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Moments of random matrices and hypergeometric orthogonal polynomials

Research output: Contribution to journalArticle

Original languageEnglish
Pages (from-to)1091-1145
Number of pages55
JournalCommunications in Mathematical Physics
Issue number3
Early online date6 Feb 2019
DateAccepted/In press - 10 Nov 2018
DateE-pub ahead of print - 6 Feb 2019
DatePublished (current) - 1 Aug 2019


We establish a new connection between moments of n× n random matrices Xn and hypergeometric orthogonal polynomials. Specifically, we consider moments ETrXn-s as a function of the complex variable s∈ 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. (J Math Phys 57:111901, 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→ ∞ 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.

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