Random matrix theory: Local laws and applications

Abstract

In this article, we provide a review of some fundamental theories concerning the local laws of Green's functions of high-dimensional sample covariance matrices, along with their pertinent applications. Employing a cumulant expansion method, we establish an almost sharp local law for the Green's function of a sample covariance matrix ensemble, valid down to the optimal scale of the spectral parameter. We further discuss its applications to the proof of the Tracy-Widom law of the largest eigenvalue and the study of spiked sample covariance matrices.