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

Bruno Schapira (Université Paris Sud 11 (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.

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Pages: 1-8

Publication Date: June 13, 2012

DOI: 10.1214/ECP.v17-2084


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