Asymptotic Behavior for a Version of Directed Percolation on the Triangular Lattice

Abstract

We consider a version of directed bond percolation on the triangular lattice such that vertical edges are directed upward with probability y, diagonal edges are directed from lower-left to upper-right or lower-right to upper-left with probability d, and horizontal edges are directed rightward with probabilities x and one in alternate rows. Let τ(M,N) be the probability that there is at least one connected-directed path of occupied edges from (0,0) to (M,N). For each x∈[0,1], y∈[0,1), d∈[0,1) but (1−y)(1−d)≠1 and aspect ratio α=M/N fixed for the triangular lattice with diagonal edges from lower-left to upper-right, we show that there is an αc=(d−y−dy)/[2(d+y−dy)]+[1−(1−d)2(1−y)2x]/[2(d+y−dy)2] such that as N→∞, τ(M,N) is 1, 0 and 1/2 for α>αc, α<αc and α=αc, respectively. A corresponding result is obtained for the triangular lattice with diagonal edges from lower-right to upper-left. We also investigate the rate of convergence of τ(M,N) and the asymptotic behavior of τ(M−N,N) and τ(M+N,N) where M−N/N↑αc and M+N/N↓αc as N↑∞.