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 [Spe07]. 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 the companion paper [AdTM22]. We also prove 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 [ABS12].

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.