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The Metastability Threshold for Modified Bootstrap Percolation in d Dimensions

By Alexander E. Holroyd


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 · · · exp λ+ɛ λ−ɛ p, and converges to 0 if L ≤ exp · · · exp p, for any ɛ> 0. Here the exponential function is iterated d − 1 times, and the threshold λ equals π2 /6 for all d

Topics: Key words, bootstrap percolation, cellular automaton, metastability, finite-size scaling 2000 Mathematics Subject Classifications, Primary 60K35, Secondary 82B43
Year: 2006
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