TAILIEUCHUNG - Đề tài " Dimers and amoebae "

We study random surfaces which arise as height functions of random perfect matchings (. dimer configurations) on a weighted, bipartite, doubly periodic graph G embedded in the plane. We derive explicit formulas for the surface tension and local Gibbs measure probabilities of these models. The answers involve a certain plane algebraic curve, which is the spectral curve of the Kasteleyn operator of the graph. For example, the surface tension is the Legendre dual of the Ronkin function of the spectral curve. The amoeba of the spectral curve represents the phase diagram of the dimer model. . | Annals of Mathematics Dimers and amoebae By Richard Kenyon Andrei Okounkov and Scott Sheffield Annals of Mathematics 163 2006 1019 1056 Dimers and amoebae By Richard KENyoN Andrei Okounkov and Scott Sheffield Abstract We study random surfaces which arise as height functions of random perfect matchings . dimer configurations on a weighted bipartite doubly periodic graph G embedded in the plane. We derive explicit formulas for the surface tension and local Gibbs measure probabilities of these models. The answers involve a certain plane algebraic curve which is the spectral curve of the Kasteleyn operator of the graph. For example the surface tension is the Legendre dual of the Ronkin function of the spectral curve. The amoeba of the spectral curve represents the phase diagram of the dimer model. Further we prove that the spectral curve of a dimer model is always a real curve of special type namely it is a Harnack curve. This implies many qualitative and quantitative statement about the behavior of the dimer model such as existence of smooth phases decay rate of correlations growth rate of height function fluctuations etc. Contents 1. Introduction 2. Definitions . Combinatorics of dimers . Periodic bipartite graphs and matchings . Height function . Gibbs measures . Definitions . Gibbs measures of fixed slope . Surface tension . Gauge equivalence and magnetic field . Gauge transformations . Rotations along cycles . Magnetic field coordinates The third author was supported in part by NSF Grant No. DMS-0403182. 1020 RICHARD KENYON ANDREI OKOUNKOV AND SCOTT SHEFFIELD 3. Surface tension . Kasteleyn matrix and characteristic polynomial . Kasteleyn weighting . Periodic boundary conditions . Characteristic polynomial . Newton polygon and allowed slopes . Asymptotics . Enlarging the fundamental domain . Partition function per fundamental domain . The amoeba and Ronkin function of a .

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