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

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 $F^{(\alpha,\beta)}$, parameterized by two distinct parameters $\alpha,\beta\in(0,\infty)$, 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 $\alpha\in\{0,\infty\}$ or $\beta\in\{0,\infty\}$ 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 $F^{(\alpha,\infty)}$. Finally, extending considerably the methodology developed for the case $(\alpha,\infty)$, we prove the characterization of $\mathrm{GB}_{II}$ distributions in the case $(\alpha,\beta)\in(0,\infty)^2$ with $\alpha\neq\beta$, which implies uniqueness in the ultimate missing case of the quadrirational Yang-Baxter hierarchy of IP models.