NYC Integrable Probability Working Group

organized by Ivan Corwin

Fridays, 3:15-5:00, Room 307

Schedule of talks for 2016-2017:

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.

Abstracts

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 a class of probability distributions on the six-vertex model, which originate from the higher spin vertex models of Borodin and Petrov- https://arxiv.org/abs/1601.05770. We develop operators, which extract various correlation functions, measuring the probability of observing different arrow configurations. For the class of models we consider, the correlation functions can be expressed in terms of multiple contour integrals, which are suitable for asymptotic analysis.

For a particular choice of parameters we analyze the limit of the correlation functions. Combining this asymptotic statement with some new results about Gibbs measures on Gelfand-Tsetlin cones and patterns, we show that the asymptotic behavior of our six-vertex model near the boundary is described by the GUE-corners process.

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.

A recent paper [Kuniba-Mangazeev-Maruyama-Okado] introduced the stochastic U_q(A_n^1) vertex model. The stochastic S-matrix is related to the R-matrix of the quantum group U_q(A_n^1) by a gauge transformation. We will show that a certain function D intertwines with the transfer matrix and its space reversal. When interpreting the transfer matrix as the transition matrix of a discrete-time totally asymmetric particle system on the one-dimensional lattice Z, the function D becomes a Markov duality function. The proof here uses that the R-matrix intertwines with the co-product, but does not explicitly use the Yang-Baxter equation. We will also show that the stochastic U_q(A_n^1) is a multi-species version of a stochastic vertex model studied in [Borodin-Petrov, Corwin-Petrov].

We will explain how a certain family of symmetric functions constructed from the representation theory of the elliptic quantum group can be used to define and study a family of IRF models. This talk is based on the recent paper https://arxiv.org/abs/1701.05239 of Borodin.

The Inverse-Beta polymer, introduced by myself and Pierre Le Doussal in 2015, is an anisotropic finite temperature Bethe-ansatz exactly solvable model of directed polymer on the square lattice. The model contains two parameters and converges, in different limits, to some previously known exactly solvable models: the Log-Gamma polymer in an isotropic limit, and the Strict-Weak polymer in an anisotropic one. In the fist part of the talk I will present the main steps needed to go from the Bethe ansatz solution of the Inverse-Beta polymer to an analysis of the asymptotic fluctuations of the free-energy of the polymer. The latter are found to scale with the usual KPZ-universality class exponents and to be distributed according to the GUE Tracy-Widom distribution, in agreement with the usual KPZ-universality conjecture. This analysis is non-rigorous in many ways that I will explain but, as I will show in the second part of the talk, parts of its conclusions can be obtained using an alternative, rigorous, route. This is based on the identification of the stationary measure of the model, an identification which is another sign of the "exactly solvable character" of the model as it is non-trivially linked to the Bethe ansatz sovability. Finally if time permits I will compare the Bethe-ansatz approach to the Inverse-Beta polymer with the Bethe-ansatz approach to the Beta polymer, another exactly solvable model of directed polymer on the square lattice introduced by Guillaume Barraquand and Ivan Corwin in 2015.

May 2 (special time -- 4:30-6:30pm, room 307)

Given two independent Markov processes with configuration spaces X and Y, a duality function is an observable valued on X x Y whose expected value in one process is equal to its expected value in the other. Such stochastic dualities can be an important computational tool, especially when one of the Markov processes is easier to analyse than its dual counterpart.

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.