Two-sided infinite self-avoiding walk in high dimensions

Abstract

In this paper, we consider self-avoiding walk (SAW) on ${\mathbb{Z}}^d$ for $d\ge 5$. We construct the \emph{two-sided infinite SAW} and use it to prove pattern theorems for the self-avoiding walk. We show that infinite two-sided SAW is the infinite-shift limit of infinite one-sided SAW and the infinite-size limit of finite two-sided SAW. We then prove that for every pattern $\zeta$, the fraction of times that $\zeta$ occurs in the SAW converges to the probability that the two-sided infinite SAW starts with $\zeta$. The convergence is in probability for the finite SAW and almost surely for the infinite SAW. For the proof, we show that infinite SAW is ergodic using a coupling technique. At the end of the paper, we pose a conjecture regarding the existence of infinite SAW in low dimensions. We show that this conjecture is true in high dimensions, thus giving a new proof for the existence of infinite SAW for $d\ge 5$. The proofs in this paper rely only on the asymptotics for the number of self-avoiding paths and the SAW two-point function. Although these results were shown by Hara and Slade using the lace expansion, the proofs in this paper do not use the lace expansion.

arXiv comment34 pages, 1 figure