Monotone Hurwitz Numbers and the HCIZ Integral
[Les nombres de Hurwitz monotones et l’intégrale HCIZ]
Annales mathématiques Blaise Pascal, Tome 21 (2014) no. 1, pp. 71-89.

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.

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.

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
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
@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, Tome 21 (2014) no. 1, pp. 71-89. doi : 10.5802/ambp.336. https://ambp.centre-mersenne.org/articles/10.5802/ambp.336/

[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

Cité par Sources :