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.
Highlights
We study the distribution of the least singular value associated to an ensemble of sparse random matrices
This paper studies the universality of the least singular value from the same dynamical viewpoint that was used to prove the Wigner–Dyson–Mehta conjecture
We prove that for a sparse matrix, the least singular value is universal after time t = N −1+ε when the singular values are evolved according to the singular value analogue of Dyson Brownian motion
Summary
Random real symmetric and complex Hermitian matrices have been intensely studied since Wigner’s discovery that, in the large N limit, their eigenvalue densities are universal and follow the semicircle distribution. Given in [37] are analogous results for complex matrices and for the joint distribution of multiple smallest singular values These results require that the entries are independent and have equal variances. 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 Erdos–Rényi digraphs, which are directed graphs on N vertices where each possible directed edge is present with probability p. Such matrices are sparse when p 1, and our result implies that the distribution of the least singular value is universal in the regime pN 1. An essential technical input is showing that these dynamics, which govern the evolution of the singular value distribution, reach local equilibrium after short times t N −1
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