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2018 The Hammersley-Welsh bound for self-avoiding walk revisited
Tom Hutchcroft
Electron. Commun. Probab. 23: 1-8 (2018). DOI: 10.1214/17-ECP94


The Hammersley-Welsh bound (Quart. J. Math., 1962) states that the number $c_n$ of length $n$ self-avoiding walks on $\mathbb{Z} ^d$ satisfies \[ c_n \leq \exp \left [ O(n^{1/2}) \right ] \mu _c^n, \] where $\mu _c=\mu _c(d)$ is the connective constant of $\mathbb{Z} ^d$. While stronger estimates have subsequently been proven for $d\geq 3$, 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 (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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Tom Hutchcroft. "The Hammersley-Welsh bound for self-avoiding walk revisited." Electron. Commun. Probab. 23 1 - 8, 2018.


Received: 31 August 2017; Accepted: 19 October 2017; Published: 2018
First available in Project Euclid: 12 February 2018

zbMATH: 1388.60162
MathSciNet: MR3771763
Digital Object Identifier: 10.1214/17-ECP94

Primary: 60K35
Secondary: 05A99

Keywords: Hammersley-Welsh , Self-avoiding walk


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