Abstract
4 We construct a nearest-neighbor process Sn 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 α>1/2. From this process, we obtain a probability measure μ on oriented paths in Z3 such that the number of intersections of two paths, chosen independently according to μ, 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.
Citation
Itai Benjamini. Robin Pemantle. Yuval Peres. "Unpredictable paths and percolation." Ann. Probab. 26 (3) 1198 - 1211, July 1998. https://doi.org/10.1214/aop/1022855749
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