Percolation on hyperbolic groups

Abstract: Many questions in probability theory concern the way the geometry of a space influences the behaviour of random processes on that space, and in particular how the geometry of a space is affected by random perturbations. One of the simplest models of such a random perturbation is percolation, in which the edges of a graph are either deleted or retained independently at random with retention probability p. We are particularly interested in phase transitions, in which the geometry of the percolated subgraph undergoes a qualitative change as p is varied through some special value. Although percolation has traditionally been studied primarily in the context of Euclidean lattices, the behaviour of percolation in more exotic settings has recently attracted a great deal of attention. In this talk, I will discuss conjectures and results concerning percolation on the Cayley graphs of nonamenable groups and hyperbolic spaces, and give a taste of the proof of our recent result that percolation in any hyperbolic graph has a non-trivial phase in which there are infinitely many infinite clusters.

Frenchprobability

Audience: researchers in the topic

( slides )

Les probabilités de demain webinar

Series comments: There will be no seminar on 1st June because it is a public holiday in France (Pentecôte). The session on 15th June is the last one.