# Decay of correlations and thermodynamic limit
for the circular Riesz gas

Jeanne Boursier  (JB) Université Paris-Dauphine, PSL University, UMR 7534,
CNRS, CEREMADE, 75016 Paris, France<EMAIL_ADDRESS>
###### Abstract.

We investigate the thermodynamic limit of the circular long-range Riesz gas, a
system of particles interacting pairwise through an inverse power kernel. We
show that after rescaling, so that the typical spacing of particles is of
order $1$, the microscopic point process converges as the number of points
tends to infinity, to an infinite volume measure $\mathrm{Riesz}_{s,\beta}$.
This convergence result is obtained by analyzing gaps correlations, which are
shown to decay in power-law with exponent $2-s$. One also proves that the
decay of correlations is much faster for the hypersingular Riesz gas, thereby
exhibiting a discontinuous transition at $s=1$. Our method is based on the
analysis of the Helffer-Sjöstrand equation in its static form and on various
discrete elliptic regularity estimates.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 The Helffer-Sjöstrand equation
  4. 4 Decay of correlations for the HS Riesz gas
  5. 5 Decay of correlations for the long-range Riesz gas
  6. 6 Uniqueness of the limiting measure
  7. A Auxiliary estimates
  8. B Well-posedness results
  9. C Local laws for the HS Riesz gas
  10. D Local laws for the interpolating measure

##  1\. Introduction

###  1.1. Setting of the problem

#### The circular Riesz gas

This paper aims to study an interacting particles system on the circle
$\mathbb{T}:=\mathbb{R}/\mathbb{Z}$, named circular Riesz gas. Let us note
that given a parameter $s>0$, the Riesz $s$-kernel on $\mathbb{T}$ is defined
by

$g_{s}:x\in\mathbb{T}\mapsto\lim_{n\to\infty}\Bigr{(}\sum_{k=-n}^{n}\frac{1}{|x+k|^{s}}-\frac{2}{1-s}n^{1-s}\Bigr{)}.$
(1.1)

Also note that for $s\in(0,1)$, $g_{s}$ is the fundamental solution of the
fractional Laplace equation

$(-\Delta)^{\frac{1-s}{2}}g_{s}=c_{s}(\delta_{0}-1),$ (1.2)

where $(-\Delta)^{\frac{1-s}{2}}$ is the fractional Laplacian on $\mathbb{T}$
. Let us now endow $\mathbb{T}$ with the natural order $x<y$ if
$x=x^{\prime}+k$, $y=y^{\prime}+k^{\prime}$ with $k,k^{\prime}\in\mathbb{Z}$,
$x^{\prime},y^{\prime}\in[0,1)$ and $x^{\prime}<y^{\prime}$, allowing one to
define the set of ordered configurations

$D_{N}=\\{X_{N}=(x_{1},\ldots,x_{N})\in\mathbb{T}^{N}:x_{2}-x_{1}<\ldots<x_{N}-x_{1}\\}.$

And let us also consider the pairwise energy

$\mathcal{H}_{N}:X_{N}\in D_{N}\mapsto N^{-s}\sum_{i\neq
j}g_{s}(x_{i}-x_{j}).$ (1.3)

Finally, the circular Riesz gas at inverse temperature $\beta>0$ corresponds
to the probability measure

$\mathrm{d}\mathbb{P}_{N,\beta}=\frac{1}{Z_{N,\beta}}e^{-\beta\mathcal{H}_{N}(X_{N})}\mathds{1}_{D_{N}}(X_{N})\mathrm{d}X_{N}.$
(1.4)

One of the main motivations for studying such an ensemble stems from random
matrix theory. For $s=0$, the Riesz kernel on $\mathbb{R}$, i.e the solution
of $(-\Delta)^{\frac{1}{2}}g=\delta_{0}$, is given up to a multiplicative
constant, by the logarithm kernel $-\log|x|$ and by $\log|\sin(x/2)|$ on the
circle. Interacting particles systems such as (1.4) on $\mathbb{R}$ with
logarithmic interaction and external potential are called _1D log-gases_ or
_$\beta$ -ensembles_ and the _circular log-gas_ or _circular $\beta$-ensemble_
corresponds to (1.4) with the log kernel on $\mathbb{T}$. As observed by Dyson
[Dys62], for some special values of $\beta$, namely $\beta\in\\{1,2,4\\}$, the
$\beta$-ensemble matches the joint law of the $N$ eigenvalues of
symmetric/hermitian/symplectic random matrices with independent Gaussian
entries and there are numerous results on $\beta$-ensembles including results
on fluctuations, correlations, infinite volume limit, edge behavior, dynamical
properties, relaxation time, etc.

The one-dimensional Riesz gas is a natural extension of $\beta$-ensembles and
a fundamental model on which to understand the properties of _long-range_
particles systems. The interaction (1.1) is indeed long-range when $s\in(0,1)$
while short-range (or hyper-singular) when $s\in(1,+\infty)$. The long-range
Riesz gas is to this extent a particularly rich model in which interesting
phenomena occur, falling off the classical theory of statistical mechanics
(Ruelle, Dobruhsin, Georgii, etc). Riesz gases, as a family of power-law
interacting particles systems on $\mathbb{R}^{d}$, have also received much
attention in the physics literature. Apart from the log and Coulomb cases,
which are ubiquitous in both mathematical and physics contexts [Ser18], Riesz
gases have been found out to be natural models in solid state physics,
ferrofluids, elasticity, see for instance [Maz11, BBDR05, CDR09, Tor16]. We
refer to the nice review [Lew22] which presents a comprehensive account of the
literature with many open problems.

The first-order asymptotic of long-range Riesz gases is governed by a mean-
field energy functional, which prescribes the macroscopic distribution of
particles [CGZ14, Ser15], corresponding in our circular setting (1.4) to the
uniform measure of the circle. In [Bou21], we have investigated the
fluctuations of the system and shown that gaps (large spacing between
particles) fluctuate much less than for i.i.d variables and much more than in
the log-gas case. Additionally we have established a central limit theorem for
linear statistics with singular test-functions, which can be applied in
particular to characteristic functions of intervals, thus proving rigorously
the predictions of the physics literature [Lew22, SKA+21]. The purpose of this
very paper is to investigate another class of problems, related to the
question of decay of correlations. More precisely we work at proving the
optimal decay of gap correlations as in [EY15] which considers this question
for $\beta$-ensembles and at proving the uniqueness of the limiting measure.
We will show that after rescaling, chosen so that the typical spacing between
particles is of order $1$, the point process converges in the large $N$ limit
to a certain point process $\mathrm{Riesz}_{s,\beta}$.

#### Infinite volume limit

Let $(x_{1},\ldots,x_{N})$ be distributed according to (1.4). Fix a centering
point on $\mathbb{T}$, say $x=0$, and consider the rescaled point
configuration

$\mathcal{C}_{N}=\sum_{i=1}^{N}\delta_{Nx_{i}}\mathds{1}_{|x_{i}|<\frac{1}{4}}.$

With a slight abuse of notation, $\mathcal{C}_{N}$ can be seen as a random
variable on point configurations on $\mathbb{R}$. The stake for us it to prove
that the law of $\mathcal{C}_{N}$ converges as $N$ tends to infinity, in a
suitable topology, to a certain point process $\mathrm{Riesz}_{s,\beta}$.
While the existence of limiting point processes is standard [GZ93, DHLM21],
the uniqueness of these accumulation points, or limiting measures, is a subtle
property which is not expected to hold for general interactions, even in
dimension one. A unique limit for Gaussian and circular $\beta$-ensembles has
been exhibited in the seminal works [VV09, KVV11] and then shown to be
universal in the bulk of $\beta$-ensembles for a large class of smooth
external potentials in [BEY12, BEY+14b], see also [BFG15]. The limiting
measure, called the $\mathrm{Sine}_{\beta}$ process, can be described using a
system of coupled stochastic differential equations [VV09] or alternatively as
the spectrum of an infinite-dimensional random operator [VV17]. In contrast,
the one-dimensional Coulomb gas, i.e with kernel $|x|^{-s}$ for $s=-1$, does
not satisfy this uniqueness property [Kun74]. As a consequence, the proof of
uniqueness for the long-range gas should use both convexity arguments and the
decay of the (effective) interaction.

#### Decay of the correlations

A proof of uniqueness for the limiting measure of the averaged microscopic
process is obtained for the log-gas in [EHL18] using a displacement convexity
argument showing that the free energy of the infinite gas has a unique
minimizer. The strategy of [EHL18] could possibly be applied to the circular
Riesz setting, but this method does not provide convergence without averaging
nor a speed of convergence. Instead, we propose to examine the rate of decay
of correlations, which is much related to this uniqueness problem. Since
points are very correlated (fluctuations being small), the appropriate
observables to look at are the nearest-neighbor variables. For 1D log-gases,
the correlation between $N(x_{i+1}-x_{i})$ and $N(x_{j+1}-x_{j})$ is proven in
[EY15] to decay in $|i-j|^{-2}$. In this paper we give for the first time a
proof of the optimal decay of gap correlations for the circular Riesz gas,
which matches the case $s=0$ found in [EY15] as well as the predictions of the
physics literature [AM85, Mar88, Lew22]. Moreover we establish that this gap
correlation exhibits a discontinuity at $s=1$ with a much faster decay for
$s=1^{+}$ than $s=1^{-}$.

#### The Helffer-Sjöstrand equation

For generic Gibbs measure on $D_{N}$ (or $\mathbb{R}^{N}$), the covariance
between two smooth enough test-functions is connected to the decay of the
solution of a partial differential equation, named the Helffer-Sjöstrand
(H.-S.) equation. This equation appears in [Sjő93a, Sjő93b, HS94] and is more
substantially studied in [Hel98b, Hel98a, NS97], where it is used to establish
correlation decay, uniqueness of the limiting measure and Log-Sobolev
inequalities for models with convex interactions. Different approaches to
obtain decay estimates on the solutions of Helffer-Sjöstrand equations have
been developed in the statistical physics literature, mainly for Gibbs measure
with convex interactions. The random walk representation of [EY15], already
pointed out in [HS94], [NS97] and used prior in [BM03, DGI00, GOS01] for
instance, corresponds to a Feynman-Kac representation of the solution of the
H.-S. equation. The work [EY15] then develops a sophisticated homogenization
theory for a system of coupled partial differential equations. There are also
more analytic methods relying on ideas from stochastic homogenization, see for
instance [NS97, AW19, DW20, Tho21].

As aforementioned, the method available in the literature [EY15] to prove the
decay of correlations for the 1D log-gas supposes one to control random walks
in random environments, which can be quite technical. The gamble of the
present paper is to develop a method relying _only on integration by parts_ to
treat the long-range Riesz gas with $s\in(0,1)$. We will first consider as a
landmark the hypersingular case $s>1$ and work with a known distortion
argument, used for instance in [Hel98a] or in older techniques to study the
decay of eigenfunctions of Schrödinger operators [CT73]. We will then adapt
the method to the long-range case using substantial new inputs including
discrete elliptic regularity estimates. Let us emphasize that as it stands,
our method cannot be applied to the logarithmic case since it requires to have
nearest-neighbor gaps _all_ bounded from above by a large $N$-dependent
constant much smaller than $N$, with overwhelming probability. Note that this
was also one of the crucial difficulty in [EY15] preventing a simple
implementation of the techniques of Caffarelli, Chan and Vasseur [CCV11].

###  1.2. Main results

Our first result, which concerns the correlations between gaps in the long-
range regime $s\in(0,1)$, is the following:

######  Theorem 1 (Decay of the correlations for the long-range Riesz gas).

Let $s\in(0,1)$. For all $\varepsilon>0$, there exists a constant $C>0$ such
that for all $\xi,\chi:\mathbb{R}\to\mathbb{R}$ in $H^{1}$ and for each
$i,j\in\\{1,\ldots,N\\}$,

$|\mathrm{Cov}_{\mathbb{P}_{N,\beta}}[\xi(N(x_{i+1}-x_{i})),\chi(N(x_{j+1}-x_{j}))]|\\\
\leq
C(\beta)(\mathbb{E}_{\mathbb{P}_{N,\beta}}[\xi^{\prime}(x_{i})^{2}]^{\frac{1}{2}}+|\xi^{\prime}|_{\infty}e^{-c(\beta)d(i,j)^{\delta}})(\mathbb{E}_{\mathbb{P}_{N,\beta}}[\chi^{\prime}(x_{j})^{2}]^{\frac{1}{2}}+|\chi^{\prime}|_{\infty}e^{-c(\beta)d(i,j)^{\delta}})\Bigr{(}\frac{1}{d(i,j)^{2-s-\varepsilon}}+\frac{1}{N}\Bigr{)}.$
(1.5)

Moreover, given $\varepsilon>0$ small enough and any $n\in\\{1,\ldots,N\\}$,
there exist $i,j$ such that $\frac{n}{2}\leq|i-j|\leq n$ such that

$|\mathrm{Cov}_{\mathbb{P}_{N,\beta}}[N(x_{i+1}-x_{i}),N(x_{j+1}-x_{j})]|\geq\varepsilon\frac{1}{d(i,j)^{2-s}}.$
(1.6)

Theorem 1 is the natural extension of [EY15], which proves that that for
$\beta$-ensembles the correlation between $N(x_{i+1}-x_{i})$ and
$N(x_{j+1}-x_{j})$ decays in $|i-j|^{-2}$. The lower bound (1.6) is obtained
by using a result from [Bou21] which gives the leading-order asymptotic of the
correlation between $N(x_{i}-x_{1})$ and $N(x_{j}-x_{i})$. Theorem 1 is in
accordance with the expected decay of the truncated correlation function in
the mathematical physics and physics literature, see [Lew22].

Let us comment on the norms appearing in (1.5). Our method is mainly based on
$L^{2}$ arguments for a distortion of the Helffer-Sjöstrand equation system
which is captured by the $L^{2}$ norm of $\xi^{\prime}$ and $\chi^{\prime}$.
Besides by assuming that that $\xi^{\prime}$ and $\chi^{\prime}$ are uniformly
bounded, we can control the solution on a bad event of exponentially small
probability by carrying out a maximum principle argument.

######  Remark 1.1 (Lagrange multiplier and finite volume correlations).

The factor $\frac{1}{N}$ reflects correlations due to fact that the total
number of points in system is fixed, see [EC81, PS17, BGSRS20]. In fact, in
the framework of Helffer-Sjöstrand equations, it can be interpreted as a
Lagrange multiplier associated to the constraint
$\sum_{j=1}^{N}N(x_{j+1}-x_{j})=N$, with the convention that $x_{N+1}=x_{1}$.

Theorem 1 should be compared to the decay of correlations in the short-range
case, that we quantify in the next theorem:

######  Theorem 2 (Decay of correlations for the short-range Riesz gas).

Let $s\in(1,+\infty)$. There exists a constant $\kappa>0$ such that for all
$\xi,\chi:\mathbb{R}\to\mathbb{R}$ in $H^{1}$ and each
$i,j\in\\{1,\ldots,N\\}$, we have

$|\mathrm{Cov}_{\mathbb{P}_{N,\beta}}[\xi(N(x_{i+1}-x_{i})),\chi(N(x_{j+1}-x_{j}))]|\\\
\leq
C(\beta)(\mathbb{E}_{\mathbb{P}_{N,\beta}}[\xi^{\prime}(x_{i})^{2}]^{\frac{1}{2}}+|\xi^{\prime}|_{\infty}e^{-c(\beta)d(i,j)^{\delta}})(\mathbb{E}_{\mathbb{P}_{N,\beta}}[\chi^{\prime}(x_{j})^{2}]^{\frac{1}{2}}+|\chi^{\prime}|_{\infty}e^{-c(\beta)d(i,j)^{\delta}})\Bigr{(}\frac{1}{d(i,j)^{1+s-\varepsilon}}+\frac{1}{N}\Bigr{)}.$
(1.7)

It would be interesting to establish the rate of decay of correlations in the
case $s=1$. Our next result concerns the limit as $N$ tends to infinity of the
law of the configuration

$\sum_{i=1}^{N}\delta_{Nx_{i}}\mathds{1}_{|x_{i}|<\frac{1}{4}},$ (1.8)

Since $\mathbb{P}_{N,\beta}$ is translation invariant, this is equivalent to
centering the configuration around any point $x\in\mathbb{T}$. Let
$\mathrm{Conf}(\mathbb{R})$ be the set of locally finite, simple point
configurations in $\mathbb{R}$. Given a borelian set $B\subset\mathbb{R}$, we
let $N_{B}:\mathrm{Conf}(\mathbb{R})\to\mathbb{N}$ be the number of points
lying in $B$. The set $\mathrm{Conf}(\mathbb{R})$ is endowed with the
$\sigma$-algebra generated by the maps $\\{N_{B}:B\ \text{Borel}\\}$. A point
process is then a probability measure on $\mathrm{Conf}(\mathbb{R})$. Let
$(x_{1},\ldots,x_{N})$ distributed according to $\mathbb{P}_{N,\beta}$. For
all $x\in\mathbb{T}$, denote

$\mathbb{Q}_{N,\beta}=\mathrm{Law}\left(\sum_{i=1}^{N}\delta_{Nx_{i}}\mathds{1}_{|x_{i}|<\frac{1}{4}}\right)\in\mathcal{P}(\mathrm{Conf}(\mathbb{R})).$
(1.9)

######  Theorem 3 (Uniqueness of the limiting measure ).

Let $s\in(0,1)\cup(1,+\infty)$. There exists a point process
$\mathrm{Riesz}_{s,\beta}$ such that the sequence of point processes
$(\mathbb{Q}_{N,\beta})$ converges to $\mathrm{Riesz}_{s,\beta}$ in the
topology of local convergence: for any bounded, Borel and local test function
$\phi:\mathrm{Conf}(\mathbb{R})\to\mathbb{R}$, we have

$\lim_{N\to\infty}\mathbb{E}_{\mathbb{Q}_{N,\beta}}[\phi]=\mathbb{E}_{\mathrm{Riesz}_{s,\beta}}[\phi].$

Theorem 3 extends the known convergence results for $\beta$-ensembles, see
[BEY12, BEY14a, VV09, Leb15, DHLM21]. Additionally we are able to give a
quantitative bound on the convergence of $\mathbb{Q}_{N,\beta}(x)$ to
$\mathrm{Riesz}_{s,\beta}$ for smooth test-functions.

######  Theorem 4 (Quantitative convergence).

Let $s\in(0,1)\cup(1,+\infty)$. Let $K\in\\{1,\ldots,\frac{N}{2}\\}$ and
$G:\mathbb{R}^{K}\to\mathbb{R}$ in $H^{1}$. Set

$F:X_{N}\to D_{N}\mapsto G(N(x_{2}-x_{1}),\ldots,N(x_{K}-x_{K-1})).$

Then for all $\varepsilon>0$, there holds

$\mathbb{E}_{\mathbb{P}_{N,\beta}}[F]=\mathbb{E}_{\mathrm{Riesz}_{s,\beta}}[G(z_{2}-z_{1},\ldots,z_{K}-z_{K-1})]+O_{\beta}\left(N^{-\frac{s}{2}+\varepsilon}\sup|\nabla
G|^{2}\right).$

Combining the CLT of [Bou21] and the convergence result of Theorem 3, we can
additionally prove a CLT for gaps and discrepancies under the
$\mathrm{Riesz}_{s,\beta}$ process. Let $\zeta(s,x)$ be the Hurwitz zeta
function.

######  Theorem 5 (Hyperuniformity of the $\mathrm{Riesz}_{s,\beta}$ process).

Let $s\in(0,1)$. Under the process $\mathrm{Riesz}_{s,\beta}$, the sequence of
random variables

$K^{-\frac{s}{2}}(z_{K}-z_{1}-K)$

converges in distribution to $Z\sim\mathcal{N}(0,\sigma^{2})$ as $K$ tends to
infinity with

$\sigma^{2}=\frac{1}{\beta\frac{\pi}{2}s}\mathrm{cotan}\left(\frac{\pi}{2}s\right).$

Moreover, the variance of $z_{K}-z_{1}$ under $\mathrm{Riesz}_{s,\beta}$ may
be expanded as

$\mathrm{Var}_{\mathrm{Riesz}(\beta)}[z_{k}-z_{1}]=K^{s}\sigma^{2}+o(K^{s}).$
(1.10)

In particular, Theorem 5 implies that the fluctuations of the number of points
in a given interval under $\mathrm{Riesz}_{s,\beta}$ is much smaller than for
the Poisson process. In the language of [Tor16], this says that
$\mathrm{Riesz}_{s,\beta}$ is hyperuniform when $s\in(0,1)$. Our techniques,
combined with the method of [Bou21], can also give a central limit theorem for
linear statistics under the $\mathrm{Riesz}_{s,\beta}$ process, as done in
[Leb18, Lam21] for $\mathrm{Sine}_{\beta}$.

We conclude this set of results by studying the repulsion of the
$\mathrm{Riesz}_{s,\beta}$ process at $0$. We show that the probability of
having two particles very close to each other decays exponentially.

######  Proposition 1.1.

Fix $\alpha\in(0,\frac{s}{2})$. Let $\varepsilon\in(0,1)$. There exist
constants $c(\beta)>0$ and $C(\beta)>0$ depending on $\alpha$ and locally
uniformly in $\beta$ such that

$\mathbb{P}_{\mathrm{Riesz}_{s,\beta}}(|z_{i+1}-z_{i}|\geq\varepsilon)\geq
1-C(\beta)e^{-c(\beta)\varepsilon^{-\alpha}}.$

###  1.3. Related questions and perspective

#### DLR equations and number-rigidity

Having proved the existence of an infinite volume limit for the circular Riesz
gas, a natural question is then to study the $\mathrm{Riesz}_{s,\beta}$
process from a statistical physics perspective. The first step in that
direction is to establish the Dubroshin-Landford-Ruelle (DLR) equations for
the $\mathrm{Riesz}_{s,\beta}$ process as was done for the
$\mathrm{Sine}_{\beta}$ process in [DHLM21]. We refer to [Geo11] for a
presentation of DLR equations in the context of lattice gases and to [Der19]
in the context of point processes. A question of interest is then to study the
number-rigidity property within the family of long-range Riesz gases. Number-
rigidity is a qualitative property, recently put forward in [GP17] which says
the following: a point process is number-rigid whenever given any compact
domain of $\mathbb{R}^{d}$, the knowledge of the exterior determines in a
deterministic fashion the number of points inside the domain. Number-rigidity
is a quite surprising phenomenon, which has been proved to occur for the 1D
log-gas independently in [CN18] and in [DHLM21] using DLR equations. The
recent work [DV21] also provides a strategy to discard number-rigidity.
Together with the local laws of [Bou21], the result of [DV21] should tell that
the $\mathrm{Riesz}_{s,\beta}$ process is not number-rigid for $s\in(0,1)$.
This reflects a major difference between the log-gas which is purely long-
range and the Riesz gas for which the effective energy is short-range.

#### Regularity of the free energy

A natural question is to investigate the regularity with respect to $\beta$ of
the infinite volume process $\mathrm{Riesz}_{s,\beta}$. A way to address this
problem is to study the regularity of the free energy of the infinite Riesz
gas, which is defined by

$f:\beta\in(0,+\infty)\mapsto\lim_{N\to\infty}\frac{1}{N}\Bigr{(}\log
Z_{N,\beta}-\beta N^{2-s}\iint g_{s}(x-y)\mathrm{d}x\mathrm{d}y\Bigr{)}.$
(1.11)

The existence of such a limit was obtained in [LS17] for Riesz gases in
arbitrary dimension $d\geq 1$ with $\max(0,d-2)<s<d$. In dimension one, one
expects that no phase transition occurs for the circular Riesz gas and that
the free energy is smooth and even analytic. To prove that $f$ is twice
differentiable, a standard approach is to prove that the rescaled variance of
the energy under (1.4) converges locally uniformly in $\beta$ as $N$ tends to
infinity. This should be an easy consequence of Theorems 1 and 3.

#### Riesz gases in higher dimensions

Because the Hamiltonian of the Riesz gas in dimension $d\geq 2$ is not convex,
it is not clear how one could obtain a result on the decay of correlations. In
fact, even showing local laws in the long-range setting is still open, except
in the Coulomb case $s=d-2$ tackled into the series of papers [LS17, Leb17]
cumulating into the optimal local law result of [AS21]. The hypersingular
Riesz gas [HLSS18] is seemingly a more tractable model to look at since it
resembles, as $s$ becomes large, the hard-core model, for which some results
are known. For the later, the translation invariance of the infinite volume
Gibbs measures has been proved in [Ric07] by adapting a well-known argument of
[Geo98].

###  1.4. Outline of the proofs

As mentioned, the heart of the paper is about the analysis of a partial
differential equation related to the correlations, in the context of long-
range Riesz gases. Given a reasonable Gibbs measure
$\mathrm{d}\mu=e^{-H(X_{N})}\mathrm{d}X_{N}$ on $D_{N}$ (or $\mathbb{R}^{N}$),
it is well-known that the covariance between any smooth functions
$F,G:D_{N}\to\mathbb{R}$ may be expressed as

$\mathrm{Cov}_{\mu}[F,G]=\mathbb{E}_{\mu}[\nabla\phi\cdot\nabla G],$ (1.12)

where $\nabla\phi$ solves

$\left\\{\begin{array}[]{ll}-\Delta\phi+\nabla
H\cdot\nabla\phi=F-\mathbb{E}_{\mu}[F]&\text{on }D_{N}\\\
\nabla\phi\cdot\vec{n}=0&\text{on }\partial D_{N},\end{array}\right.$ (1.13)

One may recognize the operator $\mathcal{L}^{\mu}=-\Delta+\nabla H\cdot\nabla$
which is the infinitesimal generator of the Markov semigroup associated to the
Langevin dynamics with energy $H$. The Helffer-Sjöstrand equation then
corresponds to the equation obtained by differentiating (1.13), which reads

$\left\\{\begin{array}[]{ll}A_{1}^{\mu}\nabla\phi=\nabla F&\text{on }D_{N}\\\
\nabla\phi\cdot\vec{n}=0&\text{on }\partial
D_{N}.\end{array}\right.\quad\text{where}\quad
A_{1}^{\mu}:=\mathrm{Hess}\,H+\mathcal{L}^{\mu}\otimes\mathrm{Id}$ (1.14)

When the Hessian of the energy is uniformly positive definite, then one can
derive by integration by parts a weighted $L^{2}$ estimate on $|\nabla\phi|$,
which yields a Brascamp-Lieb inequality. Additionally a maximum principle
argument can also give a uniform bound on $|\nabla\phi|$ as seen in [HS94].

The Hamiltonian we are interested in is rather a convex function of the gaps
than of the points. Henceforth it is very convenient to rewrite the equation
(1.14) in a new of system of coordinates. We define the change of variables

$\mathrm{Gap}_{N}:X_{N}\in
D_{N}\mapsto(N|x_{2}-x_{1}|,\ldots,N|x_{N}-x_{1}|)\in\mathbb{R}^{N}$

and work on the polyhedron

$\mathcal{M}_{N}:=\\{(y_{1},\ldots,y_{N})\in(\mathbb{R}^{+*})^{N}:y_{1}+\ldots+y_{N}=N\\}.$

Assume that the measure of interest $\mu$ can be written
$\mathrm{d}\mu=e^{-H^{\mathrm{g}}\circ\mathrm{Gap}_{N}(X_{N})}\mathrm{d}X_{N}$
and that the test-functions in (1.12) are of the form
$F=\xi\circ\mathrm{Gap}_{N}$ and $G=\chi\circ\mathrm{Gap}_{N}$. Set
$\nu=\mu\circ\mathrm{Gap}_{N}^{-1}$. Then letting

$\mathcal{L}^{\nu}=\nabla H^{\mathrm{g}}\cdot\nabla-\Delta\quad\text{and}\quad
A_{1}^{\nu}=\mathrm{Hess}\,H^{\mathrm{g}}+\mathcal{L}^{\nu}\otimes\mathrm{Id},$

one may check that the solution $\phi$ of (1.13) can be factorized into
$\phi=\psi\circ\mathrm{Gap}_{N}$ where $\nabla\psi$ solves

$\left\\{\begin{array}[]{ll}A_{1}^{\nu}\nabla\psi=\nabla\xi+\lambda(e_{1}+\ldots+e_{N})&\text{on
}\mathcal{M}_{N}\\\ \nabla\psi\cdot(e_{1}+\ldots+e_{N})=0&\text{on
}\mathcal{M}_{N}\\\ \nabla\psi\cdot\vec{n}=0&\text{on
}\partial\mathcal{M}_{N}.\end{array}\right.$ (1.15)

In addition, we also have

$\mathrm{Cov}_{\mu}[F,G]=\mathbb{E}_{\nu}[\nabla\psi\cdot\nabla\chi].$

Let us mention that the coefficient $\lambda$ in (1.15) can be seen as a
Lagrange multiplier associated to the linear constraint
$y_{1}+\ldots+y_{N}=N$. Our main problem is to understand how
$\partial_{j}\psi$ decays when $\nabla\xi=e_{1}$. A first important insight
comes from expanding the Hessian of the energy (1.3) in gap coordinates, that
we denote $\mathcal{H}_{N}^{\mathrm{g}}$. Using some rigidity estimates
obtained in [Bou21], one can show that for all $\varepsilon>0$, there exists
$\delta>0$ such that

$\mathbb{P}_{N,\beta}\Bigr{(}\Bigr{|}\partial_{ij}\mathcal{H}_{n}^{\mathrm{g}}=\frac{1}{1+d(i,j)^{s}}\Bigr{|}\geq\frac{1}{1+d(i,j)^{1+\frac{s}{2}-\varepsilon}}\Bigr{)}\leq
Ce^{-d(i,j)^{\delta}},$

where $d$ stands for the symmetric distance on $\\{1,\ldots,N\\}$, i.e
$d(i,j)=\min(|i-j|,N-|i-j|)$. In other words, the interaction matrix in the
system (1.15) concentrates around a constant long-range matrix. This already
gives a first heuristic to understand the decay of gap correlations stated in
Theorem 1, which is consistent with the decay of
$h:=(-\Delta)^{\frac{1-s}{2}}\delta_{0}$.

Due to the long-range nature of the interaction, the analysis of (1.15) is
rather delicate. Let us present an idea of the proof in the short-range case
$s>1$ as it will be a model for the long-range case also. To simplify assume
that there exist $s>1$ and $c>0$ such that uniformly

$\mathrm{Hess}\,\mathcal{H}_{N}^{\mathrm{g}}\geq
c^{-1}\mathrm{Id}\quad\text{with}\quad|\partial_{ij}\mathcal{H}_{N}^{\mathrm{g}}|\leq
C\frac{1}{d(i,j)^{s}}\text{for each $1\leq i,j\leq N$}.$ (1.16)

The matrix $\mathcal{H}_{N}^{\mathrm{g}}$ then looks like a diagonally
dominant matrix. The idea to obtain a decay estimate on the solution of (1.15)
is to multiply $\partial_{i}\psi$ by $d(i,1)^{\alpha}$ for some well-chosen
$\alpha>0$. Let
$\mathrm{L}_{\alpha}=\mathrm{diag}((1+d(j,1)^{\alpha})_{j})\in\mathcal{M}_{N}(\mathbb{R})$
be the distortion matrix and
$\psi^{\mathrm{dis}}:=\mathrm{L}_{\alpha}\nabla\psi$ which solves

$\beta(\mathrm{Hess}\,\mathcal{H}_{N}^{\mathrm{g}}+\delta_{\mathrm{L}_{\alpha}})\psi^{\mathrm{dis}}+\mathcal{L}^{\nu}\psi^{\mathrm{dis}}=e_{1}+\lambda\mathrm{L}_{\alpha}(e_{1}+\ldots+e_{N}),$

where $\delta_{\mathrm{L}_{\alpha}}$ is the commutator

$\delta_{\mathrm{L}_{\alpha}}:=\mathrm{L}_{\alpha}\mathrm{Hess}\,\mathcal{H}_{N}^{\mathrm{g}}\mathrm{L}_{\alpha}^{-1}-\mathrm{Hess}\,\mathcal{H}_{N}^{\mathrm{g}}.$
(1.17)

A first key is that the more $\mathrm{Hess}\,\mathcal{H}_{N}^{\mathrm{g}}$ is
diagonal, the more it will commute with diagonal matrices. In fact one can
check that for $\alpha<s-\frac{1}{2}$, the commutator (1.17) is small compared
to the identity, in the sense of quadratic forms. By integration by parts and
using the convexity of $\mathcal{H}_{N}^{\mathrm{g}}$, this gives an $L^{2}$
estimate on $\psi^{\mathrm{dis}}$ and therefore a hint on the global decay of
$\nabla\psi$. This idea of studying a distorted vector-field is well known in
statistical physics, see for instance [Hel98a, CT73]. By projecting (1.15) in
a smaller window we can then improve through a bootstrap argument this first
decay estimate.

In the long-range regime $s\in(0,1)$, the above argument no longer works. A
natural way of proceeding is to factorize the equation (1.15) around the
ground state by multiplying the system by a matrix $\mathsf{A}$ close to the
inverse of the Riesz matrix $\mathbb{H}_{s}:=(\frac{1}{1+d(i,j)^{s}})_{1\leq
i,j\leq N}$. A simple construction can ensure that
$\mathsf{A}\mathrm{Hess}\,\mathcal{H}_{n}^{\mathrm{g}}$ remains uniformly
positive definite but the drawback of the operation is that the differential
term $D\psi$ can no longer be controlled. The main novelty of the paper is a
method based on the comparison of the two distorted norms

$\mathbb{E}_{\mu}\Bigr{[}\sum_{i=1}^{n}d(i,1)^{2\alpha}(\partial_{i}\psi)^{2}\Bigr{]}\quad\text{and}\quad\mathbb{E}_{\mu}\Bigr{[}\sum_{i=1}^{n}d(i,1)^{2\gamma}|\nabla(\partial_{i}\psi)|^{2}\Bigr{]},$
(1.18)

for well-chosen constants $\alpha>0$ and $\gamma>0$. The first step is to
derive an elliptic regularity estimate on the solution of (1.15). We prove
that the solution has a discrete fractional primitive of order $\frac{3}{2}-s$
in $L^{2}$ (up to some $n^{\kappa\varepsilon}$ multiplicative factor) provided
$\psi_{i}$ decays fast enough. In a second step we will control
$|\mathrm{L}_{\gamma}D^{2}\psi|$ by
$|\mathrm{L}_{\frac{\gamma}{2}+\frac{1}{4}}\nabla\psi|$ (up to a residual term
that we do not comment here). The proof uses the distortion argument presented
in the short-range case, the elliptic regularity estimate and a discrete
Gagliardo-Nirenberg inequality. In a third step we control
$|\mathrm{L}_{\alpha}\nabla\psi|$ by
$|\mathrm{L}_{\alpha-\frac{1-s}{2}}D^{2}\psi|$ by implementing the
factorization trick aforementioned. Combining these two inequalities we deduce
that for $\alpha=\frac{3}{2}-s$ and $\gamma=1-\frac{s}{2}$, each of the terms
in (1.18) are small. This gives the optimal global decay on the solution of
(1.15), which we then seek to localize.

The proof of localization, which allows one to go from (1.18) to an estimate
on a single $\partial_{i}\psi$, is also quite delicate. Fix an index
$j\in\\{1,\ldots,N\\}$ and let

$J=\Bigr{\\{}i\in\\{1,\ldots,N\\}:d(i,j)\leq\frac{1}{2}d(j,1)\Bigr{\\}}.$

Projecting the equation (1.15) on the window $J$ makes an exterior field
appear, which takes the form

$\mathrm{V}_{l}:=-\beta\sum_{i\in
J^{c}}\partial_{il}\mathcal{H}_{N}^{\mathrm{g}}\partial_{i}\psi,\quad l\in J.$
(1.19)

We then break $\mathrm{V}$ into the sum of an almost constant field
$\mathrm{V}^{(1)}$ (looking like $\mathrm{V}_{j}\sum_{l\in J}e_{l})$ and a
smaller field $\mathrm{V}^{(2)}$. A key is that the equation (3.12) associated
to a vector-field proportional to $(e_{1}+\ldots+e_{N})$ is much easier to
analyze. It indeed admits a mean-field approximation, quite similar to the
mean-field approximation of (1.14) when $F$ is a linear statistics, see
[Bou21]. We then bootstrap the decay of solutions of (1.15). Applying the
induction hypothesis to bound (1.19) _and_ to bound the decay of (1.15) in the
window $J$, one finally obtains after a finite number of iterations, the
optimal result of Theorem 1.

The uniqueness of the limiting point process stated in Theorem 3 is then a
routine application of our result on decay of correlations (in fact stated for
slightly more general systems than (1.15)). Because the existence of an
accumulation point of (1.9) in the local topology is standard, the problem can
be rephrased into a uniqueness question. We will prove that the sequence (1.9)
defines, in some informal sense, a Cauchy sequence. We let
$I=\\{1,\ldots,n\\}$ be the active window and draw to exterior configurations
under $\mathbb{P}_{N,\beta}$ and $\mathbb{P}_{N^{\prime},\beta}$ for distinct
values of $N$ and $N^{\prime}$ which satisfy $n\ll N,N^{\prime}$. We then let
$\mu_{n}$ and $\nu_{n}$ the conditioned measures in gap coordinates, which we
try to compare. To allow such a comparison, the strategy is to define a
continuous path $\mu(t)$ from $\mu_{n}$ to $\nu_{n}$ by linear interpolation
of the exterior energies. Given a test-function
$F:\mathbb{R}^{n}\to\mathbb{R}$ depending on a finite number of coordinates,
we can then write

$\mathbb{E}_{\mu_{n}}[F]-\mathbb{E}_{\nu_{n}}[F]=\int_{0}^{1}\mathrm{Cov}_{\mu(t)}[\nabla
F,\nabla E(t)]\mathrm{d}t,$ (1.20)

where $E(t)$ corresponds to the exterior energy term. By applying our result
on the decay of correlations to the measure $\mu(t)$, we find that (1.20) is
small, which easily concludes the proof of Theorem 3.

###  1.5. Organization of the paper

  * •

Section 2 records various preliminary results, such as rigidity estimates on
circular Riesz gases and controls on the discrete fractional Laplacian.

  * •

Section 3 focuses on the well-posedness of the Helffer-Sjöstrand equation and
gives some basic properties.

  * •

In Section 4 we introduce our techniques to prove the decay of correlations
for the hypersingular Riesz gas.

  * •

Section 5 is the heart of the paper. It develops a more involved method to be
able to treat the decay of correlations for the long-range Riesz gas.

  * •

Section 6 concludes the proof of uniqueness of the limiting measure of Theorem
3.

###  1.6. Notation

  * •

We let $d$ be a distance on $\\{1,\ldots,N\\}$ defined for each
$i,j\in\\{1,\ldots,N\\}$ by

$d(i,j)=\min(|j-i|,N-|j-i|).$

  * •

For $A,B\geq 0$, we write $A\leq C(\beta)B$ or $A=O_{\beta}(B)$ whenever there
exists a constant $C\in\mathbb{R}^{+}$ locally uniform in $\beta$ (which might
depend on $s$) such that $A\leq CB$.

  * •

Given a probability measure $\mu$ on $D_{N}$, we denote $\mu^{\mathrm{g}}$ the
push forward of $\mu$ by the map

$X_{N}\in
D_{N}\mapsto(|N(x_{2}-x_{1})|,\ldots,|N(x_{N}-x_{1})|)\in(\mathbb{R}^{+})^{N}.$

  * •

For $x\in\mathbb{R}^{n}$, we let $|x|$ be the Euclidian norm of $x$ and for
$M\in\mathcal{M}_{n}(\mathbb{R})$, $\|M\|$ be the $L^{2}$ operator norm of
$M$, i.e

$\|M\|=\sup_{v\in\mathbb{R}^{n}\setminus\\{0\\}}\frac{|Mv|}{|v|}.$

  * •

We either use the notation $\mathrm{Hess}\,$ or the notation $D^{2}$ for the
Hessian of a real-valued function $f:\mathbb{R}^{n}\to\mathbb{R}$.

  * •

We let $(e_{1},\ldots,e_{N})$ be the standard orthonormal basis of
$\mathbb{R}^{N}$.

### Acknowledgments

The author would like to thank Sylvia Serfaty and Thomas Leblé for helpful
comments on this work. The author is supported by a grant from the “Fondation
CFM pour la Recherche”.

##  2\. Preliminaries

We begin by recording some useful preliminary results that will be used
throughout the paper.

###  2.1. Discrepancy estimates

We first give a control on the probability of having two particles very close
to each other. By [Bou21, Lemma 4.5], we have

######  Lemma 2.1.

Let $s\in(0,1)$. There exist constants $C(\beta)>0$ and $c(\beta)>0$ locally
uniform in $\beta$ such that for each $i\in\\{1,\ldots,N\\}$ and
$\varepsilon>0$

$\mathbb{P}_{N,\beta}(N(x_{i+1}-x_{i})\leq\varepsilon)\leq
C(\beta)e^{-c(\beta)\varepsilon^{-\frac{s}{4}}}.$

In addition under the long-range Riesz gas, the fluctuations of
$N(x_{i+k}-x_{i})$ are of order $k^{\frac{s}{2}+\varepsilon}$ with a
subexponential probability of deviation. In view of [Bou21], the following
holds:

######  Theorem 6 (Near optimal rigidity).

Let $s\in(0,1)$. There exists a constant $C(\beta)$ locally uniform in $\beta$
such that for all $\varepsilon>0$, setting
$\delta=\frac{\varepsilon}{4(s+2)}$, for each $i\in\\{1,\ldots,N\\}$ and
$1\leq k\leq\frac{N}{2}$, we have

$\mathbb{P}_{N,\beta}(|N(x_{i+k}-x_{i})-k|\geq
k^{\frac{s}{2}+\varepsilon})\leq C(\beta)e^{-c(\beta)k^{\delta}}.$ (2.1)

One may refer to [Bou21, Section 3] for a proof of Theorem 6.

###  2.2. Fractional Laplacian on the circle

In this subsection we justify the expression of the fundamental solution of
the fractional Laplace equation on the circle (1.2).

######  Lemma 2.2 (Fundamental solution).

Let $g_{s}$ be the solution of (1.2). Let $\zeta(s,a)$ be the Hurwitz zeta
function. Let $s\in(0,1)$. For all $x\in\mathbb{T}$,

$g_{s}(x)=\zeta(s,x)+\zeta(s,1-x)=\lim_{n\to\infty}\Bigr{(}\sum_{k=-n}^{n}\frac{1}{|k+x|^{s}}-\frac{2}{1-s}n^{1-s}\Bigr{)}.$
(2.2)

Moreover for all $p\geq 1$ and all $x\in\mathbb{T}$

$g_{s}^{(p)}(x)=(-1)^{p}s\ldots(s+p-1)\zeta(s+k,x)=(-1)^{p}s\cdots(s+p-1)\sum_{k\in\mathbb{Z}}\frac{1}{|x+k|^{s+p}}.$
(2.3)

###### Proof of Lemma 2.2.

We refer to [Bou21] for a detailed proof. Let us sketch the main arguments.
Using the Fourier characterization of the fractional Laplacian and applying
the formula

$\lambda^{-\frac{1-s}{2}}=\frac{1}{\Gamma(\frac{1-s}{2})}\int_{0}^{\infty}e^{-\lambda
t}\frac{\mathrm{d}t}{t^{1-\frac{1-s}{2}}},$ (2.4)

valid for all $\lambda>0$, one can express $g_{s}$ as

$g_{s}(x)=\frac{c_{s}}{\Gamma(\frac{1-s}{2})}\int_{0}^{\infty}(W_{t}(x)-1)\frac{\mathrm{d}t}{t^{\frac{1+s}{2}}},$

where $W_{t}$ is the heat kernel on $\mathbb{T}$, namely

$W_{t}(x)=\frac{1}{2\pi}\sum_{k\in\mathbb{Z}}e^{-t|k|^{2}}e^{ikx}=\frac{1}{\sqrt{4\pi
t}}\sum_{k\in\mathbb{Z}}e^{-\frac{|x-k|^{2}}{4t}}.$ (2.5)

The proof of (2.2) then follows from Fubini’s theorem which allows one to
invert the order of integration and summation. ∎

The kernel $g_{s}$ can be identified with a periodic function on $\mathbb{R}$
and a crucial consequence of (2.2) is that the restriction of this function to
$(0,1)$ is convex.

###  2.3. Semi-discrete Fourier transform

In the next subsection we will need to consider the discrete Fourier transform
of functions defined on the discrete circle $\mathbb{Z}/N\mathbb{Z}$. We
define the semi-discrete Fourier and inverse Fourier transforms by

$\mathcal{F}_{d}(f)(\theta)=\sum_{n=1}^{N}f(n)e^{in\theta},\quad\text{for
$f:\\{1,\ldots,N\\}\to\mathbb{R}$},\quad\theta\in[-\pi,\pi].$
$\mathcal{F}_{d}^{-1}(\phi)(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\phi(\theta)e^{-in\theta}\mathrm{d}\theta,\quad\text{for
$\phi\in L^{1}(\mathbb{R},\mathbb{C})$},\quad n\in\\{1,\ldots,N\\}.$

Recall that for all $f$ defined on $\mathbb{Z}/n\mathbb{Z}$,
$f=\mathcal{F}_{d}^{-1}\circ\mathcal{F}_{d}(f)$.

###  2.4. Inversion of the Riesz matrix

The aim of this subsection is to establish the decay of the solution of a
discrete convolution equation. Letting

$\widetilde{g}_{s}:x\in\mathbb{R}^{+*}\mapsto\frac{1}{|x|^{s}},$ (2.6)

we will be studying the inverse of the matrix defined by

$\mathbb{H}_{s}:=\Bigr{(}\widetilde{g}_{s}(1+d(i,j))\Bigr{)}_{1\leq i,j\leq
N}\in\mathcal{M}_{N}(\mathbb{R}).$ (2.7)

######  Lemma 2.3 (Decay of the inverse Riesz matrix).

Let $\mathbb{H}_{s}$ be the matrix (2.7). There exists a constant $C>0$ such
that for each $1\leq i,j\leq N$,

$|(\mathbb{H}_{s}^{-1})_{i,j}|\leq\frac{C}{1+d(j,i)^{2-s}}.$ (2.8)

In addition for each $i\in\\{1,\ldots,N\\}$, we have

$\Bigr{|}\sum_{i=1}^{N}(\mathbb{H}_{s}^{-1})_{i,1}\Bigr{|}\leq\frac{C}{N^{1-s}}.$
(2.9)

Let us observe that (2.8) is consistent with the decay of the fundamental
solution of the fractional Laplacian. Indeed the coefficient
$(\mathbb{H}_{s})^{-1}_{i,1}$ is given by the $i$-th coordinate of the
solution $v$ of a discrete convolution equation. The continuous counterpart of
this equation is $\widetilde{g}_{s}*\psi=\delta_{0}$ on the real line, for
which the solution decays in $|x|^{-(2-s)}$ away from the origin.

###### Proof of Lemma 2.3.

By symmetry one may take $i=1$. Let $v\in\mathbb{R}^{N}$ be the solution of
the discrete convolution equation

$\widetilde{g}_{s}*v=\delta_{1},\quad\text{with}\quad
v:\\{1,\ldots,N\\}\to\mathbb{R}.$ (2.10)

By Fourier inversion, the solution of (2.10) is given by

$v=\mathcal{F}_{d}^{-1}\Bigr{(}\frac{1}{\mathcal{F}_{d}(\widetilde{g}_{s})}\Bigr{)}.$

Let us denote $S_{N}$ the discrete Fourier transform of $\widetilde{g}_{s}$,
given for all $\theta\in[-\pi,\pi]$ by

$\frac{1}{2}\mathcal{F}_{d}(\widetilde{g}_{s})(\theta)=\sum_{n=1}^{\lfloor
N/2\rfloor}\widetilde{g}_{s}(n)\cos(n\theta):=S_{\lfloor N/2\rfloor}(\theta).$
(2.11)

The above sum corresponds to a truncation of the so-called periodic zeta
function [Apo97], given by

$S(\theta):=\sum_{n=1}^{\infty}\frac{1}{n^{s}}\cos(n\theta)=\frac{\Gamma(1-s)}{(2\pi)^{1-s}}\Bigr{(}e^{{i\pi\frac{1-s}{2}}}\zeta(1-s,\theta)+e^{{i\pi\frac{s-1}{2}}}\zeta(1-s,1-\theta)\Bigr{)},$
(2.12)

where $\zeta(s,a)$ stands for the Hurwitz zeta function. Let us bound from
below the modulus of (2.11). By performing a discrete integration by parts,
one obtains

$\sum_{n=1}^{N}\frac{1}{n^{s}}\cos(n\theta)=S(\theta)+R_{N}(\theta),$

with

$R_{N}(\theta)=-\sum_{n=N+1}^{\infty}\Bigr{(}\frac{1}{(n+1)^{s}}-\frac{1}{n^{s}}\Bigr{)}\frac{\cos(\frac{(n+1)\theta}{2})\sin(\frac{n\theta}{2})}{\sin(\frac{\theta}{2})}\geq
0.$ (2.13)

We deduce in particular that there exist constants $C_{0}>0$ and
$C_{0}^{\prime}>0$ such thay

$S(\theta)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}\cos(n\theta)\geq
S_{\lfloor\frac{1}{|\theta|}\rfloor}(\theta)\geq
c_{0}\sum_{n=1}^{\lfloor\frac{1}{|\theta|}\rfloor}\frac{1}{n^{s}}\geq\frac{c_{0}^{\prime}}{|\theta|^{1-s}}.$
(2.14)

Besides in view of (2.13), one can observe that $R_{N}(\theta)$ is bounded by

$|R_{N}(\theta)|\leq\frac{C_{0}}{|\theta|}\sum_{n\geq
N+1}\frac{1}{n^{1+s}}\leq\frac{C_{0}^{\prime}}{N^{s}|\theta|},\quad\text{for
some constants $C_{0}>0,C_{0}^{\prime}>0$}.$ (2.15)

In view of (2.14) and (2.15), there exists $c_{1}>0$ such that for
$|\theta|>\frac{c_{1}}{N}$, $0<\frac{R_{N}(\theta)}{S(\theta)}<\frac{1}{2}$
and

$\frac{1}{S_{N}(\theta)}=\frac{1}{S(\theta)}\Bigr{(}1+\sum_{k=1}^{\infty}(-1)^{k}\Bigr{(}\frac{R_{N}(\theta)}{S(\theta)}\Bigr{)}^{k}\Bigr{)}.$

For $k=1$ using (2.13), we have

$\int_{[-\pi,\pi]\setminus[-\frac{c_{1}}{N},\frac{c_{1}}{N}]}\frac{R_{N}(\theta)}{S(\theta)^{2}}e^{-in\theta}\mathrm{d}\theta\\\
=-\sum_{l=N+1}^{\infty}\Bigr{(}\frac{1}{(l+1)^{s}}-\frac{1}{l^{s}}\Bigr{)}\int_{[-\pi,\pi]\setminus[-\frac{c_{1}}{N},\frac{c_{1}}{N}]}\frac{1}{S(\theta)^{2}}\frac{\cos(\frac{(l+1)\theta}{2})\sin(\frac{l\theta}{2})}{\sin(\frac{\theta}{2})}e^{-in\theta}\mathrm{d}\theta.$
(2.16)

Let $l\geq N+1$. Let us define $h,G_{n,l}:[-\pi,\pi]\to\mathbb{R}$ such that
for all $\theta\in[-\pi,\pi]$

$\displaystyle h(\theta)=\frac{1}{S(\theta)^{2}\sin(\frac{\theta}{2})},\quad
G_{n,l}^{\prime\prime}(\theta)=\cos\Bigr{(}\frac{(l+1)\theta}{2}\Bigr{)}\sin\Bigr{(}\frac{l\theta}{2}\Bigr{)}e^{-in\theta}.$

Noting

$|h(\theta)|\leq
C|\theta|^{1-2s},\quad|h^{\prime\prime}(\theta)|\leq\frac{C}{|\theta|^{1+2s}},\quad|G_{n,l}|(\theta)\leq\frac{C}{l},\quad|\widetilde{G}_{n,l}|(\theta)\leq\frac{C}{l^{2}},$

one gets by integration by parts,

$\int_{[-\pi,\pi]\setminus[-\frac{c_{1}}{N},\frac{c_{1}}{N}]}\frac{1}{S(\theta)^{2}}\frac{\cos(\frac{(l+1)\theta}{2})\sin(\frac{l\theta}{2})}{\sin(\frac{\theta}{2})}e^{-in\theta}\mathrm{d}\theta=-\int_{-\pi}^{\pi}h^{\prime}(\theta)G_{n,l}^{\prime}(\theta)\mathrm{d}\theta+O\Bigr{(}\frac{1}{N^{2-2s}}\Bigr{)}.$

Integrating by parts again gives

$-\int_{-\pi}^{\pi}h_{l}^{\prime}(\theta)G_{n,l}^{\prime}(\theta)\mathrm{d}\theta=\int_{-\pi
l}^{\pi
l}h^{\prime\prime}(\theta){G}_{n,l}(\theta)\mathrm{d}\theta=\frac{1}{l}\int_{-\pi}^{\pi}h^{\prime\prime}(\theta/l){G}_{n,l}(\theta/l)\mathrm{d}\theta=O\Bigr{(}\frac{1}{l^{2(1-s)}}\Bigr{)}.$

Inserting this into (2.16) and summing this over $l$ yields

$\int_{[-\pi,\pi]\setminus[-\frac{c_{1}}{N},\frac{c_{1}}{N}]}\frac{R_{N}(\theta)}{S(\theta)^{2}}e^{-in\theta}\mathrm{d}\theta=O\Bigr{(}\frac{1}{N^{2-s}}\Bigr{)}.$

Let $2\leq k\leq\frac{2}{s}-1$. By performing iterative integration by parts
as in the foregoing computations, we find that

$\Bigr{|}\int_{[-\pi,\pi]\setminus[-\frac{c_{1}}{N},\frac{c_{1}}{N}]}\frac{R_{N}(\theta)^{k}}{S(\theta)^{k+1}}e^{-in\theta}\mathrm{d}\theta\Bigr{|}=O\Bigr{(}\frac{1}{N^{2-s}}\Bigr{)}.$

Finally if $k\geq\frac{2}{s}$, the integral at hand is convergent at infinity
and by (2.14), (2.15) we have

$\Bigr{|}\int_{[-\pi,\pi]\setminus[-\frac{c_{1}}{N},\frac{c_{1}}{N}]}\frac{R_{N}(\theta)^{k}}{S(\theta)^{k+1}}e^{-in\theta}\mathrm{d}\theta\Bigr{|}\leq\frac{C}{N^{ks}}\int_{[-\pi,\pi]\setminus[-\frac{c_{1}}{N},\frac{c_{1}}{N}]}\frac{1}{|\theta|^{s(k+1)-1}}+O\Bigr{(}\frac{1}{N^{2-s}}\Bigr{)}=O\Bigr{(}\frac{1}{N^{2-s}}\Bigr{)}.$

We conclude that

$\int_{[-\pi,\pi]\setminus[-\frac{c_{1}}{N},\frac{c_{1}}{N}]}\frac{\mathrm{d}\theta}{S_{N}(\theta)}=\int_{[-\pi,\pi]\setminus[-\frac{c_{1}}{N},\frac{c_{1}}{N}]}\frac{\mathrm{d}\theta}{S(\theta)}+O\Bigr{(}\frac{1}{N^{2-s}}\Bigr{)}.$
(2.17)

Furthermore one can easily check that

$\int_{[-\frac{c_{1}}{N},\frac{c_{1}}{N}]}\frac{1}{S_{N}(\theta)}\mathrm{d}\theta\leq
CN^{s}\int_{[-\frac{c_{1}}{N},\frac{c_{1}}{N}]}|\theta|\mathrm{d}\theta=O\Bigr{(}\frac{1}{N^{2-s}}\Bigr{)}\quad\text{and}\quad\int_{[-\frac{c_{1}}{N}\frac{c_{1}}{N}]}\frac{1}{S(\theta)}\mathrm{d}\theta=O\Bigr{(}\frac{1}{N^{2-s}}\Bigr{)}.$
(2.18)

Combining (2.17) and (2.18) we get

$\int_{-\pi}^{\pi}\frac{1}{S_{N}(\theta)}e^{-in\theta}\mathrm{d}\theta=\int_{-\pi}^{\pi}\frac{1}{S(\theta)}e^{-in\theta}\mathrm{d}\theta+O\Bigr{(}\frac{1}{N^{2-s}}\Bigr{)}.$
(2.19)

In particular, using the inverse Fourier transform, the solution $v$ of (2.10)
satisfies

$v(n)=O\Bigr{(}\frac{1}{n^{2-s}}+\frac{1}{(N-n+1)^{2-s}}\Bigr{)},$

as claimed in (2.8). The estimate (2.9) is straightforward. ∎

##  3\. The Helffer-Sjöstrand equation

In this section we introduce some standard results on Helffer-Sjöstrand
equations. We first recall basic properties valid for a certain class of
convex Gibbs measures. We then study an important change of variables and
rewrite the Helffer-Sjöstrand in gap coordinates. For the class of Gibbs
measures we are interested in, the energy is a convex function of the gaps.
This allows one to derive a maximum principle for solutions, which will be a
central tool in the rest of the paper.

###  3.1. Well-posedness

We start by explaining the principle of Helffer-Sjöstrand representation and
give some existence and uniqueness results. The subsection is similar to
[Bou21] and follows partly the presentation of [AW19]. Let $\mu$ be a
probability measure on $D_{N}$ in the form

$\mathrm{d}\mu=e^{-H(X_{N})}\mathds{1}_{D_{N}}(X_{N})\mathrm{d}X_{N},$

where $H:D_{N}\to\mathbb{R}$ is a smooth and convex function. Given a smooth
test-function $F:D_{N}\to\mathbb{R}$, we wish to rewrite its variance in a
convenient and effective way. Let us recall the integration by parts formula
for $\mu$. Let $\mathcal{L}^{\mu}$ be the operator acting on
$\mathcal{C}^{\infty}(D_{N},\mathbb{R})$ given by

$\mathcal{L}^{\mu}=\nabla H\cdot\nabla-\Delta,$

where $\nabla$ and $\Delta$ are the standard gradient and Laplace operators on
$\mathbb{T}^{N}$. The operator $\mathcal{L}$ is the generator of the Langevin
dynamics associated to the energy $H$ of which $\mu$ is the unique invariant
measure. By integration by parts under $\mu$, for any functions
$\phi,\psi\in\mathcal{C}^{\infty}(D_{N},\mathbb{R})$ such that
$\nabla\phi\cdot\vec{n}=0$ on $\partial D_{N}$, we can write

$\mathbb{E}_{\mu}[\psi\mathcal{L}^{\mu}\phi]=\mathbb{E}_{\mu}[\nabla\psi\cdot\nabla\phi].$
(3.1)

This formula may be proved by integration by parts under the Lebesgue measure
on $D_{N}$.

Assume that the Poisson equation

$\left\\{\begin{array}[]{ll}\mathcal{L}^{\mu}\phi=F-\mathbb{E}_{\mu}[F]&\text{on
}D_{N}\\\ \nabla\phi\cdot\vec{n}=0&\text{on }\partial D_{N}\end{array}\right.$
(3.2)

admits a weak solution in a certain functional space. Then, by (3.1), the
variance of $F$ under $\mu$ can be expressed as

$\mathrm{Var}_{\mu}[F]=\mathbb{E}_{\mu}[\nabla F\cdot\nabla\phi].$

The above identity is called the Helffer-Sjöstrand representation formula. Let
us differentiate (3.2). Formally, for all
$\phi\in\mathcal{C}^{\infty}(D_{N},\mathbb{R})$, we have

$\nabla\mathcal{L}^{\mu}\phi=A_{1}\nabla\phi,$

where $A_{1}^{\mu}$ is the so-called Helffer-Sjöstrand operator given by

$A_{1}^{\mu}=\mathrm{Hess}\,H+\mathcal{L}^{\mu}\otimes\mathrm{Id},$

with $\mathcal{L}^{\mu}\otimes\mathrm{Id}$ acting diagonally on
$L^{2}(\\{1,\ldots,N\\},\mathcal{C}^{\infty}(D_{N},\mathbb{R}))$. Therefore
the solution $\nabla\phi$ of (3.2) formally satisfies

$\left\\{\begin{array}[]{ll}A_{1}^{\mu}\nabla\phi=\nabla F&\text{on }D_{N}\\\
\nabla\phi\cdot\vec{n}=0&\text{on }\partial D_{N}.\\\ \end{array}\right.$
(3.3)

This partial differential equation is called the Helffer-Sjöstrand equation.
Let us now introduce the appropriate functional spaces to make these
derivations rigorous. Let us define the norm

$\|F\|_{H^{1}(\mu)}=\mathbb{E}_{\mu}[F^{2}]^{\frac{1}{2}}+\mathbb{E}_{\mu}[|\nabla
F|^{2}]^{\frac{1}{2}}.$

Let $H^{1}(\mu)$ be the completion of $\mathcal{C}^{\infty}(D_{N})$ with
respect to the norm $\|\cdot\|_{H^{1}(\mu)}$. Let also define the norm

$\|F\|_{H^{-1}(\mu)}=\sup\\{|\mathbb{E}_{\mu}[FG]|:G\in
H^{1}(\mu),\|G\|_{H^{1}(\mu)}\leq 1\\}.$

We denote $H^{-1}(\mu)$ the dual of $H^{1}(\mu)$, that is the completion of
$\mathcal{C}^{\infty}(D_{N})$ with respect to the norm
$\|\cdot\|_{H^{-1}(\mu)}$. We wish to prove that under mild assumptions on
$F$, the equation (3.3) is well-posed, in the sense of
$L^{2}(\\{1,\ldots,N\\},H^{-1}(\mu))$. Let us now make the following
assumptions on $\mu$:

######  Assumptions 3.1.

Assume that $\mu$ is a probability measure on $D_{N}$ written

$\mathrm{d}\mu=e^{-H(X_{N})}\mathds{1}_{D_{N}}(X_{N})\mathrm{d}X_{N},$

with $H:D_{N}\to\mathbb{R}$ in the form

$H:X_{N}\mapsto\sum_{i\neq j}\chi(|x_{i}-x_{j}|),$

with $\chi:\mathbb{R}^{+*}\to\mathbb{R}$ satisfying

$\chi\in\mathcal{C}^{2}(\mathbb{R}^{+*},\mathbb{R})\quad\text{and}\quad\chi^{\prime\prime}\geq
c>0.$

In our applications, $\chi$ is often given by $g_{s}$ or a variant of it and
the density of the measure $\mu$ is not necessarily bounded from below with
respect to the Lebesgue measure on $D_{N}$. Additionally, the measure $\mu$
does not satisfies a uniform Poincaré inequality. Due to these limitations, to
prove the well-posedness of (3.3), we further assume that $F$ is a function of
the gaps. We denote

$\Pi:X_{N}\in
D_{N}\mapsto(x_{2}-x_{1},\ldots,x_{N}-x_{1})\in\mathbb{T}^{N-1}.$ (3.4)

We also let $\mu^{\prime}$ be the push-forward of $\mu$ by the map $\Pi$:

$\mu^{\prime}=\mu\circ\Pi^{-1}.$ (3.5)

We can now state the following well-posedness result:

######  Proposition 3.1 (Existence and representation).

Let $\mu$ satisfying Assumptions 3.1. Let $F\in H^{1}(\mu)$. Assume that $F$
is in the form $F=G\circ\Pi$, $G\in H^{1}(\mu^{\prime})$ or that $\mu\geq
c>0$. Then there exists a unique $\nabla\phi\in
L^{2}(\\{1,\ldots,N\\},H^{1}(\mu))$ such that

$\left\\{\begin{array}[]{ll}A_{1}^{\mu}\nabla\phi=\nabla F&\text{ on }D_{N}\\\
\nabla\phi\cdot\vec{n}=0&\text{ on }\partial D_{N},\end{array}\right.$ (3.6)

with the first identity being, for each coordinate, an identity on elements of
$H^{-1}(\mu)$. Moreover the solution of (3.6) is the unique minimizer of the
functional

$\nabla\phi\mapsto\mathbb{E}_{\mu}[\nabla\phi\cdot\mathrm{Hess}\,H\nabla\phi+|D^{2}\phi|^{2}-2\nabla
F\cdot\nabla\phi],$

over maps $\nabla\phi\in L^{2}(\\{1,\ldots,N\\},H^{1}(\mu))$. The variance of
$F$ may be represented as

$\mathrm{Var}_{\mu}[F]=\mathbb{E}_{\mu}[\nabla\phi\cdot\nabla F]$ (3.7)

and the covariance between $F$ any function $G\in H^{1}(\mu)$ as

$\mathrm{Cov}_{\mu}[F,G]=\mathbb{E}_{\mu}[\nabla\phi\cdot\nabla G].$

The identity (3.7) is called the Helffer-Sjöstrand formula. The proof of
Proposition 3.1 is postponed to the Appendix, see Section B.

######  Remark 3.2 (On the boundary condition).

The boundary condition $\nabla\phi\cdot\vec{n}=0$ on $\partial D_{N}$ means
that if $x_{i}=x_{j}$, then $\partial_{i}\phi(X_{N})=\partial_{j}\phi(X_{N})$.

######  Remark 3.3 (Link to the Monge-Ampère equation).

We formally discuss the link between (3.6) and the Monge-Ampère equation. Let
$F:D_{N}\to\mathbb{R}$ be a smooth test-function. For all $t\geq 0$, consider
the measure
$\mathrm{d}\mu_{t}=\frac{e^{tF}}{\mathbb{E}_{\mu}[e^{tF}]}\mathrm{d}\mu$.
According to well-known optimal transportation results [Bre91], the measure
$\mu_{t}$ can be written $\mu_{t}=\mu\circ\nabla\Phi_{t}^{-1}$ with
$\Phi_{t}:D_{N}\to\mathbb{R}$ solution of the Monge-Ampère equation

$-\log\det
D\nabla\Phi_{t}+H\circ\nabla\Phi_{t}-H=tF-\log\mathbb{E}_{\mu}[e^{tF}].$

Formally, since $\mu(t)=\mu+t\nu+o(t)$, one expects that
$\Phi_{t}=\mathrm{Id}+t\phi+o(t)$. Linearizing the above equation in $t$
formally gives

$\mathcal{L}^{\mu}\phi=F-\mathbb{E}_{\mu}[F],$

which is the Poisson equation (3.2). The boundary condition in (3.6) reflects
the fact that for all $t\geq 0$, $\nabla\Phi_{t}$ maps $D_{N}$ on itself.

######  Proposition 3.2.

Let $\mu$ satisfying Assumptions 3.1. Let $v\in
L^{2}(\\{1,\ldots,N\\},H^{-1}(\mu^{\prime}))$ such that
$v\cdot(e_{1}+\ldots+e_{N})=0$. There exists a unique $\psi\in
L^{2}(\\{1,\ldots,N\\},H^{1}(\mu))$ such that

$\left\\{\begin{array}[]{ll}A_{1}^{\mu}\psi=v&\text{on }D_{N}\\\
\psi\cdot\vec{n}=0&\text{on }\partial D_{N}.\end{array}\right.$ (3.8)

In addition if $v=\nabla F\in L^{2}(\\{1,\ldots,N\\},H^{-1}(\mu^{\prime}))$,
then the solution of (3.8) is given by the solution of (3.6).

The proof of Proposition 3.2 is also given in the Appendix.

###  3.2. Rewriting the equation in gap coordinates

In the rest of the paper, the measure of interest satisfies Assumptions 3.1
but the convexity constant degenerates with $N$ (typically $\sim N^{-(s+2)}$).
A more natural system to look at is the gaps system. First we rewrite the
energy as a function of nearest-neighbor gaps and study the Helffer-Sjöstrand
equation in this new system of coordinates. Let us define the map

$\mathrm{Gap}_{N}:X_{N}\in
D_{N}\mapsto(N|x_{2}-x_{1}|,\ldots,N|x_{N}-x_{1}|)\in\mathbb{R}^{N}$

and the push-forward of $\mu$ by the map $\mathrm{Gap}_{N}$, i.e
$\nu=\mu\circ\mathrm{Gap}_{N}^{-1}$. The measure $\nu$ is supported on the set

$\mathcal{M}_{N}=\mathrm{Gap}_{N}(D_{N})=\\{Y_{N}\in\mathbb{R}^{N}:y_{1}+\ldots+y_{N}=N\\},$

which is not an open subset of $\mathbb{R}^{N}$. Proposition 3.1 should
therefore be slightly adapted. Let
$H^{\mathrm{g}}:\mathcal{M}_{N}\to\mathbb{R}$ be such that

$H=H^{\mathrm{g}}\circ\mathrm{Gap}_{N}.$

Let us define the generator acting on
$\mathcal{C}^{\infty}(\mathcal{M}_{N},\mathbb{R})$,

$\mathcal{L}^{\nu}=\nabla H^{\mathrm{g}}\cdot\nabla-\Delta,$ (3.9)

with $\nabla$ and $\Delta$ the standard gradient and Laplace operator on
$\mathcal{M}_{N}$. Also define the Helffer-Sjöstrand operator acting on
$L^{2}(\\{1,\ldots,N\\},\mathcal{C}^{\infty}(\mathcal{M}_{N},\mathbb{R}))$,

$A_{1}^{\nu}=\mathrm{Hess}\,H^{\mathrm{g}}+\mathcal{L}^{\nu}\otimes\mathrm{Id}.$
(3.10)

Let $F:D_{N}\to\mathbb{R}$ in the form $F=G\circ\mathrm{Gap}_{N}$ with
$G:\mathbb{R}^{N}\to\mathbb{R}$ smooth. Let us rewrite the equation (3.6) in
gap coordinates. One can expect that the solution $\nabla\phi$ of (3.6) can be
factorized into $\phi=\psi\circ\mathrm{Gap}_{N}$ with $\nabla\psi\in
L^{2}(\\{1,\ldots,N\\},H^{1}(\nu))$. Let us derive some formal computation to
conjecture the equation satisfies by $\nabla\psi$. For all $t\geq 0$, let
$\mathrm{d}\nu_{t}=\frac{e^{tG}}{\mathbb{E}_{\nu}[e^{tG}]}\mathrm{d}\nu$. In
view of Remark 3.3, we wish to find a map $\nabla\psi\in
L^{1}(\\{1,\ldots,N\\},H^{1}(\nu))$ such that, in a certain sense,

$\nu\circ(\mathrm{Id}+t\nabla\psi)=\nu_{t}+o(t).$ (3.11)

Since $\nu$ and $\nu_{t}$ are both measures on $\mathcal{M}_{N}$, one can
observe that $\sum_{i=1}^{N}\partial_{i}\psi=0$. It is standard the the Gibbs
measure $\nu_{t}$ is the minimizer of the functional

$\nu\in\mathcal{P}(\mathcal{M}_{N})\mapsto\mathbb{E}_{\nu}[H^{\mathrm{g}}+tG]+\mathrm{Ent}(P),$

where $\mathrm{Ent}$ stands for the entropy on $\mathcal{M}_{N}$. Equation
(3.11) is compatible with the variational characterization if $\nabla\psi$
minimizes

$\nabla\psi\mapsto\mathbb{E}_{\nu}[\nabla\psi\cdot\mathrm{Hess}\,\mathcal{H}^{\mathrm{g}}\nabla\psi+|D^{2}\psi|^{2}-2\nabla
G\cdot\nabla\psi],$

over maps $\nabla\psi\in L^{2}(\\{1,\ldots,N\\},H^{1}(\nu))$ such that
$\sum_{i=1}^{N}\partial_{i}\psi=0$ and $\nabla\psi\cdot\vec{n}=0$ on
$\partial\mathcal{M}_{N}$. The Lagrange equation associated for the minimality
of $\nabla\psi$ reads

$A_{1}^{\nu}\nabla\psi=\nabla G+\lambda(e_{1}+\ldots+e_{N}),$

where $\lambda:\mathcal{M}_{N}\to\mathbb{R}$ is a smooth function. We now
state this result in the following proposition:

######  Proposition 3.3.

Let $\mu$ satisfying Assumptions 3.1. Let $F\in H^{1}(\mu)$ in the form
$F=G\circ\mathrm{Gap}_{N}$ with $G\in H^{1}(\nu)$. There exists a unique
$\nabla\psi\in L^{2}(\\{1,\ldots,N\\},H^{1}(\nu))$ solution of

$\left\\{\begin{array}[]{ll}A_{1}^{\nu}\nabla\psi=\nabla
G+\lambda(e_{1}+\ldots+e_{N})&\text{on }\mathcal{M}_{N}\\\
\nabla\psi\cdot(e_{1}+\ldots+e_{N})=0&\text{on }\mathcal{M}_{N}\\\
\nabla\psi\cdot\vec{n}=0&\text{on }\partial\mathcal{M}_{N}.\end{array}\right.$
(3.12)

Moreover the Lagrange multiplier $\lambda$ is given by

$\lambda=\frac{1}{N}(e_{1}+\ldots+e_{N})\cdot\mathrm{Hess}\,H^{\mathrm{g}}\nabla\psi.$
(3.13)

Let $\nabla\psi$ be the solution of (3.6) with external vector-field $\nabla
F$. We have the identity

$\psi=\phi\circ\mathrm{Gap}_{N}.$ (3.14)

In addition we have

$\mathrm{Var}_{\mu}[F]=\mathbb{E}_{\nu}[\nabla G\cdot\nabla\psi].$

Furthermore, $\nabla\psi$ is the unique minimizer of

$\nabla\psi\mapsto\mathbb{E}_{\nu}[\nabla\psi\cdot\mathrm{Hess}\,H^{\mathrm{g}}\nabla\psi+|D^{2}\psi|^{2}-2\nabla
G\cdot\nabla\psi],$

over maps $\nabla\psi\in L^{2}(\\{1,\ldots,N\\},H^{1}(\mu^{\nu})$ such that
$\nabla\psi\cdot(e_{1}+\ldots+e_{N})=0$.

The proof of Proposition 3.3 is postponed to the Appendix, see Section B.

######  Remark 3.4.

There are several manners to factorize the energy (1.3) since we are working
on the circle. We choose the more natural one and set

$\mathcal{H}_{N}^{\mathrm{g}}:Y_{N}\in\mathcal{M}_{N}\mapsto
N^{-s}\sum_{i=1}^{N}\sum_{k=1}^{N/2}g_{s}(y_{i}+\ldots
y_{i+k})(2\mathds{1}_{k\neq N/2}+\mathds{1}_{k=N/2}).$ (3.15)

One may check that for each $i\in\\{1,\ldots,N\\}$ and
$Y_{N}\in\mathcal{M}_{N}$,

$\partial_{i}\mathcal{H}_{N}^{\mathrm{g}}(Y_{N})=\sum_{k=1}^{N/2}\sum_{l:i-k<l\leq
i}N^{-(1+s)}g_{s}^{\prime}\Bigr{(}\frac{y_{i}+\ldots+y_{i+l}}{N}\Bigr{)}(2\mathds{1}_{k\neq
N/2}+\mathds{1}_{k=N/2})$ (3.16)

and for each $i,j\in\\{1,\ldots,N\\}$ and $Y_{N}\in\mathcal{M}_{N}$,

$\partial_{ij}\mathcal{H}_{N}^{\mathrm{g}}(Y_{N})=\sum_{\begin{subarray}{c}1\leq
k,k^{\prime}\leq N/2\\\ |k-k^{\prime}|\leq
N/2\end{subarray}}N^{-(1+s)}g_{s}^{\prime\prime}\Bigr{(}\frac{y_{i-k}+\ldots+y_{j+k^{\prime}}}{N}\Bigr{)}(2\mathds{1}_{|k-k^{\prime}|\neq
N/2}+\mathds{1}_{|k-k^{\prime}|=N/2}).$ (3.17)

Recall that under the Gibbs measure (1.4), large gaps $N(x_{i+k}-x_{i})$
concentrates around the value $k$. The expression (3.17) then tells that the
Hessian of the energy in gap coordinates concentrates around a constant matrix
with off-diagonal entries decaying in $d(i,j)^{-s}$.

###  3.3. The Brascamp-Lieb inequality

We now recall the Brascamp-Lieb inequality, a basic concentration inequality
for strictly convex log-concave measures [BL02]. In our context, the measure
$\mu$ is not strictly log-concave, but its pushforward $\nu$ is, therefore
allowing one to upper bound the variance of any smooth function of the gaps in
the following way:

######  Lemma 3.4.

Let $\mathcal{A}\subset D_{N}$ be a convex domain with a piecewise smooth
boundary. Let $F=G\circ\mathrm{Gap}_{N}$ with $G\in H^{1}(\nu)$. There holds

$\mathrm{Var}_{\mu}[F\mid\mathcal{A}]\leq\mathbb{E}_{\mu}[\nabla
F\cdot(\mathrm{Hess}\,H)^{-1}\nabla F\mid\mathcal{A}].$

###  3.4. Localization

In this subsection we record a crucial convexity Lemma, which is due to
Brascamp, see [BL02]. This lemma is based on the Brascamp-Lieb inequality for
log-concave measures on $D_{N}$, originally derived in [BL76] on
$\mathbb{R}^{N}$, see also Lemma 3.4.

######  Lemma 3.5.

Let $\mu$ be a measure on $D_{N}$ in the form
$\mathrm{d}\mu=e^{-H}\mathrm{d}X_{N}$, with $H$ smooth enough. On $D_{N}$ let
us introduce the coordinates $x=(x_{1},\ldots,x_{n})$ and
$y=(x_{n+1},\ldots,x_{N})$. Assume that $H$ may be written in the form
$H(x,y)=H_{1}(x)+H_{2}(x,y)$. Let $\widetilde{\nu}$ be the push forward of
$\mu$ by the map $X_{N}\mapsto(x_{1},\ldots,x_{n})$. Then, the measure
$\widetilde{\mu}$ may be written in the form
$\widetilde{\mu}=e^{-\widetilde{H}}\mathrm{d}X_{n}$, with

$\widetilde{H}=\log\int e^{-H(x,y)}\mathrm{d}y$

and $\widetilde{H}$ satisfies

$\mathrm{Hess}\,\widetilde{H}\geq\mathrm{Hess}\,H_{1}.$

Moreover, letting $\mu^{x_{0}}$ be the measure $\mu$ conditioned on $x=x_{0}$,
we have

$\partial_{i}\widetilde{H}(x)=\partial_{i}H(x)-\mathbb{E}_{\mu(\cdot\mid
x)}[\partial_{i}H_{2}],\quad\text{for each}\quad 1\leq i\leq n,\quad x\in
D_{n},$ (3.18)
$\partial_{ij}\widetilde{H}(x)=\partial_{ij}H(x)-\mathrm{Cov}_{\mu^{x}}[\partial_{i}H_{2},\partial_{j}H_{2}],\quad\text{for
each}\quad 1\leq i,j\leq n,\quad x\in D_{n}.$ (3.19)

###  3.5. Maximum principle

In this subsection we derive a useful maximum principle, which allows one to
bound the supremum of the $L^{2}$ norm of the solution in presence of a
uniformly convex Hamiltonian. This maximum principle is fairly standard on
$\mathbb{R}^{N}$, see for instance [HS94, Section 10]. We adapt the proof to
make it work on $D_{N}$ and $\mathcal{M}_{N}$. A more subtle analysis could
perhaps permit to treat general convex domains.

######  Proposition 3.6.

Let $\mu$ satisfying Assumptions 3.1 and $\nu=\mu\circ\mathrm{Gap}_{N}^{-1}$.
Assume additionally that $\lim_{x\to 0}\chi^{\prime}(x)=-\infty$. Assume that
for all $U_{N}\in\mathbb{R}^{N}$,

$U_{N}\cdot\mathrm{Hess}\,H^{\mathrm{g}}U_{N}\geq c_{N}|U_{N}|^{2}.$ (3.20)

Let $G\in H^{1}(\nu)$ and $\nabla\psi\in L^{2}(\\{1,\ldots,N\\},H^{1}(\nu))$
be the solution of

$\left\\{\begin{array}[]{ll}A_{1}^{\nu}\nabla\psi=\lambda(e_{1}+\ldots+e_{N})+\nabla
G&\text{on }\mathcal{M}_{N}\\\ \nabla\psi\cdot(e_{1}+\ldots+e_{N})=0&\text{on
}\mathcal{M}_{N}\\\ \nabla\psi\cdot\vec{n}=0&\text{on
}\partial\mathcal{M}_{N}.\end{array}\right.$ (3.21)

Then $\nabla\psi$ satisfies the following uniform estimate:

$\sup|\nabla\psi|\leq c_{N}^{-1}\sup|\nabla G|.$

The proof of Proposition 3.6 is an adaptation in a more involved case of a
known maximum principle for the Helffer-Sjöstrand equation, see for instance
[HS94].

###### Proof of Proposition 3.6.

Let $\nabla\psi$ be the solution of (3.21), satisfying

$A_{1}^{\nu}\nabla\psi=\nabla G+\lambda(e_{1}+\ldots+e_{N}).$ (3.22)

The first step is to prove that $|\nabla\psi|$ attains its maximum in the
interior of $D_{N}$, which uses the assumption that $\lim_{x\to
0}\chi^{\prime}(x)=-\infty$. The point is that for each saturated variable
$i$, we have $\lim_{x\to 0}\partial_{ii}\psi=-\infty$, with other terms
$\partial_{ij}\psi$, $j\neq i$ well-controlled. This will prove that
$|\nabla\psi|$ does not attain its maximum at the boundary.

Let $y^{0}\in D_{N}$ be a maximum of $|\nabla\psi|$. Taking the scalar product
of (3.22) with $\nabla\psi(y^{0})$ yields

$\nabla\psi\cdot\mathrm{Hess}\,H^{\mathrm{g}}\nabla\psi(y^{0})+\mathcal{L}^{\nu}(\nabla\psi\cdot\nabla\psi(y^{0}))=\nabla
G\cdot\nabla\psi(y^{0}).$

Since $y\mapsto\psi\cdot\psi(y^{0})$ has a maximum at
$y^{0}\in\mathring{D_{N}}$, we have
$\nabla(\nabla\psi\cdot\nabla\psi(y^{0}))=0$ at $y=y^{0}$ and therefore
$\nabla H^{\mathrm{g}}\cdot\nabla(\nabla\psi\cdot\nabla\psi(y^{0}))=0$ at
$y=y^{0}$. Additionally, since $y^{0}\in\mathring{D}_{N}$ is the maximum of
$y\mapsto\psi\cdot\psi(y^{0})$, we have
$(-\Delta)(\nabla\psi\cdot\nabla\psi(y^{0}))\geq 0$ at $y=y^{0}$, which gives

$\mathcal{L}^{\nu}(\nabla\psi\cdot\nabla\psi(y^{0}))(y_{0})\geq 0$

It follows that

$\nabla\psi(y^{0})\cdot\mathrm{Hess}\,H^{\mathrm{g}}\nabla\psi(y^{0})\leq\nabla\psi(y^{0})\nabla
G(y^{0}),$

which yields using (3.20),

$\sup|\nabla\psi|\leq c_{N}^{-1}\sup|\nabla G|.$

∎

Let us emphasize that the above proof crucially relies on the fact that
$\lim_{x\to 0}\chi^{\prime}(x)=-\infty$. We now give the standard Gaussian
concentration lemma for uniformly log-concave measures on convex bodies.

######  Lemma 3.7.

Let $\mu$ satisfying Assumptions 3.1. Let $c_{N}$ be the constant in (3.20).
Let $\mathcal{A}\subset D_{N}$ be a convex domain with a piecewise smooth
boundary. Let $F=G\circ\mathrm{Gap}_{N}$ with $G\in H^{1}(\mu^{\mathrm{g}})$.
For all $t\in\mathbb{R}$, we have

$\log\mathbb{E}_{\mu}[e^{tF}\mid\mathcal{A}]\leq
t\mathbb{E}_{\mu}[F\mid\mathcal{A}]+\frac{t^{2}}{2c_{N}}\sup_{\mathcal{A}}|\nabla
G|^{2}.$

Lemma 3.7 can be derived using log-Sobolev inequality and Herbst argument.
When a measure $\mu$ is uniformly log-concave on a convex domain on
$\mathbb{R}^{n}$, it follows from the Bakry-Emery criterion [BÉ85] that $\mu$
satisfies a log-Sobolev inequality.

######  Lemma 3.8.

Let $\mu$ be a uniformly log-concave measure on a convex domain of
$\mathbb{R}^{N}$, with a convexity constant larger than $c>0$. Then $\mu$
satisfies the log-Sobolev inequality with constant $2c^{-1}$.

###  3.6. Concentration inequality for divergence free functions

If $\mu$ is of the form of Assumptions 3.1, $\mu$ is not uniformly log-concave
and on cannot apply directly Lemma 3.7. However, one can observe that

$U_{N}\cdot\mathrm{Hess}\,HU_{N}\geq c_{N}\sum_{i\neq
j}(N(u_{i}-u_{j}))^{2}=c_{N}(N-1)\sum_{i=1}^{N}u_{i}^{2},\\\ \quad\text{for
all $U_{N}\in\mathbb{R}^{N}$ such that $u_{1}+\ldots+u_{N}=0$}.$ (3.23)

Using this observation and the particular structure of $\mu$, one can give a
concentration estimate for divergence free functions $F$, i.e for $F$
verifying $\partial_{1}\phi+\ldots+\partial_{N}\phi=0$. We now state this
crucial concentration result found in [BEY12].

######  Lemma 3.9.

Let $\mu$ satisfying Assumptions 3.1. Assume that $\chi^{\prime\prime}\geq
c_{N}$. Let $I\subset\\{1,\ldots,N\\}$, $\mathrm{card}(I)=K$. Let $F\in
H^{1}(\mu)$ such that $\sum_{i=1}^{N}\partial_{i}F=0$ and $\partial_{i}F=0$
for each $i\in I^{c}$. We have

$\mathrm{Var}_{\mu}[F]\leq\frac{1}{(K-1)c_{N}}\mathbb{E}_{\mu}[|\nabla
F|^{2}].$ (3.24)

Furthermore, for all $t\in\mathbb{R}$,

$\log\mathbb{E}_{\mu}[e^{tF}]\leq
t\mathbb{E}_{\mu}[F]+\frac{t^{2}}{2(K-1)c_{N}}\sup|\nabla F|^{2}.$

We refer to [BEY12] for a proof, see also [Bou21, Lemma 3.13] for a
transcription.

##  4\. Decay of correlations for the HS Riesz gas

In this section, we consider the hypersingular Riesz gas, i.e the Riesz gas
with the kernel (2.2) for a parameter $s>1$. We show that the covariance
between $N(x_{i+1}-x_{j})$ and $N(x_{j+1}-x_{j})$ decays at least in
$d(i,j)^{-(s+1)}$. To this end we will be studying the Helffer-Sjöstrand
equation in gap coordinates (3.12). Taking advantage of the fact that the
Hessian of the energy in gap coordinates has typically summable entries, we
will implement a simple distortion argument inspired from [Hel98a] to obtain
decay estimates.

###  4.1. Study of a commutator

We begin the section with an introduction to our distortion argument. Given
$s>1$, let $\nu$ be the measure (1.4) in gap coordinates or a slight variant
of it. We will be studying the equation

$\left\\{\begin{array}[]{ll}A_{1}^{\nu}\nabla\psi=e_{1}+\lambda(e_{1}+\ldots+e_{N})&\text{on
}\mathcal{M}_{N}\\\ \nabla\psi\cdot(e_{1}+\ldots+e_{N})=0&\text{on
}\mathcal{M}_{N}\\\ \nabla\psi\cdot\vec{n}=0&\text{on
}\partial\mathcal{M}_{N}.\end{array}\right.$ (4.1)

According to Remark 3.4, there exists an event of overwhelming probability on
which the Hessian of the energy in gap coordinates decays as $d(i,j)^{-s}$
away from the diagonal. As explained in the introduction, to obtain a decay
estimate on the solution $\nabla\psi$, we study the equation satisfied by
$\mathrm{L}_{\alpha}\nabla\psi$, where $\mathrm{L}_{\alpha}$ stands for the
following distortion matrix:

$\mathrm{L}_{\alpha}=\mathrm{diag}(\gamma_{1},\ldots,\gamma_{N}),\quad\text{where
$\gamma_{i}=1+d(i,i_{0})^{\alpha}$ for each $1\leq i\leq N$}.$ (4.2)

Let us denote

$\psi^{\mathrm{dis}}=\mathrm{L}_{\alpha}\nabla\psi\quad\in
L^{2}(\\{1,\ldots,N\\},H^{1}(\nu)).$

One can check that $\psi^{\mathrm{dis}}$ is solution of the equation

$A_{1}^{\nu}\nabla\psi+\beta\delta_{\mathrm{L}_{\alpha}}\nabla\psi=e_{1}+\lambda\mathrm{L}_{\alpha}(e_{1}+\ldots+e_{N}),\quad\text{where}\quad\delta_{\mathrm{L}_{\alpha}}:=\mathrm{L}_{\alpha}\mathrm{Hess}\,{\mathcal{H}}_{N}^{\mathrm{g}}\mathrm{L}_{\alpha}^{-1}-{\mathcal{H}}_{N}^{\mathrm{g}}.$

As one may observe, when $\mathrm{M}\in\mathcal{M}_{n}(\mathbb{R})$ is a
matrix with coefficients decaying fast enough away from the diagonal, then the
commutator $\mathrm{L}_{\alpha}\mathrm{M}\mathrm{L}_{\alpha}^{-1}-\mathrm{M}$
is small compared to the identity, in the sense of quadratic forms, as shown
in the next lemma.

######  Lemma 4.1 (Commutation lemma).

Let $s>1$. Let $\mathrm{M}\in\mathcal{M}_{n}(\mathbb{R})$ such that there
exits a constant $C_{N}$ such that

$|\mathrm{M}_{i,j}|\leq C_{N}\frac{1}{1+d(i,j)^{s}},\quad\text{for each $1\leq
i,j\leq N$}.$ (4.3)

Let $\alpha\in(0,s-\frac{1}{2})$ and $\mathrm{L}$ be as in (4.2). There exist
constants $C>0$ and $\kappa>0$ such that for all $c_{0}>0$, setting

$K_{0}=\Bigr{\lfloor}C({C_{N}^{\kappa}}/c_{0})^{\frac{1}{\min(2s-2,1,2s-2\alpha-1)}}\Bigr{\rfloor},$

we have

$|U_{N}\cdot(\mathrm{L}_{\alpha}\mathrm{M}\mathrm{L}_{\alpha}^{-1}-\mathrm{M})U_{N}|\leq\frac{1}{2}c_{0}|U_{N}|^{2}+CC_{N}^{\kappa}|U_{N}|\Bigr{(}\sum_{i:d(i,1)\leq
K_{0}}u_{i}^{2}\Bigr{)}^{\frac{1}{2}},\quad\text{for all}\quad
U_{N}\in\mathbb{R}^{N}.$

###### Proof of Lemma 4.1.

Let us denote

$\delta_{\mathrm{L}_{\alpha}}=\mathrm{L}_{\alpha}\mathrm{M}\mathrm{L}_{\alpha}^{-1}-\mathrm{M}\in\mathcal{M}_{N}(\mathbb{R}).$

Fix $c_{0}>0$ (which may depend on $N$). Let $K_{0}\geq 1$. One can write

$\Bigr{|}\sum_{d(i,1)>K_{0}}\sum_{l=1}^{N}(\delta_{\mathrm{L}_{\alpha}})_{i,l}u_{i}u_{l}\Bigr{|}\leq\sum_{d(i,1)>K_{0}}u_{l}^{2}\Bigr{(}\sum_{d(l,1)>K_{0}}\sum_{i=1}^{N}(\delta_{\mathrm{L}})_{i,l}^{2}\Bigr{)}^{\frac{1}{2}}.$

Let us bound the quantity above. Note that

$\sum_{l=1}^{N}(\delta_{\mathrm{L}})_{i,l}^{2}=\sum_{l=1}^{N}(\mathrm{M})_{i,l}^{2}\Bigr{(}\frac{\gamma_{i}-\gamma_{l}}{\gamma_{l}}\Bigr{)}^{2},\quad\text{for
each }i\in\\{1,\ldots,N\\}.$

We split the sum above according to whether $d(l,i)\leq\frac{1}{2}d(i,1)$. If
$d(l,i)\leq\frac{1}{2}d(i,1)$, then

$\Bigr{|}\frac{\gamma_{i}-\gamma_{l}}{\gamma_{l}}\Bigr{|}\leq
C\frac{d(i,l)}{1+d(l,1)}.$

It follows that

$\begin{split}\sum_{l:d(l,i)\leq\frac{1}{2}d(l,1)}\mathrm{M}_{i,l}^{2}\Bigr{(}\frac{\gamma_{i}-\gamma_{l}}{\gamma_{l}}\Bigr{)}^{2}&\leq\frac{CC_{N}^{\kappa}}{d(i,1)^{2}}\sum_{k\leq
d(i,1)}\frac{1}{k^{2(s-1)}}\\\ &\leq
CC_{N}^{\kappa}\frac{1}{1+d(i,1)^{\min(2s-1,2)}}.\end{split}$ (4.4)

Let us examine the second contribution, which carries the main term and gives
the restriction $\alpha\in(0,s-\frac{1}{2})$. If
$d(i,l)\geq\frac{1}{2}d(i,1)$, one has

$\Bigr{|}\frac{\gamma_{i}-\gamma_{l}}{\gamma_{l}}\Bigr{|}\leq
C\frac{\gamma_{i}}{\gamma_{l}}.$

This gives

$\begin{split}\sum_{l:d(l,i)\geq\frac{1}{2}d(l,1)}\mathrm{M}_{i,l}^{2}\Bigr{(}\frac{\gamma_{i}-\gamma_{l}}{\gamma_{i}}\Bigr{)}^{2}&\leq
CC_{N}^{\kappa}d(i,1)^{2\alpha}\sum_{l:d(l,i)\geq\frac{1}{2}d(i,1)}\frac{1}{d(l,i)^{2s}}\frac{1}{d(l,1)^{2\alpha}}\\\
&\leq
CC_{N}^{\kappa}d(i,1)^{2\alpha-2s}\sum_{l:d(l,i)\geq\frac{1}{2}d(i,1)}\frac{1}{d(l,1)^{2\alpha}}\\\
&\leq
CC_{N}^{\kappa}\frac{1}{d(i,1)^{2(s-\alpha)}}(1+d(i,1)^{1-2\alpha}).\end{split}$
(4.5)

Combining (4.4) and (4.5), one gets

$\sum_{l=1}^{N}(\delta_{\mathrm{L}_{\alpha}})_{i,l}^{2}\leq
CC_{N}^{\kappa}\frac{1}{d(i,1)^{\min(2s-1,2,2(s-\alpha))}}.$ (4.6)

The above series is convergent whenever $\alpha<s-\frac{1}{2}$ and there
exists $K_{0}$ such that

$\sum_{i:d(i,1)\geq
K_{0}}\sum_{l=1}^{N}(\delta_{\mathrm{L}_{\alpha}})_{i,l}^{2}\leq\frac{1}{2}c_{0}.$

Moreover one can pick $K_{0}$ depending on $C_{N}$ and $c_{0}$ in the
following way:

$K_{0}\leq
C\Bigr{(}\frac{C_{N}^{\kappa}}{c_{0}}\Bigr{)}^{\frac{1}{\min(2s-2,1,2s-2\alpha-1)}},\quad\text{for
some }\kappa>0.$ (4.7)

Using (4.6) one additionally obtains

$\sum_{d(i,1)\leq
K_{0}}\sum_{l=1}^{N}(\delta_{\mathrm{L}_{\alpha}})_{i,l}^{2}\leq
CC_{N}^{\kappa}.$

This concludes the proof of Lemma 4.1. ∎

###  4.2. Localization in a smaller window

Due to the degeneracy of the interaction at infinity, the system lacks uniform
convexity and one therefore needs to restrict the system to a window with a
small number of points $n\ll N$. Let us constrain the system on the window
$\\{1,\ldots,n\\}$. For $\varepsilon>0$ and $\theta:[0,+\infty)\to(0,+\infty)$
smooth such that $\theta=0$ on $(1,+\infty)$, $\theta^{\prime\prime}\geq 1$ on
$[0,\frac{1}{2})$, $\theta^{\prime\prime}\geq 0$ on $[0,+\infty]$, let us
define

$\mathrm{F}=\sum_{i=1}^{n}\theta\Bigr{(}\frac{N(x_{i+1}-x_{i})}{n^{\varepsilon}}\Bigr{)}\quad\text{and}\quad\mathrm{F}^{\mathrm{g}}=\mathrm{F}\circ\mathrm{Gap}_{N}$
(4.8)

and the constrained measures

$\displaystyle\mathrm{d}\mathbb{Q}_{N,\beta}\propto
e^{-\beta\mathrm{F}}\mathrm{d}\mathbb{P}_{N,\beta}$
$\displaystyle\quad\mathrm{d}\mathbb{Q}_{N,\beta}^{\mathrm{g}}\propto
e^{-\beta\mathrm{F}^{\mathrm{g}}}\mathrm{d}\mathbb{P}_{N,\beta}^{\mathrm{g}}.$
(4.9)

Note that the forcing (4.8) is tuned so that the total variation distance
between $\mathbb{P}_{N,\beta}$ and $\mathbb{Q}_{N,\beta}$ decays exponentially
in $n$. Let us denote $I=\\{1,\ldots,n\\}$ and
$\pi:\mathcal{M}_{N}\to\pi(\mathcal{M}_{N})\subset\mathbb{R}^{n}$ the
projection on the coordinates $(x_{1},\ldots,x_{n})$. We now define $\mu$ the
law of $(x_{1},\ldots,x_{n})$ when $(x_{1},\ldots,x_{N})$ is distributed
according to $\mathbb{Q}_{N,\beta}^{\mathrm{g}}$:

$\mu=\mathbb{Q}_{N,\beta}^{\mathrm{g}}\circ\pi^{-1}.$ (4.10)

By Lemma 3.5, $\mu$ may be written in the form

$\displaystyle\mathrm{d}\mu(x)$ $\displaystyle\propto
e^{-\beta(\mathcal{H}_{n}^{\mathrm{g}}+\widetilde{E}_{n,N})(x)}\mathds{1}_{\pi(\mathcal{M}_{N})}(x)\mathrm{d}x$
(4.11)

with

$\displaystyle\widetilde{E}_{n,N}(x)$ $\displaystyle=-\frac{1}{\beta}\log\int
e^{-\beta(\mathcal{H}_{N-n}(y)+\mathcal{H}_{n,N}^{\mathrm{g}}(x,y)+\mathrm{F}^{\mathrm{g}}(x,y))}\mathrm{d}y,$
(4.12)

where

$\mathcal{H}_{n,N}^{\mathrm{g}}(x,y):=\mathcal{H}_{N}^{\mathrm{g}}(x,y)-\mathcal{H}_{n}^{\mathrm{g}}(x)-\mathcal{H}_{N-n}^{\mathrm{g}}(y).$
(4.13)

Furthermore as proven in Lemma 3.5, the function $\widetilde{E}_{n,N}$ is
convex. As a consequence the measure $\mu$ is uniformly log-concave with a
lower bound on the Hessian proportional to $n^{-\varepsilon(s+2)}$. Choosing
two gaps in $\\{1,\ldots,n\\}$ at distance of order $n$ will therefore give a
defect of convexity which is not too large to conclude the proof of (1.5). In
the sequel one studies the correlation between $N(x_{i+1}-x_{i})$ and
$N(x_{j+1}-x_{j})$ under $\mu$ and study the related Helffer-Sjöstrand
equation. Define the good event

$\mathcal{A}=\left\\{X_{n}\in\pi(\mathcal{M}_{N}):\forall
i\in\\{1,\ldots,n\\},n^{-\varepsilon}\leq x_{i}\leq
n^{\varepsilon}\right\\}\\\ \cap\left\\{\forall
i\in\\{1,\ldots,n\\},k\in\\{1,\ldots,n-i\\},|x_{i}+\ldots+x_{i+k-1}-k|\leq
n^{\varepsilon}k^{\frac{1}{2}}\right\\}.$ (4.14)

The projected measure $\mu$ is then a Gibbs measure with energy

$\widetilde{\mathcal{H}}_{n}^{\mathrm{g}}:=\mathrm{F}^{\mathrm{g}}+\mathcal{H}_{n}^{\mathrm{g}}+\widetilde{\mathrm{E}}_{n,N}.$
(4.15)

The term $\mathcal{H}_{n}^{\mathrm{g}}$ corresponds to the inner interaction
of $x_{1},\ldots,x_{n}$ while $\widetilde{E}_{n,N}$ reflects the interaction
with the exterior. Let us compute the second derivative of
$\widetilde{E}_{n,N}$. As checked in Lemma 3.5, for each $1\leq i,j\leq n$, we
have

$\partial_{ij}\widetilde{\mathrm{E}}_{n,N}=\mathbb{E}_{\mathbb{Q}_{N,\beta}(\cdot\mid
x)}[\partial_{ij}\mathcal{H}_{n,N}^{\mathrm{g}}]-\mathrm{Cov}_{\mathbb{Q}_{N,\beta}(\cdot\mid
x)}[\partial_{i}\mathcal{H}_{n,N}^{\mathrm{g}},\partial_{j}\mathcal{H}_{n,N}^{\mathrm{g}}].$

The above identity should be read as a Schur complement. Let us decompose the
Hessian of $\widetilde{\mathcal{H}}_{n}^{\mathrm{g}}$ into

$\mathrm{Hess}\,\widetilde{\mathcal{H}}_{n}^{\mathrm{g}}=\mathsf{M}+\widetilde{H}$
(4.16)

with

$\mathsf{M}=\mathrm{Hess}\,\mathrm{F}^{\mathrm{g}}+\mathrm{Hess}\,\mathcal{H}_{n}^{\mathrm{g}}\quad\text{and}\quad\widetilde{H}=\mathrm{Hess}\,\widetilde{\mathrm{E}}_{n,N}.$
(4.17)

In the case $n\ll N/2$, we will replace the Hessian of
$\widetilde{H}_{n}^{\mathrm{g}}$ in (4.1) by $\mathsf{M}$ and derive some
optimal decay estimates. This provides an approximate solution to (4.1) and
there will remain to control $\widetilde{H}\psi^{(1)}$. One shall then record
some useful estimate on $\widetilde{H}$. One can check that uniformly on the
event (4.14) and for each $1\leq i,j\leq n$, we have

$|\widetilde{H}_{i,j}|\leq\frac{Cn^{\kappa\varepsilon}}{d(i,\partial
I)^{\frac{s}{2}}d(j,\partial I)^{\frac{s}{2}}}.$ (4.18)

In addition, uniformly on $\mathcal{B}$ and in $1\leq i,l,j\leq n$,

$|\widetilde{H}_{i,j}-\widetilde{H}_{l,j}|\leq\frac{Cn^{\kappa\varepsilon}|i-l|}{(d(i,\partial
I)\wedge d(l,\partial I))^{\frac{1}{2}+s}d(j,\partial I)^{\frac{s}{2}}}.$
(4.19)

For the stakes of Section 6 it will be convenient to write down our estimates
for solutions of (4.1) for a general measure $\mu$ satisfying the following
assumptions:

######  Assumptions 4.1.

Let $\mu$ be a measure on $\pi(\mathcal{M}_{N})$ in the form
$\mathrm{d}\mu=e^{-\beta H}\mathrm{d}X_{n}$ with $H$ uniformly convex with a
convexity constant $c=n^{-\kappa\varepsilon}$ for some $\kappa>0$. Letting
$\mathcal{A}$ be the good event (4.14), suppose that there exist some
constants $\kappa>0$, $c(\beta)$ and $C(\beta)>0$ such that

$\mu(\mathcal{A})\geq 1-e^{-c(\beta)n^{\kappa\varepsilon}}.$

###  4.3. The initial decay estimate

In this subsection we introduce a simple perturbation argument, which gives a
first estimate on the decay of correlations for the constrained hypersingular
Riesz gas. The method can be applied to other convex models for which the
Hessian of the energy satisfies some decay assumption. This technique follows
from an adaptation of a rather classical argument in statistical physics
[HS94, CT73].

######  Lemma 4.2.

Let $s\in(1,+\infty).$ Let $\mu$ satisfying Assumptions 4.1. Let $\chi_{n}\in
H^{-1}(\mu),i_{0}\in\\{1,\ldots,n\\}$ and $\psi\in
L^{2}(\\{1,\ldots,n\\},H^{1}(\mu))$ be the solution of

$\left\\{\begin{array}[]{ll}\beta\mathsf{M}\psi+\mathcal{L}^{\mu}\psi=\chi_{n}e_{i_{0}}+\lambda(e_{1}+\ldots+e_{n})&\text{on
}\pi(\mathcal{M}_{N})\\\ \psi\cdot(e_{1}+\ldots+e_{n})=0&\text{on
}\pi(\mathcal{M}_{N})\\\ \psi\cdot\vec{n}=0&\text{on
}\partial(\pi(\mathcal{M}_{N})).\end{array}\right.$ (4.20)

Then, for all $\alpha\in(\frac{1}{2},s-\frac{1}{2})$, there exist a constant
$C(\beta)$ locally uniform in $\beta$ and $\kappa>0$ such that

$\mathbb{E}_{\mu}\Bigr{[}\sum_{i=1}^{n}d(i,1)^{2\alpha}\psi_{i}^{2}\Bigr{]}^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}})$
(4.21)

and

$\mathbb{E}_{\mu}[\lambda^{2}]^{\frac{1}{2}}\leq\frac{C(\beta)}{n^{1-\kappa\varepsilon}}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$
(4.22)

###### Proof of Lemma 4.2.

Let $\psi\in L^{2}(\\{1,\ldots,n\\},H^{1}(\mu))$ be in the solution of (4.20).

#### Step 1: a priori estimates

Taking the scalar product of (4.20) with $\psi$ and integrating by parts, one
may show that there exist constants $\kappa>0$ and $C>0$ such that

$\mathbb{E}_{\mu}[|\nabla\psi|^{2}]+\beta\mathbb{E}_{\mu}[|\psi|^{2}]\leq
C\beta^{-1}n^{\kappa\varepsilon}\mathbb{E}_{\mu}[\chi_{n}^{2}].$ (4.23)

Moreover, using the maximum principle of Proposition 3.6, we have

$\sup|\psi|\leq C\beta^{-1}n^{\kappa\varepsilon}\sup|\chi_{n}|.$ (4.24)

#### Step 2: splitting

Let $\widetilde{\theta}:\mathbb{R}\to\mathbb{R}^{+}$ smooth such that
$\widetilde{\theta}(x)=0$ if $|x|<\frac{1}{2}$, $\widetilde{\theta}(x)=1$ if
$|x|>1$. Let us define a cutoff function supported on (4.14). Let

$\eta=\prod_{i=1}^{n}\widetilde{\theta}\left(\frac{1}{x_{i}n^{\varepsilon}}\right)\prod_{i=1}^{n}\prod_{k=1}^{\frac{N}{2}\wedge
n}\widetilde{\theta}\Bigr{(}\frac{x_{i}+\ldots+x_{i+k-1}-k}{k^{\frac{1}{2}}n^{\varepsilon}}\Bigr{)}.$
(4.25)

By linearity one can split $\psi$ into $\psi=\sum_{l=1}^{n}\psi^{(l)}+w$,
where $\psi^{(l)}\in L^{2}(\\{1,\ldots,n\\},H^{1}(\mu))$ solves

$\left\\{\begin{array}[]{ll}\beta\mathsf{M}\psi^{(l)}+\mathcal{L}^{\mu}\psi^{(l)}=(\chi_{n}\mathds{1}_{l=i_{0}}+\lambda)\eta
e_{l}&\text{on }\pi(\mathcal{M}_{N})\\\ \psi^{(l)}\cdot\vec{n}=0&\text{on
}\partial(\pi(\mathcal{M}_{N}))\end{array}\right.$

and where $w\in L^{2}(\\{1,\ldots,n\\},H^{1}(\mu))$ solves

$\left\\{\begin{array}[]{ll}\beta\mathsf{M}w+\mathcal{L}^{\mu}w=(\chi_{n}e_{i_{0}}+\lambda(e_{1}+\ldots+e_{N}))(1-\eta)\\\
w\cdot\vec{n}=0&\text{on }\partial(\pi(\mathcal{M}_{N}))\end{array}\right.$

First by integration by parts and using Assumptions 4.1 and (4.24), we can see
that

$\mathbb{E}_{\mu}[|w|^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}(\mathbb{E}_{\mu}[\chi_{n}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$

#### Step 3: the distortion argument

Fix $l\in\\{1,\ldots,n\\}$. To enlighten the notation we set $\phi=\psi^{(l)}$
and $v=(\chi_{n}\mathds{1}_{l=i_{0}}+\lambda)\eta e_{l}$. Fix
$\alpha\in(\frac{1}{2},s-\frac{1}{2})$ and let us consider as in (4.2) the
distortion matrix

$\mathrm{L}_{\alpha}=\mathrm{diag}(\gamma_{1},\ldots,\gamma_{n}),\quad\text{where
$\gamma_{i}=1+d(i,l)^{\alpha}$ for each $1\leq i\leq n$}.$

Let us define $\phi^{\mathrm{dis}}$ the distorted vector-field

$\phi^{\mathrm{dis}}=\mathrm{L}_{\alpha}\phi\quad\in
L^{2}(\\{1,\ldots,n\\},H^{1}(\mu)).$ (4.26)

Observing that $\mathcal{L}_{\alpha}v=v$, we can check that
$\phi^{\mathrm{dis}}$ solves

$A_{1}^{\mu}\phi^{\mathrm{dis}}+\beta\delta_{\mathrm{L}_{\alpha}}\phi^{\mathrm{dis}}=v,$
(4.27)

where

$\delta_{\mathrm{L}_{\alpha}}:=\mathrm{L}_{\alpha}\mathsf{M}\mathrm{L}_{\alpha}^{-1}-\mathsf{M}.$

#### Step 4: integration by parts

In view of expression (3.17) and Lemma C.1, there exists an event of
overwhelming probability on which the matrix $\mathsf{M}$ satisfies the decay
property (4.3) with $\gamma:=s-1>0$. Let $\mathcal{A}$ be the event defined in
(4.14). By Assumptions 4.1

$\mu(\mathcal{A}^{c})\leq C(\beta)e^{-c(\beta)n^{\delta}}.$ (4.28)

Taking the scalar product of (4.27) with $\eta\psi^{\mathrm{dis}}$ and
integrating by parts under $\mu$ gives

$\mathbb{E}_{\mu}[\beta\eta\phi^{\mathrm{dis}}\cdot(\mathsf{M}+\delta_{\mathrm{L}_{\alpha}})\phi^{\mathrm{dis}}]+\mathbb{E}_{\mu}[\eta|\nabla\phi^{\mathrm{dis}}|^{2}]+\sum_{i=1}^{n}\mathbb{E}_{\mu}[\nabla\phi_{i}^{\mathrm{dis}}\cdot\nabla\eta\phi_{i}^{\mathrm{dis}}]=\mathbb{E}_{\mu}[\phi\cdot
v\eta],$ (4.29)

where we have used the fact that $\psi^{\mathrm{dis}}_{1}=\psi_{1}$. The third
term in the left-hand side of (4.29) is the expectation of a function
supported on the event $\mathcal{A}^{c}$ growing at most polynomially in $n$.
By Assumptions 4.1, $\mathcal{A}$ has overwhelming probability, which implies
that

$\Bigr{|}\sum_{i=1}^{n}\mathbb{E}_{\mu}[\nabla\phi_{i}^{\mathrm{dis}}\cdot\nabla\eta\phi_{i}^{\mathrm{dis}}]\Bigr{|}\leq\mathbb{E}_{\mu}[|\nabla\phi^{\mathrm{dis}}|^{2}]^{\frac{1}{2}}\mathbb{E}_{\mu}[|\phi|^{2}|\nabla\eta|^{2}]^{\frac{1}{2}}\leq\mathbb{E}_{\mu}[|\nabla\phi^{\mathrm{dis}}|^{2}]^{\frac{1}{2}}\sup|\phi|n^{\kappa}\mu(\mathcal{A}^{c}).$

Inserting (4.24) and (4.28), one gets

$\Bigr{|}\sum_{i=1}^{n}\mathbb{E}_{\mu}[\nabla\phi_{i}^{\mathrm{dis}}\cdot\nabla\eta\phi_{i}^{\mathrm{dis}}]\Bigr{|}\leq
C(\beta)\sup|v|e^{-c(\beta)n^{\delta}}.$

Together with (4.29), this gives

$\mathbb{E}_{\mu}[\beta\eta\phi^{\mathrm{dis}}\cdot(\mathsf{M}+\delta_{\mathrm{L}_{\alpha}})\phi^{\mathrm{dis}}]+\mathbb{E}_{\mu}[\eta|\nabla\phi^{\mathrm{dis}}|^{2}]\leq
C(\beta)n^{\kappa_{0}\varepsilon}\mathbb{E}_{\mu}[|\mathrm{L}_{\alpha}v|^{2}]+C(\beta)e^{-c(\beta)n^{\delta}}\sup|v|.$

#### Step 5: control on the commutator

By construction of the cutoff function (4.25), there exist a constant $C>0$
and a constant $\kappa>0$ such that uniformly on $i\neq j$,

$|\eta\mathsf{M}_{i,j}|\leq\frac{Cn^{\kappa\varepsilon}}{1+d(i,j)^{1+s}}.$

Given $X_{n}\in\\{\eta\neq 0\\}$, one may therefore apply Lemma 4.1 to the
matrix $\mathrm{M}=\mathrm{Hess}\,\mathcal{H}_{n}^{\mathrm{g}}(X_{n})$. In
particular, there exist $\kappa>0$ and $\kappa^{\prime}>0$ independent of
$X_{n}$ such that, setting

$K_{0}=\lfloor n^{\kappa\varepsilon}\rfloor,$

there holds

$|\mathbb{E}_{\mu}[\eta\phi^{\mathrm{dis}}\cdot\delta_{\mathrm{L}_{\alpha}}\phi^{\mathrm{dis}}]|\leq\frac{n^{-\varepsilon(s+2)}}{2}\mathbb{E}_{\mu}[\eta|\phi^{\mathrm{dis}}|^{2}]-C(\beta)n^{\kappa^{\prime}\varepsilon}\mathbb{E}_{\mu}[|\phi^{\mathrm{dis}}|^{2}]^{\frac{1}{2}}\mathbb{E}_{\mu}\Bigr{[}\sum_{i:d(i,1)\leq
K_{0}}(\phi^{\mathrm{dis}}_{i})^{2}\Bigr{]}^{\frac{1}{2}}.$ (4.30)

Furthermore, using the definition of $\psi^{\mathrm{dis}}$ (4.26) and the a
priori bound (4.23), we find that

$\mathbb{E}_{\mu}\Bigr{[}\sum_{i:d(i,1)\leq
K_{0}}(\phi^{\mathrm{dis}}_{i})^{2}\Bigr{]}^{\frac{1}{2}}\leq
K_{0}^{\alpha}\mathbb{E}_{\mu}[|\phi|^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa^{\prime\prime}\varepsilon}\mathbb{E}_{\mu}[|v|^{2}]^{\frac{1}{2}}.$

#### Step 6: conclusion

Combining these we deduce that there exists $\kappa>0$ such that for each
$l\in\\{1,\ldots,n\\}$,

$\frac{\beta}{2}n^{-\varepsilon(s+2)}\mathbb{E}_{\mu}\Bigr{[}\sum_{i=1}^{n}d(i,l)^{2\alpha}(\psi_{i}^{(l)})^{2}\Bigr{]}^{\frac{1}{2}}+\mathbb{E}_{\mu}\Bigr{[}\sum_{i=1}^{n}d(i,l)^{2\alpha}|\nabla\psi_{i}^{(l)}|^{2}\Bigr{]}^{\frac{1}{2}}\\\
\leq
C(\beta)n^{\kappa\varepsilon}\Bigr{(}\mathds{1}_{l=i_{0}}\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\mathbb{E}_{\mu}[\eta\lambda^{2}]^{\frac{1}{2}}+e^{-c(\beta)n^{\delta}}\sup|\chi_{n}|\Bigr{)}.$
(4.31)

Let $\eta>0$. One may then write

$\sum_{i=1}^{n}|\psi_{i}|\leq\sum_{i=1}^{n}\sum_{l=1}^{n}|\psi_{i}^{(l)}|\leq\sum_{i,l:d(i,l)\leq
n^{\eta}}|\psi_{i}^{(l)}|+\sum_{i,l:d(i,l)>n^{\eta}}|\psi_{i}^{(l)}|.$

The first term can be bounded using the a priori estimate (4.23) (independent
of $\lambda$!) as follows:

$\mathbb{E}_{\mu}\Bigr{[}\Bigr{(}\sum_{i,l:d(i,l)\leq
n^{\eta}}|\psi_{i}^{(l)}|\Bigr{)}^{2}\Bigr{]}^{\frac{1}{2}}\leq
n^{\frac{\delta}{2}}\mathbb{E}_{\mu}[|\psi|^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon+\frac{\delta}{2}}\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}.$
(4.32)

Besides from (4.31) we have

$\mathbb{E}_{\mu}\Bigr{[}\Bigr{(}\sum_{i,l:d(i,l)>n^{\eta}}|\psi_{i}^{(l)}|\Bigr{)}^{2}\Bigr{]}^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon-\eta(s-1)}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+n\mathbb{E}_{\mu}[\eta\lambda^{2}]^{\frac{1}{2}}).$
(4.33)

From (3.13), we can upper bound the Lagrange multiplier $\lambda$ by

$\mathbb{E}_{\mu}[\eta\lambda^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}\mathbb{E}_{\mu}[|\psi|^{2}]^{\frac{1}{2}}.$

Inserting (4.32) and (4.33) and choosing $\eta$ large enough with respect to
$\varepsilon$ we deduce that

$\mathbb{E}_{\mu}[\lambda^{2}]^{\frac{1}{2}}\leq\frac{C(\beta)}{n^{1-\kappa\varepsilon}}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$

We thus conclude that for $\alpha\in(\frac{1}{2},s-1)$, there holds

$\mathbb{E}_{\mu}\Bigr{[}\sum_{i=1}^{n}d(i,i_{0})^{2\alpha}\psi_{i}^{2}\Bigr{]}+\mathbb{E}_{\mu}\Bigr{[}\sum_{i=1}^{n}d(i,i_{0})^{2\alpha}|\nabla\psi_{i}|^{2}\Bigr{]}\leq
C(\beta)n^{\kappa\varepsilon}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$

∎

###  4.4. Bootstrap on the decay exponent

In this subsection we introduce a simple procedure to improve the decay
estimate of Lemma 4.2. The method consists in studying the projection of
Equation (4.20) in a small window. By controlling the field outside the window
with the a priori decay estimate, one obtains through the distortion argument
of Lemma 4.2 a better decay estimate on the solution. After a finite number of
iterations one gets the following result:

######  Proposition 4.3.

Let $s\in(1,+\infty).$ Let $\mu$ satisfying Assumptions 4.1. Let $\chi_{n}\in
H^{-1}(\mu),i_{0}\in\\{1,\ldots,n\\}$ and $\psi\in
L^{2}(\\{1,\ldots,n\\},H^{1}(\mu))$ be the solution of

$\left\\{\begin{array}[]{ll}\beta\mathsf{M}\psi+\mathcal{L}^{\mu}\psi=\chi_{n}e_{i_{0}}+\lambda(e_{1}+\ldots+e_{n})&\text{on
}\pi(\mathcal{M}_{N})\\\ \psi\cdot(e_{1}+\ldots+e_{n})=0&\text{on
}\pi(\mathcal{M}_{N})\\\ \psi\cdot\vec{n}=0&\text{on
}\partial(\pi(\mathcal{M}_{N})).\end{array}\right.$ (4.34)

There exist $\kappa>0$ and $C(\beta)>0$, $c(\beta)>0$ locally uniform in
$\beta$ such that for each $1\leq j\leq N$,

$\mathbb{E}_{\mu}[\psi_{j}^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}(\frac{1}{1+d(j,i_{0})^{1+s}}+\frac{1}{n}\Bigr{)}\Bigr{(}\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$
(4.35)

###### Proof of Proposition 4.3.

Fix a scale $\varepsilon^{\prime}\in(0,\min(1,2(s-1)))$. Assume that there
exist a number $\gamma>0$ and constants $\kappa>0$ and $C(\beta)>0$,
$c(\beta)>0$ such that for each $j\in\\{1,\ldots,n\\}$,

$\mathbb{E}_{\mu}\Bigr{[}\sum_{k:d(k,j)\leq
d(j,i_{0})^{\varepsilon^{\prime}}}\psi_{k}^{2}\Bigr{]}^{\frac{1}{2}}\leq\frac{C(\beta)n^{\kappa\varepsilon}}{d(j,i_{0})^{\gamma}}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$
(4.36)

Let us prove that (4.36) holds for some $\gamma^{\prime}>\gamma$. Fix
$j\in\\{1,\ldots,n\\}$ and define

$J=\\{i\in\\{1,\ldots,n\\}:d(j,i)\leq d(j,i_{0})^{\varepsilon^{\prime}}\\}.$

Let $\mathcal{A}$ be the event (4.14). One may study the equation satisfied by
the field $\widetilde{\psi}:=(\psi_{j})_{j\in J}\in L^{2}(J,H^{1}(\mu))$. Fix
$l\in J$. Let us denote $\pi_{J}$ the projection on the coordinates
$(x_{i})_{i\in J}$. Projecting equation (4.34) on the $l$-coordinate reads

$\beta\sum_{i\in
J}\mathsf{M}_{l,i}\psi_{i}+\mathcal{L}^{\mu}\psi_{l}=-\beta\sum_{i\in
J^{c}}\mathsf{M}_{i,l}\psi_{i}+\lambda.$

Let us denote $\mathsf{M}^{J}=(\mathsf{M}_{i,j})_{i,j\in J}$ and
$\mathrm{V}\in L^{2}(J,H^{-1}(\mu))$

$\mathrm{V}_{l}=-\beta\sum_{i\in
J^{c}}\mathsf{M}_{l,i}\psi_{i}+\lambda,\quad\text{for each $l\in J$}.$

so that $\widetilde{\psi}$ solves

$\begin{cases}\mathsf{M}^{J}\widetilde{\psi}+\mathcal{L}^{\mu}\widetilde{\psi}=\mathrm{V}&\text{on
$\pi_{J}(\mathcal{M}_{N}$)}\\\ \widetilde{\psi}\cdot\vec{n}=0&\text{on
$\partial(\pi_{J}(\mathcal{M}_{N}))$}.\end{cases}$ (4.37)

Let $\eta$ be the cutoff function (4.25). For a parameter
$\alpha\in(\frac{1}{2},s-\frac{1}{2})$, we define the distortion matrix

$\mathrm{L}_{\alpha}=\mathrm{diag}((\gamma_{i})_{i\in J}),\quad\text{with
$\gamma_{i}=1+d(i,\partial J)^{\alpha},$}\quad\text{for each $i\in J$}$ (4.38)

and the commutator

$\delta_{\mathrm{L}_{\alpha}}=\mathrm{L}_{\alpha}\mathsf{M}^{J}\mathrm{L}_{\alpha}^{-1}-\mathsf{M}^{J}.$

Observe that the vector-field

$\psi^{\mathrm{dis}}:=\mathrm{L}_{\alpha}\widetilde{\psi}\in
L^{2}(J,H^{1}(\mu))$

solves

$\beta(\mathsf{M}^{J}+\delta_{\mathrm{L}_{\alpha}})\psi^{\mathrm{dis}}+\mathcal{L}^{\mu}\psi^{\mathrm{dis}}=\mathrm{L}_{\alpha}\mathrm{V}.$
(4.39)

Taking the scalar product of (4.39) with $\eta\psi^{\mathrm{dis}}$ and
integrating by parts yields

$\beta\mathbb{E}_{\mu}[\eta\psi^{\mathrm{dis}}\cdot(\mathsf{M}^{J}+\delta_{\mathrm{L}_{\alpha}})\psi^{\mathrm{dis}}]+\mathbb{E}_{\mu}[\eta|\nabla\psi^{\mathrm{dis}}|^{2}]\leq\mathbb{E}_{\mu}[\eta\mathrm{L}_{\alpha}\mathrm{V}\cdot\psi^{\mathrm{dis}}]-\sum_{i\in
J}\mathbb{E}_{\mu}[\psi_{i}^{\mathrm{dis}}\nabla\psi_{i}^{\mathrm{dis}}\cdot\nabla\eta].$
(4.40)

Following the proof of Lemma 4.2, one can upper bound the error term in the
last display by

$\Bigr{|}\sum_{i\in
J}\mathbb{E}_{\mu}[\psi_{i}^{\mathrm{dis}}\nabla\psi_{i}^{\mathrm{dis}}\cdot\nabla\eta]\Bigr{|}\leq\mathbb{E}_{\mu}[|\psi^{\mathrm{dis}}|^{2}]^{\frac{1}{2}}\sup|\psi^{\mathrm{dis}}|\mu(\mathcal{A}^{c})\leq
C(\beta)\sup|\chi_{n}|e^{-c(\beta)n^{\delta}},$ (4.41)

where we have used (4.28) in the last line. Arguing as in (4.30), we can then
write

$\beta\mathbb{E}_{\mu}[\eta\psi^{\mathrm{dis}}\cdot(\mathsf{M}^{J}+\delta_{\mathrm{L}_{\alpha}})\psi^{\mathrm{dis}}]\geq\frac{1}{2}n^{-\varepsilon(s+2)}\mathbb{E}_{\mu}[\eta|\psi^{\mathrm{dis}}|^{2}]-Cn^{\kappa\varepsilon}\mathbb{E}_{\mu}[|\psi^{\mathrm{dis}}|^{2}]^{\frac{1}{2}}.$
(4.42)

Inserting (4.41) and (4.42) into (4.40) gives

$\mathbb{E}_{\mu}[|\psi^{\mathrm{dis}}|^{2}]\leq
C(\beta)(n^{\kappa\varepsilon}\mathbb{E}_{\mu}[\eta^{2}|\mathrm{L}_{\alpha}\mathrm{V}|^{2}]+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$
(4.43)

Let us control the external vector-field using the induction hypothesis
(4.36). Let $l\in J$. One can first split $\mathrm{V}_{l}$ into

$\mathrm{V}_{l}=\underbrace{\sum_{i\in
J^{c},d(i,i_{0})\leq\frac{1}{2}d(j,i_{0})}\mathsf{M}_{i,l}\psi_{i}}_{({I})_{l}}+\underbrace{\sum_{i\in
J^{c},d(i,i_{0})>\frac{1}{2}d(j,i_{0})}\mathsf{M}_{i,l}\psi_{i}}_{({II})_{l}}+\lambda.$
(4.44)

The first term may be bounded by

$\begin{split}\mathbb{E}_{\mu}[\eta^{2}({I})^{2}]^{\frac{1}{2}}&\leq
C(\beta)n^{\kappa\varepsilon}\frac{1}{d(j,i_{0})^{s}}\Bigr{(}\sum_{i\in
J^{c}}d(j,i_{0})\mathbb{E}_{\mu}[\psi_{i}^{2}]\Bigr{)}^{\frac{1}{2}}\\\ &\leq
C(\beta)n^{\kappa\varepsilon}\frac{1}{d(j,i_{0})^{s}}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}),\end{split}$

where we have used (4.21) in the last inequality. For the remaining term, let
us split the window $\\{i\in J^{c},d(i,i_{0})>\frac{1}{2}d(j,i_{0})\\}$ into
intervals $K$ intervals $I_{1},\ldots,I_{K}$ of length smaller than
$d(j,i_{0})^{\varepsilon^{\prime}}$:

$\Bigr{\\{}i\in
J^{c},d(i,i_{0})>\frac{1}{2}d(j,i_{0})\Bigr{\\}}=I_{1}\cup\ldots\cup I_{K}.$

One can upper bound $({II})_{l}$ by

$\eta|({II})_{l}|\leq\sum_{k=2}^{K}\Bigr{(}\sum_{i\in
I_{k}}\psi_{i}^{2}\Bigr{)}^{\frac{1}{2}}\frac{1}{k^{s}d(j,i_{0})^{\varepsilon^{\prime}(s-\frac{1}{2})}}+\Bigr{(}\sum_{i\in
I_{1}}\psi_{i}^{2}\Bigr{)}^{\frac{1}{2}}\frac{1}{d(l,\partial
J)^{s-\frac{1}{2}}}.$

Using the recursion hypothesis (4.36), we find

$\mathbb{E}_{\mu}[\eta^{2}({II})_{l}^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}\frac{1}{d(j,i_{0})^{\gamma}}\Bigr{(}\frac{1}{d(l,\partial
J)^{s-\frac{\varepsilon^{\prime}}{2}}}+\frac{1}{d(j,i_{0})^{\varepsilon^{\prime}(s-\frac{1}{2})}}\Bigr{)}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$

Since $J$ is of length $2d(j,i_{0})^{\varepsilon^{\prime}}+1$, one can absorb
the second term into the first one, which reads

$\mathbb{E}_{\mu}[\eta^{2}({II})^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}\frac{1}{d(j,i_{0})^{\gamma}}\frac{1}{d(l,\partial
J)^{s-\frac{1}{2}}}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$

Combining these two estimates and using (4.22), one gets

$\mathbb{E}_{\mu}[\eta^{2}\mathrm{V}_{l}^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}\Bigr{(}\frac{1}{d(j,i_{0})^{s}}+\frac{1}{d(j,i_{0})^{\gamma}}\frac{1}{d(l,\partial
J)^{s-\frac{1}{2}}}+\frac{1}{n}\Bigr{)}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$
(4.45)

Let $\alpha>0$ such that $2s-1-2\alpha>1$. Multiplying the last display by
$\mathrm{L}_{\alpha}$ and summing over $l\in J$ gives

$\mathbb{E}_{\mu}[\eta^{2}|\mathrm{L}_{\alpha}\mathrm{V}|^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}\Bigr{(}\frac{1}{d(j,i_{0})^{s-(\frac{1}{2}+\alpha)\varepsilon^{\prime}}}+\frac{1}{d(j,i_{0})^{\gamma}}+\frac{d(j,i_{0})^{\varepsilon^{\prime}(\frac{1}{2}+\alpha)}}{n}\Bigr{)}\\\
\times(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$

In conjunction with (4.43) this yields

$\mathbb{E}_{\mu}[|\psi^{\mathrm{dis}}|^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}\Bigr{(}\frac{1}{d(j,i_{0})^{s-(\frac{1}{2}+\alpha)\varepsilon^{\prime}}}+\frac{1}{d(j,i_{0})^{\gamma}}+\frac{d(j,i_{0})^{\varepsilon^{\prime}\frac{1}{2}}}{n}\Bigr{)}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$

As a consequence, for each $j\in\\{1,\ldots,N\\}$, we have

$\mathbb{E}_{\mu}\Bigr{[}\sum_{k:d(k,j)\leq\frac{1}{2}d(j,i_{0})^{\varepsilon^{\prime}}}\psi_{k}^{2}\Bigr{]}^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}\Bigr{(}\frac{1}{d(j,i_{0})^{s-\frac{1}{2}\varepsilon^{\prime}}}+\frac{1}{d(j,i_{0})^{\gamma+\varepsilon^{\prime}\alpha}}+\frac{1}{n}\Bigr{)}\\\
\times(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$
(4.46)

As a consequence, (4.36) is verified for
$\alpha^{\prime}=\min(s-\frac{1}{2}\varepsilon^{\prime},\gamma+\frac{1}{2}\alpha)$.
Having chosen $\varepsilon^{\prime}\in(0,\min(1,2(s-1)))$, the first term is
$O({d(j,i_{0})^{-(1+\delta)}})$ for some $\delta>0$. We deduce after a finite
number of iterations that (4.36) is satisfied for some $\gamma>1$. We now
conclude the proof of Proposition 4.3. Let $\delta>0$ be a small number. Fix a
scale $\varepsilon^{\prime}\in(0,\min(1,2\delta,s-1))$. Assume that (4.36)
holds for some $\gamma>1$. Then one can decompose the term $({I})_{l}$ in
(4.44) into

$({I})_{l}=\underbrace{\sum_{i\in
J^{c},d(i,i_{0})\leq\frac{1}{2}d(j,i_{0})}\mathsf{M}_{i_{0},l}\psi_{i}}_{({I}^{\prime})_{l}}+\underbrace{\sum_{i\in
J^{c},d(i,i_{0})\leq\frac{1}{2}d(j,i_{0})}(\mathsf{M}_{i,l}-\mathsf{M}_{i_{0},l})\psi_{i}}_{({I}^{\prime\prime})_{l}}$

The important observation is that since $\psi\cdot(e_{1}+\ldots+e_{N})=0$ and
by (4.36), the first term is bounded by

$\mathbb{E}_{\mu}[\eta^{2}({I}^{\prime})_{l}^{2}]^{\frac{1}{2}}\leq\frac{C(\beta)n^{\kappa\varepsilon}}{d(j,i_{0})^{\gamma+s-1}}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$

The bound on the second term uses (for the first time in this proof) a form of
regularity on the Hessian matrix: we have

$|\eta^{2}(\mathsf{M}_{i,l}-\mathsf{M}_{i_{0},l})|\leq\frac{C(\beta)n^{\kappa\varepsilon}d(i,i_{0})}{d(j,i_{0})^{1+s}},$

which gives

$\begin{split}\mathbb{E}_{\mu}[\eta^{2}({I}^{\prime\prime})_{l}^{2}]^{\frac{1}{2}}&\leq
C(\beta)n^{\kappa\varepsilon}\frac{1}{d(j,i_{0})^{1+s}}\sum_{k=1}^{d(j,i_{0})}\frac{1}{k^{\gamma-1}}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}})\\\
&\leq
C(\beta)n^{\kappa\varepsilon}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}})\times\begin{cases}\frac{1}{d(j,i_{0})^{1+s}}&\text{if
$\gamma\geq 2$}\\\ \frac{1}{d(j,i_{0})^{s+\gamma-1}}&\text{if
$\gamma\in(1,2)$}.\end{cases}\end{split}$

As a consequence the control (4.45) is replaced by

$\mathbb{E}_{\mu}[\eta^{2}\mathrm{V}_{l}^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}\Bigr{(}\frac{1}{d(j,i_{0})^{1+s}}+\frac{1}{d(j,i_{0})^{\gamma+s-1}}+\frac{1}{d(j,i_{0})^{\gamma}}\frac{1}{d(l,\partial
J)^{s-\frac{1}{2}}}+\frac{1}{n}\Bigr{)}\\\
\times(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$

It follows that

$\Bigr{(}\sum_{k:d(k,j)\leq\frac{1}{2}d(j,i_{0})^{\varepsilon^{\prime}}}\mathbb{E}_{\mu}[\psi_{k}^{2}]\Bigr{)}^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}\Bigr{(}\frac{1}{d(j,i_{0})^{1+s-\frac{1}{2}\varepsilon^{\prime}}}+\frac{1}{d(j,i_{0})^{\gamma+s-1-\frac{1}{2}\varepsilon^{\prime}}}+\frac{1}{d(j,i_{0})^{\gamma+\varepsilon^{\prime}\alpha_{0}}}+\frac{1}{n}\Bigr{)}\\\
\times(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$

Since $\varepsilon^{\prime}\in(0,\min(1,2\delta,s-1))$, this improves (4.36).
We conclude after a finite number of steps that (4.36) holds for
$\gamma=1+s-\delta$. By choosing $\varepsilon^{\prime}>0$ arbitrarily small,
we obtain that $\psi$ satisfies

$\mathbb{E}_{\mu}[\psi_{j}^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}\Bigr{(}\frac{1}{d(j,i_{0})^{1+s}}+\frac{1}{n}\Bigr{)}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}\Bigr{)}.$

∎

###  4.5. Estimate on the main equation

In the case $n\leq N/2$, there remains to compare the solution of (4.1) to the
solution $\psi^{(1)}$ of (4.20). This supposes to estimate the quantity
$\widetilde{H}\psi^{(1)}$ where $\widetilde{H}$ is the perturbation in (4.17).
We obtain the following:

######  Proposition 4.4.

Let $s\in(1,+\infty).$ Let $\widetilde{\mathcal{H}}_{n}^{\mathrm{g}}$ be as in
(4.15). Let $i\in\\{1,\ldots,n\\}$ such that $|i-n/2|\leq n/4$. Let
$\chi_{n}\in H^{-1}(\mu)$ and $\psi\in L^{2}(\\{1,\ldots,n\\},H^{1}(\mu))$ be
the solution of

$\left\\{\begin{array}[]{ll}\beta\mathrm{Hess}\,\widetilde{\mathcal{H}}_{n}^{\mathrm{g}}\psi+\mathcal{L}^{\mu}\psi=\chi_{n}e_{i}+\lambda(e_{1}+\ldots+e_{n})&\text{on
}\pi(\mathcal{M}_{N})\\\ \psi\cdot(e_{1}+\ldots+e_{n})=0&\text{on
}\pi(\mathcal{M}_{N})\\\ \psi\cdot\vec{n}=0&\text{on
}\partial(\pi(\mathcal{M}_{N})).\end{array}\right.$ (4.47)

Then, uniformly in $1\leq j\leq n$, we have

$\mathbb{E}_{\mu}[(\psi_{j})^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}\Bigr{(}\frac{1}{d(i,j)^{1+s}}+\frac{1}{n}\Bigr{)}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$
(4.48)

###### Proof of Proposition 4.4.

Let $\psi\in L^{2}(\\{1,\ldots,n\\},H^{1}(\mu))$ be the solution of (4.47) and
$\psi^{(1)}\in L^{2}(\\{1,\ldots,n\\},H^{1}(\mu))$ be the solution of

$\left\\{\begin{array}[]{ll}\beta\mathsf{M}\psi^{(1)}+\mathcal{L}^{\mu}\psi^{(1)}=\chi_{n}e_{i}+\lambda^{(1)}(e_{1}+\ldots+e_{n})&\text{on
}\pi(\mathcal{M}_{N})\\\ \psi^{(1)}\cdot(e_{1}+\ldots+e_{n})=0&\text{on
}\pi(\mathcal{M}_{N})\\\ \psi^{(1)}\cdot\vec{n}=0&\text{on
}\partial(\pi(\mathcal{M}_{N})).\end{array}\right.$

Let us define $\psi^{(2)}=\psi-\psi^{(1)}$. One can check that $\psi^{(2)}$ is
solution of

$\left\\{\begin{array}[]{ll}\beta\mathrm{Hess}\,\widetilde{\mathcal{H}}_{n}^{\mathrm{g}}\psi^{(2)}+\mathcal{L}^{\mu}\psi^{(2)}=-\beta\widetilde{H}\psi^{(1)}+\lambda^{(2)}(e_{1}+\ldots+e_{n})&\text{on
}\pi(\mathcal{M}_{N})\\\ \psi^{(2)}\cdot(e_{1}+\ldots+e_{n})=0&\text{on
}\pi(\mathcal{M}_{N})\\\ \psi^{(2)}\cdot\vec{n}=0&\text{on
}\partial(\pi(\mathcal{M}_{N})).\end{array}\right.$

Let us estimate the vector-field $\widetilde{H}\psi^{(1)}$. Let $1\leq j\leq
n$. Using Lemma 6.3, the fact that $\psi^{(1)}\cdot(e_{1}+\ldots+e_{n})$ and
(4.18), one finds that uniformly in $1\leq j\leq n$,

$\mathbb{E}_{\mu}[(\widetilde{H}\psi^{(1)})_{j}^{2}]^{\frac{1}{2}}\leq\frac{C(\beta)n^{\kappa\varepsilon}}{d(j,\partial
I)^{\frac{s}{2}}}\frac{1}{n}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$
(4.49)

Let us detail the computations. Fix $1\leq j\leq n$. One can split the
quantity $(\widetilde{H}\psi^{(1)})_{j}$ into

$(\widetilde{H}\psi^{(1)})_{j}=\underbrace{\sum_{k:d(k,\partial I)\leq
n/4}\widetilde{H}_{j,k}\psi_{k}}_{({I})_{j}}+\underbrace{\sum_{k:d(k,\partial
I)>n/4}\widetilde{H}_{j,k}\psi_{k}}_{({II})_{j}}.$

For the first quantity we can write

$\begin{split}\mathbb{E}_{\mu}[({I})_{j}^{2}]^{\frac{1}{2}}&\leq\frac{C(\beta)n^{\kappa\varepsilon}}{d(j,\partial
I)^{\frac{s}{2}}}\sum_{k:d(k,\partial I)\leq
n/4}\frac{1}{d(k,\frac{n}{2})^{1+s}}\frac{1}{d(k,\partial
I)^{\frac{s}{2}}}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}})\\\
&\leq\frac{C(\beta)n^{\kappa\varepsilon}}{d(j,\partial
I)^{\frac{s}{2}}}\frac{1}{n^{\min(1,\frac{s}{2})+s}}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).\end{split}$

For the second quantity, we can write

$({II})_{j}=\sum_{k:d(k,\partial
I)>n/4}(\widetilde{H}_{j,k}-\widetilde{H}_{j,i})\psi_{k}^{((1)}+\sum_{k:d(k,\partial
I)>n/4}\widetilde{H}_{j,i}\psi_{k}^{(1)}.$ (4.50)

For the first term in the last display, using the bound on the increments of
$\widetilde{H}$ given in (4.19), we find

$\mathbb{E}_{\mu}\Bigr{[}\Bigr{(}\sum_{k:d(k,\partial
I)>n/4}(\widetilde{H}_{j,k}-\widetilde{H}_{j,i})\psi_{k}^{(1)}\Bigr{)}^{2}\Bigr{]}^{\frac{1}{2}}\\\
\leq\frac{C(\beta)n^{\kappa\varepsilon}}{d(j,\partial
I)^{\frac{s}{2}}}\sum_{k:d(k,\partial
I)>n/4}\frac{1}{|k-i|^{1+s}}\frac{1}{n}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}})\\\
\leq\frac{C(\beta)n^{\kappa\varepsilon}}{d(j,\partial
I)^{\frac{s}{2}}}\frac{1}{n}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$
(4.51)

Because $\psi^{(1)}\cdot(e_{1}+\ldots+e_{n})=0$, the second term satisfies

$\mathbb{E}_{\mu}\Bigr{[}\Bigr{(}\sum_{k:d(k,\partial
I)>n/4}\widetilde{H}_{j,i}\psi_{k}^{(1)}\Bigr{)}^{2}\Bigr{]}^{\frac{1}{2}}\leq\frac{C(\beta)n^{\kappa\varepsilon}}{d(j,\partial
I)^{\frac{s}{2}}}\frac{1}{n^{\frac{3s}{2}}}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$
(4.52)

Putting (4.50), (4.51) and (4.52) together we obtain (4.49). Summing this over
$j$ yields

$\mathbb{E}_{\mu}[|\widetilde{H}\psi^{(1)}|^{2}]\leq\frac{C(\beta)n^{\kappa\varepsilon}}{n}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$

Using the uniformly convexity of $\widetilde{H}_{n}^{\mathrm{g}}$, we then
find

$\mathbb{E}_{\mu}[|\psi^{(2)}|^{2}]\leq\frac{C(\beta)n^{\kappa\varepsilon}}{n}(\mathbb{E}_{\mu}[\chi_{n}^{2}]^{\frac{1}{2}}+\sup|\chi_{n}|e^{-c(\beta)n^{\delta}}).$

In particular, together with (4.35), this yields (4.48). ∎

###  4.6. Decay of gaps correlations

We are now ready to conclude the proof of the decay of correlations for the
hypersingular Riesz gas. When $x_{i}$ and $x_{j}$ are at macroscopic or large
mesosopic distance, one can take $n=N$ and use the estimate of Proposition
4.3. Otherwise we choose $n$ to be a power of $|i-j|$ and apply the estimate
of Proposition 4.4 for such a number $n$. This will complete proof of Theorem
2.

###### Proof of of Theorem 2.

We first examine the case where there exists $\varepsilon_{0}>0$ such that
$d(i,j)\leq N^{\varepsilon_{0}}$. Let $\mu$ be the constrained measure on
$\\{1,\ldots,N\\}$ defined in (4.10) with $n=N$. Using the Pinsker inequality,
the fact that $\mu$ satisfies a log-Sobolev inequality (see Lemma 3.8) and the
local law of Lemma C.1, one can observe that

$\mathrm{TV}(\mathbb{P}_{N,\beta}^{\mathrm{g}},\mu)\leq\sqrt{2\mathrm{Ent}(\mathbb{P}_{N,\beta}^{\mathrm{g}}\mid\mu)}\leq
C(\beta)N^{\kappa\varepsilon}\mathbb{E}_{\mathbb{P}_{N,\beta}^{\mathrm{g}}}[|\nabla\mathrm{F}^{\mathrm{g}}|^{2}]^{\frac{1}{2}}\leq
C(\beta)e^{-c(\beta)N^{\delta}}.$ (4.53)

In particular, it follows that

$\mathrm{Cov}_{\mathbb{P}_{N,\beta}}[\xi(N(x_{j+1}-x_{j})),\chi(N(x_{i+1}-x_{i}))]=\mathrm{Cov}_{\mu}[\xi(x_{j}),\chi(x_{i})]+O_{\beta}(e^{-N^{\delta}}\sup|\xi|\sup|\chi|).$
(4.54)

Moreover, by Proposition 3.3, the covariance term in the last display may be
expressed as

$\mathrm{Cov}_{\mu}[\xi(x_{j}),\chi(x_{i})]=\mathbb{E}_{\mu}[\xi^{\prime}(x_{j})\partial_{j}\phi],$

with $\nabla\phi\in L^{2}(\\{1,\ldots,N\\},H^{1}(\mu))$ solution of

$\left\\{\begin{array}[]{ll}A_{1}^{\mu}\nabla\phi=\chi^{\prime}(x_{i})e_{i}&\text{on
}\pi(\mathcal{M}_{N})\\\ \nabla\phi\cdot(e_{1}+\ldots+e_{N})=0&\text{on
}\pi(\mathcal{M}_{N})\\\ \nabla\phi\cdot\vec{n}=0&\text{on
}\partial(\pi(\mathcal{M}_{N})).\end{array}\right.$

Using the estimate of Proposition 4.3, the Hölder’s inequality and (4.54), we
conclude the proof of (1.5) in the case $d(j,i)\geq N^{\varepsilon_{0}}.$ We
now consider the case where $d(j,i)$ is much smaller than a power of $N$. We
fix $n\in\\{1,\ldots,N\\}$ such that $\frac{1}{\sqrt{n}}\leq d(j,i)^{2(1+s)}$.
Without loss of generality, one can assume that $1\leq\frac{n}{3}\leq
i,j\leq\frac{2n}{3}$. Since $N(x_{i+1}-x_{i})$ and $N(x_{j+1}-x_{1})$ are
functions of $x_{1},\ldots,x_{n}$ and since $\mathcal{A}$ has overwhelming
probability, one may write

$\mathrm{Cov}_{\mathbb{P}_{N,\beta}}[\xi(N(x_{j+1}-x_{j})),\chi(N(x_{i+1}-x_{i}))]=\mathrm{Cov}_{\mu}[\xi(x_{j}),\chi(x_{i})]+O_{\beta}(e^{-c(\beta)n^{\delta}}\sup|\xi|\sup|\chi|).$
(4.55)

By Proposition 3.3 again one can express this covariance term as

$\mathrm{Cov}_{\mu}[x_{j},x_{i}]=\mathbb{E}_{\mu}[\xi^{\prime}(x_{j})\partial_{j}\phi],$
(4.56)

where $\nabla\phi\in L^{2}(\\{1,\ldots,n\\},H^{1}(\mu))$ is solution of

$\left\\{\begin{array}[]{ll}A_{1}^{\mu}\nabla\phi=\chi^{\prime}(x_{i})e_{i}&\text{on
}\pi(\mathcal{M}_{N})\\\ \nabla\phi\cdot\vec{n}=0&\text{on
}\partial(\pi(\mathcal{M}_{N})).\end{array}\right.$

We now split $\nabla\phi$ into $\nabla\phi=\nabla\psi+\nabla\widetilde{\psi}$
with $\nabla\psi$ the unique solution of the equation

$\left\\{\begin{array}[]{ll}A_{1}^{\mu}\nabla\psi=\chi^{\prime}(x_{i})e_{i}+\lambda(e_{1}+\ldots+e_{n})&\text{on
}\pi(\mathcal{M}_{N})\\\ \nabla\psi\cdot(e_{1}+\ldots+e_{n})=0&\text{on
}\pi(\mathcal{M}_{N})\\\ \nabla\psi\cdot\vec{n}=0&\text{on
}\partial(\pi(\mathcal{M}_{N})).\end{array}\right.$ (4.57)

The existence and uniqueness of the solution of (4.57) follow from the proof
of Proposition 3.3. By Proposition 4.4, the vector-field $\nabla\psi$
satisfies

$\mathbb{E}_{\mu}[(\partial_{j}\psi)^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}\Bigr{(}\frac{1}{d(j,i)^{1+s}}+\frac{1}{n}\Bigr{)}(\mathbb{E}_{\mu}[\chi^{\prime}(x_{i})^{2}]^{\frac{1}{2}}+\sup|\chi^{\prime}|e^{-c(\beta)n^{\delta}}).$
(4.58)

It remains to control the vector-field $\nabla\widetilde{\psi}$. Let $\lambda$
be the Lagrange multiplier in (4.57). First, observe that
$\nabla\widetilde{\psi}$ is solution of

$\left\\{\begin{array}[]{ll}A_{1}^{\mu}\nabla\widetilde{\psi}=\lambda(e_{1}+\ldots+e_{n})&\text{on
}\pi(\mathcal{M}_{N})\\\ \nabla\widetilde{\psi}\cdot\vec{n}=0&\text{on
}\partial(\pi(\mathcal{M}_{N})).\end{array}\right.$ (4.59)

In view of (4.22), the Lagrange multiplier is bounded by

$\mathbb{E}_{\mu}[\lambda^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}\frac{1}{n}(\mathbb{E}_{\mu}[\chi^{\prime}(x_{i})^{2}]^{\frac{1}{2}}+\sup|\chi^{\prime}|e^{-c(\beta)n^{\delta}}).$

Taking the scalar product of (4.59) and integrating by parts over $\mu$ yields

$\beta
n^{\varepsilon(s+2)}\mathbb{E}_{\mu}[|\nabla\widetilde{\psi}|^{2}]\leq\mathbb{E}_{\mu}[\lambda^{2}]^{\frac{1}{2}}\sqrt{n}\mathbb{E}_{\mu}[|\nabla\widetilde{\psi}|^{2}]^{\frac{1}{n}}.$

As a consequence,

$\mathbb{E}_{\mu}[(\partial_{j}\widetilde{\psi})^{2}]^{\frac{1}{2}}\leq\mathbb{E}_{\mu}[|\nabla\widetilde{\psi}|^{2}]^{\frac{1}{2}}\leq
C(\beta)(\mathbb{E}_{\mu}[\chi^{\prime}(x_{i})^{2}]^{\frac{1}{2}}+\sup|\chi^{\prime}|e^{-c(\beta)n^{\delta}})\frac{n^{\varepsilon(s+2)}}{\sqrt{n}}.$
(4.60)

Putting (4.58) and (4.60) together gives

$\mathbb{E}_{\mu}[(\partial_{j}\phi)^{2}]^{\frac{1}{2}}\leq
C(\beta)n^{\kappa\varepsilon}\Bigr{(}\frac{1}{d(j,i)^{1+s}}+\frac{1}{\sqrt{n}}\Bigr{)}(\mathbb{E}_{\mu}[\chi^{\prime}(x_{i})^{2}]^{\frac{1}{2}}+\sup|\chi^{\prime}|e^{-c(\beta)n^{\delta}}).$

Inserting this into (4.56) and using (4.55) completes the proof of (1.5). ∎

##  5\. Decay of correlations for the long-range Riesz gas

This section is the heart of the paper. We elaborate a method to study the
decay of correlations in the long-range case $s\in(0,1)$. Because the Hessian
of the energy in gap coordinates concentrates around the Riesz matrix (2.7)
which has slowly decaying entries, it is not clear that the strategy of
Section 4 can be adapted to this new situation. Indeed the commutation result
of Lemma 4.1 cannot be applied to (2.7). The trick is to exploit the fact that
the Hessian is much more than uniformly definite positive. It actually
controls a fractional primitive of the solution, which will help carrying out
the argument. This should be compared with the method of [Bou21, Section 4]
adapted from [BEY12] which exploits the long-range nature of the interaction
to have sharp concentration estimates.

###  5.1. Rewriting the equation

We study the decay of the covariance between $N(x_{i+1}-x_{i})$ and
$N(x_{j+1}-x_{j})$ for $1\leq i,j\leq N$ under the Gibbs measure (1.4) in the
long-range case. Until the end of the section, $s$ is a fixed parameter in
$(0,1)$. As in Section 4, we will constrain the measure on a subsystem
containing the indices $i$ and $j$. Depending on $d(i,j)$, we define an active
scale $n\in\\{1,\ldots,N\\}$ and assume without loss of generality that
$i,j\in\\{1,\ldots,n\\}$. Let $\theta:\mathbb{R}^{+}\to\mathbb{R}^{+}$ smooth
such that $\theta=0$ on $(1,+\infty)$, $\theta^{\prime\prime}\geq 1$ on
$[0,\frac{1}{2})$ and $\theta^{\prime\prime}\geq 0$ on $[0,+\infty]$. Define
the forcing

$\mathrm{F}=\sum_{i=1}^{n}\theta\Bigr{(}\frac{N(x_{i+1}-x_{i})}{n^{\varepsilon}}\Bigr{)}\quad\text{and}\quad\mathrm{F}^{\mathrm{g}}=\mathrm{F}\circ\mathrm{Gap}_{N}$

and the constrained measures

$\displaystyle\mathrm{d}\mathbb{Q}_{N,\beta}\propto
e^{-\beta\mathrm{F}}\mathrm{d}\mathbb{P}_{N,\beta}$
$\displaystyle\quad\mathrm{d}\mathbb{Q}_{N,\beta}^{\mathrm{g}}\propto
e^{-\beta\mathrm{F}^{\mathrm{g}}}\mathrm{d}\mathbb{P}_{N,\beta}^{\mathrm{g}}.$

In the sequel we will denote $I=\\{1,\ldots,n\\}$,
$\pi:\mathcal{M}_{N}\to\pi(\mathcal{M}_{N})$ the projection on
$x:=(x_{1},\ldots,x_{n})$ and $\mu$ the push-forward of
$\mathbb{Q}_{N,\beta}^{\mathrm{g}}$ by $\pi$:

$\mathrm{d}\mu(x)=\exp(-\beta\widetilde{\mathcal{H}}_{n}^{\mathrm{g}}(x))\mathds{1}_{\pi(\mathcal{M}_{N})}(x)\mathrm{d}x.$

As seen in Section 4, for all $x\in\pi(\mathcal{M}_{N})$, the Hessian of
$\widetilde{\mathcal{H}}_{n}^{\mathrm{g}}$ at $x$ is given for each $1\leq
i,j\leq n$ by

$\partial_{ij}\widetilde{\mathcal{H}}_{n}^{\mathrm{g}}(x)=\partial_{ij}\mathrm{F}^{\mathrm{g}}(x)+\partial_{ij}\mathcal{H}_{n}^{\mathrm{g}}(x)+\mathbb{E}_{\mathbb{Q}_{N,\beta}^{\mathrm{g}}(\cdot\mid
x)}[\partial_{ij}\mathcal{H}_{n,N}^{\mathrm{g}}(x,\cdot)]\\\
-\mathrm{Cov}_{\mathbb{Q}_{N,\beta}^{\mathrm{g}}(\cdot\mid
x)}[\partial_{i}\mathcal{H}_{n,N}^{\mathrm{g}}(x,\cdot),\partial_{j}\mathcal{H}_{n,N}^{\mathrm{g}}(x,\cdot)],$

where
$\mathcal{H}_{n,N}^{\mathrm{g}}(x,y)=\mathcal{H}_{N}^{\mathrm{g}}(x,y)-\mathcal{H}_{n}^{\mathrm{g}}(x)-\mathcal{H}_{N-n}^{\mathrm{g}}(y)$
for all $(x,y)\in(\mathbb{R}^{n}\times\mathbb{R}^{N-n})\cap\mathcal{M}_{N}$.
The first task is to split the Hessian of
$\widetilde{\mathcal{H}}_{n}^{\mathrm{g}}$ to allow a convenient analysis of
the Helffer-Sjöstrand equation. We will work again on a good event

$\mathcal{A}:=\left\\{X_{n}\in\pi(\mathcal{M}_{n}):\forall
i\in\\{1,\ldots,n-1\\},n^{-\varepsilon}\leq x_{i}\leq
n^{\varepsilon}\right\\}\\\ \cap\\{X_{n}\in\pi(\mathcal{M}_{n}):\forall
i\in\\{1,\ldots,n\\},k\in\\{1,\ldots,n-i\\},|x_{i}+\ldots+x_{i+k-1}-k|\leq
n^{\varepsilon}k^{\frac{s}{2}}\\}.$ (5.1)

In order to have a uniformly positive definite interaction matrix, we isolate
small gaps interactions and let for a large $K_{0}\in\\{1,\ldots,n\\}$,

$\mathcal{H}_{n}^{\mathrm{g},1}:X_{n}\in\pi(\mathcal{M}_{N})\mapsto
N^{-(s+2)}\sum_{i\neq j:|i-j|\leq
K_{0}}g_{s}^{\prime\prime}\Bigr{(}\frac{x_{i}+\ldots+x_{j}}{N}\Bigr{)}\quad\text{and}\quad\mathcal{H}_{n}^{\mathrm{g},2}=\mathcal{H}_{n}^{\mathrm{g}}-\mathcal{H}_{n}^{\mathrm{g},1}.$

Let us define the modified Riesz kernel

$g_{s,m}(i)=\sum_{k:d(k,i)\geq K_{0}}g_{s}^{\prime\prime}(k),\quad
i\in\\{1,\ldots,N\\}$

and

$\mathbb{H}_{s,m}=(1+g_{s,m}(d(i,j))_{1\leq i,j\leq
N}\in\mathcal{M}_{N}(\mathbb{R}).$ (5.2)

Consider the block decomposition of $\mathbb{H}_{s,m}$ on
$\mathbb{R}^{n}\times\mathbb{R}^{N-n}$,

$\mathbb{H}_{s,m}=\begin{pmatrix}A&B\\\ C&D\end{pmatrix},\quad
A\in\mathcal{M}_{n}(\mathbb{R}).$ (5.3)

Since $\mathbb{H}_{s,m}$ is definite positive, so is the Schur complement
$A-BD^{-1}C$. In addition, on the good event (5.1), the $L^{2}$ operator norm
of $\mathcal{H}_{n}^{\mathrm{g},2}-A$ is bounded by

$\|\mathcal{H}_{n}^{\mathrm{g},2}-A\|\leq
C(\beta)n^{\kappa\varepsilon}K_{0}^{-\frac{s}{2}}.$

and the Hessian of $\mathrm{F}^{\mathrm{g}}+\mathcal{H}_{n}^{\mathrm{g},1}$ is
bounded from below by

$\mathrm{Hess}\,(\mathrm{F}^{\mathrm{g}}+\mathcal{H}_{n}^{\mathrm{g},1})\geq
n^{-\kappa\varepsilon}\mathrm{Id}.$

Consequently by choosing $K_{0}$ to be a large power of $n^{\varepsilon}$, we
obtain that on $L^{2}(\\{1,\ldots,n\\},H^{1}(\mu))$,

$\mathrm{Hess}\,\mathrm{F}^{\mathrm{g}}+\mathrm{Hess}\,\mathcal{H}_{n}^{\mathrm{g}}-B(D+\beta^{-1}\mathcal{L}^{\mu}\otimes\mathrm{Id})^{-1}C\geq
n^{-\kappa\varepsilon}\mathrm{Id}.$ (5.4)

One can therefore split the Hessian of
$\widetilde{\mathcal{H}}_{n}^{\mathrm{g}}$ into
$\mathrm{Hess}\,\widetilde{\mathcal{H}}_{n}^{\mathrm{g}}=\mathsf{M}+\widetilde{H}$
where

$\mathsf{M}:=\mathrm{Hess}\,\mathrm{F}^{\mathrm{g}}+\mathrm{Hess}\,\mathcal{H}_{n}^{\mathrm{g}}-B(D+\beta^{-1}\mathcal{L}^{\mu}\otimes\mathrm{Id})^{-1}C$
(5.5)

and $\widetilde{H}$ given for each $1\leq i,j\leq n$ by

$\widetilde{H}_{i,j}=\mathbb{E}_{\mathbb{Q}_{N,\beta}^{\mathrm{g}}(\cdot\mid
x)}[\partial_{ij}\mathcal{H}_{n,N}^{\mathrm{g}}(x,\cdot)]-\mathrm{Cov}_{\mathbb{Q}_{N,\beta}^{\mathrm{g}}(\cdot\mid
x)}[\partial_{i}\mathcal{H}_{n,N}^{\mathrm{g}}(x,\cdot),\partial_{j}\mathcal{H}_{n,N}^{\mathrm{g}}(x,\cdot)]\\\
+e_{i}\cdot B(D+\beta^{-1}\mathcal{L}^{\mu}\otimes\mathrm{Id})^{-1}(Ce_{j}).$
(5.6)

As in Section 4, we work with a more general measure $\mu$ on $\pi(D_{N})$.

######  Assumptions 5.1.

Let $\mu$ be a measure on $\pi(\mathcal{M}_{N})$ in the form
$\mathrm{d}\mu(x)=e^{-\beta
H(x)}\mathds{1}_{\pi(\mathcal{M}_{N})}(x)\mathrm{d}x$ with
$H:\pi(\mathcal{M}_{N})\to\mathbb{R}$ uniformly convex with a lower bound on
the Hessian larger than $c:=n^{-\kappa\varepsilon}$ for some $\kappa>0$.
Letting $\mathcal{A}$ be the good event (5.1), suppose that there exist some
constants $\kappa>0$, $c(\beta)$ and $C(\beta)>0$ such that

$\mu(\mathcal{A})\geq 1-e^{-c(\beta)n^{\kappa\varepsilon}}.$

Instead of studying the inverse of $A_{1}^{\mu}$, we rather study the equation

$\left\\{\begin{array}[]{ll}\beta\mathsf{M}\psi+\mathcal{L}^{\mu}\psi=\chi_{n}e_{i_{0}}&\text{on
}\pi(\mathcal{M}_{N})\\\ \psi\cdot\vec{n}=0&\text{on
}\partial\pi(\mathcal{M}_{N}),\end{array}\right.$ (5.7)

where $\mathsf{M}$ satisfies the following assumptions:

######  Assumptions 5.2.

Let $n\in\\{1,\ldots,N\\}$. Let $A,B,C,D$ be as in (5.3).
Let $\mathsf{M}:L^{2}(\\{1,\ldots,n\\},H^{1}(\mu))\to
L^{2}(\\{1,\ldots,n\\},H^{1}(\mu))$ in the form

$\mathsf{M}=A(x)-B(D+\beta^{-1}\mathcal{L}^{\mu}\otimes\mathrm{Id})^{-1}C,$
(5.8)

where $A$ takes values into $\mathcal{M}_{n}(\mathbb{R})$ and verifies the
following:

  * •

There exists a positive constant $\kappa>0$ such that uniformly on
$\pi(\mathcal{M}_{N})$, $A(x)\geq n^{-\kappa\varepsilon}\mathrm{Id}$.

  * •

There exists a positive constant $\kappa>0$ such that uniformly on (5.1) and
for each $1\leq i,j\leq n$,

$\mathsf{A}_{i,j}(x)=A_{i,j}+O\Bigr{(}\frac{n^{\kappa\varepsilon}}{d(i,j)^{1+\frac{s}{2}}}\Bigr{)}.$
(5.9)

###  5.2. Embedding the system

The proof starts by embedding the system (5.7) in a larger system on
$\mathbb{R}^{2n}$. Let us denote $\psi^{(1)}\in
L^{2}(\\{1,\ldots,n\\},H^{1}(\mu))$ the solution of (5.7). We let
$I^{\prime}=\\{1,\ldots,2n\\}$ and
$\pi^{\prime}:\mathcal{M}_{N}\to\pi^{\prime}(\mathcal{M}_{N})$ be the
projection on the coordinates $(x_{i})_{i\in I^{\prime}}$. Let
$B,C,D\in\mathcal{M}_{n}(\mathbb{R})$ be as in (5.3) for $N=2n$. If
$\mathsf{M}$ in as in Assumptions 5.1, we denote
$\mathsf{M}^{\prime}:L^{2}(\\{1,\ldots,2n\\},H^{1}(\mu))\to
L^{2}(\\{1,\ldots,2n\\},H^{1}(\mu))$ the operator

$\mathsf{M}^{\prime}=\begin{pmatrix}A(x)&B\\\ C&D\end{pmatrix}.$ (5.10)

We consider the solution of

$\left\\{\begin{array}[]{ll}\beta\mathsf{M}^{\prime}\psi+\mathcal{L}^{\mu}\psi=\chi_{n}e_{i_{0}}&\text{on
}\pi^{\prime}(\mathcal{M}_{N})\\\ \psi\cdot\vec{n}=0&\text{on
}\partial\pi^{\prime}(\mathcal{M}_{N}).\end{array}\right.$ (5.11)

The point is that

$\psi^{(1)}_{j}=\psi_{j}\quad\text{for each $j\in\\{1,\ldots,n\\}$}.$ (5.12)

We will thus work with $\psi$ itself. In order to compare the solutions of
(5.7) and of the main equation, one shall upper bound the quantity
$B(D+\beta^{-1}\mathcal{L}^{\mu}\otimes\mathrm{Id})^{-1}C$. Let us first
emphasize that by Remark 3.4 and Theorem 6, if $\mu$ satisfies Assumptions
5.1, there exist $\kappa>0$ and $\delta>0$ such that for each $1\leq i,j\leq
n$,

$\mu\Bigr{(}\Bigr{|}(\mathsf{M}^{\prime})_{i,j}-\frac{1}{1+d(i,j)^{s}}\Bigr{|}\geq\frac{n^{\kappa\varepsilon}}{d(i,j)^{1+\frac{s}{2}}}\Bigr{)}\leq
C(\beta)e^{-c(\beta)n^{\delta}}.$ (5.13)

######  Lemma 5.1.

Let $s\in(0,1)$. Let $B,C,D$ be as in (5.3). Let $\eta$, $\phi\in L^{2}(\mu)$.
Then for each $1\leq i,j\leq n$, we have

$|\mathbb{E}_{\mu}[\eta e_{j}\cdot B(\beta
D+\mathcal{L}^{\mu}\otimes\mathrm{Id})^{-1}(Ce_{i}\phi)]|\leq\frac{C(\beta)n^{\kappa\varepsilon}}{d(i,\partial
I)^{\frac{s}{2}}d(j,\partial
I)^{\frac{s}{2}}}\mathbb{E}_{\mu}[\eta^{2}]^{\frac{1}{2}}\mathbb{E}_{\mu}[\phi^{2}]^{\frac{1}{2}}.$
(5.14)

In addition for each $1\leq i,j,l\leq n$, we have

$|\mathbb{E}_{\mu}[\eta(e_{j}-e_{l})\cdot B(\beta
D+\mathcal{L}^{\mu}\otimes\mathrm{Id})^{-1}(Ce_{i}\phi)]|\leq\frac{C(\beta)n^{\kappa\varepsilon}|j-l|}{\min(d(j,\partial
I)^{1+\frac{s}{2}},d(l,\partial I)^{1+\frac{s}{2}})}\frac{1}{d(j,\partial
I)^{\frac{s}{2}}}\mathbb{E}_{\mu}[\eta^{2}]^{\frac{1}{2}}\mathbb{E}_{\mu}[\phi^{2}]^{\frac{1}{2}}.$
(5.15)

Informally the term in the left-hand side of (5.14) is comparable to the
covariance between $\partial_{i}\mathcal{H}_{n,N}^{\mathrm{g}}$ and
$\partial_{j}\mathcal{H}_{n,N}^{\mathrm{g}}$ under a Gaussian measure. This
analogy suggests us to proceed as if we were trying to control the variances
of $\partial_{i}\mathcal{H}_{n,N}^{\mathrm{g}}$ and
$\partial_{j}\mathcal{H}_{n,N}^{\mathrm{g}}$, which supposes one to control
the fluctuations of large gaps. We will thus import a method of [BEY12] which
starts by decomposing a given gap into a sum of block averaged statistics.

###### Proof of Lemma 5.1.

First note that

$|\mathbb{E}_{\mu}[\eta e_{j}\cdot B(\beta
D+\mathcal{L}^{\mu}\otimes\mathrm{Id})^{-1}(Ce_{i}\phi)]|\\\
\leq\mathbb{E}_{\mu}[(Ce_{j}\eta)\cdot B(\beta
D+\mathcal{L}^{\mu}\otimes\mathrm{Id})^{-1}(Ce_{j}\eta)]^{\frac{1}{2}}\mathbb{E}_{\mu}[(Ce_{i}\phi)\cdot
B(\beta
D+\mathcal{L}^{\mu}\otimes\mathrm{Id})^{-1}(Ce_{i}\phi)]^{\frac{1}{2}}.$
(5.16)

These two terms can be read as the variances of
$\phi\partial_{i}\mathcal{H}_{n,N}^{\mathrm{g}}$ and
$\eta\partial_{j}\mathcal{H}_{n,N}^{\mathrm{g}}$, which motivates (5.14). Let
us first change the system of coordinates and rewrite the quantities of
interest in terms of points rather than gaps. Let
$\nu=\mu\circ\mathrm{Gap}_{N}$, $G=\mathrm{Hess}\,\mathrm{Gap}_{n}$,
$\hat{D}=GD$. Recalling that $B=C^{t}$, one may first justify that

$\mathbb{E}_{\mu}[(Ce_{i}\phi)(\beta
D+\mathcal{L}^{\mu}\otimes\mathrm{Id})^{-1}(Ce_{i}\phi)]^{\frac{1}{2}}\\\ \leq
C\sum_{k\in
I^{c}}\frac{1}{d(k,i)^{1+s}}\mathbb{E}_{\nu}[(\phi(e_{n}+\ldots+e_{k}))\cdot(\beta\hat{D}+\mathcal{L}^{\nu}\otimes\mathrm{Id})^{-1}(\phi(e_{n}+\ldots+e_{k}))]^{\frac{1}{2}}.$
(5.17)

We now control the above quantity by following the insight of [BEY12]. Let
$n\leq i\leq i+k\leq 2n$. Let us split $N(x_{i+k}-x_{i})$ into a sum of block
average statistics. For each $1\leq k\leq n-1$ and $i\in\\{n+1,\ldots,2n\\}$,
let $I_{k}(k)$ be an interval of integers in $\\{n+1,\ldots,2n\\}$ of cardinal
$k+1$ such that $i\in I_{k}(i)$. Define the block average

$x_{i}^{[k]}=\frac{1}{k+1}\sum_{j\in I_{k}(i)}x_{k}.$

Let $\alpha>0$ be a small number, $\alpha=\frac{1}{p}$ with
$p\in\mathbb{N}^{*}$. One may write

$N(x_{i}-x_{i}^{[k]})=\sum_{m=0}^{p-1}N(x_{i}^{[\lfloor
k^{m\alpha}\rfloor]}-x_{i}^{[\lfloor k^{(m+1)\alpha}\rfloor]}).$ (5.18)

For each $m\in\\{0,\ldots,p-1\\}$, denote $G_{m}=N(x_{i}^{[\lfloor
k^{m\alpha}\rfloor]}-x_{i}^{[\lfloor k^{(m+1)\alpha}\rfloor]})$ and
$I_{m}=I_{\lfloor k^{(m+1)\alpha}\rfloor}(i)$. Let
$\hat{D}^{(1)}=(\hat{D}_{i,j})_{i,j\in I_{m}}$. Using the fact that
$\hat{D}-\hat{D}^{(1)}$ is positive definite, we find that

$\mathbb{E}_{\nu}[\nabla G_{m}\eta\cdot B(\beta
D+\mathcal{L}^{\nu}\otimes\mathrm{Id})^{-1}(\nabla
G_{m}\eta)]\leq\mathbb{E}_{\nu}[\nabla G_{m}\eta\cdot B(\beta
D^{(1)}+\mathcal{L}^{\nu}\otimes\mathrm{Id})^{-1}(\nabla G_{m}\eta)].$ (5.19)

Denote $v=(\beta D^{(1)}+\mathcal{L}^{\nu}\otimes\mathrm{Id})^{-1}(\nabla
G_{m}\eta)\in L^{2}(\\{n+1,\ldots,2n\\},H^{1}(\nu))$. One can check that for
each $i\in I_{m}^{c}$, $v_{i}=0$. I addition, since $\sum_{i\in
I_{m}}\partial_{i}G_{m}=0$ and $\hat{D}^{(1)}\sum_{i\in I_{m}}e_{i}=0$, then
$\sum_{i\in I_{m}}v_{i}=0$. It follows that

$v\cdot\hat{D}^{(1)}v\geq\sum_{i\neq j\in
I_{m}}\frac{1}{|i-j|^{s+2}}(N(v_{i}-v_{j}))^{2}\geq\frac{N^{2}}{|I_{m}|^{s+1}}|v|^{2}.$

Furthermore observe that

$|\nabla G_{m}|^{2}\leq\frac{CN^{2}}{|I_{m}|}.$

The two last displays give by integrative by parts the series of inequalities

$\beta\frac{N^{2}}{|I_{m}|^{s+1}}\mathbb{E}_{\mu}[|v|^{2}]\leq\mathbb{E}_{\nu}[v\cdot\hat{D}^{(1)}v]\\\
\leq\mathbb{E}_{\nu}[\nabla G_{m}\eta\cdot B(\beta
D^{(1)}+\mathcal{L}^{\nu}\otimes\mathrm{Id})^{-1}(\nabla G_{m}\eta)]\leq
C\mathbb{E}_{\mu}[|v|^{2}]^{\frac{1}{2}}\mathbb{E}_{\mu}[\eta^{2}]^{\frac{1}{2}}\frac{N^{2}}{|I_{m}|}.$

It follows that

$\mathbb{E}_{\nu}[\nabla G_{m}\eta\cdot B(\beta
D^{(1)}+\mathcal{L}^{\nu}\otimes\mathrm{Id})^{-1}(\nabla G_{m}\eta)]\leq
C(\beta)|I_{m}|^{s}\mathbb{E}_{\mu}[\eta^{2}]^{\frac{1}{2}}.$ (5.20)

Summing (5.20) over $m$ and using (5.18), one finds that

$\mathbb{E}_{\nu}[\eta N(e_{i+k}-e_{i})\cdot B(\beta
D+\mathcal{L}^{\mu}\otimes\mathrm{Id})^{-1}(\eta N(e_{i+k}-e_{i}))]\leq
Ck^{s+\kappa\varepsilon},$

which yields (5.14) by (5.17). The proof of (5.15) is similar. ∎

Let us now control the perturbation $\widetilde{H}$ appearing in (5.6).

######  Lemma 5.2.

Let $\mathcal{A}$ stand for the good event (5.1). Uniformly in
$x\in\mathcal{A}$, $1\leq i\leq j\leq n$ and $N$, we have

$\mathrm{Var}_{\mathbb{Q}_{N,\beta}(\cdot\mid
x)}[\partial_{i}\mathcal{H}_{n,N}^{\mathrm{g}},\partial_{j}\mathcal{H}_{n,N}^{\mathrm{g}}]\leq\frac{C(\beta)n^{\kappa\varepsilon}}{d(i,\partial
I)^{\frac{s}{2}}d(j,\partial I)^{\frac{s}{2}}}.$ (5.21)

Besides if $\mu$ satisfies Assumptions 4.1, then uniformly in $1\leq i,j\leq
n$,

$\mathbb{E}_{\mu}[\phi e_{i}\cdot\widetilde{H}(\eta
e_{j})]^{\frac{1}{2}}\leq\frac{C(\beta)n^{\kappa\varepsilon}}{d(i,\partial
I)^{\frac{s}{2}}d(i,\partial
I)^{\frac{s}{2}}}\mathbb{E}_{\mu}[\eta^{2}]^{\frac{1}{2}}.$

In addition, for all $\phi,\eta\in L^{2}(\mu)$ and uniformly $1\leq i,l,j\leq
n$,

$\mathbb{E}_{\mu}[(\phi
e_{i})\cdot\widetilde{H}(\eta(e_{j}-e_{l}))]^{\frac{1}{2}}\leq\frac{C(\beta)n^{\kappa\varepsilon}}{d(j,\partial
I)^{\frac{s}{2}}}\frac{d(i,l)}{(d(i,\partial I)\wedge d(l,\partial
I))^{1+\frac{s}{2}}}\mathbb{E}_{\mu}[\phi^{2}]^{\frac{1}{2}}\mathbb{E}_{\mu}[\eta^{2}]^{\frac{1}{2}}.$

The proof of Lemma 5.2 immediately follows from a rigidity estimate under
$\mathbb{Q}_{N,\beta}(\cdot\mid x)$ that will be proved in Lemma 6.2 in the
Appendix. Note that (5.21) is far from optimal. One can actually prove that
there exists $\alpha>0$ such that

$\mathrm{Var}_{\mathbb{Q}_{N,\beta}(\cdot\mid
x)}[\partial_{i}\mathcal{H}_{n,N}^{\mathrm{g}},\partial_{j}\mathcal{H}_{n,N}^{\mathrm{g}}]=e_{i}\cdot
B(D+\beta^{-1}\mathcal{L}_{\mu}\otimes\mathrm{Id})^{-1}(Ce_{j})\\\
+n^{\kappa\varepsilon}O_{\beta}\Bigr{(}\frac{1}{1+d(i,\partial
I)^{\frac{s+\alpha}{2}}}\frac{1}{1+d(j,\partial
I)^{\frac{s+\alpha}{2}}}\Bigr{)},$

where $B$, $C$ and $D$ are as in (5.3).

###  5.3. Elliptic regularity estimate

The stake for us is to obtain a decay estimate on the solution of (5.7). We
first derive an elliptic regularity estimate and give an $L^{2}$ bound on the
discrete primitive of order $\frac{3}{2}-s$ of $\psi$ in terms of
$|\mathrm{L}_{1/2}\psi|$. We then state a straightforword control on the
$L^{1}$ norm on the discrete primitive of order $1-s$ of $\psi$ with respect
to $|\mathrm{L}_{3/2-s}\psi|$. By interpolation, this yields via a discrete
(1D) Gagliardo-Nirenberg inequality a control on the
$L^{\frac{1}{1-\frac{s}{2}}}$ norm of the fractional primitive of order
$1-\frac{s}{2}$ of $\psi$. Throughout the section, for all $\alpha>0$, we will
denote $\mathrm{L}_{\alpha}$ the distortion matrix