Optimal Delocalization for Generalized Wigner Matrices

TL;DR

This paper establishes optimal delocalization rates for eigenvectors of generalized Wigner matrices with subexponential entries, using advanced flow analysis and comparison techniques, and also provides new spectral estimates.

Contribution

It introduces a novel approach combining eigenvector moment flow analysis with comparison methods to prove optimal delocalization bounds for generalized Wigner matrices.

Findings

Eigenvectors delocalize at optimal rate with high probability.

Sharp constants achieved in delocalization bounds.

Level repulsion and eigenvalue overcrowding estimates established.

Abstract

We study the eigenvectors of generalized Wigner matrices with subexponential entries and prove that they delocalize at the optimal rate with overwhelming probability. We also prove high probability delocalization bounds with sharp constants. Our proof uses an analysis of the eigenvector moment flow introduced by Bourgade and Yau (2017) to bound logarithmic moments of eigenvector entries for random matrices with small Gaussian components. We then extend this control to all generalized Wigner matrices by comparison arguments based on a framework of regularized eigenvectors, level repulsion, and the observable employed by Landon, Lopatto, and Marcinek (2018) to compare extremal eigenvalue statistics. Additionally, we prove level repulsion and eigenvalue overcrowding estimates for the entire spectrum, which may be of independent interest.