Multispecies totally asymmetric zero range process: I. Multiline process and combinatorial $R$

Abstract

We introduce an $n$-species totally asymmetric zero range process ($n$-TAZRP) on one-dimensional periodic lattice with $L$ sites. It is a continuous time Markov process in which $n$ species of particles hop to the adjacent site only in one direction under the condition that smaller species ones have the priority to do so. Also introduced is an $n$-line process, a companion stochastic system having the uniform steady state from which the $n$-TAZRP is derived as the image by a certain projection $\pi$. We construct the $\pi$ by a combinatorial $R$ of the quantum affine algebra $U_q({\widehat{sl}}_L)$ and establish a matrix product formula of the steady state probability of the $n$-TAZRP in terms of corner transfer matrices of a $q=0$-oscillator valued vertex model. These results parallel the recent reformulation of the $n$-species totally asymmetric simple exclusion process ($n$-TASEP) by the authors, demonstrating that $n$-TAZRP and $n$-TASEP are the canonical sister models associated with the symmetric and the antisymmetric tensor representations of $U_q({\widehat{sl}}_L)$ at $q=0$, respectively.

25 pages. Minor corrections in Example 5.9

References in corpus (4)

• Yang-Baxter maps: dynamical point of view
• Multispecies TASEP and the tetrahedron equation
• Multispecies TASEP and combinatorial $R$
• Multispecies totally asymmetric zero range process: II. Hat relation and tetrahedron equation

Cited by in corpus (8)

• Exact confirmation of 1D nonlinear fluctuating hydrodynamics for a two-species exclusion process
• Limiting current distribution for a two species asymmetric exclusion process
• Transition probability and total crossing events in the multi-species asymmetric exclusion process
• Multiline queues with spectral parameters
• A $q$-boson representation of Zamolodchikov-Faddeev algebra for stochastic $R$ matrix of $U_q(A_n^{(1)})$
• Integrable Structure of Multispecies Zero Range Process
• Multispecies totally asymmetric zero range process: II. Hat relation and tetrahedron equation
• Twisted multiline queues for the steady states of TASEP and TAZRP