Dobrushin (1972) showed that the interface of a 3D Ising model with minus boundary conditions above the @math -plane and plus below is rigid (has @math -fluctuations) at every sufficiently low temperature. Since then, basic features of this interface -- such as the asymptotics of its maximum -- were only identified in more tractable random surface models that approximate the Ising interface at low temperatures, e.g., for the (2+1)D Solid-On-Solid model. Here we study the large deviations of the interface of the 3D Ising model in a cube of side-length @math with Dobrushin's boundary conditions, and in particular obtain a law of large numbers for @math , its maximum: if the inverse-temperature @math is large enough, then @math as @math , in probability, where @math is given by a large deviation rate in infinite volume. We further show that, on the large deviation event that the interface connects the origin to height @math , it consists of a 1D spine that behaves like a random walk, in that it decomposes into a linear (in @math ) number of asymptotically-stationary weakly-dependent increments that have exponential tails. As the number @math of increments diverges, properties of the interface such as its surface area, volume, and the location of its tip, all obey CLTs with variances linear in @math . These results generalize to every dimension @math .
In lieu of this approach, a coarse-graining technique of Pisztora @cite_24 enabled the establishment of the surface tension and Wulff shape scaling limit for the 3D Ising model at low-temperature: Cerf and Pisztora @cite_22 considered an Ising model on an @math box with all-plus boundary conditions, and showed that conditional on having @math minus spins (atypically many), the largest minus cluster macroscopically takes on the corresponding Wulff shape. Results of this sort are focused on the macroscopic behavior of the model (as opposed to the interface-based approach) and do not describe the fluctuations around the limiting shape. In particular, the convergence to the Wulff shape holds all the way up to @math (when combined with @cite_0 @cite_26 ), even though near @math (above the roughening transition) it is expected that the interface is not only delocalized, but that the minus cluster actually percolates all the way to the boundary of the box @cite_30 .
@cite_30 @cite_26 @cite_22 @cite_24 @cite_0
2048984514 2054432045 1993711495 2073921960 2010196640
We describe inequalities relating to the interface between coexisting phases of Ising ferromagnets. Some implications for the nature of the roughening transition are discussed. For the FK representation of the Ising model, we prove that the slab percolation threshold coincides with the critical temperature in any dimension d≥3. We study the phase separation phenomenon in the Ising model in dimensions d ≥ 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 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. We derive uniform surface order large deviation estimates for the block magnetization in finite volume Ising (or Potts) models with plus or free (or a combination of both) boundary conditions in the phase coexistence regime ford≧3. The results are valid up to a limit of slab-thresholds, conjectured to agree with the critical temperature. Our arguments are based on the renormalization of the random cluster model withq≧1 andd≧3, and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability. The results are new even for the case of independent percolation (q=1). As a byproduct of our methods, we obtain further results in the FK model concerning semicontinuity (inp andq) of the percolation probability, the second largest cluster in a box and the tail of the finite cluster size distribution. In this paper we prove the Wulff construction in three and more dimensions for an Ising model with nearest neighbor interaction.