On a Property of Random-Oriented Percolation in a Quadrant

ISSN： 0022-4715

Source： Journal of Statistical Physics, Vol.153, Iss.5, 2013-12, pp. : 751-762

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Abstract

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 $mathbb{Z}^{3}$ .