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.