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From random matrices to systems of particles in interaction

Valentin Pesce

TL;DR

The notes present a structured, accessible pathway from classical random matrix models to the dynamics of interacting particle systems, centered on Dyson Brownian motion and log-gas interpretations. They derive explicit eigenvalue laws for the Ginibre and Gaussian unitary ensembles, establishing determinantal structures and linking spectral statistics to Coulomb/Riesz gases; they also develop mean-field and large deviations viewpoints. The Wigner semicircle and circular laws are developed with mean and almost-sure convergence, accompanied by concentration results and extremal eigenvalue behavior. Finally, the Dyson Brownian motion framework is developed in depth, connecting matrix dynamics to a mean-field particle system, with rigorous results on existence, containment, and the associated Fokker-Planck description, highlighting the role of interactions and phase transitions in spectral dynamics.

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.

From random matrices to systems of particles in interaction

TL;DR

The notes present a structured, accessible pathway from classical random matrix models to the dynamics of interacting particle systems, centered on Dyson Brownian motion and log-gas interpretations. They derive explicit eigenvalue laws for the Ginibre and Gaussian unitary ensembles, establishing determinantal structures and linking spectral statistics to Coulomb/Riesz gases; they also develop mean-field and large deviations viewpoints. The Wigner semicircle and circular laws are developed with mean and almost-sure convergence, accompanied by concentration results and extremal eigenvalue behavior. Finally, the Dyson Brownian motion framework is developed in depth, connecting matrix dynamics to a mean-field particle system, with rigorous results on existence, containment, and the associated Fokker-Planck description, highlighting the role of interactions and phase transitions in spectral dynamics.

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.

Paper Structure

This paper contains 74 sections, 140 theorems, 673 equations, 1 table.

Table of Contents

  1. Introduction
  2. Ginibre and Gaussian unitary ensembles
  3. Examples of remarkable ensembles of matrices
  4. Notation and first properties
  5. Gaussian unitary ensemble
  6. Ginibre ensemble
  7. Law of the eigenvalues and eigenvectors
  8. Gaussian unitary ensemble case
  9. Eigenvectors
  10. Eigenvalues
  11. Ginibre case
  12. Determinental structure
  13. Ginibre case
  14. Gaussian unitary ensemble case
  15. Circular law
  16. ...and 59 more sections

Key Result

Theorem 1.1

Assume that there exists $c>0$ such that $d_N$ satisfies $\frac{d_N}{N}\underset{N\to+\infty}{\longrightarrow} c$. 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 e where $\rho_c$ is an explicit distribution called the Marchenko-Pastur law of parameter $c$ given b

Theorems & Definitions (328)

  • Theorem 1.1: Marchenko-Pastur, Figure 1
  • Theorem 1.2: Wigner, Figure 2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 318 more