| The Metastability Threshold for Modified Bootstrap Percolation in d Dimensions (2006) | |||||||||||
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...exp [(λ+ε)/p], and converges to 0 if L ≤ exp...exp [(λ-ε)/p], for any ε > 0. Here the exponential function is iterated d-1 times, and the threshold λ equals π2/6 for all d. | |||||||||||
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