A Generalized Dynamic Asymmetric Exclusion Process: Orthogonal Dualities and Degenerations

Abstract

In this paper, a generalized version of dynamic ASEP is introduced, and it is shown that the process has a Markov duality property with the same process on the reversed lattice. The duality functions are multivariate $q$-Racah polynomials, and the corresponding orthogonality measure is the reversible measure of the process. By taking limits in the generator of dynamic ASEP, its reversible measure, and the duality functions, we obtain orthogonal and triangular dualities for several other interacting particle systems. In this sense, the duality of dynamic ASEP sits on top of a hierarchy of many dualities. For the construction of the process, we rely on representation theory of the quantum algebra ${\mathcal{U}}_q({\mathfrak{sl}}_2)$. In the standard representation, the generator of generalized ASEP can be constructed from the coproduct of the Casimir. After a suitable change of representation, we obtain the generator of dynamic ASEP. The corresponding intertwiner is constructed from $q$-Krawtchouk polynomials, which arise as eigenfunctions of twisted primitive elements. This gives a duality between dynamic ASEP and generalized ASEP with $q$-Krawtchouk polynomials as duality functions. Using this duality, we show the (almost) self-duality of dynamic ASEP.

References in corpus (19)

• Asymmetric Simple Exclusion Process with Open Boundaries and Askey-Wilson Polynomials
• From duality to determinants for q-TASEP and ASEP
• Factorized duality, stationary product measures and generating functions
• Asymmetric stochastic transport models with ${\mathcal{U}}_q(\mathfrak{su}(1,1))$ symmetry
• An algebraic construction of duality functions for the stochastic U_q(A_n^{(1)}) vertex model and its degenerations
• A Multi-species ASEP(q,j) and q-TAZRP with Stochastic Duality
• Stationary uphill currents in locally perturbed Zero Range Processes
• Orthogonal stochastic duality functions from Lie algebra representations
• Orthogonal Dualities of Markov Processes and Unitary Symmetries
• Self-duality of Markov processes and intertwining functions
• A reverse duality for the ASEP with open boundaries
• On dualities for SSEP and ASEP with open boundary conditions
• Quantitative Boltzmann Gibbs principles via orthogonal polynomial duality
• Markov duality and Bethe ansatz formula for half-line open ASEP
• A quantum algebra approach to multivariate Askey-Wilson polynomials
• Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP$(2j)$ and Multi-Species IRW
• An Askey-Wilson Algebra of Rank 2
• Orthogonal polynomial duality of a two-species asymmetric exclusion process
• Stochastic PDE limit of the dynamic ASEP