Twenty-Vertex Model with Domain Wall Boundaries and Domino Tilings

Philippe Di Francesco (1, 2, 3) , Emmanuel Guitter (1, 2)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681
2 Université Paris-Saclay
3 Department of Mathematics, Illinois State University

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.

Domains: Mathematics [math]; Combinatorics [math.CO]

Philippe Di Francesco, Emmanuel Guitter. Twenty-Vertex Model with Domain Wall Boundaries and Domino Tilings. The Electronic Journal of Combinatorics, 2020, 27 (2), pp.P2.13. ⟨10.37236/8809⟩. ⟨cea-02932251⟩