Unpredictable paths and percolation
Unpredictable paths and percolation
复制标题
不可预测的路径和渗透
DOI：
10.1214/aop/1022855749
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发表时间：
1998
影响因子：
2.3
通讯作者：
Y. Peres
中科院分区：
文献类型：
--
作者：
I. Benjamini;Robin Pemantle;Y. Peres
我们在Z上构造一个最近邻过程{Sn}，它比简单随机游动更难预测，在这个意义上，给定这个过程直到时间n，Sn +k = x的条件概率一致有界于Ck -α，其中α > 1/2。由此得到Z3中定向路的概率测度μ，使得根据μ独立选择的两条路的交叉数具有指数尾. (For d ≥ 4，则从Zd中原点出发的定向路上的一致测度具有此性质。我们表明，在任何图中存在这样的措施的路径，定向渗流集群是短暂的，如果保留参数P是足够接近1。这产生了Grimmett，Kesten和Zhang定理的扩展，他们证明了Z d中的超临界渗流团簇对于所有d ≥ 3都是瞬态的。
We construct a nearest-neighbor process {S n } on Z that is less predictable than simple random walk, in the sense that given the process until time n, the conditional probability that S n+k = x is uniformly bounded by Ck -α for some α > 1/2. From this process, we obtain a probability measure μ on oriented paths in Z 3 such that the number of intersections of two paths, chosen independently according to μ, has an exponential tail. (For d ≥ 4, the uniform measure on oriented paths from the origin in Z d has this property.) We show that on any graph where such a measure on paths exists, oriented percolation clusters are transient if the retention parameter P is close enough to 1. This yields an extension of a theorem of Grimmett, Kesten and Zhang, who proved that supercritical percolation clusters in Z d are transient for all d ≥ 3.