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Étude de la quadrangulation infinie uniforme

Abstract : Quadrangulations are proper embeddings of finite connected graphs in the two-dimensional sphere for which all faces have degree 4. Our main object of study is the so-called uniform infinite quadrangulation. This infinite random map has been defined in two different ways. The first definition consists in taking the local limit of large random (finite) quadrangulations whose laws are uniform over the set of all quadrangulations with a given size. The second definition takes advantage of a bijection between quadrangulations and well-labelled trees. The starting point is a suitably defined uniform infinite random tree whose law is mapped to the set of all infinite quadrangulations using an extended version of the bijection. In the second chapter of this manuscript, we prove that the two previous definitions lead to the same object. This fact is not trivial as, in the infinite setting, the bijection between quadrangulations and trees is not continuous for the topology of local convergence. The proof depends on studying some combinatorial properties of the bijection and estimating the distribution of vertices with small labels in high generations of the uniform infinite tree. In chapter 3, we use the equivalence of the two definitions to compute scaling limits for the uniform infinite quadrangulation. Indeed, quantities such as the volume of balls around a distinguished point of the quadrangulation can be evaluated by studying the uniform infinite tree. The main technical ingredient of this chapter is the convergence of the rescaled contour functions of the uniform infinite tree towards a stochastic process linked to the Brownian snake.
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Contributor : Laurent Ménard Connect in order to contact the contributor
Submitted on : Friday, March 26, 2010 - 10:49:57 AM
Last modification on : Sunday, June 26, 2022 - 11:51:35 AM
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  • HAL Id : tel-00467174, version 1


Laurent Ménard. Étude de la quadrangulation infinie uniforme. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2009. Français. ⟨tel-00467174⟩



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