Unpredictable paths and percolation.(English)Zbl 0937.60070

Summary: We construct a nearest-neighbor process $$\{S_n\}$$ on $${\mathbf Z}$$ 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 $$Ck^{-\alpha}$$ for some $$\alpha>1/2$$. From this process, we obtain a probability measure $$\mu$$ on oriented paths in $${\mathbf Z}^3$$ such that the number of intersections of two paths, chosen independently according to $$\mu$$, has an exponential tail. (For $$d\geq 4$$, the uniform measure on oriented paths from the origin in $${\mathbf Z}^d$$ 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 $${\mathbf Z}^d$$ are transient for all $$d\geq 3$$.

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

Zbl 0791.60095
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