# Self-attracting self-avoiding walk

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## Abstract

This article is concerned with self-avoiding walks (SAW) on \(\mathbb {Z}^{d}\) 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 \(d\ge 5\), 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).

## Keywords

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

Primary 60K35 Secondary 60D05 82B27## Notes

### Acknowledgements

The authors would like to thank both referees for their critiques and comments, which have lead to a significantly improved article. T.H. would like to thank Gordon Slade and Remco van der Hofstad for encouraging discussions. A.H. is supported by NSF grant DMS-1512908. The majority of this work was carried out while T.H. was supported by an NSERC postdoctoral fellowship at UC Berkeley; additional support was provided by EPSRC Grant EP/P003656/1.

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