Swelling-collapse transition of self-attracting walks

We study the structural properties of self-attracting walks in d dimensions using scaling arguments and Monte Carlo simulations. We find evidence of a transition analogous to the Theta transition of polymers. Above a critical attractive interaction u(c), the walk collapses and the exponents nu and k, characterizing the scaling with time t of the mean square end-to-end distance approximately t(2nu) and the average number of visited sites approximately t(k), are universal and given by nu=1/(d+1) and k=d/(d+1). Below u(c), the walk swells and the exponents are as with no interaction, i.e., nu=1/2 for all d, k=1/2 for d=1 and k=1 for d>/=2. At u(c), the exponents are found to be in a different universality class.