Local Semicircle law and Gaussian fluctuation for Hermite $β$ ensemble

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.