Monotone Hurwitz Numbers and the HCIZ Integral

I. P. Goulden; Mathieu Guay-Paquet; Jonathan Novak

doi:10.5802/ambp.336

I. P. Goulden ¹; Mathieu Guay-Paquet ²; Jonathan Novak ³

¹ Department of Combinatorics & Optimization University of Waterloo 200 University Avenue West Waterloo, ON N2L 3G1 Canada
² LaCIM Université du Québec à Montréal 201 Avenue du Président-Kennedy Montréal, QC H2X 3Y7 Canada
³ Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Ave. Boston, MA 02114 USA

Annales mathématiques Blaise Pascal, Volume 21 (2014) no. 1, pp. 71-89.

Abstract
Résumé

In this article, we prove that the complex convergence of the HCIZ free energy is equivalent to the non-vanishing of the HCIZ integral in a neighbourhood of $z = 0$ . Our approach is based on a combinatorial model for the Maclaurin coefficients of the HCIZ integral together with classical complex-analytic techniques.

Nous démontrons que la convergence de l’énergie libre de l’intégrale HCIZ dans le plan complexe est équivalente à la non-nullité de l’intégrale HCIZ autour de $z = 0$ . Notre approche est basée sur un modèle combinatoire pour les coefficients de Maclaurin de l’intégrale HCIZ et sur des méthodes classiques d’analyse complexe.

MR: 3248222 | Zbl: 1296.05202

DOI: 10.5802/ambp.336

Classification: 05E10, 15B62, 14N10
Keywords: Matrix models, Hurwitz numbers, asymptotic analysis
Mot clés : Modèles matriciels, nombres de Hurwitz, analyse asymptotique

Author's affiliations:

I. P. Goulden ¹; Mathieu Guay-Paquet ²; Jonathan Novak ³

@article{AMBP_2014__21_1_71_0,
     author = {I. P. Goulden and Mathieu Guay-Paquet and Jonathan Novak},
     title = {Monotone {Hurwitz} {Numbers} and the {HCIZ} {Integral}},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {71--89},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {21},
     number = {1},
     year = {2014},
     doi = {10.5802/ambp.336},
     mrnumber = {3248222},
     zbl = {1296.05202},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.336/}
}

TY  - JOUR
AU  - I. P. Goulden
AU  - Mathieu Guay-Paquet
AU  - Jonathan Novak
TI  - Monotone Hurwitz Numbers and the HCIZ Integral
JO  - Annales mathématiques Blaise Pascal
PY  - 2014
SP  - 71
EP  - 89
VL  - 21
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.336/
DO  - 10.5802/ambp.336
LA  - en
ID  - AMBP_2014__21_1_71_0
ER  -

%0 Journal Article
%A I. P. Goulden
%A Mathieu Guay-Paquet
%A Jonathan Novak
%T Monotone Hurwitz Numbers and the HCIZ Integral
%J Annales mathématiques Blaise Pascal
%D 2014
%P 71-89
%V 21
%N 1
%I Annales mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.336/
%R 10.5802/ambp.336
%G en
%F AMBP_2014__21_1_71_0

I. P. Goulden; Mathieu Guay-Paquet; Jonathan Novak. Monotone Hurwitz Numbers and the HCIZ Integral. Annales mathématiques Blaise Pascal, Volume 21 (2014) no. 1, pp. 71-89. doi : 10.5802/ambp.336. https://ambp.centre-mersenne.org/articles/10.5802/ambp.336/

References
Cited by

[1] Benoît Collins Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability, Int. Math. Res. Not., Volume 2003 (2003) no. 17, pp. 953-982 | DOI | MR | Zbl

[2] Benoît Collins; Alice Guionnet; Edouard Maurel-Segala Asymptotics of unitary and orthogonal matrix integrals, Adv. Math., Volume 222 (2009) no. 1, pp. 172-215 | DOI | MR | Zbl

[3] Benoît Collins; Piotr Śniady Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Comm. Math. Phys., Volume 264 (2006) no. 3, pp. 773-795 | DOI | MR | Zbl

[4] Torsten Ekedahl; Sergei Lando; Michael Shapiro; Alek Vainshtein Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math., Volume 146 (2001) no. 2, pp. 297-327 | DOI | MR | Zbl

[5] László Erdős; Horng-Tzer Yau Universality of local spectral statistics of random matrices, Bull. Amer. Math. Soc. (N.S.), Volume 49 (2012) no. 3, pp. 377-414 | DOI | MR | Zbl

[6] I. P. Goulden; Mathieu Guay-Paquet; Jonathan Novak Toda equations and piecewise polynomiality for mixed double Hurwitz numbers (Submitted)

[7] I. P. Goulden; Mathieu Guay-Paquet; Jonathan Novak Monotone Hurwitz numbers in genus zero, Canad. J. Math., Volume 65 (2013) no. 5, pp. 1020-1042 | DOI | MR | Zbl

[8] I. P. Goulden; Mathieu Guay-Paquet; Jonathan Novak Polynomiality of monotone Hurwitz numbers in higher genera, Adv. Math., Volume 238 (2013), pp. 1-23 | DOI | MR | Zbl

[9] Ian P. Goulden; David M. Jackson Combinatorial enumeration, Dover Publications Inc., Mineola, NY, 2004, pp. xxvi+569 (With a foreword by Gian-Carlo Rota, Reprint of the 1983 original) | MR | Zbl

[10] Mathieu Guay-Paquet; Jonathan Novak A self-interacting random walk on the symmetric group (In preparation)

[11] Alice Guionnet Large deviations and stochastic calculus for large random matrices, Probab. Surv., Volume 1 (2004), pp. 72-172 | DOI | MR | Zbl

[12] A. Hurwitz Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann., Volume 39 (1891) no. 1, pp. 1-60 | DOI | MR

[13] C. Itzykson; J. B. Zuber The planar approximation. II, J. Math. Phys., Volume 21 (1980) no. 3, pp. 411-421 | DOI | MR | Zbl

[14] M. E. Kazarian; S. K. Lando An algebro-geometric proof of Witten’s conjecture, J. Amer. Math. Soc., Volume 20 (2007) no. 4, pp. 1079-1089 | DOI | MR | Zbl

[15] Sho Matsumoto; Jonathan Novak Jucys-Murphy elements and unitary matrix integrals, Int. Math. Res. Not. IMRN, Volume 2013 (2013) no. 2, pp. 362-397 | MR

[16] Jonathan I. Novak Jucys-Murphy elements and the unitary Weingarten function, Noncommutative harmonic analysis with applications to probability II (Banach Center Publ.), Volume 89, Polish Acad. Sci. Inst. Math., Warsaw, 2010, pp. 231-235 | DOI | MR | Zbl

[17] Andrei Okounkov Toda equations for Hurwitz numbers, Math. Res. Lett., Volume 7 (2000) no. 4, pp. 447-453 | DOI | MR | Zbl

[18] R. Pandharipande The Toda equations and the Gromov-Witten theory of the Riemann sphere, Lett. Math. Phys., Volume 53 (2000) no. 1, pp. 59-74 | DOI | MR | Zbl

[19] L. Pyber Enumerating finite groups of given order, Ann. of Math. (2), Volume 137 (1993) no. 1, pp. 203-220 | DOI | MR | Zbl

[20] E. C. Titchmarsh The Theory of Functions, Oxford University Press, 1939 | MR

[21] P. Zinn-Justin HCIZ integral and 2D Toda lattice hierarchy, Nuclear Phys. B, Volume 634 (2002) no. 3, pp. 417-432 | DOI | MR | Zbl

[22] P. Zinn-Justin; J.-B. Zuber On some integrals over the $U (N)$ unitary group and their large $N$ limit, J. Phys. A, Volume 36 (2003) no. 12, pp. 3173-3193 (Random matrix theory) | DOI | MR | Zbl

Cited by Sources: