Two-sided infinite self-avoiding walk in high dimensions

Abstract

We construct the two-sided infinite self-avoiding walk (SAW) on $\mathbb{Z}^d$ for $d\geq5$ 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 $\zeta$ 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. Along the way, 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 $d\geq5$. 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 and might be adapted to prove existence and ergodicity of other infinite high-dimensional lattice models.