Richard Kenyon ; Robin Pemantle - Double-dimers and the hexahedron recurrence

dmtcs:12797 - Discrete Mathematics & Theoretical Computer Science, January 1, 2013, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) - https://doi.org/10.46298/dmtcs.12797
Double-dimers and the hexahedron recurrenceArticle

Authors: Richard Kenyon 1; Robin Pemantle 2

  • 1 Department of Mathematics
  • 2 Department of Mathematics [Philadelphia]

We define and study a recurrence relation in $\mathbb{Z}^3$, 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 functions for 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.


Volume: DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)
Section: Proceedings
Published on: January 1, 2013
Imported on: November 21, 2016
Keywords: cluster algebra,urban renewal,Laurent property,Y-Delta,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • The geometry of probability generating functions; Funder: National Science Foundation; Code: 1209117

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