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

TL;DR

This paper improves the upper bound on the probability that a self-avoiding walk in any dimension $d \\geq 2$ closes, using a new method called the snake method combined with Gaussian pattern fluctuation.

Contribution

It introduces the snake method and applies Gaussian pattern fluctuation to tighten the upper bound on closing probability from $n^{-1/4+o(1)}$ to $n^{-1/2+o(1)}$ for self-avoiding walks.

Findings

Closing probability is at most $n^{-1/2+o(1)}$ in all dimensions $d \\geq 2$.

The snake method explicitly quantifies pattern fluctuation effects.

Improves previous bounds on self-avoiding walk closure probabilities.

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.