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

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.