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 with $$\mathbb {E}\,X_{ij} = 0$$ , $$\mathbb {E}\,X^2_{ij} = 1$$ and $$\mathbb {E}\,X_{ij} X_{ji} = \rho $$ . We are interested in the asymptotic distribution of eigenvalues of the matrix $$W =\frac{1}{N^2} X^2 X^{*2}$$ . We show that this distribution is determined by its moments, and we provide a recurrence relation for these moments. We prove that the (symmetrized) asymptotic distribution is determined by its free cumulants, which are Narayana polynomials of type B: $$\begin{aligned} c_{2n} = \sum _{k=0}^n {\left( {\begin{array}{c}n k\end{array}}\right) }^2 \rho ^{2k}. \end{aligned}$$