Unpredictable paths and percolation

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Abstract
We construct a nearest-neighbor processfSng on Z that is less predictable than simple random walk, in the sense that given the process until time n, the conditional probability that Sn+k = x is uniformly bounded by Ck for some > 1=2. From this process, we obtain a probability measure on oriented paths in Z3 such that the number of intersections of two paths chosen independently according to , has an exponential tail. (For d 4, the uniform measure on oriented paths from the origin in Zd 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 Zd are transient for all d 3.

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