Dimers and Circle patterns

TL;DR

This paper establishes a mathematical correspondence between dimer models on bipartite graphs and circle patterns, extending known cases and linking local moves in both frameworks, with implications for various physical models.

Contribution

It introduces a new correspondence between dimer models and circle patterns, generalizing the isoradial case and connecting local moves to the octahedron recurrence.

Findings

Urban renewal corresponds to Miquel move in circle patterns

Miquel dynamics are governed by the octahedron recurrence

Recovers harmonic embeddings and s-embeddings as special cases

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.