Integer moments of complex Wishart matrices and Hurwitz numbers

Cunden, Fabio Della, Dahlqvist, Antoine and O'Connell, Neil (2021) Integer moments of complex Wishart matrices and Hurwitz numbers. Annales De L'institut Henri Poincaré D. ISSN 2308-5827

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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, arXiv:1805.08760] 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, J. Phys. A 49 (2016)] 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.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Probability and Statistics Research Group
Subjects: Q Science > QA Mathematics > QA0150 Algebra. Including machine theory, game theory > QA0164 Combinatorics. Combinatorial analysis
Q Science > QA Mathematics > QA0273 Probabilities. Mathematical statistics
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Depositing User: Antoine Dahlqvist
Date Deposited: 06 Jun 2019 11:34
Last Modified: 12 Mar 2021 16:30

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