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

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 ${\pi }^{-2}\log n$.

arXiv comment12 pages