The ratio set of the harmonic measure of a random walk on a hyperbolic group

For every non-elementary hyperbolic group, we show that for every random walk with finitely supported admissible step distribution, the associated entropy equals the drift times the logarithmic volume growth if and only if the corresponding harmonic measure is comparable with Hausdorff measure on the boundary. Moreover, we introduce one parameter family of probability measures which interpolates a Patterson–Sullivan measure and the harmonic measure, and establish a formula of Hausdorff spectrum (multifractal spectrum) of the harmonic measure. We also give some finitary versions of dimensional properties of the harmonic measure.

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

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700008543 ◽

1995 ◽

Vol 15 (3) ◽

pp. 593-619 ◽

Cited By ~ 47

Author(s):

Russell Lyons ◽

Robin Pemantle ◽

Yuval Peres

Keyword(s):

Random Walk ◽

Ergodic Theory ◽

Harmonic Measure ◽

Branching Process ◽

Main Tool ◽

Simple Random Walk ◽

Family Tree ◽

First Order ◽

The Family ◽

Rate Of Escape

AbstractWe consider simple random walk on the family treeTof 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 ofT. Concretely, this implies that an exponentially small fraction of thenth level ofTcarries 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.

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A Discrete Analogue of a Theorem of Makarov

Combinatorics Probability Computing ◽

10.1017/s0963548300000584 ◽

1993 ◽

Vol 2 (2) ◽

pp. 181-199 ◽

Cited By ~ 7

Author(s):

Gregory F. Lawler

Keyword(s):

Random Walk ◽

Hausdorff Dimension ◽

Harmonic Measure ◽

Simple Random Walk ◽

Discrete Analogue ◽

Connected Subset ◽

Image Position ◽

Dimension One

A theorem of Makarov states that the harmonic measure of a connected subset of ℝ2 is supported on a set of Hausdorff dimension one. This paper gives an analogue of this theorem for discrete harmonic measure, i.e., the hitting measure of simple random walk. It is proved that for any 1/2 < α < 1, β < α − 1/2, there is a constant k such that for any connected subset A ⊂ ℤ2 of radius n,where HA denotes discrete harmonic measure.

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