Lectures on Integrable probability: Stochastic vertex models and symmetric functions

Abstract

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. A key role is played by Cauchy-like summation identities for these functions, which are obtained as a direct corollary of the Yang-Baxter equation for the higher spin six vertex model. These are lecture notes for a course given by A.B. at the Ecole de Physique des Houches in July of 2015. All the results and proofs presented here generalize to the setting of the fully inhomogeneous higher spin six vertex model, see arXiv:1601.05770 for a detailed exposition of the inhomogeneous case.

arXiv comment77 pages, >30 figures. arXiv admin note: substantial text overlap with arXiv:1601.05770

References in corpus (15)

• Quantum Inverse Scattering Method and Correlation Functions
• Asymptotics in ASEP with Step Initial Condition
• Stochastic six-vertex model
• Random-walk in Beta-distributed random environment
• Algebraic Bethe ansatz for the elliptic quantum group $E_{\tau ,\eta }(s{l}_2)$
• Correlation functions for a strongly correlated boson system
• On the Yang-Baxter equation for the six-vertex model
• Spectral theory for the q-Boson particle system
• Lectures on the integrability of the 6-vertex model
• Infinite wedge and random partitions
• Probabilistic measures and algorithms arising from the Macdonald symmetric functions
• The q-PushASEP: A New Integrable Model for Traffic in 1+1 Dimension
• Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram
• Shifted Schur process and asymptotics of large random strict plane partitions
• Discrete orthogonal polynomial ensembles and the Plancherel measure

Cited by in corpus (16)

• Current Fluctuations of the Stationary ASEP and Six-Vertex Model
• The ASEP and determinantal point processes
• Point-to-line polymers and orthogonal Whittaker functions
• Generalizations of TASEP in discrete and continuous inhomogeneous space
• Between the stochastic six vertex model and Hall-Littlewood processes
• Stochastic higher spin six vertex model and Madconald measures
• Interpolation Macdonald polynomials and Cauchy-type identities
• Limiting current distribution for a two species asymmetric exclusion process
• Combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice
• Transition probability and total crossing events in the multi-species asymmetric exclusion process
• Random polymers via orthogonal Whittaker and symplectic Schur functions
• Parameter Permutation Symmetry in Particle Systems and Random Polymers
• Two-parameter Littlewood identities and half-space Yang--Baxter random fields for stable spin Hall--Littlewood symmetric functions
• On the $q$-TASEP with a random initial condition
• Inhomogeneous spin $q$-Whittaker polynomials
• The q-Hahn PushTASEP