Edge universality of sparse Erdős-Rényi digraphs

2023/04/10 by Yukun He, He, Yukun
#math.PR #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.2304.04723

Abstract

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 λ₁^{\cal A},...,λ^{\cal A}_N, and |λ₁^{\cal A}|=max_i|λ_i^{\cal A}|. For N^-1+o(1)≤ p≤ 1/2, we show that max_i=2,3,...,N |\fracλ_i^{\mathcal A}√(Np(1-p))| =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/√(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.

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