Miquel Dynamics, Clifford Lattices and the Dimer Model

TL;DR

This paper explores the dynamics of circle patterns on a torus, revealing their connection to Clifford lattices and the dimer model, and introduces the Miquel move as a local transformation that preserves certain probabilities.

Contribution

It establishes that Miquel dynamics produce Clifford lattices and links circle pattern transformations to the dimer model through probability-preserving moves.

Findings

Circle centers under Miquel dynamics form Clifford lattices.

The Miquel move is a local transformation that preserves edge weights.

A new connection between circle patterns and the dimer model is demonstrated.

Abstract

Miquel dynamics were introduced by Ramassamy as a discrete time evolution of square grid circle patterns on the torus. In each time step every second circle in the pattern is replaced with a new one by employing Miquel's six circle theorem. Inspired by these dynamics we define the Miquel move, which changes the combinatorics and geometry of a circle pattern locally. We prove that the circle centers under Miquel dynamics are Clifford lattices, considered as an integrable system by Konopelchenko and Schief. Clifford lattices have the combinatorics of an octahedral lattice and every octahedron contains six intersection points of Clifford's four circle configuration. The Clifford move replaces one of these circle intersection points with the opposite one. We establish a new connection between circle patterns and the dimer model: If the distances between circle centers are interpreted as edge weights, the Miquel move preserves probabilities in the sense of of urban renewal.