Asymptotic entropy and Green speed for random walks on groups

Sébastien Blachère (1) , Peter Haïssinsky (2) , Pierre Mathieu (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 Picard

We study asymptotic properties of the Green metric associated to random walks on discrete transient groups. We prove that the rate of escape of the random walk computed in the Green metric equals its asymptotic entropy. Two proofs are given. One relies on integral representations of both quantities with the extended Martin kernel. The other proof (valid only when the volume growth of the group is superpolynomial) relies on a version of the so called fundamental inequality (relating the rate of escape, the entropy and the logarithmic volume growth) extended to random walk with unbounded support.

Mots clés: Green function; Random walks on groups

Domaines: Probabilités [math.PR]; Théorie des groupes [math.GR]

Sébastien Blachère, Peter Haïssinsky, Pierre Mathieu. Asymptotic entropy and Green speed for random walks on groups. 2006. ⟨hal-00086803v1⟩