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

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 $h$.