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Unpredictable paths and percolation. (English) Zbl 0937.60070
Summary: We construct a nearest-neighbor process {S n } on 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 -α for some α>1/2. From this process, we obtain a probability measure μ on oriented paths in 𝐙 3 such that the number of intersections of two paths, chosen independently according to μ, has an exponential tail. (For d4, the uniform measure on oriented paths from the origin in 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 𝐙 d are transient for all d3.

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