Abstract:

We study the phase separation phenomenon in the Ising model in dimensions . 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 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 domimant minus spin cluster of the Wulff droplet permeates the entire box and has a strictly positive local density everywhere.

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