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

Carel Wagenaar

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.

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

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 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.

Paper Structure

This paper contains 31 sections, 27 theorems, 303 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Outlook
  3. Outline of the paper
  4. Preliminaries and notations
  5. Dynamic ASIP
  6. Definition
  7. Duality dynamic ASIP with ASIP
  8. Duality $\textnormal{ASIP}_{\mathsf{L}}$ and $\textnormal{ASIP}_{\mathsf{R}}$
  9. Reversibility dynamic ASIP
  10. Dynamic SIP
  11. Limits of (orthogonal) dualities for discrete asymmetric processes.
  12. Duality functions which follow from Theorem \ref{['thm:AWduality']}
  13. Orthogonality relations which follow from Theorem \ref{['thm:AWdualityOrth']}
  14. Dynamic ABEP
  15. Definition dynamic ABEP
  16. ...and 16 more sections

Key Result

Theorem 2.6

Define where the factor $u(\eta,\vec{k})$ can be found in eq:defu. Then $P^{}_\mathsf{R}(\eta,\xi)$ is a duality function between $\textnormal{ASIP}(q,\vec{k})$ and $\textnormal{ASIP}_{\mathsf{R}}(q,\vec{k},\rho)$, i.e.

Theorems & Definitions (81)

  • Remark 1.1
  • Remark 2.1
  • Definition 2.2: Dynamic ASIP for $\lambda,\rho > -1$
  • Remark 2.3
  • Definition 2.4: Dynamic ASIP for $\lambda,\rho \leq -1$
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Remark 2.7
  • Corollary 2.8
  • ...and 71 more