Moments of random matrices and hypergeometric orthogonal polynomials

Deelan Cunden, Fabio, Mezzadri, Francesco, O'Connell, Neil and Simm, Nick (2019) Moments of random matrices and hypergeometric orthogonal polynomials. Communications in Mathematical Physics. ISSN 0010-3616

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We establish a new connection between moments of $n \times n$ random matrices $X_n$ and hypergeometric orthogonal polynomials. Specifically, we consider moments $\mathbb{E}\Tr X_n^{-s}$ 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 \textit{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.

Item Type: Article
Keywords: Random matrix theory, Gaussian unitary ensemble, Meixner polynomials, hypergeometric functions, Jacobi polynomials, orthogonal polynomials, zeta functions
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Probability and Statistics Research Group
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA0273 Probabilities. Mathematical statistics
Q Science > QC Physics
Depositing User: Nicholas Simm
Date Deposited: 19 Nov 2018 17:01
Last Modified: 06 Feb 2020 02:00

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