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• W2972939003 abstract "We consider products of independent [Formula: see text] non-Hermitian random matrices [Formula: see text]. Assume that their entries, [Formula: see text], are independent identically distributed random variables with zero mean, unit variance. Götze 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 [Formula: see text] converges to the limiting distribution which coincides with the distribution of the [Formula: see text]th 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 [Formula: see text] for some [Formula: see text], we prove that ESD of [Formula: see text] converges to the limiting distribution on the optimal scale up to [Formula: see text] (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." @default.
• W2972939003 created "2019-09-19" @default.
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• W2972939003 date "2019-11-15" @default.
• W2972939003 modified "2023-10-14" @default.
• W2972939003 title "Local laws for non-Hermitian random matrices and their products" @default.
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• W2972939003 doi "https://doi.org/10.1142/s2010326321500040" @default.
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