Electronic Journal of Probability

Sub-ballistic random walk in Dirichlet environment

Élodie Bouchet

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Abstract

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.

Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 58, 25 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465064283

Digital Object Identifier
doi:10.1214/EJP.v18-2109

Mathematical Reviews number (MathSciNet)
MR3068389

Zentralblatt MATH identifier
1296.60267

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Random walk in random environment Dirichlet distribution Reinforced random walks Invariant measure viewed from the particle

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Bouchet, Élodie. Sub-ballistic random walk in Dirichlet environment. Electron. J. Probab. 18 (2013), paper no. 58, 25 pp. doi:10.1214/EJP.v18-2109. https://projecteuclid.org/euclid.ejp/1465064283


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