Poisson statistics for matrix ensembles at large temperature

In this article, we consider $β$-ensembles, i.e. collections of particles with random positions on the real line having joint distribution $\frac{1}{Z_N(β)}|Δ(λ)|^βe^{- \frac{Nβ}{4}\sum_{i=1}^Nλ_i^2}d λ,$ in the regime where $β\to 0$ as $N\to\infty$. We briefly describe the global regime and then consider the local regime. In the case where $Nβ$ stays bounded, we prove that the local eigenvalue statistics, in the vicinity of any real number, are asymptotically to those of a Poisson point process. In the case where $Nβ\to\infty$, we prove a partial result in this direction.

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