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

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 in the closing probability $\mathsf {W}_n \big ( \vert \vert \Gamma _n \vert \vert = 1 \big )$ that $\Gamma$’s endpoint neighbours the origin and is at most $n^{-1/2 + o(1)}$ in any dimension $d \geq 2$. The method of proof is a reworking of that in [Ann. Probab. 44 (2016), pp. 955–983], 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 of a technique of Gaussian pattern fluctuation.