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A subdiffusive behaviour of recurrent random walk in random environment on a regular tree
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  • Published: 06 January 2007

A subdiffusive behaviour of recurrent random walk in random environment on a regular tree

  • Yueyun Hu1 &
  • Zhan Shi2 

Probability Theory and Related Fields volume 138, pages 521–549 (2007)Cite this article

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  • 24 Citations

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Abstract

We are interested in the random walk in random environment on an infinite tree. Lyons and Pemantle (Ann. Probab. 20, 125–136, 1992) give a precise recurrence/transience criterion. Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk (X n ) in random environment on a regular tree, which is closely related to Mandelbrot’s (C. R. Acad. Sci. Paris 278, 289–292, 1974) multiplicative cascade. We prove, under some general assumptions upon the distribution of the environment, the existence of a new exponent \({\nu\in \big(0, {1\over 2}\big]}\) such that \({\max_{0\le i \le n} |X_i|}\) behaves asymptotically like \({n^{\nu}}\) . The value of ν is explicitly formulated in terms of the distribution of the environment.

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Authors and Affiliations

  1. Département de Mathématiques, Université Paris XIII, 99 avenue J-B Clément, 93430, Villetaneuse, France

    Yueyun Hu

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

    Zhan Shi

Authors
  1. Yueyun Hu
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  2. Zhan Shi
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Correspondence to Yueyun Hu.

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Hu, Y., Shi, Z. A subdiffusive behaviour of recurrent random walk in random environment on a regular tree. Probab. Theory Relat. Fields 138, 521–549 (2007). https://doi.org/10.1007/s00440-006-0036-z

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  • Received: 08 March 2006

  • Revised: 20 September 2006

  • Published: 06 January 2007

  • Issue Date: July 2007

  • DOI: https://doi.org/10.1007/s00440-006-0036-z

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Keywords

  • Random walk
  • Random environment
  • Tree
  • Mandelbrot’s multiplicative cascade

Mathematics Subject Classification (2000)

  • 60K37
  • 60G50
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