Mesoscopic linear statistics of Wigner matrices of mixed symmetry class

We prove a central limit theorem for the mesoscopic linear statistics of $N\times N$ Wigner matrices $H$ satisfying $\mathbb{E}\mathrm{\mid }{H}_{ij}{\mathrm{\mid }}^2=1\mathrm{/}N$ and $\mathbb{E}{H}_{ij}^2=\sigma \mathrm{/}N$, where $\sigma \in [-1,1]$. We show that on all mesoscopic scales $\eta$ ($1\mathrm{/}N\ll \eta \ll 1$), the linear statistics of $H$ have a sharp transition at $1-\sigma \sim \eta$. 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 $t\sim \eta$.