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

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 t(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 α_c = (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 > α_c, α < α_c and α = α_c, 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 t(M^-_N, N) and τ(M⁺_N, N) where M^-_N/N ↑ α_c and M⁺_N/N ↓ α_c as N ↑ ∞.