Two-neighbour stochastic cellular automata and their planar lattice duals

Abstract

Two-neighbour stochastic cellular automata (SCA) are the set of one-dimensional discrete-time interacting particle systems with two parameters, which show non-equilibrium phase transitions from the extinction phase to the survival phase. The phase diagram was first studied by Kinzel using a numerical method called transfer-matrix scaling. For some parameter region the processes can be defined as directed percolation models on the spatio-temporal plane and the bond- and site-directed percolation models are included as special cases. Extending the argument of Dhar, Barma and Phani (1981) originally given for bond-directed percolation, we introduce diode-resistor percolation models which are the planar lattice duals of the SCA and give rigorous lower bounds for the critical line. In special cases, our results give 0.6885<or= alpha c and 0.6261<or= beta c, where alpha c and beta c denote the critical probabilities of the site- and bond-directed percolation models on the square lattice, respectively. Combining the upper bound for the critical line recently proved by Liggett (1994), we summarize the rigorous results for the phase diagram of the systems. Results of computer simulation are also shown.