Elliptic law for real random matrices
Abstract
In this paper we consider ensemble of random matrices $\X_n$ with independent identically distributed vectors $(X_{ij}, X_{ji})_{i \neq j}$ 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 $X_{12}$ and $X_{21}$. This result is called Elliptic Law. Limit distribution doesn't depend on distribution of matrix elements and the result in this sence is universal.
 Publication:

arXiv eprints
 Pub Date:
 January 2012
 arXiv:
 arXiv:1201.1639
 Bibcode:
 2012arXiv1201.1639N
 Keywords:

 Mathematics  Probability;
 Mathematics  Spectral Theory
 EPrint:
 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