Small deviation estimates for the largest eigenvalue of Wigner matrices

László Erdős, Yuanyuan Xu

Published 2021-12-22, updated 2022-03-31Version 2

We establish precise right-tail small deviation estimates for the largest eigenvalue of real symmetric and complex Hermitian matrices whose entries are independent random variables with uniformly bounded moments. The proof relies on a Green function comparison along a continuous interpolating matrix flow for a long time. Less precise estimates are also obtained in the left tail.