The Metastability Threshold for Modified Bootstrap Percolation in d Dimensions

2006/03/28 by Alexander E. Holroyd, Holroyd, Alexander E.
Mathematics · Physics and Astronomy · #60K35 #82B43 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.MP #math.PR #msc:60K35 #msc:82B43

paper · pdf · doi:10.48550/arxiv.math/0603645

20 pages, 3 figures (added discussion, corrected typo in (24))

arxiv created 2006/04/03 · arxiv updated 2009/12/01

Abstract

In the modified bootstrap percolation model, sites in the cube 1,...,L^d are initially declared active independently with probability p. At subsequent steps, an inactive site becomes active if it has at least one active nearest neighbour in each of the d dimensions, while an active site remains active forever. We study the probability that the entire cube is eventually active. For all d>=2 we prove that as L→∞ and p→ 0 simultaneously, this probability converges to 1 if L=exp^d-1 (lambda+epsilon)/p, and converges to 0 if L=exp^d-1 (lambda-epsilon)/p, for any epsilon>0. Here expⁿ denotes the n-th iterate of the exponential function, and the threshold lambda equals pi²/6 for all d.

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