Poisson statistics for Gibbs measures at high temperature

TL;DR

This paper demonstrates that in high-temperature regimes, the local particle fluctuations in a broad class of interacting particle systems converge to a Poisson point process, with applications to Coulomb, Riesz gases, and beta-ensembles.

Contribution

It establishes the Poisson nature of local fluctuations for general two-body interactions at high temperature, extending previous results to new settings.

Findings

Local fluctuations are described by a Poisson point process as N→∞

Applicable to Coulomb and Riesz gases in any dimension

Provides insights into the edge behavior of beta-ensembles

Abstract

We consider a gas of N particles with a general two-body interaction and confined by an external potential in the mean field or high temperature regime, that is when the inverse temperature satisfies $\beta N \to \kappa \ge 0$ as $N\to+\infty$. We show that under general conditions on the interaction and the potential, the local fluctuations are described by a Poisson point process in the large N limit. We present applications to Coulomb and Riesz gases on $\mathbb{R}^n$ for any $n\ge 1$, as well as to the edge behavior of $\beta$-ensembles on $\mathbb{R}$.