Martinelli F.
On the Two Dimensional Dynamical Ising
Model In the Phase Coexistence Region
(164K, plain TeX)
ABSTRACT. We consider a Glauber dynamics reversible with respect to the two
dimensional Ising model in a finite square of side L, in the absence of an
external field and at large inverse temperature $\beta$. We first consider the gap
in the spectrum of the generator of the dynamics in two
different cases: with plus and open boundary condition. We prove
that, when the symmetry under global spin flip is broken by the boundary conditions,
the gap is much larger than the case in which the symmetry is present. For this latter
we compute exactly the asymptotics of $-{1\over \beta L}\log
(\hbox{gap})$ as $L\to\infty$ and show that it coincides with the surface
tension along one of the coordinat axes. As a consequence we are able to study quite
precisely the large deviations in time of the magnetization and to obtain an upper
bound on the spin-spin time correlation in the infinite volume plus phase. Our
results establish a connection between the dynamical large deviations and those of
the equilibrium Gibbs measure studied by Shlosman in the framework of the rigorous
description of the Wulff shape for the Ising model. Finally we show that, in the case
of open boundary conditions, it is possible to rescale the time with L in such a
way that, as $L\to \infty$, the finite dimensional distributions of the time rescaled
magnetization converge to those of a symmetric continuous time Markov chain on the
two state space $\{-m^*(\beta ),m^*(\beta )\}$, $m^*(\beta )$ being the spontaneous
magnetization. Our methods rely upon a novel combination of techniques for bounding
from below the gap of symmetric Markov chains on complicate graphs, developed by
Jerrum and Sinclair in their Markov chain approach to hard computational problems, and
the idea of introducing "block Glauber dynamics" instead of the standard single site
dynamics, in order to put in evidence more effectively the effect of the boundary
conditions in the approach to equilibrium.