Universality of the least singular value for sparse random matrices

TL;DR

This paper proves that the distribution of the smallest singular value of sparse random matrices, such as Erd"H{o}s--Rényi adjacency matrices, is universal and matches that of Gaussian matrices when the average degree is sufficiently large.

Contribution

It establishes the universality of the least singular value distribution for sparse matrices, extending previous results to more general, possibly correlated and complex-valued entries.

Findings

Least singular value distribution matches Gaussian ensemble in sparse regime

Universality holds for joint distribution of multiple small singular values

Results extend to matrices with correlated, complex, and variably scaled entries

Abstract

We study the distribution of the least singular value associated to an ensemble of sparse random matrices. Our motivating example is the ensemble of $N\times N$ matrices whose entries are chosen independently from a Bernoulli distribution with parameter $p$. These matrices represent the adjacency matrices of random Erd\H{o}s--R\'enyi digraphs and are sparse when $p\ll 1$. We prove that in the regime $pN\gg 1$, 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.