Compressed self-avoiding walks, bridges and polygons

TL;DR

This paper investigates self-avoiding walks and polygons constrained in a half-plane under compressive forces, combining theoretical predictions from Schramm-Loewner evolution with series analysis to understand their partition functions.

Contribution

It introduces a novel analysis of compressed self-avoiding walks and polygons, linking them to Schramm-Loewner evolution and validating predictions through series analysis.

Findings

Predicted the form of the partition function using SLE conjectures

Validated theoretical predictions with series analysis

Provided insights into the behavior of constrained SAWs and polygons

Abstract

We study various self-avoiding walks (SAWs) which are constrained to lie in the upper half-plane and are subjected to a compressive force. This force is applied to the vertex or vertices of the walk located at the maximum distance above the boundary of the half-space. In the case of bridges, this is the unique end-point. In the case of SAWs or self-avoiding polygons, this corresponds to all vertices of maximal height. We first use the conjectured relation with the Schramm-Loewner evolution to predict the form of the partition function including the values of the exponents, and then we use series analysis to test these predictions.