On the limiting velocity of random walks in mixing random environment
Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 375-402.

Nous considérons des marches aléatoires dans un environnement Gibbsien fortement mélangeant dans d , d2. A l’aide d’arguments de renouvellement, nous donnons d’abord une preuve alternative de la loi conditionnelle des grands nombres de Rassoul-Agha (Electron. Commun. Probab. 10 (2005) 36-44) pour des environnements mélangeants. Ensuite, par des méthodes de couplage, nous montrons qu’il existe au plus une vitesse limite non nulle en grande dimension (d5).

We consider random walks in strong-mixing random Gibbsian environments in d , d2. Based on regeneration arguments, we will first provide an alternative proof of Rassoul-Agha’s conditional law of large numbers (CLLN) for mixing environment (Electron. Commun. Probab. 10 (2005) 36-44). Then, using coupling techniques, we show that there is at most one nonzero limiting velocity in high dimensions (d5).

DOI : https://doi.org/10.1214/12-AIHP534
Classification : 60K37
Mots clés : random walks, random environment, mixing, limiting speed, conditional law of large numbers
@article{AIHPB_2014__50_2_375_0,
     author = {Guo, Xiaoqin},
     title = {On the limiting velocity of random walks in mixing random environment},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {375--402},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {2},
     year = {2014},
     doi = {10.1214/12-AIHP534},
     zbl = {1291.60211},
     mrnumber = {3189076},
     language = {en},
     url = {www.numdam.org/item/AIHPB_2014__50_2_375_0/}
}
Guo, Xiaoqin. On the limiting velocity of random walks in mixing random environment. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 375-402. doi : 10.1214/12-AIHP534. http://www.numdam.org/item/AIHPB_2014__50_2_375_0/

[1] N. Berger. Limiting velocity of high-dimensional random walk in random environment. Ann. Probab. 36 (2008) 728-738. | MR 2393995 | Zbl 1145.60051

[2] F. Comets and O. Zeitouni. A law of large numbers for random walks in random mixing environments. Ann. Probab. 32 (2004) 880-914. | MR 2039946 | Zbl 1078.60089

[3] F. Comets and O. Zeitouni. Gaussian fluctuations for random walks in random mixing environments. Probability in mathematics. Israel J. Math. 148 (2005) 87-113. | MR 2191225 | Zbl 1086.60065

[4] R. Dobrushin and S. Shlosman. Completely analytical Gibbs fields. In Statistical Physics and Dynamical Systems (Köszeg, 1984) 371-403. Progr. Phys. 10. Birkhäuser, Boston, MA, 1985. | MR 821307 | Zbl 0569.46043

[5] L. Goergen. Limit velocity and zero-one laws for diffusions in random environment. Ann. Appl. Probab. 16 (2006) 1086-1123. | MR 2260058 | Zbl 1107.60070

[6] F. Rassoul-Agha. The point of view of the particle on the law of large numbers for random walks in a mixing random environment. Ann. Probab. 31 (2003) 1441-1463. | MR 1989439 | Zbl 1039.60089

[7] F. Rassoul-Agha. Large deviations for random walks in a mixing random environment and other (non-Markov) random walks. Comm. Pure Appl. Math. 57 (2004) 1178-1196. | MR 2059678 | Zbl 1051.60033

[8] F. Rassoul-Agha. On the zero-one law and the law of large numbers for random walk in mixing random environment. Electron. Commun. Probab. 10 (2005) 36-44. | MR 2119152 | Zbl 1060.60101

[9] H. Thorisson. Coupling, Stationarity, and Regeneration. Probability and Its Applications (New York). Springer, New York, 2000. | MR 1741181 | Zbl 0949.60007

[10] F. Solomon. Random walks in a random environment. Ann. Probab. 3 (1975) 1-31. | MR 362503 | Zbl 0305.60029

[11] A. S. Sznitman and M. Zerner. A law of large numbers for random walks in random environment. Ann. Probab. 27 (1999) 1851-1869. | MR 1742891 | Zbl 0965.60100

[12] O. Zeitouni. Random walks in random environment. In Lectures on Probability Theory and Statistics 189-312. Lecture Notes in Math. 1837. Springer, Berlin, 2004. | MR 2071631 | Zbl 1060.60103

[13] M. Zerner. A non-ballistic law of large numbers for random walks in i.i.d. random environment. Electron. Commun. Probab. 7 (2002) 191-197. | MR 1937904 | Zbl 1008.60107

[14] 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