Local Semicircle law and Gaussian fluctuation for Hermite $\beta$ ensemble

TL;DR

This paper proves a local semicircle law and Gaussian fluctuations for the Hermite beta ensemble, extending understanding of eigenvalue distributions at small scales and their probabilistic behavior.

Contribution

It establishes the local semicircle law for the Hermite beta ensemble and demonstrates Gaussian fluctuations of positive eigenvalues, advancing spectral analysis in random matrix theory.

Findings

Local semicircle law holds for intervals larger than √log n.

Number of positive states fluctuates normally around n/2.

Variance of fluctuations is proportional to log n.

Abstract

Let $\beta>0$ and consider an $n$-point process $\lambda_1, \lambda_2,..., \lambda_n$ from Hermite $\beta$ ensemble on the real line $\mathbb{R}$. Dumitriu and Edelman discovered a tri-diagonal matrix model and established the global Wigner semicircle law for normalized empirical measures. In this paper we prove that the average number of states in a small interval in the bulk converges in probability when the length of the interval is larger than $\sqrt {\log n}$, i.e., local semicircle law holds. And the number of positive states in $(0,\infty)$ is proved to fluctuate normally around its mean $n/2$ with variance like $\log n/\pi^2\beta$. The proofs rely largely on the way invented by Valk$\acute{o}$ and Vir$\acute{a}$g of counting states in any interval and the classical martingale argument.