The Schwarzian Octahedron Recurrence (dSKP Equation) II: Geometric Systems

Abstract

We consider nine geometric systems: Miquel dynamics, P-nets, integrable cross-ratio maps, discrete holomorphic functions, orthogonal circle patterns, polygon recutting, circle intersection dynamics, (corrugated) pentagram maps and the short diagonal hyperplane map. Using a unified framework, for each system we prove an explicit expression for the solution as a function of the initial data; more precisely, we show that the solution is equal to the ratio of two partition functions of an oriented dimer model on an Aztec diamond whose face weights are constructed from the initial data. Then, we study the Devron property (Glick in J Geom Phys 87:161–189, 2015), which states the following: if the system starts from initial data that is singular for the backwards dynamics, this singularity is expected to reoccur after a finite number of steps of the forwards dynamics. Again, using a unified framework, we prove this Devron property for all of the above geometric systems, for different kinds of singular initial data. In doing so, we obtain new singularity results and also known ones (Glick in J Geom Phys 87:161–189, 2015; Yao in Glick’s conjecture on the point of collapse of axis-aligned polygons under the pentagram maps (2014). Preprint arXiv:1410.7806). Our general method consists in proving that these nine geometric systems are all related to the Schwarzian octahedron recurrence (dSKP equation), and then to rely on the companion paper (Affolter et al. in Comb Theory 3(2), 2023), where we study this recurrence in general, prove explicit expressions and singularity results.