Integer moments of complex Wishart matrices and Hurwitz numbers
Annales de l’Institut Henri Poincaré D, Tome 8 (2021) no. 2, pp. 243-268.
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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.

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DOI : 10.4171/aihpd/103
Classification : 05-XX, 15-XX, 60-XX, 81-XX
Mots-clés : Moments of random matrices, genus expansion,Wishart distribution, Hurwitz numbers, Weingarten calculus, quantum chaotic transport 
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     title = {Integer moments of complex {Wishart} matrices and {Hurwitz} numbers},
     journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
     pages = {243--268},
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     doi = {10.4171/aihpd/103},
     mrnumber = {4261672},
     zbl = {1466.60009},
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     url = {http://www.numdam.org/articles/10.4171/aihpd/103/}
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Cunden, Fabio Deelan; Dahlqvist, Antoine; O'Connell, Neil. Integer moments of complex Wishart matrices and Hurwitz numbers. Annales de l’Institut Henri Poincaré D, Tome 8 (2021) no. 2, pp. 243-268. doi : 10.4171/aihpd/103. http://www.numdam.org/articles/10.4171/aihpd/103/

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