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June 2012 Universality and the circular law for sparse random matrices
Philip Matchett Wood
Ann. Appl. Probab. 22(3): 1266-1300 (June 2012). DOI: 10.1214/11-AAP789


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.


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Philip Matchett Wood. "Universality and the circular law for sparse random matrices." Ann. Appl. Probab. 22 (3) 1266 - 1300, June 2012.


Published: June 2012
First available in Project Euclid: 18 May 2012

zbMATH: 1250.15037
MathSciNet: MR2977992
Digital Object Identifier: 10.1214/11-AAP789

Primary: 15A52

Keywords: circular law , Random matrix , sparse matrix

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.22 • No. 3 • June 2012
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