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Hausdorff dimension of the harmonic measure on trees

Published online by Cambridge University Press:  01 June 1998

VADIM A. KAIMANOVICH
VADIM A. KAIMANOVICH
Affiliation:
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
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 18 , Issue 3 , June 1998 , pp. 631 - 660
Copyright
© 1998 Cambridge University Press

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