Mixing Times for a Constrained Ising Process on the Two-Dimensional Torus at Low Density

Abstract

We study a kinetically constrained Ising process (KCIP) associated with a graph $G$ and density parameter $p$; this process is an interacting particle system with state space $\{ 0, 1 \}^{G}$, the location of the particles. The `constraint' in the name of the process refers to the rule that a vertex cannot change its state unless it has at least one neighbour in state `1'. The KCIP has been proposed by statistical physicists as a model for the glass transition. In this note, we study the mixing time of a KCIP on the 2-dimensional torus $G = \mathbb{Z}_{L}^{2}$ in the low-density regime $p = \frac{c}{L^{2}}$ for arbitrary $0 < c < \infty$, extending our previous results for the analogous process on the torus $\mathbb{Z}_{L}^{d}$ in dimension $d \geq 3$. Our general approach is similar, but the extension requires more delicate bounds on the behaviour of the process at intermediate densities.