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

TL;DR

This paper studies the eigenvalue distribution of squared elliptic Gaussian matrices, revealing that their asymptotic moments are governed by Narayana polynomials of type B, connecting random matrix theory with combinatorial structures.

Contribution

It establishes the asymptotic eigenvalue distribution of squared elliptic Gaussian matrices and links their free cumulants to Narayana polynomials of type B, providing a recursive moment relation.

Findings

Distribution characterized by moments and free cumulants

Recursive relation for moments derived

Free cumulants expressed as Narayana polynomials of type B

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, $E X_{ij} = 0$, $E X^2_{ij} = 1$ and $E X_{ij} X_{ji} = \rho$. We are interested in the asymptotic distribution of eigenvalues of the matrix $W =\frac{1}{N^2} X^2 X^{*2}$. 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: $$c_{2n} = \sum_{k=0}^n \binom{n}{k}^2 \rho^{2k}.$$