EXTINCTION OF CONTACT AND PERCOLATION PROCESSES IN A RANDOM ENVIRONMENT

• 成果类型: Article
• 署名作者: KLEIN, A
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176988601
• 发表日期: 1994
• 页码: 1227-1251
• 关键词: transverse field; ising-model; localization; decay

摘要

We consider the (inhomogeneous) percolation process on Z(d) x R defined as follows: Along each vertical line {x} x R we put cuts at times given by a Poisson point process with intensity delta(x), and between each pair of adjacent vertical lines {x} x R and {y} x R we place bridges at times given by a Poisson point process with intensity lambda(x, y). We say that (x, t) and (y, s) are connected (or in the same cluster) if there is a path from (x, t) to (y, s) made out of uncut segments of vertical lines and bridges. If we consider only oriented percolation, we have the graphical representation of the (inhomogeneous) d-dimensional contact process. We consider these percolation and contact processes in a random environment by taking delta = {delta(x); x epsilon Z(d)} and lambda = {lambda(x, y); x, y epsilon Z(d), parallel to x-y parallel to(2) = 1} to be independent families of independent identically distributed strictly positive random variables; we use delta and lambda for representative random variables. We prove extinction (i.e., no percolation) ofthese percolation and contact processes, for almost every delta and lambda, if delta and lambda satisfy E{(log(1 + lambda))(beta)} < infinity and E{(log(1 + 1/delta))(b)eta} < infinity for some beta > 2d(2)(1 + root 1 + 1/d + 1/2d), and if E{(log(1 + lambda/delta))(<)beta} is sufficiently small.