Aspects of Coulomb gases

TL;DR

This paper provides an accessible overview of Coulomb gases, focusing on models, mathematical aspects, and their connections to random matrix theory, especially the exactly solvable Ginibre model, highlighting their significance in probability, analysis, and physics.

Contribution

It offers a concise exposition of mathematical structures and solvability in Coulomb gases, emphasizing two-dimensional models like the Ginibre ensemble, bridging multiple mathematical disciplines.

Findings

Analysis of exact solvability in Coulomb gases

Insights into global asymptotics of models

Connections between Coulomb gases and random matrix theory

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 Synth\`eses" series around the workshop "\'Etats de la recherche en m\'ecanique statistique", organized by Soci\'et\'e Math\'ematique de France, held at Institut Henri Poincar\'e, Paris, in the fall of 2018 (http://statmech2018.sciencesconf.org).