Rh. Schonmann et Sb. Shlosman, WULFF DROPLETS AND THE METASTABLE RELAXATION OF KINETIC ISING-MODELS, Communications in Mathematical Physics, 194(2), 1998, pp. 389-462
We consider the kinetic Ising models (Glauber dynamics) corresponding
to the infinite volume Ising model in dimension 2 with nearest neighbo
r ferromagnetic interaction and under a positive external magnetic fie
ld h. Minimal conditions on the flip rates are assumed, so that all th
e common choices are being considered, We study the relaxation towards
equilibrium when the system is at an arbitrary subcritical temperatur
e T and the evolution is started from a distribution which is stochast
ically lower than the (-)-phase. We show that as h SE arrow 0 the rela
xation time blows up as exp(lambda(c)(T)/h), with lambda(c)(T) = w(T)(
2)/(12Tm(T)). Here m*(T) is the spontaneous magnetization and w(T) is
the integrated surface tension of the Wulff body of unit volume. More
over, 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. Th
e difference between this state and the (-)-phase can be described in
terms of an asymptotic expansion in powers of the external held, This
expansion can be interpreted as describing a set of C-infinity continu
ations in h of the family of Gibbs distributions with the negative mag
netic fields into the region of positive fields.