On the Random-Oriented percolation

Abstract

Let Ld=(Zd, Ed) be the d-dimensional lattice, suppose that each edge of Ld be oriented in a random direction, i.e., each edge being independently oriented positive direction along the coordinate axises with probability p and negative direction otherwise. Let Pp be the percolation measure, η(p) be the probability that there exists an infinite oriented path from the origin. This paper first proves η(p) ≤ θ(p) for d ≥ 2 and ½ ≤ p ≤ 1, where θ(p) is the percolation probability of bond model; then, as corollaries, the author gets η(12)=0 for d = 2 and dc(12)=2, where dc(12)=sup{d : η(12)=0}. Next, based on BK Inequality for arbitrary events in percolation (see[2]), two inequalities are proved, which can be used as FKG Inequality in many cases (note that FKG Inequality is absent for Random-Oriented model). Finally, the author proves the uniqueness of infinite cluster and a theorem on geometry of the infinite cluster (similar to theorem (6.127) in [1] for bond percolation).