Roberto H. Schonmann, Senya B. Shlosman
Wulff droplets and the metastable relaxation of
kinetic Ising models
(264K, AmSTeX)
ABSTRACT. We consider the kinetic Ising models (Glauber dynamics)
corresponding to the infinite volume Ising model in dimension 2 with
nearest neighbor ferromagnetic interactionand under a positive external
magnetic field $h$. Minimalconditions 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 isstarted from a distribution
which is stochastically lower than the ($-$)-phase. We show that as $h\to 0$
the relaxation time blows up as $\exp(\lambda_c(T)/h)$, with
$\lambda_c(T) = w(T)^2/(12 T m^*(T))$.
Here $m^*(T)$ is the spontaneous magnetization and $w(T)$ is the integrated
surface tension of the Wulff body of unit volume.
Moreover, for $0 < \lambda < \lambda_c$, the state of the process at time
$\exp(\lambda/h)$ 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 $\Cal C^\infty$ continuations
in $h$ of the family of Gibbs distributions with the negative magnetic
fields into the region of positive fields.