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

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F20: Zero-one laws
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Self-interacting random walk Reinforced random walk $0$-$1$ law

This work is licensed under a Creative Commons Attribution 3.0 License.


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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