Poisson statistics for Gibbs measures at high temperature

We consider a gas of N particles subject to a two-body interaction and confined by an external potential in the mean field or high temperature regime, that is when the inverse temperature β > 0 satisfies βN → γ ≥ 0 as N → +∞. 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 Rn for any n ≥ 1, as well as to the edge behavior of β-ensembles on R.