The Schwarzian octahedron recurrence (dSKP equation) I: explicit solutions

TL;DR

This paper derives explicit solutions for the discrete Schwarzian octahedron recurrence using partition functions of dimer models, revealing polynomial growth, cancellation phenomena, and applications to singularity patterns and limit shapes.

Contribution

It provides a new explicit formula for solutions of the dSKP equation as ratios of partition functions, extending Speyer's approach and analyzing algebraic entropy and combinatorial structures.

Findings

Solutions expressed as ratios of partition functions of dimer models

Polynomial growth indicating zero algebraic entropy

Identification of singularity repetition patterns (Devron property)

Abstract

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].