Fluctuations of the eigenvalue number in the fixed interval for $\beta$-models with $\beta=1,2,4$

TL;DR

This paper investigates the eigenvalue count fluctuations in fixed intervals for beta-ensembles with beta=1,2,4, showing they tend to Gaussian distributions when properly normalized as the matrix size grows large.

Contribution

It provides a rigorous analysis of eigenvalue fluctuations in beta-ensembles with polynomial potentials across multi-cut supports, extending understanding of their asymptotic behavior.

Findings

Eigenvalue fluctuations are Gaussian in the limit as matrix size increases.

Fluctuations are normalized by π^{-2} log n for convergence.

Results apply to beta=1,2,4 with polynomial potentials and multi-cut supports.

Abstract

We study the fluctuation of the eigenvalue number of any fixed interval $\Delta=[a,b]$ inside the spectrum for $\beta$- ensembles of random matrices in the case $\beta=1,2,4$. We assume that the potential $V$ is polynomial and consider the cases of any multi-cut support of the equilibrium measure. It is shown that fluctuations become gaussian in the limit $n\to\infty$, if they are normalized by $\pi^{-2}\log n$.