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 ${\sum }_i\mathbb{E}|h_{ij}{|}^2=1$. 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.