PERCOLATION OF ARBITRARY WORDS IN (0,1)(N)

• 成果类型: Article
• 署名作者: BENJAMINI, I; KESTEN, H
• 署名单位: Cornell University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176988173
• 发表日期: 1995
• 页码: 1024-1060
• 关键词: probability; lattice; TREE

摘要

Let g be a (possibly directed) locally finite graph with countably infinite vertex set V. Let {X(upsilon): v is an element of V} be an i.i.d. family of random variables with P{X(upsilon) = 1} = 1- P{X(upsilon) = 0} = p. Finally, let xi = (xi(1),xi(2),...) be a generic element of {0,1}(N); such a xi is called a word. We say that the word xi is seen from the vertex upsilon if there exists a self-avoiding path (upsilon,upsilon(1),upsilon(2),...) on g starting at upsilon and such that X(upsilon(i)) = xi(i) for i greater than or equal to 1. The traditional problem in (site) percolation is whether P{(1,1,1,...) is seen from upsilon} > 0. So-called AB-percolation occurs if P{(1,0,1,0,1,0,...) is seen from upsilon} > 0. Here we investigate (a) whether P{all words are seen from upsilon} > 0 (for a fixed upsilon) and (b) whether P{all words are seen from some upsilon} = 1. We show that both answers are positive if g = Z(d), or even Z(+)(d) with all edges oriented in the ''positive direction,'' when d is sufficiently large. We show that on the oriented Z(+)(3) the answer to (a) is negative, but we do not know the answer to (b) on Z(+)(3). Various graphs d are constructed (almost all of them trees) for which the set of words xi which can be seen from a given upsilon (or from some upsilon) is large, even though it is w.p.1 not the set of all words.