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.
References
Berger, N.: Slowdown estimates for ballistic random walk in random environment. arXiv:0811.1710
Simenhaus F.: Asymptotic direction for random walks in random environments. Ann. Inst. H. Poincaré 43(6), 751–761 (2007)
Solomon F.: Random walk in a random environment. Ann. Probab. 3, 1–31 (1975)
Sznitman A.S., Zerner M.: A law of large numbers for random walks in random environment. Ann. Probab. 27(4), 1851–1869 (1999)
Sznitman A.S.: On a class of transient random walks in random environment. Ann. Probab. 29(2), 724–765 (2001)
Sznitman A.S.: An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Relat. Fields 122(4), 509–544 (2001)
Zerner M.: A non-ballistic law of large numbers for random walks in i.i.d. random environment. Electron. Comm. Probab. 7, 191–197 (2002)
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-010-0268-9
Keywords
- Random walk in random environment
- Slowdowns
- Ballisticity conditions
- Asymptotic direction
Mathematics Subject Classification (2000)
- 60K37
- 82D30