We consider the joint distribution of real and imaginary parts of eigenvalues
of random matrices with independent entries with mean zero and unit variance.
We prove the convergence of this distribution to the uniform distribution on
the unit disc without assumptions on the existence of a density for the
distribution of entries. We assume that the entries have a finite moment of
order larger than two and consider the case of sparse matrices. The results are
based on previous work of Bai, Rudelson and the authors extending those results
to a larger class of sparse matrices.Comment: Published in at http://dx.doi.org/10.1214/09-AOP522 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
We consider powers of random matrices with independent entries. Let X ij , i, j ≥ 1, be independent complex random variables with E X ij = 0 and E |X ij | 2 = 1 and let X denote an n×n matrix with [X] ij = X ij , for 1 ≤ i, j ≤ n. Denote by s (m) 1
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.