Moments of Random Matrices and Hypergeometric Orthogonal Polynomials
Abstract
We establish a new connection between moments of {n × n} random matrices X _{ n } and hypergeometric orthogonal polynomials. Specifically, we consider moments E}Tr X_n^{s as a function of the complex variable {s \in 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 timedelay 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.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 February 2019
 DOI:
 10.1007/s00220019033239
 arXiv:
 arXiv:1805.08760
 Bibcode:
 2019CMaPh.tmp...37C
 Keywords:

 Mathematical Physics;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Complex Variables
 EPrint:
 53 pages, 4 figures, 1 table