Dobrushin-koteck Y-shlosman Theorem up to the Critical Temperature

We develop a non-perturbative version of the Dobrushin-Koteck y-Shlosman theory of phase separation in the canonical 2D Ising ensemble. The results are valid for all temperatures below critical. 1. Introduction Dobrushin-Koteck y-Shlosman (DKS) Theorem DKS], DS] gives a rigorous probabilistic content to the assertion that pure phases are separated on the macroscopic scale along the boundary of the equilibrium crystal shape. Their results were formulated and proved in the context of the 2D Ising model at very low temperatures. Despite this particular setting it would be appropriate to talk in terms of the DKS Theory, rather than in terms of only one theorem with a very long proof (as the authors of DKS] modestly did). For many of their ideas and insights will certainly nd a way into both more general lattice models and higher dimensions. During several years following the publication of DKS], however, the main eeorts have been invested into attempts to relax their proof Pf] and, later on, to get rid of the \very low temperature" assumption. It has been commonly believed that one needs low temperature solely in order to have an additional technical tool of convergent cluster expansions readily available, whereas the results themselves should remain qualitatively the same in the whole of the phase transition region. And indeed, in a series of articles I1], I2], SS1], SS3], CGMS] and culminating in PV] and V] some sort of the DKS theory has been developed in the non-perturbative regime and pushed all the way to the critical temperature. These results, however, have been based on the integral type limit theorems and are, in parts, closer in spirit to the non-perturbative treatment of the 2D Bernoulli percolation ACC] than to the local limit setting of the exact canonical ensemble in the original monograph DKS]. Subsequently, the phase separation geometry has been much less pronounced, and the accent has been generally shifted to the precise leading surface order of integral estimates. Moreover, it seems that the local limit part of the DKS theory is the one to be the most robust and amenable as opposed to the skeleton coarse graining techniques, which are probably too much oriented to the two-dimensional lattice and nearest neighbour interactions. In this paper we try to ll in this gap and to extend the theory up to the critical temperature in the original setting of DKS], i.e., in the exact canonical ensemble. The …