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 coor...
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ftdatacite:10.48550/arxiv.1110.4176 2023-05-15T15:05:07+02:00 Elliptically Distributed Lozenge Tilings of a Hexagon Betea, Dan 2011 https://dx.doi.org/10.48550/arxiv.1110.4176 https://arxiv.org/abs/1110.4176 unknown arXiv https://dx.doi.org/10.3842/sigma.2018.032 Creative Commons Attribution Share Alike 4.0 International https://creativecommons.org/licenses/by-sa/4.0/legalcode cc-by-sa-4.0 CC-BY-SA Mathematical Physics math-ph Combinatorics math.CO Probability math.PR FOS Physical sciences FOS Mathematics article-journal Article ScholarlyArticle Text 2011 ftdatacite https://doi.org/10.48550/arxiv.1110.4176 https://doi.org/10.3842/sigma.2018.032 2022-04-01T14:03:31Z 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 $BC_n$-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 to such tilings is determinantal with correlation kernel given in terms of the univariate elliptic biorthogonal functions of Spiridonov and Zhedanov. Text Arctic DataCite Metadata Store (German National Library of Science and Technology) Arctic Borodin ENVELOPE(-72.627,-72.627,-71.603,-71.603) |
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DataCite Metadata Store (German National Library of Science and Technology) |
op_collection_id |
ftdatacite |
language |
unknown |
topic |
Mathematical Physics math-ph Combinatorics math.CO Probability math.PR FOS Physical sciences FOS Mathematics |
spellingShingle |
Mathematical Physics math-ph Combinatorics math.CO Probability math.PR FOS Physical sciences FOS Mathematics Betea, Dan Elliptically Distributed Lozenge Tilings of a Hexagon |
topic_facet |
Mathematical Physics math-ph Combinatorics math.CO Probability math.PR FOS Physical sciences FOS Mathematics |
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 $BC_n$-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 to such tilings is determinantal with correlation kernel given in terms of the univariate elliptic biorthogonal functions of Spiridonov and Zhedanov. |
format |
Text |
author |
Betea, Dan |
author_facet |
Betea, Dan |
author_sort |
Betea, Dan |
title |
Elliptically Distributed Lozenge Tilings of a Hexagon |
title_short |
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_sort |
elliptically distributed lozenge tilings of a hexagon |
publisher |
arXiv |
publishDate |
2011 |
url |
https://dx.doi.org/10.48550/arxiv.1110.4176 https://arxiv.org/abs/1110.4176 |
long_lat |
ENVELOPE(-72.627,-72.627,-71.603,-71.603) |
geographic |
Arctic Borodin |
geographic_facet |
Arctic Borodin |
genre |
Arctic |
genre_facet |
Arctic |
op_relation |
https://dx.doi.org/10.3842/sigma.2018.032 |
op_rights |
Creative Commons Attribution Share Alike 4.0 International https://creativecommons.org/licenses/by-sa/4.0/legalcode cc-by-sa-4.0 |
op_rightsnorm |
CC-BY-SA |
op_doi |
https://doi.org/10.48550/arxiv.1110.4176 https://doi.org/10.3842/sigma.2018.032 |
_version_ |
1766336872956559360 |