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Sub-ballistic random walk in Dirichlet environment. (English) Zbl 1296.60267

Summary: 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.

MSC:

60K37 Processes in random environments
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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