Edge universality of sparse Erdős-Rényi digraphs

Let $\mathcal{A}$ be the adjacency matrix of the Erdős-Rényi directed graph $\mathcal{G}(N,p)$. We denote the eigenvalues of $\mathcal{A}$ by ${\lambda }_1^{\mathcal{A}},\mathrm{.}\mathrm{.}\mathrm{.},{\lambda }_N^{\mathcal{A}}$, and $\mathrm{\mid }{\lambda }_1^{\mathcal{A}}\mathrm{\mid }={\max }_i\mathrm{\mid }{\lambda }_i^{\mathcal{A}}\mathrm{\mid }$. For $N^{-1+o(1)}\le p\le 1\mathrm{/}2$, we show that $$\underset{i=2,3,\mathrm{.}\mathrm{.}\mathrm{.},N}{\max }\mid \frac{{\lambda }_i^{\mathcal{A}}}{\sqrt{Np(1-p)}}\mid =1+O(N^{-1\mathrm{/}2+o(1)})$$ with very high probability. In addition, we prove that near the unit circle, the local eigenvalue statistics of $\mathcal{A}\mathrm{/}\sqrt{Np(1-p)}$ coincide with those of the real Ginibre ensemble. As a by-product, we also show that all non-trivial eigenvectors of $\mathcal{A}$ are completely delocalized. For Hermitian random matrices, it is known that the edge statistics are sensitive to the sparsity: in the very sparse regime, one needs to remove many noise random variables (which affect both the mean and the fluctuation) to recover the Tracy-Widom distribution. Our results imply that, compared to their analogues in the Hermitian case, the edge statistics of non-Hermitian sparse random matrices are more robust.