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Sharp metastability threshold for an anisotropic bootstrap percolation model. (English) Zbl 1356.60166

Ann. Probab. 41, No. 3A, 1218-1242 (2013); erratum ibid. 44, No. 2, 1599 (2016).
Summary: 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
82C43 Time-dependent percolation in statistical mechanics
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