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Tilings of convex polygons

Richard Kenyon — 1997

Annales de l'institut Fourier

Call a polygon rational if every pair of side lengths has rational ratio. We show that a convex polygon can be tiled with rational polygons if and only if it is itself rational. Furthermore we give a necessary condition for an arbitrary polygon to be tileable with rational polygons: we associate to any polygon P a quadratic form q ( P ) , which must be positive semidefinite if P is tileable with rational polygons. The above results also hold replacing the rationality...

Dimers and cluster integrable systems

Alexander B. GoncharovRichard Kenyon — 2013

Annales scientifiques de l'École Normale Supérieure

We show that the dimer model on a bipartite graph Γ on a torus gives rise to a quantum integrable system of special type, which we call a. The phase space of the classical system contains, as an open dense subset, the moduli space Ł Γ of line bundles with connections on the graph Γ . The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs Γ 1 and Γ 2 areif the Newton polygons of the corresponding partition functions coincide up to translation. We define...

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