Mesoscopic linear statistics of Wigner matrices

2015/03/11 by Asad Lodhia, Lodhia, A., Nick Simm +1
Mathematics · #60B20 #Advanced Algebra and Geometry #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1503.03533

openalex publication_date 2015/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study linear spectral statistics of N × N Wigner random matrices H on mesoscopic scales. Under mild assumptions on the matrix entries of H, we prove that after centering and normalizing, the trace of the resolvent Tr(H-z)^-1 converges to a stationary Gaussian process as N → ∞ on scales N^-1/3 ≪ Im(z) ≪ 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 \citeFKS13. 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.

Citations

• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices
• Central limit theorem for linear eigenvalue statistics of elliptic\n random matrices
• Fractional Gaussian fields: a survey
• The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos - The L²-phase

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