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Orthogonal polynomial duality of a two-species asymmetric exclusion process

Danyil Blyschak, Olivia Burke, Jeffrey Kuan, Dennis Li, Sasha Ustilovsky, Zhengye Zhou

Abstract

We examine type D ASEP, a two--species interacting particle system which generalizes the usual asymmetric simple exclusion process. For certain cases of type D ASEP, the process does not give priority for one species over another, even though there is nontrivial interaction between the two species. For those specific cases, we prove that the type D ASEP is self--dual with respect to an independent product of $q$--Krawtchouk polynomials. The type D ASEP was originally constructed in arXiv:2011.13473, using the type D quantum groups $\mathcal{U}_q(\mathfrak{so}_6)$ and $\mathcal{U}_q(\mathfrak{so}_8)$. That paper claimed that certain states needed to be "discarded'' in order to ensure non--negativity. Here, we also provide a more efficient argument for the same claim.

Orthogonal polynomial duality of a two-species asymmetric exclusion process

Abstract

We examine type D ASEP, a two--species interacting particle system which generalizes the usual asymmetric simple exclusion process. For certain cases of type D ASEP, the process does not give priority for one species over another, even though there is nontrivial interaction between the two species. For those specific cases, we prove that the type D ASEP is self--dual with respect to an independent product of --Krawtchouk polynomials. The type D ASEP was originally constructed in arXiv:2011.13473, using the type D quantum groups and . That paper claimed that certain states needed to be "discarded'' in order to ensure non--negativity. Here, we also provide a more efficient argument for the same claim.
Paper Structure (15 sections, 4 theorems, 65 equations, 1 figure)

This paper contains 15 sections, 4 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary Definitions
  3. Definition of the type $D$ ASEP
  4. Duality
  5. $q$--deformed notation
  6. Algebraic Definitions
  7. The Lie Algebra $\mathfrak{so}_{2 m}$
  8. The quantized enveloping algebra $\mathcal{U}_{q}\left(\mathfrak{s o}_{2 m}\right)$
  9. Coproduct structure of $\mathcal{U}_{q}\left(\mathfrak{s o}_{2 m}\right)$
  10. Representation theory of $\mathcal{U}_{q}\left(\mathfrak{s o}_{2 m}\right)$
  11. $*-$structure on $\mathcal{U}_{q}\left(\mathfrak{s o}_{2 m}\right)$
  12. Results
  13. Orthogonal Polynomial Duality
  14. Conjecture about Asymptotics
  15. A more efficient approach than non--negativity

Key Result

Lemma 2.2

The quantum groups $U_{q}\left(\mathfrak{s o}_{2 m}\right)$ is a Hopf $*$-algebra with $*:U_{q}\left(\mathfrak{s o}_{2 m}\right) \rightarrow U_{q}\left(\mathfrak{s o}_{2 m}\right)$ defined as follows:

Figures (1)

  • Figure :

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 3
  • Theorem 3.1
  • Remark 4
  • proof
  • Theorem 3.2
  • ...and 3 more