Random walk on barely supercritical branching random walk

• 成果类型: Article
• 署名作者: van der Hofstad, Remco; Hulshof, Tim; Nagel, Jan
• 署名单位: Eindhoven University of Technology; Dortmund University of Technology
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00942-0
• 发表日期: 2020
• 页码: 1-53
• 关键词: quenched invariance-principles; reversible markov-processes; incipient infinite cluster; central-limit-theorem; galton-watson trees; biased random-walks; einstein relation; critical-behavior; brownian-motion; Scaling Limit

摘要

Let T be a supercritical Galton-Watson tree with a bounded offspring distribution that has mean mu > 1, conditioned to survive. Let phi T be a random embedding of T into Z(d) according to a simple random walk step distribution. Let T-p be percolation on T with parameter p, and let p(c) = mu(-1) be the critical percolation parameter. We consider a random walk (X-n)(n >= 1) on T-p and investigate the behavior of the embedded process phi T-p (X-n) as n -> infinity and simultaneously, T-p becomes critical, that is, p = p(n) SE arrow p(c). We show that when we scale time by n/(p(n) - p(c))(3) and space by root(p(n) - p(c))/n, the process (phi T-p (X-n))(n >= 1) converges to a d-dimensional Brownian motion. We argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments.