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

TL;DR

This paper analyzes the mixing times of a kinetically constrained Ising process on a 2D torus at low density, extending previous high-dimensional results to two dimensions with more refined bounds.

Contribution

It provides new bounds on the mixing time of the KCIP on the 2D torus at low density, expanding understanding from higher dimensions.

Findings

Derived bounds on mixing times for the 2D case

Extended previous high-dimensional results to two dimensions

Developed more delicate bounds for intermediate densities

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.