On the Circular Law

F. Götze, A. Tikhomirov

Published 2007-02-13Version 1

We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real 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 however that the entries have sub-Gaussian tails or are sparsely non-zero.