Probability laws associated to the quadrirational Yang-Baxter maps -- the ultimate case

TL;DR

The paper addresses the problem of uniquely characterizing independence-preserving (IP) maps in the quadrirational Yang–Baxter hierarchy by focusing on the top map F^{(α,β)} acting on independent GB_{II} random variables. It develops hypergeometric-type Laplace transforms L_W^{(γ)} and uses them to translate IP into transform-identities, enabling a rigorous converse result: if U,V=F^{(α,β)}(X,Y) are independent, then X,Y must follow GB_{II} distributions with parameters (λ,a,b;α) and (−λ,a,b;β), and (U,V) follow GB_{II}(−λ,a,b;α) and GB_{II}(λ,a,b;β). The analysis extends to boundary cases β=∞ and β=0, linking the IP property to Matsumoto– Yor-type independence results and via reciprocal mappings to GB_{I}/B_{I} distributions, thereby situating the uniqueness result within a broader probabilistic framework. The authors perform a detailed, technically intricate proof that leverages difference equations, M-functions, and a hypergeometric recursion to identify the exact parameter relationships, concluding the ultimate missing case in the IP hierarchy. This work completes the IP-characterization for the quadri- rational Yang–Baxter maps and provides a robust transform toolkit that may aid future characterizations of independent-structured probabilistic models with hypergeometric-type laws.

Abstract

Recently, Sasada and Uozumi (2024) investigated connections between classical (deterministic) and random integrable models, discovering a hierarchy of quadrirational Yang-Baxter independence preserving (IP) maps together with related families of probability distributions. In view of the limiting properties of these IP maps, the newly defined generalized second kind beta ($\mathrm{GB}_{II}$) model stands at the top of the hierarchy: for independent random variables $X$ and $Y$ following a $\mathrm{GB}_{II}$ distribution, Sasada and Uozumi (2024) showed that when a special quadrirational Yang-Baxter map $F^{(α,β)}$, parameterized by two distinct parameters $α,β\in(0,\infty)$, is applied to the pair $(X,Y)$, it produces another pair $(U,V)$ of independent $\mathrm{GB}_{II}$-distributed random variables. Interestingly, the boundary cases of $α\in\{0,\infty\}$ or $β\in\{0,\infty\}$ are related to one of the Matsumoto-Yor IP maps identified in Koudou and Vallois (2012). The aim of this paper is to show the uniqueness of this IP model. To this end, we introduce specially designed Laplace-type transforms. First, we carefully explain the connection between the results from Sasada and Uozumi (2024) and Koudou and Vallois (2012). Next, we focus on the characterization of second kind beta and the generalized second kind beta distributions through the IP map $F^{(α,\infty)}$. Finally, extending considerably the methodology developed for the case $(α,\infty)$, we prove the characterization of $\mathrm{GB}_{II}$ distributions in the case $(α,β)\in(0,\infty)^2$ with $α\neqβ$, which implies uniqueness in the ultimate missing case of the quadrirational Yang-Baxter hierarchy of IP models.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• proof : Proof of Theorem \ref{['SU22']}
• Theorem 3.1
• proof : Proof of Theorem \ref{['infin']}
• Theorem 3.2
• proof : Proof of Theorem \ref{['charac1']}
• Theorem 3.3: KouVal2012
• Remark 3.1
• Remark 3.2