Dobrushin-Kotecký-Shlosman theorem up to the critical temperature
The physical theory states that the pure phases in the canonical ensemble are separated on the macroscopic scale by the boundary of the equilibrium cristal, whose shape can be described as the solution of the related variational problem. The first mathematically rigorous formulation of this result was given in the monograph by \textit{R. Dobrushin, R. Kotecký,} and \textit{S. Shlosman} [``Wulff construction. A global shape from local interaction'' (1992; Zbl 0917.60103)] that developed in detail the main ingredients of the phase separation phenomenon in the two-dimensional Ising model (i.e., the exact local limit theorems for the magnetization and the statistical theory of the phase boundary) and gave quite a detailed description of the typical configurations in the canonical ensemble. The proof was based on the method of cluster expansions, the techique that being supposingly applicable to a large class of statistical mechanical models is nevertheless limited to the low temperature region. This paper provides a mathematically rigorous justification of the general belief that the results of the Dobrushin-Kotecký-Shlosman (DKS) theory are valid for all subcritical temperatures. Using the full strength of correlation inequalities, the authors obtain results comparable to those in the DKS theory. The proof is based on the coarse graining of contours and accurate estimates of the probabilities of (small and large) moderate deviations for the magnetization. The latter in particular extends the region of validity of the theory of limit theorems (developed in DKS theory via cluster expansions) to all subcritical temperatures. The paper is essentially self-contained and presents a big step forward in understanding the phenomenon of phase separation. It should be noted, however, that the absence of detailed information about the statistical properties of the phase boundary up to the critical point forced the authors to use the ``external'' facts obtained on the base of the exact solution: the Ornstein-Zernike decay of the two-point correlation functions and some analytic properties of the surface tension. It would be interesting to recover these results without referring to the exact solution.
scientific article; zbMATH DE number 1269784