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Astérisque - Titles - 326 (2009) 299-320

Titles < 2009 < 326

Séminaire Bourbaki, volume 2007/2008, Exposés 982-996
Astérisque 326 (2009), x+409 pages
Buy the book
Presentation, Summary

Grands graphes planaires aléatoiresarte brownienne
Vincent Beffara
Astérisque 326 (2009), 299-320

Résumé :
La mécanique statistique en dimension deux a toujours été une importante source d'inspiration tant pour les mathématiciens que pour les physiciens ; un outil important dans sa compréhension, et qui est longtemps resté mystérieux, est la gravitation quantique, qui consiste à rendre le graphe sous-jacent (sur lequel le modèle est défini) lui-même aléatoire. Pour pouvoir passer à la limite thermodynamique, il est alors nécessaire de comprendre le comportement asymptotique de tels graphes planaires aléatoires quand leur volume tend vers l'infini. On présentera les travaux récents de J.-F. Le Gall dans ce domaine, et en particulier le résultat fondamental suivant : soit G_n une quadrangulation aléatoire de la sphère, choisie uniformément parmi les quadrangulations à n faces ; on munit G_n de sa distance de graphe renormalisée par un facteur n^-1/4. Alors, toute limite en loi d'une sous-suite de (G_n) (pour la topologie de Gromov-Hausdorff) est homéomorphe à une sphère.

Mots-clefs : Arbres aléatoires, arbres aléatoires continus, carte brownienne.

Abstract:
Large random planar maps, and the Brownian map
Two-dimensional statistical mechanics have long been an important source of inspiration to both mathematicians and physicists; an important tool in its understanding, and which remains quite mysterious, is known as quantum gravity and amounts to letting the underlying graph (on which a random model is defined) random itself. Taking a thermodynamical limit then involves understanding the asymptotic behavior of such a random graph as its volume tends to infinity. We will present recent work by J.-F. Le Gall in this direction, focussing on the following fundamental result (joint with F. Paulin): For each n>0, let G_n be a uniformly random quadrangulation of the sphere into n faces, equipped with its graph distance multiplied by a factor n^-1/4. Then, any subsequential limit in distribution of the sequence (G_n) (for the Gromov-Hausdorff topology) is homeomorphic to the 2-sphere.

Keywords: Random maps, random trees, continuum random trees, Brownian map.

Class. math. : 05C12, 05C80, 60B05, 60F17.


ISSN : 0303-1179
Publié avec le concours de : Centre National de la Recherche Scientifique

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