The Hammersley-Welsh bound for self-avoiding walk revisited

The Hammersley-Welsh bound (Quart. J. Math., 1962) states that the number c(n) of length n self-avoiding walks on Z(d) satisfies c(n) <= exp [O(n(1/2))]mu(n)(c), where mu(c) = mu(c) (d) is the connective constant of Z(d). While stronger estimates have subsequently been proven for d >= 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) <= exp [o(n(1/2))]mu(n)(c). The improved bound is obtained as a corollary to the sub-ballisticity theorem of Duminil-Copin and Hammond (Commun. Math. Phys., 2013). We also show that any quantitative form of that theorem would yield a corresponding quantitative improvement to the Hammersley-Welsh bound.