Ergodic Theory and Dynamical Systems



Hausdorff dimension of the harmonic measure on trees


VADIM A. KAIMANOVICH a1
a1 Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK and CNRS UMR-6625, Institut de Recherche Mathématique de Rennes, Campus Beaulieu, Rennes 35042, France

Abstract

For a large class of Markov operators on trees we prove the formula ${\bf HD}\,\nu=h/l$ connecting the Hausdorff dimension of the harmonic measure $\nu$ on the tree boundary, the rate of escape $l$ and the asymptotic entropy $h$. Applications of this formula include random walks on free groups, conditional random walks, random walks in random environment and random walks on treed equivalence relations.

(Received November 8 1996)
(Revised April 5 1997)