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 Θ transition of polymers. Above a critical attractive interaction u_c, the walk collapses and the exponents ν and k, characterizing the scaling with time t of the mean square end-to-end distance 〈R²〉∼t^2ν and the average number of visited sites 〈S〉∼t^k, are universal and given by ν=1/(d+1) and k=d/(d+1). Below u_c, the walk swells and the exponents are as with no interaction, i.e., ν=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.