Random walks on discrete point processes

Berger, Noam ; Rosenthal, Ron

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015), p. 727-755 / Harvested from Numdam

Résumé

Nous considérons un modèle de marches aléatoires en milieu aléatoire ayant pour sommets un sous-ensemble aléatoire de ${\mathbb {Z} }^{d}$ et une probabilité de transition uniforme sur $2d$ points (les plus proches voisins dans chacune des directions des coordonnées). Nous prouvons que la vitesse de ce type de marches est presque sûrement zéro, donnons une caractérisation partielle de transience et récurrence dans les différentes dimensions et prouvons un théorème central limite (CLT) pour de telles marches sous une condition concernant la distance entre plus proches voisins.

Abstract

We consider a model for random walks on random environments (RWRE) with a random subset of ${\mathbb {Z} }^{d}$ as the vertices, and uniform transition probabilities on $2d$ points (the closest in each of the coordinate directions). We prove that the velocity of such random walks is almost surely zero, give partial characterization of transience and recurrence in the different dimensions and prove a Central Limit Theorem (CLT) for such random walks, under a condition on the distance between coordinate nearest neighbors.

Détail

Publié le : 2015-01-01

MR 3335023

DOI : https://doi.org/10.1214/13-AIHP593
Classification:  60K37,  60K35

Bibliographie

[1] M. T. Barlow. Random walks on supercritical percolation clusters. Ann. Probab. 32 (4) (2004) 3024–3084. | MR 2094438 | Zbl 1067.60101

[2] N. Berger. Transience, recurrence and critical behavior for long-range percolation. Comm. Math. Phys. 226 (3) (2002) 531–558. | MR 1896880 | Zbl 0991.82017

[3] N. Berger and M. Biskup. Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 (1–2) (2007) 83–120. | MR 2278453 | Zbl 1107.60066

[4] N. Berger, M. Biskup, C. E. Hoffman and G. Kozma. Anomalous heat-kernel decay for random walk among bounded random conductances. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2) (2008) 374–392. | Numdam | MR 2446329 | Zbl 1187.60034

[5] M. Biskup and T. M. Prescott. Functional CLT for random walk among bounded random conductances. Electron. J. Probab. 12 (49) (2007) 1323–1348. | MR 2354160 | Zbl 1127.60093

[6] E. Bolthausen and I. Goldsheid. Lingering random walks in random environment on a strip. Comm. Math. Phys. 278 (1) (2008) 253–288. | MR 2367205 | Zbl 1142.82007

[7] E. Bolthausen and A. S. Sznitman. Ten Lectures on Random Media. DMV Seminar 32. Birkhäuser, Basel, 2002. | MR 1890289 | Zbl 1075.60128

[8] J. Brémont. On some random walks on $\mathbb {Z}$ in random medium. Ann. Probab. 30 (3) (2002) 1266–1312. | MR 1920108 | Zbl 1021.60034

[9] P. Caputo, A. Faggionato and A. Gaudillière. Recurrence and transience for long range reversible random walks on a random point process. Electron. J. Probab. 14 (90) (2009) 2580–2616. | MR 2570012 | Zbl 1191.60120

[10] N. Crawford and A. Sly. Simple random walk on long range percolation clusters I: Heat kernel bounds. Probab. Theory Related Fields 154 (3–4) (2012) 753–786. | MR 3000561 | Zbl 1257.82050

[11] J. D. Deuschel and A. Pisztora. Surface order large deviations for high-density percolation. Probab. Theory Related Fields 104 (4) (1996) 467–482. | MR 1384041 | Zbl 0842.60023

[12] P. G. Doyle and J. L. Snell. Random Walks and Electric Networks. Carus Mathematical Monographs 22. Mathematical Association of America, Washington, DC, 1984. | MR 920811 | Zbl 0583.60065

[13] R. Durrett. Probability: Theory and Examples, 2nd edition. Duxbury Press, Belmont, CA, 1996. | MR 1609153 | Zbl 1202.60002

[14] B. D. Hughes. Random Walks and Random Environments. Vol. 2. Random Environments. Oxford Univ. Press, New York, 1996. | MR 1420619 | Zbl 0820.60053

[15] E. S. Key. Recurrence and transience criteria for random walk in a random environment. Ann. Probab. 12 (2) (1984) 529–560. | MR 735852 | Zbl 0545.60066

[16] C. Kipnis and S. R. S. Varadhan. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 (1) (1986) 1–19. | MR 834478 | Zbl 0588.60058

[17] R. Lyons and Y. Peres. Probability on Trees and Networks. Cambridge Univ. Press, Cambridge, in preparation, 2015. Current version available at http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html.

[18] P. Mathieu and A. Piatnitski. Quenched invariance principles for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2085) (2007) 2287–2307. | MR 2345229 | Zbl 1131.82012

[19] B. Morris and Y. Peres. Evolving sets, mixing and heat kernel bounds. Probab. Theory Related Fields 133 (2) (2005) 245–266. | MR 2198701 | Zbl 1080.60071

[20] A. Nevo and E. M. Stein. A generalization of Birkhoff’s pointwise ergodic theorem. Acta Math. 173 (1) (1994) 135–154. | MR 1294672 | Zbl 0837.22003

[21] P. Révész. Random Walk in Random and Non-Random Environments, 2nd edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. | MR 2168855 | Zbl 1283.60007

[22] R. Rosenthal. Random walk on discrete point processes. Preprint, 2010. Available at arXiv:1005.1398.

[23] V. Sidoravicius and A. S. Sznitman. Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields 129 (2) (2004) 219–244. | MR 2063376 | Zbl 1070.60090

[24] A. S. Sznitman. Vacant set of random interlacements and percolation. Ann. of Math. (2) 171 (3) (2010) 2039–2087. | MR 2680403 | Zbl 1202.60160

[25] S. R. S. Varadhan. Random walks in a random environment. Proc. Indian Acad. Sci. Math. Sci. 114 (4) (2004) 309–318. | MR 2067696 | Zbl 1077.60078

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