Let \({\mathcal {A}}\) be the adjacency matrix of the Erdős-Rényi directed graph \({\mathscr {G}}(N,p)\) . We denote the eigenvalues of \({\mathcal {A}}\) by \(\lambda _1^{{\mathcal {A}}},...,\lambda ^{\mathcal {A}}_N\) , with \(|\lambda _1^{{\mathcal {A}}}|=\max _i|\lambda _i^{{\mathcal {A}}}|\) . For \(N^{-1+o(1)}\leqslant p\leqslant 1/2\) , 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 (Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral statistics of erdős-rényi graphs i: local semicircle law. Ann. Prob. 41, 2279–2375 (2013)), (Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral statistics of erdős-rényi graphs ii: eigenvalue spacing and the extreme eigenvalues. Comm. Math. Phys. 314, 587–640 (2012)), (Lee, J.O., Schnelli, K.: Local law and tracy-widom limit for sparse random matrices. Prob. Theor. Rel. Fields 171, 543–616 (2018)), (Huang, J., Landon, B., Yau, H.-T.: Transition from tracy-widom to gaussian fluctuations of extremal eigenvalues of sparse erdős-rényi graphs. Ann. Prob. 48, 916–962 (2020)), (He, Y., Knowles, A.: Fluctuations of extreme eigenvalues of sparse erdős-rényi graphs. Prob. Theor. Rel. Fields 180, 985–1056 (2021)), (Lee, J.: Higher order fluctuations of extremal eigenvalues of sparse random matrices, Preprint arXiv:2108.11634), (Huang, J., Yau, H.T.: Edge universality of sparse random matrices, Preprint arXiv: 2206.06580). Our results imply that, compared to their analogues in the Hermitian case, the edge statistics of non-Hermitian sparse random matrices are more robust.
