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Ballisticity conditions for random walk in random environment

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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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References

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Author information

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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Keywords

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

Mathematics Subject Classification (2000)

  • 60K37
  • 82D30