Exponential convergence to equilibrium in supercritical kinetically constrained models at high temperature

Abstract

Kinetically constrained models (KCMs) were introduced by physicists to model the liquid-glass transition. They are interacting particle systems on $\mathbb{Z}^d$ in which each element of $\mathbb{Z}^d$ can be in state 0 or 1 and tries to update its state to 0 at rate $q$ and to 1 at rate $1-q$, provided that a constraint is satisfied. In this article, we prove the first non-perturbative result of convergence to equilibrium for KCMs with general constraints: for any KCM in the class termed "supercritical" in dimension 1 and 2, when the initial configuration has product $\mathrm{Bernoulli}(1-q')$ law with $q' \neq q$, the dynamics converges to equilibrium with exponential speed when $q$ is close enough to 1, which corresponds to the high temperature regime.

Figures (4)

• Figure 1: Illustration of a dual path $\Gamma$ of length $t'$ starting at $(x,t)$ for $d=1$ and $\rho=2$. Each horizontal line represents the timeline of a site of $\mathds{Z}$, the $\times$ representing the clock rings. $\Gamma$ is the thick polygonal line; it starts at $t$ and ends at $t-t'$. It can jump only when there is a clock ring, and never at a distance greater than $\rho=2$.
• Figure 2: Illustration of proposition \ref{['prop_Bollobas']}. The $\ast$ represent the sites $x_1,\dots,x_m$. If we start with the sites of $R$ at zero and there are successive 0-clock rings at $x_1,\dots,x_m$ while there is no 1-clock ring in $R \cup \{x_1,\dots,x_m\}$, these clock rings will put $x_1,\dots,x_m$ at zero, hence the sites of $a_1u+R$ will be put at zero.
• Figure 3: The proof of proposition \ref{['prop_Bollobas']}. (a) The mechanism for $d=1$; the $\bullet$ represent zeroes and the $\circ$ represent ones. (b) The shape delimited by the solid line is the droplet of Bollobas_et_al2015; if it is infected in the bootstrap percolation dynamics, the infection can grow to the shape delimited by the dashed line. (c) $R$ contains the original droplet (dashed line), hence if $R$ is infected, the infection can propagate to a bigger droplet (in gray) that contains $a_1u+R$ and is contained in $R \cup (a_1u+R) \cup (2a_1u+R)$.
• Figure 4: An illustration of proposition \ref{['prop_transfer_zeroes2']} with $n^{y,k} = 3$ and $r_0=1$. The figure at the left represents the bonds of the oriented percolation process $\zeta^{y,k}$; the open bonds are the thick ones, and the path of open bonds allowing $\zeta_{n^{y,k}}^{y,k}(r_0)=1$ is outlined by arrows. The figure at the right represents the consequences on the KCM process; each vertical strip represents the state of $\bigcup_{i \in \mathds{Z}}(y+ia_1u+R)$ at a certain time. If at time $t-(k+3)K$ the rectangle $y+\frac{1-n^{y,k}}{2}a_1u+R = y-a_1u+R$ is at zero (in gray), since the bond $(0,2) \rightarrow (1,3)$ (bond $b_1$) is open, $y-a_1u+R$ is still at zero at time $t-(k+2)K$. Moreover, since $(1,1) \rightarrow (0,2)$ (bond $b_2$) is open and $y-a_1u+R$ is at zero at time $t-(k+2)K$, $a_1u+(y-a_1u+R) = y+R$ is at zero at time $t-(k+1)K$. Finally, since $(0,0) \rightarrow (1,1)$ (bond $b_3$) is open and $y+R$ is at zero at time $t-(k+1)K$, $y+R$ is still at zero at time $t-kK$.

Theorems & Definitions (31)

• Definition 1
• Definition 2
• Theorem 3
• Remark 4
• Proposition 5
• Lemma 6
• proof : Sketch of proof.
• Proposition 7
• Proposition 8
• Lemma 9