## Electronic Communications in Probability

### 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

#### 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.

#### Article information

Source
Electron. Commun. Probab., Volume 17 (2012), paper no. 22, 8 pp.

Dates
Accepted: 13 June 2012
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465263155

Digital Object Identifier
doi:10.1214/ECP.v17-2084

Mathematical Reviews number (MathSciNet)
MR2943105

Zentralblatt MATH identifier
1244.60033

Rights

#### Citation

Schapira, Bruno. A $0$-$1$ law for vertex-reinforced random walks on $\mathbb{Z}$ with weight of order $k^\alpha$, $\alpha\in[0,1/2)$. Electron. Commun. Probab. 17 (2012), paper no. 22, 8 pp. doi:10.1214/ECP.v17-2084. https://projecteuclid.org/euclid.ecp/1465263155

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