Poisson statistics for matrix ensembles at large temperature

Abstract

In this article, we consider $\beta$-ensembles, i.e. collections of particles with random positions on the real line having joint distribution $$\frac{1}{Z_N(\beta )}|\Delta (\lambda ){|}^{\beta }{e}^{-\frac{N\beta }{4}{\sum }_{i=1}^N{\lambda }_i^2}d\lambda ,$$ in the regime where $\beta \to 0$ as $N\to \infty$. We briefly describe the global regime and then consider the local regime. In the case where $N\beta$ 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 $N\beta \to \infty$, we prove a partial result in this direction.

26 pages, 1 figure. In v2: references added and minor clarifications