Electronic Journal of Probability

Universality of the least singular value for sparse random matrices

Ziliang Che and Patrick Lopatto

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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 Erdős–Rényi 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.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 9, 53 pp.

Received: 16 April 2018
Accepted: 22 January 2019
First available in Project Euclid: 15 February 2019

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Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

random matrix theory sparse universality singular value

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Che, Ziliang; Lopatto, Patrick. Universality of the least singular value for sparse random matrices. Electron. J. Probab. 24 (2019), paper no. 9, 53 pp. doi:10.1214/19-EJP269. https://projecteuclid.org/euclid.ejp/1550221265

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