Self-attracting self-avoiding walk

2017/12/20 by Hammond, Alan, Helmuth, Tyler
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1712.07673

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. 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.

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