Asymptotic behavior for a version of directed percolation 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 y, 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 is an element of (0, 1], y is an element of (0, 1] and aspect ratio alpha = M/N fixed, we show that there is a critical value alpha(c) = (1 - x + xy)(1 + x - xy)/(xy(2)) such that as N -> infinity, tau(M, N) is 1, 0 and 1/2 for alpha = alpha(c), alpha = alpha(c) and alpha = alpha(c), respectively. 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 up arrow alpha(c) and M-N(+)/N down arrow alpha(c) as N up arrow infinity. (C) 2015 Elsevier B.V. All rights reserved.