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

Random Matrices-Theory and Applications. 2020. Vol. 9. No. 4. P. 2150004.
Naumov A., Tikhomirov A., Гётце Ф.

We consider products of independent $$n \times n$$ non-Hermitian random matrices $$\X^{(1)}, \ldots, \X^{(m)}$$. Assume that their entries, $$X_{jk}^{(q)}, 1 \le j,k \le n, q = 1, \ldots, m$$, are independent identically distributed random variables with zero mean, unit variance. G\"otze -- Tikhomirov~\cite{GotTikh2011} and O'Rourke--Sochnikov~\cite{Soshnikov2011} proved that under these assumptions the empirical spectral distribution (ESD) of $$X^{(1)} \cdots X^{(m)}$$ converges to the limiting distribution which coincides with the distribution of the $$m$$-th power of random variable uniformly distributed in the unit circle.  In the current paper we provide a local vesion of this result. More precisely, assuming additionally that  $$\E |X_{11}^{(q)}|^{4+\delta} < \infty$$ for some $$\delta > 0$$, we prove that ESD of $$X^{(1)} \cdots X^{(m)}$$ 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--Yau--Yin~\cite{Bourgade2014a}, Tao--Vu~\cite{TaoVu2015a} and Nemish~\cite{nemish2017}.  We also give further development of Stein's type approach to estimate the the Stieltjes transform of ESD.