Dynamic generalizations of the Asymmetric Inclusion Process, Asymmetric Brownian Energy Process and their Dualities

TL;DR

This paper introduces dynamic variants of the asymmetric inclusion process and asymmetric Brownian energy process, establishing their dualities and reversibility using quantum algebra and orthogonal polynomials, advancing the understanding of interacting particle systems.

Contribution

It presents novel dynamic versions of ASIP and ABEP, along with their duality functions and reversibility properties, utilizing quantum algebra and special functions.

Findings

Duality functions based on Askey-Wilson and Jacobi polynomials identified.

Reversibility of the new processes proven using quantum algebra.

Hierarchies of duality functions established.

Abstract

Two new interacting particle systems are introduced in this paper: dynamic versions of the asymmetric inclusion process (ASIP) and the asymmetric Brownian energy process (ABEP). Dualities and reversibility of these processes are proven, where the quantum algebra $\mathcal{U}_q(\mathfrak{su}(1,1))$ and the Al-Salam--Chihara polynomials play a crucial role. Two hierarchies of duality functions are found, where the Askey-Wilson polynomials and Jacobi polynomials sit on top.