Hostname: page-component-7c8c6479df-94d59 Total loading time: 0 Render date: 2024-03-28T18:05:18.509Z Has data issue: false hasContentIssue false

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)