Local elliptic law

TL;DR

This paper proves that the eigenvalue distribution of elliptic random matrices converges to a uniform measure on an ellipse at mesoscopic scales, and shows eigenvector delocalization.

Contribution

It establishes optimal convergence rates for the eigenvalue distribution and demonstrates eigenvector delocalization in elliptic random matrices.

Findings

Eigenvalue distribution converges to uniform measure on an ellipse

Optimal convergence rate on mesoscopic scales

Eigenvectors are completely delocalized

Abstract

The empirical eigenvalue distribution of the elliptic random matrix ensemble tends to the uniform measure on an ellipse in the complex plane as its dimension tends to infinity. We show this convergence on all mesoscopic scales slightly above the typical eigenvalue spacing in the bulk spectrum with an optimal convergence rate. As a corollary we obtain complete delocalisation for the corresponding eigenvectors in any basis.