Universality and the circular law for sparse random matrices

TL;DR

This paper proves a universality principle for sparse random matrices, showing that their eigenvalue distribution follows the circular law under broad conditions, extending previous results to sparser matrices.

Contribution

It establishes the circular law for sparse matrices with entries that are zero with high probability, generalizing universality results to a wider class of sparse matrices.

Findings

Eigenvalue distribution follows the circular law for sparse matrices

Universality holds for matrices with entries zero with probability $1/n^{1-eta}$

Extends the most general sparse matrix universality results to date

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.