@TECHREPORT{Götze_thecircular, author = {F. Götze and A. Tikhomirov}, title = {The Circular Law for Random Matrices}, institution = {}, year = {} }

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Abstract

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 that the entries have moment of order E |Xjk | 2 ϕ(x), with some positive function ϕ(x) which is growing of order (log(1 + |x|)) 7, or that they are sparsely nonzero. The results are based on and extend previous work of Bai, Rudelson and the authors. 1