Low-Density Series Expansion for the Domany-Kinzel Model - Inui, Norio; Katori, Makoto; Bhatti, Faqir M (2001)
Domany-Kinzel (DK) model is a family of the 1+1 dimensional stochastic cellular automata with two parameters p(1) and p(2), which simulate time evolution of interacting active elements in a random medium. By identifying a set of active sites on the spatio-temporal plane with a percolation cluster, we discuss the directed percolation (DP) transitions in the DK model. We parameterize p(1) = p and p(2) = alphap with p is an element of [0, 1] and alpha is an element of [0, 2] and calculate the mean cluster size and other quantities characterizing the DP cluster as the series of p up to order 51 for several values of alpha by using a graphical expansion formula recently given by Konno and Katori. We analyze the series by the first- and second-order differential approximations and the Zinn-Justin method and study the dependence on alpha of the convergence of estimations of critical values and critical exponents. In the mixed site-bond DP region; 1 less than or equal to alpha less than or equal to 1.3553, the convergence is excellent. As alpha -> 2 slowing down of convergence and as alpha -> 0 peculiar oscillation of estimations are observed. This paper is the first report of the systematic study of DK model by series expansion method.