Asymptotic behavior for a generalized Domany–Kinzel model. (27th February 2017) - Chang, Shu-Chiuan; Chen, Lung-Chi; Huang, Chien-Hao
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 ∈ [ 0, 1 ), p 2 ∈ [ 0, 1 ), but p 1 ∨ p 2 > 0 . 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 ). Defining the aspect ratio α = M / N, we show that there is a critical value α c = ( 2 − p 1 − p 2 ) / (   p 1 + p 2 ) such that as N → ∞, τ ( M, N ) is 1, 0 and 1/2 for α > α c, α < α c and α = α c, respectively. In particular, the model reduces to the square lattice with uniform vertical probability when p 1 = p 2 [001 ], 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 α c changes between the square lattice and the honeycomb lattice as bricks. In this article, we investigate the rate of convergence of τ ( M, N ) and the asymptotic behavior of τ ( M N −, N ) and τ ( M N +, N ), where M N − / N ↑ α c and M N + / N ↓ α c as N ↑ ∞ .