NYC Integrable Probability Working Group
organized by Ivan Corwin
Fridays, 3:15-5:00, Room 307
|Sept 23||Guillaume Barraquand (Columbia)||Stochastic spin models and symmetric functions (after Borodin-Petrov)|
|Oct 7||Guillaume Barraquand (Columbia)||Stochastic spin models and symmetric functions (after Borodin-Petrov)|
|Oct 28||Evgeni Dimitrov (MIT)||GUE corners process in the six vertex model|
|Dec 5, 7 and 9||Nicolai Reshetikhin||(Minvera Lecutres) Limit shapes in integrable models in statistical mechanics|
|Dec 16||Zhipeng Liu (Courant)||Fluctuations of TASEP on a ring in relaxation time scale|
|Jan 20||Jeff Kuan (Columbia)||An algebraic construction of duality functions for the stochastic Uq(An^1) vertex model|
|Feb 3||Yi Sun (Columbia)||IRF models and elliptic symmetric functions (after Borodin)|
|Feb 10||Thimothee Thiery (Leuven)||The inverse-Beta polymer|
|Mar 3||Yi Sun (Columbia)||IRF models and elliptic symmetric functions (after Borodin)|
|May 2||Michael Wheeler (Melbourne)||Duality in multi-species ASEP and the qKZ equation (Special day of week and time 4:30-6:30pm)|
(10am-12pm in room 622)
|Alisa Knizel (MIT)||Gap probabilities for discrete orthogonal polynomial ensembles|
A note about the working group: the purpose of this group is to discuss recent papers related to integrable probability. If you would like to present some work (your own, or that of someone else), please contact Ivan Corwin. Blackboard talks are highly recommended.
September 23 and October 7
We will explain how certain properties of a family of symmetric rational functions -- Cauchy summation formula, orthogonality relation...-- can be used to study stochastic models of spins on the square lattice. This talk is based on the paper "Lectures on integrable probability: Stochastic six vertex models and symmetric functions" https://arxiv.org/abs/1605.01349 by Borodin and Petrov (see also https://arxiv.org/abs/1410.0976, https://arxiv.org/abs/1601.05770).
We consider the totally asymmetric simple exclusion process on a ring with flat and step initial conditions. We assume that the size of the ring and the number of particles tend to infinity proportionally and evaluate the fluctuations of tagged particles and currents as the time tends to infinity. The crossover from the KPZ dynamics to the equilibrium dynamics occurs when the time is proportional to the 3/2 power of the ring size. We compute the limiting distributions in this relaxation time scale. The analysis is based on an explicit formula of the finite-time one-point distribution obtained from the coordinate Bethe ansatz method. See https://arxiv.org/abs/1605.07102.
This talk will focus on the particular example in which both systems are asymmetric simple exclusion processes (ASEPs). I will explain some recently developed techniques which enable the construction of rich classes of duality functions within the ASEP. These techniques draw from the theory of non-symmetric Macdonald polynomials, the quantum Knizhnik--Zamolodchikov (qKZ) equation, and integrable lattice models.
I will talk about a way to explicitly compute gap probabilities for discrete orthogonal polynomial ensembles using discrete Riemann-Hilbert problems. The talk will be based onthe following papers: “Distribution of the first particle in discrete orthogonal polynomial ensembles,” A. Borodin, D. Boyarchenko, https://arxiv.org/pdf/math-ph/0204001.pdf; “Moduli spaces of d-connections and difference Painleve equations,” A. Borodin, D. Arinkin, https://arxiv.org/pdf/math/0411584.pdf; “Moduli spaces of q-connections and gap probabilities,” A. Knizel, https://arxiv.org/pdf/1506.06718.pdf.