Appears in collection : French Japanese Conference on Probability and Interactions

The Fredrickson-Andersen $2$-spin facilitated model (FA-$2$f) on $\mathbb Z^d$ is a paradigmatic interacting particle system with kinetic constraints (KCM) featuring cooperative and glassy dynamics. For FA-$2$f vacancies facilitate motion: a particle can be created/killed on a site only if at least $2$ of its nearest neighbors are empty. We will present sharp results for the first time, $\tau$, at which the origin is emptied for the stationary process when the density of empty sites ($q$) is small. In any dimension $d\geq 2$ it holds $$\tau\sim \exp\left(\frac{d\lambda(d,2)+o(1)}{q^{1/(d-1)}}\right)$$ w.h.p., with $\lambda(d,2)$ the threshold constant for the $2$-neighbour bootstrap percolation on $\mathbb Z^d$. We will explain the dominant relaxation mechanism leading to this result, give a flavour of the proof techniques, and discuss further results that can be obtained via our technique for more general KCM, including full universality results in two dimensions. Joint work with I.Hartarsky and F.Martinelli.