Structural properties of self-attracting walks

Self-attracting walks (SATW) with attractive interaction $u>\mathrm{}0\mathrm{}$ display a swelling-collapse transition at a critical $u_{\mathrm{c}}$ for dimensions $d>~2,$ analogous to the $\Theta$ transition of polymers. We are interested in the structure of the clusters generated by SATW below $u_{\mathrm{c}}$ (swollen walk), above $u_{\mathrm{c}}$ (collapsed walk), and at $u_{\mathrm{c}},$ 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 $u<\mathrm{}{u}_{\mathrm{c}},$ the structures are in the universality class of clusters generated by simple random walks. For $u>\mathrm{}{u}_{\mathrm{c}},$ the clusters are compact, i.e., $d_f=d$ and $d_I=d-1.$ At $u_{\mathrm{c}},$ the SATW is in a new universality class. The clusters are compact in both $d=2\mathrm{}$ and $d=3,$ but their interface is fractal: $d_I=1.50\mathrm{}\pm 0.01$ and $2.73\pm 0.03$ in $d=2\mathrm{}$ 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.