CONTINUUM RANDOM TREES AND APPLICATIONS
In this project, we aim to study several aspects of continuum random
- Tree-valued processes and random walks on random trees.
We want to define tree-valued markov processes which would be the
continuum analogous of discrete-tree-valued random walks introduced
by Aldous-Pitman. These processes would give a new definitions of
continuum random trees as limits of theses processes.
In another direction, a way to study the regularity of continuum trees
is to consider random walks on discrete approximations of these
trees. This approach has already led to interesting results and we
want to study further this idea.
Fragmentation processes model masses that undergo dislocation as time
passes, with numerous applications in physics, biology and computer
science. Some of these processes have been constructed by a random
fragmentation of continuum trees. This construction is very rich and
we hope to get interesting results on these fragmentation processes
thanks to the study of continuum trees.
Random graphs have numerous applications in physics or computer
science. This project aims to study the asymptotics of these random
graphs when thier size tends to infinity, describing in particular the
limiting objects which can be seen as "continuum graphs".
We know that the incipient infinite cluster in percolation in high
dimension converges after renormalization toward the Integrated
Super-Excursion. We hope that fine results on continuum trees will
lead to a better knowledge of the behaviour of this infinite cluster
when the percolation probability tends to the critical value.