The Annals of Probability

The circular law for sparse non-Hermitian matrices

Anirban Basak and Mark Rudelson

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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 $p_{n}$.

Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 2359-2416.

Dates
Received: July 2017
Revised: June 2018
First available in Project Euclid: 4 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1562205711

Digital Object Identifier
doi:10.1214/18-AOP1310

Mathematical Reviews number (MathSciNet)
MR3980923

Subjects
Primary: 15B52: Random matrices 60B10: Convergence of probability measures 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
Random matrix sparse matrix smallest singular value circular law

Citation

Basak, Anirban; Rudelson, Mark. The circular law for sparse non-Hermitian matrices. Ann. Probab. 47 (2019), no. 4, 2359--2416. doi:10.1214/18-AOP1310. https://projecteuclid.org/euclid.aop/1562205711


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