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May 2013 Sharp metastability threshold for an anisotropic bootstrap percolation model
H. Duminil-Copin, A. C. D. Van Enter
Ann. Probab. 41(3A): 1218-1242 (May 2013). DOI: 10.1214/11-AOP722

Abstract

Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following “anisotropic” bootstrap percolation model: the neighborhood of a point $(m,n)$ is the set

\[\{(m+2,n),(m+1,n),(m,n+1),(m-1,n),(m-2,n),(m,n-1)\}.\]

At time 0, sites are occupied with probability $p$. At each time step, sites that are occupied remain occupied, while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied. We prove that it exhibits a sharp metastability threshold. This is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model.

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H. Duminil-Copin. A. C. D. Van Enter. "Sharp metastability threshold for an anisotropic bootstrap percolation model." Ann. Probab. 41 (3A) 1218 - 1242, May 2013. https://doi.org/10.1214/11-AOP722

Information

Published: May 2013
First available in Project Euclid: 29 April 2013

zbMATH: 1356.60166
MathSciNet: MR3098677
Digital Object Identifier: 10.1214/11-AOP722

Subjects:
Primary: 60K35 , 83B43 , 83C43

Keywords: Anisotropy , Bootstrap percolation , metastability , sharp threshold

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 3A • May 2013
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