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

## TL;DR

This paper proves that supercritical kinetically constrained models in one and two dimensions rapidly reach equilibrium at high temperatures, extending understanding of their convergence behavior.

## Contribution

It provides the first non-perturbative proof of exponential convergence to equilibrium for general supercritical KCMs in low dimensions at high temperature.

## Key findings

- Convergence to equilibrium is exponential when $q$ is close to 1.
- Results apply to a broad class of supercritical KCMs in 1D and 2D.
- Initial configurations with Bernoulli distribution converge rapidly.

## Abstract

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.