Poisson statistics for Gibbs measures at high temperature

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 $\mathbb{R}$.