The weak/strong survival transition on trees and nonamenable graphs

Various stochastic processes on nonamenable graphs and manifolds of exponential volume growth exhibit phases that do not occur in the corresponding processes on amenable graphs. Examples include: (1) branching diffusion and random walk on hyperbolic space, which for intermediate branching rates may survive globally but not locally; (2) contact processes on homogeneous trees, which likewise can survive globally while dying out locally; and (3) percolation on Cayley graphs of nonamenable groups, where for certain parameter values infinitely many infinite percolation clusters may coincide. This article surveys some of what is known about the intermediate phases and the upper phase transitions for these processes. Mathematics Subject Classification (2000). Primary 00A05; Secondary 00B10.