The Annals of Probability

Sharp metastability threshold for an anisotropic bootstrap percolation model

H. Duminil-Copin and A. C. D. Van Enter

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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.

Article information

Source
Ann. Probab., Volume 41, Number 3A (2013), 1218-1242.

Dates
First available in Project Euclid: 29 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1367241499

Digital Object Identifier
doi:10.1214/11-AOP722

Mathematical Reviews number (MathSciNet)
MR3098677

Zentralblatt MATH identifier
1356.60166

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 83B43 83C43

Keywords
Bootstrap percolation sharp threshold anisotropy metastability

Citation

Duminil-Copin, H.; Van Enter, A. C. D. Sharp metastability threshold for an anisotropic bootstrap percolation model. Ann. Probab. 41 (2013), no. 3A, 1218--1242. doi:10.1214/11-AOP722. https://projecteuclid.org/euclid.aop/1367241499


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