Workshop: Doubly periodic tilings from the point of view of matrix valued orthogonal polynomials

Speaker
Kuijlaars, Katholieke Universiteit Leuven

Abstract
I will describe the use of matrix valued orthogonal polynomials (MVOP) in the study of doubly periodic tiling models.The two prototypical examples are doubly periodic domino tilings of Aztec diamond and doubly periodic lozenge tilings of a hexagon. In the large size limit these models exhibit a separation of phases, known as frozen, rough and smooth.
The Aztec diamond model was analyzed in great detail by Berggren-Duits (2019), and Berggren-Borodin (2023) on the basis of a Wiener-Hopf factorization of the matrix weight for the MVOP. The Wiener-Hopf factorization is not available for the hexagon model.
The main new result is the asymptotic analysis of the MVOP associated with a special class of 3×3 periodic weights on the hexagon. The main tool is a suitable equilibrium problem with external field on the spectral curve. The smooth phase in the tiling model has six cusps.