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The Hammersley-Welsh bound for self-avoiding walk revisited

Hutchcroft, Tom (2018) The Hammersley-Welsh bound for self-avoiding walk revisited. Electronic Communications in Probability, 23 . Art. No. 5. ISSN 1083-589X. doi:10.1214/17-ECP94.

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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.

Item Type:Article
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URLURL TypeDescription Paper
Hutchcroft, Tom0000-0003-0061-593X
Additional Information:© 2018 The Author(s). Creative Commons Attribution 4.0 International License. Submitted to ECP on August 31, 2017, final version accepted on October 19, 2017. First available in Project Euclid: 12 February 2018. The author was supported by internships at Microsoft Research and a Microsoft Research PhD Fellowship. We thank Omer Angel, Hugo Duminil-Copin, Tyler Helmuth and Gordon Slade for comments on an earlier draft. Finally, we thank the anonymous referee for catching several errors in the preprint.
Funding AgencyGrant Number
Microsoft ResearchUNSPECIFIED
Subject Keywords:self-avoiding walk; Hammersley-Welsh
Classification Code:AMS MSC 2010: Primary 60K35, Secondary 05A99
Record Number:CaltechAUTHORS:20210924-184806499
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Official Citation:Tom Hutchcroft. "The Hammersley-Welsh bound for self-avoiding walk revisited." Electron. Commun. Probab. 23 1 - 8, 2018.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111024
Deposited By: George Porter
Deposited On:27 Sep 2021 17:23
Last Modified:27 Sep 2021 17:23

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