### 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 *Q*_{n} chosen uniformly at random among
quadrangulations (maps whose face have degree 4) with *n* vertices. We
view the random quadrangulation *Q*_{n} 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 *Q*_{n} 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 *Q*_{n} is of order *n*^{1/4}. 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`.