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Title:
Triangular Ice: Combinatorics and Limit Shapes
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Abstract:
We consider the triangular lattice version of the two-dimensional ice modelwith suitable boundary conditions, leading to an integrable 20 Vertex model.Configurations give rise to generalizations of Alternating Sign Matrices, which we call Alternating Phase Matrices (APM). After reviewing a few facts on the square lattice version and the role of integrability, we compute the number of APM of any given size in the form of a determinant, which turns out to match the number of quarter-turn symmetric domino tilings of a quasi-Aztec square with a central cross-shaped hole. We also present results/conjectures for triangular Ice with other types of boundary conditions, and results on the limit shape of large APM, obtained by applying the so-called ``Tangent Method". (joint works with E. Guitter (IPhT Saclay) and B Debin (U Louvain))
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