Conformal invariance predictions for the three-dimensional self-avoiding walk

TL;DR

This paper provides strong numerical evidence supporting the conjecture that the three-dimensional self-avoiding walk is conformally invariant, by comparing theoretical predictions with advanced Monte Carlo simulations.

Contribution

It introduces a method to simulate mixed-length ensembles of SAWs using weighted single-length ensembles, enabling more accurate testing of conformal invariance predictions.

Findings

Good agreement between predictions and simulations

Enhanced simulation accuracy strengthens conformal invariance support

Method allows efficient simulation of mixed-length ensembles

Abstract

If the three dimensional self-avoiding walk (SAW) is conformally invariant, then one can compute the hitting densities for the SAW in a half-space and in a sphere. The ensembles of SAW's used to define these hitting densities involve walks of arbitrary lengths, and so these ensembles cannot be directly studied by the pivot Monte Carlo algorithm for the SAW. We show that these mixed length ensembles should have the same scaling limit as certain weighted ensembles that only involve walks with a single length, thus providing a fast method for simulating these ensembles. Preliminary simulations which found good agreement between the predictions and Monte Carlo simulations for the SAW were reported in [14]. In this paper we present more accurate simulations testing the predictions and find even stronger support for the prediction that the SAW is conformally invariant in three dimensions.