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 the notable exception of ASEP) remain integrable when we equip each particle 𝑥ᵢ with its own jump rate parameter νᵢ. It is a consequence of integrability that the distribution of each particle 𝑥ₙ(𝘵) in a system started from the step initial configuration depends on the parameters 𝑣ⱼ, j ≤ 𝑛, symmetrically. A transposition 𝑣ₙ ↔ 𝑣ₙ₊₁ of the parameters thus affects only the distribution of 𝑥ₙ(𝘵). For q-Hahn TASEP and its degenerations (q-TASEP and directed beta polymer), we realize the transposition 𝑣ₙ ↔ 𝑣ₙ₊₁ as an explicit Markov swap operator acting on the single particle 𝑥ₙ(𝘵). For a 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 Q⁽ᵗ⁾ preserving the time t distribution of the 𝑞-TASEP (with step initial configuration, where 𝘵 ∈ 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 Q⁽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.