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We consider the triangular lattice version of the two-dimensional Ice (6 Vertex) model with suitable boundary conditions, leading to an integrable 20 Vertex (20V) model. After reviewing a few facts on the square lattice version and the role of integrability, we compute the number of 20V configurations with Domain Wall boundary conditions in the form of a determinant, which turns out to match the number of quarter-turn symmetric domino tilings of a quasi-Aztec square of same size, with a central cross-shaped hole: this provides us with one more example of re-expression of an “interacting fermion” model into a “free fermion” one. We also present results/conjectures for triangular Ice with other types of boundary conditions in relation to domino tilings of certain Aztec domains. Finally we apply the so-called ``Tangent Method” of Colomo and Sportiello to the determination of the limit shape of large typical 20V model configurations. (based on works with E. Guitter and B. Debin).

Combinatorics And Limit Shapes