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Annales de l’Institut Henri Poincaré D

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Volume 8, Issue 2, 2021, pp. 243–268
DOI: 10.4171/AIHPD/103

Published online: 2021-03-08

Integer moments of complex Wishart matrices and Hurwitz numbers

Fabio Deelan Cunden[1], Antoine Dahlqvist[2] and Neil O'Connell[3]

(1) Università di Bari, Italy
(2) University of Sussex, Brighton, UK
(3) University College Dublin, Ireland

We give formulae for the cumulants of complex Wishart (LUE) and inverse Wishart matrices (inverse LUE). Their large-$N$ expansions are generating functions of double (strictly and weakly) monotone Hurwitz numbers which count constrained factorisations in the symmetric group. The two expansions can be compared and combined with a duality relation proved in [F. D. Cunden, F. Mezzadri, N. O’Connell, and N. J. Simm, Moments of random matrices and hypergeometric orthogonal polynomials, Comm. Math. Phys. 369 (2019), no. 3, 1091–1145] to obtain: i) a combinatorial proof of the reflection formula between moments of LUE and inverse LUE at genus zero and, ii) a new functional relation between the generating functions of monotone and strictly monotone Hurwitz numbers. The main result resolves the integrality conjecture formulated in [F. D. Cunden, F. Mezzadri, N. J. Simm, and P. Vivo, Correlators for the Wigner–Smith time-delay matrix of chaotic cavities, J. Phys. A 49 (2016), no. 18, 18LT01, 20 pp] on the time-delay cumulants in quantum chaotic transport. The precise combinatorial description of the cumulants given here may cast new light on the concordance between random matrix and semiclassical theories.

Keywords: Moments of random matrices, genus expansion,Wishart distribution, Hurwitz numbers, Weingarten calculus, quantum chaotic transport

Cunden Fabio Deelan, Dahlqvist Antoine, O'Connell Neil: Integer moments of complex Wishart matrices and Hurwitz numbers. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 8 (2021), 243-268. doi: 10.4171/AIHPD/103