Self-Attracting Walk on Lattices

We have studied a model of self-attracting walk proposed by Sapozhnikov using Monte Carlo method. The mean square displacement $< R^2(t) > \sim t^{2\nu}$ and the mean number of visited sites $< S(t) > \sim t^{k}$ are calculated for one-, two- and three-dimensional lattice. In one dimension, the walk shows diffusive behaviour with $\nu=k=1/2$. However, in two and three dimension, we observed a non-universal behaviour, i.e., the exponent $\nu$ varies continuously with the strength of the attracting interaction.