Quantitative estimates of discrete harmonic measures

A theorem of Bourgain states that the harmonic measure for a domain in ℝ d is supported on a set of Hausdorff dimension strictly less thand [2]. 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 allβ>0 there existsρ(d,β)N(d,β), anyx ∈ ℤ d , and anyA ⊂ {1, ,n} d •{y∈ℤ whereν A,x (y) denotes the probability thaty is the first entrance point of the simple random walk starting atx intoA. Furthermore,ρ must converge tod asβ →