# From duality to determinants for q-TASEP and ASEP

@article{Borodin2014FromDT, title={From duality to determinants for q-TASEP and ASEP}, author={Alexei Borodin and Ivan Corwin and Tomohiro Sasamoto}, journal={Annals of Probability}, year={2014}, volume={42}, pages={2314-2382} }

We prove duality relations for two interacting particle systems: the $q$-deformed totally asymmetric simple exclusion process ($q$-TASEP) and the asymmetric simple exclusion process (ASEP). Expectations of the duality functionals correspond to certain joint moments of particle locations or integrated currents, respectively. Duality implies that they solve systems of ODEs. These systems are integrable and for particular step and half-stationary initial data we use a nested contour integral…

## 164 Citations

q -TASEPS

- Mathematics
- 2013

. We introduce two new exactly solvable (stochastic) interacting particle systems which are discrete time versions of q -TASEP. We call these geometric and Bernoulli discrete time q -TASEP. We obtain…

Discrete Time q-TASEPs

- Mathematics
- 2013

We introduce two new exactly solvable (stochastic) interacting particle systems which are discrete time versions of q-TASEP. We call these geometric and Bernoulli discrete time q-TASEP. We obtain…

Spectral theory for the $q$-Boson particle system

- MathematicsCompositio Mathematica
- 2014

Abstract We develop spectral theory for the generator of the $q$-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has…

Spectral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz

- Mathematics
- 2014

We develop spectral theory for the q-Hahn stochastic particle system introduced recently by Povolotsky. That is, we establish a Plancherel type isomorphism result that implies completeness and…

Stochastic Higher Spin Vertex Models on the Line

- Mathematics
- 2015

We introduce a four-parameter family of interacting particle systems on the line, which can be diagonalized explicitly via a complete set of Bethe ansatz eigenfunctions, and which enjoy certain…

Duality relations for the ASEP conditioned on a low current

- Mathematics
- 2015

We consider the asymmetric simple exclusion process (ASEP) on a finite lattice with periodic boundary conditions, conditioned to carry an atypically low current. For an infinite discrete set of…

Duality relations for the ASEP conditioned on a low current

- Mathematics
- 2015

We consider the asymmetric simple exclusion process (ASEP) on a finite lattice with periodic boundary conditions, conditioned to carry an atypically low current. For an infinite discrete set of…

The $q$-Hahn asymmetric exclusion process

- Mathematics
- 2015

We introduce new integrable exclusion and zero-range processes on the one-dimensional lattice that generalize the $q$-Hahn TASEP and the $q$-Hahn Boson (zero-range) process introduced in [Pov13] and…

Fluctuations for stationary q-TASEP

- MathematicsProbability Theory and Related Fields
- 2018

We consider the q-totally asymmetric simple exclusion process (q-TASEP) in the stationary regime and study the fluctuation of the position of a particle. We first observe that the problem can be…

$q$-TASEP with position-dependent slowing

- Mathematics
- 2021

We introduce a new interacting particle system on Z, slowed t-TASEP. It may be viewed as a q-TASEP with additional position-dependent slowing of jump rates depending on a parameter t, which leads to…

## References

SHOWING 1-10 OF 47 REFERENCES

A Fredholm Determinant Representation in ASEP

- Mathematics
- 2008

In previous work (Tracy and Widom in Commun. Math. Phys. 279:815–844, 2008) the authors found integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer…

Discrete Time q-TASEPs

- Mathematics
- 2013

We introduce two new exactly solvable (stochastic) interacting particle systems which are discrete time versions of q-TASEP. We call these geometric and Bernoulli discrete time q-TASEP. We obtain…

Duality relations for asymmetric exclusion processes

- Mathematics
- 1997

We derive duality relations for a class ofUq[SU(2)]-symmetric stochastic processes, including among others the asymmetric exclusion process in one dimension. Like the known duality relations for…

Integral Formulas for the Asymmetric Simple Exclusion Process

- Mathematics
- 2008

In this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice $${\mathbb{Z}}$$ with nearest neighbor hopping rates p to…

Bethe ansatz solution of zero-range process with nonuniform stationary state.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2004

The eigenfunctions and eigenvalues of the master equation for zero-range process with totally asymmetric dynamics on a ring are found exactly using the Bethe ansatz weighted with the stationary weights of particle configurations using the partition function of the model.

Stochastic Burgers and KPZ Equations from Particle Systems

- Mathematics
- 1997

Abstract: We consider two strictly related models: a solid on solid interface growth model and the weakly asymmetric exclusion process, both on the one dimensional lattice. It has been proven that,…

Macdonald processes

- Mathematics
- 2014

Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters $$q,t \in…

On the numerical evaluation of Fredholm determinants

- MathematicsMath. Comput.
- 2010

It is proved that the approximation error essentially behaves like the quadrature error for the sections of the kernel, which is typical in random matrix theory, and a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nystrom method.

The Kardar-Parisi-Zhang Equation and Universality Class

- Mathematics
- 2011

Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or…