Exponential convergence to equilibrium in supercritical kinetically constrained models at high temperature

Kinetically constrained models (KCMs) were introduced by physicists to model the liquid-glass transition. They are interacting particle systems on ${\mathbb{Z}}^d$ in which each element of ${\mathbb{Z}}^d$ can be in state 0 or 1 and tries to update its state to 0 at rate $q$ and to 1 at rate $1-q$, provided that a constraint is satisfied. In this article, we prove the first non-perturbative result of convergence to equilibrium for KCMs with general constraints: for any KCM in the class termed “supercritical” in dimension 1 and 2, when the initial configuration has product $\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{o}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{i}(1-q^{\mathrm{\prime }})$ law with $q^{\mathrm{\prime }}\mathrm{\ne }q$, the dynamics converges to equilibrium with exponential speed when $q$ is close enough to 1, which corresponds to the high temperature regime.