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

• A version of directed bond percolation on the honeycomb lattice is studied.
• We derive the critical aspect ratio for the percolation in the thermodynamic limit.
• A critical exponent is determined.
• The asymptotic behavior of the percolation near the critical aspect ratio is obtained.
• A special case of our result gives the Domany–Kinzel model on the honeycomb lattice.

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 yy, and horizontal edges are directed rightward with probabilities xx and one in alternate rows. Let τ(M,N)τ(M,N) be the probability that there is at least one connected-directed path of occupied edges from (0,0)(0,0) to (M,N)(M,N). For each x∈(0,1]x∈(0,1], y∈(0,1]y∈(0,1] and aspect ratio α=M/Nα=M/N fixed, we show that there is a critical value αc=(1−x+xy)(1+x−xy)/(xy2)αc=(1−x+xy)(1+x−xy)/(xy2) such that as N→∞N→∞, τ(M,N)τ(M,N) is 11, 00 and 1/21/2 for α>αcα>αc, α<αcα<αc and α=αcα=αc, respectively. We also investigate the rate of convergence of τ(M,N)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↑∞N↑∞.