Elliptically Distributed Lozenge Tilings of a Hexagon

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2018
• Main Author: Betea, D.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2018
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/209540
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Elliptically Distributed Lozenge Tilings of a Hexagon / D. Betea // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 28 назв. — англ.

We present a detailed study of a four-parameter family of elliptic weights on tilings of a hexagon introduced by Borodin, Gorin, and Rains, generalizing some of their results. In the process, we connect the combinatorics of the model with the theory of elliptic special functions. Using canonical coordinates for the hexagon, we show how the n-point distribution function and transitional probabilities connect to the theory of BCn-symmetric multivariate elliptic special functions and of elliptic difference operators introduced by Rains. In particular, the difference operators intrinsically capture all of the combinatorics. Based on quasi-commutation relations between the elliptic difference operators, we construct certain natural measure-preserving Markov chains on such tilings, which we immediately use to obtain an exact sampling algorithm for these elliptic distributions. We present some simulated random samples exhibiting interesting and probably new arctic boundary phenomena. Finally, we show that the particle process associated with such tilings is determinantal with a correlation kernel given in terms of the univariate elliptic biorthogonal functions of Spiridonov and Zhedanov.