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On slowdown and speedup of transient random walks in random environment
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  • Published: 10 February 2009

On slowdown and speedup of transient random walks in random environment

  • Alexander Fribergh1,
  • Nina Gantert2 &
  • Serguei Popov3 

Probability Theory and Related Fields volume 147, pages 43–88 (2010)Cite this article

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  • 12 Citations

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Abstract

We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time n the particle is typically at a distance of order O(n κ) from the origin, \({\kappa \in (0, 1)}\) . We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time n, the particle is at a distance of order \({O(n^{\nu_0})}\) from the origin, \({\nu_0 \in (0, \kappa)}\)), and speedup (at time n, the particle is at a distance of order \({n^{\nu_1}}\) from the origin, \({\nu_1 \in (\kappa, 1)}\)), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time n, the particle is located around (−n ν), thus making an unusual excursion to the left. For the slowdown, our results are valid in the ballistic case as well.

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Author information

Authors and Affiliations

  1. Université de Lyon, Université Lyon 1, CNRS UMR5208, Institut Camille Jordan, 69622, Villeurbanne Cedex, France

    Alexander Fribergh

  2. CeNoS, Center for Nonlinear Science, Institut für Mathematische Statistik, Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstr. 62, 48149, Munster, Germany

    Nina Gantert

  3. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, CEP 05508-090, São Paulo, SP, Brazil

    Serguei Popov

Authors
  1. Alexander Fribergh
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  2. Nina Gantert
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  3. Serguei Popov
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Corresponding author

Correspondence to Nina Gantert.

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Fribergh, A., Gantert, N. & Popov, S. On slowdown and speedup of transient random walks in random environment. Probab. Theory Relat. Fields 147, 43–88 (2010). https://doi.org/10.1007/s00440-009-0201-2

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  • Received: 03 July 2008

  • Revised: 11 December 2008

  • Published: 10 February 2009

  • Issue Date: May 2010

  • DOI: https://doi.org/10.1007/s00440-009-0201-2

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Keywords

  • Slowdown
  • Speedup
  • Moderate deviations
  • Transience

Mathematics Subject Classification (2000)

  • Primary 60K37
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