If D is a simply connected complex domain containing the origin with boundary points a,b and A_n is an approximating lattice domain, we consider the probability that the loop-erased random walk in A_n from (points near to) a,b goes through the origin. We show that the probability is asymptotic to a universal constant times the Schramm-Loewner evolution (SLE) prediction for this quantity. In particular it is conformally covariant.
This is joint work with Christian Benes and Fredrik Viklund.