Mesoscopic linear statistics of Wigner matrices of mixed symmetry class

Abstract

We prove a central limit theorem for the mesoscopic linear statistics of $N\times N$ Wigner matrices $H$ satisfying $\mathbb{E}|H_{ij}|^2=1/N$ and $\mathbb{E} H_{ij}^2= σ/N$, where $σ\in [-1,1]$. We show that on all mesoscopic scales $η$ ($1/N \ll η\ll 1$), the linear statistics of $H$ have a sharp transition at $1-σ\sim η$. As an application, we identify the mesoscopic linear statistics of Dyson's Brownian motion $H_t$ started from a real symmetric Wigner matrix $H_0$ at any nonnegative time $t \in [0,\infty]$. In particular, we obtain the transition from the central limit theorem for GOE to the one for GUE at time $t \sim η$.