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July 2000 On the Wulff crystal in the Ising model
Raphaël Cerf, Ágoston Pisztora
Ann. Probab. 28(3): 947-1017 (July 2000). DOI: 10.1214/aop/1019160324


We study the phase separation phenomenon in the Ising model in dimensions $d \geq 3$. 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 $\hat{T}_c$ 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 propose a heuristic picture that for subcritical temperatures close enough to $T_c$, the dominant minus spin cluster of the Wulff droplet permeates the entire box and has a strictly positive local density everywhere.


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Raphaël Cerf. Ágoston Pisztora. "On the Wulff crystal in the Ising model." Ann. Probab. 28 (3) 947 - 1017, July 2000.


Published: July 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1034.82006
MathSciNet: MR1797302
Digital Object Identifier: 10.1214/aop/1019160324

Primary: 60F10 , 60K35 , 82B20

Keywords: FK , Ising model , large deviations , Phase separation , Wulff crystal

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.28 • No. 3 • July 2000
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