Asymptotic direction of random walks in Dirichlet environment
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, p. 716-726

We prove that, on d , random walks in i.i.d. Dirichlet environment – or equivalently oriented-edge reinforced random walks – have almost surely an asymptotic direction equal to the direction of the initial drift, i.e. X n X n converges to E o [X 1 ] E o [X 1 ] as n, unless this drift is zero. This is obtained by generalizing the result of directional transience from (Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 1–8). In addition, we identify the exact value or distribution of certain probabilities, answering and generalizing a conjecture of that paper.

On démontre que, dans d , les marches aléatoires en milieu aléatoire i.i.d. de Dirichlet – ou, de façon équivalente, les marches renforcées par arêtes orientées – ont presque sûrement une direction asymptotique égale à la direction de la dérive initiale, c’est-à-dire que X n X n converge vers E o [X 1 ] E o [X 1 ] quand n, à moins que cette dérive soit nulle. Ceci est obtenu en généralisant le résultat de transience directionnelle de (Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 1–8). De plus, on explicite la valeur ou la loi de certaines probabilités, ce qui démontre et généralise une conjecture de ce dernier article.

DOI : https://doi.org/10.1214/13-AIHP582
Classification:  60K37,  60K35
Keywords: random walk, random environment, Dirichlet distribution, reinforced random walk, asymptotic direction, time reversal
@article{AIHPB_2015__51_2_716_0,
     author = {Tournier, Laurent},
     title = {Asymptotic direction of random walks in Dirichlet environment},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {2},
     year = {2015},
     pages = {716-726},
     doi = {10.1214/13-AIHP582},
     mrnumber = {3335022},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2015__51_2_716_0}
}
Tournier, Laurent. Asymptotic direction of random walks in Dirichlet environment. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, pp. 716-726. doi : 10.1214/13-AIHP582. http://www.numdam.org/item/AIHPB_2015__51_2_716_0/

[1] E. Bouchet. Sub-ballistic random walk in Dirichlet environment. Electron. J. Probab. 18 (58) (2013) 1–25 (electronic). | MR 3068389 | Zbl 1296.60267

[2] J.-F. Chamayou and G. Letac. Explicit stationary distributions for compositions of random functions and products of random matrices. J. Theoret. Probab. 4 (1991) 3–36. | MR 1088391 | Zbl 0728.60012

[3] A. Drewitz and A. Ramírez. Asymptotic direction in random walks in random environment revisited. Braz. J. Probab. Stat. 24 (2) (2010) 212–225. | MR 2643564 | Zbl 1200.60092

[4] N. Enriquez and C. Sabot. Edge oriented reinforced random walks and RWRE. C. R. Math. Acad. Sci. Paris 335 (11) (2002) 941–946. | MR 1952554 | Zbl 1016.60051

[5] N. Enriquez and C. Sabot. Random walks in a Dirichlet environment. Electron. J. Probab. 11 (31) (2006) 802–817 (electronic). | MR 2242664 | Zbl 1109.60087

[6] M. Zerner and F. Merkl. A zero–one law for planar random walks in random environment. Ann. Probab. 29 (4) (2001) 1716–1732. | MR 1880239 | Zbl 1016.60093

[7] C. Sabot. Ballistic random walks in random environments at low disorder. Ann. Probab. 32 (4) (2004) 2996–3023. | MR 2094437 | Zbl 1063.60149

[8] C. Sabot. Random walks in random Dirichlet environment are transient in dimension d3. Probab. Theory Related Fields 151 (1–2) (2009) 297–317. | MR 2834720 | Zbl 1231.60121

[9] C. Sabot. Random Dirichlet environment viewed from the particle in dimension d3. Ann. Probab. 41 (2) (2013) 722–743. | MR 3077524 | Zbl 1269.60077

[10] C. Sabot and L. Tournier. Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment. Ann. Inst. Henri Poincaré Probab. Stat. 47 (1) (2011) 1–8. | Numdam | MR 2779393 | Zbl 1209.60055

[11] F. Simenhaus. Asymptotic direction for random walks in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 43 (6) (2007) 751–761. | MR 3252429 | Zbl 1172.60337

[12] L. Tournier. Integrability of exit times and ballisticity for random walks in Dirichlet environment. Electron. J. Probab. 14 (16) (2009) 431–451 (electronic). | MR 2480548 | Zbl 1192.60113

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