Moments of random matrices and hypergeometric orthogonal polynomials

TL;DR

This paper links moments of random matrices to hypergeometric orthogonal polynomials, revealing symmetries, zeros, and orthogonality, and applies these findings to conjectures in quantum chaos and classical ensembles.

Contribution

It establishes a novel connection between random matrix moments and hypergeometric orthogonal polynomials, including symmetry properties and applications to chaos theory.

Findings

Identified reflection symmetry and zeros on a critical line in moments of random matrices.

Characterized moments in classical ensembles using the Askey scheme of orthogonal polynomials.

Derived a duality formula relating moments in orthogonal/symplectic ensembles to hypergeometric polynomials.

Abstract

We establish a new connection between moments of $n \times n$ random matrices $X_n$ and hypergeometric orthogonal polynomials. Specifically, we consider moments $\mathbb{E}\mathrm{Tr} X_n^{-s}$ as a function of the complex variable $s \in \mathbb{C}$, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. An application of the theory resolves part of an integrality conjecture of Cunden et al. [F. D. Cunden, F. Mezzadri, N. J. Simm and P. Vivo, J. Math. Phys. 57 (2016)] on the time-delay matrix of chaotic cavities. In each of the classical ensembles of random matrix theory (Gaussian, Laguerre, Jacobi) we characterise the moments in terms of the Askey scheme of hypergeometric orthogonal polynomials. We also calculate the leading order $n\to\infty$ asymptotics of the moments and discuss their symmetries and zeroes. We discuss aspects of these phenomena beyond the random matrix setting, including the Mellin transform of products and Wronskians of pairs of classical orthogonal polynomials. When the random matrix model has orthogonal or symplectic symmetry, we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials.