Spectral radii of sparse non-Hermitian random matrices

TL;DR

This paper estimates the spectral radii of sparse non-Hermitian random matrices with fixed average degree, showing that the spectral radius converges to a nonzero value with high probability and providing bounds involving diverging functions.

Contribution

It introduces new bounds for the spectral radius of sparse non-Hermitian matrices using structural results, extending understanding of their spectral properties.

Findings

Spectral radius converges to a nonzero value with high probability.

Bounds on spectral radius involve functions diverging to infinity.

Provides asymptotic estimates for spectral radii in the sparse regime.

Abstract

We provide estimates for the spectral radii of an $n\times n$ sparse non-Hermitian random matrix $Z$ with general entries in the regime $p=d/n$ where $0<d<1$ is fixed. Utilizing the structural results of ({\L}uczak, '90), we show that the spectral radius $\rho (Z)$ is $0$ with probability converging to some nonzero value, and satisfies the inequality $(\phi (n))^{-1}\leq \rho (Z)\leq \phi (n)$ in the asymptotic sense for any function $\phi$ satisfying $\lim_{n\to\infty}\phi (n)=\infty$ with the remaining probability.