2017/08/23 by Götze, Friedrich, Naumov, Alexey, Tikhomirov, Alexander
#FOS: Mathematics #Probability (math.PR) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1708.06950
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 Xjk(q) of non-Hermitian random matrices \bf X(q), 1 ≤ j,k ≤ n, q = 1, …, m, m ≥ 1 are i.i.d. r.v. with \mathbb E Xjk =0, \mathbb E Xjk2 = 1 and \mathbb E |Xjk|4+δ < ∞ for some δ> 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.