Spectral radii of sparse non-Hermitian random matrices

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.