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 \(n\ge 2\) and let \(X_1,\ldots ,X_n\) be independent \(N\times N\) random matrices with independent centered entries (either real or complex with independent real and imaginary parts) with variance \(N^{-1}\). 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 \(X_1\cdots X_n\) is supported in the unit disk. We prove that if the entries of the matrices \(X_1,\ldots ,X_n\) satisfy uniform subexponential decay condition, then the spectral radius of \(X_1\cdots X_n\) converges to 1 almost surely as \(N\rightarrow \infty \).