Combinatorics of the double-dimer model. (English) Zbl 1476.05170
Summary: We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the dimer partition function was established nearly 20 years ago by E. H. Kuo [Theor. Comput. Sci. 319, No. 1–3, 29–57 (2004; Zbl 1043.05099)] and has applications to random tiling theory and the theory of cluster algebras. This work was motivated in part by the potential for applications in these areas. Additionally, we discuss an application to a problem in Donaldson-Thomas and Pandharipande-Thomas theory. The proof of our recurrence requires generalizing work of R. W. Kenyon and D. B. Wilson [Trans. Am. Math. Soc. 363, No. 3, 1325–1364 (2011; Zbl 1230.60009); Electron. J. Comb. 16, No. 1, Research Paper R112, 28 p. (2009; Zbl 1225.60020)]; specifically, lifting their assumption that the nodes of the graph are black and odd or white and even.
MSC:
| 05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
| 13F60 | Cluster algebras |
| 05A15 | Exact enumeration problems, generating functions |
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
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