Eigenvalue variance bounds for Wigner and covariance random matrices

TL;DR

This paper establishes finite range bounds on the variance of individual eigenvalues of Wigner and covariance matrices, extending results from GUE to broader classes using advanced probabilistic tools.

Contribution

It introduces new variance bounds for eigenvalues of Wigner and covariance matrices, utilizing the Four Moment Theorem and localization results to generalize from GUE.

Findings

Variance bounds for eigenvalues in the bulk and at the edge

Bounds on 2-Wasserstein distance between spectral measure and semicircle law

Extension of results to real Wigner matrices

Abstract

This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example, which needs to be investigated first, the main bounds are extended to families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erd\"os, Yau and Yin. The case of real Wigner matrices is obtained from interlacing formulas. As an application, bounds on the expected 2-Wasserstein distance between the empirical spectral measure and the semicircle law are derived. Similar results are available for random covariance matrices.