Singular Values Distribution of Squares of Elliptic Random Matrices and Type B Narayana Polynomials

Abstract

We consider Gaussian elliptic random matrices $X$ of a size $N\times N$ with parameter $\rho$, i.e., matrices whose pairs of entries $(X_{ij},X_{ji})$ are mutually independent Gaussian vectors, $E{X}_{ij}=0$, $E{X}_{ij}^2=1$ and $E{X}_{ij}{X}_{ji}=\rho$. We are interested in the asymptotic distribution of eigenvalues of the matrix $W=\frac{1}{N^2}{X}^2{X}^{\ast 2}$. We have shown that this distribution is defined by its moments and we provide a recurrent relation for these moments. We have proven that the (symmetrized) asymptotic distribution is determined by its free cumulants, which are Narayana polynomials of type B: $$c_{2n}=\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2{\rho }^{2k}.$$

arXiv comment19 pages, 8 figures