Arbres Aléatoires (continus) et Applications – A3

Continuum random trees is nowadays intensively studied, mainly because of numerous applications. The goal of this project is to study some properties of these continuum trees : - continuum tree valued processes and random walks on random trees We aim to define continuum tree valued Markov processes that would be the continuous analogue of discrete tree valued random walk introduced by Aldous-Pitman. These processes would give an alternative construction of continuum random trees as limit trees of these processes. A different approach to study the regularity of continuum random trees is to study a random walk on the discrete approximation of the tree. These techniques have already been fruitful and we want to go on in that direction. - Fragmentation Fragmentations processes model masses that undergo random dislocations, with numerous applications in physics, biology and computer science. Some of these processes are constructed by cutting randomly continuum random trees. This construction is very rich and we hope to get interesting results on these fragmentations by studying continuum random trees. - planar maps Planar maps have now numerous applications in physics and computer science. This project aims to study the asymptotic behaviour of these maps when their size tends to infinity, defining in particular the limiting objects that can be viewed as "continuum maps". - percolation We know that the incipient infinite cluster in high-dimensional percolation converges, after renormalization, to the integrated super-Brownian excursion which is easily related to continuum trees. We hope that the study of continuum random trees will give some interesting insight on the behaviour of this infinite cluster when the percolation probability is close to its critical value.