Yang-Baxter maps and independence preserving property

TL;DR

This paper explores the relationship between Yang-Baxter maps and the independence preserving property (IP), revealing that many such maps inherently possess the IP property and can generate all known IP bijections.

Contribution

It demonstrates that quadrirational Yang-Baxter maps on positive real numbers have the IP property and serve as fundamental building blocks for all known IP bijections.

Findings

All quadrirational Yang-Baxter maps in the studied subclass have the IP property.

New classes of bijections with the IP property are identified.

Most known IP bijections are derived from these Yang-Baxter maps through parameter choices or limits.

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.