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Edge-reinforced random walk on one-dimensional periodic graphs
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  • Published: 13 August 2008

Edge-reinforced random walk on one-dimensional periodic graphs

  • Franz Merkl1 &
  • Silke W. W. Rolles2 

Probability Theory and Related Fields volume 145, pages 323–349 (2009)Cite this article

  • 157 Accesses

  • 8 Citations

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Abstract

In the present paper, linearly edge-reinforced random walk is studied on a large class of one-dimensional periodic graphs satisfying a certain reflection symmetry. It is shown that the edge-reinforced random walk is recurrent. Estimates for the position of the random walker are given. The edge-reinforced random walk has a unique representation as a random walk in a random environment, where the random environment is given by random weights on the edges. It is shown that these weights decay exponentially in space. The distribution of the random weights equals the distribution of the asymptotic proportion of time spent by the edge-reinforced random walker on the edges of the graph. The results generalize work of the authors in Merkl and Rolles (Ann Probab 33(6):2051–2093, 2005; 35(1):115–140, 2007) and Rolles (Probab Theory Related Fields 135(2):216–264, 2006) to a large class of graphs and to periodic initial weights with a reflection symmetry.

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Author information

Authors and Affiliations

  1. Mathematical Institute, University of Munich, Theresienstr. 39, 80333, Munich, Germany

    Franz Merkl

  2. Zentrum Mathematik, Technische Universität München, Bereich M5, 85747, Garching bei München, Germany

    Silke W. W. Rolles

Authors
  1. Franz Merkl
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  2. Silke W. W. Rolles
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Corresponding author

Correspondence to Franz Merkl.

Additional information

Supported in part by the Swiss national science foundation grant 47102009.

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Merkl, F., Rolles, S.W.W. Edge-reinforced random walk on one-dimensional periodic graphs. Probab. Theory Relat. Fields 145, 323–349 (2009). https://doi.org/10.1007/s00440-008-0170-x

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  • Received: 23 August 2006

  • Revised: 04 July 2008

  • Published: 13 August 2008

  • Issue Date: November 2009

  • DOI: https://doi.org/10.1007/s00440-008-0170-x

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Keywords

  • Reinforced random walk
  • Recurrence
  • Random environment

Mathematics Subject Classification (2000)

  • Primary: 82B41
  • Secondary: 60K35
  • 60K37
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