Random domino tilings of the Aztec diamond shape exhibit interesting features and statistical properties related to random matrix theory. As a statistical mechanical model it can be thought of as a dimer model or as a certain random surface. We consider the Aztec diamond with a two-periodic weighting which exhibits all three possible phases that occur in these types of models, often referred to as solid, liquid and gas. This model is considerably harder to analyze than the uniform model which only has two phases, liquid and solid. I will give an overview of some of the asymptotic results for the two-periodic Aztec diamond that have been obtained in joint work with Kurt Johansson.
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