The Annals of Probability

Unpredictable paths and percolation

Itai Benjamini, Robin Pemantle, and Yuval Peres

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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 $d \geq 3$.

Article information

Ann. Probab., Volume 26, Number 3 (1998), 1198-1211.

First available in Project Euclid: 31 May 2002

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Zentralblatt MATH identifier

Primary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J65: Brownian motion [See also 58J65] 60J15 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Percolation transience electrical networks multitype branching process


Benjamini, Itai; Pemantle, Robin; Peres, Yuval. Unpredictable paths and percolation. Ann. Probab. 26 (1998), no. 3, 1198--1211. doi:10.1214/aop/1022855749.

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