On a Property of Random-Oriented Percolation in a Quadrant

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 $\frac{1}{2}$ . 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  $\mathbb{Z}^{3}$ .