We consider a version of directed bond percolation on the triangular lattice such that vertical edges are directed upward with probability y\documentclass[12pt]{minimal}
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\begin{document}$$y$$\end{document}, diagonal edges are directed from lower-left to upper-right or lower-right to upper-left with probability d\documentclass[12pt]{minimal}
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\begin{document}$$d$$\end{document}, and horizontal edges are directed rightward with probabilities x\documentclass[12pt]{minimal}
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\begin{document}$$x$$\end{document} and one in alternate rows. Let τ(M,N)\documentclass[12pt]{minimal}
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\begin{document}$$\tau (M,N)$$\end{document} be the probability that there is at least one connected-directed path of occupied edges from (0,0)\documentclass[12pt]{minimal}
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\begin{document}$$(0,0)$$\end{document} to (M,N)\documentclass[12pt]{minimal}
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\begin{document}$$(M,N)$$\end{document}. For each x∈[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$x \in [0,1]$$\end{document}, y∈[0,1)\documentclass[12pt]{minimal}
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\begin{document}$$y \in [0,1)$$\end{document}, d∈[0,1)\documentclass[12pt]{minimal}
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\begin{document}$$d \in [0,1)$$\end{document} but (1-y)(1-d)≠1\documentclass[12pt]{minimal}
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\begin{document}$$(1-y)(1-d) \ne 1$$\end{document} and aspect ratio α=M/N\documentclass[12pt]{minimal}
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\begin{document}$$\alpha =M/N$$\end{document} 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]\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _c = (d-y-dy)/[2(d+y-dy)] + [1-(1-d)^2(1-y)^2x]/[2(d+y-dy)^2]$$\end{document} such that as N→∞\documentclass[12pt]{minimal}
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\begin{document}$$N \rightarrow \infty $$\end{document}, τ(M,N)\documentclass[12pt]{minimal}
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\begin{document}$$\tau (M,N)$$\end{document} is 1\documentclass[12pt]{minimal}
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\begin{document}$$1$$\end{document}, 0\documentclass[12pt]{minimal}
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\begin{document}$$0$$\end{document} and 1/2\documentclass[12pt]{minimal}
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\begin{document}$$1/2$$\end{document} for α>αc\documentclass[12pt]{minimal}
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\begin{document}$$\alpha > \alpha _c$$\end{document}, α<αc\documentclass[12pt]{minimal}
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\begin{document}$$\alpha < \alpha _c$$\end{document} and α=αc\documentclass[12pt]{minimal}
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\begin{document}$$\alpha =\alpha _c$$\end{document}, 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)\documentclass[12pt]{minimal}
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\begin{document}$$\tau (M,N)$$\end{document} and the asymptotic behavior of τ(MN-,N)\documentclass[12pt]{minimal}
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\begin{document}$$\tau (M_N^-,N)$$\end{document} and τ(MN+,N)\documentclass[12pt]{minimal}
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\begin{document}$$\tau (M_N^+ ,N)$$\end{document} where MN-/N↑αc\documentclass[12pt]{minimal}
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\begin{document}$$M_N^-/N\uparrow \alpha _c$$\end{document} and MN+/N↓αc\documentclass[12pt]{minimal}
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\begin{document}$$M_N^+/N\downarrow \alpha _c$$\end{document} as N↑∞\documentclass[12pt]{minimal}
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\begin{document}$$N\uparrow \infty $$\end{document}.
