CRITICAL DENSITY OF ACTIVATED RANDOM WALKS ON TRANSITIVE GRAPHS
成果类型：
Article
署名作者：
Stauffer, Alexandre; Taggi, Lorenzo
署名单位：
University of Bath; Technical University of Darmstadt
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/17-AOP1224
发表日期：
2018
页码：
2190-2220
关键词：
摘要：
We consider the activated random walk model on general vertex-transitive graphs. A central question in this model is whether the critical density mu(c) for sustained activity is strictly between 0 and 1. It was known that mu c > 0 on Z(d), d >= 1, and that mu(c) < 1 on Z for small enough sleeping rate. We show that mu(c) -> 0 as lambda -> 0 in all vertex-transitive transient graphs, implying that mu(c) < 1 for small enough sleeping rate. We also show that mu(c) < 1 for any sleeping rate in any vertex-transitive graph in which simple random walk has positive speed. Furthermore, we prove that mu(c) > 0 in any vertex-transitive amenable graph, and that mu(c) ( 0, 1) for any sleeping rate on regular trees.
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