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

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) := E vertical bar H-ij vertical bar(2), satisfies Sigma(N)(i=1) s(ij) = 1, for all 1 <= j <= N and c(-1) <= Ns(ij) <= c for all 1 <= i, j <= N with some constant c >= 1. We establish Gaussian fluctuations for the linear eigenvalue statistics of HN 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.

Automated summary

We consider an N by N real or complex generalized Wigner matrix with independent centered random variables. Gaussian fluctuations for linear eigenvalue statistics are established on global scales and mesoscopic scales, up to the spectral edges, with universal mesoscopic central limit theorems obtained for statistics inside the bulk and at the edges.