Bootstrap percolation on the random regular graph

The k-parameter bootstrap percolation on a graph is a model of an interacting particle system, which can also be viewed as a variant of a cellular automaton growth process with threshold k >= 2. At the start, each of the graph vertices is active with probability p and inactive with probability I - p, independently of other vertices. Presence of active vertices triggers a bootstrap percolation process controlled by a recursive rule: an active vertex remains active forever, and a currently inactive vertex becomes active when at least k of its neighbors are active. The basic problem is to identify, for a given graph, p(-) and p(+) such that for p < p(-) (p > p(+) resp.) the probability that all vertices are eventually active is very close to 0 (1 resp.). The bootstrap percolation process is a deterministic process on the space of subsets of the vertex set, which is easy to describe but hard to analyze rigorously in general. We study the percolation on the random d-regular graph, d >= 3, via analysis of the process on the multigraph counterpart of the graph. Here, thanks to a "principle of deferred decisions," the percolation dynamics is described by a surprisingly simple Markov chain. Its generic state is formed by the counts of currently active and nonactive vertices having various degrees of activation capabilities. We replace the chain by a deterministic dynamical system, and use its integrals to show-via exponential supermartingales-that the percolation process undergoes relatively small fluctuations around the deterministic trajectory. This allows us to show existence of the phase transition within an interval [p(-)(n),p(+)(n)], such that (1)p(+/-)(n) -> p* = 1 - min(y epsilon(0,1)) y/P(Bin(d - 1, 1 -y) < k); (2)p(+)(n)-p(-)(n) is of order n(-1/2) for k < d(-1), and n(-epsilon n), (epsilon(n) down arrow 0, epsilon(n) log n -> infinity), for k = d - 1. Note that p* is the same as the critical probability of the process on the corresponding infinite regular tree. (c) 2006 Wiley Periodicals, Inc.