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2019 Universality of the least singular value for sparse random matrices
Ziliang Che, Patrick Lopatto
Electron. J. Probab. 24(none): 1-53 (2019). DOI: 10.1214/19-EJP269

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 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.

Citation

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Ziliang Che. Patrick Lopatto. "Universality of the least singular value for sparse random matrices." Electron. J. Probab. 24 1 - 53, 2019. https://doi.org/10.1214/19-EJP269

Information

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

zbMATH: 1412.60015
MathSciNet: MR3916329
Digital Object Identifier: 10.1214/19-EJP269

Subjects:
Primary: 60B20

JOURNAL ARTICLE
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