Edge universality of sparse Erd\H{o}s-R\'enyi digraphs

Let $\mathcal A$ be the adjacency matrix of the Erd\H{o}s-R\'{e}nyi directed graph $\mathscr G(N,p)$. We denote the eigenvalues of $\mathcal A$ by $\lambda_1^{\cal A},...,\lambda^{\cal A}_N$, and $|\lambda_1^{\cal A}|=\max_i|\lambda_i^{\cal A}|$. For $N^{-1+o(1)}\leq p\leq 1/2$, we show that $$\max_{i=2,3,...,N} \bigg|\frac{\lambda_i^{\mathcal A}}{\sqrt{Np(1-p)}}\bigg| =1+O(N^{-1/2+o(1)})$$ with very high probability. In addition, we prove that near the unit circle, the local eigenvalue statistics of ${\mathcal A}/\sqrt{Np(1-p)}$ 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.