Moments of Random Matrices and Hypergeometric Orthogonal Polynomials

Abstract

We establish a new connection between moments of n × n random matrices Xn and hypergeometric orthogonal polynomials. Specifically, we consider moments E Tr X −s n 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. [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 → ∞ 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. Contents 48References 50 1 arXiv:1805.08760v5 [math-ph]

Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

“…for more details see Proposition 4.4. Thus the formulae of Corollary 2.7 extend the connection between JUE moments tr X k and Wilson polynomials described in [18] (see also [42,43]) to the JUE multi-point correlators tr…”

Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals

“…x r 3 = 0, (2.85) [42]. In [23], it is pointed out that such three term recursion is a manifestation of the fact that A (N, M ) is expressible in terms of hypergeometric orthogonal polynomials; this property extends to the entries B (N, M ), as we now show. Introducing the generalized hypergeometric…”

“…, k ≥ 0, (1.16) which were already derived in the literature [23,44]. From Theorem 1.1, for example, one can deduce compact expressions for correlators of the form…”

“…We deduce this recursion together with a similar threeterm recursion for B (N, M ), in Lemma 2.11. It was pointed out in [23] that the entries A (N, M ) are hypergeometric orthogonal polynomials (in particular suitably normalized Hahn and dual Hahn polynomials [23,50]), a fact which provides another interpretation of the same three-term recursion; this interpretation extends to the entries B (N, M ), see Remark 2.12. In Lemma 3.2, we provide an alternative expression for the entries A (N, M ), B (N, M ), which makes clear that they are polynomials in N, M with integer coefficients.…”

“…Alternatively, introducing the Hahn and dual Hahn polynomials [23,50] Q j (x; μ, ν, k) : which is used in Example 2.14 to prove formulae (1.17).…”