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

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  • Stanislav Volkov

    (University of Bristol)

Abstract

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, respectively. More general results are also proven.

Suggested Citation

  • Stanislav Volkov, 2006. "Phase Transition in Vertex-Reinforced Random Walks on $${\mathbb{Z}}$$ with Non-linear Reinforcement," Journal of Theoretical Probability, Springer, vol. 19(3), pages 691-700, December.
    • Handle: RePEc:spr:jotpro:v:19:y:2006:i:3:d:10.1007_s10959-006-0033-2
    DOI: 10.1007/s10959-006-0033-2
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    References listed on IDEAS

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    1. Dai, Jack Jie, 2003. "A note on vertex-reinforced random walks," Statistics & Probability Letters, Elsevier, vol. 62(3), pages 275-280, April.
    2. Dai, Jack Jie, 2004. "Some results regarding vertex-reinforced random walks," Statistics & Probability Letters, Elsevier, vol. 66(3), pages 259-266, February.
    3. J. F. C. Kingman & S. E. Volkov, 2003. "Solution to the OK Corral Model via Decoupling of Friedman's Urn," Journal of Theoretical Probability, Springer, vol. 16(1), pages 267-276, January.
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