Rewriting History in Integrable Stochastic Particle Systems.

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.