Elliptic Determinantal Processes and Elliptic Dyson Models
We introduce seven families of stochastic systems of interacting particles in one-dimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants contr...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2017 |
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| Sprache: | English |
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Інститут математики НАН України
2017
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/149273 |
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| Zitieren: | Elliptic Determinantal Processes and Elliptic Dyson Models / M. Katori // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 43 назв. — англ. |
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Katori, M. 2019-02-19T19:40:23Z 2019-02-19T19:40:23Z 2017 Elliptic Determinantal Processes and Elliptic Dyson Models / M. Katori // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 43 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 60J65; 60G44; 82C22; 60B20; 33E05; 17B22 DOI:10.3842/SIGMA.2017.079 https://nasplib.isofts.kiev.ua/handle/123456789/149273 We introduce seven families of stochastic systems of interacting particles in one-dimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single function called the spatio-temporal correlation kernel. For the four families AN₋₁, BN, CN and DN, we identify the systems of stochastic differential equations solved by these determinantal processes, which will be regarded as the elliptic extensions of the Dyson model. Here we use the notion of martingales in probability theory and the elliptic determinant evaluations of the Macdonald denominators of irreducible reduced affine root systems given by Rosengren and Schlosser. This paper is a contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications. The full collection is available at https://www.emis.de/journals/SIGMA/EHF2017.html. The author would like to thank the anonymous referees whose comments considerably improved the presentation of the paper. A part of the present work was done during the participation of the author in the ESI workshop on “Elliptic Hypergeometric Functions in Combinatorics, Integrable Systems and Physics” (March 20–24, 2017). The present author expresses his gratitude for the hospitality of Erwin Schr¨odinger International Institute for Mathematics and Physics (ESI) of the University of Vienna and for well-organization of the workshop by Christian Krattenthaler, Masatoshi Noumi, Simon Ruijsenaars, Michael J. Schlosser, Vyacheslav P. Spiridonov, and S. Ole Warnaar. He also thanks Soichi Okada, Masatoshi Noumi, Simon Ruijsenaars, and Michael J. Schlosser for useful discussion. This work was supported in part by the Grant-in-Aid for Scientific Research (C) (No. 26400405), (B) (No. 26287019), and (S) (No. 16H06338) of Japan Society for the Promotion of Science. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Elliptic Determinantal Processes and Elliptic Dyson Models Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Elliptic Determinantal Processes and Elliptic Dyson Models |
| spellingShingle |
Elliptic Determinantal Processes and Elliptic Dyson Models Katori, M. |
| title_short |
Elliptic Determinantal Processes and Elliptic Dyson Models |
| title_full |
Elliptic Determinantal Processes and Elliptic Dyson Models |
| title_fullStr |
Elliptic Determinantal Processes and Elliptic Dyson Models |
| title_full_unstemmed |
Elliptic Determinantal Processes and Elliptic Dyson Models |
| title_sort |
elliptic determinantal processes and elliptic dyson models |
| author |
Katori, M. |
| author_facet |
Katori, M. |
| publishDate |
2017 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We introduce seven families of stochastic systems of interacting particles in one-dimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single function called the spatio-temporal correlation kernel. For the four families AN₋₁, BN, CN and DN, we identify the systems of stochastic differential equations solved by these determinantal processes, which will be regarded as the elliptic extensions of the Dyson model. Here we use the notion of martingales in probability theory and the elliptic determinant evaluations of the Macdonald denominators of irreducible reduced affine root systems given by Rosengren and Schlosser.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149273 |
| citation_txt |
Elliptic Determinantal Processes and Elliptic Dyson Models / M. Katori // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 43 назв. — англ. |
| work_keys_str_mv |
AT katorim ellipticdeterminantalprocessesandellipticdysonmodels |
| first_indexed |
2025-12-07T17:45:57Z |
| last_indexed |
2025-12-07T17:45:57Z |
| _version_ |
1850872490792845312 |