Bootstrap Percolation on Infinite Trees and Non-Amenable Groups

Abstract. 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 Zd; 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 th...

• Type: Journal
• Year: 2006
• Where: CPC
• Authors: József Balogh, Yuval Peres, Gábor Pete
• Keywords: CPC 2006, Critical Probability, Deterministic Spreading Rule, Random Initial Configuration