The Annals of Probability

On the Wulff crystal in the Ising model

Raphaël Cerf and Ágoston Pisztora

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Abstract

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.

Article information

Source
Ann. Probab., Volume 28, Number 3 (2000), 947-1017.

Dates
First available in Project Euclid: 18 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1019160324

Digital Object Identifier
doi:10.1214/aop/1019160324

Mathematical Reviews number (MathSciNet)
MR1797302

Zentralblatt MATH identifier
1034.82006

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60F10: Large deviations

Keywords
Phase separation Wulff crystal Ising model large deviations FK

Citation

Cerf, Raphaël; Pisztora, Ágoston. On the Wulff crystal in the Ising model. Ann. Probab. 28 (2000), no. 3, 947--1017. doi:10.1214/aop/1019160324. https://projecteuclid.org/euclid.aop/1019160324


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  • CNRS, Universit´e Paris Sud Math´ematique, B atiment 425 91405 Orsay Cedex France Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, Pennsylvania 15213-3890