Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915552116604928 |
|---|---|
| author | Kondo, Hiroki Nakajima, Sachiko Sasada, Makiko |
| author_facet | Kondo, Hiroki Nakajima, Sachiko Sasada, Makiko |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21359 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property Kondo, Hiroki Nakajima, Sachiko Sasada, Makiko Exactly Solvable and Integrable Systems Mathematical Physics Probability Quantum Algebra 60E05, 62E10, 37K60, 16T25 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. |
| title | Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Probability Quantum Algebra 60E05, 62E10, 37K60, 16T25 |
| url | https://arxiv.org/abs/2504.21359 |