arXiv subjects

Niklas C. Affolter

Publications and source records attributed to Niklas C. Affolter.

3 recordsLinked to original sources

arXiv2026

Möbius Invariant Dimers and Miquel Dynamics

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.

math-ph↗

arXiv2026

Multi-dimensional consistency of principal binets

Principal binets are a discretization of curvature line parametrized surfaces defined on the vertices and faces of the square lattice $\Z^2$. They generalize the previously established discretizations given by circular nets, conical nets, and principal contact element nets. We show that principal binets constitute a discrete integrable system in the sense of multi-dimensional consistency. In particular, they generalize to higher-dimensional square lattices $\Z^N$. We also discuss relations to the notion of discrete orthogonal coordinate systems as previously established for discrete confocal quadrics.

math-ph↗

arXiv2018

Miquel Dynamics, Clifford Lattices and the Dimer Model

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.

math.DS↗