The self-avoiding walk in a strip

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 $β\rightarrow β_c-$ of the probability measure on all finite length walks $ω$ with the probability of $ω$ proportional to $β_c^{|ω|}$ where $|ω|$ is the number of steps in $ω$. The self-avoiding walk in a strip $\{z : 0<\Im(z)<y\}$ is defined by considering all self-avoiding walks $ω$ in the strip which start at the origin and end somewhere on the top boundary with probability proportional to $β_c^{|ω|}$ 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$_{8/3}$.