Sharp threshold for the FA-2f kinetically constrained model

Abstract

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 \[τ_0=\exp\Big(\frac{d\cdotλ(d,2)+o(1)}{q^{1/(d-1)}}\Big)\quad \text{w.h.p.}\] 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.

41 pages, 5 figures, minor changes in section 2

References in corpus (6)

• Theoretical perspective on the glass transition and amorphous materials
• Bootstrap percolation in three dimensions
• Fredrickson-Andersen one spin facilitated model out of equilibrium
• Refined universality for critical KCM: lower bounds
• Coalescing and branching simple symmetric exclusion process
• Bisection for kinetically constrained models revisited

Cited by in corpus (6)

• Refined universality for critical KCM: lower bounds
• Coalescing and branching simple symmetric exclusion process
• Refined universality for critical KCM: upper bounds
• Sharp metastability transition for two-dimensional bootstrap percolation with symmetric isotropic threshold rules
• Locality approach to the bootstrap percolation paradox
• Fredrickson--Andersen model in two dimensions