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The Metastability Threshold for Modified Bootstrap Percolation in d Dimensions
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Alexander E Holroyd, Department of Mathematics, University of British Columbia
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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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Full text: PDF
Pages: 418-433
Published on: June 6, 2006
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Electronic Journal of Probability. ISSN: 1083-6489 |
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