Maximal displacement for bridges of random walks in a random environment
Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 3, p. 663-678

It is well known that the distribution of simple random walks on ℤ conditioned on returning to the origin after 2n steps does not depend on p=P(S1=1), the probability of moving to the right. Moreover, conditioned on {S2n=0} the maximal displacement maxk≤2n|Sk| converges in distribution when scaled by √n (diffusive scaling). We consider the analogous problem for transient random walks in random environments on ℤ. We show that under the quenched law Pω (conditioned on the environment ω), the maximal displacement of the random walk when conditioned to return to the origin at time 2n is no longer necessarily of the order √n. If the environment is nestling (both positive and negative local drifts exist) then the maximal displacement conditioned on returning to the origin at time 2n is of order nκ/(κ+1), where the constant κ>0 depends on the law on environments. On the other hand, if the environment is marginally nestling or non-nestling (only non-negative local drifts) then the maximal displacement conditioned on returning to the origin at time 2n is at least n1-ε and at most n/(ln n)2-ε for any ε>0. As a consequence of our proofs, we obtain precise rates of decay for Pω(X2n=0). In particular, for certain non-nestling environments we show that Pω(X2n=0)=exp{-Cn-C'n/(ln n)2+o(n/(ln n)2)} with explicit constants C, C'>0.

Il est bien connu que la distribution d'une marche aléatoire simple sur ℤ, conditionnée à retourner à l'origine au temps 2n est indépendante de p=P(S1=1), la probabilité d'un pas vers la droite. De plus, conditionnellement à {S2n=0}, le déplacement maximum maxk≤2n|Sk|, divisé par √n, converge en distribution. Nous considérons le même problème pour les marches transientes en environnement aléatoire sur ℤ. Nous montrons que sous la loi “quenched,” le déplacement maximum pour la marche conditionnée à retourner à l'origine au temps 2n n'est pas toujours de l'ordre de √n. Si l'environnement a des drifts locaux positifs et négatifs alors cet ordre de grandeur est nκ/(κ+1), où κ>0 dépend de la loi de l'environnement. Mais, si l'environnement n'a que des drifts locaux positifs ou nuls, alors cet ordre de grandeur est proche de n. Les preuves fournissent de plus l'ordre de grandeur de Pω(X2n=0). Dans le cas où les drifts locaux sont tous positifs nous montrons que Pω(X2n=0)=exp{-Cn-C'n/(ln n)2+o(n/(ln n)2)}.

DOI : https://doi.org/10.1214/10-AIHP378
Classification:  60K37
Keywords: random walk in random environment, moderate deviations
@article{AIHPB_2011__47_3_663_0,
     author = {Gantert, Nina and Peterson, Jonathon},
     title = {Maximal displacement for bridges of random walks in a random environment},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {3},
     year = {2011},
     pages = {663-678},
     doi = {10.1214/10-AIHP378},
     zbl = {1262.60096},
     mrnumber = {2841070},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2011__47_3_663_0}
}
Gantert, Nina; Peterson, Jonathon. Maximal displacement for bridges of random walks in a random environment. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 3, pp. 663-678. doi : 10.1214/10-AIHP378. http://www.numdam.org/item/AIHPB_2011__47_3_663_0/

[1] F. Comets, N. Gantert and O. Zeitouni. Quenched, annealed and functional large deviations for one-dimensional random walk in random environment. Probab. Theory Related Fields 118 (2000) 65-114. | MR 1785454 | Zbl 0965.60098

[2] A. Dembo, Y. Peres and O. Zeitouni. Tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 181 (1996) 667-683. | MR 1414305 | Zbl 0868.60058

[3] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Applications of Mathematics 38. Springer, New York, 1998. | MR 1619036 | Zbl 0896.60013

[4] N. Enriquez, C. Sabot and O. Zindy. Limit laws for transient random walks in random environments on ℤ. Ann. Institut Fourier 59 (2009) 2469-2508. | Numdam | MR 2640927 | Zbl 1200.60093

[5] A. Fribergh, N. Gantert and S. Popov. On slowdown and speedup of transient random walks in random environment. Probab. Theory Related Fields 147 (2010) 43-88. | MR 2594347 | Zbl 1193.60122

[6] N. Gantert. Subexponential tail asymptotics for a random walk with randomly placed one-way nodes. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 1-16. | Numdam | MR 1899228 | Zbl 1004.60043

[7] N. Gantert and O. Zeitouni. Quenched sub-exponential tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 194 (1998) 177-190. | MR 1628294 | Zbl 0982.60037

[8] A. Greven and F. Den Hollander. Large deviations for a random walk in random environment. Ann. Probab. 22 (1994) 1381-1428. | MR 1303649 | Zbl 0820.60054

[9] H. Kesten, M. V. Kozlov and F. Spitzer. A limit law for random walk in a random environment. Compos. Math. 30 (1975) 145-168. | Numdam | MR 380998 | Zbl 0388.60069

[10] A. A. Mogul'Skiĭ. Small deviations in the space of trajectories. Teor. Verojatn. Primen. 19 (1974) 755-765. | Zbl 0326.60061

[11] A. Pisztora and T. Povel. Large deviation principle for random walk in a quenched random environment in the low speed regime. Ann. Probab. 27 (1999) 1389-1413. | MR 1733154 | Zbl 0964.60056

[12] A. Pisztora, T. Povel and O. Zeitouni. Precise large deviation estimates for a one-dimensional random walk in a random environment. Probab. Theory Related Fields 113 (1999) 191-219. | MR 1676839 | Zbl 0922.60059

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

[14] 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