Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property
We study the relationship between Yang-Baxter maps and the independence preserving (IP) property, motivated by their role in integrable systems, from the perspective of ultra-discretization. Yang-Baxter maps satisfy the set-theoretic Yang-Baxter equation, while the IP property ensures independence o...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2025 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2025
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/214200 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property. Hiroki Kondo, Sachiko Nakajima and Makiko Sasada. SIGMA 21 (2025), 084, 16 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862673638807830528 |
|---|---|
| author | Kondo, Hiroki Nakajima, Sachiko Sasada, Makiko |
| author_facet | Kondo, Hiroki Nakajima, Sachiko Sasada, Makiko |
| citation_txt | Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property. Hiroki Kondo, Sachiko Nakajima and Makiko Sasada. SIGMA 21 (2025), 084, 16 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study the relationship between Yang-Baxter maps and the independence preserving (IP) property, motivated by their role in integrable systems, from the perspective of ultra-discretization. Yang-Baxter maps satisfy the set-theoretic Yang-Baxter equation, while the IP property ensures independence of transformed random variables. The relationship between these two seemingly unrelated properties has recently started to be studied by Sasada and Uozumi (2024). Ultra-discretization is a concept primarily used in the context of integrable systems and is an area of active research, serving as a method for exploring the connections between different integrable systems. However, there are few studies on how the stationary distribution for integrable systems changes through ultra-discretization. In this paper, we introduce the concept of ultra-discretization for probability distributions and prove that the properties of being a Yang-Baxter map and having the IP property are both preserved under ultra-discretization. Applying this to quadrirational Yang-Baxter maps, we confirm that their ultra-discrete versions retain these properties, yielding new examples of piecewise linear maps having the IP property. We also explore implications of our results for stationary distributions of integrable systems and pose several open questions.
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| first_indexed | 2026-03-16T18:56:41Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-214200 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T18:56:41Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kondo, Hiroki Nakajima, Sachiko Sasada, Makiko 2026-02-20T08:02:04Z 2025 Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property. Hiroki Kondo, Sachiko Nakajima and Makiko Sasada. SIGMA 21 (2025), 084, 16 pages 1815-0659 2020 Mathematics Subject Classification: 60E05; 62E10; 37K60; 16T25 arXiv:2504.21359 https://nasplib.isofts.kiev.ua/handle/123456789/214200 https://doi.org/10.3842/SIGMA.2025.084 We study the relationship between Yang-Baxter maps and the independence preserving (IP) property, motivated by their role in integrable systems, from the perspective of ultra-discretization. Yang-Baxter maps satisfy the set-theoretic Yang-Baxter equation, while the IP property ensures independence of transformed random variables. The relationship between these two seemingly unrelated properties has recently started to be studied by Sasada and Uozumi (2024). Ultra-discretization is a concept primarily used in the context of integrable systems and is an area of active research, serving as a method for exploring the connections between different integrable systems. However, there are few studies on how the stationary distribution for integrable systems changes through ultra-discretization. In this paper, we introduce the concept of ultra-discretization for probability distributions and prove that the properties of being a Yang-Baxter map and having the IP property are both preserved under ultra-discretization. Applying this to quadrirational Yang-Baxter maps, we confirm that their ultra-discrete versions retain these properties, yielding new examples of piecewise linear maps having the IP property. We also explore implications of our results for stationary distributions of integrable systems and pose several open questions. The authors would like to thank the anonymous referees for their valuable comments and suggestions, which helped to improve the quality and clarity of this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property Article published earlier |
| spellingShingle | Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property Kondo, Hiroki Nakajima, Sachiko Sasada, Makiko |
| title | Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property |
| title_full | Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property |
| title_fullStr | Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property |
| title_full_unstemmed | Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property |
| title_short | Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property |
| title_sort | ultra-discretization of yang-baxter maps, probability distributions and independence preserving property |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214200 |
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