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Dimers and circle patterns

Abstract : We establish a correspondence between the dimer model on a bipartite graph and a circle pattern with the combinatorics of that graph, which holds for graphs that are either planar or embedded on the torus. The set of positive face weights on the graph gives a set of global coordinates on the space of circle patterns with embedded dual. Under this correspondence, which extends the previously known isoradial case, the urban renewal (local move for dimer models) is equivalent to the Miquel move (local move for circle patterns). As a consequence the Miquel dynamics on circle patterns is governed by the octahedron recurrence. As special cases of these circle pattern embeddings, we recover harmonic embeddings for resistor networks and s-embeddings for the Ising model.
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Submitted on : Sunday, November 4, 2018 - 12:09:26 AM
Last modification on : Friday, October 15, 2021 - 1:37:39 PM
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Richard Kenyon, Wai Yeung Lam, Sanjay Ramassamy, Marianna Russkikh. Dimers and circle patterns. Annales Scientifiques de l'École Normale Supérieure, Société mathématique de France, In press. ⟨hal-01911855⟩



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