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

This work extends to dimension $d\ge 3$ 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 $\mathrm{\infty }$, 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 $${\kappa }_d=\frac{1}{d+1}({\mathrm{\Gamma }}_1+\cdots +{\mathrm{\Gamma }}_d),$$ where ${\mathrm{\Gamma }}_i$ is the energy of the $i$-dimensional critical droplet of the Ising model at zero temperature and magnetic field $h$.