No Outliers in the Spectrum of the Product of Independent Non-Hermitian Random Matrices with Independent Entries

Yuriy Nemish ()
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Yuriy Nemish: Université de Toulouse

Journal of Theoretical Probability, 2018, vol. 31, issue 1, 402-444

Abstract: Abstract We consider products of independent square random non-Hermitian matrices. More precisely, let $$n\ge 2$$ n ≥ 2 and let $$X_1,\ldots ,X_n$$ X 1 , … , X n be independent $$N\times N$$ N × N random matrices with independent centered entries (either real or complex with independent real and imaginary parts) with variance $$N^{-1}$$ N - 1 . In Götze and Tikhomirov (On the asymptotic spectrum of products of independent random matrices, 2011. arXiv:1012.2710 ) and O’Rourke and Soshnikov (Electron J Probab 16(81):2219–2245, 2011) it was shown that the limit of the empirical spectral distribution of the product $$X_1\cdots X_n$$ X 1 ⋯ X n is supported in the unit disk. We prove that if the entries of the matrices $$X_1,\ldots ,X_n$$ X 1 , … , X n satisfy uniform subexponential decay condition, then the spectral radius of $$X_1\cdots X_n$$ X 1 ⋯ X n converges to 1 almost surely as $$N\rightarrow \infty $$ N → ∞ .

Keywords: Random matrices; Circular law; Stieltjes transform; Outliers; 60B20 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-016-0708-2

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