Random trees and planar maps.

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/0607567

Additional references will be given during the lectures.