Mesoscopic linear statistics of Wigner matrices

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.

32 pages

References in corpus (5)

• Local semicircle law and complete delocalization for Wigner random matrices
• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• Stochastic calculus for fractional Brownian motion with Hurst exponent $H>1/4$: A rough path method by analytic extension
• Fixed energy universality for generalized Wigner matrices
• Central limit theorem for linear eigenvalue statistics of elliptic random matrices