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We consider a homogeneous stochastic higher spin six vertex model in aquadrant. For this model we derive concise integral representations formulti-point q-moments of the height function and for the q-correlationfunctions. At least in the case of the step initial condition, our formulasdegenerate in appropriate limits to many known formulas of such type forintegrable probabilistic systems in the (1+1)d KPZ universality class,including the stochastic six vertex model, ASEP, various q-TASEPs, andassociated zero range processes.  Our arguments are largely based on properties of a family of symmetricrational functions (introduced in arXiv:1410.0976) that can be defined aspartition functions of the higher spin six vertex model for suitable domains;they generalize classical Hall-Littlewood and Schur polynomials. A key role isplayed by Cauchy-like summation identities for these functions, which areobtained as a direct corollary of the Yang-Baxter equation for the higher spinsix vertex model.  These are lecture notes for a course given by A.B. at the Ecole de Physiquedes Houches in July of 2015. All the results and proofs presented heregeneralize to the setting of the fully inhomogeneous higher spin six vertexmodel, see arXiv:1601.05770 for a detailed exposition of the inhomogeneouscase.

Lectures on Integrable probability: Stochastic vertex models and symmetric functions