The spectral edge of sparse directed Erdös-Rényi graphs

Let $d>1$ be fixed and let $A_n$ be an $n\times n$ matrix with independent $\text{Ber}(d\mathrm{/}n)$ entries. For every $0<r<\sqrt{d}$, we prove that, with high probability, a positive proportion of the eigenvalues of $A_n$ have modulus larger than $r$. Together with the known upper bound, this implies that the modulus of the second largest eigenvalue converges in probability to $\sqrt{d}$. Our proof works with the Brown measure ${\mu }_d$ of the adjacency operator of the directed Poisson–Galton–Watson tree. The convergence theorem of Sah, Sahasrabudhe, and Sawhney, together with the Brown-measure identification following it, gives weak convergence of the empirical spectral measure of $A_n$ to ${\mu }_d$ in probability. We prove that the outer radius of its support is $\sqrt{d}$, using a resolvent recursion on Poisson–Galton–Watson trees.