Sub-ballistic random walk in Dirichlet environment

Élodie Bouchet (Université de Lyon)


We consider random walks in Dirichlet environment (RWDE) on $\mathbb{Z} ^d$, for $d \geq 3$, in the sub-ballistic case. We associate to any parameter $ (\alpha_1, \dots, \alpha _{2d}) $ of the Dirichlet law a time-change to accelerate the walk. We prove that the continuous-time accelerated walk has an absolutely continuous invariant probability measure for the environment viewed from the particle. This allows to characterize directional transience for the initial RWDE. It solves as a corollary the problem of Kalikow's $0-1$ law in the Dirichlet case in any dimension. Furthermore, we find the polynomial order of the magnitude of the original walk's displacement.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-25

Publication Date: May 25, 2013

DOI: 10.1214/EJP.v18-2109


  • Atkinson, Giles. Recurrence of co-cycles and random walks. J. London Math. Soc. (2) 13 (1976), no. 3, 486--488. MR0419727
  • Bolthausen, Erwin; Sznitman, Alain-Sol. Ten lectures on random media. DMV Seminar, 32. Birkhäuser Verlag, Basel, 2002. vi+116 pp. ISBN: 3-7643-6703-2 MR1890289
  • Durrett, Richard. Probability: theory and examples. Second edition. Duxbury Press, Belmont, CA, 1996. xiii+503 pp. ISBN: 0-534-24318-5 MR1609153
  • Enriquez, Nathanaël; Sabot, Christophe. Edge oriented reinforced random walks and RWRE. C. R. Math. Acad. Sci. Paris 335 (2002), no. 11, 941--946. MR1952554
  • Enriquez, Nathanaël; Sabot, Christophe. Random walks in a Dirichlet environment. Electron. J. Probab. 11 (2006), no. 31, 802--817 (electronic). MR2242664
  • Kalikow, Steven A. Generalized random walk in a random environment. Ann. Probab. 9 (1981), no. 5, 753--768. MR0628871
  • Keane, M. S.; Rolles, S. W. W. Tubular recurrence. Acta Math. Hungar. 97 (2002), no. 3, 207--221. MR1933730
  • Kesten, H.; Kozlov, M. V.; Spitzer, F. A limit law for random walk in a random environment. Compositio Math. 30 (1975), 145--168. MR0380998
  • Krengel, Ulrich. Ergodic theorems. With a supplement by Antoine Brunel. de Gruyter Studies in Mathematics, 6. Walter de Gruyter & Co., Berlin, 1985. viii+357 pp. ISBN: 3-11-008478-3 MR0797411
  • Lyons, Russell and Peres, Yuval: Probabilities on trees and networks. Cambridge University Press. In preparation, available at .
  • Pemantle, Robin. Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 (1988), no. 3, 1229--1241. MR0942765
  • Sabot, Christophe. Random walks in random Dirichlet environment are transient in dimension $d\geq 3$. Probab. Theory Related Fields 151 (2011), no. 1-2, 297--317. MR2834720
  • Sabot, Christophe: Random Dirichlet environment viewed from the particle in dimension d geq 3 . To appear in Annals of Probability. arxiv:1007.2565
  • Sabot, Christophe; Tournier, Laurent. Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), no. 1, 1--8. MR2779393
  • Sinaĭ, Ya. G. The limit behavior of a one-dimensional random walk in a random environment. (Russian) Teor. Veroyatnost. i Primenen. 27 (1982), no. 2, 247--258. MR0657919
  • Solomon, Fred. Random walks in a random environment. Ann. Probability 3 (1975), 1--31. MR0362503
  • Sznitman, Alain-Sol; Zerner, Martin. A law of large numbers for random walks in random environment. Ann. Probab. 27 (1999), no. 4, 1851--1869. MR1742891
  • Tournier, Laurent. Integrability of exit times and ballisticity for random walks in Dirichlet environment. Electron. J. Probab. 14 (2009), no. 16, 431--451. MR2480548
  • Tournier, Laurent: Asymptotic direction of random walks in Dirichlet environment. Preprint. arxiv:1205.6199
  • Zeitouni, Ofer. Random walks in random environment. Lectures on probability theory and statistics, 189--312, Lecture Notes in Math., 1837, Springer, Berlin, 2004. MR2071631
  • Zerner, Martin P. W.; Merkl, Franz. A zero-one law for planar random walks in random environment. Ann. Probab. 29 (2001), no. 4, 1716--1732. MR1880239

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.