PUB - Publikationen an der Universität Bielefeld

We consider products of independent n x n non-Hermitian random matrices X-(1), ..., X-(m). Assume that their entries, Xj(jk)((q)), 1 <= j, k <= n, q = 1, ..., m, are independent identically distributed random variables with zero mean, unit variance. Gotze and Tikhomirov [On the asymptotic spectrum of products of independent random matrices, preprint (2010), arXiv:1012.2710] and O'Rourke and Sochnikov [Products of independent non-Hermitian random matrices, Electron. J. Probab. 16 (2011) 2219-2245] proved that under these assumptions the empirical spectral distribution (ESD) of X-(1) ... X-(m) converges to the limiting distribution which coincides with the distribution of the mth power of random variable uniformly distributed in the unit circle. In this paper, we provide a local version of this result. More precisely, assuming additionally that E vertical bar X-11((q))vertical bar(4+delta) < infinity for some delta > 0, we prove that ESD of X-(1) ... X-(m) converges to the limiting distribution on the optimal scale up to n(-1+2a) , 0 < a < 1/2 (up to some logarithmic factor). Our results generalize the recent results of Bourgade et al. [Local circular law for random matrices, Probab. Theory Related Fields 159 (2014) 545-595], Tao and Vu [Random matrices: Universality of local spectral statistics of non-Hermitian matrices, Ann. Probab. 43 (2015) 782-874] and Nemish [Local law for the product of independent non-hermitian random matrices with independent entries, Electron. J. Probab. 22 (2017) 1-35]. We also give further development of Stein's type approach to estimate the Stieltjes transform of ESD.