  Simple random walks on a partially directed version of $\mathbb{Z}^2$ are
considered. More precisely, vertical edges between neighbouring vertices of
$\mathbb{Z}^2$ can be traversed in both directions (they are undirected) while
horizontal edges are one-way. The horizontal orientation is prescribed by a
random perturbation of a periodic function, the perturbation probability decays
according to a power law in the absolute value of the ordinate. We study the
type of the simple random walk, i.e.\ its being recurrent or transient, and
show that there exists a critical value of the decay power, above which the
walk is almost surely recurrent and below which is almost surely transient.
