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.


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