Dimers and circle patterns

Richard Kenyon (1) , Wai Yeung Lam (2) , Sanjay Ramassamy (3, 4, 5) , Marianna Russkikh (6)

1 Department of Mathematics [Yale University]
2 Department of Mathematics, Brown University
3 DMA - Département de Mathématiques et Applications - ENS Paris
4 CNRS - Centre National de la Recherche Scientifique
5 PSL - Université Paris sciences et lettres
6 UNIGE - Université de Genève = University of Geneva

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, we show that Miquel dynamics on circle patterns is a discrete integrable system 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.

Domaines: Physique mathématique [math-ph]; Variables complexes [math.CV]; Systèmes dynamiques [math.DS]; Topologie géométrique [math.GT]

Richard Kenyon, Wai Yeung Lam, Sanjay Ramassamy, Marianna Russkikh. Dimers and circle patterns. Annales Scientifiques de l'École Normale Supérieure, 2022, 55 (3), pp.863-901. ⟨10.24033/asens.2507⟩. ⟨hal-01911855v2⟩