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Ballisticity conditions for random walk in random environment
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  • Published: 23 February 2010

Ballisticity conditions for random walk in random environment

  • A. Drewitz1 &
  • A. F. Ramírez2 

Probability Theory and Related Fields volume 150, pages 61–75 (2011)Cite this article

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Abstract

Consider a random walk in a uniformly elliptic i.i.d. random environment in dimensions d ≥ 2. In 2002, Sznitman introduced for each \({\gamma\in (0, 1)}\) the ballisticity conditions (T) γ and (T′), the latter being defined as the fulfillment of (T) γ for all \({\gamma\in (0, 1)}\). He proved that (T′) implies ballisticity and that for each \({\gamma\in (0.5, 1)}\), (T) γ is equivalent to (T′). It is conjectured that this equivalence holds for all \({\gamma\in (0, 1)}\). Here we prove that for \({\gamma\in (\gamma_d, 1)}\), where γ d is a dimension dependent constant taking values in the interval (0.366, 0.388), (T) γ is equivalent to (T′). This is achieved by a detour along the effective criterion, the fulfillment of which we establish by a combination of techniques developed by Sznitman giving a control on the occurrence of atypical quenched exit distributions through boxes.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Sekr. MA 7-5, Str. des 17. Juni 136, 10623, Berlin, Germany

    A. Drewitz

  2. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860 Macul, Santiago, Chile

    A. F. Ramírez

Authors
  1. A. Drewitz
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  2. A. F. Ramírez
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Corresponding author

Correspondence to A. F. Ramírez.

Additional information

A. Drewitz was partially supported by the International Research Training Group “Stochastic Models of Complex Processes” and by the Berlin Mathematical School. A. F. Ramírez was partially supported by Iniciativa Científica Milenio P-04-069-F and by Fondo Nacional de Desarrollo Científico y Tecnológico grant 1060738.

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Drewitz, A., Ramírez, A.F. Ballisticity conditions for random walk in random environment. Probab. Theory Relat. Fields 150, 61–75 (2011). https://doi.org/10.1007/s00440-010-0268-9

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  • Received: 09 April 2009

  • Revised: 13 January 2010

  • Published: 23 February 2010

  • Issue Date: June 2011

  • DOI: https://doi.org/10.1007/s00440-010-0268-9

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Keywords

  • Random walk in random environment
  • Slowdowns
  • Ballisticity conditions
  • Asymptotic direction

Mathematics Subject Classification (2000)

  • 60K37
  • 82D30
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