Kinetically constrained models out of equilibrium

TL;DR

This paper proves exponential convergence to equilibrium for a broad class of kinetically constrained models out of equilibrium, including the critical class, using a unified approach that simplifies previous results.

Contribution

It provides the first out-of-equilibrium results for all models in the critical class, unifying and extending prior work with a novel coupling approach.

Findings

Exponential convergence to equilibrium in infinite volume.

Linear time precutoff in finite volume with boundary conditions.

Generalization of exponential tail results in cellular automata and bootstrap percolation.

Abstract

We study the full class of kinetically constrained models in arbitrary dimension and out of equilibrium, in the regime where the density $q$ of facilitating sites in the equilibrium measure (but not necessarily in the initial measure) is close to $1$. 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, including e.g. the Fredrickson-Andersen 2-spin facilitated model. In addition, they 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.