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

Understand

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.