# Stochastic Fusion of Interacting Particle Systems and Duality Functions

@article{Kuan2019StochasticFO, title={Stochastic Fusion of Interacting Particle Systems and Duality Functions}, author={Jeffrey Kuan}, journal={arXiv: Probability}, year={2019} }

We introduce a new method, which we call stochastic fusion, which takes an exclusion process and constructs an interacting particle systems in which more than one particle may occupy a lattice site. The construction only requires the existence of stationary measures of the original exclusion process on a finite lattice. If the original exclusion process satisfies Markov duality on a finite lattice, then the construction produces Markov duality functions (for some initial conditions) for the…

## 9 Citations

### Orthogonal dualities for asymmetric particle systems *

- Mathematics
- 2021

We study a class of interacting particle systems with asymmetric interaction showing a self-duality property. The class includes the ASEP(q, θ), asymmetric exclusion process, with a repulsive…

### Interacting particle systems and random walks on Hecke algebras

- Mathematics
- 2020

In this paper we show that a variety of interacting particle systems with multiple species can be viewed as random walks on Hecke algebras. This class of systems includes the asymmetric simple…

### Hydrodynamic limit for a $d$-dimensional open symmetric exclusion process

- Mathematics
- 2020

In this paper we focus on the open symmetric exclusion process with parameter $m$ (open SEP($m/2$)), which allows $m$ particles each site and has an open boundary. We generalize the result about…

### Two Dualities: Markov and Schur–Weyl

- Mathematics
- 2020

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…

### Orthogonal Polynomial Stochastic Duality Functions for Multi-Species SEP(2j) and Multi-Species IRW

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2021

. We obtain orthogonal polynomial self-duality functions for multi-species version of the symmetric exclusion process (SEP(2 j )) and the independent random walker process (IRW) on a finite…

### Algebraic Symmetry and Self–Duality of an Open ASEP

- MathematicsMathematical 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…

### Orthogonal polynomial duality and unitary symmetries of multi--species ASEP$(q,\boldsymbol{\theta})$ and higher--spin vertex models via $^*$--bialgebra structure of higher rank quantum groups

- Mathematics
- 2022

We propose a novel, general method to produce orthogonal polynomial dualities from the ∗ –bialgebra structure of Drinfeld–Jimbo quantum groups. The ∗ –structure allows for the construction certain…

### Coxeter group actions on interacting particle systems

- MathematicsStochastic Processes and their Applications
- 2022

### A short note on Markov duality in multi–species higher spin stochastic vertex models

- MathematicsElectronic Communications in Probability
- 2021

We show that the multi-species higher spin stochastic vertex model, also called the U_q(A_n^{(1)}) vertex model, satisfies a duality where the indicator function has the form {\eta^x_{[1,n]} \geq…

## References

SHOWING 1-10 OF 79 REFERENCES

### Consistent particle systems and duality

- MathematicsElectronic Journal of Probability
- 2021

We consider consistent particle systems, which include independent random walkers, the symmetric exclusion and inclusion processes, as well as the dual of the KMP model. Consistent systems are such…

### Duality and Hidden Symmetries in Interacting Particle Systems

- Mathematics
- 2009

In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions and symmetries of the generator. If the generator can be written 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…

### Coupling the Simple Exclusion Process

- Mathematics
- 1976

Consider the infinite particle system on the countable set $S$ with the simple exclusion interaction and one-particle motion determined by the stochastic transition matrix $p(x, y)$. In the past, the…

### Dynamical stochastic higher spin vertex models

- Mathematics
- 2017

We introduce a new family of integrable stochastic processes, called dynamical stochastic higher spin vertex models, arising from fused representations of Felder’s elliptic quantum group $$E_{\tau ,…

### A generalized asymmetric exclusion process with Uq(sl2) 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…

### 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…

### 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).…

### Multi-state Asymmetric Simple Exclusion Processes

- Mathematics
- 2015

It is known that the Markov matrix of the asymmetric simple exclusion process (ASEP) is invariant under the $$U_q(sl_2)$$Uq(sl2) algebra. This is the result of the fact that the Markov matrix of the…

### Symmetric elliptic functions, IRF models, and dynamic exclusion processes

- Mathematics
- 2017

We introduce stochastic Interaction-Round-a-Face (IRF) models that are related to representations of the elliptic quantum group $E_{\tau,\eta}(sl_2)$. For stochasic IRF models in a quadrant, we…