# Law of large numbers for the maximum of the two-dimensional Coulomb gas
potential

Gaultier Lambert  KTH Royal Institute of technology, Matematik,
Lindstedtsvägen 25, 11428 Stockholm<EMAIL_ADDRESS>,  Thomas Leblé
Université de Paris-Cité, CNRS, MAP5 UMR 8145, F-75006 Paris, France.
<EMAIL_ADDRESS>and  Ofer Zeitouni  Department of Mathematics,
Weizmann Institute of Science, Rehovot 76100, Israel.
<EMAIL_ADDRESS>
###### Abstract.

We derive the leading order asymptotics of the logarithmic potential of a two
dimensional Coulomb gas at arbitrary positive temperature. The proof is based
on precise evaluation of exponential moments, and the theory of Gaussian
multiplicative chaos.

G.L. acknowledges the supports of the Ambizione grant S-71114-05-01 from the
Swiss National Science Foundation and of the starting grant 2022-04882 from
the Swedish Research Council.

T.L. acknowledges the support of JCJC grant ANR-21-CE40-0009 from Agence
Nationale de la Recherche.

The third author was partially supported by Israel Science Foundation grant
number 421/20.

##  1\. Introduction

###  1.1. Setting and main result

We are interested in proving a law of large numbers for the maximal value of
the random electrostatic (or logarithmic) potential generated by the particles
of a two-dimensional Coulomb gas - sometimes also called a 2d log-gas, or
“two-dimensional, one-component plasma” (2DOCP). We start by recalling the
definition of the 2DOCP.

#### The 2DOCP

Let $N\geq 1$ and let $\mathrm{X}_{N}:=(x_{1},\dots,x_{N})$ be a $N$-tuple of
(distinct) points in $\mathbb{R}^{2}$. Let
$\mathbf{X}_{N}:=\sum_{i=1}^{N}\delta_{x_{i}}$ be the associated purely atomic
measure of total mass $N$ on $\mathbb{R}^{2}$. Let $\mathsf{D}(z,r)$ denote
the (closed) disk of center $z$ and radius $r$ in $\mathbb{R}^{2}$ and set
$\mathsf{D}_{r}=\mathsf{D}(0,r)$ and $\mathsf{D}=\mathsf{D}_{1}$. Let
$\mathbf{m}_{0}$ be the probability measure with uniform density, denoted
$\mathrm{m}_{0}$, on the unit disk $\mathsf{D}$.

We define the logarithmic interaction energy
$\mathsf{F}(\mathrm{X}_{N},\mathbf{m}_{0})$ as:

(1.1)
$\mathsf{F}(\mathrm{X}_{N},\mathbf{m}_{0}):=\frac{1}{2}\iint_{\left(\mathbb{R}^{2}\times\mathbb{R}^{2}\right)\setminus\triangle}-\log|x-y|(\mathrm{d}\mathbf{X}_{N}-N\mathrm{d}\mathbf{m}_{0})(x)(\mathrm{d}\mathbf{X}_{N}-N\mathrm{d}\mathbf{m}_{0})(y),$

where $\triangle$ denotes the diagonal in
$\mathbb{R}^{2}\times\mathbb{R}^{2}$. Recalling that $-\log$ is (up to a
multiplicative constant) the Coulomb kernel in $\mathbb{R}^{2}$, we can think
of $\mathsf{F}(\mathrm{X}_{N},\mathbf{m}_{0})$ as being the electrostatic
interaction energy of a neutral system made of $N$ point charges placed at
$(x_{1},\dots,x_{N})$ together with a continuous background $N\mathbf{m}_{0}$
of opposite charges.

We also introduce an auxiliary function $\zeta$, which vanishes on
$\mathsf{D}$ and is set to:

(1.2) $\zeta(x):=-\log|x|+\frac{1}{2}|x|^{2}-\frac{1}{2}\text{ on
$\mathbb{R}^{2}\setminus\mathsf{D}$.}$

Let $\beta>0$ be fixed. Throughout this paper, we will work with the
probability measure $\mathbb{P}^{\beta}_{N}$ on
$\left(\mathbb{R}^{2}\right)^{N}$ whose density is given by:

(1.3)
$\mathrm{d}\mathbb{P}^{\beta}_{N}(\mathrm{X}_{N}):=\frac{1}{\mathrm{K}_{N}^{\beta}}\exp\left(-\beta\left(\mathsf{F}(\mathrm{X}_{N},\mathbf{m}_{0})+2N\sum_{i=1}^{N}\zeta(x_{i})\right)\right)\mathrm{d}\mathrm{X}_{N},$

where $\mathrm{d}\mathrm{X}_{N}:=\mathrm{d}x_{1}\dots\mathrm{d}x_{N}$ is the
Lebesgue measure on $(\mathbb{R}^{2})^{N}$ and $\mathrm{K}_{N}^{\beta}$ is the
normalizing constant, or _partition function_ , namely the following integral:

(1.4)
$\mathrm{K}_{N}^{\beta}:=\int_{\left(\mathbb{R}^{2}\right)^{N}}\exp\left(-\beta\left(\mathsf{F}(\mathrm{X}_{N},\mathbf{m}_{0})+2N\sum_{i=1}^{N}\zeta(x_{i})\right)\right)\mathrm{d}\mathrm{X}_{N}.$

The measure $\mathbb{P}^{\beta}_{N}$ is called the canonical Gibbs measure of
a 2DOCP at inverse temperature $\beta$.

######  Remark 1.1.

Some authors define the two-dimensional log-gas in a seemingly different way.
To a $N$-tuple $\mathrm{X}_{N}$, one first associates an energy
$\mathsf{H}_{N}(\mathrm{X}_{N})$ given by the sum of all the pairwise
logarithmic interactions between points plus the effect of a strong harmonic
(“confining”) potential on each particle, namely:

(1.5) $\mathsf{H}_{N}(\mathrm{X}_{N}):=\frac{1}{2}\sum_{1\leq i\neq j\leq
N}-\log|x_{i}-x_{j}|+N\sum_{i=1}^{N}|x_{i}|^{2},$

and then one defines the canonical Gibbs measure of a 2DOCP as:

(1.6)
$\mathrm{d}\mathbb{P}^{\beta}_{N}(\mathrm{X}_{N}):=\frac{1}{\mathrm{Z}_{N,\beta}}\exp\left(-\beta\left(\mathsf{H}_{N}(\mathrm{X}_{N})\right)\right)\mathrm{d}\mathrm{X}_{N},\quad\mathrm{Z}_{N,\beta}:=\int_{\left(\mathbb{R}^{2}\right)^{N}}\exp\left(-\beta\left(\mathsf{H}_{N}(\mathrm{X}_{N})\right)\right)\mathrm{d}\mathrm{X}_{N}.$

In fact, both (1.3) and (1.6) give rise to _the same probability measure_ on
$(\mathbb{R}^{2})^{N}$, and the expression (1.3) can be obtained from (1.6) by
introducing the so-called equilibrium measure $\mathbf{m}_{0}$ and “splitting”
the interaction energy as in [SS15, Sec. 2]. For convenience, here we prefer
to work with the “next-order” formulation of the interaction energy as in
(1.1).

In the physics literature about the 2DOCP, the Gibbs measure is often written
down as in (1.3) but with an “effective confinement” $\zeta$ set to $+\infty$
outside $\mathsf{D}$ (“perfect confinement”). Our analysis in the present
paper applies to this model as well (the only differences would appear when
studying properties close to the boundary, which is not our purpose).

Henceforth, we fix $N\geq 2$ and an arbitrary value of $\beta>0$. We let
$\mathrm{X}_{N}$ be a random variable in $(\mathbb{R}^{2})^{N}$ distributed
according to the Gibbs measure $\mathbb{P}^{\beta}_{N}$, and let
$\mathbf{X}_{N}$ be the associated random point measure on $\mathbb{R}^{2}$ as
defined above. Unless specified otherwise, expectations in what follows are
taken under $\mathbb{P}^{\beta}_{N}$.

#### Main result

Define the “Coulomb gas potential” generated by
$\mathrm{X}_{N}=(x_{1},\dots,x_{N})$ as the following random scalar field on
$\mathbb{R}^{2}$:

(1.7)
$\mathrm{Pot}_{N}:z\mapsto\sum_{i=1}^{N}\log|z-x_{i}|-N\int_{\mathsf{D}}\log|z-x|\mathrm{d}\mathbf{m}_{0}(x).$

In physical terms, $\mathrm{Pot}_{N}(z)$ corresponds to the value at $z$ of
the electrostatic potential generated by the system of charges and the
background measure $\mathbf{m}_{0}$. Obviously $\mathrm{Pot}_{N}(z)$ is equal
to $-\infty$ whenever $z$ coincides with one of the point charges. Our main
result is the following description of the maximum of $\mathrm{Pot}_{N}$ over
closed disks in the interior of $\mathsf{D}$:

######  Theorem 1 (LLN for the max of the 2D Coulomb gas potential).

For all $r\in(0,1)$ we have:

(1.8) $\frac{1}{\log
N}\max_{z\in\mathsf{D}(0,r)}\mathrm{Pot}_{N}(z)\longrightarrow\frac{1}{\sqrt{\beta}}\text{
as }{N\to\infty},\text{ in probability.}$

The case $\beta=2$ of Theorem 1 corresponds to the Ginibre ensemble of
(complex) random matrices, and was obtained in [Lam20] based on its
determinantal structure.

As a byproduct of our proof, we obtain a control on a certain regularization
of $\mathrm{Pot}_{N}$ at microscopic scale (see Corollary 3.9), which allows
us to state a uniform control on the fluctuations of linear statistics for a
certain class of $C^{2}$ test functions (see Proposition 3.10).

###  1.2. Connections with the literature

Our interest in Theorem 1 is motivated by connections with the theory of
random matrices and the theory of logarithmically correlated fields.

#### The Ginibre ensemble and normal matrix models

For the specific value $\beta=2$ of the inverse temperature in (1.6),
$\mathbb{P}^{\beta}_{N}$ describes the joint distribution of eigenvalues for
Ginibre matrices, i.e. matrices whose entries are i.i.d. complex Gaussians
normalized by $1/\sqrt{N}$ (see [For10] or the recent survey [BF22] for
background). In this case, as proved (using determinantal methods) in [RV07]
following a prediction of Forrester, the counting measure of eigenvalues minus
$N\mathbf{m}_{0}$, converges to the sum of a 2d-Gaussian free field (GFF)
supported on $\mathsf{D}$ and an independent 1d-GFF on the unit circle. There
the Coulomb gas potential $\mathrm{Pot}_{N}$ can be interpreted as the
logarithm of the (absolute value of the) determinant of the associated
(complex) Ginibre matrix, and as noted above the counterpart of Theorem 1 was
proved in [Lam20].

For more general normal matrix models (still with $\beta=2$ but replacing
$|x|^{2}$ in (1.5) by suitable polynomial potentials), a CLT for the log-
characteristic polynomial (with variance $\log N/4$) has been obtained in
[AKS21] together with a description of the local (determinantal) point process
around the singularity. Such a connection with Gaussian fields motivates the
study of the maximum, as we now explain.

#### Logarithmically correlated structure at arbitrary temperature

More generally, for all $\beta>0$, the central limit theorems of [BBNY19] and
[LS18] imply that $\mathrm{Pot}_{N}$ converges in some weak sense to a
Gaussian Free Field (even in mesoscopic scales). Is it thus natural to ask
whether the maximum of $\mathrm{Pot}_{N}$ behaves like the maximum of a 2d
logarithmically correlated field, and Theorem 1 shows that the leading order
is indeed the expected one.

#### One-dimensional analogues.

The density (1.6) restricted to either the real line or to the unit circle
$S^{1}$ in the complex plane gives rise to the classical G$\beta$E and
C$\beta$E ensembles. For these, similar logarithmically correlated fields can
be constructed (see e.g. [DS94] and [HKO01] for early results concerning the
CUE). The extremes for the (logarithm of the) characteristic polynomial of
these matrix models have generated much interest (especially in the case of
the CUE, corresponding to $\beta=2$) due to a celebrated conjecture of
Fyodorov, Hiary and Keating [FHK12] that predicts the limiting form of the
fluctuations of the maximum and links these to analogous fluctuations for the
Riemann zeta-function. This has stimulated much recent work, starting with
[ABB17], [PZ18] (for the CUE), [CMN18] and [PZ22] (for the C$\beta$E); the
last article indeed proves a version of the FHK conjecture. On a related
subject, it has been established that powers of the CUE characteristic
polynomial converges to Gaussian multiplicative chaos (GMC) measures in the
so-called $L^{1}$ phase [NSW20] and that these measures describe the
fluctuations of thick points (extreme level sets) of the characteristic
polynomial [JLW22]. Weaker analogous results for the case of GUE are contained
in [LP19] and [CFLW21], see also [BMP22] and [ABZ23] for the G$\beta$E.

###  1.3. Comments on the strategy.

A general methodology to handle extremes of Gaussian logarithmically
correlated fields has been developed in the last decades; due to space
limitations, we do not review here the history, and refer instead to [DRZ17]
and [Bis20] for details. When applied outside the Gaussian context, this
methodology requires the evaluation of exponential moments and characteristic
functions of the field, together with the introduction of certain barriers.
These seem crucial in obtaining sharp results (at the level of $O(1)$
fluctuations for the extremes), and are often hard to implement outside the
Gaussian setup. The above works concerning the extremes of the electrostatic
potential of Coulomb gases on the real line or $S^{1}$ use, at different
levels of precision, variants of these methods, with much technical work going
into inserting appropriate barriers and controlling comparisons with the
Gaussian setup.

For our needs, a crucial observation was made in [LOS18] and [CFLW21], by
using the theory of Gaussian multiplicative chaos (GMC): in order to obtain
the leading order of fluctuations, one can bypass the use of barriers, at the
cost of obtaining sharp estimates on exponential moments (without need to
consider characteristic functions). This is the approach taken in this paper.

The evaluation of exponential moments in the context of Coulomb gases is at
the heart of many proofs of the central limit theorem for linear statistics.
An early application of this method is due to Johansson [Joh98] in the context
of one-dimensional log gases, and for the two-dimensional case this was done
in [LS18, BBNY19, Ser20]. Here we crucially use some key results of the latter
paper, together with the general “energy” approach to 2DOCP’s as developed by
Serfaty and co-authors, starting with [SS15] \- in particular we will rely on
the more recent [AS21]. As in [LS18, Ser20] we follow a fairly general
approach based on transportation of measures (and not on “loop equations” as
e.g. in [BBNY19], although the two methods are related), a technique that has
proven to be fruitful in the one-dimensional case - see for example [Shc14,
BFG15, BLS18]. Our method relies on computing the asymptotics of (joint)
exponential moments of $\mathrm{Pot}_{N}$, the potential with the 2d Coulomb
gas and the background measure $\mathbf{m}_{0}$. These asymptotics are also
related to the problem of determining the statistics of types of quantum
quasi-particles arising in fractional Hall experiments by considering a
certain modification of Laughlin function as a trial state. This connection is
explained in further details in [LLR22, Section 1.3], we just emphasize that
the estimates required for that problem go beyond the precision achieved here.

###  1.4. Open problems

Theorem 1 only gives the _leading order_ of the maximum of $\mathrm{Pot}_{N}$.
In order to actually see the influence of the underlying logarithmically
correlated structure, one needs to evaluate (at least) the next order
correction, which is expected to be $\frac{3}{4\sqrt{\beta}}\log\log
N(1+o(1))$ due to the log-correlated structure. The techniques for achieving
that go beyond our methods.

In a different direction, it is natural to consider replacing $|x|^{2}$ in
(1.5) by other growing (real) functions $f$. Applying our methods to that case
requires three ingredients: first, one needs regularity of the background
density $\mathrm{m}_{0}$, and to modify the electrostatic potential
$\mathfrak{h}_{0}$, see (A.15) accordingly. Second, one would need to modify
the background function $\mathrm{g}$ of (3.2), which is easy to do in the
radial case. And third, one should look for a replacement for Claim A.10,
whose proof is based on a scaling argument. In the case of monomials
$f(x)=|x|^{q}$, it is simple to carry out the adaptations, but already in the
case of a general (even) polynomial $f(|x|)$ one needs to find a replacement
for the scaling argument.

A particular case of interest, concerns the real Ginibre ensemble, see [FN07].
There, one needs to deal with the symmetry of the point configuration
$\mathrm{X}_{N}$, as well as with the special role of the real axis. This
would require significant changes in our derivation, and we leave this as an
open problem.

Finally, we expect the result of Theorem 1 to remain true if one considers the
maximum of $\mathrm{Pot}_{N}$ over the whole unit disk (or even the whole
plane). We also expect that Proposition 4.2 holds without any regularization;
see Remark 4.3.

###  1.5. Sketch of the proof and plan of the paper

In the evaluation of exponential moments, a crucial role is played by a
version of the logarithmic interaction energy (1.1) defined locally at scale
$\ell\geq N^{-1/2}$, denoted $\mathsf{EnerPts}$, see (2.5), for which a-priori
exponential moment bounds are available from [AS21], see (2.7).

Exponential moments of linear statistics for smooth enough test functions of a
single (possibly mesoscopic or microscopic) scale can be controlled either by
purely energy-based considerations (see Lemma 2.3) or in a more precise
fashion in terms of the $H^{1}$ norm of the test function and the ratio of two
partition functions corresponding to the original background measure
$\mathbf{m}_{0}$ and to a perturbed version $\mathbf{m}_{s}$. This ratio can
itself be studied using the transportation technique, see Lemmas 2.4 and 2.5.

Since the linear statistics we care about is generated by the logarithm, it is
not based on a single scale. We thus need to be able to consider test
functions living on different scales, and we develop the required tool in
Proposition 2.8. All these preliminary steps are carried out in Section 2,
with proofs (building on [Ser20]) postponed to the appendices.

The proof of Theorem 1 is split into an upper bound (provided in Section 3)
and a lower bound (provided in Section 4). Concerning the former, since the
logarithm is not smooth and not compactly supported111Technically speaking,
the real issue is not the lack of compact support, but rather the fact that
the total mass of its Laplacian is not $0$., to apply the estimates discussed
in the preliminaries we need to regularize it and center222Namely, we
substract some well-chosen test function so that the Laplacian of the
difference has total mass 0. it; the centering is constructed in Section 3.1,
and Proposition A.7 gives an a-priori control of the centering term. The
exponential moments of the regularized (at scale $\ell$) test function has
exponential moments that can be evaluated using the preliminary results of
Section 2. Those estimates are enough (using Chebyshev’s inequality and a
union bound) to yield the claimed upper bound on a lattice of stepsize of
order $N^{-1/2-\delta}$ with $\delta$ small, see Lemma 3.7. To extend the
result from a lattice to the whole of $\mathsf{D}_{r}$ requires a multi-scale
estimation of the difference between the logarithm centered at different
points, and is presented in Proposition 3.8.

The proof of the lower bound is provided in Section 4. As mentioned above, it
follows the recipe of [CFLW21], and is based on the construction of a sequence
of measures obtained from exponentiating a regularized version of the linear
statistic constructed from the (shifted) logarithm function, see (4.3). The
heart of the proof is to show that this sequence of measures converges to a
Gaussian Multiplicative Chaos (GMC), which is the limit of similar measures
constructed from a Gaussian process, see Proposition 4.5. The general recipe
from [CFLW21] establishes that these measures converge if one obtains
asymptotics of exponential moments of linear combinations of the regularized
logarithm, centered at different points, see (4.6). The proof of the latter is
done by induction, using in a crucial way the two-scale asymptotics of
Proposition 2.8.

###  1.6. Notation

  * •

If $\Omega$ is a measurable subset, $\mathbf{X}_{N}(\Omega)$ denotes the
number of points of $\mathrm{X}_{N}$ contained in $\Omega$.

  * •

We denote measures with a bold typeface (e.g. $\mathbf{m}$) and their
densities with respect to the Lebesgue measure on $\mathbb{R}^{2}$ with a
roman typeface (e.g. $\mathrm{m}$).

  * •

If $\varphi$ is a function, whenever applicable we denote by
$|\varphi|_{\mathsf{0}}$ a uniform bound on $\varphi$, and by $|\varphi|_{k}$
a uniform bound on its $k$th derivative(s).

  * •

Integrals with respect to the Lebesgue measure are often written without
explicitly mentioning the volume form, ie
$\int\varphi=\int\varphi(x)\mathrm{d}x$.

  * •

If $A,B$ are two quantities (depending on various parameters) we write
$A\preceq B$ (or $A=O(B)$) when $|A|$ is bounded by some universal constant
times $|B|$. We write $A=o(B)$ or $A\ll B$ if $A/B\to 0$ when a (sometimes
implicit) parameter goes to infinity. When no parameter is mentioned, it is
understood that the parameter is $N$. We write $A\asymp B$ when $A=O(B)$ and
$B=O(A)$.

  * •

When $\mathbf{m}$ is a probability measure on $\mathbb{R}^{2}$ with continuous
density $\mathrm{m}$, we denote its relative entropy (with respect to
Lebesgue) by
$\mathcal{E}(\mathrm{m}):=\int_{\mathbb{R}^{2}}\mathrm{m}\log\mathrm{m}$.

### Acknowledgment

_We thank Sylvia Serfaty for communicating early versions of[Ser20] to us and
making some statements thereof more easily citable for our purposes._

##  2\. Preliminaries

###  2.1. Energy at global and local scales

Let $\mathbf{m}$ be a probability measure on $\mathsf{D}$ with a density that
is continuous and bounded below by a positive constant on the interior of
$\mathsf{D}$.

#### Global energy

If $\mathrm{X}_{N}$ is a $N$-tuple of points and $\mathbf{X}_{N}$ is the
associated atomic measure of mass $N$, we extend the definition (1.1) and
define the “global energy” $\mathsf{F}(\mathrm{X}_{N},\mathbf{m})$ as:

(2.1)
$\mathsf{F}(\mathrm{X}_{N},\mathbf{m}):=\frac{1}{2}\iint_{\left(\mathbb{R}^{2}\times\mathbb{R}^{2}\right)\setminus\triangle}-\log|x-y|(\mathrm{d}\mathbf{X}_{N}-N\mathrm{d}\mathbf{m})(x)(\mathrm{d}\mathbf{X}_{N}-N\mathrm{d}\mathbf{m})(y).$

We introduce the associated Gibbs measure (where333The function $\zeta$ plays
almost no role in our analysis, as we are focused on the bulk of the system.
For simplicity we keep “the same $\zeta$” in all cases. $\zeta$ is as in
(1.2))

(2.2)
$\mathrm{d}\mathbb{P}^{\beta}_{N,\mathbf{m}}(\mathrm{X}_{N}):=\frac{1}{\mathrm{K}_{N}^{\beta}(\mathbf{m})}\exp\left(-\beta\left(\mathsf{F}(\mathrm{X}_{N},\mathbf{m})+2N\sum_{i=1}^{N}\zeta(x_{i})\right)\right)\mathrm{d}\mathrm{X}_{N},$

with the corresponding partition function:

(2.3)
$\mathrm{K}_{N}^{\beta}(\mathbf{m}):=\int_{(\mathbb{R}^{2})^{N}}\exp\left(-\beta\left(\mathsf{F}(\mathrm{X}_{N},\mathbf{m})+2N\sum_{i=1}^{N}\zeta(x_{i})\right)\right)\mathrm{d}\mathrm{X}_{N}.$

It is known (see e.g. the pioneering analysis of [SS15, Thm. 1]) that for all
$\beta>0$:

(2.4) $\log\mathrm{K}_{N}^{\beta}(\mathbf{m})=\frac{\beta}{4}N\log
N+O_{\beta}(N).$

One should thus think of the energy $\mathsf{F}(\mathrm{X}_{N},\mathbf{m})$ as
being equal to $-\frac{1}{4}N\log N$ plus a random term which is typically of
order $N$.

#### Length scales and local laws

For many interesting questions it is crucial to understand the system at
_local_ scales i.e. in squares (or disks) of size $\ell\ll 1$. One
distinguishes between _mesoscopic_ scales $\ell$ such that $N^{-1/2}\ll\ell\ll
1$ and the _microscopic_ scale $\ell\simeq N^{-1/2}$. A constant
$\rho_{\beta}\geq 1$ depending only on $\beta$ was introduced in [AS21,
(1.15)], it corresponds to the “minimal lengthscale” above which good controls
on the energy can be obtained. In this paper when considering a length scale
$\ell$ we will always assume that444Since [AS21] work with a different scaling
than us, we need to rescale their $\rho_{\beta}$ by $N^{-1/2}$.
$\ell\geq\widetilde{\rho}_{\beta}:=\rho_{\beta}N^{-1/2}$ (note that since
$\rho_{\beta}\geq 1$, we always have $N\ell^{2}\geq 1$).

The proper notion of a _local energy_ , however, is a bit subtle due to the
long-range nature of the logarithmic potential. Suitable definitions were
given in [Leb17, AS21], unfortunately they are rather involved and for
simplicity we will use them as a black box. For $z\in\mathsf{D}$ and any
length scale $\ell$, we denote by
$\mathrm{F}^{(z,\ell)}(\mathrm{X}_{N},\mathbf{m})$ the “local energy in the
disk $\mathsf{D}(z,\ell)$” as defined in [AS21, (2.24)] (they mostly consider
small squares rather than small disks but the distinction is irrelevant). In
this paper, we denote by $\mathsf{EnerPts}(z,\ell)$ the quantity

(2.5)
$\mathsf{EnerPts}(z,\ell)(\mathrm{X}_{N}):=\left(\mathrm{F}^{(z,\ell)}(\mathrm{X}_{N},\mathbf{m})+\frac{1}{4}n\log
N\right)+n,\quad n=\mathbf{X}_{N}\left(\mathsf{D}(z,\ell)\right).$

Due to a weaker understanding of the system near the boundary
$\partial\mathsf{D}$, one needs to introduce the following condition on
$(z,\ell)$ (where $z$ is a point of $\mathsf{D}$ and $\ell$ is a length
scale):

(2.6)
$\mathrm{dist}\left(\mathsf{D}(z,\ell),\partial\mathsf{D}\right)\geq\mathtt{C}_{\beta}N^{-1/4},$

where $\mathtt{C}_{\beta}$ is some constant depending only on $\beta$
introduced in the assumptions of [AS21, Thm.1]. Then [AS21] prove the
following “local laws”:

######  Lemma 2.1 (Local laws for $\mathsf{EnerPts}$).

If $(z,\ell)$ satisfies (2.6) and the parameter $t$ is smaller than some
constant depending only on $\beta$, then:

(2.7)
$\log\mathbb{E}_{\mathbb{P}^{\beta}_{N,\mathbf{m}}}\left[\exp\left(t\mathsf{EnerPts}(z,\ell)\right)\right]=O\left(tN\ell^{2}\right),$

with an implicit constant depending only on $\beta$.

Thus the local laws of [AS21] ensure that $\mathrm{F}^{(z,\ell)}$ is equal to
$-\frac{1}{4}n\log N$, where $n$ is the number of points in
$\mathsf{D}(z,\ell)$, plus a (fluctuation) term of order $N\ell^{2}$ in
exponential moments, and so _down to the microscopic scale_ $\ell\simeq
N^{-1/2}$ (a crucial improvement over [Leb17]). The number of points $n$ is
also of order $N\ell^{2}$.

######  Remark 2.2.

To be specific, the statement of [AS21, Theorem 1] treats the electric energy
and the number of points separately. They work with a different scaling so
their length scale $R$ corresponds to $N^{1/2}\ell$ for us.
Their statement concerning the electric energy ([AS21, (1.17)]) involves the
quantity $\mathsf{F}^{\square_{R}(x)}$ (their notation) which is short for
$\mathsf{F}^{\square_{R}(x)}(\mathrm{X}_{N},U)$ with $U=\mathbb{R}^{2}$ as
defined in [AS21, (2.24)]. The potential $u$ appearing there is defined in
[AS21, Sec. 2.3] but as used in the main statement of [AS21] it coincides with
the true potential (because, with their notation, $U=\mathbb{R}^{2}$ in this
case). It remains to observe that Theorem 1 in [AS21] states a control on
$\mathsf{F}^{\square_{R}(x)}$ which can be turned into a control on the
electric energy only, this is precisely the purpose of [AS21, Lemma B.2] and
in particular their equation (B.8), which shows that one can indeed control
the electric energy in terms of $\mathsf{F}^{\square_{R}(x)}$. As a last
technical comment for the careful reader: note that since here
$U=\mathbb{R}^{2}$ (in their notation) the various truncations
$\mathsf{r},\tilde{\mathsf{r}},\tilde{\tilde{\mathsf{r}}}$ all coincide.
In [AS21, Theorem 1] there are two statements concerning the number of points,
here we can use [AS21, (1.18)] which controls exponential moments of the
_discrepancy_ (which corresponds to
$\mathbf{X}_{N}\left(\mathsf{D}(z,\ell)\right)-\pi{N}\ell^{2}$). It is not
hard to see that it implies (and is in fact a lot stronger than) a bound on
the number of points of the form; under (2.6) and for $t$ smaller than a
constant depending only on $\beta$,

(2.8)
$\log\mathbb{E}_{\mathbb{P}^{\beta}_{N,\mathbf{m}}}\left[\exp\left(t\mathbf{X}_{N}({D(z,\ell))}\right)\right]=O\left(tN\ell^{2}\right).$

###  2.2. Fluctuations

When $\varphi$ is a $\mathbf{m}$-integrable test function, we define “the
fluctuation $\mathbf{L}_{N}^{\mathbf{m}}(\varphi)$ of the linear statistics
associated to $\varphi$ for the configuration $\mathrm{X}_{N}$”, as:

(2.9)
$\mathbf{L}_{N}^{\mathbf{m}}(\varphi):=\sum_{i=1}^{N}\varphi(x_{i})-N\int_{\mathsf{D}}\varphi(x)\mathrm{d}\mathbf{m}(x).$

We write $\mathbf{L}_{N}(\varphi)$ for
$\mathbf{L}_{N}^{\mathbf{m}_{0}}(\varphi)$. If $\varphi$ is Lipschitz, one can
always use the following possibly non-optimal but _uniform_ control on
$\mathbf{L}_{N}(\varphi)$:

######  Lemma 2.3.

Let $z$ be a point of $\mathsf{D}$ and $\ell$ be a length scale such that
$(z,\ell)$ satisfies (2.6). Denote by $\mathcal{L}_{z,\ell}$ the set of all
functions that are $\frac{1}{\ell}$-Lipschitz and compactly supported on
$\mathsf{D}(z,\ell)$. Then for all $t$ such that $|t|$ is smaller than some
constant depending on $\beta$, we have:

(2.10)
$\log\mathbb{E}\left[\sup_{\varphi\in\mathcal{L}_{z,\ell}}\exp\left(t\mathbf{L}_{N}(\varphi)\right)\right]=O\left(t\left(N\ell^{2}\right)^{\frac{1}{2}}\right),$

with an implicit constant depending only on $\beta$. A similar estimate holds
for $\mathcal{L}$, the set of all 1-Lipschitz function with compact support in
$\mathsf{D}(0,2)$ with $\ell=1$.

###### Proof.

This can be traced back to the analysis of [SS15, Lemma 3.9], see also [RS16,
Theorem 5]. The basic idea is that we have a configuration-wise bound of the
form:

$|\mathbf{L}_{N}(\varphi)|\leq|\varphi|_{\mathsf{1}}\times\ell\times\mathsf{EnerPts}(z,\ell)^{\frac{1}{2}}\leq|\varphi|_{\mathsf{1}}\times\ell\times\left({\frac{\mathsf{EnerPts}(z,\ell)}{(N\ell^{2})^{\frac{1}{2}}}+(N\ell^{2})^{\frac{1}{2}}}\right),$

(see e.g. [AS21, Lemma B.5] for a precise statement), and the result follows
from an application of the “local laws” as presented in the previous section,
see e.g. (2.7), and our convention that $\ell\geq N^{-1/2}$. ∎

If $\varphi$ is assumed to have more regularity, the results of [LS18, BBNY19,
Ser20] give a much better estimate on the exponential moments of
$\mathbf{L}_{N}(\varphi)$, but they are only stated _function-wise_. Our
method, which relies on those results, yields a _uniform_ control for the
fluctuations of a certain class of $C^{2}$ linear statistics; see Corollary
3.9 and Proposition 3.10 below.

###  2.3. Re-writing the Laplace transform

######  Lemma 2.4 (Laplace transforms as ratio of partition functions).

Let $\varphi$ be a $C^{2}$ function whose Laplacian is supported on
$\mathsf{D}$ and satisfies $\int\Delta\varphi=0$. Let $t,s$ be such that:

(2.11) $|t|\leq\beta N|\varphi|^{-1}_{\mathsf{2}},\quad s:=\frac{-t}{2\pi
N\beta}.$

Let $\mathbf{m}_{s}$ be the probability measure on $\mathsf{D}$ with density
$\mathrm{m}_{s}:=\mathrm{m}_{0}+s\Delta\varphi$. The following identity holds:

$\mathbb{E}\left[e^{t\mathbf{L}_{N}(\varphi)}\right]=\exp\left(\frac{t^{2}}{4\pi\beta}\int_{\mathbb{R}^{2}}-\varphi\Delta\varphi\right)\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0})}.$

Of course, if $\nabla\varphi$ is compactly supported then one can integrate by
parts and write the “variance” term as
$\int_{\mathbb{R}^{2}}|\nabla\varphi|^{2}$.

###### Proof.

This follows from elementary manipulations that can be found e.g. in [LS18,
Section 2.6]. It is, however, a much older idea see [Joh98] or [AHM11, Sec
7.2] and the references therein. ∎

###  2.4. Comparison of partition functions

######  Proposition 2.5 (Main comparison result).

Let $\ell$ be a lengthscale and let $z\in\mathsf{D}$ such that $(z,{\ell})$
satisfies (2.6). Let $\mathtt{C}$ be some positive constant. Let $\mathbf{m}$
be a probability measure on $\mathsf{D}$, whose density $\mathrm{m}$ is of
class $C^{3}$ and satisfies:

(2.12) $\frac{1}{2}\leq\mathrm{m}\leq
2,\quad|\mathrm{m}|_{\mathsf{k}}\leq\mathtt{C}\ell^{-k}\text{ for
}\mathsf{k}=1,2,3.$

Moreover, let $f$ be a function of class $C^{2}$ supported in
$\mathsf{D}(z,\ell)$, such that:

(2.13)
$\int_{\mathbb{R}^{2}}f=0,\quad|f|_{\mathsf{k}}\leq\mathtt{C}\ell^{-(\mathsf{k}+2)}\text{
for }\mathsf{k}=0,1,2.$

Then there exists a constant $\mathtt{C}^{\prime}$ depending only on
$\mathtt{C}$ and $\beta$ such that the following holds. For all
$s\in\mathbb{R}$ such that:

(2.14) $|s|\leq\frac{1}{\mathtt{C}^{\prime}}\ell^{3/2}N^{-1/4},$

let $\mathbf{m}_{s}$ be the probability measure with density
$\mathrm{m}_{s}:=\mathrm{m}+sf$. We have:

(2.15)
$\log\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s})}{\mathrm{K}_{N}^{\beta}(\mathbf{m})}=N\left(\frac{\beta}{4}-1\right)\left(\mathcal{E}(\mathrm{m}_{s})-\mathcal{E}(\mathrm{m})\right)+sO\left(N^{3/4}\ell^{-1/2}\left(1+\log\left(N\ell^{2}\right)\right)^{\frac{1}{2}}\right),$

with an implicit multiplicative constant depending only on $\mathtt{C}$ and
$\beta$.

We postpone the proof of Proposition 2.5 to Section A.2.

######  Remark 2.6.

In the statement of Proposition 2.5 we assume that the parameter $s$ is
smaller than $O\left(\ell^{3/2}N^{-1/4}\right)$, but we eventually apply this
result with $s$ of order $N^{-1}$ as in (2.11), which is always a valid choice
for $\ell\geq\rho_{\beta}N^{-1/2}$. On the other hand, the well-definiteness
of $\mathrm{m}_{s}$ as $\mathrm{m}+sf$ only requires $s$ to be $O(\ell^{-2})$.
However, for values of $s$ between $\ell^{3/2}N^{-1/4}$ and $\ell^{2}$ we do
not get an interesting estimate.

In Proposition 2.5, we assumed that $(z,{\ell})$ satisfies (2.6) to avoid the
case where the perturbation is near the boundary. The following auxiliary
lemma treats the latter case for a “macroscopic perturbation” ($\ell=1$),
which will be enough for our purposes.

######  Lemma 2.7 (Main comparison - macroscopic case).

Let $f=2\pi\left(\chi-\mathrm{m}_{0}\right)$, where $\mathrm{m}_{0}$ is the
uniform density on $\mathsf{D}$ and $\chi$ is a smooth, radially symmetric
function which is compactly supported in $\mathsf{D}_{r}$ for some $r<1$, such
that $\int\chi=1$ and $|\chi|_{\mathsf{k}}\leq\mathtt{C}$ for
$\mathsf{k}=0,1,2$. Then there exists a constant $\mathtt{C}^{\prime}$
depending only on $\mathtt{C},\beta$ and $r$ such that the following holds:
for all $s\in\mathbb{R}$ with $|s|\leq\frac{1}{\mathtt{C}^{\prime}}$, let
$\mathbf{m}_{s}^{\prime}$ be the probability measure with density
$\mathrm{m}^{\prime}_{s}:=\mathrm{m}_{0}+sf$. We have:

(2.16)
$\log\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}^{\prime}_{s})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0})}=sO(N),$

with an implicit multiplicative constant depending on $\mathtt{C},\beta$ and
$r$.

Lemma 2.7 is proven along the same lines as Proposition 2.5, the radial
symmetry of both $f$ and $\mathrm{m}_{0}$ provides a simplification which
allows us to efficiently treat the boundary case (note also that we are aiming
at a less precise estimate, compare (2.16) with (2.15)). We give the proof in
Section A.2.

The next result builds upon Proposition 2.5 and treats a situation where the
perturbation $f$ is made of two pieces living at different scales.

######  Proposition 2.8 (Comparison with mass transfer between scales).

Let $\ell_{a}<\ell_{b}$ be two length scales and let
$z_{a},z_{b}\in\mathsf{D}$ be two points such that both $(z_{a},\ell_{a})$ and
$(z_{b},\ell_{b})$ satisfy (2.6). Let $\mathtt{C}$ be some positive constant
and let $\mathbf{m}$ be a probability measure on $\mathsf{D}$ with a density
$\mathrm{m}$ is of class $C^{3}$ such that:

(2.17)
$\frac{3}{4}\leq\mathrm{m}\leq\frac{3}{2},\quad|\mathrm{m}|_{\mathsf{k}}\leq\mathtt{C}\ell_{b}^{-\mathsf{k}}\text{
for }\mathsf{k}=1,2,3.$

Moreover, let $f_{a},f_{b}$ be two functions of class $C^{2}$ supported on
$\mathsf{D}(z_{a},\ell_{a})$ (resp. $\mathsf{D}(z_{b},\ell_{b})$), such that,
with $f=f_{a}+f_{b}$,:

(2.18)
$\int_{\mathbb{R}^{2}}f=0,\quad|f_{a}|_{\mathsf{k}}\leq\mathtt{C}\ell_{a}^{-(\mathsf{k}+2)},\
|f_{b}|_{\mathsf{k}}\leq\mathtt{C}\ell_{b}^{-(\mathsf{k}+2)}\text{ for
}\mathsf{k}=0,1,2.$

Assume that $\mathsf{D}(z_{a},\ell_{a})$ and $\mathsf{D}(z_{b},\ell_{b})$ are
both contained in $\mathsf{D}_{r}$ for some $r<1$, so that for $N$ large
enough (depending on $r$), condition (2.6) is satisfied. Then there exists a
constant $\mathtt{C}^{\prime}$ depending only on $\mathtt{C}$ and $\beta$ such
that the following holds. For all $s\in\mathbb{R}$ such that:

(2.19) $|s|\leq\frac{1}{\mathtt{C}^{\prime}}\ell_{a}^{3/2}N^{-1/4},$

let $\mathbf{m}_{s}$ be the probability measure with density
$\mathrm{m}_{s}:=\mathrm{m}+sf$. We have:

(2.20)
$\log\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s})}{\mathrm{K}_{N}^{\beta}(\mathbf{m})}=N\left(\frac{\beta}{4}-1\right)\left(\mathcal{E}(\mathrm{m}_{s})-\mathcal{E}(\mathrm{m})\right)+O\left(sN^{3/4}\ell_{a}^{-1/2}\left(1+{\log(N\ell_{a}^{2})}\right)^{\frac{1}{2}}\right),$

with a multiplicative constant depending only on $\mathtt{C},\beta$ and $r$.

We postpone the proof of Proposition 2.8 to Section A.3.

######  Remark 2.9.

It is important to observe that in the conclusions of Proposition 2.5 (resp.
Proposition 2.8), if we work at the _microscopic_ scale
$\ell={\rho_{\beta}}N^{-1/2}$ (resp. $\ell_{a}={\rho_{\beta}}N^{-1/2}$), then
for $s$ of order $N^{-1}$ (which will be our choice later on) the error term
is $O(1)$, whereas as soon as $\ell$ (resp. $\ell_{a}$) is _mesoscopic_ there
is a gain and the error term becomes $o(1)$.

As we record in the next remark, Proposition 2.5 yields the classical CLT for
(arbitrary smooth mesoscopic) test functions supported inside $\mathsf{D}$,
see [LS18] and [BBNY19] for the original results. A CLT-like precision will be
required to show that the exponential of a regularization of
$\mathrm{Pot}_{N}$ converges to a certain GMC measure in Section 4, which in
turn is instrumental in obtaining a lower-bound for the maximum of
$\mathrm{Pot}_{N}$.

######  Remark 2.10.

Let $\varphi\in C^{4}(\mathbb{R}^{2}\to\mathbb{R})$ be a function (possibly
depending on $N$) and assume that $\Delta\varphi=f$, where $f$ is as in
Proposition 2.5 or Proposition 2.8 Then by combining Lemma 2.4 and
Propositions 2.5 or 2.8, we get:

$\mathbb{E}\left[e^{\mathbf{L}_{N}(\varphi)}\right]=\exp\left(\frac{-1}{4\pi\beta}\int_{\mathbb{R}^{2}}\varphi
f\right)\exp\left(N\left(\tfrac{\beta}{4}-1\right)\mathcal{E}(\mathbf{m})+o(1)\right)$

where $\mathbf{m}=\mathbf{m}_{0}-\frac{f}{2\pi N\beta}$. Here we used that
$\mathbf{m}_{0}$ satisfies the conditions (2.12) and (2.17) and that
$\mathcal{E}(\mathbf{m}_{0})=0$. Moreover, since $f$ is supported in
$\mathsf{D}$ with $\int f=0$ and $|f|_{0}\leq\mathtt{C}\ell^{-2}$, we have

$\mathcal{E}(\mathbf{m})=\int_{\mathbb{R}^{2}}\log\big{(}1-\tfrac{f(x)}{2\pi
N\beta}\big{)}\big{(}1-\tfrac{f(x)}{2\pi N\beta}\big{)}\mathrm{d}x=o(N^{-1}).$

In particular, $\mathbf{L}_{N}(\varphi)$ converges in distribution to a
Gaussian random variable with mean 0 and variance
$\frac{1}{2\pi\beta}\int_{\mathbb{R}^{2}}|\nabla\varphi|^{2}$ (this last
expression follows by an integration by parts).

##  3\. Law of large numbers: upper bound

The goal of this section is to prove the upper bound part of Theorem 1,
namely:

######  Proposition 3.1.

Recall the definition (1.7) of $\mathrm{Pot}_{N}$. For all fixed $r\in(0,1)$
and all $\alpha>1$, we have:

(3.1)
$\lim_{N\to\infty}\mathbb{P}^{\beta}_{N}\left(\max_{z\in\mathsf{D}(0,r)}\mathrm{Pot}_{N}(z)\geq\frac{\alpha\log
N}{\sqrt{\beta}}\right)=0.$

In the rest of this section we fix some $r<1$.

###  3.1. An auxiliary linear statistics

For $z\in\mathsf{D}$, let $\log_{z}:x\mapsto\log|z-x|$. The value
$\mathrm{Pot}_{N}(z)$ of the Coulomb gas potential at $z$ corresponds to the
fluctuations $\mathbf{L}_{N}(\log_{z})$ of the linear statistics associated to
$\log_{z}$, see (2.9). An important technical observation is that
$\int\Delta\log_{z}=1$ so that (even after a mesoscopic regularization), one
cannot directly apply Proposition 2.5 to control the exponential moments of
$\mathbf{L}_{N}(\log_{z})$. To fix this issue, we can consider instead the
fluctuation of the test function $\log_{z}-\mathrm{g}$ where $\mathrm{g}$ is
any nice function with $\int\Delta\mathrm{g}=1$. It is particularly convenient
to make the following choice: we take $\chi$ to be a radially symmetric smooth
function (independent of $N$) supported in $\mathsf{D}_{r}$, and we let

(3.2) $\text{$\mathrm{g}$ be a solution of Poisson's equation
$\Delta\mathrm{g}=2\pi\chi$}.$

Then, the following estimate shows that the fluctuations of
$\mathbf{L}_{N}(\mathrm{g})$ are negligible compared to the maximum of
$\mathrm{Pot}_{N}$.

######  Proposition 3.2.

One has (for $N$ large enough depending on $\beta,r$)

$\mathbb{P}^{\beta}_{N}\left[|\mathbf{L}_{N}(\mathrm{g})|\geq(\log
N)^{0.8}\right]\leq\exp\left(-\frac{1}{2}(\log N)^{1.5}\right).$

The value $0.8$ is arbitrary, the point being that $0.8<1$ while $1.5>1$ and
thus the probabilistic tail is better than algebraic in $N$. The proof of
Proposition will be given in Section A.4 and the argument actually shows that
$\mathbf{L}_{N}(\mathrm{g})$ is typically of order 1 as expected, see
Corollary A.8.

###  3.2. A regularized version of the potential

Let $\rho$ be a radial $C^{\infty}$ mollifier supported on the unit disk and
for $\varepsilon\in(0,1)$, let
$\rho_{\varepsilon}:=\varepsilon^{-2}\rho\left(\frac{\cdot}{\varepsilon}\right)$
and let $\varphi_{z,\varepsilon}$ be defined as:

$\varphi_{z,\varepsilon}:=\rho_{\varepsilon}\star\log_{z}-\mathrm{g}$

where $\mathrm{g}$ satisfying (3.2) is independent of $N,z$ and $\varepsilon$.
We think of $\varphi_{z,\varepsilon}$ as being an alternative to $\log_{z}$
which is regularized in two ways: the singularity near $z$ is removed by
convolving with a mollifier, and the Laplacian of $\varphi_{z,\varepsilon}$
has total mass $0$. Then, in view of Proposition 3.2, in order to prove
Proposition 3.1, we first focus on controlling the exponential moments of
$\mathbf{L}_{N}(\varphi_{z,\varepsilon})$.

######  Proposition 3.3.

For some constant $\mathtt{C}\geq 1$ depending only on $\beta$, for all fixed
$r\in(0,1)$, for all $N$ large enough (depending on $r$,$\beta$), for all
$\varepsilon$ such that $\mathtt{C}N^{-1/2}\leq\varepsilon\leq\frac{1-r}{2}$
and for all $t$ such that $|t|\leq\mathtt{C}^{-1}N\varepsilon^{2}$, for any
$z\in\mathsf{D}_{r}$, we have:

(3.3)
$\mathbb{E}\left[e^{t\mathbf{L}_{N}(\varphi_{z,\varepsilon})}\right]=\exp\left(\frac{t^{2}\log{\varepsilon^{-1}}}{2\beta}+t^{2}O_{\varepsilon}(1)+tO_{N}(1)\right),$

with $O_{\varepsilon}(1)$ depending only on $\beta$ and $O_{N}(1)$ depending
on $\beta,r$.

###### Proof of Proposition 3.3.

First, we impose that $\mathtt{C}$ is large enough (depending only on $\beta$)
so that the length scale $\mathtt{C}N^{-1/2}$ is larger than the minimal
length scale $\widetilde{\rho}_{\beta}$ mentioned in Section 2.1. Next,
observe that $\varphi_{z,\varepsilon}$ is a $C^{\infty}$ function whose
Laplacian is supported in $\mathsf{D}_{r+\varepsilon}$ and has mean zero.
Moreover, $|\varphi_{z,\varepsilon}|_{k}$ is of order $\varepsilon^{-k}$ for
any $k\in\mathbb{N}$. Hence, for $|t|\leq\mathtt{C}^{-1}N\varepsilon^{2}$
(with $\mathtt{C}$ large enough depending only on $\beta$) we may apply Lemma
2.4 and write:

$\mathbb{E}\left[e^{t\mathbf{L}_{N}(\varphi_{z,\varepsilon})}\right]=\exp\left(\frac{t^{2}}{4\pi\beta}\int_{\mathbb{R}^{2}}-\varphi_{z,\varepsilon}\Delta\varphi_{z,\varepsilon}\right)\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0})},$

with $s=\frac{-t}{2\pi N\beta}$ and
$\mathrm{m}_{s}:=\mathrm{m}_{0}+s\Delta\varphi_{z,\varepsilon}$. We have:

$-\int_{\mathbb{R}^{2}}\varphi_{z,\varepsilon}\Delta\varphi_{z,\varepsilon}=-2\pi\int_{\mathbb{R}^{2}}\left(\log|\cdot|\star\rho_{\varepsilon}\right)\rho_{\varepsilon}+O_{\varepsilon}(1),$

and by scaling we can see that:
$\int_{\mathbb{R}^{2}}\left(\log|\cdot|\star\rho_{\varepsilon}\right)\rho_{\varepsilon}=\log\varepsilon+O_{\varepsilon}(1)$,
so we may write:

(3.4)
$\mathbb{E}\left[e^{t\mathbf{L}_{N}(\varphi_{z,\varepsilon})}\right]\leq\exp\left(-\frac{t^{2}\log\varepsilon}{2\beta}+t^{2}O_{\varepsilon}(1)\right)\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0})}.$

The assumptions of Proposition 2.8 are fulfilled with
$\mathbf{m}=\mathbf{m}_{0}$, $f_{a}=2\pi\rho_{\varepsilon}$,
$f_{b}=-2\pi\chi$, $z_{a}=z$, $z_{b}=0$, $\ell_{a}=\varepsilon$ and $\ell_{b}$
fixed (independent of $\varepsilon$ and $N$). We obtain:

$\log\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0})}=N\left(\frac{\beta}{4}-1\right)\mathcal{E}(\mathrm{m}_{s})+O\left(sN^{3/4}\varepsilon^{-1/2}\left(1+\log(N\varepsilon^{2})\right)^{{\frac{1}{2}}}\right),$

where we used that $\mathcal{E}(\mathrm{m}_{0})=0$ and we recall that the
implicit multiplicative constant depends only on $\beta$ and $r$. A direct
computation shows that $\mathcal{E}(\mathrm{m}_{s})$ is $s=O(t/N)$ – cf.
Remark 2.10 – so we conclude that

$\log\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0})}=O(t)+O\left(sN^{3/4}\varepsilon^{-1/2}\left(1+\log(N\varepsilon^{2})\right)^{\frac{1}{2}}\right),$

Using that $\varepsilon\geq\mathtt{C}N^{-1/2}$, this completes the proof, the
dominant error term being the first one in the right-hand side. ∎

######  Corollary 3.4.

There exists $\mathtt{C}\geq 1$ depending only on $\beta$, such that for all
$\lambda\geq\mathtt{C}$, taking $\varepsilon=\lambda N^{-1/2}$ and
$|t|\leq\lambda^{2}\mathtt{C}^{-1}$ we have:

(3.5)
$\mathbb{E}\left[e^{t\mathbf{L}_{N}(\varphi_{z,\varepsilon})}\right]\leq\exp\left(\frac{t^{2}\log
N}{4\beta}+O(t^{2}+t)\right),$

with an implicit constant depending on $\beta,r,{\lambda}$.

###  3.3. Lattice approximation and proof of the LLN upper bound

We have obtained in Corollary 3.4 a control on the exponential moments of
$\mathbf{L}_{N}(\varphi_{z,\varepsilon})$ _for a fixed_ $z\in\mathsf{D}_{r}$.
In the first step below, we show that it yields an upper bound on
$\mathrm{Pot}_{N}(z)$ (still for a fixed $z$). Then we get a bound controlling
the values of $\mathrm{Pot}_{N}$ at _all points $z$ on a sub-microscopic
lattice_ contained in $\mathsf{D}_{r}$. Finally, we turn it into a uniform
control of $\mathrm{Pot}_{N}$ over $\mathsf{D}_{r}$.

#### 1\. Regularization and one-point tail estimate.

First, we observe in Lemma 3.5 below that we can regularize the $\log$ in the
definition of $\mathrm{Pot}_{N}$ at the microscopic scale with a small cost.
For $\varepsilon>0$, let $\mathrm{Pot}_{N,\varepsilon}$ be the map:

(3.6)
$\mathrm{Pot}_{N,\varepsilon}:z\mapsto\mathbf{L}_{N}\left(\rho_{\varepsilon}\star\log_{z}\right),$

with $\rho_{\varepsilon}$ as in Section 3.2. The test function
$\widetilde{\log}_{z}:=\rho_{\varepsilon}\star\log_{z}$ is smooth, and for
$z,x\in\mathbb{R}^{2}$, we have:

(3.7)
$|\nabla\widetilde{\log}_{z}(x)|\preceq\max\left(\varepsilon,|z-x|\right)^{-1},\quad\|\mathrm{D}^{2}\widetilde{\log}_{z}(x)\|\preceq\max\left(\varepsilon,|z-x|\right)^{-2}.$

######  Lemma 3.5.

For $\varepsilon>0$ and $z\in\mathsf{D}$ we have, with a universal implicit
constant:

$\mathrm{Pot}_{N}(z)\leq\mathrm{Pot}_{N,\varepsilon}(z)+O\left(N\varepsilon^{2}\right).$

###### Proof.

By subharmonicity of $\log$, we have
$\log_{z}\leq\rho_{\varepsilon}\star\log_{z}$ pointwise. On the other hand,
for the centering term, a direct computation (using e.g. Newton’s theorem)
yields:

$\left|\int_{\mathbb{R}^{2}}\log_{z}(x)\mathrm{d}\mathbf{m}_{0}(x)-\int_{\mathbb{R}^{2}}\left(\rho_{\varepsilon}\star\log_{z}\right)(x)\mathrm{d}\mathbf{m}_{0}(x)\right|=O\left(\varepsilon^{2}\right).$

∎

For the rest of the section, we fix:

(3.8) $\varepsilon=\lambda
N^{-1/2},\quad\lambda^{2}=\mathtt{C}^{2}\max\left(4\sqrt{\beta},1\right),$

where the constant $\mathtt{C}$ is as in Corollary 3.4 (depending only on
$\beta$). By Lemma 3.5 we have, with an implicit constant depending only on
$\beta$:

(3.9)
$\mathrm{Pot}_{N}(z)\leq\mathrm{Pot}_{N,\varepsilon}(z)+O\left(1\right).$

######  Lemma 3.6.

Fix a point $z\in\mathsf{D}_{r}$ and let $\alpha\in(1,2)$. We have:

(3.10)
$\mathbb{P}^{\beta}_{N}\left[|\mathrm{Pot}_{N,\varepsilon}(z)|\geq\frac{\alpha\log
N}{\sqrt{\beta}}\right]\leq\exp\left(-\alpha^{2}\log N+O(1)\right),$

with an implicit constant depending on $\beta,r$.

###### Proof.

Take $t:=2\alpha\sqrt{\beta}\leq 4\sqrt{\beta}$ so that, in view of (3.8),
$t\leq\lambda^{2}\mathtt{C}^{-2}\leq\lambda^{2}\mathtt{C}^{-1}$ with
$\mathtt{C}$ as in Corollary 3.4. Then, we may thus use (3.5) and write:

$\mathbb{E}\left[e^{t\mathbf{L}_{N}(\varphi_{z,\varepsilon})}\right]=\exp\left(\frac{t^{2}\log
N}{4\beta}+O(t^{2}+t)\right)=\exp\left(\alpha^{2}\log N+O(1)\right),$

with an implicit constant depending only on $r,\beta$. So, Markov’s inequality
yields:

(3.11)
$\displaystyle\mathbb{P}^{\beta}_{N}\left[\mathbf{L}_{N}(\varphi_{z,\varepsilon})\geq\frac{\alpha\log
N}{\sqrt{\beta}}\right]$ $\displaystyle\leq\exp\left(-\frac{t\alpha\log
N}{\sqrt{\beta}}\right)\mathbb{E}\left[e^{t\mathbf{L}_{N}(\varphi_{z,\varepsilon})}\right]$
$\displaystyle\leq\exp\left(-\alpha^{2}\log N+O(1)\right).$

Note that since (3.4) is valid for $t\in\mathbb{R}$, the same bound holds for
$-\mathbf{L}_{N}(\varphi_{z,\varepsilon})$ as well. On the other hand, by
Proposition 3.2, we have $|\mathbf{L}_{N}(\mathrm{g})|\leq(\log N)^{0.8}$ with
probability $1-\exp\left(-(\log N)^{1.5}\right)$. Since by definition
$\mathbf{L}_{N}(\varphi_{z,\varepsilon})=\mathrm{Pot}_{N,\varepsilon}(z)+\mathbf{L}_{N}(\mathrm{g})$,
we may convert the control (3.11) on $\mathbf{L}_{N}(\varphi_{z,\varepsilon})$
into a similar estimate for $\mathrm{Pot}_{N,\varepsilon}(z)$, as stated. ∎

#### 2\. Tail estimate on a lattice.

For $\alpha\in(1,2)$ and fix $\delta$ (depending on $\alpha$) such that:

(3.12) $0<\delta<\frac{\alpha^{2}-1}{2}$

Let $\ell_{\delta}:=N^{-\frac{1}{2}-\delta}$ (in particular, $\ell_{\delta}$
is sub-microscopic) and consider the square lattice
$\Lambda_{\delta}:=\left(\ell_{\delta}\mathbb{Z}\right)^{2}$.

######  Lemma 3.7 (Tail estimate on the lattice).

Let $\alpha$ be in $(1,2)$. We have:

(3.13)
$\mathbb{P}^{\beta}_{N}\left[\max_{z\in\Lambda_{\delta}\cap\mathsf{D}(0,r)}|\mathrm{Pot}_{N,\varepsilon}(z)|\geq\frac{\alpha\log
N}{\sqrt{\beta}}\right]\leq\exp\left((1+2\delta)\log N-\alpha^{2}\log
N+O(1)\right),$

with an implicit constant depending on $\beta,r$. In particular we have:

(3.14)
$\mathbb{P}^{\beta}_{N}\left[\max_{z\in\Lambda_{\delta}\cap\mathsf{D}(0,r)}|\mathrm{Pot}_{N,\varepsilon}(z)|\geq\frac{\alpha\log
N}{\sqrt{\beta}}\right]\to 0,$

as $N\to\infty$ – the decay being algebraic in $N$.

###### Proof.

The number of points of $\Lambda_{\delta}$ that fall into $\mathsf{D}_{r}$ is
bounded above and below by positive constants (depending only on $r$) times
$N^{1+2\delta}$. Thus (3.13) follows from a simple union bound, using the one-
point estimate (3.10). From our choice (3.12) for $\delta$ we easily deduce
(3.14). ∎

#### 3\. Extension to the whole disk

It remain to prove a result comparing the maximum of the Coulomb gas potential
over the lattice points versus the whole unit disk. Recall that $\varepsilon$
corresponds to the “microscopic” scale, see (3.8).

######  Proposition 3.8 (The lattice vs. the whole disk).

One has

(3.15)
$\mathbb{P}^{\beta}_{N}\left[\max_{z\in\mathsf{D}(0,r)}\mathrm{Pot}_{N,\varepsilon}(z)\geq
1+\max_{z^{\prime}\in\Lambda_{\delta}\cap\mathsf{D}(0,r)}\mathrm{Pot}_{N,\varepsilon}(z^{\prime})\right]\to
0\text{ as }N\to\infty.$

###### Proof of Proposition 3.8.

Let $z\in\mathsf{D}_{r}$, there exists a point $z^{\prime}$ in
$\Lambda_{\delta}\cap\mathsf{D}_{r}$ such that $|z-z^{\prime}|\leq
4\ell_{\delta}$. We now show that, with high probability, the values of the
regularized potential at $z$ and $z^{\prime}$ are close. Recall the definition
(3.6) of $\mathrm{Pot}_{N,\varepsilon}$. We can write the difference
$\mathrm{Pot}_{N,\varepsilon}(z)-\mathrm{Pot}_{N,\varepsilon}(z^{\prime})$ as
$\mathbf{L}_{N}(G^{z,z^{\prime}})$, where $G^{z,z^{\prime}}$ is the function

$G^{z,z^{\prime}}(x):=\rho_{\varepsilon}\star\log_{z}(x)-\rho_{\varepsilon}\star\log_{z^{\prime}}(x)=\widetilde{\log}_{x}(z)-\widetilde{\log}_{x}(z^{\prime}).$

In view of the bounds (3.7) on the derivatives of $\widetilde{\log}$, and
since $|z-z^{\prime}|\leq 4\ell_{\delta}$, we get:

$|G^{z,z^{\prime}}(x)|\preceq\ell_{\delta}\varepsilon^{-1},\quad|\nabla
G^{z,z^{\prime}}(x)|\preceq\ell_{\delta}\varepsilon^{-2}\text{ for
}x\in\mathsf{D}(z^{\prime},4\varepsilon),$
$|G^{z,z^{\prime}}(x)|\preceq\ell_{\delta}|z-x|^{-1},\quad|\nabla
G^{z,z^{\prime}}(x)|\preceq\ell_{\delta}|z-x|^{-2}\text{ for
}x\notin\mathsf{D}(z^{\prime},4\varepsilon).$

We introduce a sequence of intermediate length scales
$\ell_{0}<\dots<\ell_{n}$ by taking $c,n$ such that:

$n=\lfloor\log\left(\frac{1-r}{4\varepsilon}\right)\rfloor,\quad\log
c:=\frac{1}{n}\log\left(\frac{1-r}{4\varepsilon}\right),$

and by setting $\ell_{k}:=c^{k}\varepsilon$ for $k=0,\dots,n$. Note that
$c\in[1,2]$, that $n=O(\log N)$, that $\ell_{0}=\varepsilon$ and that
$\ell_{n}=\frac{1-r}{4}$. We also set $\ell_{n+1}=M$, with $M$ large
(depending on $\beta$ but not on $N$) to be chosen later. Next, we take a
family $(\chi_{i})_{0\leq i\leq n+1}$ of functions such that:

$\sum_{i=0}^{n+1}\chi_{i}\equiv 1\text{ on
}\mathsf{D}(0,M),\quad\sum_{i=0}^{n+1}\chi_{i}\equiv 0\text{ outside
}\mathsf{D}(0,2M),$

each $\chi_{i}$ living at scale $\ell_{i}$ around $z^{\prime}$, namely:

$|\chi_{i}|_{\mathsf{0}}\leq
1,\quad|\chi_{i}|_{\mathsf{1}}\preceq\ell_{i}^{-1},\quad\chi_{i}\equiv 0\text{
outside
}\mathsf{D}(z^{\prime},4\ell_{i})\setminus\mathsf{D}(z^{\prime},\ell_{1}/2),$

and we let $G^{z,z^{\prime}}_{i}:=\chi_{i}G^{z,z^{\prime}}$ for $0\leq i\leq
n$. Using the bounds on $G^{z,z^{\prime}},\nabla G^{z,z^{\prime}}$ and the
ones on $\chi,\nabla\chi_{i}$, a direct computation ensures that
$G^{z,z^{\prime}}_{i}$ is (up to some multiplicative constant)
$\ell_{\delta}\ell_{i}^{-2}$-Lipschitz for each $i$ \- in other words, the
function
$\widetilde{G}^{z,z^{\prime}}_{i}:=\frac{\ell_{i}}{\ell_{\delta}}G^{z,z^{\prime}}_{i}$
is $\frac{1}{\ell_{i}}$-Lipschitz and thus satisfies the assumptions of Lemma
2.3. We may write, using a convexity inequality for $\exp$:

$\exp\left(N^{\delta/2}\sum_{i=0}^{n+1}\mathbf{L}_{N}(G^{z,z^{\prime}}_{i})\right)=\exp\left(N^{\delta/2}\sum_{i=0}^{n+1}\frac{\ell_{\delta}}{\ell_{i}}\mathbf{L}_{N}(\widetilde{G}^{z,z^{\prime}}_{i})\right)\\\
\preceq\frac{1}{\log
N}\sum_{i=0}^{n+1}\exp\left(\frac{N^{\delta/2}\ell_{\delta}\log
N}{\ell_{i}}\mathbf{L}_{N}(\widetilde{G}^{z,z^{\prime}}_{i})\right),$

where we used that $n$ is of order $\log N$. Since
$\frac{N^{\delta/2}\ell_{\delta}\log N}{\ell_{i}}\ll 1$, we may use the
control in exponential moments given by Lemma 2.3 and write:

$\log\mathbb{E}\left[\exp\left(\frac{N^{\delta/2}\ell_{\delta}\log
N}{\ell_{i}}\mathbf{L}_{N}(\widetilde{G}^{z,z^{\prime}}_{i})\right)\right]=O\left(\frac{N^{\delta/2}\ell_{\delta}\log
N}{\ell_{i}}\left(N\ell_{i}^{2}\right)^{\frac{1}{2}}\right)=O\left(N^{-\delta/2}\log
N\right).$

We obtain a similar control for the (macroscopic) scale $\ell_{n+1}$, see the
last comment in Lemma 2.3. Then Markov’s inequality yields:

(3.16)
$\mathbb{P}^{\beta}_{N}\left[\sum_{i=0}^{n+1}\mathbf{L}_{N}(G^{z,z^{\prime}}_{i})\geq
1\right]\leq\exp\left(-N^{\delta/2}\right).$

We emphasize that since we are using Lemma 2.3, which provides uniform control
for fluctuations of all Lipschitz functions living at scale $\ell$ around a
given point, then for fixed $z^{\prime}$, the control (3.16) is in fact
uniform in $z$ i.e.

(3.17) $\mathbb{P}^{\beta}_{N}\left[\max_{z,|z-z^{\prime}|\leq
4\ell_{\delta}}\sum_{i=0}^{n+1}\mathbf{L}_{N}(G^{z,z^{\prime}}_{i})\geq
1\right]\leq\exp\left(-N^{\delta/2}\right).$

On the other hand, since by construction $\sum_{i=0}^{n+1}\chi_{i}\equiv
1\text{ on }\mathsf{D}(0,M)$, any difference between
$\mathrm{Pot}_{N,\varepsilon}(z)-\mathrm{Pot}_{N,\varepsilon}(z^{\prime})=\mathbf{L}_{N}(G^{z,z^{\prime}})$
and $\sum_{i=0}^{n+1}\mathbf{L}_{N}(G_{i})$ comes from hypothetical outliers
living far away from the unit disk, namely outside $\mathsf{D}(0,M)$. A simple
large deviation estimate (see e.g. [SS15, (1.48)]) shows that the probability
of any point being outside $\mathsf{D}(0,M)$ decays as $\exp\left(-\beta
NM^{2}\right)$ as $N\to\infty$ for $M$ large enough depending only on $\beta$
(indeed the non-negative confinement term $2\zeta(x)$ appearing in the
Boltzmann factor grows as $|x|^{2}$ for large $x$ as can be seen in (1.2)).

Finally, we obtain (by symmetry) for any fixed $z^{\prime}$:

(3.18) $\mathbb{P}^{\beta}_{N}\left[\max_{z,|z-z^{\prime}|\leq
4\ell_{\delta}}\left|\mathrm{Pot}_{N,\varepsilon}(z)-\mathrm{Pot}_{N,\varepsilon}(z^{\prime})\right|\geq
1\right]\leq\exp\left(-N^{\delta/2}\right).$

A union bound over $z^{\prime}\in\Lambda_{\delta}\cap\mathsf{D}$ concludes the
proof. ∎

Combining Lemma 3.7 and Lemma 3.8 proves the LLN upper bound stated in
Proposition 3.1. Moreover, using (3.18), we obtain a quantitative tail
estimate:

######  Corollary 3.9.

For all $\alpha>1$, there is a small $\nu>0$ (depending on $\alpha$) such that

$\mathbb{P}^{\beta}_{N}\left[\max_{z\in\mathsf{D}(0,r)}\left|\mathrm{Pot}_{N,\varepsilon}(z)\right|\geq\frac{\alpha\log
N}{\sqrt{\beta}}\right]\leq\exp\left(-\nu\log N+O(1)\right).$

###  3.4. Uniform control of fluctuations for smooth linear statistics

A direct consequence of the previous estimates is a uniform bound for the
fluctuations of linear statistics for a class of $C^{2}$ test functions with,
say, $|\cdot|_{2}\leq 1$. Let

$\mathcal{F}_{N}:=\left\\{f\in C^{2}({\mathrm{D}_{r}}),\text{ $f$ {is
supported in some disk} $\mathsf{D}(z,\ell)$ \text{ with } $\ell\geq\lambda
N^{-1/2}$, and $|f|_{2}\leq\ell^{-2}$}\right\\},$

where $\lambda$ is the constant depending only on $\beta$ chosen above in
(3.8).

######  Proposition 3.10.

For any $k\in\mathbb{N}$, there is a constant $C_{k}=C_{k}(\beta)$ so that if
$N$ is sufficiently large (depending on $\beta$, $k$ and $r$),

$\mathbb{P}^{\beta}_{N}\left[\sup_{f\in\mathcal{F}_{N}}\left|\mathbf{L}_{N}(f)\right|\geq
C_{k}\log N\right]\leq N^{-k}$

###### Proof.

We use again $\varepsilon=\lambda/\sqrt{N}$. Let
$(\square_{j}=\square(z_{j},\varepsilon))_{j=1}^{M}$ be a collection of
squares centered at points $z_{j}\in\varepsilon\mathbb{Z}^{2}$ such that
$\mathsf{D}(0,r)\subset\bigcup_{j=1}^{M}\square_{j}\subset\mathsf{D}(0,1-\delta)$
for a small $\delta>0$. The collection can be chosen so that
$M\leq\mathtt{C}_{\beta}N$ for some constant depending on $\beta$. Using the
local laws (2.8) and a union bound, we deduce that for all $k\in\mathbb{N}$,
there is a constant $C_{k}=C_{k}(\beta)$ such that if $N$ is sufficiently
large,

(3.19) $\mathbb{P}^{\beta}_{N}\left[\max_{j\leq
M}\mathbf{X}_{N}(\square_{j})\leq C_{k}\log N\right]\geq 1-N^{-k}.$

Let $f\in\mathcal{F}_{N}$, $\ell$ be the associated length scale, and let
$f_{\varepsilon}=f\star\rho_{\varepsilon}$. Since $\rho$ is a radial mollifier
(with compact support in $\mathsf{D}$) and $|f|_{2}\leq\ell^{-2}$, one has

$|f_{\varepsilon}-f|_{0}\leq(\varepsilon/\ell)^{2}$

and both $f,f_{\varepsilon}$ are compactly supported in $\mathsf{D}(z,2\ell)$
since $\ell\geq\varepsilon$. One can tile $\mathsf{D}(z,2\ell)$ with at most
$16(\ell/\varepsilon)^{2}$ squares of sidelength $\varepsilon$, thus on the
event introduced in (3.19) we have:

(3.20) $|\mathbf{L}_{N}(f_{\varepsilon})-\mathbf{L}_{N}(f)|\leq 16C_{k}(1+\log
N).$

Moreover, by an integration by parts we can write $\int\Delta
f\,\mathrm{Pot}_{N,\varepsilon}=\mathbf{L}_{N}(f_{\varepsilon})$, and using
again that $|f|_{2}\leq\ell^{-2}$ and that $\Delta f$ supported in
$\mathsf{D}(0,r)$, we deduce:

$|\mathbf{L}_{N}(f_{\varepsilon})|\leq\max_{z\in\mathsf{D}(0,r)}\left|\mathrm{Pot}_{N,\varepsilon}(z)\right|.$

The RHS is controlled by Corollary 3.9 and, by (3.20), on the event introduced
in (3.19) we can use this to control $\mathbf{L}_{N}(f)$ _uniformly_ for all
$f\in\mathcal{F}_{N}$. Adjusting the constants $C_{k}$, this proves the claim.
∎

##  4\. Regularized multiplicative chaos: lower bound

This section is devoted to the proof of the lower bound in Theorem 1.

###  4.1. Reduction to a regularized version of the potential

Recall that
$\varphi_{z,\varepsilon}:=\rho_{\varepsilon}\star\log_{z}-\mathrm{g}$, see
(3.2), is a regularization of $\log_{z}$ at scale $\varepsilon>0$ and let
$\mathcal{U}\subset\mathsf{D}$ be a (non-empty) open set. By Proposition 3.2,
we know that for any (small) $\delta>0$:

$\liminf_{N\to\infty}\mathbb{P}\left[\sup_{z\in\mathcal{U}}\mathbf{L}_{N}(\rho_{\varepsilon}\star\log_{z})\geq(\beta^{-1/2}-\delta)\log
N\right]\geq\liminf_{N\to\infty}\mathbb{P}\left[\sup_{z\in\mathcal{U}}\mathbf{L}_{N}(\varphi_{z,\varepsilon})\geq(\beta^{-1/2}-\delta/2)\log
N\right].$

In addition, recall that $\mathrm{Pot}_{N}(z)=\mathbf{L}_{N}(\log_{z})$ so
that

$\mathbf{L}_{N}(\rho_{\varepsilon}\star\log_{z})=\mathbf{L}_{N}\bigg{(}\int\rho_{\varepsilon}(x)\log_{z-x}(\cdot){\mathrm{d}x}\bigg{)}=\int\rho_{\varepsilon}(x)\mathrm{Pot}_{N}(z-x){\mathrm{d}x}.$

The right-hand side is the convolution of the function $\mathrm{Pot}_{N}$ with
a non-negative function of total mass $1$ and as such

$\max_{z\in\mathcal{K}}\mathrm{Pot}_{N}(z)\geq\sup_{z\in\mathcal{U}}\mathbf{L}_{N}({\rho_{\varepsilon}\star\log_{z}})$

where $\mathcal{K}$ is a closed ${\varepsilon}$-neighborhood of $\mathcal{U}$.

Altogether, this implies that for any (small) $\delta>0$,

(4.1)
$\liminf_{N\to\infty}\mathbb{P}\left[\max_{z\in\mathcal{K}}\mathrm{Pot}_{N}(z)\geq(\beta^{-1/2}-\delta)\log
N\right]\\\
\geq\liminf_{N\to\infty}\mathbb{P}\left[\sup_{z\in\mathcal{U}}\mathbf{L}_{N}(\varphi_{z,\varepsilon})\geq(\beta^{-1/2}-\delta/2)\log
N\right].$

Let $\mathcal{U}\subset\mathrm{D}$ be some fixed (small) ball, it suffices to
show that the probability on the RHS of (4.1) converges to 1 as $N\to\infty$.
It turns out that this is the case if the scale $\varepsilon(N)\leq
N^{\nu-1/2}$ for some $\nu>0$ sufficiently small (depending on $\delta>0$).
For technical reasons, we also need that $\varepsilon(N)\gg N^{-1/2}$ is
larger than the microscopic scale. We claim the following:

######  Proposition 4.1.

Let $\varepsilon(N)$ be a sequence so that $\varepsilon(N)\to 0$ as
$N\to\infty$ in such a way that $\varepsilon(N)\gg N^{-1/2}$. Then, for any
$\alpha>2$,

(4.2)
$\lim_{N\to\infty}\mathbb{P}\left[\sqrt{\beta}\sup_{z\in\mathcal{U}}\mathbf{L}_{N}(\varphi_{z,\varepsilon})\geq\alpha\log\varepsilon(N)^{-1}\right]=1.$

Combining (4.1) and (4.2), we get:
$\lim_{N\to\infty}\mathbb{P}\left[\sup_{z\in\mathcal{U}}\mathrm{Pot}_{N}(z)\geq\bar{\alpha}\log
N\right]=1$ for any value $\bar{\alpha}>\beta^{-1/2}$, which yields the lower
bound in Theorem 1.

The proof of Proposition 4.1 relies on the theory of Gaussian multiplicative
chaos and in particular on [CFLW21] and [LOS18]. We review these results in
the next section and present the main steps of the proof.

###  4.2. Multiplicative chaos.

We introduce new notations. Let $\ell(k)=e^{-k}$ for $k\in\mathbb{N}$ and set

$\Phi_{k}(z):=\sqrt{\beta}\,\mathbf{L}_{N}(\varphi_{z,\ell(k)}),\qquad
z\in\mathsf{D}.$

For $\gamma>0$, define a sequence of random measure
$(\boldsymbol{\mu}_{k}^{\gamma})_{k\in\mathbb{N}}$ with density function

(4.3)
$\mu_{k}^{\gamma}=\frac{e^{\gamma\Phi_{k}}}{\mathbb{E}e^{\gamma\Phi_{k}}}.$

This density obviously depends on the $N$ and $\beta$ even though it is not
emphasized in the notation.

According to Proposition 3.3, if $\ell(k)\geq\mathtt{C}N^{-1/2}$, it holds
uniformly for $x\in\mathcal{K}$,

(4.4)
$\mathbb{E}[e^{\gamma\Phi_{k}(x)}]=\exp\left(\frac{\gamma^{2}k}{2}+O_{N}(1)\right)\asymp\ell(k)^{-\gamma^{2}/2}.$

These asymptotics corresponds to [CFLW21, Assumptions 3.1]. In fact, using the
method developed in [CFLW21, Section 3], [LOS18, Section 2], we will obtain
the following convergence result (with respect to the vague topology for
positive measures on $\mathsf{D}$).

######  Proposition 4.2.

Let $n(N)$ be a sequence such that $n(N)\to\infty$ and
${\ell(n):=\ell(n(N))}\gg N^{-1/2}$ as $N\to\infty$. Then for any
$)<\gamma<2$, $\boldsymbol{\mu}_{n(N)}^{\gamma}\to\operatorname{GMC}_{\gamma}$
in distribution as $N\to\infty$ where $\operatorname{GMC}_{\gamma}$ is a
Gaussian multiplicative chaos measure which is defined shortly.

######  Remark 4.3.

If one consider the sequence of random measures with densities
$\hat{\mu}_{N}^{\gamma}=\frac{e^{\gamma\sqrt{\beta}\mathrm{Pot}_{N}}}{\mathbb{E}e^{\gamma\sqrt{\beta}\mathrm{Pot}_{N}}}$
for $N\in\mathbb{N}$ instead of (4.3), we expect that the result of
Proposition 4.2 remains true, that is, for any $0<\gamma<2$,
$\boldsymbol{\hat{\mu}}_{N}^{\gamma}\to\operatorname{GMC}_{\gamma}$ in
distribution as $N\to\infty$ for a slightly different GMC (associated to the
GFF with _free boundary condition_ on $\mathsf{D}$). Note that the regime
$0<\gamma<2$ corresponds to the whole GMC $L^{1}$-phase, as $\gamma=2$ is the
critical value with our normalization.

We now turn to the definitions of the random measures
$(\operatorname{GMC}_{\gamma})_{0<\gamma<2}$. We consider the following log-
correlated field.

######  Definition 4.4.

Let $\Psi$ be a (distribution-valued) Gaussian process on $\mathrm{D}$ with
mean zero and the following correlation structure: for any
$f,g\in\mathcal{C}_{c}(\mathsf{D})$ with $\int f=\int g=1$,

(4.5)
$\mathbb{E}\big{[}\Psi(f)\Psi(g)\big{]}:=\frac{1}{(2\pi)^{2}}\iint\big{(}f(x)-\chi(x)\big{)}\big{(}g(z)-\chi(z)\big{)}\log|x-z|^{-1}\mathrm{d}x\mathrm{d}z$

with $\chi$ as in (3.2). It is well-known that the RHS defines the covariance
of a Gaussian process (this also follows from the CLT of Remark 2.10 which
implies that with $\varphi=-\Delta^{-1}(f-\chi):z\mapsto
2\pi\int\log|x-z|^{-1}(f(x)-\chi(x))\mathrm{d}x$, one has
$\mathbf{L}_{N}(\varphi)\to\Psi(f)/\sqrt{\beta}$ in distribution), so that the
field $\Psi$ has correlation kernel,

$\Sigma(x,z)=\log|x-z|^{-1}+\mathrm{g}(x)+\mathrm{g}(z)+c$

where $c\in\mathbb{R}$ is a constant.

For any $k\in\mathbb{N}$, we define $\Psi_{k}:=\rho_{\ell(k)}\star\Psi$. This
is a (smooth) approximation of $\Psi$ as $k\to\infty$ and, for $\gamma>0$, we
also let $\boldsymbol{\nu}_{k}^{\gamma}$ be a random measure on $\mathrm{D}$
with density function

$\boldsymbol{\nu}_{k}^{\gamma}(x)=\frac{e^{\gamma\Psi_{k}(x)}}{\mathbb{E}e^{\gamma\Psi_{k}(x)}},\qquad
x\in\mathsf{D}.$

Then, the following convergence result follows from the general theory of
multiplicative chaos; e.g. [Ber17].

######  Proposition 4.5.

For any $\gamma<2$ (subcritical phase), the random measure
$\boldsymbol{\nu}_{k}^{\gamma}\to\operatorname{GMC}_{\gamma}$ in probability
as $k\to\infty$. Moreover, for any (non-empty) open set $A\subset\mathsf{D}$,
$\operatorname{GMC}_{\gamma}(A)>0$ almost surely.

Hence, in the _subcritical phase_ ($\gamma<2$), as a consequence of
Proposition 4.2 and [CFLW21, Theorem 3.4], we obtain Proposition 4.1. Finally,
the proof of Proposition 4.2 will be a direct application of [CFLW21, Theorem
2.4] (reproduced there from [LOS18, Theorem 1.7]). Namely, it suffices to show
that for any $j\in\mathbb{N}$, $k_{1},\cdots,k_{j}\leq n(N)$ and
$\gamma_{1},\cdots,\gamma_{j}\in\mathbb{R}$,

(4.6)
$\mathbb{E}\big{[}e^{\gamma_{1}\Phi_{k_{1}}(x_{1})+\cdots+\gamma_{j}\Phi_{k_{j}}(x_{j})}\big{]}=\mathbb{E}\big{[}e^{\gamma_{1}\Psi_{k_{1}}(x_{1})+\cdots+\gamma_{j}\Psi_{k_{j}}(x_{j})}\big{]}\big{(}1+{o(1)}\big{)}$

uniformly for $x_{1},\cdots,x_{j}\in\mathsf{D}_{r}$ (here $r<1$ is fixed). The
error term is also uniform for $k_{1},\cdots k_{j}\leq n(N)$, in which case
the condition $\ell(n)\gg N^{-1/2}$ as $N\to\infty$ is crucial. The next
section is devoted to the proof of (4.6).

###  4.3. Exponential moments asymptotics.

We deduce (4.6) from the estimates of Proposition 2.8 by a simple induction.

First observe that since
$\Phi_{k}(x)=\sqrt{\beta}\,\mathbf{L}_{N}(\varphi_{x,\ell(k)})$ with
$\Delta\varphi_{x,\ell}=\rho_{x,\ell}-\mathrm{g}$, as a consequence of Remark
2.10 (with $f_{a}=\rho_{x,\ell}=\rho((\cdot-x)\ell^{-1})\ell^{-2}$ and
$f_{b}=\mathrm{g}$ so that the conditions (2.18) hold), we have

$\mathbb{E}\big{[}e^{\gamma\Phi_{k}(x)}\big{]}=\exp\left(\frac{-\gamma^{2}}{4\pi}\int_{\mathbb{R}^{2}}\varphi_{x,\ell(k)}\Delta\varphi_{x,\ell(k)}\right)\big{(}1+o(1)\big{)}$

uniformly for $x\in\mathsf{D}_{r}$, $\ell(k)\gg N^{-1/2}$ and locally
uniformly for $\gamma\in\mathbb{R}$. Moreover (by definition of the 2d Green’s
function for $-\Delta$ and by Fubini’s theorem), according to Definition 4.4,
we have for $k,n\in\mathbb{N}$ and $x,z\in\mathsf{D}$,

$\displaystyle\frac{-1}{2\pi}\int\varphi_{x,\ell(k)}\Delta\varphi_{z,\ell(n)}$
$\displaystyle=\iint\big{(}\rho_{x,\ell(k)}(u)-\mathrm{g}(u)\big{)}\big{(}\rho_{z,\ell(n)}(v)-\mathrm{g}(v)\big{)}\log|u-v|^{-1}\mathrm{d}u\mathrm{d}v$
(4.7) $\displaystyle=\mathbb{E}[\Psi_{k}(x)\Psi_{n}(z)].$

Thus, since $\Psi$ is a (mean-zero) Gaussian process, we obtain for any
$x\in\mathsf{D}$, $\gamma\in\mathbb{R}$ and $k\ll N^{-1/2}$,

$\mathbb{E}\big{[}e^{\gamma\Phi_{k}(x)}\big{]}=\mathbb{E}\big{[}e^{\gamma\Psi_{k}(x)}\big{]}\big{(}1+{o(1)}\big{)}$

with the required uniformity. This establishes that (4.6) holds when $j=1$.

We now proceed by induction to extend these asymptotics for any
$j\in\mathbb{N}$ with $j\geq 2$. Without loss of generality, we assume that
$k_{1}\leq\cdots\leq k_{j}\leq n(N)$. Then, according to Lemma 2.4, we have

$\displaystyle\mathbb{E}\big{[}e^{\gamma_{1}\Phi_{k_{1}}(x_{1})+\cdots+\gamma_{j}\Phi_{k_{j}}(x_{j})}\big{]}$
$\displaystyle=\mathbb{E}\big{[}e^{\sqrt{\beta}\mathbf{L}_{N}(\varphi_{j})}\big{]}$
$\displaystyle=\exp\left(\frac{-1}{4\pi}\int_{\mathbb{R}^{2}}\varphi_{j}\Delta\varphi_{j}\right)\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0}+s\Delta\varphi_{j})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0})}$

where $\varphi_{j}=\sum_{i\leq j}\gamma_{i}\varphi_{x_{i},\ell(k_{i})}$ and
$s=\frac{-1/\beta^{1/2}}{2\pi N}$. Moreover, using (4.7), we obtain

$\mathbb{E}\Big{[}e^{\sum_{i\leq
j}\gamma_{i}\Phi_{k_{i}}(x_{i})}\Big{]}=\mathbb{E}\big{[}e^{\gamma_{1}\Psi_{k_{1}}(x_{1})+\cdots+\gamma_{j}\Psi_{k_{j}}(x_{j})}\big{]}\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0}+s\Delta\varphi_{j})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0})}$

and, by taking a ratio,

$\frac{\mathbb{E}\Big{[}e^{\sum_{i\leq
j}\gamma_{i}\Phi_{k_{i}}(x_{i})}\Big{]}}{\mathbb{E}\Big{[}e^{\sum_{i\leq
j}\gamma_{i}\Psi_{k_{i}}(x_{i})}\Big{]}}=\frac{\mathbb{E}\Big{[}e^{\sum_{i<j}\gamma_{i}\Phi_{k_{i}}(x_{i})}\Big{]}}{\mathbb{E}\Big{[}e^{\sum_{i<j}\gamma_{i}\Psi_{k_{i}}(x_{i})}\Big{]}}\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0}+s\Delta\varphi_{j})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0}+s\Delta\varphi_{j-1})}.$

We now apply Proposition 2.8 with $f_{a}=\gamma_{j}\rho_{x_{j},\ell(k_{j})}$
and $f_{b}=\gamma_{j}\mathrm{g}$ as above. We emphasize that the conditions
(2.18) hold while the reference measure
$\mathbf{m}=\mathbf{m}_{0}+s\Delta\varphi_{j-1}$ also satisfies
$|\mathrm{m}-1|\leq\mathtt{C}\ell(n)^{2}/N$. In particular, in the regime
$\ell(n)\ll\sqrt{N}$, we obtain

$\log\bigg{(}\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0}+s\Delta\varphi_{j})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0}+s\Delta\varphi_{j-1})}\bigg{)}=N\left(\frac{\beta}{4}-1\right)\left(\mathcal{E}(\mathrm{m}+s\Delta\varphi_{j})-\mathcal{E}(\mathrm{m}+s\Delta\varphi_{j-1})\right)+{o(1),}$

with the required uniformity. Now, repeating the argument from Remark 2.10,
the entropy $\mathcal{E}(\mathrm{m}+s\Delta\varphi_{j})=o(N^{-1})$ for any
$j\in\mathbb{N}$ which implies that

$\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0}+s\Delta\varphi_{j})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0}+s\Delta\varphi_{j-1})}=1+{o(1)}.$

By induction, this concludes the proof of the asymptotics (4.6). ∎

##  Appendix A Auxiliary proofs

###  A.1. A technical remark about partition functions

We import below several results from [Ser20], notably we use the comparison
between partition functions associated to measures that coincide outside a
small disk (see (A.8) below). It is worth noting that our definition of the
partition function $\mathrm{K}_{N}^{\beta}(\mathbf{m})$ associated to a
probability measure $\mathbf{m}$ on $\mathsf{D}$, as given above in (2.3),
differs from the one in [Ser20] by a small detail: we always integrate against
the Lebesgue measure whereas $\mathrm{K}_{N}^{\beta}(\mathbf{m})$ is defined
in [Ser20, Sec 3.3] as an integral against the product measure
$\mathbf{m}^{\otimes N}$. Thus the integrands differ by:

$\prod_{i=1}^{N}\mathrm{m}(x_{i})=\exp\left(\sum_{i=1}^{N}\log\mathrm{m}(x_{i})\right)=\exp\left(N\int\log\mathrm{m}(x)\mathrm{d}\mathbf{m}(x)+\mathbf{L}_{N}^{\mathbf{m}}(\log\mathrm{m})\right).$

In the first term we recognize $N\mathcal{E}(\mathrm{m})$, where $\mathcal{E}$
is the relative entropy as in Section 1.6, which is the reason why our
statement of e.g. Lemma A.4 has an extra relative entropy term in (A.8)
compared to the corresponding expressions in [Ser20]. On the other hand, the
fluctuation term $\mathbf{L}_{N}^{\mathbf{m}}(\log\mathrm{m})$ contributes
$O(1)$, which always gets absorbed into our main error term. This follows from
the fact that fluctuations of smooth enough test functions are bounded in
exponential moments, see e.g. [Ser20, Sec. 2.3].

###  A.2. Proof of Proposition 2.5 and Lemma 2.7

####  A.2.1. Proof of Proposition 2.5

We combine the following tools from [LS18, Ser20].

  1. (1)

Approximate transport and estimates on the approximation error.

  2. (2)

Comparison of partition functions along the approximate transport & the
anisotropy term.

  3. (3)

Relative expansion of partition functions.

  4. (4)

“Serfaty’s trick” for proving smallness of the anisotropy term.

We next review the main arguments.

##### _1\. Approximate transport._

By assumption, the perturbation $f$ is supported in $\mathsf{D}(z,\ell)$ and
thus in a square $\square(z,\ell)$ of sidelength $\ell$. Using [Ser20, Lemma
4.8] together with [Ser20, (4.31)] and applying our assumptions (2.12) and
(2.13), we can find a vector field $\psi$, supported on $\square(z,\ell)$,
such that

(A.1)
$-\mathrm{div}(\mathrm{m}\psi)=f,\quad|\psi|_{\mathsf{k}}\leq\bar{\mathtt{C}}\mathtt{C}\ell^{-1-\mathsf{k}},\text{
for }\mathsf{k}=0,1,2,3,$

where $\mathtt{C}$ is as in the statement and $\bar{\mathtt{C}}$ is a
universal constant.

For $s\in\mathbb{R}$, we define a map $\Phi_{s}$ and another probability
measure $\widetilde{\mathbf{m}}_{s}$ by:

(A.2)
$\Phi_{s}:=\mathrm{Id}+s\psi,\quad\widetilde{\mathbf{m}}_{s}:=\Phi_{s}\\#\mathbf{m}.$

In view of (A.1) we can guarantee that
$|\Phi_{s}-\mathrm{Id}|_{\mathsf{1}}\leq\frac{1}{2}$ as long as the parameter
$s$ is chosen smaller than $\frac{1}{\mathtt{C}}\ell^{2}$ for some
$\mathtt{C}$ large enough, a condition which is implied by the stronger
constraint (2.14) (recall that $N\ell^{2}$ is always larger than $1$). We
think of $\Phi_{s}$ as an _approximate transport map_ , pushing $\mathbf{m}$
forward not quite onto $\mathbf{m}_{s}$ (whose density is $\mathrm{m}+sf$) but
rather on $\widetilde{\mathbf{m}}_{s}$. The following lemma quantifies the
error in terms of partition functions.

######  Lemma A.1.

The partition functions associated to $\widetilde{\mathbf{m}}_{s}$ and
$\mathbf{m}_{s}$ are close:

$\left|\log\mathrm{K}_{N}^{\beta}(\widetilde{\mathbf{m}}_{s})-\log\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s})\right|=O\left(s^{2}N\ell^{-2}\right),$

and so are their relative entropies:

$\left|\mathcal{E}(\widetilde{\mathrm{m}}_{s})-\mathcal{E}(\mathrm{m}_{s})\right|=O\left(s^{2}N\ell^{-2}\right),$

with implicit constants depending only on (A.1) and $\beta$.

###### Proof.

The first point follows from combining [Ser20, Lemma 5.1], which bounds
$|\widetilde{\mathrm{m}}_{s}-\mathrm{m}_{s}|_{\mathsf{k}}$ for
$\mathsf{k}=1,2$ in terms of the norms of $\psi$ (controlled in (A.1)) and
$\mathbf{m}$ (controlled by assumption), and [Ser20, Lemma 4.9], which states
a direct comparison of the partition functions in terms of
$\widetilde{\mathbf{m}}_{s}-\mathbf{m}_{s}$. To prove the second point, we use
again [Ser20, Lemma 5.1], which bounds
$|\widetilde{\mathrm{m}}_{s}-\mathrm{m}_{s}|_{\mathsf{0}}$ by
$O(s^{2}\ell^{-4})$, and plug that estimate into the definition of
$\mathcal{E}$. ∎

##### _2\. Comparison of partition functions along a transport._

The so-called “anisotropy term” was introduced in [LS18], cf. also the “angle
term” in [BBNY19, Section 8]. We refer to [Ser20, Section 4] for a careful
study of its properties. It can be understood as the first-order correction to
the energy when one pushes _both the configuration and the background measure_
by a small perturbation of the identity map. Here we will not go into the
details and we treat the anisotropy as a black box. The key estimates that we
need to import are contained in the following lemma. Let
$s,\psi,\Phi_{s},\widetilde{\mathbf{m}}_{s}$ be as above (in particular $\psi$
is supported on $\square(z,\ell)$) and recall that $\mathsf{EnerPts}(z,\ell)$
(defined in (2.5)) controls both the energy and the number of points at scale
$\ell$ near $z$, and is typically of order $N\ell^{2}$.

######  Lemma A.2.

There exists a term $\mathsf{A}_{\mathsf{1}}[\psi,\mathbf{m},\mathrm{X}_{N}]$
(independent of $s$) satisfying:

(A.3)
$\mathsf{A}_{\mathsf{1}}[\psi,\mathbf{m},\mathrm{X}_{N}]=\ell^{-2}O\left(\mathsf{EnerPts}(z,\ell)\right),$

and such that provided $|s|$ is smaller than $\ell^{2}$:

(A.4)
$\mathsf{F}\left(\Phi_{s}(\mathrm{X}_{N}),\widetilde{\mathbf{m}}_{s}\right)=\mathsf{F}(\mathrm{X}_{N},\mathbf{m})+s\mathsf{Ani}[\psi,\mathrm{X}_{N},\mathbf{m}]+s^{2}\mathsf{ErrorF},$

with a “second order” error term $\mathsf{ErrorF}$ bounded by:

(A.5) $\mathsf{ErrorF}=O\left(\ell^{-4}(1+\log(\ell
N^{1/2}))\right)\mathsf{EnerPts}(z,\ell).$

with implicit constants in (A.3) and (A.5) depending only on the constant
$\mathtt{C}$ in (2.12), (2.13).

###### Proof of Lemma A.2.

We apply [Ser20, Prop. 4.2] to the vector field $\psi$ chosen above. The main
task is to check that [Ser20, (4.7)], which bounds the second derivative of
the energy along a transport, can itself be controlled by our
$\mathsf{ErrorF}$.

  * •

We use (A.1) to bound $\psi$ and its derivatives.

  * •

Using our assumptions on $\mathrm{m}$ and $f$ we can control the derivatives
of $\mathrm{m}_{s}$ by:

$|\mathrm{m}_{s}|_{\mathsf{k}}\leq|\mathrm{m}|_{\mathsf{k}}+s|f|_{\mathsf{k}}\leq\mathtt{C}\ell^{-\mathsf{k}}+s\mathtt{C}\ell^{-\mathsf{k}+2},$

but we are taking $|s|\ll\ell^{2}$ (see (2.14) and (2.6)) so we can replace
$|\mathrm{m}_{s}|_{\mathsf{k}}$ by $\mathtt{C}\ell^{-\mathsf{k}}$ for
$\mathsf{k}=1,2$. In particular,
$N^{-\mathsf{k}/2}|\mathrm{m}_{s}|_{\mathsf{k}}\preceq 1$ for
$\mathsf{k}=1,2$.

Combining these observations and studying [Ser20, (4.8)], we are left with:

$\left(\ell^{-4}\mathtt{C}+\ell^{-5}N^{-1/2}\log(\ell
N^{1/2})\mathtt{C}\right)\times\log(\ell
N^{1/2})\mathsf{EnerPts}(z,\ell)+\mathtt{C}\ell^{-5}N^{-1/2}\mathsf{EnerPts}(z,\ell).$

Simplifying a bit and using that $\frac{1}{\ell^{2}N}\leq 1$, we can indeed
take $\mathsf{ErrorF}$ as in (A.5). ∎

For convenience, we denote by $\mathsf{Ani}$ the quantity:

(A.6)
$\mathsf{Ani}[\psi,\mathbf{m},\mathrm{X}_{N}]:=\mathsf{A}_{\mathsf{1}}[\psi,\mathbf{m},\mathrm{X}_{N}]-\frac{1}{4}\sum_{i=1}^{N}\mathrm{div}\
\psi(x_{i}).$

We can now state the conclusion of this paragraph:

######  Lemma A.3 (Comparison of partition functions along a transport).

We have:

(A.7)
$\log\frac{\mathrm{K}_{N}^{\beta}(\widetilde{\mathbf{m}}_{s})}{\mathrm{K}_{N}^{\beta}(\mathbf{m})}=\left(\frac{\beta}{4}-1\right)N\left(\mathcal{E}(\widetilde{\mathrm{m}}_{s})-\mathcal{E}(\mathrm{m})\right)\\\
+\log\mathbb{E}_{\mathbb{P}^{\beta}_{N,\mathbf{m}}}\left[\exp\left(s\mathsf{Ani}[\psi,\mathrm{X}_{N},\mathbf{m}]+s^{2}\mathsf{ErrorF}\right)\right].$

We recall that the notation $\mathbb{P}^{\beta}_{N,\mathbf{m}}$ was introduced
in (2.2).

###### Proof of Lemma A.3.

It follows the same steps as [LS18, Prop. 4.3]. In short:

  1. (1)

We change variables by $\Phi_{s}$ in the integral defining
$\mathrm{K}_{N}^{\beta}(\widetilde{\mathbf{m}}_{s})$ and use Lemma A.2 to
expand the interaction energy, hence the terms $\mathsf{A}_{\mathsf{1}}$ and
$\mathsf{ErrorF}$ appear in the exponential. We replace
$\mathsf{A}_{\mathsf{1}}$ by $\mathsf{Ani}$ up to the correction mentioned in
(A.6).

  2. (2)

The Jacobian of this change of variables is
$\exp\left(\sum_{i=1}^{N}\log\mathrm{det}\mathrm{D}\Phi_{s}(x_{i})\right)$.
The exponent can be related to the difference of entropies between
$\mathbf{m}_{s}$ and $\mathbf{m}$, up to a fluctuation term (the fluctuations
of $x\mapsto\log\mathrm{det}\mathrm{D}\Phi_{s}(x)$), which itself is bounded
using Lemma 2.3 and is negligible compared to the estimate (A.5) for the
“second order” error $\mathsf{ErrorF}$.

  3. (3)

The difference between $\mathsf{Ani}$ and $\mathsf{A}_{\mathsf{1}}$ as in
(A.6) also happens to be equal to the difference of relative entropies (up to
some multiplicative factor) plus an error term that is negligible compared to
$\mathsf{ErrorF}$.

∎

##### _3\. Relative expansion of partition functions._

The following is contained in [Ser20, Prop. 6.4] (keeping in mind the remark
of Section A.1).

######  Lemma A.4 (Relative expansion of partition functions).

Let $(z,\ell)$ such that (2.6) holds, and let
$\mathbf{m},\widetilde{\mathbf{m}}$ be two probability measures on
$\mathsf{D}$ such that $\mathrm{m}=\widetilde{\mathrm{m}}$ outside
$\mathsf{D}(z,\ell)$. Assume that $\mathrm{m},\widetilde{\mathrm{m}}$ are of
class $C^{1}$ and satisfy
$\frac{1}{2}\leq\mathrm{m},\widetilde{\mathrm{m}}\leq 2$ on $\mathsf{D}$.
Assume also that
$|\mathrm{m}|_{\mathsf{1}}+|\widetilde{\mathrm{m}}|_{\mathsf{1}}\leq\mathtt{C}\ell^{-1}$
for some constant $\mathtt{C}$. Then:

(A.8)
$\log\frac{\mathrm{K}_{N}^{\beta}(\widetilde{\mathbf{m}})}{\mathrm{K}_{N}^{\beta}(\mathbf{m})}=\left(\frac{\beta}{4}-1\right)N\left(\mathcal{E}(\widetilde{\mathrm{m}})-\mathcal{E}(\mathrm{m})\right)+O\left(N\ell^{2}\right)^{\frac{1}{2}}\left(1+\log\left(N\ell^{2}\right)\right)^{\frac{1}{2}},$

with an implicit constant depending only on $\mathtt{C}$ and $\beta$.

##### _4\. Serfaty’s trick._

It is hard to prove a bound on $\mathsf{Ani}$ that is better than (A.3) and
holds _configuration-wise_. However one can improve the control on
$\mathsf{Ani}$ _in exponential moments_ , using the following trick. Recall
the assumption (2.6).

######  Lemma A.5 (Smallness of the anisotropy).

There exists $\mathtt{C}^{\prime}$ depending only on on the constant
$\mathtt{C}$ in (2.12), (2.13) such that if
$|s|\leq\frac{1}{\mathtt{C}^{\prime}}\ell^{3/2}N^{-1/4}$, we have:

(A.9)
$\log\mathbb{E}_{\mathbb{P}^{\beta}_{N,\mathbf{m}}}\left[\exp\left(s\mathsf{Ani}[\psi,\mathrm{X}_{N},\mathbf{m}]\right)\right]=O\left(sN^{3/4}\ell^{-1/2}\left(1+\log\left(N\ell^{2}\right)\right)^{\frac{1}{2}}\right),$

with an implicit constant depending only on $\mathtt{C}^{\prime}$ and $\beta$.

###### Proof of Lemma A.5.

We first take $s=s_{\star}:=\frac{1}{\mathtt{C}^{\prime}}\ell^{3/2}N^{-1/4}$
for some large enough constant $\mathtt{C}^{\prime}$ (if $\ell$ is of order
$N^{-1/2}$ then $\ell^{2}$ and $\ell^{3/2}N^{-1/4}$ are comparable, thus we
need to divide by $\mathtt{C}^{\prime}$ large enough in order to match our
previous assumptions on $s$. For mesoscopic length scales, this is
irrelevant). By comparing the two expressions (A.7) and (A.8) and discarding
negligible terms one gets:

(A.10)
$\log\mathbb{E}_{\mathbb{P}^{\beta}_{N,\mathbf{m}}}\left[\exp\left(s_{\star}\mathsf{Ani}[\psi,\mathrm{X}_{N},\mathbf{m}]+s_{\star}^{2}\mathsf{ErrorF}\right)\right]=O\left(\left(N\ell^{2}\right)^{\frac{1}{2}}\left(1+\log\left(N\ell^{2}\right)\right)^{\frac{1}{2}}\right).$

Using the expression (A.5) for $\mathsf{ErrorF}$ and the local law (2.7) we
know that:

(A.11)
$\log\mathbb{E}_{\mathbb{P}^{\beta}_{N,\mathbf{m}}}\left[\exp\left(s_{\star}^{2}\mathsf{ErrorF}\right)\right]\leq
O\left(\ell^{-1}N^{-1/2}\log(\ell N^{1/2})N\ell^{2}\right)=O\left(\ell
N^{1/2}\log(\ell N^{1/2})\right).$

Combining (A.10) and (A.11) and using Cauchy-Schwarz’s inequality we deduce
that:

$\log\mathbb{E}_{\mathbb{P}^{\beta}_{N,\mathbf{m}}}\left[\exp\left(\frac{1}{2}s_{\star}\mathsf{Ani}[\psi,\mathrm{X}_{N},\mathbf{m}]\right)\right]\leq
O\left(\left(N\ell^{2}\right)^{\frac{1}{2}}\left(1+\log\left(N\ell^{2}\right)\right)^{\frac{1}{2}}\right)+O\left(\ell
N^{1/2}\log(\ell N^{1/2})\right),$

and thus for $s=\frac{1}{2}s_{\star}$:

$\log\mathbb{E}_{\mathbb{P}^{\beta}_{N,\mathbf{m}}}\left[\exp\left(\frac{1}{2}s_{\star}\mathsf{Ani}[\psi,\mathrm{X}_{N},\mathbf{m}]\right)\right]=O\left(\left(N\ell^{2}\right)^{\frac{1}{2}}\left(1+\log\left(N\ell^{2}\right)\right)^{\frac{1}{2}}\right)$

For smaller values of $s$, we apply Hölder’s inequality and obtain (A.9). ∎

##### _5\. Conclusion_

We compare $\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s})$ and
$\mathrm{K}_{N}^{\beta}(\widetilde{\mathbf{m}}_{s})$ using Lemma A.1 and then
apply Lemma A.3 to compare
$\mathrm{K}_{N}^{\beta}(\widetilde{\mathbf{m}}_{s})$ and
$\mathrm{K}_{N}^{\beta}(\mathbf{m})$, using Lemma A.5 to control the
anisotropy term. Lemma A.1 also allows us to replace
$\mathcal{E}(\widetilde{\mathrm{m}}_{s})$ by $\mathcal{E}(\mathrm{m}_{s})$ up
to some error. Finally one can check that for $|s|\leq s_{\star}$ the dominant
error term is the one coming from (A.9), which yields (2.15). ∎

######  Remark A.6.

In view of the remark made in Section A.1, one could wonder whether a
difference of order $O(1)$ between the value of
$\log\mathrm{K}_{N}^{\beta}(\mathbf{m})$ as defined in [Ser20] and the
definition used here might not create a problem when the length scale $\ell$
is mesoscopic. Indeed in that case Proposition 2.5 is stated with an error
term that is $o(1)$, see Remark 2.6. In fact, one only needs a comparison of
partition functions as stated in [Ser20] in steps 3 and 4 of the proof of
Proposition 2.5, and we apply Lemma A.4 (which is a direct consequence of
[Ser20, Prop. 2.4]) for a “large” value of $s$ (namely $s=s_{\star}$ of order
$\ell^{3/2}N^{-1/4}\gg N^{-1}$ for $\ell$ mesoscopic). At that point the
effective error term is larger than $1$ (see (A.10)) and thus one can
harmlessly incorporate an additional $O(1)$ error.

On the other hand, the comparison result of Lemma A.3 does not rely on [Ser20]
regarding the definition of partition functions.

####  A.2.2. Proof of Lemma 2.7

Compared to the previous proof, we dispense with Step 1 as one can easily find
an exact transport, as well as Steps 3 and 4 because we are not aiming for
precise estimates on the anisotropy.
Since the reference measure is $\mathbf{m}_{0}$ and since $f$ has radial
symmetry, it is easy to construct a bijective, “radial rearrangement” map
$\Phi_{s}:\mathsf{D}\to\mathsf{D}$ that pushes $\mathbf{m}_{0}$ onto
$\mathbf{m}_{0}+sf$ and can be written as
$\Phi_{s}=\mathrm{Id}+\alpha_{s}(x)x$ on $\mathsf{D}$, for some radial
function $\alpha_{s}$ whose derivative is bounded on $\mathsf{D}$ by
$s\mathtt{C}^{\prime\prime}$ for some constant $\mathtt{C}^{\prime\prime}$
depending only on $\mathtt{C}$ and $r$. We can always extend $\alpha_{s}$ into
a $C^{1}$, compactly supported function on $\mathsf{D}_{2}$ with
$|\alpha_{s}|\leq\frac{1}{2},|\alpha_{s}|_{1}\leq s\mathtt{C}^{\prime\prime}$.
Applying [Ser20, Prop 4.2] (note that now, compared to the proof of
Proposition 2.5, we are only able to use - and in fact only need - results
where the vector field $\psi$ is simply assumed to be $C^{1}$, see in
particular [Ser20, (4.6)]) to $\Phi_{s}$ and changing variables as in the
previous proof, we obtain (see also [Ser20, (4.3)])

$\log\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}^{\prime}_{s})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0})}=\log\mathbb{E}\left[\exp\left(s\mathtt{C}^{\prime\prime}\mathsf{EnerPts}(\mathsf{D}_{2})\right)\right],$

but the number of points in $\mathsf{D}_{2}$ is always $\leq N$ and the
(fluctuations of the) global energy have exponential moments of order $N$
(this “global law” follows from (2.4)). We thus get (2.16). ∎

###  A.3. Proof of Proposition 2.8

###### Proof of Proposition 2.8.

We introduce a sequence of intermediate length scales
$\ell_{a}=:\ell_{0}<\dots<\ell_{n}:=\ell:=\min\left(\ell_{b},\frac{1}{4}\left(1-\mathrm{dist}(z_{a},\partial\mathsf{D})\right)\right)$
by defining $c,n$ as:

$n=\lfloor\log\left(\ell/\ell_{a}\right)\rfloor,\quad\log
c:=\frac{1}{n}\log\left(\ell/\ell_{a}\right),$

and by setting $\ell_{k}:=c^{k}\ell_{a}$ for $k=0,\dots,n$. Note that
$c\in[1,2]$ and $(z_{a},\ell_{i})$ satisfy (2.6) for all $i\leq n$ even if the
larger length scale $\ell_{b}$ is macroscopic. Moreover, $\ell_{b}$ and $\ell$
are always comparable.

For each $i=0,\dots,n$ we fix a smooth cut-off function $\chi^{(i)}$ such
that:

$\chi^{(i)}\equiv 1\text{ on }\mathsf{D}(z_{a},\ell_{i}),\quad\chi^{(i)}\equiv
0\text{ outside
}\mathsf{D}(z_{a},2\ell_{i}),\quad|\chi^{(i)}|_{\mathsf{k}}\leq
100\ell_{i}^{-\mathsf{k}}\text{ for }\mathsf{k}=1,2,3,$

and we define $\mathbf{m}_{s}^{(i)}$ as the probability measure with the
following density:

(A.12)
$\mathrm{m}_{s}^{(i)}:=\mathrm{m}+sf_{b}+s\left(\int_{\mathbb{R}^{2}}f_{a}\right)\frac{\chi^{(i)}}{\int_{\mathbb{R}^{2}}\chi^{(i)}}.$

In particular, in case $\int f_{a}=0$, this construction is not relevant and
we can directly apply Proposition 2.5 twice to compare
${\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s})}$ to
${\mathrm{K}_{N}^{\beta}(\mathbf{m}+sf_{b})}$ and then
${\mathrm{K}_{N}^{\beta}(\mathbf{m}+sf_{b})}$ to
${\mathrm{K}_{N}^{\beta}(\mathbf{m})}$. This is also the case if the scales
$\ell_{a}$ and $\ell_{b}$ are comparable, which corresponds to the case where
$n$ is independent of $N$.

Using our assumptions on $\mathbf{m},f_{a},f_{b},\chi^{(i)}$ and the fact that
(2.19) and (2.6) imply that $|s|$ is of order at most $\ell_{a}^{2}$ (the
minimal scale in this problem), we obtain that on $\mathsf{D}$ the density
$\mathrm{m}_{s}^{(i)}$ satisfies:

(A.13) $\frac{1}{2}\leq\mathrm{m}_{s}^{(i)}\leq
2,\quad|\mathrm{m}_{s}^{(i)}|_{\mathsf{k}}\preceq\mathtt{C}\ell_{i}^{-\mathsf{k}}\text{
for }\mathsf{k}=1,2,3.$

We decompose the ratio of partition functions as :

$\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s})}{\mathrm{K}_{N}^{\beta}(\mathbf{m})}=\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s}^{(0)})}\times\prod_{i=0}^{n-1}\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s}^{(i)})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s}^{(i+1)})}\times\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s}^{(n)})}{\mathrm{K}_{N}^{\beta}(\mathbf{m})}.$

For each $i=0,\dots,n-1$, in view of (A.12) we may write:

$\mathrm{m}_{s}^{(i)}=\mathrm{m}_{s}^{(i+1)}+sf^{(i)},\text{ with
}f^{(i)}:=s\left(\int_{\mathbb{R}^{2}}f_{a}\right)\left(\frac{\chi^{(i)}}{\int_{\mathbb{R}^{2}}\chi^{(i)}}-\frac{\chi^{(i+1)}}{\int_{\mathbb{R}^{2}}\chi^{(i+1)}}\right).$

Since $\int f_{a}$ is of order $1$ (this follows from (2.18)), while
$\int\chi^{(i)}$ is of order $\ell_{i}^{2}$, the perturbation $f^{(i)}$
satisfies:

(A.14) $|f^{(i)}|_{\mathsf{k}}\preceq\ell_{i}^{-(\mathsf{k}+2)}\text{ for
}\mathsf{k}=0,1,2.$

In view of (A.13) and (A.14) we can apply Proposition 2.5 with the length
scale chosen as $\ell_{i}$ and we obtain:

$\log\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s}^{(i+1)})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s}^{(i)})}\\\
=N\left(\frac{\beta}{4}-1\right)\left(\mathcal{E}(\mathrm{m}_{s}^{(i+1)})-\mathcal{E}(\mathrm{m}_{s}^{(i)})\right)+O\left(sN^{3/4}\ell_{i}^{-1/2}\left(1+\log\left(N\ell_{i}^{2}\right)\right)^{\frac{1}{2}}\right),$

The same reasoning applies for
$\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s})}{\mathrm{K}_{N}^{\beta}(\mathbf{m}^{(0)})}$
(which corresponds to the smallest length scale $\ell_{a}$) and for
$\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s}^{(n)})}{\mathrm{K}_{N}^{\beta}(\mathbf{m})}$
(which corresponds to the largest length scale $\ell$ – either equal to
$\ell_{b}$ or of order $1$). Summing the contributions, we obtain:

$\log\frac{\mathrm{K}_{N}^{\beta}(\mathbf{m}_{s})}{\mathrm{K}_{N}^{\beta}(\mathbf{m})}=N\left(\frac{\beta}{4}-1\right)\left(\mathcal{E}(\mathrm{m}_{s})-\mathcal{E}(\mathrm{m})\right)+O\left(s\sum_{i=0}^{n}N^{3/4}\ell_{i}^{-1/2}\left(1+\log\left(N\ell_{i}^{2}\right)\right)^{\frac{1}{2}}\right).$

The sum over dyadic scales can be compared to an integral and we get:

$\sum_{i=0}^{n}N^{3/4}\ell_{i}^{-1/2}\left(1+\log\left(N\ell_{i}^{2}\right)\right)^{\frac{1}{2}}=O\left(N^{3/4}\ell_{a}^{-1/2}\left(1+\log(N\ell_{a}^{2})\right)^{\frac{1}{2}}\right),$

which yields (2.20) as claimed. ∎

###  A.4. Proof of Proposition 3.2

Let $\mathfrak{h}_{0}$ denote the electrostatic potential generated by the
uniform background $\mathbf{m}_{0}$, namely:

$\mathfrak{h}_{0}:z\mapsto\int_{\mathsf{D}}\log|z-x|\mathrm{d}\mathbf{m}_{0}(x)=\int_{|x|\leq
1}\log|z-x|\frac{\mathrm{d}x}{\pi},$

a quantity that already appeared in the definition (1.7) of
$\mathrm{Pot}_{N}$. An explicit computation gives the following expression

(A.15) $\mathfrak{h}_{0}:z\mapsto\begin{cases}\log|z|&\text{if }|z|\geq 1\\\
-\frac{1-|z|^{2}}{2}&\text{ if }|z|\leq 1.\end{cases}$

The potential $\mathfrak{h}_{0}$ satisfies Poisson’s equation
$\Delta\mathfrak{h}_{0}=2\pi\mathbf{m}_{0}$. In particular,
$\Delta\mathfrak{h}_{0}$ does _not_ have total mass $0$. Yet, using a simple
trick that is totally unrelated to the methods of [LS18, Ser20], we are able
to control the size of $\mathbf{L}_{N}(\mathfrak{h}_{0})$ as expressed in the
following proposition:

######  Proposition A.7 (Exponential moments of $\mathfrak{h}_{0}$).

We have, for $|t|\leq\frac{N\beta}{100}$:

$\log\mathbb{E}\left[\exp\left(t\mathbf{L}_{N}(\mathfrak{h}_{0})\right)\right]=O(t+t^{2}),$

with an implicit constant depending only on $\beta$.

Proposition A.7 gives us the crucial freedom to substract a multiple of
$\mathfrak{h}_{0}$ from a given test function and to cancel out the masses of
their Laplacians, while making a (typically) bounded error on the size of the
linear statistics. We postpone the proof of Proposition A.7 for now and deduce
from it a generalization of Proposition 3.2.

######  Corollary A.8.

Let $\chi$ be a $C^{2}$, radially symmetric function which is compactly
supported in $\mathsf{D}_{r}$ for some $r<1$, such that
$|\chi|_{\mathsf{k}}\leq\mathtt{C}$ for $\mathsf{k}=0,1,2$. Let $\mathrm{g}$
be a solution of Poisson’s equation $\Delta\mathrm{g}=2\pi\chi$. Then, one has
for $|t|\ll N$,

$\log\mathbb{E}[e^{t\mathbf{L}_{N}(\mathrm{g})}]=O(t+t^{2})$

where the implied constant depends only on $\beta$.

###### Proof.

Let $c=\int\chi$ which is of order 1 (by assumptions) and let
$\varphi=\mathrm{g}-c\mathfrak{h}_{0}$. By Lemmas 2.4 and 2.7, one has for
$|t|\ll N$,

$\log\mathbb{E}[e^{t\mathbf{L}_{N}(\varphi)}]=O(t+t^{2}).$

This estimate is comparable to that of Lemma A.7, thus the claim follows
directly from Hölder’s inequality. ∎

###### Proof of Proposition A.7.

This relies on the following two claims:

######  Claim A.9.

For $|t|\leq\beta N$, we have:

(A.16) $\log\mathbb{E}\left[e^{t\sum_{i=1}^{N}\zeta(x_{i})}\right]=o(t).$

###### Proof.

This follows from the analysis of [LS18]. We return to the notation of (1.4),
(2.3) for partition functions and make them more explicit by writing:

$\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0},\zeta):=\int_{\left(\mathbb{R}^{2}\right)^{N}}\exp\left(-\beta\left(\mathsf{F}(\mathrm{X}_{N},\mathbf{m}_{0})+2N\sum_{i=1}^{N}\zeta(x_{i})\right)\right)\mathrm{d}\mathrm{X}_{N},$

where we explicitly keep track of both the background measure (as in (2.3))
_and_ the “effective confining potential” $\zeta$. By [LS18, Corollary 1.1] or
[LS18, (4.12)] one finds an expansion for
$\log\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0},\zeta)$ up to order $o(N)$ which
_does not depend on $\zeta$_ (see [LS18, Remark 4.3]). Thus in particular,
taking $t=\pm\beta N$ in the left-hand side of (A.16) we get:

$\log\mathbb{E}\left[e^{\pm\beta
N\sum_{i=1}^{N}\zeta(x_{i})}\right]=\log\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0},\zeta\mp\frac{1}{2}\zeta)-\log\mathrm{K}_{N}^{\beta}(\mathbf{m}_{0},\zeta)=o(N).$

The claim follows from Hölder’s inequality. ∎

######  Claim A.10.

For $|t|\leq\frac{N\beta}{2}$, we have:

(A.17)
$\mathbb{E}\left[e^{t\sum_{i=1}^{N}|x_{i}|^{2}}\right]=\exp\left(-\log\left(1-\frac{t}{\beta
N}\right)\left(\frac{\beta N(N-1)}{4}+N\right)\right).$

###### Proof.

It follows by a scaling argument using the equivalent expression (1.6) for the
joint law of the particles. ∎

We may now prove Proposition A.7. On the one hand, using the expression (A.15)
of $\mathfrak{h}_{0}$ and an elementary computation we get:

$\int\mathfrak{h}_{0}(x)\mathrm{d}\mathbf{m}_{0}(x)=\int\frac{|x|^{2}-1}{2}\mathrm{d}\mathbf{m}_{0}(x)=-\frac{1}{4}.$

On the other hand, a Taylor’s expansion of (A.17) gives (for $\frac{|t|}{\beta
N}$ small):

$\mathbb{E}\left[e^{t\sum_{i=1}^{N}|x_{i}|^{2}}\right]=\exp\left(\frac{tN}{4}+O(t+t^{2})\right),$

and we deduce that:

(A.18)
$\mathbb{E}\left[e^{t\left(\sum_{i=1}^{N}\frac{|x_{i}|^{2}-1}{2}\right)-tN\int\mathfrak{h}_{0}(x)\mathrm{d}\mathbf{m}_{0}(x)}\right]=\mathbb{E}\left[e^{\frac{t}{2}\sum_{i=1}^{N}|x_{i}|^{2}-\frac{tN}{2}+\frac{tN}{4}}\right]\\\
=\mathbb{E}\left[e^{\frac{1}{2}\left(\frac{tN}{4}+O(t+t^{2})\right)-\frac{tN}{8}}\right]=\mathbb{E}\left[e^{O(t+t^{2})}\right].$

Since the expression of $\mathfrak{h}_{0}(x)$ is not always
$\frac{|x|^{2}-1}{2}$ but rather $\frac{|x|^{2}-1}{2}-\zeta(x)$ in general
(compare (A.15) with (1.2)), we can write:

$\mathbb{E}\left[e^{t\mathbf{L}_{N}(\mathfrak{h}_{0})}\right]=\mathbb{E}\left[e^{t\left(\sum_{i=1}^{N}\frac{|x_{i}|^{2}-1}{2}\right)-tN\int\mathfrak{h}_{0}(x)\mathrm{d}\mathbf{m}_{0}(x)-t\sum_{i=1}^{N}\zeta(x_{i})}\right].$

Using Cauchy-Schwarz’s inequality combined with (A.16) and (A.18) we prove
Proposition A.7. ∎

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