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Main Title: Miquel dynamics, Clifford lattices and the Dimer model
Author(s): Affolter, Niklas C.
Type: Article
Abstract: Miquel dynamics was 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 this dynamics we consider the local Miquel move, which changes the combinatorics and geometry of a circle pattern. We prove that the circle centers under Miquel dynamics are Clifford lattices, an integrable system considered 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 urban renewal.
Subject(s): Miquel dynamics
circle patterns
Clifford lattices
Dimer model
urban renewal
Issue Date: 2-May-2021
Date Available: 28-May-2021
Language Code: en
DDC Class: 530 Physik
Sponsor/Funder: TU Berlin, Open-Access-Mittel – 2021
DFG, 195170736, TRR 109: Diskretisierung in Geometrie und Dynamik
Journal Title: Letters in mathematical physics
Publisher: Springer Nature
Volume: 111
Article Number: 61
Publisher DOI: 10.1007/s11005-021-01406-0
EISSN: 1573-0530
ISSN: 0377-9017
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik » FG Diskrete Mathematik / Geometrie
Appears in Collections:Technische Universität Berlin » Publications

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