The Hammersley-Welsh bound for self-avoiding walk revisited

The Hammersley-Welsh bound (1962) states that the number $c_n$ of length $n$ self-avoiding walks on $\mathbb{Z}^d$ satisfies $$c_n \leq \exp \left[ O(n^{1/2}) \right] \mu_c^n,$$ where $\mu_c=\mu_c(d)$ is the connective constant of $\mathbb{Z}^d$. While stronger estimates have subsequently been proven for $d\geq 3$, for $d=2$ this has remained the best rigorous, unconditional bound available. In this note, we give a new, simplified proof of this bound, which does not rely on the combinatorial analysis of unfolding. We also prove a small, non-quantitative improvement to the bound, namely $$c_n \leq \exp\left[ o(n^{1/2})\right] \mu_c^n.$$ The improved bound is obtained as a corollary to the sub-ballisticity theorem of Duminil-Copin and Hammond (2013). We also show that any quantitative form of that theorem would yield a corresponding quantitative improvement to the Hammersley-Welsh bound.