On the Wulff crystal in the Ising model

• 成果类型: Article
• 署名作者: Cerf, R; Pisztora, A
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite Paris Saclay; Carnegie Mellon University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2000
• 页码: 947-1017
• 关键词: order large deviations; surface-tension; phase-separation; finite cluster; free-energy; coexistence; percolation; THEOREM; fluctuations; CONSTRUCTION

摘要

We study the phase separation phenomenon ilo the Ising model in dimensions d greater than or equal to 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 (T) over cap (c) conjectured to agree with the critical point T,. 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,, the dominant minus spin cluster of the Wulff droplet permeates the entire box and has a strictly positive local density everywhere.