Sharp asymptotics for stochastic dynamics with parallel updating rule with self-interaction

In this paper we study metastability for a stochastic dynamics with a parallel updating rule in particular for a probabilistic cellular automata. The problem is addressed in the Freidlin Wentzel regime, i.e., finite volume, small magnetic field, and in the limit when temperature tends to zero. We are interested in how the system nucleates, i.e., in properties of the transition from the metastable state (the configuration with all minuses) to the stable state (the configuration with all pluses). In this paper we show that the nucleation time divided by its average converges to an exponential random variable and we express the proportionality constant for the average nucleation time in terms of parameters of the model. Our approach combines geometric and potential theoretic arguments. A special feature of parallel dynamics is the existence of many fixed points and cyclic pairs of the zero temperature dynamics, in which the system can be trapped in its way to the stable phase. These cyclic points are corresponding to chessboard kind of configurations that under the parallel dynamics alternate between even and odd.