A Discrete Analogue of a Theorem of Makarov

Source abstract

A theorem of Makarov states that the harmonic measure of a connected subset of ℝ 2 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 ⊂ ℤ 2 of radius n , where H A denotes discrete harmonic measure.