Poisson statistics for beta ensembles on the real line at high temperature

TL;DR

This paper investigates the local statistical behavior of beta ensembles on the real line at high temperature, demonstrating convergence to a Poisson point process in this regime.

Contribution

It establishes the weak convergence of local statistics of beta ensembles at high temperature to a Poisson process, extending understanding of their local behavior.

Findings

Local statistics converge to Poisson point process

High temperature regime characterized by $eta N o const$

Supports recent large deviation results for beta ensembles

Abstract

This paper studies beta ensembles on the real line in a high temperature regime, that is, the regime where $\beta N \to const \in (0, \infty)$, with $N$ the system size and $\beta$ the inverse temperature. In this regime, the convergence to the equilibrium measure is a consequence of a recent result on large deviation principle by Liu and Wu (Stochastic Processes and their Applications (2019)). This paper focuses on the local behavior and shows that the local statistics around any fixed reference energy converges weakly to a homogeneous Poisson point process.