes (see §3).
+We proceed by projecting equation (24) onto each of the three wave modes, giving rise to
+differential equations of the form (for j = 1, 2, 3)
+∂ttˆu1,j + Ljˆu1,j = −
+�
+⟨Ψj, fj⟩e−iΩjt + c.c.
+�
++ nonresonant terms,
+(25)
+where ˆu1,j = ⟨Ψj, u1⟩ is the projection of u1 onto the mode Ψj. By recalling that Lj = Ω2
+j, we
+immediately see that the term in square brackets in equation (25) is itself a solution to the linear
+operator ∂tt+Ω2
+j. It follows that the solution of equation (25) comprises of particular solutions that
+15
+
+have temporal dependence te±iΩjt, leading to an ill-posed asymptotic expansion when ϵt = O(1).
+The resolution to this problem is achieved via the solubility condition ⟨Ψj, fj⟩ = 0, which suppresses
+the secular growth.
+By applying the solubility condition ⟨Ψj, fj⟩ = 0 for j = 1, 2, 3, we conclude that the com-
+plex amplitude, Aj(τ), of each wave mode, Ψj(x)e−iΩjt, evolves according to the triad system of
+canonical form [5, 15]
+dA1
+dτ = α1A∗
+2A∗
+3,
+dA2
+dτ = α2A∗
+1A∗
+3,
+dA3
+dτ = α3A∗
+1A∗
+2,
+(26)
+where
+α1 =
+1
+2Ω1
+��
+Ω2
+�
+L2
+3 − K2
+3
+�
++ Ω3
+�
+L2
+2 − K2
+2
+�
+− 2Ω1L2L3
+�
+C − 2Ω1
+�
+Ψ1, ∇Ψ2 · ∇Ψ3
+��
+,
+(27)
+while α2 and α3 follow by cyclic coefficient of the subscripts (1, 2, 3). Furthermore, the correlation
+integral, C , is defined
+C = 1
+S
+��
+D
+Ψ1Ψ2Ψ3 dA,
+(28)
+where we recall that S is the area of the cylinder cross-section (see §2.2). As the triad equations
+(26) are valid for τ = O(1) (or t = O(1/ϵ)), their dynamics yield an informative view of the
+long-time evolution of the resonant triad.
+In order to assess the influence of the triad coefficients on the triad evolution (see §4.2), we first
+simplify the algebraic form given in equation (27). As shown by Miles [42], one may simplify the
+inner product ⟨Ψ1, ∇Ψ2 · ∇Ψ3⟩ by repeated application of the divergence theorem and utilisation
+of the relationship −∆Ψj = K2
+j Ψj; it follows that
+�
+Ψ1, ∇Ψ2 · ∇Ψ3
+�
+= 1
+2
+�
+K2
+2 + K2
+3 − K2
+1
+�
+C ,
+(29)
+where C is the correlation integral defined in equation (28). We then substitute equation (29) into
+equation (27) and simplify using the relation Ω1 + Ω2 + Ω3 = 0. After some algebra, we derive the
+reduced expression
+α1 = C
+2Ω1
+�
+Ω2L2
+3 + Ω3L2
+2 − 2Ω1L2L3 +
+3
+�
+l=1
+ΩlK2
+l
+�
+,
+(30)
+where α2 and α3 follow similarly. Finally, we demonstrate in appendix C that the algebraic form
+of the triad coefficients may be further reduced to
+αj = C β
+2Ωj
+for
+j = 1, 2, 3,
+(31)
+where
+β =
+3
+�
+l=1
+ΩlK2
+l − 1
+2Ω1Ω2Ω3
+�
+Ω2
+1 + Ω2
+2 + Ω2
+3
+�
+.
+(32)
+Equations (26), (28), (31) and (32) constitute the triad equations for resonant gravity waves
+confined to a cylinder of finite depth. Although the triad equations are of canonical form [5],
+the novelty of our investigation is the computation of the coefficients, αj, whose algebraic form is
+specific to our system.
+16
+
+Figure 4: Contours of β/K2
+3 (see equation (32)) for the case Ω1, Ω2 > 0 and Ω3 < 0 (with
+Ω1 + Ω2 + Ω3 = 0), for which β < 0 (see equation (33)).
+4.2
+Properties of the triad coefficients
+The simplified form of the coefficients, αj (equation (31)), allows for some important theoretical
+observations that were obfuscated by the more complicated expressions for αj given in equations
+(27) and (30). In particular, as exactly two of the angular frequencies, Ωj, have the same sign, we
+deduce from equation (31) that the two corresponding coefficients, αj, also have the same sign,
+with the third coefficient having the opposite sign. By utilising the well-known results pertaining
+to the canonical triad equations, we conclude that all solutions to the triad equations (26) are
+periodic in time, with solutions expressible in terms of elliptic functions [1, 5, 54, 15]. Typically,
+these solutions result in an exchange of energy between the comprising modes, although there is a
+class of periodic solution that, perhaps counter-intuitively, results in zero energy exchange for all
+time [9, 10]. Moreover, it is readily verified that the leading-order energy density, E1 + E2 + E3, is
+conserved, where Ej = Ω2
+j|Aj|2, consistent with the Hamiltonian structure of the Euler equations
+[5, 15]. The reader is directed to the work of Craik [15] for a more detailed account of the various
+properties of the canonical triad equations.
+Of particular relevance to the evolution of the triad is the quantity β (see equation (32)), which,
+together with C , determines the time scale over which energy exchange arises. In particular, we
+present the form of β in figure 4 for the case Ω1, Ω2 > 0 and Ω3 < 0. As we will demonstrate
+below, β < 0 in this case; in general, the sign of β is the same as the sign of the largest (in
+magnitude) angular frequency, Ωj. Notably, |β| decreases sharply towards zero as K1 + K2 → K3,
+corresponding to the limit hc → 0. Similarly, |β| approaches zero in the limiting cases K1 ≪
+K3 or K2 ≪ K3, corresponding to one low-oscillatory wave mode interacting with two highly-
+oscillatory wave modes. Away from these limiting cases, however, |β| depends only weakly on
+the wavenumbers, Kj, suggesting that the correlation integral, C , predominantly controls the
+time-scale of the triad evolution. Finally, we observe that β is symmetric about the line K1 = K2,
+consistent with the invariance of equation (32) under the mapping K1 ↔ K2 (and hence, Ω1 ↔ Ω2).
+17
+
+0
+-0.1
+0.8
+-0.2
+0.6
+-0.3
+K2
+K3
+-0.4
+0.4
+-0.5
+0.2
+-0.6
+-0.7
+0
+0
+0.2
+0.4
+0.6
+0.8
+1
+Ki/K3We conclude this section by proving that β < 0 in the case Ω1, Ω2 > 0 and Ω3 < 0. By
+comparing the forms of equations (32) and (30), and then permuting the subscripts (1, 2, 3) �→
+(3, 1, 2), we first note that β may be equivalently expressed as
+β = Ω1L2
+2 + Ω2L2
+1 − 2Ω3L1L2 +
+3
+�
+l=1
+ΩlK2
+l ,
+or
+β = Ω1(L2
+2 + K2
+1) + Ω2(L2
+1 + K2
+2) + |Ω3|(2L1L2 − K2
+3).
+By bounding Lj = Kj tanh(Kjhc) < Kj for 0 < hc < ∞ and utilising the relation Ω1 + Ω2 = |Ω3|,
+we obtain
+β < |Ω3|
+�
+K2
+1 + K2
+2 + 2K1K2 − K2
+3
+�
+= |Ω3|
+�
+(K1 + K2)2 − K2
+3
+�
+.
+(33)
+As resonant triads exist only when K1 + K2 < K3 (see Theorem 1), we conclude that β < 0 in this
+case.
+4.3
+Summary
+To summarise our theoretical developments, the velocity potential, u, at the fluid rest level (z = 0)
+evolves according to
+u(x, t) ∼
+3
+�
+j=1
+�
+Aj(τ)Ψj(x)e−iΩjt + c.c.
+�
++ O(ϵ),
+(34)
+where the complex amplitudes, Aj(τ), evolve over the slow time-scale, τ = ϵt, according to the triad
+equations (26). In particular, the triad coefficients, αj (see equation (31)), are defined in terms
+of the correlation integral, C (equation (28)), and the coefficient β (equation (32)). Notably, we
+assume that C is nonzero; if this condition were violated then all three of the triad coefficients, αj,
+would be equal to zero, giving rise to non-interacting wave modes at leading order (contradicting
+the notion of a triad).
+Indeed, the condition C ̸= 0 is identical to the correlation condition
+detailed in equation (8), the origins of which we have now justified. Finally, the evolution of the
+free surface, η, may be recovered by recalling that η = −ut + O(ϵ): we conclude that η(x, t) has
+a similar leading-order form to u(x, t), but each complex amplitude, Aj(τ), in (34) is replaced by
+iΩjAj(τ) (see equation (37) below).
+We briefly contrast our investigation of triad interaction with the early-time calculation of
+Michel [40], who characterised the initial linear growth of a child mode induced by the nonlinear
+interaction of two parent modes (where all three modes comprise the triad).
+If modes 1 and
+2 are the parent modes and mode 3 is the child mode, then the initial linear growth may be
+deduced directly from triad equations (26) in the limit |A3| ≪ |A1| ∼ |A2|. Specifically, the initial
+variation of A1 and A2 is slow relative to that of A3, which has the approximate early-time form
+A3(τ) ≈ α3C∗
+1C∗
+2τ + C3, where Cj = Aj(0). Notably, the linear growth rate of the child mode
+depends on the corresponding triad coefficient, α3, and the product of the initial amplitudes of the
+two parent modes. However, our result for circular cylinders differs to that of Michel; we believe
+that the author neglected some important nonlinear contributions (compare Michel’s equation
+(A2) to equations (2.4) and (2.4a) of Longuet-Higgins [33]). As Michel’s experiment verified the
+scaling of the interaction only up to a proportionality constant, this discrepancy was not captured.
+18
+
+4.3.1
+The influence of weak detuning
+As discussed earlier in §4, the analysis in §§4.1 and 4.2 does not account for weak detuning of the
+angular frequencies, as might arise when the fluid depth, h, differs slightly from the critical depth,
+hc. We now briefly consider the case of weak detuning, for which equation (22) is replaced by the
+condition Ω1 +Ω2 +Ω3 = ϵσ (see §3.3.2); here ϵ is the small parameter representative of the typical
+wave slope (see §2) and σ = O(1) determines the extent of the detuning [5, 39]. By following
+a very similar multiple-scales procedure to the case σ = 0, we obtain amplitude equations that
+are now augmented by a time-dependent modulation. Specifically, each complex amplitude now
+evolves according to
+dA1
+dτ = α1A∗
+2A∗
+3eiστ,
+dA2
+dτ = α2A∗
+1A∗
+3eiστ,
+dA3
+dτ = α3A∗
+1A∗
+2eiστ,
+where each coefficient, αj, is defined in equation (31). Although detuning yields non-autonomous
+amplitude equations, autonomous equations may be derived by mapping Aj(τ) �→ Aj(τ)eiστ/3 for
+all j = 1, 2, 3 [15]. Finally, we note that the energy, E1 + E2 + E3, is not exactly conserved when
+considering the effects of detuning; instead, the energy slowly oscillates about a constant value
+[15].
+4.3.2
+The case of a 1:2 resonance
+A 1:2 resonance is a resonant triad for which two modes comprising the triad coincide. For this
+case, we define two angular frequencies, Ω1 and Ω2, so that Ω2 = 2Ω1 [42], where the connection
+to resonant triads is clear when writing Ω1 + Ω1 = Ω2. By following a very similar multiple-scales
+procedure to that outlined in §4.1, we obtain
+u(x, t) ∼
+2
+�
+j=1
+�
+Aj(τ)Ψj(x)e−iΩjt + c.c.
+�
++ O(ϵ),
+where
+dA1
+dτ = −γA∗
+1A2
+and
+dA2
+dτ = γ
+4A2
+1.
+(35)
+In particular, the evolution of the amplitude equations (35) depends on the coefficient γ = C
+�
+K2
+2 −
+K2
+1 − 3Ω4
+1
+�
+, where C = 1
+S
+��
+D Ψ2
+1Ψ2 dA is the correlation integral. Indeed, the amplitude equations
+(35) and coefficient, γ, are consistent with the results of Miles [42] when expressing the evolution
+of each complex amplitude, Aj, in polar form (with appropriate rescaling). Finally, we note that a
+weak detuning (see §4.3.1) may also be incorporated within the amplitude equations (35), thereby
+accounting for a slight mismatch between the fluid depth, h, and the corresponding critical depth,
+hc [42].
+Of particular interest is the evolution of weakly nonlinear waves steadily propagating around
+a circular cylinder of unit radius, focusing on the case where the fluid depth is precisely equal to
+the critical depth of a 1:2 resonance [64]. For the complex-valued eigenmodes defined in equation
+(19), the correlation condition,
+��
+D Ψ2
+1Ψ∗
+2 dA ̸= 0, determines that the angular wavenumbers satisfy
+m2 = 2m1 [12, 64]. By expressing the complex wave amplitudes in polar form, Aj(τ) = aj(τ)eiθj(τ)
+(for j = 1, 2), equation (35) may be recast as [42]
+da1
+dτ = −γa1a2 cos Θ,
+da2
+dτ = γ
+4a2
+1 cos Θ,
+dΘ
+dτ = 2γa2
+�
+1 − a2
+1
+8a2
+2
+�
+sin Θ,
+19
+
+where Θ(τ) = θ2(τ) − 2θ1(τ) is the time-dependent phase shift.
+Steadily propagating waves
+correspond to time-independent solutions for a1, a2 (both nonzero) and Θ, from which we deduce
+that cos Θ = 0 and a1/a2 = 2
+√
+2.
+Indeed, it is remarkable that the amplitude ratio of the
+two dominant (normalised) wave modes is independent of the angular wavenumbers, mj, the
+radial wavenumbers, Kj, and the corresponding angular frequencies, Ωj (see §3.3.2 for details).
+Furthermore, one may readily determine the relationship between the angular velocity of the
+steady wave rotation and the corresponding wave amplitude, which may then be compared to the
+numerical solution of the full Euler equations [64]. This comparison, as well as a comparison to
+steadily propagating waves computed from various truncations of the Euler equations, will be the
+subject of future investigation.
+5
+The excitation of resonant triads
+Having established the existence and evolution of resonant triads, we now focus on the excitation
+of a particular triad via external forcing. So as to motivate the method of excitation, we first
+recall (§5.1) the well-known result that one mode in the triad may, or may not, excite the other
+two modes [16, 22, 54]; in the case of excitation, the initial mode is referred to as the pump mode
+[15]. We will then utilise the criterion of the pump mode to excite all three modes in the triad via
+a pulsating pressure source (§5.2). Throughout this section, we continue with the convention that
+the triad angular frequencies satisfy Ω1 + Ω2 + Ω3 = 0, as set forth in §4.
+5.1
+Excitation via the triad pump mode
+To first identify the triad pump mode and then characterise the resultant excitation, we consider
+the case for which A3, say, is much larger in magnitude than the other two mode amplitudes, so
+|A1|, |A2| ≪ |A3| [16, 22, 54]. By linearising the triad equations (26), we obtain
+dA1
+dτ = α1A∗
+2A∗
+3,
+dA2
+dτ = α2A∗
+1A∗
+3,
+dA3
+dτ = 0,
+(36)
+from which we immediately conclude that A3 is constant (whilst the linearisation assumption
+holds); we denote A3(τ) = C for some given complex number C. By considering second derivatives
+of A1 and A2, we deduce the linearised evolution equations [15]
+d2A1
+dτ 2 = α1α2|C|2A1
+and
+d2A2
+dτ 2 = α1α2|C|2A2,
+where α1α2 = C 2β2/(4Ω1Ω2) (see equation (31)). We conclude that A1(τ) and A2(τ) grow ex-
+ponentially in time (whilst the linearisation approximation holds) when Ω1Ω2 > 0, and exhibit
+sinusoidal oscillations when Ω1Ω2 < 0 [16, 22, 15].
+Thus, mode 3 may excite modes 1 and 2
+when Ω1 and Ω2 have the same sign (and likewise for other mode permutations). As one angular
+frequency must have a different sign from the other two (so as to satisfy Ω1 + Ω2 + Ω3 = 0), we
+conclude that the mode whose angular frequency is largest in magnitude (i.e. differs in sign) is the
+triad pump mode [15]. Equivalently, the pump mode is the mode with largest wavenumber, Kj.
+To visualise the influence of the pump mode on the resultant free-surface pattern, we present the
+solution of the triad equations (26) and the corresponding pump-mode approximation (equation
+(36)) in figure 5. By recalling that the free surface satisfies η = −ut + O(ϵ), we first deduce that
+η(x, t) ∼
+3
+�
+j=1
+�
+iΩjAj(τ)Ψj(x)e−iΩjt + c.c.
+�
++ O(ϵ).
+(37)
+20
+
+Figure 5: Excitation of a triad via its pump mode for the case of a circular cylinder. We consider
+triad 24 in table 1, but with m3 �→ −m3. We choose Ω1, Ω2 > 0 and Ω3 < 0, so that mode 3
+is the pump mode. (a) Evolution of the free-surface, η ∼ −ut, over the slow time-scale, τ = ϵt,
+with ϵ = 10−3. (b) The evolution of the wave amplitudes, |Aj|, according to the triad equations
+(equation (26), solid curves) and the pump-mode approximation (equation (36), dashed-dotted
+curves). Insets: modes 1 (blue), 2 (red) and 3 (gold) at τ = 0; all three modes rotate counter-
+clockwise. The simulations were initialised from A1(0) = 0.01 and A2(0) = 0.01i, where A3(0) was
+chosen to be the positive real number satisfying E1 + E2 + E3 = 1, with Ej = Ω2
+j|Aj|2 (see §4.2).
+21
+
+T=O
+8
+T= 16
+24
+T川
+a
+(6)
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+5
+10
+15
+20
+25
+30
+=For the case of a circular cylinder, we utilise the complex-valued eigenmodes defined in equation
+(19), corresponding to the superposition of steadily propagating waves for mj ̸= 0 (the rotation di-
+rection depends on the sign of Ωj/mj). Upon initialising the system so that the energy is primarily
+within the pump mode (mode 3), modes 1 and 2 are gradually excited due to nonlinear interaction,
+with exponential growth evident for τ ≲ 10. As time further increases, the dynamics depart from
+the pump-mode approximation: the energy in the pump mode appreciably decreases, whilst the
+energy in modes 1 and 2 saturates. The free surface varies qualitatively during this evolution, with
+an appreciable change in pattern structure visible by τ = 24 (primarily a superposition of modes 1
+and 2). Notably, the system evolution is periodic, which becomes apparent over longer time scales.
+5.2
+Excitation via an applied pressure source
+Based on the ideas of the previous section, we consider a methodology for exciting the pump
+mode of a triad, which will subsequently excite the remaining two modes (provided that the initial
+disturbance of each of the remaining modes is nonzero). Notably, several methods for exciting
+internal resonances have been considered in prior investigations, primarily focusing on imposed
+motion of the fluid vessel via horizontal [42, 45] or vertical vibration [42, 44, 46, 24]. Furthermore,
+one may, in principle, utilise sinusoidal paddles or plungers to excite a particular triad’s pump
+mode for a given geometry (similar wave makers are used in rectangular wave tanks [37, 23]).
+However, for large-scale fluid tanks, imposed motion of the vessel may be impractical (if the tank
+were set in a concrete base, for example), and it may be challenging to determine the correct
+paddle motion necessary to excite a chosen pump mode for geometrically complex cylinders. We
+choose, therefore, to consider a slightly different approach: we instead excite the pump mode via
+a pulsating pressure source located just above the free surface (e.g. an air blower).
+In order to incorporate a pressure source within our mathematical framework, we first refor-
+mulate the dimensionless dynamic boundary condition (equation (1b)) as
+φt + η + ϵ
+2
+�
+|∇φ|2 + φ2
+z
+�
++ ϵP(x, t) = 0
+for
+x ∈ D,
+z = ϵη,
+where the dimensional pressure is ϵ2aρgP for fluid density ρ (P = 0 corresponds to atmospheric
+pressure). The pressure source is chosen to be small in magnitude so that the resultant wave
+excitation arises over the slow time-scale, τ = ϵt, and may thus be saturated by weakly nonlinear
+effects. By modifying the developments outlined in §2.1, we derive the forced Benney-Luke equation
+utt + L u + ϵ
+�
+ut
+�
+L 2 + ∆
+�
+u + ∂
+∂t
+�
+(L u)2 + |∇u|2�
++ ∂tP
+�
+= O(ϵ2)
+for
+x ∈ D,
+(38)
+which will be the starting point for the asymptotic analysis.
+Before proceeding further, we first describe two forms of the pressure source relevant to our
+investigation. For a stationary pressure source oscillating periodically over the fast time-scale, t,
+we express P(x, t) = f(τ)s(x)e−iΩpt + c.c., where s(x) is a fixed spatial profile (generally spanning
+the cavity), f(τ) accounts for a slow modulation in the magnitude of the pressure, and Ωp is the
+pulsation angular frequency. We choose Ωp to be close to the angular frequency of the pump
+mode, which, without loss of generality, we assume to be mode 3 (i.e. Ω3 has the opposite sign
+from Ω1 and Ω2). We denote, therefore, Ωp = Ω3 + ϵµ, where µ = O(1) determines the extent
+of the frequency mismatch. For a pressure source orbiting the centre of a circular cylinder at a
+constant angular velocity, we instead posit that P has the form P(r, θ, t) = f(τ)s(r, θ−Ωpt), where
+22
+
+0
+5
+10
+1
+2
+3
+4
+0
+20
+40
+60
+80
+100
+0
+1
+2
+3
+4
+0
+50
+100
+150
+200
+250
+300
+1
+2
+3
+4
+5
+Figure 6: Evolution of the forced triad equations (39) for σ = µ = 0 and constant f. We consider
+the same triad as figure 5, with s3f = 0.1. In all three panels, A1(0) = 0.02i and A2(0) = 0.01.
+For A3(0) = 0.01, we observe (a) the initial excitation of the triad and (b) the resultant periodic
+dynamics (the initial growth is highlighted within the grey box). (c) For A3(0) = 0.01i, the triad
+evolution is chaotic.
+Ωp = (Ω3 + ϵµ)/m3 is the angular velocity of the pressure source (assuming that the pump mode
+is non-axisymmetric, i.e. m3 ̸= 0).
+For both standing and orbiting pressure sources, we now follow a similar multiple-scales pro-
+cedure to that outlined in §4.1, starting from the forced Benney-Luke equation (38). So as to
+discount the possibility that the pressure source excites more than one mode in the triad, we
+assume that neither |Ω1| or |Ω2| are close to |Ω3|. Furthermore, we incorporate a weak detuning
+in the triad angular frequencies, denoting Ω1 + Ω2 + Ω3 = ϵσ (see §4.3.1). It follows that each
+complex amplitude, Aj(τ), evolves according to
+dA1
+dτ = α1A∗
+2A∗
+3eiστ,
+dA2
+dτ = α2A∗
+1A∗
+3eiστ,
+dA3
+dτ = α3A∗
+1A∗
+2eiστ − Ω3s3f(τ)e−iµτ,
+(39)
+where the coefficients, αj, are defined in equation (31). Notably, the pump mode may only be
+excited provided that the corresponding eigenmode is non-orthogonal to the pressure source, cor-
+responding to a nonzero projection, i.e. s3 ̸= 0, where s3 = ⟨Ψ3, s⟩. Similar equations describing
+the evolution of forced resonant triads have been explored by McEwan et al. [35] (with the inclusion
+of linear damping) and Raupp & Silva Dias [50].
+In the special case of time-independent forcing (f constant) and no frequency detuning (σ =
+µ = 0), the dynamics of the forced triad equations has been analysed by Harris et al. [20], with
+both periodic and quasi-periodic dynamics reported. We also consider this case, leaving the effects
+of detuning and variable forcing for future investigation. In this setting, when |A1|, |A2| and |A3|
+23
+
+are initially small relative to the magnitude of the forcing, |Ω3s3f|, the initial growth in A3 is
+approximately linear (see figure 6(a)). As mode 3 is the pump mode, the growth in A3 excites A1
+and A2, thus activating the triad. The conservation laws of the forced triad equations [20] result
+in a temporary diminution of mode 3, which is later augmented by the external forcing; whence
+the process repeats. In some parameter regimes, the resulting evolution of the forced triad is
+periodic in time (see figure 6(b) and Raupp & Silva Dias [50]); in contrast to the findings of Harris
+et al. [20], however, we also identify initial conditions (with all other parameters unchanged) that
+result in hitherto unidentified chaotic dynamics (see figure 6(c)). The chaotic nature of this latter
+example may be verified via estimation of the maximal Lyapunov exponent [55], which is found
+to be positive (i.e. initially adjacent trajectories diverge exponentially in phase space); however, a
+more thorough investigation of the chaotic dynamics of the forced triad equations, and the subtle
+dependence on initial conditions, will be presented elsewhere.
+6
+Discussion
+We have performed a systematic investigation into nonlinear resonant triads of free-surface gravity
+waves confined to a cylinder of finite depth; previously studied 1:2 resonances are obtained as
+special cases. A key result of our study is Theorem 1, which determines whether there exists
+a fluid depth at which three given wave modes resonate due to the nonlinear evolution of the
+fluid. Equipped with this result, we determined the long-time fluid evolution using multiple-scales
+analysis, from which we deduced that all solutions to the triad equations are periodic in time.
+Finally, we determined that a given triad may be excited via external forcing of the triad’s pump
+mode, thereby providing a mechanism for exciting a given triad in a wave tank. All our results
+are derived for cylinders of arbitrary cross-section (barring some technical assumptions; see §2),
+thus forming a broad framework for characterising nonlinear resonance of confined free-surface
+gravity waves. In particular, our theoretical developments buttress experimental observations [40]
+and demonstrate the potential generality of confinement as a mechanism for promoting nonlinear
+resonance.
+A second fundamental component of our study is the influence of the cylinder cross-section
+on the existence of resonant triads; for example, resonant triads are impossible in rectangular
+cylinders, yet abundant within circular and annular cylinders (for particular fluid depths). Of the
+vast array of resonances arising in a circular cylinder, triads consisting of an axisymmetric pump
+mode and two identical counter-propagating waves are of notable interest. This combination of
+axisymmetric and non-axisymmetric modes possesses an interesting analogy to the excitation of
+counter-propagating subharmonic beach edge waves due to a normally incident standing wave
+[18]. Specifically, the wave crests of the standing axisymmetric mode are always parallel to the
+bounding wall of the circular cylinder, and may excite steadily propagating waves that are periodic
+in the azimuthal direction. For the special case for which the amplitudes of the two counter-
+propagating modes coincide, one observes the resonant interaction of standing axisymmetric and
+non-axisymmetric waves.
+So as to gain a deeper insight into the influence of nonlinearity on resonant triads, a primary
+focus for future investigations will be the simulation of the Euler equations within a cylindrical
+domain, with consideration of various truncated systems [14, 47, 4, 57]. From a computational
+perspective, the most natural geometry to consider is a circular cylinder [49]; this geometry has
+been previously explored in the context of steadily propagating nonlinear waves in the vicinity of
+a 1:2 resonance [8, 64], but it remains to assess the efficacy of the amplitude equations (26) for
+24
+
+predicting the evolution of nonlinear triads. Indeed, exploration of the nonlinear dynamics may
+reveal additional resonant triads arising beyond the small-wave-amplitude limit explored herein.
+Of similar interest is the fluid evolution when multiple triads are excited at a single depth, with
+the potential for energy exchange via triad-triad interactions [35, 15, 13, 11]. The simulation of
+free-surface gravity waves in non-circular cylinders presents a more formidable challenge, however,
+except for cylinder cross-sections that possess a tractable eigenmode decomposition.
+A second natural avenue for future investigation is to characterise the influence of applied
+forcing on resonant triads. For example, when the fluid bath is subjected to sufficiently vigorous
+vertical vibration, Faraday waves [17, 30] may appear on the free surface; although this scenario
+has been studied in the case of a 1:2 internal resonance [44, 46, 24], resonant triads may give rise to
+the formation of more exotic free-surface patterns, particularly at fluid depths that differ from that
+of a 1:2 resonance. In a similar vein, horizontal vibration [42, 45] or a pulsating pressure source
+at the frequency of the triad’s pump mode may lead to a wealth of periodic and quasi-periodic
+dynamics, as predicted by the forced triad equations [20].
+Our study has indicated, however,
+that chaotic dynamics are also possible in some parameter regimes, and might thus be excited in
+numerical simulation or experiments. Lastly, our study has focused on flat-bottomed cylinders; it
+seems plausible, however, that submerged topography may enhance or mitigate certain resonances,
+which may be an important consideration in the design of industrial-scale fluid tanks.
+Finally, our study has focused on the special case of a liquid-air interface, for which the dynamics
+of the air are neglected within the Euler equations. It is natural, however, to extend our formulation
+to the case of two-layer flows (in the absence of surface tension), with two immiscible fluids (e.g.
+air and water) confined within a cylinder whose lid and base are both rigid. In this setting, the
+density difference across the fluid-fluid interface has a strong influence of the system dynamics; it
+seems plausible, therefore, that additional resonances may be excited in this configuration, relative
+to the liquid-air interface considered herein. Notably, the anticipated resonances would arise across
+a single interface, in contrast to the cross-interface resonances explored in previous investigations
+[1, 54, 25, 53, 56, 11].
+Finally, exploring the influence of parametric forcing [31] on resonant
+triads arising for two-layer flows opens up exciting new vistas in nonlinear resonance induced by
+confinement.
+A
+Proof of Theorem 1
+Proof. To prove Theorem 1, we first show that there are no values of h ∈ (0, ∞) satisfying Ω1+Ω2 =
+Ω3 when K1+K2 ≥ K3 or when √K1+√K2 ≤ √K3, where we recall that Ωj(h) =
+�
+Kj tanh(Kjh)
+and Kj > 0 for j = 1, 2, 3. We then prove that there exists a solution to Ω1 + Ω2 = Ω3 when
+K1 + K2 < K3 < (√K1 + √K2)2, and that this solution is unique.
+In the case K1 + K2 ≥ K3, we first define χ(K; h) =
+�
+K tanh(Kh). For fixed h > 0, we
+observe that
+χ(K3; h) ≤ χ(K1 + K2; h) < χ(K1; h) + χ(K2; h),
+where we have utilised that χ(K; h) is a positive, monotonically increasing, concave function of
+K > 0. We conclude that Ω3 < Ω1 + Ω2 for any h > 0, so there are no values of h for which
+Ω1 + Ω2 = Ω3.
+In the case √K1 + √K2 ≤ √K3, we first note that the lower bound Kj > 0 (for j = 1, 2, 3)
+implies that K1 < K3 and K2 < K3.
+Furthermore, as tanh(x) is a monotonically increasing
+function for x > 0, we conclude that tanh(Kjh) < tanh(K3h) for j = 1, 2 and all h > 0. We now
+25
+
+utilise this property to deduce that
+�
+K1 tanh(K1h) +
+�
+K2 tanh(K2h) <
+��
+K1 +
+�
+K2
+��
+tanh(K3h) ≤
+�
+K3 tanh(K3h).
+We conclude that Ω3 > Ω1 + Ω2 for any h > 0, so there are no values of h for which Ω1 + Ω2 = Ω3.
+For the remainder of the proof, we consider the case
+K1 + K2 < K3
+and
+�
+K3 <
+�
+K1 +
+�
+K2,
+(40)
+which is equivalent to the pair of inequalities given by equation (9). Indeed, we will show that there
+exists a unique value of h > 0 satisfying Ω1 + Ω2 = Ω3 in this case. Equivalently, we demonstrate
+that F(h) =
+�
+Ω1(h) + Ω2(h)
+�
+/Ω3(h) − 1 has a unique positive root, where we express
+F(h) =
+�
+ψ1(h) +
+�
+ψ2(h) − 1,
+with the positive functions ψ1 and ψ2 defined
+ψj(h) = Kj tanh(Kjh)
+K3 tanh(K3h)
+for j = 1, 2.
+In order to show the existence of a root of F(h), we first note that
+lim
+h→0 F(h) = K1 + K2
+K3
+− 1 < 0
+and
+lim
+h→∞ F(h) =
+√K1 + √K2
+√K3
+− 1 > 0,
+where we have used the limits limx→0(tanh(x)/x) = 1 and limx→∞ tanh(x) = 1, respectively,
+and implemented the inequalities given in equation (40). As F(h) is a continuous function, the
+intermediate-value theorem determines that F(h) has at least one positive root.
+To prove that such a root is unique, we demonstrate that F(h) is a strictly monotonically
+increasing function for h > 0. Specifically, we note that (for j = 1, 2)
+dψj
+dh = 2K3ψj(h)
+�Kj
+K3
+cosech(2Kjh) − cosech(2K3h)
+�
+> 0
+for 0 < Kj < K3,
+where the inequality follows from the convexity of cosech(x) for x > 0, i.e. b cosech(bx) > cosech(x)
+for 0 < b < 1 and all x > 0 (associating x = 2K3h and b = Kj/K3). As the bounds K1 < K3
+and K2 < K3 incorporate the region determined by equation (40), we deduce that F(h) is strictly
+monotonically increasing. We conclude, therefore, that the root of F(h) must be unique, thereby
+completing the proof.
+B
+Wavenumbers in an annulus
+The no-flux condition (equation (1d)) on the inner and outer radii of an annulus requires that
+∂rΦmn(r0, θ) = 0 and ∂rΦmn(1, θ) = 0 for all θ, where Φmn(r, θ) is the cylinder function defined
+in equation (20). It follows, therefore, that the corresponding wavenumber, kmn, and weighting
+factor, γmn, satisfy the equations
+J′(kmnr0) cos(γmnπ) + Y′
+m(kmnr0) sin(γmnπ) = 0,
+(41a)
+J′(kmn) cos(γmnπ) + Y′
+m(kmn) sin(γmnπ) = 0.
+(41b)
+26
+
+By rearranging equation (41), we determine the following expressions for tan(γmnπ):
+tan(γmnπ) = − J′
+m(kmnr0)
+Y′
+m(kmnr0)
+and
+tan(γmnπ) = − J′
+m(kmn)
+Y′
+m(kmn).
+(42)
+By eliminating tan(γmnπ) and rearranging, we find that kmn > 0 satisfies equation (21). Upon
+computing kmn, one may then determine γmn ∈ [0, 1] using either of the equivalent expressions for
+tan(γmnπ) given in equation (42).
+C
+Reduction of the triad coefficients
+As motivated by the form of α1 given in equation (30), we demonstrate that
+Ω2L2
+3 + Ω3L2
+2 − 2Ω1L2L3 = −1
+2Ω1Ω2Ω3
+�
+Ω2
+1 + Ω2
+2 + Ω2
+3
+�
+,
+(43)
+where we recall that Ω1 + Ω2 + Ω3 = 0 and Lj = Ω2
+j. In fact, the equality given in equation
+(43) holds under cyclic permutation of the indices (1, 2, 3) (as is necessary when defining α2 and
+α3), where we note that the right-hand side is unchanged under such permutations. We conclude
+that α2 and α3 may be simplified in a similar manner, with the right-hand side of equation (43)
+appearing as a constant term in all three coefficients (see §4.2).
+We now detail the algebraic manipulations necessary to transform the left-hand side of equation
+(43) into the right-hand side. By substituting Lj = Ω2
+j into the left-hand side of equation (43) and
+factorising, we obtain
+Ω2L2
+3 + Ω3L2
+2 − 2Ω1L2L3 = Ω4
+2Ω3 + Ω2Ω2
+3
+�
+Ω2
+3 − 2Ω1Ω2
+�
+.
+(44)
+Next, we substitute
+Ω2
+3 = (Ω2
+1 + Ω2
+2) = Ω2
+1 + 2Ω1Ω2 + Ω2
+2
+(45)
+into equation (44), yielding
+Ω2L2
+3 + Ω3L2
+2 − 2Ω1L2L3 = Ω2Ω3
+�
+Ω3
+2 + Ω3
+�
+Ω2
+1 + Ω2
+2
+��
+.
+(46)
+We proceed by substituting Ω3 = −(Ω1 + Ω2) within the square brackets in equation (46); by
+distributing and cancelling common terms, we obtain
+Ω2L2
+3 + Ω3L2
+2 − 2Ω1L2L3 = −Ω1Ω2Ω3
+�
+Ω2
+1 + Ω1Ω2 + Ω2
+2
+�
+.
+(47)
+Finally, we rearrange equation (45) to give
+Ω1Ω2 = 1
+2
+�
+Ω2
+3 − Ω2
+1 − Ω2
+2
+�
+,
+which, upon substitution into equation (47), supplies the required result (equation (43)).
+27
+
+References
+[1] F. K. Ball. Energy transfer between external and internal gravity waves. J. Fluid Mech.,
+19(3):465–478, 1964.
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+31
+

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+Bohm - de Broglie Cycles
+To my beloved Izabel
+Olivier Piguet∗
+January 30, 2023
+Abstract
+In the de Broglie-Bohm quantum theory, particles describe trajectories determined
+by the flux associated with their wave function. These trajectories are studied here for
+relativistic spin-one-half particles. Based in explicit numerical calculations for the case of
+a massless particle in dimension three space-time, it is shown that if the wave function
+is an eigenfunction of the total angular momentum, the trajectories begin as circles of
+slowly increasing radius until a transition time at which they tend to follow straight lines.
+Arrival times at some detector, as well as their probability distribution are calculated,
+too. The chosen energy and momentum parameters are of the orders of magnitude met
+in graphene’s physics.
+Keywords: Bohm - de Broglie, Quantum Mechanics, Transport properties, Graphene.
+1
+Introduction
+Since its beginning in the first decades of XXth century [3]-[9] Quantum Mechanics and its
+extension in the form of Quantum Field Theory led to an accurate description of atomic and
+subatomic phenomena, confirmed in an extraordinarily precise way by countless experiments.
+However, there is no such a broad consensus about its interpretation. Various ones are present
+in the literature, such as the Copenhagen [10], the Many-World [11], the Relational [12] or the
+de Broglie-Bohm (dBB) one. We will deal here with the latter interpretation, first proposed by
+Louis de Broglie [6] as the ”Pilot Wave Theory”, later on formulated by David Bohm [13, 14]
+as the ”Ontological Interpretation of Quantum Theory” and finally, critically defended by John
+∗Pra¸ca Graccho Cardoso, 76/504, 45015-180 Aracaju, SE, Brazil, E-mail: [email protected]
+1
+arXiv:2301.13251v1  [quant-ph]  30 Jan 2023
+
+Bell in a series of papers reproduced in [15]. This interpretation of Quantum Mechanics differs
+essentially from the largely more widespread Copenhagen interpretation by taking particle
+trajectories as elements of reality, i.e., a particle really follows a trajectory, the latter being
+defined by “guying conditions” first proposed by de Broglie. A probabilistic interpretation is
+maintained, but now in the sense of classical statistical mechanics. The trajectory followed by
+a particle is fully defined by giving boundary conditions such as, e.g., the coordinates of its
+initial position. The probability distribution of the particle following a particular trajectory
+is then given by the value of the squared modulus of the wave function taken at the initial
+time and position.
+Since this statistical distribution is equal to the “usual” (Copenhagen)
+quantum probability distribution and in particular satisfies the same conservation conditions,
+mean values of beables1 evolve identically in both interpretations of Quantum Mechanics. On
+the other side, the possibility of observable consequences of the existence of trajectories remains
+an open question. Let us however mention an experimental proposal [17, 18] on the measure
+of the times of arrival at a detector for a particle prepared in some initial state.
+The first aim of the present paper is to introduce the reader to the dBB theory by treat-
+ing simple physical examples. Since the main peculiarity of this interpretation is the factual
+existence of trajectories, some effort will be made in the study of their trajectories, looking for
+situations in which dBB could make a difference.
+The dBB trajectories we will calculate are those of a relativistic2, massive or massless, spin
+one half particle in dimension 3 space-time. Its quantum state will be supposed to be described
+by an eigenfunction of the total angular momentum J defined relatively to some space point3.
+Stationary as well as packets of stationary wave functions will be considered. The main result
+for the latter is that the dBB trajectories consist of an initial phase of quasi circles whose
+radius increases till a critical time at which the trajectory begins tending to a straight line –
+the straight line expected for a classical particle with definite angular momentum.
+The paper begins in Section 2 with an introduction to the dBB formalism and in Section
+3 for its application to the relativistic spin one half particle described by the Dirac equation.
+Numerical computations of dBB trajectories and arrival times for electrons in the context of
+graphene’s transport properties are presented in Section 3.3. Final considerations are given in
+the Conclusion Section.
+Most analytic and numerical calculations are made with the help of the software Mathe-
+matica [22].
+1Following John Bell [16], I use here the term ”beable“ , instead of the usual term “observable” which is
+somewhat related to the Copenhagen interpretation and to its postulate of the “wave function reduction”.
+2Only the theory of a single particle is considered here. See [2, 19, 20] for a discussion of the N-particle
+relativistic case.
+3See [21] for the calculation of such states in the case of the non-relativistic free particle.
+2
+
+2
+Summary of the de Broglie-Bohm theory
+In the ”usual” (i.e., Copenhagen) interpretation of Quantum Mechanics [10], the state of a
+physical system constituted of a single particle is described, in the Schr¨odinger picture, by a
+wave equation
+iℏ∂ψ(x, t)
+∂t
+= ˆHψ(x, t),
+(2.1)
+where ˆH is a self-adjoint partial derivative operator acting on a N-components wave function
+ψ(x, t) =
+�
+�
+ψ1(x, t)
+· · ·
+ψN(x, t)
+�
+� ,
+(2.2)
+belonging to a Hilbert space, with scalar product and norm defined by4
+⟨ψ|Φ⟩ =
+�
+Rdddx ψ†(x, t)φ(x, t) =
+�
+Rdddx
+N
+�
+α=1
+ψ∗
+αφα(x, t),
+||ψ|| = ⟨ψ|ψ⟩
+1
+2 .
+(2.3)
+x = (xi, i = 1, · · · , d) are the space coordinates, d the space dimension and †, ∗ denote the
+hermitian and complex conjugate, respectively. The number of components, N, depends on the
+particle’s spin and on the space dimension d. E.g., N = 1 for a scalar (spin 0) particle, N = 2
+for a non-relativistic particle of spin 1/2 in any dimension, N = 2 for a relativistic particle of
+spin 1/2 in 2 dimensions, N = 4 for the same in 3 dimensions, etc [23].
+The wave equation (2.1) implies the existence of a non-negative density
+ρ(x, t) = ψ†(x, t)ψ(x, t),
+(2.4)
+and an associate d-current density5 j(x, t) obeying a continuity equation
+∂tρ + ∇j = 0,
+(2.5)
+which assures the constancy of the norm defined in (2.3). Normalizing the norm to 1, one
+interprets ρ as a probability density and j as a probability flux.
+In the Copenhagen theory, the state of the system is completely characterized by the wave
+function ψ, solution of the wave equation (2.1), with an arbitrarily given initial wave function
+ψ(x, 0) = ψ0(x). The de Broglie-Bohm (dBB) theory completes the characterization of the
+state of the system by postulating the existence of a trajectory of the system (the particle,
+here), determined by the de Broglie “guidance conditions” [6] for the particle’s velocity
+v(x, t) = j(x, t)/ρ(x, t),
+(2.6)
+4This holds e.g., for a non-relativistic particle in general, or a spin 1/2 relativistic one. We do not consider
+here cases such as the relativistic spin zero particle described by the Klein-Gordon equation.
+5Its explicit form will be given below for the cases studied there.
+3
+
+the dBB trajectory xB(x0, t) being then a solution of the set of differential of equations
+∂xB(x0, t)
+∂t
+= v(xB(x0, t), t),
+(2.7)
+with an arbitrarily given initial position xB(x0, 0) = x0. In other word, the possible trajectories
+are the integral lines of the dBB vector field v(x, t), labelled by their initial position x0. Since
+the flux j and the density ρ turn out to be bilinear in ψ and ψ† (see later on), the dBB
+trajectories do not depend on the “intensity” of the wave function, but only on its “form”.
+Recall that, in the quantum probabilistic interpretation of Copenhagen, the density ρ repre-
+sents the probability of experimentally finding the particle at a given place at a given time. On
+the other hand, the dBB theory treats this density in a more “classical statistical mechanics”
+way: Trajectories “really happen”, and their probability distribution, which amounts to the
+probability distribution of their initial positions x0, is given [24] by the density ρ(x0, 0). The
+latter represents our lack of knowledge of the precise initial position. Thanks to the continuity
+equation (2.5), the probability for the particle being inside a co-moving volume6 V (t) is con-
+stant in time, which sustains this statistical interpretation, called “thermal equilibrium” in the
+literature [24].
+Such a formulation is called a theory with “hidden variables”, the hidden variables of the
+present one being, e.g., the components of the initial position x0.
+A heuristic justification for the guiding condition (2.6) may be found in analogy with fluid
+mechanics, considering ρ and v as the fluid mass density and velocity field, respectively. With
+j = ρv, Eq. (2.5) has the form of the continuity equation expressing the conservation of the
+fluid mass.
+Now, granted the existence of a trajectory, one can define properties of the particle along
+it, such as energy, spin, etc. , in the following way. First, given an “observable” A represented
+by a self-adjoint operator ˆA, one defines the associated beable field (See footnote 1)
+A(x, t) = ψ†(x, t) ˆA ψ(x, t)/ρ(x, t),
+(2.8)
+where ρ is the probability density (2.4). One then defines the instantaneous value of A, i.e., its
+value when the particle is at point xB(x0, t) at time t:
+AB(x0, t) = A(xB(x0, t), t).
+(2.9)
+Note that these definitions are independent of the normalization of the wave function. Their
+justification is obvious in the case of ˆA being the multiplication by a function fA(x, t), since
+in this case A(x, t) = fA(x, t). For the general case, including that of a matricidal and/or
+differential operator, one derives from (2.8) the identity
+�
+ddx ρ(x, t)A(x, t) =
+�
+ddx ψ†(x, t) ˆA ψ(x, t) = ⟨A⟩ (t).
+(2.10)
+6I.e., a volume whose boundary points move along the dBB trajectories.
+4
+
+The first integral is an expectation value in the dBB theory sense, i.e., in the “classical statisti-
+cal mechanical” sense, whereas the second one gives it in the conventional quantum mechanical
+sense. Their equality indeed shows the equivalence of both theories for what concerns expecta-
+tion values of observables.
+To summarize, the dBB theory proposes the existence of:
+1. A wave function or ”guidance field” (2.2), solution of the Schr¨odinger-like wave equation
+(2.1).
+2. A statistical ensemble of particle’s trajectories xB(x0, t) as the integral lines of the dBB
+vector field (2.6), solutions of the differential equations (2.7) parametrized by the value
+of the initial position.
+3. Beable fields and beable instantaneous values, defined by (2.8) and (2.9). Such beables
+may be energy, momentum, angular momentum, spin, etc.
+All considerations made in this subsection generalize easily to systems of N particles, x
+denoting a point of the d × N dimensional configuration space7.
+3
+Free relativistic spin one half particle in 3-dimensional
+space-time
+3.1
+Dirac equation
+A free, spin 1/2 relativistic particle of mass m in 3-dimensional space-time is described in the
+usual theory by a 2-components spinor wave function
+ψ(x) =
+� ψ1(x)
+ψ2(x)
+�
+,
+(3.1)
+solution of the free Dirac equation
+iℏcγµ∂µψ(x) − mc2ψ(x) = 0,
+(3.2)
+where x = (xµ, µ = 0, 1, 2) are the space-time coordinates space-time, whose metric is ηµν =
+diag(1, −1, −1). The Dirac matrices obey the anticommutation rules {γµ, γν} = 2ηµν. Our
+choice for them is given in Appendix A. c, ℏ are the speed of light and the reduced Planck
+constant, respectively. The Dirac equation may be cast in the form8 of (2.1):
+iℏ∂ψ(x, t)
+∂t
+= ˆHψ(x, t),
+(3.3)
+7At least in the non-relativistic case. See footnote 2.
+8In fact Dirac’s original form, reduced to 3 space-time dimensions.
+5
+
+with
+ˆH = −iℏc αi∂i + mc2 β,
+(3.4)
+with αi = γ0γi and β = γ0 (see Appendix A). The density and the flux obeying the continuity
+equation (2.5) are given by
+ρ = ψ†ψ,
+ji = c ψ†αiψ.
+(3.5)
+The theory is relativistic: cρ and ji are the time and space components of the space-time
+3-vector jµ = ¯ψγµψ, and the continuity equation reads ∂µjµ = 0.
+Another relativistic object is the scalar density
+σ(x, t) = 1
+2
+¯ψψ = 1
+2ψ†βψ.
+(3.6)
+jµ and σ fulfil the identity
+jµjµ = 4σ2,
+(3.7)
+consequence of the Pauli matrices identity
+3
+�
+i=1
+σi
+αβσi
+γδ = 2δαδδβγ − δαβδγδ.
+(3.8)
+Let us go now to the dBB theory. The identity (3.7) allows one to define the time-like 3-velocity
+field’
+uµ = jµ
+2|σ|,
+uµuµ = 1,
+u0 > 0.
+(3.9)
+The relativistic form of the dBB guidance equation (2.7) then reads9
+dxµ(λ)
+dλ
+= uµ(x(λ)),
+(3.10)
+with λ as curve’s parameter. This equation is equivalent to (2.7) and defines the space-time
+trajectories of the particle.
+In the same way as one defines the dBB velocity field (2.6) or (3.9), one can define a dBB
+spin field
+S(x, t) = ℏ σ(x, t)/ρ(x, t),
+(3.11)
+which takes values between −ℏ/2 and ℏ/2. The beable S(x, t) can thus be interpreted as the
+field associated to the spin operator
+ˆS = ℏ
+2σz,
+(3.12)
+according to the definition (2.8). Note that, in the instantaneous rest frame of the particle
+guided by the wave, where vi = ji = 0, the identity (3.7) implies S = ±ℏ/2, in agreement with
+the interpretation of S as the intrinsic angular momentum of a spin one half particle. One then
+gets the instantaneous spin of the particle according to (2.9):
+SB(x0, t) = S(xB(x0, t), t).
+(3.13)
+9The present discussion is the reduction to 3-dimensional space-time of the one made by [1, 2] in 4 dimensions.
+6
+
+3.2
+Eigenstates of the angular momentum and of the energy
+We will look for the solutions of the Dirac equation which are eigenfunctions of the total angular
+momentum and Hamiltonian operators. It will be useful to work in polar coordinates r, φ:
+x = r cos φ,
+y = r sin φ.
+In these coordinates, the Hamiltonian operator (3.4) reads
+ˆH = −iℏc
+��
+α1 cos φ + α2 sin φ
+�
+∂r + 1
+r
+�
+α2 cos φ − α1 sin φ
+�
+∂φ
+�
++ mc2β.
+(3.14)
+One easily checks, using the algebra of the Pauli matrices, that the total angular momentum
+operator with respect to the origin, which has a single component in the two-dimensional space’s
+case,
+ˆJ = −iℏ∂φ + ℏ
+2β,
+(3.15)
+commutes with the Hamiltonian operator. Spinor eigenfunctions of ˆJ with eigenvalue j are
+readily found to be of the form
+ψ(r, φ, t) =
+� ei(j− 1
+2 )φf1(r, t),
+ei(j+ 1
+2 )φf2(r, t)
+�
+.
+(3.16)
+One directly sees that the uniformity of the wave function requires j to be half-integer. With
+the result (3.16), solving the Dirac equation (3.2) amounts to solving the two radial equations
+for the functions f1(r, t) and f2(r, t):
+i��
+�
+r (∂tf1 + c ∂rf2) + c(j + 1
+2) f2
+�
+− mc2r f1 = 0,
+iℏ
+�
+r (∂tf2 + c ∂rf1) − c(j − 1
+2) f1
+�
++ mc2r f2 = 0.
+(3.17)
+In terms of the radial functions fα, the probability density ρ, the flux j and the scalar density
+σ defined by (3.5), (3.6) read:
+ρ(r, t) = |f1(r, t)|2 + |f2(r, t)|2,
+jx(r, φ, t) = c
+�
+eiφf ∗
+1(r, t)f2(r, t) + e−iφf ∗
+2(r, t)f1(r, t)
+�
+,
+jy(r, φ, t) = ic
+�
+−eiφf ∗
+1f2(r, t) + e−iφf ∗
+2(r, t)f1(r, t)
+�
+,
+σ(r, t) = 1
+2 (|f1(r, t)|2 − |f2(r, t)|2) .
+(3.18)
+ρ and σ, as well as the radial and azimuthal components of the flux j,
+jr(r, t) = cos φ jx(r, φ, t) + sin φ jy(r, φ, t) = c (f ∗
+1(r, t)f2(r, t) + f ∗
+2(r, t)f1(r, t)) ,
+jφ(r, t) = − sin φ jx(r, φ, t) + cos φ jy(r, φ, t) = ic (−f ∗
+1(r, t)f2(r, t) + f ∗
+2(r, t)f1(r, t)) ,
+(3.19)
+turn out to be independent of the angular coordinate.
+7
+
+3.2.1
+Instantaneous values of energy, spin and orbital angular momentum
+Before going to our main task, i.e., the concrete study of the dBB trajectories, let us look at
+the expressions of the instant values of energy, spin and orbital angular momentum, namely
+EB(x0, t), SB(x0, t) and LB(x0, t) along a dBB trajectory xB(x0, t) fixed by the initial position
+x0. They are generated by the corresponding fields E(x, t), S(x, t) and L(x, t) according to
+(2.9). One obtains
+E(x, t) = ψ†(x, t) ˆHψ(x, t)/ρ(x, t) = iℏ(f ∗
+1(r, t)∂tf1(r, t) + f ∗
+2(r, t)∂tf2(r, t))/ρ(r, t),
+EB(x0, , t) = E(xB(x0, t), t),
+(3.20)
+where f1, f2 are the radial components defined by (3.16)
+and ˆH is the Hamilton operator, the validity of the Schr¨odinger-like equation (2.1) being
+assumed;
+S(x, t) = ψ†(x, t) ˆSψ(x, t)/ρ(x, t) = 1
+2(f ∗
+1(r, t)f1(r, t) − f ∗
+2(r, t)f2(r, t))/ρ(r, t),
+SB(x0, t) = S(xB(x0, t), t),
+(3.21)
+where ˆS is the spin operator (3.12);
+L(x, t) = ψ†(x, t)ˆLψ(x, t)/ρ(x, t) = ℏ(j − S(x, t)),
+LB(x0, t) = L(xB(x0, t), t) = ℏ(j − S(x0, t)),
+(3.22)
+where ˆL is the orbital angular momentum operator (in Cartesian and polar coordinates)
+ˆL = −i (x∂y − y∂x) = −iℏ ∂φ.
+(3.23)
+Use has been made in (3.22) of the fact that ˆL = ˆJ − ˆS and of the wave function being an
+eigenfunction of the total angular momentum ˆJ with eigenvalue j.
+3.2.2
+Stationary solutions:
+Since the Hamiltonian (3.14) and the total angular momentum (3.15) commute, we can impose
+the stationarity condition
+ˆHψ = ℏωp ψ.
+(3.24)
+Thus the radial wave functions fα(r, t) take the form
+f p, stat
+α
+(r, t) = e−iωpt hp, stat
+α
+(r),
+α = 1, 2,
+(3.25)
+ℏωp being the energy of the stationary state. This leads to a pair of equations for the function
+hp, stat
+α
+(r), derived from (3.17) by substituting i∂t by ωp. The general solution of these equa-
+tions is a superposition of the Bessel functions of the first and second kind, Jj±1/2(pr/ℏ) and
+Yj±1/2(pr/ℏ), with p a function of ωp defined as the positive solution for p of
+ωp = 1
+ℏ
+�
+m2c4 + p2c2.
+(3.26)
+8
+
+Square integrability of the wave function at r = 0 leads us to discard the solutions involving
+Yj±1/2 because of the latter’s singularity at the origin (see (B.3) in Appendix B). The regular
+solution thus is
+hp, stat
+1
+(r) = icp Jj−1/2(pr/ℏ),
+hp, stat
+2
+(r) = −(ℏωp − mc2)Jj+1/2(pr/ℏ).
+(3.27)
+The general stationary solution of the Dirac equation for angular momentum eigenstates, non-
+singular at the origin, then reads
+ψp, stat(r, φ, t) =
+�
+� ei(j− 1
+2 )φf p, stat
+1
+(r, t),
+ei(j+ 1
+2 )φf p, stat
+2
+(r, t)
+�
+�
+= e−iωpt
+�
+icp ei(j−1/2)φJj−1/2(pr/ℏ),
+−(ℏωp − mc2)ei(j+1/2)φJj+1/2(pr/ℏ)
+�
+,
+(3.28)
+with ωp given by (3.26). These spinors form a basis for the solutions of the Dirac equation,
+however an improper one since they are not square integrable due to the asymptotic behaviour
+of the Bessel functions shown in (B.4) of Appendix B.
+We can nevertheless apply the dBB guidance principle, expressed in Eqs, (2.6) and (2.7),
+to such a basis element. From the result (3.28) we can compute explicitly the density ρ given
+in (3.18):
+ρp, stat(r) = c2p2J2
+j−1/2(pr/ℏ) + (ℏωp − mc2)2J2
+j+1/2(pr/ℏ),
+(3.29)
+as well as the flux components (3.19) which, divided through ρ according to the dBB condition,
+yields the radial and azimuthal components of the velocity vector field:
+vp, stat
+r
+(r) = 0
+vp, stat
+φ
+(r) = 2c
+cp(ℏωp − mc2)Jj−1/2(pr/ℏ)Jj+1/2(pr/ℏ)
+(cp)2J2
+j−1/2(pr/ℏ) + (ℏωp − mc2)2J2
+j+1/2(pr/ℏ).
+(3.30)
+Obviously all these expressions are time independent due to the stationarity condition. One sees
+that the radial component of the velocity field is vanishing and that its azimuthal component
+does not depend on the polar angle. Thus the dBB trajectories of the particle, defined as the
+integral curves of the velocity vector field, are circles of radius r centred at the origin travelled
+at a constant velocity vφ and whose radius dependent value is bounded by c, in the massive
+as well as in the massless case. In all cases the bound c is effectively reached, for a discrete
+set of values of the radius. Fig. 1 in Subsection 3.3 shows a typical behaviour of the azimuthal
+velocity vp, stat
+φ
+(r) in the massless particle’s case.
+One may observe that, in the massless case, in which ℏωp = pc, changing the sign of the
+total angular momentum implies a change of sign of the velocity field: vp, stat
+φ
+(r) → −vp, stat
+φ
+(r),
+hence of the orientation of the trajectories. However this does not hold for the massive case,
+and also not for the more general wave packets examined in Subsection (3.2.3).
+9
+
+3.2.3
+Gaussian wave packet
+The general solution of the Dirac equation for angular momentum eigenstates, non-singular at
+the origin, reads
+ψ(x, t) =
+� ∞
+0
+dp a(p)ψp, stat(x, t),
+(3.31)
+where ψp, stat is the stationary solution (3.28) of energy10 ℏωp given by (3.26), and the amplitude
+a(p) is an arbitrary complex function, but constrained by the requirement of ψ(x, t) to be a
+square integrable function of x.
+The angular momentum eigenstates we will consider in the following are described by the
+spinor wave functions of the form (3.31), with a(p) the Gaussian amplitude
+a(p) = √p e
+−(p − p0)2
+2Σ2
+.
+(3.32)
+Here and in the rest of this paper we consider only positive energy solutions of the Dirac
+equations.
+3.2.4
+Mean values
+We recall that mean values of observables obtained from the dBB or from the Copenhagen
+theory coincide.
+The wave functions (3.31) are normalizable and we can calculate the square norm ||ψ||2 in
+the following way:
+∥ψ∥2 =
+� 2π
+0
+dφ
+� ∞
+0
+dr r ρ(r, t),
+with ρ(r, t) the density (3.5). From (3.31) we get
+∥ψ∥2 = 2π
+� ∞
+0
+dr r
+� ∞
+0
+dp
+� ∞
+0
+dp′ a(p)a(p′)
+2
+�
+α=1
+f p, stat
+α
+(r, t)∗f p′, stat
+α
+(r, t)
+= 2π
+� ∞
+0
+dp
+� ∞
+0
+dp′ a(p)a(p′)ei(ωp−ωp′)
+� ∞
+0
+dr r
+�
+(mc2 + ℏωp)(mc2 + ℏωp′)Jj−1/2(pr/ℏ)Jj−1/2(pr′/ℏ) + ℏ2pp′Jj+1/2(pr/ℏ)Jj+1/2(pr′/ℏ)
+�
+.
+From (3.32) and the completeness identity [25] for the Bessel functions:
+� ∞
+0
+dr r Jl(kr)Jl(k′r) = 1
+kδ(k − k′),
+one gets
+∥ψ∥2 = 2πℏ2
+� ∞
+0
+dp 1
+pa2(p)
+�
+c2p2 + (ℏωp − mc2)2�
+.
+(3.33)
+10We restrict to positive energy contributions.
+10
+
+In the same way one establishes expressions for the mean energy:
+⟨E⟩ =
+� 2π
+0
+dφ
+� ∞
+0
+dr r ψ†(r, φ, t) ˆH ψ(r, φ, t) / ∥ψ∥2,
+where ˆH is the Hamiltonian operator (3.4), and for the standard energy deviation
+∆E =
+�
+⟨E2⟩ − ⟨E⟩2.
+The result is
+⟨E⟩ = 2πℏ2
+∥ψ∥2
+� ∞
+0
+dp 1
+pa2(p)
+�
+c2p2 + (ℏωp − mc2)2�
+ℏωp,
+∆E =
+�2πℏ2
+∥ψ∥2
+� ∞
+0
+dp 1
+pa2(p)
+�
+c2p2 + (ℏωp − mc2)2�
+(ℏωp)2 − ⟨E⟩2
+� 1
+2
+.
+(3.34)
+Finally, a computation of the mean value of the spin operator (3.12) yields
+⟨S⟩ = πℏ3
+∥ψ∥2
+� ∞
+0
+dp 1
+pa2(p)
+�
+−c2p2 + (ℏωp − mc2)2�
+.
+(3.35)
+Substituting ∥ψ∥2 in the denominator by its expression (3.33), one sees that this mean value
+obeys the bounds −ℏ/2 < ⟨S⟩ < ℏ/2.
+All these integrals can be computed analytically in the massless case m = 0 for the amplitude
+given by (3.32). One gets
+∥ψ∥2 = πℏ2c2Σ3 �√π(1 + 2z2)(1 + erf(z)) + 2ze−z2�
+,
+⟨E⟩ = cp0
+√πz(3 + 2z2) (1 + erf(z)) + 2(1 + z2)e−z2
+z(√π(1 + 2z2)(1 + erf(z)) + 2ze−z2)
+,
+(3.36)
+∆E = cp0
+�
+π(3 + 4z4)(1 + erf(z))2 + 8√πz(−1 + z2)(1 + erf(z))e−z2 + 4(−2 + z2)e−2z2
+2z2(π(1 + 2z2)2(1 + erf(z))2 + 4√πz(1 + 2z2)(1 + erf(z))e−z2 + 4z2e−2z2)
+(3.37)
+where we have set
+z = p0
+Σ ,
+(3.38)
+and erf(z) is the error function [26].
+Finally, the mean spin is null in this massless case:
+⟨S⟩ = 0,
+(3.39)
+11
+
+which implies ⟨L⟩ = ℏj for the orbital angular momentum since the states considered are
+eigenstates of the total angular momentum with eigenvalue ℏj. Associated to this mean orbital
+angular momentum, we can define an L-radius
+rL = ⟨L⟩
+⟨p⟩ = c ℏ j
+⟨E⟩ ,
+(3.40)
+where ⟨p⟩ is the mean value of the momentum p, equal to ⟨E⟩ /c in the present massless
+case. This definition mimics the classical relation between angular momentum and radius for
+a uniform circular motion.
+As one may expect, in the case of a very narrow width of the amplitude (3.32), i.e., z ≫ 1,
+the energy and the energy uncertainty approximate to the values
+⟨E⟩ ≃ cp0,
+∆E ≃
+c
+√
+2Σ.
+(3.41)
+The results (3.34) and (3.35) allow us to identify the expression
+˜ρ(p) = a2(p)c2p2 + (ℏωp − mc2)2
+p2
+(3.42)
+up to a due normalization, as the probability density in p-space, conjugate to the x-space
+probability defined in (3.5).
+3.3
+j - electrons in graphene
+Let us denote free electrons in eigenstates of the total angular momentum with eigenvalue j as
+“j-electrons”. Free electrons in monolayer graphene [27] with energy less than Ecrit ≈ 160 meV
+obey approximatively a relativistic-like massless dispersion law E ≈ p c, where p is the linear
+momentum and the “velocity of light”11 c ≈ 106 m s−1. The dynamics of these electron is that
+of free massless particles in dimension three space-time with pseudo-spin 1/2 obeying the Dirac
+equation (3.2) with m = 0 [28]. Pseudo-spin is a chirality effect due to the peculiar crystalline
+structure of graphene and should not be confused with the usual spin. Nevertheless, since the
+wave function obeys the Dirac equation (3.2), the pseudo-spin adds itself to the orbital angular
+momentum yielding the conserved total angular momentum j (3.15) as explained in Subsection
+3.2.
+We will consider first stationary states and then states defined by Gaussian-like wave pack-
+ets. Conventions and units used in the following are described in Appendix A.
+3.3.1
+Stationary states in graphene
+As already shown in Subsection 3.2.2, the dBB trajectories associated to the stationary wave
+functions (3.28) are circles centred at the origin, travelled at a constant dBB velocity vp, stat
+φ
+11This is the so-called critical velocity, which we will denote by c.
+12
+
+20
+40
+60
+80
+100
+r
+1000
+2000
+3000
+ρp,stat (r)
+20
+40
+60
+80
+100
+r
+-1.0
+-0.5
+0.5
+1.0
+vϕ
+p,stat (r)/c
+Figure 1: Stationary case: density ρp, stat(r) and azimuthal velocity vp, stat
+φ
+(r)/c for j = 5/2,
+p = 10−4 meV/c (E = 100 meV).
+j
+1/2
+3/2
+5/2
+7/2
+9/2
+11/2
+13/2
+15/2
+17/2
+αj
+0
+2.19
+3.45
+4.61
+5.74
+6.84
+7.93
+9.01
+10.08
+Table 1: Coefficients αj of Eq. (3.45) in function of the angular momentum j.
+which depends on the radius r according to (3.30). For m = 0, this velocity reads
+vp, stat
+φ
+(r) = 2c Jj−1/2(pr/ℏ)Jj+1/2(pr/ℏ)
+J2
+j−1/2(pr/ℏ) + J2
+j+1/2(pr/ℏ).
+(3.43)
+The dBB velocity value oscillates between −c and c. A typical behaviour, in function of the
+radius, of the probability density and of the dBB velocity, which depend on the energy E = c p
+and on the angular momentum j, is shown in Fig. 1 for j = 5/2 and E = 100 meV. The most
+probable radius ˆrj, is given by the position of the first maximum of the probability density
+ρp, stat (3.29) (See Fig. 1), which for m = 0 reads
+ρp, stat(r) = c2p2 �
+J2
+j−1/2(pr/ℏ) + J2
+j+1/2(pr/ℏ)
+�
+.
+(3.44)
+Since r appears in the combination pr/ℏ, the most probable radius ˆrj may be written as a
+function of p:
+ˆrj(p) = αj
+ℏ
+p,
+(3.45)
+the j-dependent coefficients αj being shown in Table 1 for some values of j. These results can
+be taken as an approximation for the more realistic case of a wave packet with a very small
+energy width.
+13
+
+0.002
+0.004
+0.006
+0.008
+0.010
+t
+40
+60
+80
+100
+120
+rB
+(a)
+0.002
+0.004
+0.006
+0.008
+0.010
+t
+-250
+-200
+-150
+-100
+-50
+ϕB
+(b)
+-50
+50
+xB
+-50
+50
+100
+yB
+(c)
+Figure 2: The most probable dBB tractory xB(ˆr0, t) for Gaussian wave packet parameters
+j = 5/2, p0 = 10−4 meV/c (Peak energy E0 = 100 meV), and Σ = 10−7 meV/c.
+This
+trajectory is fixed by the initial condition parameter = ˆr0 = 22.7. ˆr0 is the position of the peak
+of the probability density ρ at t = 0 (the blue point in Fig. 4a.)
+(a) Radial coordinate rB(ˆr0, t).
+(b) Azimuthal coordinate φB(ˆr0t).
+(c) dBB trajectory in the (x, y) plane for 0 ≤ t ≤ 0.010 ns. The trajectory performs 38 loops
+before the critical time (decay time) τ ∼ 0.006 ns.
+14
+
+0.002 0.004 0.006 0.008 0.010 0.012 0.014
+t
+500
+1000
+1500
+rB
+(a)
+0.002 0.004 0.006 0.008 0.010 0.012 0.014
+t
+-250
+-200
+-150
+-100
+-50
+ϕB
+(b)
+500
+1000
+1500
+xB
+-100
+100
+200
+300
+400
+500
+yB
+(c)
+Figure 3: dBB tractories for Gaussian wave packet: same parametrization as in Fig. 2, but
+with the larger time scale 0 ≤ t ≤ 0.015 ns.
+3.3.2
+Gaussian wave packets in graphene
+We turn now to the Gaussian wave functions defined by (3.31) and (3.32), which are normal-
+izable. We shall denote by
+xB(r0, t) = (rB(r0, t), φB(r0, t)) ,
+20
+40
+60
+80
+100
+r0
+5.×10-7
+1.×10-6
+1.5×10-6
+2.×10-6
+ρ(r0,0)
+(a)
+0.002
+0.004
+0.006
+0.008
+0.010
+0.012
+t
+200
+400
+600
+800
+rB
+(b)
+0.002
+0.004
+0.006
+0.008
+0.010
+0.012
+t
+-600
+-500
+-400
+-300
+-200
+-100
+ϕB
+(c)
+200
+400
+600
+800
+xB
+-300
+-200
+-100
+100
+yB
+(d)
+Figure 4: dBB tractories for Gaussian wave packet: same parametrization as in Figs. 2 and
+3, but with five trajectories corresponding to the initial radial positions r0 = 10, 17, 22.7,
+30 and 35 nm. The respective numbers of trajectory’s closed loops are 93, 66, 38, 10 and 1.
+Their respective relative probabilities are proportional to the heights of the coloured dots in
+the subfigure (a) showing the initial probability density ρ(r0, 0) in function of the initial radial
+position r0.
+15
+
+the (polar) coordinates of the dBB trajectory solution of the trajectory equation (2.7), fixed
+by the initial position
+xB(r0, 0) = (rB(r0, 0), φB(r0, 0) = (r0, 0).
+(3.46)
+We have made explicit, in our notation, the dependence on the trajectory parameter r0.
+Figs. 2 and 3 show the most probable dBB trajectory xB(ˆr0, t) for a particular wave func-
+tion’s parametrization in which the energy dispersion is very small, i.e., Σ ≪ p0.
+“Most
+probable” means that the initial particle’s position parameter ˆr0 is the value of the radial co-
+ordinate r which maximizes the initial probability density ρ(r, 0) This value, equal to 22.7 nm
+in the present example12 corresponds to the blue dot in Fig. 4a. Thus, the behaviour of this
+trajectory, shown by Figs. 2c or 3c in the (x, y)-plane, is very similar to the circular one shown
+in the corresponding stationary solution, but only up to a certain critical “decay time” τobs,
+approximately equal to 0.006 ns in this example. For later times the trajectory tends to a
+straight line, reproducing the expected classical behaviour. This is best observed in Figs. 2b or
+3b which show the time behaviour of the azimuthal angle φ: the angular velocity is almost con-
+stant until the time τobs, and almost vanishes thereafter. This decay time marks the transition
+from the almost circular regime to an almost straight-way, classical-like, regime.
+A theoretical lower bound for the decay time τ may be computed from the quantum ”time-
+energy uncertainty principle” [29]:
+τ ≥ τmin =
+ℏ
+2 ∆E ,
+(3.47)
+where ∆E is the quantum energy uncertainty given by (3.34) taken with m = 0.
+In our
+example, τmin = 0.00465 ns: this is the order of magnitude of the observed decay time for the
+most probable trajectory, τobs ∼ 0.006 nm, and the uncertainty inequality (3.47) is obeyed.
+Table 2 displays, for certain values of the wave parameters p0, Σ and j, quantities of interest
+such as mean energy ⟨E⟩ (3.36), standard energy deviation ∆E (3.37), τmin (3.47), (which are
+usual quantum theory quantities), and also, specifically concerning the most probable dBB
+trajectory, its approximate observed decay time τobs, its L-radius rL (3.40), its initial radial
+coordinate ˆr0 (see (3.46)) which fixes it and the number of closed loops, Nloops,’ it performs
+before passing to the straight-way regime.
+One can make the following remarks about the items of Table 2.
+1. The mean values ⟨E⟩, ∆E, hence τmin, do not depend on the quantum number j, which is
+obvious from the explicit expressions (3.36) and (3.37). ⟨E⟩ and ∆E tend towards their
+limit values (3.41) as the width Σ becomes narrower, as can be seen in the Table.
+2. The observed decay time τobs seen in the behaviour of the azimuthal angle φB shown, e.g.,
+in Figs. 4c or 4d, depends on the specific trajectory: it diminishes when the value of the
+12This value is very near of the one corresponding to the stationary wave function with same j = 5/2 and p
+equal to the momentum parameter p0 = 10−4. Indeed, Eq. (3.45) together with Table 1 yield ˆr= 22.6.
+16
+
+p0( meV
+c
+)
+Σ( meV
+c
+)
+⟨E⟩ (meV)
+∆E(meV)
+τmin(ns)
+τobs(ns)
+rL(nm)
+ˆr0(nm)
+Nloops
+j
+0.3
+32.91
+0
+757
+1/2
+10−8
+10.0000
+0.00707
+0.0465
+0.06
+164.6
+227
+39
+5/2
+0.05
+362.0
+450
+20
+11/2
+10−5
+0.05
+3258.
+3450
+3
+99/2
+0.001
+32.59
+0
+3
+1/2
+10−6
+10.0995
+0.704
+0.000468
+–
+162.9
+220
+< 1
+5/2
+–
+358.4
+435
+< 1
+11/2
+–
+3226.
+3250
+< 1
+99/2
+0.027
+3.291
+0
+676
+1/2
+10−7
+100.000
+0.0707
+0.00465
+0.006
+16.46
+22.7
+39
+5/2
+0.006
+36.20
+45.0
+20
+11/2
+10−4
+0.005
+325.8
+345
+3
+99/2
+0.0001
+3.259
+0
+2
+1/2
+10−5
+100.995
+7.04
+0.0000468
+–
+16.29
+22.0
+< 1
+5/2
+–
+35.84
+43.5
+< 1
+11/2
+–
+322.6
+325
+< 1
+99/2
+Table 2: Mean energy ⟨E⟩, standard energy deviation ∆E, decay time lower bound τmin,
+observed decay time τobs, L-radius rL, initial radial coordinate ˆr0 and number of trajectory
+loops Nloops for some values of the wave packet parameters p0, Σ and j. Trajectories concerned
+in columns 6 to 9 are the most probable ones.
+initial radial coordinate rB(r0, 0) = r0 augments. On the other hand, its value does not
+depend sensibly on the quantum number j, as can be seen in the table.
+3. Except for j = 1/2, the L-radius rL (3.40) is near of the value of the initial radial
+coordinate ˆr0 of the most probable trajectory.
+This is what can be expected for the
+nearly circular motion which takes place at times t < τobs.
+4. The number of revolutions also tends to decrease with increasing initial position r0, as
+shown in the example of Fig.
+4, which shows five trajectories corresponding to five
+different initial radial positions.
+5. The behaviours observed in these examples are generic, this being confirmed by all other
+cases we have numerically studied.
+Concluding this subsection, an important observation can be made. Although the minimum
+value for the decay-time τ was inferred from the usual quantum theoretical uncertainty principle
+for time-energy (3.47), it appears difficult to interpret τ in this framework. But it looks quite
+17
+
+natural in the dBB scheme, namely as a property of the dBB trajectories.
+May one even
+imagine an experimental way of discriminating the dBB trajectories by measuring it?
+3.3.3
+Instantaneous beables
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+t
+99.9999
+100.0000
+100.0000
+100.0000
+100.0000
+E
+(a)
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+t
+-0.4
+-0.2
+0.2
+0.4
+spin
+(b)
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+t
+200000
+400000
+600000
+800000
+1×106
+|v|
+(c)
+Figure 5: Instaneous energy (subfigure (a)), spin (subfigure (b)) and absolute velocity (subfigure
+(c)) in function of t for the solution shown in Figs. 2, 3 and 4. Colors correspond to the five
+different trajectories exhibited in 4
+Fig. 5 shows the time evolution of the instantaneous energy E(r0, t) (3.20), spin S(r0, t)
+(3.21) and absolute velocity |v|(r0, t) for the state already exhibited in the figures of the former
+subsection. The quantities are shown for five dBB trajectories, caracterized by their initial
+position parameter r0, and graphically by colours as in Fig. 4.
+Note the striking similarity between the time behaviours of both energy and spin.
+One further observes that, for a given trajectory, all three quantities tend to constant values
+above a certain threshold. E.g., for the blue one, which corresponds to the initial radial position
+r0 = 22.7, the threshold is at t ∼ 0.019 ns for the energy and spin, and at at t ∼ 0.017 ns for
+the velocity. This threshold is substantially higher than the decay time τ (∼ 0.006 ns here).
+The energy tends to its mean value ⟨E⟩ (∼ 100 meV here), the spin to its mean value 0 and the
+absolute velocity to the ”velocity of light” c = 106 ms−1. Note in particular that the energy is
+not conserved13 and that the velocity stays inferior to c before the time threshold, after which
+it goes rapidly to its asymptotic value c.
+3.3.4
+Times of flight
+The dBB theory offers a very natural way to define the time of flight of a particle which has
+followed a dBB trajectory xB(x0, t) from its initial position x0 to some target, e.g., consisting
+13This is a general feature of the dBB theory.
+Think of the obvious time dependence of the ”quantum
+potential” which defines the quantum contribution to the particle’s motion of a non-relativistic particle (see Eq.
+(3.6) of Ref. [14]).
+18
+
+of a detector. In our case, one can think of a detector occupying a circle centred at the origin
+and of radius R. This time of flight is then the solution tflight(R, r0) of the equation
+rB(r0, tflight) − R = 0,
+(3.48)
+where rB(r0, t) is the radial coordinate of the considered trajectory, characterized by its initial
+radial coordinate r0. In case the solution is not unique, one has to take the lowest one, corre-
+sponding to the first hit of the particle to the target [17, 18].However, this precaution is not
+needed in all cases we have investigated, where rB(r0, t) is a monotonically increasing function
+of t.
+The outcome of such an experiment is a probability distribution Π(τ), in terms of the time
+of flight tflight = τ, given by Eq. (9) of [17] and taking the form, in our context:
+Π(τ) = N2π
+� ∞
+0 dr0 r0 ρ(r0, 0) δ (tflight(R, r0) − τ)
+= N2πr0(R, τ) ρ(r0(R, τ), 0) |∂r0tflight(R, r0(R, τ))| ,
+(3.49)
+where r0(R, τ) is the inverse of the time of flight function tflight(R, r0), i.e., the solution (unique,
+here) of (3.48) for r0 in terms of R and tflight = τ. Recall that ρ(r0, 0) represents the probability
+distribution for the trajectory defined by its initial radial coordinate r0. N is a normalization
+factor ensuring the normaliztion condition
+� τmax
+0
+dτ Π(τ) = 1.
+(3.50)
+If the probability flux through the detector’s entry is always positive, which is the case in our
+examples, an alternative expression for the probability distribution is given by [30]
+ΠFlux(τ) = N
+�
+Σ
+ds · j(x, τ)
+=
+(here) N2πRjr(R, τ),
+(3.51)
+where j is the probability flux and Σ the detector entry’s surface. This result was proved in a
+scattering context by [30], and more generally, but in the one-dimensional case, by [31], and
+by [32] in the case of a spinless non-relativistic particle. We have checked numerically the
+equivalence of both formulae (3.49) and (3.51) in our specific situation for various parametriza-
+tions of the wave function.
+Figs. 6 and 7 show the time of flight in function of the trajectory
+parameter r0 and the corresponding probability distribution (3.49) at a circular target of radius
+30 and 500 nm, respectively.
+4
+Conclusion
+The trajectories predicted by the de Broglie-Bohm (dBB) quantum theory were calculated for
+the case of a guiding wave function being solution of the two-dimensional free Dirac equation,
+a solution constrained to be an eigenfunction of the total angular momentum operator relative
+19
+
+5
+10
+15
+20
+25
+30
+r0
+0.005
+0.010
+0.015
+0.020
+0.025
+tflight
+(a)
+0.005
+0.010
+0.015
+0.020
+0.025
+tflight
+20
+40
+60
+80
+100
+120
+Π
+(b)
+Figure 6: Times of flight tflight solutions of (3.48) and values of their probability density Π(tflight)
+(3.49) for the dBB trajectories shown in Fig. 4. The wave function parameters are the same
+as those in Figs. 2 to 5. The target is a circle centred at the origin, with radius R = 30 nm.
+The initial radial coordinate r0 varies between 2 and 30 nm.
+(a) Values of the time of flight for each dBB trajectory. The dots represent the numerically
+calculated values, and the continuous line an interpolation used for the calculation of the
+probability distribution.
+(b) Values of the corresponding probability density. Use of Eq. (3.51) has been made.
+100
+200
+300
+400
+500
+r0
+0.005
+0.010
+0.015
+tflight
+(a)
+0.005
+0.010
+0.015
+tflight
+20
+40
+60
+80
+100
+120
+Π
+(b)
+Figure 7: Same as Fig. 6, but with target’s radius R = 500 nm and initial radial coordinate r0
+in the interval 10 to 500 nm.
+to a given origin point of space. Numerical results have being provided for the case of massless
+particles with momentum-energy specifications corresponding to those of free electrons in mono-
+layer graphene.
+The trajectories corresponding to stationary wave functions turn out to be circles travelled
+at a constant speed. For Gaussian-like wave packets, the trajectories begin as quasi circles of
+20
+
+slowly increasing radius till a critical time at which they tend to straight lines approximating
+the behaviour expected for a classical free particle. This transition time decreases when the
+value of the initial radial coordinate which labels a particular trajectory increases, but appears
+to be insensible to the chosen value of the total angular momentum. It is worth noting that
+the transition time obtained in each example is of the order of magnitude of, but greater than,
+the lower bound given by the ”time-energy uncertainty principle”. Although the nature of this
+lower bound is of course purely quantum mechanical, a theory such as the dBB one appears
+necessary in order to interpret it. More, it is the use of the dBB theory which has allowed us
+to evidenciate this phenomenon.
+Given a wave function, the possible times of arrival of the particle at some region have
+also been calculated in function of its initial position for the same examples, taking profit
+of the objective reality of the trajectories in the dBB theory. The corresponding probability
+distribution of these arrival times has been calculated using the Das-D¨urr formula based on the
+dBB theory and also using the conventional quantum theory formula involving the probability
+flux. Both calculation’s results coincide, as can be expected from the equivalence’s proof given
+in [31] for the spin one-half particle in one-dimensional space and by [32] for the non-relativistic
+spinless particle. Note that this equivalence holds if the flux on any target is always positive
+– which is true in our examples. The importance of this probability distribution is that the
+latter may in principle be measured in a suitable physical context such as, e.g., the monolayer
+graphene.
+Acknowledgements
+I would like to thank Siddhant Das for the indication of interesting references and for his
+valuable comments.
+Appendices
+A
+Notations and conventions
+Units used in this paper are adapted to the physics of graphene. Length, time and energy are
+given in nm, ns and meV, respectively. The critical velocity and the Planck constant take the
+values
+c = 106 nm ns−1,
+ℏ = 6.5821 × 10−4 meV ns.
+(A.1)
+Space-time coordinate are denoted by xµ, µ = 0, 1, 2, space coordinates by x = (x, y), or (r, φ).
+Space-time metric is ηµν = diag(1, −1, −1)
+Dirac matrices are chosen in terms of the Pauli matrices as
+γ0 = σz,
+γ1 = γ0σx,
+γ2 = γ0σy.
+(A.2)
+21
+
+The Dirac matrices used in the non-relativistic formulation are
+α1 = σx,
+α2 = σy,
+β = σz.
+(A.3)
+B
+Some useful properties of the Bessel functions
+The general solution of the Bessel equation [33]
+z2f ′′(z) + zf ′(z) + (z2 − n2)f(z) = 0,
+(B.1)
+has the form
+f(z) = C1Jn(z) + C2Yn(z),
+(B.2)
+where Jn and Yn are the Bessel functions of the first [33], resp. second [33] kind, and C1, C2
+are two arbitrary complex constants. We shall restrict ourselves to an integer index n.
+The asymptotic behaviors of the Bessel functions at the origin are given by
+Jn(x) ∼ 1
+n!
+�x
+2
+�n
+(0 < x ≪ 1, n ≥ 0),
+Yn(x) ∼ −(n − 1)!
+π
+�2
+x
+�n
+(0 < x ≪ 1, n ≥ 1),
+Y0(x) ∼ 2
+π log
+�x
+2
+�
+(0 < x ≪ 1),
+(B.3)
+and at infinity by
+Jn(x) ∼
+�
+2
+πx cos
+�
+x − (n + 1
+2)π
+2
+�
+(x ≫ 1, n ≥ 0),
+Yn(x) ∼
+�
+2
+πx sin
+�
+x − (n + 1
+2)π
+2
+�
+(x ≫ 1, n ≥ 0).
+(B.4)
+Functions with a negative index are related to those with a positive one by the identities
+J−n(z) = (−1)nJn(z),
+Y−n(z) = (−1)nYn(z).
+(B.5)
+Under parity z → −z, the function Jn transforms as
+Jn(−z) = (−1)nJn(z).
+(B.6)
+An interesting orthogonality property is given by [33]
+� R
+0
+dr r Jn
+�zn,α r
+R
+�
+Jn
+�zn,β r
+R
+�
+= R2
+2 (Jn+1(zn,α)2 δαβ,
+(B.7)
+for n ≥ 0, where zn,α is the αth positive zero of the Bessel function Jn(z) [34]. Moreover, any
+function f(r) defined in the interval 0 ≤ r ≤ R with bounded variation and vanishing at the
+end point r = R can be represented as a “Fourier Bessel series” [35] as
+f(r) =
+∞
+�
+α=1
+cαJn
+�zn,α r
+R
+�
+,
+(B.8)
+for any n ≥ 0. The coefficients cα can be calculated using the orthogonality formula (B.7).
+22
+
+References
+[1] P.R. Holland, “The Dirac equation in the de Broglie-Bohm theory of motion”, Found.
+Phys. 22 (1992) 1287.
+[2] Peter R. Holland, “The quantum theory of motion”, Revised ed., Cambridge University
+Press (1995).
+[3] Max Planck, “Ueber das Gesetz der Energieverteilung im Normalspectrum” (English
+translation), Annalen der Physik 4 (1901) 553.
+[4] Niels Bohr, “On the Constitution of Atoms and Molecules”, Philos. Mag. 26 (1913) 1 and
+476.
+[5] Albert Einstein, “Concerning an Heuristic Point of View Toward the Emission and Trans-
+formation of Light”, Annalen der Physik 17 (1905) 132.
+[6] Louis de Broglie, “Recherches sur la th´eorie des quanta”, Thesis (Paris), 1924;
+Louis de Broglie, Ann. Phys. (Paris) 3, 22 (1925). Reprint in Ann. Found. Louis de
+Broglie 17 (1992) p. 22;
+Louis De Broglie, “La m´ecanique ondulatoire et la structure atomique de la mati`ere et du
+rayonnement”, J. Phys. Radium 8 (1927) 225, DOI 10.1051/jphysrad:0192700805022500.
+[7] Erwin Schr¨odinger, “Quantisierung als Eigenwertproblem”, Annalen der Physik 79 (1926),
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+Physik 81 (1926) 109.
+[8] Werner Heisenberg, ҬUber quantentheoretische Umdeutung kinematischer und mechanis-
+cher Beziehungen”, Z. Phys. 33 (1925) 879.
+[9] Paul A.M. Dirac, “The quantum theory of the electron”, Proc. R. Soc. A 117 (1928) 610
+and 118 (1928) 351.
+[10] Niels Bohr, “The Quantum Postulate and the Recent Development of Atomic Theory”,
+Supplement to ”Nature April 14 (1928) 580;
+Werner Heisenberg, “Physics and Philosophy”, Harper, New York (1958),
+[11] Hugh Everett, “Relative State Formulation of Quantum Mechanics”,
+Rev. Mod. Phys. 29 (1957) 454.
+[12] Carlo Rovelli, “Relational quantum mechanics”,
+Int. J. Theor. Phys. 35 (1996) 1637 e-Print: quant-ph/9609002 [quant-ph];
+Andrea Di Biagio and Carlo Rovelli, “Stable Facts, Relative Facts”,
+Found. Phys. (2021) 51:30.
+[13] David Bohm, “A Suggested interpretation of the quantum theory in terms of hidden
+variables 1, 2.”, Phys. Rev. 85 (1952) 166, 180.
+23
+
+[14] D. Bohm and B.J. Hiley, ”The Undivided Universe”, Routledge, London and New York
+(1995).
+[15] John S. Bell, “Speakable and Unspeakable in Quantum Mechanics”, Cambridge University
+Press, New York (2010).
+[16] John S. Bell, “Beables for quantum field theory”, preprint CERN TH-4035/84 (1984),
+reprinted in Cap. 19 of [15].
+[17] Siddhant Das and Detlef D¨urr, ”Arrival time distributions of spin-1/2 particles, Nature
+Scient. Rep. 9 (2019) 2242.
+[18] Siddhant Das, Markus N¨oth and Detlef D¨urr, “Exotic arrival times of spin-1/2 particles
+I - An analytical treatment”, Phys. Rev. A99 (2019) 052124.
+[19] Detlef D¨urr, Sheldon Goldstein, Travis Norsen, Ward Struyve and Nino Zangh`ı, “Can
+Bohmian mechanics be made relativistic?” Proc. R. Soc. A 470 (2013) 20130699
+[20] Roderich Tumulka, “On Bohmian Mechanics, Particle Creation, and Relativistic Space-
+Time: Happy 100th Birthday, David Bohm!”, Entropy 20 (2018) 462.
+[21] Dario Bressanini and Alessandro Ponti, “Angular Momentum and the Two-Dimensional
+Free Particle´´, J. Chem. Educ. 75 (1998) 916.
+[22] Wolfram Research, Inc., Mathematica, Champaign, IL.
+[23] Abraham Pais, “On Spinors in n Dimensions”, J. Math. Phys. 3 (1962) 1135; doi:
+10.1063/1.1703856.
+[24] D. D¨urr D., S. Goldstein and N. Zanghi, “Quantum equilibrium and the origin of absolute
+uncertainty”, J. Stat. Phys. 67 (1992) 843.
+[25] The Wolfram Functions Site,
+https://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/21/02/02/
+[26] WolframMathWorld,
+https://functions.wolfram.com/GammaBetaErf/Erf/
+[27] Mikhail I. Katsnelson, “The Physics of Graphene” 2nd Edition, Cambridge University
+Press, Cambridge (2020).
+[28] S. Das Sarma, Shaffique Adam, E. H. Hwang and Enrico Rossi, “Electronic transport
+in two-dimensional graphene”, Rev. Mod. Phys. 83 (2011) 407, e-Print: arXiv:1003.4731
+(cond-mat.mtrl-sci).
+[29] Albert Messiah, “Quantum Mechanics”, Vol. 1, Section VIII-13, Dover Publications, New
+York (2014) (English translation of “M´ecanique Quantique”, Dunod, Paris, (1962)).
+[30] M. Daumer, D. D¨urr, S. Goldstein and N. Zanghi, “On the Quantum Probability Flux
+Through Surfaces”, J. Stat. Phys. 88 (1997) 967.
+24
+
+[31] C. Richard Leavens, “Bohm Trajectory Approach to Timing Electrons”, p. 129 of “Time
+in Quantum Mechanics - Vol.1”, 2d Ed., J.G. Muga, R. Sala Mayato, ´I.L. Egusquiza
+(Eds.), Lecture Notes in Physics 734, Springer, Heidelberg, 2008.
+[32] Siddhant Das and Markus N¨oth, Times of arrival and gauge invariance, Proc. R. Soc. A
+477 (2021) 20210101, https://doi.org/10.1098/rspa.2021.0101.
+[33] WolframMathWorld, “Bessel function”,
+https://mathworld.wolfram.com/topics/BesselFunctions.html.
+[34] WolframMathWorld,
+https://mathworld.wolfram.com/BesselFunctionZeros.html
+[35] Eric W. Weisstein, “Fourier-Bessel Series.” From MathWorld–A Wolfram Web Resource.
+https://mathworld.wolfram.com/Fourier-BesselSeries.html
+25
+

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+Examination of saturation coverage of short polymers using 
+random sequential adsorption algorithm  
+Aref Abbasi Moud1* 
+1 Department of Chemical and Petroleum Engineering, University of Calgary, 2500 University Dr NW, 
+Calgary, AB, T2N 1N4, Canada 
+*Author to whom correspondence should be addressed; electronic mail:  [email protected] 
+Abstract: We filled a void with a regular or asymmetric pattern without overlap using a time-dependent 
+packing method termed random sequential adsorption (RSA). In the infinite-time limit, the density of 
+coverage frequently hits a limit. This study focused on the saturation packing of squares and their dimers, 
+trimers, tetramers, pentamers, and hexamers, all of which were orientated in two randomly chosen 
+orientations (vertical and horizontal). Our results concurred with those of previous extrapolation-based 
+research1. We used the "separating axis theorem" to detect if freshly added polygons and previously put 
+ones overlapped. When RSA insertion became disproportionately sluggish, we concluded that saturation 
+had been attained. We also discovered that the system's capacity to fill the area decreased as squares were 
+stretched into dimers and trimmers. The microstructure of the resultant saturation was also thoroughly 
+investigated, including block function and the structural arrangement of dimers and trimers.  
+Keywords: RSA, short polymers, separating axis theorem, trimers, and dimers 
+1. Introduction 
+ 
+One of the most popular nonequilibrium packing models is the random sequential addition (RSA) packing 
+method, which is a time-dependent process. Unpredictable sphere packings are created using a method 
+similar to earlier established techniques; see References2-3. The RSA packing process in the three spatial 
+dimensions4 has been used to represent a range of scenarios, including protein adsorption5, polymer 
+oxidation, particles in cell membranes, and ion implantation in semiconductors2.  
+Euclidean space (d-dimensional) particles with certain shapes are randomly and progressively introduced 
+into the volume subject with the restriction that they do not overlap to carry out the simulation starting from 
+a big, vacant zone. The freshly created particles are only maintained if they do not overlap any already 
+existing particles; otherwise, they are deleted. After the simulation has begun, this procedure can be stopped 
+at any time, making the density that has been acquired time dependent. Density reaches a "saturation" or 
+"jamming" limit as time goes on6.  
+In its simplest form, an RSA sphere packing may be obtained by randomly, irreversibly, and sequentially 
+adding nonoverlapping objects into a huge volume that is initially empty of spheres. A further attempt is 
+made until the sphere can be added without doing so if an attempt to add a sphere (or any other polygon) at 
+time t overlaps with a sphere that is already present in the packing. An RSA configuration with a time-
+dependent packing fraction can be created by selecting any moment in finite time t as the process' endpoint. 
+The maximum saturation packing fraction, which occurs at the infinite-time and thermodynamic limits, 
+prevents this figure from becoming higher.  
+RSA implementations fall into two groups. The two fundamental categories into which the RSA models 
+may be split are continuum and lattice models. Based on object kinds, they are further split into two groups: 
+RSA of finite (nonzero) area objects and RSA of zero area objects. Things having finite area in this sense 
+are those that enclose a certain amount of space, whereas those with zero area lack this geometric feature 
+
+of enclosure. Therefore, upon adsorption on the substrate, items with finite area take up some space, 
+whereas those with zero area take up no space. Lattice models are defined by the realisation of the jammed 
+state7, regardless of the item types.  
+In general, in case of RSA of finite area objects, the approach of instantaneous coverage 𝜃(𝑡) to the jammed 
+state coverage 𝜃𝑚𝑎𝑥 is found to follow a power law 𝜃𝑚𝑎𝑥 − 𝜃(𝑡)~𝑡−𝑝. Researchers have proposed certain 
+laws about the value of the exponents p by researching the RSA of objects with a variety of geometries, 
+including circular, elliptical, rectangular, and sphero-cylindrical. Feder 8 is credited with being the first to 
+link the observed value of the exponent p in the RSA of circular objects in two dimensions to the object 
+dimensionality and to propose the general rule that p = 1/D for d-dimensional circles on a two-dimensional 
+continuum platform. Swendsen 9 subsequently demonstrated that, if the items are placed with random 
+orientations, the same ought to apply for RSA of items of any arbitrary form.  
+Since its invention by Feder 8 , Random Sequential Adsorption (RSA) has gained widespread acceptance 
+as a technique for simulating adsorption characteristics, particularly for spherical molecules. However, 
+employing RSA to replicate the adsorption of more complex particles like polymers or proteins raises 
+concerns about how RSA's inherent features alter when non-spherical molecules are involved. For simple 
+forms like spheroids, spherocylinders, rectangles, needles, and others, the subject has already been 
+addressed10-12. Recent research, however, indicates that these geometries are insufficient for simulating the 
+adsorption of common proteins, such as fibrinogen, for instance13. As a result, researchers' focus has 
+recently turned to coarse-grained modelling of complex biomolecules and polymers14-16.  
+In 1-D case, the saturation packing fraction can be obtained analytically as 0.747597920 17 however for 2-
+D and 3-D the saturation packing fraction for discs and spheres has been estimated only through numerical 
+simulations; the most precise ones are 0.5470735 ± 0.0000028 for 2-d and 0.3841307 ± 0.000002 for 3-D 
+cases2. 
+Other 
+figures 
+reported 
+for 
+2-D 
+simulations 
+of 
+discs 
+are 
+0.54707352, 0.54706718, 0.54707019, 0.547069020, 0.5470021, 
+0.5471122, 0.547223, 
+0.5478, 
+0.547924. 
+Similarly for 2-D aligned squares saturation coverage reported in the literature is 0.56200925, 0.562324, 
+0.5628, 0.556526, 0.562527, 0.544428, 0.562929, 0.56230. 
+In this study, we used the RSA technique to determine the saturation packing limit for squares and its dimer 
+and trimers. As the length of square increases, the results indicated that samples eventually generated 
+structures with less and less packing.  In this study, the "separating axis theorem" approach was used to 
+detect whether there was a collision between two polygons; more information on the procedure is provided 
+in the following sections. Polygons here refer to squares that encompasses its polymers as well as a 
+constructing monomer. Our preliminary findings on RSA packing with respect to polygons, which we just 
+published31, are the basis for this study, which extends that work by extending the polygon (square) into its 
+polymers.  
+2. Model and simulation procedure  
+ 
+When colloidal particles or molecules are being adsorbed, they frequently diffuse close to the surface. This 
+process might lead to the formation of a film consisting of molecules that were randomly adsorbed because 
+of adhesion. Here, we focus on adsorbate monolayers formed by irreversible adsorption. The most 
+straightforward technique for quantitatively modelling these processes is molecular dynamics (MD). The 
+advantages of MD include accurate forecasting and management of most environmental factors, such as 
+temperature and the diffusion constant. The main issue is the performance deficiency. As a result, we 
+decided to utilise a new method, continuum Random Sequential Adsorption (RSA), which has been 
+successfully employed to investigate colloidal and other systems32.  
+
+To simulate, a virtual particle was created (square, its polymers), and its location on an area was chosen at 
+random based on a uniform probability distribution. 
+- The overlap with the previously adsorbed nearest neighbours of a virtual particle was tested (the topic of 
+the next section). The result of this test tells us whether the surface-to-surface distance of a particle is greater 
+than zero. 
+- If there was no overlap, the virtual particle was adsorbed and added to an existing covering layer. 
+- If there was overlap, the virtual particle was dropped and abandoned.  
+2.1 Proposed algorithm 
+ 
+Numerous methods may be used to determine if two polygons intersect or not. One method for determining 
+if two polygons are overlapping uses mathematical equations and is known as the "separating axis 
+theorem"33. 
+The separating axis theorem states that if a line divides two convex polygons, they cannot intersect. The 
+separation axis, which is a line, may be thought of as the normal to one of the edges of each polygon. 
+Using the separating axis theorem, the following procedures can be used to determine if two polygons cross:  
+• 
+Determine each polygon A edge's edge normal, then project both polygons onto that value. 
+ 
+• 
+Establish the minimum and maximum projections of each polygon onto the normal. 
+ 
+• 
+If the maximum projection of polygon A is less than the minimum projection of polygon B, or the 
+other way around, the polygons do not overlap. 
+ 
+• 
+If the projections overlap, repeat the procedure for each edge in polygon B. 
+ 
+• 
+If the projections of the polygons onto the separating axes do not overlap, the polygons are 
+connected.  
+It is likely that non-convex polygons or polygons with holes will not be covered by this theorem, even 
+though the separating axis theory may be used to determine if two convex polygons overlap. In some 
+situations, it could be necessary to verify for intersection using alternative techniques.  
+3. Results and discussion 
+ 
+We construct saturated RSA configurations of polymers (dimer to hexamer) and compare saturation 
+packing with other findings reported in the literature, particularly in ref2, where authors employed a 
+different approach and orientation was random, to show the precision and utility of our algorithm. We 
+generate 1000 variations for each particle form using the system size that results in the fastest and densest 
+packing.  
+
+ 
+Figure 1. Typical monolayer samples (from trimer’s sample) for three different coverages: θ = 0.1, θ = 0.3, 
+θ = 0.4 and θ = 0.5 for trimers. The collector side length was equal to 50[-]. Fixed boundary conditions 
+were used. Figure shows truncated images of the distribution for a better visibility (20 by 20). 
+Using the greatest area (50 by 50) with fixed bounds, most of the results presented later in the study were 
+achieved. We made sure that the adoption of periodic boundary conditions had no discernible impact on 
+the results that were made (Equivalent of periodic and fixed boundary condition). Figure 1 shows the 
+outcomes of one of the simulations, in which trimers are placed in an area measuring 20 by 20 with a 
+monomer having a side length of 0.5 units. Particles are gradually added to the surface as the simulation 
+progresses.  
+ 
+
+0=10%
+0=30%
+0=40%
+0=50% 
+Figure 2. Hexamer units put irrevocably inside a 50 by 50 space are the focus of the RSA algorithm. (a) 
+Asymptotic observation of coverage as a function of total simulation duration indicated by t. (b) The 
+instantaneous time τ determined by based on coverage. The line represents an exponential fit.  
+To look at the development in more details, surface coverage was depicted as a function of simulation time 
+to the power of 𝑡−1/3 and results are shown in Figure 2a. Clearly at long simulation times of ~104 coverage 
+very slowly reaches its asymptotic limit that is 0.4968±0.0011.  Similarly using same schemes, we arrived 
+at 0.5631±0.0002 surface coverage for squares. This surface coverage corresponds very well with the results 
+reported in the literature for squares 8, 24-30.  
+The number of attempts needed to add a new particle to the grid (or collector both terms in congruency with 
+literature has been used here interchangeably) can be known, and this information can be used to model the 
+blocking of further adsorption through monitoring time. Clearly, as more of the surface is covered, adding 
+new particles should be more difficult, which can be represented by a lower probability (See Figure 2b) 
+that is it takes considerably more amount of time for a particle to be added. Adsorption kinetics in a real 
+experiment typically depends on two variables: the effectiveness of the transport process (primarily 
+diffusion or convection depending on the experimental setup) that moves the adsorbate from the bulk to the 
+surface, and the likelihood of catching particles that are nearby 34-39. Authors in other reports40 have 
+concentrated on the second aspect in this case, which is defined by the blocking function, also known as 
+the available surface function (ASF). The simulation makes it simple to obtain it as a ratio of successful 
+attempts to all RSA attempts. Equivalent to available surface function that is represented in shape of time 
+simulation is presented in Figure 2b. Figure 2b shows simulation time as a function of coverage; 
+statistically, it is evident that more trials are required to attain adsorption because the surface is already 
+rather packed. The exponential fit is, τ = 0.0006 exp (24.03 𝜃), thus describing increasing time required to 
+place an additional point onto the grid. Discussion on ASF is subject of next section.  
+The main objectives of this work were to determine the maximum random adsorption ratio for squares and 
+their polymers and compare it to the results for hard circles (spheres). That ratio ought to be provided for 
+an infinite grid area and adsorption duration. Although one must deal with constrained simulation durations, 
+one must also manage the accuracy issue brought on by the finite grid size. Because it is unclear if there 
+would be any possibility of adsorption after the simulation time, particularly in the case of large grids, the 
+determination of maximal coverage depends on the RSA kinetics model. As a result of prior research in the 
+region 9, 41-42, there have been a number of works in the area, and asymptotically:  
+𝜃𝑚𝑎𝑥 − 𝜃(𝑡)~𝑡−1
+𝐷     eq.1 
+
+0.5
+4000
+0.45
+3000
+0.4
+2000
+0.35
+1000
+0.3
+0.01
+0.02
+0.03Regarding the irreversible deposition of discs or squares that are not orientated (formerly known as p = -
+1/D). Despite controversy, D here specifies the grid's dimension9. When adsorbed particles are organised, 
+the situation is altered9, 42. Figure 2a previously in this post showed an example of the results of fitting 
+Equation 1. Asymptotic observations of coverage for squares seem to neatly match Equation 1. Although 
+Equation 1 hasn't been definitively proved, its validity has been vigorously defended12 by analytical and 
+numerical grounds. It should be noted that Equation 1 simplifies to the standard Feder's law8 for isotropic 
+objects since n equals the number of dimensions.  
+For instance, RSA of discs on a two-dimensional plane has D = 2, whereas RSA of rectangles, ellipses, and 
+other rigid but noticeably anisotropic structures has D = 3 37, 43. It appears that parameter D generally 
+correlates to the degrees of freedom of a number of packed objects, which has been validated for the random 
+packing of hyperspheres in higher dimensions 2, 21, not only the integral ones44-45. The power law (Equation 
+1) is satisfied for the RSA of polymers examined here, however the exponent -1/D strongly relies on a 
+polymer length. The parameter D is about equivalent to 3, which is the value recognised for anisotropic 
+molecules, for a small number of vertexes such as pentagon and squares. However, as number of vertices 
+increases parameter D converges to 2. This finding is consistent with those made for the RSA of spherical 
+beads examined in Ref.46. However, unlike what was shown in the cases of spherical beads46  or generalized 
+dimers 40, there is no abrupt transition between these two limitations.  
+The results are averaged across 10 simulation runs with time t in the order of 5 × 108 for each run in order 
+to determine parameter D for different polymers. These runs' data are not displayed here, and we will go 
+into more depth about the outcomes in our upcoming paper.  
+ 
+3.1 RSA for polymers 
+ 
+In the last part, we laid the foundation for using the RSA approach to create oriented squares and trimmers. 
+Results showing the behaviour of adsorption at asymptotic limits, the kinetics of the adsorption index (p), 
+and the relationship between simulation time and coverage were given. Additionally, results and discs were 
+compared. Utilizing the extrapolation method shown in Figure 2a previously, Table 1 generates 
+saturation densities for various polymer lengths. Figure 3 displays a sample of saturation densities for 
+various forms.  
+To arrive at the values reported in Table 1 following equation has been used: 
+𝜃(𝑡) = 𝜃𝑚𝑎𝑥 + 𝑏/𝑡𝑝     eq.2 
+When arriving at the values shown in table 1, we gave the data from longer simulation times more weight. 
+The approach's possible downside is that each data point is given the same weighting factor, assuming all 
+values are given the same weight. Because there are a lot more of these points in the higher part of the 
+asymptotic area, it is sort of underweighted. Therefore, we investigated a novel strategy that introduces a 
+bias favouring the longer durations. These changes are in line with the accounts in ref12.  
+Table 1. Saturation density, index, for square-based polymers with a monomer to simulation box length 
+ratio of 0.01 and their respective 95% confidence intervals.  
+Shape (oriented) 
+𝜃𝑚𝑎𝑥 [-] (95% confidence 
+bounds)  
+p [-] 
+(95% 
+confidence 
+bounds) 
+b [-] 
+(95% 
+confidence 
+bounds) 
+Square  
+0.5631(0.5629, 0.5633) 
+0.5138 (0.5125, 0.5151) 
+-4.226 (-4.475, -4.561) 
+
+Dimer 
+0.57 (0.5697, 0.5704) 
+0.4599 (0.4595, 0.4603) 
+-4.805 (-4.819, -4.792) 
+Trimers 
+0.5621 (0.5612, 0.5629) 
+0.466 (0.4652, 0.4668) 
+-8.003 (-8.066, -7.941) 
+Tetramer  
+0.5558 (0.5539, 0.5577)  
+0.4998 (0.499, 0.5007) 
+-6.046 (-6.15, -5.942) 
+Pentamer 
+0.5504 (0.5481, 0.5527) 
+0.46 (0.4576, 0.4624) 
+-5.007 (-5.122, -4.892) 
+Hexamer 
+0.4968 (0.4949, 0.4987) 
+0.5 (0.499, 0.501) 
+-6.264 (-6.458, -6.069) 
+Discs 
+0.5470732 
+- 
+- 
+ 
+As outlined in introduction section, similarly for 2-D aligned squares saturation coverage reported in the 
+literature is 0.56200925, 0.562324, 0.5628, 0.556526, 0.562527, 0.544428, 0.562929, 0.56230. Our values for 
+square are very well within range of values reported elsewhere. However, as particles get longer and become 
+Trimers, saturation has dropped since longer particles require more accessible area for deposition. For dimer 
+and trimer, the greater aspect ratio of the dimer is projected to result in somewhat higher saturation for 
+dimers. These findings are crucial because they suggest that it gets progressively harder for molecules to 
+adhere to surfaces as they become longer; an example of superiority of simulation over experiment in giving 
+researcher a tool to examine parameters hard to measure through experiments.  
+Results from this study can also be extrapolated to higher dimensions. For instance, the efficiency of a 
+sequential adsorption process with hard materials decreases with increasing size. It is noteworthy to note 
+that, as a general rule, the saturation coverage in D dimensions is very well estimated by that in one 
+dimension raised to power D (for the RSA of spherical particles, 𝜃𝑚𝑎𝑥≃ 0.75 for D = 1, 𝜃𝑚𝑎𝑥≃ 0.55 for D 
+= 2, 𝜃𝑚𝑎𝑥 ≃ 0.38 for D = 3, etc)47. Therefore, results obtained here can be extended to 1-D and 3-D cases 
+with good approximation, for instance for squares for cubes is predicted to lie around 0.38 and in 1-D case 
+around 0.73.  
+
+ 
+Figure 3. Square, dimers, and trimers near saturation points for samples with monomer’s side length of 0.5 
+and distributed within area of 50 by 50. Fixed boundary condition has been applied. Figure shows truncated 
+images of the distribution for a better visibility (20 by 20). For improved visibility, the horizontally oriented 
+polymers have been coloured blue, while the vertically oriented ones have been painted red.  
+Clearly visually samples experience a bit higher coverage for dimers and less coverage for trimers.   
+ 
+
+Square
+Dimers
+Trimers 
+Figure 4. RSA saturation density for polymers, with a line created to guide the viewer's eyes. Each data 
+point has an almost imperceptible error bar. For discs, RSA saturation density (red dotted line). All the 
+polymers in Table 1's saturation density changes as a function of simulation duration (a) The dependence 
+of 𝜃𝑚𝑎𝑥 on length of the polymer (c) Kinetic index as a function of the monomer length.  
+Figure 4 shows the RSA saturation density as a function of polymer length together with an eye-guiding 
+line. The saturation density coverage first increases somewhat as the polymer's length rises, but as it 
+continues to grow, it starts to decline. This outcome is in line with the outcomes for rectangles with various 
+aspect ratios and ellipses that have been reported in the literature5, 48. Additionally, we discovered that 
+saturation is somewhat lower for polymers with aspect ratios longer than 2, and we hypothesise that this is 
+because, as was already noted, longer particles are more difficult to pack efficiently.  
+The random coverage ratio dropped exponentially with polymer size in earlier investigations refs43, 49. On 
+a continuous surface, at least two competing variables can affect the maximum random coverage ratio. 
+First, there is less chance of finding a large enough uncovered section to place on a collector, making it 
+harder to separate larger particles than in the lattice case. The second point is that a polymer globule is more 
+likely to form a cluster when necessary because it has a greater monomer packing ratio than a group of 
+individual monomers. For continuous collectors as opposed to lattice ones, this second element is more 
+important.  
+
+0.6
+0.65
+0.5
+0.6
+0.4
+0.55
+Sguare
+0.3
+0.5
+Dimer
+Trimer
+0.2
+0.45
+Tetramer
+A
+Hexame
+0.4
+a
+b
+0.35
+3
+4
+5
+2
+X10 4
+0.65
+Monomerlength
+0.6
+DiscS
+0.55
+0.5
+0.45
+0.4
+0.35 
+3.2 Block function 
+ 
+Knowing how many tries are necessary to add a new particle to the grid allows us to simulate how further 
+adsorption is blocked over time. It is obvious that when more of the surface is covered, adding more 
+particles should become more challenging, which may be represented by a decreased probability (as shown 
+in Figure 2b, which depicts the same trend with time as the dependent variable)  
+ 
+ 
+ 
+Figure 5. The ratio of successful attempts to blocking attempts versus coverage. The green line represents 
+simulation data in b and the dotted line represent polynomial fit.; details of the fits are given in table 2.   
+ 
+Equation 3 describe the function with a simple second order polynomial perfectly describing the decreasing 
+trend in probability of a successful adsorption.  
+ 
+𝐴𝑆𝐹(𝜑) = 𝐶1𝜃2 + 𝐶2𝜃 + 𝐶3 eq.3 
+ 
+In which C1, C2 and C3 are pre-factors. Details of the fit of equation 3 is given in the Table 2. Figure 5 
+shows that the likelihood of adsorption for trimers drops more quickly than for dimers and squares, and the 
+dimer with respect to the square exhibits the same pattern. This is because consecutive adsorption is much 
+less likely to occur quickly in trimers and dimers than in squares due to their greater aspect ratio and unusual 
+orientation.  
+ 
+Table 2. Represents polynomial fit to the data in Figure 5 along with corresponding 95% confidence 
+bounds. 
+ 
+Shape (oriented) 
+C1 [-] (95% confidence 
+bounds)  
+C2 [-] 
+(95% 
+confidence 
+bounds) 
+C3 [-] 
+(95% 
+confidence 
+bounds) 
+Square  
+0.5439 (0.5296, 0.5582) 
+-2.244 (-2.252, -2.237) 
+1.008 (1.007, 1.009) 
+Dimer 
+2.508 (2.484, 2.532) 
+-3.18 (-3.194, -3.167) 
+0.9796 (0.978, 0.9811) 
+Trimers 
+5.483 (5.382, 5.585) 
+-4.463 (-4.511, -4.414) 
+0.9623 (0.9574, 0.9671) 
+Tetramer  
+5.359 (5.176, 5.543) 
+-4.338 (-4.437, -4.238) 
+0.8921 (0.8809, 0.9034) 
+Pentamer  
+6.352 (6.063, 6.641) 
+-4.828 (-4.94, -4.715) 
+0.97 (0.9621, 0.9779) 
+
+1.5
+b
+a
+Square
+Square
+ • Dimer
+Fit
+Trimer
+0.5
+0.5
+0.2
+0.4
+0.6
+0.2
+0.4
+0.6Hexamer 
+6.572 (6.359, 6.785) 
+-5.038 (-5.116, -4.961) 
+0.9933 (0.9881, 0.9986) 
+ 
+In the case of square, dimers, trimers simulations show that C1 = 0.5439-6.572 and C2 =-2.244-5.038, 
+whereas those parameters for hard circles adsorption are analytically derived as C2 =-4 and C1 = 3.3150 (we 
+obtained coefficients of C1=2.426  (1.071, 3.782) and C2=-2.907  (-3.644, -2.169) with aid of our 
+simulation) . Contrary to discs, the coefficient for squares suggests that they have an easier time adhering 
+to the surface. Higher saturation coverage for squares is another manifestation of this phenomenon. 
+Therefore, dimers values are very close to the values reported for hard discs.  
+ 
+ 
+According to Figure 5, as coverage levels increase, we get closer to the asymptotic phase, where dynamics 
+are well understood, and finally the jamming limit. Thus, the RSA procedure has now been completely 
+explained. Since it is challenging to conduct adsorption investigations near to the jamming limit51, 
+measuring the terms of the RSA process is the most effective technique to show that adsorption follows an 
+RSA process. It's crucial to remember that words up to 𝜃2 don't reveal anything about the properties of the 
+adsorption process (i.e., the degree of irreversibility). This implies that any experiment that involves the 
+adsorption of particles that resemble hard discs is susceptible to such an extension (to second order). Our 
+strategy is also applicable to combinations and non-circular particles.  
+ 
+3.3 Ordering and orientation 
+ 
+ 
+We study the presence of any orientational order in a monolayer using the amorphous form of a polymer. 
+Although most of these studies have used a collector surface lattice structure, as in ref. 52, such ordering has 
+been well investigated. It could also have an impact on the RSA kinetics mentioned before. Based on the 
+polymer structure, we offer the following function to measure orientational order in our continuous system:  
+ 
+𝑆(𝜑) =
+1
+𝑁 ∑
+(𝑥𝑖 cos(𝜑) + 𝑦𝑖 sin(𝜑))
+𝑁
+𝑖=1
+ eq.4 
+ 
+where (xi, yi) are positions along the i-th molecule in a layer for a unit vector. It is clear that 𝑆(𝜑) is an 
+average scalar product between both the orientation of molecules and the direction determined by an angle. 
+As a result, for a perfectly aligned layer, 𝑆(𝜑) will swing between 0 and 1, with highest values for angles 
+parallel to molecules and minimum values for angles perpendicular to the alignment direction. 𝑆(𝜑) will 
+always be a constant and equal to 0.5 for pure random alignment. 
+ 
+For trimers as coverage increases across simulation time, ordering hovered 0.49, 0.53,0.50,0.51 as surface 
+coverage increased from 10 to 30, 40 and 50%. Clearly ordering in trimer population is very close to random 
+due to simulation being designed to give equal chance to parallel or vertical orientation of trimers as shown 
+in Figure 3. Situation is very similar for dimers as well.  
+ 
+Figure 4 illustrates how ordering in trimers may be further examined as a function of the radius of the 
+particle neighbours. Figure 4 was made using a similar idea to the pair correlation function by treating the 
+trimer centre as a circle with a radius of 0.25. 𝑆(𝜑)  fluctuates in small regions because dense clusters of 
+horizontally or vertically oriented trimers are more likely to form, but as the sweeping radius grows, this 
+fluctuation decreases to a value that is very similar to a randomly oriented arrangement. Dimers also face a 
+similar set of circumstances. As a result, at r5, local order in each system vanishes. Small amplitude 
+fluctuations continue after r=5, although their amplitudes and frequency are higher for trimers.  
+ 
+
+ 
+Figure 4. Local ordering as a function of radius for two RSA packings made with dimers and trimers near 
+their saturation coverage.  (a) dimers (b) trimers. 
+ 
+ 
+Similar trends are expected for tetramers due to similarity of behavior we have refrained from exploring 
+them further here.  As an example, liquid crystals are one type of orientationally organised structure that an 
+elongated particle (such as polymers here with aspect ratio>2) can produce. When particle orientations are 
+chosen at random using a uniform probability distribution for RSA on an infinite collector, the global 
+orientational order is not expected to exist. However, since parallelly aligned particles take up less space, 
+the formation of local ordered domains is feasible53-54.  
+ 
+3.4 Radial distribution function  
+ 
+The radial distribution function (G(r)), also known as the pair correlation function, in a system of particles 
+(such as atoms, molecules, colloids, etc.) explains how density changes in response to distance from a 
+reference particle. G(r) is the radial distribution function 55 obtained from following equation:  
+ 
+𝐺(𝑟) =
+1
+𝜌 〈∑
+𝛿(𝑟 − 𝑟𝑖)
+𝑖≠0
+〉 eq.5 
+ 
+ 
+The monolayer's first crucial characteristic is the particle autocorrelation. Squares are assumed to have a 
+radius of 0.25 and to be treated equally regardless of whether they are made of the same polymer or a 
+different one in order to compute G(r). Figure 5 displays the average structures seen by various RSA 
+packings. We consistently saw a peak at a distance of r=0.5 (right on the edge of the particles). In other 
+words, the function reaches its maximum for the closest neighbour, r = 0.5, and then begins to degrade 
+because of the volume that is lacking. The similar trend is seen in the trimer and hexamer, but there are 
+more peaks. Hexamer contains additional peaks, for instance, at r=1, 1.5, and 2, while the trimer exhibits 
+an additional peak at 1.  
+ 
+As the radius gets bigger, these peaks get weaker. Due to the coverage's randomness, these oscillations 
+superexponentially vanish21, and after normalisation, the function stabilises at a value of 1. In addition, 
+when the number of monomers inside the polymer rises, the first peak corresponding to the nearest 
+neighbour grows progressively sharper. According to this behaviour, particles that may be seen as a chain 
+of squares pack more well even if the saturation coverage is smaller for monomers (squares) and short 
+oligomers (dimers).  
+
+.5
+1.5
+b
+a
+S( Φ),Dimers
+.- S( Φ),Trimers
+0.5
++
+0.5
+:
+.
+L
+10
+15
+20
+25
+5
+10
+15
+20
+25 
+Figure 5. Functions of autocorrelation. Behavior autocorrelation function is depicted as a function of radius 
+for square, dimer, trimer, tetramer, pentamer and hexamer.  
+ 
+The average structure seen by a generic particle of the system described by G(r) displayed in Figure 5, 
+shows a full agreement with the predicted theoretical regimes found in literature 56-57. In all cases, we 
+observe a pronounced peak at a distance r~0.5, with the sphere diameter that corresponds to the distance of 
+the nearest neighbors in contact. For r larger than the diameter, the probability to find neighbors decreases. 
+In fluid-like systems, theoretically for 𝜑 ≲ 0.55, 56-58 the G(r) is known to oscillate with decreasing 
+amplitude. 
+ 
+Conclusions  
+ 
+For a range of stiff polymers produced using squared monomers, we show the maximum random coverage 
+or saturation coverage in this paper and contrast our findings with those reported in the literature. In order 
+to do this, we enhanced an algorithm that was described in Ref 32. We prove the validity of our method by 
+calculating the RSA saturation densities of polymers (dimer, trimer, and tetramer) and showing their 
+consistency with prior findings in the literature. 
+The RSA model shown here may be extended to squares that may change into rectangles with larger aspect 
+ratios to incorporate anisotropic particles in future research. Moreover, like ref43 it can also include branched 
+or more flexible polymers. Biological molecules are usually non-spherical, as seen by the previous example, 
+and when their surface area in contact with the substrate is greatest, they firmly cling. According to 
+experimental results, Schaaf et al. 59 discovered that the maximum substrate coverage they were able to 
+achieve during the adsorption of fibrinogen—a non-spherical protein with an aspect ratio of roughly 7.5—
+was only about 40%, which was lower than the absorption coverage predicted by the RSA of hard discs—
+which is around 55%—and seen in experiments involving reasonably spherical globular proteins8 (A similar 
+impact was noted for albumin adsorption 60 ).  
+Here are some pertinent queries: 
+How does increasing the aspect ratio affect the saturation coverage of the substrate? 
+How does the particle shape impact the kinetics over both short and long time periods? 
+
+Square
+5
+. Trimer
+  .Hexamer
+6
+8What are the similarities and differences between equilibrium configurations produced by RSA and 
+configurations with equivalent surface coverage? 
+These questions will get their solutions in upcoming publications. This study's findings are pertinent since 
+they considered a variety of particle morphologies, including those of asphaltene, graphene, cellulose 
+nanocrystals, and kaolinite, among others 61-64.  
+The findings are important because they might help to understand how polymers behave when they are 
+close to surfaces. For instance, numerous biological processes depend on proteins adhering to different 
+surfaces. Understanding and having control over how protein molecules attach to surfaces and interact with 
+them is essential when creating biomaterials. For instance, among other things, the production of 
+biocompatible materials requires decreasing the adsorption of blood proteins to the material's surface. It is 
+generally known that platelet adhesion followed by blood protein adsorption can result in surface-induced 
+thrombosis. When protein adsorption is prevented or diminished, there is very little platelet adhesion to the 
+surface. Eliminating lysozyme buildup from the surface of contact lenses is another illustration. In other 
+circumstances, we would like to promote the adsorption of some proteins while inhibiting the adsorption 
+of others.  
+Conflict of interest statement: Author declares no conflict of interest 
+ 
+Data availability statement: The datasets generated during and/or analysed during the current study are 
+not publicly available but are available from the corresponding author on reasonable request. 
+ 
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+ 
+ 
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+Broadband three-mode converter and multiplexer based on 
+cascaded symmetric Y-junctions and subwavelength engineered 
+MMI and phase shifters 
+David González-Andradea,*, Irene Olivaresb, Raquel Fernández de Caboc, Jaime Vilasb, Antonio 
+Diasb, Aitor V. Velascoc 
+a Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, Palaiseau 91120, France 
+b Alcyon Photonics S.L., Madrid 28004, Spain 
+c Instituto de Óptica Daza de Valdés, Consejo Superior de Investigaciones Científicas (CSIC), Madrid 28006, Spain 
+* Corresponding author: [email protected] 
+ARTICLE INFO 
+ 
+Keywords: 
+Silicon photonics 
+Mode-division 
+multiplexing 
+Subwavelength 
+metamaterial 
+MMI coupler 
+Phase shifter 
+Y-junction 
+ABSTRACT
+ 
+Mode-division multiplexing has emerged as a promising route for increasing 
+transmission capacity while maintaining the same level of on-chip integration. Despite 
+the large number of on-chip mode converters and multiplexers reported for the silicon-
+on-insulator platform, scaling the number of multiplexed modes is still a critical 
+challenge. In this paper, we present a novel three-mode architecture based on 
+multimode interference couplers, passive phase shifters and cascaded symmetric Y-
+junctions. This architecture can readily operate up to the third-order mode by including 
+a single switchable phase shifter. Moreover, we exploit subwavelength grating 
+metamaterials to overcome bandwidth limitations of multimode interference couplers 
+and phase shifters, resulting in a simulated bandwidth of 161 nm with insertion loss 
+and crosstalk below 1.18 dB and -20 dB, respectively.
+1. Introduction 
+The relentless growth of global Internet traffic has been 
+driven in recent years by the emergence of data-hungry 
+services and their mass adoption by an increasingly 
+interconnected society [1-3]. Moreover, the cloud nature 
+of many new applications such as machine learning or 
+artificial intelligence require large data sets to be 
+processed on internal servers or transferred between data 
+centers. This resource-intensive paradigm for accessing, 
+computing, and storing data has led to the creation of 
+hyperscale data centers consisting of thousands of 
+servers located in the same physical facility [4]. To cope 
+with the resulting zetta scale of annual data flow, modern 
+data centers have been relying on optical technologies for 
+both long-haul and few-meter interconnects. Compared 
+to 
+their 
+electronic 
+counterparts, 
+these 
+optical 
+technologies offer higher processing speeds, broader 
+bandwidths and lower latency and energy consumption. 
+Silicon photonics, leveraging the mature fabrication 
+facilities of the microelectronics industry, plays a key 
+role in the optical interconnect industry due to its 
+capacity for high-yield and low-cost mass production of 
+high-performance optoelectronic circuits [5,6]. 
+However, the development of next-generation 
+datacenters for Tbps communications and exascale 
+computing systems 
+is not feasible by scaling 
+infrastructures alone and requires increasingly efficient 
+optical interconnects for short-reach distances [7]. As 
+single-mode transmission approaches its fundamental 
+limits, space-division multiplexing has emerged as a 
+promising way to further improve the transmission 
+capacity of optical interconnects through the use of 
+multicore or multimode waveguides [8]. The latter, 
+which is also called mode-division multiplexing (MDM), 
+has attracted an increasing interest as it leverages the 
+orthogonality of the eigenmodes supported by a single 
+multimode waveguide, thus allowing to maintain the 
+same level of on-chip integration [9,10]. That is, MDM 
+enables encoding different data channels into specific 
+spatial modes, increasing capacity proportionally to the 
+number of modes used. 
+Numerous 
+on-chip 
+mode 
+converters 
+and 
+multiplexers/demultiplexers 
+(MCMD) 
+have 
+been 
+proposed for the silicon-on-insulator (SOI) platform to 
+date. Asymmetric Y-junctions are based on the principle 
+of mode evolution in adiabatic structures, which results 
+in broad operating bandwidths but also in long device 
+lengths [11-13]. The minimum feature size of current 
+lithography processes also has a significant impact in 
+these devices, since the finite resolution at which the tip 
+can be fabricated severely hampers their performance. 
+Asymmetric directional couplers (ADCs) [14], relying 
+on evanescent coupling between adjacent waveguides, 
+are well suited for implementing high-channel count 
+MDM systems, but they typically exhibit narrow 
+bandwidths, and their performance is highly susceptible 
+to fabrication errors. Adiabatic tapers have been 
+employed in the coupling region of ADCs to improve the 
+bandwidth and the resilience against fabrication 
+deviations [15]. MCMDs building upon multimode 
+interference (MMI) couplers and other auxiliary 
+
+components such as phase shifters (PSs) and symmetric 
+Y-junctions have been proposed as well [16,17], yielding 
+low losses and low crosstalk over a relatively broad 
+wavelength range (~100 nm). 
+The patterning of silicon at the subwavelength scale 
+has proven to be a simple yet powerful tool to tailor the 
+medium optical properties while inhibiting diffractive 
+effects [35]. More specifically, subwavelength (SWG) 
+metamaterials 
+can 
+behave 
+as 
+a 
+homogeneous 
+metamaterial that provides flexible dispersion and 
+anisotropy engineering, non-feasible in conventional 
+strip and rib waveguides. These properties have led to the 
+realization 
+of 
+Si 
+devices 
+with 
+unprecedented 
+performance over the past 15 years [36-38]. In the MDM 
+field, MCMDs based on subwavelength pixelated 
+structures have demonstrated ultra-compact footprints 
+[18]. SWGs have also been applied to ADCs and triple-
+waveguide couplers to improve fabrication tolerances 
+and extend the operation bandwidth of conventional 
+counterparts [19-21]. Furthermore, low losses and low 
+crosstalk within ultra-broad bandwidths have also been 
+reported using subwavelength engineered MMI couplers 
+and PSs, and SWG-slot-assisted adiabatic couplers [22-
+26]. 
+Despite the large number of available two-mode 
+MCMDs, scaling the number of multiplexed modes 
+beyond the fundamental and first-order modes is of great 
+importance to multiply the capacity of next-generation 
+datacom systems. Although it is fairly straightforward to 
+extend operation to a larger number of modes in 
+asymmetric Y-junctions and conventional and tapered 
+ADCs [27], three- and four-mode MCMD based on MMI 
+couplers have only recently been reported [28-32]. 
+However, the proposed architectures are still limited by 
+narrow operating bandwidths and high crosstalk values. 
+In this work, we propose a novel MCMD architecture 
+based on a 4×4 MMI, three phase shifters and four 
+symmetric 1×2 Y-junctions arranged in a conventional 
+cascaded configuration. The device operates as a three-
+mode MCDM with passive phase shifters but can readily 
+convert up to the third-order mode by including a single 
+switchable phase shifter. Moreover, we demonstrate loss 
+and crosstalk reduction in a broad bandwidth by SWG-
+engineering of both the MMI coupler and phase shifters. 
+Simulations show operation bandwidth of 161 nm with 
+insertion loss and crosstalk below 1.18 dB and -20 dB, 
+respectively. 
+2. Principle of operation and device design 
+To explain the operation principle and the device design, 
+let us first focus on the nanophotonic structure shown in 
+Fig. 1(a) consisting of a conventional 4×4 MMI, three 
+phase shifters (PS1, PS2 and PS3) and four symmetric 
+1×2 Y-junctions (three identical Y1 and a different one 
+Y2). SWG enhancement of the proposed architecture, 
+shown in Fig. 1(b) and Fig. 1(d) will be discussed in 
+epigraphs 4 and 5. An SOI platform with a thin Si wire 
+surrounded by SiO2 bottom layer and upper cladding are 
+considered. A schematic view of the waveguide cross-
+section is shown in Fig. 1(c) for clarity. 
+In order to illustrate the operation of the MCMD, let 
+us focus on the mode evolution and phase relations in 
+each individual constituent of the MCMD. Here, we aim 
+ 
+Fig. 1. Three-dimensional schematic of the proposed three-mode converter and multiplexer/demultiplexer comprising a 4×4 MMI, three 
+phase shifters and four symmetric Y-junctions implemented with (a) conventional homogeneous and (b) SWG metamaterial waveguides. 
+(c) Cross-sectional view of the SOI strip waveguides with a SiO2 cladding. (c) Top view of the SWG waveguides with their main 
+geometrical parameters. 
+
+(a)
+Wi
+MMI
+PS2
+W
+Y1
+3.
+2W,
+WA
+Y2
+Wps
+tWs
+Y1
+2W,
+P
+2
+PS1<Lpsi
+WMMI
+Y1
+4W1
+LpS2
+4
+2 W,
+PS3
++ Lyl
+LpS3
+1
+Ly2
+Wi
+Wi
+Lyl
+Li
+LMMI
+PS3
+Z
+X
+(b)
+SWG MMI Ws
+WI+
+SPS2
+SPS3
+WR3
+Y1
+D2
+2Wi
+3
+Y2
+Y1
+D
+2W,
+2
+WR2
+Y1
+4WI
+WRI
+4
+2Wi
+Lyl
+Ly2
+Wi
+Ly1
+SPS1
+LsT LsMMI
+LSPS3
+1
+(c)
+(d)
+D
+Si
+H
+Z4
+yt
+SiO2
+y
+W
+xat mode conversion and multiplexing of the first four 
+modes for transverse-electric (TE) polarization, that is, 
+the fundamental mode (TE0), the first-order mode (TE1), 
+the second-order mode (TE2) and the third-order mode 
+(TE3). 
+Our MCMD includes two types of symmetric 
+multimode 1×2 Y-junctions: Y1, with a stem supporting 
+up to two modes; and Y2, with a wider stem supporting 
+up to four modes. In general, multimode symmetric 1×2 
+Y-junctions transform the two in-phase 𝑚𝑡ℎ-order modes 
+in the arms into the (2𝑚)𝑡ℎ-order mode in the stem when 
+𝑚 is even, and into the (2𝑚 + 1)𝑡ℎ-order mode in the 
+steam when 𝑚 is odd [33]. Likewise, two anti-phase 
+𝑚𝑡ℎ-order modes in the arms are transformed into the 
+(2𝑚 + 1)𝑡ℎ-order mode in the stem when 𝑚 is even, and 
+into the (2𝑚)𝑡ℎ-order mode in the stem when 𝑚 is odd. 
+Figure 2(a) illustrates how this principle affects Y1 
+operation. Since only two modes are supported by the Y1 
+stem, a TE0 (red) mode at the stem results in two in-phase 
+TE0 modes at the arms, whereas TE1 (orange) mode at 
+the stem results in two anti-phase TE0 modes at the arms. 
+Figure 2(b) shows the extension of this behavior to four 
+mode operation in Y2. Operation for TE0 (red) and TE1 
+(orange) is the same as in Y1, whereas injection of TE2 
+(green) and TE3 (purple) modes through the stem 
+waveguide generates two anti-phase TE1 or two in-phase 
+TE1 modes at the arms, respectively. Therefore, by 
+cascading Y1 and Y2, and judiciously tailoring the phase 
+relations induced by the rest of the MCMD, mode 
+conversion and multiplexing between up to four modes 
+can be achieved. We will hence study the phase shift 
+induced by the 4×4 MMI coupler, and subsequently 
+design a phase shifter architecture that satisfies the phase 
+distributions imposed by the cascaded Y-junctions. 
+Bachmann et al. already derived a set of equations to 
+calculate the phase relations of 𝑁×𝑁 MMI couplers [34]. 
+At this point, it is important to mention that the definition 
+of the phase in this work is 𝜑 = 𝛽𝑥 − 𝜔𝑡, where 𝛽 is the 
+phase constant (also known as propagation constant), 𝑥 
+is the propagation direction and the term −𝜔𝑡 
+corresponds to the temporal dependence. As in [34] the 
+authors used the opposite phase convention, i.e., = 𝜔𝑡 −
+𝛽𝑥, equations can be rewritten as follows: 
+𝑖 + 𝑗 even: 𝜑𝑖𝑗 = −𝜑0 − 𝜋 −
+𝜋(𝑗−𝑖)(2𝑁−𝑗+𝑖)
+4𝑁
+, 
+(1) 
+𝑖 + 𝑗 odd: 𝜑𝑖𝑗 = −𝜑0 −
+𝜋(𝑗+𝑖−1)(2𝑁−𝑗−𝑖+1)
+4𝑁
+, 
+(2) 
+where 𝜑0 is a constant phase, 𝑖 and 𝑗 are the indices of 
+the 𝑁 inputs and outputs, respectively. Using Eqs. (1) and 
+(2), the phase relations of a 4×4 MMI coupler can be 
+calculated as shown in Table 1. Please note the input and 
+output numbering in Fig. 3. 
+Table 1 
+Calculated phase relations 𝝋𝒊𝒋 of a 4×4 MMI coupler. 
+𝒋 
+𝒊 
+1 
+2 
+3 
+4 
+1 
+−𝜋 
+−3𝜋 4
+⁄  
+−7𝜋 4
+⁄  
+−𝜋 
+2 
+−3𝜋 4
+⁄  
+−𝜋 
+−𝜋 
+−7𝜋 4
+⁄  
+3 
+𝜋 4
+⁄  
+−𝜋 
+−𝜋 
+−3𝜋 4
+⁄  
+4 
+−𝜋 
+𝜋 4
+⁄  
+−3𝜋 4
+⁄  
+−𝜋 
+ 
+We then calculate, for each input port, the resulting 
+phase difference at the two upper output ports (∆𝜑12) and 
+the two lower ports (∆𝜑34) as: 
+∆𝜑12 = 𝜑𝑖1 − 𝜑𝑖2, 
+(3) 
+∆𝜑34 = 𝜑𝑖3 − 𝜑𝑖4, 
+(4) 
+Calculated phase differences are shown in Table 2. 
+Since phase evolution at both the MMIs and Y-splitters 
+are fixed, we then need to design a combination of PSs 
+(placement and phase shift values), that results in the 
+required phase relations. As shown in Figure 1(a), we 
+achieve this goal by including a first phase shifter (PS1) 
+between inputs 3 and 2 of the MMI, with a phase shift of 
+𝜋 2
+⁄ ; a second phase shifter (PS2) between outputs 2 and 
+1, with a phase shift of − 𝜋 4
+⁄ ; and a third phase shifter 
+(PS3) between outputs 3 and 4 with a phase shift of 
+3 𝜋 4
+⁄ . An additional two-mode Y-junction (Y1) is 
+included at MCMD port 2 to satisfy even-order modes 
+phase conditions, as discussed hereunder. 
+Figure 4 shows the operation of the device working 
+in multiplexer configuration, including the value of the 
+phase relations at different locations for clarity. It should 
+be noted that phase values have been calculated with  
+ 
+ 
+Fig. 2. Schematic and principle of operation of a multimode 
+symmetric 1×2 Y-junction for (a) a two-mode stem, and (b) a 
+four-mode stem. 
+ 
+Fig. 3. Schematic of a 4×4 MMI coupler, illustrating port and 
+phase notations. 
+
+(a)
+TEo TEo
+TEo
+Y1
+TE1
+TEo TEo
+y4
+x
+(b)
+TE, TEo
+TE2 TEo
+TE TEo
+Y2
+TE, TEo
+TE,
+TE1
+TE, TEo
+yt
+xInputs (i)
+Outputs (i)
+4
+3
+4×4 MMI
+2
+△34Table 2 
+Calculated phase differences between MMI output pairs for each input. 
+𝒊 
+∆𝝋𝟏𝟐 
+∆𝝋𝟑𝟒 
+1 
+−𝜋 4
+⁄  
+−3𝜋 4
+⁄  
+2 
+𝜋 4
+⁄  
+3𝜋 4
+⁄  
+3 
+5𝜋 4
+⁄  
+−𝜋 4
+⁄  
+4 
+−5𝜋 4
+⁄  
+𝜋 4
+⁄  
+ 
+with respect to the mode phase at the input ports, which 
+is considered to be zero for simplicity. When light is 
+injected through MCMD port 1 [Fig. 4(a)], the 
+combination of all the aforementioned phase relations 
+results in all modes arriving in-phase at the arms of the 
+cascaded Y-junctions. Thus, two in-phase TE0 modes are 
+coupled into Y1 stems, which subsequently generate the 
+desired TE0 mode at the multimode stem waveguide of 
+Y2 (MCMD port 4). 
+When light is injected through MCMD port 2 [Fig. 
+4(b)], the combination of Y1 and PS1 results in 
+simultaneous light coupling to MMI input ports 2 and 3, 
+but with a 𝜋 2
+⁄  phase difference. This in turn generates 
+in-phase modes in the upper arms that are in anti-phase 
+with the two in-phase modes in the lower arms at their 
+arrival at the cascaded Y-junctions. This combination 
+results in TE1 generation at the MCMD output. 
+Finally, when light is injected through MCMD port 3 
+[Fig. 4(c)], that is, MMI input port 4, in-phase modes are 
+generated in the middle arms, which are in anti-phase 
+with the two in-phase modes generated in the top and 
+bottom arms, before the cascaded Y-junctions. This 
+results in anti-phase TE1 modes at the output of Y1 
+stems, which subsequently generate the TE2 mode at the 
+MCDM output. 
+So far, we have only considered passive PSs, that is, 
+PSs with a fixed phase shift. However, if the phase 
+introduced by PS1 is 3𝜋 2
+⁄  instead of 𝜋 2
+⁄ , it is possible 
+to generate the TE3 mode at MCMD output [Fig. 4(d)]. 
+For illustrative purposes, we represent this phase shift by 
+switching the position of the tapers in PS1. This feature 
+opens the possibility of extending MCMD operation to 
+four modes using a single switchable PS. 
+3. Proof-of-concept results 
+To verify the principle of operation explained in the 
+previous section, we firstly optimized each constituent 
+(i.e., MMI, phase shifters and Y-junctions) for a design 
+wavelength of 1550 nm. We chose a standard silicon 
+thickness of 𝐻 = 220 nm and an interconnection 
+waveguide width of 𝑊𝐼 = 400 nm. Thus, symmetric Y-
+junctions are designed with stem widths of 2𝑊𝐼 =
+800 nm for Y1 and 4𝑊𝐼 = 1600 nm for Y2. 
+Geometrical parameters of the 4×4 MMI coupler, the 
+phase shifters and the symmetric Y-junction are 
+summarized in Table 3. 
+In order to evaluate the performance of each 
+constituent element, the figures of merit for the MMI are 
+the excess loss (EL), imbalance (IB) and phase error 
+(PE): 
+EL𝑖 [dB] = −10log10 (∑ |S𝑗𝑖|
+2
+𝑗
+), 
+(5) 
+IB𝑖
+𝑗𝑘 [dB] = 10log10 (|S𝑗𝑖|
+2 |S𝑘𝑖|2
+⁄
+), 
+(6) 
+PE𝑖
+𝑗𝑘[°] = [∠(S𝑗𝑖 S𝑘𝑖
+⁄
+) − 𝜑𝑖𝑑𝑒𝑎𝑙] · 180 π
+⁄ , 
+(7) 
+where S𝑗𝑖 and S𝑘𝑖 are the scattering parameters for input 
+𝑖 and outputs 𝑗 and 𝑘, and 𝜑𝑖𝑑𝑒𝑎𝑙 is the ideal phase 
+relation depending on selected input and output ports as 
+shown in Table 1. The designed 4×4 MMI exhibits EL <  
+ 
+Table 3 
+Geometrical parameters of the three-mode converter and 
+multiplexer/demultiplexer with homogeneous waveguides. 
+Constituent 
+Parameter 
+ 
+Value 
+Waveguides 
+Width 
+𝑊𝐼 
+400 nm 
+MMI 
+Separation 
+𝑊𝑆 
+500 nm 
+Access width 
+𝑊𝐴 
+1.3 µm 
+Taper length 
+𝐿𝑇 
+6 µm 
+MMI width 
+𝑊𝑀𝑀𝐼 7.2 µm 
+MMI length 
+𝐿𝑀𝑀𝐼 91 µm 
+Y1 
+Arm width 
+𝑊𝐼 
+400 nm 
+Arm length 
+𝐿𝑌1 
+5 µm 
+Stem width 
+2𝑊𝐼 
+800 nm 
+Y2 
+Arm width 
+2𝑊𝐼 
+800 nm 
+Arm length 
+𝐿𝑌2 
+20 µm 
+Stem width 
+4𝑊𝐼 
+1.6 µm 
+PS1 
+PS width 
+𝑊𝑃𝑆1 600 nm 
+PS length 
+𝐿𝑃𝑆1 
+2.41 µm 
+PS2 
+PS width 
+𝑊𝑃𝑆2 600 nm 
+PS length 
+𝐿𝑃𝑆2 
+8.38 µm 
+PS3 
+PS width 
+𝑊𝑃𝑆3 600 nm 
+PS length 
+𝐿𝑃𝑆3 
+3.61 µm 
+ 
+Fig. 4. Principle of operation of the proposed three-mode 
+converter and multiplexer/demultiplexer for (a) TE0, (b) TE1, 
+(c) TE2 and (d) TE3 mode multiplexing. 
+
+(a)
+TE, multiplexing
+3
+TEo
+-T元
+一元
+T
+2
+3元/4-元
+f
+F7元/4
+TEo
+T
+.
+(b)
+TE, multiplexing
+3
+T
+TE,
+元
+T
+元/2
+3元/4 一元
+TEo
+0
+0
+3元/4
+0
+1
+0
+0
+(c)
+TE, multiplexing
+TE2
+3
+TEo
+一元
+2
+元/4
+0
+0
+→
+3元／40
+1
+T
+(d)
+TE, multiplexing
+0
+TE3
+3
+0
+0
+0
+F3元/4—元
+TEo
+0
+0
+元/2
+3元/4
+1
+元
+yt
+x0.54 dB, IB < ±0.4 dB 
+and PE < ±0.32° 
+at 
+the 
+wavelength of 𝜆0 = 1550 nm. Regarding the spectral 
+response, 
+EL < 2.15 dB, 
+IB < ±8.1 dB 
+and PE <
+±46.03° are attained in the entire simulated wavelength 
+range (1.45 – 1.65 µm). 
+Designed phase shifters introduce small phase 
+deviations of only 0.12° for PS1, 0.13° for PS2 and 
+0.16° for PS3, with respect to their target phase 
+difference at 1550 nm. However, considering the 
+simulated bandwidth of 200 nm, phase errors increase up 
+to 9.84° for PS1, 22.28° for PS2, and 12.12° for PS3. 
+Symmetric Y-junctions Y1 and Y2 were also 
+designed showing negligible excess losses and power 
+imbalance between output ports at the design 
+wavelength. More specifically, Y1 losses are lower than 
+0.01 dB for both TE0 and TE1 mode operation in the 1.45 
+– 1.65 µm wavelength range. Conversely, calculated 
+excess losses for Y2 are below 0.15 dB for TE0, TE1, TE2 
+and TE3 mode operation within the same bandwidth. 
+Once all elements were optimized, two-dimensional 
+finite-difference time-domain (FDTD) simulations of the 
+whole MCMD were performed by applying the effective 
+index method to the original three-dimensional structure 
+[see Fig. 1(a)]. The simulated field distribution of the 
+three-mode MCMD is shown in Fig. 5(a)-(d), 
+demonstrating the successful implementation of the 
+phase relations described in section 2. 
+Some ripples can be observed for TE0 and TE2 mode 
+multiplexing in the stem waveguide of Y-junction Y2 
+[see Figs. 5(a) and 5(c)], which we attribute to a higher 
+crosstalk between both modes compared to TE1 and TE3 
+mode multiplexing. 
+The transmittance as a function of the wavelength 
+was computed for the complete MCMD [see Fig.5(e)-
+(h)]. At the central wavelength of 𝜆0 = 1550 nm, 
+insertion losses are lower than 0.53 dB, 0.79 dB and 0.59 
+dB for the generation of TE0, TE1 and TE2 modes in the 
+stem waveguide, respectively. Our device also exhibits a 
+low crosstalk at the same wavelength with values below 
+-21.61 dB for TE0, -28.94 dB for TE1 and -21.11 dB for 
+TE2. 
+By tuning the value of PS1 to 3 𝜋 2
+⁄ , TE3 mode 
+(instead of TE1 mode) can be generated. In this case, 
+insertion losses are below 0.75 dB, and the crosstalk is 
+better than -28.79 dB, both at 1550 nm. These results 
+corroborate the higher crosstalk for TE2 mode operation, 
+which leads to a slight ripple in the field distribution at 
+port 4. 
+Regarding performance across the spectrum, insertion 
+losses lower than 1 dB are attained for a 55 nm 
+bandwidth (1542 – 1597 nm), whereas the crosstalk is 
+below -20 dB for a 60 nm bandwidth (1537 –1597 nm) 
+as shown with vertical lines in Fig. 5. These results prove 
+the correct operation of the proposed architecture, but the 
+overall bandwidth is significantly limited by the narrow 
+spectral response of both the MMI and PSs. 
+4. SWG performance enhancement 
+To overcome these bandwidth limitations, we 
+propose the MCMD with SWG metamaterials shown in 
+Fig. 1(b). The design of each of the constituents of the 
+SWG MCMD was performed by individual three-
+dimensional FDTD simulations. The three symmetric Y-
+junction labeled Y1 maintain the same geometrical 
+dimensions as those used for the conventional 
+multiplexer for the arm and stem widths (see Table 3), 
+but arm length was shortened to 𝐿𝑌1 = 2 µm. Y-junction 
+Y2 was slightly redesigned to reduce the crosstalk 
+between TE0 and TE2 modes by increasing the length of 
+the arms to 𝐿𝑌2  =  40 μm. 
+A procedure similar to those already reported in 
+[39,40] was followed for the optimization of the 4×4 
+SWG MMI. We restrict the value of the duty cycle (DC =
+𝑎 Λ
+⁄ ) to 0.5 in order to maximize the minimum feature 
+size for a given period (Λ) [see Fig.1(d)]. We explored 
+then different periods and found that Λ = 222 nm 
+significantly flattens the beat length across the spectrum. 
+Compared to the conventional MMI section design, the 
+width 𝑊𝑆𝑀𝑀𝐼 is increased by 0.8 µm but the length 𝐿𝑆𝑀𝑀𝐼 
+is reduced by more than half to ~41.3 µm. To increase 
+the quality of the interferometric patterns formed in the 
+MMI, the access width is 𝑊𝐵 = 1.7 µm and the 
+ 
+Fig. 5. Electric field amplitude |𝐸| in the XY plane at the middle 
+of the silicon layer for (a) TE0, (b) TE1, (c) TE2 and (d) TE3 
+mode multiplexing. Simulated transmittance to output port 4 as 
+a function of the wavelength when TE0 mode is launched into 
+(e) input port 1, (f) input port 2 with PS1 = 𝜋 2
+⁄ , (g) input port 
+3 and (h) input port 2 with PS1 = 3𝜋 2
+⁄ . Vertical lines indicate 
+the bandwidth where IL < 1 dB (55 nm) and XT < −20 dB (60 
+nm) are achieved for all modes simultaneously. 
+
+(a)
+(b)
+(c)
+(d)
+200
+0.8
+150
+0.6
+X
+0.4
+50
+0.2
+0
+420-2-4 420-2-4420-2-4420-2-4
+y (μm)
+y (μm)
+y (μm)
+y (μm)
+(e)
+Input 1
+(G)
+Input 2 (PS1 = π/2)
+(dB)
+0
+(dB)
+55 nm
+55nm
+Transmittance
+Transmittance
+60 nm
+60 nm
+-20
+-20
+40
+TE
+40
+TE
+1.45
+1.5
+1.55
+1.6
+1.65
+1.45
+1.5
+1.551.61.65
+Wavelength (um)
+Wavelength (um)
+(g)
+(h)
+Input 3
+Input2(PS1=3元/2)
+(dB)
+0
+(dB)
+55 nm
+55 nm
+Transmittance
+Transmittance
+60 nm
+60 nm
+-20
+20
+TE,
+TE
+-40
+40
+TE
+TE
+TE3
+1.45
+1.5
+1.55
+1.6
+1.65
+1.45
+1.5
+1.551.6
+51.65
+Wavelength(um
+Wavelength (um)separation is reduced to 𝑊𝑅 = 0.3 µm. The transition 
+between 
+the 
+interconnection 
+waveguides 
+(𝑊𝐼 =
+400 nm) and the access to the MMI section (𝑊𝐵) is 
+performed by means of adiabatic SWG tapers with a 
+length 𝐿𝑆𝑇 = 13.32 µm. The performance of the 4×4 
+SWG MMI is shown in Fig. 6(a)-(c). Owing to the 
+symmetry of the structure, only the results obtained when 
+injecting light into input ports 1 and 2 are depicted. It is 
+observed that the device exhibits EL < 0.77 dB, IB <
+±1 dB and PE < ±8.02° within a broad bandwidth of 
+200 nm (1.45 – 1.65 µm). 
+To drastically extend the operating bandwidth of the  
+ 
+nanophotonic phase shifters, we build upon the strategy 
+we recently reported in [41] to develop SWG phase 
+shifters SPS1, SPS2 and SPS3. Notwithstanding, here we 
+employ four parallel SWG waveguides of two different 
+widths to implement SPS2 and SPS3. That is, each PS 
+has three identical reference SWG waveguides with 
+width 𝑊𝐷, and one dissimilar SWG waveguide with 
+width 𝑊𝑅. Both the reference and dissimilar waveguides 
+have a length of LSPS. Note that for SPS1 this 
+configuration is not necessary as only two MMI inputs 
+are illuminated for TE1 and TE3 mode generation. 
+Analogous to the 4×4 SWG MMI, a flat phase shift can 
+be achieved by judicious selecting the SWG period and 
+duty cycle. A duty cycle of 0.5 was fixed to maximize 
+MFS, while a period of 200 nm resulted in minimum 
+phase shift deviation. In order to induce 𝜋 4
+⁄ , 𝜋 2
+⁄ , and 
+3𝜋 4
+⁄  phase shifts, we selected respectively 𝑊𝐷2 =
+1.8 µm, 𝑊𝑅2 = 1.6 µm, 𝐿𝑆𝑃𝑆2 = 6.2 µm and 𝐿𝑆𝑇2 =
+3.0 µm for SPS2; 𝑊𝐷1 = 1.8 µm, 𝑊𝑅1 = 1.6 µm, 
+𝐿𝑆𝑃𝑆1 = 16.8 µm and 𝐿𝑆𝑇1 = 3.0 µm for SPS1; and 
+𝑊𝐷3 = 1.8 µm, 𝑊𝑅3 = 1.6 µm, 𝐿𝑆𝑃𝑆3 = 28.2 µm and 
+𝐿𝑆𝑇3 = 3.0 µm for SPS3. The simulated phase shifts are 
+shown in Fig. 6(d). Negligible deviations can be 
+appreciated with phase shift errors as small as 2.29° for 
+SPS1, and 1.15° for SPS2 and SPS3 within the entire 
+1.45 – 1.65 µm wavelength range. 
+5. SWG results 
+The simulation of the entire MCMD is quite 
+resource-intensive and time-consuming due to the device 
+footprint and the need for a fine mesh to simulate SWG-
+based devices. Thus, instead of performing the full 
+device simulation, we leverage the S-parameter matrices 
+calculated during the design process and concatenate all 
+of them using a circuit simulator to obtain the S-
+ 
+Fig. 7. Simulated transmittance as a function of the wavelength of the MCMD with SWG metamaterials when TE0 mode is launched 
+into (a) input port 1, (b) input port 2 with SPS1 = 𝜋 2
+⁄ , (c) input port 3 and (d) input port 2 with SPS1 = 3𝜋 2
+⁄ . Vertical lines indicate 
+the bandwidth where IL < 1 dB (183 nm) and XT < −20 dB (161 nm) are achieved for all modes simultaneously. 
+ 
+Fig. 6. Simulated performance of the 4×4 SWG MMI including 
+(a) excess loss, (b) imbalance and (c) phase error between 
+output ports. (d) Phase error of each SWG PSs as a function of 
+the wavelength. 
+
+(a)
+Input 1
+(b)
+Input 2 (SPS1 = π/2)
+0
+(dB)
+(dB)
+0
+183 nm
+183 nm
+161 nm
+161 nm
+20
+Transmittance
+Transmittance 
+20
+TE
+TEo
+-40
+40
+TE,
+TE,
+-60
+TE,
+-60
+T-80
+TE
+-80
+TE
+1.45
+1.5
+1.55
+1.6
+1.65
+1.45
+1.5
+1.55
+1.6
+1.65
+Wavelength (μm)
+Wavelength (um)
+(c)
+Input 3
+(d)
+Input 2 (SPS1 = 3π/2)
+(dB)
+183 nm
+(dB)
+0
+183 nm
+161 nm
+161 nm
+-20
+20
+Transmittance 
+40
+4
+TE
+-60
+TE
+60
+TH
+TE
+TE
+TE.
+-80
+1.45
+1.5
+1.55
+1.6
+1.65
+1.45
+1.5
+1.55
+1.6
+1.65
+Wavelength (um)
+Wavelength (um)(a)
+(b)
+(dB)
+Input1 (EL)
+(dB)
+—Input 1: 0,/02 (IB12)
+Input2(EL2)
+Input 1: O3/O4 (IB24)
+Imbalance (
+Input 2: O,/0, (IB22)
+2
+Input 2:0,/04 (IB34)
+0
+1.45
+1.5
+1.55
+1.6
+1.65
+1.45
+1.5
+1.551.6
+1.65
+Wavelength (μm)
+Wavelength (um)
+(c)
+(d)
+10
+Input1:012
+10
+SPS2 (元/4rad)
+Input 1: Ap
+SPS1 (元/2rad)
+error
+error
+5
+0
+Phase 
+Input 2: A934
+-5
+SPS1 (3元/2rad)
+Input2:012
+-SPS3 (3元/4rad)
+-10
+-10
+1.45
+1.5
+1.551.6
+1.65
+1.45
+1.5
+1.551.61.65
+Wavelength (um)
+Wavelength (um)parameter matrix and hence the spectral response of the 
+complete device. The circuit simulator enables 
+bidirectional signals to be accurately simulated, 
+including coupling of modes in the single elements. 
+Figure 7 shows the overall transmittance of the SWG 
+MCMD. Insertion losses (ILs) are lower than 0.37 dB, 
+0.47 dB and 0.37 dB for TE0, TE1 and TE2 multiplexing, 
+respectively, at the central wavelength of 𝜆0 =
+1550 nm. Moreover, low crosstalk (XT) is achieved at 
+the same wavelength with values below -21.54 dB for 
+TE0, -32.89 dB for TE1 and -21.24 dB for TE2 
+multiplexing. 
+When SPS1 takes the value of 3 𝜋 4
+⁄ , insertion losses 
+for TE3 multiplexing reach a low value of 0.47 dB at 
+1550 nm, while crosstalk values are lower than -39.48 dB 
+for the same wavelength. 
+This design also shows an excellent performance 
+over a broad bandwidth (BW) of 200 nm with insertion 
+loss lower than 1.18 dB and crosstalk below -16.53 dB. 
+Insertion losses decrease to 1 dB when the bandwidth is 
+restricted to 183 nm (1450 – 1633 nm), whereas a 
+crosstalk below -20 dB is achieved over a 161 nm 
+bandwidth (1489 – 1650 nm). For the sake of 
+comparison, Table 4 summarizes the performance of 
+other three- and four-mode MCMD that are based on 
+MMI couplers and have been reported in the state of the 
+art. To the best of our knowledge, it is the first time such 
+low losses and crosstalk are achieved in an outstanding 
+161 nm wavelength range. 
+ 
+6. Conclusions 
+In this work, we have proposed a novel architecture to 
+scale the number of multiplexed modes of mode 
+converters and multiplexer based on MMI couplers. 
+Unlike 
+other 
+reported 
+architectures 
+that 
+use 
+unconventional 1×4 Y-junctions or 1×3 Ψ-junctions, 
+here we employ symmetric 1×2 Y-junctions arranged in 
+a conventional cascaded configuration. The design 
+methodology was proposed on the basis of a two-
+dimensional model with conventional homogenous 
+components (i.e., without patterning the silicon 
+waveguide). The conventional mode converter and 
+multiplexer features sub-decibel insertion loss and 
+crosstalk better than -20 dB in the 1542 – 1597 nm 
+wavelength range. Once the principle of operation was 
+verified, we redesigned and optimized the mode 
+converter 
+and 
+multiplexer 
+by 
+incorporating 
+subwavelength grating metamaterials to leverage the 
+additional degrees of freedom they introduced into the 
+design. A broad design bandwidth of 161 nm for 
+insertion losses below 1.18 dB and crosstalk lower than 
+-20 dB was confirmed by 3D FDTD simulations, 
+comparing very favorably to state-of-the-art three- and 
+four-mode converters and multiplexers. The crosstalk 
+between TE0 and TE1 modes could be further reduced by 
+including optimized Y-junction geometries that mitigate 
+the effect of the non-perfect tip at the junction [42-44]. 
+We believe that our design strategy will open promising 
+prospects for the development of high-performance 
+mode converters and multiplexer based on MMI couplers 
+with a high channel count. 
+Credit authorship contribution statement 
+David 
+González-Andrade: 
+Conceptualization, 
+Methodology, Software, Validation, Formal analysis, 
+Investigation, Data curation, Writing – original draft, 
+Visualization. Irene Olivares: Methodology, Software, 
+Validation, Formal analysis, Data curation, Writing – 
+review & editing. Raquel Fernández de Cabo: 
+Software, Validation, Data curation, Writing – review & 
+editing. Jaime Vilas: Writing – review & editing. 
+Antonio Dias: Resources, Writing – review & editing, 
+Project administration, Funding acquisition. Aitor V. 
+Velasco: Resources, Writing – review & editing, 
+Supervision, 
+Project 
+administration, 
+Funding 
+acquisition. 
+Declaration of Competing Interest 
+The authors declare that they have no known competing 
+financial interests or personal relationships that could 
+have appeared to influence the work reported in this 
+paper. 
+Acknowledgements 
+This work has been funded in part by the Spanish 
+Ministry of Science and Innovation (MICINN) under 
+grants RTI2018-097957-B-C33, PID2020-115353RA-
+I00; 
+the 
+Spanish 
+State 
+Research 
+Agency 
+(MCIN/AEI/10.13039/501100011033); the Community 
+of Madrid – FEDER funds (S2018/NMT-4326); the 
+European Union – NextGenerationEU through the 
+Recovery, 
+Transformation 
+and 
+Resilience 
+Plan 
+(DIN2020-011488, 
+PTQ2021-011974); 
+and 
+the 
+European Union's Horizon Europe research and 
+innovation program under the Marie Sklodowska-Curie 
+grant agreement Nº 101062518. 
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+Ref. 
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+[dB] 
+XT 
+[dB] 
+BW 
+[nm] 
+Length 
+[µm] 
+No. of 
+modes 
+[28] 
+<0.71 
+<-18 
+100 
+400 
+3 
+[29] 
+<0.9 
+<-17 
+40 
+400 
+3 
+[30] 
+<1.1 
+<-10 
+90 
+17.05 
+3 
+[31] 
+<1.3 
+<-15 
+40 
+120 
+3 
+[32] 
+<3.3 
+<-14 
+70 
+173 
+4 
+[32] 
+<2.25 
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+140 
+123 
+4 
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+Code-based Cryptography in IoT:
+A HW/SW Co-Design of HQC
+Maximilian Sch¨offel
+Microelectronic Design Research Group
+University of Kaiserslautern
+Kaiserslautern, Germany
+[email protected]
+Johannes Feldmann
+Microelectronic Design Research Group
+University of Kaiserslautern
+Kaiserslautern, Germany
+[email protected]
+Norbert Wehn
+Microelectronic Design Research Group
+University of Kaiserslautern
+Kaiserslautern, Germany
+[email protected]
+Abstract—Recent advances in quantum computing pose a
+serious threat on the security of widely used public-key cryp-
+tosystems. Thus, new post-quantum cryptographic algorithms
+have been proposed as part of the associated US NIST process to
+enable secure, encrypted communication in the age of quantum
+computing. Many hardware accelerators for structured lattice-
+based algorithms have already been published to meet the strict
+power, area and latency requirements of low-power IoT edge de-
+vices. However, the security of these algorithms is still uncertain.
+Currently, many new attacks against the lattice structure are
+investigated to judge on their security. In contrast, code-based
+algorithms, which rely on deeply explored security metrics and
+are appealing candidates in the NIST process, have not yet been
+investigated to the same depth in the context of IoT due to the
+computational complexity and memory footprint of state-of-the-
+art software implementations.
+In this paper, we present to the best of our knowledge
+the first HW/SW co-design based implementation of the code-
+based Hamming Quasi Cyclic Key-Encapsulation Mechanism.
+We profile and evaluate this algorithm in order to explore
+the trade-off between software optimizations, tightly coupled
+hardware acceleration by instruction set extension and modular,
+loosely coupled accelerators. We provide detailed results on
+the energy consumption and performance of our design and
+compare it to existing implementations of lattice- and code-based
+algorithms. The design was implemented in two technologies:
+FPGA and ASIC. Our results show that code-based algorithms
+are valid alternatives in low-power IoT from an implementation
+perspective.
+Index Terms—Post Quantum Cryptography; Key Encapsu-
+lation Mechanism; IoT; Security; RISC-V; ASIC; Hardware
+Implementation; HW/SW co-design; HQC
+I. INTRODUCTION
+Privacy and data integrity are a key requirement in the In-
+ternet of Things (IoT). In many applications such as industrial
+IoT (IIoT), medical and healthcare, online banking, and even
+smart homes, highly sensitive data that should not be altered
+or made available to the public is transmitted over the Internet.
+In the vast majority of cases, the required security is provided
+by a combination of symmetric cryptography and Public Key
+Cryptography (PKC). However, recent advances in quantum
+computing severely compromise the security of the State-of-
+the-Art (SoA) PKC. While they are intractable on conven-
+tional computers, the underlying mathematical problems can
+be solved in polynomial time using Shor’s Algorithms [1]
+once large scale quantum computers become available. This is
+expected to be the case by the end of this decade [2] and thus,
+the US NIST is currently conducting a standardization process
+to find new post-quantum cryptographic (PQC) algorithms.
+The Key Encapsulation Mechanisms (KEMs) in the current,
+third round of the US NIST PQC standardization process rely
+on assumptions about the computational hardness of lattice-,
+code-, or isogeny-based problems. Among these, the structured
+lattice-based algorithms are considered as most promising can-
+didates for future standardization and for IoT applications due
+to their low-complexity computations. However, the structure
+of the lattices used is still the subject of cryptanalysis, and the
+security claims of the developers remain controversial [3].
+Due to the novelty of these algorithms, crypto-agility, i.e.
+the ability to seamlessly replace cryptographic algorithms in
+case that they are vulnerable, is even more important for PQC
+than for SoA cryptography. Code-based algorithms are based
+on different, well studied security assumptions, but have a
+higher computational complexity and larger memory footprints
+than lattice-based algorithms in state-of-the-art implementa-
+tions [4]. To determine if they are a viable alternative in low-
+power IoT environments in case that lattice-based algorithms
+turn out to be vulnerable, hardware implementations are essen-
+tial for a conclusive evaluation and have also been requested
+by the US NIST [5].
+Therefore, in this work we present to the best of our
+knowledge the first HW/SW co-design based implementation
+of the code-based Hamming Quasi Cyclic KEM (HQC) [4].
+Our design deploys a custom RISC-V processor and was im-
+plemented as an application-specific integrated circuit (ASIC)
+and field programmable gate array (FPGA). In summary, the
+new contributions of this work are:
+1) We provide the first ASIC implementation of a code-
+based KEM from the US NIST standardization process,
+which is fully compatible with the NIST C reference
+implementation.
+2) We identify the bottlenecks of the PQC algorithm and
+investigate for each bottleneck the best implementation
+method. We develop and implement software optimiza-
+tions, instruction set extensions, and loosely coupled
+accelerators and provide detailed information about their
+individual benefits and overhead.
+arXiv:2301.04888v1  [cs.CR]  12 Jan 2023
+
+3) We compare the energy consumption, hardware require-
+ments, and latency of our design to SoA implementa-
+tions of lattice- and code-based primitives.
+The results show that our implementation is the most
+efficient design. Furthermore, we show that HQC can be
+implemented with a similar resource utilization as lattice-based
+algorithms while achieving viable performance.
+This paper is structured as follows. In Section II, we briefly
+introduce the working principle of KEMs in general and HQC.
+In Section III, we provide an overview of the related work
+and of the SoA. In Section IV, we identify and evaluate the
+computational bottlenecks of HQC in software. In Section
+V, we present the IoT processing system and the hardware
+implementation of the different accelerators. In Section VI, we
+compare our results with the SoA. In Section VII, we draw a
+conclusion.
+II. BACKGROUND
+KEMs form a public key cryptosystem that is build out
+of three algorithms, Key-Generation (KeyGen), Encapsula-
+tion (Encaps) and Decapsulation (Decaps). Unlike general
+purpose Public Key Encryption Schemes (PKEs), KEMs are
+not thought to perform any application data encryption, but
+are designed to establish a randomly generated shared secret
+between communication partners in cryptographic protocols
+like Transport Layer Security (TLS) similar to the state-of-the-
+art Diffie-Hellmann Key-Exchange. Afterwards, this shared
+secret is used to derive a secret key for de- and encryption of
+application data with fast symmetric cryptographic algorithms
+like Advanced Encryption Standard (AES). KEMs are often
+build out of existing PKEs using transformations like the
+Fujisaki-Okamoto Transform.
+The first code-based PKE was introduced by McEliece in
+1978 and is based on the assumption that the error-correction
+code used is indistinguishable from random codes [6]. Al-
+though the original McEliece cryptosystem, which relied on
+Goppa codes, remains secure to this day, the method of hiding
+the generator matrix of the code in the public key carries
+a potential vulnerability. Attempts to reduce the key size
+by using more structured codes than the original McEliece
+approach have shown that this vulnerability can be exploited
+to crack the cryptosystems in 0.06 seconds [7].
+Therefore, the authors of HQC proposed a novel approach
+which combines two different types of codes:
+1) A decodable [n, k] code C with a fixed, publicly known
+generator matrix G ∈ Fk×n
+2
+and the error correction
+capability δ based on concatenated Reed-Muller (RM)
+and Reed-Solomon (RS) codes.
+2) A random double-circulant [2n, n] code with a publicly
+known parity check matrix h.
+This design rational allows HQC to use significantly smaller
+keys than the other code-based KEM Classic McEliece (2 KB
+vs 255 KB public key size) while still achieving the same
+security metrics.
+Fig. 1 shows how the shared secret ss is established between
+the communication partners using the HQC KEM. HQC uses
+Alice
+Bob
+KeyGen():
+1. h
+$
+←
+− R
+2. sk = (x, y)
+$
+←
+− R2
+3. pk = (h, s = x + h · y)
+Send pk
+Encaps(pk):
+4. m
+$
+←
+− Fk
+2
+5. θ ← G(m)
+6. e
+$
+←
+− R
+7. (r1, r2)
+$
+←
+− R2
+8. u = r1 + h · r2
+9. v = mG + s · r2 + e
+10. c ← (u, v)
+11. d ← H(m)
+12. ss ← K(m, c)
+Send ct = (c, d)
+Decaps(sk,ct)
+13. m′ = C.Decode(v − u · y)
+14. θ′ ← G(m′)
+15. c′ = Encrypt(pk, m′, θ′)
+16. If c ̸= c′or d ̸= H(m′)abort
+17. ss ← K(m′, c)
+Encrypt(pk, m, θ)
+Fig. 1. HQC KEM as defined in [4] with R = F2[X]/(Xn − 1), the hash
+functions G, H, K, the sampling operator
+$
+←− and the KEM’s public key pk,
+private key sk, ciphertext ct and shared secret ss. θ is the seed for the pseudo-
+random number generation during the encryption in Encaps() and Decaps().
+the Keccak-based extendable output function SHAKE as a
+seedexpander of a random generated seed as the scheme
+requires a large amount of random bytes (n = 17669 for the
+smallest parameter set HQC-128). Furthermore, the Keccak-
+based Secure Hash Alorithm 3 (SHA3) [8] is used for the G, H
+and K functions which are required due to the KEM-DEM
+transformation in HQC to construct an IND-CCA2 secure
+KEM.
+The procedure of HQC in short is as follows, a detailed
+description can be found in [4]. First, Alice randomly gener-
+ates the parity check matrix h and the private key sk, from
+which the public key pk is constructed. Here, the polynomials
+x and y which build the secret key are hidden in the public
+key by multiplying h with y and adding x in R. Bob uses
+pk to encrypt his randomly generated message m, which is
+the basis for the shared secret ss. During this encryption, the
+randomly generated vectors r1, r2 and e which have a fixed,
+predefined hamming weight are used to disguise m further.
+The hamming weights are selected in a way such that they
+still allow a correct decryption of m by Alice with respect to
+δ with a very high probability. The ciphertext ct is sent back to
+Alice, who decrypts the message m′ and calculates ss based
+on it.
+The HQC algorithm is available in 3 different parameter
+sets. This paper is focused on the NIST level 1 parameter set
+HQC-128.
+
+III. STATE OF THE ART
+Many works have been published which deal with hard-
+ware accelerations of new PQC primitives. Among these
+publications, the vast majority is focused on accelerators for
+lattice-based algorithms. A cryptographic co-processor was
+implemented in [9] as an ASIC to support various lattice
+based NIST schemes. Fritzmann et al. developed a HW/SW
+based co-design on a RISCV core for the lattice-based scheme
+NewHope [10]. In [11], FrodoKEM, an algorithm which has
+a high security confidentiality due to its less structured lattice,
+was accelerated by using a HW/SW co-design approach.
+In contrast, the code-based KEMs have not yet been in-
+vestigated to the same depth. For BIKE, another code-based
+candidate with a comparable key size to HQC, an FPGA
+implementation has been proposed in [12]. So far, the only
+hardware implementation for HQC was presented by the
+original authors of HQC in [4] and is based on FPGA HLS.
+Therefore, in this work, we present the first HW/SW co-design
+approach of HQC and implement our design both as ASIC and
+FPGA.
+IV. HW/SW CO-DESIGN
+There are three possibilities for implementation:
+1) Software.
+2) Custom processor instructions.
+3) Loosely coupled accelerators.
+In a first step, the execution of the NIST reference software
+was profiled to determine the computational bottlenecks and
+the memory footprint. In a second step, we investigated for
+each bottleneck the most suitable approach to find the optimum
+trade-off between the area, latency, memory footprint, and
+energy consumption. The highest priority was assigned to
+software optimization, as it offers high flexibility without
+additional costs. Then, if this is not efficient, custom processor
+instructions were considered as a second option, since they are
+still flexible and require little additional hardware. Only when
+these two approaches were found to be ineffective a loosely-
+coupled accelerator was considered.
+A. IoT Processing System (IoT-PS)
+Our methodology requires a processing system that allows
+instruction set extensions and the efficient interfacing of
+loosely-coupled accelerators. Therefore, we chose an adaptive
+platform that includes a RISC-V core whose instruction set
+architecture provides the ability to add custom instructions.
+Fig. 2 shows the final architecture of the IoT-PS. Our custom,
+area optimized RISC-V core supports the RV32IC instruction
+set which features additional compressed instructions and,
+therefore, significantly reduces the program size. The Direct
+Memory Access (DMA) controller features a memory copy
+(memcpy) and memory initialization (memset) function, of
+which both are able to operate on byte, half-word, and
+word granularity. The JTAG module provides access to the
+memories and the register file of the RISC-V core. It also can
+be used to start, stop, and reset the IoT-PS. Depending on
+the target platform, the data memory module was either based
+on an SRAM hard macro cell (ASIC) or Block RAM (Xilinx
+FPGA). Block RAM was also used for the instruction memory
+in case of an FPGA implementation. However, for the ASIC
+implementation we used a ROM macro cell, thus the program
+code is available after reset and does not need to be loaded
+via JTAG. The IoT-PS features no peripheral units except the
+I/O controller which is used to communicate via pin toggling.
+RISC-V
+JTAG
+DMA
+HQC Accelerator
+20 KB Instruction Memory
+32 KB Data Memory
+I/O Controller
+AXI4-Lite Interconnect
+Fig. 2. Architecture Overview
+B. Profiling of HQC-128
+The US NIST C reference implementation was used to
+identify the bottlenecks of the HQC-128 execution in our
+setup. The code was compiled with optimization level 2 (O2)
+and simulated cycle-accurately with the RTL model of our
+IoT-PS. Compared to Fig. 2, the size of ROM and RAM had
+to be increased for the analysis due to the large requirements
+of the reference implementation.
+The simulation results are shown in Table I. The specified
+clock cycles in the table refer to the processor cycles that
+the RISC-V core spends within the respective C function
+and excluding the time spent in sub-functions, e.g., gf mul is
+called during the computation of RS-Encode, but not included
+in its reported cycles. For all three KEM-functions, (1) the
+arithmetic in R, (2) the SHAKE-based hashing and (3) mem-
+ory operations are the main contributors to the total execution
+time. On top of that, the sampling operation, the RM-Decoding
+algorithm (4) and the finite field multiplication (5) are further
+contributors to the computation time. The unsigned division,
+which is performed in software, is mostly used during the
+polynomial multiplication in (1).
+In (1), the largest part is accounted by the multiplication of
+the large polynomials (n = 17669 for HQC-128), which are
+represented as bit vectors. This includes the subsequent reduc-
+tion by Xn − 1 of the intermediate result, and is performed,
+for example, in steps 3., 8. and 9. in Fig. 1. The multiplication
+complexity is reduced by the fact that one of the vectors is
+sparse and has a small, known hamming weight w ≤ 75,
+which allows to consider only the non-zero coefficients in
+the sparse polynomial during processing. The execution of the
+multiplication consists mostly of XOR operations for adding
+the binary coefficients of the same degrees and SHIFT / AND
+operations to determine the degree of the intermediate results.
+Due to the high degree of the polynomials, a large number
+of LOAD and STORE instructions is required during the
+computation.
+
+TABLE I
+TOTAL CYCLE COUNT AND SHARE OF IMPORTANT FUNCTIONS OF THE NIST C REFERENCE IMPLEMENTATION ON THE RISC-V, N.A. (NOT
+APPLICABLE) REFERS TO FUNCTIONS WHICH ARE NOT USED IN THIS STEP.
+Function
+Keygen Cycles
+Encaps Cycles
+Decaps Cycles
+Total
+5609k
+13850k
+19903k
+Arithmetic in R
+1540k (27.46%)
+3448k (24.9%)
+4989k (25.1%)
+- Vect Mul
+1528k
+3413k
+4942k
+- Vect Add
+12k
+35k
+47k
+SHAKE
+1854k (33.05%)
+5007k (36.15%)
+5414k (27.2%)
+- Keccak State Permute
+1744k
+4626k
+5005k
+- Keccak Inc Squeeze
+103k
+131k
+154k
+- Keccak Inc Absorb
+7k
+250k
+255k
+RS-RM Code
+n.A.
+26k (0.18%)
+1440k (7.24%)
+- RS-Encode
+n.A.
+26k
+26k
+- RS-Decode
+n.A.
+n.A.
+56k
+- RM-Decode
+n.A.
+n.A.
+1358k
+Sampling
+81k (1.44%)
+155k (1.1%)
+236k (1.18%)
+Memory-Operation
+2071k (36.92%)
+5068k (36.59%)
+7175k (36.05%)
+- memcpy
+2045k
+5021k
+7092k
+- memset
+26k
+47k
+83k
+Rest
+63k (1.12%)
+146k (2.17%)
+649k (3.26%)
+- unsigned division
+49k
+100k
+151k
+- gf mul
+n.A.
+20k
+162k
+Keccak’s permutation function in (2) is the computational
+core of the sponge construction in SHA3 and consists of
+bitwise AND, XOR, and rotate operations on the 25 lanes
+of 64 bits each. The major bottleneck in this permutation is
+the interdependence of the intermediate results which causes
+the contents of the processor registers to be swapped with the
+main memory multiple times during the execution of one of
+the 24 rounds.
+The large overhead of the memory operations in (3) is driven
+by two reasons. First, the RISC-V core supports only one
+outstanding memory read or write access at a time. It waits
+for a slave response before continuing the program execution.
+Second, the reference implementation is not optimized for
+low memory usage, e.g., it often stores multiple copies of
+temporary results, initializes a larger number of arrays, or
+copies parts of arrays to different memory locations.
+TABLE II
+STACK MEMORY AND CODE-SIZE OF THE NIST C REFERENCE
+IMPLEMENTATION OF HQC-128 ON OUR RISC-V CORE.
+Keygen
+Encaps
+Decaps
+Code Size
+10.798 KB
+17.015 KB
+22.378 KB
+Stack Memory
+53.018 KB
+68.714 KB
+77.762 KB
+C. Software Optimization
+In multiple functions, the reference implementation uses
+non-optimal data-types which increases the number of required
+memory accesses and processor instructions. An example of
+this is the comparisons in Step 16 of Fig. 1, which are
+performed on a byte boundary rather than a processor word
+boundary. The memory footprint and computation time was
+further improved by removing redundant arrays which often
+get initialized with zeroes or are the target of memcpy
+operations. The operations are performed with pointers in-
+stead. For the remaining memory operations, the time required
+for memcpy and for array initialization via memset were
+accelerated by using the DMA controller of the platform.
+D. Instruction Set Extension
+The bottleneck of RS-Encoding and -Decoding is caused
+by the multiplication in F28. This operation has only three
+operands, including the generator polynomial, and one return
+value with the size of one byte each. A F28-Unit is added to
+the RISC-V core which is able to perform the operation shown
+in Equation 1, where a = (a15, · · · , a0) and b = (b7, · · · , b0)
+are the input operands, and d = (d14, · · · , d0) is the output.
+This unit is made accessible via both an R-type and an I-type
+custom instruction, where register rs1 is used as operand a,
+register rs2 respectively the immediate value imm is used as
+operand b, and the output d is stored in register rd.
+(a15 · x7 + · · · + a8 · x0) · (b7 · x7 + · · · + b0 · x0)
++(a7 · x7 + · · · + a0 · x0) ⇒ (d14 · x14 + · · · + d0 · x0) (1)
+Using these custom instructions, a multiplication in F28 is
+performed within four clock cycles.
+E. Loosely-Coupled Accelerators
+Fig. 4 shows the block diagram of the loosely-coupled
+HQC accelerators. To enable parallel calculations between
+the processor and the accelerator, the accelerator has both
+an AXI slave and an AXI master interface and fetches its
+calculation inputs (e.g. the polynomials) from the processor’s
+
+Dense Polynomial
+Sparse Polynomial
+coord0
+coord1
+coordw-1
+. . .
+. . .
+word0
+0
+63
+word1
+64
+127
+wordN
+17664
+17228
+. . .
+word0
+coord0
+coord0+63
+wordN
+coord0
++ 17664
+coord0
++17228
+. . .
+word0
+coord1
+coord1+63
+wordN
+coord1
++ 17664
+coord1
++17228
+. . .
+word0
+coordw-1
+Coordw-1+63
+wordN
+coordw-1
++ 17664
+coordw-1
++17228
+. . .
+XOR
+XOR
+XOR
+Intermediate Result
+Fig. 3. Working principle of the polynomial multiplication in R with n = 17669 and 64-bit memory words.
+main memory via the master interface, according to the
+processor command which was previously received via the
+slave interface. Due to the data dependencies between the steps
+in the HQC-KEM, and based on the previous observation that
+the bottleneck in the HQC is driven by memory accesses, we
+decided that enabling parallel read/read or read/write accesses
+is more beneficial than running the dedicated compute units
+in parallel. Therefore, only two SRAMs are used and shared
+between the compute units, and only one of the compute units
+is processing at the same time. The access to the SRAMs and
+the operation mode of the compute units are managed by one
+control unit.
+AXI4-Lite Interconnect
+HQC Control Unit
+SRAM0
+(288 x
+64 Bit)
+SRAM1
+(566 x
+64 Bit)
+R-Unit
+Sampling-Unit
+RM-Decoder
+Keccak-IP
+AXI-Slave
+AXI-Master
+Fig. 4. HQC hardware accelerator.
+The R-Unit implements the addition, multiplication and
+reduction of the polynomials in R. Fig. 3 shows the working
+principle of the polynomial multiplication. The values of
+the sparse polynomial contain the locations of its non-zero
+coordinates. In our implementation, we iterate through the
+multiplication by a word-by-word shift of the dense polyno-
+mial by the coordinates given in the sparse polynomial.
+After the shift, the interim result is XOR-ed with the word
+that is currently stored at the respective location in memory
+and the carry out with respect to the word alignment of the
+memory is calculated. The R-Unit is designed such that only
+two cycles are necessary for calculating a resulting word. In
+the first cycle the address of the dense polynomial and the
+intermediate polynomial are calculated and read based on the
+coordinate, while in the second cycle the new value and the
+carry-out are calculated and written to memory.
+The Sampling-Unit combines the sponge functions squeeze
+and absorb of the incremental version of SHAKE, the permu-
+tation function of Keccak and the rejection-based sampling,
+during which SHAKE functions are used as extendable output
+functions (XOF). For the permutation function, a highspeed
+open-source implementation by the original authors of Kec-
+cak was used, which executes the permutation in 24 clock
+cycles [13].
+As shown in Table I, the vast majority of decoding time
+is spent on the RM-Codes, which employs a Maximum
+Likelihood (ML) algorithm based on the Hadamard Transform
+and a subsequent peak-search for the highest value in the
+transformed codeword. Because HQC uses duplicated RM
+codes, the decoding algorithm must be preceded by another
+transformation function [4]. This transform requires many
+single-bit operations, and thus, the implementation in soft-
+ware is not efficient. Therefore, we adapted the transform
+and the subsequent decoding steps to an efficient hardware-
+implementation.
+V. RESULTS AND COMPARISON
+The IoT-PS presented was implemented in a 22 nm FD-SOI
+technology from GlobalFoundries under worst case Process,
+Voltage and Temperature (PVT) conditions (125 °C, 0.72 V for
+timing; 25 °C, 0.8 V for power). Synthesis is performed with
+the Synopsys DesignCompiler, Place&Route is carried out
+with the Synopsys IC-Compiler. The SRAMs were generated
+by the INVECAS Memory Compiler. Power numbers are
+calculated with back-annotated wiring data. The layout of
+ASIC IP core presented in this work can be seen in Figure 5
+has a size of 0.12 mm2 with an aspect ratio of 1.77 and a
+maximum frequency of 700 MHz. The IoT-PS presented was
+also implemented on a Xilinx Artix xc7a100tcsg324-3 using
+Xilinx Vivado for a better comparison to existing work.
+
+TABLE III
+CYCLE COUNT OF THE HQC-KEM IN OUR SETUP WITH THE DIFFERENT HARDWARE MODULES. THE IMPROVEMENT REFERS TO SPEEDUP WITH
+RESPECT TO THE NIST C REFERENCE IMPLEMENTATION WITHOUT ANY HARDWARE ACCELERATORS.
+Keygen
+Encaps
+Decaps
+Cycles
+Improvement
+Cycles
+Improvement
+Cycles
+Improvement
+Reference
+5609k
+-
+13850k
+19903k
+-
+DMA + SW OPT
+3587k
+36.0%
+7044k
+49.1%
+10851k
+45.5%
++ R-Unit
+1862k
+66.8%
+5183k
+62.6%
+7245k
+63.6%
++ Sampling-Unit
+1623k
+71.1%
+1955k
+85.9%
+5176k
+74.0%
++ RM-Decoder
+n.A.
+n.A.
+n.A.
+n.A.
+9636k
+51.6%
++ F28-Instruction
+n.A.
+n.A.
+7028k
+49.3%
+10722k
+46.1%
++ All Modules
+56k
+98.9%
+131k
+99%
+557k
+97.2%
+Code Size
+1.5 KB
+86.1%
+6.6 KB
+61.2%
+13.362 KB
+40.3%
+Stack Memory
+10 KB
+81.1%
+24 KB
+65.7%
+31 KB
+60.1%
+A. Impact of individual optimizations on the overall run time
+Table III shows the extent to which the presented hardware
+modules accelerate the computation time of the three KEM
+functions. As illustrated, the use of a DMA and the software
+optimization already provide a significant speed up between
+36% and 49% over the reference implementation. This shows
+that the NIST C reference implementation is not meant to
+be used in IoT devices without optimizations. The loosely
+coupled Sampling-Unit is the accelerator that provides the a
+runtime reduction of at least 50% compared to the optimized
+software implementation with DMA in all KEM functions. The
+R-Unit, however, reduces the runtime only between 26.5%
+and 48% depending on the KEM function. The RM-Decoder
+has the least impact on the performance since it is only used
+in Decaps. Also it is able to achieve a runtime reduction of
+11.2%. The F28-Instructions give only a minor speedup on its
+own, but shows its potential in combination with all loosely
+coupled accelerators.
+B. Resource distribution of the individual hardware modules
+Figure 5 shows a qualitative area distribution of the distinct
+modules for ASIC, while Table IV shows a quantitative
+distribution for FPGA. The R-Unit shows the highest area
+efficiency among all loosely coupled accelerators. Although
+this unit has a lower runtime reduction compared to the
+Fig. 5.
+Layout; RISC-V - blue; JTAG - cyan; Interconnect - lime; I/O
+Controller - purple; DMA - pink; Sampling Unit - red; RM-Decoder - orange;
+R-Unit - yellow; HQC Control Unit, SRAM0, SRAM1 - green; ROM - white,
+horizontal strips; RAM - white, cross pattern
+Sampling-Unit, it needs less than 10% of its FPGA resources.
+The RM-Decoder, however, requires fewer resources than the
+R-Unit, but also offers the least performance gain and is only
+used in Decaps. Therefore, it is far less efficient compared to
+both R-Unit and Sampling-Unit. The F28-Unit, which is used
+by the custom instructions, requires only negligible resources.
+However, the decoding of these instructions as well as the
+controlling of the unit requires additional resources which are
+hidden inside the RISC-V core.
+TABLE IV
+RESOURCE UTILIZATION ON FPGA (ARTIX7)
+LUTs
+Registers
+Block RAM
+RISC-V
+2210
+1682
+0
+⌞ F28-Unit
+27
+0
+0
+Interconnect
+2775
+1919
+0
+Memories
+53
+6
+24
+HQC Accelerator
+7920
+2370
+3
+⊢ R-Unit
+565
+117
+0
+⊢RM-Decoder
+435
+63
+0
+⌞Sampling Unit
+5610
+1814
+0
+⌞Keccak Permute
+4685
+1622
+0
+DMA
+489
+412
+0
+JTAG
+452
+546
+0
+I/O Controller
+41
+68
+0
+IoT-PS
+13934
+7003
+27
+C. Comparison to State of the Art
+Table V presents the required clock cycles for the different
+KEM functions of SoA implementations and of our work. We
+use the number of clock cycles as metric rather than absolute
+computation time. This is, for our our work, a pessimistic
+comparison due to the high achievable clock frequency. How-
+ever, even under this assumption, our implementation requires
+a comparable number of clock cycles as the low-latency HQC
+implementation, while it requires less hardware than its low
+area hardware implementation. Compared to BIKE, the other
+code-based KEM, our implementation requires about the same
+amount of clock cycles like the low latency implementation
+while using significantly less hardware resources. The imple-
+mentation of FrodoKEM, which would be an alternative if
+structured lattice-based KEMs like Kyber and NewHope are
+
+.TABLE V
+COMPARISON OF CLOCK CYCLES, FREQUNCY AND FPGA RESOURCES FOR DIFFERENT STATE-OF-THE-ART IMPLEMENTATIONS. HW/SW REFERS TO
+IMPLEMENTATIONS BASED ON HW/SW CO-DESIGN, FULL REFERS TO FULL HARDWARE IMPLEMENTATIONS OF THE RESPECTIVE SCHEME. NEWHOPE
+AND KYBER ARE STRUCTURED-LATTICE BASED ALGORITHMS, FRODOKEM IS BASED ON LESS-STRUCTURED LATTICES, AND THE REMAINING
+IMPLEMENTATIONS ARE BASED ON CODES.
+Implementation
+Keygen
+Encaps
+Decaps
+Frequency
+FPGA Resources
+Target Plattform
+Cycles
+Cycles
+Cycles
+MHz
+LUTs
+FFs
+BRAMs
+HQC (low area, HW) [4]
+630k
+1500k
+2100k
+132
+8.9k
+4k
+14
+FPGA (Xilinx Artix-7)
+HQC (low latency, HW) [4]
+40k
+89k
+190k
+148
+20k
+16k
+12.5
+FPGA (Xilinx Artix-7)
+BIKE (low area, HW) [12]
+2671k
+153k
+1628k
+121
+13k
+5k
+17
+FPGA (Xilinx Artix-7)
+BIKE (low latency, HW) [12]
+259k
+12k
+189k
+96
+53k
+7k
+49
+FPGA (Xilinx Artix-7)
+NewHope (HW/SW) [10]
+357k
+590k
+167k
+n.A.
+11k
+5k
+1
+FPGA (Xilinx Zynq-7000)
+FrodoKEM (HW/SW) [11]
+23.4M
+25.5M
+25.3M
+100
+5.6k
+1.1k
+0
+FPGA (Xilinx Zynq Ultrascale+)
+Kyber (HW/SW) [9]
+75k
+132k
+142k
+72
+n.A.
+n.A.
+n.A.
+ASIC (40nm)
+HQC on Cortex M4 (SW)
+1048k
+2436k
+4001k
+64
+n.A.
+n.A.
+n.A.
+nRF52840
+This Work HQC (DMA+SW OPT)
+3587k
+7044k
+10851k
+700
+n.A.
+n.A.
+n.A.
+ASIC (22nm)
+This Work HQC (HW/SW)
+56k
+131k
+557k
+700
+n.A.
+n.A.
+n.A.
+ASIC (22nm)
+This Work HQC (HW/SW)
+56k
+131k
+557k
+100
+8k
+2.4k
+3
+FPGA (Xilinx Artix-7)
+proven to be vulnerable to attacks, is overall 100 times slower
+than our work.
+Table VI shows the energy consumption of our implementa-
+tion, of HQC on a Cortex M4 processor and of the structured
+lattice-based KYBER, also implemented as an ASIC. As
+can be seen, our implementation requires considerably less
+energy than the pure software on the Cortex M4 and also less
+than KYBER’s ASIC implementation, which, however, was
+implemented on a larger technology node.
+TABLE VI
+COMPARISON OF ENERGY CONSUMPTION. FOR A TRADE-OFF BETWEEN
+POWER AND LATENCY, OUR DESIGN WAS IMPLEMENTED AND SIMULATED
+WITH A 200 MHZ CLOCK.
+Implementation
+Keygen
+Encaps
+Decaps
+µJ
+µJ
+µJ
+Kyber (HW/SW) [9]
+5.97
+9.37
+11.25
+HQC on Cortex M4 (SW)
+500
+1184
+1872
+This Work HQC (HW/SW)
+1.02
+2.41
+7.1
+VI. CONCLUSION
+In this work, we investigated the performance of the code-
+based post-quantum KEM HQC in the context of low power
+IoT system. We presented the first ASIC implementation of a
+code-based US NIST PQC candidate. With a combination of
+software optimizations, instruction set extensions, and loosely
+coupled hardware accelerators, we achieve similar perfor-
+mance to the full SoA hardware implementation of HQC, but
+require significantly less hardware resources and provide more
+flexibility. Compared to SoA implementations of lattice-based
+algorithms, we have shown that code-based algorithms are
+promising alternatives in IoT based applications in terms of
+energy efficiency, computation time, and required hardware.
+ACKNOWLEDGEMENT
+This paper was partly founded by the German Federal
+Ministry of Education and Research as part of the project
+“SIKRIN-KRYPTOV” (16KIS1069).
+REFERENCES
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+and factoring,” in Proceedings of the 35th Annual Symposium on
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+[2] M. Mosca, “Cybersecurity in an Era with Quantum Computers: Will We
+Be Ready?” IEEE Security & Privacy, vol. 16, no. 5, pp. 38–41, 2018.
+[3] Peikert,
+Christopher
+J.
+and
+Bernstein,
+D.J.,
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+KYBER:
+Round
+3
+Official
+Comments,”
+https://csrc.nist.gov/
+CSRC/media/Projects/post-quantum-cryptography/documents/round-3/
+official-comments/CRYSTALS-KYBER-round3-official-comment.pdf,
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+figurable Crypto-Processor for Post-Quantum Lattice-Based Protocols,”
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+

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+arXiv:2301.02661v1  [q-bio.QM]  5 Jan 2023
+Evaluating Evasion Strategies in Zebrafish Larvae
+Yusheng Jiaoa, Brendan Colverta,b, Yi Mana,c, Matthew J. McHenryd, and Eva Kanso∗a,*
+aAerospace and Mechanical Engineering, University of Southern California, 854 Downey way, Los Angeles, California 90089, USA
+bDepartment of Bioengineering, University of California, San Diego, 9500 Gilman Dr, La Jolla, CA 92093, USA
+cDepartment of Mechanics and Engineering Science at College of Engineering and LTCS, Peking University, Beijing 100871, P. R. China.
+dDepartment of Ecology and Evolutionary Biology, University of California, Irvine, 321 Steinhaus Hall, Irvine, CA 92697, USA
+January 10, 2023
+Abstract
+An effective evasion strategy allows prey to survive encoun-
+ters with predators.
+Prey are generally thought to escape
+in a direction that is either random or serves to maximize
+the minimum distance from the predator.
+Here we intro-
+duce a comprehensive approach to determine the most likely
+evasion strategy among multiple hypotheses and the role of
+biomechanical constraints on the escape response of prey fish.
+Through a consideration of six strategies with sensorimo-
+tor noise and previous kinematic measurements, our analy-
+sis shows that zebrafish larvae generally escape in a direc-
+tion orthogonal to the predator’s heading. By sensing only
+the predator’s heading, this orthogonal strategy maximizes
+the distance from fast-moving predators, and, when operating
+within the biomechanical constraints of the escape response,
+it provides the best predictions of prey behavior among all
+alternatives. This work demonstrates a framework for resolv-
+ing the strategic basis of evastion in predator-prey interac-
+tions, which could be applied to a broad diversity of animals.
+Keywords— Predator-prey interactions | Probabilistic modeling |
+Inference | Fish C-start | Fluid-structure interactions | Hydrody-
+namics
+Author contributions: EK secured funds and designed and supervised
+research.
+YJ, BC, YM, and EK performed research.
+YJ, BC, YM,
+MJM, and EK analyzed results. YJ, YM, and EK wrote the paper and
+YJ, BC, YM, MJM, and EK revised and edited it.
+Author declaration: The authors declare no conflict of interest.
+Introduction
+The abilities to sense and evade predators are central to the survival
+of a diversity of prey species. The timing, speed, and direction of
+a prey’s escape reflect the animal’s evasion strategy, which is form-
+ulated by its neurophysiology and biomechanics [1]. Despite the
+fundamental importance of predator encounters, resolving a prey’s
+strategy is experimentally challenging due to the variability inher-
+ent to animal behavior. Predators vary in their approach toward
+prey and the ability of the prey to respond is filtered through the
+environment and the animal’s physiology, which may additionally
+introduce noise in sensing, integration, and motor response. The
+aims of the present study are to develop an analytical approach
+that is capable of resolving prey strategy from kinematic measure-
+ments and to use that approach to test classic theory on strategy
+in fish predator-prey interactions.
+The interactions between an individual predator fish and prey
+fish offer a classic system for the study of evasion strategy. Fish
+evade predators with a stereotypical ‘C-start’ response, character-
+ized by the fish body bending into a preparatory ‘C’ shape, followed
+by a rapid acceleration as the body unfolds with largely planar mo-
+tion [2]. Fish escape behavior inspired an application of differential
+game theory to determine the optimal strategy of prey [3]. The
+distance-optimal strategy is the solution to the ‘homicidal chauf-
+feur’ game where prey move in the direction that maximizes the
+∗To
+whom
+correspondence
+should
+be
+addressed.
+E-mail:
+[email protected]
+closest distance achieved by a predator that maintains a constant
+velocity [4]. The distance-optimal strategy has been invoked to ex-
+plain the escape responses in animals as divergent as cockroaches
+[5], crickets [6], shrimp [7], frogs [8], salamanders [9], crabs [10],
+and a variety of fish species [11, 12]. This strategy is generally con-
+sidered the primary alternative to escaping in a random direction,
+known as the protean strategy, which offers the tactical benefit of
+confusing the predator [13, 14, 15, 16].
+The present study con-
+siders whether previously-measured escape kinematics in zebrafish
+larvae [17, 18] are consistent with distance-optimal, pure-protean,
+or alternative strategies.
+The zebrafish (Danio rerio) larva is a compelling system for
+investigating evasion because it has served as a model for the neu-
+rophysiology and biomechanics of the C-start. Its small size, lack
+of pigmentation, and amenability to genetic manipulation have fa-
+cilitated applications of functional imaging and optogenetics in ze-
+brafish to observe and manipulate the sensory and motor circuits
+responsible for visually-mediated escapes [19, 20, 21].
+A combi-
+nation of high-speed kinematics, flow visualization, and computa-
+tional fluid dynamics have revealed a comprehensive accounting of
+the fluid forces that propel the escape response of zebrafish lar-
+vae [22, 23, 24, 25, 26]. We incorporate these findings into a con-
+sideration of the biomechanical constraints on the escape strategy.
+We adopt a multi-pronged approach for testing the evasion
+strategy in larval zebrafish. Using a strong-inference technique [27],
+we mathematically define models for six strategies, both with and
+without sensorimotor noise (Fig. 1). We then proceed to evaluate
+these model predictions against previous measurements of escape
+kinematics [17] to determine the strategy that most-likely describes
+those observations. Finally, a consideration of the escape hydro-
+dynamics and fluid-structure interactions allows us to evaluate the
+constraints on these strategies. These measures combine to offer
+a general framework for evaluating evasion strategies in predator-
+prey encounters.
+Results
+Our description of the major results is organized around four main
+themes: (1) experimental data of the evasion kinematics of larval
+zebrafish and their descriptive statistics, (2) mathematical defini-
+tions of the evasion strategies, (3) formulation of the analytical
+approach and testing of evasion strategies, with and without sen-
+sorimotor noise, and (4) evaluation of the effects of biomechanical
+constraints on evasion.
+Experimental measurements of escape kinematics
+We analyzed anew a large experimental dataset for the kinematics
+of escape responses in zebrafish larvae that were previously pub-
+lished [17, 18]. Larvae were exposed to a robotic predator, consist-
+ing of a sacrificed adult zebrafish (of fixed size) controlled with a
+motor to move through an aquarium of otherwise still water. As
+detailed previously [17, 18], larvae were largely motionless prior to
+the escape response that was stimulated by the presentation of the
+predator.
+The recorded responses of larvae were compiled from
+numerous experiments, each of which elicited a modest number
+1
+
+V
+−λ
+d
+φ
+prey frame of reference
+predator
+prey
+Contralateral
+θ
+Orthogonal
+θ
+Distance-optimal
+θ
+Parallel
+θ
+Antipodal
+θ
+ψ
+B
+A
+Fig. 1. Evasion strategies. (A) Schematic shows the predator position (d, φ) and heading ψ in the prey’s frame of reference. (B) The change θ in prey heading direction
+at evasion as predicted by five evasion strategies: distance-optimal, prey makes a turn that maximizes the shortest distance from predator; orthogonal, prey turns to the
+direction orthogonal to the predator heading in order to flee the path of the predator; parallel, prey turns to align with the predator heading direction; antipodal, prey turns in the
+opposite direction of the predator angular position; and contralateral, prey turns left or right by 90◦ depending on the predator angular position. Strategies are distinguished
+by color.
+of responses. Larvae were excluded from the analysis if they re-
+sponded within a few body lengths, or a few seconds after, another
+responding larva. The 3D kinematics of larvae were compiled in the
+predator’s frame-of-reference to yield a cloud of responses anterior
+to the robotic predator.
+The speed of the robotic predator was set to a constant equal to
+2, 11, or 20 cm·s−1 to reflect the speed range of a typical foraging
+predator [22]. This ensured a repeatable stimulus that elicited a
+fast C-start response from the larvae [18, 17]. High-speed kinemat-
+ics recorded a total of 699 evasion instances: Nslow = 251 for the
+slow-moving predator, Nmid = 233 for the mid-speed predator, and
+Nfast = 215 for the fast-moving predator (Fig. 2).
+Experimental analysis
+From the previous kinematic measurements, we presently calcu-
+lated the predator distance d, angular position φ ∈ [0, 2π), and
+heading ψ ∈ [−π, π) in the prey’s frame of reference at the onset of
+the C-start escape response, and we calculated the change in the
+prey’s orientation θ ∈ [−π, π) as it completed the C-start escape
+response (Fig. 1A and SI, Fig. S1B). In our analysis, θ captures the
+rotation of the entire fish body, that is, the change in prey heading,
+which is not the same as the change in the body angular position
+employed previously [17, 18]. We clearly distinguish between the
+prey’s sensing of the predator angular position φ and heading ψ,
+which are often confused in empirical studies of evasion [28, 29, 12].
+In addition to the predator’s actual heading direction ψ, we con-
+sidered that the prey perceives λ, the deviation of the predator’s
+heading from the angular position φ, given by λ = ψ − (φ + π),
+λ ∈ [−π, π) (see Fig. 1A and SI, Fig. S1B, C). The predatory stim-
+ulus is said to be sinistral if λ > 0, that is, predator is headed to
+the left of where it appears in the prey’s visual field, and dextral
+otherwise.
+Table 1. Sensory Requirements of Evasion Strategies
+Predator state
+angular
+heading
+Complexity
+position
+heading
+deviation
+speed
+of sensing
+φ
+ψ
+λ
+V
+Distance-optimal
+·
+⃝
+◦
+⃝
+most
+Orthogonal
+·
+⃝
+◦
+·
+Parallel
+·
+⃝
+·
+·
+�
+Antipodal
+⃝
+·
+·
+·
+Contralateral
+◦
+·
+·
+·
+least
+⃝ exact value
+◦ interval value
+· not needed
+Descriptive statistics of kinematic measurements
+We found no correlation between the prey’s escape direction θ and
+its distance d from the predator at the onset of evasion (SI, Fig.
+S3). However, we did find a clear correlation between the escape
+direction θ and the angular position φ in instances where the preda-
+tor appears in the prey’s visual field (SI, Fig. S5). The data also
+showed a correlation between θ and the predator heading ψ, when
+partitioned based on whether the predator’s heading is sinistral
+(λ > 0) or dextral (λ < 0), relative to its angular position φ (SI,
+Fig. S7). Importantly, although the distributions of φ, ψ, and θ
+varied with predator speed V , the correlations between θ and φ
+and between θ and ψ were qualitatively similar for all V (SI, Figs.
+S5 and S7), suggesting that for the range of speeds considered, V
+can be treated as a model parameter, rather than a variable that
+fundamentally changed the evasion behavior.
+In sum, our statistical analysis (SI, Figs. S2-S7, Table S1) in-
+dicates that the escape direction θ depends on the prey’s sensing
+of the predator’s angular position φ, heading ψ, and deviation be-
+tween them λ, but does not disambiguate which stimuli determine
+the escape direction and the behavioral rules that best explain the
+data.
+Definition of evasion strategies
+We next define the six fish evasion strategies: distance-optimal, or-
+thogonal, parallel, antipodal, contralateral, and pure-protean. We
+index these strategies with an integer n = 1, . . . , 6 in the order listed
+above. In all strategies, we ignore the prey biomechanics and treat
+both the predator and prey as point masses equipped with head-
+ing directions. Therefore, the prey’s strategy is demonstrated by
+the direction of its escape θ. However, these escape direction vary
+among the strategies depending on the relative position and head-
+ing of predator (Fig. 1B). We rate the strategies by their complexity
+of sensing (Table 1), which is a relative measure that increases with
+the number of geometric parameters that must be accurately de-
+termined to execute the escape in the direction predicted by the
+strategy. By this metric, the sensing of exact quantities is more
+complex than interval quantities.
+Distance-optimal evasion strategy
+A distance-optimal evasion strategy considers that the prey’s ob-
+jective, once it detects the predator, is to maximize its minimum
+future distance from the predator [3, 30].
+Accordingly, the prey
+should head in the direction θ relative to its pre-evasion heading
+(see SI, section 2),
+θ = f(1)(ψ, λ; χ) =
+� ψ − χ,
+sinistral: λ ∈ [0, π),
+ψ + χ,
+dextral: λ ∈ (−π, 0),
+(1)
+where χ = cos−1(U/V ) is an angle that depends on the ratio U/V
+of prey speed U to predator speed V . For U > V , χ = 0. We treat χ
+2
+
+2cm/s
+11cm/s
+20cm/s
+head
+tail
+before
+after
+1cm
+slow predator 
+mid-speed predator 
+fast predator
+Fig. 2. Experimental measurements of zebrafish larvae evasion in response to robotic predator. Zebrafish larvae were randomly placed in a tank with an approaching robotic
+predator driven at three speeds: V = 2, 11 and 20 cm·s−1. They were mostly straight and motionless until exhibiting a fast C-start evasion response to the predator [17, 18].
+Each experiment involved a single predator-prey encounter. The experiment was repeated to collect three large datasets of size Nslow = 251, Nmid = 233, Nfast = 215
+for the slow, mid-speed, and fast moving predator, respectively. Evasion instances are superimposed for visualization purposes. For each evasion instance, we calculated,
+in the predator frame of reference, the position and orientation of the prey at the onset of evasion (gray macebells where the head represents the prey’s position and spike
+represents its orientation). The change in prey’s orientation θ induced by the C-start evasion response (see inset) is shown in colored macebells. Color is used only for
+illustration purposes and not to be confused with the color code used in Figs. 1, 3, 4, 6 to distinguish between evasion strategies.
+as a model parameter rather than a variable. This distinction may
+not be important for the prey, but it is relevant for our subsequent
+analysis of this strategy.
+Orthogonal evasion strategy
+We propose a simpler evasion strategy where the prey turns 90o
+away from the heading direction ψ of the predator,
+θ = f(2)(ψ, λ) =
+� ψ − π/2,
+sinistral: λ ∈ [0, π),
+ψ + π/2,
+dextral: λ ∈ (−π, 0).
+(2)
+This strategy is equivalent to the distance-optimal strategy in the
+fast predator limit U/V → 0, but may determine θ without the
+need to sense the predator speed V .
+Parallel evasion strategy
+For a slow predator U/V
+≥ 1, the optimal strategy is for the
+prey to reorient itself in the direction of the predator heading,
+θ = f(3)(ψ) = ψ, which can be readily deduced by setting χ = 0
+in (1).
+The major disadvantage of this strategy is that it could
+place the predator in the blind spot of the prey’s visual field.
+Antipodal evasion strategy
+Empirical observations [31, 28] suggest that the prey might follow
+an antipodal strategy by reorienting its heading θ in the direction
+opposite to the angular position φ where the predator appears in
+its visual field, without any account for the predator heading ψ,
+θ = f(4)(φ) =
+� φ + π,
+left stimulus: φ ∈ [0, π),
+φ − π,
+right stimulus: φ ∈ (π, 2π).
+(3)
+Contralateral evasion strategy
+A similar but simpler strategy, called contralateral, was suggested
+in [17] when the prey is approached by the predator from either
+side. Accordingly, the prey escapes by turning 90o either to the
+‘left’ or ‘right’ of its own pre-evasion heading,
+θ = f(5)(φ) =
+� −π/2,
+left stimulus: φ ∈ [0, π),
+π/2,
+right stimulus: φ ∈ (π, 2π).
+(4)
+Pure-protean evasion strategy
+The pure-protean strategy suggests that the evasion response θ is
+random, independent of the predator state, with a uniform prob-
+ability of moving in any particular direction.
+This strategy is
+best expressed in a probabilistic manner, where the probability
+density function (PDF) is uniform with equal probability density
+p(6)(θ) = 1/(2π) of obtaining any change in orientation θ.
+Testing evasion strategies
+We developed a method for evaluating evasion strategies in terms
+of their ability to explain the experimental observations (Fig. 2).
+Although the pure-protean strategy is not supported by our exper-
+imental data (see SI, Figs. S5-S7), the data exhibits some level of
+randomness, as indicated by the variability in the location of the
+predator at the onset of evasion, but potentially also due to inher-
+ent sensorimotor noise in the prey’s perception of the predator and
+its execution of the evasion response. Our approach accounts for
+this variation in evaluating, comparing, and ranking the hypotheti-
+cal evasion strategies. To emphasize the generality of our approach,
+we express it in terms of a generic stimulus s and response r, with-
+out reference to the specific degrees of freedom that these vectors
+encompass.
+For the zebrafish larvae, r is simply θ, but s varies
+depending on the strategy; theoretically, it could encompass all or
+any combination of the variables that define the predator state d,
+φ, ψ, λ and V .
+To examine how well the probabilistic strategy models fit the
+experimental data, we interpreted the latter from a probabilistic
+perspective. An experimental dataset generates N samples (si, ri),
+i = 1, . . . N, from a joint PDF, denoted by po(s, r), whose exact
+form is unknown. An evasion behavior follows a conditional PDF
+po(r|s) = po(s, r)/po(s), which is related to the joint PDF po(s, r)
+and the PDF po(s) of stimuli that elicit an escape response via the
+Law of Total Probability [32]. Unfortunately, po(s, r) and po(s) are
+unknown, and only discrete samples of these PDFs are available
+from experiments, thus the need for further modeling and analysis.
+Probabilistic models under precise vs. noisy sensing
+and response
+We distinguish between the actual predator state s and the prey’s
+sensing ˆs of the predator state. Similarly, we distinguish between
+3
+
+strategy
+r=f(n)(s)
+perception
+noise σS
+response
+noise σR
+predator
+state
+s
+prey
+response
+r
+ˆs
+ˆr
+ˆ
+ˆ
+prey response θ
+predator angle φ
+predator angle φ
+predator angle φ
+predator angle φ
+predator angle φ
+prey response θ
+180◦
+−180◦
+0◦
+360◦
+0◦
+180◦
+φ
+Experiment
+frequency
+0
+max
+Probablistic models
+180◦
+−180◦
+0◦
+0◦
+360◦
+180◦
+0◦
+360◦
+180◦
+0◦
+360◦
+180◦
+0◦
+360◦
+180◦
+0◦
+360◦
+180◦
+Contralateral
+Antipodal
+Parallel
+Distance-optimal
+Orthogonal
+B
+C
+A
+D
+Fig. 3. Model predictions in response to experimentally observed predator states. (A) Bivariate histogram of (φ, θ) from experimental data. Darker color means larger
+fraction of data points in that area of the (φ, θ) space. (B) Bivariate histogram based on the evasion models (Eqs 1–4) with no noise; model predictions θi in response to
+experimentally observed predator states φi, ψi, λi, i = 1 . . . , N, where N = 699 is the size of the combined data. (C) Schematic illustration of how noise in sensing and
+response is built into the evasion models. (D) Bivariate histogram using realizations from the noisy evasion models (Eq. 5) under optimal noise levels.
+the actual escape heading r and the prey’s desired escape heading
+ˆr. If the prey’s sensing and response are precise, we get ˆs = s and
+ˆr = r. However, the sensorimotor modalities underlying evasion
+are often noisy: the prey may perceive a noisy version ˆs of the
+predator’s state s and its desired response ˆr may be altered by
+noisy execution or environmental conditions to yield r.
+Each evasion strategy n, save the pure-protean, defines a desired
+escape response ˆr given a perceived predatory stimulus ˆs and can
+be expressed as a conditional PDF using the Dirac-delta function
+p(n)(ˆr|ˆs) = δ �
+ˆr − f (n)(ˆs)�
+. The joint PDF p(n)(s, r) formed based
+on evasion strategy n follows from the Law of Total Probability
+p(n)(s, r) =
+� �
+p(r|ˆr)p(n)(ˆr|ˆs)p(ˆs|s)po(s)dˆs dˆr.
+(5)
+Here, p(ˆs|s) and p(r|ˆr) model the noise in the prey’s sensing and
+response. In the case of precise sensing and response, (5) reduces
+to
+p(n)(s, r) = δ �
+r − f (n)(s)�
+po(s).
+(6)
+In the following, we treat each case separately.
+Evaluating evasion strategies under precise sensing
+and response
+To obtain samples of the evasion response predicted by (6), we use
+as input the distribution of the empirically-observed stimuli si, and
+we construct a dataset (si, r(n)
+i
+= f (n)(si)) for each strategy. For
+each predator speed, we arrive at five datasets representing theo-
+retical predictions of the prey’s evasion response according to the
+distance-optimal, orthogonal, parallel, antipodal, and contralat-
+eral strategies.
+Bivariate histograms in the (φ, θ)-plane for each
+strategy based on the dataset combining all predator speeds are
+shown in Fig. 3B. The histograms represent discrete cross-sections
+of p(n)(s, r) and can be used to estimate the joint probability of
+obtaining a predatory stimulus φi and prey response θi.
+Com-
+pared to the histogram obtained from experiments (Fig. 3A), the
+contralateral and antipodal strategies form straight lines because
+the predicted θ(n)
+i
+are uniquely determined by the predator angu-
+lar position φi, while the other distributions are spread out due to
+their dependency on the predator heading ψi and λi.
+To measure the difference between model predictions and exper-
+imental data, we estimated numerically the Kullback-Leibler (K-L)
+divergence DKL, which quantifies the entropy of p(n)(s, r) relative
+to po(s, r), using the method in [33]; see SI, S5.
+Results of the
+K-L divergence are shown in Fig. 4A for all five strategies applied
+to the slow, mid-speed, and fast predator, as well as the combined
+data. The actual K-L divergence is always non-negative; the nega-
+tive values are due to discrete estimation of the PDF. In each of the
+four datasets, the distance-optimal and orthogonal strategies yield
+the lowest estimates of the K-L divergence, implying that, of all
+five evasion strategies, they give the closest predictions of the prey
+escape response. The distance-optimal strategy performs slightly
+better for the slow and mid-speed predator while the orthogonal is
+more advantageous for the fast predator and when considering all
+data combined. The antipodal strategy also gives relatively low K-
+L divergence estimates. The parallel and contralateral strategies,
+whose K-L divergence estimates are significantly higher than the
+other strategies, have the worst fit to experimental data across all
+predator speeds.
+Modeling noise in sensing and response
+We next introduced sensing and response noise according to (5).
+To model sensing noise, we considered ˆs to be normally-distributed
+around the actual state of the predator s, with dispersion σS, and
+to model response noise, we considered r to be normally-distributed
+around the desired response ˆr, with dispersion σR. Substituting
+the noise models p(ˆs|s; σS) and p(r|ˆr; σR) into (5), and recalling
+that p(n)(ˆr|ˆs) = δ �
+ˆr − f (n)(ˆs)�
+, we arrived, for each evasion strat-
+egy n, at a probabilistic model that depends on the noise parame-
+ters σ = {σS, σR} (see SI, section 4). Specifically, we used a von
+Mises distribution (normal distribution on the circle) for θ, φ and λ
+with noise parameters σΘ, σΦ and σΛ; we let the noise on ψ follow
+from ψ = φ + λ + π (see SI, section 4).
+At zero noise, the von
+Mises distribution converges to a Dirac-delta function at the mean
+value; when the noise level is high, it approaches a circular uniform
+distribution with constant PDF 1/(2π) in any escape direction.
+Limit of high noise levels
+If the response noise σΘ is large, any evasion direction is predicted
+with equal probability density 1/(2π), irrespective of the strat-
+egy or the sensing noise, that is, all strategies become essentially
+equivalent to the pure-protean strategy. On the other hand, if the
+response is precise σΘ = 0, but the noise in sensing the preda-
+tor’s angular position σΦ is large, all strategies, except the con-
+tralateral, converge to the pure-protean strategy; the contralateral
+strategy predicts θ = ±π/2 with equal probability. If the prey’s
+4
+
+all
+slow
+mid-speed
+fast
+K-L divergence estimate
+0
+1
+2
+1.5
+0.5
+∆AIC/N
+all
+slow
+mid-speed
+fast
+−0.2
+0.8
+0.6
+0.4
+0.2
+0
+Contralateral
+Orthogonal
+Antipodal
+Parallel
+Distance-optimal
+A
+B
+Fig. 4. Evaluation of precise and noisy evasion strategies. (A) K-L divergence estimate from precise model predictions to experiment data is computed separately for each
+dataset (slow, mid-speed and fast predator) and for all data combined. The K-L values for the distance-optimal and orthogonal strategies are the lowest, indicating better fit
+to data. (B) AIC difference (∆AIC = AIC-AICmin), normalized by the respective sample size of each dataset. For each dataset, we used bootstrap method to construct 200
+distinct datasets (by sampling with repetition) of equal size to the original dataset. We optimized each of 200 sets, evaluated the corresponding AIC, and computed the mean
+and standard deviation of ∆AIC. The orthogonal strategy has the lowest ∆AIC, indicating that it is the most parsimonious strategy and best explains the data.
+α1
+−α2
+β
+b2 b1
+A
+C
+Nair et al. 2015
+model (massless)
+model (neutral)
+time
+0
+0.25T
+0.5T
+0.75T
+T
+0◦
+middle link orientation β
+20◦
+40◦
+60◦
+80◦
+−20◦
+head/tail angles
+0◦
+α2
+α1
+90◦
+135◦
+45◦
+−45◦
+−90◦
+stage 1
+stage 2
+stage 3
+B
+maximum
+bending αmax
+D
+Voesenek et al. 2019
+model (massless)
+model (neutral)
+prey response θ
+maximum bending angle αmax
+100◦
+50◦
+0◦
+25◦
+75◦
+125◦
+Nair et al. 2015
+E
+180◦
+−180◦
+180◦
+−180◦
+deformation angle α1
+deformation angle α2
+0◦
+0◦
+stage 2
+stage 1
+αmax
+A
+B
+F
+0
+1
+−1
+Fig. 5. Biomechanics of fish C-start response. (A) Larval zebrafish bends its body into a C-shape to initiate a fast start. (B) Three fish model, with α1 and α2 representing
+the fish body shape and β the overall body orientation relative to the straight pre-evasion direction. (C) Experimental data of shape changes of larval zebrafish during evasion
+taken from Ref.[34] and processed to represent body deformations in terms of head and tail rotations α1, α2 (gray dots) then fitted by third-order fourier series (black lines).
+(D) Experimental data of overall body rotation for the same evasion instance shown in C (gray dots and black line). Predictions based on fish model, taking as input the
+shape changes in C are shown in blue lines (solid line for massless and dashed line for neutrally-buoyant fish). (E) The sequence of shape changes in C forms a curve C
+in the shape space (α1, α2) (black line). The experimental curve C is approximated by an ellipse (red) of axes A, B along α1 = ±α2 directions. Colormap represents
+curl2[A1, A2] of fish model, which predicts larger turns for curves that encompass solely positive (orange) or negative (blue) values. (F) By varying A and calculating B that
+maximizes the turn, we get a mapping from maximum bending angle αmax =
+√
+2A to turning angle θ for massless and neutrally buoyant fish (blue lines) that form an upper
+and lower bounds on the experimental data set of Ref. [23]. Both numerical and experimental data show that the C-start mechanics limits larval zebrafish to turning angles
+θ ≲ 100◦.
+response and sensing of the predator angular position are both pre-
+cise σΘ = σΦ = 0, but the prey’s sensing of the predator’s heading
+is very noisy (σΛ large), the antipodal and contralateral strategies
+do not get affected while the parallel strategy becomes protean.
+Interestingly, in this case, the distance-optimal strategy predicts
+higher probability of evasion in directions opposite to the predator
+location spanning a range of 2χ (see SI, section 4, Fig. S9). That
+is, the distance-optimal strategy becomes a noisy variant of the an-
+tipodal strategy. For χ = π/2, the orthogonal strategy with large
+noise on λ converges to the antipodal strategy with uniform noise
+spanning a range of π on either φ or θ.
+Optimizing noise levels in sensing and response
+For each noisy evasion strategy, we calculated the noise parame-
+ters σ = {σS, σR} that maximize the total likelihood L of the
+model given an experimental dataset, or equivalently minimize the
+negative log-likelihood function NLL (see SI, section 6)
+NLL = − ln L (σ|(r|s); n) = −
+�
+i
+ln p(n)(ri|si; σ),
+(7)
+where p(n)(ri|si; σ) is the conditional PDF of obtaining a response
+ri given stimulus si for strategy n at noise level σ. The optimal
+noise parameters σ∗ are given by
+σ∗ = arg min
+σ NLL.
+(8)
+5
+
+We solved this optimization problem numerically in the range
+σΦ, σΛ, σΘ ∈ (0, π) (SI, Fig. S6). In Fig. 3D, we plot realizations
+generated from the five probabilistic evasion models p(n)(s, r; σ∗)
+at the optimal noise values corresponding to the dataset of all
+predator speeds combined.
+Compared to the deterministic pre-
+dictions in Fig. 3B, all five distributions appear closer to the ex-
+perimental data in Fig. 3A.
+Evaluating strategies under optimal noise parameters
+To evaluate how well each optimized strategy describes the exper-
+imental data, we applied the Akaike information criterion (AIC)
+defined as [35]
+AIC = 2K − 2 ln L(σ∗|(s, r); n)
+(9)
+where K is the number of model parameters in each strategy. AIC
+considers both the goodness of fit represented by the likelihood
+function, and the complexity of the model: if two models have
+the same likelihood to explain the data, the criterion favors the
+simpler model. For example, for the antipodal strategy, we have
+two noise parameters σS ≡ {σΦ} and σR ≡ {σΘ}, thus K = 2;
+whereas for the orthogonal strategy, we have three noise parameters
+σS ≡ {σΦ, σΛ} and σR ≡ {σΘ}, and the orthogonal strategy is
+deemed more complex than the antipodal strategy.
+We used bootstrapping to probe the accuracy of our evaluation
+of the noisy strategies.
+Starting from each dataset (e.g., that of
+the fast predator), we constructed 200 distinct datasets of equal
+size to the original dataset (e.g., Nfast) by random sampling with
+repetition. We solved the optimization problem 200 times and ob-
+tained 200 values of σ∗ per strategy for each dataset. We calculated
+the likelihood value L and evaluated the AIC for all 200 optimal
+noise values, thus obtaining a distribution of AIC values for each
+strategy and predator speed. The mean and standard deviation of
+the distributions of AIC values, minus the lowest mean value and
+normalized by the size of the respective dataset (Fig. 4B) show
+that strategies with lower mean values of the AIC better fit the
+experimental data.
+The results based on the AIC evaluation of the probabilis-
+tic strategies in the presence of sensory and response noise are
+mostly consistent with the results based on the K-L divergence
+(Fig. 4A) for precise sensing and response, but with marked differ-
+ences. The orthogonal strategy ranks the highest in every dataset;
+the distance-optimal strategy is slightly behind, in second place, in
+all but the slow predator dataset where the antipodal strategy ranks
+second. The difference between the orthogonal, distance-optimal,
+and antipodal strategies is most distinguishable in the case of the
+fast predator. The contralateral and parallel strategies come last
+in all datasets and are least descriptive of experimental data.
+Further analysis of distance-optimal strategy
+While the predator speed was controlled at V = 2, 11, 20 cm s−1,
+the zebrafish larvae were almost identical in all experiments, im-
+plying that the speed ratio U/V varied drastically between evasion
+instances: for the fast predator, this ratio is up to 10 times that of
+the slow predator. If the prey were to sense and use the speed ratio
+to implement the distance-optimal strategy, we would expect the
+best performance to appear at different values of χ = cos−1(U/V )
+depending on predator speed. To test this, we evaluated this strat-
+egy for the slow, mid, and fast predator as a function of χ ∈ [0, 90◦]
+under both precise and noisy sensing and response (SI, section 7,
+Fig.
+S14).
+We found that the K-L divergence decreased as χ
+increased and reached a minimum near χ = 75◦ independent of
+predator speed. Similarly, the NLL dropped as χ increased until it
+reached a minimum at, or close to, χ = 90◦. These results suggest
+that, even if following the distance-optimal strategy, the prey does
+not rely on real-time and accurate measurements of the speed ratio
+U/V , but favors the limit of large predator speed (χ → 90◦), where
+the distance-optimal strategy converges to the orthogonal strategy.
+The same conclusion can be reached by examining the values
+of the optimized noise parameters. In the range 20◦ ≲ χ ≲ 75◦,
+the optimizer mostly selects the largest possible value of σΛ = π to
+best fit the data. This high level of optimized noise indicates that
+λ is not an effective sensory cue in the distance-optimal strategy,
+and that the prey is unlikely to use this strategy at moderate χ
+values (see SI, section7, Fig. S14).
+Evaluating the biomechanical constraints on es-
+cape strategy
+To complete our evaluation of fish evasion strategies, we consid-
+ered the biomechanics of the C-start response. In [34], the motion
+of a zebrafish larvae undergoing a C-start maneuver starting from
+a straight motionless configuration was recorded using high-speed
+photography, and the time evolution of each segment of the fish
+body from the onset of evasion at time t = 0 to after the com-
+pletion of the C-start response at t = T = 25ms was measured.
+We developed a mathematical model of the biomechanics of these
+events and incorporated that model into our analysis.
+We reinterpreted the experimental measurements in the context
+of a three-link fish, head, middle, and tail (Fig. 5B), and we ex-
+tracted from experimental measurements the fish orientation β(t)
+and rotations α1(t) and α2(t) of the head and tail relative to the
+middle segment (see SI, section 8-9).
+The time evolution of the
+zebrafish body during evasion follows the three archetypal stages
+of the C-start response: in stage 1, the fish curls its body to one
+side, rapidly unfurls its body in stage 2, and begins its undulatory
+swimming in stage 3 (Fig. 5C-D).
+A larger number of C-start maneuvers were recorded in [23],
+albeit only measuring the maximum degree of body bending αmax
+and the net change in heading θ = β(T) − β(0) induced by the
+C-start maneuver (Fig. 5F). These results show that the change
+in heading direction θ correlates strongly with the degree of body
+bending [23]. In all recorded maneuvers, the change in body orien-
+tation barely reaches 100◦.
+Physics-based modeling of the C-start response
+To shed light on the relationship between body deformations and
+change in heading θ during evasion, we employed a physics-based
+model of a three-link fish in potential flow [36, 37]. Experimen-
+tal and computational flow analysis had shown that the C-start
+maneuver is dominated by unsteady, pressure-based exchange of
+momentum between the fish and surrounding fluid, with negligible
+contributions from fluid viscosity and shed vorticity [26, 38]. The
+potential flow model captures these unsteady pressure forces via
+the added mass effect (see SI, S8). The fish model is composed of
+three identical prolate spheroids (of major and minor axes a and b)
+such that the head and tail are free to rotate relative to the middle
+link (Fig. 5B); as before, body deformations are described by the
+angles α1(t), α2(t) representing the relative head and tail rotations
+as a function of time t, and body orientation β(t) is the angle be-
+tween the middle section and an inertial direction taken along the
+direction of the initially-straight fish.
+From consideration of momentum balance on the fish-fluid sys-
+tem, we arrived at an equation governing the rate of change of body
+orientation [39, 40, 37] (see SI, section 8)
+˙β = A1(α1, α2) ˙α1 + A2(α1, α2) ˙α2,
+(10)
+where A1 and A2 are nonlinear functions of body deformations
+α1(t), α2(t); A1 and A2 also depend on fish geometry and fluid
+and body densities (ρf and ρb). For ρb = ρf, the fish is neutrally-
+buoyant. When the fluid forces are dominate, the fish can be con-
+sidered massless and ρb is set to zero. Body rotations are propor-
+tional to the line integral of (10) over a curve C describing body
+deformations in the shape space (α1, α2). For a closed curve C,
+this line integral can be rewritten, using Stokes theorem, as an
+area integral over the region of the (α1, α2) space enclosed by C,
+θ = β(T) − β(0) =
+� � �∂A2
+∂α1
+− ∂A1
+∂α2
+�
+dα1dα2.
+(11)
+The scalar field curl2([A1, A2]) ≡ ∂A2/∂α1 − ∂A1/∂α2 is shown
+in Fig. 5E as a colormap over the entire shape space (α1, α2). To
+maximize the turning angle θ, a straight fish should deform its body
+following a closed curve C that encompasses either non-positive or
+non-negative values of curl2([A1, A2]), i.e., either blue or orange
+regions of the shape space. Closed curves in the orange region lead
+6
+
+−0.2
+0.6
+0.2
+0
+−0.4
+0.4
+0.8
+K-L divergence estimate
+combined
+slow
+mid-speed
+fast
+∆AIC/N
+−0.2
+0.5
+0.4
+0.2
+0.1
+0
+0.3
+−0.1
+combined
+slow
+mid-speed
+fast
+Orthogonal
+Distance-optimal
+original
+constrained
+Antipodal
+B
+A
+Fig. 6. Evaluation of the constrained strategies that consider the physical constraint on turning. Results are shown for the three best-performing models. (A) The
+K-L divergence estimates of the constrained models with precise sensing and response (hollow bars), shown with the results of the original models (solid bars, from Fig. 4A).
+In all four datasets, the constrained models provide discernibly lower K-L divergence estimates, thus better fit to experimental data than the original models, except the
+distance-optimal strategy for the mid-speed predator. The orthogonal strategy improved the most after imposing the constraint, making it fit the data best in all datasets.
+(B) The normalized relative AIC for the constrained models with optimized noise in sensing and response, compared to results using the original models in Fig. 4B. The
+orthogonal strategy still provides best fit to all datasets, marked by the lowest AIC scores, and its advantage over the second best model is more noticeable after imposing
+the constraint. The constrained antipodal strategy performs comparable to or even better than the constrained distance-optimal strategy.
+to turning counter-clockwise.
+By symmetry, diagonally-opposite
+curves in the blue region lead to turning clockwise. Theoretically,
+the simplest curve for turning is a circle or an ellipse in the shape
+space of major axis A aligned with α1 = α2, for which the maxi-
+mum bending angle is αmax =
+√
+2A (Fig. 5E). Corresponding fish
+shape deformations and body rotations β(t) are discussed in SI
+(S8-9, Figs. S15-S16).
+Comparing
+model predictions
+to
+C-start
+induced
+turning of the fish body
+We represented the empirical time evolution of shape deformations
+(α1(t), α2(t)) (Fig. 5C) onto the shape space (Fig. 5E). Interest-
+ingly, the curve C (black line) traced by the actual fish follows
+closely the elliptic curve (red line) predicted by the model as best
+for turning. Moreover, when taking the empirical values of α1(t)
+and α2(t) as input to the physics-based model in (10), the resulting
+predictions of β(t) (blue lines in Fig. 5D) follow closely the empiri-
+cal values of β(t) (black line), especially during the first stage of the
+C-start response, where vorticity is negligible; note that while the
+neutrally buoyant model (dashed blue line) deviates slightly from
+empirical observations in stage 2, the massless fish model (solid blue
+line) performs remarkably well way into stage 3, indicating that in-
+deed unsteady pressure forces dominate the C-start maneuver, as
+previously predicted [26].
+We next considered a family of shape changes following the el-
+liptic curve in Fig. 5E by varying A such that αmax =
+√
+2A varied
+from 0 to 120◦. This upper limit on αmax corresponds to a maxi-
+mum bending angle without causing the head and tail of the model
+fish to cross each other, and is consistent with the experimental
+observations of [23]. Using (11), we computed, for each αmax, the
+value of B that optimizes the change in orientation θ, thus creating
+a map from αmax to θ. We compare these model-predictions (blue
+lines) to experimental data [23] (black dots) in Fig. 5F. As before,
+we considered massless and neutrally-buoyant fish. The theoretical
+predictions behave nearly as upper and lower limits to experimen-
+tal data.
+As observed previously [23], turning in the model fish
+barely reaches 100◦ even when the three-link fish bends its body
+to the extreme of the head and tail touching. This indicates that
+the biomechanics of the C-start maneuver imposes an upper limit
+on achievable heading directions θ.
+Constrained evasion strategies
+We next incorporated the physical constraints on θ imposed by the
+C-start biomechanics into our evasion strategies. To this end, we
+mapped the response angle θ(n)
+i
+predicted by evasion strategy n
+onto the interval [0, 100◦] using the quadratic mapping
+θ →
+�
+1 −
+�
+1 − θmax
+π
+� |θ|
+π
+�
+θ.
+(12)
+Small turns get less constrained whereas large turns are limited
+to the maximum angle θmax = 100◦ allowable by the fish biome-
+chanics.
+We applied this constraint to the three most plausible
+strategies: distance-optimal, orthogonal, and antipodal. For each
+constrained strategy, we repeated the analysis presented above un-
+der precise and noisy sensing and response. Results of the K-L di-
+vergence and AIC analysis for the constrained strategies are shown
+in Fig. 6.
+Compared to the unconstrained strategies, penalizing
+large turns makes all three strategies fit better the experimental
+data across all datasets, with or without added noise, with the ex-
+ception of the distance-optimal strategy for the mid-speed preda-
+tor. Under precise sensing and response, the relative ranking of the
+constrained strategies (Fig. 6A) is similar to the original ranking
+(Fig. 4A), with the distinction that the orthogonal strategy at slow
+and mid-speed predator speed surpasses the distance-optimal strat-
+egy and becomes the best ranking model. Under noisy sensing and
+response, the antipodal strategy ranks higher than the distance-
+optimal strategy in all but the slow predator dataset (Fig. 6B).
+Importantly, whether precise or noisy, the orthogonal strategy fits
+the experimental data better than the other two in all datasets.
+Discussion
+We developed a comprehensive framework for resolving evasion
+strategy from kinematic measurements.
+Our approach considers
+multiple hypotheses, each defined mathematically (Fig. 1), that
+address the role of sensorimotor noise (Fig.
+3) and incorporate
+the effects of biomechanical constraints (Figs. 5–6). Importantly,
+our approach provides a rigorous methodology, rooted in strong-
+inference principles [27], for revealing the strategy that best fits pre-
+vious kinematic measurements of zebrafish larvae. This approach
+eliminates bias towards a particular hypothetical strategy, as done
+in a previous study that favored the contralateral strategy from the
+dataset presently analyzed [17].
+We found that the responses of zebrafish larvae to evade a preda-
+tor are best-characterized by the orthogonal strategy (Fig. 4). This
+finding challenges the notion that a prey aims either to solely con-
+fuse, or maximize its distance from, the predator with its escape
+[16]. The kinematics of zebrafish do not exhibit the uniform distri-
+bution of escape direction θ characteristic of a pure-protean strat-
+egy (SI, Fig. S2E) [13, 14, 15]. Instead, larvae exhibited correla-
+tions between θ and predator state, including angular position φ
+(SI, Fig. S5) and heading ψ (SI, Fig. S7). The distance-optimal
+strategy is more predictive of zebrafish kinematics, but is inferior
+to the orthogonal strategy, based on K-L divergence and the AIC
+scores (Figs. 4 and 6). Therefore, zebrafish larvae do not conform
+to the classic dichotomy of models for prey strategy. Although the
+prevailing patterns favor an orthogonal strategy, variation about
+the predictions for this hypothesis allows for the possibility of a
+mixed strategy that could hinder a predator’s ability to anticipate
+7
+
+the prey’s direction.
+These results are relevant to predator-prey
+encounters, and hence the ecology, of fish species and reflect the ad-
+vantages and constraints of the prey’s neurophysiology and biome-
+chanics.
+The distance-optimal strategy requires sensing that may exceed
+the abilities of larval fish. This strategy requires detection of the
+speed of the approaching predator (Table 1), but larval fish possess
+poor visual acuity, compared to adult fish, due to a relatively small
+number of retinal cells [41].
+It has been demonstrated that the
+escape is triggered by a threshold diameter of a looming visual
+stimulus, which may be simulated as a circle with an expanding
+diameter [19]. A looming stimulus alone does not offer the means
+to differentiate between threats that are small and fast or large and
+slow. Therefore, the visual system of larval fish may offer a sensory
+constraint on its ability to perform the distance-optimal strategy.
+A more sophisticated visual system could allow for additional cues
+to gauge the speed or size of a predator, but the processing time
+necessary to formulate a distance-optimal response may still pose
+a liability in evasion speed compared with the orthogonal strategy.
+The orthogonal strategy merely requires an estimate of the
+predator’s heading and offers tactical benefits relative to many of
+the alternatives. This strategy is equivalent to the distance-optimal
+strategy for a high-speed approach (U/V ≪ 1) and therefore suc-
+ceeds in maximizing the prey’s distance from a fast predator at
+reduced sensing requirements (Table 1). It is the fastest predators
+that likely present the greatest threat to the prey. The orthogo-
+nal strategy offers an additional tactical advantage by evading in
+a direction that is challenging for a fast-approaching predator to
+follow because, in order for the predator to execute such large turn
+at high speed, it needs a large turning radius, which could increase
+its distance from the prey even further.
+The predictions of the orthogonal strategy improved in their fit
+to measured kinematics when we considered constraints imposed by
+the biomechanics of the C-start (Fig. 6). In particular, our model
+of a three-link fish in potential flow accurately describes the rela-
+tionship between the change in fish shape and its turning motion
+during evasion (Fig. 5). By mapping maximum bending angle to
+turning angle, the model predicted an upper limit (around 100◦) on
+achievable turning motion, consistent with the maximum angle ob-
+served in zebrafish exposed to a lateral looming stimulus [19, 23].
+The improvement in model predictions that included mechanics
+demonstrates the influence of the constraints imposed by the prey
+biomechanics and its interaction with the fluid environment on the
+evasion strategy of zebrafish larvae.
+Comparing the K-L divergence and AIC values across slow, in-
+termediate, and fast predators, the prevalence of the orthogonal
+strategy is clearest in the case of the fast predator (Fig. 4 and
+Fig. 6). This feature can be related to the fact that a weak stimu-
+lus (slow predator) is more likely to trigger an escape response via
+the less predictable, long-latency neural pathway [42, 43], as op-
+posed to the fast pathway with minimal latency between perceived
+danger and motor response [44]. An untangling of these features
+requires a deeper investigation of how our analytical framework
+relates to the neurophysiology underlying zebrafish evasion.
+Our study combined tools from information theory and proba-
+bilistic methods with behavioral evasion models and physics-based
+models of the C-start biomechanics to develop a comprehensive
+analytical approach and thereby determine the evasion strategy of
+zebrafish larvae. Aside from the details of the biomechanics model,
+nothing about our approach is specific to the study of fish. Our
+analysis could be applied to the myriad of studies that have mea-
+sured escape responses relative to a predator’s approach in a diver-
+sity of animals [5, 6, 7, 8, 9, 10]. This approach may therefore be
+applied broadly to the study of predator-prey encounters to reveal
+the strategic basis of this fundamental aspect of animal behavior.
+Acknowledgement
+E.K. acknowledges support from the Of-
+fice of Naval Research (ONR) Grants N00014-22-1-2655, N00014-
+19-1-2035, N00014-17-1-2062, and N00014-14-1-0421; the National
+Science Foundation (NSF) Grants RAISE IOS-2034043, CBET-
+2100209, and INSPIRE MCB-1608744; the National Institutes of
+Health (NIH) Grant R01 HL 153622-01A1; the Army Research Of-
+fice (ARO) Grant W911NF-16-1-0074.
+This research started in
+summer 2018 at the Summer Graduate School on Mathematical
+Analysis of Behavior organized by Ann Hermundstad, Vivek Ja-
+yaraman, Eva Kanso, and L. Mahadevan. The school was jointly
+supported by the Mathematical Science Research Institute (MSRI)
+and the Howard-Hugh Medical Institute (HHMI) Janelia Research
+Campus, and was held at Janelia. BC, YM, and EK acknowledge
+support from Janelia and would like to thank Sashank Pisupati and
+Ann Hermundstad for helpful discussions.
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+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL
+NOTES
+CHENGYANG SHAO
+In this note, we propose several unsolved problems concerning the irrotational oscillation of a water
+droplet under zero gravity. We will derive the governing equation of this physical model, and convert it
+to a quasilinear dispersive partial differential equation defined on the sphere, which formally resembles the
+capillary water waves equation but describes oscillation defined on curved manifold instead. Three types of
+unsolved mathematical problems related to this model will be discussed in observation of hydrodynamical
+experiments under zero gravity1: (1) Strichartz type inequalities for the linearized problem (2) existence of
+periodic solutons (3) normal form reduction and generic lifespan estimate. It is pointed out that all of these
+problems are closely related to certain Diophantine equations, especially the third one.
+1. Capillary Spherical Water Waves Equation: Derivation
+1.1. Water Waves Equation for a Bounded Water Drop. Comparing to gravity water waves problems,
+the governing equation for a spherical droplet of water under zero gravity takes a very different form. At
+a first glance it looks similar to the water waves systems as mentioned above, but some crucial differences
+do arise after careful analysis. To the author’s knowledge, besides those dealing with generic free-boundary
+Euler equation ( [14], [36]), the only reference on this problem is Beyer-G¨unther [9], in which the local
+well-posedness of the equation is proved using a Nash-Moser type implicit function theorem. We will briefly
+discribe known results for gravity water waves problems in the next subsection.
+To start with, let us pose the following assumptions on the fluid motion that we try to describe:
+• (A1) The perfect, irrotational fluid of constant density ρ0 occupies a smooth, compact region in R3.
+• (A2) There is no gravity or any other external force in presence.
+• (A3) The air-fluid interface is governed by the Young-Laplace law, and the effect of air flow is
+neglected.
+We assume that the boundary of the fluid region has the topological type of a smooth compact orientable
+surface M, and is described by a time-dependent embedding ι(t, ·) : M → R3. We will denote a point on M
+by x, the image of M under ι(t, ·) by Mt, and the region enclosed by Mt by Ωt. The outer normal will be
+denoted by N(ι). We also write ¯∇ for the flat connection on R3.
+Adopting assumption (A3), we have the Young-Laplace equation:
+σ0H(ι) = pi − pe,
+where H(ι) is the (scalar) mean curvature of the embedding, σ0 is the surface tension coefficient (which is
+assumed to be a constant), and pi, pe are respectively the inner and exterior air pressure at the boundary;
+they are scalar functions on the boundary and we assume that pe is a constant. Under assumptions (A1) and
+(A2), we obtain Bernoulli’s equation, sometimes referred as the pressure balance condition, on the evolving
+1There are numbers of visual materials on such experiments conducted by astronauts. See for example https://www.youtube.
+com/watch?v=H_qPWZbxFl8&t or https://www.youtube.com/watch?v=e6Faq1AmISI&t.
+1
+arXiv:2301.00115v1  [math.AP]  31 Dec 2022
+
+2
+CHENGYANG SHAO
+surface:
+(1.1)
+∂Φ
+∂t
+����
+Mt
++ 1
+2| ¯∇Φ|Mt|2 − pe = −σ0
+ρ0
+H(ι),
+where Φ is the velocity potential of the velocity field of the air. Note that Φ is determined up to a function in
+t, so we shall leave the constant pe around for convenience reason that will be explained shortly. According
+to assumption (A1), the function Φ is a harmonic function within the region Ωt, so it is uniquely determined
+by its boundary value, and the velocity field within Ωt is ¯∇Φ. The kinematic equation on the free boundary
+Mt is naturally obtained as
+(1.2)
+∂ι
+∂t · N(ι) = ¯∇Φ|Mt · N(ι).
+Finally, we would like to discuss the conservation laws for (1.1)-(1.2). The preservation of volume Vol(Ωt) =
+Vol(Ω0) is a consequence of incompressibility. The system describes an Eulerian flow without any external
+force, so the center of mass moves at a uniform speed along a fixed direction, i.e.
+(1.3)
+1
+Vol(Ω0)
+�
+Ωt
+PdVol(P) = V0t + C0,
+with Vol being the Lebesgue measure, P marking points in R3, V0 and C0 being the velocity and starting
+position of center of mass respectively. Furthermore, the total momentum is conserved, and since the flow is
+a potential incompressible one, the conservation of total momentum is expressed as
+(1.4)
+�
+Mt
+ρ0ΦN(ι)dArea(Mt) ≡ ρ0Vol(Ω0)V0.
+Most importantly, it is not surprising that (1.1)-(1.2) is a Hamilton system (the Zakharov formulation for
+water waves; see Zakharov [43]), with Hamiltonian
+(1.5)
+σ0Area(ι) + 1
+2
+�
+Ωt
+ρ0| ¯∇Φ|2dVol = σ0Area(Mt) + 1
+2
+�
+Mt
+ρ0Φ|Mt
+� ¯∇Φ|Mt · N(ι)
+�
+dArea,
+i.e. potential proportional to surface area plus kinetic energy of the fluid.
+1.2. Converting to a Differential System. It is not hard to verify that the system (1.1)-(1.2) is invariant
+if ι is composed with a diffeomorphism of M; we may thus regard it as a geometric flow. If we are only
+interested in perturbation near a given configuration, we may reduce system (1.1)-(1.2) to a non-degenerate
+dispersive differential system concerning two scalar functions defined on M, just as Beyer and G¨unther did
+in [9]. In fact, during a short time of evolution, the interface can be represented as the graph of a function
+defined on the initial surface: if ι0 : M → R3 is a fixed embedding close to the initial embedding ι(0, x), we
+may assume that ι(t, x) = ι0(x) + ζ(t, x)N0(x), where ζ is a scalar “height” function defined on M0 and N0
+is the outer normal vector field of M0.
+With this observation, we shall transform the system (1.1)&(1.2) into a non-local system of two real scalar
+functions (ζ, φ) defined on M, where ζ is the “height” function described as above, and φ(t, x) = Φ(t, ι(t, x))
+is the boundary value of the velocity potential, pulled back to the underlying manifold M.
+The operator
+Bζ : φ → ¯∇Φ|Mt
+maps the pulled-back Dirichlet boundary value φ to the boundary value of the gradient of Φ. We shall write
+(D[ζ]φ)N(ι)
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+3
+Figure 1. The shape of the surface
+for its normal part, where D[ζ] is the Dirichlet-Neumann operator corresponding to the region enclosed by
+the image of ι0 + ζN0. Thus
+∂ζ
+∂t N0 · N(ι) = D[ζ]φ.
+We also need to calculate the restriction of ∂tΦ on Mt in terms of φ and ι. By the chain rule,
+∂Φ
+∂t
+����
+Mt
+= ∂φ
+∂t − ¯∇Φ|Mt · ∂ι
+∂t
+= ∂φ
+∂t −
+� ¯∇Φ|Mt · N0
+� ∂ζ
+∂t
+= ∂φ
+∂t −
+1
+N0 · N(ι)
+� ¯∇Φ|Mt · N0
+�
+· D[ζ]φ.
+We thus arrive at the following nonlinear system:
+(EQ(M))
+�
+�
+�
+�
+�
+�
+�
+∂ζ
+∂t =
+1
+N0 · N(ι)D[ζ]φ,
+∂φ
+∂t =
+1
+N0 · N(ι) (Bζφ · N0) · D[ζ]φ − 1
+2|Bζφ|2 − σ0
+ρ0
+H(ι) + pe,
+where ι = ι0 + ζN0.
+Remark 1.
+We may obtain an explicit expression of Bζφ = ¯∇Φ|Mt in terms of φ (together with the
+connection ∇0 on the fixed embedding ι0(M)), just as standard references did for Euclidean or periodic
+water waves, but that is not necessary for our discussion at the moment. It is important to keep in mind
+that the preservation of volume and conservation of total momentum (1.3)-(1.4) convert to integral equalities
+of (ζ, φ). These additional restrictions are not obvious from the differential equations (EQ(M)), though they
+can be deduced from (EQ(M)) since they are just rephrase of the original physical laws (1.1)-(1.2).
+For M = S2, the case that we shall discuss in detail, we refer to the system as the capillary spherical water
+waves equation. To simplify our discussion, we shall be working under the center of mass frame, and require
+the eigenmode Π(0)φ to vanish for all t. This could be easily accomplished by absorbing the eigenmode
+into φ since the equation is invariant by a shift of φ. In a word, from now on, we will be focusing on the
+
+NS+ 0
+1
+1
+1
+-
+-
+-
+-
+1
+-
+-
+y4
+CHENGYANG SHAO
+non-dimensional capillary spherical water waves equation
+(EQ)
+�
+�
+�
+�
+�
+�
+�
+∂ζ
+∂t =
+1
+N0 · N(ι)D[ζ]φ,
+∂φ
+∂t =
+1
+N0 · N(ι) (Bζφ · N0) · D[ζ]φ − 1
+2|Bζφ|2 − H(ι),
+where ι = (1 + ζ)ι0, and Π(0)φ ≡ 0. We assume that the total volume of the fluid is 4π/3, so that the
+preservation of volume is expressed as
+(1.6)
+1
+3
+�
+S2(1 + ζ)3dµ0 ≡ 4π
+3 ,
+where µ0 is the standard measure on S2. The inertial movement of center of mass (1.3) and conservation of
+total momentum (1.4) under our center of mass frame are expressed respectively as
+(1.7)
+�
+S2(1 + ζ)4N0dµ0 = 0,
+�
+S2 φN(ι)dµ(ι) = 0,
+where µ(ι) is the induced surface measure. Further, the Hamiltonian of the system is
+(1.8)
+H[ζ, φ] = Area(ι) + 1
+2
+�
+S2 φ · D[ζ]φ · dµ(ι),
+and for a solution (ζ, φ) there holds H[ζ, φ] ≡ 4π.
+Up to this point, we are still working within the realm of well-established frameworks. We already know
+that the general free-boundary Euler equation is locally well-posed due to the work of [14] or [36], and due
+to the curl equation the curl free condition persists during the evolution. On the other hand, the Cauchy
+problem of system (1.1) and (1.2) is known to be locally well-posed, due to Beyer and G¨unther in [9].
+They used an iteration argument very similar to a Nash-Moser type argument in the sense that it involves
+multiple scales of Banach spaces and “tame” maps in a certain sense. Finally, it is not hard to transplant
+the potential-theoretic argument of Wu [40] to prove the local well-posedness.
+To sum up, we already know that the system (EQ(M)) for a compact orientable surface M (hence (EQ)
+specifically) is locally well-posed. But this is all we can assert for the motion of a water droplet under zero
+gravity. In the following part of this note, we will propose several questions and conjectures concerning the
+long-time behaviour of water droplets under zero gravity.
+1.3. Previous Works on Water Waves. There has already been several different approaches to describe
+the motion of perfect fluid with free boundary. One is to consider the motion of perfect fluid occupying an
+arbitrary domain in R2 or R3, either with or without surface tension. The motion is described by a free
+boundary value problem of Euler equation. This generic approach was employed by Countand-Shkoller [14]
+and Shatah-Zeng [36]. Both groups proved the local-wellposedness of the problem. This approach has the
+advantage of being very general, applicable to all geometric shapes of the fluid.
+On the other hand, when coming to potential flows of a perfect fluid, the curl free property results in dis-
+persive nature of the problem. The motion of a curl free perfect fluid under gravity and a free boundary value
+condition is usually referred to as the gravity water waves problem. The first breakthroughs in understanding
+local well-posedness were works of Wu [39] [40], who proved local well-posedness of the gravity water waves
+equation without any smallness assumption. Lannes [27] extended this to more generic bottom shapes. Taking
+surface tension into account, the problem becomes gravity-capillary water waves. Schweizer [32] proved local
+well-posedness with small Cauchy data of the gravity-capillary water waves problem, and Ming-Zhang [30]
+proved local well-posedness without smallness assumption. Alazard-Metevier [2] and Alazard-Burq-Zuily [3]
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+5
+used para-differential calculus to obtain the optimal regularity for local well-posedness of the water waves
+equation, either with or without surface tension.
+For discussion of long time behavior, it is important to take into account the dispersive nature of the
+problem. For linear dispersive properties, there has been work of Christianson-Hur-Staffilani [13]. For gravity
+water waves living in R2, works on lifespan estimate include Wu [41] and Hunter-Ifrim-Tataru [22] (almost
+global result), Ionescu-Pusateri [24] and Alazard-Delort [4] and Ifrim-Tataru [23] (global result). For gravity
+water waves living in R3, there are works of Germain-Masmoudi-Shatah [18] and Wu [42] (no surface tension),
+Germain-Masmoudi-Shatah [19] (no gravity), Deng-Ionescu-Pausader-Pusateri [15] (gravity-capillary water
+waves) and Wang [38] (gravity-capillary water waves with finite depth). These results all employed different
+forms of decay estimates derived from dispersive properties.
+As for long time behavior of periodic water waves, Berti-Delort [6] considered gravity-capillary water waves
+defined on T1, Berti-Feola-Pusateri [7] considered gravity water waves defined on T1, Ionescu-Pusateri [26]
+considered gravity-capillary water waves defined on T2, and obtained an estimate on the lifespan beyond
+standard energy method. All of the three groups used para-differential calculus and suitable normal form
+reduction; the results of Berti-Delort and Ionescu-Pusateri were proved for physical data of full Lebesgue
+measure.
+To sum up, all results on the gravity water waves problem listed above are concerned with an equation for
+two scalar functions defined on a fixed flat manifold, being one of the following: R1, R2, T1, T2, sometimes
+called the “bottom” of the fluid. These two functions represents the geometry of the liquid-gas interface and
+the boundary value of the velocity potential, respectively. The manifold itself is considered as the bottom
+of the container in which all dynamics are performed. We observe that the differential equation (EQ(M)) is,
+mathematically, fundamentally different from the water waves equations that have been well-studied.
+2. Initial Notes on Unsolved Problems
+In this section, we propose unsolved problems related to the spherical capillary water waves system
+(EQ(M)) with M = S2. Not surprisingly, these problems all have deep backgrounds in number theory.
+2.1. Linearization Around the Static Solution. We are mostly interested in the stability of the static
+solution of (EQ(M)). A static solution should be a fluid region whose shape stays still, with motion being
+a mere shift within the space.
+In this case, we have pe = 0 since the reference is relatively static with
+respect to the air. Moreover, the velocity field ¯∇Φ and “potential of acceleration” ∂Φ/∂t must both be
+spatially uniform, so that the left-hand-side of the pressure balance condition (1.1) is a function of t alone.
+It follows that ι(M) is always a compact embedded surface of constant mean curvature, hence in fact always
+an Euclidean sphere by the Alexandrov sphere theorem (see [29]), and we may just take M = S2. Moreover,
+since ι(S2) should enclose a constant volume by incompressibility, the radius of that sphere does not change.
+After suitable scaling, we may assume that the radius is always 1, and ρ0 = 1, σ0 = 1, to make (1.1)-(1.2)
+non-dimensional. Finally, by choosing the center of mass frame, we may simply assume that the spatial shift
+is always zero, so that the velocity potential Φ ≡ Φ0, a real constant. It is harmless to fix it to be zero.
+Thus, under our convention, a static solution of (1.1)-(1.2) takes the form
+(2.1)
+�
+ι(t, x)
+Φ(t, x)
+�
+=
+�
+ι0(x)
+0
+�
+,
+where a ∈ R3 is a constant vector, and ι0 is the standard embedding of S2 as ∂B(0, 1) ⊂ R3. Equivalently,
+this means that a static solution of (EQ(M)) under our convention must be (ζ, φ) = (0, 0). Note here that
+the Gauss map of ι0 coincides with itself.
+
+6
+CHENGYANG SHAO
+We can now start our perturbation analysis around a static solution at the linear level. Let E(n) be the
+space of spherical harmonics of order n, normalized according to the standard surface measure on S2. In
+particular, E(1) is spanned by three components of N0. Let Π(n) be the orthogonal projection on L2(S2)
+onto E(n), Π≤n be the orthogonal projection on L2(S2) onto �
+k≤n E(k), Π≥n be the orthogonal projection
+on L2(S2) onto �
+k≥n E(k). For ι = (1 + ζ)ι0, the linearization of −H(ι) around the sphere ζ ≡ 0 is ∆ζ + 2ζ,
+where ∆ is the Laplacian on the sphere S2; cf. the standard formula for the second variation of area in [5].
+Then H′(ι0) acts on E(n) as the multiplier −(n − 1)(n + 2). Note that even if we consider the dimensional
+form of (EQ), there will only be an additional scaling factor σ0/(ρ0R2), where R is the radius of the sphere.
+On the other hand, the following solution formula for the Dirichlet problem on B(0, 1) is well-known: if
+f ∈ L2(S2), then the harmonic function in B(0, 1) with Dirichlet boundary value f is determined by
+u(r, ω) =
+�
+n≥0
+rn(Π(n)f)(ω),
+where (r, ω) is the spherical coordinate in R3. Thus the Dirichlet-Neumann operator D[ι0] acts on E(n) as the
+multiplier n. Note again that even if we consider the dimensional form (EQ), there will only be an additional
+scaling factor R−1.
+Thus, setting
+u = Π(0)ζ + Π(1)ζ +
+�
+n≥2
+�
+(n − 1)(n + 2) · Π(n)ζ + i
+�
+n≥1
+√n · Π(n)φ,
+we find that the linearization of (EQ) around the static solution (2.1) is a linear dispersive equation
+(2.2)
+∂u
+∂t + iΛu = 0,
+where the 3/2-order elliptic operator Λ is given by a multiplier
+Λ =
+�
+n≥2
+�
+n(n − 1)(n + 2) · Π(n) =:
+�
+n≥0
+Λ(n)Π(n).
+Note that (ζ, φ) is completely determined by u. At the linear level, there must hold Π(0)u ≡ 0 because the
+first variation of volume must be zero; and Π(1)u ≡ 0 because of the conservation laws (1.7).
+Let us also re-write the original nonlinear system (EQ) into a form that better illustrates its perturbative
+nature. For simplicity, we use O(u⊗k) to abbreviate a quantity that can be controlled by k-linear expressions
+in u, and disregard its continuity properties for the moment. For example, ∥u∥2
+H1 + ∥u∥4
+H2 is an expression
+of order O(u⊗2) when u → 0.
+Since the operator Λ acts degenerately on E(0) ⊕ E(1), we should be more careful about the eigenmodes
+Π(0)u and Π(1)u. The volume preservation equation (1.6) implies ∂tΠ(0)ζ = O(u⊗2). Projecting (EQ) to
+E(1), which is spanned by the components of N0, we obtain ∂tΠ(1)ζ = Π(1)φ + O(u⊗2) = O(u⊗2) since the
+conservation law (1.7) implies Π(1)φ = O(u⊗2); and ∂tΠ(1)φ = O(u⊗2) since H′(ι0) = −∆ − 2 annihilates
+E(1). We can thus formally re-write the nonlinear system (EQ) as the following:
+(2.3)
+∂u
+∂t + iΛu = N(u),
+with N(u) = O(u⊗2) vanishing quadratically as u → 0. Note that we are disregarding all regularity problems
+at the moment.
+2.2. Question at Linear Level. At the linear level, our first unanswered question is
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+7
+Question 1. Does the solution of the linear capillary spherical water waves equation (2.2) satisfy a Strichartz
+type estimate of the form
+∥eitΛf∥Lp
+T Lq
+x ≲T ∥f∥Hs,
+where Lp
+T Lq
+x = Lp([0, T]; Lq(S2)), and the admissible indices (p, q) and s should be determined?
+Answer to Question 1 should be important in understanding the dispersive nature of linear capillary
+spherical water waves. For Schr¨odinger equation on a compact manifold, a widly cited result was obtained
+by Burq-G´erard-Tzvetkov [12]:
+Theorem 2.1. On a general compact Riemannian manifold (M d, g), there holds
+∥eit∆gf∥Lp
+t Lq
+x([0,T ]×M) ≲T ∥f∥H1/p(M),
+where
+2
+p + d
+q = d
+2,
+p > 2.
+The authors used a time-localization argument for the parametrix of ∂t − i∆g to prove this result. For the
+sphere Sd, this inequality is not optimal. The authors further used a Bourgain space argument to obtain the
+optimal Strichartz inequality:
+Theorem 2.2. Let (Sd, g) be the standard n-dimensional sphere. For a function f ∈ C∞(Sd), there holds
+the Strichartz inequality
+∥eit∆gf∥Lp
+t Lq
+x([0,T ]×M) ≲T ∥f∥Hs(M),
+s > s0(d)
+where
+s0(2) = 1
+8,
+s0(d) = d
+4 − 1
+2, d ≥ 3.
+Furthermore these inequalities are optimal in the sense that the Sobolev index s cannot be less than or equal
+to s0(d).
+The proof is the consequence of two propositions. The first one is the “decoupling inequality on compact
+manifolds”, in particular the following result proved by Sogge [33]:
+Proposition 2.1. Let Πk be the spectral projection to eigenspaces with eigenvalues in [k2, (k + 1)2] on Sd.
+Then there holds
+∥Πk∥L2→Lq ≤ Cqns(q),
+where
+s(q) =
+�
+d−1
+2
+�
+1
+2 − 1
+2q
+�
+,
+2 ≤ q ≤ 2(d+1)
+d−1
+d−1
+2
+− d
+q ,
+2(d+1)
+d−1
+≤ q ≤ ∞.
+These estimates are sharp in the following sense: if hk is a zonal spherical harmonic function of degree k on
+Sd, then as k → ∞,
+∥hk∥Lq ≃ Cqks(q)∥hk∥L2.
+The second one is a Bourgain space embedding result:
+Proposition 2.2. For a function f ∈ C∞
+0 (R × Sd), define the Bourgain space norm
+∥f(t, x)∥Xs,b :=
+��⟨∂t + i∆g⟩bf(t, x)
+��
+L2
+t Hsx .
+Then for b > 1/2 and s > s0(d), there holds
+∥f∥L4(R×Sd) ≤ Cs,b∥f∥Xs,b.
+
+8
+CHENGYANG SHAO
+The key ingredient for proving this proposition is the following number-theoretic result:
+#{(p, q) ∈ N2 : p2 + q2 = A} = O(Aε).
+As for optimality of the Strichartz inequality, the authors of [12] implemented standard results of Gauss
+sums.
+The parametrix and Bourgain space argument can be repeated without essential change for the linear
+capillary spherical water waves equation (2.2), but this time the Bourgain space argument would be more
+complicated: the Bourgain space norm is now
+∥f(t, x)∥Xs,b :=
+��⟨∂t + iΛ⟩bf(t, x)
+��
+L2
+t Hsx ,
+and the embedding result becomes
+∥f∥L4(R×S2) ≤ Cs,b∥f∥Xs,b
+for f ∈ C∞
+0 (R × S2) and all s > 3ρ/8 + 1/8, b > 1/2, where ρ is the infimum of all exponents ρ′ such that
+when A → ∞, the number
+#
+�
+(n1, n2) ∈ N2 : 1
+2 ≤ n2
+n1
+≤ 2, |Λ(n1) + Λ(n2) − A| ≤ 1
+2
+�
+≤ Cρ′Aρ′.
+Some basic analytic number theory implies ρ = 1/3, and thus the range of s is s > 1/4. Surprisingly this
+index is not better than that predicted by the parametrix method. It remains unknown whether this index
+could be further optimized.
+To close this subsection, we note that the capillary spherical water wave lives on a compact region, so the
+dispersion does not take away energy from a locality to infinity. This is a crucial difference between waves
+on compact regions and waves in Euclidean spaces. In particular, we do not expect decay estimate for eitΛf.
+For the nonlinear problem (EQ), techniques like vector field method (Klainerman-Sobolev type inequalities)
+do not apply.
+2.3. Rotationally Symmetric Solutions: Bifurcation Analysis. Illuminated by observations in hydro-
+dynamical experiments under zero gravity, and suggested by the existence of standing gravity capillary water
+waves due to Alazard-Baldi [1], we propose the following conjecture:
+Conjecture 2.1. There is a Cantor family of small amplitude periodic solutions to the spherical capillary
+water waves system (EQ).
+Let us conduct the bifurcation analysis that suggests why this conjecture should be true. By time rescaling,
+we aim to find solution (ζ, φ, ω0) of the following system that is 2π-peiodic in t:
+(2.4)
+�
+�
+�
+�
+�
+�
+�
+ω0
+∂ζ
+∂t =
+1
+N0 · N(ι)D[ζ]φ,
+ω0
+∂φ
+∂t =
+1
+N0 · N(ι) (Bζφ · N0) · D[ζ]φ − 1
+2|Bζφ|2 − H(ι),
+together with the conservation laws (1.6)-(1.8). Here we refer ω0 > 0 as the fundamental frequency. The
+linearization of this system at the equilibrium (ζ, φ) = (0, 0) is
+(2.5)
+Lω0
+�
+ζ
+φ
+�
+:=
+�
+ω0∂t
+−D[0]
+−∆ − 2
+ω0∂t
+� �
+ζ
+φ
+�
+= 0,
+Π(0)ζ = Π(1)ζ = Π(1)φ = 0.
+We restrict to rotationally symmetric solutions of the system: that is, water droplets which are always
+rotationally symmetric with a fixed axis. In addition, we require ζ to be even in t and φ to be odd in t. The
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+9
+solution thus should take the form
+ζ(t, x) =
+�
+j,n≥0
+ζjn cos(jt)Yn(x),
+φ(t, x) =
+�
+j≥1,n≥0
+φjn sin(jt)Yn(x),
+where Yn is the n’th zonal spherical harmonic, i.e. the (unique) normalized spherical harmonic of degree n
+that is axially symmetric. In spherical coordinates this means that Yn(θ, ϕ) = Pn(cos θ), where Pn is the
+n’th Legendre polynomial. Since φ0n are irrelevant we fix them to be 0. Then
+Lω0
+�
+ζ
+φ
+�
+=
+�
+j,n≥0
+�
+(−ω0jζjn − nφjn) sin(jt)Yn(x)
+((n − 1)(n + 2)ζjn + ω0jφjn) cos(jt)Yn(x)
+�
+.
+In order that (ζ, φ)T ∈ KerLω0, at the level n = 0, we must have ζj0 = φj0 = 0 for all j ≥ 0. At the level
+n = 1, we have ζ01 = φ01 = 0, and for j ≥ 1 there holds ω0jζj1 − φj1 = 0 and ω0jφj1 = 0, so ζj1 = φj1 = 0
+for all j ≥ 0. Hence ζjn, φjn can be nonzero only for j ≥ 1 and n ≥ 2.
+Consequently, Lω0 has a one-dimensional kernel if and only if the Diophantine equation
+(2.6)
+ω2
+0j2 = n(n − 1)(n + 2),
+j ≥ 1, n ≥ 2
+has exactly one solution (j0, n0). If ω0 has this property, then at the linear level, the lowest frequency of
+oscillation is
+ω0j0 =
+�
+n0(n0 − 1)(n0 + 2).
+We look into this Diophantine equation. Obviously ω2
+0 has to be a rational number. The equation is
+closely related to a family of elliptic curves over Q:
+Ec : y2 = x(x − c)(x + 2c) = x3 + c2x2 − 2c2x,
+c ∈ N.
+If we set a/b = ω2
+0 (irreducible fraction), then integral solutions of (2.6) are in 1-1 correspondence with
+integral points with natural number coordinates on the elliptic curve Eab, under the following map:
+(j0, n0) → (abn0, a2bj0) ∈ Eab.
+Thus we just need to find natural numbers a, b such that there is a unique (up to negation) integral point
+(x, y) ∈ Eab, where x > 0 is divided by ab, and y is divided by a2b. We seek for n0 as small as possible with
+such property, which gives lowest frequency of oscillation as small as possible. For ab = 1, · · · , 50, we find
+that if ab = 15, then the integral points on elliptic curve E15 (up to negation of Mordel-Weil group) are
+(−30, 0), (−5, 50), (0, 0), (15, 0), (24, 108), (90, 900).
+The only point (x, y) with ab|x and ab2|y is (90,900), which gives n0 = 6, and the lowest frequency Λ(n0) =
+�
+n0(n0 − 1)(n0 + 2) of oscillation is 4
+√
+15 ≃ 15.49 · · · .
+There are other choices of a, b. We list down the value of ab below 50, the corresponding n0 and the lowest
+frequency Λ(n0):
+ab
+n0
+Λ(n0)
+15
+6
+4
+√
+15
+17
+49
+4
+√
+323
+22
+9
+6
+√
+22
+26
+50
+15
+√
+78
+42
+7
+3
+√
+42
+46
+576
+2040
+√
+46
+50
+25
+90
+√
+2
+
+10
+CHENGYANG SHAO
+See Appendix A for the MAGMA code used to find these values. This list suggests that n0 = 6 might
+be the smallest order that meets the requirement, but this remains unproved. We summarize these into the
+following number-theoretic question:
+Question 2. For the family of elliptic curves
+Eab : y2 = x(x − ab)(x + 2ab),
+a, b ∈ N,
+how many choices of a, b ∈ N are there such that, there is exactly one integral point (x, y) ∈ Eab with x, y > 0
+and ab|x, a2b|y? For such a, b and integral point (x, y), is the minimal value of x/(ab) exactly 6, or is it
+smaller?
+Of course, a complete answer of Question 2 should imply very clear understanding of periodic solutions
+of the spherical capillary water waves equation constructed using bifurcation analysis. But at this moment
+we are satisfied with existence, so we may pick any ω0 = a/b and (j0, n0) that meets the requirement, for
+example the simplest case n0 = 6, and any of the following choices of ω0 and j0:
+ω0 =
+√
+15, j0 = 4;
+ω0 =
+�
+1
+15, j0 = 60;
+ω0 =
+�
+3
+5, j0 = 20;
+ω0 =
+�
+5
+3, j0 = 12.
+We thus refine our conjecture as follows:
+Conjecture 2.2. Let (n0, j0) be a pair of natural numbers with n0 ≥ 2, j0 ≥ 1, and set ω0 =
+�
+Λ(n0)/j0.
+Suppose that the only natural number solution of the Diophantine equation
+ω2
+0j0 = Λ(n0)2 = n0(n0 − 1)(n0 + 1)
+is (j0, n0). Then there is a Cantor set with positive measure of parameters ω, clustered near ω0, such that
+the spherical capillary water waves equation (2.4) admits small amplitude periodic solution with frequency ω.
+The counterpart of Conjecture 2.2 for gravity capillary standing water waves was proved by Alazard-
+Baldi [1] using a Nash-Moser type theorem. The key technique in their proof was to find a conjugation of
+the linearized operator of the gravity capillary water waves system on T1 to an operator of the form
+ω∂t + iT + iλ1|Dx|1/2 + iλ−1|Dx|−1/2 + Operator of order ≤ −3
+2,
+where T is an elliptic Fourier multiplier of order 3/2, and λ1, λ−1 are real constants. The frequency ω lives
+in a Cantor type set that clusters around a given frequency so that the kernel of the linearized operator
+is 1-dimensional. With this conjugation, they were able to find periodic solutions of linearized problems
+required by Nash-Moser iteration.
+It is expected that this technique could be transplanted to the equation (2.4), since our analysis for (2.6)
+suggests that the 1-dimensional kernel requirement for bifurcation analysis is met. It seems that the greatest
+technical issue is to find a suitable conjugation that takes the linearized operator of (2.4) to an operator of
+the form
+ω∂t + i(T3/2 + T1/2 + T−1/2) + Operator of order ≤ −3
+2,
+where each Tk is a real Fourier multiplier acting on spherical harmonics. The difficulty is that, since we are
+working with pseudo-differential operators on S2, the formulas of symbolic calculus are not as neat as those
+on flat spaces. It seems necessary to implement some global harmonic analysis for compact homogeneous
+spaces, e.g. extension of results collected in Ruzhansky-Turunen [31]. Unfortunately, those results do not
+include pseudo-differential operators with “rough coefficients” and para-differential operators, so it seems
+necessary to re-write the whole theory.
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+11
+2.4. Number-Theoretic Obstruction with Normal Form Reduction. As pointed out in Section 1,
+the system (2.3) is locally well-posed due to a result in [9]. General well-posedness results [14], [36] for free
+boundary value problem of Euler equation also apply. As for lifespan estimate for initial data ε-close to
+the static solution (2.1), it should not be hard to conclude that the lifespan should be bounded below by
+1/ε. The result relates to the fact that the sphere is a stable critical point of the area functional, cf. [5].
+This is nothing new: a suitable energy inequality should imply it. However, although the clue is clear, the
+implementation is far from standard since we are working on a compact manifold. A rigorous proof still calls
+for hard technicalities.
+Now we will be looking into the nonlinear equation (2.3) for its longer time behavior. Although appearing
+similar to the well-studied water waves equation in e.g. [26], [27], [39], [40], there is a crucial difference between
+the dispersive relation in (2.3) and the well-studied water waves equations: the dispersive relation exhibits
+a strong rigidity property, i.e. the arbitrary physical constants enter into the dispersive relation Λ only as
+scaling factors. For the gravity-capillary water waves, the linear dispersive relation reads
+�
+g|∇| + σ|∇|3,
+where g is the gravitational constant and σ is the surface tension coefficient. For the Klein-Gordon equation
+on a Riemannian manifold, the linear dispersive relation reads
+�
+−∆ + m2,
+where m is the mass. In [26], Ionescu and Pusateri referred such dispersive relations as having non-degenerate
+dependence on physical parameter, while in our context it is appropriate to refer to the dependence as
+degenerate. We will see that this crucial difference brings about severe obstructions for the long-time well-
+posedness of the system.
+Following the idea of Delort and Szeftel [16], we look for a normal form reduction of (2.3) and explain
+why the rigidity property could cause obstructions. Not surprisingly, the obstruction is due to resonances,
+and strongly relates to the solvablity of a Diophantine equation. Delort and Szeftel cast a normal form
+reduction to the small-initial-data problem of quasilinear Klein-Gordon equation on the sphere and obtained
+an estimate on the lifespan longer than the one provided by standard energy method. After their work, normal
+form reduction has been used by mathematicians to understand water waves on flat tori, for example [6], [7]
+and [26]. The idea was inspired by the normal form reduction method introduced by Shatah [35]: for a
+quadratic perturbation of a linear dispersive equation
+∂tu + iLu = N(u) = O(u⊗2),
+using a new variable u + B(u, ¯u) with a suitably chosen quadratic addendum B(u, ¯u) can possibly eliminate
+the quadratic part of N, thus extending the lifespan estimate beyond the standard 1/ε.
+So we shall write the quadraticr part of N(u) as
+�
+n3≥0
+Π(n3)
+�
+� �
+n1≥0
+�
+n2≥0
+M1
+�
+Π(n1)u, Π(n2)u
+�
++ M2
+�
+Π(n1)u, Π(n2)¯u
+�
++ M3
+�
+Π(n1)¯u, Π(n2)¯u
+�
+�
+� ,
+where M1, M2, M3 are complex bi-linear operators, following the argument of Section 4 in [16]. They are
+independent of t since the right-hand-side of the equation does not depend on t explicitly. Let’s look for a
+diffeomorphism
+u → v := u + B[u, u]
+
+12
+CHENGYANG SHAO
+in the function space C∞(S2), where B is a bilinear operator, so that the equation (2.3) with quadratic
+nonlinearity reduces to an equation with cubic nonlinearity.
+The B[u, u] is supposed to take the form
+B[u, u] = B1[u, u] + B2[u, u] + B3[u, u], with
+B1[u, u] =
+�
+n3≥0
+�
+n1,n2≥0
+b1(n1, n2, n3)Π(n3)M1
+�
+Π(n1)u, Π(n2)u
+�
+,
+B2[u, u] =
+�
+n3≥0
+�
+n1,n2≥0
+b2(n1, n2, n3)Π(n3)M2
+�
+Π(n1)u, Π(n2)¯u
+�
+,
+B3[u, u] =
+�
+n3≥0
+�
+n1,n2≥0
+b3(n1, n2, n3)Π(n3)M3
+�
+Π(n1)¯u, Π(n2)¯u
+�
+,
+where the bj(n1, n2, n3)’s are complex numbers to be determined. Implementing (2.3), we find
+(∂t + iΛ)(u + B[u, u])
+= N(u) +
+�
+n3≥0
+�
+n1,n2≥0
+b1(n1, n2, n3)Π(n3)M1
+�
+Π(n1)∂tu, Π(n2)u
+�
++
+�
+n3≥0
+�
+n1,n2≥0
+b1(n1, n2, n3)Π(n3)M1
+�
+Π(n1)u, Π(n2)∂tu
+�
++ (similar terms)
++
+�
+n3≥0
+�
+n1,n2≥0
+iΛ(n3)b1(n1, n2, n3)Π(n3)M1
+�
+Π(n1)u, Π(n2)u
+�
+= N(u) +
+�
+n3≥0
+�
+min(n1,n2)≤1
+i [Λ(n3) − Λ(n1) − Λ(n2)] b1(n1, n2, n3)Π(n3)M1
+�
+Π(n1)u, Π(n2)u
+�
++
+�
+n3≥0
+�
+n1,n2≥2
+i [Λ(n3) − Λ(n1) − Λ(n2)] b1(n1, n2, n3)Π(n3)M1
+�
+Π(n1)u, Π(n2)u
+�
++ (similar terms) + O(u⊗3).
+We aim to eliminate most of the second order portions of N(u). The coefficients bj(n1, n2, n3) are fixed as
+follows:
+(2.7)
+b1(n1, n2, n3) = i [Λ(n3) − Λ(n1) − Λ(n2)]−1 ,
+n1, n2, n3 ≥ 2
+b2(n1, n2, n3) = i [Λ(n3) − Λ(n1) + Λ(n2)]−1 ,
+n1, n2, n3 ≥ 2
+b3(n1, n2, n3) = i [Λ(n3) + Λ(n1) + Λ(n2)]−1 ,
+n1, n2, n3 ≥ 2
+b1,2,3(n1, n2, n3) = 0,
+if Λ(n3) ± Λ(n1) ± Λ(n2) = 0 or min(n1, n2, n3) ≤ 1,
+then a large portion of the second order part of Π≥2N (u) will be eliminated. In fact, for n1, n2, n3 ≥ 2, if
+Λ(n3) ± Λ(n1) ± Λ(n2) ̸= 0, then the term
+Π(n3)
+�
+� �
+n1,n2≥2
+M1
+�
+Π(n1)u, Π(n2)u
+�
++ M2
+�
+Π(n1)u, Π(n2)¯u
+�
++ M3
+�
+Π(n1)¯u, Π(n2)¯u
+�
+�
+�
+is cancelled out. On the other hand, by the volume preservation equality (1.6) and conservation law (1.7),
+there holds Π(0)u = O(u⊗2), Π(1)u = O(u⊗2), so the low-low interaction
+Π≥2
+�
+�
+�
+min(n1,n2)≤1
+M1
+�
+Π(n1)u, Π(n2)u
+�
++ M2
+�
+Π(n1)u, Π(n2)¯u
+�
++ M3
+�
+Π(n1)¯u, Π(n2)¯u
+�
+�
+�
+is automatically O(u⊗3).
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+13
+Thus the existence and continuity of the normal form B[u, u] depends on the property of the 3-way
+resonance equation
+(2.8)
+Λ(n3) − Λ(n1) − Λ(n2) = 0,
+n1, n2, n3 ≥ 2.
+which is equivalent to the Diophantine equation
+(2.9)
+[F(n1) + F(n2) − F(n3)]2 − 4F(n1)F(n2) = 0,
+n1, n2, n3 ≥ 2,
+where F(X) = X(X − 1)(X + 2).
+If the tuple (n1, n2, n3) is non-resonant, i.e. it is such that b1,2,3(n1, n2, n3) ̸= 0, then some elementary
+number theoretic argument will give a lower bound on |b1,2,3(n1, n2, n3)| in terms of a negative power (can
+be fixed as −9/2) of n1, n2, n3. This is usually referred as small divisor estimate.
+To study the distribution of resonant frequencies, we propose the following unsolved question:
+Question 3. Does the Diophantine equation (2.9) have finitely many solutions?
+However, the Diophantine equation (2.9) does admit non-trivial solutions (5, 5, 8) and (10, 10, 16). In other
+words, the second order terms e.g.
+Π(8)M1(Π(5)u · Π(5)u),
+Π(5)M1(Π(8)u · Π(5)¯u),
+in the quadratic part of N(u) cannot be eliminated by normal form reduction. On the other hand, it seems
+to be very hard to determine whether (2.9) still admits any other solution. We have the following proposition
+(the author would like to thank Professor Bjorn Poonen for the proof):
+Proposition 2.3. The Diophantine equation (2.9) has no solution with n1 ≤ 104 other than (5, 5, 8) and
+(10, 10, 16).
+The proof of this proposition is computer-aided. The key point is to use the so-called Runge’s method to
+show that if (n1, n2, n3) is a solution, then there must hold n2 = O(n2
+1). For a given n1, this reduces the proof
+to numerical verification for finitely many possibilities. The algorithm can of course be further optimized,
+but due to some algebraic geometric considerations, it is reasonable to conjecture that the solution of (2.9)
+should be very rare. In fact, there are two ways of viewing the problem. We observe that if (n1, n2, n3) is
+a solution, then F(n1)F(n2) must be a square, and the square free part of F(n1), F(n2), F(n3) must be the
+same. Further reduction turns the problem into finding integral points on a family of elliptic curves
+Y 2 = cF(X),
+c is square-free,
+which is of course difficult, but since Siegel’s theorem asserts that there are only finitely many integer points
+on an elliptic curve over Q, it is reasonable to conjecture that there are not “too many” solutions to (2.9).
+We may also view the problem as finding integral (rational) points on a given algebraic surface. The complex
+projective surface corresponding to (2.9) is given by
+V : [X(X − W)(X + 2W) + Y (Y − W)(Y + 2W) − Z(Z − W)(Z + 2W)]2
+= 4X(X − W)(X + 2W)Y (Y − W)(Y + 2W),
+where [X, Y, Z, W] is the homogeneous coordinate on CP3. With the aid of computer, we obtain
+Proposition 2.4. The complex projective surface V ⊂ CP3 has Kodaira dimension 2 (i.e. it is of general
+type under Kodaira-Enriquez classification), and its first Betti number is 0.
+See Appendix A for the code.
+
+14
+CHENGYANG SHAO
+The first part of the proposition suggests that the rational points of V should be localized on finitely
+many algebraic curves laying on V; nevertheless, this seemingly simple suggestion is indeed a special case of
+the Bomberi-Lang conjecture, a hard problem in number theory (its planar case is known as the celebrated
+Faltings’s theorem). The second part suggests that the rational points of V should be rare since its Albanese
+variety is a single point. But these are just heuristics that solutions to (2.9) should be rare. In general,
+determining the solvability of a given Diophantine equation is very difficult 2, as number theorists and
+arithmetic geometers generally believe.
+The reason that such issues do not occur for water waves in the flat setting or nonlinear Klein-Gordon
+equations is twofold. First of all, the resonance equation is easily understood even in the degenerate case in
+the flat setting. For example, the capillary water waves without gravity on T2 has dispersive relation |∇|1/2,
+and the 3-way resonance equation is
+4�
+k2
+1 + k2
+2 +
+4�
+l2
+1 + l2
+2 =
+4�
+m2
+1 + m2
+2,
+with the additional requirement m = k + l. We already know that the resonance equation has no non-trivial
+solution at all, cf. [7]. But even without using m = k +l, we would be able to conclude that there are at most
+finitely many non-trivial solutions from the celebrated Faltings’s theorem on rational points on high-genus
+algebraic projective curves (although this is like using a sledge hammer to crack a nut). Secondly, for the non-
+degenrate case, for example the gravity-capillary waves, the dispersive relation reads
+�
+g|∇| + σ|∇|3, so if the
+ratio σ/g is a transcedental number then the 3-way resonance equation has no solution. Furthermore, using
+some elementary calculus and a measure-theoretic argument, it can be shown, not without technicalities, that
+the resonances admit certain small-divisor estimates for almost all parameters. This is exactly the argument
+employed by Delort-Szeftle [16], Berti-Delort [6] and Ionescu-Pusateri [26], so that their results were stated for
+almost all parameters. These parameters are, roughly speaking, badly approximated by algebraic numbers.
+However, the resonance equation (2.8) is inhomogeneous and allows no arbitrary physical parameter at
+all. Furthermore, since product of spherical harmonics are no longer spherical harmonics in general, Fourier
+series techniques employed by [7] [6] [26] that works for the torus are never valid for S2; for example, we
+cannot simply assume n3 = n1 + n2 in (2.8), as already illustrated by the solutions (5,5,8) (10,10,16). These
+are the crucial differences between the capillary spherical water waves and all known results for water waves
+in the flat setting.
+2.5. Heuristics for Lifespan Estimate. To summarize, almost global lifespan estimate of (2.3) depends
+on the difficult number theoretic question 3. Before it is fully resolved, we can only expect partial results
+regarding the normal form transformation.
+If there are only finitely many solutions to the Diophantine equation (2.9), then under the normal form
+reduction u → v = u + B[u, u] with coefficients given by (2.7), the equation (2.3) is transformed into the
+following system:
+∂
+∂tΠcv = O(v⊗2),
+∂
+∂t(1 − Πc)v = O(v⊗3),
+where Πc is the orthogonal projection to �
+n3 E(n3) ⊂ L2(S2), with n3 being either 0 or 1, or exhausting the
+third component of all nontrivial solutions of (2.9).
+2For example, the seemingly simple Diophantine equation x3 + y3 + z3
+=
+42 is in fact a puzzle of more than 60
+years, and its first solution was found recently by Booker-Sutherland [11].
+It is of extremely large magnitude:
+42 =
+(−80 538 738 812 075 974)3 + 80 435 758 145 817 5153 + 12 602 123 297 335 6313.
+Another example is the equation of
+same type x3 + y3 + z3 = 3. Beyond the easily found solutions (1, 1, 1), (4, 4, -5), (4, -5, 4), (-5, 4, 4), the next solution reads
+(569 936 821 221 962 380 720, −569 936 821 113 563 493 509, −472 715 493 453 327 032).
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+15
+There is no reasonable assertion to be made if the conjecture fails. However, if the conjecture does hold
+true, then we can expect that the lifespan estimate for ε-Cauchy data of (EQ) goes beyond ε−1, as what we
+expect for gravity water waves in the periodic setting, e.g. in [26]:
+Conjecture 2.3. If the Diophantine equation (2.9) has only finitely many solutions, then there is some
+α > 0 such that for ε-Cauchy data of (EQ), the lifespan goes beyond ε−(1+α) as ε → 0.
+Let’s explain the heuristic as follows. The argument we aim to implement is the standard “continuous
+induction method”, i.e. for some suitably large s and K and suitable α > 0, assuming T = ε−(1+α) and
+supt∈[0,T ] ∥v∥Hs(g0) ≤ Kε, we try to prove a better bound supt∈[0,T ] ∥v∥Hs(g0) ≤ Kε/2.
+Here g0 is the
+standard metric on S2. It is intuitive to expect such a result for the cubic equation ∂t(1−Πc)v = O(v⊗3). As
+for the quadratic equation ∂tΠcv = O(v⊗2), it is crucial to implement the conservation of energy H[ζ, φ] ≡ 4π
+for a solution. We summarize it as
+Proposition 2.5. Fix T > 0. Let u be a smooth solution of (2.3) and v = u + B[u, u] be as above. Suppose
+for some suitably large s and K, there holds
+sup
+t∈[0,T ]
+∥v∥Hs(g0) ≤ Kε
+with ε sufficiently small. Then there is in fact a better bound for the low frequency part Πcv:
+sup
+t∈[0,T ]
+∥Πcv∥Hs(g0) ≤ Kε/4.
+Proof. We consider the “approximate” energy functional
+H0[ζ, φ] = 4π +
+�
+S2 2ζ · dµ0 + 1
+2
+�
+S2(|∇0ζ|2 + 2|ζ|2)dµ0 + 1
+2
+�
+S2
+���|∇1/2
+0
+|φ
+���
+2
+dµ0,
+where µ0 is the standard area measure on S2 and ∇0 is the standard connection on S2. This is nothing but
+the quadratic approximation to H[ζ, φ] in (1.8) at (0, 0), so there holds H0[ζ, φ] = H[ζ, φ] + O(u⊗3). Using
+volume preservation (1.6), we obtain
+�
+S2 ζdµ0 = −∥ζ∥2
+L2(g0) + O(ζ⊗3), so summarizing we have
+(2.10)
+1
+2
+�
+S2(|∇0ζ|2 − 2|ζ|2)dµ0 + 1
+2
+�
+S2
+���|∇1/2
+0
+|φ
+���
+2
+dµ0 = O(u⊗3).
+Note that we used the conservation law H[ζ, φ] ≡ 4π. By spectral calculus on S2, we have
+∥ζ∥2
+H1(g0) ≃ ∥Π(0)ζ��2
+L2(g0) + ∥Π(1)ζ∥2
+L2(g0) +
+�
+S2(|∇0ζ|2 − 2|ζ|2)dµ0,
+and by volume preservation and conservation of momentum (1.7) we find
+(2.11)
+∥ζ∥2
+H1(g0) ≃
+�
+S2(|∇0ζ|2 − 2|ζ|2)dµ0 + O(ζ⊗4).
+Now, for some N0 > 0 relating to the loss of regularity caused by B, we may choose s >> 2N0. Then if
+∥v∥Hs(g0) ≤ Kε, it follows that ∥u∥Hs−N0(g0) ≤ K′ε, so by (2.10) and (2.11) we have
+∥u∥2
+L2(g0) ≃ ∥ζ∥2
+H1(g0) + ∥φ∥2
+H1/2(g0) ≤ K′ε3.
+Thus
+∥v∥L2(g0) ≤ C∥u∥L2(g0) + C∥u∥2
+HN0(g0)
+≤ Cε3/2 + K′ε2.
+Since the spectrum of Πcv is bounded, by Bernstein type inequality we have
+∥Πcv∥Hs(g0) ≤ C∥v∥L2(g0) ≤ K′ε3/2(1 + ε1/2).
+
+16
+CHENGYANG SHAO
+If ε is sufficiently small then this implies ∥Πcv∥Hs(g0) ≤ Kε/4.
+□
+We point out that the above proof is independent from the magnitude of the lifespan T, so it is always
+applicable as long as the cubic equation ∂t(1 − Πc)v = O(v⊗3) is well-understood. There are two crucial
+points in the proof of Proposition 2.5: the conservation of energy, and that the projection Πc is of finite
+rank, so that a Bernstein type inequality holds. The last fact holds only if there are finitely many 3-way
+resonances, i.e. there are only finitely many solutions to the Diophantine equation (2.8).
+Finally, we propose an even more ambitious conjecture concerning global dynamical properties of spherical
+water droplets, which is again illuminated by observation in hydrodynamical experiments under zero gravity,
+and also suggested by the results of Berti-Montalto [8]:
+Conjecture 2.4. If the Diophantine equation (2.9) has only finitely many solutions, then a KAM type result
+holds for (EQ): there is a family of infinitely many quasi-periodic solutions of (EQ), depending on a parameter
+which takes value in a Cantor-type set.
+Appendix A. MAGMA Code
+MAGMA is a large, well-supported software package designed for computations in algebra, number theory,
+algebraic geometry, and algebraic combinatorics. In this appendix, we give the MAGMA code used to conduct
+computations on Diophantine equations related to the spherical capillary water waves system.
+A.1. Integral Points on Elliptic Curve. We can find all integral points on a given elliptic curve over
+Q using MAGMA. For a monic cubic polynomial f(x), the function EllipticCurve(f) creates the elliptic
+curve
+E : y2 = f(x),
+and the function IntegralPoints(E) returns a sequence containing all the integral points on E under the
+homogeneous coordinate of QP2, modulo negation. We use this to find out all integral points on the elliptic
+curve
+Ec : y2 = x3 + cx2 − 2c2x = x(x − c)(x + 2c).
+for natural number c ≤ 50. The MAGMA code is listed below, which excludes all the c’s such that there are
+only trivial integral points {(−c, 0), (0, 0), (c, 0)} on Ec.
+> Qx<x> := PolynomialRing(Rationals());
+> for c in [1..50] do
+> E := EllipticCurve(x^3+c*x^2-2*c^2*x);
+> S, reps := IntegralPoints(E);
+> if # S gt 3 then
+> print c, E;
+> print S;
+> end if;
+> end for;
+2 Elliptic Curve defined by y^2 = x^3 + 2*x^2 - 8*x over Rational Field
+[ (-4 : 0 : 1), (-2 : 4 : 1), (-1 : -3 : 1), (0 : 0 : 1), (2 : 0 : 1), (4 : 8 : 1), (8 : -24
+: 1), (50 : 360 : 1) ]
+8 Elliptic Curve defined by y^2 = x^3 + 8*x^2 - 128*x over Rational Field
+[ (-16 : 0 : 1), (-8 : 32 : 1), (-4 : -24 : 1), (0 : 0 : 1), (8 : 0 : 1), (9 : -15 : 1), (16
+: 64 : 1), (32 : -192 : 1), (200 : 2880 : 1) ]
+
+LONGTIME DYNAMICS OF IRROTATIONAL SPHERICAL WATER DROPS: INITIAL NOTES
+17
+13 Elliptic Curve defined by y^2 = x^3 + 13*x^2 - 338*x over Rational Field
+[ (-26 : 0 : 1), (0 : 0 : 1), (13 : 0 : 1), (121 : 1386 : 1) ]
+15 Elliptic Curve defined by y^2 = x^3 + 15*x^2 - 450*x over Rational Field
+[ (-30 : 0 : 1), (-5 : -50 : 1), (0 : 0 : 1), (15 : 0 : 1), (24 : 108 : 1), (90 : -900 : 1)
+]
+17 Elliptic Curve defined by y^2 = x^3 + 17*x^2 - 578*x over Rational Field
+[ (-34 : 0 : 1), (-32 : -56 : 1), (0 : 0 : 1), (17 : 0 : 1), (833 : 24276 : 1) ]
+18 Elliptic Curve defined by y^2 = x^3 + 18*x^2 - 648*x over Rational Field
+[ (-36 : 0 : 1), (-32 : -80 : 1), (-18 : 108 : 1), (-9 : -81 : 1), (0 : 0 : 1), (18 : 0 : 1),
+(36 : 216 : 1), (72 : -648 : 1), (450 : 9720 : 1) ]
+22 Elliptic Curve defined by y^2 = x^3 + 22*x^2 - 968*x over Rational Field
+[ (-44 : 0 : 1), (-32 : -144 : 1), (0 : 0 : 1), (22 : 0 : 1), (198 : 2904 : 1) ]
+23 Elliptic Curve defined by y^2 = x^3 + 23*x^2 - 1058*x over Rational Field
+[ (-46 : 0 : 1), (0 : 0 : 1), (23 : 0 : 1), (50 : -360 : 1) ]
+26 Elliptic Curve defined by y^2 = x^3 + 26*x^2 - 1352*x over Rational Field
+[ (-52 : 0 : 1), (-49 : -105 : 1), (0 : 0 : 1), (26 : 0 : 1), (1300 : 47320 : 1) ]
+30 Elliptic Curve defined by y^2 = x^3 + 30*x^2 - 1800*x over Rational Field
+[ (-60 : 0 : 1), (-50 : 200 : 1), (-45 : -225 : 1), (-24 : -216 : 1), (-20 : 200 : 1), (-6
+: 108 : 1), (0 : 0 : 1), (30 : 0 : 1), (36 : 144 : 1), (40 : -200 :
+1), (75 : -675 : 1), (90 : 900 : 1), (300 : 5400 : 1), (324 : -6048 : 1), (480 : -10800 : 1),
+(7290 : 623700 : 1), (10830 : -1128600 : 1), (226875 : 108070875 : 1) ]
+32 Elliptic Curve defined by y^2 = x^3 + 32*x^2 - 2048*x over Rational Field
+[ (-64 : 0 : 1), (-32 : 256 : 1), (-16 : -192 : 1), (0 : 0 : 1), (32 : 0 : 1), (36 : -120 :
+1), (64 : 512 : 1), (128 : -1536 : 1), (800 : 23040 : 1) ]
+33 Elliptic Curve defined by y^2 = x^3 + 33*x^2 - 2178*x over Rational Field
+[ (-66 : 0 : 1), (0 : 0 : 1), (33 : 0 : 1), (81 : -756 : 1) ]
+35 Elliptic Curve defined by y^2 = x^3 + 35*x^2 - 2450*x over Rational Field
+[ (-70 : 0 : 1), (-49 : 294 : 1), (-45 : -300 : 1), (-40 : 300 : 1), (-14 : -196 : 1), (0 :
+0 : 1), (35 : 0 : 1), (50 : 300 : 1), (175 : -2450 : 1), (224 : 3528 : 1), (280 : -4900 : 1),
+(4410 : -294000 : 1), (14450 : -1739100 : 1) ]
+39 Elliptic Curve defined by y^2 = x^3 + 39*x^2 - 3042*x over Rational Field
+[ (-78 : 0 : 1), (0 : 0 : 1), (39 : 0 : 1), (147 : -1890 : 1) ]
+42 Elliptic Curve defined by y^2 = x^3 + 42*x^2 - 3528*x over Rational Field
+[ (-84 : 0 : 1), (-56 : -392 : 1), (-12 : 216 : 1), (0 : 0 : 1), (42 : 0 : 1), (63 : -441 :
+1), (294 : 5292 : 1) ]
+43 Elliptic Curve defined by y^2 = x^3 + 43*x^2 - 3698*x over Rational Field
+[ (-86 : 0 : 1), (-32 : -360 : 1), (0 : 0 : 1), (43 : 0 : 1) ]
+46 Elliptic Curve defined by y^2 = x^3 + 46*x^2 - 4232*x over Rational Field
+[ (-92 : 0 : 1), (0 : 0 : 1), (46 : 0 : 1), (26496 : 4316640 : 1) ]
+50 Elliptic Curve defined by y^2 = x^3 + 50*x^2 - 5000*x over Rational Field
+[ (-100 : 0 : 1), (-50 : 500 : 1), (-25 : -375 : 1), (-4 : 144 : 1), (0 : 0 : 1), (50 : 0 :
+1), (100 : 1000 : 1), (200 : -3000 : 1), (1250 : 45000 : 1) ]
+Running Magma V2.27-7.
+Seed: 821911319; Total time: 2.430 seconds; Total memory usage: 85.16MB.
+
+18
+CHENGYANG SHAO
+A.2. Classification of Projective Surface. Some basic geometric parameters of the complex projective
+surface
+V : [X(X − W)(X + 2W) + Y (Y − W)(Y + 2W) − Z(Z − W)(Z + 2W)]2
+= 4X(X − W)(X + 2W)Y (Y − W)(Y + 2W)
+in CP3 can be computed using MAGMA. The author would like to thank Professor Bjorn Poonen for intro-
+ducing MAGMA and providing the code listed below.
+> Q:=Rationals();
+> P<x,y,z,t>:=ProjectiveSpace(Q,3);
+> fx:=x*(x-t)*(x+2*t);
+> fy:=y*(y-t)*(y+2*t);
+> fz:=z*(z-t)*(z+2*t);
+> V:=Surface(P,(fz-fx-fy)^2-4*fx*fy);
+> KodairaEnriquesType(V);
+2 0 General type
+Running Magma V2.27-7.
+Seed: 989287753; Total time: 0.650 seconds; Total memory usage: 32.09MB.
+Note that the Kodaira dimension is invariant regardless of the choice of base field, so it is legitimate to
+choose the base field to be Q in the above code. The variable t is used to homogenize the equation. The
+function KodairaEnriquesType(V) returns three values for the given projective surface V : the first is the
+Kodaira dimension, the second is irrelevant when the Kodaira dimension is not −∞, 1 or 0, and the third is
+the Kodaira-Enriquez classification of the surface X.
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+
+20
+CHENGYANG SHAO
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