The local relaxation flow approach to universality of the local statistics for random matrices

TL;DR

This paper extends the local relaxation flow method to prove that the local spectral statistics of broad classes of large random matrices match those of Gaussian ensembles, under mild conditions on eigenvalue locations.

Contribution

It generalizes the local relaxation flow approach to establish universality for a wide class of random matrices with minimal eigenvalue location assumptions.

Findings

Proves local spectral universality for Wishart matrices.

Shows eigenvalue distributions match Gaussian ensembles under smoothness conditions.

Requires only average eigenvalue location estimates for universality proof.

Abstract

We present a generalization of the method of the local relaxation flow to establish the universality of local spectral statistics of a broad class of large random matrices. We show that the local distribution of the eigenvalues coincides with the local statistics of the corresponding Gaussian ensemble provided the distribution of the individual matrix element is smooth and the eigenvalues ${x_j}_{j=1}^N$ are close to their classical location ${\gamma_j}_{j=1}^N$ determined by the limiting density of eigenvalues. Under the scaling where the typical distance between neighboring eigenvalues is of order 1/N, the necessary apriori estimate on the location of eigenvalues requires only to know that $\E |x_j - \gamma_j |^2 \le N^{-1-\e}$ on average. This information can be obtained by well established methods for various matrix ensembles. We demonstrate the method by proving local spectral universality for Wishart matrices.