Sharp threshold for the FA-2f kinetically constrained model
Résumé
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. The process, while reversible w.r.t. a product Bernoulli measure, has degenerate jumps rates because of the constraints and it is non-attractive, with an anomalous divergence of characteristic time scales as the density $q$ of the empty sites tends to zero. 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 thresold 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.