# Self-Duality for the Two-Component Asymmetric Simple Exclusion Process

@article{Belitsky2015SelfDualityFT,
title={Self-Duality for the Two-Component Asymmetric Simple Exclusion Process},
journal={arXiv: Probability},
year={2015}
}
• Published 20 April 2015
• Mathematics
• arXiv: Probability
We study a two-component asymmetric simple exclusion process (ASEP) that is equivalent to the ASEP with second-class particles. We prove self-duality with respect to a family of duality functions which are shown to arise from the reversible measures of the process and the symmetry of the generator under the quantum algebra $U_q[\mathfrak{gl}_3]$. We construct all invariant measures in explicit form and discuss some of their properties. We also prove a sum rule for the duality functions.
Quantum Algebra Symmetry of the ASEP with Second-Class Particles
• Mathematics
• 2015
We consider a two-component asymmetric simple exclusion process (ASEP) on a finite lattice with reflecting boundary conditions. For this process, which is equivalent to the ASEP with second-class
Algebraic Symmetry and Self–Duality of an Open ASEP
• Jeffrey Kuan
• Mathematics
Mathematical Physics, Analysis and Geometry
• 2019
We consider the asymmetric simple exclusion process (ASEP) with open boundary condition at the left boundary, where particles exit at rate {\gamma} and enter at rate {\alpha} = {\gamma}{\tau}^2, and
Stochastic duality of ASEP with two particle types via symmetry of quantum groups of rank two
We study two generalizations of the asymmetric simple exclusion process (ASEP) with two types of particles, which will be called type A2 ASEP and type C2 ASEP. Particles of type 1 force particles of
A generalized asymmetric exclusion process with Uq ( sl 2 ) stochastic duality
• Mathematics
• 2015
We study a new process, which we call ASEP(q, j), where particles move asymmetrically on a one-dimensional integer lattice with a bias determined by q ∈ (0, 1) and where at most 2 j ∈ N particles per
Self-duality and shock dynamics in the $n$-component priority ASEP
• Mathematics
• 2016
We study the $n$-component priority asymmetric simple exclusion process ($n$-ASEP) with reflecting boundaries. We obtain all invariant measures in explicit form and prove reversibility. Using the
Interacting particle systems with type D symmetry and duality.
• Mathematics
• 2020
We construct a two-class asymmetric interacting particle system with $U_q(so_6)$ or $U_q(so_8)$ symmetry, in which up to two particles may occupy a site if the two particles have different class. The
Duality, supersymmetry and non-conservative random walks
• Mathematics
Journal of Statistical Mechanics: Theory and Experiment
• 2019
We derive a probabilistic interpretation of the observation that the quantum XY chain is supersymmetric in the sense that the Hamiltonian commutes with the generators of a subalgebra of the universal
Duality relations for the ASEP conditioned on a low current
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
Two Dualities: Markov and Schur–Weyl
We show that quantum Schur-Weyl duality leads to Markov duality for a variety of asymmetric interacting particle systems. In particular, we consider three cases: (1) Using a Schur-Weyl duality

## References

SHOWING 1-10 OF 29 REFERENCES
Quantum Algebra Symmetry of the ASEP with Second-Class Particles
• Mathematics
• 2015
We consider a two-component asymmetric simple exclusion process (ASEP) on a finite lattice with reflecting boundary conditions. For this process, which is equivalent to the ASEP with second-class
Duality relations for asymmetric exclusion processes
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
On the Asymmetric Simple Exclusion Process with Multiple Species
• Mathematics
• 2013
In previous work the authors, using the Bethe Ansatz, found for the N-particle asymmetric simple exclusion process on the integers a formula—a sum of multiple integrals—for the probability that a
A generalized Asymmetric Exclusion Process with $U_q(\mathfrak{sl}_2)$ stochastic duality
• Mathematics
• 2014
We study a new process, which we call ASEP$(q,j)$, where particles move asymmetrically on a one-dimensional integer lattice with a bias determined by $q\in (0,1)$ and where at most $2j\in\mathbb{N}$
From duality to determinants for q-TASEP and ASEP
• Mathematics
• 2014
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).
Determinant Representation for Some Transition Probabilities in the TASEP with Second Class Particles
• Mathematics
• 2010
We study the transition probabilities for the totally asymmetric simple exclusion process (TASEP) on the infinite integer lattice with a finite, but arbitrary number of first and second class
On the notion(s) of duality for Markov processes
• Mathematics
• 2012
We provide a systematic study of the notion of duality of Markov processes with respect to a function. We discuss the relation of this notion with duality with respect to a measure as studied in
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
Quantum Operators in Classical Probability Theory: II. The Concept of Duality in Interacting Particle Systems
• Mathematics
• 1995
Duality has proved to be a powerful tool in the theory of interacting particle systems. The approach in this paper is algebraic rather than via Harris diagrams. A form of duality is found which
Current Moments of 1D ASEP by Duality
• Mathematics
• 2010
We consider the exponential moments of integrated currents of 1D asymmetric simple exclusion process using the duality found by Schütz. For the ASEP on the infinite lattice we show that the nth