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 $\tau (M,N)$ 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 $M_N^-/N\uparrow \alpha _c$ and $M_N^+/N\downarrow \alpha _c$ as $N\uparrow \infty$.