From random matrices to systems of particles in interaction

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.

💡 Research Summary

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The manuscript “From random matrices to systems of particles in interaction” is an extensive expository work that aims to introduce the fundamental concepts of random matrix theory (RMT) to readers without a deep background in the field, while emphasizing the deep connections between random matrices and interacting particle systems. The author, Valentin Pesce, structures the paper into four major parts, each of which builds on the previous one to develop a coherent narrative from static eigenvalue distributions to dynamic stochastic processes and finally to large‑deviation principles.

Part I – Classical Ensembles and Their Eigenvalue Laws
The first part focuses on two of the most celebrated matrix ensembles: the Ginibre ensemble (non‑Hermitian complex Gaussian matrices) and the Gaussian Unitary Ensemble (GUE, Hermitian complex Gaussian matrices). For each ensemble the joint probability density of eigenvalues is derived explicitly. In the Ginibre case the density contains a Vandermonde factor (\prod_{i<j}|\lambda_i-\lambda_j|^2) multiplied by a Gaussian weight, which leads to the celebrated circular law: as the matrix size (N) tends to infinity, the empirical spectral measure converges weakly to the uniform distribution on the unit disk in the complex plane. In the GUE case the same Vandermonde factor appears, but the eigenvalues are real and the limiting law is the Wigner semicircle distribution. The author explains how these joint densities have a determinantal structure, which allows the computation of all (k)-point correlation functions via an explicit kernel. This determinantal point process viewpoint is crucial for later sections because it makes the connection with log‑gases (Coulomb or Riesz gases) transparent: the Vandermonde factor corresponds to a pairwise logarithmic repulsion between particles.

Part II – Dyson Brownian Motion (DBM)
The second part introduces the dynamical counterpart of the static ensembles: Dyson’s Brownian motion. By adding independent Brownian motions to each matrix entry and letting the matrix evolve in time, the eigenvalues themselves follow a system of stochastic differential equations (SDEs) of the form
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