99-133 Raphael Cerf, Agoston Pisztora
On the Wulff crystal in the Ising model (1124K, Postscript) 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 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.

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