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Dobrushin interfaces via reflection positivity. (English) Zbl 1138.82005
For low temperature 3D Ising model an example of a pure state, with a coexistence of phases separated by an interface, was given by R. L. Dobrushin [Theory Probab. Appl. 17, 582–600 (1972; Zbl 0275.60119)]. Since then a conjecture has been built that at critical temperature, such system apart from the translation invariant states posesses as well states that are not translation invariant. They are to describe a coexistence of ordered states and a chaotic state, with the rigid interface separating them.
The present paper primarily aimed at the proof of this conjecture. This program has not been satisfactorily completed, but a novel method of reflection positivity [this, originally due to S. Shlosman, Russ. Math. Surv. 41, 83–134 (1986)] has been adopted to tackle the problem. The method has been proved to work for non-linear sigma models, Ising and Potts models, large entropy systems of continuous spins, with a hope for a generalization for systems with continuous symmetry.

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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