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A Discrete Analogue of a Theorem of Makarov

Published online by Cambridge University Press:  12 September 2008

Gregory F. Lawler
Gregory F. Lawler
Affiliation:
Department of Mathematics, Duke University, Box 90320, Durham, NC 27708-0320
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Abstract

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.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 2 , June 1993 , pp. 181 - 199
Copyright
Copyright © Cambridge University Press 1993

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