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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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

Permanent link to this document
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

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

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

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

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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