Electronic Journal of Probability

Sub-ballistic random walk in Dirichlet environment

Élodie Bouchet

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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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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