Self-attracting self-avoiding walk

TL;DR

This paper investigates self-avoiding walks with self-attraction on integer lattices, establishing the existence of the connective constant for weak attraction and analyzing mean-field behavior in higher dimensions using lace expansion.

Contribution

It extends the analysis of self-attracting self-avoiding walks to bounded step distributions and proves mean-field behavior in dimensions five and above.

Findings

Connective constant exists for weak self-attraction.

Mean-field behavior of the critical two-point function in dimensions ≥ 5.

Addresses a problem posed by den Hollander.

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. This article considers the case of bounded step distributions. For weak self-attractions we show that the connective constant exists, and, in $d\geq 5$, 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.