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Moments of random matrices and hypergeometric orthogonal polynomials

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Version 2 2023-06-12, 07:30
Version 1 2023-06-09, 15:55
journal contribution
posted on 2023-06-12, 07:30 authored by Fabio Deelan Cunden, Francesco Mezzadri, Neil O'Connell, Nicholas SimmNicholas Simm
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.

History

Publication status

  • Published

File Version

  • Published version

Journal

Communications in Mathematical Physics

ISSN

0010-3616

Publisher

Springer

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Probability and Statistics Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2018-11-19

First Open Access (FOA) Date

2019-04-05

First Compliant Deposit (FCD) Date

2018-11-19

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