Ars Inveniendi Analytica (2023), Paper No. 7, 63 pp.

DOI 10.15781/63wy-ad98

ISSN: 2769-8505

00footnotetext: © Guillaume Dubach, Pierre Germain, and Benjamin Harrop-
Griffiths00footnotetext:  Licensed under a Creative Commons Attribution
License (CC-BY).

On the derivation of the homogeneous kinetic wave equation for a nonlinear
random matrix model

Guillaume Dubach

École Polytechnique

Pierre Germain

Imperial College London

Benjamin Harrop-Griffiths

Georgetown University

Communicated by Zaher Hani

Abstract. We consider a nonlinear system of ODEs, where the underlying linear
dynamics are determined by a Hermitian random matrix ensemble. We prove that
the leading order dynamics in the weakly nonlinear, infinite volume limit are
determined by a solution to the corresponding kinetic wave equation on a non-
trivial timescale. Our proof relies on estimates for Haar-distributed unitary
matrices obtained from Weingarten calculus, which may be of independent
interest.

Keywords. Wave turbulence, wave kinetic equation, random matrix, Weingarten
calculus.

##  1\. Introduction

###  1.1. Wave turbulence, old and new

When considering a dynamical system involving a large number of nonlinearly
interacting waves, it is often preferable to consider ensembles of solutions
instead of individual trajectories. This statistical mechanics approach to the
dynamics is referred to as the theory of _wave turbulence_.

A central question of this theory, which lies at the forefront of current
research, is to rigorously derive a _kinetic wave equation_ for the non-
equilibrium dynamics of the underlying microscopic system. Such kinetic wave
equations first appeared in work of Peierls [26] and Hasselmann [20, 21], and
have seen renewed interest after Zakharov, L’vov, and Falkovich [33]
discovered power-law type stationary solutions analogous to the Kolmogorov
spectra of hydrodynamic turbulence [22, 23].

In recent years, significant progress has been made on the problem of
rigorously deriving the kinetic wave equation in the context of nonlinear
dispersive equations. The most popular model has been the nonlinear
Schrödinger equation

(1.1) $i\partial_{t}u+\Delta u=\mu^{2}|u|^{2}u,$

set on a torus, with spatially homogeneous random initial data. For this
model, a full derivation of the kinetic limit was obtained in [15, 14],
following earlier results in [5, 12, 13, 11], and formal derivations in [16,
17, 18]. Other models have also been considered: The foundational work [24]
dealt with discrete NLS at statistical equilibrium, [30] focused on
multiplicative white noise forcing (both reaching the kinetic time scale), and
[1] treated the case of spatially inhomogeneous data.

All of the aforementioned rigorous derivations consider PDEs whose linear
dynamics enjoy strong algebraic structures, e.g., the eigenvalues and
eigenvectors of the Laplacian on a Euclidean torus. In the present article we
consider the opposite case, where the linear dynamics are so disordered that
they are best understood in probabilistic terms and modeled as a random
ensemble. We prove that the leading order dynamics (in a suitable scaling
regime) are once again determined by the solution to a corresponding kinetic
wave equation. This testifies to the universality of the kinetic description
of wave turbulence.

###  1.2. The model and its relevance

We consider solutions $u\colon\varmathbb{R}\to\varmathbb{C}^{2N+1}$ of the ODE

(1.2) $i\frac{d}{dt}u-Hu=\mu^{2}\bigl{(}u\odot\bar{u}\odot u\bigr{)},$

where $H$ is a $(2N+1)\times(2N+1)$ Hermitian random matrix, $\mu>0$ is a
constant, and

$(u\odot v)_{j}:=u_{j}v_{j}$

defines the Hadamard product. Here, and throughout, we index
$\varmathbb{C}^{2N+1}$ by $j\in\llbracket-N,N\rrbracket$, where $\llbracket
a,b\rrbracket=[a,b]\cap\varmathbb{Z}$.

The equation (1.2) is readily seen to be globally well-posed for all Hermitian
matrices $H$ by combining Picard’s Theorem with conservation of the mass,

$\mathcal{M}(t):=\|u(t)\|_{\ell^{2}}^{2}=\sum_{j=-N}^{N}|u_{j}(t)|^{2}.$

We remark that (1.2) has one further conservation law: the Hamiltonian,

$\mathcal{H}(t):=\langle u,Hu\rangle+\tfrac{\mu^{2}}{2}\|u\|_{\ell^{4}}^{4},$

where $\langle u,v\rangle=\sum_{j=-N}^{N}\bar{u}_{j}v_{j}$ and
$\|u\|_{\ell^{p}}^{p}=\sum_{j=-N}^{N}|u_{j}|^{p}$.

Our motivation for investigating the model (1.2) is twofold. First, the
Bohigas–Giannoni–Schmit conjecture [3] suggests that the spectral properties
of quantum systems with chaotic classical limits should be well-approximated,
in a statistical sense, by random matrix ensembles. In this way, one may view
(1.2) as a toy model for (1.1) posed on a domain with a chaotic geodesic flow
or, more generally, for a nonlinear interaction in a strongly mixing medium.
Second, the derivation of the kinetic wave equation appears to require a very
fine spectral resolution of the associated linear equation. This is currently
only possible when the linear flow is integrable, and as discussed above,
deterministic flows have already seen significant progress. In this paper, we
turn to an example with random dynamics.

We subsequently take $H=(h_{jk})$ to be distributed according to the _Gaussian
Unitary Ensemble_ (GUE), defined as follows: The $h_{jk}$ are independent for
$-N\leqslant j\leqslant k\leqslant N$, and the rescaled entry
$\sqrt{2N+1}\,h_{jk}$ is either a complex standard normal random variable when
$j<k$, or a real standard normal random variable when $j=k$. GUE is an
integrable ensemble in the sense of random matrix theory. Moreover, its
eigenvectors are independent from its eigenvalues and form a uniformly
distributed orthonormal basis. This allows one to study them using the
Weingarten calculus [7], which will prove to be an indispensable ingredient in
the ensuing analysis.

We expect that the results presented below can be generalized to other
Gaussian random matrix ensembles using the corresponding Weingarten calculus
for their eigenvectors [10, 7]. Moreover, the universality paradigm in random
matrix theory suggests one should be able to extend our results from the
archetypal Gaussian ensembles to more general random matrix ensembles, e.g.,
Wigner ensembles.

In addition to the space-homogeneous, delocalized regime we consider here, we
believe the model (1.2) may be well-suited to a rigorous investigation of the
effects of localization, or space-dependence. In [25], a heuristic derivation
of a kinetic wave equation was proposed, starting from the nonlinear
Schrödinger equation on the lattice with a random potential. In the diffusion
limit, a porous medium equation was derived, the degenerate diffusion stemming
from Anderson localization. In the Physics literature, understanding the
combined effects of interaction (nonlinearity) and a disordered medium has
received much attention. A kinetic description seems appropriate in the case
of weak disorder, see [32, 29, 28] for recent publications on this subject.
While a fully rigorous approach to these derivations seems out of reach, we
believe that considering band matrices in (1.2) could provide an entry point
to addressing questions about the effects of localization.

###  1.3. The kinetic limit in the weakly turbulent regime

Rather than working directly with the formulation (1.2), we instead
diagonalize the matrix $H$, taking $\Lambda=\operatorname{diag}(\lambda)$ with

$\lambda=(\lambda_{-N},\dots,\lambda_{N})\in\varmathbb{R}_{\geqslant}^{2N+1}:=\bigl{\\{}\lambda\in\varmathbb{R}^{2n+1}:\lambda_{-N}\leqslant\dots\leqslant\lambda_{N}\bigr{\\}},$

and $\Psi\in\varmathbb U(2N+1)$ so that

$H=\Psi\Lambda\Psi^{*}.$

When $H$ is a GUE matrix, the vector of eigenvalues $\lambda$ and the matrix
of eigenvectors $\Psi$ are independent. The vector of eigenvalues $\lambda$ is
distributed with joint density

(1.3) $\frac{1}{Z_{N}}\prod\limits_{-N\leqslant j<k\leqslant
N}|\lambda_{k}-\lambda_{j}|^{2}e^{-\frac{1}{2}(2N+1)\,|\lambda|^{2}}\,d\lambda,$

where the partition function $Z_{N}>0$ is a normalizing constant and
$d\lambda$ is the Lebesgue measure on $\varmathbb{R}_{\geqslant}^{2N+1}$. The
matrix of eigenvectors $\Psi$ is distributed according to the _Circular
Unitary Ensemble_ (CUE), i.e., the Haar probability measure on $\varmathbb
U(2N+1)$.

We now introduce

(1.4) $a:=\sqrt{N}\Psi^{*}u,$

and write the equation (1.2) in the form

(1.5) $i\frac{d}{dt}a-\Lambda a=\frac{\mu^{2}}{N}\Psi^{*}\left(\Psi
a\odot\overline{\Psi a}\odot\Psi a\right).$

We consider initial data for (1.5) given by

(1.6) $a_{k}(t=0)=A(\tfrac{k}{N}),$

where $A\in C^{1}[-1,1]$ is a continuously differentiable, complex-valued
function. We note that with this choice of data:

  * •

The vector $u(t=0)$ has size $\sim 1$ in $\ell^{2}$;

  * •

Its entries have typical size $N^{-1/2}$, or equivalently $a(t=0)$ has entries
of typical size $1$.

It is helpful to keep in mind these orders of magnitude, which will also hold
for positive time.

Taking $a(t)$ to be the solution of (1.5) with initial data (1.6), we consider
the regime where:

  * •

$N\to\infty$ (infinite volume limit);

  * •

$\mu/\sqrt{N}\to 0$ (weakly nonlinear limit).

A heuristic derivation, whose validity we will investigate below, shows that
in this regime

(1.7) $\varmathbb
E|a_{k}(t)|^{2}\sim\rho(\tfrac{t}{T_{\mathrm{kin}}},\tfrac{k}{N}),\quad\text{where}\quad
T_{\mathrm{kin}}:=\frac{N^{2}}{\mu^{4}}$

and $\rho\colon[0,\infty)\times[-1,1]\to[0,\infty)$ is a solution of the
kinetic wave equation

(1.8) $\begin{cases}\partial_{t}\rho(t,k)=\mathbf{C}[\rho(t)](k),\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\ \rho(0,k)=|A(k)|^{2}.\end{cases}$

Here, the collision operator $\mathbf{C}$ is given by

$\mathbf{C}[\rho](k):=\tfrac{\pi}{2}\int_{[-1,1]^{3}}\delta\bigl{(}\Theta(k,\ell,m,n)\bigr{)}\rho(k)\rho(\ell)\rho(m)\rho(n)\Bigl{[}\tfrac{1}{\rho(k)}-\tfrac{1}{\rho(\ell)}+\tfrac{1}{\rho(m)}-\tfrac{1}{\rho(n)}\Bigr{]}\,d\ell\,dm\,dn,$

with phase function

(1.9) $\Theta(k,\ell,m,n):=\nu(k)-\nu(\ell)+\nu(m)-\nu(n),$

where $\nu:[-1,1]\to[-2,2]$ is defined by

(1.10) $\int_{0}^{\nu(k)}d\sigma_{\mathrm{sc}}=\tfrac{k}{2},$

and the semicircle density

(1.11)
$d\sigma_{\mathrm{sc}}(x):=\tfrac{1}{2\pi}(4-x^{2})_{+}^{\frac{1}{2}}\,dx\quad\text{for}\quad
x_{+}=\max\\{x,0\\}.$

We remark that $\nu(k/N)$ represents the deterministic location of the $k$th
eigenvalue of the random matrix $H$: see Lemma 3.14 below.

###  1.4. Timescales and parameter ranges

Assuming that the solution $a(t)$ of (1.5) remains roughly equidistributed in
its entries, the relevant timescales for the problem are:

  * •

The linear timescale, $T_{\mathrm{lin}}=1$, on which nontrivial linear effects
occur;

  * •

The nonlinear timescale, $T_{\mathrm{nonlin}}=N/\mu^{2}$, on which nontrivial
nonlinear effects occur;

  * •

The kinetic timescale, $T_{\mathrm{kin}}=N^{2}/\mu^{4}$, on which we expect
(1.7) to be valid.

The weakly nonlinear regime $\mu/\sqrt{N}\ll 1$ corresponds to the case that

$T_{\mathrm{lin}}\ll T_{\mathrm{nonlin}}\ll T_{\mathrm{kin}}.$

On timescales $T\ll T_{\mathrm{lin}}$, we expect the linear dynamics to
dominate. This is readily seen from the fact that the resonance modulus,

(1.12)
$\Omega(k,\ell,m,n)=\lambda_{k}-\lambda_{\ell}+\lambda_{m}-\lambda_{n},$

satisfies $|\Omega|\lesssim 1$ with overwhelming probability. This suggests
that at best we can hope (1.7) will hold for timescales $T$ satisfying

$T_{\mathrm{lin}}=1\ll T\lesssim T_{\mathrm{kin}}.$

As we will discuss in Section 3.3, the eigenvalues of $H$ in the bulk of the
spectrum fluctuate about their deterministic positions at scales $\gg 1/N$. If
$T\gtrsim N$, the resonance modulus will not be equidistributed at scale
$1/T$, which prevents us from passing from the empirical distribution of the
eigenvalues to the semicircle distribution. As a consequence, we only expect
(1.7) to hold at times $T\sim T_{\mathrm{kin}}$ if we have

$T_{\mathrm{kin}}\ll N.$

Combining these heuristics, we conjecture that (1.7) will hold in the regime
where

$1\ll T\lesssim T_{\mathrm{kin}}\quad\text{and}\quad N^{\frac{1}{4}}\ll\mu\ll
N^{\frac{1}{2}}.$

###  1.5. The main result

Our main result proves that (1.7) does indeed describe the leading order
asymptotic behavior in the regime where

$1\ll T\ll T_{\mathrm{kin}}^{\frac{2}{3}}\quad\text{and}\quad
N^{\frac{1}{4}}\ll\mu\ll N^{\frac{1}{2}}.$

######  Theorem 1.1.

Let $1/4<\beta<1/2$ be fixed and $A\in C^{1}[-1,1]$. For each choice of
integer $N\geqslant 1$, vector of eigenvalues
$\lambda\in\varmathbb{R}^{2N+1}_{\geqslant}$, and matrix of eigenvectors
$\Psi\in\varmathbb U(2N+1)$, let $a=a(N,\lambda,\Psi)\in
C^{\infty}(\varmathbb{R};\varmathbb{C}^{2N+1})$ be the smooth global solution
of (1.5) corresponding to the choice of initial data (1.6) and parameter
$\mu=N^{\beta}$.

Taking $\lambda$ and $\Psi$ to be distributed according to (1.3) and the Haar
measure on $\varmathbb{U}(2N+1)$, respectively, for any $\varepsilon>0$ and
times $N^{\varepsilon}\leqslant t\leqslant
N^{-\varepsilon}T_{\mathrm{kin}}^{2/3}$, we have

(1.13) $\varmathbb
E|a_{k}(t)|^{2}=|A\bigl{(}\tfrac{k}{N}\bigr{)}|^{2}+\tfrac{t}{T_{\mathrm{kin}}}\mathbf{C}\bigl{[}|A|^{2}\bigr{]}\bigl{(}\tfrac{k}{N}\bigr{)}+R_{k}(t),$

where

(1.14)
$\tfrac{1}{N}\|R(t)\|_{\ell^{1}}\lesssim_{\beta,A,\varepsilon}N^{-\frac{\varepsilon}{4}}\tfrac{t}{T_{\mathrm{kin}}},$

and $T_{\mathrm{kin}}=N^{2}/\mu^{4}$ is defined as in (1.7).

The timescale $T_{\mathrm{kin}}^{2/3}$ obtained in Theorem 1.1 is nontrivial,
in the sense that $T_{\mathrm{nonlin}}=T_{\mathrm{kin}}^{1/2}\ll
T_{\mathrm{kin}}^{2/3}$. While our proof follows a similar strategy to
previous work, e.g., [5, 13, 15, 1, 12, 11], the model (1.5) requires multiple
innovations to obtain estimates in the random matrix setting. Indeed, as the
reader will soon appreciate, some significantly involved analysis is required
to make even a modest step towards the conjectured range on which we expect
(1.7) to hold.

We begin our proof of Theorem 1.1 by iterating Duhamel’s formula to obtain an
approximate solution to (1.5). The first few iterates in this construction are
well-approximated by the kinetic wave equation, and are analyzed in detail
Section 4. The remainder is bounded by writing each iterate in terms of the
initial data. These higher order iterates are analyzed in Section 5.

A key point of departure from previous results is that all frequencies in our
model interact together during each nonlinear interaction. To control the
complex behavior of this soup of interacting waves we require several tools
from random matrix theory, which are discussed in Section 3.

For the eigenvalues, we rely on rigidity estimates derived from the local
semicircle law. These enable us to treat the eigenvalues as being located at
their deterministic positions, up to small fluctuations.

For the eigenvectors, we crucially rely on the Weingarten calculus: an analog
of the Wick calculus in the setting of random unitary matrices. To implement
the Weingarten calculus, we introduce a graph-theoretic approach to obtaining
bounds. Several of our estimates for CUE matrices appear to be new and we
believe they may be of independent interest.

Once we have constructed our approximate solution, we complete the proof of
Theorem 1.1 by estimating the remainder using the contraction principle. This
relies on estimates for the linearization of (1.5) about the approximate
solution, which are proved in Section 6. Here, we use a $T^{*}T$ argument to
reduce the problem to similar estimates to those obtained for the approximate
solution itself.

The remainder of the paper is structured as follows: In the final part of this
introduction we present some properties of the kinetic wave equation (1.8). In
Section 2 we discuss the construction of the approximate solution, state our
main estimates, and then prove Theorem 1.1. In Section 3 we introduce the
tools we require from random matrix theory and prove several estimates for
random unitary matrices. In Section 4 we consider the leading order terms in
our approximate solution, from which the kinetic wave equation arises. In
Section 5 we prove estimates for the remaining terms in our approximate
solution. Finally, in Section 6 we consider the linearization of (1.5) about
the approximate solution.

###  1.6. Properties of the kinetic wave equation

We conclude this introduction by discussing some properties of the kinetic
wave equation (1.8). From (1.10), we see that

(1.15)
$\nu^{\prime}(k)=\pi\bigl{(}4-\nu(k)^{2}\bigr{)}_{+}^{-\frac{1}{2}}\quad\text{for}\quad
k\in(-1,1).$

In particular, denoting $L^{p}=L^{p}[-1,1]$ and taking $q\in L^{\infty}$ and
$f,g,h\in L^{1}$, we have

$\displaystyle\left|\tfrac{\pi}{2}\int_{[-1,1]^{4}}\delta\bigl{(}\Theta(k,\ell,m,n)\bigr{)}q(k)f(\ell)g(m)h(n)\,dk\,d\ell\,dm\,dn\right|$
$\displaystyle=\left|\tfrac{1}{2}\int_{[-1,1]^{3}}\\!q\bigl{(}\nu^{-1}\big{[}\nu(\ell)-\nu(m)+\nu(n)\bigr{]}\bigr{)}f(\ell)g(m)h(n)\,\bigl{(}4\\!-\\!\bigl{[}\nu(\ell)\\!-\\!\nu(m)\\!+\\!\nu(n)\bigr{]}^{2}\bigr{)}^{\frac{1}{2}}_{+}\,d\ell\,dm\,dn\right|$
$\displaystyle\lesssim\|q\|_{L^{\infty}}\|f\|_{L^{1}}\|g\|_{L^{1}}\|h\|_{L^{1}}.$

By symmetry and duality, we obtain an estimate for the the collision operator,

$\|\mathbf{C}[\rho]-\mathbf{C}[\phi]\|_{L^{1}}\lesssim\bigl{(}\|\rho\|_{L^{1}}+\|\phi\|_{L^{1}}\bigr{)}^{2}\|\rho-\phi\|_{L^{1}}.$

As a consequence, the map $\rho\mapsto\mathbf{C}[\rho]$ is locally Lipschitz
on $L^{1}$ and Picard’s Theorem gives local well-posedness of (1.8) for any
initial data in $L^{1}$. We note that the collision operator maps continuous
functions to continuous functions, and hence the flow may be restricted to the
subspace $C[-1,1]\subset L^{1}$.

Choosing a test function $f\in C[-1,1]$ and symmetrizing gives us the formula

$\displaystyle\frac{d}{dt}\int_{-1}^{1}f(k)\rho(t,k)\,dk$
$\displaystyle=\tfrac{\pi}{8}\int_{[-1,1]^{4}}\delta\bigl{(}\Theta(k,\ell,m,n)\bigr{)}\rho(t,k)\rho(t,\ell)\rho(t,m)\rho(t,n)$
$\displaystyle\qquad\qquad\qquad\times\Bigl{[}\tfrac{1}{\rho(t,k)}-\tfrac{1}{\rho(t,\ell)}+\tfrac{1}{\rho(t,m)}-\tfrac{1}{\rho(t,n)}\Bigr{]}$
$\displaystyle\qquad\qquad\qquad\qquad\times\Bigl{[}f(k)-f(\ell)+f(m)-f(n)\Bigr{]}\,dk\,d\ell\,dm\,dn.$

Choosing successively $f(k)=1$ and $f(k)=\nu(k)$, we obtain the conservation
of mass and energy,

$\frac{d}{dt}\int_{-1}^{1}\rho(t,k)\,dk=0\quad\text{and}\quad\frac{d}{dt}\int_{-1}^{1}\nu(k)\rho(t,k)\,dk=0.$

Writing (1.8) in the form

$\partial_{t}\rho(t,k)=g(t,k)\rho(t,k)+h(t,k),$

where

$\displaystyle g(t,k)$
$\displaystyle=\tfrac{\pi}{2}\int_{[-1,1]^{3}}\delta\bigl{(}\Theta(k,\ell,m,n)\bigr{)}\rho(t,\ell)\rho(t,m)\rho(t,n)\Bigl{[}-\tfrac{1}{\rho(t,\ell)}+\tfrac{1}{\rho(t,m)}-\tfrac{1}{\rho(t,n)}\Bigr{]}\,d\ell\,dm\,dn,$
$\displaystyle h(t,k)$
$\displaystyle=\tfrac{\pi}{2}\int_{[-1,1]^{3}}\delta\bigl{(}\Theta(k,\ell,m,n)\bigr{)}\rho(t,\ell)\rho(t,m)\rho(t,n)\,d\ell\,dm\,dn,$

we see that if the initial data $\rho(t=0)$ is non-negative (respectively
positive) then $\rho(t)$ is non-negative (respectively positive) for all times
$t\geqslant 0$. In particular, conservation of mass ensures that (1.8) is
globally well-posed, forwards in time, for all non-negative initial data in
$L^{1}$.

Taking $\rho(t=0)$ to be positive and $F\in C^{1}(\varmathbb{R}_{+})$, another
symmetrization gives the expression

$\displaystyle\frac{d}{dt}\int_{-1}^{1}F\bigl{(}\rho(t,k)\bigr{)}\,dk$
$\displaystyle\qquad=\tfrac{\pi}{8}\int_{[-1,1]^{4}}\delta\bigl{(}\Theta(k,\ell,m,n)\bigr{)}\rho(t,k)\rho(t,\ell)\rho(t,m)\rho(t,n)$
$\displaystyle\qquad\quad\qquad\qquad\quad\times\Bigl{[}\tfrac{1}{\rho(t,k)}-\tfrac{1}{\rho(t,\ell)}+\tfrac{1}{\rho(t,m)}-\tfrac{1}{\rho(t,n)}\Bigr{]}$
$\displaystyle\qquad\quad\qquad\qquad\qquad\quad\quad\times\Bigl{[}F^{\prime}\bigl{(}\rho(t,k)\bigr{)}-F^{\prime}\bigl{(}\rho(t,\ell)\bigr{)}+F^{\prime}\bigl{(}\rho(t,m)\bigr{)}-F^{\prime}\bigl{(}\rho(t,n)\bigr{)}\Bigr{]}\,dk\,d\ell\,dm\,dn.$

Setting $F(\rho)=\log\rho$ results in the $H$-theorem (entropy increases in
time) for positive solutions

$\displaystyle\frac{d}{dt}\int_{-1}^{1}\log\rho(t,k)\,dk$
$\displaystyle=\tfrac{\pi}{8}\int_{[-1,1]^{4}}\delta\bigl{(}\Theta(k,\ell,m,n)\bigr{)}\rho(t,k)\rho(t,\ell)\rho(t,m)\rho(t,n)$
$\displaystyle\quad\qquad\qquad\quad\times\Bigl{[}\tfrac{1}{\rho(t,k)}-\tfrac{1}{\rho(t,\ell)}+\tfrac{1}{\rho(t,m)}-\tfrac{1}{\rho(t,n)}\Bigr{]}^{2}\,dk\,d\ell\,dm\,dn$
$\displaystyle\geqslant 0.$

The entropy dissipation vanishes on the Rayleigh–Jeans distribution

$\phi(k)=\frac{\alpha}{\beta+\nu(k)},$

where $\alpha>0$ and $\beta>2$ are constants. It is natural to conjecture that
the Rayleigh–Jeans distributions are maximizers of the entropy for fixed mass
and energy, see [27] for a proof of this fact in a closely related context,
along with a convergence result as $t\to\infty$.

### Acknowledgments.

The authors are grateful to Paul Bourgade for several very helpful discussions
and providing a number of important references. The authors also thank Benoît
Collins, Sho Matsumoto and Jonathan Novak for informative discussions about
the Weingarten calculus. The authors are very grateful to the anonymous
referee for several insightful comments and suggestions.

G. Dubach gratefully acknowledges funding from the European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie Grant
Agreement No. 754411. P. Germain was supported by the Simons collaborative
grant on weak turbulence. B. Harrop-Griffiths is grateful to the Department of
Mathematics at UCLA, which he was a member of when this work was completed.

##  2\. Statement of the main estimates and proof of Theorem 1.1

In this section, we first introduce some preliminary reductions, identities,
and estimates. We then state our main linear and nonlinear estimates, and use
these to prove Theorem 1.1.

###  2.1. Wick ordering

As in the case of the NLS equation considered in, e.g., [12, 11, 13, 15], we
introduce a “Wick ordering” to remove certain resonant terms. Precisely, we
introduce a time-dependent rotation

$\underline{u}(t)=e^{it\mu^{2}\frac{2\mathcal{M}}{2N+1}}u(t)$

so that (1.2) becomes

$i\frac{d}{dt}\underline{u}-H\underline{u}=\mu^{2}\Bigl{[}\bigl{(}\underline{u}\odot\bar{\underline{u}}\odot\underline{u}\bigr{)}-\tfrac{2\mathcal{M}}{2N+1}\underline{u}\Bigr{]},$

where we note that
$\mathcal{M}=\|u\|_{\ell^{2}}^{2}=\|\underline{u}\|_{\ell^{2}}^{2}$ is still
conserved.

Taking

(2.1) $\underline{a}:=\sqrt{N}\Psi^{*}\underline{u},$

the equation (1.5) becomes

(2.2)
$i\frac{d}{dt}\underline{a}-\Lambda\underline{a}=\frac{\mu^{2}}{N}\Bigl{[}\Psi^{*}\bigl{(}\Psi\underline{a}\odot\overline{\Psi\underline{a}}\odot\Psi\underline{a}\bigr{)}-\tfrac{2N\mathcal{M}}{2N+1}\underline{a}\Bigr{]}.$

Observing that $|a_{k}|^{2}=|\underline{a}_{k}|^{2}$, we subsequently drop the
underlines and take (2.1) to be our definition of $a$ instead of (1.4).

###  2.2. Decomposing the solution

We now write (2.2) using the Duhamel formula as

(2.3)
$a(t)=e^{-it\Lambda}a(0)-i\int_{0}^{t}e^{-i(t-s)\Lambda}\mathcal{N}[a(s),a(s),a(s)]\,ds,$

where we denote

$\displaystyle\mathcal{N}[a,b,c]$
$\displaystyle=\tfrac{\mu^{2}}{3N}\Bigl{[}\Psi^{*}\bigl{(}\Psi
a(s)\odot\overline{\Psi b(s)}\odot\Psi c(s)\bigr{)}-\tfrac{1}{2N+1}\langle
b,c\rangle a-\tfrac{1}{2N+1}\langle b,a\rangle c\bigr{]}$
$\displaystyle\quad+\tfrac{\mu^{2}}{3N}\Bigl{[}\Psi^{*}\bigl{(}\Psi
b(s)\odot\overline{\Psi c(s)}\odot\Psi a(s)\bigr{)}-\tfrac{1}{2N+1}\langle
c,a\rangle b-\tfrac{1}{2N+1}\langle c,b\rangle a\Bigr{]}$
$\displaystyle\quad+\tfrac{\mu^{2}}{3N}\Bigl{[}\Psi^{*}\bigl{(}\Psi
c(s)\odot\overline{\Psi a(s)}\odot\Psi b(s)\bigr{)}-\tfrac{1}{2N+1}\langle
a,b\rangle c-\tfrac{1}{2N+1}\langle a,c\rangle b\Bigr{]},$

and recall that our convention is that the inner product
$\langle\cdot,\cdot\rangle$ is linear in the second variable. We remark that
if

(2.4)
$\gamma(k,\ell,m,n):=\sum_{j=-N}^{N}\overline{\psi_{jk}}\,\psi_{j\ell}\,\overline{\psi_{jm}}\,\psi_{jn}-\tfrac{1}{2N+1}\varmathbb
1_{\\{k=\ell,m=n\\}}-\tfrac{1}{2N+1}\varmathbb 1_{\\{k=n,\ell=m\\}}$

then the $k$th entry

$\mathcal{N}[a,b,c]_{k}=\tfrac{\mu^{2}}{3N}\sum_{\ell,m,n}\gamma(k,\ell,m,n)\,\bigl{[}a_{\ell}\,\overline{b_{m}}\,c_{n}+b_{\ell}\,\overline{c_{m}}\,a_{n}+c_{\ell}\,\overline{a_{m}}\,b_{n}\bigr{]}.$

For a sufficiently large integer $M\geqslant 1$, we construct an approximate
solution of (2.3) by

(2.5) $a^{\mathrm{app}}(t)=\sum\limits_{m=0}^{M}a^{(m)}(t),$

where

$a^{(0)}(t)=e^{-it\Lambda}a(0),$

and for $m\geqslant 0$

(2.6)
$a^{(m+1)}(t)=-i\sum\limits_{m_{1}+m_{2}+m_{3}=m}\int_{0}^{t}e^{-i(t-s)\Lambda}\mathcal{N}\bigl{[}a^{(m_{1})}(s),a^{(m_{2})}(s),a^{(m_{3})}(s)\bigr{]}\,ds.$

We then introduce the error term

(2.7) $\mathcal{E}=-i\frac{d}{dt}a^{\mathrm{app}}+\Lambda
a^{\mathrm{app}}+\mathcal{N}[a^{\mathrm{app}},a^{\mathrm{app}},a^{\mathrm{app}}],$

so that the remainder

(2.8) $a^{\mathrm{err}}(t)=a(t)-a^{\mathrm{app}}(t)$

satisfies

(2.9) $\begin{cases}i\dfrac{d}{dt}a^{\mathrm{err}}-\Lambda
a^{\mathrm{err}}=\mathcal{L}a^{\mathrm{err}}+3\mathcal{N}[a^{\mathrm{app}},a^{\mathrm{err}},a^{\mathrm{err}}]+\mathcal{N}[a^{\mathrm{err}},a^{\mathrm{err}},a^{\mathrm{err}}]+\mathcal{E},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ a^{\mathrm{err}}(0)=0,\end{cases}$

where the real linear operator

(2.10)
$\mathcal{L}a=3\mathcal{N}\bigl{[}a^{\mathrm{app}},a^{\mathrm{app}},a\bigr{]}.$

###  2.3. Feynman diagrams and Duhamel iterates

To understand the structure of $a^{(m)}$, it will be useful to introduce
_Feynman diagrams_ that encode the expansion of the Duhamel integral using
graphs. These diagrams will play an important role in our estimates for the
approximate solution and linear operator in Sections 5 and 6.

Feynman diagrams originally appeared in derivations of the kinetic wave
equation in the work of Lukkarinen and Spohn [24]; see also [6] for a related
application. Due to the particular features of the random matrix setting, we
adopt some slightly different conventions to expedite the framework.

A Feynman diagram for $a^{(m)}$ is a graph $G$ with $(m+1)^{2}+1$ vertices and
$(m+1)^{2}$ edges. The shape of the graph is uniquely determined by an
_interaction history_ , $\ell=(\ell_{1},\dots,\ell_{m})$, where each
$\ell_{r}\in\llbracket 1,2(m-r)+1\rrbracket$.

The vertices of the graph represent the linear and nonlinear interactions of
waves. We index the vertices by $v_{r,j}$, where $r\in\llbracket
0,m+1\rrbracket$ and $j\in\llbracket 1,2(m-r)+1\rrbracket$ for $0\leqslant
r\leqslant m$, with $j=1$ when $r=m+1$. We distinguish four sets of vertices:

  * •

The output vertex, $v_{m+1,1}$, which we denote by ;

  * •

The input vertices, $v_{0,j}$, which we denote by ;

  * •

The linear interaction vertices, $v_{r,j}$ for $1\leqslant r\leqslant m$ and
$j\neq\ell_{r}$, which we denote by ;

  * •

The nonlinear interaction vertices, $v_{r,\ell_{r}}$ for $1\leqslant
r\leqslant m$, which we denote by .

The edges of the graph represent the linear propagation of waves. We index the
edges by $e_{r,j}$, where $r\in\llbracket 0,m\rrbracket$ and $j\in\llbracket
1,2(m-r)+1\rrbracket$. For $0\leqslant r\leqslant m-1$, the edge $e_{r,j}$
connects the vertex

$v_{r,j}\quad\text{to}\quad\begin{cases}v_{r+1,j}&\quad\text{if}\quad
j<\ell_{r+1},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
v_{r+1,\ell_{r+1}}&\quad\text{if}\quad\ell_{r+1}\leqslant
j\leqslant\ell_{r+1}+2,\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
v_{r+1,j-2}&\quad\text{if}\quad j>\ell_{r+1}+2.\end{cases}$

The edge $e_{m,1}$ connects the vertex $v_{m,1}$ to $v_{m+1,1}$.

Given a time $t>0$ and a frequency $k\in\llbracket-N,N\rrbracket$, we convert
each Feynman diagram $G=G(\ell)$ into an integral by imposing a set of
_Feynman rules_ that encode the contribution of each element of the graph. We
then sum over all choices of $\ell$ to obtain an expression for
$a^{(m)}_{k}(t)$.

Before imposing these rules, we first label each edge, $e_{r,j}$, of the graph
with an integer $k_{r,j}\in\llbracket-N,N\rrbracket$. This represents the
frequency of the corresponding linear wave. We denote the array of frequencies
by $\mathcal{K}=(k_{r,j})$.

Second, we choose times $s=(s_{0},\dots,s_{m})\in\varmathbb{R}_{+}^{m+1}$ so
that, with $t_{0}=0$ and $t_{r}=\sum_{j=0}^{r-1}s_{j}$ for $r\in\llbracket
1,m+1\rrbracket$, we obtain a partition $0=t_{0}<t_{1}<\dots<t_{m+1}=t$. We
will subsequently treat time as the vertical direction in our graph. For each
$r$, we locate the vertices $v_{r,j}$ at time $t_{r}$ with the index $j$
running from left to right. The edges $e_{r,j}$ then lie in the time interval
$(t_{r},t_{r+1})$ with the index $j$ again running from left to right.

We now impose the following Feynman rules for our graph:

  * •

The output vertex, $v_{m+1,1}$, contributes a factor of $\varmathbb
1_{\\{k=k_{m,1}\\}}\,\delta\left(\sum_{r=0}^{m}s_{r}-t\right)$.

  * •

The input vertices, $v_{0,j}$, contribute factors of

$\begin{cases}A\bigl{(}\tfrac{k_{0,j}}{N}\bigr{)}&\text{if }j\text{ is
odd},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\overline{A\bigl{(}\tfrac{k_{0,j}}{N}\bigr{)}}&\text{if }j\text{ is
even}.\end{cases}$

  * •

The linear interaction vertices, $v_{r,j}$ for $1\leqslant r\leqslant m$ and
$j\neq\ell_{r}$, contribute factors of

$\begin{cases}\varmathbb 1_{\\{k_{r-1,j}=k_{r,j}\\}}&\quad\text{if}\quad
j<\ell_{r},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \varmathbb
1_{\\{k_{r-1,j}=k_{r,j-2}\\}}&\quad\text{if}\quad j>\ell_{r}+2.\end{cases}$

  * •

The nonlinear interaction vertices, $v_{r,\ell_{r}}$ for $1\leqslant
r\leqslant m$, contribute factors of $-i\frac{\mu^{2}}{N}\gamma_{r,\ell_{r}}$,
where

(2.11)
$\gamma_{r,\ell_{r}}(\mathcal{K})=\begin{cases}\gamma\bigl{(}k_{r,\ell_{r}},k_{r-1,\ell_{r}},k_{r-1,\ell_{r}+1},k_{r-1,\ell_{r}+2}\bigr{)}&\quad\text{if}\quad\ell_{r}\text{
is odd},\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
-\overline{\gamma\bigl{(}k_{r,\ell_{r}},k_{r-1,\ell_{r}},k_{r-1,\ell_{r}+1},k_{r-1,\ell_{r}+2}\bigr{)}}&\quad\text{if}\quad\ell_{r}\text{
is even},\end{cases}$

and $\gamma$ is defined as in (2.4).

  * •

The edges $e_{r,j}$ contribute factors of

$\begin{cases}e^{-is_{j}\lambda_{k_{r,j}}}&\text{if }j\text{ is odd},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ e^{is_{j}\lambda_{k_{r,j}}}&\text{if }j\text{
is even}.\end{cases}$

An expression for $a^{(m)}_{k}(t)$ is now obtained by multiplying these
contributions together, integrating over $s\in\varmathbb{R}_{+}^{m+1}$, and
summing over $\ell,\mathcal{K}$.

As an example, we have the following:

######  Example 2.1 (The case $m=2$).

For $m=2$ we have 3 possible diagrams:

$t_{0}$$t_{1}$$t_{2}$$t_{3}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,1}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,2}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,3}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,4}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,5}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,1}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{2,1}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,2}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,3}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,1}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,2}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,3}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,4}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,5}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,1}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,2}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,3}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{2,1}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{3,1}}$

$t_{0}$$t_{1}$$t_{2}$$t_{3}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,1}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,2}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,3}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,4}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,5}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,1}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{2,1}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,2}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,3}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,1}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,2}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,3}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,4}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,5}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,1}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,2}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,3}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{2,1}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{3,1}}$

$t_{0}$$t_{1}$$t_{2}$$t_{3}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,1}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,2}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,3}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,4}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,5}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,3}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{2,1}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,2}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,1}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,5}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,4}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,3}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,2}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,1}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,3}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,2}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,1}}$${\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{2,1}}$${\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{3,1}}$

Figure 1. Feynman diagrams for $m=2$ with $\ell=(1,1),(2,1),(3,1)$,
respectively.

The corresponding contribution to our integral for the $\ell=(1,1)$ diagram is

$\displaystyle\underbrace{\varmathbb
1_{\\{k=k_{2,1}\\}}\delta(s_{0}+s_{1}+s_{2}-t)}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{3,1}}}\times\underbrace{e^{-is_{2}\lambda_{k_{2,1}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{2,1}}}\times\underbrace{-i\tfrac{\mu^{2}}{N}\gamma_{2,1}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{2,1}}}\times\underbrace{e^{-is_{1}\lambda_{k_{1,1}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,1}}}\times\underbrace{e^{is_{1}\lambda_{k_{1,2}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,2}}}\times\underbrace{e^{-is_{1}\lambda_{k_{1,3}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,3}}}$
$\displaystyle\qquad\times\underbrace{-i\tfrac{\mu^{2}}{N}\gamma_{1,1}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,1}}}\times\underbrace{\varmathbb
1_{\\{k_{0,4}=k_{1,2}\\}}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,2}}}\times\underbrace{\varmathbb
1_{\\{k_{0,5}=k_{1,3}\\}}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,3}}}\times\underbrace{e^{-is_{0}\lambda_{k_{0,1}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,1}}}\times\underbrace{e^{is_{0}\lambda_{k_{0,2}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,2}}}\times\underbrace{e^{-is_{0}\lambda_{k_{0,3}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,3}}}\times\underbrace{e^{is_{0}\lambda_{k_{0,4}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,4}}}\times\underbrace{e^{-is_{0}\lambda_{k_{0,5}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,5}}}$
$\displaystyle\qquad\qquad\times\underbrace{A\bigl{(}\tfrac{k_{0,1}}{N}\bigr{)}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,1}}}\times\underbrace{\overline{A\bigl{(}\tfrac{k_{0,2}}{N}\bigr{)}}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,2}}}\times\underbrace{A\bigl{(}\tfrac{k_{0,3}}{N}\bigr{)}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,3}}}\times\underbrace{\overline{A\bigl{(}\tfrac{k_{0,4}}{N}\bigr{)}}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,4}}}\times\underbrace{A\bigl{(}\tfrac{k_{0,5}}{N}\bigr{)}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,5}}},$

for the $\ell=(2,1)$ diagram, it is

$\displaystyle\underbrace{\varmathbb
1_{\\{k=k_{2,1}\\}}\delta(s_{0}+s_{1}+s_{2}-t)}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{3,1}}}\times\underbrace{e^{-is_{2}\lambda_{k_{2,1}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{2,1}}}\times\underbrace{-i\tfrac{\mu^{2}}{N}\gamma_{2,1}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{2,1}}}\times\underbrace{e^{-is_{1}\lambda_{k_{1,1}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,1}}}\times\underbrace{e^{is_{1}\lambda_{k_{1,2}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,2}}}\times\underbrace{e^{-is_{1}\lambda_{k_{1,3}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,3}}}$
$\displaystyle\qquad\times\underbrace{\varmathbb
1_{\\{k_{0,1}=k_{1,1}\\}}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,1}}}\times\underbrace{-i\tfrac{\mu^{2}}{N}\gamma_{1,2}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,2}}}\times\underbrace{\varmathbb
1_{\\{k_{0,5}=k_{1,3}\\}}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,3}}}\times\underbrace{e^{-is_{0}\lambda_{k_{0,1}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,1}}}\times\underbrace{e^{is_{0}\lambda_{k_{0,2}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,2}}}\times\underbrace{e^{-is_{0}\lambda_{k_{0,3}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,3}}}\times\underbrace{e^{is_{0}\lambda_{k_{0,4}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,4}}}\times\underbrace{e^{-is_{0}\lambda_{k_{0,5}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,5}}}$
$\displaystyle\qquad\qquad\times\underbrace{A\bigl{(}\tfrac{k_{0,1}}{N}\bigr{)}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,1}}}\times\underbrace{\overline{A\bigl{(}\tfrac{k_{0,2}}{N}\bigr{)}}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,2}}}\times\underbrace{A\bigl{(}\tfrac{k_{0,3}}{N}\bigr{)}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,3}}}\times\underbrace{\overline{A\bigl{(}\tfrac{k_{0,4}}{N}\bigr{)}}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,4}}}\times\underbrace{A\bigl{(}\tfrac{k_{0,5}}{N}\bigr{)}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,5}}},$

and for the $\ell=(3,1)$ diagram is

$\displaystyle\underbrace{\varmathbb
1_{\\{k=k_{2,1}\\}}\delta(s_{0}+s_{1}+s_{2}-t)}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{3,1}}}\times\underbrace{e^{-is_{2}\lambda_{k_{2,1}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{2,1}}}\times\underbrace{-i\tfrac{\mu^{2}}{N}\gamma_{2,1}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{2,1}}}\times\underbrace{e^{-is_{1}\lambda_{k_{1,1}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,1}}}\times\underbrace{e^{is_{1}\lambda_{k_{1,2}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,2}}}\times\underbrace{e^{-is_{1}\lambda_{k_{1,3}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{1,3}}}$
$\displaystyle\qquad\times\underbrace{\varmathbb
1_{\\{k_{0,1}=k_{1,1}\\}}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,1}}}\times\underbrace{\varmathbb
1_{\\{k_{0,2}=k_{1,2}\\}}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,2}}}\times\underbrace{-i\tfrac{\mu^{2}}{N}\gamma_{1,3}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{1,3}}}\times\underbrace{e^{-is_{0}\lambda_{k_{0,1}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,1}}}\times\underbrace{e^{is_{0}\lambda_{k_{0,2}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,2}}}\times\underbrace{e^{-is_{0}\lambda_{k_{0,3}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,3}}}\times\underbrace{e^{is_{0}\lambda_{k_{0,4}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,4}}}\times\underbrace{e^{-is_{0}\lambda_{k_{0,5}}}}_{{\color[rgb]{0.93359375,0.19921875,0.46484375}\definecolor[named]{pgfstrokecolor}{rgb}{0.93359375,0.19921875,0.46484375}e_{0,5}}}$
$\displaystyle\qquad\qquad\times\underbrace{A\bigl{(}\tfrac{k_{0,1}}{N}\bigr{)}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,1}}}\times\underbrace{\overline{A\bigl{(}\tfrac{k_{0,2}}{N}\bigr{)}}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,2}}}\times\underbrace{A\bigl{(}\tfrac{k_{0,3}}{N}\bigr{)}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,3}}}\times\underbrace{\overline{A\bigl{(}\tfrac{k_{0,4}}{N}\bigr{)}}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,4}}}\times\underbrace{A\bigl{(}\tfrac{k_{0,5}}{N}\bigr{)}}_{{\color[rgb]{0.19921875,0.6015625,0.93359375}\definecolor[named]{pgfstrokecolor}{rgb}{0.19921875,0.6015625,0.93359375}v_{0,5}}}.$

An expression for $a_{k}^{(2)}(t)$ is obtained by summing these expressions,
then summing over $\mathcal{K}$ and integrating over $s$.

It will be useful to derive an explicit expression for the Duhamel iterates.
Here, we have the following:

######  Lemma 2.2.

For $m\geqslant 1$, denote the profile

(2.12) $f^{(m)}(t)=e^{it\Lambda}a^{(m)}(t).$

Then, for any $T>0$ we have the expression

(2.13) $\displaystyle f_{k}^{(m)}(t)\varmathbb 1_{\varmathbb{R}_{\pm}}(t)$
$\displaystyle\qquad=\frac{ie^{\pm
t/T}}{2\pi}\left(\frac{\mu^{2}}{N}\right)^{m}\sum_{\ell,\mathcal{K}}\Biggl{\\{}\Delta(k,\mathcal{K},\ell)\,\mathbf{A}(\mathcal{K})\,\prod_{r=1}^{m}\gamma_{r,\ell_{r}}(\mathcal{K})\int_{\varmathbb}{R}\prod_{r=0}^{m}\frac{1}{\alpha+\omega_{r}(\mathcal{K},\ell)\pm\frac{i}{T}}e^{-i\alpha
t}\,d\alpha\Biggr{\\}},$

valid for all $t\in\varmathbb{R}$, where $\gamma_{r,\ell_{r}}(\mathcal{K})$ is
defined as in (2.11) and:

  * •

The linear interactions

(2.14) $\Delta(k,\mathcal{K},\ell)=\varmathbb
1_{\\{k=k_{m,1}\\}}\prod_{r=0}^{m-1}\left[\prod_{j=1}^{\ell_{r+1}-1}\varmathbb
1_{\\{k_{r,j}=k_{r+1,j}\\}}\prod_{j=\ell_{r+1}+3}^{2(m-r)+1}\varmathbb
1_{\\{k_{r,j}=k_{r+1,j-2}\\}}\right];$

  * •

The modulations

(2.15)
$\omega_{r}(\mathcal{K},\ell)=\sum_{j=r+1}^{m}(-1)^{\ell_{j}-1}\Omega\bigl{(}k_{j,\ell_{j}},k_{j-1,\ell_{j}},k_{j-1,\ell_{j}+1},k_{j-1,\ell_{j}+2}\bigr{)}\quad\text{for}\quad
0\leqslant r\leqslant m-1,$

with $\omega_{m}(\mathcal{K},\ell)=0$ and $\Omega(k,\ell,m,n)$ defined as in
(1.12);

  * •

The initial data

(2.16)
$\mathbf{A}(\mathcal{K})=A\bigl{(}\tfrac{k_{0,1}}{N}\bigr{)}\,\overline{A\bigl{(}\tfrac{k_{0,2}}{N}\bigr{)}}\,\dots\,\overline{A\bigl{(}\tfrac{k_{0,2m}}{N}\bigr{)}}\,A\bigl{(}\tfrac{k_{0,2m+1}}{N}\bigr{)}.$

###### Proof.

First consider the case that $t>0$. Writing (2.3) as

$f_{k}(t)=A\left(\tfrac{k}{N}\right)-i\frac{\mu^{2}}{N}\sum_{\ell,m,n}\int_{0}^{t}e^{is\Omega(k,\ell,m,n)}\gamma(k,\ell,m,n)f_{\ell}(s)\overline{f_{m}(s)}f_{n}(s)\,ds,$

the identity (2.13) follows from induction on $m$ and the identity

(2.17)
$\int_{\varmathbb{R}^{m+1}_{+}}\left[\prod_{r=0}^{m}e^{is_{r}\omega_{r}}\right]\,\delta\left(\sum_{r=0}^{m}s_{r}-t\right)\,ds_{0}\dots
ds_{r}=\frac{e^{t/T}}{2\pi}i^{m+1}\int_{\varmathbb}{R}\prod_{r=0}^{m}\frac{1}{\alpha+\omega_{r}+\frac{i}{T}}e^{-i\alpha
t}\,d\alpha,$

which is proved in [12, Lemma 4.2] by writing

$\delta\left(\sum_{r=0}^{m}s_{r}-t\right)=e^{-\tfrac{1}{T}\left(\sum_{r=0}^{m}s_{r}-t\right)}\frac{1}{2\pi}\int_{\varmathbb}{R}e^{i\alpha\left(\sum_{r=0}^{m}s_{r}-t\right)}\,d\alpha$

and then integrating in $s_{0},\dots,s_{r}$. We also note that

$\int_{\varmathbb}{R}\prod_{r=0}^{m}\frac{1}{\alpha+\omega_{r}+\frac{i}{T}}e^{-i\alpha
t}\,d\alpha$

vanishes for $t<0$ which shows that (2.13) is valid for all times
$t\in\varmathbb{R}$.

The case $t<0$ now follows from the case $t>0$ and the observation that
$\widetilde{a}(t):=-\overline{a(-t)}$ satisfies the equation

$i\frac{d}{dt}\widetilde{a}-\Lambda\widetilde{a}=\frac{\mu^{2}}{N}\Bigl{[}\overline{\Psi}^{*}\bigl{(}\overline{\Psi}\widetilde{a}\odot\overline{\overline{\Psi}\widetilde{a}}\odot\overline{\Psi}\widetilde{a}\bigr{)}-\tfrac{2N\mathcal{M}}{2N+1}\widetilde{a}\Bigr{]}.$

∎

###  2.4. The $X^{b}_{T}$ spaces

To bound our solution, we introduce a modification of Bourgain’s $X^{s,b}$
spaces that are well-adapted to our problem. Using the unitary normalization
of the Fourier transform,

$\widehat{f}(\tau)=\tfrac{1}{\sqrt{2\pi}}\int_{\varmathbb}{R}e^{-it\tau}f(t)\,dt,$

for a diagonal matrix $\Lambda\in M_{2N+1}(\varmathbb{R})$,
$b\in\varmathbb{R}$, and $T\geqslant 1$, we define the space $X^{b}_{T}$ with
norm

$\|a\|_{X^{b}_{T}}^{2}:=\left\|\left(\tfrac{1}{T}+|\tau|\right)^{b}\widehat{e^{it\Lambda}a}(\tau)\right\|_{L^{2}_{\tau}\ell^{2}}^{2}=\sum_{k=-N}^{N}\left\|\left(\tfrac{1}{T}+|\tau|\right)^{b}\widehat{a}_{k}(\tau-\lambda_{k})\right\|_{L^{2}_{\tau}}^{2}.$

The following properties follow directly from the definition, see, e.g. [31]:

######  Lemma 2.3.

Let $\Lambda\in M_{2N+1}(\varmathbb{R})$ be a diagonal matrix,
$b\in\varmathbb{R}$, $T\geqslant 1$, and $\chi_{T}(t)=\chi(\tfrac{t}{T})$ for
some $\chi\in\mathscr{S}(\varmathbb{R})$. Then we have the following
estimates:

  * (i)

(Free solution) If $f\in\varmathbb{C}^{2N+1}$ then

(2.18)
$\left\|\chi_{T}\,e^{-it\Lambda}f\right\|_{X^{b}_{T}}\lesssim_{b,\chi}T^{\frac{1}{2}-b}\|f\|_{\ell^{2}}.$

  * (ii)

(Time restriction) If $a\in X^{b}_{T}$ then

(2.19)
$\left\|\chi_{T}\,a\right\|_{X^{b}_{T}}\lesssim_{b,\chi}\|a\|_{X_{T}^{b}}.$

  * (iii)

(Time continuity) If $b>\frac{1}{2}$ and $a\in X^{b}_{T}$ then $\chi_{T}\,a\in
C(\varmathbb{R};\varmathbb{C}^{2N+1})$ and

(2.20)
$\|\chi_{T}\,a\|_{L^{\infty}_{t}\ell^{2}}\lesssim_{b,\chi}T^{b-\frac{1}{2}}\|a\|_{X^{b}_{T}}$

  * (iv)

(Hyperbolic regularity) If $a\in X^{b}_{T}$ is a solution of

$\begin{cases}\displaystyle i\frac{d}{dt}a-\Lambda a=F,\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\ a(0)=0,\end{cases}$

where $F\in X^{b-1}_{T}$ then

(2.21)
$\left\|\chi_{T}\,a\right\|_{X^{b}_{T}}\lesssim_{b,\chi}\|F\|_{X^{b-1}_{T}}.$

###  2.5. The main estimates

We are now in a position to state our main estimates.

We subsequently assume that the initial data $A\in C^{1}[-1,1]$ has been fixed
and ignore the dependence of the implicit constants on $A$. We take $\chi\in
C^{\infty}_{c}(\varmathbb{R})$ to be an even, real-valued bump function that
is identically $1$ on $[-1,1]$ and supported on $[-2,2]$. We denote the
rescaling $\chi_{T}(t)=\chi(\frac{t}{T})$ and similarly ignore the dependence
of the implicit constants on $\chi$.

It will be useful to introduce the following definition from [19]:

######  Definition 2.4 (Stochastic domination).

For random variables $X,Y$ depending on $N$ we write

$X\prec Y$

if, for any $K,\delta>0$, we have

$\varmathbb
P[X>N^{\delta}Y]=O_{\delta,K}\bigl{(}N^{-K}\bigr{)}\quad\text{as}\quad
N\rightarrow\infty,$

i.e., for any $\delta>0$ and sufficiently large $N\gg 1$ we have $X\leqslant
N^{\delta}Y$ with overwhelming probability. If the implicit constant depends
on some deterministic parameter $\alpha$, we indicate this by writing
$X\prec_{\alpha}Y$. Finally, we write $X=O_{\prec}(Y)$ if $|X|\prec Y$.

We remark that the relation $\prec$ is transitive and behaves like an
inequality with respect to arithmetic operations; see [19, Lemma 4.4].

Our first estimate considers the leading order terms in the approximation:
$a^{(0)}$, $a^{(1)}$, and $a^{(2)}$. Here, we have the following estimate,
which is proved in Section 4:

######  Proposition 2.5 (The leading order terms).

For any $\delta>0$ and $1\leqslant t\leqslant N$ we have

(2.22) $\displaystyle\tfrac{1}{N}\left\|\varmathbb
E\bigl{|}a_{k}^{(0)}(t)+a_{k}^{(1)}(t)\bigr{|}^{2}+2\operatorname{Re}\varmathbb
E\Bigl{[}\overline{a_{k}^{(0)}(t)}a_{k}^{(2)}(t)\Bigr{]}-|A(\tfrac{k}{N})|^{2}-\tfrac{t}{T_{\mathrm{kin}}}\mathbf{C}\bigl{[}|A|^{2}\bigr{]}(\tfrac{k}{N})\right\|_{\ell_{k}^{1}}$
$\displaystyle\qquad\lesssim_{\delta}\tfrac{t}{T_{\mathrm{kin}}}\left(\tfrac{1}{\sqrt{t}}+N^{\delta}\tfrac{t}{N}\right).$

Our second estimate is proved in Section 5. This gives bounds for $a^{(m)}$
that will primarily be used to control higher order error terms:

######  Proposition 2.6 (The Duhamel iterates).

For all $m\geqslant 1$, $\frac{1}{2}<b<\frac{1}{2}+m$, and $1\leqslant
T\leqslant N$ we have the estimate

(2.23)
$\tfrac{1}{N}\|a^{(m)}\|_{X^{b}_{T}}^{2}\prec_{b,m}\left(\tfrac{T^{4/3}}{T_{\mathrm{kin}}}\right)^{m}.$

Further, for all $\delta>0$ we have the refinements

(2.24) $\displaystyle\tfrac{1}{N}\varmathbb E\|a^{(1)}\|_{X^{b}_{T}}^{2}$
$\displaystyle\lesssim_{b,\delta}N^{\delta}\tfrac{T}{T_{\mathrm{kin}}},$
(2.25) $\displaystyle\tfrac{1}{N}\varmathbb E\|a^{(2)}\|_{X^{b}_{T}}^{2}$
$\displaystyle\lesssim_{b,\delta}N^{\delta}\left(\tfrac{T}{T_{\mathrm{kin}}}\right)^{2}.$

The linear term is bounded in Section 6, relying on similar estimates to the
proof of Proposition 2.6:

######  Proposition 2.7 (The linear operator).

Let $M\geqslant 0$ and $\mathcal{L}$ be defined as in (2.10). Then, if
$\frac{1}{2}<b\leqslant\frac{5}{6}$ and $1\leqslant
T\leqslant\min\\{N,T_{\mathrm{kin}}\\}$ we have the estimate

(2.26) $\|\chi_{T}\,\mathcal{L}\|_{X_{T}^{b}\to
X^{b-1}_{T}}\prec_{M,b}(TN)^{2b-1}\left(\tfrac{T^{4/3}}{T_{\mathrm{kin}}}\right)^{\frac{1}{2}}.$

For the nonlinear terms we use the following (deterministic) bound:

######  Lemma 2.8 (The nonlinear terms).

For $\frac{1}{2}<b<1$ we have the estimate

(2.27)
$\|\chi_{T}\,\mathcal{N}[u,v,w]\|_{X^{b-1}_{T}}\lesssim_{b}T^{2b}\tfrac{\mu^{2}}{N}\|u\|_{X^{b}_{T}}\|v\|_{X^{b}_{T}}\|w\|_{X^{b}_{T}}.$

###### Proof.

We first use the embedding $\ell^{2}\subset\ell^{4}$ with the fact that
$\|\Psi e^{-it\Lambda}\|_{\ell^{2}\to\ell^{2}}=1$ to estimate

(2.28) $\|\chi_{T}\Psi e^{-it\Lambda}f\|_{L^{4}_{t}\ell^{4}}^{4}\lesssim
T\|f\|_{\ell^{2}}^{4}\quad\text{for any}\quad f\in\varmathbb{C}^{2N+1}.$

Taking the inverse Fourier transform, we have

$u_{k}(t)=\tfrac{1}{\sqrt{2\pi}}\int_{\varmathbb}{R}e^{it\tau}e^{-it\lambda_{k}}\widehat{u}_{k}(\tau-\lambda_{k})\,d\tau.$

Setting $g_{k}(\tau)=\widehat{u}_{k}(\tau-\lambda_{k})$ and applying (2.28)
followed by the Cauchy–Schwarz inequality gives us

$\|\chi_{T}\,\Psi
u\|_{L^{4}_{t}\ell^{4}}\lesssim\int_{\varmathbb}{R}\|\chi_{T}\Psi
e^{-it\Lambda}g(\tau)\|_{L^{4}_{t}\ell^{4}}\,d\tau\lesssim
T^{\frac{1}{4}}\int_{\varmathbb}{R}\|g\|_{\ell^{2}}\,d\tau\lesssim_{b}T^{b-\frac{1}{4}}\|u\|_{X^{b}_{T}}.$

By duality, for any $b^{\prime}<-\frac{1}{2}$ and $h\in\varmathbb{C}^{2N+1}$
we have the bound

$\|\chi_{T}\,\Psi^{*}h\|_{X^{b^{\prime}}_{T}}\lesssim_{b^{\prime}}T^{-b^{\prime}-\frac{1}{4}}\|h\|_{L^{4/3}_{t}\ell^{4/3}}.$

Combining the above bounds and using that $\chi_{T}=\chi_{T}\chi_{2T}$, we
obtain

$\displaystyle\|\chi_{T}\,\Psi^{*}(\Psi u\odot\overline{\Psi v}\odot\Psi
w)\|_{X^{b^{\prime}}_{T}}$
$\displaystyle\lesssim_{b^{\prime}}T^{-b^{\prime}-\frac{1}{4}}\|\chi_{2T}\,\Psi
u\|_{L^{4}_{t}\ell^{4}}\|\chi_{2T}\,\Psi
v\|_{L^{4}_{t}\ell^{4}}\|\chi_{2T}\,\Psi w\|_{L^{4}_{t}\ell^{4}}$
$\displaystyle\lesssim_{b,b^{\prime}}T^{3b-b^{\prime}-1}\|u\|_{X^{b}_{T}}\|v\|_{X^{b}_{T}}\|w\|_{X^{b}_{T}}.$

As $X_{T}^{0}=L^{2}_{t}\ell^{2}$, we may use the embedding
$\ell^{2}\subset\ell^{\infty}$ and (2.20) to yield the bound

$\displaystyle\|\chi_{T}\,\Psi^{*}(\Psi u\odot\overline{\Psi v}\odot\Psi
w)\|_{X^{0}_{T}}$ $\displaystyle\lesssim\|\chi_{2T}\,\Psi
u\|_{L^{2}_{t}\ell^{2}}\|\chi_{2T}\,\Psi
v\|_{L^{\infty}_{t}\ell^{\infty}}\|\chi_{2T}\,\Psi
w\|_{L^{\infty}_{t}\ell^{\infty}}$
$\displaystyle\lesssim\|\chi_{2T}\,u\|_{L^{2}_{t}\ell^{2}}\|\chi_{2T}\,v\|_{L^{\infty}_{t}\ell^{2}}\|\chi_{2T}\,w\|_{L^{\infty}_{t}\ell^{2}}$
$\displaystyle\lesssim_{b}T^{3b-1}\|u\|_{X^{b}_{T}}\|v\|_{X^{b}_{T}}\|w\|_{X^{b}_{T}}.$

We then interpolate between these two estimates to get

$\|\chi_{T}\,\Psi^{*}(\Psi u\odot\overline{\Psi v}\odot\Psi
w)\|_{X^{b-1}_{T}}\lesssim_{b}T^{2b}\|u\|_{X^{b}_{T}}\|v\|_{X^{b}_{T}}\|w\|_{X^{b}_{T}}.$

To complete the proof of (2.27), we apply (2.19) and (2.20) to estimate

$\displaystyle\left\|\chi_{T}\,\langle v,u\rangle w\right\|_{X_{T}^{b-1}}$
$\displaystyle\lesssim\|\chi_{2T}\,u\|_{L^{\infty}_{t}\ell^{2}}\|\chi_{2T}\,v\|_{L^{\infty}_{t}\ell^{2}}\|\chi_{2T}\,w\|_{X_{T}^{b-1}}\lesssim_{b}T^{2b}\|u\|_{X_{T}^{b}}\|v\|_{X_{T}^{b}}\|w\|_{X_{T}^{b}}.$

The remaining terms in $\mathcal{N}[u,v,w]$ are bounded using symmetric
estimates. ∎

###  2.6. Proof of Theorem 1.1

We now complete the proof of Theorem 1.1. Let $\frac{1}{4}<\beta<\frac{1}{2}$
and $\varepsilon>0$ be fixed as in the hypothesis, and choose some fixed

$\tfrac{1}{2}<b<\min\bigl{\\{}1-\beta,\tfrac{1}{2}+\tfrac{\varepsilon}{24}\bigr{\\}}.$

We subsequently ignore the dependence of the implicit constants on
$\beta,\varepsilon,b$.

Denoting the sample space by $\Omega$, for each $\omega\in\Omega$, let
$\Lambda=\Lambda(\omega)$, $\Psi=\Psi(\omega)$, and take $a=a(\omega)\in
C(\varmathbb{R};\varmathbb{C}^{2N+1})$ to be the corresponding (unique,
global) solution of (2.2). Let
$M=\lceil\frac{9}{4}+\frac{6}{\varepsilon}\rceil$ and define
$a^{\mathrm{app}}$, $a^{\mathrm{err}}$ as in (2.5), (2.8), respectively.

We first prove an estimate for the remainder term $a^{\mathrm{err}}$. Given
$N\geqslant 1$ and $1\leqslant T\leqslant N$, let $E_{N}\subseteq\Omega$ be
the event that

(2.29) $\displaystyle\tfrac{1}{N}\|a^{(m)}\|_{X_{T}^{b}}^{2}\leqslant
N^{\frac{\varepsilon}{3}}\left(\tfrac{T^{4/3}}{T_{\mathrm{kin}}}\right)^{m}\text{
for all $m\in\llbracket
1,M\rrbracket$}\quad\text{and}\quad\|\chi_{T}\mathcal{L}\|_{X_{T}^{b}\to
X_{T}^{b-1}}\leqslant
N^{\frac{\varepsilon}{6}}\left(\tfrac{T^{4/3}}{T_{\mathrm{kin}}}\right)^{\frac{1}{2}}.$

Applying Propositions 2.6 and 2.7, where we use that $T\leqslant N$ and
$4b-2<\varepsilon/6$ in the latter case, we see that (2.29) holds with
overwhelming probability. In particular, for any $K>0$ we have

(2.30) $\varmathbb P\bigl{[}E_{N}^{c}\bigr{]}\lesssim_{K}N^{-K},$

provided $N\gg 1$ is sufficiently large. Let us also note that, as $T\leqslant
N$ and $\mu=N^{\beta}$, Lemma 2.8 and our choice of $b$ ensure that

(2.31) $\|\chi_{T}\mathcal{N}[u,v,w]\|_{X_{T}^{b-1}}\leqslant
N\|u\|_{X_{T}^{b}}\|v\|_{X_{T}^{b}}\|w\|_{X_{T}^{b}},$

provided $N\gg 1$ is sufficiently large.

We first consider the contribution of $E_{N}$. If $\omega\in E_{N}$ and
$1\leqslant T\leqslant N^{-\varepsilon}T_{\mathrm{kin}}^{3/4}$, we define the
operator

$\displaystyle\Upsilon_{T}(a)(t)$
$\displaystyle:=\\!-i\chi_{T}(t)\\!\int_{0}^{t}\\!\\!\\!e^{-i(t-s)\Lambda}\chi_{T}(s)\Bigl{\\{}\mathcal{L}a(s)\\!+\\!3\mathcal{N}[a^{\mathrm{app}}(s),a(s),a(s)]\\!+\\!\mathcal{N}[a(s),a(s),a(s)]\\!+\\!\mathcal{E}(s)\Bigr{\\}}\,ds.$

We claim that for sufficiently large $N\gg 1$ the map $\Upsilon_{T}$ is a
contraction on the ball

$\mathcal{B}_{T}=\Bigl{\\{}a\in X_{T}^{b}:\|a\|_{X_{T}^{b}}\leqslant
N^{-\frac{3}{2}-\frac{\varepsilon}{2}}\Bigr{\\}}.$

To prove this, first apply (2.29) to bound

$\|\chi_{T}\,\mathcal{L}a\|_{X_{T}^{b-1}}\leqslant
N^{-\frac{\varepsilon}{2}}\|a\|_{X_{T}^{b}},$

where we note that our hypothesis on $T$ ensures that
$\frac{T^{4/3}}{T_{\mathrm{kin}}}\leqslant T^{-\frac{4}{3}\varepsilon}$.
Second, by applying (2.18) to get

(2.32) $\|\chi_{2T}\,a^{(0)}\|_{X_{T}^{b}}\lesssim N^{\frac{1}{2}},$

and (2.29) with (2.19) to bound

(2.33) $\|\chi_{2T}\,a^{(m)}\|_{X_{T}^{b}}\leqslant
N^{\frac{1}{2}+\frac{1-4m}{6}\varepsilon}\quad\text{for}\quad m\in\llbracket
1,M\rrbracket,$

we obtain the estimate

$\|\chi_{2T}\,a^{\mathrm{app}}\|_{X_{T}^{b}}\lesssim N^{\frac{1}{2}}.$

As a consequence, the inequality (2.31) gives us

$\displaystyle\|\chi_{T}\,\mathcal{N}[a^{\mathrm{app}},a,a]\|_{X_{T}^{b-1}}\lesssim
N^{\frac{3}{2}}\|a\|_{X_{T}^{b}}^{2}\quad\text{and}\quad\|\chi_{T}\,\mathcal{N}[a,a,a]\|_{X_{T}^{b-1}}\lesssim
N\|a\|_{X_{T}^{b}}^{3}.$

Third, the error $\mathcal{E}$ defined in (2.7) can be written as

$\mathcal{E}=\sum_{(m_{1},m_{2},m_{3})\in
S_{M}}\mathcal{N}\bigl{[}a^{(m_{1})},a^{(m_{2})},a^{(m_{3})}\bigr{]},$

where

$S_{M}=\bigl{\\{}(m_{1},m_{2},m_{3})\in\llbracket
0,M\rrbracket^{3}:m_{1}+m_{2}+m_{3}\geqslant M\bigr{\\}}.$

Using that $\frac{5}{2}+\frac{\varepsilon}{2}-\frac{2}{3}\varepsilon
M\leqslant-\frac{3}{2}-\varepsilon$, we may apply (2.31), (2.32), and (2.33)
to bound

$\|\chi_{T}\,\mathcal{E}\|_{X_{T}^{b-1}}\lesssim
N^{-\frac{3}{2}-\varepsilon}.$

Combining these estimates with (2.21), for $a\in\mathcal{B}_{T}$ we have

$\|\Upsilon_{T}(a)\|_{X_{T}^{b}}\lesssim N^{-\frac{3}{2}-\varepsilon}.$

Applying similar estimates for the difference of two solutions, for all
$a,\widetilde{a}\in\mathcal{B}_{T}$ we have

$\|\Upsilon_{T}(a)-\Upsilon_{T}(\widetilde{a})\|_{X_{T}^{b}}\lesssim
N^{-\frac{\varepsilon}{2}}\|a-\widetilde{a}\|_{X_{T}^{b}}.$

As a consequence, for all sufficiently large $N\gg 1$ the map $\Upsilon_{T}$
is a contraction on the ball $\mathcal{B}_{T}$, as claimed.

We now apply Banach’s Fixed Point Theorem to show that for sufficiently large
$N\gg 1$, each $\omega\in E_{N}$, and each $1\leqslant T\leqslant
N^{-\varepsilon}T_{\mathrm{kin}}^{3/4}$ there exists a unique fixed point
$a^{\mathrm{err}}_{T}\in\mathcal{B}_{T}$ of $\Upsilon_{T}$. By uniqueness of
solutions to the ODE (1.5), we see that
$a^{\mathrm{err}}(t)=a_{T}^{\mathrm{err}}(t)$ for all $|t|\leqslant T$.
Consequently, recalling our choice of $b$, the estimate (2.20) shows that

(2.34) $\tfrac{1}{N}\varmathbb E\left[\varmathbb
1_{E_{N}}\|a^{\mathrm{err}}(t)\|_{\ell^{2}}^{2}\right]\leqslant
N^{-4}\quad\text{for all}\quad|t|\leqslant
N^{-\varepsilon}T_{\mathrm{kin}}^{3/4}.$

To control the contribution of the event $E_{N}^{c}$, we first use the
embedding $\ell^{2}\subset\ell^{\infty}$ to bound

$\|\mathcal{N}[a,b,c]\|_{\ell^{2}}\lesssim\tfrac{\mu^{2}}{N}\|a\|_{\ell^{2}}\|b\|_{\ell^{2}}\|c\|_{\ell^{2}}.$

Noting that $\|a^{(0)}(t)\|_{\ell^{2}}\lesssim\sqrt{N}$, we may then use the
definition (2.6) and induction on $m\geqslant 0$ to obtain the crude estimate

$\|a^{(m)}(t)\|_{\ell^{2}}\lesssim_{m}(\mu t)^{m}\sqrt{N}.$

Combining this with the conservation of mass, which gives us the bound
$\|a(t)\|_{\ell^{2}}\lesssim\sqrt{N}$, we have

$\|a^{\mathrm{err}}(t)\|_{\ell^{2}}\lesssim\langle\mu^{2}t\rangle^{M}\sqrt{N}.$

In particular, with $\mu=N^{\beta}$, the estimate (2.30) gives us the bound

$\tfrac{1}{N}\varmathbb E\left[\varmathbb
1_{E_{N}^{c}}\|a^{\mathrm{err}}(t)\|_{\ell^{2}}^{2}\right]\lesssim_{K}N^{-K}\quad\text{for
all}\quad|t|\leqslant N^{-\varepsilon}T_{\mathrm{kin}}^{3/4}\quad\text{and
any}\quad K>0.$

After possibly increasing the size of $N\gg 1$, we may combine this bound with
(2.34) to get

(2.35) $\tfrac{1}{N}\varmathbb
E\|a^{\mathrm{err}}(t)\|_{\ell^{2}}^{2}\leqslant N^{-4}\quad\text{for
all}\quad|t|\leqslant N^{-\varepsilon}T_{\mathrm{kin}}^{3/4},$

i.e., we no longer need to condition on the event $E_{N}$.

Let us now take $N^{\varepsilon}\leqslant t\leqslant
N^{-\varepsilon}T_{\mathrm{kin}}^{3/4}$. From Proposition 2.6 and (2.20), for
any $\delta>0$ we may bound

(2.36) $\sum_{m=3}^{M}\tfrac{1}{N}\varmathbb
E\|a^{(m)}(t)\|_{\ell^{2}}^{2}\lesssim_{\delta}N^{\delta}\frac{t^{4}}{T_{\mathrm{kin}}^{3}}.$

Similarly, applying Propositions 2.5, 2.6, and (2.20), for any $\delta>0$ we
have

(2.37) $\frac{1}{N}\left\|\varmathbb
E\bigl{|}a_{k}^{(0)}(t)+a_{k}^{(1)}(t)+a_{k}^{(2)}(t)\bigr{|}^{2}-\bigl{|}A\bigl{(}\tfrac{k}{N}\bigr{)}\bigr{|}^{2}-\tfrac{t}{T_{\mathrm{kin}}}\mathbf{C}\bigl{[}|A|^{2}\bigr{]}\bigl{(}\tfrac{k}{N}\bigr{)}\right\|_{\ell_{k}^{1}}\lesssim_{\delta}\frac{t}{T_{\mathrm{kin}}}N^{\delta-\frac{\varepsilon}{2}}.$

Combining the estimates (2.35), (2.36), and (2.37) with the fact that
$\frac{t}{T_{\mathrm{kin}}}\geqslant\frac{1}{N}$, we obtain

$\mathrm{LHS}\eqref{R}\lesssim_{\delta}\frac{t}{T_{\mathrm{kin}}}\left[N^{\delta-\frac{\varepsilon}{2}}+N^{\delta}\frac{t^{3}}{T_{\mathrm{kin}}^{2}}+N^{-3}\right].$

The estimate (1.14) now follows from taking $\delta=\varepsilon/4$.∎

##  3\. Random Matrix estimates

In this section we establish estimates concerning the random matrix $H$. A
property of the Gaussian Unitary Ensemble (GUE) is that its eigenvectors and
eigenvalues form two independent systems, which we can study separately. The
vector of eigenvalues, $\lambda$, is distributed according to (1.3). The
matrix of eigenvectors, $\Psi=(\psi_{jk})$, is distributed according to the
Haar measure on $\varmathbb U(2N+1)$, also known as the circular unitary
ensemble ($\rm{CUE}$). Section 3.1 presents a method to deal with general
expressions involving entries of a ${\rm{CUE}}(d)$ matrix, and can be read
independently from the rest of the paper. This method is then applied to
particular expressions involving entries of $\Psi$ in Section 3.2. Finally,
Section 3.3 presents the application of the local semicircle law to our
estimates involving the eigenvalues of $H$.

###  3.1. Weingarten calculus

Given two integers $q,d\geqslant 1$, the expectation of a product of $2q$
terms from a Haar-distributed unitary matrix of size $d\times d$ is given
explicitly by the following result of Collins [7, Theorem 2.1] in the so
called stable range $d\geqslant q$:

######  Theorem 3.1 (Fundamental Theorem of Weingarten calculus).

If $\Psi=(\psi_{jk})_{j,k=1}^{d}$ is a Haar-distributed unitary matrix, then
for any choice of indices
$(j_{l},k_{l},j_{l}^{\prime},k_{l}^{\prime})_{l=1}^{q}$ with $1\leqslant
q\leqslant d$,

(3.1) $\varmathbb E\left[\psi_{j_{1}k_{1}}\cdots\psi_{j_{q}k_{q}}\
\overline{\psi_{j_{1}^{\prime}k_{1}^{\prime}}}\cdots\overline{\psi_{j_{q}^{\prime}k_{q}^{\prime}}}\right]=\\!\sum_{\sigma,\tau\in\mathfrak{S}_{q}}\delta_{j_{1},j_{\tau(1)}^{\prime}}\\!\cdots\delta_{j_{q},j_{\tau(q)}^{\prime}}\
\delta_{k_{1},k_{\sigma(1)}^{\prime}}\\!\cdots\delta_{k_{q},k_{\sigma(q)}^{\prime}}\operatorname{Wg}(\sigma\tau^{-1},d).$

The rational function $\operatorname{Wg}(\omega,\cdot)$ depends on the cycle
structure of the permutation $\omega\in\mathfrak{S}_{q}$, and thus also
implicitly on $q$; we refer the reader to [7, 9] for a definition. Of
particular interest is the leading order term, which is separated from lower
order terms by a $d^{-2}$ gap. Indeed,

(3.2) $\mathrm{Wg}(\omega,d)=d^{-2q+\mathscr{C}(\omega)}\text{M\oe
b}(\omega)\left(1+O(d^{-2})\right)$

where $\mathscr{C}(\omega)$ denotes the number of cycles111Note that equation
(3.2) is equivalent to [10, Corollary 2.7] with the relation
$|\omega|=q-\mathscr{C}(\omega)$ between $\mathscr{C}(\omega)$ and $|\omega|$,
the number of transpositions needed to generate $\omega$. of $\omega$, and the
Mœbius function is defined as

(3.3) ${\text{M\oe
b}}(\omega):=\prod_{i=1}^{\mathscr{C}(\omega)}(-1)^{C_{i}-1}\frac{(2C_{i}-2)!}{C_{i}!(C_{i}-1)!},$

with $C_{1},\dots,C_{\mathscr{C}(\omega)}$ the respective sizes of the cycles
of $\omega$. In particular, the case $\sigma=\tau$ in (3.1) corresponds to an
exact pairing of edges, and yields a contribution of the same leading order as
one would obtain from Wick’s formula when considering Gaussian variables of
variance $1/d$, namely:

(3.4)
$\operatorname{Wg}({\mathrm{Id}}_{\mathfrak{S}_{q}},d)=d^{-q}+O(d^{-q-2}).$

To facilitate the computation of general bounds, we introduce a structure of
directed bipartite graphs that match the elements of Weingarten’s formula
(3.1). For a term of the form

(3.5) $\psi_{j_{1}k_{1}}\dots\psi_{j_{q}k_{q}}\
\overline{\psi_{j_{1}^{\prime}k_{1}^{\prime}}}\dots\overline{\psi_{j_{q}^{\prime}k_{q}^{\prime}}},$

we represent each distinct row index from the set
$\\{j_{1},\dots,j_{q},j_{1}^{\prime},\dots,j_{q}^{\prime}\\}$ as a black
vertex  and each distinct column index from the set
$\\{k_{1},\dots,k_{q},k_{1}^{\prime},\dots,k_{q}^{\prime}\\}$ as a white
vertex . The random variables $\psi_{j_{r}k_{r}}$ are represented as edges
directed from the black vertex $j_{r}$ to the white vertex $k_{r}$, whereas
the conjugated random variables
$\overline{\psi_{j_{r}^{\prime}k_{r}^{\prime}}}$ are represented as edges
directed from the white vertex $k_{r}^{\prime}$ to the black vertex
$j_{r}^{\prime}$. We refer to the corresponding directed graph $G$ as the
_Weingarten graph_ 222This is not to be confused with the different notion of
Weingarten graphs used in [8], where vertices represent permutations. of the
expression (3.5). Figure 2 gives three basic examples. Reciprocally, we will
use the shorthand $\psi_{G}$ to denote a term such as (3.5), where the
structure of the indices is encoded in the graph $G$.

$\ell$$j$$\psi_{j\ell}$$\overline{\psi_{j\ell}}$$\ell$$k$$j$$\psi_{j\ell}$$\psi_{jk}$$\overline{\psi_{j\ell}}$$\overline{\psi_{jk}}$$i$$j$$\ell$$k$$\psi_{ik}$$\psi_{j\ell}$$\overline{\psi_{jk}}$$\overline{\psi_{i\ell}}$
Figure 2. Weingarten graphs associated to $|\psi_{jl}|^{4}$,
$|\psi_{jk}|^{2}|\psi_{j\ell}|^{2}$ and
$\psi_{ik}\overline{\psi_{jk}}\psi_{j\ell}\overline{\psi_{i\ell}}$, where
$i\neq j$ and $k\neq\ell$. Note that identities between row and column
vertices, such as $j=l$, do not appear on the graph – nor do they make a
difference to the corresponding computation.

Given a Weingarten graph $G$, we define a _circuit covering_ of $G$ to be a
partition of the edges of $G$ in directed circuits. The reason for considering
circuit coverings is that they are equivalent to the pairs of permutations
$(\sigma,\tau)\in\mathfrak{S}_{q}^{2}$ that give a nonzero contribution in
(3.1).

######  Fact 3.2.

Let $G$ be the Weingarten graph of (3.5). Then, the pairs of permutations
$(\sigma,\tau)\in\mathfrak{S}_{q}^{2}$ meeting the admissibility conditions

(3.6) $j_{l}=j_{\tau(l)}^{\prime},\qquad
k_{l}=k_{\sigma(l)}^{\prime}\qquad\text{for all}\quad l=1,\dots,q,$

are in a natural one-to-one correspondence with the circuit coverings of $G$,
in such a way that each $n$-cycle of $\sigma\tau^{\\!-1}$ corresponds to a
$2n$-circuit in $G$.

###### Proof.

The bipartite structure of the Weingarten graph forces the $\psi$-edges and
$\overline{\psi}$-edges of a circuit to alternate. As illustrated on Figure 3,
$\sigma$ and $\tau$ encode the two kinds of steps: if
$\overline{\psi_{j_{r}^{\prime}k_{r}^{\prime}}}$ follows $\psi_{j_{l}k_{l}}$
in the circuit covering, then the admissibility condition
$k_{l}=k_{r}^{\prime}$ is implied and the step is encoded by $\sigma(l)=r$;
similarly, if $\psi_{j_{m}k_{m}}$ follows
$\overline{\psi_{j_{r}^{\prime}k_{r}^{\prime}}}$, then the admissibility
condition $j_{m}=j_{r}^{\prime}$ is implied and is the step is encoded by
$r=\tau(m)$. Reciprocally, the sequences of edges obtained by applying
$\sigma$ and $\tau^{-1}$ alternatively forms a circuit covering if and only if
the admissibility conditions are met. It follows by construction that each
cycle of size $n$ in $\sigma\tau^{\\!-1}$ corresponds to a circuit of size
$2n$ on the graph $G$. ∎

$\sigma$$\tau^{-1}$$\ddots$$j_{l}$$\psi_{j_{l}k_{l}}$$k_{l}=k^{\prime}_{r}$$\overline{\psi_{j_{r}^{\prime}k_{r}^{\prime}}}$$j_{r}^{\prime}=j_{m}$$\psi_{j_{m}k_{m}}$$k_{m}$$\ddots$
Figure 3. Illustration of the correspondence between circuits and admissible
pairs $(\sigma,\tau)$: the edge $\psi_{j_{l}k_{l}}$ can be followed by
$\overline{\psi_{j_{r}^{\prime}k_{r}^{\prime}}}$ in the circuit if and only if
$k_{l}=k_{r^{\prime}}$, and so on.

We call pairs $(\sigma,\tau)\in\mathfrak{S}_{q}^{2}$ that verify the
admissibility conditions (3.6) covering pairs of $\psi_{G}$, and denote their
set by ${\mathscr{C}\mathscr{P}}(\psi_{G})$. In light of Fact 3.2, we will
simultaneously think of these as pairs of permutations and as circuit
coverings of the graph $G$.

######  Proposition 3.3.

Let $G$ be a Weingarten graph with $E=2q$ edges, $V$ vertices, and $c$
connected components. Then, we have the general bounds

(3.7) $\varmathbb
E\left[\psi_{G}\right]=O_{q}\left(d^{-q}\right)\qquad\text{and}\qquad\varmathbb
E\left[\psi_{G}\right]=O_{q}\left(d^{c-V}\right)$

###### Proof.

It follows from Theorem 3.1, (3.2) and Fact 3.2 that the leading order is
given by $d^{-2q+n}$ with $n$ being the maximal number of circuits in a
circuit covering, or equivalently the maximal number of cycles of
$\sigma\tau^{\\!-1}$ with
$(\sigma,\tau)\in{\mathscr{C}\mathscr{P}}(\psi_{G})$. This is at most
$d^{-q}$, because $n=\mathscr{C}(\sigma\tau^{\\!-1})\leqslant q$. We now prove
that this is also bounded by $d^{c-V}$. In a connected graph, the inequality
$V-E\leqslant 1$ holds, so that more generally we obtain

(3.8) $V-E\leqslant c$

by summing over the connected components. Now, consider the graph
$\widetilde{G}$ obtained by removing one edge in each circuit. The fact that
each edge appears in exactly one circuit ensures, by inspection, that
$\widetilde{G}$ has $\widetilde{c}=c$ connected components, $\widetilde{V}=V$
vertices and $\widetilde{E}=E-n$ edges. Applying (3.8) to $\widetilde{G}$,

(3.9) $V-E+n=\widetilde{V}-\widetilde{E}\leqslant\widetilde{c}=c.$

This implies that $-2q+n\leqslant c-V$, which concludes the proof of the
second bound. ∎

Circuits of size $2$ (corresponding to fixed points of $\sigma\tau^{-1}$) play
a particular role in maximizing the number of circuits in a circuit covering.
We refer to such a circuit between vertex $a$ and vertex $b$ as the
$2$-circuit $(a,b)$. The next lemma simplifies the identification of the
leading order by allowing us to choose $2$-circuits first, at no risk to the
final estimate. Note that no such statement holds for circuits of size $4$ or
more, so that the greedy strategy (i.e., choosing the smallest possible
circuits first) is not optimal in general.

######  Proposition 3.4.

If the expression $|\psi_{ab}|^{2}\psi_{G}$ has at least one covering pair,
then it has a covering pair with largest possible number of circuits that
includes the $2$-circuit $(a,b)$. Equivalently: if
${\mathscr{C}\mathscr{P}}(|\psi_{ab}|^{2}\psi_{G})\neq\emptyset$, then

(3.10)
$\max_{(\sigma,\tau)\in\mathrm{{\mathscr{C}\mathscr{P}}}(|\psi_{ab}|^{2}\psi_{G})}\mathscr{C}(\sigma\tau^{\\!-1})=1+\max_{(\sigma,\tau)\in\mathrm{{\mathscr{C}\mathscr{P}}}(\psi_{G})}\mathscr{C}(\sigma\tau^{\\!-1}).$

###### Proof.

As illustrated in Figure 4, if $(\sigma,\tau)$ is a maximal covering pair of
$|\psi_{ab}|^{2}\psi_{G}$ which does not include the $2$-circuit $(a,b)$, then
$\psi_{ab}$ is part of a circuit $(a,b,L_{1})$ and $\overline{\psi_{ab}}$ is
part of another circuit $(b,a,L_{2})$. But then, considering instead the
$2$-circuit $(a,b)$ and the concatenation of $L_{1}$ and $L_{2}$ yields a
circuit-covering with same number of circuits (assumed maximal) that includes
the $2$-circuit $(a,b)$. ∎

$b$$a$$\psi_{ab}$$\overline{\psi_{ab}}$$\vdots$$L_{2}$$\vdots$$L_{1}$$\Rightarrow$

$b$$a$$\psi_{ab}$$\overline{\psi_{ab}}$$\vdots$$L_{2}$$\vdots$$L_{1}$

Figure 4. Illustration of the proof of Proposition 3.4: if the first
configuration is realizable, so is the second one, which has the same number
of circuits and includes the $2$-circuit $(a,b)$.

######  Remark 3.5.

In the rest of this section, we use the following notations to relate cycle
structures of permutations of different sizes:

  * •

if $\sigma\in\mathfrak{S}_{q}$, then ${\scriptstyle\circ}\hskip
1.084pt\sigma\in\mathfrak{S}_{q+1}$ stands for the permutation obtained by
extending $\sigma$ by a fixed point;

  * •

$\mathbin{\rotatebox[origin={c}]{-90.0}{ $\cancer$}}\hskip
1.4457pt\sigma\in\mathfrak{S}_{q+2}$ stands for the extension of
$\sigma\in\mathfrak{S}_{q}$ by a transposition;

  * •

such notations are used with multiplicative convention, so that
${\scriptstyle\circ}\hskip
1.084pt^{\\!k}\mathbin{\rotatebox[origin={c}]{-90.0}{ $\cancer$}}\hskip
1.4457pt^{\\!l}\sigma\in\mathfrak{S}_{q+k+2l}$ stands for the extension of
$\sigma\in\mathfrak{S}_{q}$ by $k$ fixed points and $l$ transpositions;

  * •

in the same spirit, $(i,{\scriptstyle\circ}\hskip 1.084pt)$ stands for the
transposition between $i$ and one extra point added to the domain, so that
$(i,{\scriptstyle\circ}\hskip 1.084pt)\sigma$ means that this extra point is
inserted between $\sigma^{-1}(i)$ and $i$.

We use the shorthand $i\in\sigma$ for sums over all indices in the domain of
$\sigma$. An extra point is sometimes denoted by
${\scriptstyle\hat{\circ}}\hskip 1.084pt$ to avoid an ambiguity. Note that
$\operatorname{Wg}(\sigma,\cdot)$ only depends on the cycle structure of
$\sigma$, and not on any particular labelling of indices.

The next theorem deals with expressions involving centered terms of order $2$,
i.e., factors $|\psi_{ab}|^{2}-\frac{1}{d}$, which we call atoms. Having these
terms centered amounts to ‘penalizing’ circuit coverings that contain the
corresponding $2$-circuits. In this context, we take
$\mathfrak{A}(\sigma,\tau)$ to be the set of $2$-circuits corresponding to
atoms implied by $(\sigma,\tau)$, and
$\mathfrak{a}(\sigma,\tau)=|\mathfrak{A}(\sigma,\tau)|$. We also define the
functions $\rho_{K}(\cdot)$ on permutations with at least $K$ fixed points, by
the following formulas. For odd terms,

(3.11) $\rho_{2k+1}({\scriptstyle\circ}\hskip
1.084pt^{\\!2k+1}\omega)=(2k)!!\sum_{l=0}^{k}\frac{(2k-2l-1)!!}{(2k-2l)!!}\left(4l+\sum_{i=1}^{\mathscr{C}(\omega)}\frac{4C_{i}^{2}-2C_{i}}{C_{i}+1}\right),$

and for even terms,

(3.12) $\rho_{2k}({\scriptstyle\circ}\hskip 1.084pt^{\\!2k}\omega)=(2k-1)!!$

where $n!!$ is the semifactorial of the natural number $n$ (product of all
smaller integers of same parity), and the convention that $(-1)!!=1$.

######  Theorem 3.6.

If
$(a_{l},b_{l})_{l=1}^{L},(j_{r},k_{r},j_{r}^{\prime},k_{r}^{\prime})_{r=1}^{R}$
are indices and
$\psi_{G}=\prod_{r}\psi_{j_{r}k_{r}}\overline{\psi_{j_{r}^{\prime}k_{r}^{\prime}}}$,
then

(3.13) $\varmathbb
E\left[\prod_{l=1}^{L}\left(|\psi_{a_{l}b_{l}}|^{2}-\frac{1}{d}\right)\psi_{G}\right]=\sum_{(\sigma,\tau)\in{\mathscr{C}\mathscr{P}}(\prod_{l}|\psi_{a_{l}b_{l}}|^{2}\psi_{G})}d^{-2\left\lfloor\frac{\mathfrak{a}(\sigma,\tau)+1}{2}\right\rfloor}\rho_{\mathfrak{a}(\sigma,\tau)}(\sigma\tau^{\\!-1})\operatorname{Wg}(\sigma\tau^{\\!-1},d)\left(1+O(d^{-1})\right).$

To the best of our knowledge, the first result of this kind was obtained in
the recent work of Bordenave and Collins [4] for products of centered terms.
In principle, their approach could be extended to also include a non-centered
term $\psi_{G}$, which would lead to a generalization of Theorem 3.6 with
atoms of any size, and to a combinatorial understanding of the multiplicative
penalty, which in this case reduces to the exponent
$-2\left\lfloor\frac{\mathfrak{a}+1}{2}\right\rfloor$ and the constant
$\rho_{\mathfrak{a}}(\sigma\tau^{\\!-1})$.

######  Lemma 3.7.

For any integer $K\geqslant 1$ and any permutation
$\sigma\in\mathfrak{S}_{q}$,

(3.14)
$\sum_{k=0}^{K}\binom{K}{k}\left(-\frac{1}{d}\right)^{k}\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!K-k}\sigma,d)=d^{-2\left\lfloor\frac{K+1}{2}\right\rfloor}\rho_{K}({\scriptstyle\circ}\hskip
1.084pt^{\\!K}\sigma)\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!K}\sigma,d)\left(1+O\left(d^{-1}\right)\right),$

with $\rho_{K}(\cdot)$ defined by (3.11), (3.12).

###### Proof.

The essential input is a particular case of [8, Proposition 2.2], which we
rewrite as

(3.15) $\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt\sigma,d)-\frac{1}{d}\operatorname{Wg}(\sigma,d)=-\frac{1}{d}\sum_{i\in\sigma}\operatorname{Wg}((i,{\scriptstyle\circ}\hskip
1.084pt)\sigma,d)$

with the notations introduced above. We first treat the initial cases. For
$K=1$, the left hand side can directly be rewritten using (3.15), as

(3.16) $\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt\sigma,d)-\frac{1}{d}\operatorname{Wg}(\sigma,d)=-\frac{1}{d}\sum_{i\in\sigma}\operatorname{Wg}((i,{\scriptstyle\circ}\hskip
1.084pt)\sigma)=d^{-2}\rho_{1}({\scriptstyle\circ}\hskip
1.084pt\sigma)\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt\sigma,d)\left(1+O(d^{-1})\right)$

where the last estimate comes from comparing leading order terms with (3.2),
which gives the value

(3.17) $\rho_{1}({\scriptstyle\circ}\hskip
1.084pt\sigma)=-\sum_{j\in\sigma}\frac{\text{M\oe
b}((j,{\scriptstyle\circ}\hskip 1.084pt)\sigma)}{\text{M\oe
b}({\scriptstyle\circ}\hskip
1.084pt\sigma)}=\sum_{i=1}^{\mathscr{C}(\sigma)}C_{i}\times\frac{4C_{i}-2}{C_{i}+1},$

noting that a cycle of size $C_{i}$ corresponds to $C_{i}$ terms with the same
contribution in the first sum. For $K=2$, the expression at stake is

(3.18) $\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!2}\sigma,d)-\frac{2}{d}\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt\sigma,d)+\frac{1}{d^{2}}\operatorname{Wg}(\sigma,d),$

which we rewrite as a difference of differences

(3.19) $\left(\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!2}\sigma,d)-\frac{1}{d}\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt\sigma,d)\right)-\frac{1}{d}\left(\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt\sigma,d)-\frac{1}{d}\operatorname{Wg}(\sigma,d)\right).$

Each term can be simplified according to (3.15), yielding

(3.20) $-\frac{1}{d}\operatorname{Wg}(\mathbin{\rotatebox[origin={c}]{-90.0}{
$\cancer$}}\hskip
1.4457pt\sigma,d)-\frac{1}{d}\sum_{i\in\sigma}\left(\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt(i,{\scriptstyle\hat{\circ}}\hskip
1.084pt)\sigma,d)-\frac{1}{d}\operatorname{Wg}((i,{\scriptstyle\hat{\circ}}\hskip
1.084pt)\sigma,d)\right),$

and these last differences can be dealt with using (3.15) again, yielding a
double sum of terms with low contribution. The final form of the expression is

(3.21) $-\frac{1}{d}\operatorname{Wg}(\mathbin{\rotatebox[origin={c}]{-90.0}{
$\cancer$}}\hskip
1.4457pt\sigma,d)+\frac{1}{d^{2}}\sum_{i\in\sigma}\sum_{j\in(i,{\scriptstyle\hat{\circ}}\hskip
1.084pt)\sigma}\operatorname{Wg}((j,{\scriptstyle\circ}\hskip
1.084pt)(i,{\scriptstyle\hat{\circ}}\hskip
1.084pt)\sigma,d)=d^{-2}\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!2}\sigma,d)\left(1+O\left(d^{-1}\right)\right),$

which comes from comparing leading order terms with (3.2). This gives
$\rho_{2}({\scriptstyle\circ}\hskip 1.084pt^{\\!2}\sigma)=1$. Note that we
have only lost a factor $d^{-2}$ in leading order, which is the same penalty
as for $K=1$. We will see that this exponent then decreases every two steps,
so that the the resulting exponent is
$-2\left\lfloor\frac{K+1}{2}\right\rfloor$. We proceed by strong induction on
$K$, with the induction hypothesis that the claim holds for $K-2$ and $K-1$
(and for all permutations in any case). We will use the property of binomial
coefficients through the formula

(3.22)
$\sum_{k=0}^{K}\binom{K}{k}a_{k}=\sum_{k=0}^{K-1}\binom{K-1}{k}(a_{k}+a_{k+1}),$

which yields

$\sum_{k=0}^{K}\binom{K}{k}\left(-\frac{1}{d}\right)^{k}\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!K-k}\sigma,d)=\sum_{k=0}^{K-1}\binom{K-1}{k}\left(-\frac{1}{d}\right)^{k}\left(\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!K-k}\sigma,d)-\frac{1}{d}\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!K-k-1}\sigma,d)\right),$

in which we replace the differences using (3.15), which creates two kinds of
terms:

(3.23)
$\sum_{k=0}^{K-1}\binom{K-1}{k}\left(-\frac{1}{d}\right)^{k+1}\left((K-k-1)\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!K-k-1}\mathbin{\rotatebox[origin={c}]{-90.0}{ $\cancer$}}\hskip
1.4457pt\sigma,d)+\sum_{i\in\sigma}\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!K-k-1}(i,{\scriptstyle\hat{\circ}}\hskip
1.084pt)\sigma,d)\right).$

For the first terms, we rewrite

(3.24) $(K-k-1)\binom{K-1}{k}=(K-1)\binom{K-2}{k},$

and then use the induction hypotheses at step $K-2$ for the permutation
$\mathbin{\rotatebox[origin={c}]{-90.0}{ $\cancer$}}\hskip 1.4457pt\sigma$.
This part of (3.23) becomes

(3.25)
$-\frac{1}{d}(K-1)\sum_{k=0}^{K-2}\binom{K-2}{k}\left(-\frac{1}{d}\right)^{k}\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!K-2-k}\mathbin{\rotatebox[origin={c}]{-90.0}{ $\cancer$}}\hskip
1.4457pt\sigma,d)\\\
=d^{-2\left\lfloor\frac{K+1}{2}\right\rfloor}(K-1)\rho_{K-2}({\scriptstyle\circ}\hskip
1.084pt^{\\!K-2}\mathbin{\rotatebox[origin={c}]{-90.0}{ $\cancer$}}\hskip
1.4457pt\sigma)\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!K}\sigma,d)\left(1+O\left(d^{-1}\right)\right).$

The second terms in (3.23) are dealt with using the induction hypotheses at
step $K-1$ for the permutations $(i,{\scriptstyle\hat{\circ}}\hskip
1.084pt)\sigma$, yielding

(3.26)
$-\frac{1}{d}\sum_{i\in\sigma}\sum_{k=0}^{K-1}\binom{K-1}{k}\left(-\frac{1}{d}\right)^{k}\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!K-k-1}(i,{\scriptstyle\hat{\circ}}\hskip 1.084pt)\sigma,d)\\\
=-d^{-2\left\lfloor\frac{K}{2}\right\rfloor-1}\sum_{i\in\sigma}\rho_{K-1}({\scriptstyle\circ}\hskip
1.084pt^{\\!K-1}(i,{\scriptstyle\hat{\circ}}\hskip
1.084pt)\sigma)\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!K-1}(i,{\scriptstyle\hat{\circ}}\hskip
1.084pt)\sigma,d)\left(1+O\left(d^{-1}\right)\right),$

which is a
$O\left(d^{-2\left\lfloor\frac{K+2}{2}\right\rfloor}\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!K}\sigma,d)\right)$, so either the same order as (3.25) if $K$ is
odd, or strictly smaller if $K$ is even. Accordingly, the general formula for
the coefficients $\rho_{K}$ is found by distinguishing the odd and even terms.
When $K=2k$, only the first term (3.25) contributes, yielding

(3.27) $\rho_{2k}({\scriptstyle\circ}\hskip
1.084pt^{\\!2k}\sigma)=(2k-1)\rho_{2k-2}({\scriptstyle\circ}\hskip
1.084pt^{\\!2k-2}\mathbin{\rotatebox[origin={c}]{-90.0}{ $\cancer$}}\hskip
1.4457pt\sigma).$

Together with the fact that $\rho_{2}\equiv 1$, it follows immediately that
$\rho_{2k}({\scriptstyle\circ}\hskip 1.084pt^{\\!2k}\sigma)=(2k-1)!!$ and does
not depend on the structure of $\sigma$, whence (3.12). When $K=2k+1$, both
terms (3.25) and (3.26) contribute, which implies

(3.28) $\rho_{2k+1}({\scriptstyle\circ}\hskip
1.084pt^{\\!2k+1}\sigma)=2k\rho_{2k-1}({\scriptstyle\circ}\hskip
1.084pt^{\\!2k-1}\mathbin{\rotatebox[origin={c}]{-90.0}{ $\cancer$}}\hskip
1.4457pt\sigma)-\sum_{i\in\sigma}\rho_{2k}({\scriptstyle\circ}\hskip
1.084pt^{\\!2k}(i,{\scriptstyle\hat{\circ}}\hskip
1.084pt)\sigma)\frac{\text{M\oe b}((i,{\scriptstyle\hat{\circ}}\hskip
1.084pt)\sigma)}{\text{M\oe b}({\scriptstyle\circ}\hskip 1.084pt\sigma)}.$

Replacing the even terms $\rho_{2k}$ by their value, and then recognizing
$\rho_{1}$ from (3.17), we write:

$\displaystyle\rho_{2k+1}({\scriptstyle\circ}\hskip 1.084pt^{\\!2k+1}\sigma)$
$\displaystyle=2k\rho_{2k-1}({\scriptstyle\circ}\hskip
1.084pt^{\\!2k-1}\mathbin{\rotatebox[origin={c}]{-90.0}{ $\cancer$}}\hskip
1.4457pt\sigma)-\sum_{i\in\sigma}(2k-1)!!\frac{\text{M\oe
b}((i,{\scriptstyle\hat{\circ}}\hskip 1.084pt)\sigma)}{\text{M\oe
b}({\scriptstyle\circ}\hskip 1.084pt\sigma)}$
$\displaystyle=2k\rho_{2k-1}({\scriptstyle\circ}\hskip
1.084pt^{\\!2k-1}\mathbin{\rotatebox[origin={c}]{-90.0}{ $\cancer$}}\hskip
1.4457pt\sigma)+(2k-1)!!\rho_{1}({\scriptstyle\circ}\hskip 1.084pt\sigma).$

The solution of this recurrence is given by the formula

(3.29) $\rho_{2k+1}({\scriptstyle\circ}\hskip
1.084pt^{\\!2k+1}\sigma)=(2k)!!\sum_{l=0}^{k}\frac{(2k-2l-1)!!}{(2k-2l)!!}\rho_{1}({\scriptstyle\circ}\hskip
1.084pt\mathbin{\rotatebox[origin={c}]{-90.0}{ $\cancer$}}\hskip
1.4457pt^{\\!l}\sigma),$

and the definition (3.11) follows from the fact that
$\rho_{1}({\scriptstyle\circ}\hskip
1.084pt\mathbin{\rotatebox[origin={c}]{-90.0}{ $\cancer$}}\hskip
1.4457pt^{\\!l}\sigma)=4l+\rho_{1}({\scriptstyle\circ}\hskip 1.084pt\sigma)$.
∎

We can now complete the proof of Theorem 3.6, which follows directly from
Lemma 3.7 once the terms are regrouped appropriately.

###### Proof of Theorem 3.6.

We illustrate the argument by treating the first few cases in details. For
$L=1$, one may distinguish between covering pairs
$(\sigma,\tau)\in{\mathscr{C}\mathscr{P}}(|\psi_{ab}|^{2}\psi_{G})$ that do
not include the $2$-circuit $(a,b)$ and those that do. The latter can be
decomposed as a circuit covering of $\Psi_{G}$ and the $2$-circuit $(a,b)$
(which yields a fixed point that we denote as ${\scriptstyle\circ}\hskip
1.084pt$, according to Remark 3.5), and then compared to the centered terms:

$\displaystyle\varmathbb
E\left[\left(|\psi_{ab}|^{2}-\frac{1}{d}\right)\Psi_{G}\right]$
$\displaystyle=\varmathbb
E\left(|\psi_{ab}|^{2}\Psi_{G}\right)-\frac{1}{d}\varmathbb
E\left(\Psi_{G}\right)$
$\displaystyle=\sum_{(\sigma,\tau)\in{\mathscr{C}\mathscr{P}}(|\psi_{ab}|^{2}\Psi_{G})}\operatorname{Wg}(\sigma\tau^{\\!-1},d)-d^{-1}\sum_{(\sigma,\tau)\in{\mathscr{C}\mathscr{P}}(\Psi_{G})}\operatorname{Wg}(\sigma\tau^{\\!-1},d)$
$\displaystyle=\sum_{\begin{subarray}{c}(\sigma,\tau)\in{\mathscr{C}\mathscr{P}}(|\psi_{ab}|^{2}\Psi_{G})\\\
(a,b)\notin\mathfrak{A}(\sigma,\tau)\end{subarray}}\operatorname{Wg}(\sigma\tau^{\\!-1},d)+\sum_{(\sigma,\tau)\in{\mathscr{C}\mathscr{P}}(\Psi_{G})}\left(\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt\sigma\tau^{\\!-1},d)-\frac{1}{d}\operatorname{Wg}(\sigma\tau^{\\!-1},d)\right)$

The first terms do not suffer a penalty and are kept untouched (which
corresponds to $\rho_{0}(\omega)=1$). The second terms suffer a multiplicative
penalty of $d^{-2}$ from Lemma 3.7, so that the total contribution is

(3.30)
$\sum_{\begin{subarray}{c}(\sigma,\tau)\in{\mathscr{C}\mathscr{P}}(|\psi_{ab}|^{2}\Psi_{G})\\\
(a,b)\notin\mathfrak{A}(\sigma,\tau)\end{subarray}}\operatorname{Wg}(\sigma\tau^{\\!-1},d)\
+\sum_{\begin{subarray}{c}(\sigma,\tau)\in{\mathscr{C}\mathscr{P}}(|\psi_{ab}|^{2}\Psi_{G})\\\
(a,b)\in\mathfrak{A}(\sigma,\tau)\end{subarray}}d^{-2}\rho_{1}(\sigma\tau^{\\!-1})\operatorname{Wg}(\sigma\tau^{\\!-1},d)\left(1+O\left(d^{-1}\right)\right),$

which is consistent with the claim, as $\mathfrak{a}=0$ in the first sum and
$\mathfrak{a}=1$ in the second. We now treat the case $L=2$, by first
expanding:

$\displaystyle\varmathbb
E\left[\left(|\psi_{a_{1}b_{1}}|^{2}-\frac{1}{d}\right)\left(|\psi_{a_{2}b_{2}}|^{2}-\frac{1}{d}\right)\Psi_{G}\right]$
$\displaystyle\qquad=\varmathbb
E\left[\left(|\psi_{a_{1}b_{1}}|^{2}|\psi_{a_{2}b_{2}}|^{2}-\frac{1}{d}|\psi_{a_{1}b_{1}}|^{2}-\frac{1}{d}|\psi_{a_{2}b_{2}}|^{2}+\frac{1}{d^{2}}\right)\Psi_{G}\right]$

and then rearranging the subsequent Weingarten terms depending on whether the
structure of the circuit covering explicitly excludes the presence of such
atoms. In other words, we consider a covering pair to include an atom
$(a_{l},b_{l})$ if it either implies it explicitly, or allows it implicitly
(i.e., the atom not being in the support). This leads to the following
decomposition:

$\displaystyle\sum_{\begin{subarray}{c}(\sigma,\tau)\in{\mathscr{C}\mathscr{P}}(|\psi_{a_{1}b_{1}}|^{2}|\psi_{a_{2}b_{2}}|^{2}\Psi_{G})\\\
(a_{1},b_{1}),(a_{2},b_{2})\notin\mathfrak{A}(\sigma,\tau)\end{subarray}}\operatorname{Wg}(\sigma\tau^{\\!-1},d)$
(terms excluding both atoms)
$\displaystyle+\sum_{\begin{subarray}{c}(\sigma,\tau)\in{\mathscr{C}\mathscr{P}}(|\psi_{a_{2}b_{2}}|^{2}\Psi_{G})\\\
(a_{2},b_{2})\notin\mathfrak{A}(\sigma,\tau)\end{subarray}}\left(\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt\sigma\tau^{\\!-1},d)-\frac{1}{d}\operatorname{Wg}(\sigma\tau^{\\!-1},d)\right)$
(terms excluding $(a_{2},b_{2})$ only)
$\displaystyle+\sum_{\begin{subarray}{c}(\sigma,\tau)\in{\mathscr{C}\mathscr{P}}(|\psi_{a_{1}b_{1}}|^{2}\Psi_{G})\\\
(a_{1},b_{1})\notin\mathfrak{A}(\sigma,\tau)\end{subarray}}\left(\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt\sigma\tau^{\\!-1},d)-\frac{1}{d}\operatorname{Wg}(\sigma\tau^{\\!-1},d)\right)$
(terms excluding $(a_{1},b_{1})$ only)
$\displaystyle+\sum_{(\sigma,\tau)\in{\mathscr{C}\mathscr{P}}(\Psi_{G})}\left(\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt^{\\!2}\sigma\tau^{\\!-1},d)-\frac{2}{d}\operatorname{Wg}({\scriptstyle\circ}\hskip
1.084pt\sigma\tau^{\\!-1},d)+\frac{1}{d^{2}}\operatorname{Wg}(\sigma\tau^{\\!-1},d)\right)$
(terms excluding no atom)

Note that these four groups correspond to a number of atoms
$\mathfrak{a}(\sigma,\tau)$ of $0$, $1$, $1$ and $2$ respectively. The first
sum is kept unchanged. The second and third are treated as above, using Lemma
3.7 for $K=\mathfrak{a}(\sigma,\tau)=1$. The last sum is treated with Lemma
3.7 for $K=\mathfrak{a}(\sigma,\tau)=2$, which brings down the leading order
contributions by a factor $d^{-\lfloor(K+1)/2\rfloor}=d^{-2}$. More generally,
for any choice of $K$ atoms (among $L$ possible atoms), the terms that do not
strictly exclude the corresponding circuits but do exclude the $L-K$ others
are grouped together and treated using Lemma 3.7, getting a multiplicative
penalty of $d^{-\lfloor(K+1)/2\rfloor}\rho_{K}(\sigma\tau^{\\!-1})$ to the
leading order term, where $K=\mathfrak{a}(\sigma,\tau)$, which is the claim. ∎

For practical purposes, the results stated above can be summed up as the
following method, which can be applied when it comes to identifying the
leading order of an expression involving $\psi$ and $\overline{\psi}$ terms,
as well as centered terms of order $2$:

  1. (1)

The leading order term of $\varmathbb E\psi_{G}$ is obtained by maximizing the
number of circuits in a circuit-covering of $G$ (Theorem 3.1, (3.2), and Fact
3.2); an elementary bound is given by Proposition 3.3.

  2. (2)

If $\psi_{G}$ contains terms $|\psi_{ab}|^{2}$, it is always ‘safe’ to take
the corresponding $2$-circuits, in the sense that the total number of circuits
could not be strictly increased by not taking them (Proposition 3.4);

  3. (3)

but, if the expression at stake includes atoms, i.e., centered terms
$|\psi_{ab}|^{2}-1/d$, taking the corresponding $2$-circuits implies a
multiplicative penalty of at least $d^{-1}$ for each such circuit taken
(Theorem 3.6).

To put it in one sentence: in order to identify the leading order, one wants
to maximize the number of circuits, taking as many non-centered $2$-circuits
as possible, but avoiding atoms (unless it is impossible to do so). In
principle, the remaining combinatorics in the choice of larger circuits can
still be extremely complex. However, in the cases at stake here, these
elementary rules will be sufficient to identify the leading order with
certainty.

###  3.2. Application to bounds on moments of $\gamma$ terms

We will now apply the general results of Section 3.1 to the particular
expressions at stake here, namely, the expectation of products of terms such
as

(3.31)
$\gamma(k,l,m,n):=\sum_{j=1}^{d}\overline{\psi_{jk}}\psi_{jl}\overline{\psi_{jm}}\psi_{jn}-\frac{1}{d}\delta_{kl}\delta_{mn}-\frac{1}{d}\delta_{kn}\delta_{lm}.$

Note that, as $d$ goes to infinity, we can always assume we are in the stable
range of Weingarten calculus and use the above asymptotics. For simplicity, we
will refer to an expression such as (3.31) as a gamma term ‘of type $klmn$’,
where the letters $k,l,m,n$ are chosen so as to indicate relevant identities
of indices.

A preliminary remark is that any $2$-circuit that can be obtained within a
single gamma term can be disregarded, as coming from an atom in the sense of
section 3.1 (see Theorem 3.6). Indeed, if at most one $2$-circuit can be
formed, then the gamma term is of type $kkmn$ with $m\neq n$ (up to reordering
and renaming indices), and therefore it can be written as

(3.32)
$\gamma(k,k,m,n)=\sum_{j=1}^{d}|\psi_{jk}|^{2}\overline{\psi_{jm}}\psi_{jn}=\sum_{j=1}^{d}\left(|\psi_{jk}|^{2}-\frac{1}{d}\right)\overline{\psi_{jm}}\psi_{jn}$

thanks to orthogonality of columns $m$ and $n$. Else, if two $2$-circuits can
be formed, then (still up to a choice of indices) the gamma term is either of
type $kkmm$ with $k\neq m$, or of type $kkkk$. In the first case,

(3.33)
$\gamma(k,k,m,m)=\sum_{j=1}^{d}|\psi_{jk}|^{2}|\psi_{jm}|^{2}-\frac{1}{d}=\sum_{j=1}^{d}\left(|\psi_{jk}|^{2}-\frac{1}{d}\right)\left(|\psi_{jm}|^{2}-\frac{1}{d}\right),$

thanks to columns $k$ and $m$ having norm $1$. The last case is that of a term
of type $kkkk$, in which case the term is not exactly, but still essentially
centered, in the sense that it is centered up to a small constant. Namely:

(3.34)
$\gamma(k,k,k,k)=\sum_{j=1}^{d}|\psi_{jk}|^{4}-\frac{2}{d}=\sum_{j=1}^{d}\left(|\psi_{jk}|^{4}-\varmathbb
E|\psi_{jk}|^{4}\right)-\frac{2}{d(d+1)}.$

For the sake of simplicity, we introduce the exactly centered terms

(3.35)
$\widetilde{\gamma}(k,k,k,k):=\sum_{j=1}^{d}\left(|\psi_{jk}|^{4}-\varmathbb
E|\psi_{jk}|^{4}\right).$

Our proof strategy regarding these centered $\widetilde{\gamma}$ terms can be
summarized as follows.

######  Remark 3.8.

Expectation of products involving terms such as (3.35) can be bounded in the
same spirit thanks to the result of Bordenave and Collins (Proposition 8 and
Theorem 10 in [4]). Similarly to Theorem 3.6, the consequence is that a
multiplicative penalty is inflicted to coverings pairs that isolate a part of
the graph corresponding to a centered term $|\psi_{jk}|^{4}-\varmathbb
E(|\psi_{jk}|^{4})$ from the rest of the graph. Our proofs will proceed by
first estimating expressions involving $\widetilde{\gamma}(k,k,k,k)$ terms,
and then checking that replacing them by the actual value $\gamma(k,k,k,k)$ as
in (3.34) does not affect the result, owing to the small difference (of order
$d^{-2}$) between the two terms.

We first illustrate our method by evaluating moments of gamma terms of types
$klmn$ and $kkmn$, with distinct $k,l,m,n$.

######  Proposition 3.9.

Let $\gamma$ be defined as in (3.31). Then, for distinct $k,l,m,n$ we have,
when $d\rightarrow\infty$:

(3.36) $\varmathbb
E\left(\gamma(k,l,m,n)^{p}\overline{\gamma(k,l,m,n)^{q}}\right)=\begin{cases}p!\
d^{-3p}+O_{p}(d^{-3p-1})&\quad\text{if}\quad p=q,\vskip 3.0pt plus 1.0pt minus
1.0pt\\\ 0&\quad\text{otherwise.}\end{cases}$

and the same holds if $l=k$.

It follows in particular that $d^{3/2}\gamma(k,l,m,n)$ converges in
distribution to a complex normal variable with mean $0$ and variance $1$, and
likewise for $d^{3/2}\gamma(k,k,m,n)$.

###### Proof.

We apply the method outlined in Section 3.1 to the expression

$\varmathbb
E\left(\gamma(k,l,m,n)^{p}\overline{\gamma(k,l,m,n)^{q}}\right)=\\!\\!\\!\\!\sum_{j_{1},\dots,j_{p},j_{1}^{\prime},\dots,j_{q}^{\prime}}\\!\\!\\!\\!\varmathbb
E\left[\prod_{i=1}^{p}\overline{\psi_{j_{i}k}}{\psi_{j_{i}l}}\overline{\psi_{j_{i}m}}{\psi_{j_{i}n}}\times\prod_{i=1}^{q}{\psi_{j_{i}^{\prime}k}}\overline{\psi_{j_{i}^{\prime}l}}{\psi_{j_{i}^{\prime}m}}\overline{\psi_{j_{i}^{\prime}n}}\right].$

The associated Weingarten graphs are represented in Figure 5. First, note that
if $p\neq q$ then this expectation is zero, for instance by considering that
vertex $n$ has $p$ incoming edges and $q$ outgoing edges, which imposes $p=q$
for a circuit covering of the whole graph to exist.

$k$$l$$m$$n$$j_{p}$$\dots$$j_{1}$$j_{1}^{\prime}$$\dots$$j_{q}^{\prime}$

$m$$n$$l=k$$j_{p}$$\dots$$j_{1}$$j_{1}^{\prime}$$\dots$$j_{q}^{\prime}$

$\otimes$

$\otimes$

$\otimes$

$\otimes$

Figure 5. Weingarten graphs considered in the proof of Proposition 3.9. The
crossed out circles in the second graph indicate that the $2$-circuits
corresponding to atoms involve a penalty, according to Theorem 3.6.

Then, note that a $2$-circuit that is not an atom is only made possible by an
equality $j_{i}=j_{r}^{\prime}$, and each such equality enables four
$2$-circuits in the corresponding terms. Proposition 3.4 ensures that these
can be taken first (and in any order) when building an optimal circuit
covering. Let $\ell$ be the number of pairs $j=j^{\prime}$ used in picking the
$2$-circuits (yielding $4\ell$ of them). It follows that the number of
distinct $j$-indices (corresponding to distinct vertices in the graph) is
bounded by $2p-\ell$. The remaining edges can at best be used in $4$-circuits,
and a quick count yields at most $2p-2l$ of these. The total contribution is
therefore bounded by:

$\underbrace{d^{2p-\ell}}_{\text{free indices
}j}\times\underbrace{d^{-8p}}_{-2q}\times\underbrace{d^{4\ell}}_{2\text{-circuits}}\times\underbrace{d^{2p-2\ell}}_{\text{other
circuits}}=d^{-4p+\ell}\leqslant d^{-3p}.$

The only situation in which $d^{-3p}$ is reached is with $\ell=p$, exactly $p$
free $j$-indices, and the other edges all optimally used in $4$-circuits. This
is the case if and only if the edges are matched in exact pairs, and such a
configuration only depends on the way we pair the indices $j$ and
$j^{\prime}$, so that the overall contribution is

(3.37) $\varmathbb E\left|\gamma\right|^{2p}=p!\ d^{-3p}+O(d^{-3p-1}),$

as claimed. ∎

It is actually true of all gamma terms that their fluctuations are of order
$O(d^{-3/2})$, and it can be checked in the same way that once renormalized,
these terms converge in distribution when $d\rightarrow\infty$ to either a
real or a complex normal distribution, depending on the values of $k,l,m,n$,
with a variance that also depends on the type.

Our purpose is now to bound the expectation of a product of gamma terms when
it contains terms of different types. The general idea is that such an
expectation will be higher when terms are combined in certain ways (typically:
paired). This motivates the following definition of what we consider to be
‘smart combinations’.

######  Definition 3.10.

Given a particular identification of $j$-indices (i.e. specifying which
indices are equal and which are not) and a circuit covering of the resulting
Weingarten graph, we say that

  1. (1)

a smart pair is a configuration where a pair of gamma terms form four
$2$-circuits with one common index $j$;

  2. (2)

a smart triplet is when a triple $\gamma_{1},\gamma_{2},\gamma_{3}$ forms six
$2$-circuits with one index $j$;

  3. (3)

a smart $2$-chain is when a pair $\gamma_{1},\gamma_{2}$ that forms three
circuits (two $2$-circuits and one $4$-circuits) with two distinct $j$’s.

  4. (4)

a smart $3$-chain is a triple $\gamma_{1}\gamma_{2}\gamma_{3}$ that allows $4$
circuits (two $2$-circuits and two $4$-circuits) with three distinct $j$’s.

By inspection, smart pairs only occur when pairing a $\gamma$ term with its
corresponding $\overline{\gamma}$ term, and smart $2$-chains only occur with
terms of type $kkkk$. Figure 6 represents a possible configuration for a smart
triplet and the only possible configurations for smart $2$-chain and smart
$3$-chain respectively.

$k$$o$$m$$l$$n$$r$$j$

$j_{1}$$j_{2}$$k$

$\otimes$

$\otimes$

$j_{1}$$k$$j_{2}$$l$$j_{3}$$\otimes$ $\otimes$

$\otimes$

$\otimes$

Figure 6. From left to right: a smart triplet involving terms of type $klmn$,
$nmor$ and $rolk$; a smart $2$-chain involving terms of type $kkkk$; a smart
$3$-chain involving terms of type $kkkk$, $kkll$ and $llll$. Note that
imposing $j_{1}=j_{2}=j_{3}$ transforms the smart $3$-chain into another smart
triplet.

######  Theorem 3.11 (Moment bound for gamma terms: general case).

We consider a product $\gamma_{1}\cdots\gamma_{p}$ of gamma terms, denoting by
$p_{2}^{*}$ the largest number of smart pairs that can be obtained, and by
$p_{3}^{*}$ the largest number of smart triplets that can be obtained once
$p_{2}^{*}$ smart pairs are removed. Then,

(3.38) $\varmathbb
E\left[\gamma_{1}\cdots\gamma_{p}\right]=O_{p}\left(d^{-\frac{3}{2}p-\frac{1}{6}(3p_{3}^{*})-\frac{1}{4}(p-2p_{2}^{*}-3p_{3}^{*})}\right).$

This is to say that the best possible contribution (of order
$d^{-\frac{3}{2}p}$) can be reached with exact pairs only; in other
situations, smart triplets terms yield a $-\frac{1}{6}$ penalty to the
exponent, and all other terms a penalty of $-\frac{1}{4}$.

###### Proof.

As explained above (after the definition (3.35)), we perform the computation
with terms $\gamma(k,k,k,k)$ replaced by $\widetilde{\gamma}(k,k,k,k)$, and
check in the end that this does not make a difference. After expanding the
product as a sum over indices $j_{1},\dots,j_{p}$ and applying Theorem 3.6, we
consider the order of the contribution of a given circuit-covering for a
specific partition of the $j$-indices. For such configuration, we denote by
$\ell_{n}$ the number of groups of $n$ equal $j$-indices, so that $\ell_{1}$
corresponds to isolated $j$’s, $\ell_{2}$ corresponds to terms being paired
(but not necessarily optimally so), etc. As this is a partition of all
$j$-indices, we have

(3.39) $\sum_{n\geqslant 1}n\ell_{n}=p.$

Note that a $2$-circuit can be created without penalty only if either two
$j$’s are equal, or else in a term of type $kkkk$ — these terms represent the
main difficulty here and require careful treatment (see last bullet point
below). We review the different ways in which $2$-circuits can appear,
referring to smart pairs, smart triplets and smart chains as introduced in
Definition 3.10:

  * •

Among $\ell_{2}$ pairs of $j$-indices, we denote by $p_{2}$ those that
correspond to smart pairs, i.e., two gamma terms yielding four $2$-circuits.
The other $\ell_{2}-p_{2}$ pairs allow at most three $2$-circuits each.

  * •

Among $\ell_{3}$ triplets of $j$-indices, we denote by $p_{3}$ those that
correspond to smart triplets, leading to six $2$-circuits for each smart
triplet. The other $\ell_{3}-p_{3}$ triplets allow at most five $2$-circuits
each.

  * •

Groups of $n$ $j$-indices with $n\geqslant 4$ induce at most $2\ell_{n}$
circuits (this bound is achieved in the optimal situation where all circuits
are $2$-circuits); the gain we expect here comes solely from the missing
$j$-indices.

  * •

The last possibility of a $2$-circuit is in a term of type $kkkk$ with
isolated $j$-index. We call $m$ the number of such terms, and distinguish them
according to what happens to the other two edges.

    * –

If the remaining two edges also form a $2$-circuit, this is cancelled by a
penalty, and so we say these two edges are ‘wasted’.

    * –

If they form an $n$-circuit with a non-isolated term and $n\geqslant 4$ (or an
isolated term and $n\geqslant 6$), the contribution is equivalent to that of
an $(n-2)$-circuit and a (penalized) $2$-circuit. Again, we say these two
edges are wasted.

    * –

The only case in which these edges are not considered wasted is if they form a
$4$-circuit with two other edges from another isolated term, which is
necessarily a $kkmn$-type term; $2$-chains and $3$-chains are possible
examples of such a configuration, which happen to be optimal and near-optimal,
respectively. We denote by $m_{2}$ the number of $2$-chains and by $m_{3}$ the
number of $3$-chains. By inspection, in any other possible case, such $kkkk$
terms cannot exceed a half of the total number of isolated terms.

We denote by $m_{0}$ the number of isolated terms of type $kkkk$ that lead, or
are equivalent, to ‘wasting two edges’, and by $m_{1}$ those that are linked
to other isolated terms. The above considerations imply

(3.40) $m=m_{0}+m_{1}+2m_{2}+2m_{3},\quad\text{ and }\quad
m_{1}\leqslant\frac{1}{2}(\ell_{1}-m_{0}-2m_{2}-3m_{3}).$

We now compute the order of the contributions:

  * •

The number of free $j$-indices is $\sum_{n\geqslant 1}\ell_{n}$, the number of
parts in the partition. Using (3.39), we obtain

$\sum_{n\geqslant 4}\ell_{n}\leqslant\frac{1}{4}\sum_{n\geqslant
4}n\ell_{n}=\frac{1}{4}(p-\ell_{1}-2\ell_{2}-3\ell_{3}),$

and thus the bound

(3.41) $\\#\\{\text{free }j\text{ indices}\\}=\sum_{n\geqslant
1}\ell_{n}\leqslant\frac{1}{4}p+\frac{3}{4}\ell_{1}+\frac{1}{2}\ell_{2}+\frac{1}{4}\ell_{3}.$

  * •

The number of (unpenalized) $2$-circuits is bounded by

$4p_{2}+3(\ell_{2}-p_{2})+6p_{3}+5(\ell_{3}-p_{3})+2\sum_{n\geqslant
4}n\ell_{n}+m,$

which yields after simplifications, again using (3.39):

(3.42) $\\#\\{2\text{-circuits}\\}\leqslant
2p+p_{2}-\ell_{2}+p_{3}-\ell_{3}-2\ell_{1}+m.$

  * •

The edges that are neither wasted nor part of any $2$-circuits can at best
form $4$-circuits. By inspection, starting with $4p$ edges and discounting
$2m_{0}$ wasted edges, the total number of circuits is bounded by

(3.43) $\\#\\{\text{circuits}\\}\leqslant
p+\frac{1}{2}\\#\\{2\text{-circuits}\\}-\frac{1}{2}m_{0}\leqslant
2p+\frac{1}{2}(p_{2}-\ell_{2})+\frac{1}{2}(p_{3}-\ell_{3})-\ell_{1}+\frac{1}{2}m-\frac{1}{2}m_{0}.$

According to the method outlined in Section 3.1, the contribution of such a
configuration is bounded by

(3.44)
$\underbrace{d^{\frac{1}{4}p+\frac{3}{4}\ell_{1}+\frac{1}{2}\ell_{2}+\frac{1}{4}\ell_{3}}}_{\text{free
indices
}j}\times\underbrace{d^{-4p}}_{-2q}\times\underbrace{d^{2p+\frac{1}{2}(p_{2}-\ell_{2})+\frac{1}{2}(p_{3}-\ell_{3})-\ell_{1}+\frac{1}{2}m-\frac{1}{2}m_{0}}}_{\text{total
number of circuits}}.$

After rearranging, the exponent becomes

(3.45)
$-\frac{3}{2}p-\frac{1}{4}(p-\ell_{1}-2\ell_{2}-3\ell_{3})-\frac{1}{4}(2\ell_{2}-2p_{2})-\frac{1}{3}(3\ell_{3}-3p_{3})-\frac{1}{6}(3p_{3})+\frac{1}{2}(m-m_{0})-\frac{1}{2}\ell_{1}.$

Using (3.40), we get the following bound on the exponent:

$\displaystyle-\frac{3}{2}p-\frac{1}{6}\underbrace{(3p_{3}+3m_{3})}_{\text{
\tiny smart triplets and 3-chains}}$
$\displaystyle-\frac{1}{4}\underbrace{(p-\ell_{1}-2\ell_{2}-3\ell_{3})}_{\text{
\tiny terms in groups of size }n\geqslant 4}$
$\displaystyle-\frac{1}{4}\underbrace{(2\ell_{2}-2p_{2})}_{\text{\tiny non-
optimal pairs}}-\frac{1}{3}\underbrace{(3\ell_{3}-3p_{3})}_{\text{\tiny non-
optimal
triplets}}-\frac{1}{4}\underbrace{(\ell_{1}-2m_{2}-3m_{3})}_{\text{\tiny
isolated terms outside smart chains}},$

which means, in concrete terms, the following penalties from the optimal
$-\frac{3}{2}p$ exponent:

  * –

no penalty for smart pairs and smart $2$-chains;

  * –

penalties of $-1/6$ for smart triplets and smart $3$-chains;

  * –

penalties of $-1/4$ for unpaired and untripled terms, as well as for non-
optimally paired terms;

  * –

a larger penalty of $-1/3$ for terms in non-optimal triplets, which we simply
bound by $-\frac{1}{4}$.

The claim follows directly, with the following remarks: smart $2$-chains are
automatically counted as smart pairs when maximizing $p_{2}$ (so that
$p_{2}^{*}$ includes both $p_{2}$ and $m_{2}$); similarly, smart $3$-chains
are counted as smart triplets when maximizing $p_{3}$, so that $p_{3}^{*}$
includes both $p_{3}$ and $m_{3}$.

The above argument proves the result holds when replacing $\gamma(k,k,k,k)$
terms by the corresponding $\widetilde{\gamma}(k,k,k,k)$. To conclude the
proof (according to the strategy outlined in Remark 3.8), we check that this
replacement does not affect the result, owing to the small constant difference
(of order $d^{-2}$) between $\gamma$ and $\widetilde{\gamma}$. Indeed,
replacing each $\widetilde{\gamma}(k,k,k,k)$ by the true value
$\gamma(k,k,k,k)$ in the product $\gamma_{1}\cdots\gamma_{p}$ amounts, after
expanding, to adding a certain number of products where the same
$\widetilde{\gamma}(k,k,k,k)$ terms are simply replaced by a constant of order
$d^{-2}$. All of these terms (which number is bounded in a way that depends
only on $p$) have smaller order than the estimate: indeed, the contribution of
each gamma term to the exponent is between $-\frac{3}{2}$ and
$-\frac{3}{2}-\frac{1}{4}=-\frac{14}{8}$, so is always larger than $-2$. ∎

We note that Theorem 3.11 implies the following two estimates on the moments
of any gamma term and correlation of two gamma terms, respectively:

######  Corollary 3.12.

For any values of $k,l,m,n$ we have the following moment estimates, when
$d\rightarrow\infty$:

(3.46) $\varmathbb E|\gamma(k,l,m,n)|^{2p}=O_{p}(d^{-3p}).$

######  Corollary 3.13.

We have

$\varmathbb
E\left[\gamma(k,l,m,n)\gamma(k^{\prime},l^{\prime},m^{\prime},n^{\prime})\right]=\begin{cases}\sigma_{k,l,m,n}^{2}d^{-3}+O(d^{-4})&\quad\text{if}\quad\gamma(k^{\prime},l^{\prime},m^{\prime},n^{\prime})=\overline{\gamma(k,l,m,n)},\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ O(d^{-4})&\quad\text{otherwise.}\end{cases}$

where $\sigma_{k,l,m,n}^{2}>0$, with $\sigma_{k,l,m,n}^{2}=1$ if $k,l,m,n$ are
distinct.

###  3.3. The semicircle law

A fundamental result from random matrix theory is the _semicircle law_ , which
is to say that, under generic assumptions, and in particular for GUE, the
empirical distribution of the eigenvalues converges almost surely as
$N\to\infty$ to the semicircle distribution (1.11).

In fact, we have some rather stronger results, the proof of which can be found
in, e.g., [2]:

######  Lemma 3.14 (Local semicircle law and eigenvalue rigidity).

Let the vector of eigenvalues, $\lambda$, be distributed according to the
density (1.3). Then:

  * •

We have the local semicircle law

(3.47) $\\#\bigl{\\{}\lambda_{k}\in
I\bigr{\\}}=(2N+1)\int_{I}\,d\sigma_{\mathrm{sc}}+O_{\prec}(1),$

uniformly for all intervals $I\subseteq\varmathbb{R}$.

  * •

If we define $\nu(\kappa)$ for $\kappa\in[-1,1]$ as in (1.10) then we have the
eigenvalue rigidity estimate

(3.48)
$\lambda_{k}=\nu\bigl{(}\tfrac{k}{N}\bigr{)}+O_{\prec}\left(N^{-\frac{2}{3}}\bigl{(}N+1-|k|\bigr{)}^{-\frac{1}{3}}\right).$

Our proof of Theorem 1.1 will rely on the following corollary of Lemma 3.14:

######  Corollary 3.15.

Let $\alpha\in\varmathbb{R}$ and $1\leqslant T\leqslant N$. Then,

(3.49)
$\displaystyle\sum_{k=-N}^{N}\frac{1}{\frac{1}{T}+|\lambda_{k}-\alpha|}$
$\displaystyle\prec N\log\bigl{(}16+|\alpha|\bigr{)},$

uniformly in $\alpha$.

###### Proof.

Choose a partition of unity $1=\sum_{j\in\varmathbb{Z}}\eta(\omega-j),$ where
$\eta\in C^{\infty}_{c}(\varmathbb{R})$ is a non-negative function supported
in $[-2,2]$. We then set $\eta_{j,N}(x)=\eta(Nx-j)$, which localizes to the
region $x=\frac{j}{N}+O(\frac{1}{N})$. Setting $x_{k}=\lambda_{k}-\alpha$, we
estimate the small, medium, and large choices of $|j|$ as

$\frac{1}{\frac{1}{T}+|x_{k}|}\lesssim\sum_{|j|\leqslant
4\frac{N}{T}}T\,\eta_{j,N}(x_{k})+\sum_{4\frac{N}{T}<|j|\leqslant(16+|\alpha|)N}\frac{N}{|j|}\,\eta_{j,N}(x_{k})+\sum_{|j|>(16+|\alpha|)N}\eta_{j,N}(x_{k}).$

Let us note that

$\sum_{k}\eta_{j,N}(x_{k})\lesssim\\#\Bigl{\\{}\lambda_{k}\in\bigl{[}\alpha+\tfrac{j-2}{N},\alpha+\tfrac{j+2}{N}\bigr{]}\Bigr{\\}},$

From (3.47) and the fact that $\sigma_{\mathrm{sc}}$ is supported on $[-2,2]$,
we see that the contribution of the large choices of $|j|$ should be
negligible. Precisely,

$\sum_{k}\sum_{|j|>(16+|\alpha|)N}\eta_{j,N}(x_{k})\lesssim\\#\bigl{\\{}|\lambda_{k}|\geqslant
8\bigr{\\}}\prec 1.$

For the small and medium choices of $|j|$, where $|j|\leqslant(16+|\alpha|)N$,
we again apply (3.47) to estimate

$\sum_{k}\eta_{j,N}(x_{k})\prec 1.$

Combining these bounds, we may sum to obtain

$\sum_{k}\frac{1}{\frac{1}{T}+|x_{k}|}\prec N\Bigl{[}1+\log
T+\log\bigl{(}16+|\alpha|\bigr{)}\Bigr{]},$

and the estimate (3.49) now follows from the fact that, as $1\leqslant
T\leqslant N$, we have $\log T\prec 1$. ∎

##  4\. The leading order terms

In this section we investigate the leading order terms in our expansion and
prove Proposition 2.5. We subsequently denote the conditional expectation
$\varmathbb E_{\lambda}[\,\cdot\,]=\varmathbb E[\,\cdot\,|\,\lambda\,]$.

Recalling the definition of the profile (2.12), the modulation (1.12), and the
interaction coefficient (2.4), we may use the Duhamel formula (2.3) to obtain
the expressions

(4.1) $\displaystyle f_{k}^{(0)}(t)$
$\displaystyle=a_{k}^{(0)}=A\left(\tfrac{k}{N}\right),$ (4.2) $\displaystyle
f_{k}^{(1)}(t)$
$\displaystyle=-i\frac{\mu^{2}}{N}\sum_{\ell,m,n}\left(\int_{0}^{t}e^{is\Omega(k,\ell,m,n)}\,ds\right)\gamma(k,\ell,m,n)a_{\ell}^{(0)}\overline{a_{m}^{(0)}}a_{n}^{(0)},$
(4.3) $\displaystyle f_{k}^{(2)}(t)$
$\displaystyle=-2\frac{\mu^{4}}{N^{2}}\sum_{\ell,m,n,o,p,q}\left(\int_{0}^{t}\int_{0}^{s}e^{is\Omega(k,\ell,m,n)}e^{i\sigma\Omega(\ell,o,p,q)}\,d\sigma\,ds\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\gamma(k,\ell,m,n)\gamma(\ell,o,p,q)a_{o}^{(0)}\overline{a_{p}^{(0)}}a_{q}^{(0)}\overline{a_{m}^{(0)}}a_{n}^{(0)}$
$\displaystyle\quad+\frac{\mu^{4}}{N^{2}}\sum_{\ell,m,n,o,p,q}\left(\int_{0}^{t}\int_{0}^{s}e^{is\Omega(k,\ell,m,n)}e^{-i\sigma\Omega(m,o,p,q)}\,d\sigma\,\,ds\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\gamma(k,\ell,m,n)\overline{\gamma(m,o,p,q)}a_{\ell}^{(0)}\overline{a_{o}^{(0)}}a_{p}^{(0)}\overline{a_{q}^{(0)}}a_{n}^{(0)}.$

We may then write

(4.4) $\displaystyle\varmathbb
E_{\lambda}\bigl{|}a_{k}^{(0)}+a_{k}^{(1)}\bigr{|}^{2}+2\operatorname{Re}\varmathbb
E_{\lambda}\Bigl{[}\overline{a_{k}^{(0)}}a_{k}^{(2)}\Bigr{]}$
$\displaystyle\qquad=\varmathbb
E_{\lambda}\bigl{|}f_{k}^{(0)}\bigr{|}^{2}+\varmathbb
E_{\lambda}\bigl{|}f_{k}^{(1)}\bigr{|}^{2}+2\operatorname{Re}\varmathbb
E_{\lambda}\Bigl{[}\overline{f_{k}^{(0)}}f_{k}^{(1)}\Bigr{]}+2\operatorname{Re}\varmathbb
E_{\lambda}\Bigl{[}\overline{f_{k}^{(0)}}f_{k}^{(2)}\Bigr{]}.$

Our first lemma rules out the quartic (in the initial data) terms in (4.4):

######  Lemma 4.1.

We have

(4.5) $2\operatorname{Re}\varmathbb
E_{\lambda}\Bigl{[}\overline{f_{k}^{(0)}}f_{k}^{(1)}\Bigr{]}=0.$

###### Proof.

Observe that

$\overline{f_{k}^{(0)}}f_{k}^{(1)}=-i\frac{\mu^{2}}{N}\sum_{\ell,m,n}\left(\int_{0}^{t}e^{is\Omega(k,\ell,m,n)}\,ds\right)\gamma(k,\ell,m,n)\overline{a_{k}^{(0)}}a_{\ell}^{(0)}\overline{a_{m}^{(0)}}a_{n}^{(0)}.$

Recalling (3.1), we see that

$\varmathbb E\gamma(k,\ell,m,n)=0,$

unless $k=\ell,m=n$ or $k=n,\ell=m$. This leads to the simplification

$\displaystyle\varmathbb
E_{\lambda}\Bigl{[}\overline{f_{k}^{(0)}}f_{k}^{(1)}\Bigr{]}$
$\displaystyle=-2i\frac{\mu^{2}t}{N}\sum_{\ell}\varmathbb
E\gamma(k,k,\ell,\ell)\bigl{|}a_{k}^{(0)}\bigr{|}^{2}\bigl{|}a_{\ell}^{(0)}\bigr{|}^{2}+i\frac{\mu^{2}t}{N}\varmathbb
E\gamma(k,k,k,k)\bigl{|}a_{k}^{(0)}\bigr{|}^{4}.$

However, this term is clearly imaginary, so taking the real part we obtain
(4.5). ∎

Next, we apply Corollary 3.13 to obtain the following:

######  Lemma 4.2.

We have

(4.6) $\displaystyle\varmathbb
E_{\lambda}\Bigl{[}\bigl{|}f_{k}^{(1)}\bigr{|}^{2}\Bigr{]}$
$\displaystyle=\frac{2\mu^{4}}{N^{2}(2N+1)^{3}}\sum_{\begin{subarray}{c}\ell,m,n\end{subarray}}\left|\int_{0}^{t}e^{is\Omega(k,\ell,m,n)}\,ds\right|^{2}\bigl{|}a_{\ell}^{(0)}\bigr{|}^{2}\bigl{|}a_{m}^{(0)}\bigr{|}^{2}\bigl{|}a_{n}^{(0)}\bigr{|}^{2}+O\left(\tfrac{t}{T_{\mathrm{kin}}}\tfrac{t}{N}\right),$
$\displaystyle 2\operatorname{Re}\varmathbb
E_{\lambda}\Bigl{[}\overline{f_{k}^{(0)}}f_{k}^{(2)}\Bigr{]}$
$\displaystyle=-\frac{8\mu^{4}}{N^{2}(2N+1)^{3}}\operatorname{Re}\sum_{\begin{subarray}{c}\ell,m,n\end{subarray}}\left(\int_{0}^{t}\int_{0}^{s}e^{i(s-\sigma)\Omega(k,\ell,m,n)}\,d\sigma\,ds\right)\bigl{|}a_{k}^{(0)}\bigr{|}^{2}\bigl{|}a_{n}^{(0)}\bigr{|}^{2}\bigl{|}a_{m}^{(0)}\bigr{|}^{2}$
(4.7)
$\displaystyle\quad+\frac{4\mu^{4}}{N^{2}(2N+1)^{3}}\operatorname{Re}\sum_{\begin{subarray}{c}\ell,m,n\end{subarray}}\left(\int_{0}^{t}\int_{0}^{s}e^{i(s-\sigma)\Omega(k,\ell,m,n)}\,d\sigma\,ds\right)\bigl{|}a_{k}^{(0)}\bigr{|}^{2}\bigl{|}a_{\ell}^{(0)}\bigr{|}^{2}\bigl{|}a_{n}^{(0)}\bigr{|}^{2}$
$\displaystyle\quad+O\left(\tfrac{t}{T_{\mathrm{kin}}}\tfrac{t}{N}\right).$

###### Proof.

We present the computation in detail for (4.6). The argument for (4.7) is
identical.

Using the expression (4.2), we compute that

$\displaystyle\bigl{|}f_{k}^{(1)}\bigr{|}^{2}$
$\displaystyle=\frac{\mu^{4}}{N^{2}}\sum_{\ell,m,n,o,p,q}\left(\int_{0}^{t}e^{is\Omega(k,\ell,m,n)}\,ds\right)\left(\int_{0}^{t}e^{-is\Omega(k,o,p,q)}\,ds\right)$
$\displaystyle\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\gamma(k,\ell,m,n)\overline{\gamma(k,o,p,q)}a_{\ell}^{(0)}\overline{a_{m}^{(0)}}a_{n}^{(0)}\overline{a_{o}^{(0)}}a_{p}^{(0)}\overline{a_{q}^{(0)}}.$

Recalling (3.1), we see that after taking the conditional expectation we only
obtain a non-zero contribution when indices are paired, which implies that
$\\#\\{k,\ell,m,n,o,p,q\\}\leqslant 4$. In particular, we may decompose

$\varmathbb
E_{\lambda}\Bigl{[}\bigl{|}f_{k}^{(1)}\bigr{|}^{2}\Bigr{]}=E_{1}+E_{2}+E_{3}+E_{4},$

where

$\displaystyle E_{r}$
$\displaystyle=\frac{\mu^{4}}{N^{2}}\sum_{r=1}^{4}\sum_{\begin{subarray}{c}\ell,m,n,o,p,q\\\
\\#\\{k,\ell,m,n,o,p,q\\}=r\end{subarray}}\left(\int_{0}^{t}e^{is\Omega(k,\ell,m,n)}\,ds\right)\left(\int_{0}^{t}e^{-is\Omega(k,o,p,q)}\,ds\right)$
$\displaystyle\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\varmathbb
E\Bigl{[}\gamma(k,\ell,m,n)\overline{\gamma(k,o,p,q)}\Bigr{]}a_{\ell}^{(0)}\overline{a_{m}^{(0)}}a_{n}^{(0)}\overline{a_{o}^{(0)}}a_{p}^{(0)}\overline{a_{q}^{(0)}}.$

In the cases $r=1,2,3$, Corollary 3.13 ensures that

$\varmathbb
E\Bigl{[}\gamma(k,\ell,m,n)\overline{\gamma(k,o,p,q)}\Bigr{]}=O(N^{-3}).$

We then use the trivial bound

$\left|\left(\int_{0}^{t}e^{is\Omega(k,\ell,m,n)}\,ds\right)\left(\int_{0}^{t}e^{-is\Omega(k,o,p,q)}\,ds\right)\right|\leqslant
t^{2}$

to estimate

$|E_{r}|\lesssim\frac{t}{T_{\mathrm{kin}}}\frac{t}{N},$

where we note that as $k$ is fixed the sum over $\ell,m,n,o,p,q$ has dimension
$r-1$.

When $r=4$ we have

$\displaystyle E_{4}$
$\displaystyle=\frac{2\mu^{4}}{N^{2}}\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}\ell,m,n\\\
\\#\\{k,\ell,m,n\\}=4\end{subarray}}\\!\\!\\!\\!\\!\left|\int_{0}^{t}e^{is\Omega(k,\ell,m,n)}\,ds\right|^{2}\varmathbb
E|\gamma(k,\ell,m,n)|^{2}\bigl{|}a_{\ell}^{(0)}\bigr{|}^{2}\bigl{|}a_{m}^{(0)}\bigr{|}^{2}\bigl{|}a_{n}^{(0)}\bigr{|}^{2}$
$\displaystyle\quad+\frac{4\mu^{4}}{N^{2}}\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}\ell,m,p\\\
\\#\\{k,\ell,m,p\\}=4\end{subarray}}\\!\\!\\!\\!\\!\left(\int_{0}^{t}e^{is\Omega(k,\ell,m,m)}\,ds\right)\left(\int_{0}^{t}e^{-is\Omega(k,\ell,p,p)}\,ds\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\varmathbb
E\left[\gamma(k,\ell,m,m)\overline{\gamma(k,\ell,p,p)}\right]\bigl{|}a_{\ell}^{(0)}\bigr{|}^{2}\bigl{|}a_{m}^{(0)}\bigr{|}^{2}\bigl{|}a_{p}^{(0)}\bigr{|}^{2}.$

For distinct $k,\ell,m,n$, Corollary 3.13 gives us

$\varmathbb E|\gamma(k,\ell,m,n)|^{2}=(2N+1)^{-3}+O(N^{-4}),$

whereas, for distinct $k,\ell,m,p$, we have

$\varmathbb
E\left[\gamma(k,\ell,m,m)\overline{\gamma(k,\ell,p,p)}\right]=O(N^{-4}).$

These yield

$E_{4}=\frac{2\mu^{4}}{N^{2}(2N+1)^{3}}\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}\ell,m,n\\\
\\#\\{k,\ell,m,n\\}=4\end{subarray}}\\!\\!\\!\\!\\!\left|\int_{0}^{t}e^{is\Omega(k,\ell,m,n)}\,ds\right|^{2}\bigl{|}a_{\ell}^{(0)}\bigr{|}^{2}\bigl{|}a_{m}^{(0)}\bigr{|}^{2}\bigl{|}a_{n}^{(0)}\bigr{|}^{2}+O\left(\frac{t}{T_{\mathrm{kin}}}\frac{t}{N}\right),$

where we have used that

$\left|\int_{0}^{t}e^{is\Omega(k,\ell,m,n)}\,ds\right|^{2}\leqslant t^{2}$

to bound the remainder terms. Finally, we observe that the condition
$\\#\\{k,\ell,m,n\\}=4$ in the sum can be relaxed, up to an error term of size
$O\left(\tfrac{t}{T_{\mathrm{kin}}}\tfrac{t}{N}\right)$. This gives the
desired estimate. ∎

We now apply Lemmas 4.1 and 4.2 to the identity (4.4) to obtain

(4.8) $\displaystyle\varmathbb
E_{\lambda}\bigl{|}a_{k}^{(0)}+a_{k}^{(1)}\bigr{|}^{2}+2\operatorname{Re}\varmathbb
E_{\lambda}\Bigl{[}\overline{a_{k}^{(0)}}a_{k}^{(2)}\Bigr{]}$
$\displaystyle=\left|A\left(\tfrac{k}{N}\right)\right|^{2}+I_{k}+O\left(\tfrac{t}{T_{\mathrm{kin}}}\tfrac{t}{N}\right),$

where, after integrating in time and symmetrizing the middle term,

$\displaystyle I_{k}:$
$\displaystyle=\frac{2\mu^{4}}{N^{2}(2N+1)^{3}}\sum_{\ell,m,n}\left|\int_{0}^{t}e^{is\Omega(k,\ell,m,n)}\,ds\right|^{2}\bigl{|}a_{\ell}^{(0)}\bigr{|}^{2}\bigl{|}a_{m}^{(0)}\bigr{|}^{2}\bigl{|}a_{n}^{(0)}\bigr{|}^{2}$
$\displaystyle\quad-\frac{8\mu^{4}}{N^{2}(2N+1)^{3}}\operatorname{Re}\sum_{\ell,m,n}\left(\int_{0}^{t}\int_{0}^{s}e^{i(s-\sigma)\Omega(k,\ell,m,n)}\,d\sigma\,ds\right)\bigl{|}a_{k}^{(0)}\bigr{|}^{2}\bigl{|}a_{m}^{(0)}\bigr{|}^{2}\bigl{|}a_{n}^{(0)}\bigr{|}^{2}$
$\displaystyle\quad+\frac{4\mu^{4}}{N^{2}(2N+1)^{3}}\operatorname{Re}\sum_{\ell,m,n}\left(\int_{0}^{t}\int_{0}^{s}e^{i(s-\sigma)\Omega(k,\ell,m,n)}\,d\sigma\,ds\right)\bigl{|}a_{k}^{(0)}\bigr{|}^{2}\bigl{|}a_{\ell}^{(0)}\bigr{|}^{2}\bigl{|}a_{n}^{(0)}\bigr{|}^{2}$
$\displaystyle=\frac{8t^{2}\mu^{4}}{N^{2}(2N+1)^{3}}\sum_{\ell,m,n}\phi\bigl{(}t\Omega(k,\ell,m,n)\bigr{)}f\bigl{(}\tfrac{k}{N},\tfrac{\ell}{N},\tfrac{m}{N},\tfrac{n}{N}\bigr{)},$

with

$\displaystyle\phi(x)$
$\displaystyle=\begin{cases}\dfrac{\sin^{2}(x/2)}{x^{2}}&\quad\text{if}\quad
x\neq 0,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\ 1/4&\quad\text{if}\quad
x=0,\end{cases}$

and

$\displaystyle f(k,\ell,m,n)$
$\displaystyle=|A(k)|^{2}|A(\ell)|^{2}|A(m)|^{2}|A(n)|^{2}\Bigl{[}\tfrac{1}{|A(k)|^{2}}-\tfrac{1}{|A(\ell)|^{2}}+\tfrac{1}{|A(m)|^{2}}-\tfrac{1}{|A(n)|^{2}}\Bigr{]}.$

We now approximate the eigenvalues by their deterministic locations using the
following:

######  Lemma 4.3.