We prove an estimate for the probability that a simple random walk in a
simply connected subset A of Z^2 starting on the boundary exits A at another
specified boundary point. The estimates are uniform over all domains of a given
inradius. We apply these estimates to prove a conjecture of S. Fomin in 2001
concerning a relationship between crossing probabilities of loop-erased random
walk and Brownian motion.Comment: 26 pages, 0 figure