Integrable stochastic particle systems in one space dimension, like the Totally Asymmetric Simple Exclusion process (TASEP), have been studied for over 50 years (introduced simultaneously in biology and mathematics in 1969-70). They strike a balance between being simple enough to be mathematically tractable and complicated enough to describe many interesting phenomena. Many natural questions about these systems can be generalized by introducing multiple parameters, and interpreting the time evolution of a particle system as a vertex model transfer matrix. I will discuss applications of vertex models and their associated Yang-Baxter equations to stochastic systems.
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