Parameter Permutation Symmetry in Particle Systems and Random Polymers

TL;DR

This paper explores parameter permutation symmetry in integrable stochastic particle systems, introduces Markov swap operators, and constructs processes that preserve distributions, connecting to known TASEP results and extending understanding of particle system symmetries.

Contribution

It introduces explicit Markov swap operators for parameters in integrable particle systems and constructs a process preserving the TASEP distribution, extending symmetry understanding.

Findings

Markov swap operators realize parameter transpositions in particle systems.

Constructed a process preserving the $q$-TASEP distribution over time.

Connected results to classical TASEP when $q=0$.

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.