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Unpredictable paths and percolation. (English) Zbl 0937.60070
Summary: We construct a nearest-neighbor process $\left\{{S}_{n}\right\}$ 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 $C{k}^{-\alpha }$ 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 $d\ge 4$, 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 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 $d\ge 3$.

##### MSC:
 60J45 Probabilistic potential theory 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60J65 Brownian motion 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G50 Sums of independent random variables; random walks