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Almost sure convergence for stochastically biased random walks on trees
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  • Published: 08 July 2011

Almost sure convergence for stochastically biased random walks on trees

  • Gabriel Faraud1,
  • Yueyun Hu1 &
  • Zhan Shi2 

Probability Theory and Related Fields volume 154, pages 621–660 (2012)Cite this article

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Abstract

We are interested in the biased random walk on a supercritical Galton–Watson tree in the sense of Lyons (Ann. Probab. 18:931–958, 1990) and Lyons, Pemantle and Peres (Probab. Theory Relat. Fields 106:249–264, 1996), and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random; the resulting random walk is then a tree-valued random walk in random environment. We investigate the recurrent case, and prove, under suitable general integrability assumptions, that upon the system’s non-extinction, the maximal displacement of the walk in the first n steps, divided by (log n)3, converges almost surely to a known positive constant.

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Authors and Affiliations

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

    Gabriel Faraud & Yueyun Hu

  2. Laboratoire de Probabilités UMR 7599, Université Paris VI, 4 place Jussieu, 75252, Paris Cedex 05, France

    Zhan Shi

Authors
  1. Gabriel Faraud
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  2. Yueyun Hu
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  3. Zhan Shi
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Correspondence to Gabriel Faraud.

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Faraud, G., Hu, Y. & Shi, Z. Almost sure convergence for stochastically biased random walks on trees. Probab. Theory Relat. Fields 154, 621–660 (2012). https://doi.org/10.1007/s00440-011-0379-y

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  • Received: 21 October 2010

  • Revised: 13 May 2011

  • Published: 08 July 2011

  • Issue Date: December 2012

  • DOI: https://doi.org/10.1007/s00440-011-0379-y

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Keywords

  • Biased random walk on a Galton–Watson tree
  • Branching random walk
  • Slow movement
  • Random walk in a random environment

Mathematics Subject Classification (2010)

  • 60J80
  • 60G50
  • 60K37
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