Abstract
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.
Highlights
In this paper we present a novel approach to the moments of the classical ensembles of random matrices.Much of random matrix theory is devoted to moments k n (k N)of random matrices of finite or asymptotically large size n
We show that for the classical matrix ensembles, the Mellin transform ρn(2),∗(s) of a Wronskian of two adjacent wavefunctions is a hypergeometric OP
Once the analytic structure of the Mellin transform of Wr(ψn(x), ψn+1(x)) was established, it became natural for us to look for similar polynomial properties for Wronskians of nonadjacent wavefunctions Wr(ψn(x), ψn+k(x)), k > 1
Summary
In this paper we present a novel approach to the moments of the classical ensembles of random matrices. Once the analytic structure of the Mellin transform of Wr(ψn(x), ψn+1(x)) was established, it became natural for us to look for similar polynomial properties for Wronskians of nonadjacent wavefunctions Wr(ψn(x), ψn+k(x)), k > 1 Such Wronskians do not have a random matrix interpretation. It turns out that for the classical ensembles of random matrices with orthogonal and symplectic symmetries, certain combinations of moments satisfy three term recursion formulae which, again, correspond to the S-L equations defining families of hypergeometric OP’s. This combination of moments plays the role of the single moments in the unitary case: they satisfy three term recursions, have hypergeometric OP factors, reflection symmetries, zeros on a vertical line, etc.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
