Open Access
2013 Sub-ballistic random walk in Dirichlet environment
Élodie Bouchet
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Electron. J. Probab. 18: 1-25 (2013). DOI: 10.1214/EJP.v18-2109


We consider random walks in Dirichlet environment (RWDE) on $\mathbb{Z} ^d$, for $d \geq 3$, in the sub-ballistic case. We associate to any parameter $ (\alpha_1, \dots, \alpha _{2d}) $ of the Dirichlet law a time-change to accelerate the walk. We prove that the continuous-time accelerated walk has an absolutely continuous invariant probability measure for the environment viewed from the particle. This allows to characterize directional transience for the initial RWDE. It solves as a corollary the problem of Kalikow's $0-1$ law in the Dirichlet case in any dimension. Furthermore, we find the polynomial order of the magnitude of the original walk's displacement.


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Élodie Bouchet. "Sub-ballistic random walk in Dirichlet environment." Electron. J. Probab. 18 1 - 25, 2013.


Accepted: 25 May 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 1296.60267
MathSciNet: MR3068389
Digital Object Identifier: 10.1214/EJP.v18-2109

Primary: 60K37
Secondary: 60K35

Keywords: Dirichlet distribution , invariant measure viewed from the particle , Random walk in random environment , reinforced random walks

Vol.18 • 2013
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