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


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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Limit shapes in the discrete model interpolating out from the geometric corner growth

Generalizations of TASEP in discrete and continuous inhomogeneous space


We investigate a rich new class of exactly solvable particle systems generalizing the Totally Asymmetric Simple Exclusion Process (TASEP). Our particle systems evolve in discrete or continuous space and can be thought of as new exactly solvable examples of tandem queues, directed first- or last-passage percolation models, or Robinson-Schensted-Knuth type systems with random input. One of the features of the particle systems we consider is the presence of spatial inhomogeneity which can lead to the formation of traffic jams.

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Front of percolation

MATH 3100 • Introduction to Probability (2 sections)

[Fall 2018 semester]