A maximum principle for the Coulomb gas: microscopic density bounds, confinement estimates, and high temperature limits

TL;DR

This paper establishes a maximum principle for the Coulomb gas, providing microscopic density bounds, confinement estimates, and analyzing high temperature limits, with implications for the behavior of the gas across different regimes.

Contribution

It introduces a maximum principle for the Coulomb gas's correlation functions and derives new bounds and limits, especially at high temperatures, extending understanding beyond mean-field theory.

Findings

Confines Coulomb gas to the droplet using an effective potential

Provides upper bounds for particle density at any temperature

Shows that high temperature limits lead to homogeneous Poisson processes

Abstract

We introduce and prove a maximum principle for a natural quantity related to the $k$-point correlation function of the classical one-component Coulomb gas. As an application, we show that the gas is confined to the droplet by a well-known effective potential in dimensions two and higher. We also prove new upper bounds for the particle density in the droplet that apply at any temperature. In particular, we give the first controls on the microscopic point process for high temperature Coulomb gases beyond the mean-field regime, proving that their laws are uniformly tight in the particle number $N$ for any inverse temperatures $\beta_N$. Furthermore, we prove that limit points are homogeneous mixed Poisson point processes if $\beta_N\to 0$.