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Random walks in random Dirichlet environment are transient in dimension d ≥ 3
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  • Published: 22 May 2010

Random walks in random Dirichlet environment are transient in dimension d ≥ 3

  • Christophe Sabot1 

Probability Theory and Related Fields volume 151, pages 297–317 (2011)Cite this article

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Abstract

We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On \({\mathbb{Z}^d}\), RWDE are parameterized by a 2d-uplet of positive reals. We prove that for all values of the parameters, RWDE are transient in dimension d ≥ 3. We also prove that the Green function has some finite moments and we characterize the finite moments. Our result is more general and applies for example to finitely generated symmetric transient Cayley graphs. In terms of reinforced random walks it implies that directed edge reinforced random walks are transient for d ≥ 3.

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Authors and Affiliations

  1. CNRS UMR5208, Institut Camille Jordan, Université de Lyon, Université Lyon 1, 43 bd du 11 nov., 69622, Villeurbanne Cedex, France

    Christophe Sabot

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  1. Christophe Sabot
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Correspondence to Christophe Sabot.

Additional information

This work was partly supported by the ANR project MEMEMO.

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Cite this article

Sabot, C. Random walks in random Dirichlet environment are transient in dimension d ≥ 3. Probab. Theory Relat. Fields 151, 297–317 (2011). https://doi.org/10.1007/s00440-010-0300-0

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  • Received: 09 April 2009

  • Revised: 26 April 2010

  • Published: 22 May 2010

  • Issue Date: October 2011

  • DOI: https://doi.org/10.1007/s00440-010-0300-0

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Keywords

  • Random walk in random environment
  • Dirichlet distribution
  • Reinforced random walks
  • Max-Flow Min-Cut theorem

Mathematics Subject Classification (2000)

  • Primary 60K37
  • 60K35
  • Secondary 5C20
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