Moments of Generalized Cauchy Random Matrices and continuous-Hahn Polynomials

TL;DR

This paper proves that the moments of generalized Cauchy random matrices, after rescaling, form continuous-Hahn polynomials, extending known results for classical ensembles and deriving eigenvalue density asymptotics.

Contribution

It demonstrates that the moments of generalized Cauchy matrices are continuous-Hahn polynomials and develops a new proof strategy suitable for ensembles with finite moments.

Findings

Moments form continuous-Hahn polynomials in the variable k.

Derived a differential equation for the eigenvalue density.

Established large N asymptotics of the moments.

Abstract

In this paper we prove that, after an appropriate rescaling, the sum of moments $\mathbb{E}_{N}^{(s)} \left( Tr \left( |\mathbf{H}|^{2k+2}+|\mathbf{H}|^{2k}\right) \right)$ of an $N\times N$ Hermitian matrix $\mathbf{H}$ sampled according to the generalized Cauchy (also known as Hua-Pickrell) ensemble with parameter $s>0$ is a continuous-Hahn polynomial in the variable $k$. This completes the picture of the investigation that began by Cunden, Mezzadri, O'Connell and Simm who obtained analogous results for the other three classical ensembles of random matrices, the Gaussian, the Laguerre and Jacobi. Our strategy of proof is somewhat different from the one employed previously due to the fact that the generalized Cauchy is the only classical ensemble which has a finite number of integer moments. Our arguments also apply, with straightforward modifications, to the Gaussian, Laguerre and Jacobi cases as well. We finally obtain a differential equation for the one-point density function of the eigenvalue distribution of this ensemble and establish the large $N$ asymptotics of the moments.