Multispecies TASEP and combinatorialR

Abstract

We identify the algorithm for constructing steady states of the n-species totally asymmetric simple exclusion process (TASEP) on an L site periodic chain by Ferrari and Martin with a composition of combinatorial R for the quantum affine algebra U sl ( ) in crystal base theory. Based on this connection and the factorized form of the R matrix derived recently from the tetrahedron equation, we establish a new matrix product formula for the steady state of the TASEP, which is expressed in terms of corner transfer matrices of the q-oscillator valued five-vertex model at q = 0.

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

“…(v) The whole story about the n-TAZRP in this paper and [19] is closely parallel with the recent result on the n-TASEP by the authors [17,18]. In fact the n-TAZRP and the n-TASEP turn out to be the canonical sister models associated with the symmetric and the antisymmetric tensor representations of U q ( sl L ) at q = 0, respectively.…”

Inhomogeneous Generalization of a Multispecies Totally Asymmetric Zero Range Process

“…The proposal of the n-TAZRP was preceded by the discovery [14,15] of the quite parallel features in the n-species totally asymmetric simple exclusion process (n-TASEP). We refer to [21,3,23] for results on more general n-ASEP that have been established for arbitrary n. In our approach, the n-TAZRP and the n-TASEP on the length L periodic chain turn out to be the sister models associated with crystals [20] of the symmetric and the anti-symmetric tensor representation of the quantum affine algebra U q ( sl L ) [6,10], respectively.…”

Multiline Queues with Spectral Parameters

“…There is a well-known (in slightly different terminology) bijection Ξ r relating r-queues with an element of the KR crystal B r,1 in type A (1) n−1 by considering q = {b 1 < · · · < b r } as a single-column Young tableau of height r. In [KMO15], this was extended to a bijection Ξ between multiline queues of type m with ℓ + 1 classes and B p 1 ,1 ⊗ B p 2 ,1 ⊗ · · · ⊗ B p ℓ ,1 by…”

“…Next, we describe how to interpret our action from looking at the corner transfer matrix described in [KMO15], which can be given diagrammatically by…”

“…This is a generalization of 2-line queues used by Angel [Ang06] and the work of Ferrari, Fontes, and Kohayakawa [FFK94]. In [KMO15,KMO16b], the FM algorithm was reformulated in terms of the combinatorial R-matrix [NY97,Shi02] and using type A (1) n−1 Kirillov-Reshetikhin crystals [KKM + 92]. This interpretation gives a connection with five-vertex models, corner transfer matrices [Bax89], 3D integrable lattice models, and the tetrahedron equation [Zam80], yielding a matrix product formula for the steady state distribution different than [CdGW15,EFM09,PEM09].…”