The self-avoiding walk in a strip

TL;DR

This paper explores the properties of the self-avoiding walk in a strip, establishing its connection to bridges and the half-plane model, and discusses implications for simulations and conjectures related to SLE$_{8/3}$.

Contribution

It demonstrates that the self-avoiding walk in a strip can be derived by conditioning the half-plane SAW to have a bridge at a certain height, linking it to SLE$_{8/3}$.

Findings

The probability measure for the strip SAW is obtained by conditioning the half-plane SAW.

The measure is also the limit of finite walks with probability proportional to $eta_c^{| ext{walk}|}$.

This framework facilitates simulations to test conjectures on SAW endpoint distribution and SLE$_{8/3}$.

Abstract

We review the existence of the infinite length self-avoiding walk in the half plane and its relationship to bridges. We prove that this probability measure is also given by the limit as $\beta \rightarrow \beta_c-$ of the probability measure on all finite length walks $\omega$ with the probability of $\omega$ proportional to $\beta_c^{|\omega|}$ where $|\omega|$ is the number of steps in $\omega$. The self-avoiding walk in a strip $\{z : 0<\Im(z)<y\}$ is defined by considering all self-avoiding walks $\omega$ in the strip which start at the origin and end somewhere on the top boundary with probability proportional to $\beta_c^{|\omega|}$ We prove that this probability measure may be obtained by conditioning the SAW in the half plane to have a bridge at height $y$. This observation is the basis for simulations to test conjectures on the distribution of the endpoint of the SAW in a strip and the relationship between the distribution of this strip SAW and SLE$_{8/3}$.