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 $X_1,\dots ,X_n$ be independent $N\times N$randommatriceswithindependentcenteredentrieswithvariance$N^{-1}$. It was shown by G"otze and Tikhomirov and by Soshnikov and O’Rourke 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 $X_1,\dots ,X_n$ satisfy uniform subexponential decay condition, then the spectral radius of $X_1\cdots X_n$ converges to 1 almost surely as $N\to \mathrm{\infty }$.