The Hammersley-Welsh bound for self-avoiding walk revisited

Abstract

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))]μ^n_c, 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^(n^(1/2))] μ^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.

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