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

Abstract

For $d\ge 2$ 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 ${\lim }_{n\in 2\mathbb{N}}{p}_n^{1/n}\in (0,\infty )$ is called the connective constant and denoted by $\mu$. Madras [J. Statist. Phys. 78 (1995) no. 3--4, 681--699] has shown that $p_n{\mu }^{-n}\le C{n}^{-1/2}$ in dimension $d=2$. Here we establish that $p_n{\mu }^{-n}\le n^{-3/2+o(1)}$ 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 $d\ge 3$, 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.

32 pages and four figures. This article corresponds to Part II of arXiv:1504.05286, a submission giving a unified treatment of three articles

References in corpus (4)

• Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis
• Lectures on Self-Avoiding Walks
• A renormalisation group method. II. Approximation by local polynomials
• On self-avoiding polygons and walks: counting, joining and closing