Swelling-collapse transition of self-attracting walks

Abstract

We study the structural properties of self-attracting walks in d dimensions using scaling arguments and Monte Carlo simulations. We find evidence for 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, characterising the scaling with time t of the mean square end-to-end distance <R^2> ~ t^{2 \nu} and the average number of visited sites <S> ~ 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.