Bounding the number of self-avoiding walks: Hammersley–Welsh with polygon insertion

Abstract

Let $c_{n}=c_{n}(d)$ denote the number of self-avoiding walks of length $n$ starting at the origin in the Euclidean nearest-neighbour lattice $\mathbb{Z}^{d}$. Let $\mu=\lim_{n}c_{n}^{1/n}$ denote the connective constant of $\mathbb{Z}^{d}$. In 1962, Hammersley and Welsh (Quart. J. Math. Oxford Ser. (2) 13 (1962) 108–110) proved that, for each $d\geq2$, there exists a constant $C>0$ such that $c_{n}\leq\exp(Cn^{1/2})\mu^{n}$ for all $n\in\mathbb{N}$. While it is anticipated that $c_{n}\mu^{-n}$ has a power-law growth in $n$, the best-known upper bound in dimension two has remained of the form $n^{1/2}$ inside the exponential.

The natural first improvement to demand for a given planar lattice is a bound of the form $c_{n}\leq\exp(Cn^{1/2-\varepsilon})\mu^{n}$, where $\mu$ denotes the connective constant of the lattice in question. We derive a bound of this form for two such lattices, for an explicit choice of $\varepsilon>0$ in each case. For the hexagonal lattice $\mathbb{H}$, the bound is proved for all $n\in\mathbb{N}$; while for the Euclidean lattice $\mathbb{Z}^{2}$, it is proved for a set of $n\in\mathbb{N}$ of limit supremum density equal to one.

A power-law upper bound on $c_{n}\mu^{-n}$ for $\mathbb{H}$ is also proved, contingent on a nonquantitative assertion concerning this lattice’s connective constant.