CRITICAL RANDOM-WALK IN RANDOM ENVIRONMENT ON TREES

• 成果类型: Article
• 署名作者: PEMANTLE, R; PERES, Y
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176988379
• 发表日期: 1995
• 页码: 105-140
• 关键词: reinforced random-walk; percolation; capacity; BEHAVIOR; graphs; times

摘要

We study the behavior of random walk in random environment (RWRE) on trees in the critical case left open in previous work. Representing the random walk by an electrical network, we assume that the ratios of resistances of neighboring edges of a tree Gamma are i.i.d. random variables whose logarithms have mean zero and finite variance. Then the resulting RWRE is transient if simple random walk on Gamma is transient, but not vice versa We obtain general transience criteria for such walks, which are sharp for symmetric trees of polynomial growth. In order to prove these criteria, we establish results on boundary crossing by tree-indexed random walks. These results rely on comparison inequalities for percolation processes on trees and on some new estimates of boundary crossing probabilities for ordinary mean-zero finite variance random walks in one dimension, which are of independent interest.