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

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

2011-07-13

arxiv.org/abs/1107.2505v1

Mathematics

Probability

Scientific paper

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.

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