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 $\mathit{N}\times \mathit{N}$ Wigner matrices, in Kolmogorov–Smirnov distance. For all test functions $\mathit{f}\in {\mathit{C}}^5(\mathbb{R})$, we show that the convergence rate is either ${\mathit{N}}^{-1/2\mathbf{+}\mathit{\epsilon }}$ or ${\mathit{N}}^{-1\mathbf{+}\mathit{\epsilon }}$, 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, nonuniversal contribution in the linear eigenvalue statistics, which is responsible for the slow rate ${\mathit{N}}^{-1/2\mathbf{+}\mathit{\epsilon }}$ for non-Gaussian ensembles. By removing this nonuniversal part, we show that the shifted linear eigenvalue statistics have the unified convergence rate ${\mathit{N}}^{-1\mathbf{+}\mathit{\epsilon }}$ for all test functions.