The dimer model is a model from statistical mechanics corresponding to random perfect matchings on graphs. Circle patterns are a class of embeddings of planar graphs such that every face admits a circumcircle. We describe how to construct a 't-embedding' (or a circle pattern) of a dimer planar graph using its Kasteleyn weights, and discuss algebro-geometric properties of these embeddings.
This new class of embeddings is the key for studying Miquel dynamics, a discrete integrable system on circle patterns: we identify Miquel dynamics on the space of square-grid circle patterns with the Goncharov-Kenyon dimer dynamics and deduce the integrability of the former one and show that the evolution is governed by cluster algebra mutations.
