We study the full class of kinetically constrained models in arbitrary
dimension and out of equilibrium, in the regime where the density
of
facilitating sites in the equilibrium measure (but not necessarily in the initial measure) is
close to
.
For these models, we establish exponential convergence to equilibrium in infinite
volume and linear time precutoff in finite volume with appropriate boundary
condition. Our results are the first out-of-equilibrium results that hold for any model
in the so-called critical class, which is covered in its entirety by our treatment. It
includes, e.g., the Fredrickson–Andersen 2-spin facilitated model, in which a site is
updated only when at least two neighbouring sites are in the facilitating state. In
addition, these results generalise, unify and sometimes simplify several previous
works in the field. As byproduct, we recover and generalise exponential tails for the
connected component of the origin in the upper invariant trajectory of perturbed
cellular automata and in the set of eventually infected sites in subcritical bootstrap
percolation models. Our approach goes through the study of cooperative
contact processes, last passage percolation, Toom contours, as well as a very
convenient coupling between contact processes and kinetically constrained
models.
