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

Bruno Schapira (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: 0-1 law; Vertex Reinforced Random Walk; 0-1 law.

Domaines: Probabilités [math.PR]

Bruno Schapira. A $0-1$ law for vertex-reinforced random walks on $\mathbb{Z}$ with weight of order $k^\alpha$, $\alpha<1/2$.. 2012. ⟨hal-00608462v2⟩