A 0-1 law for vertex-reinforced random walks on $\mathbb{Z}$ with weight of order $k^{\alpha }$, $\alpha <1/2$

Abstract

We prove that Vertex Reinforced Random Walk on $\mathbb{Z}$ with weight of order $k^{\alpha }$, with $\alpha \in [0,1/2)$, is either almost surely recurrent or almost surely transient. This improves a previous result of Volkov who showed that the set of sites which are visited infinitely often was a.s. either empty or infinite.

arXiv commentFinal version, to appear in ECP, 8p