Elliptic law for real random matrices

Alexey Naumov

Elliptic law for real random matrices

Probability 2012-08-07 v5 Spectral Theory

Authors: Alexey Naumov

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.

Keywords

random matrix theory limit theorems probability theory

Cite

@article{arxiv.1201.1639,
  title  = {Elliptic law for real random matrices},
  author = {Alexey Naumov},
  journal= {arXiv preprint arXiv:1201.1639},
  year   = {2012}
}

Comments

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

Elliptic law for real random matrices

Abstract

Keywords

Cite

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