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 than d [2]. We a pply Bourgain's method to the discrete case, i.e., to the distribution of t he first entrance point of a random walk into a subset of Z(d), d greater t han or equal to 2. By refining the argument, we prove that for all beta > 0 there exists rho (d, beta) < d and N(d, beta), such that for any n > N(d, beta), any x is an element of Z(d), and any A subset of {1, ..., n}(d) \{y is an element of Z(d): nu (A,x)(y) greater than or equal to n(-beta)}\ less than or equal to n(rho (d,beta)), where nu (A,x)(y) denotes the probability that y is the first entrance poin t of the simple random walk starting at x into A. Furthermore, rho must con verge to d as beta --> infinity.