Lectures on Integrable probability: Stochastic vertex models and symmetric functions

Understand

We consider a homogeneous stochastic higher spin six vertex model in a quadrant.

• For this model we derive concise integral representations for multi-point q-moments of the height function and for the q-correlation functions.
• At least in the case of the step initial condition, our formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six vertex model, ASEP, various q-TASEPs, and associated zero range processes.
• Our arguments are largely based on properties of a family of symmetric rational functions (introduced in arXiv:1410.0976) that can be defined as partition functions of the higher spin six vertex model for suitable domains; they generalize classical Hall-Littlewood and Schur polynomials.