The East process, a well known reversible linear chain of spins, represents the prototype of a general class of interacting particle systems with constraints modeling the dynamics of real glasses. In this paper we consider a generalization of the East process living in the d-dimensional lattice and we establish new progresses on the out- of-equilibrium behavior. In particular we prove a form of (local) exponential ergodicity when the initial distribution is far from the stationary one and we prove that the mixing time in a finite box grows linearly in the side of the box.
Paul, C., Alessandra, F., Martinelli, F. (2015). Mixing time and local exponential ergodicity of the East-like process in $mathbf Z^d$. ANNALES DE LA FACULTÉ DES SCIENCES DE TOULOUSE., 24(4), 717-743.