The true reinforced random walk with bias

TL;DR

This paper studies a one-dimensional self-attracting random walk biased towards a target, revealing that any bias induces ballistic behavior and dominates memory effects, contrasting with static models.

Contribution

It provides a detailed analysis of the dynamic self-attracting walk with bias, showing the dominance of bias over memory effects and establishing a connection to trapping problems.

Findings

Bias induces ballistic behavior for any s>0

Memory effects are overshadowed by bias in the dynamic model

Contrasts with static models where behavior differs

Abstract

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