# Aspects of Coulomb gases

Djalil Chafaï CEREMADE, Université Paris-Dauphine, Université PSL, CNRS, 75016
Paris, France http://djalil.chafai.net/

(Date: Summer 2021, compiled .)

###### Abstract.

Coulomb gases are special probability distributions, related to potential
theory, that appear at many places in pure and applied mathematics and
physics. In these short expository notes, we focus on some models, ideas, and
structures. We present briefly selected mathematical aspects, mostly related
to exact solvability and first and second order global asymptotics. A
particular attention is devoted to two-dimensional exactly solvable models of
random matrix theory such as the Ginibre model. Thematically, these notes lie
between probability theory, mathematical analysis, and statistical physics,
and aim to be very accessible. They form a contribution to a volume of the
_Panoramas et Synthèses_ series around the workshop _États de la recherche en
mécanique statistique_ , organized by Société Mathématique de France, held at
Institut Henri Poincaré, Paris, in the fall of 2018
(https://statmech2018.sciencesconf.org/).

###### Key words and phrases:

Asymptotic analysis; Exactly solvable model; High dimensional phenomenon;
Mean-field particle system; Random matrix; Spectral analysis; Determinantal
point process; Coulomb gas; Potential theory; Wigner jellium; Langevin
dynamics; Large deviations principle; Boltzmann – Gibbs measure.

###### 2000 Mathematics Subject Classification:

Primary: 82B21; Secondary: 82D05; 81V45; 60F05.

###### Contents

  1. 1 Coulomb electrostatics and equilibrium measures
  2. 2 Coulomb gases
    1. 2.1 One-dimensional log-gases as Coulomb gases
    2. 2.2 Beta-Ginibre gas
    3. 2.3 From multivariate statistics to atomic physics
    4. 2.4 The Wigner jellium and electrons in metals
    5. 2.5 Random polynomials
  3. 3 First order global asymptotics and large deviations
    1. 3.1 The large deviations principle
    2. 3.2 First order global asymptotics
    3. 3.3 Weakly confining versus strongly confining potential
    4. 3.4 Concentration of measure
  4. 4 Edge behavior
  5. 5 Global fluctuations and Gaussian free field
  6. 6 Aspects of general exact solvability
  7. 7 Exactly solvable two-dimensional gases from random matrix theory
    1. 7.1 Ginibre model
    2. 7.2 More determinantal models
  8. 8 Comments and open problems

There are several introductory texts around Coulomb gases. We refer for
instance to [ER05, DG09, Dei99, AGZ10, ABDF11, For10] for the relation to
random matrices, to [Ser15] for the relation to analysis and Ginzburg – Landau
vortices, to [Bou15, GZ19b] and references therein for a relation to geometry,
to [But17] and references therein for a relation to random polynomials, to
[Rou15] for a relation to Fock – Hartree quantum theory and Bose – Einstein
condensates, to [Ser18b] and [Lew21] for an overview from a mathematical
analysis/physics perspective, and to [LACTMS19] for a statistical physics
point of view.

## Notation

The Euclidean norm of $x=(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}$ is
$|x|=\sqrt{x_{1}^{2}+\cdots+x_{d}^{2}}$. It is the modulus if $d=2$ with the
identification $\mathbb{C}=\mathbb{R}^{2}$. We set
$\mathrm{i}=(0,1)\in\mathbb{C}$. The real and imaginary parts of
$z\in\mathbb{C}$ are denoted $\Re z$ and $\Im z$. The Lebesgue measure is
denoted $\mathrm{d}x$. Let $(E,\tau)$ be a topological space with Borel
$\sigma$-field $\mathcal{B}(E)$. We denote by $\mathcal{C}_{b}(E,\mathbb{R})$
the set of bounded continuous functions $E\to\mathbb{R}$, and by
$\mathcal{M}_{1}(E)$ the set of probability measures on $(E,\mathcal{B}(E))$.
If $\mu_{1},\mu_{2},\ldots,\mu$ are in $\mathcal{M}_{1}(\mathbb{R}^{d})$, then
$\lim_{n\to\infty}\mu_{n}=\mu$ weakly, denoted

$\mu_{n}\underset{n\to\infty}{\overset{\mathcal{C}_{b}}{\longrightarrow}}\mu,\text{
when for all }f\in\mathcal{C}_{b}(E,\mathbb{R})\text{ we have
}\lim_{n\to\infty}\int f\mathrm{d}\mu_{n}=\int f\mathrm{d}\mu.$

This defines a sequential topology on $\mathcal{M}_{1}(E)$, giving a Borel
$\sigma$-field $\mathcal{B}(\mathcal{M}_{1}(E))$. A random probability measure
on $E$ is a random variable taking values in $\mathcal{M}_{1}(E)$. By
$X\sim\mu$ we mean that the random variable $X$ has law $\mu$. We denote by
$\overset{\mathrm{d}}{=}$ and $\overset{\mathrm{d}}{\longrightarrow}$ the
equality and the convergence in law respectively.

##  1\. Coulomb electrostatics and equilibrium measures

The Coulomb kernel is identical to the Newton kernel. Mathematically,
potential theory deals with the analysis of the Laplacian and its Green
function and the behavior of harmonic functions. In some sense, it emerges
naturally from the gravitation theory of Johannes Kepler and Isaac Newton, as
well as from the modeling of electrostatics, namely the study of the
distribution of static electric charges on conductors and their interactions.
From this last point of view, it takes its historical roots in the works of
Charles-Augustin de Coulomb, Joseph-Louis Lagrange, and Carl Friedrich Gauss.
Some mathematical parts of potential theory were developed later on by – among
others – Johann Peter Gustav Lejeune Dirichlet, Victor Gustave Robin, Henri
Poincaré, David Hilbert, Charles-Jean de La Vallée Poussin [de 37], Marcel
Brelot [Bre67], Otto Frostman [Fro35], Oliver Dimon Kellogg [Kel67], Gustave
Choquet [Cho54], and their followers. Deep links with Markov processes and
probability theory were explored in particular by Joseph Leo Doob in [Doo01],
Gilbert Hunt, Claude Delacherie and Paul-André Meyer [DM78], and their
followers. Nowadays the basic objects and structures of potential theory
appear at many places in mathematics and physics, providing in general a very
useful electrostatic modeling or interpretation.

_Cet aspect probabiliste, que Brelot regrettait tant d’être arrivé trop tard
pour maîtriser, est sans doute le plus bel exemple d’interaction féconde entre
deux théories : La théorie du potentiel, née dans le ciel (Kepler 1618, Newton
1665) et la théorie des probabilités née d’un coup de dés (Pascal 1654), donc
presque simultanément, devaient après trois siècles et des petits pas l’une
vers l’autre (Wiener 1923, puis P. Levy, Doob), prendre avec G. Hunt (1957)
pleinement conscience que leurs parties les plus vivaces ne sont que deux
faces complémentaires d’un même bel objet, et qu’on ne peut bien comprendre
l’une sans connaître l’autre (le traité Dellacherie-P.A. Meyer veut en donner
la preuve)._

Gustave Choquet, _La vie et l’oeuvre de Marcel Brelot (1903 – 1987)_ [Cho90].

$\cdots$ | $\cdots$
---|---
1660 | Newton, …
1770 | Coulomb, …
1800 | Gauss, …
1870 | Boltzmann, Gibbs, …
1900 | Thomson, de la Vallée Poussin, …
1920 | Fock, Hartree, …
1930 | Wigner, Wishart, …
1940 | Doob, Onsager, …
1950 | Choquet, Hunt, Wigner, …
1960 | Dyson, Ginibre, Mehta, Selberg, …
1970 | Kosterlitz, Landkof, Pastur, Thouless, …
1980 | Deift, Laughlin, Lebowitz, Saff, Voiculescu, …
1990 | Ben Arous, Edelman, Guionnet, Johansson, …
2000 | Forrester, Erdős, Lewin, Serfaty, Yau, …
$\cdots$ | $\cdots$
Table 1.1. The arrow of time and some of the main actors mentioned in the text
or in the references. As mentioned in [Cha15], the _Stigler law of eponymy_
states that “ _No scientific discovery is named after its original
discoverer._ ”, attributed by Stephen Stigler to Robert K. Merton. This is
also known as the _Arnold principle_ by some people.

Let $d\geq 1$. The _Coulomb kernel_ $g$ in $\mathbb{R}^{d}$ is given by
$g(0)=+\infty$, and, for all $x\in\mathbb{R}^{d}\setminus\\{0\\}$,

$g(x)=\begin{cases}\displaystyle\log\frac{1}{|x|}&\text{if
$d=2$,}\\\\[15.00002pt] \displaystyle\frac{1}{(d-2)|x|^{d-2}}&\text{if
not}.\end{cases}$ (1.1)

We say that $(x,y)\mapsto G(x,y)=g(x-y)$ is the Green function of the Laplace
operator $\Delta=\partial^{2}_{1}+\cdots+\partial^{2}_{d}$, and $g$ is the
fundamental solution of the Poisson equation, see for instance [LL01, Theorem
6.20]. Indeed, denoting $\delta_{0}$ the Dirac mass at the origin, we have, in
the sense of Schwartz distributions,

$-\Delta g=c_{d}\delta_{0}\quad\text{and}\quad
c_{d}=d\omega_{d}=\frac{2\pi^{d/2}}{\Gamma(d/2)}$ (1.2)

where $\omega_{d}$ is the volume of the unit ball (its surface is
$d\omega_{d}$). The case $d=3$ is physical for electrostatics modeling in the
ambient space. The case $d=2$ also appears at many places in mathematical
physics. The case $d=1$ serves historically as a toy model, less singular but
exactly solvable.

For simplicity, we suppose from now on that $d\geq 2$.

$|x|$$g(x)$$1$$0$$1$$-1$$d=3$$d=2$$d=1$ Figure 1.1. Coulomb kernel in
dimension $1$ (solid line) $2$ (dotted line) and $3$ (dashed line).

For all $\mu\in\mathcal{M}_{1}(\mathbb{R}^{d})$ such that
$\log(1+\left|\cdot\right|)\mathbf{1}_{d=2}\in\mathrm{L}^{1}(\mu)$, the
_Coulomb energy_ of $\mu$ is

$\mathcal{E}(\mu)=\frac{1}{2}\iint
g(x-y)\mathrm{d}\mu(x)\mathrm{d}\mu(y)\in(-\infty,+\infty].$ (1.3)

Note that if $d=2$ then $\mathcal{E}(\mu)=+\infty$ if $\mu$ has a Dirac mass.
If $\mu$ models the distribution of unit charges (say electrons) in
$\mathbb{R}^{d}$ then $\mathcal{E}(\mu)$ is the electrostatic self-interaction
energy of the configuration $\mu$.

We say that a Borel set $B\in\mathcal{B}(\mathbb{R}^{d})$ is of _positive
capacity_ when $\mathrm{supp}(\mu)\subset B$ and $\mathcal{E}(\mu)<\infty$ for
some $\mu\in\mathcal{M}_{1}(\mathbb{R}^{d})$, and is of _zero capacity_ when
it does not carry a probability measure $\mu$ with $\mathcal{E}(\mu)<\infty$.

For all $\mu\in\mathcal{M}_{1}(\mathbb{R}^{d})$ with
$\log(1+\left|\cdot\right|)\mathbf{1}_{d=2}\in\mathrm{L}^{1}(\mu)$, the
_Coulomb potential_ of $\mu$ at $x\in\mathbb{R}^{d}$ is defined by

$U_{\mu}(x)=\int g(x-y)\mathrm{d}\mu(y)=(g*\mu)(x).$

We have $U_{\mu}(x)\in(-\infty,+\infty]$, and $U_{\mu}(x)=+\infty$ if $\mu$
has a Dirac mass at point $x$. We also have

$\mathcal{E}(\mu)=\frac{1}{2}\int U_{\mu}(x)\mathrm{d}\mu(x).$ (1.4)

Since $g\in\mathrm{L}^{1}_{\mathrm{loc}}(\mathbb{R}^{2},\mathrm{d}x)$, the
Fubini – Tonelli theorem gives
$U_{\mu}\in\mathrm{L}^{1}_{\mathrm{loc}}(\mathbb{R}^{2},\mathrm{d}x)$, hence
$U_{\mu}<+\infty$ almost everywhere. Moreover $U_{\mu}=g*\mu$, and, in the
sense of Schwartz distributions, we get from (1.2) that

$\Delta U_{\mu}=-c_{d}\mu.$ (1.5)

In particular, this gives the formula

$\mathcal{E}(\mu)=\frac{1}{2}\int U_{\mu}\mathrm{d}\mu=-\frac{1}{2c_{d}}\int
U_{\mu}\Delta U_{\mu}\mathrm{d}x.$ (1.6)

When $d\geq 3$, the functional $\mathcal{E}$ does not take negative values on
probability measures because $g\geq 0$. However, when $d=2$, the functional
$\mathcal{E}$ may take negative values on compactly supported probability
measures, due to the change of sign of $g$ when $d=2$ inside and outside the
unit ball. For instance if $\mu_{r}$ is the uniform law on the circle
$\\{x\in\mathbb{C}:|x|=r\\}$ of radius $r>0$ then, for all
$x\in\mathbb{C}^{2}$,

$U_{\mu_{r}}(x)=-\log(r)\mathbf{1}_{|x|\leq
r}-\log|x|\mathbf{1}_{|x|>r},\quad\text{and}\quad\mathcal{E}(\mu_{r})=-\frac{\log(r)}{2},$
(1.7)

which is negative if $r>1$. See for instance [ST97, (0.5.5) and (I.1.6)] for
these computations. Similary, if $\mu_{R}$ is the uniform law on the disc
$\\{x\in\mathbb{C}:|x|\leq R\\}$ of radius $R>0$ then we find that for all
$x\in\mathbb{C}^{2}$,

$U_{\mu_{R}}(x)=-\frac{1}{2}\Big{(}\frac{|x|^{2}}{R^{2}}-1+2\log
R\Bigr{)}\mathbf{1}_{|x|\leq
R}-\log|x|\mathbf{1}_{|x|>R},\quad\text{and}\quad\mathcal{E}_{\mu_{R}}=\frac{1}{4}-\log(R).$
(1.8)

The functionals $U$ and $\mathcal{E}$ extend to signed measures. If
$\eta=\mu-\nu$ where $\mu$ and $\nu$ are two compactly supported probability
measures on $\mathbb{R}^{d}$, then $U_{\eta}$ vanishes at infinity and an
integration by parts gives

$\mathcal{E}(\eta)=\frac{1}{2}\int
U_{\eta}\mathrm{d}\eta=-\frac{1}{2c_{d}}\int U_{\eta}\Delta
U_{\eta}\mathrm{d}x=\frac{1}{2c_{d}}\int|\nabla U_{\eta}|^{2}\mathrm{d}x,$
(1.9)

see [Ser15]. This shows that $\mathcal{E}$ does not take negative values on
signed measures with total mass zero.

The right hand side of (1.9) is the “carré du champ” in potential theory
[Rot76, Hir78] while $-\nabla U_{\mu}$ is the electric field – “champ
électrique” in French – generated by the configuration of charges $\mu$.

Let us introduce now $V:\mathbb{R}^{d}\to(-\infty,+\infty]$ such that (we say
then that $V$ is an _admissible potential_):

  * •

the function $V$ is lower semi-continuous;

  * •

the set $\\{x\in\mathbb{R}^{d}:V(x)<+\infty\\}$ has positive capacity;

  * •

the function $V$ is not beaten by the Coulomb kernel at infinity, namely

$\varliminf_{|x|\to\infty}\left(V(x)-\log\left|x\right|\mathbf{1}_{d=2}\right)>-\infty.$
(1.10)

The _electrostatic energy_ with _external potential_ $V$ is defined from
$\mathcal{M}_{1}(\mathbb{R}^{d})$ to $(-\infty,+\infty]$ by

$\mathcal{E}_{V}(\mu)=\frac{1}{2}\iint\left(g(x-y)+V(x)+V(y)\right)\mu(\mathrm{d}x)\mu(\mathrm{d}y).$
(1.11)

This makes sense since the function under the double integral is bounded below
on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ thanks to (1.10). Finally, for all
$\mu\in\mathcal{M}_{1}(\mathbb{R}^{d})$, if both
$\log(1+\left|\cdot\right|)\mathbf{1}_{d=2}$ and $V$ are in
$\mathrm{L}^{1}(\mu)$, then

$\mathcal{E}_{V}(\mu)=\mathcal{E}(\mu)+\int V(x)\mathrm{d}\mu(x).$ (1.12)

The external potential plays typically the role of a confinement.

The convexity of the quadratic form $\mathcal{E}_{V}$ is related to a Bochner
positivity of the kernel $g$, see [Lan72, HP00, CGZ14, BHS19]. Indeed for all
$\lambda\in(0,1)$ and $\mu,\nu\in\mathcal{M}_{1}(\mathbb{R}^{d})$ with
$\mathcal{E}(\mu)<+\infty$ and $\mathcal{E}(\nu)<+\infty$ and
$V\in\mathrm{L}^{1}(\mu)\cap\mathrm{L}^{1}(\nu)$,

$\frac{\lambda\mathcal{E}_{V}(\mu)+(1-\lambda)\mathcal{E}_{V}(\nu)-\mathcal{E}_{V}(\lambda\mu+(1-\lambda)\nu)}{\lambda(1-\lambda)}=\mathcal{E}(\mu-\nu)=\frac{1}{2c_{d}}\int|\nabla
U_{\mu-\nu}|^{2}\mathrm{d}x\geq 0.$

We are now ready for the general concept of equilibrium measure and its
properties. The following couple of theorems is a classic in potential theory.
For a proof, we refer for instance to the books [Lan72, Hel14, ST97, Ser15]
and to the articles [BAG97, CGZ14, Ser18b].

######  Theorem 1.1 (Equilibrium measure).

The following properties hold true:

  1. (1)

$\mathcal{E}_{V}$ is strictly convex on its domain, is lower semi-continuous,
with compact level sets;

  2. (2)

$\inf_{\mathcal{M}_{1}(\mathbb{R}^{d})}\mathcal{E}_{V}<+\infty$;

  3. (3)

there exists a unique $\mu_{V}\in\mathcal{M}_{1}(\mathbb{R}^{d})$, called the
_equilibrium measure_ , such that

$\mathcal{E}_{V}(\mu_{V})=\inf_{\mu\in\mathcal{M}_{1}(\mathbb{R}^{d})}\mathcal{E}_{V}(\mu)\quad\text{in
other
words}\quad\mu_{V}=\arg\min_{\mathcal{M}_{1}(\mathbb{R}^{d})}\mathcal{E}_{V}.$

Some examples of equilibrium measures are gathered in Table 1.2.

Dimension $d$ | Potential $V$ | Equilibrium measure $\mu_{V}$
---|---|---
$\geq 1$ | $\infty\mathbf{1}_{\left|\cdot\right|>r}$ | Uniform on sphere $\\{x\in\mathbb{R}^{d}:|x|=r\\}$
$\geq 1$ | finite and $\mathcal{C}^{2}$ | With density $\frac{\Delta V}{c_{d}}$ on the interior of its support
|  | (absolutely continuous part of $\mu_{*}$)
$\geq 1$ | $\frac{1}{2}\left|\cdot\right|^{2}$ | Uniform on unit ball with density $\frac{\mathbf{1}_{\left|\cdot\right|\leq 1}}{\omega_{d}}$
(Ginibre) $2$ | $\frac{1}{2}\left|\cdot\right|^{2}$ | Uniform on unit disc with density $\frac{\mathbf{1}_{\left|\cdot\right|\leq 1}}{\pi}$
(Spherical) $2$ | $\frac{1}{2}\log(1+\left|\cdot\right|^{2})$ | Heavy-tailed with density $\frac{1}{\pi(1+\left|\cdot\right|^{2})^{2}}$
(CUE) $2$ | $\infty\mathbf{1}_{([a,b]\times\\{0\\})^{c}}$ | Arcsine on $[a,b]\times\\{0\\}$, density $s\mapsto\frac{\mathbf{1}_{s\in[a,b]}}{\pi\sqrt{(s-a)(b-s)}}$
(GUE) $2$ | $\frac{\left|\cdot\right|^{2}}{2}\mathbf{1}_{\mathbb{R}\times\\{0\\}}+\infty\mathbf{1}_{(\mathbb{R}\times\\{0\\})^{c}}$ | Semicircle on $[-2,2]\times\\{0\\}$, density $s\mapsto\frac{\sqrt{4-s^{2}}}{2\pi}\mathbf{1}_{s\in[-2,2]}$
Table 1.2. Basic examples of equilibrium measures. Some other examples are
given in Section 7.2. The last four examples appear as limiting spectral
distributions of random matrices. The last two examples are singular in the
sense that the potential is infinite outside the real line. They appear as
one-dimensional log-gases from random matrices. In this case, the equilibrium
measure cannot be deduced as a specialization of the second example, and its
computation is a bit more subtle, see for instance [ST97].

######  Theorem 1.2 (Properties of the equilibrium measure).

The following properties hold true:

  1. (1)

the equilibrium measure $\mu$ is compactly supported if

$\lim_{|x|\to\infty}\left(V(x)-\log\left|x\right|\mathbf{1}_{d=2}\right)=+\infty;$
(1.13)

  2. (2)

the equilibrium measure $\mu_{V}$ has finite Coulomb energy
$\mathcal{E}(\mu_{V})\in\mathbb{R}$;

  3. (3)

we have $\mathrm{supp}(\mu_{V})\subset\\{x\in\mathbb{R}^{d}:V(x)\leq R\\}$ for
some constant $R<\infty$;

  4. (4)

the following Euler – Lagrange equations hold:

     * •

$U_{\mu_{V}}(x)+V(x)\leq c_{V}$ for all $x\in\mathrm{supp}(\mu_{V})$,

     * •

$U_{\mu_{V}}(x)+V(x)\geq c_{V}$ for all $x\in\mathbb{R}^{d}$ except on a set
of zero capacity,

where $c_{V}$ is a quantity called the _modified Robin constant_ defined by

$c_{V}=\mathcal{E}(\mu_{V})-\int V\mathrm{d}\mu_{V}.$

In particular, for all $x\in\mathrm{supp}(\mu_{V})$ except on a set of zero
capacity, we have

$U_{\mu_{V}}(x)+V(x)=c.$

In particular, we have the equality in the sense of distributions

$\mu_{V}=\frac{\Delta V}{c_{d}},$

and the interior of $\mathrm{supp}(\mu_{V})$ does not intersect $\\{\Delta
V<0\\}$.

######  Remark 1.3 (Logarithmic kernels and Riesz kernels).

  * •

The logarithmic kernel in dimension $d$ is given by

$-\log\left|x\right|,\quad x\in\mathbb{R}^{d},x\neq 0;$

  * •

The Riesz kernel $k_{s}$ in $\mathbb{R}^{d}$ with parameter $s>0$ is given by

$\frac{1}{s|x|^{s}},\quad x\in\mathbb{R}^{d},x\neq 0.$

  * •

The Coulomb kernel in dimension $d\neq 2$ matches the Riesz kernel with
$s=d-2$;

  * •

The logarithmic kernel for all $d\geq 1$ can be seen as the Riesz kernel with
$s=0$. Indeed, it suffices to remove the singularity in the sense that for all
$x\in\mathbb{R}^{d}$ with $x\neq 0$,

$\lim_{s\to 0}\frac{|x|^{-s}-1}{s}=\partial_{s=0}|x|^{-s}=-\log|x|.$

In particular the Coulomb kernel in dimension $d=2$ is the Riesz kernel with
$s\to 0$.

  * •

For all $\alpha\in(0,d)$, the Riesz kernel with $s=d-\alpha$ is the
fundamental solution of the fractional Laplace operator
$\Delta_{\alpha}=\Delta^{\frac{\alpha}{2}}$, a Fourier multiplier, non-local
operator if $\alpha\neq 2$, see [CGZ14, RS16].

We refer for instance to [BHS19] for more analytic properties of these kernels
and various applications.

##  2\. Coulomb gases

Let $d\geq 2$, $n\geq 1$, $\beta>0$, and let $g$ and $V$ as before. Suppose
moreover that $V$ is such that

$\int_{\mathbb{R}^{d}}\mathrm{e}^{-n\beta(V(x)-\log(1+|x|)\mathbf{1}_{d=2})}\mathrm{d}x<\infty.$
(2.1)

By using the fact that $g\geq 0$ when $d\geq 3$ and $|x-y|\leq(1+|x|)(1+|y|)$
when $d=2$, we get then

$Z_{n}=\int_{(\mathbb{R}^{d})^{n}}\mathrm{e}^{-\beta
E_{n}(x_{1},\ldots,x_{n})}\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}<\infty$

where

$E_{n}(x_{1},\ldots,x_{n})=n\sum_{i=1}^{n}V(x_{i})+\frac{1}{2}\sum_{i\neq
j}g(x_{i}-x_{j}).$

The _Coulomb gas_ $P_{n}$ is the Boltzmann – Gibbs probability measure on
$(\mathbb{R}^{d})^{n}$ given by

$\mathrm{d}P_{n}(x_{1},\ldots,x_{n})=\frac{\mathrm{e}^{-\beta
E_{n}(x_{1},\ldots,x_{n})}}{Z_{n}}\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}.$ (2.2)

It models a “gas of electrons” in $\mathbb{R}^{d}$ of charge $1/n$, at
positions $x_{1},\ldots,x_{n}$, inverse temperature $\beta n^{2}$, energy
$(1/n^{2})E_{n}(x_{1},\ldots,x_{n})$, subject to Coulomb pair interaction and
external field of potential $V$, namely

$\beta E_{n}(x_{1},\ldots,x_{n})=\beta
n^{2}\Bigr{(}\frac{1}{n}\sum_{i=1}^{n}V(x_{i})+\frac{1}{n^{2}}\sum_{i<j}g(x_{i}-x_{j})\Bigr{)}.$
(2.3)

Beware that we should interpret $P_{n}$ as a way to model a random static
configuration of charged particles. We deal here with electrostatics rather
than with electrodynamics. The charged particles do not move and there is no
magnetic field. We have only an electric field.

In view of Remark 1.3, we could also define log-gases and Riesz gases. We do
not follow this idea in these notes, for simplicity and because the Coulomb
case is by far the most important in applications.

###  2.1. One-dimensional log-gases as Coulomb gases

The formula (2.2) makes sense provided that $Z_{n}>0$. Actually the
integration in (2.1) should be interpreted as with respect to the trace of the
Lebesgue measure or Hausdorff measure on
$\\{V<+\infty\\}\subset\mathbb{R}^{d}$. Similarly the integration in (2.2)
should be interpreted as with respect to the trace of the Lebesgue measure or
Hausdorff measure on $\\{V<+\infty\\}^{n}\subset(\mathbb{R}^{d})^{n}$. This
allows to incorporate in the Coulomb gas model (2.2) the one-dimensional log-
gases of random matrix theory, by taking $d=2$ and $V=+\infty$ on $S^{c}$
where $S$ is a one-dimensional subset of $\mathbb{R}^{2}$, typically
$S=\\{x\in\mathbb{R}^{2}:x_{2}=0\\}$ or $S=\\{x\in\mathbb{R}^{2}:|x|=1\\}$.
This includes all beta Hermite/Laguerre/Jacobi ensembles, Gaussian
Unitary/Orthogonal/Simplectic Ensembles, etc. For instance, the famous
Gaussian Unitary Ensemble (GUE) corresponds to take $d=2$ and

$x\in\mathbb{R}^{2}\mapsto
V(x)=\begin{cases}\displaystyle\frac{|x|^{2}}{2}&\text{if $x\in
S=\mathbb{R}\times\\{0\\}$}\\\ +\infty&\text{ if not}\end{cases}.$

For simplicity, we do not study further the one-dimensional log-gases, in
particular the ones coming from random matrix theory. We refer to the books
[Dei99, Meh04, ER05, DG09, For10, AGZ10, PS11]. Actually, most of the models
that we consider in the sequel are fully dimensional in the sense that $V$ is
finite everywhere. The simplest models that we focus on are two-dimensional:
beta-Ginibre gases.

###  2.2. Beta-Ginibre gas

The case $d=2$ is known as the _two-dimensional one-component plasma_. We call
it the _beta gas_ for short. Its density with respect to the Lebesgue measure
on $(\mathbb{R}^{2})^{n}=\mathbb{C}^{n}=\mathbb{R}^{2n}$ is

$(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}\mapsto\frac{\mathrm{e}^{-n\beta\sum_{j=1}^{n}V(z_{j})}}{Z_{n}}\prod_{i<j}|z_{i}-z_{j}|^{\beta}.$
(2.4)

The quadratic potential case $V=\frac{1}{2}\left|\cdot\right|^{2}$ is
sometimes referred to as the _beta-Ginibre gas_. In the special case $\beta=2$
and $V=\frac{1}{2}\left|\cdot\right|^{2}$, that we call the _Ginibre gas_ ,
the density of $P_{n}$ can be written as

$(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}\mapsto
n^{n}\varphi_{n}(\sqrt{n}z_{1},\ldots,\sqrt{n}z_{n})\quad\text{with}\quad\varphi_{n}(z_{1},\ldots,z_{n})=\frac{\mathrm{e}^{-\sum_{j=1}^{n}|z_{j}|^{2}}}{\pi^{n}\prod_{k=1}^{n}k!}\prod_{i<j}|z_{i}-z_{j}|^{2}.$
(2.5)

The beta gas (2.4) with $V=\frac{1}{2}\left|\cdot\right|^{2}$ and
$\beta\in\\{2,4,6,\ldots\\}$ matches the squared modulus of the Laughlin wave
function of the fractional quantum hall effect [Lau87, Gir05]. The Ginibre gas
(2.5) matches the density of the eigenvalues of Gaussian random matrices
[Gin65] (see Section 7.1 for more details), the distribution of vortices in
the Ginzburg – Landau modeling of superconductivity [Ser15], and rotating
trapped fermions in two dimensions [LACTMS19]. The beta gas (2.4) with
$\beta=2$ such as the Ginibre gas (2.5) has a determinantal structure which
provides exact solvability (see Section 7.1 for more details).

###  2.3. From multivariate statistics to atomic physics

Historically, Coulomb gases emerged in mathematical statistics in the years
1920/30 in the study of the spectral decomposition of empirical covariance
matrices of Gaussian samples. We speak nowadays about Laguerre ensembles and
Wishart random matrices. In the 1950s, Eugene P. Wigner discovered by accident
this model when reading a statistics textbook, and this led him to use random
matrices for the modeling of energy levels of heavy nuclei in atomic physics,
see for instance [Cha14, BW11]. His work generated an enormous trend of
activity in statistical physics in the 1960s, with the works of Gaudin, Mehta,
Dyson, Ginibre, Marchenko, Pastur, among others. The term _Coulomb gas_ is
already in the abstract of the first seminal article of Dyson [Dys62b] and of
Ginibre [Gin65]. The terms _Fermi gas_ and _one-component plasma_ are also
used.

###  2.4. The Wigner jellium and electrons in metals

It turns out that Coulomb gases are related to another famous model of
mathematical physics also due to Wigner. Let $S\subset\mathbb{R}^{d}$ be
compact and let $\mu$ be a positive measure on $\mathbb{R}^{d}$ with
$\mu(\mathbb{R}^{d})=\alpha>0$. The Coulomb gas with potential

$V=\begin{cases}-\frac{1}{n}U_{\mu}&\text{on $S$},\\\ +\infty&\text{outside
$S$}\end{cases}$

is known as a _Wigner jellium_ with background $\mu$, and is said to be
_charge neutral_ when $n=\alpha$. The background $\mu$ models a positive
charge $\alpha$ smeared out on $\mathrm{supp}(\mu)$. This model, or more
precisely its thermodynamic limit as $|S|\to+\infty$, was derived by Wigner in
1938 as an approximation of the Hartree – Fock quantum model in order to model
electrons in metals [Wig38], see also the 1904 pre-quantum work by Thomson
[Tho04] on electrons and the structure of atoms. Conversely, a Coulomb gas
with smooth potential $V$ can be seen as a jellium with background $\mu$ of
density $\frac{\Delta V}{c_{d}}$. From this point of view, by looking at
(1.8), the complex Ginibre ensemble can be seen as a Jellium with full space
Lebesgue background. The measure $\mu$ is positive when $V$ is sub-harmonic
(meaning that $\Delta V\geq 0$). If $V$ is not sub-harmonic then $\mu$ is no-
longer positive but we may interpret it as an opposite charge on the subset
$\\{\Delta V<0\\}$. We refer for instance to [CGZJ20a, CGZJ20b] for a
bibliography and a discussion. The term _jellium_ was apparently coined by
Conyers Herring, the smeared charge being viewed as a positive “jelly”, see
[Hug06].

###  2.5. Random polynomials

The Coulomb or log gases emerging from random matrix theory describe the law
of the eigenvalues of a random matrix, the roots of the characteristic
polynomial. This random polynomial has random dependent coefficients. We could
study the distribution of the roots of random polynomials with random
independent coefficients. Actually this question emerged from various fields
of research including algebraic and geometric analysis and number theory, for
instance with the works of Littlewood and Offord in the 1920s, independently
of the works of the statisticians on the spectral analysis of empirical
covariance matrices. The simplest model that we could imagine is a random
polynomial with independent and identically distributed coefficients. This
model is known as Kac polynomials, and the distribution of the roots was
computed in the Gaussian case by John Hammersley in 1956. There are several
other natural models of random polynomials and plenty of works on such models.
The gases emerging from these models are two-dimensional but differ from
Coulomb gases due to the presence in the energy of an additional non-quadratic
term with respect to the empirical measure. For more details, we refer for
instance to [BBL96, BBLu92, KZ13, But17] and references therein. When the
degree tends to infinity, such models give rise to random analytic functions,
see for instance [KZ14, HKPV09, But17] and references therein.

##  3\. First order global asymptotics and large deviations

Let $P_{n}$ be the Coulomb gas as in (2.2). If $x_{1},\ldots,x_{n}$ are
pairwise distinct elements of $\mathbb{R}^{d}$, which holds almost everywhere
with respect to $P_{n}$ in $(\mathbb{R}^{d})^{n}$, we get from (2.3) that

$E_{n}(x_{1},\ldots,x_{n})=n^{2}\mathcal{E}^{\neq}_{V}(\mu_{x_{1},\ldots,x_{n}})$

where

$\mathcal{E}^{\neq}_{V}(\mu)=\int
V\mathrm{d}\mu+\frac{1}{2}\iint\mathbf{1}_{u\neq
v}g(u-v)\mathrm{d}\mu(u)\mathrm{d}\mu(v)\quad\text{and}\quad\mu_{x_{1},\ldots,x_{n}}=\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}.$

The probability measure $P_{n}$ is _exchangeable_ in the sense that it is
invariant by permutation of the $n$ particles. The system is _mean-field_ in
the sense that each particle interacts with all the other particles via their
empirical measure. The density of $P_{n}$ at $(x_{1},\ldots,x_{n})$ is a
function of $\mu_{x_{1},\ldots,x_{n}}$ and rewrites

$\frac{\exp\Bigr{(}-\beta
n^{2}\mathcal{E}^{\neq}_{V}(\mu_{x_{1},\ldots,x_{n}})\Bigr{)}}{Z_{n}}.$ (3.1)

In terms of asymptotic analysis, we expect that
$\mathcal{E}^{\neq}_{V}\approx\mathcal{E}_{V}$ as $n\to\infty$, and the
Laplace method suggests that under $P_{n}$, the empirical measure
$\mu_{x_{1},\ldots,x_{n}}$ concentrates as $n\to\infty$ around the minimizers
of $\mathcal{E}_{V}$. Since there is a unique minimizer known as the
equilibrium measure $\mu_{V}$, we expect that the empirical measure
$\mu_{x_{1},\ldots,x_{n}}$ under $P_{n}$ converges towards $\mu_{V}$ as
$n\to\infty$. More precisely, for all $n$, let us define

$X_{n}=(X_{n,1},\ldots,X_{n,n})\sim
P_{n}\quad\text{and}\quad\mu_{n}=\frac{1}{n}\sum_{k=1}^{n}\delta_{X_{n,k}}.$

###  3.1. The large deviations principle

For all Borel subsets $A\subset\mathcal{M}_{1}(\mathbb{R}^{d})$,
$\mathbb{P}(\mu_{n}\in A)=P_{n}(\mu_{x_{1},\ldots,x_{n}}\in A)$. The following
theorem translates mathematically the intuition above based on the Laplace
principle: $\mathbb{P}(\mu_{n}\in A)\approx_{n\to\infty}\mathrm{e}^{-\beta
n^{2}\inf_{A}(\mathcal{E}_{V}-\mathcal{E}_{V}(\mu_{V}))}$. The difficulty lies
in the singularity of the Coulomb interaction.

######  Theorem 3.1 (Large deviations principle).

We have

$\lim_{n\to\infty}\frac{\log Z_{n}}{\beta n^{2}}=-\mathcal{E}_{V}(\mu_{V}).$

Moreover the sequence ${(\mu_{n})}_{n}$ satisfies to a large deviations
principle of speed $n^{2}$ and good rate function
$\mathcal{E}_{V}-\mathcal{E}_{V}(\mu_{V})$, in other words for all Borel
subset of $A\subset\mathcal{M}_{1}(\mathbb{R}^{d})$, we have

$\mathcal{E}_{V}(\mu_{V})-\inf_{\mathrm{int}(A)}\mathcal{E}_{V}\leq\varliminf_{n\to\infty}\frac{\log\mathbb{P}(\mu_{n}\in
A)}{\beta n^{2}}\leq\varlimsup_{n\to\infty}\frac{\log\mathbb{P}(\mu_{n}\in
A)}{\beta
n^{2}}\leq\mathcal{E}_{V}(\mu_{V})-\inf_{\mathrm{clo}(A)}\mathcal{E}_{V}$

where $\mathrm{int(A)}$ and $\mathrm{clo}(A)$ are the interior and closure of
$A$ respectively.

###### About the proof.

The first proof of such a result dates back to [BAG97] and concerns one-
dimensional log-gases. It is inspired by the work of Voiculescu on a Boltzmann
point of view over free entropy and random matrices. Later contributions
include [PH98, Har12, CGZ14]. The approach developed in [DLR20, Ber18b, Gar19]
is very efficient. ∎

Theorem 3.1 remains valid when $\beta=\beta_{n}$ provided that

$\lim_{n\to\infty}n\beta_{n}=+\infty.$

This can be called the “low temperature regime”. In the “high temperature
regime” $\beta=\beta_{n}$ with

$\lim_{n\to\infty}n\beta_{n}=\kappa\in(0,+\infty),$

then Theorem 3.1 remains valid provided that we replace $\mathcal{E}_{V}$ by
the new functional

$\mathcal{E}+\frac{1}{\kappa}\mathrm{Entropy}(\cdot\mid\nu_{V,\kappa})=\mathcal{E}_{V}+\frac{1}{\kappa}\mathrm{Entropy}(\cdot\mid\mathrm{d}x)+c_{V,\kappa}$

where $\nu_{V,\kappa}$ has density proportional to $\mathrm{e}^{-\kappa V}$,
and where $\mathrm{Entropy}$ is the Kullback – Leibler divergence or relative
entropy. Note that $-\mathrm{Entropy}(\cdot\mid\mathrm{d}x)$ is by definition
the Boltzmann – Shannon entropy. We should also replace $\mu_{V}$ in Theorem
3.1 by the minimizer of this new functional. This is also known as the
_crossover regime_ , interpolating between $\mu_{V}$ and $\nu_{V,\kappa}$.
Formally, if we turn off the interaction by taking $g=0$ and if we take
$\beta_{n}=\kappa/n$ then $P_{n}$ is the product probability measure
$\nu_{V,\kappa}^{\otimes n}$ and the large deviations principle becomes the
classical Sanov theorem associated to the law of large numbers for independent
random variables. The crossover regime is considered for instance in [CLMP92,
BG99, Gar19, AB19, AS19].

###  3.2. First order global asymptotics

The (weak) convergence in $\mathcal{M}_{1}(\mathbb{R}^{d})$ is metrized by the
bounded-Lipschitz distance defined by

$\mathrm{d}_{\mathrm{BL}}(\mu,\nu)=\sup\Bigr{\\{}\int
f\mathrm{d}(\mu-\nu):\left\|f\right\|_{\infty}\leq
1,\left\|f\right\|_{\mathrm{Lip}}\leq 1\Bigr{\\}}$

where the supremum runs over all measurable functions
$f:\mathbb{R}^{d}\to\mathbb{R}$ and where

$\left\|f\right\|_{\infty}=\sup_{x}|f(x)|\quad\text{and}\quad\left\|f\right\|_{\mathrm{Lip}}=\sup_{x\neq
y}\frac{|f(x)-f(y)|}{|x-y|}.$

Now for all $r\geq 0$, by Theorem 3.1 with
$A=A_{r}=\\{\mu\in\mathcal{M}_{1}(\mathbb{R}^{d}):\mathrm{d}_{\mathrm{BL}}(\mu,\mu_{V})\geq
r\\}$, for $n$ large enough,

$\mathrm{e}^{-c_{r}\beta
n^{2}}\leq\mathbb{P}(\mathrm{d}_{\mathrm{BL}}(\mu_{n},\mu_{V})\geq
r)\leq\mathrm{e}^{-C_{r}\beta n^{2}},$ (3.2)

where $c_{r},C_{r}>0$ are constants depending on $A_{r}$ and $\mathcal{E}_{V}$
but not on $n$. In particular, for all $\varepsilon>0$,

$\sum_{n}\mathbb{P}(\mathrm{d}_{\mathrm{BL}}(\mu_{n},\mu_{V})>\varepsilon)<\infty.$

By the Borel – Cantelli lemma, it follows that regardless of the way we choose
a common probability space to define the sequence of random vectors
${(X_{n})}_{n}$, we have, almost surely,

$\lim_{n\to\infty}\mathrm{d}_{\mathrm{BL}}(\mu_{n},\mu_{V})=0.$ (3.3)

This is a sort of law of large numbers for our system of exchangeable
particles. They are not independent due to the Coulomb interaction, and the
information about the interaction remains in $\mu_{V}$. We refer to [Ser15,
Ber18a, Gar19] for the relation to the notion of “Gamma convergence”.

###  3.3. Weakly confining versus strongly confining potential

We could say that $V$ is _weakly confining_ when (1.10) holds, and that $V$ is
_strongly confining_ when (1.13) holds. The integrability condition (2.1) may
hold for weakly confining potentials. An example of a two dimensional Coulomb
gas with a weakly confining potential is given by the Forrester – Krishnapur
spherical ensemble considered in the sequel, for which the equilibrium measure
is not compactly supported and is heavy-tailed.

###  3.4. Concentration of measure

The proof of Theorem 3.1 can be adapted in order to provide quantitative
(meaning non-asymptotic) estimates for deviation probabilities. Namely, for
all $r\geq 0$,

$\mathbb{P}(\mathrm{d}_{\mathrm{BL}}(\mu_{n},\mu_{V})\geq
r)=\frac{1}{Z_{n}}\int_{\mathrm{d}_{\mathrm{BL}}(\mu_{n},\mu_{V})\geq
r}\mathrm{e}^{-\beta
n^{2}\mathcal{E}_{V}^{\neq}(\mu_{x_{1},\ldots,x_{n}})}\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}.$

Now if we could approximate $\mathcal{E}_{V}^{\neq}(\mu_{x_{1},\ldots,x_{n}})$
with $\mathcal{E}_{V}(\mu_{x_{1},\ldots,x_{n}})$ and use an inequality of the
form

$\mathrm{d}_{\mathrm{BL}}(\mu,\mu_{V})\leq
c(\mathcal{E}_{V}(\mu)-\mathcal{E}_{V}(\mu_{V})),$

and use a bound of the form

$\log Z_{n}\geq
n^{2}\mathcal{E}_{V}(\mu_{V})+n(\beta\mathcal{E}(\mu_{V})+c_{V}),$ (3.4)

then we would obtain a concentration of measure inequality of the form

$\mathbb{P}(\mathrm{d}(\mu_{n},\mu_{V})\geq
r)\leq\mathrm{e}^{-cn^{2}r^{2}+o(n^{2})}.$

for all $n$ and all $r\geq r_{n}$ for some threshold $r_{n}$. Actually the
quantity $\mathcal{E}_{V}(\mu_{x_{1},\ldots,x_{n}})$ is infinite due to the
atomic nature of $\mu_{x_{1},\ldots,x_{n}}$ and the method requires then a
regularization procedure. The details are in [CHM18]. The method, inspired by
[MMS14], is related to [RS16]. See also [GZ19a, MS19, Ber19a, PG20] for other
variations on this topic. Moreover we could replace the bounded-Lipschitz
distance by a Kantorovich – Wasserstein distance, provided a growth assumption
on $V$.

Such concentration inequalities around the equilibrium measure provide
typically an upper bound on the speed of the almost sure convergence. More
precisely if $r_{n}$ is such that
$\sum_{n}\mathrm{e}^{-cn^{2}r_{n}^{2}+o(n^{2})}<\infty$, then by the Borel –
Cantelli lemma, we get that almost surely, for $n$ large enough,

$\mathrm{d}_{\mathrm{BL}}(\mu_{n},\mu_{V})\leq r_{n}.$

On the other hand, in the case of one-dimensional log-gases with strongly
convex potential $V$ such as the Gaussian unitary ensemble, another approach
is possible for concentration of measure, related to logarithmic Sobolev
inequalities, see for instance [CL20] and references therein.

##  4\. Edge behavior

We suppose in this section that $V$ is strongly confining, in particular the
equilibrium measure $\mu_{V}$ is compactly supported (Theorem 1.2). The
convergence (3.3) holds in a weak sense, which does not imply the convergence
of the support. The most general result about the convergence of the support
is probably [Ame21], and appears as a refinement of [CHM18, Theorem 1.12].
When $V$ is rotationally invariant, this provides constants $c,r_{*},p>0$ such
that for all $n$ and $r\geq r_{*}$,

$\mathbb{P}\Bigr{(}\max_{1\leq k\leq n}|X_{n,k}|\geq
r\Bigr{)}\leq\mathrm{e}^{-cnr^{p}}.$

The fluctuation at the edge is a difficult subject which is well understood
for one-dimensional log-gases, for which it gives rise to Tracy – Widom laws.
For strongly confined rotationally invariant determinantal two-dimensional
Coulomb gases, it gives rise to Gumbel laws. An explicit analysis of the
Ginibre Coulomb gas is presented in the sequel (Theorem 7.14), see also [CP14,
JQ17, CGZJ20b, Seo20, Ame21, BGZ18, GZ18, BGZNW21, CGZJ20a] for more results
in the same spirit.

##  5\. Global fluctuations and Gaussian free field

Formally, from (1.9) we could write

$\mathcal{E}(\mu)=\frac{1}{2}\langle-c_{d}^{-1}\Delta
U_{\mu},U_{\mu}\rangle+\langle-c_{d}^{-1}\Delta V,U_{\mu}\rangle,$

and thus

$\beta n^{2}\mathcal{E}(\mu)=\frac{1}{2}\langle-\beta c_{d}^{-1}\Delta
U_{n\mu},U_{n\mu}\rangle+\langle-n\beta c_{d}^{-1}\Delta V,U_{n\mu}\rangle.$

In view of the Coulomb gas formula (3.1), this suggests to interpret as
$n\to\infty$ the random function $U_{n\mu_{x_{1},\ldots,x_{n}}}$ under $P_{n}$
as a Gaussian with covariance operator $K=c_{d}(-\beta\Delta)^{-1}$. Actually
such an object is known as a _Gaussian Free Field_ (GFF). Next, again from
(1.9), this suggests to interpret formally as $n\to\infty$ the random measure
$n\mu_{x_{1},\ldots,x_{n}}=-c_{d}^{-1}\Delta
U_{\mu_{x_{1},\ldots,x_{n}}}=AU_{\mu_{x_{1},\ldots,x_{n}}}$ under $P_{n}$ as a
Gaussian random measure with covariance operator
$A^{2}K=(-c_{d}^{-1}\Delta)^{2}(c_{d}(-\beta\Delta)^{-1})=-(\beta
c_{d})^{-1}\Delta$. This argument would involve in principle a change of
variable and a Jacobian, that we do not consider here. This leads naturally to
conjecture that for a smooth enough test function
$f:\mathbb{R}^{d}\to\mathbb{R}$,

$n\Bigr{(}\int f\mathrm{d}\mu_{n}-\mathbb{E}\int
f\mathrm{d}\mu_{n}\Bigr{)}=\sum_{k=1}^{n}f(X_{n,k})-\mathbb{E}(f(X_{n,k}))\underset{n\to\infty}{\overset{\mathrm{d}}{\longrightarrow}}\mathcal{N}\Bigr{(}0,\frac{1}{\beta
c_{d}}\int|\nabla f|^{2}\mathrm{d}x\Bigr{)}.$ (5.1)

This can be seen as the “central limit theorem” statement associated to the
“law of large numbers” statement (3.3). The limiting variance could be
perturbed by edge effects depending on the relative position of the support
and regularity of $f$ and $\mu_{V}$. We could have also an additional bias
correction.

The Coulomb interaction together with the confinement produces a rigidity of
the global configuration and reduces the variance of linear statistics. Indeed
(5.1) comes with an $n$ scaling that differs from the usual $\sqrt{n}$ scaling
for independent random variables (no interaction).

The covariance of the limiting Gaussian in (5.1) is easily guessed from the
Hessian at the minimizer of the rate function in the large deviations
principle of Theorem 3.1. This CLT – LDP link is well known. The GFF is an
example of a log-correlated Gaussian field [DRSV17], a fashionable subject.

A statement similar to (5.1) is proved rigorously in [RV07] for the complex
Ginibre ensemble by using its exact solvability (determinantal structure). See
also [AHM15, AHM11]. Extensions to non-exactly solvable two-dimensional
Coulomb gases are considered in [BBNY19, LS18, Ser20a, LZ20].

For one-dimensional log-gases emerging from random matrix theory, central
limit theorems such as (5.1) were established using the Laplace transform and
“loop equations” in [Joh98]. See also [PS11, BGG17, BLS18, HL21] and
references therein for extensions and generalizations.

##  6\. Aspects of general exact solvability

Theorem 6.1 is taken from [Cha19a] and [CL20] (see also [CFS21]).

######  Theorem 6.1 (Exact distributions for special linear statistics of
general gases).

Let $X=(X_{1},\ldots,X_{n})$ be a random vector of $(\mathbb{R}^{d})^{n}$,
$n,d\geq 1$, with density proportional to

$\mathrm{e}^{-\sum_{i}V(x_{i})}\prod_{i<j}W(x_{i}-x_{j})$

where $V:\mathbb{R}^{d}\to[0,+\infty]$ and $W:\mathbb{R}^{d}\to[0,+\infty]$
are measurable.

  * •

If $V$ and $W$ are homogeneous in the sense that for some $a,b\geq 0$, and for
all $\lambda\geq 0$ and $x\in\mathbb{R}^{d}$,

$V(\lambda x)=\lambda^{a}V(x)\quad\text{and}\quad W(\lambda
x)=\lambda^{b}W(x),$

then

$V(X_{1})+\cdots+V(X_{n})\sim\mathrm{Gamma}\Bigr{(}\frac{nd}{a}+\frac{n(n-1)b}{2a},1\Bigr{)}.$

  * •

If $V=\gamma\left|\cdot\right|^{2}$ for some $\gamma>0$ then

$X_{1}+\cdots+X_{n}\sim\mathcal{N}\Bigr{(}0,\frac{n}{2\gamma}I_{d}\Bigr{)},$

and moreover the orthogonal projection $\pi$ on the subspace
$\\{(z,\ldots,z):z\in\mathbb{R}^{d}\\}$ of $(\mathbb{R}^{d})^{n}$ satisfies
$\pi(X)=\frac{X_{1}+\cdots+X_{n}}{n}(1,\ldots,1)$, and furthermore $\pi(X)$
and $\pi^{\perp}(X)=X-\pi(X)$ are independent.

###### Proof of Theorem 6.1.

Recall that linear change of variable is valid for integrals of measurable
functions. _First formula._ For all $\theta>0$, we have, with the substitution
$x_{i}=\bigr{(}\frac{1}{1+\theta}\bigr{)}^{1/a}y_{i}$,

$\int_{(\mathbb{R}^{d})^{n}}\mathrm{e}^{-\theta\sum_{i}V(x_{i})}\mathrm{e}^{-\sum_{i}V(x_{i})}\prod_{i<j}W(x_{i}-x_{j})\mathrm{d}x=\Bigr{(}\frac{1}{1+\theta}\Bigr{)}^{\frac{nd}{a}+\frac{(n^{2}-n)}{2}\frac{b}{2a}}\int_{(\mathbb{R}^{d})^{n}}\mathrm{e}^{-\sum_{i}V(y_{i})}\prod_{i<j}W(y_{i}-y_{j})\mathrm{d}y.$

We recognize the Laplace transform of
$\mathrm{Gamma}\bigr{(}\frac{nd}{a}+\beta\frac{n(n-1)b}{2a},1\bigr{)}$, namely

$\int_{0}^{\infty}\mathrm{e}^{-\theta x}x^{\alpha-1}\mathrm{e}^{-\lambda
x}\mathrm{d}x=\int_{0}^{\infty}x^{\alpha-1}\mathrm{e}^{-(\lambda+\theta)x}\mathrm{d}x=\Bigr{(}\frac{\lambda}{\lambda+\theta}\Bigr{)}^{\alpha}\frac{\Gamma(\alpha)}{\lambda^{\alpha}},$

therefore
$\sum_{i}V(X_{i})\sim\mathrm{Gamma}\bigr{(}\frac{nd}{a}+\frac{n(n-1)b}{2a},1\bigr{)}$.

_Second formula._ For all $\theta\in\mathbb{R}^{d}$, we have, with the
substitution $y_{i}=x_{i}+\frac{1}{2\gamma}\theta$ (a translation or shift),

$\int_{(\mathbb{R}^{d})^{n}}\mathrm{e}^{-\theta\cdot\sum_{i}x_{i}}\mathrm{e}^{-\sum_{i}V(x_{i})}\prod_{i<j}W(x_{i}-x_{j})\mathrm{d}x=\mathrm{e}^{\frac{n}{2\gamma}\frac{|\theta|^{2}}{2}}\int_{(\mathbb{R}^{d})^{n}}\mathrm{e}^{-\sum_{i}V(y_{i})}\prod_{i<j}W(y_{i}-y_{j})\mathrm{d}y,$

and we recognize the Laplace transform of the Gaussian law
$\mathcal{N}\bigr{(}0,\frac{n}{2\gamma}I_{d}\bigr{)}$. Finally the properties
related to $\pi(X)$ follow from the quadratic nature of $V$ and the shift
invariance of $W$, and correspond to a factorization of the law of $X$,
namely, denoting $\pi^{\perp}(x)=x-\pi(x)$ and using
$|x|^{2}=|\pi(x)|^{2}+|\pi^{\perp}(x)|^{2}$ (Pythagoras theorem) and
$x_{i}-x_{j}=\pi^{\perp}(x)_{i}-\pi^{\perp}(x)_{j}$ (from the definition of
$\pi$), we get

$\mathrm{e}^{-\sum_{i}V(x_{i})}\prod_{i<j}W(x_{i}-x_{j})=\mathrm{e}^{-\gamma|\pi(x)|^{2}}\times\mathrm{e}^{-\gamma|\pi^{\perp}(x)|^{2}}\prod_{i<j}W(\pi^{\perp}(x)_{i}-\pi^{\perp}(x)_{j}).$

This provides the independence of $\pi(X)$ and $\pi^{\perp}(X)$ as well as the
fact that $\pi(X)\sim\mathcal{N}(0,\frac{1}{2\gamma}I_{d})$. ∎

######  Corollary 6.2 (Exact laws for beta-Ginibre gases).

Let us consider $X_{n}=(X_{n,1},\ldots,X_{n,n})\sim P_{n}$ where $P_{n}$ is as
in (2.4) with $\beta>0$ and $V=\frac{1}{2}\left|\cdot\right|^{2}$. In other
words, the density of $P_{n}$ in $\mathbb{C}^{n}$ is given by

$(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}\mapsto\frac{\mathrm{e}^{-n\frac{\beta}{2}(|z_{1}|^{2}+\cdots+|z_{n}|^{2})}}{Z_{n}}\prod_{i<j}|z_{i}-z_{j}|^{\beta}.$

Then

$X_{n,1}+\cdots+X_{n,n}\sim\mathcal{N}\Bigr{(}0,\frac{I_{2}}{\beta}\Bigr{)}\quad\text{and}\quad|X_{n}|^{2}=|X_{n,1}|^{2}+\cdots+|X_{n,n}|^{2}\sim\mathrm{Gamma}\Bigr{(}n+\beta\frac{n(n-1)}{4},\beta\frac{n}{2}\Bigr{)},$

and in particular

$\mathbb{E}(|X_{n,1}+\cdots+X_{n,n}|^{2})=\frac{2}{\beta}\quad\text{and}\quad\mathbb{E}(|X_{n}|^{2})=\mathbb{E}(|X_{n,1}|^{2}+\cdots+|X_{n,n}|^{2})=\frac{2}{\beta}+\frac{n-1}{2}.$

When $\beta=2$ we recover the Ginibre gas (2.5). Beyond this case, and up to
our knowledge, it seems that there is no useful matrix model with independent
entries for which the spectrum follows this $\beta$ gas.

With $\beta=\frac{1}{n}$, we get as $n\to\infty$ that the variance of the
Gauss – Ginibre crossover is $2+\frac{1}{2}=\frac{5}{2}$.

###### Proof of Corollary 6.2.

It suffices to use Theorem 6.1 with $d=2$,
$V=n\frac{\beta}{2}\left|\cdot\right|^{2}$, $W=\left|\cdot\right|^{\beta}$,
for which $a=2$ and $b=\beta$, and the scaling property $\sigma
Z\sim\mathrm{Gamma}\bigr{(}\alpha,\frac{\lambda}{\sigma}\bigr{)}$ when
$Z\sim\mathrm{Gamma}(\alpha,\lambda)$, for any $\sigma>0$.

Note that in the determinantal case $\beta=2$, Theorem 7.13 gives that
$n|X|^{2}\sim\mathrm{Gamma}(1+2+\cdots+n,1)$ since it has the law of a sum of
$n$ independent random variables of law
$\mathrm{Gamma}(1,1),\ldots,\mathrm{Gamma}(n,1)$. ∎

######  Remark 6.3 (Real case).

For all $\beta>0$, $n\geq 2$, let $P_{n}$ be the law on $\mathbb{R}^{n}$ with
density

$(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}\mapsto\frac{\mathrm{e}^{-n\frac{\beta}{4}(x_{1}^{2}+\cdots+x_{n}^{2})}}{Z_{n}}\prod_{i<j}|x_{i}-x_{j}|^{\beta}.$

The normalization $Z_{n}$ can be explicitly computed via a Mehta – Selberg
integral [FW08]. It is a quadratically confined one-dimensional log-gas known
as the real beta Hermite gas. The case $\beta=2$ corresponds to GUE. If
$X_{n}=(X_{n,1},\ldots,X_{n,n})\sim P_{n}$, then the proof of Theorem 6.2
provides

$X_{n,1}+\cdots+X_{n,n}=\mathcal{N}\left(0,\frac{2}{\beta}\right)\quad\text{and}\quad
X_{n,1}^{2}+\cdots+X_{n,n}^{2}\sim\mathrm{Gamma}\left(\frac{n}{2}+\frac{\beta
n(n-1)}{4},\frac{\beta n}{4}\right),$

and in particular,

$\mathbb{E}((X_{n,1}+\cdots+X_{n,n})^{2})=\frac{2}{\beta}\quad\text{and}\quad\mathbb{E}(X_{n,1}^{2}+\cdots+X_{n,n}^{2})=\frac{2}{\beta}+n-1.$

Alternatively, these formulas can also be derived by using the tridiagonal
random matrix model of Dumitriu and Edelman [DE02] valid for all real beta
Hermite gases, see for instance [CL20].

######  Remark 6.4 (Langevin dynamics).

The Boltzmann – Gibbs measure $P_{n}$ defined in (2.2) is the invariant law of
the Kolmogorov diffusion process ${(X_{t})}_{t\geq 0}$ solution of the
stochastic differential equation

$\mathrm{d}X_{t}=\sqrt{2\frac{\alpha}{\beta}}\mathrm{d}B_{t}-\alpha\nabla
E_{n}(X_{t})\mathrm{d}t$ (6.1)

where ${(B_{t})}_{t\geq 0}$ is a standard Brownian motion on
$(\mathbb{R}^{d})^{n}$, and where $\alpha>0$ is an arbitrary parameter which
corresponds to a deterministic time change. The infinitesimal generator of the
associated Markov semi-group is the second order linear differential operator
without constant term

$L=\alpha\Bigr{(}\frac{1}{\beta}\Delta-\nabla E_{n}\cdot\nabla\Bigr{)}.$ (6.2)

The most standard parametrizations are $\alpha=1$, which allows to interpret
$1/\beta$ as the temperature of the Brownian part, and $\alpha=\beta$. See for
instance [Cha15, Roy07, BGL14]. Since $E_{n}$ is a two-body interaction
energy, the operator $L$ can be seen as a mean-field particle approximation of
a McKean – Vlasov dynamics, see for instance [BCF18, Ser20b]. The singularity
of $g$ makes non-obvious the well-posedness or absence of explosion of (6.1),
and we refer to [RS93, AGZ10, EY17] for one-dimensional log-gases, and to
[BCF18] for the (two-dimensional) beta-Ginibre gas. See also [BD20] and [AB19]
for more recent results on such dynamics. Historically (6.1) emerges as the
description of the dynamics of the spectrum of Hermitian Ornstein – Uhlenbeck
processes, and is nowadays called a Dyson process, named after [Dys62a]. We
say that (6.1) is a gradient dynamics because the drift is the gradient of a
function. From the point of view of statistical physics a stochastic
differential equation such as (6.1) is also known as an overdamped Langevin
dynamics, which is a degenerate version of the true (kinetic under-damped)
Langevin dynamics, see for instance [CF19]. Langevin dynamics can be used for
the numerical simulation of $P_{n}$, see for instance [CF19, CFS21] and
references therein. Dynamics such as (6.1) can be used as an interpolation
device between $X_{0}$ and $X_{\infty}\sim P_{n}$, possibly by using
conservation laws related to eigenfunctions. In this spirit, and following
[BCF18, CL20], we could prove Corollary 6.2 by using the fact that
$\sum_{i}x_{i}$ and $\sum_{i}|x_{i}|^{2}$ are essentially eigenfunctions of
(6.2), producing Ornstein – Uhlenbeck and Cox – Ingersoll – Ross processes for
which the targeted Gaussian and Gamma laws are invariant.

##  7\. Exactly solvable two-dimensional gases from random matrix theory

The spectrum of several random matrix models are gases in dimension
$d\in\\{1,2\\}$ with $W=g$. The cases $\beta\in\\{1,2,4\\}$ play often a
special role related to algebra. We refer to [Meh04, ER05, For10] for more
details on the zoology of random matrices. It is natural to ask if there
exists a random matrix model with independent entries for which the spectrum
is distributed according to the beta gas (2.4). Up to our knowledge, the
answer is negative for (2.4) in general but positive for the Ginibre gas
(2.5).

###  7.1. Ginibre model

A (complex) Ginibre random matrix $\mathbf{M}$ is an $n\times n$ complex
matrix such that

$\Bigr{\\{}\Re\mathbf{M}_{i,j},\Im\mathbf{M}_{i,j}:1\leq i,j\leq n\Bigr{\\}}$
(7.1)

are independent and Gaussian random variables of law
$\mathcal{N}(0,\frac{1}{2})$. In other words the complex random variables
$\\{\mathbf{M}_{i,j}:1\leq i,j\leq n\\}$ are independent and Gaussian of law
$\mathcal{N}(0,\frac{1}{2}I_{2})$. Note that
$\mathbb{E}(|\mathbf{M}_{i,j}|^{2})=1$.

Let $(\boldsymbol{\lambda}_{1},\ldots,\boldsymbol{\lambda}_{n})$ be the
eigenvalues of $\mathbf{M}$ seen as an exchangeable random vector of
$\mathbb{C}^{n}$. This means that we randomize the numbering of the
eigenvalues with an independent uniform random permutation of
$\\{1,\ldots,n\\}$. Equivalently, this corresponds to consider the random
multi-set encoding the spectrum, keeping by this way the possible
multiplicities but discarding the numbering of the eigenvalues.

######  Theorem 7.1 (From the Ginibre random matrix to the Ginibre gas).

The exchangeable random vector

$\Bigr{(}\frac{\boldsymbol{\lambda}_{1}}{\sqrt{n}},\ldots,\frac{\boldsymbol{\lambda}_{n}}{\sqrt{n}}\Bigr{)}$

is distributed according to the Ginibre gas (2.5). In other words
$(\boldsymbol{\lambda}_{1},\ldots,\boldsymbol{\lambda}_{n})$ has density
$\varphi_{n}$ as in (2.5).

###### Idea of the proof.

The set of $n\times n$ complex matrices with multiple eigenvalues has zero
Lebesgue measure. Since the law of $\mathbf{M}$ is absolutely continuous, it
follows that almost surely $\mathbf{M}$ is diagonalizable with distinct
eigenvalues. The density is proportional to ($M^{*}=\overline{M}^{\top}$ is
the conjugate-transpose of $M$)

$M\mapsto\mathrm{e}^{-\sum_{i,j=1}^{n}|M_{i,j}|^{2}}=\mathrm{e}^{-\mathrm{Trace}(MM^{*})}.$

In order to compute the law of the spectrum of $\mathbf{M}$, an idea is to use
for instance the Schur unitary decomposition as a change of variable. Namely,
if $M$ is diagonalizable, then the Schur decomposition is the matrix
factorization $M=U(D+N)U^{*}$ where $U$ is unitary, $D$ is diagonal, and $N$
is upper triangular with null diagonal (nilpotent). The matrix $D$ carries the
eigenvalues of $M$. We have the decoupling

$\mathrm{Trace}(MM^{*})=\mathrm{Trace}(DD^{*})+\mathrm{Trace}(NN^{*}).$

This allows to integrate out $(N,U)$ in the density and to get that the law of
the eigenvalues of $\mathbf{M}$ is given by (2.5). The term
$\prod_{i<j}|x_{i}-x_{j}|^{2}$ is the modulus of the determinant of the
Jacobian of the change of variable. We obtain that for every symmetric bounded
(or positive) measurable function $F:\mathbb{C}^{n}\to\mathbb{R}$,

$\mathbb{E}[F(\boldsymbol{\lambda}_{1},\ldots,\boldsymbol{\lambda}_{n})]=\int_{\mathbb{C}^{n}}\\!F(z_{1},\ldots,z_{n})\varphi_{n}(z_{1},\ldots,z_{n})\mathrm{d}z_{1}\cdots\mathrm{d}z_{n}$

where $\mathrm{d}z_{1}\cdots\mathrm{d}z_{n}$ stands for the Lebesgue measure
on $\mathbb{C}^{n}=\mathbb{R}^{2n}$. The result goes back to [Gin65]. The
scheme of proof that we follow here can be found in [KS11], see also [Meh04,
For10, Ch. 15]. ∎

######  Remark 7.2 (Immediate properties of Ginibre random matrices).

  * •

Since the law of $\mathbf{M}$ is absolutely continuous, almost surely
$\mathbf{M}\mathbf{M}^{*}\neq\mathbf{M}^{*}\mathbf{M}$ (non-normality);

  * •

By the law of large numbers, almost surely, as $n\to\infty$,
$\frac{1}{\sqrt{n}}\mathbf{M}$ has orthonormal rows/columns;

  * •

The law of $\mathbf{M}$ is bi-unitary invariant: if $U$ and $V$ are unitary
then $U\mathbf{M}V$ and $\mathbf{M}$ have same law;

  * •

The Hermitian random matrices $\frac{1}{\sqrt{2}}(\mathbf{M}+\mathbf{M}^{*})$
and $\frac{1}{\sqrt{2}\mathrm{i}}(\mathbf{M}-\mathbf{M}^{*})$, the matrix real
and imaginary parts of $\mathbf{M}$, are independent and belong to the
Gaussian Unitary Ensemble (GUE): their density is proportional to
$H\mapsto\mathrm{e}^{-\frac{1}{2}\mathrm{Trace}(H^{2})}$; Conversely, if
$\mathbf{H}_{1}$ and $\mathbf{H}_{2}$ are independent copies of the Gaussian
Unitary Ensemble then the random matrices
$\frac{1}{\sqrt{2}}(\mathbf{H}_{1}+\mathrm{i}\mathbf{H}_{2})$ and $\mathbf{M}$
have same law.

The exact solvability of the Ginibre gas (2.5) is largely due to a
determinantal structure studied below, itself related to the fact that
$\beta=2$ and $W=g$. More precisely, first of all, from (2.5) we have

$\varphi_{n}(z_{1},\ldots,z_{n})=\frac{\prod_{k=1}^{n}\gamma(z_{k})}{\prod_{k=1}^{n}k!}\prod_{i<j}|z_{i}-z_{j}|^{2}$
(7.2)

where $\gamma$ is the density of $\mathcal{N}(0,\frac{1}{2}I_{2})$ given for
all $z\in\mathbb{C}$ by

$\gamma(z)=\frac{\mathrm{e}^{-|z|^{2}}}{\pi}.$

######  Theorem 7.3 (Determinantal structure and marginals).

For all $n\geq 1$ and $(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$,

$\varphi_{n}(z_{1},\ldots,z_{n})=\frac{1}{n!}\det\left[K_{n}(z_{i},z_{j})\right]_{1\leq
i,j\leq n}$

where the kernel $K_{n}$ is given for all $z,w\in\mathbb{C}$ by

$K_{n}(z,w)=\sqrt{\gamma(z)\gamma(w)}\sum_{\ell=0}^{n-1}\frac{(z\overline{w})^{\ell}}{\ell!}.$

More generally, for all $1\leq k\leq n$, the marginal density

$(z_{1},\ldots,z_{k})\in\mathbb{C}^{k}\mapsto\varphi_{n,k}(z_{1},\ldots,z_{k})=\int_{\mathbb{C}^{n-k}}\varphi_{n}(z_{1},\ldots,z_{n})\mathrm{d}z_{k+1}\cdots\mathrm{d}z_{n}$

satisfies, for all $(z_{1},\ldots,z_{k})\in\mathbb{C}^{k}$,

$\varphi_{n,k}(z_{1},\ldots,z_{k})=\frac{(n-k)!}{n!}\det\left[K_{n}(z_{i},z_{j})\right]_{1\leq
i,j\leq k}.$

In particular for $k=n$ we get $\varphi_{n,n}=\varphi_{n}$, while for $k=1$ we
get, for all $z\in\mathbb{C}$,

$\varphi_{n,1}(z)=\frac{\gamma(z)}{n}\sum_{\ell=0}^{n-1}\frac{|z|^{2\ell}}{\ell!}.$

We say that the spectrum of $\mathbf{M}$ is a Gaussian determinantal point
process, see [HKPV09, Ch. 4].

The “$k$-point correlation” is
$R_{n,k}(z_{1},\ldots,z_{k})=\frac{n!}{(n-k)!}\varphi_{n,k}(z_{1},\ldots,z_{k})=\det[K_{n}(z_{i},z_{j})]_{1\leq
i,j\leq k}$.

###### Idea of proof.

Following for instance [Meh04, Sec. 5.2 and Ch. 15], we get, starting with
(7.2),

$\displaystyle\varphi_{n}(z_{1},\ldots,z_{n})$
$\displaystyle=\frac{\prod_{k=1}^{n}\gamma(z_{k})}{\prod_{k=1}^{n}k!}\prod_{1\leq
i{<}j\leq n}(z_{i}-z_{j})\prod_{1\leq i{<}j\leq n}\overline{(z_{i}-z_{j})}$
$\displaystyle=\frac{\prod_{k=1}^{n}\gamma(x_{k})}{n!}\det\Bigr{[}\frac{z_{j}^{i-1}}{\sqrt{(i-1)!}}\Bigr{]}_{1\leq
i,j\leq
n}\det\Bigr{[}\frac{\overline{z_{j}}^{i-1}}{\sqrt{(i-1)}!}\Bigr{]}_{1\leq
i,j\leq n}$
$\displaystyle=\frac{1}{n!}\det\Bigr{[}K_{n}(z_{i},z_{j})\Bigr{]}_{1\leq
i,j\leq n}.$

On the other hand, the orthogonality of
$\\{\frac{z^{\ell}}{\sqrt{\ell!}}:0\leq\ell\leq n-1\\}$ in
$\mathrm{L}^{2}(\mathbb{C},\gamma)$ gives the identities

$\int_{\mathbb{C}}K_{n}(x,x)\mathrm{d}x=n\quad\text{and}\quad\int_{\mathbb{C}}K_{n}(x,y)K_{n}(y,z)\mathrm{d}y=K_{n}(x,z),\quad
x,z\in\mathbb{C}.$

Finally the formula for $\varphi_{n,k}$ follows by expanding the determinant
in $\varphi_{n}$ and using these identities. ∎

######  Theorem 7.4 (Mean circular Law).

Let $\boldsymbol{\lambda}_{1},\ldots,\boldsymbol{\lambda}_{n}$ be as in
Theorem 7.1 and let us define

$\mu_{n}=\frac{1}{n}\sum_{k=1}^{n}\delta_{\frac{\boldsymbol{\lambda}_{k}}{\sqrt{n}}}.$

Let $\mu_{\infty}$ be the uniform distribution on the unit disc
$\\{z\in\mathbb{C}:|z|\leq 1\\}$ with density
$z\in\mathbb{C}\mapsto\frac{\mathbf{1}_{|z|\leq 1}}{\pi}$. Then

$\mathbb{E}\mu_{n}\underset{n\to\infty}{\overset{\mathcal{C}_{b}}{\longrightarrow}}\mu_{\infty}.$

###### Proof.

Let $\varphi_{n,1}$ be as in Theorem 7.3. For all
$f\in\mathcal{C}_{b}(\mathbb{C},\mathbb{R})$, we have, using Theorem 7.1,

$\mathbb{E}\int
f\mathrm{d}\mu_{n}=\frac{1}{n}\sum_{k=1}^{n}\int_{\mathbb{C}^{n}}f\Bigr{(}\frac{z_{k}}{\sqrt{n}}\Bigr{)}\varphi_{n}(z_{1},\ldots,z_{n})\mathrm{d}z_{1}\cdots\mathrm{d}z_{n}=n\int_{\mathbb{C}}f(z)\varphi_{n,1}(\sqrt{n}z)\mathrm{d}z.$

Thus $\mathbb{E}\mu_{n}$ has density $n\varphi_{n,1}(\sqrt{n}\bullet)$. By
Theorem 7.3 and Lemma 7.5, if $K\subset\\{z\in\mathbb{C}:|z|\neq 1\\}$ is
compact,

$\lim_{n\to\infty}\sup_{z\in
K}\Bigr{|}n\varphi_{n,1}(\sqrt{n}z)-\frac{\mathbf{1}_{|z|\leq
1}}{\pi}\Bigr{|}=\frac{1}{\pi}\lim_{n\to\infty}\sup_{z\in
K}\Bigr{|}\mathrm{e}^{-n|z|^{2}}\mathrm{e}_{n}(n|z|^{2})-\mathbf{1}_{|z|\leq
1}\Bigr{|}=0.$

It follows then by dominated convergence that
$\mathbb{E}\mu_{n}\underset{n\to\infty}{\overset{\mathcal{C}_{b}}{\longrightarrow}}\mu_{\infty}$.
∎

######  Lemma 7.5 (Exponential series).

For every $n\geq 1$ and $z\in\mathbb{C}$,

$|\mathrm{e}_{n}(nz)-\mathrm{e}^{nz}\mathbf{1}_{|z|\leq 1}|\leq r_{n}(z)$

where $\mathrm{e}_{n}(z)=\sum_{\ell=0}^{n-1}\frac{z^{\ell}}{\ell!}$ is the
truncated exponential series and

$r_{n}(z)=\frac{\mathrm{e}^{n}}{\sqrt{2\pi
n}}|z|^{n}\Bigr{(}\frac{n+1}{n(1-|z|)+1}\mathbf{1}_{|z|\leq
1}+\frac{n}{n(|z|-1)+1}\mathbf{1}_{|z|>1}\Bigr{)}.$

###### Proof of Lemma 7.5.

As in Mehta [Meh04, Ch. 15], for every $n\geq 1$, $z\in\mathbb{C}$, if
$|z|\leq n$ then

$\Bigr{|}\mathrm{e}^{z}-\mathrm{e}_{n}(z)\Bigr{|}=\Bigr{|}\sum_{\ell=n}^{\infty}\frac{z^{\ell}}{\ell!}\Bigr{|}\leq\frac{|z|^{n}}{n!}\sum_{\ell=0}^{\infty}\frac{|z|^{\ell}}{(n+1)^{\ell}}=\frac{|z|^{n}}{n!}\frac{n+1}{n+1-|z|},$

while if $|z|>n$ then

$|\mathrm{e}_{n}(z)|\leq\sum_{\ell=0}^{n-1}\frac{|z|^{\ell}}{\ell!}\leq\frac{|z|^{n-1}}{(n-1)!}\sum_{\ell=0}^{n-1}\frac{(n-1)^{\ell}}{|z|^{\ell}}\leq\frac{|z|^{n-1}}{(n-1)!}\frac{|z|}{|z|-n+1}.$

Therefore, for every $n\geq 1$ and $z\in\mathbb{C}$,

$|\mathrm{e}_{n}(nz)-\mathrm{e}^{nz}\mathbf{1}_{|z|\leq
1}|\leq\frac{n^{n}}{n!}\Bigr{(}|z|^{n}\frac{n+1}{n+1-|nz|}\mathbf{1}_{|z|\leq
1}+|z|^{n-1}\frac{|nz|}{|nz|-n+1}\mathbf{1}_{|z|>1}\Bigr{)}.$

It remains to use the Stirling bound $\sqrt{2\pi n}n^{n}\leq n!\mathrm{e}^{n}$
to get the first result. ∎

######  Remark 7.6 (Probabilistic view).

There is a probabilistic interpretation of Lemma 7.5. For all
$z\in\mathbb{C}$,

$\lim_{n\to\infty}n\varphi_{n,1}(\sqrt{n}z)=\frac{1}{\pi}\Bigr{(}\mathbf{1}_{|z|<1}+\frac{1}{2}\mathbf{1}_{|z|=1}\Bigr{)}.$

Namely, by rotational invariance, it suffices to consider the case $z=r>0$.
Next, if $Y_{1},\dots,Y_{n}$ are independent and identically distributed
random variables following the Poisson law of mean $r^{2}$, then

$\mathrm{e}^{-nr^{2}}\mathrm{e}_{n}(nr^{2})=\mathbb{P}(Y_{1}+\cdots+Y_{n}<n)=\mathbb{P}\Bigr{(}\frac{Y_{1}+\cdots+Y_{n}}{n}<1\Bigr{)}.$

Now $\lim_{n\to\infty}\frac{Y_{1}+\dots+Y_{n}}{n}=r^{2}$ almost surely by the
strong law of large numbers, and thus the probability in the right-hand side
above tends as $n\to\infty$ to $0$ if $r>1$ and to $1$ if $r<1$. In other
words, for all $r\neq 1$,

$\lim_{n\to\infty}\mathrm{e}^{-nr^{2}}\mathrm{e}_{n}(nr^{2})=\mathbf{1}_{r<1}.$

It remains to note that for $r=1$ by the central limit theorem we get

$\mathbb{P}\Bigr{(}\frac{Y_{1}+\cdots+Y_{n}}{n}<1\Bigr{)}=\mathbb{P}\Bigr{(}\frac{Y_{1}+\cdots+Y_{n}-n}{\sqrt{n}}<0\Bigr{)}\underset{n\to\infty}{\longrightarrow}\frac{1}{2}.$

######  Remark 7.7 (Incomplete gamma function).

It is well known that the Gamma and the Poisson laws are connected. Namely, if
$X\sim\mathrm{Gamma}(n,\lambda)$ with $n\geq 1$ and $\lambda>0$ and
$Y\sim\mathrm{Poisson}(r)$ with $r>0$ then

$\mathbb{P}(X\geq\lambda
r)=\frac{1}{(n-1)!}\int_{r}^{\infty}x^{n-1}\mathrm{e}^{-x}\mathrm{d}x=\mathrm{e}^{-r}\sum_{\ell=0}^{n-1}\frac{r^{\ell}}{\ell!}=\mathbb{P}(Y\geq
n).$

Also we could use Gamma random variables instead of Poisson random variables
in Remark 7.6. Note also that the integral in the middle of the formula above
is the incomplete Gamma function $\Gamma(n,r)$. This allows to benefit from
the asymptotic analysis of this special function, see [KS11] and references
therein.

######  Theorem 7.8 (Strong circular law).

With the notations of Theorem 7.4, almost surely,

$\mu_{n}\underset{n\to\infty}{\overset{\mathcal{C}_{b}}{\longrightarrow}}\mu_{\infty}.$

Note that this convergence holds regardless of the way we define the random
matrices on the same probability space when $n$ varies. This is an instance of
the concept of _complete convergence_ , see [Yuk98].

###### Idea of the proof.

The argument, due to Jack Silverstein, is in [Hwa86]. It is similar to the
quick proof of the strong law of large numbers for independent random
variables with bounded fourth moment. It suffices to establish the result for
an arbitrary compactly supported $f\in\mathcal{C}_{b}(\mathbb{C},\mathbb{R})$.
Let us define

$S_{n}=\int_{\mathbb{C}}\\!f\,\mathrm{d}\mu_{n}\quad\text{and}\quad
S_{\infty}=\frac{1}{\pi}\int_{|z|\leq 1}\\!f(z)\mathrm{d}z.$

Suppose for now that we have

$\mathbb{E}[\left(S_{n}-\mathbb{E}S_{n}\right)^{4}]=\mathcal{O}\Bigr{(}\frac{1}{n^{2}}\Bigr{)}.$
(7.3)

By monotone convergence or by the Fubini – Tonelli theorem,

$\mathbb{E}\sum_{n=1}^{\infty}\left(S_{n}-\mathbb{E}S_{n}\right)^{4}=\sum_{n=1}^{\infty}\mathbb{E}[\left(S_{n}-\mathbb{E}S_{n}\right)^{4}]<\infty$

and thus $\sum_{n=1}^{\infty}\left(S_{n}-\mathbb{E}S_{n}\right)^{4}<\infty$
almost surely, which implies $\lim_{n\to\infty}S_{n}-\mathbb{E}S_{n}=0$ almost
surely. Since $\lim_{n\to\infty}\mathbb{E}S_{n}=S_{\infty}$ by Theorem 7.4, we
get that almost surely

$\lim_{n\to\infty}S_{n}=S_{\infty}.$

Finally, one can swap the universal quantifiers on $\omega$ and $f$ thanks to
the separability of the set of compactly supported continuous bounded
functions $\mathbb{C}\to\mathbb{R}$ equipped with the supremum norm. To
establish the fourth moment bound (7.3), we set

$S_{n}-\mathbb{E}S_{n}=\frac{1}{n}\sum_{k=1}^{n}Z_{k}\quad\text{with}\quad
Z_{k}=f\Bigr{(}\frac{\boldsymbol{\lambda}_{k}}{\sqrt{n}}\Bigr{)}-\mathbb{E}f\Bigr{(}\frac{\boldsymbol{\lambda}_{k}}{\sqrt{n}}\Bigr{)}.$

Next, we obtain, with $\sum_{k_{1},\ldots}$ running over distinct indices in
$1,\ldots,n$,

$\displaystyle\mathbb{E}\left[\left(S_{n}-\mathbb{E}S_{n}\right)^{4}\right]$
$\displaystyle=\frac{1}{n^{4}}\sum_{k_{1}}\mathbb{E}[Z_{k_{1}}^{4}]$
$\displaystyle\quad+\frac{4}{n^{4}}\sum_{k_{1},k_{2}}\mathbb{E}[Z_{k_{1}}Z_{k_{2}}^{3}]$
$\displaystyle\quad+\frac{3}{n^{4}}\sum_{k_{1},k_{2}}\mathbb{E}[Z_{k_{1}}^{2}Z_{k_{2}}^{2}]$
$\displaystyle\quad+\frac{6}{n^{4}}\sum_{k_{1},k_{2},k_{3}}\mathbb{E}[Z_{k_{1}}Z_{k_{2}}Z_{k_{3}}^{2}]$
$\displaystyle\quad+\frac{1}{n^{4}}\sum_{k_{1},k_{2},k_{3},k_{3},k_{4}}\\!\\!\\!\\!\\!\\!\mathbb{E}[Z_{k_{1}}Z_{k_{3}}Z_{k_{3}}Z_{k_{4}}].$

The first three terms of the right are $\mathcal{O}(n^{-2})$ since
$\max_{1\leq k\leq n}|Z_{k}|\leq\|f\|_{\infty}$. The expressions of
$\varphi_{n,3}$ and $\varphi_{n,4}$ from Theorem 7.3 allow to show that the
remaining two terms are also $\mathcal{O}(n^{-2})$, see [Hwa86]. ∎

The following theorem includes Theorem 7.4, which corresponds to the case
$k=1$.

######  Theorem 7.9 (Chaoticity).

Let $\mu_{\infty}$ be the uniform distribution on the unit disc
$\\{z\in\mathbb{C}:|z|\leq 1\\}$. For all $1\leq k\leq n$, denoting by
$P_{n,k}$ the $k$-dimensional marginal distribution of the Ginibre gas (2.5),
we have

$P_{n,k}\underset{n\to\infty}{\overset{\mathcal{C}_{b}}{\longrightarrow}}\mu_{\infty}^{\otimes
k}.$

###### Idea of the proof.

The measures $P_{n,k}$ and $\mu_{\infty}$ have densities $\varphi_{n,k}$ and
$z\in\mathbb{C}\mapsto\varphi_{\infty}(z)=\pi^{-1}\mathbf{1}_{|z|\leq 1}$.

The case $k=1$ is nothing else but Theorem 7.4, namely
$P_{n,1}\underset{n\to\infty}{\overset{\mathcal{C}_{b}}{\longrightarrow}}\mu_{\infty}$.
This comes via dominated convergence from the fact that
$\lim_{n\to\infty}\varphi_{n,k}=\varphi_{\infty}$ uniformly on compact subsets
of $\\{z\in\mathbb{C}:|z|\neq 1\\}$.

Let us consider now the case $k=2$. Here again, by dominated convergence, it
suffices to show that

$\lim_{n\to\infty}\varphi^{n,2}=\varphi_{\infty}^{\otimes 2}$

uniformly on compact subsets of $\\{(z_{1},z_{2})\in\mathbb{C}^{2}:|z_{1}|\neq
1,|z_{2}|\neq 1,z_{1}\neq z_{2}\\}$.

By Theorem 7.3, for all $z_{1},z_{2}\in\mathbb{C}$,

$\displaystyle\varphi^{n,2}(z_{1},z_{2})$
$\displaystyle=\frac{n}{n-1}\frac{\mathrm{e}^{-n(|z_{1}|^{2}+|z_{2}|^{2})}}{\pi^{2}}\big{(}\mathrm{e}_{n}(n|z_{1}|^{2})\mathrm{e}_{n}(n|z_{2}|^{2})-|\mathrm{e}_{n}(nz_{1}\overline{z}_{2})|^{2}\big{)}$
$\displaystyle=\frac{n}{n-1}\varphi^{n,1}(z_{1})\varphi^{n,1}(z_{2})-\frac{n}{n-1}\frac{\mathrm{e}^{-n(|z_{1}|^{2}+|z_{2}|^{2})}}{\pi^{2}}|\mathrm{e}_{n}(nz_{1}\overline{z}_{2})|^{2}$
(7.4)

where $\mathrm{e}_{n}$ is as in Lemma 7.5. It follows that for any $n\geq 2$
and $z_{1},z_{2}\in\mathbb{C}$,

$\displaystyle\Delta_{n}(z_{1},z_{2})$
$\displaystyle=\varphi^{n,2}(z_{1},z_{2})-\varphi^{n,1}(z_{1})\varphi^{n,1}(z_{2})$
$\displaystyle=\frac{1}{n-1}\varphi^{n,1}(z_{1})\varphi^{n,1}(z_{2})-\frac{n}{n-1}\frac{\mathrm{e}^{-n(|z_{1}|^{2}+|z_{2}|^{2})}}{\pi^{2}}|\mathrm{e}_{n}(nz_{1}\overline{z}_{2})|^{2}.$
(7.5)

In particular, using $\varphi^{n,2}\geq 0$ for the lower bound,

$-\varphi^{n,1}(z_{1})\varphi^{n,1}(z_{2})\leq\Delta_{n}(z_{1},z_{2})\leq\frac{1}{n-1}\varphi^{n,1}(z_{1})\varphi^{n,1}(z_{2}).$

From this and Lemma 7.5 we first deduce that for any compact subset $K$ of
$\\{z\in\mathbb{C}:|z|>1\\}$

$\lim_{n\to\infty}\sup_{\begin{subarray}{c}z_{1}\in\mathbb{C}\\\ z_{2}\in
K\end{subarray}}|\Delta_{n}(z_{1},z_{2})|=\lim_{n\to\infty}\sup_{\begin{subarray}{c}z_{1}\in
K\\\ z_{2}\in\mathbb{C}\end{subarray}}|\Delta_{n}(z_{1},z_{2})|=0.$

It remains to show that $\Delta_{n}(z_{1},z_{2})\to 0$ as $n\to\infty$ when
$z_{1}$ and $z_{2}$ are in compact subsets of
$\\{(z_{1},z_{2})\in\mathbb{C}^{2}:|z_{1}|<1,|z_{2}|<1\\}$. In this case
$|z_{1}\overline{z}_{2}|\leq 1$, and Lemma 7.5 gives

$|\mathrm{e}_{n}(nz_{1}\overline{z}_{2})|^{2}\leq
2\mathrm{e}^{2n\Re(z_{1}\overline{z}_{2})}+2r_{n}^{2}(z_{1}\overline{z}_{2}).$

Next, using the elementary identity
$2\Re(z_{1}\overline{z}_{2})=|z_{1}|^{2}+|z_{2}|^{2}-|z_{1}-z_{2}|^{2}$, we
get

$\mathrm{e}^{-n(|z_{1}|^{2}+|z_{2}|^{2})}|\mathrm{e}_{n}(nz_{1}\overline{z}_{2})|^{2}\leq
2\mathrm{e}^{-n|z_{1}-z_{2}|^{2}}+2\mathrm{e}^{-n(|z_{1}|^{2}+|z_{2}|^{2})}r_{n}^{2}(z_{1}\overline{z}_{2}).$
(7.6)

Since $|z_{1}\overline{z}_{2}|\leq 1$, the formula for $r_{n}$ in Lemma 7.5
gives

$\mathrm{e}^{-n(|z_{1}|^{2}+|z_{2}|^{2})}r^{2}_{n}(z_{1}\overline{z}_{2})\leq\mathrm{e}^{-n(|z_{1}|^{2}+|z_{2}|^{2}-2-\log|z_{1}|^{2}-\log|z_{2}|^{2})}\frac{(n+1)^{2}}{2\pi
n}.$

Using (7.1), (7.6) and the bounds $\varphi^{n,1}\leq\pi^{-1}$ and $u-1-\log
u>0$ for $0<u<1$, it follows that $\Delta_{n}(z_{1},z_{2})$ tends to $0$ as
$n\to\infty$ uniformly in $z_{1},z_{2}$ on compact subsets of

$\\{(z_{1},z_{2})\in\mathbb{C}^{2}:|z_{1}|<1,|z_{2}|<1,z_{1}\neq z_{2}\\}.$

This finishes the proof of the case $k=2$. The case $k\geq 3$ follows from the
case $k=2$ by Lemma 7.11. ∎

######  Remark 7.10 (Impossibility of global uniform convergence of
densities).

The convergence of $\varphi_{n,1}$ cannot hold uniformly on arbitrary compact
sets of $\mathbb{C}$ since the point-wise limit is not continuous on the unit
circle. Similarly, the convergence of $\varphi_{n,2}$ cannot hold on
$\\{(z,z):z\in\mathbb{C},|z|<1\\}$ since $\varphi^{n,2}(z,z)=0$ for any $n\geq
2$ and $z\in\mathbb{C}$ while
$\varphi_{\infty}(z)\varphi_{\infty}(z)=\pi^{-2}\neq 0$ when $|z|<1$.

######  Lemma 7.11 (Chaoticity).

Let $E$ be a Polish space. For all $n\geq 1$, let
$P_{n}\in\mathcal{M}_{1}(E^{n})$ be exchangeable, and for all $1\leq k\leq n$,
let $P_{n,k}\in\mathcal{M}_{1}(E^{k})$ be its $k$-dimensional marginal
distribution. For all $n\geq 1$, let us pick a random vector
$X_{n}=(X_{n,1},\ldots,X_{n,n})\sim P_{n}$ and let us define the random
empirical measure

$\mu_{n}=\displaystyle\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{n,i}}.$

For all $\mu\in\mathcal{M}_{1}(E)$, the following properties are equivalent:

  1. (1)

$\mu_{n}\underset{n\to\infty}{\overset{\mathcal{C}_{b}}{\longrightarrow}}\delta_{\mu}$
in $\mathcal{M}_{1}(\mathcal{M}_{1}(E))$ (here we see $\mu_{n}$ a random
variable taking values in $\mathcal{M}_{1}(E)$);

  2. (2)

$P_{n,k}\underset{n\to\infty}{\overset{\mathcal{C}_{b}}{\longrightarrow}}\mu^{\otimes
k}$ for any fixed $k\geq 1$ (note that $P_{n,k}$ has a meaning as soon as
$n\geq k$);

  3. (3)

$P_{n,2}\underset{n\to\infty}{\overset{\mathcal{C}_{b}}{\longrightarrow}}\mu^{\otimes
2}$;

where these weak convergences are with respect to continuous and bounded test
functions.

###### Proof.

Folkloric in the domain of mean field particle systems. We refer to [BCF18]
and references therein. ∎

######  Theorem 7.12 (Central limit phenomenon).

Let $\mu_{n}$ be as in Theorem 7.4. Then, for all measurable
$f:\mathbb{C}\to\mathbb{R}$ which are $\mathcal{C}^{1}$ in a neighborhood of
the unit disc $D=\\{z\in\mathbb{C}:|z|\leq 1\\}$ of the complex plane,

$n\Bigr{[}\int f\mathrm{d}\mu_{n}-\mathbb{E}\int
f\mathrm{d}\mu_{n}\Bigr{]}=\sum_{k=1}^{n}\Bigr{[}f\Bigr{(}\frac{\boldsymbol{\lambda}_{k}}{\sqrt{n}}\Bigr{)}-\mathbb{E}f\Bigr{(}\frac{\boldsymbol{\lambda}_{k}}{\sqrt{n}}\Bigr{)}\Bigr{]}\overset{\mathrm{d}}{\underset{n\to\infty}{\longrightarrow}}\mathcal{N}\Bigr{(}\frac{1}{4\pi}\|f\|_{\mathrm{H}^{1}(D)}^{2}+\frac{1}{2}\|f\|_{\mathrm{H}^{1/2}(\partial
D)}^{2}\Bigr{)}$

where

$\|f\|_{\mathrm{H}^{1}(D)}^{2}=\int_{D}|\nabla
f|^{2}\mathrm{d}z\quad\text{and}\quad\|f\|_{\mathrm{H}^{1/2}(\partial
D)}^{2}=\sum_{k\in\mathbb{Z}}|k||\widehat{f}(k)|^{2}$

where $\widehat{f}(k)$ is the $k$-th Fourier coefficient of $f$ on $\partial
D=\\{z\in\mathbb{C}:|z|=1\\}$, namely

$\widehat{f}(k)=\frac{1}{2\pi}\int_{0}^{2\pi}f(\mathrm{e}^{\mathrm{i}\theta})\mathrm{e}^{-\mathrm{i}k\theta}\mathrm{d}\theta.$

Note that $\|f\|_{\mathrm{H}^{1/2}(\partial D)}=0$ if $f$ is analytic on a
neighborhood of $\partial D$.

###### About the proof.

It is known [CL95] that the cumulants of linear statistics of determinantal
processes have a nice form that can be used to prove a central limit theorem.
This idea is followed in [RV07] in order to produce the result, via
combinatorial identities, and via reduction to polynomial test functions. The
method is used for more general two-dimensional determinantal gases in [AHM11,
AHM15]. ∎

######  Theorem 7.13 (Distribution of the moduli in the Ginibre model).

Let $(\boldsymbol{\lambda}_{1},\ldots,\boldsymbol{\lambda}_{n})$ be the
exchangeable random vector considered in Theorem 7.1. Then the following
equality in distribution holds

$(|\boldsymbol{\lambda}_{1}|,\ldots,|\boldsymbol{\lambda}_{n}|)\overset{\mathrm{d}}{=}(Z_{\sigma(1)},\ldots,Z_{\sigma(n)})$

where $Z_{1},\ldots,Z_{n}$ are independent non-negative random variables
with111The law $\mathrm{Gamma}(a,\lambda)$ has density
$x\in\mathbb{R}\mapsto\frac{\lambda^{a}}{\Gamma(a)}x^{a-1}\mathrm{e}^{-\lambda
x}\mathbf{1}_{x\geq 0}$, and
$\mathrm{Gamma}(a,\lambda)*\mathrm{Gamma}(b,\lambda)=\mathrm{Gamma}(a+b,\lambda)$.222Note
that
$(\sqrt{2}Z_{k})^{2}\sim\mathrm{Gamma}(k,\frac{1}{2})=\mathrm{Exponential}(\frac{1}{2})^{*k}=\chi^{2}(2k)$
since $\chi^{2}(n)=\mathrm{Gamma}(\frac{n}{2},\frac{1}{2})$ for all $n\geq
1$.333For $n=1$ we recover the Box – Muller formula
$|X|^{2}\sim\chi^{2}(2)=\mathrm{Gamma}(1,\frac{1}{2})=\mathrm{Exponential}(\frac{1}{2})$
with $X\sim\mathcal{N}(0,I_{2})$.

$Z_{k}^{2}\sim\mathrm{Gamma}(k,1),\quad 1\leq k\leq n,$

and where $\sigma$ is a uniform random permutation of $\\{1,\ldots,n\\}$
independent of $Z_{1},\ldots,Z_{n}$. Equivalently, for all symmetric bounded
measurable $F:\mathbb{R}^{n}\to\mathbb{R}$, we have
$\mathbb{E}(F(|\boldsymbol{\lambda}_{1}|,\ldots,|\boldsymbol{\lambda}_{n}|))=\mathbb{E}(F(Z_{1},\ldots,Z_{n})).$

This is an equality between two exchangeable laws on $\mathbb{R}^{n}$, in
other words an equality in law between two configurations of unlabeled random
points in $\mathbb{R}$ (multi-sets). Note in particular that for all $1\leq
k\leq n$, taking
$F(x_{1},\ldots,x_{n})=\sum_{\begin{subarray}{c}i_{1},\ldots,i_{k}\\\ \text{
distinct}\end{subarray}}f(x_{i_{1}})\cdots f(x_{i_{k}})$ gives equality of
$k$-point correlation functions.

###### Proof.

From Theorem 7.1, the exchangeable random vector
$(\boldsymbol{\lambda}_{1},\ldots,\boldsymbol{\lambda}_{n})$ has density
$\varphi_{n}$. It follows that the density of the exchangeable random vector
$(|\boldsymbol{\lambda}_{1}|,\ldots,|\boldsymbol{\lambda}_{n}|)$ is obtained
from $\varphi_{n}$ by integrating the phases in polar coordinates. In polar
coordinates $x_{k}=r_{k}\mathrm{e}^{\mathrm{i}\theta_{k}}$, the density
$\varphi_{n}$ writes

$(r_{1},\ldots,r_{n},\theta_{1},\ldots,\theta_{n})\mapsto\frac{\mathrm{e}^{-\sum_{j=1}^{n}r_{j}^{2}}}{\pi^{n}\prod_{k=1}^{n}k!}\prod_{j<k}|r_{j}\mathrm{e}^{\mathrm{i}\theta_{j}}-r_{k}\mathrm{e}^{\mathrm{i}\theta_{k}}|^{2}.$

Now we have, denoting $\Sigma_{n}$ the symmetric group of permutations of
$\\{1,\ldots,n\\}$,

$\displaystyle\prod_{j<k}|r_{j}\mathrm{e}^{\mathrm{i}\theta_{j}}-r_{k}\mathrm{e}^{\mathrm{i}\theta_{k}}|^{2}$
$\displaystyle=\prod_{j<k}(r_{j}\mathrm{e}^{\mathrm{i}\theta_{j}}-r_{k}\mathrm{e}^{\mathrm{i}\theta_{k}})\prod_{j<k}\overline{(r_{j}\mathrm{e}^{\mathrm{i}\theta_{j}}-r_{k}\mathrm{e}^{\mathrm{i}\theta_{k}})}$
$\displaystyle=\det\Bigr{[}r_{j}^{k-1}\mathrm{e}^{\mathrm{i}(k-1)\theta_{j}}\Bigr{]}_{1\leq
j,k\leq
n}\det\Bigr{[}r_{j}^{k-1}\mathrm{e}^{-\mathrm{i}(k-1)\theta_{j}}\Bigr{]}_{1\leq
j,k\leq n}$
$\displaystyle=\Bigr{(}\sum_{\sigma\in\Sigma_{n}}(-1)^{\mathrm{sign}(\sigma)}\prod_{j=1}^{n}r_{j}^{\sigma(j)-1}\mathrm{e}^{\mathrm{i}(\sigma(j)-1)\theta_{j}}\Bigr{)}\Bigr{(}\sum_{\sigma^{\prime}\in\Sigma_{n}}(-1)^{\mathrm{sign}(\sigma^{\prime})}\prod_{j=1}^{n}r_{j}^{\sigma^{\prime}(j)-1}\mathrm{e}^{-\mathrm{i}(\sigma^{\prime}(j)-1)\theta_{j}}\Bigr{)}$
$\displaystyle=\sum_{\sigma,\sigma^{\prime}\in\Sigma_{n}}(-1)^{\mathrm{sign}(\sigma)+\mathrm{sign}(\sigma^{\prime})}\prod_{j=1}^{n}r_{j}^{\sigma(j)+\sigma^{\prime}(j)-2}\mathrm{e}^{\mathrm{i}((\sigma(j)-\sigma^{\prime}(j))\theta_{j})}.$

If we integrate the phases, we note that only the terms with
$\sigma=\sigma^{\prime}$ contribute to the result, namely

$\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}\prod_{j<k}|r_{j}\mathrm{e}^{\mathrm{i}\theta_{j}}-r_{k}\mathrm{e}^{\mathrm{i}\theta_{k}}|^{2}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{n}=(2\pi)^{n}\sum_{\sigma\in\Sigma_{n}}\prod_{j=1}^{n}r_{j}^{2(\sigma(j)-1)}=(2\pi)^{n}\mathrm{per}\Bigr{[}r_{j}^{2k}\Bigr{]}_{1\leq
j,k\leq n}$

where “per” stands for “permanent”. Therefore, the (exchangeable) density of
the moduli is given by

$\int_{0}^{2\pi}\cdots\int_{0}^{2\pi}\frac{\mathrm{e}^{-\sum_{j=1}^{n}r_{j}^{2}}}{\pi^{n}\prod_{k=1}^{n}k!}\prod_{j<k}|r_{j}\mathrm{e}^{\mathrm{i}\theta_{j}}-r_{k}\mathrm{e}^{\mathrm{i}\theta_{k}}|^{2}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{n}=\mathrm{perm}\Bigr{[}\frac{2}{k!}r_{j}^{2k}\mathrm{e}^{-r_{j}^{2}}\Bigr{]}_{1\leq
j,k\leq n}.$

But if $f_{1},\ldots,f_{n}:\mathbb{R}\to\mathbb{R}$ are probability density
functions then $(x_{1},\ldots,x_{n})\mapsto\mathrm{perm}[f_{j}(x_{k})]_{1\leq
j,k\leq n}$ is the density of the random vector
$(X_{\sigma(1)},\ldots,X_{\sigma(n)})$ where $X_{1},\ldots,X_{n}$ are
independent real random variables with densities $f_{1},\ldots,f_{n}$ and
where $\sigma$ is a uniform random permutation of $\\{1,\ldots,n\\}$
independent of $X_{1},\ldots,X_{n}$. Also the desired result follows from the
formula above and the fact that for all $1\leq k\leq n$, a non-negative random
variable $Z_{k}$ has density
$r\mapsto\frac{2}{k!}r^{2k}\mathrm{e}^{-r^{2}}\mathbf{1}_{r\geq 0}$ if and
only if $Z_{k}^{2}\sim\mathrm{Gamma}(k,1)$.

This proof, essentially due to Kostlan [Kos92], see also [Rid03], relies on
the determinantal nature of $\varphi_{n}$ in (2.5), and remains usable for
general determinantal processes, see for instance [HKPV09]. ∎

######  Theorem 7.14 (Spectral radius).

With the notation of Theorem 7.1, almost surely

$\rho_{n}=\max_{1\leq k\leq
n}\frac{|\boldsymbol{\lambda}_{k}|}{\sqrt{n}}\underset{n\to\infty}{\longrightarrow}1.$

Moreover, denoting $\kappa_{n}=\log\frac{n}{2\pi}-2\log(\log(n))$,

$\sqrt{4n\kappa_{n}}\Bigr{(}\rho_{n}-1-\sqrt{\frac{\kappa_{n}}{4n}}\Bigr{)}\underset{n\to\infty}{\overset{\mathrm{d}}{\longrightarrow}}\mathrm{Gumbel}.$

Note the the second statement (Gumbel fluctuation) implies that
$\rho_{n}\underset{n\to\infty}{\longrightarrow}1$ in probability.

Regarding the convergence, see [BCGZ20] for a random analytic function point
of view, related to the central limit theorem. Regarding the fluctuation, see
[Joh07, Ben10] for an interpolation with the Tracy – Widom fluctuation at the
edge of GUE.

###### Idea of proof.

By Theorem 7.13 Since $Z_{k}^{2}\overset{\mathrm{d}}{=}E_{1}+\cdots+E_{k}$
where $E_{1},\ldots,E_{k}$ are independent and identically distributed
exponential random variables of unit mean, we get, for every $r>0$,

$\mathbb{P}(\rho_{n}\leq\sqrt{n}r)=\prod_{1\leq k\leq
n}\mathbb{P}\left(\frac{E_{1}+\cdots+E_{k}}{n}\leq r^{2}\right).$

By the law of large numbers, this tends as $n\to\infty$ to $0$ or $1$
depending on the position of $r$ with respect to $1$. Moreover the central
limit theorem suggests that $\rho_{n}$ behaves as $n\to\infty$ as the maximum
of independent and identically distributed Gaussian random variables, a
situation for which it is known that the fluctuation follows the Gumbel law.
The full proof is in [Rid03] and involves crucially a quantitative central
limit theorem and the Borel – Cantelli lemma. The approach is robust and
remains valid beyond the Ginibre gas, for determinantal gases, see for
instance [CP14, JQ17, GZ18] and references therein. ∎

######  Remark 7.15 (Real or quatertionic Ginibre model).

How about an analogue of Theorem 7.1 when the entries of $\mathbf{M}$ are real
Gaussian or real quaternionic Gaussian instead of complex Gaussian? Some
answers are already in [Gin65]. In these cases, the density of the eigenvalues
can be computed but it is not the beta gas (2.4) with $\beta\in\\{1,4\\}$.
This is in contrast with the G(O|U|S)E triplet of the Hermtian random matrix
Dysonian universe [Dys62c]. See for instance [Ede97, Dub18b] and references
therein.

######  Remark 7.16 (Large deviations).

The large deviations principle for the beta Ginibre gas (2.5) was established
in [HP00, PH98], using a method inspired from [BAG08], itself inspired from
[Voi93, Voi94]. It does not rely on the determinantal structure, and allows to
extend Theorem 7.8 to all $\beta>0$.

###  7.2. More determinantal models

It is well known that the ratio of two independent real standard Gaussian
random variables follows a Cauchy distribution. The following theorem can be
seen as a matrix version of this phenomenon.

######  Theorem 7.17 (Forrester – Krishnapur spherical ensemble).

Let $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$ be independent copies of the Ginibre
random matrix defined in (7.1). Then as an exchangeable random vector of
$\mathbb{C}^{n}$, the eigenvalues of $\mathbf{M}_{1}{\mathbf{M}_{2}}^{-1}$
have density

$(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}\mapsto\frac{1}{Z_{n}}\frac{\prod_{j<k}|z_{j}-z_{k}|^{2}}{\prod_{j=1}^{n}(1+|z_{j}|^{2})^{n+1}}.$

This corresponds to the beta gas (2.4) with
$V=\frac{1}{2}\frac{n+1}{n}\log(1+\left|\cdot\right|^{2})$ and $\beta=2$.
Moreover its push-forward on the Riemann sphere using inverse stereographic
projection is the uniform law on the sphere.

See [Kri09] and [HKPV09, FK09, For10] for a proof. The set of singular
$n\times n$ complex matrices is a hyper-surface of zero Lebesgue measure in
$\mathbb{C}^{n^{2}}$ and therefore, almost surely, the Ginibre random matrix
$\mathbf{M}$ in (7.1) is invertible (its law is absolutely continuous with
respect to the Lebesgue measure on $\mathbb{C}^{n^{2}}$).

From Theorem 1.1, the equilibrium measure of the gas is heavy tailed with
density

$z\in\mathbb{C}\mapsto\frac{1}{\pi(1+|z|^{2})^{2}}.$

A large deviations principle for the empirical measure associated to the
Coulomb gas is proved in [Har12] in relation with the sphere. The convergence
of the empirical measure is also considered in [Bor11]. The fluctuation at the
edge is discussed in [CP14] and studied in [JQ17] by using the idea of Kostlan
behind Theorem 7.13 thanks to the determinantal structure. A beta version of
the model is considered in [CMMOC18] and studied using transportation of
measure.

######  Theorem 7.18 (Życzkowski – Sommers ensemble).

Let $\mathbf{U}={(\mathbf{U}_{j,k})}_{1\leq j,k\leq m}$ be a random $m\times
m$ unitary matrix following the (Haar) uniform law on this compact group of
matrices. Then, for all $1\leq n<m$, as an exchangeable random vector of
$\mathbb{C}^{n}$, the eigenvalues of the truncation
${(\mathbf{U}_{j,k})}_{1\leq j,k\leq n}$ have density

$(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}\mapsto\frac{\prod_{j=1}^{n}(1-|z_{j}|^{2})^{m-n-1}}{Z_{n}}\prod_{1\leq
j<k\leq n}|z_{j}-z_{k}|^{2}.$

This corresponds to the beta gas (2.4) with
$V=V_{n,m}=\frac{m-n-1}{n}\log\frac{1}{1-|z|^{2}}$ and $\beta=2$.

See [ZS00] for a proof, and [FK09] for the special case $m\geq 2n$ and a link
with the pseudo-sphere and Schur transformation. Following [PR05] and
references therein, if $\lim_{m,n\to\infty}\frac{n}{m}=\alpha\in(0,1)$ then
the empirical measure converges towards the heavy tailed probability measure
with density

$z\in\mathbb{C}\mapsto\frac{(1-\alpha)}{\pi\alpha(1-|z|^{2})^{2}}\mathbf{1}_{|z|\leq\sqrt{\alpha}}.$

In a sense this law interpolates between the uniform law on the unit disc
($\alpha\to 0$ after scaling by $\sqrt{\alpha}$) and the uniform law on the
unit circle ($\alpha\to 1$). A large deviations principle is obtained in
[PR05], concentration inequalities are derived in [MS19], while the
fluctuation at the edge is studied in [JQ17].

######  Theorem 7.19 (Product of Ginibre random matrices).

Let $m\geq 1$ and let $\mathbf{M}_{1},\ldots,\mathbf{M}_{m}$ be independent
and identically distributed copies of the $n\times n$ in (7.1). Then, as an
exchangeable random vector of $\mathbb{C}^{n}$, the eigenvalues of the scaled
product $n^{-\frac{m}{2}}\mathbf{M}_{1}\cdots\mathbf{M}_{m}$ have density

$\frac{\prod_{j=1}^{n}w_{m}(\sqrt{n}|z_{j}|)}{Z_{n}}\prod_{j<k}|z_{j}-z_{k}|^{2}$

where $w_{k}$ is the Meijer G-function given by the recursive formula

$w_{1}(z)=\mathrm{e}^{-|z|^{2}}\quad\text{and}\quad
w_{k}(z)=2\pi\int_{0}^{\infty}w_{k-1}\Bigr{(}\frac{z}{r}\Bigr{)}\frac{\mathrm{e}^{-r^{2}}}{r}\mathrm{d}r.$

This corresponds to the beta gas (2.4) with $V=V_{n,m}=-\frac{1}{n}\log
w_{m}(\sqrt{n}\bullet)$ and $\beta=2$.

See [AB12] for a proof. Following [GT11, Bor11], its converges to the
equilibrium measure with density

$z\in\mathbb{C}\mapsto\frac{|z|^{\frac{2}{m}-2}}{m\pi}\mathbf{1}_{|z|\leq 1}.$

We recover the uniform law on the unit disc when $m=1$. The edge fluctuation
is considered in [JQ17].

######  Remark 7.20 (Determinantal gases and random normal matrices).

An $n\times n$ complex matrix is _normal_ when $MM^{*}=M^{*}M$. The random
matrices in theorems 7.1,7.17,7.18,7.19 are not normal. Let us comment now on
models of normal random matrices. Let $\mathcal{N}_{n}$ be the hyper-surface
of $\mathbb{C}^{n^{2}}$ of all $n\times n$ normal matrices. Let
$V:\mathbb{C}\to\mathbb{R}$ be $\mathcal{C}^{2}$ and such that $V(z)\geq
c\log(1+|z|^{2})$ for some constant $c>0$. Following [CZ98, EF05], let us
consider the probability measure on $\mathcal{N}_{n}$ with density
proportional to $M\mapsto\mathrm{e}^{-n\mathrm{Trace}(V(M))}$ with respect to
the Hausdorff measure on $\mathcal{N}_{n}$. This produces random normal
matrices, and their eigenvalues, seen as an exchangeable random vector, have
density given by the gas (2.4) with $\beta=2$. This random (normal) matrix
model is referred to as the _random normal matrix model_. The fluctuation of
the empirical measure is studied in [AHM15, AHM11], while the fluctuation at
the edge is studied in [CP14, JQ17, GZ18].

The power of a Ginibre matrix has also a nice determinantal structure, see
[Dub18a].

##  8\. Comments and open problems

We have skipped several important old and new results on Coulomb gases. The
main themes are local versus global, first versus second order, macroscopics
versus microscopics, non-universal versus universal.

Universality. The first order global convergence
$\lim_{n\to\infty}\mu_{n}=\mu_{V}$, that we call macroscopics, is not
universal in the sense that the limit $\mu_{V}$ still depends on $V$. The
second order convergence provided by the central limit theorem (5.1) is
universal in the sense that the limit should not depend on $V$. Similarly, for
a two-dimensional Coulomb gas with radial confining potential $V$, the limit
of the edge depends on $V$ but its fluctuation does not and is universal.
Universality emerges often in a second order asymptotic analysis, as for
classical limit theorems of probability theory.

Microscopics. A second order analysis corresponds to the asymptotic analysis
of $n(\mu_{n}-\nu_{V})$ as $n\to\infty$. This can be seen as a microscopic
analysis while the convergence $\mu_{n}\to\mu_{V}$ is a macroscopic analysis.
This corresponds to a second order Taylor formula for the quadratic form
$\mathcal{E}_{V}$, in other words in a special factorization, leading to a new
object called the _renormalized energy_. This was the subject of an series of
works by Étienne Sandier and Sylvia Serfaty, and by Sylvia Serfaty and other
co-authors. See for instance [Ser18b, Ser18a, LS17, Ser15] and references
therein. An outcome of this refined analysis is a second order asymptotics for
the free energy. More precisely, recall that the Boltzmann – Shannon entropy
of the Boltzmann – Gibbs measure $P_{n}$ in (2.2) is defined by

$\mathcal{S}(P_{n})=-\int_{(\mathrm{R}^{d})^{n}}f_{n}(x_{1},\cdots,x_{n})\log
f_{n}(x_{1},\ldots,x_{n})\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

where $f_{n}$ is the density of $P_{n}$. Its Helmholtz free energy is given by

$\int E_{n}\mathrm{d}P_{n}-\frac{\mathcal{S}(P_{n})}{\beta}=-\frac{\log
Z_{n}}{\beta},$

see [Cha15]. Now following [LS17], if $\mu_{V}$ has density $f_{V}$ with a
finite Boltzmann – Shannon entropy

$\mathcal{S}(\mu_{V})=-\int_{\mathbb{R}^{d}}f_{V}\log f_{V}\mathrm{d}x,$

then we have an asymptotic expansion of the free energy as $n\to\infty$ as

$-\frac{\log
Z_{n}}{\beta}=\begin{cases}\displaystyle\frac{n^{2}}{2}\mathcal{E}_{V}(\mu_{V})-\frac{n\log
n}{4}+n(c_{\beta}+c_{\beta}^{\prime}\mathcal{S}(\mu_{V}))+no_{n}(1)&\text{if
$d=2$}\\\\[10.00002pt]
\displaystyle\frac{n^{2}}{2}\mathcal{E}_{V}(\mu_{V})+n(c_{\beta,d,V}+c^{\prime}_{\beta}\mathcal{S}(\mu_{V}))+no_{n}(1)&\text{if
$d\neq 2$}\end{cases},$

where $c_{\beta},c_{\beta}^{\prime},c_{\beta,d,V}$ are constants which can be
made explicit.

Edge. The most elementary open question related to Coulomb gases is perhaps
the law of fluctuation at the edge, even in the case of rotationally invariant
confining potential for arbitrary values of $d$ and $\beta$. The Gumbel
fluctuation is known for instance to be universal for a class of two
dimensional ($d=2$) determinantal ($\beta=2$) Coulomb gases with radial
confining potential, see [CP14]. The same question for arbitrary $\beta$ is
open, and the same question for arbitrary dimension $d\geq 3$ and $\beta>0$ is
also open.

Crystallization. A conjecture related to Coulomb gases is the emergence of
rigid structures at low temperatures. This is known as crystallization and was
proved in special cases, for instance for one-dimensional Coulomb gases. See
for instance [Ser15, BL15, Ser18b, Ser18a, PS20] and references therein.

More. Among all the important results on Coulomb gases that we have not yet
mentioned, we may cite the approximate transport maps for universality
considered in [FG16, BFG15], the rigidity analysis for hierarchical Coulomb
gases considered in [Cha19b], the local density for two-dimensional Coulomb
gases considered in [BBNY17], the Dobrushin – Lanford – Ruelle equations
considered in [DHLM21], the Coulomb gas properties on the sphere considered in
[BH19], the local laws and rigidity considered in [AS21], the quasi-Monte-
Carlo method on the sphere considered in [Ber19b], and the Berezinskii –
Kosterlitz – Thouless transition [KP17, GS20].

## Acknowledgments

We would like to thank David García-Zelada and Kilian Rashel for their helpful
remarks.

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