(with Hugo Duminil-Copin, Alan Hammond, Ioan Manolescu) Arxiv

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 ℤ^d. Let μ denote the connective constant of ℤd. In 1962, Hammersley and Welsh [HW62] proved that, for each d≥2, there exists a constant C>0 such that c_n≤exp(Cn^1/2)μ^n for all n∈ℕ. While it is anticipated that c_nμ^(−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. We consider two planar lattices and prove that c_n≤exp(Cn^(1/2−ϵ))μ^n for an explicit constant ϵ>0 (where here μ 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 ℍ for all n, and subsequentially in n for ℤ2. A power-law upper bound on cnμ−n for ℍ is also proved, contingent on a non-quantitative assertion concerning this lattice's connective constant.