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.
"Estimates of Random Walk Exit Probabilities and Application to Loop-Erased Random Walk." Electron. J. Probab. 10 1442 - 1467, 2005. https://doi.org/10.1214/EJP.v10-294