We study the distribution of the least singular value associated to an ensemble of sparse random matrices. Our motivating example is the ensemble of N × N matrices whose entries are chosen independently from a Bernoulli distribution with parameter p. These matrices represent the adjacency matrices of random Erdős-Rényi digraphs and are sparse when p ≪ 1. We prove that in the regime pN ≫ 1, the distribution of the least singular value is universal in the sense that it is independent of p and equal to the distribution of the least singular value of a Gaussian matrix ensemble. We also prove the universality of the joint distribution of multiple small singular values. Our methods extend to matrix ensembles whose entries are chosen from arbitrary distributions that may be correlated, complex valued, and have unequal variances.

We establish eigenvector delocalization and bulk universality for Lévy matrices, which are real, symmetric, N \times N random matrices \mathbf{H} whose upper triangular entries are independent, identically distributed \alpha -stable laws. First, if \alpha\in(1,2) and E\in\mathbb{R} is bounded away from 0, we show that every eigenvector of \mathbf{H} corresponding to an eigenvalue near E is completely delocalized and that the local spectral statistics of \mathbf{H} around E converge to those of the Gaussian Orthogonal Ensemble as N tends to \infty . Second, we show for almost all \alpha\in(0,2) , there exists a constant c(\alpha)>0 such that the same statements hold if |E|<c(\alpha) .

No abstract

In this paper we establish eigenvector delocalization and bulk universality for Lévy matrices, which are real, symmetric, N × N random matrices H whose upper triangular entries are independent, identically distributed α-stable laws. First, if α ∈ (1, 2) and E ∈ R is any energy bounded away from 0, we show that every eigenvector of H corresponding to an eigenvalue near E is completely delocalized and that the local spectral statistics of H around E converge to those of the Gaussian Orthogonal Ensemble (GOE) as N tends to ∞. Second, we show for almost all α ∈ (0, 2), there exists a c(α) > 0 such that the same statements hold if |E| < c(α).

We prove the first eigenvalue repulsion bound for sparse random matrices. As a consequence, we show that these matrices have simple spectrum, improving the range of sparsity and error probability from work of the second author and Vu. We also show that for sparse Erdős-Rényi graphs, weak and strong nodal domains are the same, answering a question of Dekel, Lee, and Linial.