Aspects of Coulomb gases
Djalil Chafaï
TL;DR
This article surveys the Coulomb gas framework, bridging potential theory, probability, and random matrix theory to study $n$ interacting particles with Coulomb interaction in a confining field $V$. It develops the macroscopic picture via the equilibrium measure $\mu_V$ and the large deviations principle at speed $n^2$, and it analyzes fluctuations, edge behavior, and exact solvability in two dimensions, with emphasis on the Ginibre ensemble. It highlights how determinantal structures and Gaussian free field connections yield precise results for linear statistics and radii, while also outlining open problems in universality, edge fluctuations for general $\beta$, and crystallization. The notes emphasize the interplay between variational energy methods, probabilistic limit theorems, and exactly solvable random-matrix models as a coherent framework for Coulomb gases in mathematics and physics.
Abstract
Coulomb gases are special probability distributions, related to potential theory, that appear at many places in pure and applied mathematics and physics. In these short expository notes, we focus on some models, ideas, and structures. We present briefly selected mathematical aspects, mostly related to exact solvability, and to first and second order global asymptotics. A particular attention is devoted to two-dimensional exactly solvable models of random matrix theory such as the Ginibre model. Thematically, these notes lie between probability theory, mathematical analysis, and statistical physics, and aim to be very accessible. They form a contribution to a volume of the "Panoramas et Synthèses" series around the workshop "États de la recherche en mécanique statistique", organized by Société Mathématique de France, held at Institut Henri Poincaré, Paris, in the fall of 2018 (http://statmech2018.sciencesconf.org).