Fédération de recherche Math-STIC de l'université Paris 13

FR3734

Journée "Cartes" (14 novembre 2013)

"Journee cartes combinatoire"

Contact: Adrian Tanasa

Lieu: Salle B107, Institut Galilée

(PEPS INS2I "Cartes 3D", BQR UP13 "Combinatoire algebrique)

10h-11h Omid Amini (DMA, ENS Paris), Coloriage des graphes sur les surfaces

11h15-12h15 Vincent Vargas (DMA, ENS Paris), Critical and dual Gaussian multiplicative chaos

14h-15h Gilles Schaeffer (LIX, Ecole Polytechnique), Regular colored graphs of positive degree

15h15-16h15 Matti Raasakka (LIPN), Next-to-leading order in the large N expansion of the multi-orientable tensor model

16h15-17h15 James Ryan (Inst. Max Planck, Golm, Allemagne), Melons are branched polymers

 

 

Résumés:

- Coloriage des graphes sur les surfaces, Omid Amini

Je commencerai par introduire le concept de S-coloriage de graphe :
étant donné un sous-ensemble S(v) de voisins de v pour tout sommet v d'un
graphe G, c'est un coloriage propre des sommets
de G tel que, en outre, les sommets qui appartiennent ensemble à un même
S(v) pour un sommet v reçoivent des couleurs différentes.
Ceci généralise à la fois le concept de coloriage du carré de graphe et
le coloriage cyclique des graphes plongés.

Je présenterai un résultat structurel fort pour les graphes plongés
dans une surface fixe, qui permet par exemple de démontrer que la taille
maximum d'une clique dans le carré d'un tel graphe
de degré maximum D est au plus 3D / 2 plus une constante.

En utilisant ce résultat de structure, le S-coloriage d'un graphe plongé
peut se réduire au coloriage d'arêtes d'un multigraphe.
Je donnerai ensuite un aperçu général du travail de Jeff Kahn sur
arête-coloriage d'un multigraphe H, basé sur l'utilisation des
distributions hardcores sur les couplages, définies par des points à
l’intérieur du polytope des couplages de H.

En combinant ces résultats, on peut établir un résultat général sur le
S-coloriage des graphes plongés, impliquant notamment les versions
asymptotiques des conjectures de
 Wegner et de Borodin sur le coloriage du carré et le coloriage cyclique
des graphes planaires. Mon exposé est basé sur un travail en commun avec
L. Esperet et J. van den Heuvel.

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Critical and dual Gaussian multiplicative chaos, Vincent Vargas

The theory of Gaussian multiplicative chaos enables to make
sense of random measures defined formally by the exponential of a Gaussian
Free Field (and more generally any logarithmically correlated field in all
dimensions). These random measures are conjectured to be the scaling limit
of random planar maps coupled to a CFT and conformally mapped to the
sphere or the upper half plane.
The construction of the measures involve a parameter gamma related to the
central charge c of the CFT. The classical construction does not settle
the question of constructing the measure for c=1 or equivalently gamma=2
as it yields a vanishing measure. In this talk, I will explain the
appropriate construction and some properties of Gaussian multiplicative
chaos at criticality, i.e. for gamma=2. If time permits, I will also
explain the construction of the dual measure (gamma larger than 2) and how
to make sense of the KPZ equation in this context.


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Regular colored graphs of positive degree, Gilles Schaeffer

Regular colored graphs are dual representations of pure colored D-dimensional complexes. These graphs can be classified with respect to an integer, their degree, much like maps are characterized by the genus. We analyse the structure of regular colored graphs of fixed positive degree and perform their exact and asymptotic enumeration. In particular we show that the generating function of the family of graphs of fixed degree is an algebraic series with a positive radius of convergence, independant of the degree. We describe the singular behavior of this series near its dominant singularity, and use the results to establish the double scaling limit of colored tensor models

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Next-to-leading order in the large N expansion of the multi-orientable tensor model, Matti Raasakka

After providing some background and motivation for random tensor models and their large N expansion, I will discuss recent results on the next-to-leading order of the large N expansion for the multi-orientable tensor model. I will describe the class of Feynman tensor graphs contributing to this order in the expansion, and the characteristic properties of the next-to-leading order series for this model. These results represent the first step towards the larger goal of defining an appropriate double-scaling limit for the multi-orientable tensor model.

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Melons are branched polymers, James Ryan

Melonic graphs constitute the family of graphs arising at leading order in the 1/N expansion of tensor models. They were shown to lead to a continuum phase, reminiscent of branched polymers. We show here that they are in fact precisely branched polymers, that is, they possess Hausdorff dimension 2 and spectral dimension 4/3.