Wulff Droplets and the Metastable Relaxation of Kinetic Ising Models

Abstract

We consider the kinetic Ising models (Glauber dynamics) corresponding to the infinite volume Ising model in dimension 2 with nearest neighbor ferromagnetic interaction and under a positive external magnetic field h. Minimal conditions on the flip rates are assumed, so that all the common choices are being considered. We study the relaxation towards equilibrium when the system is at an arbitrary subcritical temperature T and the evolution is started from a distribution which is stochastically lower than the (−)-phase. We show that as h↘ 0 the relaxation time blows up as exp(λc(T)/h), with lgr;c(T) =w(T)2/(12 T m*(T)). Here m*(T) is the spontaneous magnetization and w(T) is the integrated surface tension of the Wulff body of unit volume. Moreover, for 0 < λ < λc, the state of the process at time exp(λ/h) is shown to be close, when h is small, to the (−)-phase. The difference between this state and the (−)-phase can be described in terms of an asymptotic expansion in powers of the external field. This expansion can be interpreted as describing a set of ?∞ continuations in h of the family of Gibbs distributions with the negative magnetic fields into the region of positive fields.