Asymptotic behavior for a generalized Domany–Kinzel model

Abstract

We consider a version of directed bond percolation on the square lattice such that horizontal edges are directed rightward with probabilities one, and vertical edges are directed upward with probabilities p1, p2 alternatively in even rows and probabilities p2, p1 alternatively in odd rows, where , , but . Let be the probability that there is at least one connected-directed path of occupied edges from (0, 0) to (M,N). Defining the aspect ratio , we show that there is a critical value such that as , is 1, 0 and 1/2 for , and , respectively. In particular, the model reduces to the square lattice with uniform vertical probability when [], and the model reduces to the honeycomb lattice when one of p1 and p2 is equal to 0. We study how the critical value changes between the square lattice and the honeycomb lattice as bricks. In this article, we investigate the rate of convergence of and the asymptotic behavior of and , where and as .