A Periodicity Theorem for the Octahedron Recurrence

TL;DR

This paper proves that a variant of the octahedron recurrence in a 3D lattice exhibits periodicity of period n+m, using a non-recursive formula based on perfect matchings, extending previous results.

Contribution

It provides an explicit non-recursive formula for the recurrence and establishes its periodicity, generalizing prior work by Speyer and relating to Fomin and Zelevinsky's Y-systems.

Findings

The recurrence is periodic with period n+m.

An explicit formula in terms of perfect matchings is derived.

The result connects to known periodicity theorems in cluster algebra theory.

Abstract

We investigate a variant of the octahedron recurrence which lives in a 3-dimensional lattice contained in [0,n] x [0,m] x R. Generalizing results of David Speyer math.CO/0402452, we give an explicit non-recursive formula for the values of this recurrence in terms of perfect matchings. We then use it to prove that the octahedron recurrence is periodic of period n+m. This result is reminiscent of Fomin and Zelevinsky's theorem about the periodicity of Y-systems.