SHARP METASTABILITY THRESHOLD FOR AN ANISOTROPIC BOOTSTRAP PERCOLATION MODEL

• 成果类型: Article
• 署名作者: Duminil-Copin, H.; Van Enter, A. C. D.
• 署名单位: University of Geneva; University of Groningen
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP722
• 发表日期: 2013
• 页码: 1218-1242
• 关键词: behavior

摘要

Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following anisotropic boot-strap 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.