Random matrices: Universality of local eigenvalue statistics up to the edge

TL;DR

This paper extends the universality of local eigenvalue statistics for Wigner matrices from the bulk to the spectral edge, including the largest eigenvalues, under moment conditions.

Contribution

It generalizes previous universality results to the spectral edge and introduces a new technical approach involving bias in the Cauchy interlacing law.

Findings

Universality holds at the spectral edge for Wigner matrices.

Largest eigenvalues follow universal distribution under moment conditions.

Eigenvector delocalization is maintained near the spectrum edge.

Abstract

This is a continuation of our earlier paper on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in that paper from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors.