Sharp threshold for the FA-2f kinetically constrained model

TL;DR

This paper establishes a precise threshold for the time it takes the origin to become empty in the FA-2f kinetically constrained model, revealing critical behavior and confirming long-standing physics conjectures.

Contribution

It provides the first sharp threshold result for a critical kinetically constrained model, connecting it to bootstrap percolation and advancing universality understanding.

Findings

Sharp threshold for the first empty time  in FA-2f model

Connection between  and bootstrap percolation thresholds

Completion of universality program for critical KCMs

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 \[\tau_0=\exp\Big(\frac{d\cdot\lambda(d,2)+o(1)}{q^{1/(d-1)}}\Big)\quad \text{w.h.p.}\] 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.