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Mesoscopic linear statistics of Wigner matrices (1503.03533v1)

Published 11 Mar 2015 in math.PR

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.

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