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Zbl 0937.60070
Benjamini, Itai; Pemantle, Robin; Peres, Yuval
Unpredictable paths and percolation. (English)
[J] Ann. Probab. 26, No.3, 1198-1211 (1998). ISSN 0091-1798

Summary: We construct a nearest-neighbor process $\{S_n\}$ on ${\bold Z}$ 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 $Ck^{-\alpha}$ for some $\alpha>1/2$. From this process, we obtain a probability measure $\mu$ on oriented paths in ${\bold Z}^3$ such that the number of intersections of two paths, chosen independently according to $\mu$, has an exponential tail. (For $d\ge 4$, the uniform measure on oriented paths from the origin in ${\bold Z}^d$ 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 {\it G. R. Grimmet}, {\it H. Kesten} and {\it Y. Zhang} [Probab. Theory Relat. Fields 96, No. 1, 33-44 (1993; Zbl 0791.60095)], who proved that supercritical percolation clusters in ${\bold Z}^d$ are transient for all $d\ge 3$.

MSC 2000:: *60J45 Probabilistic potential theory
60J10 Markov chains with discrete parameter
60J65 Brownian motion
60K35 Interacting random processes
60G50 Sums of independent random variables

Keywords: percolation; transience; electrical networks; multitype branching process

Citations: Zbl 0791.60095

Cited in: Zbl 1143.62006 Zbl 0947.60094 Zbl 0957.60097 Zbl 0937.60071

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