Nearest-neighbor walks with low predictability profile and percolation in 2+ε dimensions

• 成果类型: Article
• 署名作者: Haggstrom, O; Mossel, E
• 署名单位: Chalmers University of Technology; Hebrew University of Jerusalem
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1998
• 页码: 1212-1231
• 关键词: ising-model

摘要

A few years ago, Grimmett, Resten and Zhang proved that for supercritical bond percolation on Z(3), simple random walk on the infinite cluster is a.s. transient. We generalize this result to a class of wedges in Z(3) including, for any epsilon is an element of (0, 1), the wedge W-epsilon = {(x, y, z) is an element of Z(3): x greater than or equal to 0, \z\ less than or equal to x(epsilon)} which can be thought of as representing a (2 + epsilon)-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.