 99133 Raphael Cerf, Agoston Pisztora
 On the Wulff crystal in the Ising model
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Apr 28, 99

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Abstract. We study the phase separation phenomenon in the Ising model in
dimensions $d\ge 3$. To this end we work in a large box with
plus boundary conditions and we condition the system
to have an excess amount of negative spins so that the empirical
magnetization is smaller than the spontaneous magnetization $m^*$.
We confirm the prediction of the phenomenological theory by proving
that with high probability a single droplet of the minus phase emerges
surrounded by the plus phase. Moreover, the rescaled droplet is
asymptotically close to a definite deterministic shape  the Wulff
crystal  which minimizes the surface free energy. In the course
of the proof we establish a surface order large deviation principle
for the magnetization. Our results are valid for temperatures $T$
below a limit of slabthresholds $\tchat$ conjectured to
agree with the critical point $T_c$. Moreover, $T$ should be such that
there exist only two extremal translation invariant Gibbs states
at that temperature; a property which can fail for at most countably
many values and which is conjectured to be true for every $T$.
The proofs are based on the FortuinKasteleyn representation of
the Ising model along with coarsegraining techniques. To handle
the emerging macroscopic objects
we employ tools from geometric measure theory which
provide an adequate framework for the large deviation analysis.
Finally, we give a heuristic argument that for subcritical temperatures
close enough to $T_c$, the dominant minus spin cluster
of the Wulff droplet {\it permeates} the entire box and and has
a strictly positive local density everywhere.
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