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July 2019 The circular law for sparse non-Hermitian matrices
Anirban Basak, Mark Rudelson
Ann. Probab. 47(4): 2359-2416 (July 2019). DOI: 10.1214/18-AOP1310

Abstract

For a class of sparse random matrices of the form $A_{n}=(\xi_{i,j}\delta_{i,j})_{i,j=1}^{n}$, where $\{\xi_{i,j}\}$ are i.i.d. centered sub-Gaussian random variables of unit variance, and $\{\delta_{i,j}\}$ are i.i.d. Bernoulli random variables taking value $1$ with probability $p_{n}$, we prove that the empirical spectral distribution of $A_{n}/\sqrt{np_{n}}$ converges weakly to the circular law, in probability, for all $p_{n}$ such that $p_{n}=\omega({\log^{2}n}/{n})$. Additionally if $p_{n}$ satisfies the inequality $np_{n}>\exp(c\sqrt{\log n})$ for some constant $c$, then the above convergence is shown to hold almost surely. The key to this is a new bound on the smallest singular value of complex shifts of real valued sparse random matrices. The circular law limit also extends to the adjacency matrix of a directed Erdős–Rényi graph with edge connectivity probability $p_{n}$.

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Anirban Basak. Mark Rudelson. "The circular law for sparse non-Hermitian matrices." Ann. Probab. 47 (4) 2359 - 2416, July 2019. https://doi.org/10.1214/18-AOP1310

Information

Received: 1 July 2017; Revised: 1 June 2018; Published: July 2019
First available in Project Euclid: 4 July 2019

zbMATH: 07114719
MathSciNet: MR3980923
Digital Object Identifier: 10.1214/18-AOP1310

Subjects:
Primary: 15B52 , 60B10 , 60B20

Keywords: circular law , Random matrix , smallest singular value , sparse matrix

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 4 • July 2019
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