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

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 $h$.