No 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.

We introduce a family of layer to layer transfer matrices in a three-dimensional (3D) lattice model which can be viewed as partition functions of the q-oscillator valued six-vertex model on m × n square lattice. By invoking the tetrahedron equation we establish their commutativity and bilinear relations mixing various boundary conditions. At q = 0 and m = n, they ultimately yield a new proof of the steady state formula for the n-species totally asymmetric simple exclusion process (TASEP) obtained recently by the authors, revealing the 3D integrability in the matrix product construction.

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.

We consider a three-dimensional (3D) lattice model associated with the intertwiner of the quantized coordinate ring A q (sl 3 ), and introduce a family of layer to layer transfer matrices on m × n square lattice. By using the tetrahedron equation we derive their commutativity and bilinear relations mixing various boundary conditions. At q = 0 and m = n, they lead to a new proof of the steady state probability of the n-species totally asymmetric zero range process obtained recently by the authors, revealing the 3D integrability in the matrix product construction.