Asymptotic behavior for a version of directed percolation on a square lattice

We consider a version of directed bond percolation on a square lattice whose vertical edges are directed upward with probabilities pv and horizontal edges are directed rightward with probabilities ph and 1 in alternate rows. Let τ(M,N) be the probability that there is a connected directed path of occupied edges from (0,0) to (M,N). For each ph∈[0,1],pv=(0,1) and aspect ratio α=M/N fixed, it was established (Chen and Wu, 2006) [9] that there is an αc=[1−pv2−ph(1−pv)2]/2pv2 such that, as N→∞, τ(M,N) is 1, 0, and 1/2 for α>αc, α<αc, and α=αc, respectively. In particular, for ph=0 or 1, the model reduces to the Domany–Kinzel model (Domany and Kinzel, 1981 [7]). In this article, we investigate the rate of convergence of τ(M,N) and the asymptotic behavior of τ(Mn−,N) and τ(Mn+,N), where Mn−/N↑αc and Mn+/N↓αc as N↑∞. Moreover, we obtain a susceptibility on the rectangular net {(m,n)∈Z+×Z+:0≤m≤M and 0≤n≤N}. The proof is based on the Berry–Esseen theorem.