Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T14:22:37.471Z Has data issue: false hasContentIssue false
  • English
  • Français

Transient random walk in ${\mathbb Z}^2$ with stationary orientations

Published online by Cambridge University Press:  22 September 2009

Françoise Pène
Françoise Pène*
Affiliation:
Université Européenne de Bretagne, France. Université de Brest, Laboratoire de Mathématiques, UMR CNRS 6205, Brest, France; Francoise.Pene@univ-brest.fr
Get access

Abstract

In this paper, we extend a result of Campanino and Pétritis [Markov Process. Relat. Fields 9 (2003) 391–412]. We study a random walk in ${\mathbb Z}^2$ with random orientations.We suppose that the orientation of the kth flooris given by $\xi_k$ , where $(\xi_k)_{k\in\mathbb Z}$ isa stationary sequence of random variables.Once the environment fixed, the random walk can goeither up or down or can stay in the present floor (but moving with respect to its orientation).This model was introduced by Campanino and Pétritisin [Markov Process. Relat. Fields 9 (2003) 391–412] whenthe $(\xi_k)_{k\in\mathbb Z}$ is a sequence ofindependent identically distributed random variables. In [Theory Probab. Appl. 52 (2007) 815–826], Guillotin-Plantard and Le Ny extend thisresult to a situation where the orientations of the floors are independentbut chosen with stationary probabilities (not equal to 0and to 1). In the present paper, we generalize the result of [Markov Process. Relat. Fields 9 (2003) 391–412]to some cases when $(\xi_k)_k$ is stationary. Moreover we extend slightlya result of [Theory Probab. Appl.52 (2007) 815–826].

Keywords

Transiencerandom walkMarkov chainoriented graphsstationary orientations.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 13 , July 2009 , pp. 417 - 436
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bunimovich, L.A. and Sinai, Ya.G., Markov partitions for dispersed billiards. Commun. Math. Phys. 78 (1980) 247–280. CrossRef
Bunimovich, L.A. and Sinai, Ya.G., Statistical properties of Lorentz gas with periodic configuration of scatterers. Commun. Math. Phys. 78 (1981) 479–497. CrossRef
Bunimovich, L.A., Sinai, Ya.G. and Chernov, N.I., Markov partitions for two-dimensional hyperbolic billiards. Russ. Math. Surv. 45 (1990) 105–152. CrossRef
Bunimovich, L.A., Sinai, Ya.G. and Chernov, N.I., Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. 46 (1991) 47–106. CrossRef
Campanino, M. and Pétritis, D., Random walks on randomly oriented lattices. Markov Process. Relat. Fields 9 (2003) 391–412.
Chernov, N.I., Advanced statistical properties of dispersing billiards. J. Stat. Phys. 122 (2006) 1061–1094. CrossRef
Gallavotti, G. and Ornstein, D., Billiards and Bernoulli schemes. Commun. Math. Phys. 38 (1974) 83–101. CrossRef
G. Grimmett, Percolation, second edition. Springer, Berlin (1999).
Guillotin-Plantard, N. and Transient, A. Le Ny random walks on 2d-oriented lattices. Theory Probab. Appl. 52 (2007) 815–826.
B.D. Hughes, Random walks and random environments. Vol. 2: Random environments. Oxford Science Publications, Clarendon Press, Oxford. (1996) xxiv.
Ibragimov, I.A., Some limit theorems for stationary processes. Th. Probab. Appl. 7 (1962) 349–382. CrossRef
Jan, C., Vitesse de convergence dans le TCL pour des chaînes de Markov et certains processus associés à des systèmes dynamiques. C. R. Acad. Sci. Paris Ser. I Math. 331 (2000) 395–398. CrossRef
C. Jan, Vitesse de convergence dans le TCL pour des processus associés à des systèmes dynamiques et aux produits de matrices aléatoires. Thèse, Université de Rennes 1, 2001.
Kesten, H. and Spitzer, F., A limit theorem related to a new class of self similar processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 50 (1979) 5–25. CrossRef
Le Borgne, S., Exemples de systèmes dynamiques quasi-hyperboliques à décorrélations lentes. C. R. Acad. Sci. Paris Ser. I Math. 343 (2006) 125–128. CrossRef
Le Borgne, S. and Pène, F., Vitesse dans le théorème limite central pour certains systèmes dynamiques quasi-hyperboliques. Bull. Soc. Math. France 133 (2005) 395–417. CrossRef
Sinai, Ya.G., Dynamical systems with elastic reflections. Russ. Math. Surv. 25 (1970) 137–189. CrossRef
F. Spitzer, Principles of random walk. Univ. Ser. Higher Math., Van Nostrand, Princeton (1964).
Young, L.-S., Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147 (1998) 585–650. CrossRef