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Triangular Ice: Combinatorics and Limit Shapes

Mathematical Physics & Probability

Speaker: P. Di Francesco, University of Illinois
Location: 2112 MSB
Start time: Wed, Sep 29 2021, 4:10PM

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 typical configurations, obtained by applying the so-called ``Tangent Method” of Colomo and Sportiello.