  Grimmett's random-orientation percolation is formulated as follows. The
square lattice is used to generate an oriented graph such that each edge is
oriented rightwards (resp. upwards) with probability $p$ and leftwards (resp.
downwards) otherwise. We consider a variation of Grimmett's model proposed by
Hegarty, in which edges are oriented away from the origin with probability $p$,
and towards it with probability $1-p$, which implies rotational instead of
translational symmetry. We show that both models could be considered as special
cases of random-oriented percolation in the NE-quadrant, provided that the
critical value for the latter is 1/2. As a corollary, we unconditionally obtain
a non-trivial lower bound for the critical value of Hegarty's
random-orientation model. The second part of the paper is devoted to higher
dimensions and we show that the Grimmett model percolates in any slab of height
at least 3 in $\mathbb{Z}^3$.
