Low-Density Series Expansion for the Domany-Kinzel Model

抄録

Domany-Kinzel (DK) model is a family of the 1+1 dimensional stochastic cellular automata with two parameters p₁ and p₂, 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 p₁=p and p₂= α p with p ∈ [0,1] and α ∈ [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 α 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 α of the convergence of estimations of critical values and critical exponents. In the mixed site-bond DP region, 1 ≤ α ≤ 1.3553, the convergence is excellent. As α → 2 slowing down of convergence and as α → 0 peculiar oscillation of estimations are observed. This paper is the first report of the systematic study of DK model by series expansion method.