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.

Dimer model and holomorphic functions on t‐embeddings of planar graphs

“…A more geometric viewpoint on “nice” gauge functions, the so‐called Coulomb gauges , was suggested in [33]. These gauges have many remarkable algebraic properties (see also [1]) and are also closely related to T‐graphs mentioned above.…”

“…In parallel, a notion of s‐embeddings of graphs carrying the planar Ising model was suggested in [9]. As explained in [33, section 7], the latter are a particular case of the former under the combinatorial bosonization correspondence of the two models [19]. The notion of t‐embeddings discussed in our paper is fully equivalent to Coulomb gauges of [33] except that we focus on embeddings of the dual graphs $$(\mathcal {G}^\delta )^*$$ from the very beginning.…”

“…As explained in [33, section 7], the latter are a particular case of the former under the combinatorial bosonization correspondence of the two models [19]. The notion of t‐embeddings discussed in our paper is fully equivalent to Coulomb gauges of [33] except that we focus on embeddings of the dual graphs $$(\mathcal {G}^\delta )^*$$ from the very beginning. The appearance of another name for the same object is caused by the fact that we were not aware of the research of [33] at the beginning of this project and arrived at the same concept aiming to generalize results obtained for the Ising model observables on s‐embeddings to dimers.…”

“…We believe that s‐ and t‐embeddings of abstract weighted planar graphs are the right tools to attack these questions. This perspective is somehow similar to the idea of using square tilings to study random planar maps weighted by uniform spanning trees; in this case an alternative option could be to use Tutte's harmonic embeddings that are also related to the context of t‐embeddings via [33, section 6.2]. On deterministic graphs, we believe that the discrete complex analysis viewpoint on Kasteleyn equations provided in our paper is flexible enough to be applied in rather general situations, both in terms of the underlying lattice and of the limit shape surfaces; for example, see [13] where the case of the classical Aztec diamond is discussed from this perspective.…”

“…We now briefly recall the setup of t‐embeddings or Coulomb gauges, see Section 2 and [33] for more details. Let $$\mathcal {G}$$ be a weighted bipartite graph carrying the dimer model; the latter is a random choice of a perfect matching of vertices of $$\mathcal {G}$$ with a probability proportional to the product of the corresponding positive weights.…”