Low-Density Series Expansion for the Domany-Kinzel Model

Abstract

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 = 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 estim...