Recurrence of edge-reinforced random walk on a two-dimensional graph

Abstract

We consider a linearly edge-reinforced random walk on a class of two-dimensional graphs with constant initial weights. The graphs are obtained from ${\mathbb{Z}}^2$ by replacing every edge by a sufficiently large, but fixed number of edges in series. We prove that the linearly edge-reinforced random walk on these graphs is recurrent. Furthermore, we derive bounds for the probability that the edge-reinforced random walk hits the boundary of a large box before returning to its starting point.

Published in at http://dx.doi.org/10.1214/08-AOP446 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• A survey of random processes with reinforcement

Cited by in corpus (2)

• An asymptotic result for Brownian polymers
• Limit theorems for vertex-reinforced jump processes on regular trees