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 $tau (M,N)$ be the probability that there is at least one connected-directed path of occupied edges from $(0,0)$ to $(M,N)$ . For each $x in [0,1]$ , $y in [0,1)$ , $d in [0,1)$ but $(1-y)(1-d) ne 1$ and aspect ratio $alpha =M/N$ fixed for the triangular lattice with diagonal edges from lower-left to upper-right, we show that there is an $alpha _c = (d-y-dy)/[2(d+y-dy)] + [1-(1-d)^2(1-y)^2x]/[2(d+y-dy)^2]$ such that as $N rightarrow infty $ , $tau (M,N)$ is $1$ , $0$ and $1/2$ for $alpha > alpha _c$ , $alpha < alpha _c$ and $alpha =alpha _c$ , 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 $tau (M,N)$ and the asymptotic behavior of $tau (M_N^-,N)$ and $tau (M_N^+ ,N)$ where $M_N^-/Nuparrow alpha _c$ and $M_N^+/Ndownarrow alpha _c$ as $Nuparrow infty $ .