Transient Random Walks on 2D-Oriented Lattices

We study the asymptotic behavior of the simple random walk on oriented versions of $Z^2$. The considered lattices are not directed on the vertical axis but unidirectional on the horizontal one, with random orientations whose distributions are generated by a dynamical system. We find a sufficient condition on the smoothness of the generation for the transience of the simple random walk on almost every such oriented lattices, and as an illustration we provide a wide class of examples of inhomogeneous or correlated distributions of the orientations. For ergodic dynamical systems, we also prove a strong law of large numbers and, in the particular case of independent identically distributed orientations, we solve an open problem and prove a functional limit theorem in the space $\mathscr{D}([0,\infty[,R^2)$ of càdlàg functions, with an unconventional normalization.