Schapira, Bruno A 0-1 law for vertex-reinforced random walks on \(\mathbb{Z}\) with weight of order \(k^\alpha,\;\alpha\in[0,1/2)\). (English) Zbl 1244.60033 Electron. Commun. Probab. 17, Paper No. 22, 8 p. (2012). Summary: We prove that a 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 S. Volkov [J. Theor. Probab. 19, No. 3, 691–700 (2006; Zbl 1107.60068)] who showed that the set of sites which are visited infinitely often is a.s. either empty or infinite. Cited in 2 ReviewsCited in 4 Documents MSC: 60F20 Zero-one laws 60K35 Interacting random processes; statistical mechanics type models; percolation theory Keywords:self-interacting random walk; reinforced random walk; 0-1 law Citations:Zbl 1107.60068 × Cite Format Result Cite Review PDF Full Text: DOI arXiv