Self-attracting self-avoiding walk

Alan Hammond, Tyler Helmuth*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)
79 Downloads (Pure)


This article is concerned with self-avoiding walks (SAW) on Zd that are subject to a self-attraction. The attraction, which rewards instances of adjacent parallel edges, introduces difficulties that are not present in ordinary SAW. Ueltschi has shown how to overcome these difficulties for sufficiently regular infinite-range step distributions and weak self-attractions (Ueltschi in Probab Theory Relat Fields 124(2):189–203, 2002). This article considers the case of bounded step distributions. For weak self-attractions we show that the connective constant exists, and, in d≥ 5 , carry out a lace expansion analysis to prove the mean-field behaviour of the critical two-point function, hereby addressing a problem posed by den Hollander (Random Polymers, vol. 1974. Springer-Verlag, Berlin, 2009).

Original languageEnglish
Pages (from-to)677-719
Number of pages43
JournalProbability Theory and Related Fields
Issue number3-4
Early online date24 Jan 2019
Publication statusPublished - 1 Dec 2019


  • Critical phenomena
  • Hammersley-Welsh argument
  • Lace expansion
  • Linear polymers
  • Self-attracting walk
  • Self-avoiding walk
  • Self-interacting random walk

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