Self-avoiding walks in a rectangle

A celebrated problem in numerical analysis is to consider Brownian motion originating at the centre of a (Formula presented.) rectangle and to evaluate the probability of a Brownian path hitting the short ends of the rectangle before hitting one of the long sides. For Brownian motion this probability can be calculated exactly (The SIAM 100-digit challenge: a study in high-accuracy numerical computing. SIAM, Philadelphia, 2004). Here we consider instead the more difficult problem of a self-avoiding walk (SAW) in the scaling limit and pose the same question. Assuming that the scaling limit of SAW is conformally invariant, we evaluate, asymptotically, the same probability. For the SAW case we find the probability is approximately 200 times greater than for Brownian motion.