Excited against the tide: a random walk with competing drifts
Holmes, Mark
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012), p. 745-773 / Harvested from Numdam
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Nous étudions des marches aléatoires excitées dans un environnement de cookies indépendants en grande dimension, où le kième cookie d’un site détermine le taux de transition (vers la droite ou la gauche) pour le kième départ de ce site. Nous montrons qu’en grande dimension, quand le taux de saut moyen vers la droite du premier cookie est suffisamment grand, la vitesse est strictement positive, quelque soit l’amplitude et le signe des cookies suivants. Sous des conditions supplémentaires sur l’environnement des cookies, nous montrons que la vitesse est une fonction continue des divers paramètres du modèle et est monotone en la force moyenne du cookie à l’origine. Nous donnons aussi des examples non-triviaux où la dérive du premier cookie est dans le sens opposé à toutes les autres et où la vitesse est nulle. Les preuves se basent sur un résultat de temps de coupure de Bolthausen, Sznitman et Zeitouni, le développement en lacets de marches aléatoires auto-interagissantes de van der Hofstad et Holmes, et un argument de couplage.

We study excited random walks in i.i.d. random cookie environments in high dimensions, where the kth cookie at a site determines the transition probabilities (to the left and right) for the kth departure from that site. We show that in high dimensions, when the expected right drift of the first cookie is sufficiently large, the velocity is strictly positive, regardless of the strengths and signs of subsequent cookies. Under additional conditions on the cookie environment, we show that the limiting velocity of the random walk is continuous in various parameters of the model and is monotone in the expected strength of the first cookie at the origin. We also give non-trivial examples where the first cookie drift is in the opposite direction to all subsequent cookie drifts and the velocity is zero. The proofs are based on a cut-times result of Bolthausen, Sznitman and Zeitouni, the lace expansion for self-interacting random walks of van der Hofstad and Holmes, and a coupling argument.

Publié le : 2012-01-01
DOI : https://doi.org/10.1214/11-AIHP434
Classification:  60K35,  60K37,  82B41 
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     author = {Holmes, Mark},
     title = {Excited against the tide: a random walk with competing drifts},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {48},
     year = {2012},
     pages = {745-773},
     doi = {10.1214/11-AIHP434},
     mrnumber = {2976562},
     zbl = {1255.60179},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2012__48_3_745_0}
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Holmes, Mark. Excited against the tide: a random walk with competing drifts. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) pp. 745-773. doi : 10.1214/11-AIHP434. http://gdmltest.u-ga.fr/item/AIHPB_2012__48_3_745_0/

[1] T. Antal and S. Redner. The excited random walk in one dimension. J. Phys. A: Math. Gen. 38 (2005) 2555-2577. | MR 2132073 | Zbl 1113.82024

[2] A.-L. Basdevant and A. Singh. On the speed of a cookie random walk. Probab. Theory Related Fields 141 (2008) 625-645. | MR 2391167 | Zbl 1141.60383

[3] A.-L. Basdevant and A. Singh. Rate of growth of a transient cookie random walk. Electron. J. Probab. 13 (2008) 811-851. | MR 2399297 | Zbl 1191.60107

[4] I. Benjamini and D. B. Wilson. Excited random walk. Electron. Commun. Probab. 8 (2003) 86-92. | MR 1987097 | Zbl 1060.60043

[5] J. Bérard and A. Ramírez. Central limit theorem for excited random walk in dimension d≥2. Electron. Commun. Probab. 12 (2007) 300-314. | MR 2342709 | Zbl 1128.60082

[6] E. Bolthausen, A.-S. Sznitman and O. Zeitouni. Cut points and diffusive random walks in random environment. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 527-555. | Numdam | MR 1978990 | Zbl 1016.60094

[7] B. Davis. Brownian motion and random walk perturbed at extrema. Probab. Theory Related Fields 113 (1999) 501-518. | MR 1717528 | Zbl 0930.60041

[8] T. Hara. Private communication, 2007.

[9] T. Hara and G. Slade. The lace expansion for self-avoiding walk in five or more dimensions. Rev. Math. Phys. 4 (1992) 235-327. | MR 1174248 | Zbl 0755.60054

[10] R. Van Der Hofstad and M. Holmes. An expansion for self-interacting random walks. Brazilian J. Probab. Statist. 26 (2012) 1-55. | MR 2871279 | Zbl 1238.60116

[11] R. Van Der Hofstad and M. Holmes. A monotonicity property for excited random walk in high dimensions. Probab. Theory Related Fields 147 (2010) 333-348. | MR 2594356 | Zbl 1193.60123

[12] M. Holmes and T.S. Salisbury. A combinatorial result with applications to self-interacting random walks. Preprint, 2011. | MR 2860606 | Zbl 1232.60055

[13] M. Holmes and R. Sun. A monotonicity property for random walk in a partially random environment. Available at arXiv:1005.0927v1, 2010. | MR 2914756 | Zbl 1254.60096

[14] E. Kosygina and M. Zerner. Positively and negatively excited random walks on integers, with branching processes. Electron. J. Probab. 13 (2008) 1952-1979. | MR 2453552 | Zbl 1191.60113

[15] R Development Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2008. ISBN 3-900051-07-0.

[16] W. Rudin. Principles of Mathematical Analysis, 3rd edition. McGraw-Hill, New York, 1976. | MR 385023 | Zbl 0052.05301

[17] 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

[18] M. Zerner. Multi-excited random walks on integers. Probab. Theory Related Fields 133 (2005) 98-122. | MR 2197139 | Zbl 1076.60088