The circular law for random matrices

Friedrich Götze, Alexander Tikhomirov

Published 2007-09-25, updated 2010-10-18Version 4

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.