  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 \emph{revisits}
locations irrespective of the number of visits. Previous numerical simulations
[A. Ordemann {\it et al.}, Phys. Rev. E {\bf 64}, 046117 (2001)] 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 2 and 3 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 suggest that site and bond
reinforcement are in the same universality class, and that the purported phase
transition in 2 dimensions actually occurs at zero coupling constant. We also
show that a quasi-static 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.
