Low-density series expansion for the Domany-Kinzel model

Domany-Kinzel (DK) model is a family of the 1+1 dimensional stochastic cell ular automata with two parameters p(1) and p(2), which simulate time evolut ion of interacting active elements in a random medium. By identifying a set of active sites on the spatio-temporal plane with a percolation cluster, w e discuss the directed percolation (DP) transitions in the DK model. We par ameterize p(1) = p and p(2) = alphap with p is an element of [0, 1] and alp ha is an element of [0, 2] and calculate the mean cluster size and other qu antities characterizing the DP cluster as the series of p up to order 51 fo r several values of alpha by using a graphical expansion formula recently g iven by Konno and Katori. We analyze the series by the first- and second-or der differential approximations and the Zinn-Justin method and study the de pendence 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 alp ha -> 2 slowing down of convergence and as alpha -> 0 peculiar oscillation of estimations are observed. This paper is the first report of the systemat ic study of DK model by series expansion method.