Continuous limits of random planar maps
Speaker: Olivier Bernardi
Affiliation: Universite Paris-Sud, France
Abstract: Planar maps are connected planar graphs embedded in the sphere and considered up to continuous deformations. Planar maps are often easier to handle than planar graphs. For instance, the enumeration of a great variety of classes of planar maps was already achieved by Tutte in the sixties (this is in contrast with planar graphs whose precise asymptotic enumeration was only achieved in 2005 by Gimenez and Noy). More recently, and notably with the works by Schaeffer, a bijective approach was developed for enumerating maps. This approach gave access to new informations on maps, and in particular to their metric properties (here and in the following, the metric refers to the graph distance between vertices of the maps).
In these lectures, we will present recent results by Le Gall et al. concerning the continuous limit of random maps. More precisely, we shall consider a map Qn chosen uniformly at random among quadrangulations (maps whose face have degree 4) with n vertices. We view the random quadrangulation Qn as a random metric space and consider its convergence in distribution (for the Gromov-Hausdorff distance) as n goes to infinity. The ultimate goal of the talks will be to explain that Qn converges (at least along subsequences) to a random compact metric space closely related to the Brownian map introduced by Marckert and Mokkadem. This parallels the fact that random plane trees converge in distribution to the continuous random tree of Aldous.
We will not focus on the details of the proofs but rather try to introduce the main concepts and ideas. In particular, we shall present a fundamental bijection of Schaeffer between quadrangulations and so-called well-labelled trees. We will see how this bijection can be used to count quadrangulations and show that the typical graph distance between two vertices in Qn is of order n1/4. We will define continuous trees and the continuous random tree. And we will discuss convergence of (random) metric spaces in the sense of the Gromov-Hausdorff distance.
The main reference for the talk is the paper of Jean-Francois Le Gall: The topological structure of scaling limits of large planar maps, Inventiones mathematicae 169 (2007) 621-670. This paper is available at http://www.dma.ens.fr/~legall/Maps2.pdf.