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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 )

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    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 combinatoires
    Organisateurs de la rencontre : Haas, Bénédicte ; Goldschmidt, Christina ; Miermont, Grégory
    Dates : 06/06/2016 - 10/06/2016
    Année de la rencontre : 2016
    URL Congrès : http://conferences.cirm-math.fr/1384.html

    Citation 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


    Voir aussi

    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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