Abstract

In Bernoulli bond percolation, the edges of a graph are either deleted or retained independently at random, with retention probability p. As p changes, the geometry of the retained subgraph is expected to undergo one or more abrupt changes at special values of p, known as phase transitions. Although traditionally studied primarily on Euclidean lattices, the study of percolation on more general graphs, and in particular on general Cayley graphs, has been popular since the 90’s and has revealed several connections between probability and geometric group theory. A central conjecture in the area, due to Benjamini and Schramm, is that if G is a Cayley graph of a nonamenable group, then there exists an interval of values of p for which the open subgraph contains infinitely many infinite connected components almost surely. The goal of my talk is to survey what has been done on this problem and to discuss my recent proof that the conjecture is true for hyperbolic groups.