Statistics Papers
Document Type
Journal Article
Date of this Version
1998
Publication Source
The Annals of Probability
Volume
26
Issue
3
Start Page
1212
Last Page
1231
DOI
10.1214/aop/1022855750
Abstract
A few years ago, Grimmett, Kesten and Zhang proved that for supercritical bond percolation on Z3, simple random walk on the infinite cluster is a.s. transient. We generalize this result to a class of wedges in Z3 including, for any ε ϵ (0,1), the wedge Wε={(x,y,z) ϵ Z3 : x ≥ 0, |z| ≤ x ε} which can be thought of as representing a (2 + ε)-dimensional lattice. Our proof builds on recent work of Benjamini, Pemantle and Peres, and involves the construction of finite-energy flows using nearest-neighbor walks on Z with low predictability profile. Along the way, we obtain some new results on attainable decay rates for predictability profiles of nearest-neighbor walks.
Keywords
percolation, random walk, transience, predictability profile, Ising model
Recommended Citation
Häggström, O., & Mossel, E. (1998). Nearest-Neighbor Walks With Low Predictability Profile and Percolation in 2+ϵ Dimensions. The Annals of Probability, 26 (3), 1212-1231. http://dx.doi.org/10.1214/aop/1022855750
Date Posted: 27 November 2017
This document has been peer reviewed.