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 .
While cluster expansion only converges at sufficiently large @math , it is natural to ask if the rigidity of the interface, and our new results, hold for all @math . This is not believed to be the case, as the Ising model is widely believed to undergo a for @math (and no other dimension): Much like the SOS and DG approximations, which exhibit phase transitions in @math ---whereby they roughen and resemble the discrete Gaussian free field @cite_7 @cite_33 for small @math ---it is conjectured that for the 3D Ising model there exists a point @math such that, for @math , the model has long-range order, yet the typical fluctuations of its horizontal interface diverge with @math ; proving this transition is a longstanding open problem (see, e.g, @cite_38 @cite_30 ).
@cite_30 @cite_38 @cite_33 @cite_7
2048984514 1979774889 2048117538 2095138947
We describe inequalities relating to the interface between coexisting phases of Ising ferromagnets. Some implications for the nature of the roughening transition are discussed. We rigorously establish the existence of a Kosterlitz-Thouless transition in the rotator, the Villain, the solid-on-solid, and the ℤ n models, forn large enough, and in the Coulomb lattice gas, in two dimensions. Our proof is based on an inductive expansion of the Coulomb gas in the sine-Gordon representation, extending over all possible distance scales, which expresses that gas as a convex superposition of dilute gases of neutral molecules whose activities are small if β is sufficiently large. Such gases are known not to exhibit screening. Abelian spin systems are related to a Coulomb gas by means of a duality transformation. A convergent low-temperature expansion for a variety of models of twodimensional surfaces is presented. It yields existence of the thermodynamic limit for the pressure and correlation functions as well as analyticity inz =e−β In addition, the estimates give exponential decay of truncated correlations, which proves the existence of a gap in the spectrum of the transfer matrix below the ground state eigenvalue. Two particular examples included in the general framework are the solid-on-solid and discrete Gaussian models.