We give formulae for the cumulants of complex Wishart (LUE) and inverse Wishart matrices (inverse LUE). Their large- 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.
Publié le :
DOI : 10.4171/aihpd/103
Mots-clés : Moments of random matrices, genus expansion,Wishart distribution, Hurwitz numbers, Weingarten calculus, quantum chaotic transport
@article{AIHPD_2021__8_2_243_0, author = {Cunden, Fabio Deelan and Dahlqvist, Antoine and O'Connell, Neil}, title = {Integer moments of complex {Wishart} matrices and {Hurwitz} numbers}, journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D}, pages = {243--268}, volume = {8}, number = {2}, year = {2021}, doi = {10.4171/aihpd/103}, mrnumber = {4261672}, zbl = {1466.60009}, language = {en}, url = {http://archive.numdam.org/articles/10.4171/aihpd/103/} }
TY - JOUR AU - Cunden, Fabio Deelan AU - Dahlqvist, Antoine AU - O'Connell, Neil TI - Integer moments of complex Wishart matrices and Hurwitz numbers JO - Annales de l’Institut Henri Poincaré D PY - 2021 SP - 243 EP - 268 VL - 8 IS - 2 UR - http://archive.numdam.org/articles/10.4171/aihpd/103/ DO - 10.4171/aihpd/103 LA - en ID - AIHPD_2021__8_2_243_0 ER -
%0 Journal Article %A Cunden, Fabio Deelan %A Dahlqvist, Antoine %A O'Connell, Neil %T Integer moments of complex Wishart matrices and Hurwitz numbers %J Annales de l’Institut Henri Poincaré D %D 2021 %P 243-268 %V 8 %N 2 %U http://archive.numdam.org/articles/10.4171/aihpd/103/ %R 10.4171/aihpd/103 %G en %F AIHPD_2021__8_2_243_0
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://archive.numdam.org/articles/10.4171/aihpd/103/
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