We study the hydrodynamic limit for a periodic 1-dimensional exclusion process with a dynamical constraint, which prevents a particle at site x from jumping to sitex ± 1 unless site x ∓ 1 is occupied. This process with degenerate jump rates admits transient states, which it eventually leaves to reach an ergodic component, assuming that the initial macroscopic density is larger than 1 2 , or one of its absorbing states if this is not the case. It belongs to the class of conserved lattice gases (CLG) which have been introduced in the physics literature as systems with active-absorbing phase transition in the presence of a conserved field. We show that, for initial profiles smooth enough and uniformly larger than the critical density 1 2 , the macroscopic density profile for our dynamics evolves under the diffusive time scaling according to a fast diffusion equation (FDE). The first step in the proof is to show that the system typically reaches an ergodic component in subdiffusive time.

The East model is a one-dimensional, non-attractive interacting particle system with Glauber dynamics, in which a flip is prohibited at a site x if the right neighbour x + 1 is occupied. Starting from a configuration entirely occupied on the left half-line, we prove a law of large numbers for the position of the left-most zero (the front), as well as ergodicity of the process seen from the front. For want of attractiveness, the one-dimensional shape theorem is not derived by the usual coupling arguments, but instead by quantifying the local relaxation to the non-equilibrium invariant measure for the process seen from the front. This is the first proof of a shape theorem for a kinetically constrained spin model.

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 paper we prove the convergence to the stochastic Burgers equation from one-dimensional interacting particle systems, whose dynamics allow the degeneracy of the jump rates. To this aim, we provide a new proof of the second order Boltzmann-Gibbs principle introduced in [7]. The main technical difficulty is that our models exhibit configurations that do not evolve under the dynamics -the blocked configurations -and are locally non-ergodic. Our proof does not impose any knowledge on the spectral gap for the microscopic models. Instead, it relies on the fact that, under the equilibrium measure, the probability to find a blocked configuration in a finite box is exponentially small in the size of the box. Then, a dynamical mechanism allows to exchange particles even when the jump rate for the direct exchange is zero. 1 arXiv:1603.08975v3 [math.PR]