From random matrices to systems of particles in interaction

TL;DR

This paper introduces the connection between random matrix theory and interacting particle systems, focusing on eigenvalue laws, Dyson Brownian motion, and large deviations, providing foundational insights for non-specialists.

Contribution

It offers an accessible overview of how random matrices relate to particle systems, emphasizing continuous-time approaches and large deviation principles.

Findings

Eigenvalues of certain random matrices behave as log gases.

Dyson Brownian motion models particle interactions over time.

Large deviations describe rare eigenvalue configurations.

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.