The Fredrickson-Andersen one spin facilitated model (FA-1f) on Z belongs to the class of kinetically constrained spin models (KCM). Each site refreshes with rate one its occupation variable to empty (respectively occupied) with probability q (respectively p = 1 − q), provided at least one nearest neighbor is empty. Here, we study the non equilibrium dynamics of FA-1f started from a configuration entirely occupied on the left half-line and focus on the evolution of the front, namely the position of the leftmost zero. We prove, for q larger than a thresholdq < 1, a law of large numbers and a central limit theorem for the front, as well as the convergence to an invariant measure of the law of the process seen from the front.Let Ω = {0, 1} Z be the space of configurations andbe the subspace of configurations with a leftmost zero at the origin. For a configuration σ for which there exists x ∈ Z such that for every y < x, σ(y) = 1 and σ(x) = 0, we call x the front of configuration σ and we denote it by X(σ). For Λ ⊂ Z and σ ∈ Ω, we denote by σ Λ the restriction of σ to the set Λ. For σ ∈ Ω and x ∈ Z, let σ x be the configuration σ

In this note, we generalize the asymptotic shape theorem proved in [Des14a] for a class of random growth models whose growth is at least and at most linear. In this way, we obtain asymptotic shape theorems conjectured for several models: the contact process in a randomly evolving environment [SW08], the oriented percolation with hostile immigration [GM12b] and the bounded modified contact process [DS00].

In this paper we study the subcritical contact process on Z d for large times, starting with all sites infected. The configuration is described in terms of the macroscopic locations of infected regions in space and the relative positions of infected sites in each such region.

No abstract