Yang-Baxter maps and independence preserving property

[2212.00963] Yang-Baxter maps and independence preserving property Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Nonlinear Sciences > Exactly Solvable and Integrable Systems arXiv:2212.00963 (nlin) [Submitted on 2 Dec 2022 ( v1 ), last revised 15 Dec 2022 (this version, v2)] Title: Yang-Baxter maps and independence preserving property Authors: Makiko Sasada , Ryosuke Uozumi View a PDF of the paper titled Yang-Baxter maps and independence preserving property, by Makiko Sasada and Ryosuke Uozumi View PDF HTML (experimental) Abstract: We study a surprising relationship between two properties for bijective functions $F : \mathcal{X} \times \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ for a set $\mathcal{X}$ which are introduced from very different backgrounds. One of the property is that $F$ is a Yang-Baxter map, namely it satisfies the "set-theoretical" Yang-Baxter equation, and the other property is the independence preserving property (IP property for short), which means that there exist independent (non-constant) $\mathcal{X}$-valued random variables $X,Y$ such that $U,V$ are also independent with $(U,V)=F(X,Y)$. Recently in the study of invariant measures for a discrete integrable system, a class of functions having these two properties were found. Motivated by this, we analyze a relationship between the Yang-Baxter maps and the IP property, which has never been studied as far as we are aware, focusing on the case $\mathcal{X}=\mathbb{R}_+$. Our first main result is that all quadrirational Yang-Baxter maps $F : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+ \times \mathbb{R}_+$ in the most interesting subclass have the independence preserving property. In particular, we find new classes of bijections having the IP property. Our second main result is that these newly introduce bijections are fundamental in the class of (known) bijections with the IP property, in the sense that most of known bijections having the IP property are derived from these maps by taking special parameters or performing some limiting procedure. This reveals that the IP property, which has been investigated for specific functions individually, can be understood in a unified manner. Comments: 19 pages Subjects: Exactly Solvable and Integrable Systems (nlin.SI) ; Mathematical Physics (math-ph); Probability (math.PR); Quantum Algebra (math.QA) Cite as: arXiv:2212.00963 [nlin.SI] (or arXiv:2212.00963v2 [nlin.SI] for this version) https://doi.org/10.48550/arXiv.2212.00963 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Electron. J. Probab. 29: 1-21 (2024) Related DOI : https://doi.org/10.1214/24-EJP1107 Focus to learn more DOI(s) linking to related resources Submission history From: Makiko Sasada [ view email ] [v1] Fri, 2 Dec 2022 04:37:08 UTC (20 KB) [v2] Thu, 15 Dec 2022 03:17:09 UTC (19 KB) Full-text links: Access Paper: View a PDF of the paper titled Yang-Baxter maps and independence preserving property, by Makiko Sasada and Ryosuke Uozumi View PDF HTML (experimental) TeX Source view license Current browse context: nlin.SI < prev | next > new | recent | 2022-12 Change to browse by: math math-ph math.MP math.PR math.QA nlin References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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