Asymptotic behavior for a version of directed percolation on a square lattice

作者：Chen Lung Chi^*

来源：Physica A: Statistical Mechanics and Its Applications , 2011, 390(3): 419-426.

DOI：10.1016/j.physa.2010.09.039

• Domany-Kinzel model
• Directed percolation
• Compact directed percolation
• Asymptotic behavior
• Critical behavior
• Susceptibility
• Exact solutions
• Berry-Esseen theorem

We consider a version of directed bond percolation on a square lattice whose vertical edges are directed upward with probabilities p(v) and horizontal edges are directed rightward with probabilities p(h) and 1 in alternate rows. Let tau (M, N) be the probability that there is a connected directed path of occupied edges from (0,0) to (M, N). For each p(h) is an element of [0, 1], p(v) = (0, 1) and aspect ratio alpha = M/N fixed, it was established (Chen and Wu, 2006) [9] that there is an alpha(c) = [1 - p(v)(2) - p(h) (1 - p(v))(2)]/2p(v)(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. In particular, for p(h) = 0 or 1, the model reduces to the Domany-Kinzel model (Domany and Kinzel, 1981 [7]). In this article, we 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. Moreover, we obtain a susceptibility on the rectangular net {(m, n) is an element of Z(+) x z(+) : 0 <= m <= M and 0 <= n <= N}. The proof is based on the Berry-Esseen theorem.

• 出版日期2011-2-1