Unpredictable paths and percolation

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 d ≥ 4, 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 Grimmett, Kesten and Zhang, who proved that supercritical percolation clusters in Z d are transient for all d ≥ 3.