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Limit law for transition probabilities and moderate deviations for Sinai's random walk in random environment
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  • Published: 22 May 2003

Limit law for transition probabilities and moderate deviations for Sinai's random walk in random environment

  • Francis Comets1 &
  • Serguei Popov2 

Probability Theory and Related Fields volume 126, pages 571–609 (2003)Cite this article

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Abstract

We consider a one-dimensional random walk in random environment in the Sinai's regime. Our main result is that logarithms of the transition probabilities, after a suitable rescaling, converge in distribution as time tends to infinity, to some functional of the Brownian motion. We compute the law of this functional when the initial and final points agree. Also, among other things, we estimate the probability of being at time t at distance at least z from the initial position, when z is larger than ln2 t, but still of logarithmic order in time.

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Author information

Authors and Affiliations

  1. Université Paris 7, UFR de Mathématiques, case 7012, 2, place Jussieu, 75251, Paris Cedex 05, France

    Francis Comets

  2. Instituto de Matemática e Estatıstica, Universidade de São Paulo, Rua do Matão 1010, CEP, 05508--090, São Paulo SP, Brasil

    Serguei Popov

Authors
  1. Francis Comets
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  2. Serguei Popov
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Corresponding author

Correspondence to Francis Comets.

Additional information

Partially supported by CNRS (UMR 7599 ``Probabilités et Modéles Aléatoires''), and by the ``Réseau Mathématique France-Brésil''

Partially supported by CNPq (300676/00–0 and 302981/02–0), COFECUB, and by the ``Rede Matemática Brasil-França''

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Comets, F., Popov, S. Limit law for transition probabilities and moderate deviations for Sinai's random walk in random environment. Probab. Theory Relat. Fields 126, 571–609 (2003). https://doi.org/10.1007/s00440-003-0273-3

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  • Received: 28 July 2002

  • Revised: 28 February 2003

  • Published: 22 May 2003

  • Issue Date: August 2003

  • DOI: https://doi.org/10.1007/s00440-003-0273-3

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Keywords

  • Random environment
  • Sinai's regime
  • Elevation
  • Moments of return
  • t-stable points
  • Spectral gap
  • Metastability
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