Elliptic law for real random matrices
arXiv:1201.1639
Abstract
In this paper we consider ensemble of random matrices $\X_n$ with independent identically distributed vectors of entries. Under assumption of finite fourth moment of matrix entries it is proved that empirical spectral distribution of eigenvalues converges in probability to a uniform distribution on the ellipse. The axis of the ellipse are determined by correlation between and . This result is called Elliptic Law. Limit distribution doesn't depend on distribution of matrix elements and the result in this sence is universal.
Submitted for publication in Vestnik Moskovskogo Universiteta. Vychislitel'naya Matematika i Kibernetika. Paper contains 30 pages, 4 figures. Several misprints were corrected. Introduction and some proofs were rewritten. It is the final version