Jean-Francois Le Gall (Ecole normale superieure et Universite Paris-Sud)

In the first part of the lectures, we will describe connections between

discrete and continuous random trees. In particular we will introduce the

Continuum Random Tree (CRT), which arises as the scaling limit of critical

Galton-Watson trees conditioned to have a large fixed number of vertices.

We will in particular explain how the CRT is coded from the normalized

Brownian excursion, and how this coding allows one to make explicit

calculations for various functionals of this random tree.

The CRT and related probabilistic objects such as the Brownian snake

and the Integrated Super-Brownian Excursion (ISE) appear in a number of limit

theorems for models from statistical physics or from the theory of

interacting particle systems.In the second part of the lectures, we will use random trees to understand

scaling limits of large random planar maps. Planar maps are connected

graphs embedded in the plane. They play an important role both in combinatorics and

in theoretical physics where they serve as models of random surfaces.

Thanks to certain correspondences between planar maps and labelled trees,

we will explain how the scaling limit of large bipartite planar maps is

described in terms of the CRT equipped with Brownian labels (or equivalently

the Brownian snake driven by a normalized Brownian excursion). This

makes it possible to derive several interesting limit distributions for

random planar maps.References.

J.F. Le Gall. Random trees and applications.

Probability Surveys 2, 245-311 (2005) (electronic)P. Chassaing, G. Schaeffer. Random planar lattices and

integrated super-Brownian excursion. Probab. Th. Related

Fields 128, 161-212 (2004)J.F. Le Gall. The topological structure of scaling limits of

large planar maps. Preprint.

arXiv: math.PR/0607567Additional references will be given during the lectures.