Integer moments of complex Wishart matrices and Hurwitz numbers

Abstract

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.

19 pages, 1 figure

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• Ergodic Decomposition for Inverse Wishart Measures on Infinite Positive-Definite Matrices
• Time delay statistics for finite number of channels in all symmetry classes
• Grothendieck's Dessins d'Enfants in a Web of Dualities. III
• Symmetric Function Theory and Unitary Invariant Ensembles
• Schur expansion of random-matrix reproducing kernels
• Jacobi Beta Ensemble and $b$-Hurwitz Numbers
• $\beta$-ensembles and higher genera Catalan numbers
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