Elliptically Distributed Lozenge Tilings of a Hexagon
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 coo...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2018 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2018
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/209540 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Elliptically Distributed Lozenge Tilings of a Hexagon / D. Betea // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 28 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862728018717310976 |
|---|---|
| author | Betea, D. |
| author_facet | Betea, D. |
| citation_txt | Elliptically Distributed Lozenge Tilings of a Hexagon / D. Betea // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 28 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2025-12-07T19:06:03Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-209540 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T19:06:03Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Betea, D. 2025-11-24T10:49:46Z 2018 Elliptically Distributed Lozenge Tilings of a Hexagon / D. Betea // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33E05; 60C05; 05E05 arXiv: 1110.4176 https://nasplib.isofts.kiev.ua/handle/123456789/209540 https://doi.org/10.3842/SIGMA.2018.032 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. The author would like to thank Alexei Borodin, Fokko van de Bult, Vadim Gorin, and Eric Rains for their help through numerous conversations. He is also indebted to Igor Pak and Greta Panova for putting the tiling picture herein described into perspective, and to three anonymous referees for improving the clarity of the manuscript. This article was written while the author was a graduate student in the Department of Mathematics at the California Institute of Technology, to which many remerciements are due for all its support during the five years the author spent there. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Elliptically Distributed Lozenge Tilings of a Hexagon Article published earlier |
| spellingShingle | Elliptically Distributed Lozenge Tilings of a Hexagon Betea, D. |
| title | Elliptically Distributed Lozenge Tilings of a Hexagon |
| title_full | Elliptically Distributed Lozenge Tilings of a Hexagon |
| title_fullStr | Elliptically Distributed Lozenge Tilings of a Hexagon |
| title_full_unstemmed | Elliptically Distributed Lozenge Tilings of a Hexagon |
| title_short | Elliptically Distributed Lozenge Tilings of a Hexagon |
| title_sort | elliptically distributed lozenge tilings of a hexagon |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209540 |
| work_keys_str_mv | AT betead ellipticallydistributedlozengetilingsofahexagon |