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Möbius Invariant Dimers and Miquel Dynamics
Abstract
As is already known, there is a correspondence between circle patterns and the dimer model, which is invariant under Euclidean transformations. We establish a new correspondence that is invariant under the group of Möbius transformations PO(3,1), which is the natural symmetry group of circle patterns. We show that Miquel dynamics preserve the partition function, and that convexity of the t-embedding guarantees positivity of the face weights. We also show that the correspondence has a somewhat hidden symmetry group PO(3,3), which relates to the Lorentz lift of the t-embedding. Finally, we consider isoradial graphs, Doyle spirals, the once-punctured torus and isothermic circle patterns.
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BibTeX
@misc{affolter2026möbius,
title = {{Möbius Invariant Dimers and Miquel Dynamics}},
author = {Niklas C. Affolter and Luis Mühlhoff},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07485},
eprint = {2609.07485},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07485},
}
RIS
TY - UNPB
TI - Möbius Invariant Dimers and Miquel Dynamics
AU - Niklas C. Affolter
AU - Luis Mühlhoff
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07485
UR - https://arxiv.org/abs/2609.07485
AB - As is already known, there is a correspondence between circle patterns and the dimer model, which is invariant under Euclidean transformations. We establish a new correspondence that is invariant under the group of Möbius transformations PO(3,1), which is the natural symmetry group of circle patterns. We show that Miquel dynamics preserve the partition function, and that convexity of the t-embedding guarantees positivity of the face weights. We also show that the correspondence has a somewhat hidden symmetry group PO(3,3), which relates to the Lorentz lift of the t-embedding. Finally, we consider isoradial graphs, Doyle spirals, the once-punctured torus and isothermic circle patterns.
ER -