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 .
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.