1999/08/31 by E. Bolthausen, K. Muench-Berndl
Mathematics · #math.PR #math.CA
published as Israel Journal of Mathematics 124, 125-141 (2001) · 16 pages, 2 figures. Part (B) of the theorem is new
arxiv created 2000/05/05 · arxiv updated 2009/11/30
A theorem of Bourgain states that the harmonic measure for a domain in \Rd is supported on a set of Hausdorff dimension strictly less than d \citeBourgain. We apply Bourgain's method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of \Z d, d≥ 2. By refining the argument, we prove that for all \b>0 there exists ρ(d,\b)<d and N(d,\b), such that for any n>N(d,\b), any x ∈ \Zd, and any A⊂ \1,..., n\d | \y∈\Zd\colon νA,x(y) ≥ n-\b \| ≤ nρ(d,\b), where νA,x (y) denotes the probability that y is the first entrance point of the simple random walk starting at x into A. Furthermore, ρ must converge to d as \b → ∞.