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This article is cited in 1 scientific paper (total in 1 paper)
Parameter Permutation Symmetry in Particle Systems and Random Polymers
Leonid Petrovab a University of Virginia, Department of Mathematics,
141 Cabell Drive, Kerchof Hall, P.O. Box 400137, Charlottesville, VA 22904, USA
b Institute for Information Transmission Problems,
Bolshoy Karetny per. 19, Moscow, 127994, Russia
Abstract:
Many integrable stochastic particle systems in one space dimension (such as TASEP – totally asymmetric simple exclusion process – and its various deformations, with a notable exception of ASEP) remain integrable when we equip each particle $x_i$ with its own jump rate parameter $\nu_i$. It is a consequence of integrability that the distribution of each particle $x_n(t)$ in a system started from the step initial configuration depends on the parameters $\nu_j$, $j\le n$, in a symmetric way. A transposition $\nu_n \leftrightarrow \nu_{n+1}$ of the parameters thus affects only the distribution of $x_n(t)$. For $q$-Hahn TASEP and its degenerations ( $q$-TASEP and directed beta polymer) we realize the transposition $\nu_n \leftrightarrow \nu_{n+1}$ as an explicit Markov swap operator acting on the single particle $x_n(t)$. For beta polymer, the swap operator can be interpreted as a simple modification of the lattice on which the polymer is considered. Our main tools are Markov duality and contour integral formulas for joint moments. In particular, our constructions lead to a continuous time Markov process $\mathsf{Q}^{(\mathsf{t})}$ preserving the time $\mathsf{t}$ distribution of the $q$-TASEP (with step initial configuration, where $\mathsf{t}\in \mathbb{R}_{>0}$ is fixed). The dual system is a certain transient modification of the stochastic $q$-Boson system. We identify asymptotic survival probabilities of this transient process with $q$-moments of the $q$-TASEP, and use this to show the convergence of the process $\mathsf{Q}^{(\mathsf{t})}$ with arbitrary initial data to its stationary distribution. Setting $q=0$, we recover the results about the usual TASEP established recently in [arXiv: 1907.09155] by a different approach based on Gibbs ensembles of interlacing particles in two dimensions.
Keywords:
$q$-TASEP, stochastic $q$-Boson system, stationary distribution, coordinate Bethe ansatz, $q$-Hahn TASEP.
Received: October 26, 2020; in final form February 20, 2021; Published online March 6, 2021
Citation:
Leonid Petrov, “Parameter Permutation Symmetry in Particle Systems and Random Polymers”, SIGMA, 17 (2021), 021, 34 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1704 https://www.mathnet.ru/eng/sigma/v17/p21
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