Asymptotic entropy and Green speed for random walks on countable groups

• 成果类型: Article
• 署名作者: Blachere, Sebastien; Haissinsky, Peter; Mathieu, Pierre
• 署名单位: Aix-Marseille Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/07-AOP356
• 发表日期: 2008
• 页码: 1134-1152
• 关键词: discrete-groups; BOUNDARY; GROWTH; drift

摘要

We study asymptotic proper-ties of the Green metric associated with transient random walks on countable groups. We prove that the rate of escape of the random walk computed in the Green metric equals its asymptotic entropy. The proof relies on integral representations of both quantities with the extended Martin kernel. In the case of finitely generated groups, where this result is known (Benjamini and Peres [Probab. Theory Related Fields 98 (1994) 91-112]), we give an alternative proof relying on a version of the so-called fundamental inequality (relating the rate of escape, the entropy and the logarithmic volume growth) extended to random walks with unbounded support.