# The two-dimensional one-component plasma is hyperuniform

Thomas Leblé Université de Paris-Cité, CNRS, MAP5 UMR 8145, F-75006 Paris,
France

(September 3, 2024)

###### Abstract

We prove that at all positive temperatures in the bulk of a classical two-
dimensional one-component plasma (also called Coulomb or log-gas, or jellium)
the variance of the number of particles in large disks grows (strictly) more
slowly than the area. In other words the system is hyperuniform.

##  1 Introduction

Let $N\geq 1$ be an integer, let $\Sigma_{N}$ be the disk of center $0$ and
radius $\sqrt{\frac{N}{\pi}}$, let $\mathrm{X}_{N}:=(x_{1},\dots,x_{N})$
denote a $N$-tuple of points in $\Sigma_{N}$ and let
$\mathbf{X}_{N}:=\sum_{i=1}^{N}\delta_{x_{i}}$ be the associated atomic
measure. We let $\mathbf{f}_{N}$ be the signed “fluctuation” measure on
$\Sigma_{N}$ defined by $\mathbf{f}_{N}:=\mathbf{X}_{N}-\mathbf{m}_{0}$, where
$\mathbf{m}_{0}$ is the Lebesgue measure on $\Sigma_{N}$. We think of
$\mathbf{X}_{N}$ as a collection of _point particles_ in $\Sigma_{N}$ and of
$\mathbf{m}_{0}$ as the _background measure_. We define the logarithmic
interaction energy $\mathsf{F}_{N}(\mathbf{X}_{N})$ as:

$\mathsf{F}_{N}(\mathbf{X}_{N}):=\frac{1}{2}\iint_{(x,y)\in\Sigma_{N}\times\Sigma_{N},x\neq
y}-\log|x-y|\mathtt{d}\mathbf{f}_{N}(x)\mathtt{d}\mathbf{f}_{N}(y).$ (1.1)

Let $\beta$ be a positive real number that will be fixed throughout the paper.
We define a probability density $\mathbb{P}^{\beta}_{N}$ on the space of
$N$-tuples of points in $\Sigma_{N}$ by setting:

$\mathtt{d}\mathbb{P}^{\beta}_{N}(\mathrm{X}_{N}):=\frac{1}{\mathrm{K}^{\beta}_{N}}\exp\left(-\beta\mathsf{F}_{N}(\mathbf{X}_{N})\right)\mathtt{d}\mathrm{X}_{N},$
(1.2)

with a normalizing constant
$\mathrm{K}^{\beta}_{N}:=\int_{\Sigma_{N}\times\dots\times\Sigma_{N}}\exp\left(-\beta\mathsf{F}_{N}(\mathbf{X}_{N})\right)\mathtt{d}\mathrm{X}_{N}$
called the _partition function_. Here and below, $\mathtt{d}\mathrm{X}_{N}$
denotes the Lebesgue measure on the Cartesian product
$\Sigma_{N}\times\dots\times\Sigma_{N}\subset(\mathbb{R}^{2})^{N}$. The
probability measure $\mathbb{P}^{\beta}_{N}$ is the canonical Gibbs measure of
the two-dimensional one-component plasma (2DOCP) at inverse temperature
$\beta$. We denote the expectation under $\mathbb{P}^{\beta}_{N}$ by
$\mathbb{E}^{\beta}_{N}$.

Let $\Omega$ some (Borel measurable) subset of $\Sigma_{N}$. We denote by
$\mathsf{Pts}(\mathbf{X}_{N},\Omega)$ the number of points of $\mathbf{X}_{N}$
in $\Omega$ and by $\mathsf{Dis}(\mathbf{X}_{N},\Omega)$ the discrepancy (of
$\mathbf{X}_{N}$) in $\Omega$, defined by:

$\mathsf{Dis}(\mathbf{X}_{N},\Omega):=\mathsf{Pts}(\mathbf{X}_{N},\Omega)-\mathbf{m}_{0}(\Omega).$
(1.3)

By construction we have $\mathsf{Dis}(\mathbf{X}_{N},\Sigma_{N})=0$, which
corresponds to the fact that the system is globally neutral. However it cannot
be perfectly locally neutral, and $\mathsf{Dis}(\mathbf{X}_{N},\Omega)$ is
meant to measure charge fluctuations in $\Omega$. The number variance in
$\Omega$ is defined as the variance under $\mathbb{P}^{\beta}_{N}$ of
$\mathsf{Pts}(\mathbf{X}_{N},\Omega)$, or equivalently of
$\mathsf{Dis}(\mathbf{X}_{N},\Omega)$.

For all $x$ in $\mathbb{R}^{2}$ and $R>0$, we let $\mathfrak{D}(x,R)$ be the
disk of center $x$ and radius $R$.

###  1.1 Main result

Hyperuniformity of a system is defined by the fact that:

> [Tor18, Section 1] (…) the number variance of particles within a spherical
> observation window of radius $R$ grows more slowly than the window volume in
> the large-$R$ limit.

Our main conclusion is the following:

######  Theorem (The 2DOCP is hyperuniform).

For each $N\geq 1$, let $x=x(N)$ be a point in the bulk of $\Sigma_{N}$ and
let $R=R(N)$ be such that $R(N)\to\infty$. Then the number variance in
$\mathfrak{D}(x,R)$ is $o(R^{2})$ as $N\to\infty$.

We give a more precise statement in Section 1.5, explaining what we mean by
“the bulk of $\Sigma_{N}$” and specifying a quantitative upper bound on the
number variance of the form
$\mathcal{O}\left(\frac{R^{2}}{\log^{c}(R)}\right)$ for some $c>0$.

Hyperuniformity of the 2DOCP at all positive temperatures is a forty year old
prediction111Those results are rigorous to the extent that authors make use of
so-called “clustering assumptions”, i.e. they assume properties of the two-
point correlation function at large distances in order to derive certain
identities (called “sum rules”) which, among other things, imply
hyperuniformity. As explained in [Mar88]: “The results obtained in this way
are exact (i.e. do not follow from approximations), but not all of them are
rigorously proven, in so far as some reasonable properties (e.g. the type of
decay of the correlations) are assumed to hold a priori.”. Unfortunately,
obtaining mathematically rigorous statements about the large-distance
properties of the 2DOCP’s two-point correlation function (for $\beta\neq 2$)
is extremely challenging. from statistical physics, see [MY80, Leb83, Mar88,
JLM93, LWL00] (which use a different terminology), or [Tor18] again: “OCP
fluid phases at [all] temperatures must always be hyperuniform”. In fact, the
full physical prediction says not only that the number variance is negligible
with respect to the area of the disk, but that it should even be _comparable
to the perimeter_ ($\mathcal{O}(R)$ and not only $o(R^{2})$). We give here the
first mathematical proof of hyperuniformity at all temperatures, however our
upper bound on the number variance remains far from the conjectured sharp
estimate.

###  1.2 The 2DOCP

The two-dimensional _one-component plasma_ is a well-studied model of
statistical physics. Besides the papers cited above, let us refer to [For10,
Chapter 15] for a presentation of exact and approximate results. When defining
the energy $\mathsf{F}_{N}$ in (1.1) and the canonical Gibbs measure
$\mathbb{P}^{\beta}_{N}$ in (1.2), we think of $(x_{1},\dots,x_{N})$ as the
positions of point particles in $\Sigma_{N}$ which all carry the same electric
charge $+1$, and which are immersed in a uniform neutralizing background of
constant density on $\Sigma_{N}$. The logarithmic potential is then the
electrostatic interaction potential in dimension $2$. Instead of directly
placing a background measure, one sometimes imposes an harmonic confining
potential. This would not change our analysis, see Appendix A for a
discussion.

#### Mathematical aspects of Coulomb gases

As mathematical objects, Coulomb gases (under various forms: one-, two- or
three-dimensional, with one or two components…) have attracted much interest.
Concerning the 2DOCP itself, topics that have been investigated in the last
years alone are very diverse and include: lower bounds on the minimal distance
between points [Ame18], concentration inequalities for the empirical measure
of the particles [CHM18, GZ19a], upper bounds for the local density of points
[LRY19], generalizations to Riemannian manifolds [Ber19], Wegner’s and
clustering estimates [Tho22]… to quote only a few among many others. We refer
to [Ser18] for an overview of motivations, ranging from constructive
approximation to the study of the Quantum Hall Effect, via random matrix
theory. The long-range and singular nature of the pairwise potential raises
considerable analytic challenges.

In [LS18, BBNY19, Ser20] it was shown that the 2DOCP exhibits strong forms of
rigidity at all scales and for all values of the temperature as far as
fluctuations of _smooth_ linear statistics are concerned. Regarding
fluctuations of charges, i.e. the statistics of an indicator function, [AS21]
(see also [Leb17, BBNY17] for weaker controls) imply that the discrepancy
within a disk of radius $R$ is $\mathcal{O}(R)$ with high probability. Our
goal here is improve such bounds to $o(R)$ and to thus prove hyperuniformity.

#### The Ginibre ensemble

When the inverse temperature parameter $\beta$ is equal to $2$, the
probability measure $\mathbb{P}^{\beta}_{N}$ admits an interpretation as the
joint law of the complex eigenvalues of a $N\times N$ non-Hermitian random
matrix, known as the (finite) Ginibre ensemble, after [Gin65]. The model
belongs to the class of determinantal processes (see e.g. [HKPV06, Section
4.3.7]) and is amenable to exact computations. In that specific case, the
standard deviation of the number of points in a disk of radius $R$ is known:
it scales like $R^{\frac{1}{2}}$ (in accordance with predictions from
physics), see [Shi06] and [OS08]. Thus for $\beta=2$ our result is not new,
and our bound is far from the optimal one.

#### The hierarchical model

In [Cha19], S. Chatterjee studied Coulomb gases in dimensions $\\{1,2,3\\}$
(see [GS20] for an extension to higher dimensions), giving sharp estimates on
the number variance up to logarithmic corrections. His model is a hierarchical
Coulomb gas, where the physical space is decomposed into a tree-like structure
following an old recipe of Dyson. The hierarchical model has two built-in
properties: conditional independence of sub-systems and a self-similar nature.
A large part of the present analysis is devoted to finding approximate
analogues of those features for the “true” (non-hierarchical) model.

###  1.3 Hyperuniformity

The term “hyperuniform(ity)” has been coined in the theoretical chemistry
literature by S. Torquato (see [TS03, Tor18] for surveys), an alternative
terminology due to J. Lebowitz is “superhomogeneous/ity”. As we already
mentioned, a system is hyperuniform when the number variance in a large ball
is asymptotically negligible with respect to the volume of said ball. Of
course the definition of hyperuniformity needs an adaptation for finite
systems, as one should take both the size of the system and of the “spherical
window” to infinity. We refer to [Cos21] for a mathematical presentation of
hyperuniformity and a survey of related results.

#### Examples of hyperuniform systems

For non-interacting random particles (forming a Bernoulli point process or a
Poisson point process with constant intensity) the number variance grows
exactly like the volume, hence hyperuniformity can be thought of as the
property of systems that are “much more rigid” than independent particles
regarding discrepancy at large scales. Examples of two-dimensional objects
that are proven to be hyperuniform include: the Ginibre ensemble (as mentioned
above) and its generalizations (see [CE19] for a recent result in connection
with the Quantum Hall Effect), averaged and perturbed lattices (see [GS75]),
the zeroes of random polynomials with i.i.d Gaussian coefficients (see [FH99,
SZ08])…

In some cases (see [Tor18, Section 5.3.2]) the number variance grows as the
perimeter (in fact that was initially chosen as the definition of
hyperuniformity in [TS03] but is now called “class I hyperuniformity”), which
is the slowest possible growth (see [Bec87]). The 2DOCP is predicted to be
class I hyperuniform at all temperatures, with good tails on the probability
of large charge fluctuations, as we explain next.

#### The Jancovici-Lebowitz-Manificat law

In [JLM93], Jancovici, Lebowitz and Manificat made precise predictions
concerning the probability of observing large charge fluctuations within the
2DOCP (and its three-dimensional version). Their statement is significantly
more precise222Tail estimates as in (1.4) readily imply the strongest type of
hyperuniformity (Type I). than hyperuniformity, as they argue that for all
$\alpha>\frac{1}{2}$:

$\mathbb{P}\left[\text{Discrepancy of size }R^{\alpha}\text{ in a disk of
radius }R\right]\sim\exp(-R^{\varphi(\alpha)})\quad\textbf{(“JLM law”)}$ (1.4)

where the rate $\varphi(\alpha)>0$ is an explicit piecewise affine function of
$\alpha$. This was later checked for $\beta=2$ through explicit computations,
see [Shi06, FL21], while the general case is open. Remarkably, although the
original prediction of [JLM93] deals with Coulomb gases, it was first verified
in [NSV08] for a different model, namely the zeros of the Gaussian Entire
Function (see [GN18] for a survey of related results).

#### Fekete points and the low temperature regime

A situation of interest that we do not consider here is the case of energy
minimizers, which formally corresponds to taking $\beta=+\infty$ i.e. a zero
temperature. A long-standing conjecture posits that minimizers of
$\mathsf{F}_{N}$ form a lattice as $N\to\infty$ and in that regard Fekete
points should exhibit excellent rigidity properties at finite $N$. For studies
of such questions we refer to [AOC12, Ame17] and, in a different line of work,
to [NS15, Thm. 3]. See also [AR22, MR22] or the $\beta$-dependent statements
of [AS21] for a more general “low-temperature” regime where $\beta$ is large
but not necessarily infinite.

###  1.4 Open questions

Besides the obvious natural goal of obtaining the sharp number variance
estimate as well as proving the aforementioned “JLM laws”, let us mention
three other open directions of research.

##### The 3DOCP.

The claims of [JLM93] are made in dimension $2$ _and_ $3$, moreover sharp
hyperuniformity estimates were obtained for the _hierarchical_ model in
dimension $3$ in [Cha19], it is thus natural to ask whether one can prove
hyperuniformity for the 3DOCP. Unfortunately, in dimension $3$ there is no
(known) value of $\beta$ for which the model would be integrable and
predictions could be tested. Some understanding of fluctuations of smooth
linear statistics was recently obtained in [Ser20] but with considerable new
difficulties compared to the two-dimensional case.

##### Riesz interactions.

The 2DOCP can be seen as a member of one-parameter family of two-dimensional
systems called _Riesz gases_ for which the interaction potential is taken as a
certain power $|x-y|^{-\mathsf{s}}$ of the distance (the case $\mathsf{s}=0$
corresponds by convention to the logarithmic interaction). Rigidity properties
of those systems depend on the value of $\mathsf{s}$ as it governs both the
level of repulsiveness at $0$ and, most importantly, the long- or short-range
nature of the interaction. Rough bounds on the discrepancies were given in
[LS17, Lemma 3.2] in the infinite-volume limit, which say that the number
variance in a disk of radius $R$ grows at most as $R^{2+\mathsf{s}}$. We
believe the sharp estimate to be $R^{1+\mathsf{s}}$ and thus that two-
dimensional Riesz gases should remain hyperuniform for $\mathsf{s}<1$. For
one-dimensional systems, such questions were recently answered in [Bou21].

##### High temperature regime.

There has been recent interest in studying the “high-temperature” regime where
$\beta\to 0$ as $N\to\infty$ for Coulomb gases (and other related systems),
see e.g. [AB19, Lam21] in the two-dimensional setting. Those results have put
forward that the inverse temperature regime where $\beta\sim\frac{c}{N}$ is
the threshold that distinguishes between “Ginibre-like” ($c\to+\infty$) and
“Poisson-like” ($c\to 0$) properties. It is thus tempting to ask whether the
2DOCP stays hyperuniform as long as $\beta N\to\infty$, and whether one can
observe an interesting transition along $\beta=\frac{c}{N}$, as $c$ moves down
from $+\infty$ to $0$.

###  1.5 Precise statement of the result, proof strategy and tools

######  Theorem 1.

Let $\delta>0$ be fixed. For all $N$ and $R$ large enough (both depending on
$\beta$ and $\delta$), for all $x$ in $\Sigma_{N}$ such that $x$ is “in the
bulk” in the following sense:

$\operatorname{dist}(\mathfrak{D}(x,R),\partial\Sigma_{N})\geq\delta\sqrt{N},$
(1.5)

we have:

$\mathbb{P}^{\beta}_{N}\left(\left\\{\left|\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{D}(x,R))\right|\geq
R(\log R)^{-0.3}\right\\}\right)\leq\exp\left(-\log^{1.5}R\right).$ (1.6)

Since good exponential tails for $\mathsf{Dis}$ at values higher than $R$ are
already known (see e.g. [AS21, Thm 1]), this new sub-algebraic tail valid for
values of the discrepancy between $R\log^{-0.3}R$ and $R$ imply that:

$\mathrm{Var}[\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{D}(x,R))]\leq\mathbb{E}^{\beta}_{N}\left[\mathsf{Dis}^{2}(\mathbf{X}_{N},\mathfrak{D}(x,R))\right]=\mathcal{O}\left(\frac{R^{2}}{\log^{0.6}R}\right)=o(R^{2}),$
(1.7)

and the 2DOCP is thus indeed _hyperuniform_. The proof will be given in
Section 7.

#### General strategy

Our main source of inspiration is [NSV08, Section 4], in which Nazarov-Sodin-
Volberg prove JLM-like estimates on the probability of having large “charge”
fluctuations in the disk $\mathfrak{D}(0,R)$ _for zeros of the Gaussian Entire
Function_. Their argument can be roughly summarized as follows: fix
$\alpha\in(\frac{1}{2},1)$ and assume there is a discrepancy of size
$R^{\alpha}$ in the disk $\mathfrak{D}(0,R)$.

  1. 1.

Show that the discrepancy can be captured along the boundary
$\partial\mathfrak{D}(0,R)$.

  2. 2.

Split that boundary into $\approx R$ pieces of size $\approx 1$. Take $M$
large and apply a basic pigeonhole argument: there exists a family of
$\approx\frac{R}{M}$ pieces that capture a discrepancy of size at least
$\frac{R^{\alpha}}{M}$ and which are “well-separated” (distances between
pieces are multiples of $M$).

  3. 3.

Then comes the main probabilistic work:

    1. (a)

Show that these well-separated pieces are _approximately independent_.

    2. (b)

Show that the discrepancy on each piece (of size $\approx 1$) is typically
$\mathcal{O}(1)$.

    3. (c)

Show that the discrepancy on each piece is centered.

  4. 4.

Apply Bernstein’s concentration inequality: if $\\{D_{i}\\}_{i}$ is a family
of $\approx\frac{R}{M}$ independent centered random variables of size
$\mathcal{O}(1)$ then:

$\mathbb{P}\left(\left\\{\sum_{i}D_{i}\geq
R^{\alpha}\right\\}\right)\lesssim\exp\left(-\frac{R^{2\alpha-1}}{M}\right).$

This gives them _sharp_ bounds on the number variance _and_ the correct tail
probabilities. Whilst not aiming so high, we follow here a similar strategy,
with much effort to translate it to the context of the 2DOCP (besides point
(2) and (4), which are easy).

#### Electric approach, sub-systems

The technical core is the the “electric approach” to Coulomb gases as
developed by S. Serfaty and co-authors. In that regard, our main imports are:

  * •

The general spirit of controlling fluctuations through the electric energy
(see Section 2.3).

  * •

Local laws (up to microscopic scale) as in [AS21] (see Proposition 2.4 and
Proposition 5.8).

  * •

Optimal bounds for fluctuations of smooth linear statistics as in [LS18,
Ser20, BBNY19]. Point (1), for example follows fairly easily from such bounds.

  * •

“Smallness of the anisotropy”, for which we refer to [LS18, Ser20] (see also
[BBNY19]).

In addition, we put forward the role of so-called “sub-systems”, which arise
as restrictions of the full system to some region and which we view as two-
dimensional Coulomb systems in their own right, possibly with a small global
non-neutrality, and (most importantly) feeling the effect of some harmonic
exterior potential. As such, this object is not new - it has been sometimes
called a “conditional” or “local” measure in the literature (see e.g. [BBNY19,
BEY14]). We provide here a thorough study of its behavior, through global and
local laws, and show that with high probability the sub-systems, although
under the influence of an external potential, retain most of the rigidity
properties of the full system - as is the case “for free” in the hierarchical
model studied in [Cha19]. This requires a precise study of the external
potential “felt” by a sub-system and of the perturbation of the equilibrium
measure that it induces within the sub-system (for which we rely on the
analysis of [BBNY17, Section 2.], with some modifications). The proof of local
laws given in [AS21, Leb17] extends to sub-systems. This allows one for
example to treat Point (3) (b), which seem unsurprising, but which requires to
consider objects of size $\sim 1$, hence to control the system _down to the
smallest scales_. Such local laws are also crucial for the approximate
translation-invariance argument which we present now.

#### Approximate translation-invariance

Step (3) (c) is void in the context of [NSV08] because the underlying point
process is infinite and translation-invariant (which implies that linear
statistics are centered). However, it turns out to be a major roadblock when
adapting the proof to a Coulomb gas context. Indeed, in sub-systems (or even
in the full system if one does not impose artificial boundary conditions)
there is absolutely no obvious translation-invariance. We introduce an
“approximate translation-invariance” result, valid for both the full system
and sub-systems, which is crucial for Step (3) (c). The point being that
shifting a function (or more precisely, averaging over shifts) acts as a
mollification and enables us to compare the expectation of a discrepancy, or
of any non-smooth linear statistic, to that of a smoother one.

There is a series of results in mathematical statistical mechanics _à la_
H.-O. Georgii devoted to prove translation invariance of infinite volume Gibbs
measures in contexts where stationarity is not built-in, see e.g. [Geo99,
Ric07], following earlier works by Fröhlich-Pfister [FP81, FP86]. The basic
idea is to construct suitable “localized translations” in the form of
diffeomorphisms acting as a given translation in a large box while leaving the
majority of the system unchanged, and to control the effect on the energy of
such changes of variables. Following the wisdom found in a remark of [Sim14,
Chap. 3, Sec. III.7], we construct a localized translation that varies very
slowly (in terms of its $H^{1}$ norm), and a careful revisit of a computation
done in [Ser20], together with Serfaty’s “smallness of anisotropy” trick,
allows us to conclude that this localized translation can be chosen to have an
arbitrarily small effect on the energy.

#### Approximate independence

Finally, one needs to find some kind of independence between the sub-systems.
We introduce a new “approximate independence” argument (which we believe to be
of interest on its own) for sub-systems that are well-separated, conditionally
on the _number of points_ in each of them. The simple idea is that two domains
$\Lambda_{i},\Lambda_{j}$ with $\mathsf{n}_{i},\mathsf{n}_{j}$ points
contribute an interaction energy given to first order by:

$-\left(\mathsf{n}_{i}-|\Lambda_{i}|\right)\left(\mathsf{n}_{j}-|\Lambda_{j}|\right)\log\operatorname{dist}(\Lambda_{i},\Lambda_{j}),$

while the precise arrangements of the points inside each domain should matter
only to a lower order.

######  Remark 1.1.

In the rest of the paper we often use results found in [AS21] and [Ser20]. It
is worth noting that although both papers deal (among other things) with the
same 2DOCP model, they do not use the same scaling convention. In [AS21] the
authors work with the so-called “blown-up scaling” (as in the present paper),
which is more common in the physics literature and for which lengthscales
range from $\sim 1$ (the nearest-neighbor scale) to $\sim N^{1/2}$ (the
diameter of the system), whereas [Ser20] uses the random matrix theory
convention where the local scale is $\sim N^{-1/2}$ and the global scale is
$\sim 1$. In particular, a length scale $\ell$ in the present paper translates
into $N^{1/2}\ell$ in [Ser20], and the corresponding modifications must be
applied when quoting results.

### Acknowledgements

We thank Alon Nishry for discussions about the JLM law, out of which this
project grew. We thank New York University, Tel Aviv University and the
Institute for Advanced Study for their hospitality. Part of this work was done
while the author was a member of the School of Mathematics at the IAS with the
support of the Florence Gould foundation. We thank Prof. S. Torquato and Prof.
T. Spencer for stimulating discussions.

A previous version contained a serious error which invalidated the proof. We
are grateful to Eric Thoma for quickly spotting the dubious passage and
mentioning it to us.

We thank Sylvia Serfaty for helpful discussions about various aspects of the
proof, and Fréderic Marbach for his advice concerning the “spin wave”
construction.

##  2 Preliminary results

###  2.1 Some notation.

We denote indicator functions by $\mathsf{1}$. We denote by $|\Omega|$ the
Lebesgue measure of a measurable subset $\Omega$. For $x\in\mathbb{R}^{2}$ and
$r>0$ we let $\mathfrak{D}(x,r)$ be the (closed) disk of center $x$ and radius
$r$. We let $\square(x,r)$ be the square of center $x$ and sidelength $r$
(with sides parallel to the axes of $\mathbb{R}^{2}$). We recall the notation
$\mathsf{Dis}$ for discrepancies: if $\mathbf{X}$ is a point configuration and
$\Omega$ is a measurable subset, we let
$\mathsf{Dis}(\mathbf{X},\Omega):=\mathsf{Pts}(\mathbf{X},\Omega)-|\Omega|$.

Size of derivatives. If $M$ is a matrix we let $\|M\|$ be its Euclidean norm.
If $\varphi$ is a bounded map we write $|\varphi|_{\mathsf{0}}$ for its sup-
norm. If $\varphi$ is differentiable, we let $\mathsf{D}\varphi$ be its
differential and introduce the following notation:

  * •

$|\varphi|_{\mathsf{1},\star}(x)$ is the size of $\mathsf{D}\varphi$ at a
given point $x$, $|\varphi|_{\mathsf{1},\star}(x):=\|\mathsf{D}\varphi(x)\|$.

  * •

$|\varphi|_{\mathsf{1},\mathrm{loc}}(x)$ is the size of $\mathsf{D}\varphi$
around a given point $x$,
$|\varphi|_{\mathsf{1},\mathrm{loc}}(x):=\sup_{|x^{\prime}-x|\leq
1}|\varphi|_{\mathsf{1},\star}(x^{\prime})$

  * •

$|\varphi|_{\mathsf{1},\Omega}$ is the size of $\mathsf{D}\varphi$ in a given
domain $\Omega$,
$|\varphi|_{\mathsf{1},\Omega}:=\sup_{x\in\Omega}|\varphi|_{\mathsf{1},\star}(x)$.

  * •

Finally, $|\varphi|_{\mathsf{1}}$ is the sup-norm of $\mathsf{D}\varphi$.

We define similarly $|\varphi|_{\mathsf{k},\star}$,
$|\varphi|_{\mathsf{k},\mathrm{loc}}$, $|\varphi|_{\mathsf{k},\Omega}$ and
$|\varphi|_{\mathsf{k}}$ for $\mathsf{k}\geq 2$.

Point configurations. For all Borel subsets $\Lambda$ of $\mathbb{R}^{2}$, we
let $\mathrm{Conf}(\Lambda)$ be the space of locally finite333In fact all of
the point configurations considered in this paper are finite. point
configurations on $\Lambda$, endowed with the vague topology of Radon measures
and the associated Borel $\sigma$-algebra. When $\Lambda$ is not specified, we
use the notation $\mathrm{Conf}$ for point configurations on $\mathbb{R}^{2}$.
We will denote by $\mathbf{X}\mapsto\mathbf{X}\cap\Lambda$ the natural
projection $\mathrm{Conf}\twoheadrightarrow\mathrm{Conf}(\Lambda)$. We may
write “$x\in\mathbf{X}$” to express the fact that $\mathbf{X}$ has an atom at
a given point $x\in\mathbb{R}^{2}$. We say that a measurable function
$\mathsf{G}$ on $\mathrm{Conf}$ is $\Lambda$-local when for all $\mathbf{X}$
in $\mathrm{Conf}$ we have
$\mathsf{G}(\mathbf{X})=\mathsf{G}(\mathbf{X}\cap\Lambda)$. We say that a
measurable subset (an “event”) $\mathcal{E}$ of $\mathrm{Conf}$ is
$\Lambda$-local when its indicator function $\mathtt{1}_{\mathcal{E}}$ is
$\Lambda$-local.

Constants. Unless specified otherwise, $\mathtt{C}$ denotes a universal
constant (which may change from line to line) and $\mathtt{C}_{\beta}$ a
constant that depends only on $\beta$. We will use $\bar{\mathtt{C}}$ for
constants that may depend on $\beta$ and the parameter $\delta$ (as in (1.5)).
We write $A\preceq B$ or $A=\mathcal{O}(B)$ if $|A|\leq\mathtt{C}|B|$.

Fluctuations. If $\varphi$ is a piece-wise continuous function on
$\mathbb{R}^{2}$ we define the _fluctuation of (the linear statistics
associated to) $\varphi$_ as the following random variable:

$\mathrm{Fluct}[\varphi]:=\int_{\Sigma_{N}}\varphi(x)\mathtt{d}\mathbf{f}_{N}(x).$
(2.1)

###  2.2 Electric fields

We recall that $-\log$ satisfies $-\Delta(-\log)=2\pi\delta_{0}$ on
$\mathbb{R}^{2}$ in the sense of distributions. In particular for all smooth
enough test functions $f$, the following identity holds:

$f(x)=\frac{-1}{2\pi}\int_{\mathbb{R}^{2}}-\log|x-y|\Delta f(y).$ (2.2)

Let $\Lambda\subset\mathbb{R}^{2}$ and let $\mathbf{X}$ be a point
configuration in $\Lambda$.

######  Definition 2.1 (True electric potential and electric field).

We let $\mathfrak{h}^{\mathbf{X}}$ (resp. $\nabla\mathfrak{h}^{\mathbf{X}}$)
be the true electric potential (resp. true electric field) generated by
$\mathbf{X}$ (in $\Lambda$), namely the map (resp. vector field) defined on
$\mathbb{R}^{2}$ by:

$\mathfrak{h}^{\mathbf{X}}(x):=\int_{\Lambda}-\log|x-y|\mathtt{d}\left(\mathbf{X}-\mathbf{m}_{0}\right)(y),\quad\text{resp.
}\nabla\mathfrak{h}^{\mathbf{X}}(x)=\int_{\Lambda}-\nabla\log|x-y|\mathtt{d}\left(\mathbf{X}-\mathbf{m}_{0}\right)(y).$

It is easy to check that $\nabla\mathfrak{h}^{\mathbf{X}}$ is in
$\cap_{p\in[1,2)}\mathrm{L}_{\mathrm{loc}}^{p}(\mathbb{R}^{2},\mathbb{R}^{2})$
but fails to be in $\mathrm{L}^{2}$ around each point charge, and that the
following identity is satisfied on $\Lambda$ in the sense of distributions:

$-\Delta\mathfrak{h}^{\mathbf{X}}=-\operatorname{div}\nabla\mathfrak{h}^{\mathbf{X}}=2\pi\left(\mathbf{X}-\mathbf{m}_{0}\right).$

######  Definition 2.2 (Compatible electric fields).

Let $\mathrm{E}$ be a vector field in
$\cap_{p\in[1,2)}\mathrm{L}_{\mathrm{loc}}^{p}(\mathbb{R}^{2},\mathbb{R}^{2})$.
We say that $\mathrm{E}$ is an electric field compatible with $\mathbf{X}$ on
$\Lambda$ whenever we have:
$-\operatorname{div}\mathrm{E}=2\pi\left(\mathbf{X}-\mathbf{m}_{0}\right)$ on
$\Lambda$ in the sense of distributions.

Obviously the true electric field is a compatible electric field, however it
is not the only one as can be seen by adding any divergence-free smooth vector
field on $\Lambda$ to $\nabla\mathfrak{h}^{\mathbf{X}}$.

In order to “take care” of the singularities, one often proceeds to a
truncation of the fields near each point charge.

######  Definition 2.3 (Truncation and spreading out Dirac masses).

For $\eta>0$ we let $\mathsf{f}_{\eta}$ be the function:

$\mathsf{f}_{\eta}(x):=\max\left(-\log\frac{|x|}{\eta},0\right)=\begin{cases}-\log|x|+\log|\eta|&\text{if
}x\leq\eta\\\ 0&\text{if }x\geq\eta\end{cases}.$

For each point $x$ of $\mathbf{X}$, let $\eta(x)$ be a non-negative real
number. The data of $\vec{\eta}=\\{\eta(x),x\in\mathbf{X}\\}$ is called a
truncation vector. If $\mathfrak{h}^{\mathbf{X}}$ is the true electric
potential, we let $\mathfrak{h}^{\mathbf{X}}_{\vec{\eta}}$ (resp.
$\nabla\mathfrak{h}^{\mathbf{X}}_{\vec{\eta}}$) be the (true) truncated
electric potential (resp. field) given by:

$\mathfrak{h}^{\mathbf{X}}_{\vec{\eta}}:=\mathfrak{h}^{\mathbf{X}}-\sum_{x\in\mathbf{X}}\mathsf{f}_{\eta(x)}(\cdot-x),\quad\text{resp.
}\nabla\mathfrak{h}^{\mathbf{X}}_{\vec{\eta}}=\nabla\mathfrak{h}^{\mathbf{X}}-\sum_{x\in\mathbf{X}}\nabla\mathsf{f}_{\eta(x)}(\cdot-x).$

We are thus effectively replacing $-\log|x-\cdot|$ by $-\log\eta$ near each
point charge. Another way to think about $\mathsf{f}_{\eta}$ is that we are
truncating the singularities by smearing out the point charge $\delta_{x}$ à
la Onsager. Indeed when computing the divergence of
$\nabla\mathfrak{h}^{\mathbf{X}}_{\vec{\eta}}$ one finds that the atom at each
point $x\in\mathbf{X}$ has been replaced by a uniform measure of mass $1$ on
the circle of center $x$ and radius $\eta(x)$. We refer to [AS21, Section 2.2
& Appendix B.1] or to [Ser20, Sec 3.1] for more details. The truncation
procedure can be extended to define $\mathrm{E}_{\vec{\eta}}$, where
$\mathrm{E}$ is any electric field compatible with $\mathbf{X}$, by setting:

$\mathrm{E}_{\vec{\eta}}:=\mathrm{E}-\sum_{x\in\mathbf{X}}\nabla\mathsf{f}_{\eta(x)}(\cdot-x).$

For every point $x$ of $\mathbf{X}$ we define the “nearest-neighbor” distance
$\mathsf{r}(x)$ as:

$\mathsf{r}(x):=\frac{1}{4}\min\left(\min_{y\in\mathbf{X},y\neq
x}|x-y|,1\right).$ (2.3)

In particular $\mathsf{r}(x)$ is always smaller than $1/4$. We let
$\vec{\mathsf{r}}=(\mathsf{r}(x),x\in\mathbf{X})$ be the associated truncation
vector. We will sometimes use instead the vector $s\vec{\mathsf{r}}$ with
$s<1$.

###  2.3 Logarithmic energy and electric fields

The logarithmic interaction energy $\mathsf{F}_{N}$ defined in (1.1) can be
expressed in terms of the true electric field generated by $\mathbf{X}_{N}$
(Definition 2.1) as follows.

  * •

Taking a uniform truncation vector $\eta(x)=\eta>0$ for all $x$ in
$\mathbf{X}_{N}$, we have the following equality in the limit as $\eta\to 0$:

$\mathsf{F}_{N}(\mathbf{X}_{N})=\frac{1}{2}\lim_{\eta\to
0}\left(\frac{1}{2\pi}\int_{\mathbb{R}^{2}}|\nabla\mathfrak{h}^{\mathbf{X}_{N}}_{\vec{\eta}}|^{2}+N\log\eta\right).$

(This quantity is “almost” non-decreasing as $\eta\to 0$, see e.g. [AS21,
Lemma B.1.].)

  * •

On the other hand, taking for each $x$ a truncation
$\eta(x)\leq\mathsf{r}(x)$, we get a non-asymptotic identity:

$\mathsf{F}_{N}(\mathbf{X}_{N})=\frac{1}{2}\left(\frac{1}{2\pi}\int_{\mathbb{R}^{2}}|\nabla\mathfrak{h}^{\mathbf{X}_{N}}_{\vec{\eta}}|^{2}+\sum_{x\in\mathbf{X}_{N}}\log\eta(x)\right)-\sum_{x\in\mathbf{X}_{N}}\int_{\mathfrak{D}(x,\eta(x))}\mathsf{f}_{\eta(x)}(t-x)\mathtt{d}t.$
(2.4)

The first formulation can be found in [SS12, RS16] and the second one in
[LS18] or [AS21, Lemma 2.2.]. Of course, expressing electrostatic interactions
in terms of the electric field and “smearing out” point charges are both old
ideas. One sees that in the small truncation limit there is a compensation
between the explosion of
$\int_{\mathbb{R}^{2}}|\nabla\mathfrak{h}^{\mathbf{X}_{N}}_{\vec{\eta}}|^{2}$
and the very negative terms $\sum_{x\in\mathbf{X}_{N}}\log\eta(x)$, hence the
name “renormalized energy” given in [SS15] (this renormalization procedure
appeared in [BBH94] and the idea of using nearest-neighbor distances was
borrowed from [GP77]). Despite their apparent mutual cancellation, in general
the “positive” and “negative” parts can both be compared to
$\mathsf{F}_{N}(\mathbf{X}_{N})$ as explained e.g. in [AS21, Lemma B.2].

In the sequel we use the expression “electric energy” when referring to
quantities of the type $\int_{\Omega}|\mathrm{E}_{\vec{\eta}}|^{2}$ where
$\mathrm{E}$ is an electric field, $\vec{\eta}$ a truncation vector and
$\Omega$ is some subset of $\mathbb{R}^{2}$, and we write:

$\mathrm{Ener}(\mathbf{X}_{N},\Omega):=\int_{\Omega}|\nabla\mathfrak{h}^{\mathbf{X}_{N}}_{\vec{\mathsf{r}}}|^{2},\quad\mathrm{Ener}_{s}(\mathbf{X}_{N},\Omega):=\int_{\Omega}|\nabla\mathfrak{h}^{\mathbf{X}_{N}}_{s\vec{\mathsf{r}}}|^{2}\
(\text{for }s\in(0,1)).$ (2.5)

From the identity (2.4) we deduce that for $s<1$:

$\mathrm{Ener}_{s}(\mathbf{X}_{N},\Omega)=\mathrm{Ener}(\mathbf{X}_{N},\Omega)-\mathsf{Pts}(\mathbf{X}_{N},\Omega)\log
s,$ (2.6)

For convenience, we will sometimes write $\mathsf{EnerPts}$ for the sum of
$\mathrm{Ener}$ and $\mathsf{Pts}$ in a given domain.

###  2.4 Global and local laws

The following properties of $\mathsf{F}_{N}$ are well-known (see [AS21, Lemma
3.7]):

  * •

It is bounded below: there exists a universal constant $\mathtt{C}$ such that
for all $N\geq 2$ and for all $\mathbf{X}_{N}$, we have
$\mathsf{F}_{N}(\mathbf{X}_{N})\geq-\mathtt{C}N$.

  * •

It is typically of order $N$ in the sense that for some constant
$\mathtt{C}_{\beta}$ depending only on $\beta$:

$\mathbb{E}^{\beta}_{N}\left[\exp\left(\frac{\beta}{2}\mathsf{F}_{N}(\mathbf{X}_{N})\right)\right]\leq\exp\left(\mathtt{C}_{\beta}N\right).$
(2.7)

We refer to (2.7) as a “global law”, controlling the system at macroscopic
scale. The next proposition expresses the fact that the system is also well-
behaved down to some large microscopic scale. Let us first introduce two
important distances that will play a role in Proposition 2.4 and in the rest
of the paper.

The smallest microscopic scale $\rho_{\beta}$. We refer to a certain _minimal
lengthscale_ introduced in [AS21] and denoted by $\rho_{\beta}$. It is a
positive constant that depends only on $\beta$, and corresponds to the length-
scale above which good rigidity properties are proven.

Distance to the boundary. For technical (and possibly physical) reasons,
properties of the full-system are easier to understand when one looks “away
from the edge”, namely at some non-trivial distance of the boundary
$\partial\Sigma_{N}$. For $x$ a point in $\Sigma_{N}$ and $\ell>0$, we will
say that “$(x,\ell)$ satisfies (2.8)” when:

$\operatorname{dist}(\square(x,\ell),\partial\Sigma_{N})\geq\mathtt{C}_{\beta}N^{1/4},$
(2.8)

where $\mathtt{C}_{\beta}$ is some large enough constant introduced in [AS21,
(1.16)].

Let us note that in the statement of Theorem 1 we assume that
$\operatorname{dist}(\mathfrak{D}(x,R),\partial\Sigma_{N})\geq\delta\sqrt{N}$
which is clearly a stronger assumption, at least for $N$ large enough
depending only on $\delta,\beta$.

######  Proposition 2.4 (Local laws).

There exists some universal constant $\mathtt{C}$, and a “local laws” constant
CLL depending only on $\beta$ such that the following holds. Let $x$ be a
point in $\Sigma_{N}$ and $\ell$ a lengthscale such that:

  1. 1.

$\ell\geq\rho_{\beta}$ (the length-scale $\ell$ is larger than the “minimal”
one)

  2. 2.

$(x,\ell)$ satisfy (2.8) (we are sufficiently “far from the edge”).

Then we control the electric energy in $\square(x,\ell)$ in exponential
moments:

$\log\mathbb{E}^{\beta}_{N}\left(\exp\left(\frac{\beta}{2}\mathrm{Ener}\left(\mathbf{X}_{N},\square(x,\ell)\right)\right)\right)\leq\hyperref@@ii[prop:local_laws]{\mathtt{C}_{\mathtt{LL}}}\beta\ell^{2}.$
(2.9)

Moreover, we have the following control on the number of points:

$\log\mathbb{E}^{\beta}_{N}\left(\exp\left(\frac{\beta}{\mathtt{C}}\mathsf{Pts}\left(\mathbf{X}_{N},\square(x,\ell)\right)\right)\right)\leq\hyperref@@ii[prop:local_laws]{\mathtt{C}_{\mathtt{LL}}}\beta\ell^{2},$
(2.10)

together with a discrepancy bound:

$\log\mathbb{E}^{\beta}_{N}\left(\exp\left(\frac{\beta}{\mathtt{C}}\frac{\mathsf{Dis}^{2}\left(\mathbf{X}_{N},\square(x,\ell)\right)}{\ell^{2}}\right)\right)\leq\hyperref@@ii[prop:local_laws]{\mathtt{C}_{\mathtt{LL}}}\beta.$
(2.11)

###### Proof of Proposition 2.4.

This is a subset of the statements in [AS21, Theorem 1], see Section B.1 for a
technical discussion. ∎

As can be seen from Proposition 2.4,
$\mathrm{Ener}(\mathbf{X}_{N},\square(x,\ell)),\mathsf{Pts}(\mathbf{X}_{N},\square(x,\ell))$
and $\mathsf{EnerPts}(\mathbf{X}_{N},\square(x,\ell))$ are all expected to be
of the same order as the area $\ell^{2}$.

######  Remark 2.5.

For our purposes, we will repeatedly use the local laws under the following
form: for $\mathtt{C}_{\beta}$ large enough, the probability of having more
than $\mathtt{C}_{\beta}\ell^{2}$ points (or an energy higher than
$\mathtt{C}_{\beta}\ell^{2}$) in a given square $\square(x,\ell)$ is smaller
than $\exp\left(-\ell^{2}/\mathtt{C}_{\beta}\right)$.

One can replace squares by disks (or by any “reasonable” shape) in the
previous statement, however it is worth observing that local laws do not
directly yield interesting controls the energy (or number of points) on very
thin strips, rectangles with diverging aspect ratio, boundaries of squares,
thin annuli etc. In those situations, one must resort to splitting the region
into squares, applying the local laws to each square and then using a union
bound.

###  2.5 The electric energy controls fluctuations

The next lemma expresses how the electric energy controls linear statistics of
Lipschitz functions.

######  Lemma 2.6 (Bounds on fluctuations - Lipschitz case).

Let $\mathbf{X}$ be a point configuration in $\mathbb{R}^{2}$, and let
$\varphi$ be a function in $\mathrm{C}^{1}(\mathbb{R}^{2})$ with compact
support. Let $\mathrm{E}$ be any electric field compatible with $\mathbf{X}$
on $\operatorname{supp}\varphi$ in the sense of Definition 2.2. Let $\Omega$
be a domain containing a $1$-neighborhood of
$\operatorname{supp}\nabla\varphi$. We have:

$\left|\int\varphi(x)\mathtt{d}\left(\mathbf{X}-\mathbf{m}_{0}\right)(x)\right|\leq\left(\int_{\mathbb{R}^{2}}|\nabla\varphi|^{2}\right)^{\frac{1}{2}}\left(\int_{\Omega}|\mathrm{E}_{\vec{\mathsf{r}}}|^{2}\right)^{\frac{1}{2}}+|\varphi|_{\mathsf{1},\Omega}\mathsf{Pts}(\mathbf{X},\Omega)$
(2.12)

##### Localized case

Assume that $\tilde{\Omega}_{1},\dots,\tilde{\Omega}_{m}$ cover
$\operatorname{supp}\nabla\varphi$, and that for each $i$ the domain
$\Omega_{i}$ contains a $1$-neighborhood of $\tilde{\Omega}_{i}$, then we can
replace the right-hand side of (2.12) by:

$\sum_{k=1}^{m}|\varphi|_{\mathsf{1},\Omega_{i}}\times\left(\left(\int_{\Omega_{i}}|\mathrm{E}_{\vec{\mathsf{r}}}|^{2}\right)^{\frac{1}{2}}\times\left|\Omega_{i}\right|^{\frac{1}{2}}+\mathsf{Pts}(\mathbf{X},\Omega_{i})\right)$
(2.13)

######  Remark 2.7.

Controls of the type (2.12) have appeared under various forms in previous
works see e.g. [SS15, Lemma 5.1], [LS18, Proposition 2.5] or [AS21, Lemma
B.5], they are usually phrased as: “the electric energy controls the
fluctuations”. The electric energy in a given domain $\Omega$ is typically of
order $|\Omega|$ and so is the number of points in $\Omega$, cf. Proposition
2.4, thus (2.12) bounds the typical fluctuations of $\varphi$ by
$|\varphi|_{\mathsf{1}}|\operatorname{supp}\nabla\varphi|$ whereas a naive
$\mathrm{L}^{\infty}$ bound would rather give
$|\varphi|_{\mathsf{0}}\times|\operatorname{supp}\varphi|$. Since our test
functions often live on some large lengthscale $\ell$ with
$|\varphi|_{\mathsf{1}}$ comparable to $\ell^{-1}|\varphi|_{\mathsf{0}}$,
there is indeed an improvement.

We give the proof of Lemma 2.6 in Section B.2. Compared to existing results,
here we simply emphasize the role played by the support of the gradient
(instead of the whole support of the test function), which yields more
accurate estimates when $\varphi$ is a sharp cut-off function. We will
repeatedly use its “localized version” (2.13), whose proof is a
straightforward of (2.12).

###  2.6 Finer bound on fluctuations for smooth test functions

For test functions with higher regularity the bound of Lemma 2.6 on
fluctuations of linear statistics can be improved (see [LS18, BBNY19, Ser20]
as well as [RV07, AHM11] for the $\beta=2$ case). In particular if
$\varphi_{\ell}(x):=\bar{\varphi}(x/\ell)$ for a fixed reference smooth test
function $\bar{\varphi}$, then $\varphi_{\ell}$ lives at scale $\ell\gg 1$ yet
its fluctuations remain bounded as $\ell\to\infty$ (with high probability).
One can refer e.g. to [Ser20, Thm. 1] and [Ser20, Cor. 2.1] for such results.

In this paper we will occasionally need a more specific statement valid in the
radially symmetric case.

######  Proposition 2.8 (Finer bound, the $C^{2}$ radially symmetric case).

There exists a constant $\mathtt{C}_{\beta}$ depending only on $\beta$ such
that the following holds. Let $x$ be a point in $\Sigma_{N}$ and let $\varphi$
be a test function which is radially symmetric around $x$, with compact
support. Assume that $\varphi$ is in $\mathrm{C}^{2}(\mathbb{R}^{2})$ and let
$\mathcal{A}$ be an annulus containing (a $1$-neighborhood of) the support of
$\Delta\varphi$. Let $s$ be a real number satisfying:

$|s|\leq\frac{\pi\beta}{4|\varphi|_{\mathsf{2}}}.$ (2.14)

Then the exponential moments of the fluctuations of $\varphi$ satisfy:

$\log\mathbb{E}^{\beta}_{N}\left[\exp\left(s\mathrm{Fluct}[\varphi]\right)\right]=\frac{s^{2}}{4\pi\beta}\int_{\mathbb{R}^{2}}|\nabla\varphi|^{2}+\log\mathbb{E}^{\beta}_{N}\left[\exp\left(s|\varphi|_{\mathsf{2}}\mathtt{C}_{\beta}\left(\mathsf{EnerPts}(\mathbf{X}_{N},\mathcal{A})\right)\right)\right].$
(2.15)

Although it does not appear as such in the literature, Proposition 2.8 can be
easily deduced from the tools of [Ser20], we give a proof in Section B.3.

###  2.7 Wegner’s estimates

The recent paper [Tho22] provides upper bounds on the $k$-point correlation
functions of Coulomb gases (in dimension $2$ and higher). Among many other
things, it states so-called “Wegner’s estimates” i.e. uniform controls of the
form $\rho_{1}\leq\mathtt{C}_{\beta}$, where $\rho_{1}$ is the one-point
correlation function. We will use [Tho22, Theorem 1.6], which is valid even
for $r\ll 1$ (sub-microscopic scale):

######  Lemma 2.9.

There exists $\mathtt{C}_{\beta}$ such that for all $x$ in $\Sigma_{N}$ and
_for all $r>0$_, if $(x,r)$ satisfies (2.8) we have:

$\mathbb{P}^{\beta}_{N}\left(\left\\{\mathsf{Pts}(\mathfrak{D}(x,r))\geq
1\right\\}\right)\leq\mathtt{C}_{\beta}r^{2}.$ (2.16)

##  3 Locating discrepancies near the boundary

Let $0\leq\varepsilon_{R}\leq 1$ (to be chosen later). The goal of this
section to show that any “large” discrepancy $\varepsilon_{R}R$ within a disk
of radius $R$ can be found (with high probability) _near the boundary of the
disk_.

###  3.1 Cornering the discrepancy in an annulus close to the boundary

Let $L\geq 100$ be a lengthscale to be chosen later. Let $z$ be a point and
$R\geq\rho_{\beta}$ such that $\mathfrak{D}(z,R)$ satisfies (2.8). In this
section we sometimes simply write $\mathfrak{D}_{r}$ (for $r>0$) instead of
$\mathfrak{D}(z,r)$. Assume that the disk $\mathfrak{D}(z,R)$ contains
$\varepsilon_{R}\cdot R$ too many444We mean
$\mathsf{Pts}(\mathbf{X}_{N},\mathfrak{D}_{R})\geq\pi
R^{2}+\varepsilon_{R}\cdot R$. points points. Compare two idealized
situations: in the first one, the excess of charges is spread uniformly over
$\mathfrak{D}_{R}$, in the sense that
$\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{D}_{r})$ behaves like
$\frac{r^{2}}{R^{2}}\times\varepsilon_{R}\cdot R$ for $r\leq R$. In the second
situation, the excess is concentrated in a thin strip near
$\partial\mathfrak{D}_{R}$ and immediately compensated by a default on the
other side of the boundary. According to [JLM93], the physically realistic
picture is the second one: “Macroscopic electrostatics of conductors implies
that, for a given value of [the discrepancy] $Q$, the dominant configurations
are such that $Q$ is concentrated in a layer on the inner side of the boundary
of the disk, while a charge $-Q$ accumulates in a layer on the outer side.”.
In order to give a corresponding mathematical statement, observe that the
event

$\bigcap_{R-2L\leq r\leq
R-L}\left\\{\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{D}(z,r))\geq\frac{1}{2}\varepsilon_{R}\cdot
R\right\\}$

expresses that some excess of charges is found in the disks $\mathsf{D}_{r}$
for a fairly wide region $r\in[R-2L,R-L]$ and that the discrepancy is thus not
“concentrated in a layer on the inner side of the boundary of the disk”.

######  Proposition 3.1.

There exists a constant $\mathtt{C}_{\ref{prop:chase_the_goat}}(\beta)$
depending only on $\beta$ such that the following holds. Let $s$ be a real
parameter. Assume that $R,L,\varepsilon_{R},s$ satisfy:

$\begin{cases}&\mathtt{C}_{\ref{prop:chase_the_goat}}\frac{1}{\varepsilon_{R}}\leq
L\leq\frac{R}{10}\\\ &0\leq
s\leq\frac{1}{\mathtt{C}_{\ref{prop:chase_the_goat}}}\min\left(\frac{L^{3}}{R},L\varepsilon_{R}\right).\end{cases}$
(3.1)

Then we have:

$\mathbb{P}^{\beta}_{N}\left[\bigcap_{R-2L\leq r\leq
R-L}\left\\{\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{D}(z,r))\geq\frac{1}{2}\varepsilon_{R}\cdot
R\right\\}\right]\leq\exp\left(-\frac{s\varepsilon_{R}\cdot R}{4}\right).$
(3.2)

We postpone the proof of Proposition 3.1 to Section B.4. It relies on
Proposition 2.8.

Next, we argue that if the complementary event occurs, then one can find an
annulus of width $\approx L$ near the boundary of $\mathfrak{D}_{R}$ that
carries a large discrepancy.

######  Lemma 3.2 (Finding discrepancy in an annulus).

If $\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{D}_{R})\geq\varepsilon_{R}\cdot R$
and if, in contrast to the event considered in (3.2), there exists a radius
$r$ with $R-2L\leq r\leq R-L$ such that:

$\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{D}_{r})<\frac{1}{2}\varepsilon_{R}\cdot
R$ (3.3)

then there exists an integer $k$ with $0\leq k\leq R^{2}$ such that the
discrepancy in the annulus
$\mathfrak{D}_{R}\setminus\mathfrak{D}_{R-2L+\frac{kL}{R^{2}}}$ is larger than
$\frac{1}{4}\varepsilon_{R}\cdot R$.

###### Proof.

Let $r$ be such that (3.3) holds and let $0\leq k\leq R^{2}$ be the integer
such that:

$r_{k}:=R-2L+\frac{kL}{R^{2}}\leq r<R-2L+\frac{(k+1)L}{R^{2}}.$

It is easy to see that
$\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{D}_{r_{k}})\leq\frac{3}{4}\varepsilon_{R}\cdot
R$ (a rough bound), indeed we have:

$\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{D}_{r_{k}})\leq\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{D}_{r})+|\mathfrak{D}_{r}\setminus\mathfrak{D}_{r_{k}}|,$

with equality if and only if there is no point in the annulus
$\mathfrak{D}_{r}\setminus\mathfrak{D}_{r_{k}}$, whose area is of order
$\frac{RL}{R^{2}}\leq 1$.

Since the total discrepancy in $\mathfrak{D}_{R}$ is at least
$\varepsilon_{R}\cdot R$ and the one inside $\mathfrak{D}_{r_{k}}$ is at most
$\frac{3}{4}\varepsilon_{R}\cdot R$ then the annulus
$\mathfrak{D}_{R}\setminus\mathfrak{D}_{r_{k}}$ must carry a discrepancy of
size at least $\frac{1}{4}\varepsilon_{R}\cdot R$. ∎

######  Remark 3.3.

If there is a negative discrepancy i.e. a default of points instead of an
excess, one proceeds the same way, except that in Proposition 3.1 we bound the
probability of
$\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{D}_{r})\leq\frac{-\varepsilon_{R}\cdot
R}{2}$ for all $R-2L\leq r\leq R-L$ instead. The complementary event is that
for some $r$ with $R-2L\leq r\leq R-L$, we have
$\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{D}_{r})>-\frac{-\varepsilon_{R}\cdot
R}{2}$. Then in Lemma 3.2 instead of taking $r_{k}$ a bit smaller than $r$ we
take it a bit larger and observe that the discrepancy cannot go down too much
between $\mathfrak{D}_{r}$ and $\mathfrak{D}_{r_{k}}$. Hence
$\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{D}_{r_{k}})\geq\frac{-3\varepsilon_{R}\cdot
R}{4}$, so the annulus $\mathfrak{D}_{R}\setminus\mathfrak{D}_{r_{k}}$ carries
a default of points at least $\frac{\varepsilon_{R}\cdot R}{4}$ (in absolute
value).

###  3.2 A well-separated family of boxes carrying the discrepancy

Let $r$ be in $[R-2L,R-L]$.

##### Decomposition in boxes

We split the annulus $\mathfrak{D}_{R}\setminus\mathfrak{D}_{r}$ into smaller
parts that we call _boxes_.

######  Definition 3.4 (Decomposition in boxes).

For $i\in\\{0,\dots,\frac{R}{L}-1\\}$, we let the $i$-th “box”
$\mathfrak{B}_{i}$ be the intersection of the annulus
$\mathfrak{D}_{R}\setminus\mathfrak{D}_{r}$ with a certain angular sector of
center $z$:

$\mathfrak{B}_{i}:=\left(\mathfrak{D}_{R}\setminus\mathfrak{D}_{r}\right)\cap\left\\{z^{\prime}\in\mathbb{R}^{2}\simeq\mathbb{C},\
i\leq\frac{\arg(z^{\prime}-z)}{2\pi}\times\frac{R}{L}\leq i+1\right\\}.$

The boundary of each such box is made of two line segments of equal length in
$(L,2L)$ and two concentric circular arcs which subtend the same angle at the
center and whose arclengths are different but both in
$\left(\frac{L}{2},L\right)$. The shape is symmetric with respect to the
straight line joining the midpoints of both arcs. We sometimes call “a box of
size $L$” any domain $\mathfrak{B}$ that fits the previous description.

We let $\omega_{i}$ be the center of mass of $\mathfrak{B}_{i}$, which only
serves as a convenient reference point.

##### Introducing the parameters $M$ and $T$

Let $T$ be a lengthscale, and $M$ be an integer, both to be chosen at the end,
such that:

$T\geq 10L,\quad 100\leq M\leq\frac{R}{L},\quad T\leq\frac{ML}{100},\quad
T^{2}\log T\leq ML.$ (3.4)

(The fourth condition (which could be weakened), implies the third one but for
clarity we write them all down.)

######  Lemma 3.5 (Some well-separated boxes carry the discrepancy).

Assume that the discrepancy in $\mathfrak{D}_{R}\setminus\mathfrak{D}_{r}$ is
larger than $\frac{1}{4}\varepsilon_{R}\cdot R$. Then there exists
$l\in\\{0,\dots,M-1\\}$ such that:

$\sum_{i\equiv l\text{ mod
}M}\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{B}_{i})\geq\frac{\varepsilon_{R}\cdot
R}{4M}.$ (3.5)

###### Proof of Lemma 3.5.

It follows from a straightforward pigeonhole argument. ∎

From now on, we assume to have chosen such a $l\in\\{0,\dots,M-1\\}$ and we
write the corresponding boxes as
$\mathfrak{B}_{1},\dots,\mathfrak{B}_{\mathcal{N}}$, where
$\mathcal{N}=\mathcal{O}\left(\frac{R}{ML}\right)$ is the number of box in
each “well-separated” family. Moreover, we see each box $\mathfrak{B}_{i}$ as
being contained in a large disk $\Lambda_{i}:=\mathfrak{D}(\omega_{i},T)$,
where $T$ is as above.

Let $\mathrm{d}_{ij}$ be the distance between $\Lambda_{i}$ and $\Lambda_{j}$.

######  Claim 3.6.

We have for each fixed $i$:

$\sum_{j,j\neq i}\frac{1}{\mathrm{d}_{ij}}=\mathcal{O}\left(\frac{\log
R}{ML}\right),\quad\sum_{j,j\neq
i}\frac{1}{\mathrm{d}_{ij}^{2}}=\mathcal{O}\left(\frac{1}{M^{2}L^{2}}\right)$
(3.6)

###### Proof.

Let us observe that between two “consecutive” boxes in the family $\\{i\equiv
l\text{ mod }M\\}$ considered in (3.5), there is a distance of order $ML$
(since $ML\geq 100T$ by assumption, this is also comparable to the distance
between two consecutive _disks_). We can compare the sum to an harmonic sum
(in the first case) or a converging Riemann series (in the second case). ∎

#### Plan for the next two sections

We now want to treat each box $\mathfrak{B}_{i}$ as living in its own smaller
version of a 2DOCP contained in $\Lambda_{i}$, which leads us to the next two
sections.

Think of a “sub-system” as the random collection of particles contained in a
given sub-domain $\Lambda\subset\Sigma_{N}$ with a reasonable shape (e.g. a
square or a disk). These particles feel the influence of each other, but also
of the full system in $\Sigma_{N}$ because the logarithmic interaction is
long-range. Hence sub-systems are typically not isolated and not independent
from each other.

  1. 1.

In Section 4 we observe that if we condition on the values of the
discrepancies (or equivalently of the number of particles) in domains that are
well-separated, then the corresponding sub-systems acquire a form of
independence.

  2. 2.

In Section 5 we show that “typical” sub-systems, seen as slight
generalizations of the 2DOCP model introduced earlier (in (1.1), (1.2)),
retains most of the properties of the _full_ system mentioned in Section 2.

##  4 Approximate conditional independence for sub-systems

In this section, we consider a family $\\{\Lambda_{i},1\leq
i\leq\mathcal{N}\\}$ of $\mathcal{N}\geq 2$ disjoint disks in $\Sigma_{N}$
(forming our “sub-systems”). We will eventually apply the results below to the
$\Lambda_{i}$’s chosen in Section 3.2, but the statements in the present
section are general.

Let $\mathsf{Ext}$ be the complement
$\mathsf{Ext}:=\Sigma_{N}\setminus\cup_{i=1}^{\mathcal{N}}\Lambda_{i}$. Let
$\mathrm{V}^{\mathrm{ext}}$ be the logarithmic potential generated by the
system in $\mathsf{Ext}$, namely:

$\mathrm{V}^{\mathrm{ext}}(x):=\int_{\mathsf{Ext}}-\log|x-y|\mathtt{d}\mathbf{f}_{N}(y).$
(4.1)

The potential $\mathsf{Ext}$ is harmonic on all the $\Lambda_{i}$’s and
depends only on the configuration in $\mathsf{Ext}$.

###  4.1 Decomposition of the interaction, conditional independence error

We let $\mathsf{Int}[\Lambda_{1},\dots,\Lambda_{\mathcal{N}}]$ be the true
logarithmic interaction between the sub-systems, namely:

$\mathsf{Int}[\Lambda_{1},\dots,\Lambda_{\mathcal{N}}]:=\frac{1}{2}\sum_{1\leq
i\neq
j\leq\mathcal{N}}\iint_{\Lambda_{i}\times\Lambda_{j}}-\log|x-y|\mathtt{d}\mathbf{f}_{N}(x)\mathtt{d}\mathbf{f}_{N}(y).$
(4.2)

By expanding the double integral defining $\mathsf{F}_{N}$, we may write:

$\mathsf{F}_{N}(\mathbf{X}_{N})=\mathsf{F}_{\mathsf{Ext}}(\mathbf{X}_{N}\cap\mathsf{Ext})+\mathsf{Int}[\Lambda_{1},\dots,\Lambda_{\mathcal{N}}]+\sum_{i=1}^{\mathcal{N}}\left(\mathsf{F}_{\Lambda_{i}}(\mathbf{X}_{N}\cap\Lambda_{i})+\int_{\Lambda_{i}}\mathrm{V}^{\mathrm{ext}}(x)\mathtt{d}\mathbf{f}_{N}(x)\right)$
(4.3)

with $\mathsf{F}_{\mathsf{Ext}},\mathsf{F}_{\Lambda_{i}}$ defined the obvious
way (see (4.9)). On the other hand, for each $1\leq i\leq\mathcal{N}$, let
$\mathsf{D}_{i}$ be the discrepancy of $\mathbf{X}_{N}$ in $\Lambda_{i}$, and
let $\widetilde{\mathsf{Int}}[\mathsf{D}_{1},\dots,\mathsf{D}_{\mathcal{N}}]$
be the approximation of
$\mathsf{Int}[\Lambda_{1},\dots,\Lambda_{\mathcal{N}}]$ given by:

$\widetilde{\mathsf{Int}}[\mathsf{D}_{1},\dots,\mathsf{D}_{\mathcal{N}}]:=\frac{1}{2}\sum_{1\leq
i\neq
j\leq\mathcal{N}}-\mathsf{D}_{i}\mathsf{D}_{j}\log|\omega_{i}-\omega_{j}|,$
(4.4)

where $\omega_{i}$ denotes the center of $\Lambda_{i}$. We define the quantity
$\mathrm{ErrorCI}$ as follows:

$\mathrm{ErrorCI}\left[\mathbf{X}_{N}|\left(\Lambda_{1},\dots,\Lambda_{\mathcal{N}}\right)\right]:=\left|\mathsf{Int}[\Lambda_{1},\dots,\Lambda_{\mathcal{N}}]-\widetilde{\mathsf{Int}}[\mathsf{D}_{1},\dots,\mathsf{D}_{\mathcal{N}}]\right|,$
(4.5)

and we use it below in order to measure a “conditional independence error”.

###  4.2 Bounds on the conditional independence error

For $1\leq i\neq j\leq\mathcal{N}$, define the distance $\mathrm{d}_{ij}$ as
$\mathrm{d}_{ij}:=\operatorname{dist}(\Lambda_{i},\Lambda_{j})$. Assume that
for all $1\leq i\leq\mathcal{N}$ the disk $\Lambda_{i}$ has radius $T$ and
that:

$\max_{1\leq i\leq\mathcal{N}}\mathsf{n}_{i}\leq 10T^{2},\quad\min_{1\leq
i\neq j\leq\mathcal{N}}\mathrm{d}_{ij}\geq 10T.$ (4.6)

######  Lemma 4.1 (The size of $\mathrm{ErrorCI}$).

We have, if (4.6) holds:

$\left|\mathrm{ErrorCI}\left[\mathbf{X}_{N}|\left(\Lambda_{1},\dots,\Lambda_{\mathcal{N}}\right)\right]\right|=\mathcal{O}(T^{5})\sum_{1\leq
i\neq j\leq\mathcal{N}}\frac{1}{\mathrm{d}_{ij}}.$ (4.7)

###### Proof of Lemma 4.1.

For $x\in\Lambda_{i},y\in\Lambda_{j}$ (with $i\neq j$) since the diameter of
the squares is $\mathcal{O}(T)$ and the mutual distances satisfy (4.6), a
Taylor’s expansion yields:

$\log|x-y|=\log|\omega_{i}-\omega_{j}|+\mathcal{O}\left(\frac{T}{|\omega_{i}-\omega_{j}|}\right),$
(4.8)

with a universal implicit constant. Integrating (4.8) against the fluctuation
measures in $\Lambda_{i}$ and $\Lambda_{j}$ yields:

$\left|\iint_{\Lambda_{i}\times\Lambda_{j}}\log|x-y|\mathtt{d}\mathbf{f}_{N}(x)\mathtt{d}\mathbf{f}_{N}(y)-\mathsf{D}_{i}\mathsf{D}_{j}\log|\omega_{i}-\omega_{j}|\right|\leq(|\Lambda_{i}|+\mathsf{n}_{i})\cdot(|\Lambda_{j}|+\mathsf{n}_{j})\cdot\mathcal{O}\left(\frac{T}{\mathrm{d}_{ij}}\right).$

Using the bound on $\mathsf{n}_{i},\mathsf{n}_{j}$ given by (4.6) and summing
over $i\neq j$, we get (4.7). ∎

######  Remark 4.2.

It is possible to reduce the order of magnitude of $\mathrm{ErrorCI}$ by
expanding the interaction in a more precise way.

###  4.3 2DOCP’s with harmonic external field

In the following we consider two-dimensional one-component plasmas whose
energy takes into account the effect of an external field
$\mathrm{V}^{\mathrm{ext}}$ on each particle.

  * •

Let $\mathrm{V}^{\mathrm{ext}}$ (the external field on $\Lambda$) be a lower
semi-continuous function on $\mathbb{R}^{2}$ that is harmonic on (the interior
of) $\Lambda$. Let us emphasize that $\mathrm{V}^{\mathrm{ext}}$ is harmonic
hence very regular in the interior of $\Lambda$ but we do not _a priori_
control $\mathrm{V}^{\mathrm{ext}}$ or its derivatives near $\partial\Lambda$.
The situation is thus different from the choice of an external
weight/potential as frequently found in the literature on log-gases.

  * •

Let $\mathrm{X}_{\mathsf{n}}$ denote a $\mathsf{n}$-tuple of points
$\mathrm{X}_{\mathsf{n}}=(x_{1},\dots,x_{\mathsf{n}})$ in $\Lambda$, let
$\mathbf{X}_{\mathsf{n}}:=\sum_{i=1}^{\mathsf{n}}\delta_{x_{i}}$ be the
associated atomic measure, and let
$\mathbf{f}_{\Lambda}:=\mathbf{X}_{\mathsf{n}}-\mathbf{m}_{0}\mathtt{1}_{\Lambda}$
be the signed fluctuation measure on $\Lambda$. Let
$\mathsf{F}_{\Lambda}(\mathbf{X}_{\mathsf{n}})$ be the logarithmic interaction
energy:

$\mathsf{F}_{\Lambda}(\mathbf{X}_{\mathsf{n}}):=\frac{1}{2}\iint_{(x,y)\in\Lambda\times\Lambda,x\neq
y}-\log|x-y|\mathtt{d}\mathbf{f}_{\Lambda}(x)\mathtt{d}\mathbf{f}_{\Lambda}(y).$
(4.9)

We define a probability density
$\mathbb{P}^{\beta}_{\mathsf{n},\Lambda,\mathrm{V}^{\mathrm{ext}}}$ on the
space of $\mathsf{n}$-tuples of points in $\Lambda$ by setting:

$\mathtt{d}\mathbb{P}^{\beta}_{\mathsf{n},\Lambda,\mathrm{V}^{\mathrm{ext}}}(\mathrm{X}_{\mathsf{n}}):=\frac{\exp\left(-\beta\left(\mathsf{F}_{\Lambda}(\mathbf{X}_{\mathsf{n}})+\int_{\Lambda}\mathrm{V}^{\mathrm{ext}}(x)\mathtt{d}\mathbf{f}_{\Lambda}(x)\right)\right)}{\mathrm{K}^{\beta}_{\mathsf{n},\Lambda,\mathrm{V}^{\mathrm{ext}}}}\mathtt{d}\mathrm{X}_{\mathsf{n}},$
(4.10)

where $\mathrm{K}^{\beta}_{\mathsf{n},\Lambda,\mathrm{V}^{\mathrm{ext}}}$ is
the partition function, namely the normalizing constant:

$\mathrm{K}^{\beta}_{\mathsf{n},\Lambda,\mathrm{V}^{\mathrm{ext}}}:=\int_{\Lambda^{\mathsf{n}}}\exp\left(-\beta\left(\mathsf{F}_{\Lambda}(\mathbf{X}_{\mathsf{n}})+\int_{\Lambda}\mathrm{V}^{\mathrm{ext}}(x)\mathtt{d}\mathbf{f}_{\Lambda}(x)\right)\right)\mathtt{d}\mathrm{X}_{\mathsf{n}},$
(4.11)

and $\mathtt{d}\mathrm{X}_{\mathsf{n}}$ is the Lebesgue measure on
$\Lambda^{\mathsf{n}}$. We may now state the main result of this section.

###  4.4 Approximate conditional independence

######  Proposition 4.3 (Approximate conditional independence).

  * •

For $1\leq i\leq\mathcal{N}$, let $\mathsf{G}_{i}$ be a measurable function on
$\mathrm{Conf}$ with _non-negative_ real values and let $\mathcal{E}_{i}$ be a
measurable subset of $\mathrm{Conf}$. Assume that $\mathsf{G}_{i}$,
$\mathcal{E}_{i}$ are $\Lambda_{i}$-local in the sense of Section 2.1.

  * •

Let $\mathcal{E}_{\mathrm{ext}}$ be a measurable subset of $\mathrm{Conf}$,
assume that $\mathcal{E}_{\mathrm{ext}}$ is $\mathsf{Ext}$-local.

  * •

Denote by $\mathcal{E}_{N}$ the following event:
$\mathcal{E}_{N}:=\mathcal{E}_{\mathrm{ext}}\cap\bigcap_{i=1}^{\mathcal{N}}\mathcal{E}_{i}.$

  * •

Finally, we say that a family $\\{\mathsf{n}_{i}\\}_{1\leq i\leq\mathcal{N}}$
of integers is “admissible” (we write “$\\{\mathsf{n}_{i}\\}\ \mathrm{adm.}$”
below) when there exists $\mathbf{X}_{N}\in\mathcal{E}_{N}$ such that
$\mathsf{Pts}(\mathbf{X}_{N},\Lambda_{i})=\mathsf{n}_{i}$ for all $1\leq
i\leq\mathcal{N}$.

We have:

$\mathbb{E}^{\beta}_{N}\left[\prod_{i=1}^{\mathcal{N}}\mathsf{G}_{i}(\mathbf{X}_{N})\mathtt{1}_{\mathcal{E}_{N}}\right]\leq\exp\left(2\beta\sup_{\mathbf{X}_{N}\in\mathcal{E}_{N}}\mathrm{ErrorCI}(\mathbf{X}_{N})\right)\\\
\times\sup_{\mathbf{X}^{\mathrm{ext}}\in\mathcal{E}_{\mathrm{ext}},\\{\mathsf{n}_{i}\\}\
\mathrm{adm.}}\prod_{i=1}^{\mathcal{N}}\mathbb{E}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}}\left[\mathsf{G}_{i}(\mathbf{X})\Big{|}\mathcal{E}_{i}\right].$
(4.12)

Moreover, with the same assumptions, the following lower bound holds:

$\mathbb{E}^{\beta}_{N}\left[\prod_{i=1}^{\mathcal{N}}\mathsf{G}_{i}(\mathbf{X}_{N})\mathtt{1}_{\mathcal{E}_{N}}\right]\geq\exp\left(-2\beta\sup_{\mathbf{X}_{N}\in\mathcal{E}_{N}}\mathrm{ErrorCI}(\mathbf{X}_{N})\right)\times\mathbb{P}^{\beta}_{N}(\mathcal{E}_{N})\\\
\times\inf_{\mathbf{X}^{\mathrm{ext}}\in\mathcal{E}_{\mathrm{ext}},\\{\mathsf{n}_{i}\\}\
\mathrm{adm.}}\prod_{i=1}^{\mathcal{N}}\mathbb{E}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}}\left[\mathsf{G}_{i}(\mathbf{X})\Big{|}\mathcal{E}_{i}\right].$
(4.13)

Notice that one $\sup$ is now an $\inf$, the error term $\sup\mathrm{ErrorCI}$
now appears in the exponent with a minus sign, and there is an extra factor
$\mathbb{P}^{\beta}_{N}(\mathcal{E}_{N})$.

###### Proof of Proposition 4.3.

Let us start by using the definition (1.2) of $\mathbb{P}^{\beta}_{N}$ and the
decomposition (4.3) of the logarithmic interaction
$\mathsf{F}_{N}(\mathrm{X}_{N})$. We obtain:

$\mathbb{E}^{\beta}_{N}\left[\prod_{i=1}^{\mathcal{N}}\mathsf{G}_{i}(\mathbf{X}_{N})\mathtt{1}_{\mathcal{E}_{N}}\right]=\frac{1}{\mathrm{K}^{\beta}_{N}}\int_{(\Sigma_{N})^{N}}\exp\left(-\beta\mathsf{F}_{\mathsf{Ext}}(\mathbf{X}_{N}\cap\mathsf{Ext})\right)\exp\left(-\beta\mathsf{Int}[\Lambda_{1},\dots,\Lambda_{\mathcal{N}}]\right)\\\
\times\left[\prod_{i=1}^{\mathcal{N}}\exp\left(-\beta\left(\mathsf{F}_{\Lambda_{i}}(\mathbf{X}_{N}\cap\Lambda_{i})+\int_{\Lambda_{i}}\mathrm{V}^{\mathrm{ext}}(x)\mathtt{d}\mathbf{f}_{N}(x)\right)\right)\times\mathsf{G}_{i}(\mathbf{X}_{N})\right]\mathtt{1}_{\mathcal{E}_{N}}(\mathbf{X}_{N})\mathtt{d}\mathrm{X}_{N}.$

Next:

  * •

We use a complete system of events by fixing the number $\mathsf{n}_{i}$ of
points in each $\Lambda_{i}$ (and thus also the number
$\mathsf{n}_{\mathrm{ext}}$ of points in $\mathsf{Ext}$). The knowledge of
$\mathsf{n}_{i}$ is equivalent to fixing the discrepancy $\mathsf{D}_{i}$ in
$\Lambda_{i}$. Since we are working under the event $\mathcal{E}_{N}$, the
$\\{\mathsf{n}_{i}\\}$ must be admissible as defined in the statement of
Proposition 4.3.

  * •

Up to a combinatorial factor, we may then decompose the $N$-tuple
$\mathrm{X}_{N}$ into a $\mathsf{n}_{\mathrm{ext}}$-tuple
$\mathrm{X}^{\mathrm{ext}}$ of points in $\mathsf{Ext}$, and $\mathrm{X}_{i}$
($1\leq i\leq\mathcal{N}$), where each $\mathrm{X}_{i}$ is a
$\mathsf{n}_{i}$-tuple of points in $\Lambda_{i}$.

  * •

We decompose the Lebesgue measure $\mathtt{d}\mathrm{X}_{N}$ accordingly.

  * •

We write $\mathbf{X}^{\mathrm{ext}}$ and $\mathbf{X}_{i}$ ($1\leq
i\leq\mathcal{N}$) for the associated atomic measures. We have the identities
$\mathbf{X}^{\mathrm{ext}}=\mathbf{X}_{N}\cap\mathsf{Ext}$ and
$\mathbf{X}_{i}=\mathbf{X}_{N}\cap\Lambda_{i}$ ($1\leq i\leq\mathcal{N}$).

  * •

By our locality assumptions:
$\mathtt{1}_{\mathcal{E}_{N}}(\mathbf{X}_{N})=\mathtt{1}_{\mathcal{E}_{\mathrm{ext}}}(\mathbf{X}^{\mathrm{ext}})\times\prod_{i=1}^{\mathcal{N}}\mathsf{1}_{\mathcal{E}_{i}}(\mathbf{X}_{i})$.

  * •

We use our locality assumption on $\mathsf{G}_{i}$ to write
$\mathsf{G}_{i}(\mathbf{X}_{N})=\mathsf{G}_{i}(\mathbf{X}_{i})$.

  * •

We introduce the measures
$\mathbf{f}_{\mathsf{Ext}}:=\mathbf{X}^{\mathrm{ext}}-\mathbf{m}_{0}\mathtt{1}_{\mathsf{Ext}}$
and
$\mathbf{f}_{\Lambda_{i}}:=\mathbf{X}_{i}-\mathbf{m}_{0}\mathtt{1}_{\Lambda_{i}}$.

  * •

Finally, using the definition (4.5) we may replace
$\mathsf{Int}[\Lambda_{1},\dots,\Lambda_{\mathcal{N}}]$ by
$\widetilde{\mathsf{Int}}[\mathsf{D}_{1},\dots,\mathsf{D}_{\mathcal{N}}]$, up
to an error quantified by $\mathrm{ErrorCI}$. The quantity
$\exp\left(-\beta\widetilde{\mathsf{Int}}[\mathsf{D}_{1},\dots,\mathsf{D}_{\mathcal{N}}]\right)$
can be taken outside the integrals because it only depends on the data of
$\\{\mathsf{n}_{i}\\}_{1\leq i\leq\mathcal{N}}$.

We obtain the following upper bound:

$\mathbb{E}^{\beta}_{N}\left[\prod_{i=1}^{\mathcal{N}}\mathsf{G}_{i}(\mathbf{X}_{N})\mathtt{1}_{\mathcal{E}_{N}}\right]\leq\frac{1}{\mathrm{K}^{\beta}_{N}}\sum_{\\{\mathsf{n}_{i}\\}\
\mathrm{adm.}}\binom{N}{\mathsf{n}_{1}\dots\mathsf{n}_{\mathcal{N}}}\exp\left(-\beta\widetilde{\mathsf{Int}}[\mathsf{D}_{1},\dots,\mathsf{D}_{\mathcal{N}}]\right)\\\
\times\exp\left(\beta\sup_{\mathbf{X}_{N}\in\mathcal{E}_{N}}\mathrm{ErrorCI}(\mathbf{X}_{N})\right)\times\int_{(\mathsf{Ext})^{\mathsf{n}_{\mathrm{ext}}}}\exp\left(-\beta\mathsf{F}_{\mathsf{Ext}}(\mathbf{X}^{\mathrm{ext}})\right)\mathtt{1}_{\mathcal{E}_{\mathrm{ext}}}(\mathbf{X}^{\mathrm{ext}})\\\
\times\left[\prod_{i=1}^{\mathcal{N}}\int_{(\Lambda_{i})^{\mathsf{n}_{i}}}\exp\left(-\beta\left(\mathsf{F}_{\Lambda_{i}}(\mathbf{X}_{i})+\int_{\Lambda_{i}}\mathrm{V}^{\mathrm{ext}}(x)\mathtt{d}\mathbf{f}_{\Lambda_{i}}(x)\right)\right)\times\mathsf{G}_{i}(\mathbf{X}_{i})\mathtt{1}_{\mathcal{E}_{i}}(\mathbf{X}_{i})\mathtt{d}\mathrm{X}_{i}\right]\mathtt{d}\mathrm{X}^{\mathrm{ext}}.$
(4.14)

We may conveniently condense (4.14) using our notation. For $1\leq
i\leq\mathcal{N}$, in view of the definitions (4.10), (4.11), we write:

$\int_{(\Lambda_{i})^{\mathsf{n}_{i}}}\exp\left(-\beta\left(\mathsf{F}_{\Lambda_{i}}(\mathbf{X}_{i})+\int_{\Lambda_{i}}\mathrm{V}^{\mathrm{ext}}(x)\mathtt{d}\mathbf{f}_{\Lambda_{i}}(x)\right)\right)\mathsf{G}_{i}(\mathbf{X}_{i})\mathtt{1}_{\mathcal{E}_{i}}(\mathbf{X}_{i})\mathtt{d}\mathrm{X}_{i}\\\
=\mathbb{E}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}}\left[\mathsf{G}_{i}(\mathbf{X})\Big{|}\mathcal{E}_{i}\right]\times\mathbb{P}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}}\left[\mathcal{E}_{i}\right]\times\mathrm{K}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}},$

and we may thus re-write (4.14) as:

$\mathbb{E}^{\beta}_{N}\left[\prod_{i=1}^{\mathcal{N}}\mathsf{G}_{i}(\mathbf{X}_{N})\mathtt{1}_{\mathcal{E}_{N}}\right]\leq\frac{1}{\mathrm{K}^{\beta}_{N}}\sum_{\\{\mathsf{n}_{i}\\}\
\mathrm{adm.}}\binom{N}{\mathsf{n}_{1}\dots\mathsf{n}_{\mathcal{N}}}\exp\left(-\beta\widetilde{\mathsf{Int}}[\mathsf{D}_{1},\dots,\mathsf{D}_{\mathcal{N}}]\right)\\\
\times\exp\left(\beta\sup_{\mathbf{X}_{N}\in\mathcal{E}_{N}}\mathrm{ErrorCI}(\mathbf{X}_{N})\right)\times\int_{(\mathsf{Ext})^{\mathsf{n}_{\mathrm{ext}}}}\exp\left(-\beta\mathsf{F}_{\mathsf{Ext}}(\mathbf{X}^{\mathrm{ext}})\right)\mathtt{1}_{\mathcal{E}_{\mathrm{ext}}}(\mathbf{X}^{\mathrm{ext}})\\\
\times\left[\prod_{i=1}^{\mathcal{N}}\mathbb{E}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}}\left[\mathsf{G}_{i}(\mathbf{X})\Big{|}\mathcal{E}_{i}\right]\times\mathbb{P}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}}\left[\mathcal{E}_{i}\right]\times\mathrm{K}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}}\right]\mathtt{d}\mathrm{X}^{\mathrm{ext}}.$
(4.15)

In (4.14), (4.15), the converse inequalities hold up to adding a minus sign in
front of $\mathrm{ErrorCI}$ in the exponent (recall that the
$\mathsf{G}_{i}$’s have non-negative values by assumption). Now, for all
$\mathbf{X}^{\mathrm{ext}}\in\mathcal{E}_{\mathrm{ext}}$ and all admissible
$\\{\mathsf{n}_{i}\\}$, we have:

$\prod_{i=1}^{\mathcal{N}}\mathbb{E}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}}\left[\mathsf{G}_{i}(\mathbf{X})\Big{|}\mathcal{E}_{i}\right]\leq\sup_{\mathbf{X}^{\mathrm{ext}}\in\mathcal{E}_{\mathrm{ext}}\\{\mathsf{n}_{i}\\}\
\mathrm{adm.}}\prod_{i=1}^{\mathcal{N}}\mathbb{E}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}}\left[\mathsf{G}_{i}(\mathbf{X})\Big{|}\mathcal{E}_{i}\right],$

and the converse inequality holds up to trading the $\sup$ for an $\inf$. We
get:

$\mathbb{E}^{\beta}_{N}\left[\prod_{i=1}^{\mathcal{N}}\mathsf{G}_{i}(\mathbf{X}_{N})\mathtt{1}_{\mathcal{E}_{N}}\right]\leq\sup_{\mathbf{X}^{\mathrm{ext}}\in\mathcal{E}_{\mathrm{ext}},\\{\mathsf{n}_{i}\\}\mathrm{adm.}}\prod_{i=1}^{\mathcal{N}}\mathbb{E}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}}\left[\mathsf{G}_{i}(\mathbf{X})\Big{|}\mathcal{E}_{i}\right]\times\exp\left(\beta\sup_{\mathbf{X}_{N}\in\mathcal{E}_{N}}\mathrm{ErrorCI}(\mathbf{X}_{N})\right)\\\
\times\frac{1}{\mathrm{K}^{\beta}_{N}}\sum_{\\{\mathsf{n}_{i}\\}\
\mathrm{adm.}}\binom{N}{\mathsf{n}_{1}\dots\mathsf{n}_{\mathcal{N}}}\exp\left(-\beta\widetilde{\mathsf{Int}}[\mathsf{D}_{1},\dots,\mathsf{D}_{\mathcal{N}}]\right)\int_{(\mathsf{Ext})^{\mathsf{n}_{\mathrm{ext}}}}\exp\left(-\beta\mathsf{F}_{\mathsf{Ext}}(\mathbf{X}^{\mathrm{ext}})\right)\mathtt{1}_{\mathcal{E}_{\mathrm{ext}}}(\mathbf{X}^{\mathrm{ext}})\\\
\times\left[\prod_{i=1}^{\mathcal{N}}\int_{(\Lambda_{i})^{\mathsf{n}_{i}}}\mathbb{P}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}}\left[\mathcal{E}_{i}\right]\times\mathrm{K}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}}\right]\mathtt{d}\mathrm{X}^{\mathrm{ext}},$
(4.16)

and the converse inequality holds up to trading the first $\sup$ for an $\inf$
and writing $-\sup\mathrm{ErrorCI}$ instead of $\sup\mathrm{ErrorCI}$ in the
exponent. We now establish a bound on the partition function
$\mathrm{K}^{\beta}_{N}$ in a similar fashion. By definition, we have:

$\begin{cases}\frac{1}{\mathrm{K}^{\beta}_{N}}&\leq\left(\int_{(\Sigma_{N})^{N}}\mathtt{1}_{\mathcal{E}_{N}}\exp\left(-\beta\mathsf{F}_{N}(\mathbf{X}_{N})\right)\mathtt{d}\mathrm{X}_{N}\right)^{-1}\\\
\frac{1}{\mathrm{K}^{\beta}_{N}}&=\left(\int_{(\Sigma_{N})^{N}}\mathtt{1}_{\mathcal{E}_{N}}\exp\left(-\beta\mathsf{F}_{N}(\mathbf{X}_{N})\right)\mathtt{d}\mathrm{X}_{N}\right)^{-1}\times\mathbb{P}^{\beta}_{N}(\mathcal{E}_{N}).\end{cases}$
(4.17)

Arguing as above (taking all the $\mathsf{G}_{i}$’s equal to the constant
$1$), we get:

$\int_{(\Sigma_{N})^{N}}\mathtt{1}_{\mathcal{E}_{N}}\exp\left(-\beta\mathsf{F}_{N}(\mathbf{X}_{N})\right)\mathtt{d}\mathrm{X}_{N}\\\
\leq\exp\left(\beta\sup_{\mathbf{X}_{N}\in\mathcal{E}_{N}}\mathrm{ErrorCI}(\mathbf{X}_{N})\right)\times\sum_{\\{\mathsf{n}_{i}\\}\
\mathrm{adm.}}\binom{N}{\mathsf{n}_{1}\dots\mathsf{n}_{\mathcal{N}}}\exp\left(-\beta\widetilde{\mathsf{Int}}[\mathsf{D}_{1},\dots,\mathsf{D}_{\mathcal{N}}]\right)\\\
\times\int_{\mathsf{Ext}^{\mathsf{n}_{\mathrm{ext}}}}\exp\left(-\beta\mathsf{F}_{\mathsf{Ext}}(\mathbf{X}^{\mathrm{ext}})\right)\mathtt{1}_{\mathcal{E}_{\mathrm{ext}}}(\mathbf{X}^{\mathrm{ext}})\left[\prod_{i=1}^{\mathcal{N}}\int_{(\Lambda_{i})^{\mathsf{n}_{i}}}\mathbb{P}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}}\left[\mathcal{E}_{i}\right]\times\mathrm{K}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}}\right]\mathtt{d}\mathrm{X}^{\mathrm{ext}},$
(4.18)

and the converse inequality holds with $-\sup\mathrm{ErrorCI}$ instead of
$\sup\mathrm{ErrorCI}$. Combining (4.16), (4.17) and (4.18) we obtain (4.12)
(note that many terms cancel out in the ratio). We have also proven the
converse inequality (4.13), with an extra factor
$\mathbb{P}^{\beta}_{N}(\mathcal{E}_{N})$ coming from (4.17). ∎

##  5 Generalized 2DOCP’s arising as sub-systems

Let $\Lambda$ be a disk of center $\omega\in\mathbb{R}^{2}$ and radius $T$,
and let $\mathsf{n}\geq 1$ be an integer, corresponding to the number of
points in $\Lambda$. In general we may have $\mathsf{n}\neq|\Lambda|$.

###  5.1 Good external potentials, good sub-systems

Let $\mathrm{V}^{\mathrm{ext}}$ be an external field as in Section 4.3 (we
will eventually use this for the particular choice (4.1)). We introduce two
definitions to pinpoint “good situations” for the generalized 2DOCP measure
$\mathbb{P}^{\beta}_{\mathsf{n},\Lambda,\mathrm{V}^{\mathrm{ext}}}$ introduced
in (4.10).

######  Definition 5.1 (Good potential).

We say that $\mathrm{V}^{\mathrm{ext}}$ is a “ _good external potential_ on
$\Lambda$ with constant $\bar{\mathtt{C}}$” when the following holds:

1\. Control up to the edge.

There exists a function $\tilde{\mathrm{V}}^{\mathrm{ext}}$ on $\Lambda$
satisfying:

$\tilde{\mathrm{V}}^{\mathrm{ext}}(x)=\mathrm{V}^{\mathrm{ext}}(x)\text{ if
}\operatorname{dist}(x,\partial\Lambda)\geq
1,\quad\tilde{\mathrm{V}}^{\mathrm{ext}}(x)\leq\mathrm{V}^{\mathrm{ext}}(x)+100\text{
for all }x\in\Lambda,$ (5.1)

such that:

$\left|\tilde{\mathrm{V}}^{\mathrm{ext}}(x)-\tilde{\mathrm{V}}^{\mathrm{ext}}(\omega)\right|\leq\bar{\mathtt{C}}\times
T\times\log^{3}T.$ (5.2)

2\. A technical decomposition.

$\mathrm{V}^{\mathrm{ext}}$ can be decomposed as the sum

$\mathrm{V}^{\mathrm{ext}}=\mathfrak{h}^{\nu}+R,$ (5.3)

where $\mathfrak{h}^{\nu}$ is the logarithmic potential generated by a
positive measure supported on an annulus of width $\hat{T}:=\log T$ right
outside $\Lambda$, and $R$ is harmonic in $\Lambda$, such that we control the
derivative of $R$ up to the edge by:

$|R|_{1,\Lambda}\leq\bar{\mathtt{C}}\times\log^{2}T,$ (5.4)

and moreover we control the mass of $\nu$ locally at scale $\hat{T}=\log T$:

$\sup_{x\in\partial\Lambda}\nu(\mathfrak{D}(x,\log
T))\leq\bar{\mathtt{C}}\log^{2}T.$ (5.5)

######  Definition 5.2 (Good sub-system).

We consider the event “$\Lambda$ is a good-system” defined as the sub-set of
all point configurations $\mathbf{X}\in\mathrm{Conf}$ such that:

  1. 1.

The discrepancy $\mathsf{Dis}(\mathbf{X},\Lambda)$ satisfies:
$|\mathsf{Dis}(\mathbf{X},\Lambda)|\leq T\log^{2}T$.

  2. 2.

$\mathbf{X}$ belongs to the event $\mathcal{E}_{\Lambda}$ defined by:

    1. (a)

There is absolutely no point in $\Lambda$ at distance $\leq e^{-\log^{2}T}$
from $\partial\Lambda$.

    2. (b)

There is no more than $T\log T$ points in $\Lambda$ at distance $\leq 1$ from
$\partial\Lambda$.

This event is of course $\Lambda$-local and even
$\\{x\in\Lambda,\operatorname{dist}(x,\partial\Lambda)\leq 1\\}$-local.

#### The effective external potential is often good.

Let $\Lambda_{i}:=\mathfrak{D}(\omega_{i},T)$ ($1\leq i\leq\mathcal{N}$) be
the disks introduced at the end of Section 3.2. We define a common external
potential for all $\Lambda_{i}$’s by setting (cf. (4.1)):

$\mathrm{V}^{\mathrm{ext}}(x):=\int_{\Sigma_{N}\setminus\cup_{i=1}^{\mathcal{N}}\Lambda_{i}}-\log|x-y|\mathtt{d}\mathbf{f}_{N}(y).$
(5.6)

######  Proposition 5.3 (The effective external potential is often good).

There exists a constant $\bar{\mathtt{C}}$ depending only on $\beta$ (and on
the choice of $\delta$ as in Assumption 1.5) such that the following holds.
With $\mathbb{P}^{\beta}_{N}$-probability greater than
$1-\mathcal{N}\exp\left(-\frac{\log^{2}T}{\bar{\mathtt{C}}}\right)$, for all
$i=1,\dots,\mathcal{N}$ the external potential $\mathrm{V}^{\mathrm{ext}}$ is
a good external potential on $\Lambda_{i}$ with constant $\bar{\mathtt{C}}$
(in the sense of Definition 5.1).

The proof of Proposition 5.3 is elementary but cumbersome. We postpone it to
Section C. Since the constant $\bar{\mathtt{C}}$ given by Proposition 5.3
depends only on $\beta,\delta$, let us keep in mind that if we say that some
constant depends on the “good external potential constant” $\bar{\mathtt{C}}$
then in fact it itself only depends on $\beta,\delta$.

#### The sub-systems are often good

######  Lemma 5.4.

For $\mathtt{C}_{\beta}$ large enough, if $T\geq\mathtt{C}_{\beta}$, then with
$\mathbb{P}^{\beta}_{N}$-probability $\geq
1-\mathcal{N}\exp\left(-\frac{\log^{2}T}{\mathtt{C}_{\beta}}\right)$, for all
$i=1,\dots,\mathcal{N}$, the conditions of Definition 5.2 (expressing the fact
that “$\Lambda_{i}$ is a good sub-system”) are satisfied.

###### Proof.

Using the “discrepancy” part (2.11) of the local laws (Proposition 2.4) we see
that if $T$ is large enough (depending on $\beta$) then for any fixed $i$ we
have:

$\mathbb{P}^{\beta}_{N}\left(|\mathsf{Dis}(\mathbf{X}_{N},\Lambda_{i})|\geq
T\log T\right)\leq\exp\left(-\frac{\log^{2}T}{\mathtt{C}_{\beta}}\right).$

Checking the conditions of $\mathcal{E}_{\Lambda}$ requires more care.

######  Claim 5.5 (The conditions of $\mathcal{E}_{\Lambda}$ are often met).

For any fixed $i$ we have:

$\mathbb{P}^{\beta}_{N}\left(\mathcal{E}_{\Lambda_{i}}\right)\geq
1-\exp\left(-\frac{\log^{2}T}{\mathtt{C}_{\beta}}\right).$

###### Proof of Claim 5.5.

From the local laws, it is easy to see that the second condition in Definition
5.2 is often satisfied. Indeed we can cover the $1$-neighborhood of the
boundary $\partial\Lambda_{i}$ by $\mathtt{C}T$ squares of sidelength $1$,
each of which contains at most $\frac{\log T}{100\mathtt{C}}$ points with
probability $\geq 1-\exp\left(-\frac{\log^{2}T}{\mathtt{C}_{\beta}}\right)$
(for $T$ large enough). We conclude with a union bound, which does not hinder
the estimate.

The first condition of Definition 5.2 is more subtle and we rely on the “one-
particle cluster” bound from [Tho22] mentioned in Lemma 2.9. Cover the region
$\Gamma_{i}:=\\{z\in\Lambda,\operatorname{dist}(z,\partial\Lambda_{i})\leq
e^{-\log^{2}T}\\}$ by $\mathcal{O}(Te^{\log^{2}T})$ disks of radius
$r=10e^{-\log^{2}T}$. For each disk, we know from (2.16) that the probability
of it being occupied by at least one particle is smaller than
$\mathtt{C}_{\beta}r^{2}$. Then an union bound shows that the probability of
at least one point falling anywhere in $\Gamma_{i}$ is bounded by
$\mathtt{C}_{\beta}Te^{-\log^{2}T}$, which concludes the proof of the claim. ∎

We use a union bound over $\mathcal{N}$ such events to handle all the
$\Lambda_{i}$’s at once. ∎

Until the end of Section 5, we consider a “good external potential”
$\mathrm{V}^{\mathrm{ext}}"$ with constant $\bar{\mathtt{C}}$ as in Definition
5.1 and we assume that $|\mathsf{n}-|\Lambda||\leq T\log T$.

###  5.2 2DOCP’s with non-uniform neutralizing background

Instead of adding $\mathrm{V}^{\mathrm{ext}}$, let us consider 2DOCP’s in
which the “neutralizing background” is no longer the uniform one, but a
perturbation thereof.

  * •

Let $\mathbf{m}$ be a (non-negative) measure on $\Lambda$ and let $\zeta$ be
some non-negative function on $\mathbb{R}^{2}$. Assume that $\mathbf{m}$ is
supported in $\Lambda$ and that $\zeta$ vanishes on the support of
$\mathbf{m}$.

  * •

Assume that the measure $\mathbf{m}$ can be written as the sum of a measure
which has a bounded density with respect to the Lebesgue measure
$\mathbf{m}_{0}$ on $\Lambda$ and of a singular measure which has a bounded
density with respect to the arc-length measure $\mathtt{d}s$ on
$\partial\Lambda$. This assumption will be justified later in Section D.2. In
particular, it implies that
$\iint-\log|x-y|\mathtt{d}\mathbf{m}(x)\mathtt{d}\mathbf{m}(y)$ is finite.

  * •

Let $\mathrm{X}_{\mathsf{n}}$ be a $\mathsf{n}$-tuple of points in $\Lambda$
and $\mathbf{X}_{\mathsf{n}}$ be the associated point configuration, we let
$\mathsf{F}_{\Lambda}(\mathbf{X}_{\mathsf{n}},\mathbf{m})$ be the logarithmic
interaction energy _computed with respect to_ $\mathbf{m}$, namely:

$\mathsf{F}_{\Lambda}(\mathbf{X}_{\mathsf{n}},\mathbf{m}):=\frac{1}{2}\iint_{(x,y)\in\Sigma_{N}\times\Sigma_{N},x\neq
y}-\log|x-y|\mathtt{d}(\mathbf{X}_{\mathsf{n}}-\mathbf{m})(x)\mathtt{d}(\mathbf{X}_{\mathsf{n}}-\mathbf{m})(y).$
(5.7)

We define a probability density
$\mathbb{P}_{\mathsf{n},\Lambda}^{\beta}(\cdot,\mathbf{m},\zeta)$ on the space
of $\mathsf{n}$-tuples of points in $\Lambda$ by setting:

$\mathtt{d}\mathbb{P}_{\mathsf{n},\Lambda}^{\beta}(\mathrm{X}_{\mathsf{n}},\mathbf{m},\zeta):=\frac{\exp\left(-\beta\left(\mathsf{F}_{\Lambda}(\mathbf{X}_{\mathsf{n}},\mathbf{m})+\mathsf{n}\sum_{i=1}^{\mathsf{n}}\zeta(x_{i})\right)\right)}{\mathrm{K}_{\mathsf{n},\Lambda}^{\beta}(\mathbf{m},\zeta)}\mathtt{d}\mathrm{X}_{\mathsf{n}},$
(5.8)

where $\mathrm{K}_{\mathsf{n},\Lambda}^{\beta}(\mathbf{m},\zeta)$ is the
partition function, namely the normalizing constant:

$\mathrm{K}^{\beta}_{\mathsf{n},\Lambda,\mathrm{V}^{\mathrm{ext}}}(\mathbf{m},\zeta):=\int_{\Lambda^{\mathsf{n}}}\exp\left(-\beta\left(\mathsf{F}_{\Lambda}(\mathbf{X}_{\mathsf{n}},\mathbf{m})+\mathsf{n}\sum_{i=1}^{\mathsf{n}}\zeta(x_{i})\right)\right)\mathtt{d}\mathrm{X}_{\mathsf{n}}.$
(5.9)

It is well-known that there is a way to pass from an external potential
$\mathrm{V}^{\mathrm{ext}}$ (which should be here treated as a _perturbation_
of a reference potential) as in Section 4.3 to the appropriate “equilibrium”
measure and vice-versa (we refer e.g. to the lecture notes [Ser15, Sec. 2&3]).
However, there is here a specific difficulty due to the behavior of
$\mathrm{V}^{\mathrm{ext}}$ near $\partial\Lambda$ and we postpone the
necessary discussion (inspired by similar concerns in [BBNY17]) to Section
D.1.

#### Electric formalism for
$\mathsf{F}_{\Lambda}(\mathbf{X}_{\mathsf{n}},\mathbf{m})$

We extend here some of the formalism from Sections 2.2 and 2.3 in a fairly
straightforward way. Define the “true electric potential/field” associated to
a point configuration $\mathbf{X}_{\mathsf{n}}$ and the background measure
$\mathbf{m}$ as (cf. Definition 2.1):

$\mathfrak{h}^{\mathbf{X}_{\mathsf{n}},\mathbf{m}}:=-\log\ast(\mathbf{X}_{\mathsf{n}}-\mathbf{m}),\quad\nabla\mathfrak{h}^{\mathbf{X}_{\mathsf{n}},\mathbf{m}}=-\nabla\log\ast(\mathbf{X}_{\mathsf{n}}-\mathbf{m}),$

their truncated versions being defined as in Definition 2.3. Then we have the
following identity, which extends (2.4):

######  Lemma 5.6.

Assume that the total mass of $\mathbf{m}$ is equal to $\mathsf{n}$. Let
$\\{\eta(x),x\in\mathbf{X}_{\mathsf{n}}\\}$ be a truncation vector with
$\eta(x)\leq\mathsf{r}(x)$ (the nearest-neighbor distance introduced in (2.3))
for all $x\in\mathbf{X}_{\mathsf{n}}$. We have:

$\mathsf{F}_{N}(\mathbf{X}_{\mathsf{n}},\mathbf{m})=\frac{1}{2}\left(\frac{1}{2\pi}\int_{\mathbb{R}^{2}}|\nabla\mathfrak{h}^{\mathbf{X}_{\mathsf{n}},\mathbf{m}}_{\vec{\eta}}|^{2}+\sum_{x\in\mathbf{X}_{\mathsf{n}}}\log\eta(x)\right)-\sum_{x\in\mathbf{X}_{\mathsf{n}}}\int_{\mathfrak{D}(x,\eta(x))}\mathsf{f}_{\eta(x)}(t-x)\mathtt{d}\mathbf{m}(t),$
(5.10)

provided the disks $\mathfrak{D}(x,\eta(x))$ do not intersect
$\partial\Lambda$.

###### Proof of Lemma 5.6.

Since we place ourselves away from the singular part of $\mathbf{m}$, the
proof works exactly as when $\mathbf{m}$ has a bounded density, see [AS21,
Lemma 2.2]. ∎

We also extend the notation $\mathrm{Ener}$ from (2.5) by setting:
$\mathrm{Ener}(\mathbf{X}_{\mathsf{n}},\mathbf{m},\Omega):=\int_{\Omega}|\nabla\mathfrak{h}_{\vec{\eta}}^{\mathbf{X}_{\mathsf{n}},\mathbf{m}}|^{2}$.

###  5.3 Good properties of sub-systems with good external potentials

If $\mathrm{V}^{\mathrm{ext}}$ is a good external potential, there exists a
probability measure $\mu_{\mathrm{W}}$ on $\Lambda$ and a function
$\zeta_{\mathrm{W}}$ such that:

$\mathbb{P}^{\beta}_{\mathsf{n},\Lambda,\mathrm{V}^{\mathrm{ext}}}(\cdot)=\mathbb{P}_{\mathsf{n},\Lambda}^{\beta}\left(\cdot,\mathsf{n}\mu_{\mathrm{W}},\zeta_{\mathrm{W}}\right),$
(5.11)

we refer to Section D.2 for a definition and precise study of
$\mu_{\mathrm{W}}$ and to Lemma D.6 for a proof of (5.11). The key features of
$\mu_{\mathrm{W}}$ are that:

  1. 1.

It might be singular on $\partial\Lambda$.

  2. 2.

It might have “holes” near $\partial\Lambda$ (and $\zeta_{\mathrm{W}}>0$ on
these holes).

  3. 3.

It has a constant, positive density $\frac{1}{\mathsf{n}}$ as soon as one
looks at distance $\geq\bar{\mathtt{C}}^{\prime}\log T$ from $\partial\Lambda$
(with $\bar{\mathtt{C}}^{\prime}$ depending on the “good external potential”
constant $\bar{\mathtt{C}}$). On that region we have $\zeta_{\mathrm{W}}\equiv
0$.

For good external potentials, it is thus equivalent to consider the “2DOCP
with external background”
$\mathbb{P}^{\beta}_{\mathsf{n},\Lambda,\mathrm{V}^{\mathrm{ext}}}$ (as in
(4.10)) or the “2DOCP with background measure”
$\mathbb{P}_{\mathsf{n},\Lambda}^{\beta}(\cdot,\mathbf{m},\zeta)$ (with
$\mathbf{m}:=\mathsf{n}\mu_{\mathrm{W}}$) and we will simply write
$\mathbb{P}_{\mathsf{n},\Lambda}^{\beta}$ for the corresponding Gibbs measure
(and $\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}$ for expectations under
$\mathbb{P}_{\mathsf{n},\Lambda}^{\beta}$). We now compare the properties of
this “generalized 2DOCP” to the ones of the original Gibbs measure
$\mathbb{P}^{\beta}_{N}$. The “good properties” are easier to obtain away from
the boundary of $\Lambda$ and for simplicity we will often work in “the bulk”
$\Lambda^{\mathrm{bulk}}$ defined as:

$\Lambda^{\mathrm{bulk}}:=\mathfrak{D}(\omega,T/2)\subset\mathfrak{D}(\omega,T)=\Lambda.$

#### Global law

Recall that the event $\mathcal{E}_{\Lambda}$ was introduced in Definition
5.2.

######  Proposition 5.7 (Global law for sub-systems with good external
potential).

There exists a constant CGlobal depending only on $\beta$ and the “good
potential” constant $\bar{\mathtt{C}}$ such that:

$\log\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\exp\left(\frac{\beta}{2}\mathsf{F}_{\Lambda}(\mathbf{X},\mathsf{n}\mu_{\mathrm{W}})\right)\Big{|}\mathcal{E}_{\Lambda}\right]\leq\hyperref@@ii[prop:global_law_SUBSYS]{\mathtt{C}_{\mathtt{Global}}}T^{2}\log^{5}T.$
(5.12)

We prove Proposition 5.7 in Section D. The bound (5.12) should of course be
compared with (2.7) which is valid for the full system. When considering sub-
systems with good external potentials we are (only) losing some power of $\log
T$, which we have not tried to optimize.

#### Local laws in the bulk

######  Proposition 5.8 (Local laws for sub-systems with good external
potential).

There exists a universal constant $\mathtt{C}$ and a constant CLocal
(depending only on $\beta$ and the “good potential” constant
$\bar{\mathtt{C}}$) such that if $T$ is large enough (depending on
$\beta,\bar{\mathtt{C}}$) then for all $\ell\geq\rho_{\beta}$, for all $x$ in
$\Lambda$, provided that the square $\square(x,\ell)$ is included in
$\Lambda^{\mathrm{bulk}}$ we have:

$\log\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\exp\left(\frac{\beta}{2}\mathrm{Ener}\left(\mathbf{X}_{\mathsf{n}},\square(x,\ell)\right)\right)\Big{|}\mathcal{E}_{\Lambda}\right]\leq\hyperref@@ii[prop:local_law_SUBSYS]{\mathtt{C}_{\mathtt{Local}}}\beta\ell^{2},$
(5.13)

where the “electric energy” is computed with respect to the background measure
$\mathsf{n}\mu_{\mathrm{W}}$, and also:

$\log\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\exp\left(\frac{\beta}{\mathtt{C}}\mathsf{Pts}\left(\mathbf{X}_{\mathsf{n}},\square(x,\ell)\right)\right)\Big{|}\mathcal{E}_{\Lambda}\right]\leq\hyperref@@ii[prop:local_law_SUBSYS]{\mathtt{C}_{\mathtt{Local}}}\beta\ell^{2}.$

This should be compared to Proposition 2.4 for the full system.

###### Proof of Proposition 5.8.

This follows from [Leb17, AS21], but it requires some explanation.

We wish to obtain local laws, namely good controls on the electric energy
(controls on the number of points are obtained as a byproduct) that are
proportional to the volume down to large enough microscopic scales as in
[AS21, Thm. 1]. Only this time we are in presence of an external potential / a
non-uniform background measure $\mathsf{n}\mu_{\mathrm{W}}$, and we were only
able to derive a _global bound_ as in Proposition 5.7, with a global energy
estimate that is slightly larger than the total volume. Our goal is thus
twofold:

  1. 1.

Extend the bootstrap in scales of [Leb17, AS21] to a situation where the
background measure is not constant.

  2. 2.

Show that the bootstrap in scale not only _propagates_ estimates to smaller
scales, but in fact _improves_ (if needed) the estimates at each step, which
allows to get rid of the logarithmic correction in (5.12)

The first point is, in fact, very simple. Indeed, although the methods of
[Leb17, AS21] are _not_ suited to situations where the background measure has
singularities or holes (in fact this is the reason why local laws are not
proven near the edge for the full system), they are local by design and work
as soon as we look at distances $\geq\mathtt{C}\log T$ from the boundary,
because then the background measure $\mathsf{n}\mu_{\mathrm{W}}$ is equal to
$1$ (see Section 5.3). Since we only care about the bulk of $\Lambda$, this is
fine.

The second point is more interesting. The basic tool for the proof of local
laws is the fact that thanks to the screening construction of Serfaty _et al._
one can, up to some errors, decouple the system in a given region
$\Omega\subset\Lambda$ of characteristic length $\mathcal{R}$ from the system
in $\Lambda\setminus\Omega$. The main price to pay is the first energy error
term in [AS21, (4.7)] which reads “$\frac{\ell}{\tilde{\ell}}S$”. In their
framework the quantity $\ell$ must be such that
$\ell^{3}\geq\frac{S}{\tilde{\ell}}$ ([AS21, (4.4)]) and $\tilde{\ell}$ can be
chosen555The interested reader can find a list of conditions on
$\ell,\tilde{\ell}$ at the end of the proof of [AS21, Prop. 4.5]. The choice
of $\tilde{\ell}$ mentioned in [AS21, (4.27)] correspond to the _smallest_
possible choice, but increasing $\tilde{\ell}$ up to $\mathcal{R}$ still gives
a valid choice. of order $\mathcal{R}$, so in fact we are paying a price of
order $\left(\frac{S}{\mathcal{R}}\right)^{4/3}$. Moreover the quantity $S$
can always be bounded by the energy in a domain $2\Omega$ twice as large. So
_if we know_ that the energy $E$ at scale $2\mathcal{R}$ is typically
$E(2\mathcal{R})$, then we pay a price
$\mathcal{O}\left(\frac{E(2\mathcal{R})}{\mathcal{R}}\right)^{4/3}$. In
[Leb17, AS21] (see also [BBNY17]) one starts from an estimate
$E(2\mathcal{R})\sim(2\mathcal{R})^{2}$ (the energy scales like the volume)
and gets an error of order $\mathcal{O}(\mathcal{R}^{4/3})\ll\mathcal{R}^{2}$
which is indeed negligible (this can in fact be improved further down by
chosing $\ell,\tilde{\ell}$ more cleverly, see the proof of [AS21, Prop.
2.5]).

The key observation666Having a poorer estimate is equivalent to having the
parameter $\mathcal{C}$ in [AS21, (4.25)] depend on $\mathcal{R}$. The
conditions written at the end of the proof of [AS21, Proposition 4.5] can
still be satisfied as long as $\mathcal{C}$ is much smaller than
$\mathcal{R}$, which corresponds to an initial energy estimate in
$o(\mathcal{R}^{3})$. If $\mathcal{C}$ is smaller than $\mathcal{R}^{3/2}$
then the error term will become smaller than $\mathcal{R}^{2}$ _in one step_ ,
otherwise one would need to apply the bootstrap in scales for some time (i.e.
go down in scales) before reaching the desired local laws. Our situation
corresponds to $\mathcal{C}=\log^{5}\mathcal{R}$, cf. (5.12). is that even if
we start from a poorer estimate, say with logarithmic corrections like
$E(2\mathcal{R})\sim\left(2\mathcal{R}\right)^{2}\log^{100}(\mathcal{R})$ (cf.
the one implied by our global law (5.12)) then the error
$\mathcal{O}\left(\frac{E(2\mathcal{R})}{\mathcal{R}}\right)^{4/3}$ remains
much smaller than $\mathcal{R}^{2}$. In fact, a careful inspection of the
proof (which we will not need here) shows that one could start with a “global
law” as bad as $\mathcal{O}(T^{5/2})$, or even $o(T^{3})$ instead of (5.12)
and still recover good local laws in the bulk. ∎

###  5.4 Sub-systems with good external potentials: consequences

#### Discrepancy bounds

Once local laws hold (in the bulk), we retrieve all statements that rely
purely on energy considerations. In particular the analogous of [AS21, (1.18)]
is valid, namely:

######  Lemma 5.9 (Discrepancy bounds in sub-systems).

If $\square(x,\ell)\subset\Lambda^{\mathrm{bulk}}$ then:

$\log\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\exp\left(\frac{\beta}{\mathtt{C}}\frac{\mathsf{Dis}^{2}(\mathbf{X}_{\mathsf{n}},\square(x,\ell))}{\ell^{2}}\right)\Big{|}\mathcal{E}_{\Lambda}\right]\leq\hyperref@@ii[prop:local_law_SUBSYS]{\mathtt{C}_{\mathtt{Local}}}.$
(5.14)

In particular, we have an a priori Poisson-like bound on the number variance:

$\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\mathsf{Dis}^{2}(\mathbf{X}_{\mathsf{n}},\square(x,\ell))\Big{|}\mathcal{E}_{\Lambda}\right]\leq\mathtt{C}\hyperref@@ii[prop:local_law_SUBSYS]{\mathtt{C}_{\mathtt{Local}}}\ell^{2},$

together with a tail estimate: if $\ell$ is larger than some constant
depending on $\beta,\bar{\mathtt{C}}$

$\mathbb{P}_{\mathsf{n},\Lambda}^{\beta}\left[|\mathsf{Dis}(\mathbf{X}_{\mathsf{n}},\square(x,\ell))|\geq\ell\log\ell\Big{|}\mathcal{E}_{\Lambda}\right]\leq\exp\left(-\frac{\log^{2}\ell}{\mathtt{C}_{\beta}}\right).$

###### Proof of Claim 5.9.

The proof is as in [AS21], using an inequality that relates the presence of
discrepancy to a certain energy cost, e.g. [AS21, Lemma B.4]. ∎

######  Remark 5.10.

As an inspection of the (short) proof of [AS21, Lemma B.4] quickly reveals,
there is nothing specific to a square in the previous claim, and it also
applies to a disk of radius $\ell$, or to any “reasonable” shape like the
boxes introduced in Definition 3.4.

#### Treating smooth linear statistics

Following exactly the same proof as in [LS18, Ser20] (or alternatively as in
[BBNY19]), one would obtain a control on linear statistics of smooth enough
test functions supported in the bulk $\Lambda^{\mathrm{bulk}}$. We do not need
it here, however it will be crucial for us to retrieve a specific property
(the “smallness of the anisotropy”), but since this only serves as a tool for
another result (the quantitative translation-invariance estimate presented in
Section 6) we postpone the corresponding discussion to an appendix (see
Section E.4). The only result that we will quote directly is one about
_expectations_ for fluctuations of linear statistics.

######  Lemma 5.11 (Expectation of linear statistics in the bulk of
subsystems).

There exists a constant $\bar{\mathtt{C}}$ depending only on $\beta$, and on
the “local laws” constant CLocal (thus on the “good potential” constant) such
that the following holds.

Let $\varphi$ be a function of class $C^{2}$, compactly supported on a disk of
radius $\ell\geq\rho_{\beta}$ included in $\Lambda^{\mathrm{bulk}}$. Then:

$\left|\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\mathrm{Fluct}[\varphi]\Big{|}\mathcal{E}_{\Lambda}\right]\right|\leq\bar{\mathtt{C}}|\varphi|_{\mathsf{2}}\ell^{2}.$
(5.15)

###### Proof of Lemma 5.11.

This follows from the proofs of [LS18, Ser20] but is not explicitly written as
such. Let $t$ be a small parameter, we know from [LS18, Prop. 2.10] that:

$\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[e^{t\mathrm{Fluct}[\varphi]}\Big{|}\mathcal{E}_{\Lambda}\right]=\frac{\mathrm{K}_{\mathsf{n},\Lambda}(\mathbf{m}_{s})}{\mathrm{K}_{\mathsf{n},\Lambda}(\mathbf{m})}e^{\mathcal{O}(t^{2})},$

where $\mathbf{m}_{s}$ is the measure
$\mathbf{m}_{s}:=\mathbf{m}-s\Delta\varphi$, where $s=\frac{t}{2\pi\beta}$.
This is valid because $\varphi$ is assumed to be supported in
$\Lambda^{\mathrm{bulk}}$, where $\mathbf{m}$ has density $1$. By e.g. [LS18,
Lemma 3.6] we know that we can replace $\mathbf{m}_{s}$ by the approximate
measure
$\tilde{\mathbf{m}}_{s}:=(\operatorname{Id}+s\nabla\varphi)\\#\mathbf{m}$ up
to quadratic terms, i.e.:

$\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[e^{t\mathrm{Fluct}[\varphi]}\Big{|}\mathcal{E}_{\Lambda}\right]=\frac{\mathrm{K}_{\mathsf{n},\Lambda}(\tilde{\mathbf{m}}_{s})}{\mathrm{K}_{\mathsf{n},\Lambda}(\mathbf{m})}e^{\mathcal{O}(t^{2})}.$

Next, a consequence of [Ser20, Prop. 4.2] is that the ratio of partition
functions can be written as:

$\frac{\mathrm{K}_{\mathsf{n},\Lambda}(\tilde{\mathbf{m}}_{s})}{\mathrm{K}_{\mathsf{n},\Lambda}(\mathbf{m})}=\exp\left(s\mathcal{O}\left(|\varphi|_{\mathsf{2}}\mathsf{EnerPts}(\mathbf{X}_{\mathsf{n}},\operatorname{supp}\nabla\varphi)\right)+\mathcal{O}(s^{2})\right),$

and thus taking the limit $t\to 0$ (or equivalently $s\to 0$) and using the
local laws we get (5.15) after identifying the first order terms. ∎

##  6 Quantitative estimate on translation-invariance

In this section, we denote by $\vec{u}$ be the vector $\vec{u}:=(0,1)$ in
$\mathbb{R}^{2}$.

###  6.1 The “spin wave” and its properties

##### An auxiliary function

For $\varepsilon\in(0,1)$, let
$f^{(\varepsilon)}:x=(x_{1},x_{2})\in\mathbb{R}^{2}\to\mathbb{R}$ be defined
as:

$f^{(\varepsilon)}(x):=\begin{cases}x_{1}&\text{ if }|x|\leq 1\\\
x_{1}(1-\varepsilon\log|x|)&\text{ if }1\leq|x|\leq e^{1/\varepsilon}\\\
0&\text{ if }|x|\geq e^{1/\varepsilon}.\end{cases}$

The function $f^{(\varepsilon)}$ is continuous, piecewise $C^{2}$ and
compactly supported on the disk of radius $e^{1/\varepsilon}$. We have, by
direct computations:

  * •

On (the interior of) the unit disk, $\partial_{1}f^{(\varepsilon)}\equiv 1$,
$\partial_{2}f^{(\varepsilon)}\equiv 0$ and the second (and third) derivatives
of $f^{(\varepsilon)}$ vanish.

  * •

For $1\leq|x|\leq e^{1/\varepsilon}$, we have :

    * –

$\partial_{1}f^{(\varepsilon)}(x)=1-\varepsilon\log|x|-\varepsilon\frac{x_{1}^{2}}{|x|^{2}}$,
$\partial_{2}f^{(\varepsilon)}(x)=-\varepsilon\frac{x_{1}x_{2}}{|x|^{2}}$.

    * –

The second derivatives of $f^{(\varepsilon)}$ satisfy the pointwise bound
$|f^{(\varepsilon)}|_{\mathsf{2},\star}(x)\preceq\frac{\varepsilon}{|x|}$.

    * –

The third derivatives of $f^{(\varepsilon)}$ satisfy the pointwise bound
$|f^{(\varepsilon)}|_{\mathsf{3},\star}(x)\preceq\frac{\varepsilon}{|x|^{2}}$.

In particular, observe that the first partial derivatives of
$f^{(\varepsilon)}$ are bounded by $1$ with a jump of size
$\mathcal{O}(\varepsilon)$ along both $\partial\mathfrak{D}(0,1)$ and
$\partial\mathfrak{D}(0,e^{1/\varepsilon})$, while the second partial
derivatives have a jump of size $\mathcal{O}(\varepsilon)$ along
$\partial\mathfrak{D}(0,1)$ and of size
$\mathcal{O}(\varepsilon/e^{1/\varepsilon})$ along
$\partial\mathfrak{D}(0,e^{1/\varepsilon})$.

Thus after applying a mollification to $f^{(\varepsilon)}$ at scale
$\frac{1}{2}$ near $\partial\mathfrak{D}(0,1)$ and at scale
$\frac{1}{2}e^{1/\varepsilon}$ near
$\partial\mathfrak{D}(0,e^{1/\varepsilon})$ we may consider a function
$\bar{f}^{(\varepsilon)}$ which is smooth, compactly supported in
$\mathfrak{D}(0,2e^{1/\varepsilon})$ and such that (for some universal
constant $\mathtt{C}$):

  * •

$\bar{f}^{(\varepsilon)}(x)=x_{1}$ for $|x|\leq\frac{1}{2}$.

  * •

The first derivatives of $\bar{f}^{(\varepsilon)}$ are bounded by $2$ on
$\mathbb{R}^{2}$.

  * •

$|\bar{f}^{(\varepsilon)}|_{\mathsf{2},\star}(x)\leq\mathtt{C}\varepsilon$ for
$|x|\leq 2$ and
$|\bar{f}^{(\varepsilon)}|_{\mathsf{2},\star}(x)\leq\mathtt{C}\frac{\varepsilon}{|x|}$
for $2\leq|x|\leq 2e^{1/\varepsilon}$.

  * •

$|\bar{f}^{(\varepsilon)}|_{\mathsf{3},\star}(x)\leq\mathtt{C}\varepsilon$ for
$|x|\leq 2$ and
$|\bar{f}^{(\varepsilon)}|_{\mathsf{3},\star}(x)\leq\mathtt{C}\frac{\varepsilon}{|x|^{2}}$
for $2\leq|x|\leq 2e^{1/\varepsilon}$.

##### Definition of the “spin wave”

Next, we define our “spin wave”777This terminology alludes to similar
constructions used in the theory of continuous spin systems to prove so-called
“Mermin-Wagner” theorems, see e.g. [FV17, Sec. 9.2] or [Sim14, Chapter 3].
$\mathtt{W}^{(\varepsilon)}$ as the vector field

$\mathtt{W}^{(\varepsilon)}:=\nabla^{\perp}\bar{f}^{(\varepsilon)}=\left(-\partial_{2}\bar{f}^{(\varepsilon)},\partial_{1}\bar{f}^{(\varepsilon)}\right).$

######  Lemma 6.1.

The following properties of $\mathtt{W}^{(\varepsilon)}$ are straightforward:

  1. 1.

$\mathtt{W}^{(\varepsilon)}$ is smooth, compactly supported on
$\mathfrak{D}(0,2e^{1/\varepsilon})$ (because so is
$\bar{f}^{(\varepsilon)}$).

  2. 2.

$\mathtt{W}^{(\varepsilon)}(x)=\vec{u}=(0,1)$ for $|x|\leq\frac{1}{2}$
(because then $\bar{f}^{(\varepsilon)}(x_{1},x_{2})=x_{1}$).

  3. 3.

$\operatorname{\mathrm{div}}\mathtt{W}^{(\varepsilon)}=0$ on $\mathbb{R}^{2}$
(by definition of $\mathtt{W}^{(\varepsilon)}$ as the perpendicular gradient
of a smooth function).

  4. 4.

$|\mathtt{W}^{(\varepsilon)}|_{\mathsf{0}}\leq 2$ (it is bounded by the first
derivative of $\bar{f}^{(\varepsilon)}$),
$|\mathtt{W}^{(\varepsilon)}|_{\mathsf{1}}\leq\mathtt{C}\varepsilon$ and more
precisely:

$\left|\mathtt{W}^{(\varepsilon)}\right|_{\mathsf{1},\star}(x)\leq\mathtt{C}\times\begin{cases}\varepsilon&\text{
for }|x|\leq 2\\\ \frac{\varepsilon}{|x|}&\text{ for }2\leq|x|\leq
2e^{1/\varepsilon}.\end{cases}$ (6.1)

  5. 5.

$|\mathtt{W}^{(\varepsilon)}|_{\mathsf{2},\star}(x)\leq\mathtt{C}\varepsilon$
for $|x|\leq 2$ and
$|\mathtt{W}^{(\varepsilon)}|_{\mathsf{2},\star}(x)\leq\mathtt{C}\frac{\varepsilon}{|x|^{2}}$
for $2\leq|x|\leq 2e^{1/\varepsilon}$.

We have thus constructed a smooth, _divergence-free_ vector field which is
constant near the origin _and has an arbitrary small $\mathrm{H}^{1}$ norm_
(of order $\varepsilon$). The downside is that the size of the support of
$\mathtt{W}^{(\varepsilon)}$ is _exponential_ with respect to the parameter
$1/\varepsilon$.

###  6.2 Slowly varying localized translations

For $\ell>0$, let the vector field $\mathtt{W}^{(\varepsilon,\ell)}$ be
defined for $x\in\mathbb{R}^{2}$ as
$\mathtt{W}^{(\varepsilon,\ell)}(x):=\mathtt{W}^{(\varepsilon)}(x/\ell)$.
Since $\mathtt{W}^{(\varepsilon,\ell)}$ is continuous and compactly supported,
it generates a global flow $\\{\Phi_{t}^{\varepsilon,\ell}\\}_{t}$ with the
following properties:

######  Lemma 6.2.

For all $|t|\leq\frac{\ell}{10}$ we have:

  1. 1.

$\Phi_{t}^{\varepsilon,\ell}$ is an area-preserving diffeomorphism of
$\mathbb{R}^{2}$.

  2. 2.

We have $\Phi_{t}^{\varepsilon,\ell}(x)=x+t\vec{u}$ for $|x|\leq\ell/4$.

  3. 3.

We have $\Phi_{t}^{\varepsilon,\ell}(x)=x$ for $|x|\geq 2\ell
e^{1/\varepsilon}$.

Thus for fixed $|t|\leq\frac{\ell}{10}$ the diffeomorphism
$\Phi_{t}^{\varepsilon,\ell}$ coincides with the translation by $t\vec{u}$ on
the disk $\mathfrak{D}(0,\ell/4)$ and with the identity outside
$\mathfrak{D}(0,2e^{1/\varepsilon})$, we call it a _localized translation_ as
in [Geo99]. The main difference with the construction of [Geo99] is that we
have
$\int|\Phi_{t}^{\varepsilon,\ell}-\operatorname{Id}|_{\mathsf{1},\star}^{2}=\mathcal{O}(\varepsilon)$
instead of $\mathcal{O}(1)$. Interestingly enough, a _bounded (but not small)_
$H^{1}$ norm for $\Phi_{t}^{\varepsilon,\ell}-\operatorname{Id}$ (which
induces a bounded, but not small energy cost, as we will show in Section E) is
enough to prove translation-invariance _in the infinite-volume setting_ , but
fails to give anything valuable in finite-volume. However, according to a
remark in [Sim14, Sec. III.7] _“it appears that any model in which this weaker
property is valid, the [possibility of finding a construction with arbitrarily
small energy cost exists]”_. This remains very intriguing to us.

###### Proof of Lemma 6.2.

The fact that $\Phi_{t}^{\varepsilon,\ell}$ is area-preserving follows from
Liouville’s theorem, since the vector field $\mathtt{W}^{(\varepsilon)}$ (and
thus $\mathtt{W}^{(\varepsilon,\ell)}$) is divergence-free by construction.
Moreover, since we ensured that $\mathtt{W}^{(\varepsilon)}\equiv\vec{u}$ on
$\mathfrak{D}(0,\frac{1}{2})$, the rescaled vector field
$\mathtt{W}^{(\varepsilon,\ell)}$ coincides with $\vec{u}$ on the disk
$\mathfrak{D}(0,\ell/2)$ hence we have
$\Phi_{t}^{\varepsilon,\ell}(x)=x+t\vec{u}$ as long as $x+t\vec{u}$ remains in
$\mathfrak{D}(0,\ell/2)$. In particular, if $|x|\leq\ell/4$ and since
$|t|\leq\frac{\ell}{10}$ by assumption, we have
$\Phi_{t}^{\varepsilon,\ell}(x)=x+t\vec{u}$. On the other hand,
$\mathtt{W}^{(\varepsilon,\ell)}$ vanishes identically outside
$\mathfrak{D}(0,2\ell e^{1/\varepsilon})$ and thus the flow there coincides
with the identity map. ∎

We will study $\Phi_{t}^{\varepsilon,\ell}$ a bit more closely in Section E.1
for technical purposes.

###  6.3 Effect of localized translations on the energy

We now apply the “spin wave” / “localized translation” construction to a sub-
system with good external potential. Let $\Lambda$ (a disk of radius $T$),
$\mathsf{n}$, $\mathrm{V}^{\mathrm{ext}}$ be as in Section 5, with
$\mathrm{V}^{\mathrm{ext}}$ a good external potential, let
$\mathbf{m}=\mathsf{n}\mu_{\mathrm{W}}$ be the corresponding non-uniform
background and assume that the properties listed in Section 5.3 hold. Let $L$
be as in Section 3 and let us choose $\varepsilon,\ell$ in such a way that:

$e^{\sqrt{\log\ell}}\leq L\leq\frac{\ell}{10},\quad 5\ell
e^{1/\varepsilon}\leq T\leq 10\ell
e^{1/\varepsilon},\quad\log\ell\leq\varepsilon^{-1},\quad\varepsilon^{-1}\leq\ell^{2}\leq\varepsilon^{-3}.$
(6.2)

Let $\mathrm{ErrAve}(t,\varepsilon,\ell,\mathbf{X})$ denote the following
“averaging error”:

$\mathsf{F}_{\Lambda}(\mathbf{X},\mathbf{m})=\frac{1}{2}\left(\mathsf{F}_{\Lambda}(\Phi_{t}\mathbf{X},\mathbf{m})+\mathsf{F}_{\Lambda}(\Phi_{-t}\mathbf{X},\mathbf{m})\right)+\mathrm{ErrAve}(t\vec{u},\varepsilon,\ell,\mathbf{X}),$
(6.3)

where $\Phi_{t}\mathbf{X}$ denotes the configuration obtained after applying
$\Phi_{t}$ to all the points of $\mathbf{X}$.

######  Proposition 6.3.

There exists a constant $\mathtt{C}$ depending on $\beta$ on the “local laws”
constant CLocal (and thus on the good external potential constant) such that:

$\mathbb{P}_{\mathsf{n},\Lambda}^{\beta}\left[\sup_{|t|\leq\frac{\ell}{10}}|\mathrm{ErrAve}(t\vec{u},\varepsilon,\ell,\mathbf{X}_{\mathsf{n}})|\leq\mathtt{C}t^{2}\varepsilon\log\varepsilon\Big{|}\mathcal{E}_{\Lambda}\right]\geq
1-\exp\left(-\ell^{2}\right).$ (6.4)

We prove Proposition 6.3 in Section E and use it next to “mollify” observables
before taking expectations.

###  6.4 Effect of localized translations on expectations

######  Proposition 6.4.

Let $\mathsf{G}$ be a measurable function on $\mathrm{Conf}(\Lambda)$, let
$\mathcal{E}\subset\mathrm{Conf}(\Lambda)$ be an event. Assume that the
function $\mathsf{G}$ is $\mathfrak{D}(0,\ell/10)$-local and that
$\mathcal{E}$ is $\Lambda\setminus\Lambda^{\mathrm{bulk}}$-local. Recall that
$|t|\leq\ell/10$.

For all $\tau\in(0,1)$ and for all $\sigma>0$, we have:

$\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\mathsf{G}(\mathbf{X})\Big{|}\mathcal{E}\right]=\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\frac{1}{2}\left(\mathsf{G}(\mathbf{X}+t\vec{u})+\mathsf{G}(\mathbf{X}-t\vec{u})\right)\Big{|}\mathcal{E}\right]+\mathsf{ErrorQI}(t\vec{u},\varepsilon,\ell,\tau,\sigma,\mathsf{G},\mathbf{X}),$
(6.5)

with an error term $\mathsf{ErrorQI}$ bounded as follows:

$\left|\mathsf{ErrorQI}(t\vec{u},\varepsilon,\ell,\tau,\sigma,\mathsf{G},\mathbf{X})\right|\leq
2\left(\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\mathsf{G}^{2}(\mathbf{X}\pm
t\vec{u})\Big{|}\mathcal{E}\right]+\sigma^{2}\right)^{\frac{1}{2}}\times\left(e^{\beta\tau}-1\right)\\\
+2e^{\beta\tau}\left(\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\mathsf{G}^{2}(\mathbf{X}\pm\vec{u})\Big{|}\mathcal{E}\right]+\sigma^{2}\right)^{\frac{1}{2}}\times\left(\mathbb{P}_{\mathsf{n},\Lambda}^{\beta}\left[|\mathsf{G}(\mathbf{X}\pm
t\vec{u})|\geq\sigma\right]\right)^{\frac{1}{2}}\\\
+\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\mathsf{G}^{2}(\mathbf{X})\Big{|}\mathcal{E}\right]\times\left(\mathbb{P}_{\mathsf{n},\Lambda}^{\beta}\left[\mathrm{ErrAve}(t\vec{u},\varepsilon,\ell,\mathbf{X})\geq\tau\big{|}\mathcal{E}\right]\right)^{\frac{1}{2}}$
(6.6)

We postpone the proof of Proposition 6.4 to Section E.5. There is of course
nothing special about the vector $\vec{u}$, and we may replace it by any unit
vector.

###  6.5 Application: expectation of discrepancies in sub-systems

######  Proposition 6.5.

Let $\mathfrak{B}$ be a box of size $L$ as introduced in Definition 3.4.
Assume that it is contained in the disk $\mathfrak{D}(0,\ell/10)$. Then we
have:

$\left|\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\mathsf{Dis}(\mathbf{X}_{\mathsf{n}},\mathfrak{B})\Big{|}\mathcal{E}_{\Lambda}\right]\right|\leq\bar{\mathtt{C}}L\left(\varepsilon\log\varepsilon\log
L\right)^{1/3}.$ (6.7)

The point of (6.7) is that if $\varepsilon\log\varepsilon\log L$ is $\ll 1$
then the bound on the average discrepancy is $\ll L$, and thus _much better
than the crude estimate via the standard deviation_ using Claim 5.9. This is
crucial for us.

###### Proof of Proposition 6.5.

Let us take $\mathsf{G}(\mathbf{X}):=\mathsf{Dis}(\mathbf{X},\mathfrak{B})$,
which is clearly $\mathfrak{D}(0,\ell/10)$-local in view of our assumption on
$\mathfrak{B}$. Moreover $\mathcal{E}_{\Lambda}$ (by its Definition 5.2) is
clearly $\Lambda\setminus\Lambda^{\mathrm{bulk}}$-local. Fix the parameter
$\bar{t}$ as:

$\bar{t}:=\left(\varepsilon\log\varepsilon\log L\right)^{-1/3}.$ (6.8)

We first check that $\bar{t}$ has the correct range (using (6.2)):

$\bar{t}\leq\varepsilon^{-1/3}\leq\ell^{2/3}\leq\frac{\ell}{10}.$

In particular, for all vector $v\in\mathbb{R}^{2}$ with $\|v\|\leq\bar{t}$ we
may apply the result of Proposition 6.4 to a localized translation in the
direction $\frac{v}{\|v\|}$ instead of $\vec{u}$.

Introduce a smooth cut-off function $\overline{\chi}$ equal to $1$ for
$|x|\leq\frac{\bar{t}}{2}$ and to $0$ for $|x|\geq\bar{t}$, such that
$|\overline{\chi}|_{\mathsf{k}}\leq\mathtt{C}\bar{t}^{-k}$ (for
$\mathsf{k}=1,2$). We impose that $\overline{\chi}$ be an even function. For
all $|v|\leq\bar{t}$ we have:

$\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\mathsf{Dis}(\mathbf{X},\mathfrak{B})\Big{|}\mathcal{E}\right]=\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\frac{1}{2}\left(\mathfrak{B}(\mathbf{X}+v,\mathfrak{B})+\mathsf{Dis}(\mathbf{X}-v,\mathfrak{B})\right)\Big{|}\mathcal{E}\right]+\mathsf{ErrorQI}(v,\varepsilon,\ell,\tau,\sigma,\mathsf{Dis}(\cdot,\mathfrak{B}),\mathbf{X}).$

Integrating this against $\frac{\overline{\chi}}{|\overline{\chi}|_{L^{1}}}$
(which has mass $1$) and using the fact that $\overline{\chi}$ is even we
obtain:

$\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[]\mathsf{Dis}(\mathbf{X}_{\mathsf{n}},\mathfrak{B})\Big{|}\mathcal{E}_{\Lambda}\right]=\frac{1}{|\overline{\chi}|_{L^{1}}}\int\overline{\chi}(v)\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left(\mathsf{Dis}(\mathbf{X}+v,\mathfrak{B})\Big{|}\mathcal{E}_{\Lambda}\right)+\mathsf{ErrorQI}(v)\mathtt{d}v.$

Observe that
$\mathsf{Dis}(\mathbf{X}+v,\mathfrak{B})=\mathrm{Fluct}[\mathtt{1}_{\mathfrak{B}}(\cdot+v)](\mathbf{X})$,
so that we may re-write the integral in the right-hand side as a convolution.
Introducing the function
$\varphi:=\mathtt{1}_{\mathfrak{B}}\ast\frac{\overline{\chi}}{|\overline{\chi}|_{L^{1}}}$
we get:

$\left|\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\mathsf{Dis}(\mathbf{X}_{\mathsf{n}},\mathfrak{B})\Big{|}\mathcal{E}_{\Lambda}\right]-\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[]\mathrm{Fluct}[\varphi]\Big{|}\mathcal{E}_{\Lambda}\right]\right|\leq\sup_{|v|\leq\bar{t}}\mathsf{ErrorQI}\left(v,\varepsilon,\ell,\tau,\sigma,\mathsf{Dis}(\cdot,\mathfrak{B}),\mathbf{X}_{\mathsf{n}}\right).$
(6.9)

By construction, the function $\varphi$ is now a smooth cut-off function,
equal to $1$ on
$\\{x\in\mathfrak{B},\operatorname{dist}(x,\partial\mathfrak{B})\geq\bar{t}\\}$
and to $0$ for
$\\{x\notin\mathfrak{B},\operatorname{dist}(x,\partial\mathfrak{B})\geq\bar{t}\\}$.
In particular, the support of $\nabla\varphi$ has an area
$\mathcal{O}(L\bar{t})$, and moreover we have
$|\varphi|_{\mathsf{k}}\leq\mathtt{C}\bar{t}^{-\mathsf{k}}$ for
$\mathsf{k}=1,2$ (by Young’s convolution inequality). From Lemma 5.11, we thus
know:

$\left|\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[]\mathrm{Fluct}[\varphi]\right]\Big{|}\mathcal{E}_{\Lambda}\right|\preceq\frac{1}{\bar{t}^{2}}\times
L\bar{t}=\mathcal{O}\left(\frac{L}{\bar{t}}\right),$ (6.10)

with multiplicative constants depending on $\beta,\delta$.

On the other hand, we know by Proposition 6.3 that choosing $\bar{\mathtt{C}}$
large enough we can ensure:

$\sup_{|v|\leq\bar{t}}\left(\mathbb{P}_{\mathsf{n},\Lambda}^{\beta}\left(\left\\{\left|\mathrm{ErrAve}(v,\varepsilon,\ell,\mathbf{X})\right\\}\right|\geq\bar{\mathtt{C}}\varepsilon\log\varepsilon\bar{t}^{2}\big{|}\mathcal{E}_{\Lambda}\right]\right)\leq\exp\left(-\ell^{2}/\bar{\mathtt{C}}\right).$

Moreover
$\mathsf{Dis}(\mathbf{X}+v,\mathfrak{B})=\mathsf{Dis}(\mathbf{X},\mathfrak{B}-v)$
and we have by Lemma 5.9:
$\sup_{|v|\leq\bar{t}}\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\mathsf{Dis}^{2}(\mathbf{X},\mathfrak{B}-v)\Big{|}\mathcal{E}_{\Lambda}\right]\leq\bar{\mathtt{C}}L^{2}$
(because the translated object $\mathfrak{B}-v$ remains a box within
$\Lambda^{\mathrm{bulk}}$) and, still by Lemma 5.9, for $\bar{\mathtt{C}}$
large enough we have:

$\sup_{|v|\leq\bar{t}}\mathbb{P}_{\mathsf{n},\Lambda}^{\beta}\left(\left\\{|\mathsf{Dis}(\mathbf{X}+v,\mathfrak{B})|\geq
L\log L\right\\}\right)\leq e^{-\frac{1}{\bar{\mathtt{C}}}\log^{2}L}.$

(Remark that the $\sup$ is outside
$\mathbb{P}_{\mathsf{n},\Lambda}^{\beta},\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}$
in those bounds - it would be significantly more challenging otherwise.)

Thus, choosing $\tau=\bar{\mathtt{C}}\varepsilon\log\varepsilon\bar{t}^{2}$,
$\sigma=L\log L$ in the statement of Proposition 6.4, the error term
$\mathsf{ErrorQI}$ reduces to:

$\mathsf{ErrorQI}(v,\varepsilon,\ell,\tau,\sigma,\mathsf{Dis}(\mathbf{X},\mathfrak{B}),\mathbf{X})\preceq\bar{\mathtt{C}}L\log
L\left(\varepsilon\log\varepsilon\bar{t}^{2}+\exp\left(-\ell^{2}/\mathtt{C}\right)+\exp\left(-\frac{1}{\mathtt{C}}\log^{2}L\right)\right),$

uniformly for $|v|\leq\bar{t}$ and thus (using (6.2) and keeping only the
dominant term):

$\sup_{|v|\leq\bar{t}}\mathsf{ErrorQI}\leq\bar{\mathtt{C}}L\log
L\varepsilon\log\varepsilon\bar{t}^{2}.$ (6.11)

Combining (6.9) with (6.10) and (6.11), and using our choice (6.8) for
$\bar{t}$, we obtain (6.7). ∎

######  Remark 6.6.

One can use the same argument to study expectations of discrepancies _in the
full system_. Then one does not need to use $\mathcal{E}_{\Lambda}$ since
local laws are know to hold unconditionally, and one can take $\varepsilon$ as
small as $\log^{-1}N$.

######  Corollary 6.7 (An application).

Assume that $L,T$ satisfy the following relation:

$T=100L\exp(10L).$ (6.12)

Then we have:

$\left|\mathbb{E}_{\mathsf{n},\Lambda}^{\beta}\left[\mathsf{Dis}(\mathbf{X},\mathfrak{B})\right]\right|\leq\bar{\mathtt{C}}L^{0.67}.$
(6.13)

###### Proof.

We can take $\ell=10L$ and $\varepsilon=\ell^{-1}=\frac{1}{10L}$, and
$T=100L\exp(10L)=10\ell e^{1/\varepsilon}$. The conditions of (6.2) are
clearly satisfied. The right-hand side of (6.7) is then bounded by
$\bar{\mathtt{C}}L^{2/3}\log L=\mathcal{O}(L^{0.67})$ (for $L$ large enough).
∎

##  7 Conclusion: proof of Theorem 1

Let $\delta>0$ be fixed. Let $x,R$ be such that
$\operatorname{dist}(\mathfrak{D}(x,R),\partial\Sigma_{N})\geq\delta\sqrt{N}$
as assumed in (1.5). Let $\varepsilon_{R}$ be chosen as:

$\varepsilon_{R}:=\log^{-0.3}(R),$ (7.1)

For simplicity, we will focus on the case of an _excess_ of points, i.e. a
positive discrepancy, the other case being treated similarly. Define the event
$\mathcal{A}_{R}$:

$\mathcal{A}_{R}:=\\{\mathsf{Dis}\left(\mathbf{X}_{N},\mathfrak{D}(x,R)\right)\geq\varepsilon_{R}R\\}$

The conclusion that we want to reach (as stated in (1.6)) is that for
$R,\bar{\mathtt{C}}$ large enough (depending on $\beta$ and on the parameter
$\delta$, but not on $x$):

$\mathbb{P}^{\beta}_{N}\left(\mathcal{A}_{R}\right)\leq\exp\left(-\log^{1.5}R\right).$
(7.2)

We have not tried to optimize $\varepsilon_{R}$ or the exponent in (7.2), what
matters for us is that $\varepsilon_{R}\to 0$ and that $1.5>1$ so our
probabilistic tail is better than algebraic. With the methods of the present
paper, there is a hard limit on the smallness of $\varepsilon_{R}$ \- it has
to be at least $\log^{-1}R$.

#### Step 1. Choosing $L$ and cornering the discrepancy

Let $L$ be chosen as:

$L:=\log^{0.99}R.$ (7.3)

In particular, _for $R$ greater than some constant_ (depending only on
$\beta$) we have:

$\mathtt{C}_{\ref{prop:chase_the_goat}}\frac{1}{\varepsilon_{R}}=\mathtt{C}_{\ref{prop:chase_the_goat}}\log^{0.3}R\leq\log^{0.99}R=L\leq
R/10,$ (7.4)

(where $\mathtt{C}_{\ref{prop:chase_the_goat}}$ is the constant depending only
on $\beta$ introduced in Proposition 3.1) so the first condition of (3.1) is
satisfied. Moreover let $s$ be chosen as
$\frac{1}{\mathtt{C}_{\ref{prop:chase_the_goat}}}\min\left(\frac{L^{3}}{R},L\varepsilon_{R}\right)$,
namely (for $R$ large enough):
$s:=\frac{1}{\mathtt{C}_{\ref{prop:chase_the_goat}}}\frac{L^{3}}{R}$. By
definition, the second condition of (3.1) is then satisfied. Let us compute:

$\exp\left(-\frac{s\varepsilon_{R}R}{4}\right)=\exp\left(-\frac{\varepsilon_{R}L^{3}}{4\mathtt{C}_{\ref{prop:chase_the_goat}}}\right)\leq\exp\left(-\frac{\log^{-2.67}R}{\mathtt{C}_{\beta}}\right).$

For each $k$ with $0\leq k\leq R^{2}$, let $\mathcal{B}_{R}(k)$ be the event:

$\mathcal{B}_{R}(k):=\left\\{\text{The discrepancy in the annulus
$\mathfrak{D}_{R}\setminus\mathfrak{D}_{R-2L+\frac{kL}{R^{2}}}$ is larger than
$\frac{1}{4}\varepsilon_{R}\cdot R$}\right\\}$

Combining Proposition 3.1 and Lemma 3.2 we obtain:

$\mathbb{P}^{\beta}_{N}(\mathcal{A}_{R})\leq\sum_{k=1}^{R^{2}}\mathbb{P}^{\beta}_{N}(\mathcal{B}_{R}(k))+\exp\left(-\frac{\log^{-2.67}R}{\mathtt{C}_{\beta}}\right),$
(7.5)

and we now focus on bounding $\mathcal{B}_{R}(k)$ (the index $k$ plays no
particular role, and for simplicity we forget about it).

For each $k$, as explained in Section 3.2, we can decompose the annulus
$\mathfrak{D}_{R}\setminus\mathfrak{D}_{R-2L+\frac{kL}{R^{2}}}$ into boxes
$\left\\{\mathfrak{B}_{i},i\in\\{0,\dots,\frac{R}{L}-1\\}\right\\}$ of size
$L$ as in Definition 3.4.

#### Step 2. Choosing $T,M$ and a well-separated family

Let $T$ be chosen (as in (6.7)) as:

$T:=100\times\log^{0.99}R\times\exp\left(10\log^{0.99}R\right)=100L\exp\left(10L\right),$
(7.6)

and let $M$ be chosen as:

$M:=T^{6}.$ (7.7)

Since $L=\log^{0.99}R$ (according to (7.3)), the conditions of (3.4) are
clearly satisfied for $R$ large enough.

Now, for $l\in\\{0,\dots,M-1\\}$, let $\mathcal{C}(l)$ be the event:

$\mathcal{C}(l):=\left\\{\sum_{i=l\textrm{ mod
}M}\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{B}_{i})\geq\frac{\varepsilon_{R}R}{4M}.\right\\}$
(7.8)

By Lemma 3.5 we know that:

$\mathbb{P}^{\beta}_{N}(\mathcal{B}_{R})\leq\sum_{l=0}^{M-1}\mathbb{P}^{\beta}_{N}\left(\mathcal{C}(l)\right),$
(7.9)

and we now focus on bounding $\mathcal{C}(l)$ (again, the index $l$ plays no
role in the sequel). The index $l$ being fixed, we only consider the boxes
$\mathfrak{B}_{i}$ for $i\equiv l\mod M$ and forget about the other boxes. We
relabel those boxes as $\mathfrak{B}_{i}$ for $i\in\\{1,\dots,\mathcal{N}\\}$
where $\mathcal{N}$ is the cardinality of that family of boxes, with:

$\mathcal{N}=\mathcal{O}\left(\frac{R}{ML}\right).$ (7.10)

As in Section 3.2 we let $\Lambda_{i}$ be the disk
$\mathfrak{D}(\omega_{i},T)$, where $\omega_{i}$ is the “center” of the box
$\mathfrak{B}_{i}$, and we recall that $\mathrm{d}_{ij}$ denotes the distance
between $\Lambda_{i}$ and $\Lambda_{j}$. Using (3.6) and summing over
$i=1,\dots,\mathcal{N}=\mathcal{O}(\frac{R}{ML})$, we get:

$\max_{1\leq i\leq\mathcal{N}}\sum_{j\neq
i}\frac{1}{\mathrm{d}_{ij}}=\mathcal{O}\left(\frac{\log
R}{ML}\right),\quad\sum_{1\leq i\neq
j\leq\mathcal{N}}\frac{1}{\mathrm{d}_{ij}}=\mathcal{O}\left(\frac{R\log
R}{(ML)^{2}}\right).$ (7.11)

#### A “good event”

Let $\mathrm{V}^{\mathrm{ext}}$ be the logarithmic potential generated by the
system outside the $\Lambda_{i}$’s as in (5.6). For $\bar{\mathtt{C}}>0$, let
$\mathcal{E}_{\mathrm{ext}}(\bar{\mathtt{C}})$ be the event:

$\mathcal{E}_{\mathrm{ext}}(\bar{\mathtt{C}}):=\left\\{\text{$\mathrm{V}^{\mathrm{ext}}$
is a “good external potential” on each $\Lambda_{i}$ with constant
$\bar{\mathtt{C}}$.}\right\\}$ (7.12)

(See Definition 5.1). By Proposition 5.3 we know that if $\bar{\mathtt{C}}$ is
chosen large enough and if $T$ is large enough (i.e. $R$ is large enough)
depending only on $\beta$ and on the parameter $\delta$ from (1.5), then we
have:

$\mathbb{P}^{\beta}_{N}\left(\mathcal{E}_{\mathrm{ext}}(\bar{\mathtt{C}})\right)\geq
1-\mathcal{N}\exp\left(-\log^{2}T/\bar{\mathtt{C}}\right).$

Since $\mathcal{N}$ is always smaller than $R=e^{\log R}$ (see (7.10)) and
since, by our choice (7.6) of $T$ we have:

$\exp\left(-\log^{2}T/\bar{\mathtt{C}}\right)\leq\exp\left(-\log^{1.98}R/\bar{\mathtt{C}}\right)\ll\exp(-\log
R),$

we deduce that (for $R,\bar{\mathtt{C}}$ large enough depending on
$\beta,\delta$):

$\mathbb{P}^{\beta}_{N}\left(\mathcal{E}_{\mathrm{ext}}(\bar{\mathtt{C}})\right)\geq
1-\exp\left(-\log^{2}T/\bar{\mathtt{C}}\right).$ (7.13)

On the other hand, for each $i=1,\dots,\mathcal{N}$, let $\mathcal{E}_{i}$ be
the event: “$\Lambda_{i}$ is a good sub-system”, as in Definition 5.2. By
Lemma 5.4, we know that:

$\mathbb{P}^{\beta}_{N}\left(\bigcap_{i=1}^{\mathcal{N}}\mathcal{E}_{i}\right)\geq
1-\mathcal{N}\exp\left(-\log^{2}T/\mathtt{C}_{\beta}\right),$

and by the same parameter comparison as above we get:
$\mathbb{P}^{\beta}_{N}\left(\bigcap_{i=1}^{\mathcal{N}}\mathcal{E}_{i}\right)\geq
1-\exp\left(-\log^{2}T/\mathtt{C}_{\beta}\right)$. We thus have (for
$R,\bar{\mathtt{C}}$ large enough):

$\mathbb{P}^{\beta}_{N}\left(\mathcal{C}\right)\leq\mathbb{P}^{\beta}_{N}\left(\mathcal{C}\cap\mathcal{E}_{\mathrm{ext}}(\bar{\mathtt{C}})\bigcap_{i=1}^{\mathcal{N}}\mathcal{E}_{i}\right)+\exp\left(-\log^{2}T/\bar{\mathtt{C}}\right).$
(7.14)

#### Using approximate conditional independence

Let $\omega$ be chosen as:

$\omega:=L^{-1.33}.$ (7.15)

For $1\leq i\leq\mathcal{N}$, let $\mathsf{G}_{i}$ be the following function
on $\mathrm{Conf}$, which is clearly non-negative and $\Lambda_{i}$-local:

$\mathsf{G}_{i}(\mathbf{X}):=e^{\omega\mathsf{Dis}(\mathbf{X},\mathfrak{B}_{i})}.$
(7.16)

If $\mathbf{X}_{N}$ is in $\mathcal{C}$ then by definition (see (7.8)) we
have:

$\prod_{i=1}^{\mathcal{N}}\mathsf{G}_{i}(\mathbf{X}_{N})=e^{\omega\sum_{i=1}^{\mathcal{N}}\mathsf{Dis}(\mathbf{X}_{N},\mathfrak{B}_{i})}\geq\exp\left(\frac{\omega\varepsilon_{R}R}{4M}\right),$

and thus by Markov’s inequality (dropping the $\bar{\mathtt{C}}$-dependency):

$\mathbb{P}^{\beta}_{N}\left(\mathcal{C}\cap\mathcal{E}_{\mathrm{ext}}\bigcap_{i=1}^{\mathcal{N}}\mathcal{E}_{i}\right)\leq\exp\left(-\frac{\omega\varepsilon_{R}R}{4M}\right)\mathbb{E}^{\beta}_{N}\left[\prod_{i=1}^{\mathcal{N}}\mathsf{G}_{i}(\mathbf{X}_{N})\mathtt{1}_{\mathcal{E}_{\mathrm{ext}}\cap_{i=1}^{\mathcal{N}}\mathcal{E}_{i}}\right].$
(7.17)

We are in a position to use Proposition 4.3 with
$\mathcal{E}_{N}:=\mathcal{E}_{\mathrm{ext}}\cap_{i=1}^{\mathcal{N}}\mathcal{E}_{i}$.
We obtain:

$\mathbb{E}^{\beta}_{N}\left[\prod_{i=1}^{\mathcal{N}}\mathsf{G}_{i}(\mathbf{X}_{N})\mathtt{1}_{\mathcal{E}_{\mathrm{ext}}\cap_{i=1}^{\mathcal{N}}\mathcal{E}_{i}}\right]\leq\exp\left(2\beta\sup_{\mathbf{X}_{N}\in\mathcal{E}_{N}}\mathrm{ErrorCI}(\mathbf{X}_{N})\right)\\\
\times\sup_{\mathbf{X}^{\mathrm{ext}}\in\mathcal{E}_{\mathrm{ext}},\\{\mathsf{n}_{i}\\}\
\mathrm{adm.}}\prod_{i=1}^{\mathcal{N}}\mathbb{E}^{\beta}_{\mathsf{n}_{i},\Lambda_{i},\mathrm{V}^{\mathrm{ext}}}\left[\mathsf{G}_{i}(\mathbf{X})\Big{|}\mathcal{E}_{i}\right]$
(7.18)

Let us note that the event $\bigcap_{i=1}^{\mathcal{N}}\mathcal{E}_{i}$
implies that the “admissible” number of points in each $\Lambda_{i}$ is
bounded by $10T^{2}$ (see Definition 5.2) which checks the first condition of
(4.6), and that the second condition of (4.6) is implied by the third
condition888The distance between two centers $\omega_{i},\omega_{j}$ is
bounded below by $ML$ and thus
$\operatorname{dist}(\Lambda_{i},\Lambda_{j})\geq ML-2T\geq 98T$. of (3.4). We
may thus use Lemma 4.1 which, together with (3.6), implies that the
conditional independence error $\mathrm{ErrorCI}$ between the $\Lambda_{i}$’s
is bounded by:

$\sup_{\mathbf{X}_{N}\in\bigcap_{i=1}^{\mathcal{N}}\mathcal{E}_{i}}|\mathrm{ErrorCI}[\mathbf{X}_{N}|\Lambda_{1},\dots,\Lambda_{\mathcal{N}}]|=\mathcal{O}\left(\frac{T^{5}R\log
R}{(ML)^{2}}\right),$ (7.19)

which controls the first term in the right-hand side of (7.18), and we now
focus on the second one.

#### Controlling expectations in each “sub-system”

Let us fix $1\leq i\leq\mathcal{N}$ and work in $\Lambda_{i}$. We can assume
that $\mathrm{V}^{\mathrm{ext}}$ is $\bar{\mathtt{C}}$-good on $\Lambda_{i}$
because of (7.12).

######  Claim 7.1.

Let $\mathsf{D}_{i}:=\mathsf{Dis}(\mathbf{X}_{\mathsf{n}},\mathfrak{B}_{i})$.
We have:

$\mathbb{E}^{\beta}_{\mathsf{n}_{i},\Lambda_{i}}\left[e^{\omega\mathsf{D}_{i}}\big{|}\mathcal{E}_{\Lambda_{i}}\right]=1+\mathcal{O}\left(L^{-0.66}\right)\leq
e^{\mathcal{O}(L^{-0.66})}$ (7.20)

###### Proof.

We start by decomposing the expectation as:

$\mathbb{E}^{\beta}_{\mathsf{n}_{i},\Lambda_{i}}\left[e^{\omega\mathsf{D}_{i}}\big{|}\mathcal{E}_{\Lambda_{i}}\right]=\mathbb{E}^{\beta}_{\mathsf{n}_{i},\Lambda_{i}}\left[e^{\omega\mathsf{D}_{i}}\mathtt{1}_{|\omega\mathsf{D}_{i}|\leq\frac{1}{2}}\big{|}\mathcal{E}_{\Lambda_{i}}\right]+\mathbb{E}^{\beta}_{\mathsf{n}_{i},\Lambda_{i}}\left[e^{\omega\mathsf{D}_{i}}\mathtt{1}_{|\omega\mathsf{D}_{i}|>\frac{1}{2}}\big{|}\mathcal{E}_{\Lambda_{i}}\right].$
(7.21)

Using a Taylor’s expansion, we may write the first term in the right-hand side
of (7.21) as:

$\mathbb{E}^{\beta}_{\mathsf{n}_{i},\Lambda_{i}}\left[e^{\omega\mathsf{D}_{i}}\mathtt{1}_{\omega|\mathsf{D}_{i}|\leq\frac{1}{2}}\big{|}\mathcal{E}_{\Lambda_{i}}\right]=\mathbb{E}^{\beta}_{\mathsf{n}_{i},\Lambda_{i}}\left[\left(1+\omega\mathsf{D}_{i}+\mathcal{O}\left(\omega^{2}\mathsf{D}_{i}^{2}\right)\right)\mathtt{1}_{\omega|\mathsf{D}_{i}|\leq\frac{1}{2}}\big{|}\mathcal{E}_{\Lambda_{i}}\right]\\\