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 $\mu = \lim_n c_n^{1/n}$ denote the connective constant of $\mathbb{Z}^d$. In 1962, Hammersley and Welsh [HW62] proved that, for each $d \geq 2$, there exists a constant $C > 0$ such that $c_n \leq \exp(C n^{1/2}) \mu^n$ 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 $c_n \leq \exp (C n^{1/2 - \epsilon})\mu^n$, 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 $\epsilon > 0$ 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 non-quantitative assertion concerning this lattice's connective constant.