Universality and the circular law for sparse random matrices

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 $1/n^{1-\alpha}$ where $0<\alpha\le1$ 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.Comment: Published in at http://dx.doi.org/10.1214/11-AAP789 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Some Remarks on the Dozier–silverstein Theorem for Random Matrices With Dependent Entries

“…The above corollary generalizes Lemma 5.5 in [29] where the case of δ n = n −1+α with 0 < α ≤ 1 was considered. See also [12,22] for related models of sparse random matrices.…”

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Spectral Theory for Dynamics on Graphs Containing Attractive and Repulsive Interactions

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“…This means that for any , if is the corresponding random matrix, then the associated random graph follows the uniform distribution on . In some sense, this random graph model lies between the oriented Erdős‐Rényi random graph model , and the uniform random oriented regular graph model . Suppose now that m = m n depends on n and set .…”