Elliptic Combinatorics and Markov Processes
We present combinatorial and probabilistic interpretations of recent results in the theory of elliptic special functions (due to, among many others, Frenkel, Turaev, Spiridonov, and Zhedanov in the case of univariate functions, and Rains in the multivariate case). We focus on elliptically distribute...
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ftcaltechdiss:oai:thesis.library.caltech.edu:7115 2023-09-05T13:17:26+02:00 Elliptic Combinatorics and Markov Processes Betea, Dan Dumitru 2012 application/pdf https://thesis.library.caltech.edu/7115/ https://thesis.library.caltech.edu/7115/1/betea_thesis.pdf https://thesis.library.caltech.edu/7115/2/betea_thesis_4_print.pdf https://resolver.caltech.edu/CaltechTHESIS:05312012-201348939 en eng https://thesis.library.caltech.edu/7115/1/betea_thesis.pdf https://thesis.library.caltech.edu/7115/2/betea_thesis_4_print.pdf Betea, Dan Dumitru (2012) Elliptic Combinatorics and Markov Processes. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/MMZG-5G61. https://resolver.caltech.edu/CaltechTHESIS:05312012-201348939 <https://resolver.caltech.edu/CaltechTHESIS:05312012-201348939> other Thesis NonPeerReviewed 2012 ftcaltechdiss https://doi.org/10.7907/MMZG-5G61 2023-08-14T17:28:25Z We present combinatorial and probabilistic interpretations of recent results in the theory of elliptic special functions (due to, among many others, Frenkel, Turaev, Spiridonov, and Zhedanov in the case of univariate functions, and Rains in the multivariate case). We focus on elliptically distributed random lozenge tilings of the hexagon which we analyze from several perspectives. We compute the N-point function for the associated process, and show the process as a whole is determinantal with correlation kernel given by elliptic biorthogonal functions. We furthermore compute transition probabilities for the Markov processes involved and show they come from the multivariate elliptic difference operators of Rains. Properties of difference operators yield an efficient sampling algorithm for such random lozenge tilings. Simulations of said algorithm lead to new arctic circle behavior. Finally we introduce elliptic Schur processes on bounded partitions analogous to the Schur process of Reshetikhin and Okounkov ( and to the Macdonald processes of Vuletic, Borodin, and Corwin). These give a somewhat different (and faster) sampling algorithm from these elliptic distributions, but in principle should encompass more than just tilings of a hexagon. Thesis Arctic CaltechTHESIS (California Institute of Technology Arctic Borodin ENVELOPE(-72.627,-72.627,-71.603,-71.603) |
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Open Polar |
collection |
CaltechTHESIS (California Institute of Technology |
op_collection_id |
ftcaltechdiss |
language |
English |
description |
We present combinatorial and probabilistic interpretations of recent results in the theory of elliptic special functions (due to, among many others, Frenkel, Turaev, Spiridonov, and Zhedanov in the case of univariate functions, and Rains in the multivariate case). We focus on elliptically distributed random lozenge tilings of the hexagon which we analyze from several perspectives. We compute the N-point function for the associated process, and show the process as a whole is determinantal with correlation kernel given by elliptic biorthogonal functions. We furthermore compute transition probabilities for the Markov processes involved and show they come from the multivariate elliptic difference operators of Rains. Properties of difference operators yield an efficient sampling algorithm for such random lozenge tilings. Simulations of said algorithm lead to new arctic circle behavior. Finally we introduce elliptic Schur processes on bounded partitions analogous to the Schur process of Reshetikhin and Okounkov ( and to the Macdonald processes of Vuletic, Borodin, and Corwin). These give a somewhat different (and faster) sampling algorithm from these elliptic distributions, but in principle should encompass more than just tilings of a hexagon. |
format |
Thesis |
author |
Betea, Dan Dumitru |
spellingShingle |
Betea, Dan Dumitru Elliptic Combinatorics and Markov Processes |
author_facet |
Betea, Dan Dumitru |
author_sort |
Betea, Dan Dumitru |
title |
Elliptic Combinatorics and Markov Processes |
title_short |
Elliptic Combinatorics and Markov Processes |
title_full |
Elliptic Combinatorics and Markov Processes |
title_fullStr |
Elliptic Combinatorics and Markov Processes |
title_full_unstemmed |
Elliptic Combinatorics and Markov Processes |
title_sort |
elliptic combinatorics and markov processes |
publishDate |
2012 |
url |
https://thesis.library.caltech.edu/7115/ https://thesis.library.caltech.edu/7115/1/betea_thesis.pdf https://thesis.library.caltech.edu/7115/2/betea_thesis_4_print.pdf https://resolver.caltech.edu/CaltechTHESIS:05312012-201348939 |
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://thesis.library.caltech.edu/7115/1/betea_thesis.pdf https://thesis.library.caltech.edu/7115/2/betea_thesis_4_print.pdf Betea, Dan Dumitru (2012) Elliptic Combinatorics and Markov Processes. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/MMZG-5G61. https://resolver.caltech.edu/CaltechTHESIS:05312012-201348939 <https://resolver.caltech.edu/CaltechTHESIS:05312012-201348939> |
op_rights |
other |
op_doi |
https://doi.org/10.7907/MMZG-5G61 |
_version_ |
1776198606515077120 |