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H 1 Nesting statistics in the $O(n)$ loop model on random planar maps

Auteurs : Bouttier, Jérémie (Auteur de la Conférence)
CIRM (Editeur )

Résumé : The $O(n)$ model can be formulated in terms of loops living on the lattice, with n the fugacity per loop. In two dimensions, it is known to possess a rich critical behavior, involving critical exponents varying continuously with n. In this talk, we will consider the case where the model is ”coupled to 2D quantum gravity”, namely it is defined on a random map.
It has been known since the 90’s that the partition function of the model can be expressed as a matrix integral, which can be evaluated exactly in the planar limit. A few years ago, together with G. Borot and E. Guitter, we revisited the problem by a combinatorial approach, which allows to relate it to the so-called Boltzmann random maps, which have no loops but faces of arbitrary (and controlled) face degrees. In particular we established that the critical points of the $O(n)$ model are closely related to the ”stable maps” introduced by Le Gall and Miermont.
After reviewing these results, I will move on to a more recent work done in collaboration with G. Borot and B. Duplantier, where we study the nesting statistics of loops. More precisely we consider loop configurations with two marked points and study the distribution of the number of loops separating them. The associated generating function can be computed exactly and, by taking asymptotics, we show that the number of separating loops grows logarithmically with the size of the maps at a (non generic) critical point, with an explicit large deviation function. Using a continuous generalization of the KPZ relation, our results are in full agreement with those of Miller, Watson and Wilson concerning nestings in Conformal Loop Ensembles.

Codes MSC :
05Axx - Enumerative combinatorics
60K35 - Interacting random processes; statistical mechanics type models; percolation theory
81T40 - Two-dimensional field theories, conformal field theories, etc.

 Informations sur la Vidéo Langue : Anglais Date de publication : 23/06/2016 Date de captation : 09/06/2016 Collection : Research talks ; Combinatorics ; Mathematical Physics ; Probability and Statistics Format : MP4 Durée : 01:10:19 Domaine : Probability & Statistics ; Combinatorics ; Mathematical Physics Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2016-06-09_Bouttier.mp4 Informations sur la rencontre Nom de la rencontre : Random trees and maps: probabilistic and combinatorial aspects / Arbres et cartes aléatoires : aspects probabilistes et combinatoiresOrganisateurs de la rencontre : Haas, Bénédicte ; Goldschmidt, Christina ; Miermont, GrégoryDates : 06/06/2016 - 10/06/2016 Année de la rencontre : 2016 URL Congrès : http://conferences.cirm-math.fr/1384.htmlCitation Data DOI : 10.24350/CIRM.V.18993703 Cite this video as: Bouttier, Jérémie (2016). Nesting statistics in the $O(n)$ loop model on random planar maps. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18993703 URI : http://dx.doi.org/10.24350/CIRM.V.18993703

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Bibliographie

1. Borot, G., Bouttier, J., & Duplantier, B. (2016). Nesting statistics in the $O(n)$ loop model on random planar maps. - https://arxiv.org/abs/1605.02239

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