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Transient random walks in random environment on a Galton–Watson tree
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  • Published: 23 January 2008

Transient random walks in random environment on a Galton–Watson tree

  • Elie Aidékon1 

Probability Theory and Related Fields volume 142, pages 525–559 (2008)Cite this article

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Abstract

We consider a transient random walk (X n ) in random environment on a Galton–Watson tree. Under fairly general assumptions, we give a sharp and explicit criterion for the asymptotic speed to be positive. As a consequence, situations with zero speed are revealed to occur. In such cases, we prove that X n is of order of magnitude n Λ, with \({\Lambda \in (0,1)}\). We also show that the linearly edge reinforced random walk on a regular tree always has a positive asymptotic speed, which improves a recent result of Collevecchio (Probab Theory Related 136(1):81–101, 2006).

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Authors and Affiliations

  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, 4 Place Jussieu, 75252, Paris Cedex 05, France

    Elie Aidékon

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  1. Elie Aidékon
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Correspondence to Elie Aidékon.

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Aidékon, E. Transient random walks in random environment on a Galton–Watson tree. Probab. Theory Relat. Fields 142, 525–559 (2008). https://doi.org/10.1007/s00440-007-0114-x

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  • Received: 14 March 2007

  • Revised: 17 October 2007

  • Published: 23 January 2008

  • Issue Date: November 2008

  • DOI: https://doi.org/10.1007/s00440-007-0114-x

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Keywords

  • Random walk in random environment
  • Reinforced random walk
  • Law of large numbers
  • Galton–Watson tree

Mathematics Subject Classification (2000)

  • 60K37
  • 60J80
  • 60F15
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