Asymptotic behavior for a version of directed percolation on the honeycomb lattice

Abstract

We consider a version of directed bond percolation on the honeycomb lattice as a brick lattice such that vertical edges are directed upward with probability y, and horizontal edges are directed rightward with probabilities x and one in alternate rows. Let τ(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] and aspect ratio α=M/N fixed, we show that there is a critical value αc=(1−x+xy)(1+x−xy)/(xy2) such that as N→∞, τ(M,N) is 1, 0 and 1/2 for α>αc, α<αc and α=αc, respectively. We also investigate the rate of convergence of τ(M,N) and the asymptotic behavior of View the MathML source and View the MathML source where View the MathML source and View the MathML source as N↑∞.