Edge universality of sparse Erdős-Rényi digraphs

Abstract

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 $\hat{I}{»}_1^{\mathcal{A}},...,\hat{I}{»}_N^{\mathcal{A}}$, and $|\hat{I}{»}_1^{\mathcal{A}}|={\max }_i|\hat{I}{»}_i^{\mathcal{A}}|$. For $N^{-1+o(1)}\le p\le 1/2$, we show that \[ \max_{i=2,3,...,N} \bigg|\frac{λ_i^{\mathcal A}}{\sqrt{Np(1-p)}}\bigg| =1+O(N^{-1/2+o(1)}) \] 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.

42 pages, 4 diagrams