Open Access
July 1998 Unpredictable paths and percolation
Itai Benjamini, Robin Pemantle, Yuval Peres
Ann. Probab. 26(3): 1198-1211 (July 1998). DOI: 10.1214/aop/1022855749


4 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 $Ck^{-\infty}$ for some $\alpha > 1/2$. From this process, we obtain a probability measure $\mu$ on oriented paths in $\mathbf{Z}^3$ such that the number of intersections of two paths, chosen independently according to $\mu$, has an exponential tail. (For $d \geq 4$, the uniform measure on oriented paths from the origin in $\mathbf{Z}^d$ 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 $\mathbf{Z}^d$ are transient for all $d \geq 3$.


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Itai Benjamini. Robin Pemantle. Yuval Peres. "Unpredictable paths and percolation." Ann. Probab. 26 (3) 1198 - 1211, July 1998.


Published: July 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0937.60070
MathSciNet: MR1634419
Digital Object Identifier: 10.1214/aop/1022855749

Primary: 60J10 , 60J45
Secondary: 60J15 , 60J65 , 60K35

Keywords: Electrical networks , multitype branching process , percolation , transience

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.26 • No. 3 • July 1998
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