Localization of reinforced random walks

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.

5 figures

References in corpus (3)

• Vertex-reinforced random walk on Z eventually gets stuck on five points
• Attracting edge and strongly edge reinforced walks
• Dynamics of vertex-reinforced random walks

Cited by in corpus (5)

• Strongly Vertex-Reinforced-Random-Walk on the complete graph
• Edge-reinforced random walk, Vertex-Reinforced Jump Process and the supersymmetric hyperbolic sigma model
• Recurrence for vertex-reinforced random walks on Z with weak reinforcements
• Stuck walks: A conjecture of Erschler, Tóth and Werner
• Localization on 5 sites for vertex reinforced random walks: Towards a characterization