Self-attracting self-avoiding walk

Alan Hammond; Tyler Helmuth

doi:10.1007/s00440-018-00898-7

Self-attracting self-avoiding walk

Alan Hammond, Tyler Helmuth^*

^*Corresponding author for this work

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

3 Citations (Scopus)

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Abstract

This article is concerned with self-avoiding walks (SAW) on Z^d 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 language	English
Pages (from-to)	677-719
Number of pages	43
Journal	Probability Theory and Related Fields
Volume	175
Issue number	3-4
Early online date	24 Jan 2019
DOIs	https://doi.org/10.1007/s00440-018-00898-7
Publication status	Published - 1 Dec 2019

Keywords

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

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https://arxiv.org/abs/1712.07673

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KW - Critical phenomena

KW - Hammersley-Welsh argument

KW - Lace expansion

KW - Linear polymers

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KW - Self-avoiding walk

KW - Self-interacting random walk

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Self-attracting self-avoiding walk

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