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Twisted symmetric exclusion processes and set-theoretical $R$-matrices

Mathieu Dabrowski, Loïc Poulain d'Andecy, Eric Ragoucy

Abstract

We investigate periodic integrable Markov models, constructed from set-theoretical solutions of the Yang-Baxter equation. We first focus on the simplest class of solutions, called Lyubashenko solutions. We show that the resulting models are equivalent to some twisted Symmetric Simple Exclusion Process (SSEP), which are usual periodic SSEP models where a twist is added on a bond of the ring. We also provide various possible interpretations for these Markov models. Then, we study the long time dynamics of the twisted SSEP, characterising its different stationary states and counting them. Allowing the twist to vary, we examine the possible transitions between the different stationary states. Finally, we extend our construction of Markov models to set-theoretical solutions that are more general than Lyubashenko solutions and show that such models are not equivalent to a twisted SSEP in general.

Twisted symmetric exclusion processes and set-theoretical $R$-matrices

Abstract

We investigate periodic integrable Markov models, constructed from set-theoretical solutions of the Yang-Baxter equation. We first focus on the simplest class of solutions, called Lyubashenko solutions. We show that the resulting models are equivalent to some twisted Symmetric Simple Exclusion Process (SSEP), which are usual periodic SSEP models where a twist is added on a bond of the ring. We also provide various possible interpretations for these Markov models. Then, we study the long time dynamics of the twisted SSEP, characterising its different stationary states and counting them. Allowing the twist to vary, we examine the possible transitions between the different stationary states. Finally, we extend our construction of Markov models to set-theoretical solutions that are more general than Lyubashenko solutions and show that such models are not equivalent to a twisted SSEP in general.
Paper Structure (39 sections, 6 theorems, 115 equations)

This paper contains 39 sections, 6 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Markov models from set-theoretical solutions
  3. Markov models on a periodic lattice
  4. Lyubashenko solutions of the Yang-Baxter equation
  5. Integrability of the model
  6. Interpretations of the models
  7. Symmetric exclusion process with oscillations
  8. Species with internal states
  9. Rolling polygons
  10. Filling and emptying colored boxes
  11. Twisted SSEP
  12. Definition of the model
  13. Interpretations
  14. Integrability
  15. Correspondence with set-theoretical models
  16. ...and 24 more sections

Key Result

Proposition 3.2

Let $f=g^L$. We have: where $V$ is the bijection of $\mathfrak{C}$ given by

Theorems & Definitions (38)

  • Remark 2.1
  • Remark 2.2
  • Example 2.3
  • Remark 2.4
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Definition 4.1: sector
  • ...and 28 more