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Open Access
July 1998 Unpredictable paths and percolation
Itai Benjamini, Robin Pemantle, Yuval Peres
Ann. Probab. 26(3): 1198-1211 (July 1998). DOI: 10.1214/aop/1022855749

Abstract

4 We construct a nearest-neighbor process Sn on Z that is less predictable than simple random walk, in the sense that given the process until time n, the conditional probability that Sn+k=x is uniformly bounded by Ck for some α>1/2. From this process, we obtain a probability measure μ on oriented paths in Z3 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 Zd 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 Zd are transient for all d3.

Citation

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Itai Benjamini. Robin Pemantle. Yuval Peres. "Unpredictable paths and percolation." Ann. Probab. 26 (3) 1198 - 1211, July 1998. https://doi.org/10.1214/aop/1022855749

Information

Published: July 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0937.60070
MathSciNet: MR1634419
Digital Object Identifier: 10.1214/aop/1022855749

Subjects:
Primary: 60J10 , 60J45
Secondary: 60J15 , 60J65 , 60K35

Keywords: Electrical networks , multitype branching process , percolation , transience

Rights: Copyright © 1998 Institute of Mathematical Statistics

Vol.26 • No. 3 • July 1998
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