2013/08/14 by Richard Kenyon, Kenyon, Richard, Robin Pemantle +1 · 3 citations
Chemistry · Mathematics · Physics and Astronomy · #82B20 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Molecular spectroscopy and chirality #math-ph #math.CO #math.MP #msc:82B20
paper · pdf · doi:10.48550/arxiv.1308.2998
arxiv created 2013/08/14 · openalex publication_date 2013/08/14 · arxiv updated 2013/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define and study a recurrence relation in \mathbb Z3, called the hexahedron recurrence, which is similar to the octahedron recurrence (Hirota bilinear difference equation) and cube recurrence (Miwa equation). Like these examples, solutions to the hexahedron recurrence are partition sums for edge configurations on a certain graph, and have a natural interpretation in terms of cluster algebras. We give an explicit correspondence between monomials in the Laurent expansions arising in the recurrence with certain double-dimer configurations of a graph. We compute limit shapes for the corresponding double-dimer configurations. The Kashaev difference equation arising in the Ising model star-triangle relation is a special case of the hexahedron recurrence. In particular this reveals the cluster nature underlying the Ising model. The above relation allows us to prove a Laurent phenomenon for the Kashaev difference equation.