Simulating self-avoiding walks in bounded domains

TL;DR

This paper introduces a new ensemble of self-avoiding walks in bounded domains, enabling efficient simulation and analysis of their boundary endpoint distribution, which supports predictions from Schramm-Loewner Evolution (SLE).

Contribution

It proposes a novel ensemble of SAWs that can be simulated efficiently using the pivot algorithm, facilitating the study of SLE predictions in bounded domains.

Findings

The ensemble accurately reproduces the boundary endpoint distribution.

Lattice effects in the distribution are characterized by a local function.

Simulations confirm the scaling limit predictions of SLE.

Abstract

Let D be a domain in the plane containing the origin. We are interested in the ensemble of self-avoiding walks (SAW's) in D which start at the origin and end on the boundary of the domain. We introduce an ensemble of SAW's that we expect to have the same scaling limit. The advantage of our ensemble is that it can be simulated using the pivot algorithm. Our ensemble makes it possible to accurately study SLE predictions for the SAW in bounded simply connected domains. One such prediction is the distribution along the boundary of the endpoint of the SAW. We use the pivot algorithm to simulate our ensemble and study this density. In particular the lattice effects in this density that persist in the scaling limit are seen to be given by a purely local function.