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Wulff construction. A global shape from local interaction. Transl. from the Russian by the authors. (English) Zbl 0917.60103
Translations of Mathematical Monographs. 104. Providence, RI: American Mathematical Society (AMS). ix, 204 p. (1992).
Consider two phases $$A$$ and $$B$$ in equilibrium within a box, the walls of which favor one of them, say $$A$$. Then if you impose a certain volume of phase $$B$$, it will try to concentrate in the center of the container. The shape that this volume will take at equilibrium is an old problem solved at the beginning of the century by Wulff. This shape is in fact intimately related to the interfacial free energy between the two phases $$A$$ and $$B$$. The solution proposed by Wulff was based on the classical assumptions that the interfacial free energies are additive and that the equilibrium shape should minimise the total interfacial free energy. This book is in fact devoted to a rigorous proof of this remarkable property starting from microscopic considerations within a two-dimensional Ising model. More precisely, the authors establish the validity of the Wulff method for a two-dimensional Ising ferromagnet with periodic boundary condition and at sufficiently low temperatures. The method of proof is based on limit theorems and large deviations. A key argument is also given by the fact that fixing the volume of material imposes the requirement that one work with a fixed magnetisation, or in other words, in a canonical ensemble. The extension of this work to take into account higher temperatures is still an open problem.

##### MSC:
 60K40 Other physical applications of random processes 82B05 Classical equilibrium statistical mechanics (general) 82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 60-02 Research exposition (monographs, survey articles) pertaining to probability theory