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 probabilit...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 1/2 p 1, where θ(p) is the percolation probability of bond model; then, as corollaries, the author gets η(1/2) = 0 for d = 2 and dc(1/2) = 2, where dc(1/2) = sup{d: η(1/2) = 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).展开更多
基金Research supported by the National Natural Science Foundation of China (1977100819571011)Doctoral Programm Fundation of Ins
文摘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 1/2 p 1, where θ(p) is the percolation probability of bond model; then, as corollaries, the author gets η(1/2) = 0 for d = 2 and dc(1/2) = 2, where dc(1/2) = sup{d: η(1/2) = 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).
