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

This article is cited in 12 scientific papers (total in 12 papers)

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-а
Russian Academy of Sciences - Federal Agency for Scientific Organizations 19-П


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

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Received: 11.09.2014

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
\by J.~Ambj{\o}rn, L.~O.~Chekhov
\paper A~matrix model for hypergeometric Hurwitz numbers
\jour TMF
\yr 2014
\vol 181
\issue 3
\pages 421--435
\jour Theoret. and Math. Phys.
\yr 2014
\vol 181
\issue 3
\pages 1486--1498

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    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  crossref  mathscinet  zmath  isi  elib  scopus
    2. J. Harnad, “Quantum Hurwitz numbers and Macdonald polynomials”, J. Math. Phys., 57:11 (2016), 113505  crossref  mathscinet  zmath  isi  scopus
    3. A. Yu. Orlov, “Hurwitz numbers and products of random matrices”, Theoret. and Math. Phys., 192:3 (2017), 1282–1323  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    4. M. Guay-Paquet, J. Harnad, “Generating functions for weighted Hurwitz numbers”, J. Math. Phys., 58:8 (2017), 083503  crossref  mathscinet  zmath  isi  scopus
    5. S. M. Natanzon, A. Yu. Orlov, “BKP and projective Hurwitz numbers”, Lett. Math. Phys., 107:6 (2017), 1065–1109  crossref  mathscinet  zmath  isi  scopus
    6. J. Harnad, J. Ortmann, “Asymptotics of quantum weighted Hurwitz numbers”, J. Phys. A-Math. Theor., 51:22 (2018), 225201  crossref  mathscinet  isi
    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  crossref  mathscinet  zmath  isi  scopus
    8. A. Alexandrov, G. Chapuy, B. Eynard, J. Harnad, “Weighted Hurwitz numbers and topological recursion: an overview”, J. Math. Phys., 59:8 (2018), 081102  crossref  mathscinet  zmath  isi  scopus
    9. J. Ambjorn, L. O. Chekhov, “Spectral curves for hypergeometric Hurwitz numbers”, J. Geom. Phys., 132 (2018), 382–392  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    11. N. Do, P. Norbury, “Topological recursion for irregular spectral curves”, J. Lond. Math. Soc.-Second Ser., 97:3 (2018), 398–426  crossref  mathscinet  zmath  isi  scopus
    12. S. M. Natanzon, A. Yu. Orlov, “Hurwitz numbers from Feynman diagrams”, Theoret. and Math. Phys., 204:3 (2020), 1166–1194  mathnet  crossref  crossref  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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