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Unpredictable paths and percolation. (English) Zbl 0937.60070
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:
60J45Probabilistic potential theory
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
60J65Brownian motion
60K35Interacting random processes; statistical mechanics type models; percolation theory
60G50Sums of independent random variables; random walks
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References:
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