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Ergodic theory on Galton—Watson trees: speed of random walk and dimension of harmonic measure

Published online by Cambridge University Press:  19 September 2008

Russell Lyons ,
Robin Pemantle  and
Yuval Peres
Russell Lyons
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405-5701, USA
Robin Pemantle
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA
Yuval Peres
Affiliation:
Department of Statistics, University of California, Berkeley, CA 94720, USA
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Abstract

We consider simple random walk on the family tree T of a nondegenerate supercritical Galton—Watson branching process and show that the resulting harmonic measure has a.s. strictly smaller Hausdorff dimension than that of the whole boundary of T. Concretely, this implies that an exponentially small fraction of the nth level of T carries most of the harmonic measure. First-order asymptotics for the rate of escape, Green function and the Avez entropy of the random walk are also determined. Ergodic theory of the shift on the space of random walk paths on trees is the main tool; the key observation is that iterating the transformation induced from this shift to the subset of ‘exit points’ yields a nonintersecting path sampled from harmonic measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 3 , June 1995 , pp. 593 - 619
Copyright
Copyright © Cambridge University Press 1995

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