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2006 The Metastability Threshold for Modified Bootstrap Percolation in $d$ Dimensions
Alexander Holroyd
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Electron. J. Probab. 11: 418-433 (2006). DOI: 10.1214/EJP.v11-326


In the modified bootstrap percolation model, sites in the cube $\{1,\ldots,L\}^d$ 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 $d\geq 2$ we prove that as $L\to\infty$ and $p\to 0$ simultaneously, this probability converges to $1$ if $L\geq\exp \cdots \exp \frac{\lambda+\epsilon}{p}$, and converges to $0$ if $L\leq\exp \cdots \exp \frac{\lambda-\epsilon}{p}$, for any $\epsilon>0$. Here the exponential function is iterated $d-1$ times, and the threshold $\lambda$ equals $\pi^2/6$ for all $d$.


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Alexander Holroyd. "The Metastability Threshold for Modified Bootstrap Percolation in $d$ Dimensions." Electron. J. Probab. 11 418 - 433, 2006.


Accepted: 6 June 2006; Published: 2006
First available in Project Euclid: 31 May 2016

zbMATH: 1112.60080
MathSciNet: MR2223042
Digital Object Identifier: 10.1214/EJP.v11-326

Primary: 60K35
Secondary: 82B43

Keywords: Bootstrap percolation , cellular automaton , finite-size scaling , metastability


Vol.11 • 2006
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