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Seminar

Combinatorics and Limit Shapes November 12, 2021 (11:00 AM PST - 12:00 PM PST)
Parent Program: Universality and Integrability in Random Matrix Theory and Interacting Particle Systems MSRI: Simons Auditorium, Online/Virtual
Speaker(s) Philippe Di Francesco (University of Illinois at Urbana-Champaign)
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Video

Combinatorics And Limit Shapes

Abstract/Media

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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).