Nucleation and growth for the Ising model in $d$ dimensions at very low temperatures

TL;DR

This paper analyzes the nucleation and growth process in the high-dimensional Ising model at very low temperatures, extending previous results to dimensions three and higher, and characterizes the relaxation time behavior.

Contribution

It extends the understanding of the Ising model's nucleation and growth phenomena to higher dimensions, providing an explicit formula for the relaxation time at low temperatures.

Findings

Relaxation time behaves like exp(βκ_d) as β→∞.

κ_d is given by the average of the energies of critical droplets.

The result generalizes previous two-dimensional findings to higher dimensions.

Abstract

This work extends to dimension $d\geq3$ the main result of Dehghanpour and Schonmann. We consider the stochastic Ising model on ${\mathbb{Z}}^d$ evolving with the Metropolis dynamics under a fixed small positive magnetic field $h$ starting from the minus phase. When the inverse temperature $\beta$ goes to $\infty$, the relaxation time of the system, defined as the time when the plus phase has invaded the origin, behaves like $\exp({{\beta}{\kappa}_d})$. The value $\kappa_d$ is equal to \[{\kappa}_d=\frac{1}{d+1}({\Gamma}_1+\cdots+{\Gamma}_d),\] where ${\Gamma}_i$ is the energy of the $i$-dimensional critical droplet of the Ising model at zero temperature and magnetic field $h$.