Orthogonal Dualities and Asymptotics of Dynamic Stochastic Higher Spin Vertex Models, using the Drinfeld Twister

TL;DR

This paper develops an algebraic method using the Drinfeld twister to construct duality functions for dynamic stochastic higher spin vertex models, revealing their asymptotic behavior aligns with Tracy--Widom distribution.

Contribution

It introduces a universal algebraic approach to derive duality functions for dynamic integrable models, including explicit functions for higher spin vertex models.

Findings

Duality functions are expressed as $_3 \varphi_2$ functions.

Degeneration leads to orthogonal polynomial dualities.

Asymptotic fluctuations follow Tracy--Widom distribution.

Abstract

We introduce a new, algebraic method to construct duality functions for integrable dynamic models. This method will be implemented on dynamic stochastic higher spin vertex models, where we prove the duality functions are the $ _3 \varphi_2$ functions. A degeneration of these duality functions are orthogonal polynomial dualities of Groenevelt--Wagener arXiv:2306.12318. The method involves using the universal twister of $\mathcal{U}_q(\mathfrak{sl}_2)$, viewed as a quasi--triangular, quasi--$^*$--Hopf algebra. The algebraic method is presented very generally and is expected to produce duality functions for other dynamic integrable models. As an application of the duality, we prove that the asymptotic fluctuations of the dynamic stochastic six vertex model with step initial conditions are governed by the Tracy--Widom distribution.