The Annals of Probability

Asymptotic entropy and Green speed for random walks on countable groups

Sébastien Blachère, Peter Haïssinsky, and Pierre Mathieu

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Abstract

We study asymptotic properties 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.

Article information

Source
Ann. Probab., Volume 36, Number 3 (2008), 1134-1152.

Dates
First available in Project Euclid: 9 April 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1207749092

Digital Object Identifier
doi:10.1214/07-AOP356

Mathematical Reviews number (MathSciNet)
MR2408585

Zentralblatt MATH identifier
1146.60008

Subjects
Primary: 34B27: Green functions 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

Keywords
Green function random walks on groups

Citation

Blachère, Sébastien; Haïssinsky, Peter; Mathieu, Pierre. Asymptotic entropy and Green speed for random walks on countable groups. Ann. Probab. 36 (2008), no. 3, 1134--1152. doi:10.1214/07-AOP356. https://projecteuclid.org/euclid.aop/1207749092


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