Transforming fixed-length self-avoiding walks into radial SLE_8/3

TL;DR

This paper conjectures a relationship between the scaling limit of fixed-length self-avoiding walks and radial SLE_8/3, supported by non-rigorous derivation and Monte Carlo simulations estimating critical exponents.

Contribution

It introduces a conjecture linking fixed-length SAW scaling limits to radial SLE_8/3 and provides numerical evidence supporting this relationship.

Findings

Estimated interior and boundary scaling exponents closely match conjectured values.

Monte Carlo simulations support the conjectured relationship between SAW and radial SLE.

The approach offers a new perspective on the conformal invariance of self-avoiding walks.

Abstract

We conjecture a relationship between the scaling limit of the fixed-length ensemble of self-avoiding walks in the upper half plane and radial SLE with kappa=8/3 in this half plane from 0 to i. The relationship is that if we take a curve from the fixed-length scaling limit of the SAW, weight it by a suitable power of the distance to the endpoint of the curve and then apply the conformal map of the half plane that takes the endpoint to i, then we get the same probability measure on curves as radial SLE. In addition to a non-rigorous derivation of this conjecture, we support it with Monte Carlo simulations of the SAW. Using the conjectured relationship between the SAW and radial SLE, our simulations give estimates for both the interior and boundary scaling exponents. The values we obtain are within a few hundredths of a percent of the conjectured values.