Spectral radii of sparse non-Hermitian random matrices

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