BOOTSTRAP PERCOLATION ON THE PRODUCT OF THE TWO-DIMENSIONAL LATTICE WITH A HAMMING SQUARE

• 成果类型: Article
• 署名作者: Gravner, Janko; Sivakoff, David
• 署名单位: University of California System; University of California Davis; University System of Ohio; Ohio State University; University System of Ohio; Ohio State University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1497
• 发表日期: 2020
• 页码: 145-174
• 关键词: spatial epidemics; threshold; BEHAVIOR; graph

摘要

Bootstrap percolation on a graph is a deterministic process that iteratively enlarges a set of occupied sites by adjoining points with at least. occupied neighbors. The initially occupied set is random, given by a uniform product measure with a low density p. Our main focus is on this process on the product graph Z(2) x K-n(2), where K-n is a complete graph. We investigate how p scales with n so that a typical site is eventually occupied. Under critical scaling, the dynamics with even. exhibits a sharp phase transition, while odd. yields a gradual percolation transition. We also establish a gradual transition for bootstrap percolation on Z(2) x K-n. The community structure of the product graphs connects our process to a heterogeneous bootstrap percolation on Z(2). This natural relation with a generalization of polluted bootstrap percolation is the leading theme in our analysis.