Mesoscopic linear statistics of Wigner matrices of mixed symmetry class

TL;DR

This paper establishes a central limit theorem for mesoscopic linear statistics of Wigner matrices with mixed symmetry, revealing a sharp transition depending on the symmetry parameter and time evolution.

Contribution

It introduces a CLT for mesoscopic linear statistics of Wigner matrices with mixed symmetry class, identifying a transition at the symmetry parameter and time.

Findings

Sharp transition in linear statistics at symmetry parameter

Identification of mesoscopic statistics for Dyson's Brownian motion

Transition from GOE to GUE behavior over time

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 $\mathbb{E} H_{ij}^2= \sigma /N$, where $\sigma \in [-1,1]$. We show that on all mesoscopic scales $\eta$ ($1/N \ll \eta \ll 1$), 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 $t \sim \eta$.