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We introduce intertwining relations for integrable stochastic particle systems, generalizing previous work. This approach uses the Yang-Baxter equation to unify the study of multipoint observables for the Totally Asymmetric Simple Exclusion Process (TASEP) and its q-deformation (q-TASEP).

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Area of Science:

  • Probability Theory
  • Statistical Mechanics
  • Mathematical Physics

Background:

  • Integrable stochastic particle systems, like the Totally Asymmetric Simple Exclusion Process (TASEP) and its q-deformation (q-TASEP), are fundamental models in statistical mechanics.
  • These systems retain their integrability when individual particle speed parameters are introduced.
  • Previous research established intertwining relations for specific particle systems.

Purpose of the Study:

  • To generalize intertwining relations between Markov transition operators of particle systems with permuted speed parameters.
  • To develop a novel approach using the Yang-Baxter equation for the higher spin stochastic six vertex model.
  • To explore probabilistic consequences, including new differential equations and trajectory couplings.

Main Methods:

  • Application of the Yang-Baxter equation to the higher spin stochastic six vertex model.
  • Construction of intertwining relations as Markov transition operators.
  • Derivation of a Lax-type differential equation for continuous-time Markov transition semigroups.
  • Development of couplings between probability measures on particle system trajectories.

Main Results:

  • A new Lax-type differential equation unifying the time evolution of multipoint observables for q-TASEP and TASEP.
  • Intertwining relations leading to couplings between probability measures of systems with permuted speed parameters.
  • A 'rewriting history' random walk for resampling particle trajectories within a defined chamber.
  • A novel coupling for standard Poisson processes with different rates as a byproduct.

Conclusions:

  • The novel approach provides a unified framework for studying integrable stochastic particle systems with varying speeds.
  • The derived Lax equation offers insights into the asymptotic analysis of multipoint observables.
  • The developed couplings and random walk mechanisms provide new tools for analyzing particle system dynamics and related stochastic processes.