Sharp threshold for the FA-2f kinetically constrained model

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 site may change its state only if at least two of nearest neighbours are empty. Although the process reversible w.r.t. product Bernoulli measure, it not attractive and features degenerate jump rates anomalous divergence characteristic time scales as density q empty sites tends to 0. A natural random variable encoding above $$\tau _0$$ , first which origin becomes for stationary process. Our main result sharp threshold $$\begin{aligned} \tau _0=\exp \Big (\frac{d\cdot \lambda (d,2)+o(1)}{q^{1/(d-1)}}\Big )\quad \text {w.h.p.} \end{aligned}$$ $$\lambda (d,2)$$ constant 2-neighbour bootstrap percolation monotone deterministic automaton counterpart FA-2f. This critical KCM compares Holroyd’s 2003 subsequent improvements. It also settles various controversies accumulated physics literature over last four decades. Furthermore, our novel techniques enable completing recent ambitious program universality phenomenon establishing thresholds other two-dimensional KCM.