Microscopic description of Log and Coulomb gases

TL;DR

This paper reviews recent mathematical results on the statistical mechanics of point systems with logarithmic and Coulomb interactions, focusing on their microscopic behavior and probabilistic properties.

Contribution

It introduces the electric approach to analyze microscopic states, leading to new concentration, CLT, and large deviations results for these systems.

Findings

Established concentration results for point configurations

Proved Central Limit Theorems for fluctuations

Derived Large Deviations Principles based on microscopic states

Abstract

These are the lecture notes of a course taught at the Park City Mathematics Institute in June 2017. They are intended to review some recent results, obtained in large part with Thomas Lebl\'e, on the statistical mechanics of systems of points with logarithmic or Coulomb interactions. After listing some motivations, we describe the "electric approach" which allows to get concentration results, Central Limit Theorems for fluctuations, and a Large Deviations Principle expressed in terms of the microscopic state of the system.