Exact sampling of self-avoiding paths via discrete Schramm-Loewner evolution

TL;DR

This paper introduces an algorithm based on conformal maps to generate independent, parametrized samples of self-avoiding paths in the plane, enabling better analysis of their geometric properties.

Contribution

It presents a novel discrete sampling method approximating radial Schramm-Loewner evolution, specifically addressing parametrization issues in self-avoiding walks.

Findings

Successfully reproduces lattice model parametrization

Allows analysis of asphericity in self-avoiding chains

Enables estimation of correction-to-scaling exponents

Abstract

We present an algorithm, based on the iteration of conformal maps, that produces independent samples of self-avoiding paths in the plane. It is a discrete process approximating radial Schramm-Loewner evolution growing to infinity. We focus on the problem of reproducing the parametrization corresponding to that of lattice models, namely self-avoiding walks on the lattice, and we propose a strategy that gives rise to discrete paths where consecutive points lie an approximately constant distance apart from each other. This new method allows us to tackle two non-trivial features of self-avoiding walks that critically depend on the parametrization: the asphericity of a portion of chain and the correction-to-scaling exponent.