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

Zhigang Bao, Zhonggen Su

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…