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

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 $\mu ={\lim }_n{c}_n^{1\mathrm{/}n}$ denote the connective constant of ${\mathbb{Z}}^d$. In 1962, Hammersley and Welsh [HW62] proved that, for each $d \geq 2$,thereexistsaconstant$C > 0$suchthat$c_n \leq \exp(C n^{1/2}) \mu^n$forall$n \in $\mathbb{N}$$\mathrm{.}Whileitisanticipatedthat$c_n \mu^{-n}$hasapower-lawgrowthin$n$, the best known upper bound in dimension two has remained of the form $n^{1\mathrm{/}2}$ inside the exponential. The natural first improvement to demand for a given planar lattice is a bound of the form $c_n\le \exp (C{n}^{1\mathrm{/}2-ϵ}){\mu }^n$, where $\mu$ denotes the connective constant of the lattice in question. We derive a bound of this form for two such lattices, for an explicit choice of $ϵ>0$ in each case. For the hexagonal lattice $\mathbb{H}$, the bound is proved for all $n \in $\mathbb{N}$$;whilefortheEuclideanlattice$$\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 non-quantitative assertion concerning this lattice’s connective constant.