Whole-plane self-avoiding walks and radial Schramm-Loewner evolution: a numerical study

TL;DR

This paper provides numerical evidence supporting the conjecture that the scaling limit of self-avoiding walks in the plane corresponds to Schramm-Loewner evolution with k=8/3, using a novel discrete-time approximation method.

Contribution

It introduces a new discrete-time process approximating SLE in the exterior of the unit disc and compares it with SAW, confirming their distributional similarity.

Findings

Good agreement between SAW and SLE distribution functions

Efficient algorithm for computing internal points in SAW

Numerical support for the SAW-SLE conjecture

Abstract

We numerically test the correspondence between the scaling limit of self-avoiding walks (SAW) in the plane and Schramm-Loewner evolution (SLE) with k=8/3. We introduce a discrete-time process approximating SLE in the exterior of the unit disc and compare the distribution functions for an internal point in the SAW and a point at a fixed fractal variation on the SLE, finding good agreement. This provides numerical evidence in favor of a conjecture by Lawler, Schramm and Werner. The algorithm turns out to be an efficient way of computing the position of an internal point in the SAW.