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

Abstract

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 $\infty$, 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 \[κ_d=\frac{1}{d+1}(Γ_1+\cdots+Γ_d),\] where ${\Gamma }_i$ is the energy of the $i$-dimensional critical droplet of the Ising model at zero temperature and magnetic field $h$.

Published in at http://dx.doi.org/10.1214/12-AOP801 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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