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, (p) be the probability that there exists an infinite oriented path from the origin. In this paper, we first prove (p)  (p) for d  da12  p  1, where (p) is the percolation probability of bond model; then, as corollaries,we get ( 12 ) = 0 for d = 2 and dc( 12 ) = 2, where dc( 12 ) = sup{d : ( 12 ) = 0}. Next, basedon 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 absentfor Random-Oriented model). Finally, we prove the uniqueness of infinite cluster and a theorem on geometry of the infinite cluster (similar to theorem (6.127) in [1] for bond percolation).

Key words: Random-Oriented percolation, infinite cluster, BK Inequality