Yang-Baxter equation and its bijectivisation

Yang-Baxter random fields and stochastic vertex models

[2019/05/15]

Bijectivization refines the Yang-Baxter equation into a pair of local Markov moves which randomly update the configuration of the vertex model. Employing this approach, we introduce new Yang-Baxter random fields of Young diagrams based on spin $q$-Whittaker and spin Hall-Littlewood symmetric functions. We match certain scalar Markovian marginals of these fields with (1) the stochastic six vertex model; (2) the stochastic higher spin six vertex model; and (3) a new vertex model with pushing which generalizes the $q$-Hahn PushTASEP introduced recently by Corwin-Matveev-Petrov (2018). Our matchings include models with two-sided stationary initial data, and we obtain Fredholm determinantal expressions for the $q$-Laplace transforms of the height functions of all these models. Moreover, we also discover difference operators acting diagonally on spin $q$-Whittaker or (stable) spin Hall-Littlewood symmetric functions.

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A simulation of the square ice, due to Shreyas Balaji (MIT UROP)

From infinite random matrices over finite fields to square ice

Asymptotic representation theory of symmetric groups is a rich and beautiful subject with deep connections with probability, mathematical physics, and algebraic combinatorics. A one-parameter deformation of this theory related to infinite random matrices over a finite field leads to a randomization of the classical Robinson-Schensted correspondence between words and Young tableaux. Exploring such randomizations we find unexpected applications to six vertex (square ice) type models and traffic systems on a 1-dimensional lattice.

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Chebyshev polynomials

MATH 7310 • Real Analysis and Linear Spaces I

[Spring 2019 semester]
An illustration of the Gibbs property

Gibbs measures, arctic curves, and random interfaces

This talk outlines connections between 2-dimensional Gibbs measures with a height function and particle systems in the Kardar-Parisi-Zhang universality class.

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The q-Hahn PushTASEP

The q-Hahn PushTASEP

[2018/11/15]

We introduce the $q$-Hahn PushTASEP — an integrable stochastic interacting particle system which is a 3-parameter generalization of the PushTASEP, a well-known close relative of the TASEP (Totally Asymmetric Simple Exclusion Process). The transition probabilities in the $q$-Hahn PushTASEP are expressed through the $_4\phi_3$ basic hypergeometric function. Under suitable limits, the $q$-Hahn PushTASEP degenerates to all known integrable (1+1)-dimensional stochastic systems with a pushing mechanism. One can thus view our new system as a pushing counterpart of the $q$-Hahn TASEP introduced by Povolotsky. We establish Markov duality relations and contour integral formulas for the $q$-Hahn PushTASEP. We also take a $q\to1$ limit of our process arriving at a new beta polymer-like model.

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Simulation of the six vertex model with domain wall boundary conditions and gaseous phase (simulation due to Shreyas Balaji)

Virginia Integrable Probability Summer School

Virginia Integrable Probability Summer School will be held at University of Virginia from May 27 to June 8, 2019

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