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

TL;DR

This paper proves that for products of independent non-Hermitian random matrices with subexponentially decaying entries, the spectral radius converges to 1 almost surely as the matrix size grows, confirming no outliers outside the unit disk.

Contribution

It establishes almost sure convergence of the spectral radius to 1 for such matrix products under subexponential decay conditions, extending previous spectral distribution results.

Findings

Spectral radius converges to 1 almost surely as matrix size increases.

No outliers in the spectrum outside the unit disk.

Supports the universality of spectral behavior for these matrices.

Abstract

We consider products of independent square random non-Hermitian matrices. More precisely, let $n\geq 2$ and let $X_1,\ldots,X_n$ be independent $N\times N$ random matrices with independent centered entries with variance $N^{-1}$. 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 $X_1,\ldots,X_n$ satisfy uniform subexponential decay condition, then the spectral radius of $X_1\cdots X_n$ converges to 1 almost surely as $N\rightarrow \infty$.