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Phase Transition in Vertex-Reinforced Random Walks on \({\mathbb{Z}}\) with Non-linear Reinforcement

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Vertex-reinforced random walk is a random process which visits a site with probability proportional to the weight w k of the number k of previous visits. We show that if w k ∼ k α, then there is a large time T 0 such that after T 0 the walk visits 2, 5, or ∞ sites when α < 1, = 1, or > 1, respectively. More general results are also proven.

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  1. Department of Mathematics, University of Bristol, BS8 1TW, Bristol, UK

    Stanislav Volkov

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  1. Stanislav Volkov
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Correspondence to Stanislav Volkov.

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Volkov, S. Phase Transition in Vertex-Reinforced Random Walks on \({\mathbb{Z}}\) with Non-linear Reinforcement. J Theor Probab 19, 691–700 (2006). https://doi.org/10.1007/s10959-006-0033-2

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  • DOI: https://doi.org/10.1007/s10959-006-0033-2

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