Quantitative Tracy-Widom laws for the largest eigenvalue of generalized Wigner matrices

TL;DR

This paper proves that the largest eigenvalue of generalized Wigner matrices follows Tracy-Widom laws with a convergence rate close to O(N^{-1/3}), extending previous results to more general variance profiles.

Contribution

It establishes a nearly optimal convergence rate for the Tracy-Widom law for generalized Wigner matrices without requiring second moment matching.

Findings

Convergence rate of nearly O(N^{-1/3}) for the largest eigenvalue.

Applicable to matrices with non-uniform variance profiles.

Improves previous convergence rate results.

Abstract

We show that the fluctuations of the largest eigenvalue of any generalized Wigner matrix $H$ converge to the Tracy-Widom laws at a rate nearly $O(N^{-1/3})$, as the matrix dimension $N$ tends to infinity. We allow the variances of the entries of $H$ to have distinct values but of comparable sizes such that $\sum_{i} \mathbb{E}|h_{ij}|^2=1$. Our result improves the previous rate $O(N^{-2/9})$ by Bourgade [8] and the proof relies on the first long-time Green function comparison theorem near the edges without the second moment matching restriction.