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

Pène, Françoise

ESAIM: Probability and Statistics, Tome 13 (2009), pp. 417-436

Résumé

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 $k$th floor is given by ${\xi }_k$, where ${\left({\xi }_k\right)}_{k\in \mathbb{Z}}$ is a stationary sequence of random variables. Once the environment fixed, the random walk can go either up or down or can stay in the present floor (but moving with respect to its orientation). This model was introduced by Campanino and Pétritis in [Markov Process. Relat. Fields 9 (2003) 391-412] when the ${\left({\xi }_k\right)}_{k\in \mathbb{Z}}$ is a sequence of independent identically distributed random variables. In [Theory Probab. Appl. 52 (2007) 815-826], Guillotin-Plantard and Le Ny extend this result to a situation where the orientations of the floors are independent but chosen with stationary probabilities (not equal to 0 and to 1). In the present paper, we generalize the result of [Markov Process. Relat. Fields 9 (2003) 391-412] to some cases when ${\left({\xi }_k\right)}_k$ is stationary. Moreover we extend slightly a result of [Theory Probab. Appl. 52 (2007) 815-826].

Détail

DOI : 10.1051/ps:2008019

Classification : 60J10
Keywords: transience, random walk, Markov chain, oriented graphs, stationary orientations