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Hausdorff dimension of the harmonic measure on trees

Published online by Cambridge University Press:  01 June 1998

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

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
Copyright
© 1998 Cambridge University Press

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