Local microscopic behavior for 2D Coulomb gases

TL;DR

This paper establishes a large deviation principle and local law for 2D Coulomb gases, revealing microscopic behavior and convergence properties of particle configurations near bulk points.

Contribution

It introduces a detailed large deviation principle for empirical fields in 2D Coulomb gases, connecting microscopic behavior with energy and entropy considerations.

Findings

Large deviation principle for empirical fields

Quantitative local law for particle measures

Connection between microscopic behavior and energy/entropy

Abstract

The study of two-dimensional Coulomb gases lies at the interface of statistical physics and non-Hermitian random matrix theory. In this paper we give a large deviation principle (LDP) for the empirical fields obtained, under the canonical Gibbs measure, by zooming around a point in the bulk of the equilibrium measure, up to the finest averaging scale. The rate function is given by the sum of the "renormalized energy" of Serfaty et al. weighted by the inverse temperature, and of the specific relative entropy. We deduce a local law which quantifies the convergence of the empirical measures of the particles to the equilibrium measure, up to the finest scale.