Self-attracting self-avoiding walk

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 $d\ge 5$, 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.