Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment
Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 1, p. 1-8

We consider random walks in a random environment given by i.i.d. Dirichlet distributions at each vertex of ℤd or, equivalently, oriented edge reinforced random walks on ℤd. The parameters of the distribution are a 2d-uplet of positive real numbers indexed by the unit vectors of ℤd. We prove that, as soon as these weights are nonsymmetric, the random walk is transient in a direction (i.e., it satisfies Xn ⋅ n +∞ for some ) with positive probability. In dimension 2, this result is strenghened to an almost sure directional transience thanks to the 0-1 law from [Ann. Probab. 29 (2001) 1716-1732]. Our proof relies on the property of stability of Dirichlet environment by time reversal proved in [Random walks in random Dirichlet environment are transient in dimension d ≥ 3 (2009), Preprint]. In a first part of this paper, we also give a probabilistic proof of this property as an alternative to the change of variable computation used initially.

On s'intéresse aux marches aléatoires dans un environnement défini par des variables de Dirichlet i.i.d. en chaque sommet de ℤd ou, de façon équivalente, aux marches aléatoires renforcées par arêtes orientées sur ℤd. Les paramètres de ce modèle sont un 2d-uplet de réels positifs indexé par les vecteurs unitaires de ℤd. On démontre que, dès que ces poids ne sont pas symétriques, la marche aléatoire est transiente dans une direction (c'est-à-dire qu'elle satisfait Xn ⋅ n +∞ pour un certain ) avec probabilité positive. En dimension 2, la loi du 0-1 de [Ann. Probab. 29 (2001) 1716-1732] permet de renforcer ce résultat en transience directionnelle presque-sûre. La preuve repose sur la propriété de stabilité des environnements de Dirichlet par renversement temporel introduite dans [Random walks in random Dirichlet environment are transient in dimension d≥3 (2009), Preprint] et dont on donne une nouvelle démonstration, de nature plus probabiliste, en première partie du présent article.

DOI : https://doi.org/10.1214/09-AIHP344
Classification:  60K37,  60K35
Keywords: random walk, random environment, Dirichlet distribution, directional transience, time reversal
@article{AIHPB_2011__47_1_1_0,
     author = {Sabot, Christophe and Tournier, Laurent},
     title = {Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {1},
     year = {2011},
     pages = {1-8},
     doi = {10.1214/09-AIHP344},
     zbl = {1209.60055},
     mrnumber = {2779393},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2011__47_1_1_0}
}
Sabot, Christophe; Tournier, Laurent. Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 1, pp. 1-8. doi : 10.1214/09-AIHP344. http://www.numdam.org/item/AIHPB_2011__47_1_1_0/

[1] E. Bolthausen and O. Zeitouni. Multiscale analysis of exit distributions for random walks in random environments. Probab. Theory Related Fields 138 (2007) 581-645. | MR 2299720 | Zbl 1126.60088

[2] N. Enriquez and C. Sabot. Edge oriented reinforced random walks and RWRE. C. R. Math. Acad. Sci. Paris 335 (2002) 941-946. | MR 1952554 | Zbl 1016.60051

[3] N. Enriquez and C. Sabot. Random walks in a Dirichlet environment. Electron. J. Probab. 11 (2006) 802-817 (electronic). | MR 2242664 | Zbl 1109.60087

[4] S. Kalikow. Generalized random walk in a random environment. Ann. Probab. 9 (1981) 753-768. | MR 628871 | Zbl 0545.60065

[5] R. Pemantle. Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 (1988) 1229-1241. | MR 942765 | Zbl 0648.60077

[6] C. Sabot. Random walks in random Dirichlet environment are transient in dimension d≥3. Preprint, 2009. Available at arXiv:0905.3917v1[math.PR].

[7] A.-S. Sznitman. Topics in random walks in random environment. In School and Conference on Probability Theory. ICTP Lect. Notes XVII 203-266 (electronic). Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004. | MR 2198849 | Zbl 1060.60102

[8] L. Tournier. Integrability of exit times and ballisticity for random walks in Dirichlet environment. Electron. J. Probab. 14 (2009) 431-451 (electronic). | MR 2480548 | Zbl 1192.60113

[9] M. Zerner. The zero-one law for planar random walks in i.i.d. random environments revisited. Electron. Comm. Probab. 12 (2007) 326-335 (electronic). | MR 2342711 | Zbl 1128.60090

[10] M. Zerner and F. Merkl. A zero-one law for planar random walks in random environment. Ann. Probab. 29 (2001) 1716-1732. | MR 1880239 | Zbl 1016.60093