Yang-Baxter maps and independence preserving property

Abstract

We raise a question of whether there might be deep mathematical connections 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, and study the supporting evidence that suggests this. 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, 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 independence preserving property, which has never been studied as far as we are aware. We focus 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 an important subclass have the independence preserving property. In particular, we found new classes of bijections having the independence preserving property. Our second main result is that these newly introduced bijections are fundamental in the class of (known) bijections with the independence preserving property, in the sense that many known bijections having the independence preserving property are derived from these maps by taking special parameters or performing some limiting procedure. This reveals that the independence preserving property, which has been investigated for specific functions individually, can be understood in a more unified manner.