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

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.