SPACE-TIME PERCOLATION AND DETECTION BY MOBILE NODES

• 成果类型: Article
• 署名作者: Stauffer, Alexandre
• 署名单位: University of Bath
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/14-AAP1052
• 发表日期: 2015
• 页码: 2416-2461
• 关键词: connectivity properties; phase-transitions; geometric graphs; SPREAD

摘要

Consider the model where nodes are initially distributed as a Poisson point process with intensity lambda over R-d and are moving in continuous time according to independent Brownian motions. We assume that nodes are capable of detecting all points within distance r of their location and study the problem of determining the first time at which a target particle, which is initially placed at the origin of R-d, is detected by at least one node. We consider the case where the target particle can move according to any continuous function and can adapt its motion based on the location of the nodes. We show that there exists a sufficiently large value of lambda so that the target will eventually be detected almost surely. This means that the target cannot evade detection even if it has full information about the past, present and future locations of the nodes. Also, this establishes a phase transition for lambda since, for small enough lambda, with positive probability the target can avoid detection forever. A key ingredient of our proof is to use fractal percolation and multi-scale analysis to show that cells with a small density of nodes do not percolate in space and time.