The Annals of Applied Probability

Universality and the circular law for sparse random matrices

Philip Matchett Wood

Full-text: Open access

Abstract

The universality phenomenon asserts that the distribution of the eigenvalues of random matrix with i.i.d. zero mean, unit variance entries does not depend on the underlying structure of the random entries. For example, a plot of the eigenvalues of a random sign matrix, where each entry is +1 or −1 with equal probability, looks the same as an analogous plot of the eigenvalues of a random matrix where each entry is complex Gaussian with zero mean and unit variance. In the current paper, we prove a universality result for sparse random n by n matrices where each entry is nonzero with probability 1/n1−α where 0 < α ≤ 1 is any constant. One consequence of the sparse universality principle is that the circular law holds for sparse random matrices so long as the entries have zero mean and unit variance, which is the most general result for sparse random matrices to date.

Article information

Source
Ann. Appl. Probab., Volume 22, Number 3 (2012), 1266-1300.

Dates
First available in Project Euclid: 18 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1337347545

Digital Object Identifier
doi:10.1214/11-AAP789

Mathematical Reviews number (MathSciNet)
MR2977992

Zentralblatt MATH identifier
1250.15037

Subjects
Primary: 15A52

Keywords
Random matrix sparse matrix circular law

Citation

Wood, Philip Matchett. Universality and the circular law for sparse random matrices. Ann. Appl. Probab. 22 (2012), no. 3, 1266--1300. doi:10.1214/11-AAP789. https://projecteuclid.org/euclid.aoap/1337347545


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