The matrix model for dessins d'enfants
Annales de l’Institut Henri Poincaré D, Tome 1 (2014) no. 3, pp. 337-361.

We present the matrix models that are the generating functions for branched covers of the complex projective line ramified over 0, 1, and (Grotendieck’s dessins d’enfants) of fixed genus, degree, and the ramification profile at infinity. For general ramifications at other points, the model is the two-logarithm matrix model with the external field studied previously by one of the authors (L.Ch.) and K.Palamarchuk. It lies in the class of the generalised Kontsevich models (GKM) thus being the Kadomtsev–Petviashvili (KP) hierarchy tau function and, upon the shift of times, this model is equivalent to a Hermitian one-matrix model with a general potential whose coefficients are related to the KP times by a Miwa-type transformation. The original model therefore enjoys a topological recursion and can be solved in terms of shifted moments of the standard Hermitian one-matrix model at all genera of the topological expansion. We also derive the matrix model for clean Belyi morphisms, which turns out to be the Kontsevich–Penner model introduced by the authors and Yu. Makeenko. Its partition function is also a KP hierarchy tau function, and this model is in turn equivalent to a Hermitian one-matrix model with a general potential. Finally we prove that the generating function for general two-profile Belyi morphisms is a GKM thus proving that it is also a KP hierarchy tau function in proper times.

Accepté le :
Publié le :
DOI : 10.4171/aihpd/10
Classification : 05-XX, 14-XX, 15-XX
Mots-clés : Belyi function, topological recursion, tau function, Miwa transform
@article{AIHPD_2014__1_3_337_0,
     author = {Ambj{\o}rn, Jan and Chekhov, Leonid},
     title = {The matrix model for dessins d'enfants},
     journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
     pages = {337--361},
     volume = {1},
     number = {3},
     year = {2014},
     doi = {10.4171/aihpd/10},
     mrnumber = {3239275},
     zbl = {1304.81130},
     language = {en},
     url = {http://archive.numdam.org/articles/10.4171/aihpd/10/}
}
TY  - JOUR
AU  - Ambjørn, Jan
AU  - Chekhov, Leonid
TI  - The matrix model for dessins d'enfants
JO  - Annales de l’Institut Henri Poincaré D
PY  - 2014
SP  - 337
EP  - 361
VL  - 1
IS  - 3
UR  - http://archive.numdam.org/articles/10.4171/aihpd/10/
DO  - 10.4171/aihpd/10
LA  - en
ID  - AIHPD_2014__1_3_337_0
ER  - 
%0 Journal Article
%A Ambjørn, Jan
%A Chekhov, Leonid
%T The matrix model for dessins d'enfants
%J Annales de l’Institut Henri Poincaré D
%D 2014
%P 337-361
%V 1
%N 3
%U http://archive.numdam.org/articles/10.4171/aihpd/10/
%R 10.4171/aihpd/10
%G en
%F AIHPD_2014__1_3_337_0
Ambjørn, Jan; Chekhov, Leonid. The matrix model for dessins d'enfants. Annales de l’Institut Henri Poincaré D, Tome 1 (2014) no. 3, pp. 337-361. doi : 10.4171/aihpd/10. http://archive.numdam.org/articles/10.4171/aihpd/10/

Cité par Sources :