Double-dimers and the hexahedron recurrence
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...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2013-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/12797/pdf |
| _version_ | 1797270315766644736 |
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| author | Richard Kenyon Robin Pemantle |
| author_facet | Richard Kenyon Robin Pemantle |
| author_sort | Richard Kenyon |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-25T02:02:19Z |
| format | Article |
| id | doaj.art-e2e57b81fff74221860f2c99ea1cc6db |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:02:19Z |
| publishDate | 2013-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-e2e57b81fff74221860f2c99ea1cc6db2024-03-07T14:52:35ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502013-01-01DMTCS Proceedings vol. AS,...Proceedings10.46298/dmtcs.1279712797Double-dimers and the hexahedron recurrenceRichard Kenyon0Robin Pemantle1Department of MathematicsDepartment 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.https://dmtcs.episciences.org/12797/pdfcluster algebraurban renewallaurent propertyy-delta[info.info-dm]computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Richard Kenyon Robin Pemantle Double-dimers and the hexahedron recurrence Discrete Mathematics & Theoretical Computer Science cluster algebra urban renewal laurent property y-delta [info.info-dm]computer science [cs]/discrete mathematics [cs.dm] |
| title | Double-dimers and the hexahedron recurrence |
| title_full | Double-dimers and the hexahedron recurrence |
| title_fullStr | Double-dimers and the hexahedron recurrence |
| title_full_unstemmed | Double-dimers and the hexahedron recurrence |
| title_short | Double-dimers and the hexahedron recurrence |
| title_sort | double dimers and the hexahedron recurrence |
| topic | cluster algebra urban renewal laurent property y-delta [info.info-dm]computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/12797/pdf |
| work_keys_str_mv | AT richardkenyon doubledimersandthehexahedronrecurrence AT robinpemantle doubledimersandthehexahedronrecurrence |