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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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


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

First available in Project Euclid: 29 April 2013

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

Bootstrap percolation sharp threshold anisotropy metastability


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.

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