Bootstrap Percolation on Infinite Trees and Non-Amenable Groups

Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability p, independently of each other, and a deterministic spreading rule with a fixed parameter k: if a vacant site has at least k occupied neighbors at a certain time step, then it becomes occupied in the next step. This process is well-studied on Z; here we investigate it on regular and general infinite trees and on non-amenable Cayley graphs. The critical probability is the infimum of those values of p for which the process achieves complete occupation with positive probability. On trees we find the following discontinuity: if the branching number of a tree is strictly smaller than k, then the critical probability is 1, while it is 1 − 1/k on the k-ary tree. A related result is that in any rooted tree T there is a way of erasing k children of the root, together with all their descendants, and repeating this for all remaining children, and so on, such that the remaining tree T ′ has branching number br(T ) ≤ max{br(T ) − k, 0}. We also prove that on any 2k-regular non-amenable graph, the critical probability for the k-rule is strictly positive.