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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
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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

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