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 $〈R^2〉\sim t^{2\nu }$ and the average number of visited sites $〈S〉\sim t^k,$ are universal and given by $\nu =1/(d+1\mathrm{})\mathrm{}$ and $k=d/(d+1\mathrm{}).$ Below $u_c,$ the walk swells and the exponents are as with no interaction, i.e., $\nu =1/2\mathrm{}$ for all d, $k=1/2\mathrm{}$ for $d=1\mathrm{}$ and $k=1\mathrm{}$ for $d>~2.$ At $u_c,$ the exponents are found to be in a different universality class.