Asymptotic behavior for a generalized Domany Kinzel mode

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 p 1, p 2 alternatively in even rows and probabilities p 2, p 1 alternatively in odd rows, where ${{p}_{1}}\\in \\left[0,1\\right)$ , ${{p}_{2}}\\in \\left[0,1\\right)$ , but ${{p}_{1}}\\vee {{p}_{2}}>0$ . 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). Defining the aspect ratio $\\alpha =M/N$ , we show that there is a critical value ${{\\alpha}_{c}}=\\left(2-{{p}_{1}}-{{p}_{2}}\\right)/\\left(\\,{{p}_{1}}+{{p}_{2}}\\right)$ such that as $N\\to \\infty $ , $\\tau (M,N)$ is 1, 0 and 1/2 for $\\alpha >{{\\alpha}_{c}}$ , $\\alpha <{{\\alpha}_{c}}$ and $\\alpha ={{\\alpha}_{c}}$ , respectively. In particular, the model reduces to the square lattice with uniform vertical probability when ${{p}_{1}}={{p}_{2}}$ [1], and the model reduces to the honeycomb lattice when one of p 1 and p 2 is equal to 0. We study how the critical value ${{\\alpha}_{c}}$ changes between the square lattice and the honeycomb lattice as bricks. In this article, we investigate the rate of convergence of $\\tau (M,N)$ and the asymptotic behavior of $\\tau \\left(M_{N}^{-},N\\right)$ and $\\tau \\left(M_{N}^{+},N\\right)$ , where $M_{N}^{-}/N\\uparrow {{\\alpha}_{c}}$ and $M_{N}^{+}/N\\downarrow {{\\alpha}_{c}}$ as $N\\uparrow \\infty $ .