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Unpredictable paths and percolation. (English) Zbl 0937.60070

Summary: We construct a nearest-neighbor process \(\{S_n\}\) on \({\mathbf Z}\) 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 \({\mathbf Z}^3\) 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 G. R. Grimmet, H. Kesten and Y. Zhang [Probab. Theory Relat. Fields 96, No. 1, 33-44 (1993; Zbl 0791.60095)], who proved that supercritical percolation clusters in \({\mathbf Z}^d\) are transient for all \(d\geq 3\).

MSC:

60J45 Probabilistic potential theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J65 Brownian motion
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks

Citations:

Zbl 0791.60095
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References:

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