The Fredrickson-Andersen 2-spin facilitated model on $\mathbb{Z}^d$ (FA-2f)
is a paradigmatic interacting particle system with kinetic constraints (KCM)
featuring dynamical facilitation, an important mechanism in condensed matter
physics. In FA-2f a site may change its state only if at least two of its
nearest neighbours are empty. Although the process is reversible w.r.t. a
product Bernoulli measure, it is not attractive and features degenerate jump
rates and anomalous divergence of characteristic time scales as the density $q$
of empty sites tends to $0$. A natural random variable encoding the above
features is $\tau_0$, the first time at which the origin becomes empty for the
stationary process. Our main result is the sharp threshold
\[\tau_0=\exp\Big(\frac{d\cdot\lambda(d,2)+o(1)}{q^{1/(d-1)}}\Big)\quad
\text{w.h.p.}\] with $\lambda(d,2)$ the sharp threshold constant for
2-neighbour bootstrap percolation on $\mathbb{Z}^d$, the monotone deterministic
automaton counterpart of FA-2f. This is the first sharp result for a critical
KCM and it compares with Holroyd’s 2003 result on bootstrap percolation and its
subsequent improvements. It also settles various controversies accumulated in
the physics literature over the last four decades. Furthermore, our novel
techniques enable completing the recent ambitious program on the universality
phenomenon for critical KCM and establishing sharp thresholds for other
two-dimensional KCM.
