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

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Cited by in corpus (11)

• Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes
• Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices
• Mesoscopic linear statistics of Wigner matrices of mixed symmetry class
• On fluctuations of global and mesoscopic linear eigenvalue statistics of generalized Wigner matrices
• Global fluctuations for 1D log-gas dynamics
• Mesoscopic eigenvalue statistics of Wigner matrices
• Central Limit Theorem for Linear Eigenvalue Statistics for Submatrices of Wigner Random Matrices
• Pair Dependent Linear Statistics for Circular Beta Ensemble
• Fluctuations of the Stieltjes transform of the empirical spectral distribution of selfadjoint polynomials in Wigner and deterministic diagonal matrices
• On Cramér-von Mises statistic for the spectral distribution of random matrices
• Central limit theorems for the real eigenvalues of large Gaussian random matrices