No outliers in the spectrum of the product of independent non-Hermitian random matrices with independent entries

2014/12/07 by Nemish, Yuriy
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1412.2410

Abstract

We consider products of independent square random non-Hermitian matrices. More precisely, let n≥ 2 and let X₁,…,X_n be independent N× N random matrices with independent centered entries with variance N^-1. It was shown by Götze and Tikhomirov and by Soshnikov and O'Rourke that the limit of the empirical spectral distribution of the product X₁⋯ X_n is supported in the unit disk. We prove that if the entries of the matrices X₁,…,X_n satisfy uniform subexponential decay condition, then the spectral radius of X₁⋯ X_n converges to 1 almost surely as N→ ∞.

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