An upper bound on the number of self-avoiding polygons via joining

Abstract

For $d\geq2$ and $n\in\mathbb{N}$ even, let $p_{n}=p_{n}(d)$ denote the number of length $n$ self-avoiding polygons in $\mathbb{Z}^{d}$ up to translation. The polygon cardinality grows exponentially, and the growth rate $\lim_{n\in2\mathbb{N}}p_{n}^{1/n}\in(0,\infty)$ is called the connective constant and denoted by $\mu$. Madras [J. Stat. Phys. 78 (1995) 681–699] has shown that $p_{n}\mu^{-n}\leq Cn^{-1/2}$ in dimension $d=2$. Here, we establish that $p_{n}\mu^{-n}\leq n^{-3/2+o(1)}$ for a set of even $n$ of full density when $d=2$. We also consider a certain variant of self-avoiding walk and argue that, when $d\geq3$, an upper bound of $n^{-2+d^{-1}+o(1)}$ holds on a full density set for the counterpart in this variant model of this normalized polygon cardinality.