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 TMF, 2014, Volume 181, Number 3, Pages 421–435 (Mi tmf8791)

A matrix model for hypergeometric Hurwitz numbers

J. Ambjørnab, L. O. Chekhovcde

a Niels Bohr Institute, Copenhagen University, Copenhagen Denmark
b IMAPP, Radboud University, Nijmengen, The Netherlands
c Steklov Mathematical Institute, RAS, Moscow, Russia
d Laboratoire Poncelet, Independent University of Moscow, Moscow, Russia
e Center for Quantum Geometry of Moduli Spaces, Århus University, Århus, Denmark

Abstract: We present multimatrix models that are generating functions for the numbers of branched covers of the complex projective line ramified over $n$ fixed points $z_i$, $i=1,…,n$ (generalized Grothendieck's dessins d'enfants) of fixed genus, degree, and ramification profiles at two points $z_1$ and $z_n$. We sum over all possible ramifications at the other $n-2$ points with a fixed length of the profile at $z_2$ and with a fixed total length of profiles at the remaining $n-3$ points. All these models belong to a class of hypergeometric Hurwitz models and are therefore tau functions of the Kadomtsev–Petviashvili hierarchy. In this case, we can represent the obtained model as a chain of matrices with a (nonstandard) nearest-neighbor interaction of the type $\operatorname{tr} M_iM_{i+1}^{-1}$. We describe the technique for evaluating spectral curves of such models, which opens the way for obtaining $1/N^2$-expansions of these models using the topological recursion method. These spectral curves turn out to be algebraic.

Keywords: Hurwitz number, random complex matrix, Kadomtsev–Petviashvili hierarchy, matrix chain, bipartite graph, spectral curve

 Funding Agency Grant Number European Research Council Advance Grant 291092 Independent Research Fund Denmark Russian Foundation for Basic Research 14-01-00860-à13-01-12405-îôè_ì Russian Academy of Sciences - Federal Agency for Scientific Organizations 19-Ï

DOI: https://doi.org/10.4213/tmf8791

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English version:
Theoretical and Mathematical Physics, 2014, 181:3, 1486–1498

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Citation: J. Ambjørn, L. O. Chekhov, “A matrix model for hypergeometric Hurwitz numbers”, TMF, 181:3 (2014), 421–435; Theoret. and Math. Phys., 181:3 (2014), 1486–1498

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. L. O. Chekhov, “The Harer–Zagier recursion for an irregular spectral curve”, J. Geom. Phys., 110 (2016), 30–43
2. J. Harnad, “Quantum Hurwitz numbers and Macdonald polynomials”, J. Math. Phys., 57:11 (2016), 113505
3. A. Yu. Orlov, “Hurwitz numbers and products of random matrices”, Theoret. and Math. Phys., 192:3 (2017), 1282–1323
4. M. Guay-Paquet, J. Harnad, “Generating functions for weighted Hurwitz numbers”, J. Math. Phys., 58:8 (2017), 083503
5. S. M. Natanzon, A. Yu. Orlov, “BKP and projective Hurwitz numbers”, Lett. Math. Phys., 107:6 (2017), 1065–1109
6. J. Harnad, J. Ortmann, “Asymptotics of quantum weighted Hurwitz numbers”, J. Phys. A-Math. Theor., 51:22 (2018), 225201
7. G. Akemann, E. Strahov, “Product matrix processes for coupled multi-matrix models and their hard edge scaling limits”, Ann. Henri Poincare, 19:9 (2018), 2599–2649
8. A. Alexandrov, G. Chapuy, B. Eynard, J. Harnad, “Weighted Hurwitz numbers and topological recursion: an overview”, J. Math. Phys., 59:8 (2018), 081102
9. J. Ambjorn, L. O. Chekhov, “Spectral curves for hypergeometric Hurwitz numbers”, J. Geom. Phys., 132 (2018), 382–392
10. P. Dunin-Barkowski, N. Orantin, A. Popolitov, S. Shadrin, “Combinatorics of loop equations for branched covers of sphere”, Int. Math. Res. Notices, 2018, no. 18, 5638–5662
11. N. Do, P. Norbury, “Topological recursion for irregular spectral curves”, J. Lond. Math. Soc.-Second Ser., 97:3 (2018), 398–426
12. S. M. Natanzon, A. Yu. Orlov, “Hurwitz numbers from Feynman diagrams”, Theoret. and Math. Phys., 204:3 (2020), 1166–1194
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