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

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Date Posted: 27 November 2017

This document has been peer reviewed.