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 $\mathrm{Bernoulli}(1-q')$ law with $q' \neq q$, the dynamics converges to equilibrium with exponential speed when $q$ is close enough to 1, which corresponds to the high temperature regime.