Concentration for Coulomb gases and Coulomb transport inequalities

TL;DR

This paper establishes new concentration inequalities for Coulomb gases in multiple dimensions, improving understanding of their empirical distributions and connecting to Coulomb transport inequalities, with applications to random matrix theory.

Contribution

It introduces simple, new Coulomb transport inequalities and extends concentration results for Coulomb gases without using renormalized energy techniques.

Findings

Proves concentration of measure for Coulomb gases around equilibrium measures.

Derives macroscopic and mesoscopic convergence in Wasserstein and Lipschitz distances.

Improves known concentration inequalities for Ginibre random matrices.

Abstract

We study the non-asymptotic behavior of Coulomb gases in dimension two and more. Such gases are modeled by an exchangeable Boltzmann-Gibbs measure with a singular two-body interaction. We obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure, with respect to bounded Lipschitz and Wasserstein distances. This implies macroscopic as well as mesoscopic convergence in such distances. In particular, we improve the concentration inequalities known for the empirical spectral distribution of Ginibre random matrices. Our approach is remarkably simple and bypasses the use of renormalized energy. It crucially relies on new inequalities between probability metrics, including Coulomb transport inequalities which can be of independent interest. Our work is inspired by the one of Ma{\"i}da and Maurel-Segala, itself inspired by large deviations techniques. Our approach allows to recover, extend, and simplify previous results by Rougerie and Serfaty.