Reinforced walks in two and three dimensions

In probability theory, reinforced walks are random walks on a lattice (or more generally a graph) that preferentially revisit neighboring 'locations' (sites or bonds) that have been visited before. In this paper, we consider walks with one-step reinforcement, where one preferentially revisits locations irrespective of the number of visits. Previous numerical simulations (A Ordemann et al 2001 Phys. Rev. E 64 046117) suggested that the site model on the lattice shows a phase transition at finite reinforcement between a random-walk-like and a collapsed phase, in both two and three dimensions. The very different mathematical structure of bond and site models might also suggest different phenomenology (critical properties, etc). We use high statistics simulations and heuristic arguments to argue that site and bond reinforcement are in the same universality class. We find broad agreement with the phase transition results of Ordemann et al, while improving their critical parameter estimates and suggesting that the phase transition in two dimensions actually occurs at zero coupling constant. We also show that a quasistatic approximation predicts the large time scaling of the end-to-end distance in the collapsed phase of both site and bond reinforcement models, in excellent agreement with simulation results.