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

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 i 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.