# Emergence of a Poisson process in weakly interacting particle systems

David Padilla-Garza, Luke Peilen, Eric Thoma

###### Abstract

We consider the Gibbs measure of a general interacting particle system for a
certain class of “weakly interacting” kernels. In particular, we show that the
local point process converges to a Poisson point process as long as the
inverse temperature $\beta$ satisfies $N^{-1}\ll\beta\ll N^{-\frac{1}{2}}$,
where $N$ is the number of particles. This expands the temperature regime for
which convergence to a Poisson point process has been proved.

##  1 Introduction and motivation

In this paper, we will be interested in a system of particles that interact
via a pairwise interaction, and are confined by an external potential. We will
consider a system of $N$ particles that lie in $d-$dimensional Euclidean
space. This is modeled by the Hamiltonian

$\mathcal{H}_{N}(X_{N})=\sum_{i\neq j}g(x_{i}-x_{j})+N\sum_{i=1}^{N}V(x_{i}),$
(1.1)

where $g:\mathbf{R}^{d}\to\mathbf{R}$ is the pair-wise interaction,
$X_{N}:=(x_{1},...x_{N})\in(\mathbf{R}^{d})^{N}$, and
$V:\mathbf{R}^{d}\to\mathbf{R}$ is the confining potential. We will call this
an interacting particle system. We will be interested in the behaviour of such
a system for large but finite $N$.

We are particularly interested in limiting behavior of the law of the local
point process, which is defined as

$\Xi:=\sum_{i=1}^{N}\delta_{N^{1/d}(x_{i}-x^{\ast})}$ (1.2)

for an artibrary centering point $x^{\ast}$. The scaling of $N^{1/d}$ will
reflect that the particles $\\{x_{i}\\}_{i=1}^{N}$ typically occupy a volume
of order $1$, so the local point process (in the weak topology) captures
microscopic behavior of the system near $x^{\ast}$ as $N\to\infty$.

Our main goal will be to prove that the law of $\Xi$ is asymptotically
Poissonian in certain temperature scalings $\beta=\beta_{N}$ and for a broad
class of “weak interaction” kernels $g$. Before moving to precise statements
in Section 2, we discuss more broadly the study of interacting particle
systems with Hamiltonian of the form (1.1).

A very frequent form of the interaction $g$ for $d\geq 2$ is given by the
Coulomb kernel:

$g(x)=\begin{cases}\frac{1}{|x|^{d-2}}&\text{ if }d\geq 3,\\\
-\log(|x|)&\text{ if }d=2.\end{cases}$ (1.3)

Note that $g$ satisfies that

$-\Delta g=c_{d}\delta_{0},$ (1.4)

where $\delta_{x}$ denotes a Dirac delta at $x$, for all $d\geq 2$ and for
some constant $c_{d}$ depending only on $d$.

In $d=1$, a very frequent form of $g$ is given by

$g(x)=-\log(|x|)\text{ if }d=1.$ (1.5)

This is the same formula as in $d=2$, but in dimension $1$ $g$ is not the
fundamental solution of Laplacian (i.e. it does not satisfy equation (1.4));
instead it is a solution kernel for a fractional Laplacian.

In dimension $d\geq 1$, a generalization of the Coulomb kernel is given by the
Riesz kernel:

$g(x)=\frac{1}{|x|^{d-2s}},$ (1.6)

with $0<s<\min\\{\frac{d}{2},1\\}$. In this case $g$ satisfies that

$(-\Delta)^{s}g=c_{d,s}\delta_{0}.$ (1.7)

For some constant $c_{d,s}$ depending only on $d,s$. We will refer to this
setting as the Riesz case.

The study of minimizers of (1.1) when $g$ is either a Coulomb or Riesz
interaction is an active field of research, with possible applications in
approximation theory [15, 21, 29, 36, 5, 22, 16, 11, 12]. However, in this
paper we will take a different approach, and focus on the positive-temperature
regime. In other words, we will look at the Gibbs measure associated to
Hamiltonian (1.1), instead of focusing only on minimizers. This Gibbs measure
is given by:

$\mathrm{d}\mathbb{P}_{N,\beta}(X_{N})=\frac{1}{Z^{V}_{N,\beta}}\exp\left(-\beta\mathcal{H}_{N}(X_{N})\right)\,\mathrm{d}X_{N},$
(1.8)

where

$Z^{V}_{N,\beta}=\int_{\mathbf{R}^{d\times
N}}\exp\left(-\beta\mathcal{H}_{N}(X_{N})\right)\,\mathrm{d}X_{N}$ (1.9)

is the partition function, and $\beta>0$ is the inverse temperature which may
depend on $N$.

The Gibbs measure (1.8) is linked to Random Matrix Theory if $g$ is either a
Coulomb, or a $1d$ log interaction. In $d=1$ and if $\beta=1,2$ or $4$ and if
$V$ is quadratic, (1.8) gives the density of eigenvalues of matrices in the
Gaussian Orthogonal, Unitary or Symplectic ensembles, respectively. These
ensembles are well-studied because they are determinantal ($\beta=2$) or
Pfaffian ($\beta=1,4$), which allows one to explicitly write down the
correlation kernels. In the specific case $d=2,\beta=2,V(x)=|x|^{2}$, equation
(1.8) corresponds to the Ginibre ensemble, which also has a determinantal
structure. Higher dimensional Coulomb gases, and Riesz gases are also an
active field of research [48, 2, 27, 26, 34, 40, 7, 8, 33, 50].

At a macroscopic level, an interacting particle system is well-described by
its empirical measure, defined as

$\mathrm{emp}_{N}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}.$ (1.10)

The behaviour of the empirical measure is very different depending on the
scaling of the temperature. If the temperature is not too large
($\frac{1}{N}\ll\beta$), the empirical measure converges to the equilibrium
measure, denoted $\mu_{V}$, and defined as the minimizer of the mean-field
functional:

$\mu_{V}:=\mathrm{argmin}_{\mu}\mathcal{E}(\mu)+\int_{\mathbf{R}^{d}}V\,\mathrm{d}\mu,$
(1.11)

where

$\mathcal{E}(\mu):=\iint_{\mathbf{R}^{d}\times\mathbf{R}^{d}}g(x-y)\,\mathrm{d}\mu_{x}\,\mathrm{d}\mu_{y}.$
(1.12)

This convergence happens almost surely with respect to the Gibbs measure [17].
This measure has compact support as long as $V$ grows fast enough at infinity
(see section 3).

On the other hand, for large temperatures ($\beta\simeq\frac{1}{N}$), the
empirical measure does not converge to the equilibrium measure. Instead, the
effect of temperature and entropy is large enough that particles are not
confined to a compact set. In this case, the empirical measure converges to
the thermal equilibrium measure, denoted $\mu_{\theta}$ and defined as

$\mu_{\theta}:=\mathrm{argmin}_{\mu}\mathcal{E}(\mu)+\int_{\mathbf{R}^{d}}V\,\mathrm{d}\mu+\frac{1}{\theta}\mathrm{ent}[\mu],$
(1.13)

where $\theta:=N\beta$ and $\mathrm{ent}[\cdot]$ is the Boltzmann-Gibbs
entropy:

${\rm
ent}[\mu]=\begin{cases}\int_{\mathbb{R}^{d}}\log({\frac{\mathrm{d}\mu}{\mathrm{d}x}})\,\mathrm{d}{\mu}\quad\text{
if }\mu\text{ is absolutely continuous w.r.t. Lebesgue measure}\
\mathrm{d}x\\\ \infty\quad\text{ o.w.}\end{cases}$ (1.14)

Unlike the empirical measure, the thermal equilibrium measure is everywhere
positive (see section 3). In the case $g=0$ and $V(x)=|x|^{2}$, which
corresponds to independent Gaussian particles, the thermal equilibrium measure
is a Gaussian, with variance determined by $\theta$. Thus, the thermal
equilibrium measure can be seen as an interpolation between a Gaussian
probability density, and the equilibrium measure.

The empirical measure (1.10) and local point process (1.2) exhibit very
different behavior in the large $N$ limit. Due to equally weighting the $N$
particles, the empirical measure typically exhibits a law of large numbers
effect and converges (weakly) to a deterministic limit. The local point
process is sensitive to (and, in the weak topology, effectively determined by)
each particle in a large microscopic neighborhood of the centering point. It
exhibits random, temperature dependent behavior even in the $N\to\infty$
limit, except at very low temperature.

At these low temperatures, the local point process is expected to be well
approximated by minimizers of energies associated to the Gibbs measure
Hamiltonian, an extremely delicate family of variational problems. For
example, in certain dimensions, the local point process is conjectured to
converge to a periodic lattice minimizing a next-order (renormalized) energy
[35, 45] in some generality. In dimensions $8$ and $24$, this conjecture has
also been proved, in this case due to a link to the Cohn-Kumar conjecture
[35]. Given the difficulty in working with the local point process, it is
often convenient to work instead with the empirical and tagged empirical
fields, which are less refined but more manageable observables describing the
microscopic behaviour of the particle system [46, 40, 34].

On the other extreme, in the large temperature regime
($\beta\simeq\frac{1}{N}$), the law of the local point process completely
thermalizes and is given by a Poisson process in the large $N$ limit. This was
proved in the context of general integrable interactions in [24], generalizing
the work of [4, 28] for $\beta-$ensembles, which correspond to interacting
particle systems with a Coulomb interaction. The temperature regime for
Poissonian behavior is expected to be typically much wider, an expectation we
confirm for our class of interactions.

The microscopic behaviour of a particle system isn’t necessarily
asymptotically deterministic or Poissonian. For example, in the Coulomb case,
if $\beta\simeq N^{1-\frac{2}{d}}$ then the microscopic behaviour is
stochastic, but governed by a probability measure that favours low-
renormalized-energy configurations [26, 25, 40]. There are also interesting
rigidity and tolerance properties that emerge in the $N\to\infty$ limit [20,
14, 13, 49].

This paper will be about an interacting particle system at high temperature,
with a general pair-wise interaction. The study of interacting particle
systems with a general interaction is a classical subject in statistical
mechanics [19, 42, 43], and also an active field of study [17, 24, 10, 18].
Particle systems at high temperature have also recently drawn attention, and
this subject occasionally overlaps with the analysis of general interactions
[1, 10, 17, 23, 30, 31].

The analysis of interacting particle systems with a general pair-wise
interaction is also linked to AI and machine learning, more specifically to
neural networks. Neural networks can be used to accurately represent high-
dimensional functions: given a high-dimensional function $f$, it may be
represented as

$f(x):=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}{\varphi}_{i}(x,\theta_{i}),$
(1.15)

where ${\varphi}_{i}(x,\theta_{i})$ are given functions, depending on the
parameter $\theta_{i}$. One of the most frequent algorithms used to determine
the parameters $\theta_{i}$ is stochastic gradient descent (SGD). Despite its
ubiquity, very few rigorous results for convergence existed for SGD until
recently. The approach in [39, 37, 38, 51] is to model the evolution of the
parameters $\theta_{i}$ under the SGD as a particle system with an evolution
given by an SPDE. By modelling parameters as a particle system, the energy
landscape for the empirical measure becomes convex, and the authors are able
to show a convergence rate of $O(n^{-1})$ to the mean-field limit. In this
model, the interaction between the particles depends on the error between the
measurements and the approximating function, and on the functions
${\varphi}_{i}$. In general, however, it is not a Coulomb or Riesz
interaction. If the ${\varphi}_{i}$ are radial basis function networks, the
interaction between the particles may be given by a Gaussian kernel, which is
weakly interacting (see Definition 2.1). Given that the function $f$ is high
dimensional, this approach is linked to particle systems for large $d$ as
well.

The main result in this paper is that, for a specific class of interaction
kernels, which we call weakly interacting, the local point process converges
to a Poisson process for $N^{-1}\ll\beta\ll N^{-\frac{1}{2}}$, a wider
temperature than the one considered in [24] (which, to our knowledge is
currently the most general result in that direction), namely $\beta\simeq
N^{-1}$. Hence, we prove that the hypothesis on the temperature scaling may be
weakened for this specific class of interaction kernels. Our “weakly
interacting” kernels are essentially kernels with an integrable positive
Fourier transform, see Definition 2.1.

##  2 Main results

We will now state precisely the main results of the paper. We begin by
defining exactly the class of confining potentials and interaction kernels
that we deal with.

######  Definition 2.1.

An interaction kernel $g$ is called weakly interacting if $g$ satisfies:

  * 1.

$g(x)=g(-x)$.

  * 2.

$\widehat{g}>0$ a.e.

  * 3.

$\widehat{g}\in L^{1}$.

Note that, as a consequence of item $3$, $g$ is bounded below (and above).
Without loss of generality, we assume that this lower bound is $0$. Also as a
consequence of item $3$, $g$ is continuous. Hence, we may assume without loss
of generality the following:

  * 4.

$g>0$ a.e.

  * 5.

$g$ is continuous.

A confining potential $V$ is called admissible if $V$ satisfies:

  * 1.

$V$ is l.s.c.

  * 2.

$\lim_{x\to\infty}V(x)=\infty$.

  * 3.

$\int_{|x|\geq 1}e^{-\alpha V(x)}\,\mathrm{d}x<+\infty$ for large enough
$\alpha\geq\alpha_{0}$.

######  Definition 2.2.

A key quantity throughout the paper will be the potential field
$h^{\mu}:=g\ast\mu$ associated to a finite measure $\mu$. We often take
$\mu=\mathsf{emp}_{N}$.

We will now state our results for weakly interacting kernels and admissible
confining potentials. As mentioned in the introduction, the main goal of the
paper is to prove that the local point process converges to a Poisson point
process. Along the way, we prove concentration bounds for the potential field
of $\mathsf{emp}_{N}$ at given points, and asymptotes for the Laplace
transform of fluctuations. These are interesting results in their own right,
and so we state them in this section.

We recall the definition of the equilibrium measure $\mu_{V}$ in (1.11) and
thermal equilibrium measure $\mu_{\theta}$ in (1.13). Throughout the paper, we
assume $\theta\geq\alpha_{0}$ from Definition 2.1.

######  Proposition 2.3 (Concentration bounds for the potential field).

Assume that $g$ is weakly interacting, and $V$ is admissible. For any
$k\in\mathbf{N}$, $\epsilon>0$, and points
$y_{1},y_{2},...y_{k}\in\mathbf{R}^{d}$, we have

$\mathbb{P}_{N,\beta}\left(\left|\sum_{i=1}^{k}h^{\rm
emp_{N}-\mu_{\theta}}(y_{i})\right|>\epsilon\right)\leq\exp\left(-N^{2}\beta\left[\frac{\Lambda\epsilon^{2}}{k^{2}}-\frac{1}{N}g(0)\right]\right),$
(2.1)

for some $\Lambda>0$ depending only on $g$.

The proof is found in Section 4.

######  Proposition 2.4 (Asymptotics of the Laplace transform of field
fluctuations).

Assume that $g$ is weakly interacting, and $V$ is admissible. Let
$k\in\mathbf{N}$ and $y_{1},y_{2},...y_{k}\in\mathbf{R}^{d}$. Then

$\mathbb{E}_{N,\beta}\left[\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\rm
emp_{N}}(y_{i})\right)\right)\right]=M_{N}\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(y_{i})\right)\right)+A_{N},$
(2.2)

where

$\begin{split}|\log M_{N}|&\leq
C_{1}N^{\frac{1}{2}}\beta\left(C_{2}+\sum_{i=1}^{k}V(y_{i})\right)^{\frac{1}{2}}\\\
|A_{N}|&\leq\exp\left(-N\beta\left(g(0)+\sum_{i=1}^{k}V(y_{i})\right)\right),\end{split}$
(2.3)

for some constants $C_{1},C_{2}$ depending only on $V,g,d$.

The proof is found in Section 5.

We now state the main result of this paper: that the local point process
converges to a Poisson point process.

######  Theorem 1 (Convergence to Poisson point process).

Assume that $g$ is weakly interacting, and $V$ is admissible. Assume also that
$\mathrm{ent}[\mu_{V}]<\infty$. Let $x^{*}\in\mathbf{R}^{d}$ and define the
local point process $\Xi$ by

$\Xi:=\sum_{i=1}^{N}\delta_{N^{\frac{1}{d}}\left(x_{i}-x^{*}\right)}.$ (2.4)

Then, for a.e. $x^{*}$, if $N^{-1}\ll\beta\ll N^{-\frac{1}{2}}$, $\Xi$
converges to a Poisson point process of intensity $\mu_{V}(x^{*})$ as
$N\to\infty$.

The proof is found in Section 6.

Our proof also gives precise a asymptotic for the first marginal of the Gibbs
measure, defined as

$\rho(x):=\int_{\mathbf{R}^{d-1}}\frac{1}{Z^{V}_{N,\beta}}\exp\left(-\beta\mathcal{H}_{N}(x,X_{N-1})\right)\,\mathrm{d}X_{N-1}.$
(2.5)

The problem of finding an asymptotic for the first marginal was also addressed
in [41], motivated by a link to the fractional quantum Hall effect.

######  Corollary 2.5 (First marginal and confinement).

More specifically, our proof shows the estimate

$\rho(x)=M^{\prime\prime}_{N}\mu_{\theta}(x)+A^{\prime\prime}_{N},$ (2.6)

for some $M^{\prime\prime}_{N},A^{\prime\prime}_{N}$ satisfying

$\begin{split}|\log M^{\prime\prime}_{N}|&\leq
C_{1}^{\prime\prime}N^{\frac{1}{2}}\beta\left(C_{2}^{\prime\prime}+V(x)\right)^{\frac{1}{2}}\\\
|A^{\prime\prime}_{N}|&\leq\exp\left(-N\beta
C^{\prime\prime}\left(g(0)+V(x)\right)\right),\end{split}$ (2.7)

for some constants
$C^{\prime\prime},C_{1}^{\prime\prime},C_{2}^{\prime\prime}$ depending only on
$V,g,d$.

If $V$ is bounded below by some polynomial, and if $\beta=N^{-s}$ for some
$s\in\left(\frac{1}{2},1\right)$, then equation (2.7) implies a confinement
bound, or a bound on the probability that there is one particle outside of a
compact set $K$: Let $K\subset\mathbf{R}^{d}$ be a compact set such that
$V(x)+h^{\mu_{V}}(x)-c_{\infty}$ (see Lemma 3.1) is bounded below by a
positive constant outside of $K$. Then:

$\begin{split}\mathbb{P}_{N,\beta}\left(\mathsf{emp}_{N}\left(\mathbf{R}^{d}\setminus
K\right)\geq\frac{1}{N}\right)&\leq N\int_{\mathbf{R}^{d}\setminus
K}\rho(x)\,\mathrm{d}x\\\ &\leq\exp\left(-cN^{\gamma}\right),\end{split}$
(2.8)

where $c,\gamma>0$ depend on $V,g,$ and $K$.

In the Coulomb case, results from [3] imply that any compact set $K$ that
contains a neighborhood of the support of $\mu_{V}$ satisfies that
$V(x)+h^{\mu_{V}}(x)-c_{\infty}$ is bounded below by a positive constant
outside of $K$.

######  Remark 2.6.

Physical considerations lead one to expect that the optimal regime for
convergence of the local point process to a Poisson process is $\beta\to 0$ as
$N\to\infty$ at any speed. Proving this in a mathematically rigorous way is an
interesting direction for future work.

######  Remark 2.7.

Our analysis does not cover Coulomb, Riesz, or $1d-$log gases, since their
Fourier transform is not integrable. Proving that the local point process of a
Coulomb, Riesz, or $1d-$log gas converges to a Poisson process in a wider
temperature regime than $\beta\simeq\frac{1}{N}$ is work in progress.

##  3 Preliminaries

Before giving the proof of the main results, we establish some foundational
results in interacting particle systems. Some of these results may exist in
the literature, but since we were unable to find a source corresponding to our
specific case we present arguments for them here. To our knowledge, Lemma 3.6
and Remark 3.7 are new.

During the rest of the paper, we will commit the abuse of notation of not
distinguishing between a measure and its density.

######  Lemma 3.1 (Equilibrium measure).

Assume that $g$ is weakly interacting, and $V$ is admissible. Then the
functional

$\mathcal{E}_{V}(\mu):=\mathcal{E}(\mu)+\int_{\mathbf{R}^{d}}V\,\mathrm{d}\mu$
(3.1)

has a unique minimizer in the space of probability measures which we call the
_equilibrium measure_ and denote $\mu_{V}$. The equilibrium measure has
compact support, denoted $\Sigma$, and satisfies the Euler-Lagrange equation

$\begin{split}h^{\mu_{V}}+\frac{V}{2}-c_{\infty}\geq&0\text{ in
}\mathbf{R}^{d}\\\ h^{\mu_{V}}+\frac{V}{2}-c_{\infty}=&0\>\mu_{V}-\text{a.e.
in }\\{x\in\mathbf{R}^{d}|\mu_{V}(x)>0\\},\end{split}$ (3.2)

where

$c_{\infty}=\mathcal{E}(\mu_{V})+\frac{1}{2}\int_{\mathbf{R}^{d}}V\,\mathrm{d}\mu.$
(3.3)

###### Proof.

The analogous argument can be found for the two dimensional log-gas in [44,
Chapter 1], for general interactions on compact sets in [6, Chapter 4] and for
a general Riesz gas in [47, Chapter 2].

Step 1: Existence and Uniqueness.

First, one notes that $\mathcal{E}_{V}$ is strictly convex on the space of
probability measures $\mathcal{P}(\mathbf{R}^{d})$; this follows from the
linearity of the map

$\mu\mapsto\int_{\mathbf{R}^{d}}V\,\mathrm{d}\mu$ (3.4)

and the positivity of the quadratic map

$\mu\mapsto\mathcal{E}(\mu),$ (3.5)

where positivity is a result of the positivity of the Fourier transform of
$g$:

$\iint_{\mathbf{R}^{d}\times\mathbf{R}^{d}}g(x-y)\,\mathrm{d}\mu_{x}d\mu_{y}=\int_{\mathbf{R}^{d}}(\mathsf{g}\ast\mu)\,\mathrm{d}\mu=\langle(\hat{g}\hat{\mu})^{\vee},\mu\rangle=\langle\hat{g}\hat{\mu},\overline{\hat{\mu}}\rangle>0$
(3.6)

for $\mu\neq 0$ of finite measure. (To be completely rigorous above, one
should approximate $g$ by smooth $g_{n}$ in the $L^{\infty}$ norm and send
$n\to\infty$.)

Next, we claim that $\mathcal{E}_{V}$ is lower semicontinuous with compact
level sets, i.e. any sequence $\\{\mu_{n}\\}\in\mathcal{P}(\mathbf{R}^{d})$
with bounded $\\{\mathcal{E}_{V}(\mu_{n})\\}$ has a weakly convergent
subsequence $\mu_{n_{k}}\rightharpoonup\mu\in\mathcal{P}(\mathbf{R}^{d})$ and

$\liminf_{k\rightarrow\infty}\mathcal{E}_{V}(\mu_{n_{k}})\geq\mathcal{E}_{V}(\mu).$
(3.7)

To prove this claim, note that (3.7) would follow immediately from the weak
convergence $\mu_{n_{k}}\rightharpoonup\mu\in\mathcal{P}(\mathbf{R}^{d})$
since $V$ and $g$ are both lower semicontinuous and bounded from below. One
demonstrates the weak convergence by showing that the sequence $\\{\mu_{n}\\}$
is tight and using Prokhorov’s theorem. Tightness follows from the growth of
$V$ at infinity coupled with the boundedness of $\mathcal{E}_{V}$. In fact, a
close inspection of [47, Lemma 2.10] shows we can directly apply [47, Lemma
2.10] as soon as

$\lim_{|x|,|y|\rightarrow+\infty}g(x-y)+\frac{V(x)}{2}+\frac{V(y)}{2}=+\infty$
(3.8)

and $g(x-y)+V(x)+V(y)$ is bounded below on
$\mathbf{R}^{d}\times\mathbf{R}^{d}$; both are true for us because of the
coercivity of $V$ and the boundedness of $g$.

Therefore, (3.7) is true and existence of the minimizer $\mu_{V}$ follows
immediately from the direct method in the calculus of variations; uniqueness
is guaranteed by strict convexity of $\mathcal{E}_{V}$.

Step 2: Compact support.

The contradiction argument in [47, Lemma 2.14] goes through exactly to show
compact support once one has

$\lim_{|x|,|y|\rightarrow+\infty}g(x-y)+\frac{V(x)}{2}+\frac{V(y)}{2}=+\infty,$
(3.9)

which we do in our case.

Step 3: Euler-Lagrange equation.

The Euler-Lagrange equation is obtained by taking convex perturbations. We
follow the argument in [47, Lemma 2.14], with a few simplifications in our
case.

Fix $\nu\in\mathcal{P}(\mathbf{R}^{d})$ such that
$\mathcal{E}_{V}(\nu)<+\infty$. If we consider $(1-t)\mu_{V}+t\nu$ for $t>0$,
then

$\begin{split}0&\leq\mathcal{E}_{V}((1-t)\mu_{V}+t\nu)-\mathcal{E}_{V}(\mu_{V})\\\
&=(1-t)^{2}\mathcal{E}(\mu_{V})+t^{2}\mathcal{E}(\nu)+2t(1-t)\iint_{\mathbf{R}^{d}\times\mathbf{R}^{d}}g(x-y)\,\mathrm{d}\mu_{V}(x)\,\mathrm{d}\nu(y)\\\
&\ \ \ \ \ \
+(1-t)\int_{\mathbf{R}^{d}}V\,\mathrm{d}\mu_{V}+t\int_{\mathbf{R}^{d}}V\,\mathrm{d}\nu-\mathcal{E}_{V}(\mu_{V}).\end{split}$
(3.10)

Expanding and rearranging, we find

$-2t\mathcal{E}(\mu_{V})-t\int_{\mathbf{R}^{d}}V\,\mathrm{d}\mu_{V}+2t\iint_{\mathbf{R}^{d}\times\mathbf{R}^{d}}g(x-y)\,\mathrm{d}\mu_{V}(x)\,\mathrm{d}\nu(y)+t\int_{\mathbf{R}^{d}}V\,\mathrm{d}\nu+O(t^{2})\geq
0$ (3.11)

or

$2t\int_{\mathbf{R}^{d}}\left(h^{\mu_{V}}+\frac{V}{2}\right)\,\mathrm{d}\nu+O(t^{2})\geq
2tc$ (3.12)

with
$c_{\infty}:=\mathcal{E}(\mu_{V})+\int_{\mathbf{R}^{d}}\frac{V}{2}\,\mathrm{d}\mu_{V}=\int_{\mathbf{R}^{d}}\left(h^{\mu_{V}}+\frac{V}{2}\right)\,\mathrm{d}\mu_{V}$.
Dividing by $2t$ and taking the limit as $t\downarrow 0$ yields

$\int_{\mathbf{R}^{d}}\left(h^{\mu_{V}}+\frac{V}{2}\right)\,\mathrm{d}\nu\geq
c_{\infty}.$ (3.13)

Since Diracs have finite $\mathcal{E}_{V}$-energy in our case, we can simplify
the rest of [47, Lemma 2.14] and conclude that

$h^{\mu_{V}}+\frac{V(x)}{2}\geq c_{\infty}$ (3.14)

everywhere. Finally, choosing $\nu=\mu_{V}$ yields

$c_{\infty}\leq\int_{\mathbf{R}^{d}}\left(h^{\mu_{V}}+\frac{V}{2}\right)\,\mathrm{d}\mu_{V}=c_{\infty}$
(3.15)

and since $h^{\mu_{V}}+\frac{V(x)}{2}\geq c_{\infty}$ everywhere we must have

$h^{\mu_{V}}+\frac{V(x)}{2}=c_{\infty}$ (3.16)

almost everywhere on $\Sigma$, as required. ∎

######  Lemma 3.2 (Thermal equilibrium measure).

Assume that $g$ is weakly interacting, $V$ is admissible, and let $\theta>0$.
Then the functional

$\mathcal{E}_{\theta}(\mu):=\mathcal{E}(\mu)+\int_{\mathbf{R}^{d}}V\,\mathrm{d}\mu+\frac{1}{\theta}\mathrm{ent}[\mu]$
(3.17)

has a unique minimizer in the space of probability measures which we call
_thermal equilibrium measure_ and denote $\mu_{\theta}$. The thermal
equilibrium measure is everywhere positive, and satisfies the Euler-Lagrange
equation

$\mu_{\theta}=L_{\theta}^{-1}\exp\left(-\theta\left(2h^{\mu_{\theta}}+V\right)\right),$
(3.18)

where

$L_{\theta}=\int_{\mathbf{R}^{d}}\exp\left(-\theta\left(2h^{\mu_{\theta}}(x)+V(x)\right)\right)\,\mathrm{d}x.$
(3.19)

###### Proof.

The existence of the thermal equilibrium measure follows for instance from [3,
Lemma 2.1], which only uses that $g$ is bounded from below and that
$V\rightarrow+\infty$ and satisfies the exponential integrability assumption

$\int_{|x|\geq 1}e^{-\frac{\theta}{2}V(x)}\,\mathrm{d}x<+\infty.$ (3.20)

(3.18) then follows immediately from the form of the thermal equilibrium
measure given in [24, Proposition A.2], since our kernel $g$ satisfies all of
the hypotheses of that proposition. ∎

Once we have existence of the thermal equilibrium measure,
$N^{2}\mathcal{E}_{\theta}$ gives the leading order asymptotics of the
Hamiltonian. To analyze next-order behavior, it is then useful to split off
this deterministic term and consider the remainder. This can be accomplished
by the following splitting formula and its corollary; the version stated below
is exactly [2, Lemma 2.1], which only relies on the form of the Hamiltonian
$\mathcal{H}_{N}$.

######  Lemma 3.3 (Thermal splitting formula).

Let $\theta>0$, and define

$\zeta_{\theta}:=-\frac{1}{\theta}\log(\mu_{\theta}).$ (3.21)

Then for any point configuration $X_{N}\in\mathbf{R}^{d\times N}$ the
Hamiltonian $\mathcal{H}_{N}$ can be rewritten (split) as

$\mathcal{H}_{N}(X_{N})=N^{2}\left(\mathcal{E}_{\theta}(\mu_{\theta})+\mathsf{F}_{N}(X_{N},\mu_{\theta})+N\sum_{i=1}^{N}\zeta_{\theta}(x_{i})\right),$
(3.22)

where, given a measure $\mu$,

$\mathsf{F}_{N}(X_{N},\mu):=\frac{1}{N^{2}}\sum_{i\neq
j}g(x_{i}-x_{j})+\mathcal{E}(\mu)-\frac{2}{N}\sum_{i=1}^{N}h^{\mu}(x_{i}).$
(3.23)

A computation then immediately yields the following equivalent definition of
the Gibbs measure.

######  Corollary 3.4.

The Gibbs measure may be rewritten as

$\mathrm{d}\mathbb{P}_{N,\beta}(X_{N})=\frac{1}{K_{N,\beta}^{\theta}}\exp\left(-N^{2}\beta\left[\mathsf{F}_{N}(X_{N},\mu_{\theta})\right]\right)\prod_{i=1}^{N}\mu_{\theta}(x_{i})\,\mathrm{d}X_{N},$
(3.24)

where

$K_{N,\beta}^{\theta}=\frac{Z^{V}_{N,\beta}}{\exp\left(N^{2}\beta\mathcal{E}_{\theta}(\mu_{\theta})\right)}.$
(3.25)

We call $K_{N,\beta}^{\theta}$ the _next-order partition function_. A useful
fact is that its logarithm is always non-negative.

######  Lemma 3.5.

Assume that $g$ is weakly interacting, and $V$ is admissible. Then the next-
order partition function is greater than $1$, i.e.

$\log K_{N,\beta}^{\theta}>0.$ (3.26)

###### Proof.

The proof is as in [32, Proposition 5.10], which considered the analogous
question for the Coulomb gas. The result follows immediately from the
definition of the Gibbs measure as the probability measure that minimizes the
free energy functional

$\mathbb{Q}\mapsto\int_{\mathbf{R}^{d\times
N}}\mathcal{H}_{N}(X_{N})\,\mathrm{d}\mathbb{Q}\left(X_{N}\right)+\frac{1}{\beta}\mathrm{ent}[\mathbb{Q}]$
(3.27)

over all $\mathbb{Q}\in\mathcal{P}\left(\mathbf{R}^{d\times N}\right)$.
Inserting $\mathbb{Q}=\mathbb{P}_{N,\beta}$ yields

$-\frac{\log
Z_{N,\beta}}{\beta}=\inf_{\mu\in\mathcal{P}\left(\mathbf{R}^{d\times
N}\right)}\int_{\mathbf{R}^{d\times
N}}\mathcal{H}_{N}(X_{N})\,\mathrm{d}\mu\left(X_{N}\right)+\frac{1}{\beta}\mathrm{ent}[\mu].$
(3.28)

Inserting instead $\mu_{\theta}^{\otimes N}$ and computing yields

$-\frac{\log Z_{N,\beta}}{\beta}\leq
N^{2}\mathcal{E}_{\theta}(\mu_{\theta})-\frac{N}{2}\mathcal{E}(\mu_{\theta})<N^{2}\mathcal{E}_{\theta}(\mu_{\theta})$
(3.29)

by positivity of $g$. The splitting formula (3.22) then yields

$-\frac{\log Z_{N,\beta}}{\beta}=\frac{-\log K_{N,\beta}^{\theta}+\beta
N^{2}\mathcal{E}(\mu_{\theta})}{\beta}\leq
N^{2}\mathcal{E}_{\theta}(\mu_{\theta})\implies\log K_{N,\beta}^{\theta}>0.$
(3.30)

∎

From this point forward, we present results on the thermal equilibrium measure
that are not essentially taken from existing literature.

######  Lemma 3.6 (Convergence).

Assume that $g$ is weakly interacting, and $V$ is admissible. Suppose also
that $\mathrm{ent}[\mu_{V}]<\infty$. Then, as $\theta\to\infty$,
$\mu_{\theta}$ converges to $\mu_{V}$ in the $L^{1}$ topology, and
$h^{\mu_{\theta}}$ converges to $h^{\mu_{V}}$ in the $L^{\infty}$ topology.
Furthermore, for a.e. $x\in\mathbf{R}^{d}$ and any sequence $\delta(\theta)$
such that $\lim_{\theta\to\infty}\delta=0$,

$\lim_{\theta\to\infty}\fint_{B(x,\delta)}\left|\mu_{\theta}(s)-\mu_{V}(x)\right|\,\mathrm{d}s=0.$
(3.31)

###### Proof.

In the case where $g$ is the Coulomb interaction, very precise convergence is
established by connections with the classical obstacle problem (cf. [3,
Theorem 1]). However, that machinery is not available to us in the general
interaction case.

Step 1: We will first prove that $\mu_{\theta}\to\mu_{V}$ in the topology of
weak convergence of probability measures. We will use the language of
$\Gamma$-convergence (see [9]).

Step 1.1: $\Gamma-\liminf$ inequality.

The lack of positivity of the entropy presents a difficulty, which we
circumvent by a trick as in [3]; namely, we can rewrite the function
$\mathcal{E}_{\theta}$ as

$\mathcal{E}_{\theta}(\mu)=\mathcal{E}(\mu)+\left(1-\frac{\alpha_{0}}{\theta}\right)\int_{\mathbf{R}^{d}}V\,\mathrm{d}\mu+\left(\int\frac{\alpha_{0}}{\theta}V\,\mathrm{d}\mu+\mathrm{ent}[\mu]\right),$
(3.32)

where $\alpha_{0}$ is as in Definition 2.1. Consider a sequence
$\theta_{n}\rightarrow+\infty$ and suppose that $\mu_{n}\rightharpoonup\mu$.
Then, since $g$ and $V$ are both lower semicontinuous and bounded from below,
we have for any $\epsilon>0$ that

$\displaystyle\liminf_{n\rightarrow\infty}\left(\mathcal{E}(\mu_{n})+\int_{\mathbf{R}^{d}}\left(1-\frac{\alpha_{0}}{\theta_{n}}\right)V\,\mathrm{d}\mu_{n}\right)$
$\displaystyle\geq\liminf_{n\rightarrow\infty}\left(\mathcal{E}(\mu_{n})+\int_{\mathbf{R}^{d}}\left(1-\epsilon\right)V\,\mathrm{d}\mu_{n}\right)$
(3.33)
$\displaystyle\geq\mathcal{E}(\mu)+\int_{\mathbf{R}^{d}}(1-\epsilon)V\leavevmode\nobreak\
d\mu.$

Now, if $\mu_{n}$ is not absolutely continuous with respect to Lebesgue
measure for all $n\geq N$ for some $N$, then $\mathrm{ent}[\mu_{n}]$ is
infinite for all $n\geq N$ and so

$\liminf_{n\rightarrow\infty}\left(\int_{\mathbf{R}^{d}}\frac{\alpha_{0}}{\theta_{n}}V\,\mathrm{d}\mu_{n}+\frac{1}{\theta_{n}}\mathrm{ent}[\mu_{n}]\right)=+\infty\geq
0.$ (3.34)

Otherwise, extracting the subsequence $\mu_{n_{k}}$ that are absolutely
continuous to Lebesgue measure yields

$\displaystyle\liminf_{n\rightarrow\infty}\left(\int_{\mathbf{R}^{d}}\frac{\alpha_{0}}{\theta_{n}}V\,\mathrm{d}\mu_{n}+\frac{1}{\theta_{n}}\mathrm{ent}[\mu_{n}]\right)$
$\displaystyle\geq\liminf_{k\rightarrow\infty}\int_{\mathbf{R}^{d}}\left(\frac{\alpha_{0}V}{\theta_{n_{k}}}+\frac{1}{\theta_{n_{k}}}\log\mu_{n_{k}}\right)\leavevmode\nobreak\
d\mu_{n_{k}}$
$\displaystyle\geq\liminf_{k\rightarrow\infty}\frac{-1}{\theta_{n_{k}}}\int_{\mathbf{R}^{d}}e^{-\alpha_{0}V(x)-1}\leavevmode\nobreak\
dx=0$

where we have used the inequality

$\frac{c\gamma}{\theta}+\frac{\gamma\log\gamma}{\theta}\geq-\frac{1}{\theta}e^{-c-1},$

with $c=\alpha_{0}V$ and $\gamma=\mu_{n_{k}}$, which can be seen by minimizing
the above as a function of $\gamma$.

In either case,

$\liminf_{n\rightarrow\infty}\mathcal{E}_{\theta_{n}}(\mu_{n})\geq\mathcal{E}(\mu)+\int_{\mathbf{R}^{d}}(1-\epsilon)V\leavevmode\nobreak\
d\mu.$

Since $\epsilon>0$ was arbitrary, we conclude by monotone convergence that
$\liminf_{n\rightarrow+\infty}\mathcal{E}_{\theta_{n}}(\mu_{n})\geq\mathcal{E}_{V}(\mu)$.

Step 1.2: $\Gamma-\limsup$ inequality.

Under the hypothesis that $\mathrm{ent}[\mu_{V}]<\infty$, it is trivially true
that

$\lim_{\theta\to\infty}\mathcal{E}_{\theta}(\mu_{V})=\mathcal{E}_{V}(\mu_{V}).$
(3.35)

Step 1.3: Uniform coercivity and conclusion of step 1.

Now we show that $\\{\mu_{\theta_{n}}\\}$ is a tight sequence of probability
measures. This will follow from the boundedness of the sequence
$\\{\mathcal{E}_{\theta_{n}}(\mu_{\theta_{n}})\\}$ in the same way as Lemma
4.1 (following [47, Lemma 2.10]). Without loss of generality, assume
$\theta_{1}=\inf\theta_{n}$. First, the boundedness follows from
$\frac{1}{\theta_{n}}\rightarrow 0$ and the minimality of $\mu_{\theta_{n}}$:

$\mathcal{E}_{\theta_{n}}(\mu_{\theta_{n}})\leq\mathcal{E}_{\theta_{n}}(\mu_{\theta_{1}})\leq\mathcal{E}_{\theta_{1}}(\mu_{\theta_{1}}):=C_{1}<+\infty$
(3.36)

for all $n$. Now, for any $C_{2}>0$, there is a compact set $K\times K$
outside of which

$g(x-y)+\gamma_{n}\frac{V(x)}{2}+\gamma_{n}\frac{V(y)}{2}>C_{2}$ (3.37)

with $\gamma_{n}=\left(1-\frac{\alpha_{0}}{\theta_{n}}\right)$ independently
of $n$ by our growth assumption on $V$ (we can choose this independently of
$n$ since $\theta_{n}\rightarrow+\infty$). Then, moving some of the potential
onto the entropy as above, we have

$\displaystyle C_{1}$
$\displaystyle\geq\mathcal{E}(\mu_{\theta_{n}})+\left(1-\frac{\alpha_{0}}{\theta_{n}}\right)\int_{\mathbf{R}^{d}}V\,\mathrm{d}\mu_{\theta_{n}}+\int_{\mathbf{R}^{d}}\left(\frac{\alpha_{0}}{\theta_{n}}V+\frac{1}{\theta_{n}}\log\mu_{\theta_{n}}\right)\,\mathrm{d}\mu_{\theta_{n}}$
$\displaystyle\geq\iint_{\mathbf{R}^{d}\times\mathbf{R}^{d}}\left(g(x-y)+\gamma_{n}\frac{V(x)}{2}+\gamma_{n}\frac{V(y)}{2}\right)\,\mathrm{d}\mu_{\theta_{n}}(x)\,\mathrm{d}\mu_{\theta_{n}}(y)-\frac{1}{\theta_{n}}\int_{\mathbf{R}^{d}}e^{-\alpha_{0}V(x)-1}\,\mathrm{d}x$
$\displaystyle\geq-C_{3}+C_{2}\mu_{\theta_{n}}\otimes\mu_{\theta_{n}}((K\times
K)^{c})$ $\displaystyle\geq-C_{3}+C_{2}\mu_{\theta_{n}}(K^{c}),$

where we have used that
$g(x-y)+\gamma_{n}\frac{V(x)}{2}+\gamma_{n}\frac{V(y)}{2}$ is everywhere
bounded below independently of $n$ since $|\gamma_{n}-1|$ is small, and
$-\frac{1}{\theta_{n}}\int_{\mathbf{R}^{d}}e^{-\alpha_{0}V(x)-1}\,\mathrm{d}x$
is bounded below as well since $\theta_{n}\rightarrow\infty$. Since $C_{2}$
can be made arbitrarily large, this gives us the tightness of the sequence
$\\{\mu_{\theta_{n}}\\}$.

We can now conclude as in [9, Theorem 1.21], or as in [47, Theorem 2.2]. Since
$\\{\mu_{\theta_{n}}\\}$ is a tight sequence, it has a weak limit $\mu$.
Furthermore, since $\mu_{V}$ has a density, we have

$\mathcal{E}_{V}(\mu_{V})\geq\limsup_{n\rightarrow+\infty}\mathcal{E}_{\theta_{n}}(\mu_{V})\geq\limsup_{n\rightarrow+\infty}\mathcal{E}_{\theta_{n}}(\mu_{\theta_{n}})\geq\mathcal{E}_{V}(\mu).$
(3.38)

The second inequality follows from the fact that $\mu_{\theta_{n}}$ minimizes
$\mathcal{E}_{\theta_{n}}$, and the third inequality is a result of lower
semicontinuity. Since the equilibrium measure is unique, we must have
$\mu=\mu_{V}$ and we see that $\mu_{\theta_{n}}$ converges weakly to $\mu_{V}$
under the topology of weak convergence.

Step 2: Now, we prove that $\mu_{\theta}\to\mu_{V}$ in the $L^{1}$ topology.

Note that, by definition of $\mu_{\theta}$, we have that, for any $\theta>0$,

$\mathrm{ent}[\mu_{\theta}]\leq\mathrm{ent}[\mu_{V}]<\infty.$ (3.39)

In particular,

$\overline{\lim}_{\theta\to\infty}\mathrm{ent}[\mu_{\theta}]\leq\mathrm{ent}[\mu_{V}].$
(3.40)

On the other hand, by l.s.c.,

$\mathrm{ent}[\mu_{V}]\leq\underline{\lim}_{\theta\to\infty}\mathrm{ent}[\mu_{\theta}],$
(3.41)

which implies that

${\lim}_{\theta\to\infty}\mathrm{ent}[\mu_{\theta}]=\mathrm{ent}[\mu_{V}].$
(3.42)

Given $\epsilon>0$, we define the probability measure $\mu_{V}^{\epsilon}$ as

$\mu_{V}^{\epsilon}:=\frac{\mu_{V}+\epsilon\exp\left(-\alpha_{0}V\right)}{\int_{\mathbf{R}^{d}}\mu_{V}+\epsilon\exp\left(-\alpha_{0}V\right)},$
(3.43)

where $\alpha_{0}$ is as in Definition 2.1. Then, by Pinsker’s inequality,

$\begin{split}\limsup_{\theta\to\infty}\left\|\mu_{\theta}-\mu_{V}^{\epsilon}\right\|_{L^{1}}&\leq\limsup_{\theta\to\infty}\sqrt{\int_{\mathbf{R}^{d}}\log\left(\frac{\mu_{\theta}}{\mu_{V}^{\epsilon}}\right)\mu_{\theta}}\\\
&=\limsup_{\theta\to\infty}\sqrt{\mathrm{ent}[\mu_{\theta}]-\int_{\mathbf{R}^{d}}\log\left({\mu_{V}^{\epsilon}}\right)\mu_{\theta}}\\\
&=\sqrt{\mathrm{ent}[\mu_{V}]-\int_{\mathbf{R}^{d}}\log\left({\mu_{V}^{\epsilon}}\right)\mu_{V}}.\end{split}$
(3.44)

Furthermore, by triangle inequality, there holds

$\limsup_{\theta\to\infty}\left\|\mu_{\theta}-\mu_{V}\right\|_{L^{1}}\leq\sqrt{\mathrm{ent}[\mu_{V}]-\int_{\mathbf{R}^{d}}\log\left({\mu_{V}^{\epsilon}}\right)\mu_{V}}+\left\|\mu_{V}-\mu_{V}^{\epsilon}\right\|_{L^{1}}.$
(3.45)

Letting $\epsilon$ tend to $0$, we may conclude that $\mu_{\theta}\to\mu_{V}$
in the $L^{1}$ topology.

Note that $\left\|g\right\|_{L^{\infty}}<\infty$ by property 3 of Definition
2.1. Hence, Young’s convolution inequality, implies that $h^{\mu_{\theta}}\to
h^{\mu_{V}}$ in $L^{\infty}$.

Step 3: Finally, we prove that for a.e. $x\in\mathbf{R}^{d}$ and any sequence
$\delta(\theta)$ such that $\lim_{\theta\to\infty}\delta=0$,

$\lim_{\theta\to\infty}\fint_{B(x,\delta)}\left|\mu_{\theta}(s)-\mu_{V}(x)\right|\,\mathrm{d}s=0.$
(3.46)

To prove this claim, note that, for any $\theta>0$ and a.e. $x$, we have

$\displaystyle\fint_{B(x,\delta)}\left|\mu_{\theta}(s)-\mu_{V}(x)\right|\,\mathrm{d}s$
(3.47)
$\displaystyle\leq\fint_{B(x,\delta)}\left|\mu_{\theta}(s)-\mu_{V}(s)\right|\,\mathrm{d}s+\fint_{B(x,\delta)}\left|\mu_{V}(s)-\mu_{V}(x)\right|\,\mathrm{d}s$
$\displaystyle\leq
M(\mu_{\theta}-\mu_{V})(x)+\fint_{B(x,\delta)}\left|\mu_{V}(s)-\mu_{V}(x)\right|\,\mathrm{d}s.$

where $M(\cdot)$ denotes the Hardy-Littlewood maximal function.

Let $\epsilon>0$. Then

$\displaystyle\overline{\lim}_{\theta\to\infty}\left|\left\\{x\in\mathbf{R}^{d}:\fint_{B(x,\delta)}\left|\mu_{\theta}(s)-\mu_{V}(x)\right|\,\mathrm{d}s>\epsilon\right\\}\right|$
(3.48)
$\displaystyle\leq\overline{\lim}_{\theta\to\infty}\left|\left\\{x\in\mathbf{R}^{d}:\left|M(\mu_{\theta}-\mu_{V})(x)\right|>\epsilon\right\\}\right|$
$\displaystyle\quad+\overline{\lim}_{\theta\to\infty}\left|\left\\{x\in\mathbf{R}^{d}:\fint_{B(x,\delta)}\left|\mu_{V}(s)-\mu_{V}(x)\right|\,\mathrm{d}s>\epsilon\right\\}\right|.$

By the Hardy-Littlewood maximal inequality, and step 2, there holds

$\begin{split}\overline{\lim}_{\theta\to\infty}\left|\left\\{x\in\mathbf{R}^{d}:\left|M(\mu_{\theta}-\mu_{V})(x)\right|>\epsilon\right\\}\right|&\leq\overline{\lim}_{\theta\to\infty}\frac{C_{d}}{\epsilon}\left\|\mu_{\theta}-\mu_{V}\right\|_{L^{1}}=0.\end{split}$
(3.49)

On the other hand, Lebesgue’s Theorem implies that

$\overline{\lim}_{\theta\to\infty}\left|\left\\{x\in\mathbf{R}^{d}:\fint_{B(x,\delta)}\left|\mu_{V}(s)-\mu_{V}(x)\right|\,\mathrm{d}s>\epsilon\right\\}\right|=0.$
(3.50)

Since $\epsilon$ is arbitrary, we may conclude.

∎

######  Remark 3.7 (Qualitative behaviour).

The constant $L_{\theta}$ defined in (3.19) satisfies that

$\lim_{\theta\to\infty}-\frac{1}{\theta}\log L_{\theta}=2c_{\infty},$ (3.51)

where $c_{\infty}$ is given by (3.3). Furthermore, for all $\theta>0$ there
exists a constant $c_{\theta}$ such that, for any $\theta^{\prime}\geq\theta$
there holds

$\mu_{\theta^{\prime}}(x)\leq\exp\left(-\theta^{\prime}\left(h^{\mu_{V}}(x)+V(x)-2c_{\theta}\right)\right).$
(3.52)

Also, $\lim_{\theta\to\infty}c_{\theta}=c_{\infty}$.

###### Proof.

Using Lemma 3.6, and equations (3.2) and (3.18) we have that

$\begin{split}\lim_{\theta\to\infty}-\frac{1}{\theta}\log
L_{\theta}&=\inf_{x\in\mathbf{R}^{d}}2h^{\mu_{V}}(x)+V(x)\\\
&=2c_{\infty}.\end{split}$ (3.53)

The rest of the remark follows from equations (3.18), (3.51) by taking

$c_{\theta}:=\sup_{\theta^{\prime}\geq\theta}-\frac{1}{2\theta^{\prime}}\log
L_{\theta^{\prime}}+\left\|h^{\mu_{V}}-h^{\mu_{\theta^{\prime}}}\right\|_{L^{\infty}}.$
(3.54)

∎

##  4 Proof of Proposition 2.3

We now prove Proposition 2.3, restated here for convenience.

######  Proposition 4.1 (Concentration bounds for the potential field).

Assume that $g$ is weakly interacting, and $V$ is admissible. Let
$k\in\mathbf{N}$, $\epsilon>0$, and $y_{1},y_{2},...y_{k}\in\mathbf{R}^{d}$.
Then

$\mathbb{P}_{N,\beta}\left(\left|\sum_{i=1}^{k}h^{\rm
emp_{N}-\mu_{\theta}}(y_{i})\right|>\epsilon\right)\leq\exp\left(-N^{2}\beta\left[\frac{\Lambda\epsilon^{2}}{k^{2}}-\frac{1}{N}g(0)\right]\right),$
(4.1)

for some $\Lambda>0$ depending only on $g$.

###### Proof.

Step 1: Starting point.

Using the splitting formula (Lemma 3.3), we have that

$\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\mathbb{P}_{N,\beta}\left(\left|\sum_{i=1}^{k}h^{\rm
emp_{N}-\mu_{\theta}}(y_{i})\right|>\epsilon\right)$\mbox{}\hfil\quad&\\\
&=\frac{1}{K_{N,\beta}}\int_{\left\\{X_{N}:\left|\sum_{i=1}^{k}h^{\rm
emp_{N}-\mu_{\theta}}(y_{i})\right|>\epsilon\right\\}}\exp\left(-N^{2}\beta\mathsf{F}_{N}(\mathsf{emp}_{N},\mu_{\theta})\right)\mathsf{d}\mu_{\theta}^{\otimes
N}\\\ &\leq\frac{1}{K_{N,\beta}}\sup_{\left\\{X_{N}:\left|\sum_{i=1}^{k}h^{\rm
emp_{N}-\mu_{\theta}}(y_{i})\right|>\epsilon\right\\}}\exp\left(-N^{2}\beta\mathsf{F}_{N}(\mathsf{emp}_{N},\mu_{\theta})\right)\\\
&\leq\frac{1}{K_{N,\beta}}\sup_{\left\\{X_{N}:\left|\sum_{i=1}^{k}h^{\rm
emp_{N}-\mu_{\theta}}(y_{i})\right|>\epsilon\right\\}}\exp\left(-N^{2}\beta\left[\mathcal{E}(\mathsf{emp}_{N}-\mu_{\theta})-\frac{1}{N}g(0)\right]\right)\\\
&\leq\frac{1}{K_{N,\beta}}\exp\left(-N^{2}\beta\left[\Phi_{y_{1},...y_{k}}(\epsilon)-\frac{1}{N}g(0)\right]\right),\end{split}$
(4.2)

where

$\Phi_{y_{1},...y_{k}}(\epsilon):=\inf_{\left\\{\mu\in\mathcal{M}(\mathbf{R}^{d}):\left|\sum_{i=1}^{k}h^{\mu}(y_{i})\right|>\epsilon\right\\}}\mathcal{E}(\mu).$
(4.3)

We will also use the obvious variants of $\Phi$ with different values of $k$.

Step 2: Solving the variational problem.

We claim that for any $y_{1},...y_{k}\in\mathbf{R}^{d}$,

$\Phi_{y_{1},...y_{k}}(\epsilon)\geq\frac{\Lambda\epsilon^{2}}{k^{2}},$ (4.4)

for some $\Lambda>0$.

Substep 2.1: We start by proving that

$\Lambda:=\Phi_{0}(1)>0.$ (4.5)

To prove this claim, note that for any $\mu\in\mathcal{M}(\mathbf{R}^{d})$,

$\begin{split}h^{\mu}(0)=\int_{\mathbf{R}^{d}}\widehat{h^{\mu}}(\xi)\,\mathrm{d}\xi=\int_{\mathbf{R}^{d}}\widehat{\mu}(\xi)\widehat{g}(\xi)\,\mathrm{d}\xi.\end{split}$
(4.6)

On the other hand,

$\mathcal{E}(\mu)=\int_{\mathbf{R}^{d}}\left|\widehat{\mu}(\xi)\right|^{2}\widehat{g}(\xi)\,\mathrm{d}\xi.$
(4.7)

By Cauchy-Schwartz, and since $\widehat{g}$ is positive a.e., we find

$\left(\int_{\mathbf{R}^{d}}\widehat{\mu}(\xi)\widehat{g}(\xi)\,\mathrm{d}\xi\right)^{2}\leq\left(\int_{\mathbf{R}^{d}}\left|\widehat{\mu}(\xi)\right|^{2}\widehat{g}(\xi)\,\mathrm{d}\xi\right)\left(\int_{\mathbf{R}^{d}}\widehat{g}(\xi)\,\mathrm{d}\xi\right),$
(4.8)

and therefore for any $\mu\in\mathcal{M}(\mathbf{R}^{d})$ such that
$h^{\mu}(0)=1$, there holds

$\mathcal{E}(\mu)\geq\left(\int_{\mathbf{R}^{d}}\widehat{g}(\xi)\,\mathrm{d}\xi\right)^{-1},$
(4.9)

which finishes the substep.

Substep 2.2: We now prove the claim stated at the beginning of step 2.

Note that if $\left|\sum_{i=1}^{k}h^{\mu}(y_{i})\right|>\epsilon$ then
$\left|h^{\mu}(y_{i})\right|>\frac{\epsilon}{k}$ for some $i\in\\{1,...k\\}$
by a pigeonhole argument. Clearly, $\Phi_{y_{i}}=\Phi_{0}$ by translation
invariance and $\epsilon\mapsto\Phi_{0}(\epsilon)$ is quadratic. It follows

$\begin{split}\Phi_{y_{1},...y_{k}}(\epsilon)\geq\Phi_{0}\left(\frac{\epsilon}{k}\right)=\frac{\Phi_{0}(1)\epsilon^{2}}{k^{2}}\geq\frac{\Lambda\epsilon^{2}}{k^{2}}.\end{split}$
(4.10)

By plugging equation (4.4) into the last line of equation (4.2). we may
conclude Proposition 2.3. ∎

##  5 Proof of Proposition 2.4

In this section, we prove Proposition 2.4, restated here for convenience.

######  Proposition 5.1 (Asymptotics of the Laplace transform of field
fluctuations).

Assume that $g$ is weakly interacting, and $V$ is admissible. Let
$k\in\mathbf{N}$ and $y_{1},y_{2},...y_{k}\in\mathbf{R}^{d}$. Then

$\mathbb{E}_{N,\beta}\left[\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\rm
emp_{N}}(y_{i})\right)\right)\right]=M_{N}\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(y_{i})\right)\right)+A_{N},$
(5.1)

where

$\begin{split}|\log M_{N}|&\leq
C_{1}N^{\frac{1}{2}}\beta\left(C_{2}+\sum_{i=1}^{k}V(y_{i})\right)^{\frac{1}{2}}\\\
|A_{N}|&\leq\exp\left(-N\beta\left(g(0)+\sum_{i=1}^{k}V(y_{i})\right)\right),\end{split}$
(5.2)

for some constants $C_{1},C_{2}$ depending only on $V,g,d$.

###### Proof.

We will only show a complete proof of the lower bound in equation (5.1). The
proof of the upper bound is analogous.

Step 1: First simplification.

Using the cake-slicing formula, we have that

$\mathbb{E}_{N,\beta}\left[\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\rm
emp_{N}}(y_{i})\right)\right)\right]=\int_{0}^{1}\mathbb{P}_{N,\beta}\left(\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\rm
emp_{N}}(y_{i})\right)\right)\geq x\right)\mathsf{d}x.$ (5.3)

We have used the hypothesis that $g>0$ (item 5 of Definition 2.1) to limit the
domain of integration to the $(0,1)$ interval.

We will split the integral at the point
$\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(y_{i})-T\right)\right)$,
for $T$ to be determined later. The reason for this is to optimize the
application of the concentration bound, Proposition 2.3. Using this splitting
of the integral, we have that

$\displaystyle\mathbb{E}_{N,\beta}\left[\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\rm
emp_{N}}(y_{i})\right)\right)\right]$ (5.4)
$\displaystyle=\int_{0}^{\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(y_{i})-T\right)\right)}\mathbb{P}_{N,\beta}\left(\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\rm
emp_{N}}(y_{i})\right)\right)\geq x\right)\mathsf{d}x$
$\displaystyle\quad+\int_{\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(y_{i})-T\right)\right)}^{1}\mathbb{P}_{N,\beta}\left(\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\rm
emp_{N}}(y_{i})\right)\right)\geq x\right)\mathsf{d}x$
$\displaystyle\leq\exp\left(N\beta
T\right)\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(y_{i})\right)\right)$
$\displaystyle\quad+\int_{\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(y_{i})-T\right)\right)}^{1}\mathbb{P}_{N,\beta}\left(\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\rm
emp_{N}}(y_{i})\right)\right)\geq x\right)\mathsf{d}x.$

The rest of the proof will be about estimating the last term in equation
(5.4).

Step 2: Estimating the remaining integral.

Using Proposition 2.3, we have that

$\displaystyle\int_{\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(y_{i})-T\right)\right)}^{1}\mathbb{P}_{N,\beta}\left(\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\rm
emp_{N}}(y_{i})\right)\right)\geq x\right)\mathsf{d}x$ (5.5)
$\displaystyle=\int_{\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(y_{i})-T\right)\right)}^{1}\mathbb{P}_{N,\beta}\left(\sum_{i=1}^{k}h^{\rm
emp_{N}}(y_{i})\geq\frac{-\log x}{N\beta}\right)\mathsf{d}x$
$\displaystyle=\int_{\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(y_{i})-T\right)\right)}^{1}\exp\left(-N^{2}\beta\left[\frac{\Lambda\left(\frac{\log
x}{N\beta}-\sum_{i=1}^{k}h^{\mu_{\theta}}(y_{i})\right)^{2}}{k^{2}}-\frac{1}{N}g(0)\right]\right)\mathsf{d}x.$

Note that, if we define $T:=\sqrt{\frac{k^{2}}{\Lambda
N}\left(2g(0)+\sum_{i=1}^{k}V(y_{i})\right)}$, then for
$x\in\left({\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(y_{i})-T\right)\right)},1\right)$,
there holds

$\frac{\Lambda\left(\frac{\log
x}{N\beta}-\sum_{i=1}^{k}h^{\mu_{\theta}}(y_{i})\right)^{2}}{k^{2}}-\frac{1}{N}g(0)\geq\frac{1}{N}\left(g(0)+\sum_{i=1}^{k}V(y_{i})\right).$
(5.6)

Since the integration interval is bounded in measure by $1$, we have that

$\displaystyle{\int_{\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(y_{i})-T\right)\right)}^{1}\mathbb{P}_{N,\beta}\left(\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\rm
emp_{N}}(y_{i})\right)\right)\geq x\right)\mathsf{d}x}\ $ (5.7)
$\displaystyle\quad\leq\exp\left(-N\beta\left(g(0)+\sum_{i=1}^{k}V(y_{i})\right)\right).$

This concludes the proof of the lower bound of equation (5.1).

Step 3: Upper bound sketch.

In order to prove the upper bound of equation (5.1), we proceed similarly:

$\displaystyle\mathbb{E}_{N,\beta}\left[\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\rm
emp_{N}}(y_{i})\right)\right)\right]$ (5.8)
$\displaystyle=\int_{0}^{1}\left[1-\mathbb{P}_{N,\beta}\left(\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\rm
emp_{N}}(y_{i})\right)\right)\leq x\right)\right]\mathsf{d}x$
$\displaystyle\geq\int_{0}^{\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(y_{i})+T\right)\right)}\left[1-\mathbb{P}_{N,\beta}\left(\exp\left(-N\beta\left(\sum_{i=1}^{k}h^{\rm
emp_{N}}(y_{i})\right)\right)\leq x\right)\right]\mathsf{d}x.$

The rest of the proof is analogous and hence omitted. ∎

##  6 Proof of Theorem 1

In this section, we prove the main result of this paper, namely Theorem 1. We
will use some background on point processes, which can be reviewed in the
Appendix. The main result that we will use is the following lemma:

######  Lemma 6.1 (Weak convergence and correlation functions).

Let $\mathfrak{P}_{N}$ be a sequence of point processes, and denote their $k-$
point correlation functions by $R_{k}^{N}$. Then $\mathfrak{P}_{N}$ converges
weakly to a point process $\mathfrak{P}$, with $k-$ point correlation
functions denoted by $R_{k}^{N}$ if:

  * 1.

$R_{k}^{N}\to R_{k}$ pointwise almost everywhere as $N\to\infty$.

  * 2.

For any compact set $\Omega\subset\mathbf{R}^{d}$ there holds

$\sup_{N\in\mathbf{N}}\sum_{k=1}^{\infty}\frac{1}{k!}\int_{K}R_{k}^{N}\,\mathrm{d}y_{1}...\,\mathrm{d}y_{k}<\infty.$
(6.1)

Equipped with Lemma 6.1, we will prove Theorem 1, restated here for
convenience.

######  Theorem 1 (Convergence to Poisson point process).

Assume that $g$ is weakly interacting, and $V$ is admissible. Assume also that
$\mathrm{ent}[\mu_{V}]<\infty$. Let $x^{*}\in\mathbf{R}^{d}$ and define the
local point process $\Xi$ by

$\Xi:=\sum_{i=1}^{N}\delta_{N^{\frac{1}{d}}\left(x_{i}-x^{*}\right)}.$ (6.2)

Then, for a.e. $x^{*}$, if $N^{-1}\ll\beta\ll N^{-\frac{1}{2}}$, $\Xi$
converges to a Poisson point process of intensity $\mu_{V}(x^{*})$.

###### Proof.

Step 1: Formula for the correlation functions.

Let $Y_{N}:=(y_{1},...y_{N})\in\mathbf{R}^{d\times k}$,
$x_{i}:=x^{*}+N^{-\frac{1}{d}}(y_{i}-x^{*})$, and $X_{N}=(x_{1},...x_{N})$.
Using Proposition 2.4 and Remark 7.4, the $k-$point correlation function can
be written as

$\begin{split}R^{N}_{k}(Y_{k})&=\frac{N!}{(N-k)!}\frac{1}{Z_{N,\beta}}\int_{\mathbf{R}^{d(N-k)}}\exp\left(-\beta\mathcal{H}_{N}(X_{k},x_{k+1},...x_{N})\right)\,\mathrm{d}y_{k+1}...\,\mathrm{d}y_{N}\\\
&=\mathrm{err}_{N}\frac{1}{Z_{N,\beta}}\int_{\mathbf{R}^{d(N-k)}}\exp\left(-\beta\left(\sum_{1\leq
i<j\leq
N}g(x_{i}-x_{j})+N\sum_{i=1}^{N}V(x_{i})\right)\right)\,\mathrm{d}x_{k+1}...\,\mathrm{d}x_{N}\\\
&=\mathrm{err}_{N}\frac{Z_{N-k,\beta}}{Z_{N,\beta}}\exp\left(-\beta\left(\sum_{1\leq
i<j\leq k}g(x_{i}-x_{j})+N\sum_{i=1}^{k}V(x_{i})\right)\right)\\\
&\quad\times\mathbb{E}_{N-k,\beta}\left[\exp\left(-(N-k)\beta\left(\sum_{i=1}^{k}h^{{\rm
emp}_{N-k}}(x_{i})\right)\right)\right]\\\
&=\mathrm{err}_{N}\frac{Z_{N-k,\beta}}{Z_{N,\beta}}\exp\left(-\beta\left(\sum_{1\leq
i<j\leq k}g(x_{i}-x_{j})+N\sum_{i=1}^{k}V(x_{i})\right)\right)\\\
&\quad\times\left(M_{N-k}\exp\left(-(N-k)\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(x_{i})\right)\right)+A_{N-k}\right),\end{split}$
(6.3)

where $\mathrm{err}_{N}:=\frac{N!}{(N-k)!}N^{-k}$.

Step 2: Estimating the ratio of partition functions.

Integrating equation (6.3), and using that the k-th marginal is a probability
measure, we get

$\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\frac{Z_{N,\beta}}{Z_{N-k,\beta}}$\mbox{}\hfil\quad&\\\
&=\mathrm{err}_{N}\int_{\mathbf{R}^{d\times
k}}\exp\left(-\beta\left(\sum_{1\leq i<j\leq
k}g(x_{i}-x_{j})+N\sum_{i=1}^{k}V(x_{i})\right)\right)\\\
&\quad\times\left(M_{N-k}\exp\left(-(N-k)\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(x_{i})\right)\right)+A_{N-k}\right)\,\mathrm{d}x_{1}...\,\mathrm{d}x_{k}\\\
&=M_{N}^{\prime}\int_{\mathbf{R}^{d\times
k}}\exp\left(-N\beta\left(\sum_{i=1}^{k}V(x_{i})+h^{\mu_{\theta}}(x_{i})\right)\right)\,\mathrm{d}x_{1}...\,\mathrm{d}x_{k}+A_{N}^{\prime}\\\
&=M_{N}^{\prime}L_{\theta}^{k}+A_{N}^{\prime},\end{split}$ (6.4)

where

$\begin{split}|\log M_{N}^{\prime}|&\leq C_{1}^{\prime}N^{\frac{1}{2}}\beta\\\
|A_{N}^{\prime}|&\leq\exp\left(-C^{\prime}N\beta\left(g(0)\right)\right),\end{split}$
(6.5)

for some new constants $C^{\prime},C_{1}^{\prime},C_{2}^{\prime}$ depending
only on $V,g,d$.

Recall that, by Remark 3.7, $L_{\theta}$ is of order
$\exp\left(-N\beta\right)$. More specifically,

$\lim_{N\to\infty}\frac{\log L_{\theta}}{N\beta}=-2c_{\infty}.$ (6.6)

Doing a Taylor expansion of $x\mapsto\frac{1}{x}$, we have that

$\frac{Z_{N-k,\beta}}{Z_{N,\beta}}=M^{\prime}_{N}L_{\theta}^{-k}+A^{\prime}_{N},$
(6.7)

for a new additive error term $A^{\prime}_{N}$ satisfying equation (6.5) with
a different constant $C^{\prime}$.

Step 3: Conclusion.

Subtep 3.1: Pointwise convergence (item 1. of Lemma 6.1).

Plugging in equation (6.7) into equation (6.3), and using equation (3.18) we
obtain

$\begin{split}R^{N}_{k}(Y_{k})&=\left(M^{\prime}_{N}L_{\theta}^{k}+A^{\prime}_{N}\right)\exp\left(-\beta\left(\sum_{1\leq
i<j\leq k}g(x_{i}-x_{j})+N\sum_{i=1}^{k}V(x_{i})\right)\right)\\\
&\quad\times\left(M_{N-k}\exp\left(-(N-k)\beta\left(\sum_{i=1}^{k}h^{\mu_{\theta}}(x_{i})\right)\right)+A_{N-k}\right)\\\
&=M^{\prime\prime}_{N}L_{\theta}^{-k}\exp\left(-N\beta\left(\sum_{i=1}^{k}V(x_{i})+h^{\mu_{\theta}}(x_{i})\right)\right)+A^{\prime\prime}_{N}\\\
&=M^{\prime\prime}_{N}\prod_{i=1}^{k}\mu_{\theta}(x_{i})+A^{\prime\prime}_{N},\end{split}$
(6.8)

where

$\begin{split}|\log M^{\prime\prime}_{N}|&\leq
C_{1}^{\prime\prime}N^{\frac{1}{2}}\beta\left(C_{2}^{\prime\prime}+\sum_{i=1}^{k}V(x_{i})\right)^{\frac{1}{2}}\\\
|A^{\prime\prime}_{N}|&\leq\exp\left(-N\beta
C^{\prime\prime}\left(g(0)+\sum_{i=1}^{k}V(x_{i})\right)\right),\end{split}$
(6.9)

for some new constants
$C^{\prime\prime},C_{1}^{\prime\prime},C_{2}^{\prime\prime}$.

Using Lemma 3.6, we have that $R^{N}_{k}(Y_{k})\to(\mu_{V}(x^{*}))^{k}$ for
a.e. $Y_{k}\in\mathbf{R}^{d\times k}$.

Substep 3.1: Summability condition (item 2. of Lemma 6.1).

Let $K\subset\mathbf{R}^{d}$ be a compact set. Then by Lemma 3.6, there holds

$\begin{split}\sup_{N\in\mathbf{N}}\sum_{k=1}^{\infty}\frac{1}{k!}\int_{K}R^{N}_{k}(Y_{k})\,\mathrm{d}y_{1}...\,\mathrm{d}y_{k}&\leq
C\sum_{k=1}^{\infty}\frac{1}{k!}\left(|K|\mu_{V}(x^{*})\right)^{k}<\infty.\end{split}$
(6.10)

Using Lemma 6.1, we may conclude the proof.

##  7 Appendix: Point processes

In this appendix, we will review the theory on point processes necessary to
prove Theorem 1. This appendix is based on [24].

######  Definition 7.1 (Point process).

A point process $\mathfrak{P}$ is a probability measure on the space of
locally finite point configurations on $\mathbf{R}^{d}$. Alternatively, we may
think of a point process $\mathfrak{P}$ as a probability measure on the space
of Radon measures consisting of a sum of Dirac deltas.

######  Definition 7.2 (Laplace functional).

We associate to a point process $\mathfrak{P}$ its Laplace functional $\psi$,
defined by

$\psi(f):=\mathbb{E}_{\mathfrak{P}}\left[\exp\left(-\int_{\mathbf{R}^{d}}f\,\mathrm{d}X\right)\right],$
(7.1)

for a Borel-measurable $f:\mathbf{R}^{d}\to[0,\infty)$.

######  Definition 7.3 (Correlation function).

Given a point process, we define the $k-$point correlation functions
$\\{R_{k}\\}_{k=1}^{\infty}$ by the condition that

$\psi(f):=1+\sum_{i=1}^{\infty}\frac{1}{k!}\int_{\mathbf{R}^{d}}\prod_{i=1}^{k}\left(\exp(-f(x_{i}))-1\right)R_{k}(x_{1},...x_{k})\,\mathrm{d}x_{1}...\,\mathrm{d}x_{k}.$
(7.2)

######  Remark 7.4 (Correlation function of a joint distribution).

If $\mathfrak{P}$ is supported on point configurations of $N$ points, and if
these $N$ points have a symmetric joint distribution $\mathbb{P}$, then we can
verify that

$\psi(f):=1+\sum_{i=1}^{\infty}{N\choose
k}\int_{\mathbf{R}^{d}}\prod_{i=1}^{k}\left(\exp(-f(x_{i}))-1\right)\mathbb{P}(X_{N})\,\mathrm{d}X_{N},$
(7.3)

which implies that the correlation functions are given by

$R_{k}(x_{1},...x_{k})=\frac{N!}{(N-k)!}\int_{\mathbf{R}^{d\times(N-k)}}\mathbb{P}(x_{1},...x_{k},x_{k+1},...x_{N})\,\mathrm{d}x_{k+1}...\,\mathrm{d}x_{N}$
(7.4)

for any $k\leq N$.

######  Remark 7.5 (Correlation function of a Poisson point process).

If $\mathfrak{P}$ is a Poisson process of intensity $\lambda$, then its
Laplace functional is given by

$\psi(f)=\exp\left(\int_{\mathbf{R}^{d}}\left(\exp(-f(x))-1\right)\lambda(x)\,\mathrm{d}x\right).$
(7.5)

Furthermore, the correlation functions of $\mathfrak{P}$ are given by

$R_{k}(x_{1},...x_{k})=\prod_{i=1}^{k}\lambda(x_{i}).$ (7.6)

######  Definition 7.6 (Weak convergence).

Let $\mathfrak{P}_{N}$ be a sequence of point processes, and denote their
Laplace functional by $\psi_{N}$. Then $\mathfrak{P}_{N}$ converges weakly to
a point process $\mathfrak{P}$ with Laplace functional $\psi$ if for any
continuous and compactly supported $f:\mathbf{R}^{d}\to\mathbf{R}^{+}$ there
holds

$\lim_{N\to\infty}\psi_{N}(f)=\psi(f).$ (7.7)

######  Lemma 7.7 (Weak convergence and correlation functions).

Let $\mathfrak{P}_{N}$ be a sequence of point processes, and denote their $k-$
point correlation functions by $R_{k}^{N}$. Then $\mathfrak{P}_{N}$ converges
weakly to a point process $\mathfrak{P}$, with $k-$ point correlation
functions denoted by $R_{k}^{N}$ if:

  * 1.

$R_{k}^{N}\to R_{k}$ pointwise almost everywhere as $N\to\infty$.

  * 2.

For any compact set $\Omega\subset\mathbf{R}^{d}$ there holds

$\sup_{N\in\mathbf{N}}\sum_{k=1}^{\infty}\frac{1}{k!}\int_{K}R_{k}^{N}\,\mathrm{d}y_{1}...\,\mathrm{d}y_{k}<\infty.$
(7.8)

∎

##  8 Acknowledgements

DPG acknowledges the support of the ERC Starting Grant “Bridging Scales in
Random Materials” ERC StG RandSCALES 948819. ET was supported by NSF grant
DMS-2303318.

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David Padilla-Garza
Institute of Science and Technology Austria.
Email<EMAIL_ADDRESS>

Luke Peilen
Department of Mathematics, Temple University.
Email<EMAIL_ADDRESS>

Eric Thoma
Department of Mathematics, Stanford University.
Email<EMAIL_ADDRESS>