NUCLEATION SCALING IN JIGSAW PERCOLATION

• 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
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/16-AAP1206
• 发表日期: 2017
• 页码: 395-438
• 关键词: bootstrap percolation; sharp threshold

摘要

Jigsaw percolation is a nonlocal process that iteratively merges connected clusters in a deterministic puzzle graph by using connectivity properties of a random people graph on the same set of vertices. We presume the Erdos-Renyi people graph with edge probability p and investigate the probability that the puzzle is solved, that is, that the process eventually produces a single cluster. In some generality, for puzzle graphs with N vertices of degrees about D (in the appropriate sense), this probability is close to 1 or small depending on whether pD log N is large or small. The one dimensional ring and two dimensional torus puzzles are studied in more detail and in many cases the exact scaling of the critical probability is obtained. The paper strengthens several results of Brummitt, Chatterjee, Dey, and Sivakoff who introduced this model.