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On the anisotropic walk on the supercritical percolation cluster. (English) Zbl 1087.82011
Summary: We investigate in this work the asymptotic behavior of an anisotropic random walk on the supercritical cluster for bond percolation on ${ℤ}^{d}$, $d\ge 2$. In particular we show that for small anisotropy the walk behaves in a ballistic fashion, whereas for strong anisotropy the walk is sub-diffusive. For arbitrary anisotropy, we also prove the directional transience of the walk and construct a renewal structure.

##### MSC:
 82B43 Percolation (equilibrium statistical mechanics) 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G50 Sums of independent random variables; random walks 82B41 Random walks, random surfaces, lattice animals, etc. (statistical mechanics)