From Macdonald Processes to
Hecke Algebras and Quantum Integrable Systems

Institut Henri Poincaré, May 26-30, 2014

Recently, it has become clear that there is much that can be learned from increased interactions between people working on integrability from a probabilistic perspective, those working from a more algebraic perspective, and those working from a mathematical physics perspective. It particular, it seems that there exist many connections between the theories of Macdonald processes, double affine Hecke algebras, and the algebraic / coordinate Bethe ansatz which are not understood.

The purpose of this workshop is to bring these three groups together with the aim of sharing methods, fostering collaborations and ultimately understanding these connections.

This workshop is jointly funded by the Clay Mathematics Institute, the Institut Henri Poincaré, the Poincaré Chair.

Invited participants
  • Natan Andrei (Rutgers)
  • Alexei Borodin (MIT)
  • Victor Dotsenko (Paris)
  • Erdal Emsiz (Santiago)
  • Vadim Gorin (MIT)
  • Christian Korff (Glasgow)
  • Pierre Le Doussal (Paris)
  • Eric Opdam (Amsterdam)
  • Leonid Petrov (Northeastern)
  • Alexander Povolotsky (Moscow)
  • Jeremy Quastel (Toronto)
  • Tomohiro Sasamoto (Tokyo)
  • Gunter Schutz (Julich)
  • Timo Seppalainen (Wisconsin)
  • Yoshihiro Takeyama (Tsukuba)
  • Craig Tracy (Davis)
  • Jan Felipe Van Diejen (Talca)
  • Nikos Zygouras (Warwick)


Monday May 26, 2014

Alexei Borodin [From Macdonald processes to Hecke algebras and quantum integrable systems]
Alexander Povolotsky [Integrable particle models with factorized steady states]
Leonid Petrov [Spectral theory for integrable interacting particle systems]
Vadim Gorin [Random matrix asymptotics for the six-vertex model]

Tuesday May 27, 2014

Gunter Schutz [Microscopic structure of the ASEP conditioned on an atypical current]
Jan Felipe Van Diejen [Integrable boundary interactions for Ruijsenaars' difference Toda chain]
Erdal Emsiz [Discrete harmonic analysis on a Weyl alcove and the double affine Hecke algebra at critical level]
Short talks:
Yi Sun [A new integral formula for the Heckman-Opdam hypergeometric functions];
Alexei Bufetov [Asymptotics of representations of classical Lie groups];
Jeff Kuan ["Three-dimensional Gaussian fluctuations of noncommutative random surfaces]

Wednesday May 28, 2014

Yoshihiro Takeyama [A deformation of affine Hecke algebra and integrable stochastic particle system]
Christian Korff [The Bethe ansatz in quantum Schubert calculus]
Eric Opdam [Dirac induction and Hecke algebra modules]
Ivan Corwin [Beyond the Gaussian universality class]
(Location: Séminaire Des Mathématiques, ENS)

Thursday May 29, 2014

Pierre Le Doussal [Exact results for the Kardar-Parisi-Zhang equation and directed path models, from replica Bethe ansatz and sine-Gordon quantum field theory]
Natan Andrei [Quench dynamics of quantum interacting many body systems in 1-d]
Nikos Zygouras [Tropical combinatorics, Whittaker functions and random polymers.]

Friday May 30, 2014

Open discussion

Practical Information

The talks will take place at Institut Henri Poincaré in the Amphitheater Darboux from May 26 - 29, 2014. Space is limited, so please contact Ivan Corwin to inquire as to remaining space availability.

There is some funding for junior researchers who wish to attend. To apply for funding contact Ivan Corwin with a brief description of your research interests and arrange a short recommendation letter (from a senior researcher) to be sent as well.

Directions to Institut Henri Poincaré. Event poster.

For further information, please contact the organizer Ivan Corwin (Clay Mathematics Institute, Columbia University, Institut Henri Poincaré, Massachusetts Institute of Technology).

Titles and Abstracts

Alexei Borodin
From Macdonald processes to Hecke algebras and quantum integrable systems

Abstract: The goal of the talk is to offer a brief informal introduction into some of the subjects represented at the workshop with an emphasis on connections between them.

Alexander Povolotsky
Integrable particle models with factorized steady states

Abstract: Interacting particle models serve as a testing ground for the low-dimensional non-equilibrium statistical physics. Many of them share two important features, which make them amenable to exact analytic description. First one is the simplest structure of traslationally invariant stationary measure that has a form of product of one-site factors. This allows for complete characterization of the stationary correlations by the use of the toolbox of equilibrium statistical mechanics, developed for systems with noninteracting degrees of freedom. The second one, which makes the full dynamical problem also solvable, is the matrix of transition probabilities of the Markov process being the transfer-matrix of some quantum integrable system that, in particular, can be diagonalized by the Bethe anstaz. In the talk we show how to obtain very general interacting particle model, which possesses both the product stationary measure and the Bethe anstaz solvability. Then we give a brief review of its particular limiting cases interesting for physicists and mathematicians, some of which were studied before and some are new.

Leonid Petrov
Spectral theory for integrable interacting particle systems

Abstract: I will present Plancherel isomorphism theorems for Bethe ansatz eigenfunctions of the q-Hahn Boson particle system introduced by Povolotsky in 2013. This is the most general zero-range stochastic particle system with product-form steady state which is solvable by the coordinate Bethe ansatz and depends on three parameters q,mu and nu. When nu=0, the q-Hahn Boson system becomes the q-Boson system, and a similar spectral theory for this was constructed recently in my joint work with Borodin, Corwin, and Sasamoto. This in particular lead to explicit nested contour integral formulas for moments of the q-TASEP started from an arbitrary initial configuration. Limits of these formulas as q->1 lead to previously known formulas for various delta Bose gases due to van Diejen and Emsiz (2013) and Heckman and Opdam (1997).
When q=1/nu, the eigenfunctions of the q-Hahn Boson system degenerate to the Bethe ansatz eigenfunctions for the ASEP (which are almost the same as those of the XXZ quantum spin chain). The Plancherel isomorphism theorems provide a different approach to earlier results of Babbitt, Thomas, and Gutkin, and also give a structural explanation of ASEP symmetrization formulas of Tracy and Widom.

Vadim Gorin
Random matrix asymptotics for the six-vertex model

Abstract: The six-vertex (or "square-ice") model is one of the most well-studied examples of exactly-solvable lattice models of statistical mechanics. The developments of the last 15 years suggest that the asymptotic behavior of this model should be governed by the probability distributions of random matrix origin. However, until recently the rigorous mathematical results in this direction were restricted to the so-called free fermion case, when the model can be analyzed via determinantal point processes. In my talk we will discuss two results outside the free fermions, and see how the random matrix distributions, namely the Gaussian Unitary Ensemble and the Tracy-Widom distribution F_2, emerge.

Gunter Schutz
Microscopic structure of the ASEP conditioned on an atypical current

Abstract: We consider the asymmetric simple exclusion processes (ASEP) on a ring of L sites,conditioned to produce an atypical current.
(a) For very large current we find that the particles at lattice positions n_1,n_2,...,n_N experience an effective long-range potential which is repulsive and which in the limit of very large flux takes the form U= -2 \sum_{i\neq j} log|sin(pi(n_i/L-n_j/L))|, similar to the effective potential between the eigenvalues of the circular unitary ensemble in random matrices. The stationary density correlations decay algebraically. We compute the longest relaxation time and the exact dynamical structure factor. From this we find that the dynamical exponent in the extreme current regime is z=1 rather than the KPZ exponent z=3/2 which characterizes the ASEP in the regime of typical currents.
(b) For low currents we prove that at a specific value of the conditioned global current there is a one-parameter family of shock initial measures such that the shock position performs a biased random walk and that the measure seen from the shock position remains invariant. The density profile seen from the shock position is a hyperbolic tangent. We point out some tentative links of these results to conformal invariance and to the U_q[SU(2)]-symmetry of the Heisenberg spin-1/2 quantum chain.

Jan Felipe Van Diejen
Integrable boundary interactions for Ruijsenaars' difference Toda chain

Abstract: We endow Ruijsenaars' open difference Toda chain with a one-sided boundary interaction of Askey-Wilson type and diagonalize the quantum Hamiltonian by means of deformed hyperoctahedral q-Whittaker functions that arise as a t=0 degeneration of the Macdonald-Koornwinder multivariate Askey-Wilson polynomials. This immediately entails the quantum integrability, the bispectral dual system, and the n-particle scattering operator for the chain in question. Based on joint work with Erdal Emsiz.

Erdal Emsiz
Discrete harmonic analysis on a Weyl alcove and the double affine Hecke algebra at critical level

Abstract: I will speak about recent work on a unitary representation of the affine Hecke algebra given by discrete difference-reflection operators acting in a Hilbert space of complex functions on the weight lattice of a reduced crystallographic root system. I will indicate why the action of the center under this representation is diagonal on the basis of Macdonald spherical functions (also referred to as generalized Hall-Littlewood polynomials associated with root systems).
I will also discuss a periodic counterpart of the above mentioned model related to a representation of the double affine Hecke algebra at critical level q = 1 in terms of difference-reflection operators. We use this representation to construct an explicit integrable discrete Laplacian on the discrete Weyl alcove corresponding to an element in the center. The Bethe Ansatz method is employed to show that our discrete Laplacian and its commuting integrals are diagonalized by a finite-dimensional basis of periodic Macdonald spherical functions.
This is joint work in progress with Jan Felipe van Diejen.

Yi Sun
A new integral formula for the Heckman-Opdam hypergeometric functions

Abstract: We provide integral formulas for the Heckman-Opdam hypergeometric functions which appear in recent work of Borodin-Gorin (2013) on the general beta Jacobi ensembles. Our expression takes the form of an integral over the dressing orbit and is related to known expressions as integrals over Gelfand-Tsetlin polytopes by an integration over the Liouville tori of the Gelfand-Tsetlin integrable system and an integration by parts. Our techniques are motivated by the orbit method and the representation theory of the rational Cherednik algebra.

Alexei Bufetov
Asymptotics of representations of classical Lie groups

Abstract: We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition. Connections of this result with free probability, random lozenge tilings, and extreme characters of the infinite-dimensional unitary group will be explained. The talk is based on joint works with A. Borodin, V. Gorin, and G. Olshanski.

Jeff Kuan
"Three-dimensional Gaussian fluctuations of noncommutative random surfaces

Abstract: We construct a quantum random walk on the universal enveloping algebra U(gl_N). The corresponding Markov operator is a linear operator on U(gl_N) which preserves the centre. By using the Harish--Chandra isomorphism to identify the centre with shifted symmetric polynomials in N variables, the quantum random walk reduces to a classical random walk, which turns out to be the (2+1)--dimensional interacting particle system from Borodin and Ferrari (2008). Explicit descriptions of the generators of the centre allow for a computation of the three--dimensional Gaussian field describing the limiting behaviour of the fluctuations of the height function. This Gaussian field is the stationary distribution of the Anisotropic Kardar-Parisi-Zhang equation.
We also discuss extending the construction to the quantum group U_q(gl_N) and relationships to Macdonald Processes with parameters (q,q^k), where k is a positive integer.

Yoshihiro Takeyama
A deformation of affine Hecke algebra and integrable stochastic particle system

Abstract: We consider a three-parameter deformation of affine Hecke algebra (of type GL). Making use of its representation we construct a stochastic discrete Hamiltonian system, which is a generalization of the q-Boson system.

Christian Korff
The Bethe ansatz in quantum Schubert calculus

Abstract: Quantum cohomology arose from fusion rings through the works of Gepner, Vafa, Intriligator and Witten. In this talk I will give an overview of a recent formulation of the fusion rings for the type A case which employs quantum integrable lattice models. This formulation shows new connections between these rings and Lie theory and provides a simple combinatorial formalism to compute Gromov—Witten invariants by using a hopping algorithm for particles on a discrete circular lattice. Time permitting I will sketch the extension to K-theory. This is joint work with Vassily Gorbounov, Aberdeen.

Eric Opdam
Dirac induction and Hecke algebra modules

Abstract: In this talk I will report on work with Dan Ciubotaru and Peter Trapa on the analytic and algebraic Dirac induction for graded affine Hecke algebras. The analytic aspect arises from the action of the graded affine Hecke algebra on a quantum integrable model which I studied with Heckman and later with Stokman. We will discuss various applications of Dirac induction towards a uniform classification of tempered representations of affine Hecke algebras.

Ivan Corwin
Beyond the Gaussian universality class

Abstract: The Gaussian central limit theorem says that for a wide class of stochastic systems, the bell curve (Gaussian distribution) describes the statistics for random fluctuations of important observables. In this talk I will look beyond this class of systems to a collection of probabilistic models which include random growth models, polymers, particle systems, matrices and stochastic PDEs, as well as certain asymptotic problems in combinatorics and representation theory. I will explain in what ways these different examples all fall into a single new universality class with a much richer mathematical structure than that of the Gaussian.

Pierre Le Doussal
Exact results for the Kardar-Parisi-Zhang equation and directed path models, from replica Bethe ansatz and sine-Gordon quantum field theory

Abstract: The replica method, combined with methods of integrable systems has recently allowed to obtain exact, although non rigorous, results for the KPZ equation and its equivalent formulation in terms of directed polymers in a random potential. I will review these methods and results and as time permits, present some of the most recent ones.

Natan Andrei
Quench dynamics of quantum interacting many body systems in 1-d

Abstract: I will describe a formulation for studying the quench dynamics of integrable systems generalizing an approach by Yudson. I'll discuss the evolution Lieb-Liniger system, a gas of interacting bosons moving on the continuous infinite line and interacting via a short range potential. The formalism allows us to quench the system from any initial state. Considering first a finite number of bosons on the line we find that for any value of repulsive coupling the system asymptotes towards a strongly repulsive gas for any initial state, while for an attractive coupling, the system forms a maximal bound state that dominates at longer times. In either case the system equilibrates but does not thermalize, an effect that is consistent with prethermalization. Then considering the system in the thermodynamic limit - with the number of bosons and the system size sent to infinity at a constant density with the long time limit taken subsequently- I'll discuss the equilibration of the density and density-density correlation functions for strong but finite and show they are described by GGE for translationally invariant initial states with short range correlations. For initial states with long range correlations a generalized GGE emerges. If the initial state is strongly non translationally invariant the system does not equilibrate.
If time permits I shall discuss also the quench dynamics of the XXZ Heisenberg chain and of a mobile impurity in an interacting Bose gas.

Nikos Zygouras
Tropical combinatorics, Whittaker functions and random polymers.

Abstract: Whittaker functions are special functions, which have a central position in representation theory and integrable systems. Surprisingly, they turn out to play a central role in the ?uctuation analysis of Random Polymer Models (modelling a random walk in a random potential). In this talk I will explain the emergence of Whittaker functions in the Random Polymer Model, via the use of Tropical Combinatorics. An important role is played by the tropical (or geometric) Robinson-Schensted-Knuth correspodence and its volume preserving properties, which, as a byproduct leads to a new interpretation of Giventals integral formula for Whittaker functions. Based on joint work with N. O'Connell and T.Sepallainen.