Sharp asymptotics for stochastic dynamics with parallel updating rule

In this paper we study the metastability problem for a stochastic dynamics with a parallel updating rule; in particular we consider a ¿nite volume Probabilistic Cellular Automaton (PCA) in a small external ¿eld at low temperature regime. We are interested in the nucleation of the system, i.e., the typical excursion from the metastable phase (the con¿guration with all minuses) to the stable phase (the con¿guration with all pluses), triggered by the formation of a critical droplet. The main result of the paper is the sharp estimate of the nucleation time: we show that the nucleation time divided by its average converges to an exponential random variable and that the rate of the exponential random variable is an exponential function of the inverse temperature ß times a prefactor that does not scale with ß. Our approach combines geometric and potential theoretic arguments.