The true reinforced random walk with bias

We consider a self-attracting random walk in dimension d = 1, in the presence of a field of strength s, which biases the walker toward a target site. We focus on the dynamic case (the true reinforced random walk), where memory effects are implemented at each time step, in contrast with the static case, where memory effects are accounted for globally. We analyze in details the asymptotic long-time behavior of the walker through the main statistical quantities (e.g. distinct sites visited and end-to-end distance) and we discuss a possible mapping between such a dynamic self-attracting model and the trapping problem for a simple random walk, in analogy to the static model. Moreover, we find that, for any s > 0, the random walk behavior switches to ballistic and that field effects always prevail over memory effects without any singularity, already in d = 1; this is in contrast with the behavior observed in the static model.