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

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 ${\bf X}^{(q)}, 1 \le j,k \le n, q = 1, \ldots, m, m \geq 1$ 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.