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. https://resolver.caltech.edu/CaltechAUTHORS:20210924-184806499
![[img]](https://authors.library.caltech.edu/style/images/fileicons/application_pdf.png) |
PDF
- Published Version
Creative Commons Attribution. 249kB |
|
PDF
- Accepted Version
See Usage Policy. 149kB |
Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:20210924-184806499
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.
| Item Type: | Article | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Related URLs: |
| |||||||||
| ORCID: |
| |||||||||
| 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. | |||||||||
| Funders: |
| |||||||||
| Subject Keywords: | self-avoiding walk; Hammersley-Welsh | |||||||||
| Classification Code: | AMS MSC 2010: Primary 60K35, Secondary 05A99 | |||||||||
| DOI: | 10.1214/17-ECP94 | |||||||||
| Record Number: | CaltechAUTHORS:20210924-184806499 | |||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20210924-184806499 | |||||||||
| Official Citation: | Tom Hutchcroft. "The Hammersley-Welsh bound for self-avoiding walk revisited." Electron. Commun. Probab. 23 1 - 8, 2018. https://doi.org/10.1214/17-ECP94 | |||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||
| ID Code: | 111024 | |||||||||
| Collection: | CaltechAUTHORS | |||||||||
| Deposited By: | George Porter | |||||||||
| Deposited On: | 27 Sep 2021 17:23 | |||||||||
| Last Modified: | 27 Sep 2021 17:23 |
Repository Staff Only: item control page
