On one generalization of the elliptic law for random matrices

TL;DR

This paper extends the elliptic law to products of multiple large real random matrices with correlated entries, showing the spectral distribution converges to the distribution of the mth power of a uniform variable on the unit disc, regardless of correlation.

Contribution

It introduces a generalization of the elliptic law for products of correlated random matrices, revealing the spectral distribution's independence from correlation parameter.

Findings

Spectral distribution converges to the mth power of a uniform on the unit disc.

Distribution is independent of the correlation parameter .

Generalizes the elliptic law to matrix products with correlated entries.

Abstract

We consider the products of $m\ge 2$ independent large real random matrices with independent vectors $(X_{jk}^{(q)},X_{kj}^{(q)})$ of entries. The entries $X_{jk}^{(q)},X_{kj}^{(q)}$ are correlated with $\rho=\mathbb E X_{jk}^{(q)}X_{kj}^{(q)}$. The limit distribution of the empirical spectral distribution of the eigenvalues of such products doesn't depend on $\rho$ and equals to the distribution of $m$th power of the random variable uniformly distributed on the unit disc.