Structural properties of self-attracting walks.

Abstract

Self-attracting walks (SATW) with attractive interaction u>0 display a swelling-collapse transition at a critical u(c) for dimensions d>or=2, analogous to the Theta transition of polymers. We are interested in the structure of the clusters generated by SATW below u(c) (swollen walk), above u(c) (collapsed walk), and at u(c), which can be characterized by the fractal dimensions of the clusters d(f) and their interface d(I). Using scaling arguments and Monte Carlo simulations, we find that for u<u(c), the structures are in the universality class of clusters generated by simple random walks. For u>u(c), the clusters are compact, i.e., d(f)=d and d(I)=d-1. At u(c), the SATW is in a new universality class. The clusters are compact in both d=2 and d=3, but their interface is fractal: d(I)=1.50+/-0.01 and 2.73+/-0.03 in d=2 and d=3, respectively. In d=1, where the walk is collapsed for all u and no swelling-collapse transition exists, we derive analytical expressions for the average number of visited sites <S> and the mean time <t> to visit S sites.