Quantitative CLT for linear eigenvalue statistics of Wigner matrices

Abstract

In this article, we establish a near-optimal convergence rate for the CLT of linear eigenvalue statistics of Wigner matrices, in Kolmogorov-Smirnov distance. For all test functions $f\in C^5(\mathbb{R})$, we show that the convergence rate is either $N^{-1/2+\epsilon }$ or $N^{-1+\epsilon }$, depending on the first Chebyshev coefficient of $f$ and the third moment of the diagonal matrix entries. The condition that distinguishes these two rates is necessary and sufficient. For a general class of test functions, we further identify matching lower bounds for the convergence rates. In addition, we identify an explicit, non-universal contribution in the linear eigenvalue statistics, which is responsible for the slow rate $N^{-1/2+\epsilon }$ for non-Gaussian ensembles. By removing this non-universal part, we show that the shifted linear eigenvalue statistics have the unified convergence rate $N^{-1+\epsilon }$ for all test functions.

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References in corpus (13)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• On Asymptotic Properties of Large Random Matrices with Independent Entries
• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices
• Global spectrum fluctuations for the $\beta$-Hermite and $\beta$-Laguerre ensembles via matrix models
• Second Order Freeness and Fluctuations of Random Matrices, III. Higher order freeness and free cumulants
• Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices
• Gaussian fluctuations for non-Hermitian random matrix ensembles
• Eigenstate Thermalization Hypothesis for Wigner Matrices
• Thermalisation for Wigner matrices
• On fluctuations of eigenvalues of random band matrices
• Fluctuations Of Linear Spectral Statistics Of Deformed Wigner Matrices
• On the Rate of Convergence in the Central Limit Theorem for Linear Statistics of Gaussian, Laguerre, and Jacobi Ensembles

Cited by in corpus (3)

• Speed of convergence in the Central Limit Theorem for the determinantal point process with the Bessel kernel
• Fluctuations of the Stieltjes transform of the empirical spectral distribution of selfadjoint polynomials in Wigner and deterministic diagonal matrices
• Multi-Point Functional Central Limit Theorem for Wigner Matrices