The Hammersley-Welsh bound for self-avoiding walk revisited

The Hammersley-Welsh bound (Quart. J. Math., 1962) states that the number cₙ of length n self-avoiding walks on Z ᵈ satisfies \[ c_n ≤ exp [ O(n1/2) ] μ _c^n, \] where μ c=μ 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 \[ c_n ≤ exp [ o(n1/2) ] μ _c^n. \] 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.