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# From random matrices to systems of particles in interaction

Valentin Pesce

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Valentin Pesce. From random matrices to systems of particles in interaction. 2025. hal-05143134

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Preprint submitted on 3 Jul 2025

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From random matrices to systems
of particles in interaction

Valentin Pesce *

July 3, 2025

Abstract

The goal of these expository notes is to give an introduction to random matrices for
non-specialist of this topic focusing on the link between random matrices and systems
of particles in interaction. We first recall some general results about the random matrix
theory that create a link between random matrices and systems of particles through
the knowledge of the law of the eigenvalues of certain random matrices models. We
next focus on a continuous in time approach of random matrices called the Dyson
Brownian motion. We detail some general methods to study the existence of system
of particles in singular interaction and the existence of a mean field limit for these
systems of particles. Finally, we present the main result of large deviations when
studying the eigenvalues of random matrices. This method is based on the fact that
the eigenvalues of certain models of random matrices can be viewed as log gases in
dimension 1 or 2.

***Keywords*** — Random matrices, Ginibre ensemble, Gaussian unitary ensemble, Wigner theorem, Circular law, Dyson Brownian motion, systems of particles in interaction, large deviations, Coulomb and Riesz gases

<footnote>*address: CMAP, Ecole Polytechnique, UMR 7641, 91120 Palaiseau, France
mailto:valentin.pesce@polytechnique.edu</footnote>
# Contents

1 Introduction 5

I Ginibre and Gaussian unitary ensembles 9

2 Examples of remarkable ensemble of matrices 9

2.1 Notations and first properties 9
2.1.1 Gaussian unitary ensemble 9
2.1.2 Ginibre ensemble 12

2.2 Law of the eigenvalues and eigenvectors 14
2.2.1 Gaussian unitary ensemble case 14
2.2.2 Ginibre case 20

2.3 Determinantal structure 20
2.3.1 Ginibre case 20
2.3.2 Gaussian unitary ensemble case 27

3 Circular law 30

3.1 Mean circular law 30

3.2 Strong circular law 32

3.3 Localisation of the spectrum 34

4 The Wigner theorem 40

4.1 The mean Wigner theorem 40

4.2 A concentration inequality 42

4.3 The strong Wigner theorem 48

4.4 Almost sure convergence of the smallest and the largest eigenvalue 49

II Dynamical approach to random matrices: the Dyson Brownian motion 52

5 Dynamic on the eigenvalues 52

5.1 Dyson Brownian motion 52

5.2 A differential approach of the spectrum of matrices 53

5.3 Particles in mean field interaction 54

6 Study of the system of particles 59

6.1 Notations 59

6.2 Formulas 60

6.3 Existence of solutions for the Dyson case 62

6.3.1 Existence thanks to a containment function 62

6.3.2 Solution of the Fokker Planck equation of the Dyson system 65

6.4 Interpretation of this system of particle 69

6.4.1 h-transform 70

6.4.2 Law of Brownian motions conditioned on the fact they do not intersect 71

7 Mean field limit of the system of particles 73

7.1 Empirical distribution 73

7.2 Compact subsets of $C([0, T], \mathcal{P}(\mathbb{R}))$ 75

7.3 Compactness of $(\mu_N)_{N \ge 1}$ 76

7.4 Convergence of $\mu_N$ 79

7.4.1 Characterization of the limit 79

7.4.2 Uniqueness of the limit and convergence 80

7.4.3 Application: the Wigner theorem 85

7.5 Case of Ornstein-Uhlenbeck process 88

7.5.1 Convergence of the system of particles 88

7.5.2 Long time behaviour and stationary solutions 90

III Large deviations 95
<table><thead><tr><th></th><th></th></tr></thead><tbody><tr><td>8 Notations and motivations</td><td>95</td></tr><tr><td>8.1 Recalls and notations</td><td>95</td></tr><tr><td>8.2 Motivation</td><td>97</td></tr><tr><td>8.3 Few definitions of large deviations</td><td>98</td></tr><tr><td>9 Free entropy and rate function</td><td>100</td></tr><tr><td>9.1 General study of the rate function</td><td>100</td></tr><tr><td>9.2 Equilibrium measure.</td><td>107</td></tr><tr><td>10 Large deviations</td><td>110</td></tr><tr><td>10.1 Large deviations principle of the normalizing factor</td><td>110</td></tr><tr><td>10.2 Weak large deviations principle</td><td>111</td></tr><tr><td>10.2.1 Upper bound (10.8)</td><td>114</td></tr><tr><td>10.2.2 Lower bound (10.7)</td><td>115</td></tr><tr><td>10.3 Exponential tightness and complete large deviations</td><td>117</td></tr><tr><td>10.4 Consequences of the large deviations principle</td><td>119</td></tr><tr><td>10.5 Computation of the equilibrium measure</td><td>120</td></tr><tr><td>11 Applications and further results</td><td>122</td></tr><tr><td>11.1 Convergence of the smallest and the largest eigenvalue</td><td>122</td></tr><tr><td>11.2 Extensions of the result</td><td>125</td></tr><tr><td>11.2.1 Coulomb and Riesz gases</td><td>125</td></tr><tr><td>11.2.2 Extension to other random matrices models and large deviations for the largest and smallest eigenvalue</td><td>127</td></tr><tr><td>Appendix</td><td>129</td></tr><tr><td>A The Wigner semicircle distribution</td><td>129</td></tr><tr><td>A.1 Moments</td><td>129</td></tr><tr><td>A.2 Stieltjes transformation</td><td>133</td></tr></tbody></table>
<table><caption>Table 1: Main frequently used notations.</caption>
  <tr>
    <td>x := y</td>
    <td>x is equal to y as a definition</td>
  </tr>
  <tr>
    <td>&#x211D;</td>
    <td>set of non negative integers {0, 1, ...}</td>
  </tr>
  <tr>
    <td>&#x211D;<sub>&ge;1</sub></td>
    <td>set of positive integers {1, 2, ...}</td>
  </tr>
  <tr>
    <td>&#x2121;<sub>A</sub></td>
    <td>indicator function of the set A</td>
  </tr>
  <tr>
    <td>A[X]</td>
    <td>set of polynomials with coefficients in the ring A</td>
  </tr>
  <tr>
    <td>M<sub>n,m</sub>(A)</td>
    <td>set of matrices of size n &times; m with coefficients in the ring A</td>
  </tr>
  <tr>
    <td>M<sub>N</sub>(A)</td>
    <td>set of matrices of size N &times; N with coefficients in the ring A</td>
  </tr>
  <tr>
    <td>&Delta;<sub>i,j</sub></td>
    <td>matrix with a 1 in (i, j) and 0 otherwise</td>
  </tr>
  <tr>
    <td>Tr, det</td>
    <td>trace and determinant of a matrix</td>
  </tr>
  <tr>
    <td>det[a<sub>i,j</sub>]<sub>1&le;i,j&le;N</sub></td>
    <td>determinant of the matrix A = (a<sub>i,j</sub>)<sub>1&le;i,j&le;N</sub></td>
  </tr>
  <tr>
    <td>&#x2122;, &#x2123;</td>
    <td>conjugate of a complex number z, conjugate matrix of matrix A</td>
  </tr>
  <tr>
    <td>A<sup>-1</sup>, A<sup>T</sup>, A<sup>*</sup></td>
    <td>inverse matrix of matrix A, transpose matrix of matrix A, conjugate transpose matrix of matrix A</td>
  </tr>
  <tr>
    <td>I<sub>N</sub></td>
    <td>identity matrix of size N &times; N</td>
  </tr>
  <tr>
    <td>&chi;<sub>A</sub>(X)</td>
    <td>characteristic polynomial of A</td>
  </tr>
  <tr>
    <td>&#x212C;<sub>N</sub>(C)</td>
    <td>set of hermitian complex matrices of size N &times; N</td>
  </tr>
  <tr>
    <td>S<sub>N</sub>(R)</td>
    <td>set of symmetric real matrices of size N &times; N</td>
  </tr>
  <tr>
    <td>GL<sub>N</sub>(C), U<sub>N</sub>(C)</td>
    <td>set of invertible matrices of M<sub>N</sub>(C), set of unitary matrices of M<sub>N</sub>(C)</td>
  </tr>
  <tr>
    <td>H, C &ndash; R</td>
    <td>set of complex numbers with positive imaginary part, set of complex numbers that are not real</td>
  </tr>
  <tr>
    <td>&lang;u|v&rang;, |||&rang;|&lang;2</td>
    <td>canonical scalar product between two vectors of &#x2122;<sup>N</sup> or &#x2123;<sup>N</sup> and its associated norm</td>
  </tr>
  <tr>
    <td>C<sup>k</sup>(I, E)</td>
    <td>set of functions from an interval I to a Banach space E with k continuous derivatives</td>
  </tr>
  <tr>
    <td>L<sup>k</sup>(E, &mu;)</td>
    <td>set of complex valued functions such that |f|<sup>k</sup> is integrable on E for the measure &mu;</td>
  </tr>
  <tr>
    <td>d&phi;<sub>a</sub></td>
    <td>Frechet derivative of &phi; in a</td>
  </tr>
  <tr>
    <td>&part;<sub>i</sub>f, &part;<sub>i</sub>f/&part;x<sub>i</sub></td>
    <td>partial derivative according to the i<sup>th</sup> vector of the canonical basis of the function f: &#x2122;<sup>N</sup> &rarr; &#x2122;</td>
  </tr>
  <tr>
    <td>&part;<sub>t</sub>f</td>
    <td>partial derivative of f: I &times; &#x2122;<sup>N</sup> with I an interval, with respect to the time variable</td>
  </tr>
  <tr>
    <td>&nabla;f, &nabla;<sup>2</sup>f</td>
    <td>gradient vector of f: &#x2122;<sup>N</sup> &rarr; &#x2122;, Hessian matrix of f: &#x2122;<sup>N</sup> &rarr; &#x2122;</td>
  </tr>
  <tr>
    <td>&Delta;f</td>
    <td>Laplacian of f (define as the trace of the Hessian matrix of f)</td>
  </tr>
  <tr>
    <td>F(f)</td>
    <td>Fourier transform of the function f: &#x2122;<sup>N</sup> &rarr; &#x2123;</td>
  </tr>
  <tr>
    <td>P(&#x2122;), P(A)</td>
    <td>set of probability measures on &#x2122;, set of probability measures with support included in A &sub; &#x2122;</td>
  </tr>
  <tr>
    <td>Leb</td>
    <td>Lebesgue measure on &#x2122;<sup>N</sup></td>
  </tr>
  <tr>
    <td>&mu;(dx) or d&mu;(x)</td>
    <td>integration with respect to the measure &mu;</td>
  </tr>
  <tr>
    <td>supp(f), supp(&mu;)</td>
    <td>support of the function f or the measure &mu;</td>
  </tr>
  <tr>
    <td>&mu; &otimes; &nu;</td>
    <td>product measure of &mu; and &nu;</td>
  </tr>
  <tr>
    <td>&mu;<sub>n</sub> &xrightarrow[n-&rarr;+&infin;]{} &mu;</td>
    <td>the sequence of probability measures on a metric space &#x2122; (&mu;<sub>n</sub>)<sub>n&ge;1</sub> converges in law (or weakly)</td>
  </tr>
  <tr>
    <td>iid</td>
    <td>to &mu; which means that for all f continuous and bounded on &#x2122; we have &int;<sub>&#x2122;</sub>fd&mu;<sub>n</sub> &rarr; &int;<sub>&#x2122;</sub>fd&mu;</td>
  </tr>
  <tr>
    <td>P</td>
    <td>independent and identically distributed</td>
  </tr>
  <tr>
    <td>&#x212E;, Var</td>
    <td>a probability</td>
  </tr>
  <tr>
    <td>L<sup>p</sup></td>
    <td>in the context of random variables: random variables such that &#x212E;(|X|<sup>p</sup>) &lt; +&infin;</td>
  </tr>
  <tr>
    <td>X ~ &mu;</td>
    <td>the law of the random variable X is &mu;</td>
  </tr>
  <tr>
    <td>&#x212N;(m, &sigma;<sup>2</sup>)</td>
    <td>Normal law of mean m and variance &sigma;<sup>2</sup></td>
  </tr>
  <tr>
    <td>&#x212E;(X|F)</td>
    <td>conditional expectation of the random variable X according to the filtration F</td>
  </tr>
  <tr>
    <td>GUE<sub>N</sub>/GOE<sub>N</sub></td>
    <td>Gaussian Unitary/Orthogonal ensemble of N &times; N matrix</td>
  </tr>
  <tr>
    <td>B(t)</td>
    <td>most of the time a standard Brownian motion</td>
  </tr>
  <tr>
    <td>|A|</td>
    <td>cardinal of the set A</td>
  </tr>
  <tr>
    <td>P(E)</td>
    <td>in the context of set: the set of subsets of E</td>
  </tr>
  <tr>
    <td>A<sup>c</sup></td>
    <td>complement of the subset A</td>
  </tr>
  <tr>
    <td>&#x212B;, &#x212B;<sup>&circ;</sup>, &part;A</td>
    <td>closure of the set A, interior of the set A, frontier of the set A</td>
  </tr>
  <tr>
    <td>i</td>
    <td>the element (0,1) of &#x2123; or an index of summation if it is an index</td>
  </tr>
  <tr>
    <td>&#x212E;(z), &#x212F;(z)</td>
    <td>real and imaginary part of the complex number z</td>
  </tr>
  <tr>
    <td>&Gamma;</td>
    <td>Euler Gamma function</td>
  </tr>
  <tr>
    <td>log</td>
    <td>natural Neperian logarithm function</td>
  </tr>
  <tr>
    <td>x<sub>+</sub>, x<sub>-</sub></td>
    <td>positive part, negative part of the real number x</td>
  </tr>
  <tr>
    <td>&#x2122;lim or lim sup<sub>N</sub></td>
    <td>superior limit</td>
  </tr>
  <tr>
    <td>&#x2122;lim or lim inf<sub>N</sub></td>
    <td>inferior limit</td>
  </tr>
  <tr>
    <td>&Delta;<sub>N</sub>(&lambda;<sub>1</sub>, ..., &lambda;<sub>N</sub>)</td>
    <td>most of the times Vandermonde determinant of the family (&lambda;<sub>1</sub>, ..., &lambda;<sub>N</sub>)</td>
  </tr>
  <tr>
    <td>&#x212C;<sub>N</sub></td>
    <td>symmetric group of cardinal N!</td>
  </tr>
  <tr>
    <td>&epsilon;(&sigma;)</td>
    <td>signature of the permutation &sigma; &isin; &#x212C;<sub>N</sub></td>
  </tr>
</table>
# 1 Introduction

These notes are intended for beginners in random matrices or for people interested in systems of particles in interaction as the Dyson Brownian motion or Coulomb and Riesz gases. It is recommended to have a graduate level in probability and to be familiar with the Itô calculus for Part II. There are many reference textbooks on the general topic of random matrices as for instance [5, 75, 64, 63] and the literature is already very wide. These notes shall focus on a very particular aspect of the study of large random matrices which is their links with systems of particles in interaction. For instance the Dyson Brownian motion is often presented as a related topic of random matrices but here we wanted to present the most possible self contained and accessible approach on this topic for people that are non familiar with the background of random matrices.

We can consider the works of Wishart in 1928 [86] as the starting point of the theory of large random matrices. The original motivation of Wishart was a statistical one. Given independent and identically distributed centred random vectors $X_1, ..., X_N$ of $\mathbb{R}^d$, he wanted to estimate the covariance matrix $\Sigma = \mathbb{E}[X_1X_1^T]$. Wishart proposed $\Sigma_N = \frac{1}{N} \sum_{k=1}^N X_kX_k^T := \frac{1}{N} \mathcal{X}_N \mathcal{X}_N^T$ as an estimator the empirical covariance matrix where $\mathcal{X}_N = (X_1, ..., X_N) \in M_{d,N}(\mathbb{R})$. As an application of the law of large numbers, $\Sigma_N$ converges almost surely towards $\Sigma$ when $N$ becomes large. In order to understand the behaviour of $\Sigma_N$ when for example the size $d_N$ of the data also increases with $N$, one can for instance study the behaviour of its spectrum as in the principal component analysis which is an important topic in statistics. In this sense Wishart was the first one to ask questions about large random matrices.

To understand the distribution of the spectrum of matrices a natural quantity to introduce is the so-called empirical distribution of the eigenvalues. This is the probability measure given by $\mu_N := \frac{1}{d_N} \sum_{k=1}^{d_N} \delta_{\lambda_k^N}(dx)$ where $(\lambda_k^N)_{1 \le k \le d_N}$ is the spectrum of $\Sigma_N$ and $\delta_z(dx)$ is the Dirac measure in $z$. Indeed, if we want to know how many eigenvalues of $\Sigma_N$ are in an interval $I = [a, b] \subset \mathbb{R}$, we have

$$ \int_a^b \mu_N(dx) = \frac{|\{1 \le i \le d_N, \lambda_i^N \in [a, b]\}|}{d_N}. $$

In Figure 1, we illustrate the histogram of the eigenvalues of $\Sigma_N$ when $N$ and $d_N$ are large.

<figure><img src="image_2.png" /><figcaption>Figure 1: Illustration of the Marchenko-Pastur theorem.</figcaption></figure>
The notion of convergence of probability measures that corresponds to the convergence of the histogram is the convergence in law (or also called the weak convergence of measures). We recall that $\mu_N$ converges in law towards a probability measure $\mu \in \mathcal{P}(\mathbb{R})$ if for all $f$ continuous and bounded real function

$$ \int_{\mathbb{R}} fd\mu_N \xrightarrow[N \to +\infty]{} \int_{\mathbb{R}} fd\mu $$

and we shall write $\mu_N \xrightarrow[N \to +\infty]{} \mu$ in this case. A central result on this topic was obtained by Marchenko and Pastur at the end of the 1960s [62]. For the following theorem assume that the entries of $\mathcal{X}_N$ are independent, identically distributed, centred and of variance 1.

**Theorem 1.1** (Marchenko-Pastur, Figure 1). Assume that there exists $c > 0$ such that $d_N$ satisfies $\frac{d_N}{N} \xrightarrow[N \to +\infty]{} c$. Let $(\lambda_1^N, ..., \lambda_{d_N}^N)$ be the $d_N$ eigenvalues of $\Sigma_N$ and $\mu_N := \frac{1}{d_N} \sum_{k=1}^{d_N} \delta_{\lambda_k^N}(dx)$ be the empirical distribution of the eigenvalues of $\Sigma_N$. Then, almost surely we have the following convergence

$$ \mu_N \xrightarrow[N \to +\infty]{} \rho_c(dx), $$

where $\rho_c$ is an explicit distribution called the Marchenko-Pastur law of parameter $c$ given by

$$ \rho_c(dx) = \left(1 - \frac{1}{c}\right)_+ \delta_0(dx) + \frac{1}{2\pi c x} \sqrt{(c^+ - x)(x - c^-)} \mathbf{1}_{[c^-, c^+]}(x)dx, \quad (1.1) $$

$$ \text{with } c^- = (1 - \sqrt{c})^2 \text{ and } c^+ = (1 + \sqrt{c})^2. $$

Then, in the 1950s, random matrices became an active field particularly of physic at the instigation of for instance Wigner and Dyson [85, 37]. Random matrices were used to model the levels of energies of complex quantum systems or systems of particles in interactions as Coulomb gases. In order to understand the energy levels of atoms physicists study the eigenvalues of Hamiltonian operators that are self-adjoint operators (in finite or infinite dimensions). An idea that emerged in the 1950s is that we could try to approximate the spectrum of such operators by the spectrum of large random matrices. This leads for instance to the question of the behaviour of the spectrum of large random symmetric matrices. The main result obtained by Wigner in [85] in this direction is the following one.

**Theorem 1.2** (Wigner, Figure 2). For all $N \ge 1$, let $M^N \in S_N(\mathbb{R})$ be a symmetric matrix such that $(M_{i,j}^N)_{1 \le i \le j \le N}$ is a family of independent centred random variables of variance 1. Let $(\lambda_1^N, ..., \lambda_N^N)$ be the real spectrum of $\frac{M^N}{\sqrt{N}}$ and $\mu_N := \frac{1}{N} \sum_{k=1}^N \delta_{\lambda_k^N}(dx)$ be the empirical distribution of the eigenvalues of $\frac{M^N}{\sqrt{N}}$. Then, almost surely we have the following convergence

$$ \mu_N \xrightarrow[N \to +\infty]{} \sigma(dx), $$

where $\sigma$ is an explicit distribution called the Wigner semicircle law given by

$$ \sigma(dx) = \frac{1}{2\pi} \sqrt{(4 - x^2)_+} \mathbf{1}_{[-2, 2]}(x)dx. \quad (1.2) $$
<figure><img src="image_3.png" /><figcaption>Figure 2: Illustration of the Wigner theorem.</figcaption></figure>

Random matrix theory is at the intersection of linear algebra and probability explain-
ing its mathematical richness. Moreover, random matrix theory gives new points of view
on many topics in mathematics and creates a path between some topics that were a priori
not connected. Since the seminal questions the random matrix theory has been used in
various fields. Here is a non exhaustive list of some active fields in which random matrices
appear:

• Ecology and large Lotka–Volterra models [2, 3]

* Coulomb and Riesz gases as the eigenvalues of some random matrices models can be considered as log gases [37, 74, 25]

• Zeros of random polynomials and random power series [54, 21]

* Physics [44] since for instance the distributions of fermions trapped in a harmonic potential is the same as the distribution the eigenvalues of random matrices [32] or in the study of levels of energy of atoms as in the original work of Wigner [85] or for complex systems as many-body quantum systems [84]

* Number theory through a deep connexion between the zeros of the Riemann zeta func-
tion and the eigenvalues of random matrices originally discovered by Montgomery and
Dyson [66, 19, 63]

• Random permutations [10]

* Last passage percolation (which is also linked with random permutations) [12, 43], the Brownian percolation which is connected with the Dyson Brownian motion [67] and more recently the so-called directed landscape which can be constructed as a limit object of last passage percolation [11, 31]

• Random graphs through the adjacency matrix [38, 8]

• Combinatorics of maps [48, 35]

* Free probability or non commutative probability introduced by Voiculescu to deal with the free group factors isomorphism problem in operator algebras theory [81, 65, 5]

• Statistics as in the original work of Wishart to estimate covariance matrices [86]

In these notes, we decided to focus on the link between random matrices and sys-
tem of particles in interaction.

In Part I, we shall mainly focus on two ensembles of matrices that are the so-called
Ginibre ensemble and the Gaussian unitary ensemble. From these models one can com-
pute the law of their eigenvalues. This will allow us to consider the eigenvalues as Riesz
or Coulomb log gases in dimension 1 and 2 subjected to an external quadratic potential.
Then, we shall detail how we can study the eigenvalues of Ginibre ensemble and Gaussian unitary ensemble as point processes in $\mathbb{R}$ or $\mathbb{C}$, in order to compute the correlation functions of the eigenvalues. From these computations, we shall obtain the so-called circular law and Wigner theorem that gives the statistics of the eigenvalues when the size of random matrices of the Ginibre ensemble or the Gaussian unitary ensemble becomes large.

In Part II, we deal with the so-called Dyson Brownian motion which is a continuous model of random matrices introduced by Dyson in 1962 [37]. From this continuous model of random matrices, one can obtain a system of stochastic differential equations for the dynamics of the eigenvalues, so that the eigenvalues can be considered as particles in interaction through a repulsive force and Brownian noise. As we will see, one can interpret the law of the process of the eigenvalues of the Dyson Brownian motion as the law of $N$ independent Brownian motions conditioned by the fact that they do not intersect. This approach gives another point of view and make some other proofs available for some results of random matrices viewed in Part I as the law of the eigenvalues of matrices of the Gaussian unitary ensemble or the Wigner theorem.

Finally, we study the large deviations of the empirical mean of the eigenvalues of matrices of the Gaussian unitary ensemble. This result was proved by Ben Arous and Guionnet in [6]. The method is quite robust and actually only uses the fact that the eigenvalues of random matrices of the Gaussian unitary ensemble is a log gas in dimension 1. The result obtained by Ben Arous and Guionnet is today extended for Coulomb and Riesz gases and we shall explain those results in Part III. This gives another proof for the Wigner theorem and the circular law presented in Part I.

Before starting, let us mention that notations between parts are independent except for the notations of table 1.
# Part I

## Ginibre and Gaussian unitary ensembles

In this section, we shall focus on the two most famous ensembles of matrices which are the Gaussian unitary ensemble and the Ginibre ensemble. They correspond to hermitian and non hermitian matrices with independent Gaussian entries.

These models were particularly studied because they have a certain structure that help doing explicit computations as we shall explain. Moreover, in random matrix theory it is expected a so-called "universality" phenomenon which morally says that the behaviour of spectral statistics of a random matrix is similar with the behaviour with independent Gaussian entries when the size of the matrix becomes large. Typically, the circular law or the Wigner theorem, that shall be proved in this section for Ginibre matrices and matrices of the Gaussian unitary ensemble, are relevant examples.

The remarkable point about matrices of the Gaussian unitary ensemble or the Ginibre ensemble is that the law of the eigenvalues of such matrices can be computed explicitly and has a certain form. Indeed, the eigenvalues of such matrix models are real or complex determinantal point processes. Thanks to this remark we can compute the correlation functions of the eigenvalues of matrices in the Ginibre or in the Gaussian unitary ensemble. There are many references for these standard results as for instance [5, 50, 64, 75, 25, 63].

Then, we prove that up to a re normalization, the eigenvalues of Ginibre matrices are distributed uniformly on the disk when the size of the matrix becomes large. This is called the circular law and was originally proved by Mehta and Ginibre [64]. About the circular law, we refer to the detailed revue of Bordenave and Chafaï [18] and the revue of Chafaï [25].

We finally show the Wigner theorem for matrices of the Gaussian unitary ensemble which is the counterpart of the circular law. We refer to [75, 50, 5] for this result.

All the random variables are defined on a same probability space $(\Omega, \mathcal{F}, \mathbb{P})$.

## 2 Examples of remarkable ensemble of matrices

In this part we introduce the Ginibre ensemble and the Gaussian unitary ensemble and we point out some basic properties they have in common.

### 2.1 Notations and first properties

#### 2.1.1 Gaussian unitary ensemble

Let $N \ge 1$ be a positive integer.

**Definition 2.1.** A random variable $X$ is said to be a $N$ complex (resp. real) Wigner matrix if $X$ takes value in $\mathcal{H}_N(\mathbb{C})$ (resp. $S_N(\mathbb{R})$) and $(X_{i,i}, \sqrt{2}\mathfrak{Re}(X_{i,j}), \sqrt{2}\mathfrak{Im}(X_{i,j}))_{1\le i<j\le N}$ is a family of independent Gaussian random variables of law $\mathcal{N}(0, 1)$ (resp. $(X_{i,i}, \sqrt{2}X_{i,j})_{1\le i<j\le N}$ is a family of independent Gaussian random variables of law $\mathcal{N}(0, 1)$).

If $X$ is a $N$ complex (resp. real) Wigner matrix, we say that $X$ is in the Gaussian Unitary Ensemble (resp. Gaussian Orthogonal Ensemble). We shall write $X \sim \text{GUE}_N$ (resp. $X \sim \text{GOE}_N$).

For a random variable in the $\text{GUE}_N$ or in the $\text{GOE}_N$, we denote $(\lambda_1, \dots, \lambda_N) \in \mathbb{R}^N$ the vector of its eigenvalues viewed as an exchangeable vector of $\mathbb{R}^N$.
Let us mention that in the literature what we call here Wigner matrices are most of the times called Gaussian Wigner matrices since the entries of the matrix are Gaussian. Since, in these notes we shall only focus on the case when the entries are Gaussian we decided to use the terminology of Wigner matrices instead of Gaussian Wigner matrices. We will now focus on $N$ complex Wigner matrix. To compute the law of a $N$ complex Wigner matrix, we first compute the dimension of $\mathcal{H}_N(\mathbb{C})$ as a $\mathbb{R}$ vector space.

**Lemma 2.2.** $\mathcal{H}_N(\mathbb{C})$ is a $\mathbb{R}$ vector space of dimension $N^2$ such that $((\Delta_{k,k})_{k \le N}, (\Delta_{k,j} + \Delta_{j,k})_{1 \le k < j \le N}, (i(\Delta_{k,j} - \Delta_{j,k})_{1 \le k < j \le N}))$ is a basis.

*Proof.* Let $M = (M_{k,j})_{1 \le k,j \le N} \in \mathcal{H}_N(\mathbb{C})$. By definition of $\mathcal{H}_N(\mathbb{C})$, for all $1 \le k \le N$, $M_{k,k} \in \mathbb{R}$ and for all $1 \le k < j \le N$, $\mathfrak{Re}(M_{k,j}) = \mathfrak{Re}(M_{j,k})$ and $\mathfrak{Im}(M_{k,j}) = -\mathfrak{Im}(M_{j,k})$. So we can write:

$$M = \sum_{1 \le k \le N} M_{k,k} \Delta_{k,k} + \sum_{1 \le k < j \le N} \mathfrak{Re}(M_{k,j}) (\Delta_{k,j} + \Delta_{j,k}) + \sum_{1 \le k < j \le N} \mathfrak{Im}(M_{k,j}) i (\Delta_{k,j} - \Delta_{j,k}).$$

This gives the result.

<figure><img src="image_4.png" /></figure>

We shall sometimes identify $\mathcal{H}_N(\mathbb{C})$ and $\mathbb{R}^{N^2}$ by expressing a matrix of $\mathcal{H}_N(\mathbb{C})$ in the basis of Lemma 2.2.

**Proposition 2.3.** Let $X$ be a $N$ complex Wigner matrix, then $X$ has a density with respect to the Lebesgue measure on $\mathbb{R}^{N^2}$ which is proportional to $M \mapsto \exp(-\mathrm{Tr}(M^2)/2)$.

*Proof.* Let $f \ge 0$ be a measurable function defined on $\mathbb{R}^{N^2} = \mathcal{H}_N(\mathbb{C})$. By definition of a Wigner matrix, there exists a normalizing constant $c_N$ such that

$$\mathbb{E}[f(X)] = c_N \int_{\mathbb{R}^{N^2}} f((Y_{i,j})_{1 \le i, j \le N}) \exp \left( - \sum_{1 \le i \le N} \frac{Y_{i,i}^2}{2} - \sum_{1 \le i < j \le N} (\mathfrak{Re}(Y_{i,j})^2 + \mathfrak{Im}(Y_{i,j})^2) \right) d \mathrm{Leb}(Y).$$

So we get:

$$\mathbb{E}[f(X)] = c_N \int_{\mathbb{R}^{N^2}} f((Y_{i,j})_{1 \le i, j \le N}) \exp \left( -\frac{1}{2} \sum_{1 \le i,j \le N} |Y_{i,j}|^2 \right) d \mathrm{Leb}(Y).$$

We write:

$$\mathbb{E}(f(X)) = c_N \int_{\mathcal{H}_N(\mathbb{C})} f((Y_{i,j})_{1 \le i, j \le N}) \exp \left( -\frac{1}{2} \mathrm{Tr}(Y^2) \right) d \mathrm{Leb}(Y).$$

Since this formula is true for all $f \ge 0$ measurable, we get the result.

<figure><img src="image_5.png" /></figure>

The next proposition explains the terminology of the Gaussian Unitary Ensemble and show an invariance of the law with respect to multiplication by a unitary matrices. A similar result holds for the Gaussian orthogonal ensemble with orthogonal matrices instead of unitary matrices. We first state a lemma in order do a change a variable.

**Lemma 2.4.** Let $V \in \mathcal{U}_N(\mathbb{C})$ and $\psi_V : \mathcal{H}_N(\mathbb{C}) \to \mathcal{H}_N(\mathbb{C})$ defined by $\psi_V(X) = VXV^*$. Then we have $|\det(\psi_V)| = 1$.
*Proof.* We define $d_{V'} = |\det(\psi_{V'})| \in \mathbb{R}^+$ for all $V' \in \mathcal{U}_N(\mathbb{C})$. Then for all $V' \in \mathcal{U}_N(\mathbb{C})$ and $V'' \in \mathcal{U}_N(\mathbb{C})$ we have $d_{V'}d_{V''} = d_{V'V''}$ since $\psi_{V'} \circ \psi_{V''} = \psi_{V'V''}$. This gives that $d_V d_{V*} = 1$ and so $d_V \neq 0$ and for all $k \in \mathbb{Z}$, $d_{Vk} = d_V^k$.

Since $(d_{Vk})_{k \in \mathbb{N}}$ is a sequence of matrices of $\mathcal{U}_N(\mathbb{C})$, which is a compact group, we can extract a subsequence that converges to $\tilde{V} \in \mathcal{U}_N(\mathbb{C})$. By continuity, there exists a subsequence of $(d_{Vk})_{k \in \mathbb{N}}$ that converges to $d_{\tilde{V}} \neq 0$. Since $(d_{Vk})_{k \in \mathbb{N}}$ is a non negative real geometric sequence of reason $d_V$, it converges to an element of $\mathbb{R}^{+*}$ if and only if $d_V = 1$ which proves the result. $\square$

**Proposition 2.5.** Let $U \in \mathcal{U}_N(\mathbb{C})$ be a deterministic unitary matrix and $X$ be a $N$ complex Wigner matrix, then $U^*XU$ is also a $N$ complex Wigner matrix.

*Proof.* The application $\phi : \mathcal{H}_N(\mathbb{C}) \to \mathcal{H}_N(\mathbb{C})$ defined by $\phi(X) = U^*XU$ has a Jacobian by Lemma 2.4. Using the law of a $N$ complex Wigner matrix obtained in Proposition 2.3 and doing a change of variables yields that the density of $U^*XU$ is proportional to $Y \mapsto \exp\left(-\frac{1}{2}\text{Tr}((UYU^*)^2)\right)$.

By property of the trace $\text{Tr}((UYU^*)^2) = \text{Tr}(UY^2U^*) = \text{Tr}(Y^2U^*U) = \text{Tr}(Y^2)$, which yields that the density of $U^*XU$ is proportional to $Y \mapsto \exp\left(-\frac{1}{2}\text{Tr}(Y^2)\right)$. Hence $U^*XU$ and $X$ have the same law. $\square$

An important property about the spectrum of a $N$ complex Wigner matrix is that the spectrum is almost surely simple.

**Proposition 2.6.** Let $M$ be a $N$ complex Wigner matrix. Then almost surely the spectrum of $M$ is simple (all the eigenvalues are distinct) and $\det(M) \neq 0$.

*Proof.* Let $\Lambda^0 = (\lambda_1^0, ..., \lambda_N^0) \in \mathbb{R}^N$ be the random spectrum of $M$. Let us introduce a quantity linked with the so-called discriminant of a polynomial. For $\Lambda = (\lambda_1, ..., \lambda_n) \in \mathbb{R}^N$, let $\mathcal{D}(\Lambda) := \prod_{i \neq j}(\lambda_i - \lambda_j)$. We notice that the spectrum of $M$ is simple if and only if $\mathcal{D}(\Lambda^0) = \prod_{i \neq j}(\lambda_i^0 - \lambda_j^0) \neq 0$.

Since $\mathcal{D}$ is a symmetric polynomial in the $\lambda_i$ (which means that for all permutation $\sigma \in \mathfrak{S}_N$, for all $(\lambda_1, ..., \lambda_N) \in \mathbb{R}^N$ we have that $\mathcal{D}(\lambda_1, ..., \lambda_N) = \mathcal{D}(\lambda_{\sigma(1)}, ..., \lambda_{\sigma(N)})$), the fundamental theorem of the symmetric polynomial (the so-called Newton theorem) gives that $\mathcal{D} \in \mathbb{R}[\Sigma_1, ..., \Sigma_N]$ with $(\Sigma_i)_{1 \le i \le N}$ the symmetric polynomials associated to the $\lambda_i$. Given $\Lambda \in \mathbb{R}^N$, let us consider a matrix $M^\Lambda$ in $\mathcal{M}_N(\mathbb{C})$ such that the set of its eigenvalues is exactly $\Lambda$. By the fundamental properties of the determinant, the coefficients of $M^\Lambda$ and its characteristic polynomial are in the same ring. Since the coefficients of the characteristic polynomial of $M^\Lambda$ are up to the sign the $\Sigma_i$ we have that the $\Sigma_i$ are in the same ring as the coefficients of $M^\Lambda$.

Hence, we deduce that there exists $R \in \mathbb{R}[(X_{i,j})_{1 \le i,j \le N}]$ such that for all $\Lambda \in \mathbb{R}^N$, $\mathcal{D}(\Lambda) = R((M_{i,j}^\Lambda)_{1 \le i,j \le N})$. So we get that the spectrum of $M$ is simple if and only if $\mathcal{D}(\Lambda^0) = R((M_{i,j}^0)_{1 \le i,j \le N}) \neq 0$. Let us recall a basic lemma which can be proved by induction on the dimension.

**Lemma 2.7.** Let $X \in \mathbb{R}^d$ be a random variable with a density with respect to the Lebesgue measure of $\mathbb{R}^d$. Then, if $P \in \mathbb{R}[Z_1, ..., Z_d]$ is not null, we almost surely have $P(X) \neq 0$.

*Proof.* We sketch the proof. We prove the result by induction on $d$.

If $d=1$, if we denote $\{x_1, ..., x_k\} \in \mathbb{R}^k$ the real roots of $P$ (with possibly no real roots if $k=0$) then $P(X) = 0$ if and only if $X \in \{x_1, ..., x_k\}$. Since $X$ has a density with respect
to the Lebesgue measure, $\mathbb{P}(X \in \{x_1, ..., x_k\}) = 0$ proving the result.

We suppose that the result is established for $d \ge 1$. We write $X = (X_1, ..., X_{d+1})$. We consider $P \in \mathbb{R}[Z_1, ..., Z_{d+1}]$ a non null polynomial. We can suppose that $P$ depends on all the $(Z_i)_{1 \le i \le d+1}$. Indeed if it is not the case, we can for instance suppose that $P \in \mathbb{R}[Z_1, ..., Z_d]$ and we can use the induction hypothesis to $P$ and $(X_1, ..., X_d)$ which also has a density with respect to the Lebesgue measure in $\mathbb{R}^d$ giving the result. We now suppose that $P$ depends on all the variables. Conditioning by $X_1$ yields

$$ \mathbb{P}(P(X) = 0) = \mathbb{E}(\mathbb{P}(P(X) = 0 | X_1)). \qquad (2.1) $$

Conditionally on $X_1$, $X$ has a density with respect to the Lebesgue measure of $\mathbb{R}^d$ and for all $\omega \in \Omega$, we can define the polynomial $Q_\omega(Z_2, ..., Z_{d+1})$ of $\mathbb{R}[Z_2, ..., Z_{d+1}]$ as $Q_\omega(Z_2, ..., Z_{d+1}) = P(X_1(\omega), Z_2, ..., Z_{d+1})$. Applying the induction hypothesis for all $\omega \in \Omega$ gives that $\mathbb{P}(P(X) = 0 | X_1)(\omega) = \mathbb{P}(P(X_1, X_2, ..., X_{d+1}) = 0 | X_1)(\omega)$ is equal to 0 if $Q_\omega(Z_2, ..., Z_{d+1})$ is not the null polynomial and is equal to 1 if $Q_\omega(Z_2, ..., Z_{d+1})$ is the null polynomial. So we obtain

$$ \mathbb{P}(P(X) = 0 | X_1)(\omega) = \mathbb{1}_{Q_\omega(Z_2, ..., Z_{d+1})=0} = \mathbb{1}_{P(X_1(\omega), Z_2, ..., Z_{d+1})=0}. $$

Taking the expectation and using (2.1) yields

$$ \mathbb{P}(P(X) = 0) = \mathbb{P}(P(X_1(\omega), Z_2, ..., Z_{d+1}) = 0). $$

Writing $P \in \mathbb{R}[Z_1, ..., Z_{d+1}]$ as an element of $\mathbb{R}[Z_1][Z_2, ..., Z_{d+1}]$, as $P$ is not the null polynomial, we can find $k \ge 1$ non null polynomials of $\mathbb{R}[Z_1]$ denoted $\tilde{P}_1, ..., \tilde{P}_k$ such that $P(x_1, Z_2, ..., Z_{d+1}) = 0$ if and only if $\tilde{P}_1(x_1) = \tilde{P}_2(x_1) = ... = \tilde{P}_k(x_1) = 0$. So we have

$$ \mathbb{P}(P(X) = 0) = \mathbb{P}(P(X_1(\omega), Z_2, ..., Z_{d+1}) = 0) = \mathbb{P}(\tilde{P}_1(X_1(\omega)) = ... = \tilde{P}_k(X_1(\omega)) = 0). $$

Using the case $d = 1$ we obtain the result. $\square$

Thanks to this lemma we deduce that almost surely the spectrum of $M$ is simple. Similarly, the determinant of a matrix is a polynomial in its entries and so Lemma 2.7 gives that $\det(M) \ne 0$ almost surely. $\square$

Let us mention that in the previous proof we do not use the explicit law of the entries of the matrix. The only important points are the independence of the entries and the fact that the entries have a density with respect to the Lebesgue measure.

### 2.1.2 Ginibre ensemble

Let $N \ge 1$ be a positive integer.

**Definition 2.8.** A $N$ complex Ginibre random matrix is a random variable with value in $M_N(\mathbb{C})$ such that for all $(\mathfrak{R}c(M_{i,j}), \mathfrak{I}c(M_{i,j}))_{1 \le i,j \le N}$ is a family of independent Gaussian random variable of law $N(0, 1/2)$ (for all $1 \le i, j \le N$, $M_{i,j}$ is said to be a standard complex Gaussian random variable).

For a $N$ complex Ginibre matrix, we denote $(\lambda_1, ..., \lambda_N) \in \mathbb{C}^N$ the vector of its complex eigenvalues viewed as an exchangeable vector of $\mathbb{C}^N$.

Now we shall show that a $N$ complex Ginibre random matrix has a density with respect to the Lebesgue measure of $\mathbb{C}^{N^2}$. For all $k \ge 0$, we define the Lebesgue measure on $\mathbb{C}^k$,
which shall be written $dz_1...dz_k$, as the Lebesgue measure on $\mathbb{R}^{2k}$ with the identification of $\mathbb{C}$ as $\mathbb{R}^2$. More exactly it means that for suitable $f$ defined on $\mathbb{C}^k$:

$$ \int_{\mathbb{C}^k} f(z_1, \dots, z_k) dz_1 \dots dz_k = \int_{\mathbb{R}^{2k}} f(x_1, y_1, x_2, y_2, \dots, x_k, y_k) dx_1 dy_1 dx_2 dy_2 \dots dx_k dy_k, $$

where $z_j = x_j + iy_j$ and $dz_j := dx_j dy_j$ for all $1 \le j \le k$.

**Proposition 2.9.** Let $M$ be a $N$ complex Ginibre random matrix. The law of $M$ has a density with respect to the Lebesgue measure of $\mathbb{C}^{N^2}$ which is proportional to

$$ M \mapsto \exp \left( - \sum_{i,j=1}^{N} |M_{i,j}|^2 \right) = \exp(-\mathrm{Tr}(MM^*)) $$

with $M^* = \overline{M}^T$ the transconjugate of $M$.

*Proof.* By definition of a $N$ complex Ginibre random matrix, for all $1 \le i, j \le N$, $\Re(M_{i,j})$ and $\Im(M_{i,j})$ are real Gaussian random variables with densities with respect to the real Lebesgue measure proportional to $x \mapsto \exp(-x^2)$. By independence, we get that for all $1 \le i, j \le N$, $M_{i,j}$ (as an element of $\mathbb{C}$ viewed as $\mathbb{R}^2$) has a density with respect to the complex Lebesgue measure proportional to

$$ z = (x, y) \mapsto \exp(-(x^2 + y^2))dxdy = \exp(-|z|^2)dz. $$

By independence of $(M_{i,j})_{1 \le i,j \le N}$ we have the result. $\square$

Let us state some basics properties of Ginibre random matrices. First, the law of a $N$ Complex Ginibre matrix is invariant by unitary change of basis. We first state a counterpart of Lemma 2.4 whose proof is totally similar.

**Lemma 2.10.** Let $(U, V) \in \mathcal{U}_N(\mathbb{C})^2$ and $\psi_V : \mathcal{M}_N(\mathbb{C}) \to \mathcal{M}_N(\mathbb{C})$ defined by $\psi_V(X) = VX$ and $\tilde{\psi}_U : \mathcal{M}_N(\mathbb{C}) \to \mathcal{M}_N(\mathbb{C})$ defined by $\tilde{\psi}_U(X) = XU$. Then we have $|\det(\psi_V)| = |\det(\tilde{\psi}_U)| = 1$.

**Proposition 2.11.** If $M$ is a $N$ complex Ginibre random matrix and $U, V \in \mathcal{U}_N(\mathbb{C})$ are deterministic matrices, then $UMV$ has the law of a $N$ complex Ginibre matrix.

*Proof.* Using Proposition 2.9 and doing a change of variables using Lemma 2.10, we get that $UMV$ has a density with respect to the complex Lebesgue measure which is proportional to

$$ \begin{aligned} M &\mapsto \exp(-\mathrm{Tr}(U^{-1}MV^{-1}(U^{-1}MV^{-1})^*)) = \exp(-\mathrm{Tr}(U^{-1}MM^*(U^{-1})^*)) \\ &= \exp(-\mathrm{Tr}(MM^*)) \end{aligned} $$

where we used that $UU^* = I_N$, $VV^* = I_N$ and the properties of the trace. $\square$

A direct computation gives us a formal interpretation of a $N$ complex Ginibre matrix when $N$ is large.

**Proposition 2.12.** Let $(M_{i,j})_{(i,j) \in \mathbb{N}^2}$ be a family of independent standard complex Gaussian random variables. For all $N \ge 1$, let $M_N = (M_{i,j})_{1 \le i,j \le N}$ be a $N$ complex Ginibre. Then almost surely $M_N/\sqrt{N}$ has orthonormal rows/columns when $N$ goes to $+\infty$.

*Proof.* The result is an immediate application of the law of large number since for all $1 \le i, j, k, l \le N$ with $(i, j) \ne (k, l)$ we have that $\mathbb{E}(|M_{i,j}|^2) = 1$ and $\mathbb{E}(M_{i,j} \overline{M_{k,l}}) = 0$. $\square$
Formally, Proposition 2.12 explains that given a $N$ complex Ginibre matrix $M_N$ when $N$ is large, $M_N/\sqrt{N}$ is like a unitary matrix whose law is invariant by left and right multiplication by a unitary matrix of $U_N(\mathbb{C})$ by Proposition 2.11. This shall be linked with the notion of Haar measure on $U_N(\mathbb{C})$ as we shall see in next section. Moreover, since the spectrum of a unitary matrix is located on the circle we can formally expect that the spectrum of $M_N/\sqrt{N}$ will be located near the circle when $N$ becomes large. This shall be proved in next section proving the so-called circular law. Finally, as for the Wigner case, one can show that the spectrum of a $N$ complex Ginibre matrix is simple almost surely.

**Proposition 2.13.** Let $M$ be a $N$ complex Ginibre matrix. Then almost surely the spectrum of $M$ is simple (all the eigenvalues are distinct) and $\det(M) \neq 0$. Hence almost surely, $M$ is diagonalizable.

*Proof.* The exact same proof as for complex Wigner matrices can be done. $\square$

## 2.2 Law of the eigenvalues and eigenvectors

Let $N \ge 1$ be a positive integer. In this section we shall explicitly give the law of the eigenvalues of a $N$ complex Wigner and Ginibre matrix. We first give the details for the Wigner case and then give the counterpart results for the Ginibre case.

### 2.2.1 Gaussian unitary ensemble case

**Definition 2.14.** A function $f: M_N(\mathbb{C}) \to \mathbb{C}$ is said to only depend on the spectrum if for all $M \in M_N(\mathbb{C})$ and $U \in GL_N(\mathbb{C})$ we have $f(UMU^{-1}) = f(M)$.

To give the law of the eigenvalues, the difficult point is that we can not associate canonically to a matrix its eigenvalues and eigenvectors in order to do properly a change of variable. It is actually possible by doing the so-called Weyl integration formula and integrating in quotient spaces. See Section 4.1 of [5] or Section 3.1 of [63]. We will not do this method.

To overcome this problem, in this section we shall find a law on eigenvectors and eigenvalues that matches with the law of a complex Wigner matrix.

**Eigenvectors** Let us recall a general result about the notion of Haar measures on compact groups.

**Definition 2.15.** Let $G$ be a compact metrizable group. A measure is said to be a Haar measure on $G$ if and only if for all $g \in G$, for all $A$ measurable $\mu(gA) = \mu(A)$ (this property is called left invariance).

**Remark 2.16.** The Haar measure could be more generally define for a topological group and generalizes the Lebesgue measure in $\mathbb{R}$.

The following theorem gives the existence of a unique Haar probability measure.

**Theorem 2.17 (Haar).** Let $G$ be a metrizable compact group. All Haar measures on $G$ are finite and are proportional to a unique Haar probability measure which shall be denoted $\mu_G$ which is also right invariant.

The existence and uniqueness of a Haar measure for a compact group can be obtained using the so-called Kakutani fixed point theorem. We shall not detail this general proof that does not explain how to construct explicitly a such Haar measure. However, in the particular case of $U_N(\mathbb{C})$ one can construct explicitly a Haar measure.
**Proposition 2.18.** There exists a Haar probability measure on $\mathcal{U}_N(\mathbb{C})$.

*Proof.* This construction is based on the Gram-Schmidt algorithm. Starting from a matrix $M \in GL_n(\mathbb{C})$, we write $C_1, ..., C_N$ its column. We construct an orthonormal basis associated to the basis $(C_1, ..., C_N)$ of $\mathbb{C}^N$. We first define $\tilde{C}_1 := C_1/||C_1||_2$, we then define

$$ \tilde{C}_2 := \frac{C_2 - \frac{1}{||C_1||_2^2} \langle C_1 | C_2 \rangle C_1}{||C_2 - \frac{1}{||C_1||_2^2} \langle C_1 | C_2 \rangle C_1||_2}, $$

and so on. We construct an orthonormal basis of $\mathbb{C}^N$ $(\tilde{C}_1, ..., \tilde{C}_N)$. We define the Gram-Schmidt map $\mathcal{G}: GL_N(\mathbb{C}) \to \mathcal{U}_N(\mathbb{C})$ by $\mathcal{G}(M) = (\tilde{C}_1, ..., \tilde{C}_N)$ if $M = (C_1, ..., C_N)$. An important property of the Gram-Schmidt algorithm is that it commutes with the multiplication by a matrix of $\mathcal{U}_N(\mathbb{C})$. More exactly, let $U \in \mathcal{U}_N(\mathbb{C})$ and $M = (C_1, ..., C_N) \in GL_N(\mathbb{C})$. Then $\mathcal{G}(UM) = U\mathcal{G}(M)$. Indeed let $UM = (UC_1, ...UC_N)$. Then using that $U \in \mathcal{U}_N(\mathbb{C})$ preserves the norm and the scalar product, we have

$$ \begin{aligned} \tilde{U}\tilde{C}_1 &= \frac{UC_1}{||UC_1||_2} = U \frac{C_1}{||C_1||_2} = U\tilde{C}_1 \\ \tilde{U}\tilde{C}_2 &= \frac{UC_2 - \frac{1}{||UC_1||_2^2} \langle UC_1|UC_2 \rangle UC_1}{||UC_2 - \frac{1}{||UC_1||_2^2} \langle UC_1|UC_2 \rangle UC_1||_2} = U \frac{C_2 - \frac{1}{||C_1||_2^2} \langle C_1|C_2 \rangle C_1}{||C_2 - \frac{1}{||C_1||_2^2} \langle C_1|C_2 \rangle C_1||_2} = U\tilde{C}_2, \end{aligned} $$

and so on. So we obtain that for all $1 \le k \le N$, $\tilde{U}\tilde{C}_k = U\tilde{C}_k$.

We can now construct a random variable with law a Haar measure on $\mathcal{U}_N(\mathbb{C})$. Consider a $N$ complex Ginibre random matrix $M_N$. By Proposition 2.11, we know that the law of $M_N$ is invariant by multiplication by a deterministic matrices of $\mathcal{U}_N(\mathbb{C})$. We consider the random variable $\mathcal{G}(M_N)$ which is defined almost surely by Proposition 2.13. Then, for $U \in \mathcal{U}_N(\mathbb{C})$ we have:

$$ U\mathcal{G}(M_N) = \mathcal{G}(UM_N) \stackrel{\text{Law}}{=} \mathcal{G}(M_N), $$

using the fact that the multiplication by $U$ and $\mathcal{G}$ commute. Hence the law of $\mathcal{G}(M_N)$ is a Haar probability measure on $\mathcal{U}_N(\mathbb{C})$. $\square$

**Remark 2.19.** Another more geometric point of view would be to sample uniformly on the sphere a vector, then sample an orthogonal vector to this one on the unit sphere and so on to create $N$ orthogonal vectors. We refer to the first chapter of [63] for this construction.

Since $\mathcal{U}_N(\mathbb{C})$ is a compact group one can show some general properties of its Haar probability measures.

**Lemma 2.20.** Let $\mu$ be a Haar probability measure of $G$ a compact metrizable group. Then for all $A$ measurable subset and for all $g \in G$, we have:

* $\mu(A^{-1}) = \mu(A)$ with $A^{-1} = \{g^{-1}, g \in A\}$,

* $\mu(Ag) = \mu(A)$,

* If $\nu$ is another Haar probability measure, then $\mu = \nu$.
Proof. • We directly compute using the left invariance of $\mu$ and the Fubini-Tonelli theorem:

$$
\begin{align*}
\mu(A) &= \int_G \mu(gA)d\mu(g) \\
&= \int_G \int_G 1_{gA}(z)d\mu(z)d\mu(g) \\
&= \int_G \int_G 1_{zA^{-1}}(g)d\mu(g)d\mu(z) \tag{2.2} \\
&= \int_G \mu(zA^{-1})d\mu(z) \\
&= \mu(A^{-1}).
\end{align*}
$$

• Using the previous computation and the left invariance of $\mu$ yields

$$
\mu(Ag) = \mu((Ag)^{-1}) = \mu(g^{-1}A^{-1}) = \mu(A^{-1}) = \mu(A).
$$

• We do the same computation as in (2.2) using this time $\nu$ we get that for every B measurable:

$$
\begin{align*}
\mu(B) &= \int_G \mu(gB)d\nu(g) \\
&= \int_G \int_G 1_{gB}(z)d\mu(z)d\nu(g) \\
&= \int_G \int_G 1_{zB^{-1}}(g)d\nu(g)d\mu(z) \tag{2.3} \\
&= \int_G \nu(zB^{-1})d\mu(z) \\
&= \nu(B^{-1}) = \nu(B).
\end{align*}
$$

□

As a consequence of the existence of the Haar measure on $\mathcal{U}_N(\mathbb{C})$, we can give the law of the eigenvectors of a $N$ complex Wigner matrix matrix.

**Proposition 2.21.** There exists a measure $\mu_\Lambda$ on $\mathbb{R}^N$ such that if $\Lambda$ is a diagonal matrix such that $(\Lambda_{1,1}, \dots, \Lambda_{N,N})$ has the law $\mu_\Lambda$ and $U$ is a random variable independent of $\Lambda$ whose law is $\mu_{\mathcal{U}_N(\mathbb{C})}$ then $U\Lambda U^*$ is a $N$ complex Wigner matrix.

Proof. Let $X$ be a $N$ complex Wigner matrix. By the spectral theorem we can find $W \in \mathcal{U}_N(\mathbb{C})$ and $\Lambda$ a real diagonal matrix which are random variables and such that $X = W\Lambda W^*$. Let $V$ be a matrix distributed according to $\mu_{\mathcal{U}_N(\mathbb{C})}$ independent of $W$ and $\Lambda$. Then the couple $(\Lambda, U := \Lambda, VW)$ satisfies the properties of the proposition. Indeed we first prove that $U\Lambda U^* = X$. Let $f$ be a bounded continuous function, then we have:

$$
\begin{align*}
\mathbb{E}(f(U\Lambda U^*)) &= \mathbb{E}(f(V(W\Lambda W^*)V^*)) \\
&\stackrel{\text{Independence}}{=} \int_{\mathcal{U}_N(\mathbb{C})} \mathbb{E}(f(\nu W\Lambda W^*\nu^*))d\mu_{\mathcal{U}_N(\mathbb{C})}(\nu) \\
&= \int_{\mathcal{U}_N(\mathbb{C})} \mathbb{E}(f(\nu X\nu^*))d\mu_{\mathcal{U}_N(\mathbb{C})}(\nu) \\
&= \int_{\mathcal{U}_N(\mathbb{C})} \mathbb{E}(f(X))d\mu_{\mathcal{U}_N(\mathbb{C})}(\nu) = \mathbb{E}(f(X)),
\end{align*}
$$
where we used the invariance of the law of X by a unitary change of basis. It proves the first point.

Secondly we must look at the law of $(\Lambda, U)$. Let $f, g$ two bounded continuous functions, then we have:

$$
\begin{align*}
\mathbb{E}(f(U)g(\Lambda)) &= \mathbb{E}(f(VW)g(\Lambda)) \\
&= \mathbb{E}[\mathbb{E}(f(VW)g(\Lambda)|W, \Lambda)] \\
\text{Independence} &\stackrel{\text{Haar}}{=} \mathbb{E} \left[ \int_{\mathcal{U}_N(\mathbb{C})} f(\nu W) g(\Lambda) d\mu_{\mathcal{U}_N(\mathbb{C})}(\nu) \right] \\
&= \int_{\mathcal{U}_N(\mathbb{C})} f(\nu) d\mu_{\mathcal{U}_N(\mathbb{C})}(\nu) \mathbb{E}(g(\Lambda)).
\end{align*}
$$

It yields that $\Lambda$ and $U$ are independent and that the law of $U$ is the Haar measure in $\mathcal{U}_N(\mathbb{C})$. $\square$

**Eigenvalues** We consider $(U, \Lambda)$ as in Proposition 2.21. The goal is to find the law $\mu_\Lambda$ of $\Lambda$.

Fix $U_0 \in \mathcal{U}_N(\mathbb{C})$, $\Lambda_0$ a diagonal matrix and $M_0 \in \mathcal{H}_N(\mathbb{C})$ such that $M_0 = U_0\Lambda_0 U_0^*$. We suppose that the spectrum of $M_0$ is simple (which shall be almost surely true for a $N$ complex Wigner matrix by Proposition 2.6).

To find the law of the eigenvalues we shall use a change of variables in the neighbourhood of $M_0$.

Let us give some notations and basic linear algebra results to do this change of variables. $\mathcal{H}_N(\mathbb{C})$ is a $\mathbb{R}$ vectorial space of dimension $N^2$. $\mathbb{R}^N$ is a $\mathbb{R}$ vectorial space of dimension $N$. In this section, let $\mathcal{A}_{N,0}$ be the set of anti hermitian matrices of $M_N(\mathbb{C})$ with null diagonal, it is a $\mathbb{R}$ vectorial space of dimension $N(N-1)$.

We define $\phi: \mathbb{R}^N \times A_{N,0} \to \mathcal{H}_N(\mathbb{C})$ by:

$$ \phi(\Lambda, A) = U_0 \exp(A) (\Lambda_0 + \text{Diag}(\Lambda)) \exp(-A) U_0^* $$

where $\text{Diag}(X)$ is the diagonal matrix such that $\text{Diag}(X)_{i,i} = X_i$ for all $1 \le i \le N$.

We have that $\phi(0,0) = M_0$. We shall compute the differential at $(0,0)$ of $\phi$ and then use implicit function theorem to inverse locally around this point the application $\phi$. We first observe that the dimension as a $\mathbb{R}$ vectorial space of $\mathbb{R}^N \times A_{N,0}$ is $N^2$ which is the dimension of $\mathcal{H}_N(\mathbb{C})$ as a $\mathbb{R}$ vectorial space which makes sense.

**Proposition 2.22.** $\phi$ is differentiable at the point $0 := (0,0)$ and we have:

$$ d\phi_0(\Lambda, A) = U_0(\text{Diag}(\Lambda) + A\Lambda_0 - \Lambda_0 A)U_0^* $$

*Proof.* Let $\Lambda$ and $A$ small elements of $\mathbb{R}^N$ and $A_{N,0}$. We have:

$$
\begin{align*}
\phi(\Lambda, A) &= U_0 \exp(A) (\Lambda_0 + \text{Diag}(\Lambda)) \exp(-A) U_0^* \\
&= U_0 (I_N + A) (\Lambda_0 + \text{Diag}(\Lambda)) (I_N - A) U_0^* + o((\Lambda, A)) \\
&= U_0 (\Lambda_0 + \text{Diag}(\Lambda) + A\Lambda_0) (I_N - A) U_0^* + o((\Lambda, A)) \\
&= U_0 (\Lambda_0 + \text{Diag}(\Lambda) + A\Lambda_0 - \Lambda_0 A) U_0^* + o((\Lambda, A)) \\
&= \phi(0, 0) + U_0 (\text{Diag}(\Lambda) + A\Lambda_0 - \Lambda_0 A) U_0^* + o((\Lambda, A))
\end{align*}
$$

Hence, $\phi$ is differentiable at $(0,0)$ and we have $d\phi_0(\Lambda, A) = U_0(\text{Diag}(\Lambda) + A\Lambda_0 - \Lambda_0 A)U_0^*$. $\square$

<figure><img src="image_6.png" /></figure>
Now we need to compute the determinant of the linear map $d\phi_0$ and prove that this determinant is non null so that by the implicit function theorem we can inverse this map locally around 0.

Let us remark that $d\phi_0$ is the composition of two linear maps: $\psi_{U_0} := \mathcal{H}_N(\mathbb{C}) \to \mathcal{H}_N(\mathbb{C})$ defined by $\psi_{U_0}(X) = U_0XU_0^*$ and $\Psi_{\Lambda_0} := \mathbb{R}^N \times A_{N,0} \to \mathcal{H}_N(\mathbb{C})$ defined by $\gamma(\Lambda, A) = \text{Diag}(\Lambda) + A\Lambda_0 - \Lambda_0 A$. The notations $\psi_{U_0}$ and $\Psi_{\Lambda_0}$ will only be used in this section. We already proved that $|\det(\psi_{U_0})| = 1$ in Lemma 2.4. It remains to compute the determinant of $\Psi_{U_0}$.

**Lemma 2.23.** The determinant of $\Psi_{\Lambda_0}$ is:

$$ \det(\Psi_{\Lambda_0}) = \prod_{k<j} (\lambda_0(j) - \lambda_0(k))^2 = \Delta_N(\Lambda_0)^2, $$

with $\Delta_N(\Lambda_0) = \prod_{k<j} (\lambda_0(j) - \lambda_0(k))$ the Vandermonde determinant.

*Proof.* This proof is a basic linear algebra argument. Let $(e_k)_{1\le k\le N}$ the canonical basis of $\mathbb{R}^N$, we consider $((\Delta_{k,j} - \Delta_{j,k})_{1\le k<j\le N}, (i(\Delta_{k,j} + \Delta_{j,k}))_{1\le k<j\le N})$ as a basis of $A_{N,0}$ and $((\Delta_{k,k})_{k\le N}, (\Delta_{k,j} + \Delta_{j,k})_{1\le k<j\le N}, (i(\Delta_{k,j} - \Delta_{j,k}))_{1\le k<j\le N})$ a basis of $\mathcal{H}_N(\mathbb{C})$.

First, for all $1 \le k \le N$ we have that $\Psi_{\Lambda_0}((e_k, 0)) = \Delta_{k,k}$.

Moreover we can compute for $1 \le k < j \le N$:

$$ \Psi_{\Lambda_0}(0, \Delta_{k,j} - \Delta_{j,k}) = (\Delta_{k,j} - \Delta_{j,k})\Lambda_0 - \Lambda_0(\Delta_{k,j} - \Delta_{j,k}) = (\Lambda_0(j) - \Lambda_0(k))(\Delta_{k,j} + \Delta_{j,k}), $$
and

$$ \Psi_{\Lambda_0}(0, i(\Delta_{k,j}+\Delta_{j,k})) = (i(\Delta_{k,j}+\Delta_{j,k}))\Lambda_0 - \Lambda_0(i(\Delta_{k,j}+\Delta_{j,k})) = i(\Delta_{k,j}-\Delta_{j,k})(\Lambda_0(j)-\Lambda_0(k)). $$

In these basis $\Psi_{\Lambda_0}$ is diagonal and we can directly compute its determinant:

$$ \det(\Psi_{\Lambda_0}) = \prod_{k<j} (\lambda_0(j) - \lambda_0(k))^2. $$

□

We can now compute the Jacobian of $\phi$ near 0.

**Proposition 2.24.** $\phi$ is $C^1$ local diffeomorphism around 0 and $|\det(d\phi_0)| = \Delta_N(\Lambda_0)^2$.

*Proof.* This is an immediate consequence of the fact $d\phi_0 = \psi_{U_0} \circ \Psi_{\Lambda_0}$ and Lemma 2.4 and Lemma 2.23. Since $\Delta_N(\Lambda_0) \neq 0$ by hypothesis on $M_0$ (simple spectrum), the implicit function theorem yields that $\phi$ is locally invertible around 0. □

Thanks to this change of variable we can compute the law of the eigenvalues.

**Theorem 2.25.** The law of the exchangeable spectrum in $\mathbb{R}^N$ of a $N$ complex Wigner matrix has a density with respect to the Lebesgue measure which is proportional to

$$ \Lambda = (\lambda_1, \dots, \lambda_N) \mapsto \Delta_N(\Lambda)^2 \exp \left( -\frac{\text{Tr}(\Lambda^2)}{2} \right) = \prod_{1 \le j < k \le N} (\lambda_k - \lambda_j)^2 \exp \left( -\frac{\sum_{k=1}^N \lambda_k^2}{2} \right). $$

**Remark 2.26.** More explicitly, the previous theorem said that there exists a constant of normalization $c_N$ such that for every symmetric, bounded or positive measurable functions $F$ we have

$$ \mathbb{E}(F(\Lambda_1, \dots, \Lambda_N)) = c_N \int_{\mathbb{R}^N} F(x_1, \dots, x_N) \prod_{i<j} (x_i - x_j)^2 \exp \left( -\frac{\sum_{k=1}^N |x_i|^2}{2} \right) dx_1 \dots dx_N, $$

if $\Lambda = (\Lambda_1, \dots, \Lambda_N)$ is the spectrum of a $N$ complex Wigner matrix.
Proof. As explained at the beginning of this section the goal is to find the law $\mu_\Lambda$ on $\mathbb{R}^N$ that matches with Proposition 2.21. We look for $\mu_\Lambda$ such that $\mu_\Lambda$ has a density $g_\Lambda$ with respect to the Lebesgue measure on $\mathbb{R}^N$. This density shall be a symmetric function since we look at the eigenvalues as an exchangeable vector of $\mathbb{R}^N$ and shall vanish when at least two of its arguments are equal since the spectrum of a $N$ complex Wigner matrix is almost surely simple.

Let $f$ be a non negative measurable function that only depends on the spectrum of a matrix. We can find $\varepsilon > 0$ such that $\phi_\varepsilon : \mathbb{R}^N \cap B(0, \varepsilon) \times A_{N,0} \cap B(0, \varepsilon) \to \mathcal{H}_N(\mathbb{C})$ defined by $\phi_\varepsilon(\Lambda, A) = \phi(\Lambda, A)$ is a $C^1$ diffeomorphism onto its image by Proposition 2.24.

We recall by Proposition 2.21 that $U$ is Haar distributed and independent of $\Lambda$ and is such that $X = U\Lambda U^*$ is a $N$ complex Wigner matrix. Now we compute:

$$ \mathbb{E} [f(\Lambda) \mathbf{1}_{||\Lambda-\Lambda_0|| \le \varepsilon}] = \mathbb{E} [f(\Lambda) \mathbf{1}_{||\Lambda-\Lambda_0|| \le \varepsilon}] \frac{\mathbb{E}(\mathbf{1}_{U \in \{U_0 \exp(A) \mid A \in A_{N,0} \cap B(0,\varepsilon)\}})}{\mathbb{P}(U \in \{U_0 \exp(A) \mid A \in A_{N,0} \cap B(0,\varepsilon)\})} \quad (2.4) \\ \stackrel{\text{Independence}}{=} c_\varepsilon \mathbb{E}(f(X) \mathbf{1}_{X \in \text{Im}(\phi_\varepsilon)}), $$

with $c_\varepsilon^{-1} := \mathbb{P}(U \in \{U_0 \exp(A) \mid A \in A_{N,0} \cap B(0, \varepsilon)\}) \stackrel{\text{Haar}}{=} \mathbb{P}(U \in \{\exp(A) \mid A \in A_{N,0} \cap B(0, \varepsilon)\})$.

Now we use the law of $X$ to compute this term. It gives:

$$ \mathbb{E} [f(\Lambda) \mathbf{1}_{||\Lambda-\Lambda_0|| \le \varepsilon}] = \frac{c_\varepsilon}{Z_N} \int_{\mathcal{H}_N(\mathbb{C})} f(X) \mathbf{1}_{X \in \text{Im}(\phi_\varepsilon)} \exp(-\text{Tr}(X^2)/2) dX, $$

with $Z_N$ the constant of normalization of the law of a $N$ complex Wigner matrix. Now we can do the change of variable to have:

$$ \mathbb{E} [f(\Lambda) \mathbf{1}_{||\Lambda-\Lambda_0|| \le \varepsilon}] = c'_\varepsilon \int_{\mathbb{R}^N \cap B(0,\varepsilon) \times A_{N,0} \cap B(0,\varepsilon)} f(\Lambda+\Lambda_0) e^{-\text{Tr}(\Lambda+\Lambda_0)^2/2} |\det(\phi_{\Lambda,A})| d\Lambda d\text{Leb}(A), $$

with $c'_\varepsilon := c_\varepsilon/Z_N$.

To find the exact expression of $g_\Lambda$ we shall let $\varepsilon$ goes to 0.

On one side we have by definition of $g_\Lambda$:

$$ \mathbb{E} [f(\Lambda) \mathbf{1}_{||\Lambda-\Lambda_0|| \le \varepsilon}] = \int_{||\Lambda-\Lambda_0|| \le \varepsilon} f(\Lambda) g_\Lambda(\Lambda) d\text{Leb}(\Lambda) \stackrel{\sim}{\longrightarrow}_{\varepsilon \to 0} \widetilde{C}_N \varepsilon^N f(\Lambda_0) g_\Lambda(\Lambda_0), \quad (2.5) $$

with $\widetilde{C}_N$ a constant that only depend on $N$.

On the other hand, we have:

$$ \begin{aligned} & c'_\varepsilon \int_{\mathbb{R}^N \cap B(0,\varepsilon) \times A_{N,0} \cap B(0,\varepsilon)} f(\Lambda + \Lambda_0) e^{-\text{Tr}(\Lambda+\Lambda_0)^2/2} |\det(\phi_{\Lambda,A})| d\Lambda d\text{Leb}(A) \stackrel{\sim}{\longrightarrow}_{\varepsilon \to 0} \\ & c'_\varepsilon f(\Lambda_0) \exp\left(-\frac{\text{Tr}(\Lambda_0)^2}{2}\right) |\det(\phi_{0,0})| \widetilde{C}'_N \varepsilon^N \text{Leb}(A_{N,0} \cap B(0,\varepsilon)) =: \\ & \qquad \widetilde{C}'_{N,\varepsilon} \varepsilon^N f(\Lambda_0) \exp\left(-\frac{\text{Tr}(\Lambda_0)^2}{2}\right) \Delta_N(\Lambda_0)^2, \end{aligned} \quad (2.6) $$

where $\widetilde{C}'_{N,\varepsilon}$ is a constant that only depend on $N$ and $\varepsilon$.

Comparing (2.5) and (2.6), since the result holds for every non positive measurable functions $f$, we deduce that $\widetilde{C}'_{N,\varepsilon}$ necessarily converges when $\varepsilon$ goes to 0 to a certain quantity that only depends on $N$ and we have that the density $g_\Lambda$ is proportional to

$$ \Lambda_0 \mapsto \exp\left(-\frac{\text{Tr}(\Lambda_0)^2}{2}\right) \Delta_N(\Lambda_0)^2. $$
### 2.2.2 Ginibre case

We first recall the Schür decomposition that can be viewed as the counterpart for matrices of $M_N(\mathbb{C})$ of the spectral theorem for Hermitian matrices.

**Proposition 2.27 (Schür decomposition).** Let $M \in M_N(\mathbb{C})$, there exists $U \in U_N(\mathbb{C})$, $N \in M_N(\mathbb{C})$ a nilpotent upper triangular matrix and $D \in D_N(\mathbb{C})$ a diagonal matrix such that $M = U(D+N)U^*$. Moreover we have that $\mathrm{Tr}(MM^*) = \mathrm{Tr}(DD^*) + \mathrm{Tr}(NN^*)$.

To prove the theorem we can triangularize a matrix in $M_N(\mathbb{C})$ in a certain base and then orthogonalize this base using the Gram Schmidt algorithm. Thanks to the Schür decomposition we can do a similar proof to compute the density of the eigenvalues of a $N$ complex Ginibre matrix as we did for a $N$ complex Wigner matrix.

**Theorem 2.28.** The law of the spectrum in $\mathbb{C}^N$ of a random matrix distributed according to a $N$ complex Ginibre matrix has a density with respect to the complex Lebesgue measure proportional to

$$ (z_1, \dots, z_N) \mapsto \exp(-\sum_{i=1}^{N} |z_i|^2) \prod_{i<j} |z_i - z_j|^2. $$

## 2.3 Determinantal structure

Viewing the eigenvalues of $N$ complex Ginibre or Wigner matrices as point processes in $\mathbb{C}$ or $\mathbb{R}$, we shall compute the correlation functions of these point processes. These computations are based on the remark that both densities of the eigenvalues of the two models can be written in a so-called determinantal form that make the computations easier using an orthogonal family of polynomials. For the links between random matrices and point processes see for instance [5, 54, 63, 64].

We first detail completely the Ginibre case and then state the same property giving some proofs for the Wigner case.

### 2.3.1 Ginibre case

We recall that the Vandermonde determinant of $(\mu_1, \dots, \mu_N)$ is given by $\Delta_N(\mu_1, \dots, \mu_N) = \prod_{k<j}(\mu_j - \mu_k) = \det[\mu_j^{k-1}]_{1 \le j, k \le N}$.

Let $c_N$ be the constant of normalization for the density of the eigenvalues of a matrix distributed according to a $N$ complex Ginibre matrix given in Theorem 2.28. We first compute its value:

$$ \begin{aligned} c_N^{-1} &= \int_{\mathbb{C}^N} \exp(-\sum_{i=1}^N |z_i|^2) \prod_{k<j} |z_k - z_j|^2 dz_1 \dots dz_N \\ &= \int_{\mathbb{C}^N} \exp(-\sum_{i=1}^N |z_k|^2) |\Delta_N(z_1, \dots, z_N)|^2 dz_1 \dots dz_N \\ &= \int_{\mathbb{C}^N} \exp(-\sum_{i=1}^N |z_k|^2) |\det[z_j^{k-1}]_{1 \le j, k \le N}|^2 dz_1 \dots dz_N. \end{aligned} $$
Using the multi linearity of the determinant and a family of $N$ monic polynomial $(P_k)_{0 \le k \le N-1}$ such that for all $0 \le k \le N-1$ the degree of $P_k$ is $k$, we can write:

$$
\begin{align*}
c_N^{-1} &= \int_{\mathbb{C}^N} \exp(-\sum_{i=1}^N |z_i|^2) |\Delta_N(z_1, \dots, z_N)|^2 dz_1 \dots dz_N \\
&= \int_{\mathbb{C}^N} \exp(-\sum_{i=1}^N |z_i|^2) \det[P_{k-1}(z_j)]_{1 \le j, k \le N} |dz_1 \dots dz_N| \\
&= \int_{\mathbb{C}^N} \exp(-\sum_{i=1}^N |z_i|^2) \det[P_{k-1}(z_j)]_{1 \le j, k \le N} \overline{\det[P_{k-1}(z_j)]_{1 \le j, k \le N}} |dz_1 \dots dz_N| \quad (2.7) \\
&= \sum_{\sigma, \sigma' \in \mathfrak{S}_N} (-1)^{\varepsilon(\sigma) + \varepsilon(\sigma')} \int_{\mathbb{C}^N} \prod_{i=1}^N \exp(-|z_i|^2) \prod_{i=1}^N P_{\sigma(i)-1}(z_i) \overline{P_{\sigma'(i)-1}(z_i)} |dz_1 \dots dz_N| \\
&\stackrel{\text{Fubini}}{=} \sum_{\sigma, \sigma' \in \mathfrak{S}_N} (-1)^{\varepsilon(\sigma) + \varepsilon(\sigma')} \prod_{i=1}^N \int_{\mathbb{C}} \exp(-|z|^2) P_{\sigma(i)-1}(z) \overline{P_{\sigma'(i)-1}(z)} |dz.
\end{align*}
$$

We see that a good idea would be to choose a family of polynomials that are orthogonal for the scalar product $\langle f|g \rangle = \int_{\mathbb{C}} \exp(-|z|^2) \bar{f}(z)g(z)dz$. A such family can be found explicitly.

**Lemma 2.29.** Let $\gamma(z) := \pi^{-1} \exp(-|z|^2)$. Then for all $k, l \in \mathbb{N}$, we have:

$$
\int_{\mathbb{C}} \frac{z^k \bar{z}^l}{\sqrt{k!} \sqrt{l!}} \gamma(z) dz = \mathbf{1}_{k=l}.
$$

*Proof.* To do these computations, we use the polar coordinate. Indeed, we have:

$$
\begin{align*}
\int_{\mathbb{C}} z^k \bar{z}^l \gamma(z) dz &= \frac{1}{\pi} \int_{\mathbb{R}^2} (x + iy)^k (x - iy)^l \exp(-x^2 - y^2) dxdy \\
&= \frac{1}{\pi} \int_{\mathbb{R}^+} \int_{[0,2\pi]} (r(\cos\theta) + i\sin\theta))^k (r(\cos\theta) - i\sin\theta))^l \exp(-r^2)rdrd\theta \\
&= \frac{1}{\pi} \int_{\mathbb{R}^+} r^{k+l} r \exp(-r^2) dr \int_{[0,2\pi]} \exp(i\theta(k-l)) d\theta
\end{align*}
$$

We observe that the angular integral is null if $k \neq l$ and is equal to $2\pi$ if $k = l$. When $k = l$ the radial part can be computed without difficulty by integrating by parts and is equal to

$$
\int_{\mathbb{R}+} r^{2k+1} \exp(-r^2) dr = \frac{k!}{2}.
$$

It yields:

$$
\int_{\mathbb{C}} \frac{z^k \bar{z}^l}{\sqrt{k!} \sqrt{l!}} \gamma(z) dz = \mathbf{1}_{k=l}.
$$

□

**Remark 2.30.** For $j \ge 0$, if $\mu_j(z) := \sqrt{\gamma(z)} z^j / j!$, Lemma 2.29 says that these functions are orthonormal in $L_2(\mathbb{C})$ for the Lebesgue measure.
Now we can continue the computation (2.7) with the specific polynomials $P_j(X) = X^j$ for all $j \in \mathbb{N}$. We have:

$$
\begin{align}
c_N^{-1} &= \sum_{\sigma, \sigma' \in \mathfrak{S}_N} (-1)^{\epsilon(\sigma) + \epsilon(\sigma')} \prod_{i=1}^{N} \int_{\mathbb{C}} \exp(-|z|^2) P_{\sigma(i)-1}(z) \overline{P_{\sigma'(i)-1}(z)} dz \nonumber \\
&= \sum_{\sigma \in \mathfrak{S}_N} \prod_{i=1}^{N} \int_{\mathbb{C}} \exp(-|z|^2) P_{\sigma(i)-1}(z) \overline{P_{\sigma(i)-1}(z)} dz \tag{2.8} \\
&= \sum_{\sigma \in \mathfrak{S}_N} \prod_{i=1}^{N} \pi(\sigma(i) - 1)! = \pi^N N! \prod_{k=0}^{N-1} k!. \nonumber
\end{align}
$$

Hence, we can now give the complete expression of the density of the eigenvalues of a $N$ complex Ginibre matrix.

**Theorem 2.31.** The law of the spectrum in $\mathbb{C}^N$ of a random matrix distributed according to a $N$ complex Ginibre matrix has a density with respect to the complex Lebesgue measure which is equal to:

$$ \phi_N(z_1, \dots, z_N) := \frac{\prod_{k=1}^{N} \gamma(z_k)}{\prod_{k=1}^{N} k!} \prod_{i<j} |z_i - z_j|^2, \quad (2.9) $$

$$ \text{with } \gamma(z) := \exp(-|z|^2)\pi^{-1}. $$

We can use the Vandermonde formula to give another form of the formula (2.9). Indeed we have:

$$
\begin{align}
\phi_N(z_1, \dots, z_N) &= \frac{\prod_{k=1}^{N} \gamma(z_k)}{\prod_{k=1}^{N} k!} \Delta_N(z_1, \dots, z_N) \Delta_N(\overline{z_1}, \dots, \overline{z_N}) \nonumber \\
&= \frac{\prod_{k=1}^{N} \gamma(z_k)}{\prod_{k=1}^{N} k!} \det[z_i^{j-1}]_{1 \le i,j \le N} \det[\overline{z_i^{j-1}}]_{1 \le i,j \le N} \nonumber \\
&= \frac{\prod_{k=1}^{N} \gamma(z_k)}{\prod_{k=1}^{N} k!} \det[z_i^{j-1}]_{1 \le i,j \le N} \det[\overline{z_j^{i-1}}]_{1 \le i,j \le N} \tag{2.10} \\
&= \frac{\prod_{k=1}^{N} \gamma(z_k)}{N!} \det\left[\frac{z_i^{j-1}}{\sqrt{(j-1)!}}\right]_{1 \le i,j \le N} \det\left[\frac{\overline{z_j^{i-1}}}{\sqrt{(i-1)!}}\right]_{1 \le i,j \le N} \nonumber \\
&= \frac{1}{N!} \det[K_N(z_i, z_j)]_{1 \le i,j \le N}, \nonumber
\end{align}
$$

with for all $z \in \mathbb{C}$ and $w \in \mathbb{C}$:

$$ K_N(z, w) := \sqrt{\gamma(z)\gamma(w)} \sum_{l=0}^{N-1} \frac{(zw)^l}{l!}. $$

$K_N$ shall be called the kernel of the point process of the eigenvalues of a $N$ complex Ginibre matrix.

Let us give some definitions and basic properties about point processes.

**Definition 2.32.** The set of locally finite part of $\mathbb{C}$ is the subset of the part of $\mathbb{C}$ defined by:

$$ P_{lf}^{\mathbb{C}} = \{Z \subset \mathbb{C}, |Z \cap K| < +\infty \forall K \text{ compact of } \mathbb{C}\}. $$

We endow $P_{lf}^{\mathbb{C}}$ with a sigma algebra $\mathcal{F}_{lf}^{\mathbb{C}}$ which is the smallest sigma algebra that makes measurable the maps $\phi_A : P_{lf}^{\mathbb{C}} \to (\mathbb{N}, \mathcal{P}(\mathbb{N}))$ such that $\phi_A(Z) = |Z \cap A|$ for $A$ compact subset of $\mathbb{C}$.
We can now define a point process.

**Definition 2.33.** Z is said to be a point process if it is a random variable with value in $(P_{lf}^{\mathbb{C}}, \mathcal{F}_{lf}^{\mathbb{C}})$.

**Remark 2.34.** A function Z defined on a probability space with value in $(P_{lf}^{\mathbb{C}}, \mathcal{F}_{lf}^{\mathbb{C}})$ is a random variable if and only if $|Z \cap A|$ is a $(\mathbb{N}, \mathcal{P}(\mathbb{N}))$ valued random variable for all A compact subset of $\mathbb{C}$. The law of Z is determined by the law of $(|Z \cap A_1|, ..., |Z \cap A_k|)$ for all $(A_i)_{i \le k}$ compact subsets of $\mathbb{C}$ for all $k \in \mathbb{N}$. By some standard arguments as the Dynkin lemma we can restrict the previous fact on disjoint compact subsets of $\mathbb{C}$.

Let us give a standard example of point process.

**Example 2.35.** The Poisson point process of parameter $\lambda > 0$ is the point process such that for all $k \ge 1$ for all $(A_i)_{1 \le i \le k}$ disjoint compact subsets of $\mathbb{C}$ we have that: $(|Z \cap A_1|, ..., |Z \cap A_k|) = \text{Poisson}(\lambda \text{Leb}(A_1)) \otimes \dots \otimes \text{Poisson}(\lambda \text{Leb}(A_k))$, where in this example Poisson($\nu$) is a standard Poisson random variable of parameter $\nu > 0$.

**Definition 2.36.** We say that the correlation functions of a point process Z are $(\phi_k)_{k \ge 1}$ if for all $k \ge 1$, $\phi_k : \mathbb{C}^k \to \mathbb{R}^+$ is such that for all $B_1, ..., B_k$ disjoint compact subsets of $\mathbb{C}$ we have:

$$ \mathbb{E}(|Z \cap B_1|...|Z \cap B_k|) = \int_{B_1 \times \dots \times B_k} \phi_k(z_1, ..., z_k) dz_1 \dots dz_k. $$

For the Example 2.35 we can compute explicitly the correlation functions.

**Example 2.37.** For a Poisson point process of parameter $\lambda > 0$ the correlation functions are given by: $\phi_k(z_1, ..., z_k) = \lambda^k$ for all $k \ge 1$.

**Remark 2.38.** Pay attention to the fact that correlation functions are not sure to exist. For instance, let $Z = \{0\}$ then $\mathbb{E}(|Z \cap \{0\}|) = 1 \ne \int_{\{0\}} \phi_1(z) dz = 0$.

When the point process is given by a random variable in $\mathbb{C}^N$ with a symmetric density with respect to the Lebesgue measure one can express the correlation functions of this point process.

**Proposition 2.39.** Let $Z = (X_1, ..., X_N)$ be $N$ complex random variables such that the law of $(X_1, ...X_N)$ has a density with respect to the Lebesgue measure $f(z_1, ..., z_N)dz_1...dz_N$ that we suppose symmetric, then the correlation functions of Z are given by $\psi_k = 0$ if $k > N$ and if $k \le N$,

$$ \psi_k(z_1, ..., z_k) = \frac{N!}{(N-k)!} \int_{\mathbb{C}^{N-k}} f(z_1, ..., z_N) dz_{k+1}...dz_N. $$

*Proof.* Let $B_1, ..., B_k$ be disjoint compact subsets of $\mathbb{C}$, we compute the correlations functions:

$$ \begin{aligned} \mathbb{E}(|Z \cap B_1|, ..., |Z \cap B_k|) &= \mathbb{E} \left[ \sum_{i_1=1}^{N} 1_{X_{i_1} \in B_1} \dots \sum_{i_k=1}^{N} 1_{X_{i_k} \in B_k} \right] \\ &= \sum_{i_1, ..., i_k=1}^{N} \mathbb{E} \left[ 1_{X_{i_1} \in B_1} \dots 1_{X_{i_k} \in B_k} \right]. \end{aligned} $$
Since the $B_i$ are disjoint subsets, an $X_k$ can not be in two different $B_i$. Hence, we can rewrite the sum and then use the symmetry of the law to have:

$$ 
\begin{aligned}
\mathbb{E}(|Z \cap B_1|...|Z \cap B_k|) &= \sum_{i_1,...,i_k \in \{1,...,N\}^k \text{ disjoint}} \mathbb{E}[\mathbb{1}_{X_{i_1} \in B_1}...\mathbb{1}_{X_{i_k} \in B_k}] \\
&= \sum_{i_1,...,i_k \in \{1,...,N\}^k \text{ disjoint}} \mathbb{E}[\mathbb{1}_{X_1 \in B_1}...\mathbb{1}_{X_k \in B_k}].
\end{aligned}
\quad (2.11) $$

Hence, by (2.11) $\mathbb{E}(|Z \cap B_1|...|Z \cap B_k|)$ is equal to 0 if $k > N$ (we could directly have said it because by definition of $Z$ if $k > N$ we necessarily have a set $B_i$ with no points of $Z$ and so almost surely $|Z \cap B_1|...|Z \cap B_k| = 0$ and so its expectation is also 0) and otherwise is equal to:

$$ \frac{N!}{(N-k)!} \int_{B_1 \times \dots \times B_k} \left( \int_{\mathbb{C}^{N-k}} f(z_1, \dots, z_N) dz_{k+1} \dots dz_N \right) dz_1 \dots dz_k. $$

It proves the result.

<figure><img src="image_7.png" /></figure>

We recall that in the case of $N$ complex Ginibre matrices we are interested in the point process of its eigenvalues $Z = \{\lambda_1, \dots, \lambda_N\}$ for which the density is the symmetric function obtained in Theorem 2.31 and that was written in (2.10) as

$$ \phi_N(z_1, \dots, z_N) = \frac{1}{N!} \det[K_N(z_i, z_j)]_{1 \le i,j \le N}. $$

Thanks to the orthogonality property of the monic polynomials, $K_N$ shall satisfied a semi group property making the computations of the correlation functions easier.

**Lemma 2.40.** For all $N \ge 1$ and all $x, z \in \mathbb{C}$, we have:

$$ \int_{\mathbb{C}} K_N(z, z) dz = N, \quad \int_{\mathbb{C}} K_N(x, y) K_N(y, z) dy = K_N(x, z). $$

*Proof.* This is a quite immediate consequence of the property of orthogonality of the monic polynomial for the measure $\gamma(z)dz$.

Let for all $j \ge 0$, let $\mu_j(z) = \sqrt{\gamma(z)}z^j/j!$. These functions are orthonormal in $L_2(\mathbb{C})$ for the Lebesgue measure by Lemma 2.29.

Hence we have that:

$$ \int_{\mathbb{C}} K_N(z, z) dz = \sum_{l=0}^{N-1} \int_{\mathbb{C}} |\mu_j(z)|^2 dz = N $$

and:

$$ 
\begin{aligned}
\int_{\mathbb{C}} K_N(x, y) K_N(y, z) dy &= \sum_{k,j=0}^{N-1} \int_{\mathbb{C}} \mu_j(x) \overline{\mu_j(y)} \mu_k(y) \overline{\mu_k(z)} dy \\
&= \sum_{k,j=0}^{N-1} \mu_j(x) \overline{\mu_k(z)} \mathbb{1}_{j=k} \\
&= \sum_{k,j=0}^{N-1} \mu_k(x) \overline{\mu_k(z)} = K_N(x, z)
\end{aligned}
$$

<figure><img src="image_8.png" /></figure>
Thanks to this lemma we can now prove the following inductive formula that shall give the explicit value of correlation functions obtained in Proposition 2.39.

**Proposition 2.41.** For all $k < N$ and $z_1, ..., z_k \in \mathbb{C}$ we have the following inductive formula:

$$ \int_{\mathbb{C}} \det[K_N(z_i, z_j)]_{1 \le i, j \le k+1} dz_{k+1} = (N-k) \det[K_N(z_i, z_j)]_{1 \le i, j \le k}. $$

As a corollary, we have that for all $k \le N$,

$$ \int_{\mathbb{C}^{N-k}} \det[K_N(z_i, z_j)]_{1 \le i, j \le N} dz_{k+1} \dots dz_N = (N-k)! \det[K_N(z_i, z_j)]_{1 \le i, j \le k}. $$

*Proof.* We expand the determinant along the last column and we write $A_{k+1}^{l,k+1}$ the matrix obtain from $[K_N(z_i, z_j)]_{1 \le i, j \le k+1}$ by removing the line $l$ and the column $k+1$:

$$ \det[K_N(z_i, z_j)]_{1 \le i, j \le k+1} = \sum_{l=1}^{k+1} (-1)^{l+k+1} K_N(z_l, z_{k+1}) \det(A_{k+1}^{l,k+1}) $$

We shall distinguish two cases. First when $l = k+1$ we have that $A_{k+1}^{k+1,k+1} = [K_N(z_i, z_j)]_{1 \le i, j \le k}$ and so this term does not depend on $z_{k+1}$. Hence, when we integrate the term of the sum that corresponds to $l = k+1$ against $dz_{k+1}$, we only have

$$ \det[K_N(z_i, z_j)]_{1 \le i, j \le k} \int_{\mathbb{C}} K_N(z_{k+1}, z_{k+1}) dz_{k+1} = N \det[K_N(z_i, z_j)]_{1 \le i, j \le k}, $$

thanks to Lemma 2.40. For all the other terms, we again expand the determinant along the last line to have that:

$$ \det(A_{k+1}^{l,k+1}) = \sum_{p=1}^{k} (-1)^{p+k} K_N(z_{k+1}, z_p) \det[K_N(z_i, z_j)]_{1 \le i, j \le k, i \ne l, j \ne p}. $$

We notice that in the sum the last determinant does not depend on $z_{k+1}$ so when we integrate with respect to $z_{k+1}$ the only important contribution in this expression is $K_N(z_{k+1}, z_p)$. So using Lemma 2.40 again we have that for all $l \le k$:

$$ \int_{\mathbb{C}} (-1)^{l+k+1} K_N(z_l, z_{k+1}) \det(A_{k+1}^{l,k+1}) dz_{k+1} = \sum_{p=1}^{k} (-1)^{l+p-1} K_N(z_l, z_p) \det[K_N(z_i, z_j)]_{1 \le i, j \le k, i \ne l, j \ne p}. $$

We recognize that the last sum is:

$$ - \det[K_N(z_i, z_j)]_{1 \le i, j \le k} $$

Since we sum all these terms for $l \le k$, we get that:

$$ \det[K_N(z_i, z_j)]_{1 \le i, j \le k+1} = (N-k) \det[K_N(z_i, z_j)]_{1 \le i, j \le k}. $$

The last point is clear by induction. $\square$

<figure><img src="image_9.png" /></figure>
Let us also remark that for $k > N$ we have that:

$$[K_N(z_i, z_j)]_{1 \le i, j \le k} = [(\mu_l(z_i))_{1 \le i \le k, 1 \le l \le N} \overline{(\mu_i(z_l))}_{1 \le i \le k, 1 \le l \le N}$$

with the $\mu_j$ which are the orthogonal elements of $L_2(\mathbb{C})$ introduced in Remark 2.30. Since the matrix $[(\mu_l(z_i))_{1 \le i \le k, 1 \le l \le N}]$ has $N$ columns the rank of $[K_N(z_i, z_j)]_{1 \le i, j \le k}$ is smaller than $N$ and so strictly smaller than $k$. Hence, the determinant of $[K_N(z_i, z_j)]_{1 \le i, j \le k}$ is equal to 0 if $k > N$.

Let us summarize the results obtained for the correlation functions of the eigenvalues of $N$ complex Ginibre matrices.

**Definition 2.42.** A determinantal point process of kernel $K: \mathbb{C}^2 \to \mathbb{C}$ is a point process such that its correlation functions are of the form:

$$\psi_k(z_1, \dots, z_k) = \det[K(z_i, z_j)]_{1 \le i, j \le k}, \text{ for almost all } (z_1, \dots, z_k) \in \mathbb{C}^k, \forall k \in \mathbb{N}_{\ge 1}.$$

**Example 2.43.** A Poisson point process of parameter $\lambda > 0$ is a determinantal point process with kernel $K(z, z') = \lambda \mathbb{1}_{z=z'}$.

**Theorem 2.44.** The eigenvalues of a $N$ complex Ginibre matrix is a determinantal point process of kernel $K_N$.

The next theorem gives the formulas obtained in this section and the notations we shall use in Section 3.

**Theorem 2.45.** For all $N \ge 1$, the density with respect to the complex Lebesgue measure of the eigenvalues of a $N$ complex Ginibre matrix is given by:

$$\phi_N(z_1, \dots, z_N) := \frac{1}{N!} \det[K_N(z_i, z_j)]_{1 \le i, j \le N},$$

with $K_N$ the kernel defined by:

$$K_N(z, w) := \sqrt{\gamma(z)\gamma(w)} \sum_{l=0}^{N-1} \frac{(z\bar{w})^l}{l!}, \quad \forall (z, w) \in \mathbb{C}^2.$$

Moreover we have an expression of the marginal densities of this law defined for $1 \le k \le N$ by:

$$(z_1, \dots, z_k) \in \mathbb{C}^k \mapsto \phi_{N,k}(z_1, \dots, z_k) := \int_{\mathbb{C}^{N-k}} \phi_N(z_1, \dots, z_N) dz_{k+1} \dots dz_N = \frac{(N-k)!}{N!} \psi_k(z_1, \dots, z_k),$$

with $\psi_k$ the correlation functions of the point process of the eigenvalues of a $N$ complex Ginibre matrix introduced in Definition 2.36.

Indeed for all $1 \le k \le N$, and for all $(z_1, \dots, z_k) \in \mathbb{C}^k$, we have:

$$\phi_{N,k}(z_1, \dots, z_k) = \frac{(N-k)!}{N!} \det[K_N(z_i, z_j)]_{1 \le i, j \le k}.$$

In particular, we have the following explicit formulas for the marginal $\phi_{N,N} = \phi_N$ and for $k = 1$, for all $z \in \mathbb{C}$,

$$\phi_{N,1}(z) = \frac{\gamma(z)}{N} \sum_{l=0}^{N-1} \frac{|z|^{2l}}{l!}.$$
### 2.3.2 Gaussian unitary ensemble case

As for the Ginibre case, we introduce the weight $\gamma_H(x) = \exp(-x^2/2)$ that appears in the formula of the density of the eigenvalues in Theorem 2.25. As in the Ginibre case, the key point is to find a family of orthogonal polynomials for the weight $\gamma_H(x)$. These polynomials are called Hermite polynomials and can be deduced from the so-called Rodrigues formula [57].

**Definition 2.46 (Hermite polynomials).** For all $k \ge 0$ we define the $k^{th}$ Hermite polynomial by

$$H_k(x) = (-1)^n \exp(x^2/2) \left(\frac{d}{dx}\right)^k \left[\exp(-x^2/2)\right].$$

Let us mention that a priori it is not clear that $H_k$ is a polynomial. The main important properties that we shall use about Hermite polynomials are given in the next proposition.

**Proposition 2.47.** The Hermite polynomials satisfy the following properties.

1. For all $k \ge 0$, $H_k$ is a monic polynomial of degree $k$ and has the same parity as $k$.

2. They satisfy the following orthogonal property: for all $m, n \in \mathbb{N}$

$$\int_{\mathbb{R}} H_m(x) H_n(x) \exp(-x^2/2) dx = \sqrt{2\pi n!} 1_{m=n}. \qquad (2.12)$$

3. For all $k \ge 1$ we have the following inductive formula:

$$H_{k+1}(X) = X H_k(X) - k H_{k-1}(X). \qquad (2.13)$$

4. For all $k \ge 1$, we have $H'_k = kH_{k-1}$ and $H_k$ satisfies the following differential equation:

$$H_k'' - xH_k' + kH_k = 0. \qquad (2.14)$$

5. Hermite polynomials satisfy the so-called Christoffel-Darboux formula: for all $n \ge 1$, for all $x \ne y$

$$\sum_{k=0}^{n-1} \frac{H_k(x) H_k(y)}{k!} = \frac{H_n(x) H_{n-1}(y) - H_{n-1}(x) H_n(y)}{(x-y)(n-1)!} \qquad (2.15)$$

Let us mention that these properties are general for families of orthogonal polynomials.

*Proof.* We sketch the proof since the arguments are quite straightforward. The first point can be easily proved by induction.

We detail the second point since it is the main reason for the use of Hermite polynomial in our context.

First suppose that $m > n$ then

$$\int_{\mathbb{R}} H_m(x) H_n(x) \exp(-x^2/2) dx = (-1)^m \int_{\mathbb{R}} \left(\frac{d}{dx}\right)^m \left[\exp(-x^2/2)\right] H_n(x) dx.$$

Doing $m$ integrations by parts yields

$$(-1)^m \int_{\mathbb{R}} \left(\frac{d}{dx}\right)^m \left[\exp(-x^2/2)\right] H_n(x) dx = 0$$
since the $m^{th}$ derivative of $H_n$ is 0 as $H_n$ is a polynomial of degree $n$ (let us also mention that at each integration by parts the bracket term vanishes since for all polynomial $P \in \mathbb{R}[X]$, $\exp(-x^2/2)P(x) \underset{|x| \to +\infty}{\longrightarrow} 0$).

In the case $m = n$, doing $n$ integrations by parts as before yields

$$ 
\begin{align*}
\int_{\mathbb{R}} H_n(x) H_n(x) \exp(-x^2/2) dx &= (-1)^n \int_{\mathbb{R}} \left(\frac{d}{dx}\right)^n \left[\exp(-x^2/2)\right] H_n(x) dx \\
&= (-1)^n (-1)^n \int_{\mathbb{R}} \exp(-x^2/2) \left(\frac{d}{dx}\right)^n [H_n(x)] dx \\
&= n! \int_{\mathbb{R}} \exp(-x^2/2) dx = \sqrt{2\pi n!}
\end{align*}
$$

This gives the second point.

For the third point we can notice that $[\exp(-x^2/2)]' = -x \exp(-x^2/2)$ and differentiating $n$ times this relation, using the Leibniz rule for the right term gives the result.

For the point 4, by definition of $H_k$ we have $H'_k(X) = X H_k(X) - H_{k+1}(X)$ which gives the result using (2.13). The formula (2.14) is a consequence of the fact that $H'_k = k H_{k-1}$ and the formula (2.13).

Finally, the formula (2.15) can be directly proved by induction. $\square$

Thanks to these properties and especially the orthogonality of Hermite polynomials we can do the same computations as in (2.7) and (2.8). We obtain the counterpart of Theorem 2.31.

**Theorem 2.48.** The law of the spectrum in $\mathbb{R}^N$ of a random matrix distributed according to a $N$ complex Wigner matrix has a density with respect to the complex Lebesgue measure which is equal to:

$$ \phi_N^H(x_1, \dots, x_N) := \frac{\prod_{k=1}^N \gamma_H(x_k)}{(2\pi)^{N/2} \prod_{k=1}^N k!} \prod_{i<j} (x_i - x_j)^2, $$

with $\gamma_H(x) := \exp(-x^2/2)$.

As for the Ginibre case in (2.10), this expression is called determinantal because thanks to the Vandermonde formula we can give another expression of the density. Indeed we have:

$$ 
\begin{align*}
\phi_N^H(x_1, \dots, x_N) &= \frac{\prod_{k=1}^N \gamma_H(x_k)}{(2\pi)^{N/2} \prod_{k=1}^N k!} \Delta_N(x_1, \dots, x_N) \Delta_N(x_1, \dots, x_N) \\
&= \frac{\prod_{k=1}^N \gamma_H(x_k)}{(2\pi)^{N/2} \prod_{k=1}^N k!} \det[x_i^{j-1}]_{1\le i,j \le N} \det[x_i^{j-1}]_{1\le i,j \le N} \\
&= \frac{\prod_{k=1}^N \gamma_H(x_k)}{(2\pi)^{N/2} \prod_{k=1}^N k!} \det[x_i^{j-1}]_{1\le i,j \le N} \det[x_j^{i-1}]_{1\le i,j \le N} \tag{2.16} \\
&= \frac{\prod_{k=1}^N \gamma_H(x_k)}{(2\pi)^{N/2} \prod_{k=1}^N k!} \det[H_{j-1}(x_i)]_{1\le i,j \le N} \det[H_{i-1}(x_j)]_{1\le i,j \le N} \\
&= \frac{1}{N!} \det[K_N^H(x_i, x_j)]_{1\le i,j \le N},
\end{align*}
$$

with for all $x \in \mathbb{R}$ and $y \in \mathbb{R}$:

$$ K_N^H(x, y) := \sqrt{\gamma_H(x)\gamma_H(y)} \sum_{l=0}^{N-1} \frac{H_l(x)H_l(y)}{\sqrt{2\pi l!}}. $$
We give the exact same definition of locally finite part in $\mathbb{R}$ that we did in $\mathbb{C}$ for the Ginibre case. We are not very exhausting since it has already been discussed in the part about the Ginibre case.

**Definition 2.49.** We call (real) locally finite part the subset of the part of $\mathbb{R}$:

$$P_{lf}^{\mathbb{R}} = \{Z \subset \mathbb{R}, |Z \cap K| < +\infty \forall K \text{ compact of } \mathbb{R}\}.$$

We endow $P_{lf}^{\mathbb{R}}$ with a sigma algebra $\mathcal{F}_{lf}^{\mathbb{R}}$ which is the smallest sigma algebra that makes measurable the maps $\phi_A : P_{lf} \to (\mathbb{N}, \mathcal{P}(\mathbb{N}))$ such that $\phi_A(Z) = |Z \cap A|$ for $A$ compact subset of $\mathbb{R}$.

**Definition 2.50.** $Z$ is said to be a (real) point process if it is a random variable with value in $(P_{lf}^{\mathbb{R}}, \mathcal{F}_{lf}^{\mathbb{R}})$.

**Definition 2.51.** We say that the correlation functions of a point process are $\phi_k$ for $k \ge 1$ if $\phi_k : \mathbb{R}^k \to \mathbb{R}^+$ is such that for all $B_1, \dots, B_k$ disjoint compact subsets of $\mathbb{R}$ we have:

$$\mathbb{E}(|Z \cap B_1| \dots |Z \cap B_k|) = \int_{B_1 \times \dots \times B_k} \phi_k(x_1, \dots, x_k) dx_1 \dots dx_k.$$

The counterpart of Proposition 2.39 is the following proposition.

**Proposition 2.52.** Let $Z = (X_1, \dots, X_N)$ be $N$ real random variables such that the law of $(X_1, \dots, X_N)$ has a density with respect to the Lebesgue measure $f(x_1, \dots, x_N)dx_1 \dots dx_N$ that we suppose symmetric, then the correlation functions of $Z$ are $\psi_k = 0$ if $k > N$ and if $k \le N$,

$$\psi_k(x_1, \dots, x_k) = \frac{N!}{(N-k)!} \int_{\mathbb{R}^{N-k}} f(x_1, \dots, x_N) dx_{k+1} \dots dx_N.$$

Doing the same computations as for the Ginibre case we can state the counterpart of Theorem 2.44 and Theorem 2.45.

**Definition 2.53.** A (real) determinantal point process of kernel $K : \mathbb{R}^2 \to \mathbb{R}$ is a point process such that its correlation functions are of the form:

$$\psi_k(x_1, \dots, x_k) = \det[K(x_i, x_j)]_{1 \le i,j \le k}, \text{ for almost all } (x_1, \dots, x_k) \in \mathbb{R}^k, \forall k \in \mathbb{N}_{\ge 1}.$$

**Theorem 2.54.** The eigenvalues of a $N$ complex Wigner matrix is a real determinantal point process of kernel $K_N^H$.

**Theorem 2.55.** For all $N \ge 1$, the density with respect to the real Lebesgue measure of the eigenvalues of a $N$ complex Wigner matrix is given by:

$$\phi_N^H(x_1, \dots, x_N) = \frac{1}{N!} \det[K_N^H(x_i, x_j)]_{1 \le i,j \le N},$$

with $K_N^H$ the kernel defined by:

$$K_N^H(x, y) = \sqrt{\gamma_H(x)\gamma_H(y)} \sum_{l=0}^{N-1} \frac{H_l(x)H_l(y)}{\sqrt{2\pi} l!}, \quad \forall (x, y) \in \mathbb{R}^2.$$

Moreover we have an expression of the marginal densities of this law defined for $1 \le k \le N$, by:

$$(x_1, \dots, x_k) \in \mathbb{R}^k \mapsto \phi_{N,k}^H(x_1, \dots, x_k) := \int_{\mathbb{R}^{N-k}} \phi_N^H(x_1, \dots, x_N) dx_{k+1} \dots dx_N = \frac{(N-k)!}{N!} \psi_k^H(x_1, \dots, x_k),$$
with $\psi_k^H$ the correlation functions of this point process introduced in Definition 2.51. Indeed for all $1 \le k \le N$, and for all $(x_1, ..., x_k) \in \mathbb{R}^k$, we have:

$$ \phi_{N,k}^{H}(x_1, ..., x_k) = \frac{(N-k)!}{N!} \det[K_{N}^{H}(x_i, x_j)]_{1 \le i,j \le k}. $$

In particular we have the following explicit formulas for the marginal $\phi_{N,N}^H = \phi_N$ and for $k = 1$, for all $x \in \mathbb{R}$,

$$ \phi_{N,1}^{H}(x) = \frac{\gamma_{H}(x)}{N} \sum_{l=0}^{N-1} \frac{H_{l}(x)^{2}}{\sqrt{2\pi l!}}. $$

# 3 Circular law

The goal of this section is to establish the circular law for the Ginibre ensemble which states that the empirical distribution of the eigenvalues of a $N$ complex Ginibre random matrix up to a re normalization converges to the uniform law on the unit disk of the complex plan.

Historically, the circular law was obtained for Ginibre random matrices by Ginibre and Mehta [64]. It was expected a phenomenon of universality which means that the result should not depend on the distribution of the entries of the random matrix. There were a lot of works in order to generalize the circular law in this direction. A general proof was obtained for random matrices with independent and identically distributed complex random variables with few hypothesis on the moments by Tao and Vu in [78]. Tao and Vu published many others papers on this topic as [77], [76]. The key tools are the hermitization method introduced by Girko, potential theory and good estimates for the singular values of random matrices.

For readers interested in this topic, we refer to the review on the circular law of Bordenave and Chafaï [18] for a very detailed work on these methods and for historic comments on the circular law.

## 3.1 Mean circular law

First we shall prove the circular law in mean and then in the following section we shall explain why the convergence is actually almost sure.

**Theorem 3.1 (Mean circular law).** Let $\Lambda_N = (\lambda_1^N, ..., \lambda_N^N) \in \mathbb{C}^N$ be the random spectrum of a $N$ complex Ginibre matrix. We write:

$$ \mu_N = \frac{1}{N} \sum_{k=1}^{N} \delta_{\lambda_k^N / \sqrt{N}} $$

the empirical distribution associated to $\Lambda_N/\sqrt{N}$. Moreover, let $\mu_\infty$ be the uniform distribution on the unit disk given by:

$$ \mu_\infty(dz) = \frac{1_{|z| \le 1}}{\pi} dz. $$

Then for every $f: \mathbb{C} \to \mathbb{R}$ bounded continuous function we have:

$$ \mathbb{E} \left[ \int_{\mathbb{C}} f d\mu_N \right] \xrightarrow{N \to +\infty} \int_{\mathbb{C}} f d\mu_\infty. $$
Before doing the proof, let us recall that with the notations of Theorem 3.1, for every $N \ge 1$, we can define a probability measure that shall be written $\mathbb{E}(\mu_N)$ defined by:

$$ \int_{\mathbb{C}} f d\mathbb{E}(\mu_N) := \mathbb{E} \left[ \int_{\mathbb{C}} f d\mu_N \right], $$

for all functions $f$ continuous and bounded. The circular law in mean state that the sequence of probability measures $(\mathbb{E}(\mu_N))_{N \ge 1}$ converges in law towards the uniform distribution on the circle.

*Proof.* We follow the arguments of [25]. Let $f: \mathbb{C} \to \mathbb{R}$ be a continuous compact supported function. Using the law of the eigenvalues obtained in Theorem 2.45 and the symmetry of the density, we have:

$$
\begin{align*}
\mathbb{E} \left[ \int_{\mathbb{C}} f d\mu_N \right] &= \frac{1}{N} \sum_{k=1}^{N} \int_{\mathbb{C}^N} f \left( \frac{z_k}{\sqrt{N}} \right) \phi_N(z_1, \dots, z_N) dz_1 \dots dz_N \\
&= \frac{1}{N} \sum_{k=1}^{N} \int_{\mathbb{C}^N} f \left( \frac{z_k}{\sqrt{N}} \right) \phi_N(z_k, z_2, \dots, z_1, \dots, z_N) dz_k \dots dz_1 \dots dz_N \\
&= \frac{1}{N} \sum_{k=1}^{N} \int_{\mathbb{C}} f \left( \frac{z_k}{\sqrt{N}} \right) \left[ \int_{\mathbb{C}^{N-1}} \phi_N(z_k, z_2, \dots, z_1, \dots, z_N) dz_2 \dots dz_1 \dots dz_N \right] dz_k \\
&= \frac{1}{N} \sum_{k=1}^{N} \int_{\mathbb{C}} f \left( \frac{z_k}{\sqrt{N}} \right) \phi_{N,1}(z_k) dz_k \\
&= \int_{\mathbb{C}} f \left( \frac{z}{\sqrt{N}} \right) \phi_{N,1}(z) dz \\
&= N \int_{\mathbb{C}} f(z) \phi_{N,1}(\sqrt{N}z) dz \tag{3.1}
\end{align*}
$$

where in the last equality we did the change of variable $z \to \sqrt{N}z$ (pay attention that the Jacobian of this transformation is $N$ since we must view $\mathbb{C}$ as $\mathbb{R}^2$ in the integration). Hence, to conclude it remains to prove that for all $K$ compact included in $\{z \in \mathbb{C} : |z| \ne 1\}$ we have:

$$ \limsup_{N \to \infty} \sup_{z \in K} \left| N \phi_{N,1}(\sqrt{N}z) - \frac{1}{\pi} \mathbf{1}_{|z| \le 1} \right| = 0. $$

We recall the explicit expression of $\phi_{N,1}$:

$$ \phi_{N,1}(z) = \frac{1}{\pi N} \exp(-|z|^2) e_N(|z|^2), $$

with $e_N(z) = \sum_{k=0}^{N-1} z^k/k!$ the truncation of the exponential series up to the $N$ term. To finish we just need a lemma to estimate the previous quantity.

**Lemma 3.2.** For every $N \in \mathbb{N}$, and $z \in \mathbb{C}$,

$$ |e_N(Nz) - \exp(Nz) \mathbf{1}_{|z| \le 1}| \le r_N(z), $$

with $r_N(z)$ an error term of the form

$$ r_N(z) = \frac{e^N}{\sqrt{2\pi N}} |z|^N \left( \frac{N+1}{N(1-|z|)+1} \mathbf{1}_{|z| \le 1} + \frac{N}{N(|z|-1)+1} \mathbf{1}_{|z| > 1} \right). $$
Proof. We consider two cases. First, for $|z| \le N$, we have:

$$|e_N(z) - \exp(z)1_{|z| \le N}| \le \sum_{k \ge N} \frac{|z|^k}{k!} \le \frac{|z|^N}{N!} \sum_{k \ge 0} \frac{|z|^k}{(N+1)^k} = \frac{|z|^N}{N!} \frac{N+1}{N+1-|z|}$$

Secondly if $|z| > N$, we get:

$$|e_N(z)| \le \sum_{k=0}^{N-1} \frac{|z|^k}{k!} \le \frac{|z|^{N-1}}{(N-1)!} \sum_{k=0}^{N-1} \frac{(N-1)^k}{|z|^k} \le \frac{|z|^N}{(N-1)!N+1-|z|}$$

To conclude we use the classical Stirling bound $\sqrt{2\pi N}N^N \le N!e^N$ in the two previous inequalities. $\square$

Using Lemma 3.2 and the dominated convergence theorem in (3.1) yields the result. $\square$

**Remark 3.3.** Let us mention a probabilistic point of view given in [25] that gives the pointwise convergence of $N\phi_{N,1}(\sqrt{N}\bullet)$ to $\frac{1}{\pi} \left(1_{|z|<1} + \frac{1}{2}1_{|z|=1}\right)$ (but for the previous application it is not sufficient since we need to have a uniform convergence on every compact included in $\{z \in \mathbb{C}, |z| \ne 1\}$.)

To prove this pointwise convergence, we only have to prove the result for $z = r > 0$ by invariance by rotation of the formula.

Next, we remark that if $Y_1, \dots, Y_N$ are independent Poisson random variables of mean $r^2$, then

$$\exp(-Nr^2)e_N(Nr^2) = \mathbb{P}(Y_1 + \dots + Y_N < N) = \mathbb{P}\left(\frac{Y_1 + \dots + Y_N}{N} < 1\right),$$

since $Y_1 + \dots + Y_N$ is a Poisson random variable of mean $Nr^2$.

Now, using the strong law of large numbers we see that almost surely $\frac{Y_1 + \dots + Y_N}{N}$ converges to $r^2$. We immediately deduce that if $r < 1$

$$\lim_{N \to \infty} \exp(-Nr^2)e_N(Nr^2) = 1$$

and if $r > 1$,

$$\lim_{N \to \infty} \exp(-Nr^2)e_N(Nr^2) = 0.$$

Finally, for the case $r = 1$ we use the central limit theorem to have that:

$$\exp(-Nr^2)e_N(Nr^2) = \mathbb{P}\left(\sqrt{N}\left(\frac{Y_1 + \dots + Y_N}{N} - 1\right) < 0\right) \xrightarrow{N \to \infty} \int_{\mathbb{R}^-} \frac{\exp(-x^2/2)}{\sqrt{2\pi}} dx = \frac{1}{2}.$$

## 3.2 Strong circular law

To prove an almost sure convergence thanks to the mean convergence we will first that the empirical mean is near its expectation almost surely. We follow the arguments of [25]. Let us also mention that the strong circular law can be viewed as a corollary of large deviations of Coulomb gases in dimension 2 that shall be mentioned in Section 11.2 giving another proof of the strong circular law stated in this section.

**Lemma 3.4.** For all $f$ continuous bounded function from $\mathbb{C}$ to $\mathbb{R}$, we can find a constant $C > 0$ that only depends on $f$ such that for all $N \ge 1$:

$$\mathbb{E}\left[\left(\int_{\mathbb{C}} fd\mu_N - \mathbb{E}\left(\int_{\mathbb{C}} fd\mu_N\right)\right)^4\right] \le \frac{C}{N^2}.$$
We shall not give the proof. The idea of the proof is based on the same arguments as the proof of the law of large numbers with hypothesis that the fourth moment is bounded. The proof is quite painful. We refer to [55] for a complete proof of this point.

**Corollary 3.5.** For all $f$ continuous bounded function from $\mathbb{C}$ to $\mathbb{R}$, almost surely we have:

$$ \lim_{N} \left( \int_{\mathbb{C}} f d\mu_{N} - \mathbb{E} \left( \int_{\mathbb{C}} f d\mu_{N} \right) \right) = 0. $$

*Proof.* With the notations of Lemma 3.4, we get that:

$$ \sum_{N \ge 1} \mathbb{E} \left[ \left( \int_{\mathbb{C}} f d\mu_{N} - \mathbb{E} \left( \int_{\mathbb{C}} f d\mu_{N} \right) \right)^{4} \right] < \infty. $$

Thanks to Fubini theorem, we get that:

$$ \mathbb{E} \left[ \sum_{N \ge 1} \left( \int_{\mathbb{C}} f d\mu_{N} - \mathbb{E} \left( \int_{\mathbb{C}} f d\mu_{N} \right) \right)^{4} \right] < \infty. $$

So almost surely:

$$ \sum_{N \ge 1} \left( \int_{\mathbb{C}} f d\mu_{N} - \mathbb{E} \left( \int_{\mathbb{C}} f d\mu_{N} \right) \right)^{4} < \infty. $$

It implies directly that almost surely

$$ \lim_{N} \left( \int_{\mathbb{C}} f d\mu_{N} - \mathbb{E} \left( \int_{\mathbb{C}} f d\mu_{N} \right) \right) = 0. $$

□

**Remark 3.6.** We shall see in Section 4 that we can obtain this result by a concentration inequality in the Wigner case. We shall explain at this moment why we can not do these methods for the Ginibre case.

We can now state the strong circular law.

**Theorem 3.7.** With the notations of the mean circular law, we almost surely have the following convergence:

$$ \mu_N \xrightarrow[N \to +\infty]{} \mu_\infty. $$

*Proof.* Let $f$ be a continuous bounded function from $\mathbb{C}$ to $\mathbb{R}$. On the one hand, thanks to the mean circular law we have that:

$$ \mathbb{E} \left[ \int_{\mathbb{C}} f d\mu_N \right] \xrightarrow[N \to +\infty]{} \int_{\mathbb{C}} f d\mu_\infty. $$

Thanks to Corollary 3.5 we almost surely have that:

$$ \lim_{N \to \infty} \left( \int_{\mathbb{C}} f d\mu_{N} - \mathbb{E} \left[ \int_{\mathbb{C}} f d\mu_{N} \right] \right) = 0. $$

So we deduce that for all $f$ continuous and bounded functions from $\mathbb{C}$ to $\mathbb{R}$, almost surely:

$$ \int_{\mathbb{C}} f d\mu_{N} \xrightarrow[N \to +\infty]{} \int_{\mathbb{C}} f d\mu_{\infty}. $$
Then, using a countable sequence of functions dense in the set of continuous and compactly supported functions from $\mathbb{C}$ to $\mathbb{R}$, standard arguments in measure theory (see for instance [16]) give that almost surely for all $f$ continuous bounded functions from $\mathbb{C}$ to $\mathbb{R}$,

$$ \int_{\mathbb{C}} f d\mu_N \underset{N \to +\infty}{\longrightarrow} \int_{\mathbb{C}} f d\mu_{\infty}. $$

Hence, we get that almost surely:

$$ \mu_N \xrightarrow[N \to +\infty]{\mathcal{L}} \mu_{\infty}. $$

□

### 3.3 Localisation of the spectrum

First we shall give the law of the modulus of eigenvalues of a $N$ complex Ginibre random matrix.

**Proposition 3.8.** Let $\Lambda_N = (\lambda_1, ..., \lambda_N)$ be the exchangeable vector of eigenvalues of a $N$ complex Ginibre random matrix. Then we have the following equality in law:

$$ (|\lambda_1|, ..., |\lambda_N|) = (Z_{\sigma(1)}, ..., Z_{\sigma(N)}), $$

with $Z_1, ..., Z_N$ is a family of independent random variables such that for all $k \ge 1$ the law of $Z_k^2$ is $\text{Gamma}(k, 1)$ and $\sigma$ is a random variable whose law is uniform in the group $\mathfrak{S}_N$ independent of $(Z_i)_{1\le i\le N}$. More explicitly we have that for all $F: \mathbb{R}_+^N \to \mathbb{R}$ bounded symmetric function:

$$ \mathbb{E}(F(|\lambda_1|, ..., |\lambda_N|)) = \mathbb{E}(F(Z_1, ..., Z_N)). $$

**Remark 3.9.** We recall that a random variable has a $\text{Gamma}(k, \lambda)$ distribution if its density with respect to Lebesgue measure is given by $x \mapsto \lambda^k x^{k-1} \exp(-\lambda x) \mathbb{1}_{x \ge 0}/\Gamma(k)$ with $\Gamma(x)$ the Gamma function of Euler.

*Proof.* We follow the arguments of [25] that can also be found in [58]. Using the law of the eigenvalues of $N$ complex Ginibre matrix we shall go in polar coordinate and integrate with respect to the angles to have the distribution of the modulus of the eigenvalues. In polar coordinates we can write the density of the eigenvalues as follow:

$$ (r_1, ..., r_N, \theta_1, ..., \theta_N) \mapsto \frac{\exp(-\sum_{i=1}^N r_i^2)}{\pi^N \prod_{i=1}^N k!} \prod_{j<k} |r_j \exp(i\theta_j) - r_k \exp(i\theta_k)|^2. $$

To integrate this we shall linearise the product.

To do so we interpret the product as the product of the determinant of two matrices:

$$ \begin{aligned} \prod_{j<k} |r_j \exp(i\theta_j) - r_k \exp(i\theta_k)|^2 &= \prod_{j<k} (r_j \exp(i\theta_j) - r_k \exp(i\theta_k)) \prod_{j<k} (r_j \exp(-i\theta_j) - r_k \exp(-i\theta_k)) \\ &= \det[r_j^{k-1} \exp(i(k-1)\theta_j)]_{1\le j, k \le N} \det[r_j^{k-1} \exp(-i(k-1)\theta_j)]_{1\le j, k \le N} \\ &= \sum_{\sigma, \sigma' \in \mathfrak{S}_N} (-1)^{\varepsilon(\sigma) + \varepsilon(\sigma')} \prod_{j=1}^N r_j^{\sigma(j) + \sigma'(j) - 2} \exp(i(\sigma(j) - \sigma'(j))\theta_j). \end{aligned} $$

Integrating this term with respect to the angles, the only term that shall not vanish is when $\sigma = \sigma'$. Hence it gives:

$$ \int_{[0,2\pi]^N} \prod_{j<k} |r_j \exp(i\theta_j) - r_k \exp(i\theta_k)|^2 d\theta_1 \dots d\theta_N = (2\pi)^N \sum_{\sigma \in \mathfrak{S}_N} \prod_{j=1}^N r_j^{2\sigma(j)-2}. $$
Now let us recall a linear algebra notation. Let $A \in M_N(C)$, we call permanent of A the following quantity:

$$ \mathrm{per}(A) = \sum_{\sigma \in \mathfrak{S}_N} \prod_{j=1}^{N} A_{j, \sigma(j)}. $$

Thanks to this notation we can write a compact formula for the density of the modulus of the eigenvalues:

$$ \int_{[0,2\pi]^N} \frac{e^{-\sum_{i=1}^N r_i^2}}{\pi^N \prod_{i=1}^N k!} \prod_{j<k} |r_j \exp(i\theta_j) - r_k \exp(i\theta_k)|^2 d\theta_1 \dots d\theta_N = \frac{2^N \exp(-\sum_{i=1}^N r_i^2)}{\prod_{i=1}^N k!} \mathrm{per}[r_j^{2k}]_{1\le j,k \le N} \\ = \frac{1}{N!} \mathrm{per} \left[ \frac{2}{(k-1)!} r_j^{2k} \exp(-r_j^2) \right]_{1\le j,k \le N}. $$

We recognise inside the permanent the expression of the expected density. Indeed, if $Z_k$ has a density with respect to Lebesgue measure given by $g_k(r) = \frac{2}{(k-1)!} r^{2k} \exp(-r^2) 1_{r \ge 0}$ then an easy computation show that $Z_k^2$ has a Gamma$(k, 1)$ law. Finally we just need to notice that if $X_1, \dots, X_N$ are independent random variables with there respective densities given by $(f_k)_{1 \le k \le N}$ then $(x_1, \dots, x_N) \mapsto \frac{1}{N!} \mathrm{per}[f_k(x_j)]_{1 \le j,k \le N}$ is the density of the random variable $(X_{\sigma(1)}, \dots, X_{\sigma(N)})$ with $\sigma$ a uniform permutation in $\mathfrak{S}_N$ independent of the $(X_i)_{1 \le i \le N}$. It proves the result. $\square$

We first mention an application of this result to study the hole probability of the point process of the eigenvalues of a $N$ complex Ginibre matrices with $N$ large, that is the probability that a disk of radius $r$ contains no points of the point process when $r$ becomes large.

We shall define the infinite Ginibre ensemble as the point process with kernel $K_\infty$ obtained as the limit of the kernel of $N$ complex Ginibre matrices:

$$ \forall (z, w) \in \mathbb{C}^2, \lim_{N \to +\infty} K_N(z, w) =: K_\infty(z, w). $$

A direct computation prove that for all $(z, w) \in \mathbb{C}^2$

$$ K_\infty(z, w) = \frac{1}{\pi} \exp \left( z \bar{w} - \frac{1}{2} |z|^2 - \frac{1}{2} |w|^2 \right), $$

and the convergence is uniform on every compact of $\mathbb{C}^2$. We refer to [1, 54] for more details about the infinite Ginibre ensemble.

**Definition 3.10 (Infinite Ginibre ensemble).** We define the infinite Ginibre ensemble $\Lambda_\infty$ as the point process in $\mathbb{C}$ with kernel

$$ K_\infty(z, w) = \frac{1}{\pi} \exp \left( z \bar{w} - \frac{1}{2} |z|^2 - \frac{1}{2} |w|^2 \right). $$

We can also define the hole probability of a point process.

**Definition 3.11 (Hole probability of a point process).** Let $Z$ be a point process in $\mathbb{C}$. For $r > 0$, we define $n(r) := |Z \cap \bar{B}(0, r)|$ as the number of points of the point process $Z$ in $\bar{B}(0, r)$. We call hole probability of the point process the probability that the point process has no points in $\bar{B}(0, r)$ which is $\mathbb{P}(n(r) = 0)$.
We give an easy example to understand this quantity.

**Example 3.12.** Let $Z$ be a Poisson point process on $\mathbb{C}$ of parameter $\lambda > 0$ introduced in Example 2.35. By definition of a Poisson point process, $n(r) = |Z \cap \overline{B}(0, r)|$ is a Poisson random variable of parameter $\pi r^2 \lambda$. So the hole probability is given by $\mathbb{P}(n(r) = 0) = \exp(-\pi r^2 \lambda)$.

We give another example that is an immediate consequence of Proposition 3.8.

**Example 3.13.** For $N \ge 1$, let $\Lambda_N = (\lambda_1, ..., \lambda_N)$ be the set of eigenvalues of a $N$ complex Ginibre matrix. The hole probability of $\Lambda_N$ is given by

$$ \mathbb{P}(n(r) = 0) = \mathbb{P}(\forall i \in [1, N], |\lambda_i| > R) = \mathbb{P}(\forall i \in [1, N], Z_i > R), $$

where $Z_i$ are independent random variables such that for all $1 \le k \le N$, $Z_k^2$ has a Gamma$(k, 1)$ distribution. By independence, we get

$$ \mathbb{P}(n(r) = 0) = \prod_{i=1}^{N} \mathbb{P}(Z_k^2 > r^2). $$

Let $(E_i)_{i \ge 1}$ be a family of iid exponential random variables of parameter 1 (that can also be viewed as Gamma$(1, 1)$ random variables), then by the stability property of Gamma distributions, for all $k \ge 1$, we know that the law of $E_1 + ... + E_k$ is Gamma$(k, 1)$. Hence the hole probability can be written

$$ \mathbb{P}(n(r) = 0) = \prod_{i=1}^{N} \mathbb{P}(E_1 + ... + E_k > r^2), $$

where $(E_i)_{i \ge 1}$ be a family of iid exponential law of parameter 1.

As a consequence of the uniform convergence on every compact of $K_N$ toward $K_\infty$ we have the following result

**Lemma 3.14.** Let $\Lambda_\infty$ be the infinite Ginibre ensemble, then the hole probability of $\Lambda_\infty$ is given by

$$ \mathbb{P}(n(r) = 0) = \lim_{N \to +\infty} \mathbb{P}(|\Lambda_N \cap \overline{B(0, r)}| = 0) = \prod_{k=1}^{+\infty} \mathbb{P}(Z_k^2 > r^2) = \prod_{k=1}^{+\infty} \mathbb{P}(E_1 + ... + E_k > r^2), $$

where $(E_k)_{k \ge 1}$ is a family of independent exponential law of parameter 1 and $(Z_k)_{k \ge 1}$ is a family of independent random variables whose respective law are Gamma$(k, 1)$.

**Proof.** We give the main idea but not get into details. Since $K_N$ converges uniformly to $K_\infty$ on every compacts, we have that

$$ \mathbb{P}(n(r) = 0) = \mathbb{P}(|\Lambda_\infty \cap \overline{B(0, r)}|) = \lim_{N \to +\infty} \mathbb{P}(|\Lambda_N \cap \overline{B(0, r)}|). \quad (3.2) $$

This can be obtained by using the so-called Fredholm determinant. These determinants naturally appear when we look at the probability that a set does not contain any eigenvalues of a $N$ complex Ginibre matrix (more generally when we have a determinantal point process). The Fredholm determinant theory is really useful when we have to pass to the limit when $N$ becomes large to study the limit of the probability that a set does not contain any eigenvalues of a $N$ complex Ginibre matrix. We refer to the Section 3.4 of [5] for a short introduction to Fredholm determinant and its use for these questions in determinantal models of random matrices. Using (3.2) and Example 3.13 give the result. $\square$
The goal of the next theorem is to estimate for the infinite Ginibre ensemble how fast the hole probability goes to 0 when $r$ becomes large. The following result can be viewed as large deviations principle and actually we shall use in the proof the Cramer theorem that is recall in the introduction of Section III.

**Theorem 3.15.** Let $\Lambda_\infty$ be the infinite Ginibre point process. Then

$$ \lim_{r \to +\infty} \frac{1}{r^4} \log (\mathbb{P}(n(r) = 0)) = -\frac{1}{4}. $$

*Proof.* We give the main arguments of the proof. A complete proof can be found in Section 7.2 of [54]. From Lemma 3.14, we have an explicit expression for the hole probability

$$ \mathbb{P}(n(r) = 0) = \prod_{k=1}^{+\infty} \mathbb{P}(Z_k^2 > r^2) = \prod_{k=1}^{+\infty} \mathbb{P}(E_1 + \dots + E_k > r^2), \quad (3.3) $$

where $(E_k)_{k \ge 1}$ is a family of independent exponential law of parameter 1. As explained before this expression come from the fact that $E_1 + \dots + E_k$ has a $\text{Gamma}(k, 1)$ distribution for all $k \ge 1$.

We first lower bound this probability using the Markov inequality. Indeed, we obtain by independence of the $(E_i)_{i \ge 1}$ that for $k \ge 1$, for all $\theta < 1$,

$$ \begin{aligned} \mathbb{P}(E_1 + \dots + E_k > r^2) &\le \exp(-\theta r^2) \exp(\theta(E_1 + \dots + E_k)) \\ &= \exp(-\theta r^2) (1 - \theta)^{-k}. \end{aligned} \quad (3.4) $$

We see that the optimal $\theta$ to optimize the previous bound is $\theta = 1 - \frac{k}{r^2}$. We now do as if $r^2$ is an integer (if not we should consider the integer part of $r^2$ instead which does not change the estimate we shall do). We can upper bound the hole probability using (3.4):

$$ \begin{aligned} \mathbb{P}(n(r) = 0) &= \prod_{k=1}^{+\infty} \mathbb{P}(E_1 + \dots + E_k > r^2) \\ &\le \prod_{k=1}^{r^2} \mathbb{P}(E_1 + \dots + E_k > r^2) \\ &\le \prod_{k=1}^{r^2} \exp \left( - \left( 1 - \frac{k}{r^2} \right) r^2 - k \log \left( \frac{k}{r^2} \right) \right) \quad (3.5) \\ &\le \exp \left( - \frac{r^2}{2} (r^2 - 1) - r^4 \left( \int_0^1 x \log(x) dx \right) + o(r^4) \right) \\ &= \exp \left( - \frac{r^4}{4} + o(r^4) \right), \end{aligned} $$

where we used that the Riemann sum $\sum_{k=1}^{r^2} \frac{k}{r^2} \log(\frac{k}{r^2}) = r^2(\int_0^1 x \log(x) dx + o(1))$. For the lower bound, we start by noticing that for all $k \ge 1$, for all $\lambda > 0$,

$$ \mathbb{P}(\text{Gamma}(k, 1) > \lambda) = \mathbb{P}(\text{Poisson}(\lambda) < k) $$

where $\text{Poisson}(\lambda)$ is the standard Poisson law of parameter $\lambda$. Using this remark we obtain that for all $k \ge 1$,

$$ \mathbb{P}(Z_k^2 > r^2) = \mathbb{P}(\text{Poisson}(r^2) < k - 1) \ge \exp(-r^2) \frac{r^{2(k-1)}}{(k-1)!} $$
Using this bound, we lower bound the terms for $k \le r^2$

$$
\begin{align}
\prod_{k=1}^{r^2} \mathbb{P}(Z_k^2 > r^2) &\ge \prod_{k=1}^{r^2} \exp(-r^2) \frac{r^{2(k-1)}}{(k-1)!} \nonumber \\
&= \exp\left(-r^4 + \sum_{k<r^2} [k \log(r^2) - \log(k!)]\right) \tag{3.6} \\
&= \exp\left(-r^4 + \sum_{k<r^2} \left[(r^2-k) \log\left(\frac{k}{r^2}\right)\right]\right). \nonumber
\end{align}
$$

Again using a Riemann sum, we obtain that

$$
\prod_{k=1}^{r^2} \mathbb{P}(Z_k^2 > r^2) \geq \exp\left(-\frac{r^4}{4} + o(r^4)\right). \quad (3.7)
$$

For the other terms of the product, we can first notice that since $\mathbb{P}(\text{Poisson}(\lambda) > \lambda) \xrightarrow{\lambda \to +\infty} 1/2$, for $r$ large enough, for all $k > r^2$ we get

$$
\begin{align}
\mathbb{P}(Z_k^2 > r^2) &= \mathbb{P}(E_1 + \dots + E_k > r^2) \ge \mathbb{P}(E_1 + \dots + E_{r^2} > r^2) \nonumber \\
&= \mathbb{P}(Z_{r^2}^2 > r^2) = \mathbb{P}(\text{Poisson}(r^2) > r^2 - 1) \ge \frac{1}{4}. \tag{3.8}
\end{align}
$$

This gives that for $r$ large enough

$$
\prod_{k=r^2+1}^{2r^2} \mathbb{P}(E_1 + \dots + E_k > r^2) \geq \exp(-r^2 \log(4)). \quad (3.9)
$$

Finally, using the Cramer theorem, there exists $c > 0$ such that for all $k \ge 1$,

$$
\mathbb{P} \left( E_1 + \dots + E_k < \frac{k}{2} \right) \leq \exp(-ck).
$$

Hence we obtain that

$$
\begin{align}
\sum_{k=2r^2+1}^{+\infty} \mathbb{P} (Z_k^2 < r^2) &= \sum_{k=2r^2+1}^{+\infty} \mathbb{P} (E_1 + \dots + E_k < r^2) \le \sum_{k=2r^2+1}^{+\infty} \mathbb{P} (E_1 + \dots + E_k < \frac{k}{2}) \nonumber \\
&\le \sum_{k=2r^2+1}^{+\infty} \exp(-ck). \tag{3.10}
\end{align}
$$

Taking $r$ large enough we get

$$
\sum_{k=2r^2+1}^{+\infty} \mathbb{P} (Z_k^2 < r^2) < \frac{1}{2}.
$$

By independence of the $(Z_i)_{i \ge 1}$, we obtain that for $r$ large enough:

$$
\begin{align}
\prod_{k=2r^2+1}^{+\infty} \mathbb{P}(Z_k^2 > r^2) &= \mathbb{P}\left(\bigcap_{k>2r^2} Z_k^2 > r^2\right) = 1 - \mathbb{P}\left(\bigcup_{k>2r^2} Z_k^2 \le r^2\right) \ge 1 - \sum_{k=2r^2+1}^{+\infty} \mathbb{P}(Z_k^2 \le r^2) \nonumber \\
&\ge \frac{1}{2}. \tag{3.11}
\end{align}
$$
Using (3.5), (3.9) (3.11) we have that for $r$ large enough

$$ \mathbb{P}(n(r) = 0) = \prod_{k=1}^{+\infty} \mathbb{P}(Z_k^2 > r^2) \geq \exp\left(-\frac{r^4}{4} + o(r^4)\right). \quad \square $$

Let us also mention that this result has also been proved using a potential theory approach and proving a large deviations principle for the hole probability of $N$ complex Ginibre matrix in [1]. Finally, let us also say that determinantal point processes appear when we study the roots of power series with random coefficients. Similar results about the hole probability of such determinantal point processes can be obtained. See Section 7 of [54] for instance.

We now mention some results about the spectral radius of $N$ complex Ginibre matrices.

**Theorem 3.16.** Let $\Lambda_N = (\lambda_1, ..., \lambda_N)$ be the exchangeable vector of eigenvalues of a $N$ complex Ginibre random matrix. Then we have the following convergence in probability:

$$ \rho_N := \max_{1 \leq k \leq N} \frac{\lambda_k}{\sqrt{N}} \xrightarrow{N \to +\infty} 1. $$

*Proof.* We just here give the principal idea based on Proposition 3.8. We use the same notations as in the Proposition 3.8. By the additive property of Gamma distribution, for all $k \geq 1$ we write $Z_k^2 = E_1 + ... + E_k$ with $(E_i)_{i \in \mathbb{Z}}$ a family of iid random variable with exponential law of parameter 1. Then by independence, for all $r > 0$ we have

$$ \mathbb{P}(\rho_N \leq r) = \prod_{1 \leq k \leq N} \mathbb{P} \left( \frac{E_1 + \dots + E_k}{N} \leq r^2 \right). $$

By the strong law of large numbers we know that if $r < 1$ each term in the product converges to 0 and if $r > 1$ each term converges to 1. From that we can deduce that $\mathbb{P}(\rho_N \leq r)$ converges to 0 if $r < 1$ and 1 if $r > 1$. So $\rho_N$ converges to 0 in law and so in probability since 0 is a constant random variable. $\square$

Actually we can prove that the convergence is not only in probability but also almost surely and we can compute the fluctuation of this convergence, see [70].

**Theorem 3.17.** Let $\Lambda_N = (\lambda_1, ..., \lambda_N)$ be the exchangeable vector of eigenvalues of a $N$ complex Ginibre random matrix. Then we have almost surely:

$$ \rho_N = \max_{1 \leq k \leq N} \frac{\lambda_k}{\sqrt{N}} \xrightarrow{N \to +\infty} 1. $$

Moreover, if we let $\kappa_N = \log(N/2\pi) - 2\log(\log(N))$, then we have

$$ \sqrt{4N\kappa_N} \left( \rho_N - 1 - \sqrt{\frac{\kappa_N}{4N}} \right) \xrightarrow{N \to +\infty} \text{Gumbel,} $$

where Gumbel is the law of $-\log(X)$ with $X$ an exponential random variable of parameter 1.
# 4 The Wigner theorem

## 4.1 The mean Wigner theorem

**Theorem 4.1** (Wigner theorem in mean). Let $(M_N)_{N \ge 1}$ be a sequence of random matrices such that for every $N \ge 1$, $M_N$ is a $N$ complex Wigner matrix. For every $N \ge 1$, let $\Lambda^N = (\lambda_1^N, ..., \lambda_N^N) \in \mathbb{R}^N$ be the random spectrum of the $N$ complex Wigner matrix $M_N$. We write:

$$ \mu_N = \frac{1}{N} \sum_{k=1}^{N} \delta_{\lambda_k^N / \sqrt{N}} $$

the empirical distribution associated to the eigenvalues of $M_N/\sqrt{N}$.

Then for every bounded and continuous function $f: \mathbb{R} \to \mathbb{R}$ we have:

$$ \mathbb{E} \left[ \int_{\mathbb{R}} f d\mu_N \right] \underset{N \to +\infty}{\longrightarrow} \int_{\mathbb{R}} f d\sigma, $$

where $\sigma$ is the so-called Wigner semicircle distribution introduced in Appendix A which is the measure on $\mathbb{R}$ whose density with respect to the Lebesgue measure is given by:

$$ \sigma(x) := \frac{\sqrt{4 - x^2}}{2\pi} \mathbb{1}_{[-2,2]}(x). $$

Before doing the proof, let us recall that with the notations of Theorem 4.1, for every $N \ge 1$, we can define a probability measure that shall be written $\mathbb{E}(\mu_N)$ defined by:

$$ \int_{\mathbb{R}} f d\mathbb{E}(\mu_N) := \mathbb{E} \left[ \int_{\mathbb{R}} f d\mu_N \right], $$

for all functions $f$ continuous and bounded. The Wigner theorem in mean state that the sequence of probability measures $(\mathbb{E}(\mu_N))_{N \ge 1}$ converges in law toward the semicircular distribution $\sigma$.

*Proof.* We follow in this proof the arguments given in [5] in Section 3.3 which are up to our knowledge originally from [50].

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous and bounded function. Using the density of the eigenvalues of a $N$ Wigner matrix obtained in Theorem 2.55 and the symmetry of the distribution, we have:

$$ \begin{aligned} \mathbb{E} \left[ \int_{\mathbb{R}} f d\mu_N \right] &= \frac{1}{N} \sum_{k=1}^{N} \int_{\mathbb{R}^N} f \left( \frac{x_k}{\sqrt{N}} \right) \phi_N^H(x_1, \dots, x_N) dx_1 \dots dx_N \\ &= \frac{1}{N} \sum_{k=1}^{N} \int_{\mathbb{R}^N} f \left( \frac{x_k}{\sqrt{N}} \right) \phi_N^H(x_k, x_2, \dots, x_1, \dots, x_N) dx_k \dots dx_1 \dots dx_N \\ &= \frac{1}{N} \sum_{k=1}^{N} \int_{\mathbb{R}} f \left( \frac{x_k}{\sqrt{N}} \right) \left[ \int_{\mathbb{R}^{N-1}} \phi_N^H(x_k, x_2, \dots, x_1, \dots, x_N) dx_2 \dots dx_1 \dots dx_N \right] dx_k \\ &= \frac{1}{N} \sum_{k=1}^{N} \int_{\mathbb{R}} f \left( \frac{x_k}{\sqrt{N}} \right) \phi_{N,1}^H(x_k) dx_k \\ &= \int_{\mathbb{R}} f \left( \frac{x_k}{\sqrt{N}} \right) \phi_{N,1}^H(x) dx \\ &= \int_{\mathbb{R}} f \left( \frac{x_k}{\sqrt{N}} \right) \frac{K_N^H(x, x)}{N} dx. \end{aligned} \quad (4.1) $$
We now use properties of the Hermite polynomials stated in Proposition 2.47 to obtain another expression of $K_N^H(x, x)$. In this proof only, we define for $k \ge 0$,

$$ \mathcal{H}_k(x) := \frac{\sqrt{\gamma_H(x) H_k(x)}}{\sqrt{2\pi k!}} $$

such that we can rewrite $K_N^H$ as

$$ K_N^H(x, y) = \sum_{l=0}^{N-1} \mathcal{H}_l(x) \mathcal{H}_l(y). $$

Using the Christoffel–Darboux formula for $x \neq y$, we get

$$ K_N^H(x, y) = \sqrt{N} \frac{\mathcal{H}_N(x)\mathcal{H}_{N-1}(y) - \mathcal{H}_N(y)\mathcal{H}_{N-1}(x)}{x - y}. $$

Letting $y$ goes to $x$ yields that for all $x \in \mathbb{R}$,

$$ K_N^H(x, x) = \sqrt{N} [\mathcal{H}_N'(x)\mathcal{H}_{N-1}(x) - \mathcal{H}_N(x)\mathcal{H}_{N-1}'(x)]. $$

Using the formula (2.13) or (2.14) we obtain that for all $N \ge 0$,

$$ \mathcal{H}_N''(x) + \left(n + \frac{1}{2} - \frac{x^2}{4}\right) \mathcal{H}_N(x) = 0. $$

This gives the identity

$$ \frac{d}{dx} K_N^H(x, x) = -\sqrt{N} \mathcal{H}_N(x) \mathcal{H}_{N-1}(x). \quad (4.2) $$

Thanks to these identities we shall do an integration by parts in (4.1).

Since the measure $\sigma$ is compactly supported we only need to prove the convergence of the expectation of the moments of the empirical measure toward the moments of the semicircular distribution.

To do so, we shall compute for every $N \ge 1$ the moment generating function of $\mathbb{E}[\mu_N]$. Given a measure $\mu \in \mathcal{P}(\mathbb{R})$ we introduce its Laplace transform defined on $\mathbb{R}$ as

$$ \mathcal{L}_{\mu}(s) := \int_{\mathbb{R}} \exp(sx) \mu(dx). $$

The Laplace transform is also called the moment generating function since for $\mu \in \mathcal{P}(\mathbb{R})$

$$ \mathcal{L}_{\mu}(s) = \sum_{n \ge 0} \int_{\mathbb{R}} x^n \mu(dx) \frac{s^n}{n!}, $$

when the sum converges absolutely.

For $N \ge 1$, using (4.1), for all $s \in \mathbb{R}^*$,

$$ \begin{aligned} \mathcal{L}_{\mathbb{E}[\mu_N]}(s) &= \mathbb{E} \left[ \int_{\mathbb{R}} \exp(st) \mu_N(dt) \right] \\ &= \int_{\mathbb{R}} \exp \left( \frac{sx}{\sqrt{N}} \right) \frac{K_N^H(x, x)}{N} dx \quad (4.3) \\ &= \frac{1}{s} \int_{\mathbb{R}} \exp \left( \frac{sx}{\sqrt{N}} \right) \mathcal{H}_N(x) \mathcal{H}_{N-1}(x) dx, \end{aligned} $$
where we integrated by parts for the last identity and use (4.2). Using the definition of $\mathcal{H}_N$, we compute for $s \in \mathbb{R}^*$ and $N \ge 1$,

$$ 
\begin{aligned}
\frac{1}{s} \int_{\mathbb{R}} \exp\left(\frac{sx}{\sqrt{N}}\right) \mathcal{H}_N(x) \mathcal{H}_{N-1}(x) dx &= \frac{\sqrt{N}}{N!\sqrt{2\pi s}} \int_{\mathbb{R}} \exp\left(-\frac{x^2}{2} + \frac{sx}{\sqrt{N}}\right) H_N(x) H_{N-1}(x) dx \\
&= \frac{\sqrt{N} \exp\left(\frac{s^2}{2N}\right)}{N!\sqrt{2\pi s}} \int_{\mathbb{R}} \exp\left(-\frac{x^2}{2}\right) H_N\left(x + \frac{s}{\sqrt{N}}\right) H_{N-1}\left(x + \frac{s}{\sqrt{N}}\right) dx.
\end{aligned}
\quad (4.4) $$

We finally want to use the orthogonality of the Hermite polynomials with respect to the weight $\gamma_H$ to compute this integral. To do so, let us recall that since $H'_n = nH_{n-1}$ for all $n \ge 1$, a Taylor expansion in $s$ gives that for all $n \ge 0$, for all $x \in \mathbb{R}$, for all $s \in \mathbb{R}$,

$$ H_n\left(x + \frac{s}{\sqrt{N}}\right) = \sum_{k=0}^{n} \binom{n}{k} H_{n-k}(x) \left(\frac{s}{\sqrt{N}}\right)^k = \sum_{k=0}^{n} \binom{n}{k} H_k(x) \left(\frac{s}{\sqrt{N}}\right)^{n-k}. $$

Using this formula to express

$$ H_N\left(x + \frac{s}{\sqrt{N}}\right) $$

and

$$ H_{N-1}\left(x + \frac{s}{\sqrt{N}}\right) $$

in (4.4) and using the orthogonality property of the Hermite polynomials, we obtain that for all $N \ge 1$, for all $s \in \mathbb{R}$,

$$ 
\begin{aligned}
\mathcal{L}_{\mathbb{E}[\mu_N]}(s) &= \frac{\exp\left(\frac{s^2}{2N}\right)}{s} \sum_{k=0}^{N-1} \frac{k!}{N!} \binom{N}{k} \binom{N-1}{k} \left(\frac{s}{\sqrt{N}}\right)^{2N-1-2k} \\
&= \frac{\exp\left(\frac{s^2}{2N}\right)}{s} \sum_{k=0}^{N-1} \frac{(N-1-k)!}{N!} \binom{N}{N-1-k} \binom{N-1}{k} \left(\frac{s}{\sqrt{N}}\right)^{2k+1} \quad (4.5) \\
&= \exp\left(\frac{s^2}{2N}\right) \sum_{k=0}^{N-1} \frac{1}{k+1} \binom{2k}{k} \frac{(N-1)\dots(N-k)}{N^k} \frac{s^{2k}}{(2k)!}
\end{aligned}
$$

We recall that the moments of the semicircular law were computed in Appendix in Proposition A.12. Passing to the limit, we obtain that for all $s \in \mathbb{R}$,

$$ \mathcal{L}_{\mathbb{E}[\mu_N]}(s) \xrightarrow[N \to +\infty]{s \to \infty} \sum_{k=0}^{+\infty} \frac{1}{k+1} \binom{2k}{k} \frac{s^{2k}}{(2k)!} = \sum_{k=0}^{+\infty} \int_{\mathbb{R}} x^k \sigma(dx) \frac{s^k}{k!} = \mathcal{L}_{\sigma}(s), \quad (4.6) $$

This gives the result. $\square$

## 4.2 A concentration inequality

We first prove a concentration inequality for Lipschitz functions of Gaussian random variables. More generally for concentration inequalities we refer to the textbook [60].
**Theorem 4.2.** Let $X_1, ..., X_d$ be $d$ independent centered Gaussian random variable of variance smaller than $\sigma^2$. Let $F: \mathbb{R}^d \to \mathbb{R}$ which is $\lambda > 0$ Lipschitz which means that for all $(x, y) \in \mathbb{R}^d$, $|F(x) - F(y)| \le \lambda|x - y|_2$. Then for all $t > 0$ we have:

$$ \mathbb{P}(|F(X_1, ..., X_d) - \mathbb{E}(F(X_1, ..., X_d))| > t) \le 2 \exp \left( -2 \left( \frac{t}{\lambda\sigma\pi} \right)^2 \right) \quad (4.7) $$

*Proof.* We follow the arguments of Tao in [75] (Theorem 2.1.12). Up to changing $F$ and the $(X_i)_{1 \le i \le d}$ we can suppose that $\mathrm{Var}(X_i) = 1$ for all $1 \le i \le d$, $\lambda = 1$ and $\mathbb{E}(F(X_1, ..., X_d)) = 0$. Moreover using a density argument or the fact that a Lipschitz function is differentiable almost everywhere by Rademacher theorem, we can assume that $F$ is $C^1$ and so satisfies that $|\nabla F| \le 1$ in $\mathbb{R}^d$.

Fix $\lambda > 0$. As usual for concentration inequalities, we shall try to bound $\mathbb{E}(\exp(\lambda F(X_1, ..., X_d)))$. To do so, let introduce $(Y_1, ..., Y_d)$ an independent copy of $(X_1, ..., X_d)$. First by the Jensen inequality, we get that

$$ \mathbb{E}(\exp(-\lambda F(Y_1, ..., Y_d))) \ge \exp(-\lambda \mathbb{E}(F(Y_1, ..., Y_d))) = 1. $$

We obtain

$$ \mathbb{E}(\exp(\lambda F(X_1, ..., X_d))) \le \mathbb{E}(\exp(\lambda(F(X_1, ..., X_d) - F(Y_1, ..., Y_d)))) \quad (4.8) $$

Let $X = (X_1, ..., X_d)$ and $Y = (Y_1, ..., Y_d)$. $X$ and $Y$ are independent centered Gaussian vectors in $\mathbb{R}^d$ of covariance matrix $I_d$ the identity matrix of $M_d(\mathbb{R})$. We shall construct a path to rely $X$ and $Y$ which is not the affine one. We define $Z: [0, \pi/2] \to \mathbb{R}^d$ by $Z(s) = \sin(s)X + \cos(s)Y$. We get

$$ F(X) - F(Y) = F(Z(\pi/2)) - F(Z(0)) = \int_{0}^{\pi/2} \langle \nabla F(Z(s)) | Z'(s) \rangle ds. \quad (4.9) $$

For all $s \in [0, \pi/2]$, $Z(s)$ is a centred Gaussian vector in $\mathbb{R}^d$ with covariance matrix $I_d$. We remark that for all $s \in [0, \pi/2]$, $Z'(s) = \cos(s)X - \sin(s)Y$ is also a centred Gaussian vector in $\mathbb{R}^d$ with covariance matrix $I_d$. The remarkable property that justifies the choice of the path $Z$ is that for all $s \in [0, \pi/2]$, $Z(s)$ and $Z'(s)$ are independent. Indeed, fix $s \in [0, \pi/2]$, then $(Z(s), Z'(s))$ is a centred Gaussian vector in $\mathbb{R}^{2d}$ and for all $1 \le i \ne j \le d$,

$$ \begin{aligned} \mathbb{E}[Z(s)_i Z'(s)_j] &= \mathbb{E}[(\sin(s)X_i + \cos(s)Y_i)(\cos(s)X_j - \sin(s)Y_j)] = 0 \\ \mathbb{E}[Z(s)_i Z'(s)_i] &= \mathbb{E}[(\sin(s)X_i + \cos(s)Y_i)(\cos(s)X_i - \sin(s)Y_i)] = \cos(s)\sin(s) - \cos(s)\sin(s) = 0 \end{aligned} $$

since $X$ and $Y$ are independent centred Gaussian vectors of covariance matrix $I_d$. Hence, the covariance matrix of the Gaussian vector $(Z(s), Z'(s))$ is $I_{2d}$. By standard properties of Gaussian vectors it implies that $Z(s)$ and $Z'(s)$ are independent. Starting from (4.8) we get:

$$ \begin{aligned} \mathbb{E}(\exp(\lambda F(X_1, ..., X_d))) &\le \mathbb{E} \left[ \exp \left( \lambda \int_0^{\pi/2} \langle \nabla F(Z(s)) | Z'(s) \rangle ds \right) \right] \\ &= \mathbb{E} \left[ \exp \left( \lambda \frac{\pi}{2} \int_0^{\pi/2} \langle \nabla F(Z(s)) | Z'(s) \rangle \frac{2}{\pi} ds \right) \right] \tag{4.10} \\ &\le \mathbb{E} \left[ \int_0^{\pi/2} \exp \left( \lambda \frac{2}{\pi} \langle \nabla F(Z(s)) | Z'(s) \rangle \right) \frac{\pi}{2} ds \right] \\ &= \frac{2}{\pi} \int_0^{\pi/2} \mathbb{E} \left[ \exp \left( \lambda \frac{\pi}{2} \langle \nabla F(Z(s)) | Z'(s) \rangle \right) \right] ds. \end{aligned} $$
For all $s \in [0, \pi/2]$, conditioning on $Z(s)$ and using that $Z'(s)$ is a standard Gaussian vector independent of $Z(s)$, we obtain:

$$ 
\begin{aligned}
\mathbb{E} \left[ \exp \left( \lambda \frac{\pi}{2} \langle \nabla F(Z(s)) | Z'(s) \rangle \right) \right] &= \mathbb{E} \left[ \mathbb{E} \left[ \exp \left( \lambda \frac{\pi}{2} \langle \nabla F(Z(s)) | Z'(s) \rangle \right) \right] | Z(s) \right] \\
&= \mathbb{E} \left[ \exp \left( \left( \lambda \frac{\pi}{2} \right)^2 \frac{\| \nabla F(Z(s)) \|^2}{2} \right) \right] \\
&\leq \exp \left( \frac{\lambda^2 \pi^2}{8} \right)
\end{aligned}
$$

Using this estimate in (4.10) we get:

$$ \mathbb{E}(\exp(\lambda F(X_1, \dots, X_d))) \leq \exp\left(\frac{\lambda^2 \pi^2}{8}\right) \quad (4.11) $$

Let $t > 0$, using the Markov inequality we have that for all $\lambda > 0$:

$$ 
\begin{aligned}
\mathbb{P}(F(X_1, \dots, X_d) > t) &= \mathbb{P}(\exp(\lambda F(X_1, \dots, X_d)) > \exp(\lambda t)) \\
&\leq \mathbb{E}(\exp(\lambda F(X_1, \dots, X_d))) \exp(-\lambda t) \\
&\leq \exp\left(\frac{\lambda^2 \pi^2}{8}\right) \exp(-t\lambda)
\end{aligned}
$$

We optimize the right term in $\lambda$ and choosing $\lambda = 4t/\pi^2$ yields

$$ \mathbb{P}(F(X_1, \dots, X_d) > t) \leq \exp\left(-\frac{2t^2}{\pi^2}\right). $$

By symmetry we obtain the result. $\square$

To use Theorem 4.2 in the context of random matrices the key lemma is the so-called Hoffman-Wielandt inequality.

**Lemma 4.3 (Hoffman-Wielandt's lemma).** Let $A, B \in \mathcal{H}_N(\mathbb{C})$ and $\Lambda^A := (\lambda_1^A, \dots, \lambda_N^A) \in \mathbb{R}^N$ and $\Lambda^B := (\lambda_1^B, \dots, \lambda_N^B) \in \mathbb{R}^N$ be their respective ordered spectrum $\lambda_1^A \geq \dots \geq \lambda_N^A$, $\lambda_1^B \geq \dots \geq \lambda_N^B$. Then

$$ ||\Lambda^A - \Lambda^B||_2 \leq ||A - B||_2, $$

where $||C||_2 = \text{Tr}(CC^*)$ for $C \in M_N(\mathbb{C})$ is the canonical norm in $M_N(\mathbb{C})$ and $||X||_2$ for $X \in \mathbb{R}^N$ is the canonical norm in $\mathbb{R}^N$.

Before proving this result let us mention that if we do not order the eigenvalues the result does not hold, as we shall see it plays an important role in the proof. In a probabilistic point of view we can equivalently reformulate the Hoffman-Wielandt inequality saying that the map defined from $(\mathcal{H}_N(\mathbb{C}), ||\cdot||_2)$ to $\mathcal{P}(\mathbb{R})$ by

$$ A \mapsto \frac{1}{N} \sum_{i=1}^{N} \delta_{\lambda_i^A}, $$

where $(\lambda_1^A, \dots, \lambda_N^A)$ is the spectrum of $A$ is $1/\sqrt{N}$ Lipschitz if we endow $\mathcal{P}(\mathbb{R})$ with the so-called Wasserstein distance $W_2$.

The Hoffman-Wielandt inequality is actually true for normal matrices but is not true
for matrices in $M_N(\mathbb{C})$ which is a problem in order to obtain for example concentration inequalities for Ginibre matrices. However, for matrices in $M_N(\mathbb{C})$ a Hoffman-Wielandt inequality holds by substituting the eigenvalues of a matrix by its singular values. Combining it with the concentration inequality it gives good concentration inequalities for the empirical mean of singular values of Ginibre matrices.

Let us give the proof of Lemma 4.3.

*Proof.* We first compute:

$$||\Lambda^A - \Lambda^B||_2^2 = ||\Lambda^A||_2^2 - 2\langle\Lambda^A|\Lambda^B\rangle + ||\Lambda^B||_2^2.$$

Since $||\Lambda^A||_2^2 = \sum_{i=1}^N (\lambda_i^A)^2 = \mathrm{Tr}(A^2) = ||A||_2^2$, to prove the Hoffman-Wielandt inequality it remains to show that:

$$\mathrm{Tr}(AB) \le \sum_{i=1}^N \lambda_i^A \lambda_i^B.$$

First, let us notice that up to changing $A$ by $A - \lambda_N^A I_N$ and using that $\mathrm{Tr}(B) = \sum_{i=1}^N \lambda_i^B$, we can suppose that the spectrum of $A$ is in $\mathbb{R}^+$. Since now we suppose that $\lambda_N^A \in \mathbb{R}^+$. Then, using the spectral theorem, we can find $U \in \mathcal{U}_N(\mathbb{C})$ such that $A = U \mathrm{diag}(\Lambda^A)U^*$ where $\mathrm{diag}(X)$ for $X \in \mathbb{C}^N$ is the diagonal matrix of $M_N(\mathbb{C})$ with $\mathrm{diag}(X)_{i,i} = X_i$ for $1 \le i \le N$. By properties of the trace we have

$$\mathrm{Tr}(AB) = \mathrm{Tr}(U \mathrm{diag}(\Lambda^A)U^*B) = \mathrm{Tr}(\mathrm{diag}(\Lambda^A)U^*BU) = \mathrm{Tr}(\mathrm{diag}(\Lambda^A)B'),$$

where $B' := U^*BU \in \mathcal{H}_N(\mathbb{C})$ and has the same spectrum as $B$. Thanks to this transformation we reduce to the case $A$ is a diagonal matrix. Write $B' = (B'_{i,j})_{1 \le i,j \le N}$ to compute

$$\mathrm{Tr}(AB) = \mathrm{Tr}(\mathrm{diag}(\Lambda^A)B') = \sum_{i=1}^N \lambda_i^A B'_{i,i}. \qquad (4.12)$$

A usual method to obtain the type of inequalities as Hoffman-Wielandt is to use the hypothesis that $\Lambda^A$ is ordered by doing a summation by parts. For all $0 \le i \le N+1$, we introduce $s_i := \sum_{k=1}^i B'_{k,k}$ and $s_0 = 0$. (4.12) gives

$$\mathrm{Tr}(AB) = \sum_{i=1}^{N} \lambda_{i}^{A} B'_{i,i} = \sum_{i=1}^{N} \lambda_{i}^{A} (s_{i} - s_{i-1}) = \sum_{i=1}^{N} (\lambda_{i}^{A} - \lambda_{i+1}^{A}) s_{i}, \qquad (4.13)$$

where $\lambda_{N+1}^A := 0$ by convention. We notice that for all $1 \le i \le N$, $\lambda_i^A - \lambda_{i+1}^A \ge 0$ because $\Lambda^A$ is ordered and $\lambda_N^A \ge 0$.

The only thing that it remains to prove is that for all $1 \le i \le N$, $s_i \le \sum_{k=1}^i \lambda_k^B$. We shall prove these inequalities in Lemma 4.4. Indeed assume that such inequalities have been obtained, (4.13) yields

$$\mathrm{Tr}(AB) = \sum_{i=1}^{N} (\lambda_{i}^{A} - \lambda_{i+1}^{A}) s_{i} \le \sum_{i=1}^{N} \left[ (\lambda_{i}^{A} - \lambda_{i+1}^{A}) \sum_{k=1}^{i} \lambda_{k}^{B} \right] = \sum_{i=1}^{N} \lambda_{i}^{A} \lambda_{i}^{B},$$

doing again a summation by parts for the last equality. $\square$

**Lemma 4.4.** Let $B = (b_{i,j})_{1 \le i,j \le N} \in \mathcal{H}_N(\mathbb{C})$ and $\lambda_1 \ge \dots \ge \lambda_N \in \mathbb{R}^N$ be its ordered spectrum. Then for all $1 \le i \le N$ we have

$$s_i := \sum_{k=1}^{i} b_{k,k} \le \sum_{k=1}^{i} \lambda_k.$$
Proof. In this proof, let $(e_1, ..., e_n)$ be the canonical basis of $\mathbb{C}^N$. We prove the result by induction on $1 \le i \le N$. For $i = 1$, the result comes directly from the fact that

$$b_{1,1} = \langle e_1 | B e_1 \rangle \le \sup_{||x||_2=1} \langle x | B x \rangle = \lambda_1.$$

Assume that the result holds for $1 \le i \le N - 1$. Let us prove that

$$s_{i+1} \le \sum_{k=1}^{i+1} \lambda_k.$$

Let $F_i = \text{span}(e_1, ..., e_{i+1})$. By the Min-max theorem, we get that $\lambda_{i+1} \ge \inf_{x \in F_i, ||x||_2=1} \langle x | Bx \rangle$. So, we can find $v_{i+1} \in F_i$ with $||v_{i+1}||_2 = 1$ such that $\lambda_{i+1} \ge \langle v_{i+1} | Bv_{i+1} \rangle$. We complete $v_{i+1}$ by $v_1, ..., v_i$ in an orthonormal basis of $F_i$. Then $B_i := (v_1, ..., v_i, v_{i+1}, e_{i+2}, ... e_n)$ is an orthonormal basis of $\mathbb{C}^N$. Let $\tilde{B}_i := (\tilde{b}_{k,j})_{1 \le k,j \le N}$ be the matrix $B$ in the basis $B_i$. Since $B$ and $\tilde{B}_i$ are conjugate they have the same spectrum. We apply the induction to the matrix $\tilde{B}_i$. This gives that

$$\sum_{k=1}^{i} \tilde{b}_{k,k} \le \sum_{k=1}^{i} \lambda_k.$$

By construction of $v_{i+1}$ we have that:

$$\sum_{k=1}^{i+1} \tilde{b}_{k,k} = \sum_{k=1}^{i} \tilde{b}_{k,k} + \langle v_{i+1} | B v_{i+1} \rangle \le \sum_{k=1}^{i} \lambda_k + \lambda_{i+1}. \quad (4.14)$$

Let us remark that by construction of the basis $B_i$, for all $N \ge j > i + 1$ we have $\tilde{b}_{j,j} = b_{j,j} = \langle e_j | B e_j \rangle$ and since $B$ and $\tilde{B}_i$ are conjugate they have the same trace. This leads to $\sum_{k=1}^{i+1} \tilde{b}_{k,k} = \sum_{k=1}^{i+1} b_{k,k}$. (4.14) gives the result. $\square$

Let us recall that given a function $f: \mathbb{R} \to \mathbb{C}$ and $M \in \mathcal{H}_N(\mathbb{C})$ that can be written $M = UDU^*$ with $U \in \mathcal{U}_N(\mathbb{C})$ and $D$ a real diagonal matrix, we define $f(M)$ as $Uf(D)U^*$ and this choice does not depend of the choice of basis $U$. A direct application of the Hoffman-Wielandt inequality is the following result.

**Corollary 4.5.** Let $f: \mathbb{R} \to \mathbb{C}$ be a $\lambda$ Lipschitz function, then the map $X \mapsto \frac{1}{N}\text{Tr}(f(X))$ is $\lambda/\sqrt{N}$ Lipschitz on $(\mathcal{H}_N(\mathbb{C}), ||.||_2)$.

**Proof.** Let $X, Y \in \mathcal{H}_N(\mathbb{C})$, we write $\Lambda^X$ and $\Lambda^Y$ their ordered spectrum as in Lemma 4.3. We directly compute:

$$\begin{aligned}
\left| \frac{1}{N} \text{Tr}(f(X)) - \frac{1}{N} \text{Tr}(f(Y)) \right|^2 &= \frac{1}{N^2} \left| \sum_{i=1}^{N} (f(\lambda_i^X) - f(\lambda_i^Y)) \right|^2 \\
&\le \text{Cauchy-Schwarz} \quad \frac{1}{N} \sum_{i=1}^{N} \left[ |f(\lambda_i^X) - f(\lambda_i^Y)|^2 \right] \\
&\le \frac{\lambda^2}{N} \sum_{i=1}^{N} \left[ |\lambda_i^X - \lambda_i^Y|^2 \right] \\
&\le \frac{\lambda^2}{N} \|\Lambda^X - \Lambda^Y\|_2^2 \\
&\le \frac{\lambda^2}{N} \|X - Y\|_2^2,
\end{aligned}$$

where we used the Hoffman-Wielandt inequality for the last inequality. $\square$
Now using Theorem 4.2, we shall prove a concentration inequality for the spectrum of Wigner matrices.

**Theorem 4.6.** Let $f: \mathbb{R} \to \mathbb{C}$ be a $\lambda$ Lipschitz function and $M_N$ be a $N$ complex Wigner matrix then for all $t > 0$:

$$ \mathbb{P}\left(\left|\frac{1}{N}\mathrm{Tr}\,f(M_N) - \mathbb{E}\left[\frac{1}{N}\mathrm{Tr}\,f(M_N)\right]\right| \ge t\right) \le 2\exp\left(-2N \frac{t^2}{\pi^2\lambda^2}\right). $$

*Proof.* We recall that by definition of a complex Wigner matrix, its entries are centered normal law of variance 1 on the diagonal and otherwise the real and the imaginary part of its entries are independent centered normal law of variance 1/2. By Corollary 4.5, we can apply Theorem 4.2 with $F_f(X) = X \mapsto \frac{1}{N}\mathrm{Tr}(f(X))$ which is $\lambda/\sqrt{N}$ Lipschitz and $\sigma^2 = 1$ which gives the result. $\square$

**Remark 4.7.** The result of Theorem 4.6 can also be obtained by the so-called logarithmic Sobolev inequalities (in short LSI). We refer to [28, 49] for generalities on this topic and to [5] for its applications in random matrices. The concentration inequality obtained in Theorem 4.6 can be obtained by LSI in two different ways. We can first apply a LSI directly to the random matrix $M_N$ whose law satisfies a LSI from the tensorization property since all its entries are independent Gaussian random variables that satisfy a LSI. We can also rewrite $\mathrm{Tr}(f(M_N))$ in function of the eigenvalues of $M_N$ and use that the law of the eigenvalues of a $N$ complex Wigner random matrix obtained in Theorem 2.48 satisfies a LSI by the Bakry-Émery criterion.

Let us mention that none of these two approaches can be used to obtain a similar concentration inequality for Ginibre random matrices.

As a consequence of this concentration inequality we obtain a bound on the variance of $\frac{1}{N}\mathrm{Tr}\,f(M_N/\sqrt{N})$ where $M_N$ is a $N$ complex Wigner matrix that shall be used in the following part to obtain the strong Wigner theorem from the mean one.

**Proposition 4.8.** Let $f: \mathbb{R} \to \mathbb{C}$ be a $\lambda$ Lipschitz function and $M_N$ be a $N$ complex Wigner matrix then there exists $c_f > 0$ a constant that only depend on $f$ such that

$$ \mathrm{Var}\left(\frac{1}{N}\mathrm{Tr}\,f\left(\frac{M_N}{\sqrt{N}}\right)\right) \le \frac{c_f}{N^2}. $$

*Proof.* By definition of the variance, we have:

$$ \mathrm{Var}\left(\frac{1}{N}\mathrm{Tr}\,f\left(\frac{M_N}{\sqrt{N}}\right)\right) = \mathbb{E}\left[\left(\frac{1}{N}\mathrm{Tr}\,f\left(\frac{M_N}{\sqrt{N}}\right) - \mathbb{E}\left(\frac{1}{N}\mathrm{Tr}\,f\left(\frac{M_N}{\sqrt{N}}\right)\right)\right)^2\right]. $$

Using the Fubini-Tonelli theorem yields:

$$ \begin{aligned} \mathbb{E}\left[\left(\frac{1}{N}\mathrm{Tr}\,f\left(\frac{M_N}{\sqrt{N}}\right) - \mathbb{E}\left(\frac{1}{N}\mathrm{Tr}\,f\left(\frac{M_N}{\sqrt{N}}\right)\right)\right)^2\right] = \\ \int_0^{+\infty} \mathbb{P}\left[\left(\frac{1}{N}\mathrm{Tr}\,f\left(\frac{M_N}{\sqrt{N}}\right) - \mathbb{E}\left(\frac{1}{N}\mathrm{Tr}\,f\left(\frac{M_N}{\sqrt{N}}\right)\right)\right)^2 \ge t\right] dt = \\ \int_0^{+\infty} \mathbb{P}\left[\left|\frac{1}{N}\mathrm{Tr}\,f\left(\frac{M_N}{\sqrt{N}}\right) - \mathbb{E}\left(\frac{1}{N}\mathrm{Tr}\,f\left(\frac{M_N}{\sqrt{N}}\right)\right)\right| \ge \sqrt{t}\right] dt. \end{aligned} $$
Using Theorem 4.6 with the function $\tilde{f}(x) := f(x/\sqrt{N})$ which is $\lambda/\sqrt{N}$ Lipschitz we get:

$$ \mathrm{Var} \left( \frac{1}{N} \mathrm{Tr} f \left( \frac{M_N}{\sqrt{N}} \right) \right) \le 2 \int_0^{+\infty} \exp \left( -2N^2 \frac{t}{\pi^2 \lambda^2} \right) dt = \frac{\pi^2 \lambda^2}{N^2}. $$

□

## 4.3 The strong Wigner theorem

We can now state the strong Wigner theorem.

**Theorem 4.9 (Almost sure Wigner theorem).** Let $(M_N)_{N \ge 1}$ be a sequence of random matrices such that for every $N \ge 1$ $M_N$ is a $N$ complex Wigner matrix. For every $N \ge 1$, let $\Lambda^N = (\lambda_1^N, \dots, \lambda_N^N) \in \mathbb{R}^N$ be the random spectrum of the $N$ complex Wigner matrix $M_N$. We write:

$$ \mu_N = \frac{1}{N} \sum_{k=1}^{N} \delta_{\lambda_k^N / \sqrt{N}} $$

the empirical distribution associated to the eigenvalues of $M_N/\sqrt{N}$. Then almost surely

$$ \mu_N \xrightarrow[N \to +\infty]{} \sigma. $$

*Proof.* As usual for the convergence in law it suffices to show that for a countable dense subset of the compactly supported functions $\mathcal{D}$ we have that almost surely for all $f \in \mathcal{D}$,

$$ \int_{\mathbb{R}} f d\mu_N \xrightarrow[n \to +\infty]{} \int_{\mathbb{R}} f d\sigma. $$

Since $\mathcal{D}$ is countable, it suffices to prove that for all $f \in \mathcal{D}$, almost surely,

$$ \int_{\mathbb{R}} f d\mu_N \xrightarrow[n \to +\infty]{} \int_{\mathbb{R}} f d\sigma. $$

For the rest of the proof, we fix $\mathcal{D}$ a countable dense subset of the compactly supported functions and $f \in \mathcal{D}$. By the Wigner theorem in mean we have that

$$ \mathbb{E} \left[ \int_{\mathbb{R}} f d\mu_N \right] \xrightarrow[N \to +\infty]{} \int_{\mathbb{R}} f d\sigma. $$

Now, we need to be sure that almost surely the empirical mean will not be far from its expectation. We shall use the concentration inequality obtained in the previous section. We notice that by definition of the spectrum,

$$ \int_{\mathbb{R}} f d\mu_N = \frac{1}{N} \mathrm{Tr} f \left( \frac{M_N}{\sqrt{N}} \right). $$

Moreover the bound of the variance obtained in Proposition 4.8 and the Markov inequality give that for all $\varepsilon > 0$

$$ \mathbb{P} \left[ \left| \frac{1}{N} \mathrm{Tr} f \left( \frac{M_N}{\sqrt{N}} \right) - \mathbb{E} \left( \frac{1}{N} \mathrm{Tr} f \left( \frac{M_N}{\sqrt{N}} \right) \right) \right| \ge \varepsilon \right] \le \frac{\mathrm{Var} \left( \frac{1}{N} \mathrm{Tr} f \left( \frac{M_N}{\sqrt{N}} \right) \right)}{\varepsilon^2} \le \frac{c_f}{N^2 \varepsilon^2}. $$
Using the Borel-Cantelli lemma yields that for all $\varepsilon > 0$, almost surely there exists $N_0$ such that for all $N \ge N_0$,

$$ \left| \frac{1}{N} \mathrm{Tr} f \left( \frac{M_N}{\sqrt{N}} \right) - \mathbb{E} \left( \frac{1}{N} \mathrm{Tr} f \left( \frac{M_N}{\sqrt{N}} \right) \right) \right| \le \varepsilon. $$

Again using a countable dense sequence of positive real numbers, we obtain that almost surely for all $\varepsilon > 0$, there exists $N_0$ such that for all $N \ge N_0$ we have:

$$ \left| \frac{1}{N} \mathrm{Tr} f \left( \frac{M_N}{\sqrt{N}} \right) - \mathbb{E} \left( \frac{1}{N} \mathrm{Tr} f \left( \frac{M_N}{\sqrt{N}} \right) \right) \right| = \left| \int_{\mathbb{R}} f d\mu_N - \mathbb{E} \left[ \int_{\mathbb{R}} f d\mu_N \right] \right| \le \varepsilon. \quad (4.15) $$

Since $\mathbb{E}[\int_{\mathbb{R}} f d\mu_N] \xrightarrow[N \to +\infty]{} \int_{\mathbb{R}} f d\sigma$, (4.15) yields that almost surely,

$$ \int_{\mathbb{R}} f d\mu_N \xrightarrow[N \to +\infty]{} \int_{\mathbb{R}} f d\sigma. $$

This concludes the proof. $\square$

## 4.4 Almost sure convergence of the smallest and the largest eigenvalue

In this section, given a matrix $M \in \mathcal{H}_N(\mathbb{C})$ we denote $\lambda_{\max}(M)$ and $\lambda_{\min}(M)$ the largest and the smallest real eigenvalue of $M$. As a consequence of the almost sure convergence of the empirical mean of the re normalized eigenvalues of a $N$ complex Wigner matrices we obtain the first estimate on the behaviour of the largest eigenvalue of a $N$ complex Wigner matrix.

**Proposition 4.10.** Let $(M_N)_{N \ge 1}$ be a sequence of random matrices such that for every $N \ge 1$ $M_N$ is a $N$ complex Wigner matrix. Then, we almost surely have

$$ \liminf_N \lambda_{\max} \left( \frac{M_N}{\sqrt{N}} \right) \ge 2. $$

*Proof.* By the strong Wigner theorem, we know that almost surely

$$ \mu_N := \frac{1}{N} \sum_{k=1}^{N} \delta_{\lambda_k^N / \sqrt{N}} $$

converges weakly towards the semicircle distribution $\sigma$ with the same notations as in Theorem 4.9. Let $\omega \in \Omega$ such that

$$ \mu_N(\omega) \xrightarrow[N \to +\infty]{} \sigma. $$

Assume by contradiction that

$$ \liminf_N \lambda_{\max} \left( \frac{M_N(\omega)}{\sqrt{N}} \right) < 2. $$

Then we can find $\varepsilon > 0$ and a subsequence $(\phi(N))_{N \ge 1}$ (which both depend on $\omega$) such that for all $N \ge 1$,

$$ \lambda_{\max} \left( \frac{M_{\phi(N)}(\omega)}{\sqrt{\phi(N)}} \right) < 2 - \varepsilon. $$
Let $f_{\varepsilon}$ be the continuous bounded function on $\mathbb{R}$ which is null on $(-\infty, 2 - \varepsilon)$, equal to 1 on $(2, +\infty)$ and linear on $(2 - \varepsilon, 2)$. Then, by choice of $f_{\varepsilon}$ we notice that $\int_{\mathbb{R}} f_{\varepsilon} d\sigma > 0$ and for all $N \ge 1$ $\int_{\mathbb{R}} f_{\varepsilon} d\mu_{\phi(N)}(\omega) = 0$ since all the eigenvalues of $\frac{M_{\phi(N)}(\omega)}{\sqrt{\phi(N)}}$ are smaller than $2 - \varepsilon$ where $f$ is null. However by the definition of the weak convergence of measure we have

$$ \int_{\mathbb{R}} f_{\varepsilon} d\mu_{\phi(N)}(\omega) \underset{N \to +\infty}{\longrightarrow} \int_{\mathbb{R}} f_{\varepsilon} d\sigma, $$

which is not possible. This proves the result.

<figure><img src="image_10.png" /></figure>

We now prove the other bound.

**Proposition 4.11.** Let $(M_N)_{N \ge 1}$ be a sequence of random matrices such that for every $N \ge 1$ $M_N$ is a $N$ complex Wigner matrix. Then, we almost surely have

$$ \limsup_N \lambda_{\max} \left( \frac{M_N}{\sqrt{N}} \right) \le 2. $$

*Proof.* This proof is based on the arguments of [50]. The idea is to obtain a Chernoff bound on the largest eigenvalue and then to use the Borel-Cantelli lemma. Fix $\varepsilon > 0$. Now fix $t > 0$. Using that exp is an increasing function we get:

$$ \begin{aligned} \mathbb{P} \left( \lambda_{\max} \left( \frac{M_N}{\sqrt{N}} \right) \ge 2 + \varepsilon \right) &= \mathbb{P} \left( \exp \left( t \lambda_{\max} \left( \frac{M_N}{\sqrt{N}} \right) \right) \ge \exp(t(2 + \varepsilon)) \right) \tag{4.16} \\ &= \mathbb{P} \left( \lambda_{\max} \left( \exp \left( t \frac{M_N}{\sqrt{N}} \right) \right) \ge \exp(t(2 + \varepsilon)) \right). \end{aligned} $$

Let us notice that since the eigenvalues of $\exp(t \frac{M_N}{\sqrt{N}})$ are positive, we have

$$ \lambda_{\max} \left( \exp \left( t \frac{M_N}{\sqrt{N}} \right) \right) \le \mathrm{Tr} \left( \exp \left( t \frac{M_N}{\sqrt{N}} \right) \right). $$

So we can upper bound using the Markov inequality

$$ \begin{aligned} \mathbb{P} \left( \lambda_{\max} \left( \frac{M_N}{\sqrt{N}} \right) \ge 2 + \varepsilon \right) &\le \mathbb{P} \left( \mathrm{Tr} \left( \exp \left( t \frac{M_N}{\sqrt{N}} \right) \right) \ge \exp(t(2 + \varepsilon)) \right) \tag{4.17} \\ &\le \exp(-t(2 + \varepsilon)) \mathbb{E} \left[ \mathrm{Tr} \left( \exp \left( t \frac{M_N}{\sqrt{N}} \right) \right) \right]. \end{aligned} $$

Now let us recall that we computed the expectation of the right hand side in Theorem 4.1 because it is the Laplace transform of the measure denoted $N\mathbb{E}(\mu_N)$ (with the notations of the mean Wigner theorem). So (4.5) gives

$$ \begin{aligned} \mathbb{E} \left[ \mathrm{Tr} \left( \exp \left( t \frac{M_N}{\sqrt{N}} \right) \right) \right] &= N \exp \left( \frac{t^2}{2N} \right) \sum_{k=0}^{N-1} \frac{1}{k+1} \binom{2k}{k} \frac{(N-1)\dots(N-k)}{N^k} \frac{t^{2k}}{(2k)!} \\ &\le N \exp \left( \frac{t^2}{2N} \right) \sum_{k=0}^{+\infty} \frac{1}{k!(k+1)!} t^{2k} \tag{4.18} \\ &\le N \exp \left( \frac{t^2}{2N} \right) \left[ \sum_{k=0}^{+\infty} \frac{t^k}{k!} \right]^2 \\ &\le N \exp \left( \frac{t^2}{2N} + 2t \right). \end{aligned} $$
Using (4.18) in (4.17) we obtain that for all $t > 0$

$$ \mathbb{P} \left( \lambda_{\max} \left( \frac{M_N}{\sqrt{N}} \right) \geq 2 + \varepsilon \right) \leq N \exp \left( \frac{t^2}{2N} + 2t - t(2 + \varepsilon) \right) = N \exp \left( \frac{t^2}{2N} - t\varepsilon \right). $$

Optimizing in $t > 0$ gives that for all $N \geq 1$

$$ \mathbb{P} \left( \lambda_{\max} \left( \frac{M_N}{\sqrt{N}} \right) \geq 2 + \varepsilon \right) \leq N \exp \left( - \frac{\varepsilon^2 N}{2} \right). $$

Since the right hand side is summable, the Borel-Cantelli lemma yields that for all $\varepsilon > 0$ almost surely $\limsup_N \left( \lambda_{\max} \left( \frac{M_N}{\sqrt{N}} \right) \right) \leq 2 + \varepsilon$. Using a countable dense family of $\varepsilon > 0$ we obtain that almost surely for all $\varepsilon > 0$ $\limsup_N \left( \lambda_{\max} \left( \frac{M_N}{\sqrt{N}} \right) \right) \leq 2 + \varepsilon$ giving the result. $\square$

Combining the two previous propositions and using a symmetric argument for the smallest eigenvalue of a $N$ complex Wigner matrix we get the following result.

**Theorem 4.12.** Let $(M_N)_{N \geq 1}$ be a sequence of random matrices such that for every $N \geq 1$ $M_N$ is a $N$ complex Wigner matrix. Then, we almost surely have

$$ \lim_{N \to +\infty} \lambda_{\max} \left( \frac{M_N}{\sqrt{N}} \right) = 2 \text{ and } \lim_{N \to +\infty} \lambda_{\min} \left( \frac{M_N}{\sqrt{N}} \right) = -2. $$
## Part II

# Dynamical approach to random matrices: the Dyson Brownian motion

In 1962 [37], Dyson introduced continuous models that defined new types of Coulomb gases through the spectrum of random matrices. We shall recover from these new models for instance the Wigner theorem stated in Theorem 4.1. It gives a new approach for the study of the eigenvalues of random matrices. We shall first introduced some models of random matrices and explain how we can obtain from these models a dynamic on their eigenvalues. This will define a system of stochastic differential equations (in short SDE) for the study of the eigenvalues of random matrices. We study these eigenvalues as a system of particles in interaction through a pair correlation interaction. Then we will prove the convergence of the empirical mean of these particles to a deterministic process, solution of a partial differential equation. Finally, we shall give some properties of solutions of this partial differential equation. We will emphasize on how the study of the process of the eigenvalues viewed as a system of particles in interaction can help us to recover some result presented in the first section as the Wigner theorem or the law of the eigenvalues of a N complex Wigner matrix.

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a standard filtered probability space on which all the random variables shall be defined.

# 5 Dynamic on the eigenvalues

## 5.1 Dyson Brownian motion

**Definition 5.1.** Fix $\beta \in \{1, 2\}$ and $N \ge 1$ a positive integer. Let $(B_{k,j}, \tilde{B}_{k,j})_{1\le k\le j\le N}$ be a family of independent Brownian motions. We call $\beta$ Dyson Brownian motion of size $N \ge 1$, the random process which is valued in $S_N(\mathbb{R})$ for $\beta = 1$ or $\mathcal{H}_N(\mathbb{C})$ for $\beta = 2$ defined by:

$$ 
\left\{
\begin{aligned}
D_N^\beta(t) &:= \frac{1}{\sqrt{N}} (D_{k,j}^\beta(t))_{1\le k,j\le N} \\
\forall 1 \le k \le N, D_{k,k}^\beta(t) &:= \frac{\sqrt{2}}{\sqrt{\beta}} B_{k,k}(t) \\
\forall 1 \le k \le j \le N, D_{k,j}^\beta(t) &:= \frac{1}{\sqrt{\beta}} [B_{k,j}(t) + i(\beta - 1)\tilde{B}_{k,j}]
\end{aligned}
\right.
\qquad (5.1) $$

**Remark 5.2.** For $\beta = 1$, $\sqrt{N} D_N^1(1/2)$ is a $N$ real Wigner matrix as it was defined in Definition 2.1. For $\beta = 2$, $\sqrt{N} D_N^2(1)$ is a $N$ complex Wigner matrix as it was defined in Definition 2.1.

**Definition 5.3.** Fix $N \ge 1$ and $M \ge 1$ be two non negative integers. Let $(B_{k,j}, \tilde{B}_{k,j})_{1\le k\le N, 1\le j\le M}$ be a family of independent Brownian motions. We call Wishart Brownian motion of size $N \times M$, the random process which is valued in $\mathcal{H}_N(\mathbb{C})$ defined by:

$$ 
\left\{
\begin{aligned}
W_N(t) &= \frac{1}{N} A_t A_t^* \\
A_t &= \left( \frac{B_{k,j}(t)}{\sqrt{2}} + i \frac{\tilde{B}_{k,j}(t)}{\sqrt{2}} \right)_{1\le k\le N, 1\le j\le M}.
\end{aligned}
\right.
\qquad (5.2) $$
## 5.2 A differential approach of the spectrum of matrices

We present the approach for symmetric matrices but the approach is also valid without any difference for hermitian matrices. This method is called in Physic the perturbation theory. Formally the idea is given a matrix whose spectrum and eigenvectors are known, could we get an approximation of the eigenvalues and eigenvectors of a small perturbation of the initial matrix. This method is for instance used in quantum mechanic for solving the Schrödinger equation as in the hydrogen atom case [52].

We recall that $S_N(\mathbb{R})$ is the set of $N \times N$ symmetric matrices whose characteristic polynomial has simple roots. $S_N(\mathbb{R})$ is an open and dense subset of $S_N(\mathbb{R})$. Moreover, for $A \in S_N(\mathbb{R})$ we order its eigenvalues

$$ \lambda_1(A) \le \dots \le \lambda_N(A), $$

and we shall write $(u_i(A))_{1 \le i \le N}$ an orthonormal basis of eigenvectors associated respectively to the $(\lambda_i(A))_{1 \le i \le N}$. Let us recall that given $A \in S_N(\mathbb{R})$ we can find a neighbourhood $V_A$ of $A$ in $S_N(\mathbb{R})$ such that for all $1 \le i \le N$ the map $\lambda_i$ defined from $S_N(\mathbb{R})$ to $\mathbb{R}$ by $\lambda_i(A)$ is the $i^{th}$ eigenvalue of $A$ is smooth on $V_A$.

Indeed, this result is a classical application of the implicit function theorem by looking at the map $\phi : S_N(\mathbb{R}) \times \mathbb{R} \to \mathbb{R}$ defined by

$$ \phi(A, \lambda) = \chi_A(\lambda) $$

and using that $\lambda$ is an eigenvalue of $A$ if and only if $\chi_A(\lambda) = 0$ and for $A \in S_N(\mathbb{R})$ and $\lambda$ one of its eigenvalue we have $\chi'_A(\lambda) \ne 0$.

We can do the same for the normalised eigenvector (even if they are defined up to a sign) by writing that for $A \in S_N(\mathbb{R})$ and $\lambda_i(A)$ an eigenvalue of $A$, an eigenvector of $A$ for $\lambda_i(A)$ is defined by the equation $(A - \lambda_i(A))x = 0$ on $\mathbb{R}^N - \{0\}$.

So, if we consider a smooth map $A$ from an open interval $I \subset \mathbb{R}$ to $S_N(\mathbb{R})$ then we can find a parametrization of the eigenvectors such that for all $1 \le i \le N$, $t \mapsto \lambda_i(A(t))$ and $t \mapsto u_i(A(t))$ are smooth on $I$.

**Example 5.4.** Let $A \in S_N(\mathbb{R})$ and $B \in S_N(\mathbb{R})$ then there exists a small open interval centred in 0, $I \subset \mathbb{R}$, such that $t \mapsto \lambda_i(A + tB)$ and $t \mapsto u_i(A + tB)$ are smooth on $I$.

Until the end of this section, we fix a smooth map $A$ from $I \subset \mathbb{R}$ an open interval to $S_N(\mathbb{R})$.

**Proposition 5.5 (Hadamard).** For all $1 \le i \le N$, the maps defined on $I$ by $\phi_i(t) = \lambda_i(A(t))$ are smooth and we have the following formulas for their derivatives:

$$ \phi_i'(t) = \langle u_i(A(t)) | A'(t) u_i(A(t)) \rangle \qquad (5.3) $$

$$ \phi_i''(t) = \langle u_i(A(t)) | A''(t) u_i(A(t)) \rangle + 2 \sum_{1 \le k \ne i \le N} \frac{|\langle u_k(A(t)) | A'(t) u_i(A(t)) \rangle|^2}{\lambda_i(A(t)) - \lambda_k(A(t))} \quad (5.4) $$

*Proof.* We fix $1 \le i \le N$ and $t \in I$. We shall just write $A$ for $A(t)$ and $\lambda_i$ for $\phi_i$ to lighten the notations in the proof.
By definition, we have $Au_i = \lambda_i u_i$ and $\langle u_i|u_i \rangle = 1$. We take the derivative of these two equalities. We get:

$$ \mathcal{A}'u_i + \mathcal{A}u'_i = \lambda'_i u_i + \lambda_i u'_i \quad (5.5) $$

$$ \langle u'_i|u_i \rangle = 0 \quad (5.6) $$

We shall take the scalar product against $u_i$ in (5.5). Using the fact that $u_i$ and $u'_i$ are orthogonal by (5.6), the symmetry of $\mathcal{A}$ yields

$$ \langle u_i|\mathcal{A}u'_i \rangle = \langle \mathcal{A}u_i|u'_i \rangle = \lambda_i \langle u_i|u'_i \rangle = 0 $$

We get:

$$ \langle u_i|\mathcal{A}'u_i \rangle = \lambda'_i. \quad (5.7) $$

This gives the first derivative. For the second derivative, we take the derivative of the formula (5.7):

$$ \lambda_i'' = \langle u_i|\mathcal{A}''u_i \rangle + 2\langle u'_i|\mathcal{A}'u_i \rangle. \quad (5.8) $$

It remains to express $u'_i$ in the basis $(u_k)_{1\le k\le N}$. We already know that $\langle u'_i|u_i \rangle = 0$ and to compute $\langle u_k|u'_i \rangle$ for $k \ne i$, we take the scalar product against $u_k$ in the formula (5.5). Using that $\mathcal{A}$ is symmetric, it gives:

$$ (\lambda_i - \lambda_k)\langle u'_i|u_k \rangle = \langle u_k|\mathcal{A}'u_i \rangle. $$

$$ \text{So we get } u'_i = \sum_{1 \le i \ne k \le N} \frac{\langle u_k|\mathcal{A}'u_i \rangle}{\lambda_i - \lambda_k} u_k. $$

Using formula (5.8), we have:

$$ \lambda_i'' = \langle u_i|\mathcal{A}''u_i \rangle + 2 \sum_{1 \le k \ne i \le N} \frac{|\langle u_k|\mathcal{A}'u_i \rangle|^2}{\lambda_i - \lambda_k}. $$

□

**Example 5.6.** With the notations of Example (5.4), $A(t) = A + tB$ we have $\phi_i'(0) = \langle u_i(A)|Bu_i(A) \rangle$ and $\phi_i''(0) = 2 \sum_{k \ne i} \frac{|\langle u_k(A)|Bu_i(A) \rangle|^2}{\lambda_i - \lambda_k}$.

## 5.3 Particles in mean field interaction

We first introduce a family of stochastic processes that could be viewed as a system of particles in interaction through a singular repulsive force and which are subject to agitations modelled by Brownian motions.

In this section, fix $N \ge 1$ and $M \ge 1$ two positive integers.

**Definition 5.7.** Fix $\beta \in \{1, 2\}$ and $N$ distinct real numbers $x_1 < \dots < x_N$. We call $\beta$ Dyson particles starting at $(x_i)_{1\le i\le N}$ the family of stochastic processes $(\lambda_i^{i,\beta})_{1\le i\le N}$ solution of the system of SDE:

$$ \left\{ \begin{array}{l} \forall 1 \le i \le N, \, d\lambda_t^{i,\beta} = \frac{1}{N} \sum_{j \ne i} \frac{1}{\lambda_t^{i,\beta} - \lambda_t^{j,\beta}} dt + \sqrt{\frac{2}{\beta N}} dB_t^i \\ \lambda_0^{i,\beta} = x_i \end{array} \right. \quad (5.9) $$

with $(B^i)_{1\le i\le N}$ a family of independent Brownian motions.
The next theorem proved by Dyson in [37] explains that the stochastic process of the eigenvalues of a Dyson Brownian motion matrix of size N introduced in Definition 5.1 has the same law as a Dyson particles system.

**Theorem 5.8 (Dyson).** Fix $\beta \in \{1, 2\}$. Let $(D_N^\beta(t))_{t \in \mathbb{R}^+}$ be a $\beta$ Dyson Brownian motion of size $N$. Let $(\lambda^{i,\beta}(t))_{1 \le i \le N, t \in \mathbb{R}^+}$ be the family of ordered eigenvalues of $(D_N^\beta(t))_{t \in \mathbb{R}^+}$. Then $(\lambda^{i,\beta})_{1 \le i \le N}$, is a weak solution of (5.9).

**Remark 5.9.** For the notion of weak and strong solutions of stochastic differential equations we refer to [68].

*Proof.* We shall not give a complete proof, focusing on the formal ideas. We want to emphasize on the methodology to go from a random matrix model to a system of SDE satisfied by the eigenvalues of this random matrix model. Here a main difficulty that shall not be treated is to prove that the process of eigenvalues is actually a semi martingale process in order for instance to use the Itô formula. A rigorous proof can be found in Theorem 4.3.2 of [5]. Nonetheless, the proof presented in [5] goes backward and so this requires to know the a priori system of SDE we should look for. That is why we present this method in order to identify the system of SDE that is satisfied by the eigenvalues of random matrix stochastic process. We follow the arguments given in [75].

We recall that $X$ is a complex centred Gaussian random variable of variance 1 if $\mathfrak{Re}(X)$ and $\mathfrak{Im}(X)$ are independent real centred Gaussian random variable of variance $1/2$ (in particular $\mathbb{E}[|X|^2] = \mathbb{E}(\mathfrak{Re}(X)^2 + \mathfrak{Im}(X)^2) = 1$).

Fix $t \ge 0, h > 0$ be non negative real numbers with $h$ a small quantity. We have the following equalities:

$$ \begin{aligned} \lambda^{i,\beta}(t+h) &= \lambda^{i,\beta}(D_N^\beta(t+h)) \\ &= \lambda^{i,\beta}(D_N^\beta(t) + D_N^\beta(t+h) - D_N^\beta(t)) \tag{5.10} \\ &=: \lambda^{i,\beta}(D_N^\beta(t) + \tilde{D}_N^\beta(h)) \end{aligned} $$

with $\tilde{D}_N^\beta$ whose law is a $\beta$ Dyson Brownian motion independent of $D_N^\beta$ by the stationary and independent property of a real Brownian motion.

We set $A(t) := D_N^\beta(t)$ and using the scaling property of a real Brownian motion we can rewrite (5.10) as:

$$ \lambda^{i,\beta}(t+h) :=: \lambda^{i,\beta} \left( A(t) + \sqrt{\frac{2h}{\beta N}} G^{\beta} \right), $$

with $G^\beta$ a $N$ real Wigner matrix in the case $\beta = 1$ or a $N$ complex Wigner matrix in the case $\beta = 2$ independent of $A(t)$.

Now we condition on $A(t)$, since law of $G^\beta$ is the same as the law of $G^\beta$ after we condition by $A(t)$ (by independence of $G^\beta$ and $A(t)$), we can formally continue the proof as if $A(t)$ is a deterministic matrix.

More exactly we shall show that almost surely for all $A \in S_N^\circ(\mathbb{R})$

$$ \forall 1 \le i \le N, \lambda_i \left( A + \sqrt{\frac{2h}{\beta N}} G^{\beta} \right) = \lambda_i(A) + \sqrt{\frac{2}{\beta N}} B_i(h) + \frac{h}{N} \sum_{j \ne i} \frac{1}{\lambda_i(A) - \lambda_j(A)} + \dots \quad (5.11) $$

with $(B_i)_{1 \le i \le N}$ a family of independent real centred Gaussian random variables of variance $h$.
Assume that we proved this result, we apply (5.11) with $A = A_N(t)$ (which is almost surely in $S_N(\mathbb{R})$ by Proposition 2.13) to have the following result:

$$ \lambda^{i,\beta}(t+h) - \lambda^{i,\beta}(t) = \sqrt{\frac{2}{\beta N}} B_i(h) + \frac{h}{N} \sum_{j \neq i} \frac{1}{\lambda^{i,\beta}(t) - \lambda^{j,\beta}(t)} + \dots $$

with $(B_i)_{1 \le i \le N}$ a family of independent real centred Gaussian random variables of variance $h$. In the Itô language it gives

$$ d\lambda_t^{i,\beta} = \frac{1}{N} \sum_{j \neq i} \frac{1}{\lambda_t^{i,\beta} - \lambda_t^{j,\beta}} dt + \sqrt{\frac{2}{\beta N}} dB_t^i. $$

So it remains to prove (5.11). The Hadamard formulas of Proposition 5.5 with the Example 5.6 yields:

$$ \lambda_i \left( A + \sqrt{\frac{2h}{\beta N}} G^\beta \right) = \lambda_i(A) + \sqrt{\frac{2h}{\beta N}} \langle u_i | G^\beta u_i \rangle + \frac{2h}{\beta N} \sum_{j \neq i} \frac{|\langle u_j | G^\beta u_i \rangle|^2}{\lambda_i(A) - \lambda_j(A)} + \dots \quad (5.12) $$

with $(u_i)_{1 \le i \le N}$ an orthonormal basis of $A$ associated to the $(\lambda_i(A))_{1 \le i \le N}$ as in Proposition 5.5. Then, we notice that by invariance of the law by a unitary change of basis of a $N$ real or complex Wigner matrix (Proposition 2.5), $(N_i := \langle u_i | G^\beta u_i \rangle)_{1 \le i \le N}$ are independent real centred Gaussian random variables of variance 1. Similarly, for all $1 \le j \neq i \le N$, $\langle u_j | G^\beta u_i \rangle$ is a real Gaussian random variable of variance 1/2 in the case $\beta = 1$ or a complex Gaussian random variable of variance 1 in the case $\beta = 2$. So for all $1 \le j \neq i \le N$, we can do the following approximation thanks to the "nice" concentration property of Gaussian random variables $|\langle u_j | G^\beta u_i \rangle|^2 \sim \mathbb{E}(|\langle u_j | G^\beta u_i \rangle|^2) = \beta/2$. Hence, we can rewrite (5.12) as:

$$ \lambda_i \left( A + \sqrt{\frac{2h}{\beta N}} G^\beta \right) = \lambda_i(A) + \sqrt{\frac{2h}{\beta N}} N_i + \frac{h}{N} \sum_{j \neq i} \frac{1}{\lambda_i(A) - \lambda_j(A)} + \dots $$

with $(N_i)_{1 \le i \le N}$ independent real centred Gaussian random variables of variance 1. Finally, defining for all $1 \le i \le N$, $(B_i(h))_{h \ge 0} := (\sqrt{h}N_i)$ we obtain (5.11). $\square$

**Definition 5.10.** Fix $N$ distinct real numbers $x_1 < \dots < x_N$. We call Wishart particles starting at $(x_i)_{1 \le i \le N}$ the family of stochastic processes $(\lambda^i)_{1 \le i \le N}$ solution of the system of SDE:

$$ \left\{ \begin{array}{l} \forall 1 \le i \le N, \, d\lambda_t^i = \left( \frac{1}{N} \sum_{j \neq i} \frac{\lambda_t^i + \lambda_t^j}{\lambda_t^i - \lambda_t^j} + \frac{M}{N} \right) dt + \sqrt{\frac{2\lambda_t^i}{N}} dB_t^i \quad (5.13) \\ \lambda_0^i = x_i \end{array} \right. $$

with $(B^i)_{1 \le i \le N}$ a family of independent Brownian motions.

We can state a similar result with the Dyson case.

**Theorem 5.11.** Let $(W_N(t))_{t \in \mathbb{R}^+}$ be a Wishart Brownian motion of size $N \times M$. Let $(\lambda^i(t))_{1 \le i \le N, t \in \mathbb{R}^+}$ be the family of eigenvalues of $(W_N(t))_{t \in \mathbb{R}^+}$. Then $(\lambda^i)_{1 \le i \le N}$ is a weak solution of (5.13).
Proof. We shall focus on how to obtain the counterpart of (5.11) in the Wishart case. We detail the case $N = M$ for simplicity but there is no difference when dealing with the general case.

By Definition 5.3, we can write $W_N(t) = \frac{1}{N}A_t A_t^*$ with $A_t$ introduced in the definition of a Wishart Brownian motion.

As for the Dyson case we define $A_{t+h} - A_t =: \sqrt{h}G$ where $G$ has the law of a $N$ complex Ginibre matrix. We compute:

$$ 
\begin{aligned}
W_N(t+h) &= \frac{1}{N} A_{t+h} A_{t+h}^* \\
&= \frac{1}{N} (A_t + \sqrt{h}G) (A_t + \sqrt{h}G)^* && (5.14) \\
&= W_N(t) + \frac{\sqrt{h}}{N} A_t G^* + \frac{\sqrt{h}}{N} G A_t^* + \frac{h}{N} G G^*.
\end{aligned}
$$

Conditioning as for the Dyson case by $A_N(t)$, formally we have to study the evolution of the spectrum of

$$ \mathcal{W}(h) := \frac{1}{N} BB^* + \frac{\sqrt{h}}{N} BG^* + \frac{\sqrt{h}}{N} GB^* + \frac{h}{N} GG^* $$

for any $B \in M_N(\mathbb{R})$ a deterministic matrix.

To do so, we introduce:

$$ \tilde{\mathcal{W}}(h) := \frac{1}{N} BB^* + \frac{h}{N} BG^* + \frac{h}{N} GB^* + \frac{h^2}{N} GG^*. $$

Using the Hadamard formulas of Proposition 5.5 yields:

$$ 
\begin{aligned}
\lambda_i(\tilde{\mathcal{W}}(h)) = \lambda_i(\tilde{\mathcal{W}}(0)) &+ \frac{h}{N} \langle u_i(BB^*) | [BG^* + GB^*] u_i(BB^*) \rangle + \frac{h^2}{N} \langle u_i(BB^*) | GG^* u_i(BB^*) \rangle \\
&+ \frac{h^2}{N^2} \sum_{i \neq k} \frac{|\langle u_k(BB^*) | [BG^* + GB^*] u_i(BB^*) \rangle|^2}{\lambda_i(\tilde{\mathcal{W}}(0)) - \lambda_k(\tilde{\mathcal{W}}(0))}
\end{aligned}
\quad (5.15) $$

with the notations of Proposition 5.5.

To continue the computations, it is not clear to understand how $B$ or $B^*$ acts on $u_i(BB^*)$. So we shall use the singular value decomposition of $B$ to construct $(u_i(BB^*))_{1 \le i \le N}$. We construct $(u_i(BB^*))_{1 \le i \le N}$ such that $B^* u_i(BB^*) = \sqrt{\lambda_i(BB^*)} v_i = \sqrt{N} \sqrt{\lambda_i(\tilde{\mathcal{W}}(0))} v_i$ with $(v_i)_{1 \le i \le N}$ an orthonormal basis of $\mathbb{R}^N$ (we can consider a such construction thanks to the singular value decomposition of $B$).

It gives:

$$ 
\begin{aligned}
\langle u_i(BB^*) | [BG^* + GB^*] u_i(BB^*) \rangle &= \sqrt{N} \sqrt{\lambda_i(\tilde{\mathcal{W}}(0))} [\langle v_i | G^* u_i(BB^*) \rangle + \langle u_i(BB^*) | G v_i \rangle] \\
&=: \sqrt{2N} \sqrt{\lambda_i(\tilde{\mathcal{W}}(0))} N_i
\end{aligned}
\quad (5.16) $$

with $(N_i := \sqrt{2} \mathfrak{Re}(\langle u_i(BB^*) | G v_i \rangle))_{1 \le i \le N}$ a family of independent centred real Gaussian random variables of variance 1 by invariance of the law of a Ginibre matrix by multiplication by unitary matrices (Proposition 2.11).
Moreover, since

$$ \langle u_i(BB^*) | GG^* u_i(BB^*) \rangle = \|G^* u_i(BB^*)\|_2^2 = \sum_{k=1}^{N} |\langle v_k | G^* u_i(BB^*) \rangle|^2 \quad (5.17) $$

and for all $1 \le i, k \le N$, $\langle v_k | G^* u_i(BB^*) \rangle$ is a complex Gaussian random variable of variance 1 (again by invariance of the law of a Ginibre matrix by right and left multiplication by a unitary matrix), we can do the following approximation at the first order:

$$ \begin{aligned} \langle u_i(BB^*) | GG^* u_i(BB^*) \rangle &= \sum_{k=1}^{N} |\langle v_k | G^* u_i(BB^*) \rangle|^2 \\ &\sim \sum_{k=1}^{N} \mathbb{E}[|\langle v_k | G^* u_i(BB^*) \rangle|^2] \quad (5.18) \\ &= \sum_{k=1}^{N} 1 = N \end{aligned} $$

Finally we also have for all $1 \le k \ne i \le N$:

$$ \begin{aligned} |\langle u_k(BB^*)| [BG^* + GB^*] u_i(BB^*) \rangle|^2 &= N \left[ \left| \sqrt{\lambda_k(\tilde{W}(0))} \langle v_k | G^* u_i(BB^*) \rangle + \sqrt{\lambda_i(\tilde{W}(0))} \langle u_k(BB^*) | Gv_i \rangle \right|^2 \right] \\ &=: N \left[ \lambda_k(\tilde{W}(0)) |\tilde{N}_{i,k}|^2 + \lambda_i(\tilde{W}(0)) |N_{k,i}|^2 + 2\sqrt{\lambda_i(\tilde{W}(0)) \lambda_k(\tilde{W}(0))} \mathfrak{Re}(\tilde{N}_{i,k} N_{k,i}) \right], \end{aligned} \quad (5.19) $$

where $N_{k,i} := \langle u_k(BB^*) | Gv_i \rangle$ and $\tilde{N}_{i,k} := \langle u_i(BB^*) | Gv_k \rangle$ are complex independent centred normal random variables of variance 1 again by the invariance of the law of $G$ by left and right multiplication by unitary matrices. Doing the same approximation as before, we have:

$$ \begin{aligned} N \left[ \lambda_k(\tilde{W}(0)) |\tilde{N}_{i,k}|^2 + \lambda_i(\tilde{W}(0)) |N_{k,i}|^2 + 2\sqrt{\lambda_i(\tilde{W}(0)) \lambda_k(\tilde{W}(0))} \mathfrak{Re}(\tilde{N}_{i,k} N_{k,i}) \right] &\sim \\ N \mathbb{E} \left[ \lambda_k(\tilde{W}(0)) |\tilde{N}_{i,k}|^2 + \lambda_i(\tilde{W}(0)) |N_{k,i}|^2 + 2\sqrt{\lambda_i(\tilde{W}(0)) \lambda_k(\tilde{W}(0))} \mathfrak{Re}(\tilde{N}_{i,k} N_{k,i}) \right] \\ &= N \left[ \lambda_k(\tilde{W}(0)) \mathbb{E}(|\tilde{N}_{i,k}|^2) + \lambda_i(\tilde{W}(0)) \mathbb{E}(|N_{k,i}|^2) + 2\sqrt{\lambda_i(\tilde{W}(0)) \lambda_k(\tilde{W}(0))} \mathfrak{Re}(\mathbb{E}(\tilde{N}_{i,k} N_{k,i})) \right] \\ &= N \left[ \lambda_k(\tilde{W}(0)) + \lambda_i(\tilde{W}(0)) \right] \end{aligned} \quad (5.20) $$

Using (5.16), (5.18) and (5.20) in (5.15) yields:

$$ \lambda_i(\mathcal{W}(h)) = \lambda_i(\mathcal{W}(0)) + \sqrt{\frac{2\lambda_i(\mathcal{W}(0))}{N}} B_i(h) + h \left[ 1 + \frac{1}{N} \sum_{j \ne i} \frac{\lambda_i(\mathcal{W}(0)) + \lambda_j(\mathcal{W}(0))}{\lambda_i(\mathcal{W}(0)) - \lambda_j(\mathcal{W}(0))} \right] + \dots \quad (5.21) $$

where $(B_i(h) := \sqrt{h}N_i)_{1 \le i \le N}$ is family of independent Gaussian random variables. This corresponds to the Itô formulation of (5.14). $\square$
This method is quite robust and can be used in many models as for instance if we consider a family of symmetric matrices with independent entries like the Dyson case but instead of having Brownian motions in the entries we have Ornstein-Uhlenbeck $(O_{i,j}(t))_{1\le i,j\le N}$ processes which are solutions of SDE

$$dO_{i,j} = -\theta O_{i,j} dt + d(D_N^1)_{i,j}$$

with $D_N^1$ a Dyson Brownian motion of size $N$ and $\theta$ a parameter. Using the same approach one can show that the system of eigenvalues $(\lambda_i(t))_{1\le i\le N}$ satisfies the followings system of SDE:

$$\forall 1 \le i \le N, d\lambda_i(t) = \frac{1}{N} \sum_{j \ne i} \frac{1}{\lambda_i(t) - \lambda_j(t)} dt + \frac{\sqrt{2}}{\sqrt{N}} dB_t^i - \theta \lambda_i(t) dt. \quad (5.22)$$

# 6 Study of the system of particles

In this section we focus on general properties (existence, tightness of the empirical measure...) of solutions of Dyson Brownian motion SDE (5.9). We could state similar results for other models like Wihsart or Ornstein-Uhlenbeck. We refer to [20, 42] for the Wishart case and [29, 71] for the Ornstein-Uhlenbeck case.

We shall follow the standard arguments to study the existence of a system of particles in interaction through a singular potential using a containment function. For instance, this method have been used for the Dyson Brownian case in [5], for Coulomb gas in dimension 2 in [17], for the so-called generalized Dyson Brownian motion and Ornstein-Uhlenbeck models in [61, 71] or for the circular Dyson Brownian motion [23].

We fix an integer $N \ge 2$.

## 6.1 Notations

We introduce the open subset on which we will work:

$$D = \mathbb{R}^N - \bigcup_{i \ne j} \{(x_1, \dots, x_N) \in \mathbb{R}^N : x_i = x_j\}$$

and its boundary

$$\partial D = \{+\infty\} \cup_{i \ne j} \{(x_1, \dots, x_N) \in \mathbb{R}^N : x_i = x_j\}.$$

Let us write $\mathbb{R}_>^N$ the subset of $\mathbb{R}^N$: $\{(x_1, \dots, x_N) \in \mathbb{R}^N : x_1 < x_2 < \dots < x_N\}$. We shall write $x = (x_1, \dots, x_N)$ for an element of $\mathbb{R}^N$.

We say that $x$ is $\varepsilon$ near of $\partial D$ there exists $i$ such that $|x_i| > \varepsilon^{-1}$ or if there exists $i \ne j$ such that $|x_i - x_j| \le \varepsilon$.

We introduce an energy associated to the Dyson model. If $x \in D$, we define its energy by

$$\mathcal{E}(x) = \frac{1}{N} \sum_{i=1}^{N} V(x_i) + \frac{1}{2N^2} \sum_{1 \le i \ne j \le N} W(x_i - x_j) =: \mathcal{E}_V(x) + \mathcal{E}_W(x)$$

where $V$ is an external confinement potential which is in our case $V(x) = x^2$ as the kinetic energy that will ensure that the system of particle is spatially confined and $W(x) = -\log(|x|^2)$ is potential energy of interaction associated to our system since $-W'(x) = 2/x$.
Let $\alpha_N$ and $\beta_N$ be two positive real numbers. In this section we shall focus on solutions of the following system of SDE:

$$ \forall 1 \le i \le N, \quad d\lambda_t^i = -\frac{\alpha_N}{2N^2} \sum_{j \ne i} W'(\lambda_t^i - \lambda_t^j) dt + 2\sqrt{\frac{\alpha_N}{\beta_N}} dB_t^i \quad (6.1) $$

with $(B_i)_{1 \le i \le N}$ a family of independent Brownian motions.

The cases of Dyson Brownian motion defined in Definition 5.1 correspond to the particular cases $\alpha_N = N$ and $\beta_N = 4N^2$ for $\beta = 2$ and $\beta_N = 2N^2$ for $\beta = 1$.

The first goal is to study under which condition on $\alpha_N$ and $\beta_N$ the system is well posed starting from an initial data in $D$ which is not clear a priori since there is a singular term in the system. On the one side the singular term is a repulsive interaction that should help the system to be well defined but if the fluctuations due to the Brownian motion parts are too important collisions may appear.

For this aim, we introduce the following stopping times for a solution of (6.1): for all $\varepsilon > 0$,

$$ T_{\varepsilon} = \inf\{t \ge 0 : \max_{1 \le i \le N} |\lambda_t^i| \ge \varepsilon^{-1} \text{ or } \min_{1 \le i,j \le N} |\lambda_t^i - \lambda_t^j| \le \varepsilon\} $$

and

$$ T_{\partial D} = \sup_{\varepsilon>0} T_{\varepsilon}. $$

## 6.2 Formulas

The goal of this section is to give formulas to explain how the generator of the system of particles solution of (6.1) acts on $\mathcal{E}$.

Firstly the generator associated to the system $(\lambda_t^i)_{1 \le i \le N}$ solution of (6.1) is

$$ Lf = 2 \frac{\alpha_N}{\beta_N} \Delta f - \frac{\alpha_N}{2} \nabla \mathcal{E}_W \cdot \nabla f, $$

for $f$ a smooth function from $D$ to $\mathbb{R}$. It means that for a smooth function $f$ from $D$ to $\mathbb{R}$, the Itô formula yields:

$$ \mathbb{E}(f(\lambda_t^1, \dots, \lambda_t^N)) = f(\lambda_0^1, \dots, \lambda_0^N) + \mathbb{E} \left( \int_0^t Lf(\lambda_t^1, \dots, \lambda_t^N) dt \right). $$

For $x \in D$ and $1 \le i, j \le N$, we have the following formulas:

$$ \frac{\partial \mathcal{E}_V}{\partial x_i}(x) = \frac{2x_i}{N} \qquad (6.2) $$

$$ \frac{\partial \mathcal{E}_W}{\partial x_i}(x) = -\frac{2}{N^2} \sum_{j \ne i} \frac{1}{x_i - x_j} \qquad (6.3) $$

$$ \nabla^2 \mathcal{E}_V(x) = \frac{2}{N} I_N \qquad (6.4) $$

$$ \nabla^2 \mathcal{E}_W(x) = \frac{1}{N^2} A \qquad (6.5) $$

with $A$ a matrix such that $A_{i,i} = 2 \sum_{j \ne i} \frac{1}{(x_i - x_j)^2}$ and $A_{i,j} = - \frac{2}{(x_i - x_j)^2}$ if $i \ne j$.

We can now compute $L\mathcal{E}_V$ and after $L\mathcal{E}_W$ to finally compute $L\mathcal{E}$.
We compute

$$LE_V(x) = \frac{2\alpha_N}{\beta_N} \times 2 - \frac{\alpha_N}{2} \sum_{1 \le i \ne j \le N} \frac{-4x_i}{N^3} \frac{1}{x_i - x_j} = \frac{4\alpha_N}{\beta_N} + \frac{2\alpha_N}{N^3} \sum_{1 \le i \ne j \le N} \frac{x_i}{x_i - x_j}.$$

Remark that

$$\sum_{1 \le i \ne j \le N} \frac{x_i}{x_i - x_j} = \sum_{1 \le i \ne j \le N} \frac{x_j}{x_j - x_i}$$

It yields:

$$2 \sum_{1 \le i \ne j \le N} \frac{x_i}{x_i - x_j} = \sum_{1 \le i \ne j \le N} \frac{x_i - x_j}{x_i - x_j} = N(N - 1).$$

This gives

$$LE_V(x) = \frac{4\alpha_N}{\beta_N} + \frac{\alpha_N(N-1)}{N^2}. \qquad (6.6)$$

Now we compute $LE_W(x)$. Start with:

$$LE_W(x) = 2 \frac{\alpha_N}{\beta_N} \sum_{i=1}^{N} \frac{2}{N^2} \sum_{j \ne i} \frac{1}{(x_i - x_j)^2} - \frac{\alpha_N}{2} |\nabla LE_W(x)|^2 \qquad (6.7)$$

$$= 4 \frac{\alpha_N}{N^2 \beta_N} \sum_{i=1}^{N} \sum_{j \ne i} \frac{1}{(x_i - x_j)^2} - \frac{2\alpha_N}{N^4} \sum_{i=1}^{N} \left( \sum_{j \ne i} \frac{1}{x_i - x_j} \right)^2 . \qquad (6.8)$$

Notice that

$$\begin{aligned} \sum_{i=1}^{N} \left[ \left( \sum_{j \ne i} \frac{1}{x_i - x_j} \right)^2 - \sum_{j \ne i} \frac{1}{(x_i - x_j)^2} \right] &= \sum_{j \ne i, k \ne i, j \ne k} \frac{1}{(x_i - x_k)(x_i - x_j)} \\ &= \sum_{j \ne i, k \ne i, j \ne k} \frac{1}{x_j - x_k} \left( \frac{1}{x_i - x_j} - \frac{1}{x_i - x_k} \right) \\ &= - \sum_{j \ne i, k \ne i, j \ne k} \frac{1}{(x_j - x_k)(x_j - x_i)} - \sum_{j \ne i, k \ne i, j \ne k} \frac{1}{(x_k - x_j)(x_k - x_i)} \\ &= -2 \sum_{j \ne i, k \ne i, j \ne k} \frac{1}{(x_i - x_k)(x_i - x_j)} \end{aligned} \qquad (6.9)$$

This gives:

$$3 \sum_{j \ne i, k \ne i, j \ne k} \frac{1}{(x_i - x_k)(x_i - x_j)} = 0 \quad \text{and} \quad \sum_{i=1}^{N} \left[ \left( \sum_{j \ne i} \frac{1}{x_i - x_j} \right)^2 - \sum_{j \ne i} \frac{1}{(x_i - x_j)^2} \right] = 0 \qquad (6.10)$$

Hence, we have:

$$LE_W(x) = \left[ \frac{4\alpha_N}{N^2\beta_N} - \frac{2\alpha_N}{N^4} \right] \sum_{i=1}^{N} \sum_{j \ne i} \frac{1}{(x_i - x_j)^2}. \qquad (6.11)$$
Computations (6.6) and (6.11) can be summarized into

$$L\mathcal{E}(x) = \frac{4\alpha_N}{\beta_N} + \frac{\alpha_N(N-1)}{N^2} + \left[ 4\frac{\alpha_N}{N^2\beta_N} - \frac{2\alpha_N}{N^4} \right] \sum_{i=1}^{N} \sum_{j \neq i} \frac{1}{(x_i - x_j)^2}. \quad (6.12)$$

We obtain the following result.

**Proposition 6.1.** Assume that $\beta_N \ge 2N^2$. Then we have the following upper bound

$$\sup_{x \in D} L\mathcal{E}(x) \le \frac{4\alpha_N}{\beta_N} + \frac{\alpha_N(N-1)}{N^2}.$$

## 6.3 Existence of solutions for the Dyson case

### 6.3.1 Existence thanks to a containment function

Let state the theorem we will prove in this section.

**Theorem 6.2.** Assume that $\beta_N \ge 2N^2$. For any initial data $\lambda_0 \in D$, there exists a unique strong solution of (6.1) defined on $[0, +\infty)$ (and $T_{\partial D} = +\infty$ almost surely).

This proof relies on the fact that $\mathcal{E}$ is a good containment function. We follow the arguments of [17].

**Lemma 6.3.** We have the following properties for $\mathcal{E}$:

1. $\mathcal{E} \ge 0$

2. $\lim_{x \to \partial D} \mathcal{E}(x) = +\infty$

3. $\exp(-\beta\mathcal{E})$ is integrable on $D$ for any $\beta > 0$.

*Proof.* Let $x$ in $D$. Start with:

$$\frac{1}{2} \sum_{1 \le i \ne j \le N} |x_i - x_j|^2 = \frac{1}{2} \sum_{1 \le i, j \le N} |x_i - x_j|^2 = N \sum_{1 \le N} x_i^2 - \left( \sum_{1 \le i \le N} x_i \right)^2 \le N \sum_{1 \le i \le N} x_i^2 = N|x|^2.$$

So we get that:

$$2N^2\mathcal{E}(x) = N|x|^2 + N|x|^2 - \sum_{1 \le i \ne j \le N} \log|x_i - x_j|^2 \ge N|x|^2 + \sum_{1 \le i \ne j \le N} \left( \frac{|x_i - x_j|^2}{2} - \log|x_i - x_j|^2 \right).$$

Since for all $u > 0$, $\frac{u}{2} - \log(u) \ge 1 - \log(2) > 1/4$, we can lower bound for $\mathcal{E}$ on $D$:

$$\mathcal{E}(x) \ge \frac{|x|^2}{2N} + \frac{N(N-1)}{8N^2} \ge \frac{|x|^2}{2N} + \frac{1}{16}. \quad (6.13)$$

It proves 1 and 3.

We now prove 2. By definition of $\partial D$, we must prove that for all $R > 0$ there exists $A > 0$ and $\varepsilon > 0$ such that $H(x) > R$ if there exists $1 \le i \le N$ such that $|x_i| > A$ or if there exists $1 \le i \le N$ and $1 \le j \le N$ such that $|x_i - x_j| \le \varepsilon$.

We fix $R > 0$. First, notice that by (6.13), we can find $A > 0$ large enough such that if there exists $1 \le i \le N$ such that $|x_i| > A$ then $\mathcal{E}(x) \ge R$. We now fix a such $A > 0$.
We can now assume furthermore that for all $1 \le l \le N$, $|x_l| < A$ because otherwise as explained $\mathcal{E}(x) > R$.

Let $\varepsilon > 0$ and assume that there exists $1 \le i \le N$ and $1 \le j \le N$ such that $|x_i - x_j| \le \varepsilon$, then

$$ 
\begin{aligned}
N^2 \mathcal{E}(x) &\ge -2 \log |x_i - x_j| - \sum_{1 \le k \ne j \le N, \{k,l\} \ne \{i,j\}} \log |x_k - x_l| \\
&\ge -2 \log(\varepsilon) - \sum_{1 \le k \ne j \le N, \{k,l\} \ne \{i,j\}} \log |x_k - x_l|.
\end{aligned} 
\quad (6.14) $$

Using the inequality $|x - y| \le (1 + |x|)(1 + |y|)$ for all $(x, y) \in \mathbb{R}^2$, we get that for all $l \ne k$:

$$ - \log |x_k - x_l| \ge - \log(1 + |x_k|) - \log(1 + |x_l|) \ge -2 \log(1 + A). $$

Hence it yields:

$$ N^2 \mathcal{E}(x) \ge -2 \log(\varepsilon) - 2N^2 \log(1 + A), $$

so that we can choose $\varepsilon > 0$ small enough to be sure that $\mathcal{E}(x)$ is greater than $R$. This concludes the proof. $\square$

We now prove Theorem 6.2.

*Proof.* This proof is quite standard when dealing with existence of SDE with a potential explosion in finite time. We regularize the singular part in the SDE, then construct a solution up to an explosion time and finally we prove that this solution is defined globally in time.

Let $\lambda_0 \in D$ be an initial data and $\varepsilon > 0$ smaller than $\min_{1 \le i \ne j \le N+1} (|\lambda_0^i - \lambda_0^j|)$ and such that $\varepsilon^{-1} > \max_i \lambda_0^i$.

We consider a smooth approximation $W_\varepsilon$ of $W$ which coincides with $W$ on $\mathbb{R} - [-\varepsilon, \varepsilon]$ and we define as before $\mathcal{E}_\varepsilon = \mathcal{E}_V + \mathcal{E}_{W_\varepsilon}$. By existence and uniqueness of the solution to the SDE we can consider $(\lambda_t^\varepsilon)_{t \ge 0}$ starting from $\lambda_0$ solution to the following problem:

$$ d\lambda_t^\varepsilon = 2\sqrt{\frac{\alpha_N}{\beta_N}} dB_t - \frac{\alpha_N}{2} \nabla \mathcal{E}_{W_\varepsilon}(\lambda_t^\varepsilon) dt. $$

For $\varepsilon' < \varepsilon$, let

$$ T_{\varepsilon, \varepsilon'} = \inf\{s > 0 : \min_{1 \le i \ne j \le N+1} |(\lambda_s^{\varepsilon'})^i - (\lambda_s^{\varepsilon'})^j| \le \varepsilon \text{ or } \max_i |(\lambda_s^{\varepsilon'})^i| \ge \varepsilon^{-1}\} $$

be a stopping time such that by definition we have that the processes $\lambda^\varepsilon$ and $\lambda^{\varepsilon'}$ coincide up to this time. Hence, this allows us to define for all $\varepsilon$ as in the beginning of the proof, a stopping time $T_\varepsilon := T_{\varepsilon, \varepsilon'}$ for $\varepsilon' < \varepsilon$ and $(\lambda_t)_{0 \le t \le T_\varepsilon} := (\lambda_t^{\varepsilon'})_{0 \le t \le T_\varepsilon}$ for $\varepsilon' < \varepsilon$. We clearly have that if $\varepsilon' < \varepsilon$, $T_{\varepsilon'} \ge T_\varepsilon$ almost surely, and so that we can uniquely defined $\lambda$ until the stopping time $T_{\partial D}$. Moreover by definition of $W_\varepsilon$, we have that $\lambda$ satisfies the SDE with $W$ until the time $T_{\partial D}$ since it satisfies it for all $\varepsilon$ on $[0, T_\varepsilon]$ and $W$ and $W_\varepsilon$ coincide on $\mathbb{R} - [-\varepsilon, \varepsilon]$. So it remains to prove that $T_{\partial D} = +\infty$ almost surely to conclude. Let us define for all $R > 0$ the stopping times:

$$ T'_R = \inf\{t \ge 0 : \mathcal{E}(\lambda_t) > R\} $$

and

$$ T' = \lim_{R \to \infty} T'_R. $$
By the previous lemma: $T' = +\infty \subset T_{\partial D} = +\infty$ because if $\lambda_t$ converges to $\partial D$ in a finite time, $\mathcal{E}(\lambda_t)$ would explode in finite time by Lemma 6.3.

Let $R > 0$, thanks to Lemma 6.3 we can find $\varepsilon$ (that depends on $R$) such that $T_{\varepsilon} \ge T'_R$. Hence, we can use Itô's formula to have for all $t > 0$:

$$ \mathbb{E}(\mathcal{E}_{\varepsilon}(\lambda_{t \wedge T'_R})) - \mathcal{E}_{\varepsilon}(\lambda_0) = \mathbb{E}\left(\int_0^{t \wedge T'_R} L \mathcal{E}_{\varepsilon}(\lambda_s) ds\right). $$

Since $T_{\varepsilon} \ge T'_R$ it gives:

$$ \mathbb{E}(\mathcal{E}(\lambda_{t \wedge T'_R})) - \mathcal{E}(\lambda_0) = \mathbb{E}\left(\int_0^{t \wedge T'_R} L \mathcal{E}(\lambda_s) ds\right). \quad (6.15) $$

Now by Proposition 6.1, $L\mathcal{E}$ is uniformly bounded on $D$ by a constant (which depends on $N$) that we will denote $K_N$.

So, we have the following upper bound for every $t > 0$,

$$ \mathbb{E}\left(\int_{0}^{t \wedge T'_{R}} L \mathcal{E}(\lambda_{s}) ds\right) \leq K_{N} t. $$

As a consequence of the Itô formula (6.15), for every $t > 0$,

$$ \sup_{R>0} \mathbb{E}(\mathcal{E}(\lambda_{t \wedge T'_R})) < +\infty. $$

Thanks to that we can use a kind of Markov estimate. For every $t > 0$ since $\mathcal{E}$ is non negative we have:

$$ R1_{T'_R \le t} \le \mathcal{E}(\lambda_{t \wedge T'_R}). $$

By taking the expectation for every $t > 0$ we get:

$$ \mathbb{P}(T'_R \le t) \le \frac{\sup_{R>0} \mathbb{E}(\mathcal{E}(\lambda_{t \wedge T'_R}))}{R}. $$

Hence, let $R \to +\infty$ to have that for all $t > 0$,

$$ \mathbb{P}(T' \le t) = 0. $$

So $T' = +\infty$ almost surely which concludes the proof. $\square$

**Remark 6.4.** The criterion $\beta_N \ge 2N^2$ is the better that can be obtained. Indeed if $\beta_N < 2N^2$ then Cépa and Lépingle proved that collisions occur almost surely [23, 22]. They also proved that actually multiple collisions can not occur [24]. It means that in the regime $\beta_N < 2N^2$ when there is a collision, it can not be strictly more than two particles that collide. Formally, this is linked with the fact that in dimension 1 two independent Brownian motions collide almost surely but the probability that strictly more than two independent Brownian motions collide at the same time is zero. The technique they used in order to study the multiple collisions is based on the properties of squared Bessel processes. Similar techniques have been used to study the existence and the collisions of more delicate systems of particles (typically when the singular term is attractive instead of repulsive as in the Dyson case). For instance, see the Keller-Seggel model for chemotaxis [40].

In the particular case of the Dyson Brownian motion, it gives the following result.
**Theorem 6.5.** Let $\beta \in \{1, 2\}$. For any initial data $\lambda_0 \in D$, there exists a unique strong solution of (5.9) defined on $[0, +\infty)$ (and $T_{\partial D} = +\infty$ almost surely.

**Remark 6.6.** Let us briefly explain how to give a sense to a solution to (5.9) when $\lambda_0$ is in $\mathbb{R}^N$ and not necessarily in $D$. The following argument is used in Theorem 4.3.2 in [5]. Fix $\beta \in \{1, 2\}$ It is known that for all $t > 0$, almost surely the spectrum of $D_N^\beta(t)$ introduced in Definition 5.1 is simple by Proposition 2.13. For all $t \ge 0$, we denote $(\lambda_t^{j,\beta})_{1\le j \le N}$ the ordered spectrum of $D_N^\beta(t)$.

Let $(t_n)_{n \ge 0}$ be a sequence of positive real numbers that goes to 0 when $n$ goes to $\infty$. Almost surely for all $n \ge 0$, $(\lambda_{t_n}^i)_{1 \le i \le N} \in \mathbb{R}_>^N$. By Theorem 6.5, for all $n$ we can consider the unique strong solution to (5.9) starting from $(\lambda_{t_n}^{i,\beta})_{1 \le i \le N}$, $(\tilde{\lambda}_t^{i,\beta})_{t \ge t_n, 1 \le i \le N}$ whose law coincides with the law of the eigenvalues of $(D_N^\beta(t))_{t \ge t_n}$ by the Dyson Theorem 5.8 [37]. By uniqueness of the trajectory obtained in Theorem 6.5, we can construct a solution $(\tilde{\lambda}_t)_{t>0}$ of the Dyson equation (5.9) whose law coincides with the law of the eigenvalues of $(D_N^\beta(t))_{t>0}$ and which is in $\mathbb{R}_>^N$ for all $t > 0$. Moreover the Hoffman-Wielandt inequality stated in Lemma 4.3 implies that $\lambda_t^\beta$ converges to $\lambda_0$ when $t$ goes to 0 by continuity in 0 of $(D_N^\beta(t))_{t \ge 0}$. This proves existence and uniqueness in law of solutions of (5.9) starting from $\lambda_0 \in D$.

For solutions of (6.1), in general we can not interpret the particles as eigenvalues of random matrices to give a sense to a solution starting from $\lambda_0 \in \mathbb{R}^N$. Nonetheless, we can define solutions using a comparison principle of the particles and more stochastic calculus. See Lemma 4.3.6 and Proposition 4.3.5 of [5].

### 6.3.2 Solution of the Fokker Planck equation of the Dyson system

In this part, we follow [75] to solve the Fokker Planck equation. To lighten the notation, we take $\alpha_N$ and $\beta_N$ such that the Dyson equation in this case is

$$ \forall 1 \le i \le N, \quad d\lambda_t^i = \sum_{j \ne i} \frac{1}{\lambda_t^i - \lambda_j^i} dt + dB_t^i \qquad (6.16) $$

**Remark 6.7.** As explained in Section 5.3, this system of SDE corresponds to the dynamic of the eigenvalues of $\sqrt{N}D_N^2(t)$ where $D_N^\beta$ is the Dyson Brownian motion of size $N \ge 1$ introduced in Definition 5.1.

**Proposition 6.8.** Let $\lambda_0$ in $\mathbb{R}_>^N$, $(\lambda_t^i)_{t \in \mathbb{R}^+}$ be the solution of (6.16) starting from $\lambda_0$ and $f: \mathbb{R}^n \to \mathbb{R}$ be a smooth function with bounded derivatives, then for any $t \ge 0$ we have:

$$ \partial_t \mathbb{E}[f(\lambda_t^1, \dots, \lambda_t^N)] = \mathbb{E}[D^* f(\lambda_t^1, \dots, \lambda_t^N)] \qquad (6.17) $$

where $D^*$ is the adjoint of the Dyson operator defined by:

$$ D\phi = \frac{1}{2} \sum_{i=1}^{N} \frac{\partial^2 \phi}{\partial \lambda_i^2} - \sum_{1 \le i,j \le N, i \ne j} \frac{\partial}{\partial \lambda_i} \left( \frac{\phi}{\lambda_i - \lambda_j} \right) \qquad (6.18) $$

and its adjoint is:

$$ D^* \phi = \frac{1}{2} \sum_{i=1}^{n} \frac{\partial^2 \phi}{\partial \lambda_i^2} + \sum_{1 \le i,j \le N, i \ne j} \frac{\partial \phi}{\partial \lambda_i} \frac{1}{\lambda_i - \lambda_j} \qquad (6.19) $$

Moreover, let $\rho_t$ be the distribution of the ordered vector $(\lambda_t^1, \dots, \lambda_N^t)$, then $\rho$ satisfies the Fokker Planck equation in the sense of distribution on the subset of $\mathbb{R}_>^N$:

$$ \partial_t \rho = D\rho. \qquad (6.20) $$
Proof. The proof is an immediate application of the Itô formula using (6.16). Indeed, since the derivatives of $f$ are bounded, the local martingale part of the Itô formula is actually a real martingale with expectation 0 given (6.17). The Fokker Planck equation (6.20) is obtained by integrations by parts as usual. □

The Fokker Planck equation seems difficult to solve a priori. However, this equation is not far from the heat equation as we shall show. We shall give an intuition for this idea in Remark 6.12.

We recall that given $\lambda = (\lambda_1, ..., \lambda_N) \in \mathbb{R}^N$ we define the Vandermonde determinant of these real numbers the quantity:

$$ \Delta_N(\lambda) = \prod_{1 \le i < j \le N} (\lambda_j - \lambda_i), $$

which is also given by

$$ \Delta_N(\lambda_1, ..., \lambda_N) = \det[(\lambda_i^{j-1})_{1 \le i,j \le N}]. $$

**Remark 6.9.** We have that $\Delta_N(\lambda_1, ..., \lambda_N) = 0$ if and only if two of the $\lambda_i$ are equal.

We can compute the derivatives of the Vandermonde determinant.

**Lemma 6.10.** For $(\lambda_1, ..., \lambda_N) \in \mathbb{R}^N$, we have the following derivatives:

$$ \frac{\partial \Delta_N}{\partial \lambda_i}(\lambda_1, ..., \lambda_N) = \sum_{1 \le j \le N, i \ne j} \frac{\Delta_N(\lambda_1, ..., \lambda_N)}{\lambda_i - \lambda_j} $$

$$ \Delta(\Delta_N)(\lambda_1, ..., \lambda_N) = 0 $$

with $\Delta$ the Laplacian operator.

Proof. The first derivative is immediate by derivation. The second one is obtained by computing the derivatives and using the identities (6.10). Indeed, using the formula for the first derivative yields:

$$ \begin{align*} \Delta(\Delta_N)(\lambda_1, ..., \lambda_N) &= \sum_{i=1}^{N} \frac{\partial^2 \Delta_N}{\partial \lambda_i^2}(\lambda_1, ..., \lambda_N) \\ &= \sum_{i=1}^{N} \left[ \Delta_N(\lambda_1, ..., \lambda_N) \sum_{1 \le k \le N, i \ne k} \frac{1}{\lambda_i - \lambda_k} \sum_{1 \le j \le N, i \ne j} \frac{1}{\lambda_i - \lambda_j} - \Delta_N(\lambda_1, ..., \lambda_N) \sum_{1 \le j \le N, i \ne j} \frac{1}{(\lambda_i - \lambda_j)^2} \right] \\ &= \Delta_N(\lambda_1, ..., \lambda_N) \sum_{i=1}^{N} \left[ \sum_{1 \le k \le N, i \ne k} \frac{1}{\lambda_i - \lambda_k} \sum_{1 \le j \le N, i \ne j} \frac{1}{\lambda_i - \lambda_j} - \sum_{1 \le j \le N, i \ne j} \frac{1}{(\lambda_i - \lambda_j)^2} \right] \\ &\stackrel{(6.10)}{=} 0 \end{align*} $$

□

The harmonic property of the Vandermonde determinant helps us to solve the Fokker Planck equation. The most important property to solve the Fokker Planck equation (6.20) is the following one.
**Proposition 6.11.** For $\rho$ a smooth solution on $\mathbb{R}_{>^N}$ of

$$ \partial_t \rho = D\rho, $$

then define $u$ by:

$$ \rho = \Delta_N u \qquad (6.21) $$

on $\mathbb{R}_{>^N}$. Then $u$ is a solution of the heat equation:

$$ \partial_t u = \frac{1}{2} \Delta u \qquad (6.22) $$

on $\mathbb{R}_{>^N}$.

*Proof.* The computation is quite straight forward. Differentiating the equality $\rho = \Delta_N u$ yields:

$$ \partial_t \rho = \Delta_N \partial_t u, $$

$$ \frac{1}{2} \Delta(\rho) = \frac{1}{2} \Delta(\Delta_N)u + \frac{1}{2} \Delta(u)\Delta_N + \sum_{i=1}^{N} \frac{\partial u}{\partial \lambda_i} \frac{\partial \Delta_N}{\partial \lambda_i}, $$

$$ \sum_{1 \le i,j \le N, i \ne j} \frac{\partial}{\partial \lambda_i} \left( \frac{\rho}{\lambda_i - \lambda_j} \right) = \sum_{1 \le i,j \le N, i \ne j} \frac{\partial}{\partial \lambda_i} \left( \frac{\Delta_N u}{\lambda_i - \lambda_j} \right) = \sum_{1 \le i \le N} \frac{\partial}{\partial \lambda_i} \left( u \frac{\partial \Delta_N}{\partial \lambda_i} \right), $$

by using the expression of the first derivative of the Vandermonde determinant for the last equality.

Finally using that $\rho$ satisfies the Fokker Planck equation (6.20) and using the harmonic property of $\Delta_N$ on $\mathbb{R}_{>^N}$, we get:

$$ \Delta_N \partial_t u = \frac{1}{2} \Delta(u) \Delta_N + \sum_{i=1}^{N} \frac{\partial u}{\partial \lambda_i} \frac{\partial \Delta_N}{\partial \lambda_i} - \sum_{1 \le i \le N} \frac{\partial}{\partial \lambda_i} \left( u \frac{\partial \Delta_N}{\partial \lambda_i} \right). \qquad (6.23) $$

However, again by the harmonic property of the Vandermonde determinant, the last term of the equation is:

$$ u \sum_{1 \le i \le N} \frac{\partial^2 \Delta_N}{\partial \lambda_i^2} + \sum_{1 \le i \le N} \frac{\partial u}{\partial \lambda_i} \frac{\partial \Delta_N}{\partial \lambda_i} = u \Delta(\Delta_N) + \sum_{1 \le i \le N} \frac{\partial u}{\partial \lambda_i} \frac{\partial \Delta_N}{\partial \lambda_i} = \sum_{1 \le i \le N} \frac{\partial u}{\partial \lambda_i} \frac{\partial \Delta_N}{\partial \lambda_i}. \qquad (6.24) $$

Using the relation (6.24) in (6.23) and simplifying by $\Delta_N$ we have that $u$ satisfies on $\mathbb{R}_{>^N}$

$$ \partial_t u = \frac{1}{2} \Delta u. $$

<figure><img src="image_11.png" /></figure>

**Remark 6.12.** The computation is quite immediate but it is very natural to ask ourselves what is the intuition behind this computation. An interpretation that shall be detailed in Section 6.4 is that the law of the solution of the Dyson SDE (6.16) can be interpreted as the law of $N$ independent Brownian motions conditioned to not intersect. The density of $N$ independent Brownian motion is solution of the heat equation and $\Delta_N$ should be viewed as the natural quantity that vanishes exactly when two real numbers are equal as mentioned in Remark 6.9. So, formally, (6.21) can be interpreted as a kind of Bayes rule.
We now look at the the eigenvalues of random matrices as an exchangeable vector of $\mathbb{C}^N$ (or $\mathbb{R}^N$ if the spectrum is real) as we did in Part I. We extend $\rho$ symmetrically across $\mathbb{R}_>^N$ on $\mathbb{R}^N$ and $u$ anti symmetrically across $\mathbb{R}^N$ (since $\Delta_N$ is antisymmetric). This extension of $u$ on $\mathbb{R}^N$ is solution of the heat equation (6.22) on $\mathbb{R}^N$

This gives a heuristic to solve the Fokker Planck equation (6.20) of the Dyson particles (6.16).

Indeed, assume that we start from an deterministic matrix $A_0 \in S_N(\mathbb{R})$ with spectrum $\lambda_0 = (\lambda_0^1, ..., \lambda_0^N)$. First we assume that $\lambda_0 \in \mathbb{R}_>^N$. So on $\mathbb{R}_>^N$ (before extended $\rho$ symmetrically on $\mathbb{R}^N$), $\rho(0, \lambda) = \delta_{\lambda_0}(\lambda)$ where $\delta_a(x)$ is the dirac measure at the point a.

For $\sigma \in \mathfrak{S}_N$, we write $\sigma(\lambda_0) = (\lambda_0^{\sigma(1)}, ..., \lambda_0^{\sigma(N)})$ and $\varepsilon(\sigma)$ the sign of the permutation $\sigma$. Hence $u(t, \lambda)$ satisfies the heat equation on $\mathbb{R}^N$ with initial condition:

$$u(0, \lambda) = \frac{1}{N! \Delta_N(\lambda_0)} \sum_{\sigma \in \mathfrak{S}_N} \varepsilon(\sigma) \delta_{\sigma(\lambda_0)}(\lambda).$$

Using the fundamental solution of the linear heat equation, we directly obtain that:

$$u(t, \lambda = (\lambda^1, ..., \lambda^N)) = \frac{1}{N!(2\pi t)^{N/2} \Delta_N(\lambda_0)} \sum_{\sigma \in \mathfrak{S}_N} \varepsilon(\sigma) \exp \left( - \frac{|\lambda - \sigma(\lambda_0)|^2}{2t} \right).$$

We can write this expression with a more compact formula thanks to the determinant identity:

$$u(t, \lambda = (\lambda^1, ..., \lambda^N)) = \frac{1}{N!(2\pi t)^{N/2} \Delta_N(\lambda_0)} \det \left[ \left( \exp \left( - \frac{|\lambda^i - \lambda_0^j|^2}{2t} \right) \right)_{1 \le i, j \le N} \right].$$

Using the formula (6.21), it gives the following result for $\rho$ originally obtained by Johansson in [56].

**Theorem 6.13** (Johansson formula). Let $A_0 \in S_N^\circ(\mathbb{R})$ and write $\lambda_0 \in \mathbb{R}_>^N$ its spectrum. For $t > 0$, let $\lambda_t \in \mathbb{R}_>^N$ be the solution of (6.16) starting from $\lambda_0$. Then the density on $\mathbb{R}^N$ of $\lambda_t$ (viewed as an exchangeable vector) is:

$$\rho(t, \lambda = (\lambda^1, ..., \lambda^N)) = \frac{1}{N!(2\pi t)^{N/2} \Delta_N(\lambda_0)} \det \left[ \left( \exp \left( - \frac{|\lambda^i - \lambda_0^j|^2}{2t} \right) \right)_{1 \le i, j \le N} \right]. \quad (6.25)$$

**Remark 6.14.** More explicitly, with the notations of Theorem 6.13, for every symmetric, bounded or positive measurable functions $F$ and for every $t > 0$ we have

$$\begin{aligned} \mathbb{E}(F(\lambda_t^1, ..., \lambda_t^N)) &= \frac{1}{N!(2\pi t)^{N/2} \Delta_N(\lambda_0)} \int_{\mathbb{R}^N} F(x_1, ..., x_N) \prod_{1 \le i < j \le N} (x_j - x_i) \times \\ & \det \left[ \left( \exp \left( - \frac{|x_i - \lambda_0^j|^2}{2t} \right) \right)_{1 \le i, j \le N} \right] dx_1 \dots dx_N \\ &= \frac{1}{(2\pi t)^{N/2} \Delta_N(\lambda_0)} \int_{\mathbb{R}_>^N} F(x_1, ..., x_N) \prod_{1 \le i < j \le N} (x_j - x_i) \times \\ & \det \left[ \left( \exp \left( - \frac{|x_i - \lambda_0^j|^2}{2t} \right) \right)_{1 \le i, j \le N} \right] dx_1 \dots dx_N. \end{aligned} \quad (6.26)$$