SUBLINEARITY OF THE TRAVEL-TIME VARIANCE FOR DEPENDENT FIRST-PASSAGE PERCOLATION

• 成果类型: Article
• 署名作者: Van den Berg, Jacob; Kiss, Demeter
• 署名单位: Centrum Wiskunde & Informatica (CWI); Vrije Universiteit Amsterdam
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP631
• 发表日期: 2012
• 页码: 743-764
• 关键词: 2-dimensional ising percolation; 1st passage percolation; greedy lattice animals; markov random-fields; finitary codings; TRANSITION

摘要

Let E be the set of edges of the d-dimensional cubic lattice Z(d), with d >= 2, and let (e), e is an element of E, be nonnegative values. The passage time from a vertex nu to a vertex omega is defined as inf(pi : nu ->omega) Sigma(e is an element of pi) t(e), where the infimum is over all paths pi from nu to omega, and the sum is over all edges e of pi. Benjamini, Kalai and Schramm [2] proved that if the t (e)'s are i.i.d. two-valued positive random variables, the variance of the passage time from the vertex 0 to a vertex nu is sublinear in the distance from 0 to nu. This result was extended to a large class of independent, continuously distributed t-variables by Benaim and Rossignol [1]. We extend the result by Benjamini, Kalai and Schramm in a very different direction, namely to a large class of models where the t(e)'s are dependent. This class includes, among other interesting cases, a model studied by Higuchi and Zhang [9], where the passage time corresponds with the minimal number of sign changes in a subcritical Ising landscape.