On fluctuations of global and mesoscopic linear eigenvalue statistics of generalized Wigner matrices

TL;DR

This paper proves Gaussian fluctuation results for linear eigenvalue statistics of generalized Wigner matrices on both global and mesoscopic scales, including at spectral edges, with universality in the bulk and edges.

Contribution

It establishes Gaussian fluctuation theorems for eigenvalue statistics of generalized Wigner matrices across all scales up to spectral edges, extending previous results to more general variance profiles.

Findings

Gaussian fluctuations for global eigenvalue statistics

Universal mesoscopic CLTs at bulk and edges

Fluctuations characterized by variance profile

Abstract

We consider an $N$ by $N$ real or complex generalized Wigner matrix $H_N$, whose entries are independent centered random variables with uniformly bounded moments. We assume that the variance profile, $s_{ij}:=\mathbb{E} |H_{ij}|^2$, satisfies $\sum_{i=1}^Ns_{ij}=1$, for all $1 \leq j \leq N$ and $c^{-1} \leq N s_{ij} \leq c$ for all $ 1 \leq i,j \leq N$ with some constant $c \geq 1$. We establish Gaussian fluctuations for the linear eigenvalue statistics of $H_N$ on global scales, as well as on all mesoscopic scales up to the spectral edges, with the expectation and variance formulated in terms of the variance profile. We subsequently obtain the universal mesoscopic central limit theorems for the linear eigenvalue statistics inside the bulk and at the edges respectively.