# Correlation Structure and Resonant Pairs for Arithmetic Random Waves

(Nov 2021)

###### Abstract

The geometry of Arithmetic Random Waves has been extensively investigated in
the last fifteen years, starting from the seminal papers [RW08, ORW08]. In
this paper we study the correlation structure among different functionals such
as nodal length, boundary length of excursion sets, and the number of
intersection of nodal sets with deterministic curves in different classes; the
amount of correlation depends in a subtle fashion from the values of the
thresholds considered and the symmetry properties of the deterministic curves.
In particular, we prove the existence of _resonant pairs_ of threshold values
where the asymptotic correlation is full, that is, at such values one
functional can be perfectly predicted from the other in the high energy limit.
We focus mainly on the $2$-dimensional case but we discuss some specific
extensions to dimension $3$.

Keywords and Phrases: Random Eigenfunctions, Arithmetic Random Waves, Nodal
Length, Nodal Intersections, Boundary Length, Resonant Pairs

AMS Classification: 60G60; 60D05; 58J50; 35P20

Valentina Cammarota⋆, Riccardo-W. Maffucci⋄,
Domenico Marinucci‡, Maurizia Rossi⋇

⋆Dipartimento di Scienze Statistiche, Università di Roma La Sapienza

⋄Department of Mathematics, University of Coventry

‡Dipartimento di Matematica, Università di Roma Tor Vergata

⋇Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca

##  1 Introduction

The geometry of nodal sets for Gaussian random eigenfunctions has been the
object of a considerable amount of attention over the last $15$ years. Most
papers have focussed on the $2$-dimensional case, either in Euclidean settings
(Berry’s random waves, see e.g. [Ber02, BCW19, NPR19, Vid21]), or on compact
manifolds, most notably the sphere $\mathbb{S}^{2}$ (see e.g. [Wig10, MRW20,
Tod20]) and the torus $\mathbb{T}^{2}$ (see e.g. [RW08, ORW08, KKW13, MPRW16,
TR23]), among others. The derivation of the expected value for the nodal
length is now standard thanks to the Gaussian Kinematic Formula by Adler and
Taylor (see [AT07]); the analysis of the variance is more challenging, and
goes back to [Wig10] for the case of the sphere (random spherical harmonics)
and to [KKW13] for the torus (arithmetic random waves); in the physics
literature, the variance for the nodal length of planar eigenfunctions
(Berry’s random waves) was earlier given in [Ber02].

The analysis of the asymptotic distribution for the nodal length of random
eigenfunctions was basically started in [MPRW16]. In that paper, the nodal
length of arithmetic random waves is expanded into orthogonal terms
corresponding to so-called Wiener chaos components; it is then shown that the
behaviour of nodal length is dominated (in the $L^{2}$ sense, as the
eigenvalues diverge) by the fourth-order chaos, whose limiting distribution is
nonGaussian in the toroidal case. A similar phenomenon takes place in the
planar and spherical cases, see [NPR19] and [MRW20], respectively, although in
both these cases the limiting behaviour is Gaussian.

The expansion into Wiener chaoses has allowed to provide an interpretation to
the so-called Berry cancellation phenomenon: namely, the fact that the
asymptotic variance of nodal length is an order of magnitude smaller than the
variance for the measure of boundary curves at any non-zero thresholds. As
argued extensively elsewhere, see [MPRW16, MRW20, NPR19, CM18], this
cancellation corresponds to the disappearance of the second-order chaos term,
which in turn corresponds to the random $L^{2}$ norms of eigenfunctions and
plays no role in the fluctuations of the nodal lines. Because this random norm
has variance which is larger than all the other terms in the orthogonal
decomposition, its disappearence in the nodal case fully explains the Berry’s
cancellation phenomenon.

The domination of this second order chaotic term yields another remarkable
consequence, which was already derived, by a different argument, in [Wig12].
In particular, the correlation between boundary lengths is asymptotically
equal to one (in absolute value) as the eigenvalues diverge. This follows
easily by the fact that the random sequences corresponding to boundary lengths
at different level are all asymptotically proportional (up to different
scaling constants) to the same sequence of random variables, namely the random
$L^{2}$ norms of the eigenfunctions. Asymptotic correlation has been noted in
the case of random spherical harmonics [Wig12, MR21b], but it obviously holds
with exactly the same argument both for planar random waves and for toroidal
eigenfunctions. For the same argument, it is also immediate to notice that
nodal lengths and the boundary of level curves have asymptotically correlation
zero; indeed nodal length is dominated by terms in the fourth-order chaos,
which are by construction orthogonal to the random norm, which belongs to the
second-order chaos.

For the reasons we mentioned above, it is clear that these
correlation$/$uncorrelation phenomena are in some sense an artifact due to the
random fluctuations in the $L^{2}$ norms; for many applications this could
sound meaningless (for instance, in a quantum mechanics framework random norms
should be normalized to unity). In [MR21b], a different issue was addressed
for random spherical harmonics, namely the existence of correlation _after the
effect of the random $L^{2}$\- norm has been removed_ (usually called partial
correlation in mathematical statistics). Surprisingly, it was shown that
partial correlation among boundary lengths at different levels persists, and
indeed the asymptotic correlation with nodal lengths switches from zero to
unity: namely, it is asymptotically possible to predict the boundary length of
level curves at every threshold $u$, once the confounding effect of the random
norm has been removed.

The purpose of this paper is to investigate an analogous question, in the case
of arithmetic random waves. We focus mainly on the $2$-dimensional case but we
discuss some specific extensions to dimension $3$. The final outcome turns out
to be rather different than spherical harmonics and planar random waves. In
particular, for dimension $2$, it turns out that, after the effect of the
$L^{2}$ norm has been removed, full correlation between boundary lengths at
any different levels does no longer hold: however, there are _resonant pairs_
along an algebraic curve (whose equation we write down explicitly), such that
the boundary lengths computed at any two levels corresponding to these pairs
have partial correlation tending to $1$ in the high-energy limit. These pairs
include the nodal case, where one of the two levels corresponds to $u=0$;
moreover, resonant pairs exist (for different algebraic curves, which we also
write down explicitly) for other functionals, such as nodal intersections with
a fixed reference curve, see [RW16, Maf19, Cam19, RWY16].

In order to formulate our results more precisely, we now need to introduce
more notation and definitions, which have all become standard in the last
decade.

##  2 Background and notation

###  2.1 Arithmetic random waves

We start by recalling the (by now standard) definition of Arithmetic Random
Waves, first introduced in [RW08, ORW08]. Let
${\mathbb{T}}^{d}:=\mathbb{R}^{d}/\mathbb{Z}^{d}$ be the standard
$d$-dimensional flat torus and $\Delta$ the Laplacian on ${\mathbb{T}}^{d}$.
We are interested in the (totally discrete) spectrum of $\Delta$ i.e.,
eigenvalues $E>0$ of the Helmholtz equation

$\Delta f+Ef=0.$ (2.1)

Let

$S:=\\{{{}n\in\mathbb{Z}:n}=\lambda_{1}^{2}+\cdots+\lambda_{d}^{2}\,\,\mbox{{}
for some}\>\lambda_{1},\dots\lambda_{d}\in\mathbb{Z}\\},$

be the collection of all numbers expressible as a sum of two squares. Then,
the eigenvalues of (2.1) (also called energy levels of the torus) are all
numbers of the form $E_{n}=4\pi^{2}n$, with $n\in S$.

In order to describe the Laplace eigenspace corresponding to $E_{n}$, denote
by $\Lambda_{n}$ the set of frequencies:

$\Lambda_{n}:=\\{\lambda\in\mathbb{Z}^{d}:||\lambda||^{2}=n\\}\ $

whose cardinality

$\mathcal{N}_{n}:=|\Lambda_{n}|$ (2.2)

equals the number of ways to express $n$ as a sum of $d$ squares.
(Geometrically, $\Lambda_{n}$ is the collection of all standard lattice points
lying on the centred circle with radius $\sqrt{n}$.) For
$\lambda\in\Lambda_{n}$ denote the complex exponential associated to the
frequency $\lambda$

$e_{\lambda}(x)=\exp(2\pi i\langle\lambda,x\rangle)$

with $x\in{\mathbb{T}}^{d}$. Of course, the collection
$\\{e_{\lambda}(x)\\}_{\lambda\in\Lambda_{n}}$ of the complex exponentials
corresponding to the frequencies $\lambda\in\Lambda_{n}$, is an
$L^{2}$-orthonormal basis of the eigenspace ${\mathcal{E}}_{n}$ of $\Delta$
corresponding to the eigenvalue $E_{n}$. In view of (2.2), we have
$\dim{\mathcal{E}}_{n}=\mathcal{N}_{n}=|\Lambda_{n}|$; the fluctuations in the
number $\mathcal{N}_{n}$ have been very widely studied starting from [Lan08],
see for instance [KKW13] and the references therein.

Following [RW08, ORW08, KKW13], we define the Arithmetic Random Waves (also
called random Gaussian toral Laplace eigenfunctions) to be the random fields

$T_{n}(x)=\frac{1}{\sqrt{\mathcal{N}_{n}}}\sum_{\lambda\in\Lambda_{n}}a_{\lambda}e_{\lambda}(x),\quad
x\in{\mathbb{T}}^{d},$ (2.3)

where the coefficients $a_{\lambda}$ are standard complex-Gaussian random
variables111From now on, we assume that every random object considered in this
paper is defined on a common probability space
$(\Omega,\mathcal{F},{\mathbb{P}})$, with ${\mathbb{E}}$ denoting mathematical
expectation with respect to ${\mathbb{P}}$. verifying the following
properties: $a_{\lambda}$ is stochastically independent of $a_{\gamma}$
whenever $\gamma\notin\\{\lambda,-\lambda\\}$, and
$a_{-\lambda}=\overline{a}_{\lambda}$ (ensuring that $T_{n}$ is real-valued).
By the definition (2.3), $T_{n}$ is a stationary, i.e. the law of $T_{n}$ is
invariant under all the translations $f(\cdot)\mapsto f(x+\cdot)$,
$x\in{\mathbb{T}}^{d}$, centered Gaussian random field with covariance
function

$r_{n}(x,y)=r_{n}(x-y):={\mathbb{E}}[T_{n}(x)\overline{T_{n}(y)}]=\frac{1}{\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}e_{\lambda}(x-y)=\frac{1}{\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}\cos\left(2\pi\langle
x-y,\lambda\rangle\right),$

$x,y\in{\mathbb{T}}^{d}$ (by the standard abuse of notation for stationary
fields). Note that $r_{n}(0)=1$, i.e. $T_{n}$ has unit variance.

The set $\Lambda_{n}$ induces a discrete probability measure $\mu_{n}$ on the
unit sphere $\mathbb{S}^{d-1}$

$\mu_{n}=\frac{1}{\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}\delta_{\frac{\lambda}{\sqrt{n}}}.$

It turns out that the behavior of $\\{\mu_{n}\\}_{n}$ strongly depends on the
dimension. Indeed, if $d=2$, ${\mathcal{N}_{n}}$ is subject to large and
erratic fluctuations, it grows on average, over integers which are sums of two
squares, as ${\rm const}\cdot\sqrt{\log n}$, but can be as small as 8 for an
infinite sequence of prime numbers. These erratic fluctuations are mirrored by
the behavior of $\\{\mu_{n}\\}_{n}$: indeed, let us denote by

$\widehat{\mu}_{n}(k)=\int_{{\mathbb{S}}^{1}}z^{-k}d\mu_{n}(z),z\in\mathbb{Z},$

the Fourier coefficients of $\mu_{n}$. (We note that
$\widehat{\mu}_{n}(4)\in\mathbb{R}$ since $\mu_{n}$ is invariant under the
transformations $z\to\bar{z}$ and $z\to i\cdot z$, and that
$|\widehat{\mu}_{n}(4)|\leq 1$ since $\mu_{n}$ is a probability measure.)
Remarkably, [KKW13, KW17] showed that the set of adherent points of
$\\{\widehat{\mu}_{n}(4)\\}_{n\in S}$ is all of $[-1,1]$. It is known that for
a density-$1$ sequence of eigenvalues, the sequence $\\{\mu_{n}\\}_{n}$
converge towards the uniform probability measure on the circle; weak-* limits
of the sequence $\\{\mu_{n}\\}_{n}$ are partially classified in [KW17].

In dimension $d=3$ instead, we have

$n^{\frac{1}{2}-o(1)}\ll\mathcal{N}_{n}\ll n^{\frac{1}{2}+o(1)},$

and the lattice points $\Lambda_{n}/\sqrt{n}$ become equidistributed with
respect to the normalized Lebesgue measure on $\mathbb{S}^{2}$, as
$n\to+\infty$ s.t. $n\not\equiv 0,4,7(\mathrm{mod}8)$ [Iwa87].

###  2.2 Geometry of level sets

We are interested in geometric properties of Arithmetic Random Waves, in
particular we study the distribution of their level sets

$\\{x\in\mathbb{T}^{d}:T_{n}(x)=u\\},\quad u\in\mathbb{R}.$

(Recall that ARW are a.s. Morse functions, hence the level sets are a.s.
smooth submanifolds of codimension $1$.) We mainly focus on dimension $d=2$
and investigate two types of functionals: (i) the $1$-dimensional Hausdorff
measure, and (ii) the number of intersection points between nodal sets and a
fixed reference curve. More precisely, we shall focus on the following
functionals:

  * •

boundary length at level $u\neq 0$

$\mathcal{L}_{n}^{u}=\mathcal{H}^{1}\\{x\in\mathbb{T}^{2}:T_{n}(x)=u\\},$

  * •

nodal length

$\mathcal{L}_{n}=\mathcal{H}^{1}\\{x\in\mathbb{T}^{2}:T_{n}(x)=0\\}=:\mathcal{L}_{n}^{0},$

  * •

number of intersections of the nodal lines with smooth curves
$\mathcal{C}\subset\mathbb{T}^{2}$ with no-where zero curvature:

$\mathcal{Z}_{n}\mathcal{(C)}=\mathcal{H}^{0}\\{x\in\mathbb{T}^{2}:T_{n}(x)=0,\
x\in\mathcal{C}\\}.$

In [KKW13, MPRW16], the variance of the boundary length have been
investigated; in particular, for $u\neq 0$, as $n\to+\infty$ s.t.
$\mathcal{N}_{n}\to+\infty$,

${\rm
Var}(\mathcal{L}_{n}^{u})\sim\frac{1}{32}u^{4}e^{-u^{2}}\frac{E_{n}}{\mathcal{N}_{n}},$
(2.4)

in the nodal case, the variance is of smaller order, indeed

${\rm
Var}(\mathcal{L}_{n})\sim\frac{1+\widehat{\mu}_{n}(4)^{2}}{512}\frac{E_{n}}{\mathcal{N}_{n}^{2}}.$
(2.5)

As for nodal intersections, we have the following [RW16]: as $n\to+\infty$
s.t. $\mathcal{N}_{n}\to+\infty$,

${\rm
Var}(\mathcal{Z}_{n}(\mathcal{C}))=(4\mathcal{B}_{\mathcal{C}}(\mu_{n})-L^{2})\frac{n}{\mathcal{N}_{n}}+O\left(\frac{n}{\mathcal{N}_{n}^{\frac{3}{2}}}\right),$
(2.6)

where $L$ is the length of the curve, and for a probability measure $\nu$ on
the unit circle ${\mathbb{S}}^{1}$

$\mathcal{B}_{{\cal
C}}(\nu):=\int_{{\mathbb{S}}^{1}}\left(\int_{0}^{L}\langle\theta,{\dot{\gamma}}(t)\rangle^{2}dt\right)^{2}d\nu(\theta),$

$\gamma=(\gamma_{1},\gamma_{2}):[0,L]\to{\mathbb{T}}^{2}$ being the arc-length
parametrization of ${\cal C}$. There are special curves for which the leading
term in the variance (2.6) vanishes, as noted in [RW16].

######  Definition 2.1 ([RW18, Definition 1]).

A smooth curve ${\cal C}\subset\mathbb{T}^{2}$ with nowhere zero curvature and
total length $L$ is _static_ if

$\mathcal{B}_{{\cal
C}}(\nu):=\int_{{\mathbb{S}}^{1}}\left(\int_{0}^{L}\langle\theta,{\dot{\gamma}}(t)\rangle^{2}dt\right)^{2}d\nu(\theta)=\frac{L^{2}}{4},$

for every probability measure $\nu$ on the unit circle ${\mathbb{S}}^{1}$.

From [RW16, Proposition 7.1], a curve is static if and only if
$\mathcal{B}_{\mathcal{C}}\big{(}\frac{d\theta}{2\pi}\big{)}=0$, where
$\frac{d\theta}{2\pi}$ denotes the uniform probability measure on
${\mathbb{S}}^{1}$.

The variance of nodal intersections with static curves has been investigated
in [RW18]. We first need some more notation: let $\delta>0$, a sequence of
energy levels is called $\delta$-separated [BR11] if

$\min_{\lambda\neq\lambda^{\prime}\in\Lambda_{n}}\|\lambda-\lambda^{\prime}\|\gg
n^{1/4+\delta}.$

For a static curve (that we denote $\mathcal{C}^{\prime}$ to avoid confusion),
for $\delta$-separated sequence of energy levels such that
$\mathcal{N}_{n}\to+\infty$

${\rm
Var}(\mathcal{Z}_{n}(\mathcal{C}^{\prime}))\sim\frac{n}{4\mathcal{N}_{n}^{2}}(16\mathcal{A}_{\mathcal{C}^{\prime}}(\mu_{n})-L^{2}),$
(2.7)

where for a smooth curve $\mathcal{C}\subset\mathbb{T}^{2}$ and a probability
measure $\nu$ on the unit circle ${\mathbb{S}}^{1}$

$\mathcal{A}_{{\cal
C}}(\nu):=\int_{{\mathbb{S}}^{1}}\int_{{\mathbb{S}}^{1}}\left(\int_{0}^{L}\langle\theta,{\dot{\gamma}}(t)\rangle^{2}\langle\theta^{\prime},{\dot{\gamma}}(t)\rangle^{2}dt\right)^{2}d\nu(\theta)\nu(\theta^{\prime}).$

Regarding higher dimensions, see Section 3.3 and Section 6.

###  2.3 Chaos expansion

The celebrated Wiener chaos expansion concerns the representation of square
integrable random variables in terms of an infinite orthogonal sum. In this
section we recall briefly some basic facts on Wiener chaotic expansion for
non-linear functionals of Gaussian fields. Denote by $\\{H_{k}\\}_{k\geq 0}$
the Hermite polynomials on $\mathbb{R}$, defined as follows

$\displaystyle H_{0}=1,\hskip
14.22636ptH_{k}(t)=(-1)^{k}\gamma^{-1}(t)\frac{d^{k}}{dt^{k}}\gamma(t),\;\;k\geq
1,$ (2.8)

where $\gamma(t)=e^{-t^{2}/2}/\sqrt{2\pi}$ is the standard Gaussian density on
the real line; $\mathbb{H}=\\{H_{k}/\sqrt{k!}:\;k\geq 0\\}$ is a complete
orthogonal system in

$L^{2}(\gamma)=L^{2}(\mathbb{R},{\cal B}(\mathbb{R}),\gamma(t)dt).$

The random eigenfunctions defined in (2.3) are a byproduct of the family of
complex-valued, Gaussian random variables $\\{a_{\lambda}\\}$, defined on some
probability space $(\Omega,{\cal F},\mathbb{P})$. Define the space ${\bf A}$
to be the closure in $L^{2}(\mathbb{P})$ generated by all real, finite, linear
combinations of random variables of the form
$za_{\lambda}+\overline{z}a_{-\lambda}$, $z\in\mathbb{C}$; the space ${\bf A}$
is a real, centred, Gaussian Hilbert subspace of $L^{2}(\mathbb{P})$.

For each integer $q\geq 0$, the $q$-th Wiener chaos ${\cal H}_{q}$ associated
with ${\bf A}$ is the closed linear subspace of $L^{2}(\mathbb{P})$ generated
by all real, finite, linear combinations of random variables of the form

$H_{q_{1}}(a_{1})\cdot H_{q_{2}}(a_{2})\cdots H_{q_{k}}(a_{k})$

for $k\geq 1$, where the integers $q_{1},q_{2},\dots,q_{k}\geq 0$ satisfy
$q_{1}+q_{2}+\cdots+q_{k}=q$ and $(a_{1},a_{2},\dots,a_{k})$ is a real,
standard, Gaussian vector extracted from ${\bf A}$. In particular ${\cal
H}_{0}=\mathbb{R}$.

As well-known Wiener chaoses $\\{{\cal H}_{q},\,q=0,1,2,\dots\\}$ are
orthogonal, i.e., ${\cal H}_{q}\perp{\cal H}_{p}$ for $p\neq q$ (the
orthogonality holds in the sense of $L^{2}(\mathbb{P})$) and the following
decomposition holds: every real-valued function $F\in{\bf A}$ admits a unique
expansion of the type

$\displaystyle F=\sum_{q=0}^{\infty}F[q],$

where the projections $F[q]\in{\cal H}_{q}$ for every $q=0,1,2,\dots$ and the
series converges in $L^{2}(\mathbb{P})$. Note that $F[0]=\mathbb{E}[F]$.

For $u\neq 0$, [CMR23, Theorem 2.4] proves that the boundary length
$\mathcal{L}_{n}^{u}$ is dominated by the second order chaos:

$\displaystyle\mathcal{L}_{n}^{u}[2]=\sqrt{\frac{\pi}{8}}u^{2}\phi(u)\sqrt{2\pi^{2}n}\frac{1}{{\mathcal{N}}_{n}/2}\sum_{\lambda\in\Lambda_{n}^{+}}(|a_{\lambda}|^{2}-1)+{\cal
R}_{1}(n,u)$

where, under [CMR23, Condition 2.2], $\mathbb{E}[{\cal
R}_{1}(n,u)^{2}]=O(4\pi^{2}n/{\mathcal{N}}_{n}^{2})$, and $\Lambda_{n}^{+}$ is
the following subset of the set of frequencies: if $n$ is not a square

$\Lambda_{n}^{+}=\\{\lambda=(\lambda_{1},\lambda_{2})\in\Lambda_{n}:\lambda_{2}>0\\},$

otherwise

$\Lambda_{n}^{+}=\\{\lambda=(\lambda_{1},\lambda_{2})\in\Lambda_{n}:\lambda_{2}>0\\}\cup\\{(\sqrt{n},0)\\}.$

Note that for every $n\in S$, $|\Lambda_{n}^{+}|=\mathcal{N}_{n}/2$.

Also, inspired by [MPRW16, Lemma 4.2], we are able to show that the fourth
chaotic component $\mathcal{L}_{n}^{u}[4]$ of the length of $u$-level curves
can be written as

$\displaystyle\mathcal{L}_{n}^{u}[4]=\phi(u)\sqrt{\frac{\pi}{2}}\frac{\sqrt{E_{n}/2}}{\mathcal{N}_{n}}\left[a(u)W_{1}(n)^{2}-\frac{1}{4}W_{2}(n)^{2}-\frac{1}{4}W_{3}(n)^{2}-\frac{1}{2}W_{4}(n)^{2}-\left(a(u)-\frac{1}{4}\right)+o_{\mathbb{P}}(1)\right],$

where

$\displaystyle
W_{1}(n)=\frac{1}{n\sqrt{{\mathcal{N}}_{n}/2}}\sum_{\lambda\in\Lambda^{+}_{n}}(|a_{\lambda}|^{2}-1)n,$
$\displaystyle
W_{2}(n)=\frac{1}{n\sqrt{{\mathcal{N}}_{n}/2}}\sum_{\lambda\in\Lambda^{+}_{n}}(|a_{\lambda}|^{2}-1)\lambda_{1}^{2},\;\;\;W_{3}(n)=\frac{1}{n\sqrt{{\mathcal{N}}_{n}/2}}\sum_{\lambda\in\Lambda^{+}_{n}}(|a_{\lambda}|^{2}-1)\lambda_{2}^{2},$
$\displaystyle
W_{4}(n)=\frac{1}{n\sqrt{{\mathcal{N}}_{n}/2}}\sum_{\lambda\in\Lambda^{+}_{n}}(|a_{\lambda}|^{2}-1)\lambda_{1}\lambda_{2},$

and

$a(u)=\frac{1}{4}H_{4}(u)+\frac{1}{2}H_{2}(u)-\frac{1}{8},$

and $o_{\mathbb{P}}(1)$ denotes a sequence of random variables converging to
zero in probability. Equivalently,

$\displaystyle\mathcal{L}_{n}^{u}[4]=$
$\displaystyle\phi(u)\sqrt{\frac{\pi}{2}}\frac{\sqrt{E_{n}/2}}{\mathcal{N}_{n}}\Big{[}\frac{1}{8}\frac{1}{\mathcal{N}_{n}/2}\sum_{\lambda,\lambda^{\prime}\in\Lambda^{+}_{n}}(|a_{\lambda}|^{2}-1)(|a_{\lambda^{\prime}}|^{2}-1)\left(8a(u)-2\left\langle\frac{\lambda}{|\lambda|},\frac{\lambda^{\prime}}{|\lambda^{\prime}|}\right\rangle^{2}\right)$
$\displaystyle\hskip
85.35826pt-\left(a(u)-\frac{1}{4}\right)+o_{\mathbb{P}}(1)\Big{]}.$

It is known, see [MPRW16, Section 1.4 and Lemma 4.2], that
$\mathcal{L}_{n}^{0}$ is dominated by the fourth chaotic projection:

$\displaystyle\mathcal{L}_{n}^{0}[4]=\frac{1}{2}\frac{\sqrt{E_{n}/2}}{\mathcal{N}_{n}}\left[\frac{1}{8}W_{1}(n)^{2}-\frac{1}{4}W_{2}(n)^{2}-\frac{1}{4}W_{3}(n)^{2}-\frac{1}{2}W_{4}(n)^{2}+\frac{1}{8}+o_{\mathbb{P}}(1)\right].$

[RW18, Section 2.1 and Section 4] shows that, in the case of a non-static
curve, the second chaotic projection dominates the chaos expansion of
$\mathcal{Z}_{n}(\mathcal{C})$, and it has the form

$\displaystyle\mathcal{Z}_{n}(\mathcal{C})[2]=\mathcal{Z}^{a}_{n}(\mathcal{C})[2]+\mathcal{Z}^{b}_{n}(\mathcal{C})[2],$

where ${\rm Var}(\mathcal{Z}^{b}_{n}(\mathcal{C})[2])=o({\rm
Var}(\mathcal{Z}^{a}_{n}(\mathcal{C})[2]))$, and

$\displaystyle\mathcal{Z}^{a}_{n}(\mathcal{C})[2]=\frac{\sqrt{2\pi^{2}n}}{2\pi}\frac{1}{\mathcal{N}_{n}/2}L\sum_{\lambda\in\Lambda^{+}_{n}}(|a_{\lambda}|^{2}-1)\big{(}2I_{\lambda,\lambda^{\prime}}(2,0)-1\big{)},$

where we have introduced the notation

$\displaystyle
I_{\lambda,\lambda^{\prime}}(k,k^{\prime}):=\frac{1}{L}\int_{0}^{L}\left\langle\frac{\lambda}{|\lambda|},{\dot{\gamma}}(t)\right\rangle^{k}\left\langle\frac{\lambda^{\prime}}{|\lambda^{\prime}|},{\dot{\gamma}}(t)\right\rangle^{k^{\prime}}dt.$

Now if the curve is static [RW18, Lemma 6.5], the leading term in the chaotic
expansion of $\mathcal{Z}_{n}(\mathcal{C}^{\prime})$ is no longer the
projection onto the second chaos, but the projection onto the fourth chaos:

$\displaystyle\mathcal{Z}_{n}(\mathcal{C}^{\prime})[4]=\mathcal{Z}^{a}_{n}(\mathcal{C}^{\prime})[4]+\mathcal{Z}^{b}_{n}(\mathcal{C}^{\prime})[4]$

where ${\rm Var}(\mathcal{Z}^{b}_{n}(\mathcal{C}^{\prime})[4])=o({\rm
Var}(\mathcal{Z}^{a}_{n}(\mathcal{C}^{\prime})[4]))$, and

$\displaystyle\mathcal{Z}^{a}_{n}(\mathcal{C}^{\prime})[4]$
$\displaystyle=\frac{\sqrt{2n}}{4\mathcal{N}_{n}}L\Big{[}\frac{1}{\mathcal{N}_{n}/2}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}^{+}}(|a_{\lambda}|^{2}-1)(|a_{\lambda^{\prime}}|^{2}-1)(-4I_{\lambda,\lambda^{\prime}}(2,2)-1+4I_{\lambda,\lambda^{\prime}}(2,0))$
$\displaystyle\hskip
48.36958pt+\frac{1}{\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}(4I_{\lambda,\lambda^{\prime}}(4,0)-1)\Big{]}$

but, for any static curve we have that $I_{\lambda,\lambda^{\prime}}(2,0)=1/2$
via Lemma 3.4, so

$\displaystyle\mathcal{Z}^{a}_{n}(\mathcal{C}^{\prime})[4]$
$\displaystyle=\frac{\sqrt{2n}}{4\mathcal{N}_{n}}L\Big{[}\frac{1}{\mathcal{N}_{n}/2}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}^{+}}(|a_{\lambda}|^{2}-1)(|a_{\lambda^{\prime}}|^{2}-1)(1-4I_{\lambda,\lambda^{\prime}}(2,2))+\frac{1}{\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}(4I_{\lambda,\lambda^{\prime}}(4,0)-1)\Big{]}.$

And in the particular case of a doubly static curve we have (using (A.31) –
see the proof of Lemma 3.9):

$I_{\lambda,\lambda^{\prime}}(4,0)=\frac{3}{8},\qquad\qquad
I_{\lambda,\lambda^{\prime}}(2,2)=\frac{1}{8}\Big{(}1+2\left\langle\frac{\lambda}{|\lambda|},\frac{\lambda^{\prime}}{|\lambda^{\prime}|}\right\rangle^{2}\Big{)},$

so

$\displaystyle\mathcal{Z}^{a}_{n}(\mathcal{C}^{\prime\prime})[4]$
$\displaystyle=\frac{\sqrt{2n}}{4\mathcal{N}_{n}}L\Big{[}\frac{1}{\mathcal{N}_{n}/2}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}^{+}}(|a_{\lambda}|^{2}-1)(|a_{\lambda^{\prime}}|^{2}-1)\Big{(}\frac{1}{2}-\Big{\langle}\frac{\lambda}{|\lambda|},\frac{\lambda^{\prime}}{|\lambda^{\prime}|}\Big{\rangle}^{2}\Big{)}+\frac{1}{2}\Big{]}.$

##  3 Main results

###  3.1 The correlation structure in the $2$-dimensional case

Let us start investigating the correlation between boundary lengths.

######  Theorem 3.1.

Let $u_{1},u_{2}\in\mathbb{R}$, for $n\subset\\{S\\}$ sequence of energies
such that $\mathcal{N}_{n}\to+\infty$,

${\rm
Corr}(\mathcal{L}_{n}^{u_{1}},\mathcal{L}_{n}^{u_{2}})\to\begin{cases}1,\qquad&u_{1},u_{2}\neq
0\textnormal{ or }u_{1}=u_{2}=0,\\\
0,\qquad&\textnormal{otherwise}.\end{cases}$

Theorem 3.1 follows immediately from [CMR23, Corollary 2.7] and [MPRW16,
Section 1.4]. The correlation structure between boundary length and the
intersection number with a fixed smooth reference curve of nowhere zero
curvature is given below.

######  Theorem 3.2.

Let $u\in\mathbb{R}$, and $\mathcal{C}\subset\mathbb{T}^{2}$ be a smooth curve
of total length $L$ with nowhere zero curvature and for which
$\\{4\mathcal{B}_{\mathcal{C}}(\mu_{n})-L^{2}\\}_{n}$ is bounded away from
zero. Then for $n\subset\\{S\\}$ such that $\mathcal{N}_{n}\to+\infty$,

${\rm Corr}(\mathcal{L}_{n}^{u},\mathcal{Z}_{n}(\mathcal{C}))\to 0.$ (3.9)

The proof of Theorem 3.2 is in Section 4. To deal with the static case, we
need to define the following

${\cal I}_{4}^{\prime}={\cal
I}_{4}^{\prime}(\mathcal{C}^{\prime}):=\frac{1}{L}\int_{0}^{L}({\dot{\gamma}}_{1}(t)^{4}+{\dot{\gamma}}_{2}(t)^{4}-6{\dot{\gamma}}_{1}(t)^{2}{\dot{\gamma}}_{2}(t)^{2})dt.$
(3.10)

######  Theorem 3.3.

Let $\mathcal{C}^{\prime}\subset\mathbb{T}^{2}$ be a static curve. For
$\delta$-separated sequences of energy levels $n\subset\\{S\\}$ such that
$\mathcal{N}_{n}\to+\infty$,

${\rm Corr}(\mathcal{L}_{n}^{u},\mathcal{Z}_{n}(\mathcal{C}^{\prime}))\to
0,\quad\textnormal{if }u\neq 0.$ (3.11)

If moreover $\widehat{\mu}_{n}(4)\to\eta$, we have that

${\rm Corr}(\mathcal{L}_{n},\mathcal{Z}_{n}(\mathcal{C}^{\prime}))\to
f_{\mathcal{C}^{\prime}}(\eta),$ (3.12)

where

$f_{\mathcal{C}^{\prime}}(\eta):=\frac{1+2\eta\mathcal{I}^{\prime}_{4}+\eta^{2}}{\sqrt{2}\sqrt{1+\eta^{2}}\sqrt{2(1-\eta^{2})(2{\cal
I}_{4}-1)+(\eta{\cal I}_{4}^{\prime}+1)^{2}}}.$ (3.13)

####  3.1.1 Doubly static curves

There are special static curves for which $f_{\mathcal{C}^{\prime}}(u)=1$. In
order to investigate this case, we need to deeply understand the geometry of
static curves.

######  Lemma 3.4.

A curve is static if and only if

$\int_{0}^{L}{\dot{\gamma}}_{1}(t)^{2}dt=\int_{0}^{L}{\dot{\gamma}}_{2}(t)^{2}dt=L/2\;\;\;{\rm
and}\;\;\;\int_{0}^{L}{\dot{\gamma}}_{1}(t){\dot{\gamma}}_{2}(t)dt=0.$

######  Remark 3.5.

If ${\cal C}$ is static, then also
$\int_{0}^{L}{\dot{\gamma}}_{1}(t)^{4}dt=\int_{0}^{L}{\dot{\gamma}}_{2}(t)^{4}dt$.

Let us now define

$\displaystyle A=A_{{\cal C}}$
$\displaystyle:=\frac{1}{L}\int_{0}^{L}{\dot{\gamma}}_{1}(t)^{2}{\dot{\gamma}}_{2}(t)^{2}dt,\qquad
B=B_{{\cal
C}}:=\frac{1}{L}\int_{0}^{L}{\dot{\gamma}}_{1}(t)^{3}{\dot{\gamma}}_{2}(t)dt,$
$\displaystyle{\cal I}_{4}$ $\displaystyle={\cal I}_{4,{\cal
C}}:=\frac{1}{L^{2}}\int_{0}^{L}\int_{0}^{L}\langle{\dot{\gamma}}(t),{\dot{\gamma}}(u)\rangle^{4}dtdu.$

Hence

$\mathcal{I}^{\prime}_{4}=1-8A.$

The following result may look technical, but it is instrumental to introduce
the notion of a doubly static curve.

######  Lemma 3.6.

If ${\cal C}$ is a static curve, then we have the relation,

${\cal I}_{4}=\frac{1}{2}+8A^{2}-2A+8B^{2},$ (3.14)

and the inequalities

$\displaystyle 0<A<\frac{1}{4},\hskip 28.45274ptB^{2}<\frac{A(1-4A)}{4},\hskip
28.45274pt\frac{3}{8}\leq{\cal I}_{4}<\frac{1}{2},\hskip 28.45274pt-1<{\cal
I}_{4}^{\prime}<1.$

The proof of Lemma 3.6 is postponed to Appendix A. We can now introduce the
notion of a doubly static curve.

######  Definition 3.7.

We say the curve $\mathcal{C}$ doubly static if ${\cal I}_{4}=3/8$.

######  Example 3.8.

Circles and semicircles are doubly static.

Our characterization of doubly static curves is given in the following lemmas,
also proved in Appendix A.

######  Lemma 3.9.

One has ${\cal I}_{4}=3/8$ if and only if ${\cal C}$ is static, $A=1/8$, and
$B=0$; this implies also ${\cal I}_{4}^{\prime}=0$.

######  Corollary 3.10.

Let $\mathcal{C}^{\prime\prime}\subset\mathbb{T}^{2}$ be a doubly static
curve. For $\delta$-separated sequences of energy levels $n\subset\\{S\\}$
such that $\mathcal{N}_{n}\to+\infty$ and $\widehat{\mu}_{n}(4)\to\eta$,

${\rm Corr}(\mathcal{L}_{n},\mathcal{Z}_{n}(\mathcal{C}^{\prime\prime}))\to
1.$ (3.15)

######  Lemma 3.11 (cf. [RW18, Appendix G]).

Let ${\cal C}\subset{\mathbb{T}}^{2}$ be a smooth closed curve with nowhere
$0$ curvature, invariant with respect to rotations by $2\pi/k$, for integer
$k=3$ or $k\geq 5$. Then ${\cal C}$ is doubly static.

####  3.1.2 Discussion

Our first main results on the asymptotic correlation structure among the
functionals that we introduced in Section 2.2 can be conveniently summarized
in the following (symmetric) correlation matrix. As mentioned above,
$\mathcal{C}^{\prime}$ (resp. $\mathcal{C}^{\prime\prime}$) denotes a static
(resp. doubly static) curve.

Asymptotic correlation structure, $d=2$.

$\begin{matrix}&\mathcal{L}_{n}^{0}&\mathcal{L}_{n}^{u_{1}}&\mathcal{L}_{n}^{u_{2}}&\mathcal{Z}_{n}\mathcal{(C)}&\mathcal{Z}_{n}\mathcal{(C}^{\prime})&\mathcal{Z}_{n}\mathcal{(C}^{\prime\prime})\\\
\mathcal{L}_{n}^{0}&1&&&&&\\\ \mathcal{L}_{n}^{u_{1}}&0&1&&&&\\\
\mathcal{L}_{n}^{u_{2}}&0&1&1&&&\\\ \mathcal{Z}_{n}\mathcal{(C)}&0&0&0&1&&\\\
\mathcal{Z}_{n}\mathcal{(C}^{\prime})&f_{\mathcal{C}^{\prime}}(\eta)&0&0&0&1&\\\
\mathcal{Z}_{n}\mathcal{(C}^{\prime\prime})&1&0&0&0&f_{\mathcal{C}^{\prime}}(\eta)&1\end{matrix}$

######  Remark 3.12.

Level curves have asymptotically full correlation at different non-zero
thresholds $u_{1},u_{2}$; this is the phenomenon noted by [Wig12] for random
spherical harmonics, using the expansion of the 2-point correlation function,
and then related to the domination of the second chaos by [MPRW16, CM18,
NPR19, MRW20, MR21b, CMR23] and others. On the other hand, similarly to what
was noted earlier in [MR21b] for eigenfunctions on the sphere, Theorem 3.1
shows that the nodal length and the boundary lengths of excursion sets at non-
zero levels are asymptotically fully uncorrelated in the high-energy regime.
This can be interpreted as a spurious effect: boundary lengths at non-zero
levels are dominated by the second-order chaos, which is proportional to the
random norm of the eigenfunctions, and the latter of course has no impact on
the nodal length (which is invariant to normalizations).

######  Remark 3.13.

It is important to note that the correlation between boundary length and the
number of nodal intersections with static or non-static curves is always zero
in the asymptotic limit, excluding the nodal case (i.e., $u=0$). However, the
mechanism here is different than what we observed in the previous remark:
indeed, the second-order chaos component in the intersection of the nodal
length with a non-static curve does _not_ vanish, although it is still
uncorrelated with the random norm of the eigenfunctions, which dominates the
behaviour of the boundary length. See the proof of Theorem 3.2 for more
details. On the other hand for intersections with static curves the second
order chaos is of lower order, (see [RW18], Section 2.1) and therefore the
asymptotic correlation with the boundary lengths is obviously zero. In some
sense, intersections with static curves have some form of invariance with
respect to normalization factors for the Arithmetic Random Waves, and this
makes their behaviour somewhat analogous to nodal lines, see our following
discussion on partial autocorrelation results.

######  Remark 3.14.

In the special case where $\eta=0$ the limiting spectral measure is Lebesgue,
i.e. lattice points are equidistributed in the limit. In these circumstances,
we have

$\mathrm{Corr}({\mathcal{L}}_{n}^{0},{\mathcal{Z}}_{n}(\mathcal{C}^{\prime}))\to\frac{1}{\sqrt{2}}\frac{1}{\sqrt{4{\cal
I}_{4}-1}};$

this result can be compared to Theorem 3.27 below for the three-dimensional
case. At the other extreme we have $\eta=\pm 1$, where the limiting measure is
Cilleruelo or tilted Cilleruelo, namely the spectral measure exhibits the
maximal concentration. In these cases
$\mathrm{Corr}({\mathcal{L}}_{n}^{0},{\mathcal{Z}}_{n}(\mathcal{C}^{\prime}))\to
1$ for any static curve ${\cal C}^{\prime}$.

######  Remark 3.15.

We have that
$\mathrm{Corr}({\mathcal{L}}_{n}^{0},{\mathcal{Z}}_{n}(\mathcal{C}^{\prime}))>0$
for any $\eta$ and any static curve ${\cal C}^{\prime}$.

######  Remark 3.16.

In the doubly static case, the dominant terms in ${\mathcal{L}}^{0}[4]$ and
${\mathcal{Z}}(\mathcal{C}^{{}^{\prime\prime}})[4]$ coincide, up to a factor
depending on the energy and $L$, so that we get full correlation for any
$\eta\in[-1,1]$. More explicitly, $\mathcal{L}_{n}^{0}$ and
$\mathcal{Z}_{n}(\mathcal{C}^{\prime\prime})$ are asymptotically the same
random variable up to a constant. Once again, we recall that circles and
semicircles are doubly static.

######  Remark 3.17.

Let ${\cal C}^{\prime}$ be a fixed toral curve, static, but not doubly static.
Then (as noted above) we have $f_{{\cal C}^{\prime}}(\eta)=1$ for $\eta=\pm
1$, and $f_{{\cal C}^{\prime}}(\eta)=0$ has exactly one solution for
$\eta\in(-1,1)$.

######  Remark 3.18.

Above we consider the case $\eta=\pm 1$ under the assumption of well separated
sequences of eigenvalues. We stress that it is possible that $\eta=\pm 1$ may
not be attainable under the well separated assumption.

###  3.2 The partial correlation structure in the $2$-dimensional case

To get deeper insights into the correlation structure for the geometry of
level sets of Arthmetic Random Waves, it is of greater interest to get rid of
the effect of the random $L^{2}$ norm of the eigenfunctions. More precisely,
it is of interest to investigate the so-called partial correlation structure,
where the effect of the fluctuations in the eigenfunctions norm is removed.

Let $X,Y,Z$ be square integrable random variables; we define the partial
correlation coefficient between $X$ and $Y$ conditional on $Z$ as

${\rm Corr}_{Z}(X,Y):={\rm Corr}(X^{*},Y^{*}),$

where $X$ and $Y$ are the residuals after projecting $X,Y$ onto the
explanatory variable $Z$.

In analogous circumstances, it was shown in [MR21b] that for random spherical
harmonics perfect autocorrelation holds in the high energy limit (see also
[CM22] for a similar result on critical points). For Arithmetic Random Waves,
the correlation structure is much more subtle, as detailed below.

######  Proposition 3.19.

Let $u_{1},u_{2}\in\mathbb{R}$, then for subsequences of energy levels
$\\{n\\}\subset S$ such that $\widehat{\mu_{n}}(4)\to\eta\in[-1,1]$ as ${\cal
N}_{n}\to+\infty$ we have

$\displaystyle{\rm
Cov\,}(\mathcal{L}_{n}^{u_{1}}[4],\mathcal{L}_{n}^{u_{2}}[4])\sim\phi(u_{1})\phi(u_{2})\frac{\pi}{4}\frac{E_{n}}{\mathcal{N}_{n}^{2}}\left(2a(u_{1})a(u_{2})-\frac{1}{4}(a(u_{1})+a(u_{2}))+\frac{3+\eta^{2}}{8^{2}}\right).$

Note that for $u_{1}=u_{2}=0$, since $a(0)=\frac{1}{8}$, we retrieve

${\rm
Var}(\mathcal{L}_{n}[4])\sim\frac{4\pi^{2}n}{\mathcal{N}_{n}^{2}}\frac{1+\eta^{2}}{512},$

as expected.

######  Theorem 3.20.

Let $u\in\mathbb{R}$, for a $\delta$-separated subsequences of energies
$\\{n\\}\subset S$ such that ${\cal N}_{n}\to\infty$ and
$\widehat{\mu}_{n}(4)\to\eta\in[-1,1]$, we have

$\displaystyle{\rm
Cov\,}(\mathcal{L}_{n}^{u}[4],\mathcal{Z}_{n}(\mathcal{C}^{\prime})[4])$
$\displaystyle\sim\phi(u)\sqrt{\frac{\pi}{2}}\frac{\sqrt{E_{n}/2}}{\mathcal{N}_{n}}\frac{\sqrt{2n}}{4\mathcal{N}_{n}}L\frac{1}{16}\Big{[}1+2\eta\mathcal{I}^{\prime}_{4}+\eta^{2}\Big{]}.$

####  3.2.1 Discussion

For 2-dimensional Arithmetic Random Waves, the following asymptotic partial
correlation structure holds

Partial Correlation structure, dimension $d=2$.

$\begin{matrix}&\mathcal{L}_{n}^{0}&\mathcal{L}_{n}^{u_{1}}&\mathcal{L}_{n}^{u_{2}}&\mathcal{Z}_{n}\mathcal{(C)}&\mathcal{Z}_{n}\mathcal{(C}^{\prime})&\mathcal{Z}_{n}\mathcal{(C}^{\prime\prime})\\\
\mathcal{L}_{n}^{0}&1&&&&&\\\ \mathcal{L}_{n}^{u_{1}}&M(0,u_{1};\eta)&1&&&&\\\
\mathcal{L}_{n}^{u_{2}}&M(0,u_{2};\eta)&M(u_{1},u_{2};\eta)&1&&&\\\
\mathcal{Z}_{n}\mathcal{(C)}&0&0&0&1&&\\\
\mathcal{Z}_{n}\mathcal{(C}^{\prime})&f_{\mathcal{C}^{\prime}}(\eta)&f_{\mathcal{C}^{\prime}}(u_{1};\eta)&f_{\mathcal{C}^{\prime}}(u_{2};\eta)&0&1&\\\
\mathcal{Z}_{n}\mathcal{(C}^{\prime\prime})&1&M(0,u_{1};\eta)&M(0,u_{2};\eta)&0&f_{\mathcal{C}^{\prime}}(\eta)&1\end{matrix}$

where for $u\in\mathbb{R}$

$\displaystyle
f_{\mathcal{C}^{\prime}}(u;\eta):=\frac{\frac{\sqrt{2}}{16}\Big{[}1+2\eta\mathcal{I}^{\prime}_{4}+\eta^{2}\Big{]}}{\sqrt{(2a(u)^{2}-\frac{1}{2}a(u)+\frac{3+\eta^{2}}{8^{2}})}\cdot\sqrt{2(1-\eta^{2})(2{\cal
I}_{4}-1)+(\eta{\cal I}_{4}^{\prime}+1)^{2}}},$

(note that $f_{\mathcal{C}^{\prime}}(\eta)=f_{\mathcal{C}^{\prime}}(0;\eta)$
as in (4.21) since $a(0)=\frac{1}{8}$). Moreover,

$M(u_{1},u_{2};\eta)=\frac{\left\\{2a(u_{1})a(u_{2})-\frac{1}{4}(a(u_{1})+a(u_{2}))+\frac{3+\eta^{2}}{8^{2}}\right\\}}{\sqrt{\left\\{2a^{2}(u_{1})-\frac{1}{2}a(u_{1})+\frac{3+\eta^{2}}{8^{2}}\right\\}\left\\{2a^{2}(u_{2})-\frac{1}{2}a(u_{2})+\frac{3+\eta^{2}}{8^{2}}\right\\}}},$

for

$a(u)=\frac{H_{4}(u)}{4}+\frac{H_{2}(u)}{2}-\frac{1}{8}\text{ .}$

Note that

$M(0,u;\eta)=f_{\mathcal{C}^{\prime\prime}}(u;\eta),$

where $\mathcal{C}^{\prime\prime}$ is a doubly static curve.

######  Remark 3.21.

The rationale behind the previous correlation table can be explained as
follows. Considering partial correlation, the second-order chaos term
disappears from the boundary lengths measure at non-zero thresholds. As a
consequence their correlation with the intersections for non static curves
becomes zero, because the latter is dominated by the second-order chaos. On
the other hand, for static or doubly static curves the second-order chaos is
lower order, hence the partial correlation becomes basically the correlation
between the fourth-chaos components of intersections and boundary lengths.

An important consequence of the previous results is the existence of resonant
pairs, that is, sets of threshold levels with asymptotically full correlation
between boundary length and/or nodal intersections; this is illustrated in the
following corollary.

######  Corollary 3.22.

We have that

$\displaystyle\lim_{n\rightarrow\infty}{\rm
Corr}(\mathcal{L}_{n}^{u_{1}},\mathcal{L}_{n}^{u_{2}})=1,$

if and only if

$u_{2}^{4}-4u_{2}^{2}=u_{1}^{4}-4u_{1}^{2}.$ (3.16)

######  Example 3.23.

For $u_{1}=0$ we obtain that the nodal lines (and the interesections with
doubly static curves) have asymptotically perfect correlations with the levels
$u_{2}=\pm 2$. For $u_{1}=1$ we obtain $u_{2}^{4}-4u_{2}^{2}+3=0,$ with
solutions $u_{2}=\sqrt{3},-\sqrt{3},-1,1,$ so that resonant pairs are given by
$(1,3),(1,-3)$ and $(1,-1)$.

We call (3.16) the _Full Correlation Curve_ for boundary lengths of Arithmetic
Random Waves. We believe that analogous algebraic curves characterize full
correlation pairs for other geometric functionals, such as Lipschitz-Killing
curvatures and critical values. We leave the investigation of this for future
research.

###  3.3 Some results in the 3-dimensional case

It is of obvious interest to investigate the existence of a similar
correlation structure for higher-dimensional arithmetic random waves (as
studied for instance in [Cam19, BM19, RWY16]). For brevity’s sake, we do not
explore fully this possibility here; we focus just on a special case, that is
in 3 dimensions the correlation between the nodal area
($\mathcal{A}_{n}=\mathcal{A}_{n}^{0}$) and the length of intersections of the
nodal area with static surfaces or doubly-static surfaces
($\mathcal{M}_{n}(\Sigma^{\prime})$,
$\mathcal{M}_{n}(\Sigma^{\prime\prime}))$. Let us first define

$\mathcal{A}=\mathcal{A}_{n}:=\mathcal{H}^{2}(\\{x\in\mathbb{T}^{3}:T_{n}(x)=0\\}),$

and, for $\Sigma\subset\mathbb{T}^{3}$ a fixed compact regular toral surface,
of finite area $A:=|\Sigma|$,

$\mathcal{M}=\mathcal{M}_{n}(\Sigma):=\mathcal{H}^{1}(\\{x\in\mathbb{T}^{3}:T_{n}(x)=0,x\in\Sigma\\}).$

From [BM19, Theorem 1.2], as $n\to+\infty$, $n\not\equiv
0,4,7(\mathrm{mod}\;8)$,

${\rm Var}(\mathcal{A}_{n})\sim\frac{32}{375}\frac{n}{\mathcal{N}_{n}^{2}}.$

Assume that $\Sigma$ admits a smooth normal vector locally, and call
$n(\sigma)$ the unit normal vector to $\Sigma$ at the point $\sigma$. For
$k\geq 0$ even, call

${\cal I}_{k}={\cal I}_{k,\Sigma}:=\frac{1}{A^{2}}\iint_{\Sigma^{2}}\langle
n(\sigma),n(\sigma^{\prime})\rangle^{k}\,d\sigma d\sigma^{\prime}.$ (3.17)

######  Definition 3.24.

We call $\Sigma$ of nowhere $0$ Gauss-Kronecker curvature static if ${\cal
I}_{2}=1/3$, and doubly static if ${\cal I}_{4}=1/5$.

######  Remark 3.25.

Also for surfaces doubly static implies static (see Lemma 6.4 below). Simple
examples of doubly static surfaces are the sphere and hemisphere.

From [MR21a], we have as $n\to+\infty$, $n\not\equiv 0,4,7(\mathrm{mod}\;8)$,
along a well separated222See Definition 1.6 in [MR21a]. sequence of
eigenvalues

${\rm
Var}(\mathcal{M}_{n})\sim\frac{\pi^{2}}{9600}\frac{n}{N^{2}}(81\mathcal{I}_{4}+35A^{2}).$
(3.18)

######  Remark 3.26.

Any surface $\Sigma$ of finite area and nowhere $0$ Gauss-Kronecker curvature,
invariant with respect to any permutation and sign change of the coordinates
is static. To see this, note that under this condition $\Sigma$ verifies the
criterion for staticity given by Lemma 6.2 below. For instance, $\Sigma$ may
be given piecewise by symmetric trivariate polynomials where all variables
appear to even powers (as long as the assumption on the curvature is met
everywhere).

Our main result is the following.

######  Theorem 3.27.

Let $\mathcal{A}$ be the nodal area and $\mathcal{M}$ the nodal intersection
length. For static surfaces $\Sigma$ of area $A$, we have as $n\to\infty$,
$n\not\equiv 0,4,7\pmod{8}$ along a well separated sequence of eigenvalues

${\rm
Cov\,}(\mathcal{A}_{n},\mathcal{M}_{n})\sim\frac{n}{\mathcal{N}^{2}}\cdot\frac{8\pi
A}{375},$ (3.19)

so that

${\rm
Corr}(\mathcal{A}_{n},\mathcal{M}_{n})\longrightarrow\frac{16}{\sqrt{405\cdot{\cal
I}_{4}+175}}.$

######  Remark 3.28.

Static surfaces verify $1/5\leq{\cal I}_{4}\leq 1/3$ (Lemma 6.4). The above
limit is $1$ for ‘doubly static’ surfaces i.e. those satisfying ${\cal
I}_{4}=1/5$, for instance sphere and hemisphere.

###  3.4 Plan of the paper

The results concerning the 2-dimensional correlation structure of Arithmetic
Random Waves are included in Section 4, whereas those for partial correlation
are given in Section 5; Section 6 is devoted to the arguments related to nodal
surfaces, whereas Appendix A collects the proofs for the technical lemmas that
we exploited to characterize static and doubly static curves.

###  3.5 Acknowledgments

We are grateful to GNAMPA-INdAM Project 2022 _Proprietà e teoremi limite per
funzionali di campi Gaussiani_ , to MUR Department of Excellence Project
2023-2027 _MatModTov_ , to MUR Prin 2023-2025 _Grafia_ , and to Swiss National
Science Foundation project 200021_184927, held by Prof. Maryna Viazovska, for
financial support.

##  4 The 2-dimensional Correlation Structure

Let us start investigating the correlation between the boundary length at
level $u$ and the number of nodal intersections with respect to a non-static
curve.

Proof of Theorem 3.2. For $u=0$ the result follows immediately from the
orthogonality of the projections in the chaos expansion. In fact in [RW18,
Section 2.1] it is shown that, in the case of a non-static curve, the second
chaotic projection dominates the chaos expansion of
$\mathcal{Z}_{n}(\mathcal{C})$, while it is known, see [MPRW16, Section 1.4],
that $\mathcal{L}_{n}^{0}$ is dominated by the fourth chaotic projection. For
$u\neq 0$ the boundary length is dominated by the second order chaos [CMR23,
Theorem 2.4], so we have

$\displaystyle{\rm Cov\,}(\mathcal{L}_{n}^{u},\mathcal{Z}_{n}(\mathcal{C}))$
$\displaystyle=\sqrt{2\pi^{2}n}\sqrt{\frac{\pi}{8}}\phi(u)u^{2}\frac{\sqrt{n}}{\sqrt{2}}L\frac{1}{{\mathcal{N}}_{n}^{2}/4}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}^{+}}\mathbb{E}[(|a_{\lambda}|^{2}-1)(|a_{\lambda^{\prime}}|^{2}-1)](2I_{\lambda^{\prime},\lambda^{\prime}}(2,0)-1)+o\left(\frac{n}{\mathcal{N}_{n}}\right)$
$\displaystyle=\sqrt{2\pi^{2}n}\sqrt{\frac{\pi}{8}}\phi(u)u^{2}\frac{\sqrt{n}}{\sqrt{2}}L\frac{1}{{\mathcal{N}}_{n}^{2}/4}\sum_{\lambda\in\Lambda_{n}^{+}}\mathbb{(}2I_{\lambda,\lambda}(2,0)-1)+o\left(\frac{n}{\mathcal{N}_{n}}\right)$

where the last step follows by observing that $2|a_{\lambda}|^{2}$ has a chi-
squared distribution with $2$ degrees of freedom, and by recalling that
$a_{\lambda}$ and $a_{\lambda}^{\prime}$ are independent for
$\lambda\neq\lambda^{\prime}$. The statement immediately follows by observing
that

$\displaystyle\sum_{\lambda\in\Lambda_{n}^{+}}\mathbb{(}-1+2I_{\lambda,\lambda}(2,0))$
$\displaystyle=\frac{1}{2}\sum_{\lambda\in\Lambda_{n}}\mathbb{(}-1+2I_{\lambda,\lambda}(2,0))=0,$

since, as shown in Lemma A.32 equation (A.32), we have

$\frac{1}{\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}I_{\lambda,\lambda}(2,0)=\frac{1}{2}.$

∎

The main result to prove Proposition 3.3, i.e. the correlation structure among
nodal length and nodal intersection numbers, is the following proposition.

######  Proposition 4.1.

Let ${\mathcal{L}}_{n}$ be the nodal length, and
${\mathcal{Z}}_{n}(\mathcal{C}^{\prime})$ the nodal intersections number with
a static curve. For a $\delta$-separated subsequence of energies
$\\{n\\}\subset S$ such that ${\cal N}_{n}\to\infty$ and
$\widehat{\mu}_{n}(4)\to\eta\in[-1,1]$, we have

${\rm
Cov\,}(\mathcal{L}_{n},\mathcal{Z}_{n}(\mathcal{C}^{\prime}))\sim\frac{\sqrt{\pi^{2}n}}{\mathcal{N}_{n}}\frac{\sqrt{n}}{4\mathcal{N}_{n}}L\frac{1}{16}\Big{[}1+2\eta\mathcal{I}^{\prime}_{4}+\eta^{2}\Big{]},$
(4.20)

and

${\rm
Corr}({\mathcal{L}}_{n},{\mathcal{Z}}_{n}(\mathcal{C}^{\prime}))\sim\frac{1+2\eta\mathcal{I}^{\prime}_{4}+\eta^{2}}{\sqrt{2}\sqrt{1+\eta^{2}}\sqrt{2(1-\eta^{2})(2{\cal
I}_{4}-1)+(\eta{\cal I}_{4}^{\prime}+1)^{2}}}.$ (4.21)

Proof of Proposition 4.1. The covariance in (4.20) follows immediately from
Proposition 3.20 with $u=0$. Now we take into account the expression for ${\rm
Var}({\mathcal{L}}_{n}[4])$ in [MPRW16, (2.20)], and, in our notation, [RW18,
(3.4)] is

${\rm
Var}({\mathcal{Z}}[4])\sim\frac{n}{4\mathcal{N}^{2}}\cdot\frac{L^{2}}{4}[2(1-\eta^{2})(2{\cal
I}_{4}-1)+(\eta{\cal I}_{4}^{\prime}+1)^{2}].$

∎

##  5 The Partial Correlation Structure in the $2$-dimensional Case

Proof of Proposition 3.19. Let us write

$\displaystyle{\rm
Cov\,}(\mathcal{L}_{n}^{u_{1}}[4],\mathcal{L}_{n}^{u_{2}}[4])\sim$
$\displaystyle\,\phi(u_{1})\phi(u_{2})\frac{\pi}{4}\frac{4\pi^{2}n}{\mathcal{N}_{n}^{2}}$
$\displaystyle\times\Big{\\{}a(u_{1})a(u_{2}){\rm Var}(W_{1}(n)^{2})$
$\displaystyle+[a(u_{1})+a(u_{2})]{\rm
Cov\,}\Big{(}W_{1}(n)^{2},-\frac{1}{4}W_{2}(n)^{2}-\frac{1}{4}W_{3}(n)^{2}-\frac{1}{2}W_{4}(n)^{2}\Big{)}$
$\displaystyle+{\rm
Var}\Big{(}-\frac{1}{4}W_{2}(n)^{2}-\frac{1}{4}W_{3}(n)^{2}-\frac{1}{2}W_{4}(n)^{2}\Big{)}\Big{\\}}.$

From Lemma 4.3 in [MPRW16] and a simple computations of Gaussian moments we
have, as $n\to+\infty$,

${\rm Var}(W_{1}(n)^{2})\to 2,$

and moreover

${\rm
Cov\,}\Big{(}W_{1}(n)^{2},-\frac{1}{4}W_{2}(n)^{2}-\frac{1}{4}W_{3}(n)^{2}-\frac{1}{2}W_{4}(n)^{2}\Big{)}\to-\frac{1}{4}.$

Finally,

$\displaystyle{\rm
Var}\Big{(}-\frac{1}{4}W_{2}(n)^{2}-\frac{1}{4}W_{3}(n)^{2}-\frac{1}{2}W_{4}(n)^{2}\Big{)}$
$\displaystyle\to\frac{1}{16}\cdot
2\left(\frac{3+\eta}{8}\right)^{2}+\frac{1}{16}\cdot
2\left(\frac{3+\eta}{8}\right)^{2}+\frac{1}{4}\cdot
2\cdot\left(\frac{1-\eta}{8}\right)^{2}+2\cdot\frac{1}{16}\cdot
2\left(\frac{1-\eta}{8}\right)^{2}$
$\displaystyle=\frac{1}{8^{2}}(3+\eta^{2}),$

thus concluding the proof.

∎

Let us now investigate partial correlation between the boundary length and the
number of intersections with a static curve. Define

$\displaystyle{\cal I}$
$\displaystyle:=\frac{1}{L}\int_{0}^{L}\Big{[}\frac{3+\eta}{8}{\dot{\gamma}}_{1}^{4}(t)+\frac{3+\eta}{8}{\dot{\gamma}}_{2}^{4}(t)+6\frac{1-\eta}{8}{\dot{\gamma}}_{1}^{2}(t){\dot{\gamma}}_{2}^{2}(t)\Big{]}dt,$
$\displaystyle{\cal J}$
$\displaystyle:=\frac{1}{L}\int_{0}^{L}\Big{[}\Big{(}\frac{3+\eta}{8}\Big{)}^{2}{\dot{\gamma}}_{1}^{4}(t)+\Big{(}\frac{1-\eta}{8}\Big{)}^{2}{\dot{\gamma}}_{1}^{4}(t)+4\Big{(}\frac{3+\eta}{8}\Big{)}\Big{(}\frac{1-\eta}{8}\Big{)}{\dot{\gamma}}_{1}^{2}(t){\dot{\gamma}}_{2}^{2}(t)$
$\displaystyle\hskip
45.52458pt+\Big{(}\frac{1-\eta}{8}\Big{)}^{2}{\dot{\gamma}}_{2}^{4}(t)+\Big{(}\frac{3+\eta}{8}\Big{)}^{2}{\dot{\gamma}}_{2}^{4}(t)+8\Big{(}\frac{1-\eta}{8}\Big{)}^{2}{\dot{\gamma}}_{1}^{2}(t){\dot{\gamma}}_{2}^{2}(t)\Big{]}dt.$

Proof of Theorem 3.20. Using the expressions of the $4$-th order chaos, see
Section 2.3, we write

$\displaystyle{\rm
Cov\,}(\mathcal{L}_{n}^{u}[4],{\mathcal{Z}}_{n}(\mathcal{C}^{\prime})[4])$
$\displaystyle\sim\phi(u)\sqrt{\frac{\pi}{2}}\frac{\sqrt{E_{n}/2}}{\mathcal{N}_{n}}\frac{\sqrt{2n}}{4\mathcal{N}_{n}}L\;\mathbb{E}\Big{\\{}\Big{[}a(u)W_{1}(n)^{2}-\frac{1}{4}W_{2}(n)^{2}-\frac{1}{4}W_{3}(n)^{2}-\frac{1}{2}W_{4}(n)^{2}\Big{]}$
$\displaystyle\hskip
153.6447pt\times\Big{[}\frac{2}{\mathcal{N}_{n}}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}}(|a_{\lambda}|^{2}-1)(|a_{\lambda^{\prime}}|^{2}-1)(1-4I_{\lambda,\lambda^{\prime}}(2,2))\Big{]}\Big{\\}}.$

The first term is

$\displaystyle A$
$\displaystyle=a(u)\mathbb{E}\left[W_{1}(n)^{2}\frac{2}{\mathcal{N}_{n}}\sum_{\lambda,\lambda^{\prime}\in\Lambda^{+}_{n}}(|a_{\lambda}|^{2}-1)(|a_{\lambda^{\prime}}|^{2}-1)(1-4I_{\lambda,\lambda^{\prime}}(2,2))\right]$
$\displaystyle=a(u)\frac{4}{{\mathcal{N}}^{2}_{n}}\mathbb{E}\Big{[}\Big{(}\sum_{\lambda_{1},\lambda_{2}\in\Lambda^{+}_{n}}(|a_{\lambda_{1}}|^{2}-1)(|a_{\lambda_{2}}|^{2}-1)\Big{)}\Big{(}\sum_{\lambda_{3},\lambda_{4}\in\Lambda^{+}_{n}}(|a_{\lambda_{3}}|^{2}-1)(|a_{\lambda_{4}}|^{2}-1)(1-4I_{\lambda_{3},\lambda_{4}}(2,2))\Big{)}\Big{]}$
$\displaystyle=a(u)\frac{4}{{\mathcal{N}}^{2}_{n}}\Big{[}\sum_{\lambda,\lambda^{\prime}\in\Lambda^{+}_{n}}(1-4I_{\lambda^{\prime},\lambda^{\prime}}(2,2))+2\sum_{\lambda,\lambda^{\prime}\in\Lambda^{+}_{n}}(1-4I_{\lambda,\lambda^{\prime}}(2,2))\Big{]}$
$\displaystyle=a(u)\frac{4}{{\mathcal{N}}^{2}_{n}}\sum_{\lambda,\lambda^{\prime}\in\Lambda^{+}_{n}}(1-4I_{\lambda^{\prime},\lambda^{\prime}}(2,2))$

where in the last step we apply (A.34). The second and third term have the
form

$\displaystyle B_{i}$
$\displaystyle=-\frac{1}{4}\frac{4}{n^{2}{\mathcal{N}}^{2}_{n}}\Big{[}\sum_{\lambda,\lambda^{\prime}\in\Lambda^{+}_{n}}\lambda_{i}^{4}(1-4I_{\lambda^{\prime},\lambda^{\prime}}(2,2))+2\sum_{\lambda,\lambda^{\prime}\in\Lambda^{+}_{n}}\lambda_{i}^{2}(\lambda^{\prime}_{i})^{2}(1-4I_{\lambda,\lambda^{\prime}}(2,2))\Big{]}$

for $i=1,2$, and the last term is given by

$\displaystyle C$
$\displaystyle=-\frac{1}{2}\frac{4}{n^{2}{\mathcal{N}}^{2}_{n}}\Big{[}\sum_{\lambda,\lambda^{\prime}\in\Lambda^{+}_{n}}\lambda_{1}^{2}\lambda_{2}^{2}(1-4I_{\lambda^{\prime},\lambda^{\prime}}(2,2))+2\sum_{\lambda,\lambda^{\prime}\in\Lambda^{+}_{n}}\lambda_{1}\lambda_{2}\lambda^{\prime}_{1}\lambda^{\prime}_{2}(1-4I_{\lambda,\lambda^{\prime}}(2,2))\Big{]}.$

We observe that

$\displaystyle B_{1}+B_{2}+C$
$\displaystyle=-\frac{1}{4}\frac{4}{n^{2}{\mathcal{N}}^{2}_{n}}\Big{[}\sum_{\lambda,\lambda^{\prime}\in\Lambda^{+}_{n}}\langle\lambda,\lambda\rangle^{2}(1-4I_{\lambda^{\prime},\lambda^{\prime}}(2,2))+2\sum_{\lambda,\lambda^{\prime}\in\Lambda^{+}_{n}}\langle\lambda,\lambda^{\prime}\rangle^{2}(1-4I_{\lambda,\lambda^{\prime}}(2,2))\Big{]}.$

In view of Lemma A.1 we write

$\displaystyle A$
$\displaystyle=a(u)\frac{1}{{\mathcal{N}}_{n}^{2}}[{\mathcal{N}}_{n}^{2}-4{\mathcal{N}}_{n}^{2}{\mathcal{I}}]=a(u)[1-4\,\mathcal{I}]$
$\displaystyle B_{1}+B_{2}+C$
$\displaystyle=-\frac{1}{4}\frac{1}{n^{2}{\mathcal{N}}_{n}^{2}}[n^{2}{\mathcal{N}}_{n}^{2}-4n^{2}{\mathcal{N}}_{n}^{2}\mathcal{I}+2\frac{1}{2}n^{2}{\mathcal{N}}_{n}^{2}-2\cdot
4n^{2}{\mathcal{N}}_{n}^{2}{\mathcal{J}}]=-1+4\mathcal{I}-2\frac{1}{2}+2\cdot
4\mathcal{J};$

i.e.

$\displaystyle
A+B_{1}+B_{2}+C=a(u)[1-4\,\mathcal{I}]-\frac{1}{4}+\mathcal{I}-\frac{1}{4}+2\mathcal{J}=\Big{[}a(u)-\frac{1}{4}\Big{]}[1-4\,\mathcal{I}]-\frac{1}{4}+2\mathcal{J}.$

Moreover,

$\displaystyle-\left(a(u)-\frac{1}{4}\right)\frac{1}{\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}(4\mathcal{I}_{\lambda,\lambda}(4,0)-1)=-\left(a(u)-\frac{1}{4}\right)(4\mathcal{I}-1),$

and

$\displaystyle-\left(a(u)-\frac{1}{4}\right)\frac{1}{\mathcal{N}_{n}/2}\sum_{\lambda,\lambda^{\prime}\in\Lambda^{+}_{n}}(|a_{\lambda}|^{2}-1)(|a_{\lambda^{\prime}}|^{2}-1)(1-4\mathcal{I}_{\lambda,\lambda^{\prime}}(2,2))=-\left(a(u)-\frac{1}{4}\right)(1-4\mathcal{I}).$

Finally,

$\displaystyle\frac{1}{\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}(4\mathcal{I}_{\lambda,\lambda}(4,0)-1)\frac{1}{\mathcal{N}_{n}/2}\sum_{\lambda,\lambda^{\prime}\in\Lambda^{+}_{n}}(|a_{\lambda}|^{2}-1)\left(|a_{\lambda^{\prime}}|^{2}-1)(a(u)-\frac{1}{4}\left\langle\frac{\lambda}{|\lambda|},\frac{\lambda^{\prime}}{|\lambda^{\prime}|}\right\rangle^{2}\right)$
$\displaystyle=(4\mathcal{I}-1)\left(a(u)-\frac{1}{4}\right).$

So we obtain that

$\displaystyle{\rm
Cov\,}(\mathcal{L}_{n}^{u}[4],\mathcal{Z}_{n}(\mathcal{C}^{\prime})[4])$
$\displaystyle\sim\phi(u)\sqrt{\frac{\pi}{2}}\frac{\sqrt{E_{n}/2}}{\mathcal{N}_{n}}\frac{\sqrt{2n}}{4\mathcal{N}_{n}}L\Big{\\{}\Big{[}a(u)-\frac{1}{4}\Big{]}[1-4\,\mathcal{I}]-\frac{1}{4}+2\mathcal{J}-(a(u)-\frac{1}{4})(1-4\mathcal{I})\Big{\\}}$
$\displaystyle=\phi(u)\sqrt{\frac{\pi}{2}}\frac{\sqrt{E_{n}/2}}{\mathcal{N}_{n}}\frac{\sqrt{2n}}{4\mathcal{N}_{n}}L\Big{\\{}-\frac{1}{4}+2\mathcal{J}\Big{\\}}.$

Observing that,

$\displaystyle{\cal J}=5\frac{2}{8^{2}}+\frac{4}{8^{2}}\eta{\cal
I}^{\prime}_{4}+\frac{2}{8^{2}}\eta^{2},$

we arrive at

$\displaystyle{\rm
Cov\,}(\mathcal{L}_{n}^{u}[4],\mathcal{Z}(\mathcal{C}^{\prime})[4])$
$\displaystyle\sim\phi(u)\sqrt{\frac{\pi}{2}}\frac{\sqrt{E_{n}/2}}{\mathcal{N}_{n}}\frac{\sqrt{2n}}{4\mathcal{N}_{n}}L\frac{1}{16}\Big{[}1+2\eta\mathcal{I}^{\prime}_{4}+\eta^{2}\Big{]}.$

∎

Proof of Corollary 3.22. We need to study the covariance expression for the
boundary lengths at thresholds $u_{1},u_{2}$, that is

$\mathrm{Cov}(\mathcal{L}_{n}^{u_{1}},\mathcal{L}_{n}^{u_{2}})=2a(u_{1})a(u_{2})-\frac{1}{4}(a(u_{1})+a(u_{2}))+\frac{3+\eta^{2}}{64}$

where we have that

$\displaystyle
a(u)=\frac{u^{4}-6u^{2}+3}{4}+\frac{u^{2}-1}{2}-\frac{1}{8}=\frac{1}{4}u^{4}-u^{2}+\frac{1}{8}.$

We have that

$\displaystyle\mathrm{Cov}(\mathcal{L}_{n}^{u},\mathcal{L}_{n}^{0})$
$\displaystyle=2a(u)a(0)-\frac{1}{4}(a(u)+a(0))+\frac{3+\eta^{2}}{64}$
$\displaystyle=2\left(\frac{1}{4}u^{4}-u^{2}+\frac{1}{8}\right)\frac{1}{8}-\frac{1}{4}\left(\frac{1}{4}u^{4}-u^{2}+\frac{1}{8}+\frac{1}{8}\right)+\frac{3+\eta^{2}}{64}$
$\displaystyle=\frac{1}{64}(2(2u^{4}-8u^{2}+1)-2(2u^{4}-8u^{2}+2)+3+\eta^{2})$
$\displaystyle=\frac{1+\eta^{2}}{64}.$

On the other hand

$\displaystyle\mathrm{Var}(\mathcal{L}_{n}^{u})$
$\displaystyle=\mathrm{Cov}(\mathcal{L}_{n}^{u},\mathcal{L}_{n}^{u})$
$\displaystyle=2\left(\frac{1}{4}u^{4}-u^{2}+\frac{1}{8}\right)^{2}-\frac{1}{2}\left(\frac{1}{4}u^{4}-u^{2}+\frac{1}{8}\right)+\frac{3+\eta^{2}}{64}$
$\displaystyle=\frac{1}{8}u^{8}-u^{6}+2u^{4}+\frac{1+\eta^{2}}{64}=\frac{u^{4}}{8}(u^{2}-4)^{2}+\frac{1+\eta^{2}}{64},$

so that the squared correlation is given by

$\displaystyle\frac{\\{\mathrm{Cov}(\mathcal{L}_{n}^{u},0)\\}^{2}}{\mathrm{Var}(\mathcal{L}_{n}^{0})\mathrm{Var}(\mathcal{L}_{n}^{u})}=\frac{1+\eta^{2}}{8u^{4}(u^{2}-4)^{2}+1+\eta^{2}},$

and there are resonance points for the nodal length at $u=\pm 2$, because at
those points obviously $\frac{u^{4}}{8}(u^{2}-4)^{2}=0.$ More generally,
considering any two threshold levels $u_{1},u_{2}$ we obtain that

$\displaystyle\mathrm{Cov}(\mathcal{L}_{n}^{u_{1}},\mathcal{L}_{n}^{u_{2}})$
$\displaystyle=2a(u_{1})a(u_{2})-\frac{1}{4}(a(u_{1})+a(u_{2}))+\frac{3+\eta^{2}}{64}$
$\displaystyle=2\left(\frac{1}{4}u_{1}^{4}-u_{1}^{2}+\frac{1}{8}\right)\left(\frac{1}{4}u_{2}^{4}-u_{2}^{2}+\frac{1}{8}\right)-\frac{1}{4}\left(\frac{1}{4}u_{1}^{4}-u_{1}^{2}+\frac{1}{8}+\frac{1}{4}u_{2}^{4}-u_{2}^{2}+\frac{1}{8}\right)+\frac{3}{64}$
$\displaystyle=\frac{1}{8}u_{1}^{4}u_{2}^{4}-\frac{1}{2}u_{1}^{4}u_{2}^{2}-\frac{1}{2}u_{1}^{2}u_{2}^{4}+2u_{1}^{2}u_{2}^{2}+\frac{1+\eta^{2}}{64},$

so that the correlation is one if and only if

$1=\frac{(\frac{1}{8}u_{1}^{4}u_{2}^{4}-\frac{1}{2}u_{1}^{4}u_{2}^{2}-\frac{1}{2}u_{1}^{2}u_{2}^{4}+2u_{1}^{2}u_{2}^{2}+\frac{1+\eta^{2}}{64})^{2}}{(\frac{1}{8}u_{1}^{8}-u_{1}^{6}+2u_{1}^{4}+\frac{1+\eta^{2}}{64})(\frac{1}{8}u_{2}^{8}-u_{2}^{6}+2u_{2}^{4}+\frac{1+\eta^{2}}{64})},$

and hence,

$\displaystyle\left(\frac{1}{8}u_{1}^{4}u_{2}^{4}-\frac{1}{2}u_{1}^{4}u_{2}^{2}-\frac{1}{2}u_{1}^{2}u_{2}^{4}+2u_{1}^{2}u_{2}^{2}+\frac{1+\eta^{2}}{64}\right)^{2}$
$\displaystyle\;\;-\left(\frac{1}{8}u_{1}^{8}-u_{1}^{6}+2u_{1}^{4}+\frac{1+\eta^{2}}{64}\right)\left(\frac{1}{8}u_{2}^{8}-u_{2}^{6}+2u_{2}^{4}+\frac{1+\eta^{2}}{64}\right)$
$\displaystyle=-\frac{1+\eta^{2}}{512}\left(-u_{1}^{4}+4u_{1}^{2}+u_{2}^{4}-4u_{2}^{2}\right)^{2}=0.$

∎

##  6 The 3-dimensional Correlation between Nodal Area and Nodal Intersections

Proof of Theorem 3.27. We may write [MR21a, (8.99) and Lemma 8.1] as

$\displaystyle\mathcal{M}[4]\sim\sqrt{\frac{4\pi^{2}m}{3}}\frac{3\cdot 2\cdot
A}{16\cdot
8\cdot\mathcal{N}}\left[\frac{32}{15}+\frac{1}{\mathcal{N}/2}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n_{j}}^{+}}(|a_{\lambda}|^{2}-1)(|a_{\lambda^{\prime}}|^{2}-1)\left(3-9I(2,2)+14I(2,0)\right.\right.$
$\displaystyle\left.\left.-6\left\langle\frac{\lambda}{|\lambda|},\frac{\lambda^{\prime}}{|\lambda^{\prime}|}\right\rangle^{2}+12\left\langle\frac{\lambda}{|\lambda|},\frac{\lambda^{\prime}}{|\lambda^{\prime}|}\right\rangle
I(1,1)\right)\right],$

with

$I(k,k^{\prime})=I_{\lambda,\lambda^{\prime}}(k,k^{\prime}):=\frac{1}{A}\int_{\Sigma}\left\langle\frac{\lambda}{|\lambda|},n(\sigma)\right\rangle^{k}\left\langle\frac{\lambda^{\prime}}{|\lambda^{\prime}|},n(\sigma)\right\rangle^{k^{\prime}}d\sigma.$

For static surfaces we have (Lemma 6.2)

$I(2,0)=\frac{1}{3},\qquad\qquad
I(1,1)=\frac{1}{3}\langle\lambda,\lambda^{\prime}\rangle$

hence

$\displaystyle\mathcal{M}[4]\sim\sqrt{\frac{4\pi^{2}m}{3}}\frac{3\cdot
2}{16\cdot
8\cdot\mathcal{N}}\frac{A}{15}\left[32+\frac{5}{\mathcal{N}/2}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n_{j}}^{+}}(|a_{\lambda}|^{2}-1)(|a_{\lambda^{\prime}}|^{2}-1)\left(5-27I(2,2)\right.\right.$
$\displaystyle\left.\left.-6\left\langle\frac{\lambda}{|\lambda|},\frac{\lambda^{\prime}}{|\lambda^{\prime}|}\right\rangle^{2}\right)\right].$
(6.22)

Starting with the case of doubly static $\Sigma$, then

$I(2,2)=\frac{1}{15}(1+2\langle\lambda,\lambda^{\prime}\rangle^{2})$

so that

$\mathcal{M}[4]\sim\sqrt{\frac{4\pi^{2}m}{3}}\frac{3\cdot 2\cdot A}{16\cdot
8\cdot\mathcal{N}}\cdot\frac{16}{15}\left[2+\frac{1}{\mathcal{N}/2}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n_{j}}^{+}}(|a_{\lambda}|^{2}-1)(|a_{\lambda^{\prime}}|^{2}-1)\left(1-3\left\langle\frac{\lambda}{|\lambda|},\frac{\lambda^{\prime}}{|\lambda^{\prime}|}\right\rangle^{2}\right)\right].$

Comparing with [Cam19]

$\mathcal{A}[4]\sim\frac{\sqrt{m}}{5\sqrt{3}\mathcal{N}}\cdot
2\left[2+\frac{1}{\mathcal{N}/2}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n_{j}}^{+}}(|a_{\lambda}|^{2}-1)(|a_{\lambda^{\prime}}|^{2}-1)\left(1-3\left\langle\frac{\lambda}{|\lambda|},\frac{\lambda^{\prime}}{|\lambda^{\prime}|}\right\rangle^{2}\right)\right],$
(6.23)

we see that in this case the two expressions are the same up to a
multiplicative factor depending on the area $A$, and in particular
$\text{Corr}(\mathcal{A},\mathcal{M})\to 1$.

In the general case, from (6.23) and (6) we compute ${\rm
Cov\,}(\mathcal{A}[4],\mathcal{M}[4])$, where many terms cancel out, leaving

$\displaystyle{\rm
Cov\,}(\mathcal{A}[4],\mathcal{M}[4])\sim\frac{m}{\mathcal{N}^{2}}\frac{1}{5\sqrt{3}}\frac{2\pi}{\sqrt{3}}\frac{3\cdot
2}{16\cdot
8}\frac{A}{15}\cdot\frac{10}{\mathcal{N}^{2}}\sum_{\lambda,\lambda^{\prime}\in\Lambda}\left(5-21\left\langle\frac{\lambda}{|\lambda|},\frac{\lambda^{\prime}}{|\lambda^{\prime}|}\right\rangle^{2}+18\left\langle\frac{\lambda}{|\lambda|},\frac{\lambda^{\prime}}{|\lambda^{\prime}|}\right\rangle^{4}\right.$
$\displaystyle\left.-27I(2,2)+81I(2,2)\left\langle\frac{\lambda}{|\lambda|},\frac{\lambda^{\prime}}{|\lambda^{\prime}|}\right\rangle^{2}\right).$
(6.24)

We exchange the order of sums and integral, and apply Lemma 6.1 to simplify
(6) to (3.19).

Lastly, we take into account the expressions for ${\rm Var}(\mathcal{A})$
[BM19, Theorem 1.2] and ${\rm Var}(\mathcal{M})$ [MR21a, (1.19)] to conclude
the proof. ∎

###  6.1 Static surfaces

######  Lemma 6.1.

For any static surface $\Sigma$, we have as $m\to\infty$, $m\not\equiv
0,4,7\pmod{8}$

$\displaystyle\frac{1}{\mathcal{N}^{2}}\sum_{\lambda,\lambda^{\prime}\in\Lambda}\left\langle\frac{\lambda}{|\lambda|},\frac{\lambda^{\prime}}{|\lambda^{\prime}|}\right\rangle^{4}\to$
$\displaystyle\frac{1}{5},$
$\displaystyle\frac{1}{\mathcal{N}^{2}}\sum_{\lambda,\lambda^{\prime}\in\Lambda}I(2,2)\to$
$\displaystyle\frac{1}{9},$
$\displaystyle\frac{1}{\mathcal{N}^{2}}\sum_{\lambda,\lambda^{\prime}\in\Lambda}I(2,2)\left\langle\frac{\lambda}{|\lambda|},\frac{\lambda^{\prime}}{|\lambda^{\prime}|}\right\rangle^{2}\to$
$\displaystyle\frac{11}{15^{2}}.$

Proof. In each of the three expressions, we expand the summands and apply
[BM19, Lemma 2.5]. Then we simply recall that the normal $n$ is of norm one to
complete the calculations in the second and third expressions. ∎

######  Lemma 6.2.

A surface $\Sigma$ is static if and only if for every $i,j$ one has

$\int_{\Sigma}n_{i}n_{j}\,d\sigma=\frac{A}{3}\delta_{ij}.$

Proof. We write

$\displaystyle{\cal I}:=\iint_{\Sigma^{2}}\langle
n(\sigma),n(\sigma^{\prime})\rangle^{2}\,d\sigma
d\sigma^{\prime}=\sum_{i,j}\left(\int_{\Sigma}n_{i}(\sigma)n_{j}(\sigma)d\sigma\right)^{2}\geq\sum_{i}\left(\int_{\Sigma}n_{i}(\sigma)^{2}d\sigma\right)^{2}.$
(6.25)

The sum of the three integrals

$\sum_{i}\int_{\Sigma}n_{i}(\sigma)^{2}=A$

is fixed, so the sum of their squares is smallest when they are all equal:

$\displaystyle{\cal
I}\geq\sum_{i}\left(\frac{A}{3}\right)^{2}=\frac{A^{2}}{3}.$

All summands in (6.25) are non-negative: then ${\cal I}$ is minimised, i.e.
$\Sigma$ is static, if and only if for every $i,j$ one has

$\int_{\Sigma}n_{i}n_{j}\,d\sigma=\frac{A}{3}\delta_{ij}.$

∎

######  Remark 6.3.

Surfaces in Remark 3.26 that satisfy the further condition $\int
n_{1}^{4}d\sigma=|\Sigma|/5$ are doubly static, due to Lemma 6.4.

######  Lemma 6.4.

One has ${\cal I}_{4}=1/5$ if and only if $\Sigma$ is static and

$\int n_{i}n_{j}n_{\ell}n_{k}\,d\sigma=\begin{cases}A/5\qquad\text{if
}i=j=\ell=k,\\\ A/15\qquad\text{if }i,j,\ell,k\text{ are pairwise equal},\\\
0\qquad\text{otherwise.}\end{cases}$ (6.26)

Generic surfaces $\Sigma$ satisfy $1/5\leq{\cal I}_{4}\leq 1$ (and the maximum
is attained by surfaces contained in a plane). If $\Sigma$ is static, then

$\frac{1}{5}\leq{\cal I}_{4}\leq\frac{1}{3}.$

Proof. The upper bounds are due to ${\cal I}_{4}\leq{\cal I}$. For generic
$\Sigma$,

$\displaystyle A^{2}{\cal I}_{4}$ $\displaystyle:=\iint_{\Sigma^{2}}\langle
n(\sigma),n(\sigma^{\prime})\rangle^{4}\,d\sigma
d\sigma^{\prime}=\sum_{i,j,k,l}\left(\int_{\Sigma}n_{i}n_{j}n_{k}n_{l}d\sigma\right)^{2}$
$\displaystyle\geq
3\left[\sum_{i}\frac{1}{3}\left(\int_{\Sigma}n_{i}^{4}d\sigma\right)^{2}+\sum_{i<j}2\left(\int_{\Sigma}n_{i}^{2}n_{j}^{2}d\sigma\right)^{2}\right]$
$\displaystyle=3\left[\left(\frac{a_{ii}^{2}}{9}+\frac{a_{ii}^{2}}{9}+\frac{a_{ii}^{2}}{9}+\frac{a_{jj}^{2}}{9}+\frac{a_{jj}^{2}}{9}+\frac{a_{jj}^{2}}{9}+a_{ij}^{2}+a_{ij}^{2}+a_{ik}^{2}+a_{jk}^{2}\right)\right.$
$\displaystyle\left.+\left(\frac{a_{kk}^{2}}{9}+\frac{a_{kk}^{2}}{9}+\frac{a_{kk}^{2}}{9}+a_{ik}^{2}+a_{jk}^{2}\right)\right]$
(6.27)

with the notation $a_{ij}:=\int_{\Sigma}n_{i}^{2}n_{j}^{2}d\sigma$, where
$\\{i,j,k\\}$ is any permutation of $\\{1,2,3\\}$. On the RHS of (6.1) there
are two brackets: the former is the sum of squares of ten terms, the latter of
another five. The latter five sum up to

$\frac{a_{kk}}{3}+\frac{a_{kk}}{3}+\frac{a_{kk}}{3}+a_{ik}+a_{jk}=\int_{\Sigma}n_{k}^{2}\,d\sigma.$

Since the sum of all fifteen is simply $A$, the former ten have a total of
$A-\int_{\Sigma}n_{k}^{2}\,d\sigma$. With the same idea as in Lemma 6.2,

${\cal I}_{4}\geq\frac{3}{10}\left[3X^{2}-2XA+A^{2}\right]$

where

$X=X_{\Sigma}:=\max_{k=1,2,3}\left(\int_{\Sigma}n_{k}^{2}\,d\sigma\right).$

Now $\sum_{k=1,2,3}\int_{\Sigma}n_{k}^{2}\,d\sigma=A$ and the three summands
are non-negative, so that $X\geq A/3$. Moreover, if $A$ is fixed, then
$3x^{2}-2xA+A^{2}$ is an increasing function of $x$ for $x>A/3$. It follows
that ${\cal I}_{4}\geq 1/5$.
In addition, if this minimum is achieved, then necessarily
$\int_{\Sigma}n_{k}^{2}\,d\sigma=A/3$ for $k=1,2,3$, and (6.26) must hold. It
then also follows that

$\int_{\Sigma}n_{i}n_{j}\,d\sigma=\int_{\Sigma}n_{i}^{3}n_{j}\,d\sigma+\int_{\Sigma}n_{i}n_{j}^{3}\,d\sigma+\int_{\Sigma}n_{i}n_{j}n_{k}^{2}\,d\sigma=0,\qquad
i\neq j.$

∎

######  Lemma 6.5.

One has ${\cal I}_{k}=A^{2}/(k+1)$ if and only if $\Sigma$ is static and

$\int
n_{1}^{x}n_{2}^{y}n_{3}^{z}d\sigma=\begin{cases}A\frac{(x-1)!!(y-1)!!(z-1)!!}{(k+1)!!}\qquad\text{if
}x,y,z\text{ are all even,}\\\ 0\qquad\text{otherwise,}\end{cases}$ (6.28)

for all $x,y,z\geq 0$ satisfying $x+y+z=k$.
Generic surfaces $\Sigma$ satisfy $A^{2}/(k+1)\leq{\cal I}_{k}\leq A^{2}$ (and
the maximum is attained by surfaces contained in a plane). If $\Sigma$ is
static, then

$\frac{A^{2}}{k+1}\leq{\cal I}_{k}\leq\frac{A^{2}}{3}.$

Proof. The upper bounds are due to ${\cal I}_{k}\leq{\cal I}$. For generic
$\Sigma$,

$\displaystyle{\cal I}_{k}$ $\displaystyle:=\iint_{\Sigma^{2}}\langle
n(\sigma),n(\sigma^{\prime})\rangle^{k}\,d\sigma
d\sigma^{\prime}=\sum_{x+y+z=k}c_{x,y,z}\left(\int_{\Sigma}n_{1}^{x}n_{2}^{y}n_{3}^{z}d\sigma\right)^{2}$
$\displaystyle\geq\sum_{\begin{subarray}{c}x+y+z=k\\\ x,y,z\text{
even}\end{subarray}}c_{x,y,z}\left(\int_{\Sigma}n_{1}^{x}n_{2}^{y}n_{3}^{z}d\sigma\right)^{2}$
$\displaystyle=(k-1)!!\left[\sum_{\begin{subarray}{c}x+y=k\\\ x,y\\\ \text{
even}\end{subarray}}\frac{c_{x,y,0}}{(k-1)!!}\left(\int_{\Sigma}n_{1}^{x}n_{2}^{y}d\sigma\right)^{2}+\sum_{\begin{subarray}{c}x+y+z=k\\\
x,y,z\text{ even}\\\ z\geq
2\end{subarray}}\frac{c_{x,y,z}}{(k-1)!!}\left(\int_{\Sigma}n_{1}^{x}n_{2}^{y}n_{3}^{z}d\sigma\right)^{2}\right],$

with

$c_{x,y,z}:=\binom{k}{x\ y\ z}$

the trinomial coefficient. In the RHS, we replace each summand with

$c^{\prime}_{x,y,z}:=\binom{k}{x\ y\
z}\frac{(x-1)!!^{2}(y-1)!!^{2}(z-1)!!^{2}}{(k-1)!!}$

copies of

$\left(\int_{\Sigma}\frac{n_{1}^{x}n_{2}^{y}n_{3}^{z}}{(x-1)!!(y-1)!!(z-1)!!}d\sigma\right)^{2}.$

The sum of all integrals is $A$, and there are a total of

$\sum_{\begin{subarray}{c}x+y+z=k\\\ x,y,z\text{
even}\end{subarray}}c^{\prime}_{x,y,z}=(k+1)!!$

integrals. The second sum contains a third of the terms, and they sum up to
$\int_{\Sigma}n_{3}^{2}\,d\sigma$. With the same idea as in Lemma 6.2,

${\cal
I}_{k}\geq(k-1)!!\left[\frac{(A-X)^{2}}{2(k+1)!!/3}+\frac{X^{2}}{(k+1)!!/3}\right]=\frac{3}{2(k+1)}\left[3X^{2}-2XA+A^{2}\right]$

where

$X=X_{\Sigma}:=\max_{k=1,2,3}\left(\int_{\Sigma}n_{k}^{2}\,d\sigma\right).$

Now $\sum_{k=1,2,3}\int_{\Sigma}n_{k}^{2}\,d\sigma=A$ and the three summands
are non-negative, so that $X\geq A/3$. Moreover, if $A$ is fixed, then
$3x^{2}-2xA+A^{2}$ is an increasing function of $x$ for $x>A/3$. It follows
that ${\cal I}_{k}\geq A^{2}/(k+1)$.
In addition, if this minimum is achieved, then necessarily
$\int_{\Sigma}n_{k}^{2}\,d\sigma=A/3$ for $k=1,2,3$, and (6.28) must hold. It
then also follows that

$\int_{\Sigma}n_{i}n_{j}\,d\sigma=\int_{\Sigma}n_{i}n_{j}(n_{1}^{2}+n_{2}^{2}+n_{3}^{2})^{(k-2)}\,d\sigma=0,\qquad
i\neq j.$

∎

##  Appendix A Proofs of auxiliary lemmas

Proof of Lemma 3.4. Let ${\cal C}$ be a curve parameterized by arc-lenght, and
$\nu$ be a probability measure on ${\mathbb{S}}^{1}$. We can assume that $\nu$
is invariant w.r.t. rotations by $\pi/2$, indeed the curve is static if and
only if $4\mathcal{B}_{\mathcal{C}}(\nu)-L^{2}=0$ holds for $\nu$ the uniform
measure. Via some manipulations, we rewrite

$\mathcal{B}_{{\cal C}}(\nu)=\frac{L^{2}}{8}(1+2{\cal
I}_{2})+\frac{L^{2}}{8}\widehat{\nu}(4)(1-2{\cal I}_{2}^{\perp});$ (A.29)

with $\widehat{\nu}(k)=\int_{{\mathbb{S}}^{1}}z^{-k}d\nu(z)$, for any
$k\in\mathbb{Z}$, Fourier transform of $\nu$; note that $\widehat{\nu}(4)$ is
real if $\nu$ is invariant under the transformations $z\to\bar{z}$ and $z\to
i\cdot z$; and

$\displaystyle{\cal I}_{2}$ $\displaystyle={\cal I}_{2,{\cal
C}}=\frac{1}{L^{2}}\int_{0}^{L}\int_{0}^{L}\langle{\dot{\gamma}}(t),{\dot{\gamma}}(u)\rangle^{2}dtdu,$
$\displaystyle{\cal I}_{2}^{\perp}$ $\displaystyle={\cal I}_{2,{\cal
C}}^{\perp}=\frac{1}{L^{2}}\int_{0}^{L}\int_{0}^{L}\langle{\dot{\gamma}}(t),({\dot{\gamma}}_{2}(u),{\dot{\gamma}}_{1}(u))\rangle^{2}dtdu.$

Then ${\cal C}$ is static if and only if ${\cal I}_{2}={\cal
I}_{2}^{\perp}=1/2$: indeed if ${\cal I}_{2}={\cal I}_{2}^{\perp}=1/2$ then
from (A.29) it holds that $4\mathcal{B}_{\mathcal{C}}(\nu)-L^{2}=0$ in
particular for $\nu$ the uniform measure. On the other hand, we may rearrange

${\cal
I}_{2}=\sum_{i,j=1,2}\left(\frac{1}{L}\int_{0}^{L}{\dot{\gamma}}_{i}(t){\dot{\gamma}}_{j}(t)dt\right)^{2}\geq\sum_{i=1,2}\left(\frac{1}{L}\int_{0}^{L}{\dot{\gamma}}_{i}(t)^{2}dt\right)^{2}.$

Since
$\int_{0}^{L}{\dot{\gamma}}_{1}(t)^{2}dt+\int_{0}^{L}{\dot{\gamma}}_{2}(t)^{2}dt=L$,
then ${\cal I}_{2}\geq 1/2$ with equality if and only if
$\int_{0}^{L}{\dot{\gamma}}_{1}(t)^{2}dt=\int_{0}^{L}{\dot{\gamma}}_{2}(t)^{2}dt=L/2$
and $\int_{0}^{L}{\dot{\gamma}}_{1}(t){\dot{\gamma}}_{2}(t)dt=0$. These
conditions also ensure that ${\cal I}_{2}^{\perp}=1/2$, and the proof of this
lemma is complete. ∎

Proof of Lemma 3.9. Similarly to the proof of Lemma 3.4, we have

${\cal
I}_{4}=\sum_{i=0}^{4}\binom{4}{i}\left[\frac{1}{L}\int_{0}^{L}{\dot{\gamma}}_{1}(t)^{i}{\dot{\gamma}}_{2}(t)^{4-i}dt\right]^{2}\geq\sum_{i=1,2}\left(\frac{1}{L}\int_{0}^{L}{\dot{\gamma}}_{i}(t)^{4}dt\right)^{2}+6\left(\frac{1}{L}\int_{0}^{L}{\dot{\gamma}}_{1}(t)^{2}{\dot{\gamma}}_{2}(t)^{2}dt\right)^{2}.$
(A.30)

Clearly
$\int_{0}^{L}{\dot{\gamma}}_{1}(t)^{4}dt+\int_{0}^{L}{\dot{\gamma}}_{2}(t)^{4}dt+2\int_{0}^{L}{\dot{\gamma}}_{1}(t)^{2}{\dot{\gamma}}_{2}(t)^{2}dt=L$,
hence ${\cal I}_{4}\geq 3/8$ with equality iff

$\frac{1}{L}\int_{0}^{L}{\dot{\gamma}}_{i}(t){\dot{\gamma}}_{j}(t){\dot{\gamma}}_{k}(t){\dot{\gamma}}_{l}(t)dt=\begin{cases}3/8\qquad\text{if
}i=j=k=l,\\\ 1/8\qquad\text{if }i=j\neq k=l,\\\ 0\qquad\text{\;\;\,
otherwise.}\end{cases}$ (A.31)

If (A.31) holds true, this clearly means that $A=1/8$, and $B=0$, and moreover
${\cal C}$ is static due to Lemma 3.4 (e.g.
$\int_{0}^{L}{\dot{\gamma}}_{1}(t)^{2}dt=\int_{0}^{L}{\dot{\gamma}}_{1}(t)^{4}dt+\int_{0}^{L}{\dot{\gamma}}_{1}(t)^{2}{\dot{\gamma}}_{2}(t)^{2}dt=3L/8+L/8=L/2$).
Vice versa, assume that ${\cal C}$ is static, $A=1/8$, and $B=0$. By staticity
we have
$\int_{0}^{L}{\dot{\gamma}}_{1}(t)^{4}dt=\int_{0}^{L}{\dot{\gamma}}_{2}(t)^{4}dt$.
Using $A=1/8$ and the fact that ${\cal C}$ is unit speed we get also the first
case of (A.31). As for the third one, it suffices to point out that
$\int_{0}^{L}{\dot{\gamma}}_{1}(t){\dot{\gamma}}_{2}(t)^{3}dt=-B=0$. ∎

Proof of Lemma 3.11. To construct a family of doubly static curves, we adapt
[RW18, Appendix G]. The condition is ${\cal I}_{4,{\cal C}}=3/8$. Bearing in
mind [RW18, (G.4)], we impose

$\sum_{j=0}^{k-1}\cos\left({\dot{\gamma}}(t)-\phi(u)+j\cdot\frac{2\pi}{k}\right)^{4}=\frac{3k}{8}$

where ${\dot{\gamma}}(u)=\exp(i\phi(u))$. Due to the identity

$\cos(x)^{4}=\frac{3}{8}+\frac{\cos(2x)}{2}+\frac{\cos(4x)}{8},$

it suffices to impose

$\frac{k}{\gcd(2,k)}\geq 2\qquad\text{and}\qquad\frac{k}{\gcd(4,k)}\geq 2,$

i.e. $k=3$ or $k\geq 5$. For $k=4$ the curve is static, but not necessarily
doubly static (for ${\cal C}$ to be static we only need $k/\gcd(2,k)\geq 2$).
∎

Proof of Lemma 3.6. To prove (3.14), we rewrite ${\cal I}_{4}$ as

${\cal
I}_{4}=\sum_{i=0}^{4}\binom{4}{i}\left[\frac{1}{L}\int_{0}^{L}{\dot{\gamma}}_{1}(t)^{i}{\dot{\gamma}}_{2}(t)^{4-i}dt\right]^{2}.$

The terms for $i=1,2$ are simply $4B^{2}$ and $6A^{2}$. Since ${\cal C}$ is
static, in light of Lemma 3.4 the term for $i=3$ is $4(-B)^{2}$, and the terms
for $i=0,4$ are each equal to $(1/2-A)^{2}$. Rearranging proves (3.14).
For static curves $(\int{\dot{\gamma}}_{i}^{4}dt/L)+A=1/2$, with $i=1,2$.
Moreover, by Cauchy-Schwartz, $A$ is always the smaller of the two summands ,
hence $0<A<1/4$ (extrema excluded else ${\cal C}$ would be a straight line
segment). Rearranging, $-1<{\cal I}_{4}^{\prime}<1$.
The inequality $3/8\leq{\cal I}_{4}$ is shown in Lemma 3.9. For the upper
bound, by Cauchy-Schwartz ${\cal I}_{4}<{\cal I}_{2}$ (as defined in Lemma
3.4), and ${\cal I}_{2}=1/2$ due to staticity (extremum excluded again else
${\cal C}$ would be a straight line segment). Lastly, we combine ${\cal
I}_{4}<1/2$ and (3.14) to find $B^{2}<A(1-4A)/4$. ∎

To state the technical results in Lemma A.1, we introduce the following
notation

$\displaystyle{\cal I}$
$\displaystyle:=\frac{1}{L}\int_{0}^{L}\Big{[}\frac{3+\eta}{8}{\dot{\gamma}}_{1}^{4}(t)+\frac{3+\eta}{8}{\dot{\gamma}}_{2}^{4}(t)+6\frac{1-\eta}{8}{\dot{\gamma}}_{1}^{2}(t){\dot{\gamma}}_{2}^{2}(t)\Big{]}dt,$
$\displaystyle{\cal J}$
$\displaystyle:=\frac{1}{L}\int_{0}^{L}\Big{[}\Big{(}\frac{3+\eta}{8}\Big{)}^{2}{\dot{\gamma}}_{1}^{4}(t)+\Big{(}\frac{1-\eta}{8}\Big{)}^{2}{\dot{\gamma}}_{1}^{4}(t)+4\Big{(}\frac{3+\eta}{8}\Big{)}\Big{(}\frac{1-\eta}{8}\Big{)}{\dot{\gamma}}_{1}^{2}(t){\dot{\gamma}}_{2}^{2}(t)$
$\displaystyle\hskip
45.52458pt+\Big{(}\frac{1-\eta}{8}\Big{)}^{2}{\dot{\gamma}}_{2}^{4}(t)+\Big{(}\frac{3+\eta}{8}\Big{)}^{2}{\dot{\gamma}}_{2}^{4}(t)+8\Big{(}\frac{1-\eta}{8}\Big{)}^{2}{\dot{\gamma}}_{1}^{2}(t){\dot{\gamma}}_{2}^{2}(t)\Big{]}dt$

and we observe that

$\displaystyle{\cal I}$ $\displaystyle=\frac{3}{8}+\frac{\eta}{8}{\cal
I}^{\prime}_{4},\hskip 28.45274pt{\cal
J}=5\frac{2}{8^{2}}+\frac{4}{8^{2}}\eta{\cal
I}^{\prime}_{4}+\frac{2}{8^{2}}\eta^{2}.$

######  Lemma A.1.

We have

$\displaystyle\frac{1}{\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}I_{\lambda,\lambda}(2,0)=\frac{1}{2},$
(A.32)
$\displaystyle\frac{1}{\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}I_{\lambda,\lambda}(2,2)={\mathcal{I}},$
(A.33)
$\displaystyle\frac{1}{\mathcal{N}_{n}^{2}}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}}I_{\lambda,\lambda^{\prime}}(2,2)=\frac{1}{4}.$
(A.34)
$\displaystyle\frac{1}{n^{2}{\mathcal{N}}_{n}^{2}}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}}\langle\lambda,\lambda^{\prime}\rangle^{2}=\frac{1}{2},$
(A.35)
$\displaystyle\frac{1}{n^{2}{\mathcal{N}}_{n}^{2}}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}}\langle\lambda,\lambda^{\prime}\rangle^{2}I_{\lambda,\lambda^{\prime}}(2,2)={\cal
J}.$ (A.36)

Proof. Equation (A.32) follows immediately by observing that

$\displaystyle\frac{1}{\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}I_{\lambda,\lambda}(2,0)$
$\displaystyle=\frac{1}{n\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}\frac{1}{L}\int_{0}^{L}[\lambda_{1}{\dot{\gamma}}_{1}(t)+\lambda_{2}{\dot{\gamma}}_{2}(t)]^{2}dt$
$\displaystyle=\frac{1}{n\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}\frac{1}{L}\int_{0}^{L}[\lambda_{1}^{2}{\dot{\gamma}}_{1}^{2}(t)+\lambda_{2}^{2}{\dot{\gamma}}_{2}^{2}(t)+2\lambda_{1}\lambda_{2}{\dot{\gamma}}_{1}(t){\dot{\gamma}}_{2}(t)]dt$
$\displaystyle=\frac{1}{L}\int_{0}^{L}\Big{[}\frac{1}{2}{\dot{\gamma}}_{1}^{2}(t)+\frac{1}{2}{\dot{\gamma}}_{2}^{2}(t)\Big{]}dt$
$\displaystyle=\frac{1}{2}\frac{1}{L}\int_{0}^{L}\langle{\dot{\gamma}}(t),{\dot{\gamma}}(t)\rangle\,dt=\frac{1}{2},$

where we used the relations

$\frac{1}{n\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}\lambda_{i}^{2}=\frac{1}{2},\;\;i=1,2,\hskip
14.22636pt{\rm and}\hskip
14.22636pt\frac{1}{n\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}\lambda_{1}\lambda_{2}=0.$
(A.37)

The proof of (A.33) is similar, in fact

$\displaystyle\frac{1}{\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}I_{\lambda,\lambda}(2,2)$
$\displaystyle=\frac{1}{n^{2}\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}\frac{1}{L}\int_{0}^{L}[\lambda_{1}{\dot{\gamma}}_{1}(t)+\lambda_{2}{\dot{\gamma}}_{2}(t)]^{4}dt$
$\displaystyle=\frac{1}{n^{2}\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}\frac{1}{L}\int_{0}^{L}[\lambda_{1}^{4}{\dot{\gamma}}_{1}^{4}(t)+\lambda_{2}^{4}{\dot{\gamma}}_{2}^{4}(t)+6\lambda_{1}^{2}\lambda_{2}^{2}{\dot{\gamma}}_{1}^{2}(t){\dot{\gamma}}_{2}^{2}(t)]dt$
$\displaystyle={\cal I},$

since

$\frac{1}{n^{2}\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}\lambda_{i}^{4}=\frac{3+\eta}{8},\;\;i=1,2,\hskip
14.22636pt{\rm and}\hskip
14.22636pt\frac{1}{n^{2}\mathcal{N}_{n}}\sum_{\lambda\in\Lambda_{n}}\lambda_{1}^{2}\lambda_{2}^{2}=\frac{1-\eta}{8}.$

To prove (A.34), we observe that

$\displaystyle\frac{1}{\mathcal{N}_{n}^{2}}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}}I_{\lambda,\lambda^{\prime}}(2,2)$
$\displaystyle=\frac{1}{n^{2}\mathcal{N}_{n}^{2}}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}}\frac{1}{L}\int_{0}^{L}[\lambda_{1}{\dot{\gamma}}_{1}(t)+\lambda_{2}{\dot{\gamma}}_{2}(t)]^{2}[\lambda^{\prime}_{1}{\dot{\gamma}}_{1}(t)+\lambda^{\prime}_{2}{\dot{\gamma}}_{2}(t)]^{2}dt$
$\displaystyle=\frac{1}{n^{2}\mathcal{N}_{n}^{2}}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}}\frac{1}{L}\int_{0}^{L}[\lambda_{1}^{2}{\dot{\gamma}}_{1}^{2}(t)+\lambda_{2}^{2}{\dot{\gamma}}_{2}^{2}(t)+2\lambda_{1}\lambda_{2}{\dot{\gamma}}_{1}(t){\dot{\gamma}}_{2}(t)]$
$\displaystyle\hskip
96.73918pt\times[(\lambda^{\prime}_{1})^{2}{\dot{\gamma}}_{1}^{2}(t)+(\lambda^{\prime}_{2})^{2}{\dot{\gamma}}_{2}^{2}(t)+2\lambda^{\prime}_{1}\lambda^{\prime}_{2}{\dot{\gamma}}_{1}(t){\dot{\gamma}}_{2}(t)]dt$
$\displaystyle=\frac{1}{n^{2}\mathcal{N}_{n}^{2}}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}}\frac{1}{L}\int_{0}^{L}[\lambda_{1}^{2}(\lambda_{1}^{\prime})^{2}{\dot{\gamma}}_{1}^{4}(t)+\lambda_{1}^{2}(\lambda_{2}^{\prime})^{2}{\dot{\gamma}}_{1}^{2}(t){\dot{\gamma}}_{2}^{2}(t)$
$\displaystyle\hskip
102.43008pt+\lambda_{2}^{2}(\lambda_{1}^{\prime})^{2}{\dot{\gamma}}_{1}^{2}(t){\dot{\gamma}}_{2}^{2}(t)+\lambda_{2}^{2}(\lambda^{\prime}_{2})^{2}{\dot{\gamma}}_{2}^{4}(t)]dt$
$\displaystyle=\frac{1}{L}\int_{0}^{L}\Big{[}\frac{1}{4}{\dot{\gamma}}_{1}^{4}(t)+\frac{1}{2}{\dot{\gamma}}_{1}^{2}(t){\dot{\gamma}}_{2}^{2}(t)+\frac{1}{4}{\dot{\gamma}}_{2}^{4}(t)\Big{]}dt$
$\displaystyle=\frac{1}{4}\frac{1}{L}\int_{0}^{L}\langle{\dot{\gamma}}(t),{\dot{\gamma}}(t)\rangle^{2}dt=\frac{1}{4}.$

Equation (A.35) is an immediate consequence of (A.37). Equation (A.36) follows
from

$\displaystyle\frac{1}{n^{2}\mathcal{N}_{n}^{2}}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}}\langle\lambda,\lambda^{\prime}\rangle^{2}I_{\lambda,\lambda^{\prime}}(2,2)$
$\displaystyle=\frac{1}{n^{4}\mathcal{N}_{n}^{2}}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}}(\lambda_{1}\lambda^{\prime}_{1}+\lambda_{2}\lambda_{2}^{\prime})^{2}\frac{1}{L}\int_{0}^{L}[\lambda_{1}{\dot{\gamma}}_{1}(t)+\lambda_{2}{\dot{\gamma}}_{2}(t)]^{2}[\lambda^{\prime}_{1}{\dot{\gamma}}_{1}(t)+\lambda^{\prime}_{2}{\dot{\gamma}}_{2}(t)]^{2}dt$
$\displaystyle=\frac{1}{n^{4}\mathcal{N}_{n}^{2}}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}}\\{\lambda^{2}_{1}(\lambda^{\prime}_{1})^{2}+\lambda_{2}^{2}(\lambda_{2}^{\prime})^{2}+2\lambda_{1}\lambda_{2}\lambda_{1}^{\prime}\lambda_{2}^{\prime}\\}$
$\displaystyle\;\;\times\frac{1}{L}\int_{0}^{L}[\lambda_{1}^{2}{\dot{\gamma}}_{1}^{2}(t)+\lambda_{2}^{2}{\dot{\gamma}}_{2}^{2}(t)+2\lambda_{1}\lambda_{2}{\dot{\gamma}}_{1}(t){\dot{\gamma}}_{2}(t)]\,[(\lambda^{\prime}_{1})^{2}{\dot{\gamma}}_{1}^{2}(t)+(\lambda^{\prime}_{2})^{2}{\dot{\gamma}}_{2}^{2}(t)+2\lambda^{\prime}_{1}\lambda^{\prime}_{2}{\dot{\gamma}}_{1}(t){\dot{\gamma}}_{2}(t)]dt$
$\displaystyle=\frac{1}{n^{4}\mathcal{N}_{n}^{2}}\sum_{\lambda,\lambda^{\prime}\in\Lambda_{n}}\frac{1}{L}\int_{0}^{L}[\lambda_{1}^{4}(\lambda_{1}^{\prime})^{4}{\dot{\gamma}}_{1}^{4}(t)+\lambda_{1}^{2}(\lambda_{1}^{\prime})^{2}\lambda_{2}^{2}(\lambda_{2}^{\prime})^{2}{\dot{\gamma}}_{1}^{4}(t)+\lambda_{1}^{4}(\lambda_{1}^{\prime})^{2}(\lambda_{2}^{\prime})^{2}{\dot{\gamma}}_{1}^{2}(t){\dot{\gamma}}_{2}^{2}(t)$
$\displaystyle\hskip
85.35826pt+\lambda_{1}^{2}\lambda_{2}^{2}(\lambda_{2}^{\prime})^{4}{\dot{\gamma}}_{1}^{2}(t){\dot{\gamma}}_{2}^{2}(t)+\lambda_{1}^{2}\lambda_{2}^{2}(\lambda_{1}^{\prime})^{4}{\dot{\gamma}}_{1}^{2}(t){\dot{\gamma}}_{2}^{2}(t)+(\lambda^{\prime}_{2})^{2}\lambda_{2}^{4}(\lambda_{1}^{\prime})^{2}{\dot{\gamma}}_{1}^{2}(t){\dot{\gamma}}_{2}^{2}(t)$
$\displaystyle\hskip
85.35826pt+\lambda_{1}^{2}(\lambda_{1}^{\prime})^{2}\lambda_{2}^{2}(\lambda_{2}^{\prime})^{2}{\dot{\gamma}}_{2}^{4}(t)+\lambda_{2}^{4}(\lambda_{2}^{\prime})^{4}{\dot{\gamma}}_{2}^{4}(t)+8\lambda_{1}^{2}\lambda_{2}^{2}(\lambda_{1}^{\prime})^{2}(\lambda_{2}^{\prime})^{2}{\dot{\gamma}}_{1}^{2}(t){\dot{\gamma}}_{2}^{2}(t)]dt$
$\displaystyle={\mathcal{J}}.$

∎

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