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

TL;DR

This paper explores integrable Markov models derived from set-theoretical solutions of the Yang-Baxter equation, focusing on twisted symmetric exclusion processes and their stationary states, with extensions beyond Lyubashenko solutions.

Contribution

It establishes a connection between Lyubashenko solutions and twisted SSEP models, analyzes their long-term dynamics, and extends the framework to more general solutions.

Findings

Twisted SSEP models are equivalent to certain set-theoretical solutions.

Stationary states of twisted SSEP are characterized and counted.

Extensions to more general solutions do not always correspond to twisted SSEP.

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.