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Analysis Seminar
Dimers and Amoebae
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This is joint work with Andrei Okounkov and Scott Sheffield. The "dimer model" is a natural model of 2D random interfaces in three dimensions. It is in fact essentially the only discrete random interface model to date which can be solved exactly. The measures on interfaces which arise come in three "phases", which we call gaseous, liquid and frozen. The corresponding phase space is the amoeba (in the sense of real algebraic geometry) of a two-variable polynomial P(z,w) which is the spectral curve of a certain operator on the underlying doubly-periodic planar graph. All physical properties of the model are encoded in properties of P. The polynomials P which arise are, moreover, all of a special type, they are so-called "Harnack curves". This fact has consequences about the possible large-scale behavior of the interfaces (fluctuations, decay rates of correlations, etc.) |