Algebraic Duality Methods in Probability

Organizers: Ivan Corwin and Jeffrey Kuan

All times below are Central Standard Time (CST)

Zoom link here Meeting ID: 970 5689 9453 Passcode: 2718281828

Schedule of Talks

Date and Time


Title and Abstract

Wednesday, June 2, 10am CST

Cristian Giardinà

Exact solution of a boundary-driven integrable particle system.

I present the results of a joint (ongoing) work with Rouven Frassek. We consider the boundary-driven interacting particle systems introduced in [1], related to the open non-compact Heisenberg model in one dimension. We show that a finite chain of N sites connected at its ends to two reservoirs can be solved exactly, i.e. the non-equilibrium steady-state has a closed-form expression for each N.   The solution relies on probabilistic arguments and techniques inspired by integrable systems. It is obtained in two steps:  i) the introduction of a dual absorbing process reducing the problem to a finite number of particles; ii) the solution of the dual dynamics exploiting a non-local symmetry obtained from the Quantum Inverse Scattering Method. The exact solution allows to prove by direct computation that, in the thermodynamic limit, the system approaches local equilibrium, whereas microscopically there are long-ranged correlations. A  by-product of the solution is the algebraic construction of a direct mapping (a conjugation) between the generator of the non-equilibrium process and the generator of the associated reversible equilibrium process. Macroscopically, this mapping was previously observed by Tailleur, Kurchan and Lecomte in the context of Macroscopic Fluctuation Theory. [1] R. Frassek, C. Giardinà, J. Kurchan, Non-compact quantum spin chains as integrable stochastic particle processes, Journal of Statistical Physics 180, 366-397 (2020).


Zoom Recording

Wednesday, June 2, 11am CST

Wolter Groenevelt

Orthogonal duality functions from Lie algebra representations

In this talk I explain how, for certain symmetric interacting particle processes associated to Lie algebras, orthogonal duality functions can be obtained from unitary intertwining operators for certain Lie algebra representations. This gives, for example, an algebraic explanation for the occurrence of Krawtchouk polynomials as self-duality functions for the symmetric exclusion process. 


Zoom Recording

Thursday, June 3, 5pm CST 

Michael Wheeler

q-deformed Knizhnik--Zamolodchikov equations, vertex models and duality

I will discuss certain solutions of the q-deformed Knizhnik--Zamolodchikov (qKZ) equations, which may be expressed via partition functions in stochastic vertex models. After recasting the qKZ equations in terms of generators of the Hecke algebra, one may interpret the resulting relations as a duality between two multi-species ASEPs. This leads to a kind of factory for producing duality observables, as partition functions within the vertex model in question. Several examples will be discussed. 


Zoom Recording

Friday, June 4, 10am CST

Alexey Bufetov

Color-position symmetry in interacting particle systems

Multi-species versions of several interacting particle systems, including ASEP, q-TAZRP, and k-exclusion processes, can be interpreted as random walks on Hecke algebras. An involution in Hecke algebra leads to an interesting symmetry of these processes which we refer to as the color-position symmetry. In the talk I will describe this symmetry and several applications to asymptotic questions. Based on joint works with A. Borodin and with P. Nejjar.  


Zoom Recording

Friday, June 4, 11am CST

Yier Lin

Markov duality for stochastic six vertex model

We prove that Schütz’s ASEP Markov duality functional is also a Markov duality functional for the stochastic six vertex model. We introduce a new method that uses induction on the number of particles to prove the Markov duality. If time permits, I will also talk about some application of this duality.  


Zoom Recording

Monday, June 7, 10am CST

Gunter Schütz

Integrability, supersymmetry and duality for vicious walkers with pair creation

We study a system of independent random walkers in one dimension that annihilate immediately when two particles meet on the same site. In addition, pairs of particles are created randomly on neighbouring sites. For periodic boundary conditions, a duality with independent two-level systems which arises from the integrability of the model is proved. The duality function is determinantal and has the form of a matrix product. We use this duality to compute the exact current distribution. For reflecting boundaries the Markov generator commutes with the generators of a subalgebra of the universal enveloping algebra of the Lie superalgebra sl(1|1) and its deformations. The supersymmetry leads to a duality between an even and odd number of particles, respectively.


Zoom Recording

Monday, June 7, 11am CST

Florian Völlering

Markov process representation of semigroups whose generators include negative rates

Generators of Markov processes on a countable state space can be represented as finite or infinite matrices. One key property is that the off-diagonal entries corresponding to jump rates of the Markov process are non-negative. I will present stochastic characterizations of the semigroup generated by a generator with possibly negative rates. This is done by considering a larger state space with one or more particles and antiparticles, with antiparticles being particles carrying a negative sign. 


Zoom Recording

Tuesday, June 8, 10am CST

Frank Redig

Self-duality in the continuum

We generalize classical and orthogonal dualities beyond the framework of lattice systems, including e.g. random walks on general state spaces and interacting Brownian motions. Using some natural concepts from point process theory together with the notion of consistency, we introduce two intertwining relations which in the case of lattice systems give the known ``classical’’ and orthogonal dualities for exclusion and inclusion processes.  We provide several examples including the inclusion process in the continuum. Based on joint work with S. Floreani (Delft), S. Jansen (Munich), S. Wagner (Munich). 


Zoom Recording

Tuesday, June 8, 11am CST

Chiara Franceschini  

Self-duality for particle systems via (q-)orthogonal polynomials  

In this talk I will present some results regarding self-duality for two families of symmetric and asymmetric interacting particle systems. The method relies on their algebraic description and give rise to self-duality functions which are families of orthogonal polynomials in case of symmetric processes and q-orthogonal polynomials in case of asymmetric processes. 


Zoom Recording