Localization of reinforced random walks

TL;DR

This paper investigates the localization behavior of reinforced random walks, providing new proofs and techniques to demonstrate that such walks eventually visit only finitely many sites, with a focus on integer lattices.

Contribution

It introduces martingale techniques and a continuous time-lines representation to offer simplified proofs of localization for vertex-reinforced random walks on integers.

Findings

Martingale methods prove localization on finitely many sites.

New proof for localization on five sites on Z.

Enhanced understanding of reinforced walk behaviors.

Abstract

We describe and analyze how reinforced random walks can eventually localize, i.e. only visit finitely many sites. After introducing vertex and edge self-interacting walks on a discrete graph in a general setting, and stating the main results and conjectures so far on the topic, we present martingale techniques that provide an alternative proof of the a.s. localization of vertex-reinforced random walks (VRRWs) on the integers on finitely many sites and, with positive probability, on five consecutive sites, initially proved by Pemantle and Volkov (1999). Next we introduce the continuous time-lines representation (sometimes called Rubin construction) and its martingale counterpart, and explain how it has been used to prove localization of some reinforced walks on one attracting edge. Then we show how a modified version of this construction enables one to propose a new short proof of the a.s. localization of VRRWs on five sites on Z.