Parameter Permutation Symmetry in Particle Systems and Random Polymers

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.

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