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

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 → ∞ .