Miquel dynamics is a discrete time dynamics for circle patterns. Previous work shows that the circle centers satisfy the dSKP equation on the octahedral lattice A³ and give rise to a real-valued Y-system / cluster structure. In this article we show that half the intersection points satisfy the dSKP equation as well, and we introduce two new real-valued Y-systems for Miquel dynamics that involve only the intersection points. Therefore, the new Y-systems are Möbius invariant. We also show that the circle centers and intersection points combined satisfy the dSKP equation on the 4-dimensional octahedral lattice. In addition, we present two more complex-valued Y-systems for Miquel dynamics, which are real-valued in and only in the case of integrable circle patterns. We also show that in the case of harmonic embeddings, the old and the new Y-patterns coincide. 

We introduce twisted triple crossing diagram maps, collections of points in projective space associated to bipartite graphs on the cylinder, and use them to provide geometric realizations of the cluster integrable systems of Goncharov and Kenyon constructed from toric dimer models. Using this notion, we provide geometric proofs that the pentagram map and the cross-ratio dynamics integrable systems are cluster integrable systems. We show that in appropriate coordinates, cross-ratio dynamics is described by geometric R-matrices, which solves the open question of finding a cluster algebra structure describing cross-ratio dynamics. 

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 [Gli15], 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 [Gli15, Yao14]. 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 [AdTM22], where we study this recurrence in general, prove explicit expressions and singularity results. 

We study a simple geometric model for local transformations of bipartite graphs. The state consists of a choice of a vector at each white vertex made in such a way that the vectors neighboring each black vertex satisfy a linear relation. The evolution for different choices of the graph coincides with many notable dynamical systems including the pentagram map, Q-nets, and discrete Darboux maps. On the other hand, for plabic graphs we prove unique extendability of a configuration from the boundary to the interior, an elegant illustration of the fact that Postnikov’s boundary measurement map is invertible. In all cases there is a cluster algebra operating in the background, resolving the open question for Q-nets of whether such a structure exists. 

We prove an explicit expression for the solutions of the discrete Schwarzian octahedron recurrence, also known as the discrete Schwarzian KP equation (dSKP), as the ratio of two partition functions. Each one counts weighted oriented dimer configurations of an associated bipartite graph, and is equal to the determinant of a Kasteleyn matrix. This is in the spirit of Speyer's result on the dKP equation, or octahedron recurrence (Journal of Alg. Comb. 2007). One consequence is that dSKP has zero algebraic entropy, meaning that the growth of the degrees of the polynomials involved is only polynomial. There are cancellations in the partition function, and we prove an alternative, cancellation free explicit expression involving complementary trees and forests. Using all of the above, we show several instances of the Devron property for dSKP, i.e., that certain singularities in initial data repeat after a finite number of steps. This has many applications for discrete geometric systems and is the subject of a companion paper (preprint 2022, Affolter, de Tillière, and Melotti). We also find limit shape results analogous to the arctic circle of the Aztec diamond. Finally, we discuss the combinatorics of all the other octahedral equations in the classification of Adler, Bobenko and Suris (IMRN 2012).

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. 

The main goal of this thesis is to find and investigate occurrences of cluster algebras in objects of discrete differential geometry (DDG). During the investigation it became apparent that there are multiple ways in which cluster algebras occur for multiple objects of DDG. In order to understand the different cluster algebras systematically, it is practical to introduce the common framework of triple crossing diagram maps (TCD maps). For TCD maps the multiple occurrences of cluster algebras can be defined and related systematically. It turns out the framework is exhaustive, in the sense that it covers a long list of examples and indeed every discrete 3D-integrable system that is defined in geometric terms that we are aware of. In particular, all the known examples of DDG, discrete integrable systems and embeddings associated to exactly solvable models that feature cluster algebras are included.

Given a triangulation, one can consider the incircles of its faces. If these all touch, they represent what is known as an orthogonal circle packing. For the combinatorical structure of the graph and up to boundary conditions, this circle packing is unique. In this thesis, we consider a generalization to circle packings such that incircles of incident faces need not touch. Instead we consider that two incircles divide the incident edge in two ratios with given cross ratio (i.e. shear). We use a generalization of Brägger’s functional to show, that given these fixed cross ratios, a triangulation is uniquely determined. But for general cross ratios we encounter length monodromies. We also study the case in which angle sums in vertices may be arbitrary, thus non-euclidean. In this case we introduce a new variational principle and find a necessary condition for cross ratios, such that they do not give rise to length monodromies.

In this thesis we consider the activity driven model for temporal networks. We study several modifications of the model as well as their effects on disease spreading. The modifications include unbounded activities allowing for true scale-free networks, directed disease spreading, bipartite networks and preferential attachment. We also apply several new methods for describing SIS fixed points and deriving thresholds on the original model as well as on its modifications. A microscopic approach allows us to describe sub-threshold behaviour. Working with the infection ratios per activity proves fruitful for providing a fast algorithm for finding the fixed points, for yielding a closed transcendental equation in the case of scale free networks, and for allowing power series approximations near the threshold. Finally the preferential attachment modification gives thresholds in close agreement with disease simulations on empirical communication networks.