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Integer moments of complex Wishart matrices and Hurwitz numbers

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posted on 2023-06-09, 17:59 authored by Fabio Della Cunden, Antoine DahlqvistAntoine Dahlqvist, Neil O'Connell
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.

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Publication status

  • Published

File Version

  • Accepted version

Journal

Annales De L'institut Henri Poincaré D

ISSN

2308-5827

Publisher

European Mathematical Society

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Probability and Statistics Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2019-06-06

First Open Access (FOA) Date

2021-03-12

First Compliant Deposit (FCD) Date

2019-06-05

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