Hausdorff Dimension of the Harmonic Measure on Trees

For a large class of Markov operators on trees we prove the formula HD = h=l connecting the Hausdorr dimension of the harmonic measure 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. 0. Introduction The Hausdorr dimension HD of a measure on a metric space (X; d) is deened as the minimal Hausdorr dimension of sets of full measure and shows the \degree of singularity" (or, of \fractalness" in the newspeak) of this measure. Even if the support of the measure is the whole space, HD does not have to coincide with HD X. The Hausdorr dimension HD characterizes the polynomial rate of decreasing of the measures of balls of the metric d around typical (with respect to) points of X, in particular, if ball measures decrease regularly, i.e., the limit lim log B(x; r)= log r = exists and is the same for-a.e. x 2 X, then HD =. The subject of the present paper is the Hausdorr dimension of harmonic measures on trees (more precisely, on their boundaries). Let P be a Markov operator on a countable tree T. The tree is endowed with a natural boundary @T (the space of asymptotic classes of geodesic rays, or, the space of ends). If for the Markov chain associated with P almost all sample paths starting from a point o 2 T converge to @T, then the corresponding limit distributions = o on @T is called the harmonic measure of the operator P with the pole at the point o. In other words, the harmonic measure is the image of the measure P in the space of the sample paths x = (o; x 1 ; x 2 ; : : :) under the boundary map bnd : x 7 ! lim x n 2 @T. Denote by = o ; o 2 T the ultrametric on the boundary @T determined by the hierarchical structure of its totally disconnected topology: (1 ; 2) = e ?n , where n is the length of the common part of the geodesic rays o; 1 ] and o; 2 ] with respect to the graph distance d. Although the harmonic measure depends on the starting point, under mild irre-ducibility assumptions all harmonic …