Fredrickson - Andersen One Spin Facilitated Model out of Equilibrium

We consider the Fredrickson and Andersen one spin facilitated model (FAlf) on an infinite connected graph with polynomial growth. Each site with rate one refreshes its occupation variable to a filled or to an empty state with probability p epsilon [0, 1] or q = 1 - p respectively, provided that at least one of its nearest neighbours is empty. We study the non-equilibrium dynamics started from an initial distribution v different from the stationary product p-Bernoulli measure mu. We assume that, under v, the distance between two nearest empty sites has exponential moments. We then prove convergence to equilibrium when the vacancy density q is above a proper threshold (q) over bar < I. The convergence is exponential or stretched exponential, depending on the growth of the graph. In particular it is exponential on Z(d) for d = 1 and stretched exponential for d > I. Our result can be generalized to other non cooperative models.