On self-avoiding polygons and walks: the snake method via pattern fluctuation

Abstract

For $d \geq 2$ and $n \in \mathbb{N}$, let $\mathsf{W}_n$ denote the uniform law on self-avoiding walks of length $n$ beginning at the origin in the nearest-neighbour integer lattice $\mathbb{Z}^d$, and write $\Gamma$ for a $\mathsf{W}_n$-distributed walk. We show that the closing probability $\mathsf{W}_n \big( \vert\vert \Gamma_n \vert\vert = 1 \big)$ that $\Gamma$'s endpoint neighbours the origin is at most $n^{-1/2 + o(1)}$ in any dimension $d \geq 2$. The method of proof is a reworking of that in [4], which found a closing probability upper bound of $n^{-1/4 + o(1)}$. A key element of the proof is made explicit and called the snake method. It is applied to prove the $n^{-1/2 + o(1)}$ upper bound by means a technique of Gaussian pattern fluctuation.