  We study the asymptotic behavior of the simple random walk on oriented
versions of $\mathbb{Z}^2$. 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 i.i.d.
orientations, we solve an open problem and prove a functional limit theorem in
a corresponding space D of cadlag functions, with an unconventional
normalization.
