Moments of Random Matrices and Hypergeometric Orthogonal Polynomials

  • Fabio Deelan Cunden
  • Francesco Mezzadri
  • Neil O’ConnellEmail author
  • Nick Simm
Open Access


We establish a new connection between moments of \({n \times n}\) random matrices Xn and hypergeometric orthogonal polynomials. Specifically, we consider moments \({\mathbb{E}{\rm Tr} X_n^{-s}}\) 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. (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 \({n \rightarrow \infty}\) 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.



The research of FDC and NO’C is supported by ERC Advanced Grant 669306. The research of FDC is partially supported by the Italian National Group of Mathematical Physics (GNFMINdAM). FM acknowledges support from EPSRC Grant No. EP/L010305/1. NS acknowledges support from a Leverhulme Trust Early Career Fellowship ECF-2014-309.We would like to thank Philippe Biane for helpful conversations at an earlier stage of this work, in particular for drawing our attention to the papers of Bump et al. [19,20]. We are grateful to Peter Forrester and Brian Winn for valuable remarks on the first version of the paper. We also thank the referees who indicated the papers of Mehta and Normand [61], and Forrester and Witte [34] on duality relations for moments of characteristic polynomials of random matrices, and made us aware that the duality in Proposition 8.8 already appears in Forrester’s book [36, Eq. (6.120)–(6.122)].


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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity College DublinDublin 4Ireland
  2. 2.School of MathematicsUniversity of BristolBristolUK
  3. 3.Mathematics DepartmentUniversity of SussexBrightonUK

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