Kevin Schnelli: Quantitative Tracy-Widom law for Wigner random matrices

Abstract

We will discuss a quantitative Tracy-Widom law for the largest eigenvalue of Wigner random matrices as well as sample covariance matrices. More precisely, we will prove that the fluctuations of the largest eigenvalue of a generalized Wigner matrix of size N converge to its Tracy-Widom limit at a rate nearly N^{-1/3}, as N tends to infinity. Our result follows from a quantitative Green function comparison theorem, originally introduced by Erdos, Yau and Yin to prove the edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function interpolation with the Gaussian invariant ensembles. Precise estimates on leading contributions from the second, third and fourth order moments of the matrix entries are obtained using iterative cumulant expansions and comparisons of correlation functions, along with uniform convergence estimates for correlation kernels of the Gaussian invariant ensembles. This is joint work with Yuanyuan Xu (IST Austria)