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

Abstract

We consider an N by N real or complex generalized Wigner matrix ${\mathit{H}}_{\mathit{N}}$, whose entries are independent centered random variables with uniformly bounded moments. We assume that the variance profile, ${\mathit{s}}_{\mathit{i}\mathit{j}}:=\mathbb{E}|{\mathit{H}}_{\mathit{i}\mathit{j}}{|}^2$, satisfies ${\sum }_{\mathit{i}=1}^{\mathit{N}}{\mathit{s}}_{\mathit{i}\mathit{j}}=1$, for all $1\le \mathit{j}\le \mathit{N}$ and ${\mathit{c}}^{-1}\le \mathit{N}{\mathit{s}}_{\mathit{i}\mathit{j}}\le \mathit{c}$ for all $1\le \mathit{i},\mathit{j}\le \mathit{N}$ with some constant $\mathit{c}\ge 1$. We establish Gaussian fluctuations for the linear eigenvalue statistics of ${\mathit{H}}_{\mathit{N}}$ 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.