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

Let $β>0$ and consider an $n$-point process $λ_1, λ_2,..., λ_n$ from Hermite $β$ 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 $\log n/π^2β$. 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.