Self-avoiding walks in a rectangle

TL;DR

This paper compares the probability ratios of hitting rectangle ends first for Brownian motion and self-avoiding walks, finding a significantly higher ratio for SAW under conformal invariance assumptions.

Contribution

It extends the analysis of hitting probabilities from Brownian motion to self-avoiding walks, assuming conformal invariance in the scaling limit.

Findings

SAW probability ratio is about 200 times greater than Brownian motion.

Asymptotic evaluation of hitting probabilities for SAW in a rectangle.

Supports the conformal invariance hypothesis for SAW scaling limit.

Abstract

A celebrated problem in numerical analysis is to consider Brownian motion originating at the centre of a $10 \times 1$ rectangle, and to evaluate the ratio of probabilities 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 \cite{BLWW04}. Here we consider instead the more difficult problem of a self-avoiding walk in the scaling limit, and pose the same question. Assuming that the scaling limit of SAW is conformally invariant, we evaluate, asymptotically, the same ratio of probabilities. For the SAW case we find the probability ratio is approximately 200 times greater than for Brownian motion.