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Dimers and amoebae. (English) Zbl 1154.82007
The authors prove that models based on perfect matchings (on any weighted doubly-periodic bipartite graph $$G$$ in the plan) are exactly solvable in a rather strong sense: the explicit formulas for the surface tension are derived; the set of Gibbs measures is explicitly classified and the local probabilities in each of them are explicitly computed. In particular it is shown that Gibbs measure come in three distinct phases: a rough phase, where the height fluctuations are on the order of $$\log n$$ for points separated by distance $$n,$$ and correlations decay quadratically in $$n$$; a frozen phase, where there are no large-scale fluctuations and the model is a Bernoulli process; and a smooth phase, where fluctuations have bounded variance, and correlations decay exponentially.

##### MSC:
 82B23 Exactly solvable models; Bethe ansatz 60B05 Probability measures on topological spaces
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