The Hammersley-Welsh bound for self-avoiding walk revisited


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Abstract

The Hammersley-Welsh bound (1962) states that the number cn of length n self-avoiding walks on Zd satisfies [ c_n \leq \exp \left[ O(n^{1/2}) \right] \mu_c^n, ] where μc=μc(d) is the connective constant of Zd. 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 \leq \exp\left[ o(n^{1/2})\right] \mu_c^n. ] The improved bound is obtained as a corollary to the sub-ballisticity theorem of Duminil-Copin and Hammond (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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Keywords
self-avoiding walk, Hammersley-Welsh
Journal Title
Electronic Communications in Probability
Conference Name
Journal ISSN
1083-589X
1083-589X
Volume Title
23
Publisher
Institute of Mathematical Statistics