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 n1/2 inside the exponential.
We consider two planar lattices and prove that c_n \leq \exp(C n^{1/2 -\epsilon}) \mu^n for an explicit constant \epsilon> 0 (where here \mu denotes the connective constant for the lattice in question). The result is conditional on a lower bound on the number of self-avoiding polygons of length n, which is proved to hold on the hexagonal lattice \mathbb{H} for all n, and subsequentially in n for \mathbb{Z}^2. 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.
