Sharp threshold for the FA-2f kinetically constrained model

TL;DR

The main result is the sharp threshold τ0, the first sharp result for a critical KCM and it compares with Holroyd’s 2003 result on bootstrap percolation and its subsequent improvements.

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 $τ_0$, 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 $λ(d,2)$ 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.

Figures (5)

• Figure 1: Black circles denote infected sites. The boundary condition $\omega$ in the figure is fully infected on $\partial_r R$ and fully healthy elsewhere. The rectangle $R$ is $\omega$-right-traversable (i.e $\mathcal{T}_\rightarrow^\omega(R)$ occurs) but it is neither $\omega$-up-traversable, nor $\omega$-left-traversable. It is also down-traversable ($\mathcal{T}_\downarrow(R)$ occurs) but not traversable in any other direction.
• Figure 2: An example of super-good configuration in the square $\Lambda^{(6)}$. The black square, of the form $\Lambda^{(2)}+x$, is completely infected and it is a super-good core for the rectangle of the form $\Lambda^{(3)}+x$ formed by it together with the two hatched rectangles. This rectangle of the form $\Lambda^{(3)}+x$ is also super-good because of the right/left-traversability of the hatched parts (arrows) and it is a super-good core for the square containing it and so on.
• Figure 3: The partition of $R^{(k+1)}$ into the rectangles $V_1,V_2,V_3$. Here we illustrate the event $\mathcal{F}_{1,2}\cap\mathcal{A}_3$. The grey region $\Lambda^{(n)}+s_k\vec{e}_1$ at the left boundary of $V_2$ is $\mathcal{S}\mathcal{G}$ and the dashed arrows in $V_1$ and $V_3$ indicate their $\omega$-traversability. The solid arrow in $V_2\setminus (\Lambda^{(n)}+s_k\vec{e}_1)$ indicates instead the $\mathbf 1$-traversability of $V_2\setminus (\Lambda^{(n)}+s_k\vec{e}_1)$. Clearly the entire configuration belongs to the events $\mathcal{H}$ and $\mathcal{K}$ defined in \ref{['E1']}, \ref{['E2']}. Indeed, the two ($\omega$-)right-traversability events together imply the $\omega$-right-traversability of $(V_2\cup V_3)\setminus (\Lambda^{(n)}+s_k\vec{e}_1)$.
• Figure 4: The partition of $\Lambda^{(n,+)}$ into the rectangle $V_2$ and the two columns $V_1$ and $V_3$. Here we illustrate the event $\overline{\mathcal{S}\mathcal{G}}(V_2)$: the grey region is a super-good rectangle of the type $\Lambda^{(n-2)}$, while the patterned rectangles are $\mathbf 1$-traversable in the arrow directions. If there is at least one infection in $I_3$ then the rectangle $V_2\cup V_3$ is super-good. Similarly, an infection in $I_1$ suffices to make $V_1\cup V_2$ super good.
• Figure 5: Illustration of Observation \ref{['obs:SG1']}. The shaded square of shape $\Lambda^{(2N_\delta)}$ is $\mathcal{S}\mathcal{G}$ and the arrows indicate the presence of an infection in each row/column, as guaranteed by $\mathcal{G}(\Lambda_{i,j})$ with $\Lambda_{i,j}$ being the larger square of shape $\Lambda^{(2N)}$. Observation \ref{['obs:SG1']} asserts that these events combined imply $\mathcal{S}\mathcal{G}(\Lambda_{i,j})$ (see Figure \ref{['fig:super-good']}). The dashed lines delimit the boxes $Q_k$.

Theorems & Definitions (49)

• Remark 1.1
• Remark 1.2
• Theorem 1.3
• Corollary 1.4
• Remark 1.5
• Remark 1.6
• Proposition 2.1
• proof
• Theorem 2.2: Eq. (5.11) of Aizenman88
• Theorem 2.3: Theorem 6.1, Lemma 3.9 and Eq. (4) of Hartarsky19