Poisson statistics for matrix ensembles at large temperature

TL;DR

This paper investigates the local eigenvalue statistics of $eta$-ensembles in the large $N$ limit as $eta$ approaches zero, showing they tend to Poisson point processes under certain conditions.

Contribution

It demonstrates that for $eta$-ensembles with $Neta$ bounded, the local eigenvalue statistics converge to Poisson processes, providing new insights into their behavior at large temperature regimes.

Findings

Eigenvalue statistics are Poissonian when $Neta$ is bounded.

Partial results obtained for the case $Neta o \infty$.

Global regime description included.

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.