Mesoscopic linear statistics of Wigner matrices

TL;DR

This paper investigates the behavior of linear spectral statistics of Wigner matrices on mesoscopic scales, showing convergence to a Gaussian process and explicitly characterizing its covariance structure.

Contribution

It establishes the convergence of mesoscopic linear spectral statistics of Wigner matrices to a Gaussian process with a detailed covariance structure, extending previous results.

Findings

Trace of the resolvent converges to a Gaussian process

Explicit covariance structure related to fractional Brownian motion

Limiting covariance given by the $H^{1/2}$-norm of test functions

Abstract

We study linear spectral statistics of $N \times N$ Wigner random matrices $\mathcal{H}$ on mesoscopic scales. Under mild assumptions on the matrix entries of $\mathcal{H}$, we prove that after centering and normalizing, the trace of the resolvent $\mathrm{Tr}(\mathcal{H}-z)^{-1}$ converges to a stationary Gaussian process as $N \to \infty$ on scales $N^{-1/3} \ll \mathrm{Im}(z) \ll 1$ and explicitly compute the covariance structure. The limit process is related to certain regularizations of fractional Brownian motion and logarithmically correlated fields appearing in \cite{FKS13}. Finally, we extend our results to general mesoscopic linear statistics and prove that the limiting covariance is given by the $H^{1/2}$-norm of the test functions.