# Elliptic law for real random matrices

@article{Naumov2012EllipticLF, title={Elliptic law for real random matrices}, author={Alexey Naumov}, journal={arXiv: Probability}, year={2012} }

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…

## 26 Citations

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A number of interesting results concerning elliptic random matrices whose entries have finite fourth moment are proved; these results include a bound on the least singular value and the asymptotic behavior of the spectral radius.

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We show that, under some general assumptions on the entries of a random complex $n \times n$ matrix $X_n$, the empirical spectral distribution of $\frac{1}{\sqrt{n}} X_n$ converges to the uniform law…

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