On Local laws for non-Hermitian random matrices and their products

Abstract

The aim of this paper is to prove a local version of the circular law for non-Hermitian random matrices and its generalization to the product of non-Hermitian random matrices under weak moment conditions. More precisely we assume that the entries $X_{jk}^{(q)}$ of non-Hermitian random matrices ${\mathbf{X}}^{(q)},1\le j,k\le n,q=1,\dots ,m,m\ge 1$ are i.i.d. r.v. with $\mathbb{E}{X}_{jk}=0,\mathbb{E}{X}_{jk}^2=1$ and $\mathbb{E}|X_{jk}{|}^{4+\delta }<\infty$ for some $\delta >0$. It is shown that the local law holds on the optimal scale $n^{-1+2a},a>0$, up to some logarithmic factor. We further develop a Stein type method to estimate the perturbation of the equations for the Stieltjes transform of the limiting distribution. We also generalize the recent results [Bourgade--Yau-Yin, 2014], [Tao--Vu, 2015] and [Nemish, 2017]. An extension to the case of non-i.i.d. entries is discussed.

38 pages