Moments of random matrices and hypergeometric orthogonal polynomials

Abstract

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

53 pages, 4 figures, 1 table

References in corpus (6)

• Efficient semiclassical approach for time delays
• Capacitance and charge relaxation resistance of chaotic cavities - Joint distribution of two linear statistics in the Laguerre ensemble of random matrices
• Oscillation theorems for the Wronskian of an arbitrary sequence of eigenfunctions of Schrödinger's equation
• A method to calculate correlation functions for $\beta =1$ random matrices of odd size
• Large $N$ expansions for the Laguerre and Jacobi $\beta$ ensembles from the loop equations
• Large-$N$ expansion for the time-delay matrix of ballistic chaotic cavities

Cited by in corpus (12)

• Differential identities for the structure function of some random matrix ensembles
• Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials
• Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles
• Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals
• The classical $\beta$-ensembles with $\beta$ proportional to $1/N$: from loop equations to Dyson's disordered chain
• Relations between moments for the Jacobi and Cauchy random matrix ensembles
• Moments of discrete orthogonal polynomial ensembles
• Moments of Generalized Cauchy Random Matrices and continuous-Hahn Polynomials
• Time delay statistics for finite number of channels in all symmetry classes
• Dip-ramp-plateau for Dyson Brownian motion from the identity on $U(N)$
• Grothendieck's Dessins d'Enfants in a Web of Dualities. III
• Symmetric Function Theory and Unitary Invariant Ensembles