Ultra-Discretization of Yang-Baxter Maps, Probability Distributions and Independence Preserving Property
Hiroki Kondo, Sachiko Nakajima, Makiko Sasada
TL;DR
This work develops a rigorous bridge between Yang–Baxter maps and the independence preserving property via ultra-discretization. It introduces an ultra-discretization framework for both rational maps and probability distributions, proving that the Yang–Baxter property and IP property are preserved in the tropical limit, and applies these results to quadrirational Yang–Baxter maps to produce a large family of piecewise-linear, IP-preserving maps. The authors explicitly tropicalize several probability distributions (notably gamma, Be', K, and GIG families) and show how these yield shifted/exponential-type distributions under the IP framework, enabling systematic construction of IP quadrirational maps in the ultra-discrete setting. These results offer new avenues for understanding stationary distributions in zero-temperature or tropicalized integrable systems and raise open questions about deeper connections between YB structures and IP properties, as well as extensions to higher dimensions and tropical geometry. The work thus provides a foundational toolkit for exploring IP in ultra-discretized integrable dynamics and their probabilistic implications.
Abstract
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.