Multispecies totally asymmetric zero range process: I. Multiline process and combinatorialR

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 π. We construct the π by a combinatorial R of the quantum affine algebra U q ( 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 ( sl L ) at q = 0, respectively.

A q-boson representation of Zamolodchikov-Faddeev algebra for stochastic R matrix of $$\varvec{U_q(A^{(1)}_n)}$$ U q ( A n ( 1 ) )

“…At q = 0, the q-boson representation of the matrix product operators in this paper coincides with the homogeneous case w 1 = · · · · · · = w n of [15] as shown in (31). At q = 0, there is another set of matrix product operators originating in the combinatorial R in crystal theory [13] and the tetrahedron equation [14]. They agree with the q = 0 case of the present paper for n = 2 upon adjustment of conventions.…”

“…We call them U q (A (1) n )-zero range processes. They include several models studied earlier for n = 1 [20,19,21,4,2] and for n general [22,13,14,15].…”

Matrix product formula for ${{U}_{q}}(A_{2}^{(1)})$ -zero range process

“…, γ n = β n . It means that larger species particles have the priority to jump out, which precisely reproduces the n-species TAZRP explored in [17,18] after reversing the labeling of the species 1, 2, . .…”

“…At q = µ = 0 (and a = 0 in (21)), the present model reduces to the n = 2 case of the n-TAZRP studied in [17,18]. The formula (42) simplifies to P(σ 1 , .…”

Multiline Queues with Spectral Parameters

“…Acknowledgements. We thank Atsuo Kuniba for explaining the results in his papers [KMO15,KMO16a,KMO16b,KMO16c,KMO16d]. We thank Olya Mandelshtam for useful discussions on the inhomogeneous TASEP.…”