Raphael Cerf, Agoston Pisztora
On the Wulff crystal in the Ising model
(1124K, Postscript)
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 slab-thresholds $\tchat$ 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 Fortuin-Kasteleyn representation of
the Ising model along with coarse-graining 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.