UNPREDICTABLE PATHS AND PERCOLATION

We construct a nearest-neighbor process {S-n} on Z that is less predic table than simple random walk, in the sense that given the process unt il time n, the conditional probability that Sn+k = x is uniformly boun ded by Ck(-alpha) for some alpha > 1/2. From this process, we obtain a probability measure mu on oriented paths in Z(3) such that the number of intersections of two paths, chosen independently according to mu, has an exponential tail. (For d greater than or equal to 4, the unifor m 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, orie nted percolation clusters are transient if the retention parameter p i s 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 greater than or equal to 3.