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Twenty-Vertex Model with Domain Wall Boundaries and Domino Tilings

Abstract : We consider the triangular lattice ice model (20-Vertex model) with four types of domain-wall type boundary conditions. In types 1 and 2, the configurations are shown to be equinumerous to the quarter-turn symmetric domino tilings of an Aztec-like holey square, with a central cross-shaped hole. The proof of this statement makes extensive use of integrability and of a connection to the 6-Vertex model. The type 3 configurations are conjectured to be in same number as domino tilings of a particular triangle. The four enumeration problems are reformulated in terms of four types of Alternating Phase Matrices with entries 0 and sixth roots of unity, subject to suitable alternation conditions. Our result is a generalization of the ASM-DPP correspondence. Several refined versions of the above correspondences are also discussed.
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Submitted on : Tuesday, September 8, 2020 - 9:59:51 AM
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Philippe Di Francesco, Emmanuel Guitter. Twenty-Vertex Model with Domain Wall Boundaries and Domino Tilings. The Electronic Journal of Combinatorics, Open Journal Systems, 2020, 27 (2), pp.P2.13. ⟨10.37236/8809⟩. ⟨cea-02932251⟩

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