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

This paper studies beta ensembles on the real line in a high temperature regime, that is, the regime where $βN \to const \in (0, \infty)$, with $N$ the system size and $β$ 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.