Multispecies totally asymmetric zero range process: II. Hat relation and tetrahedron equation

Abstract

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.

Density and current profiles inUq(A2(1))Kuniba1, Mangazeev2 2017Nuclear Physics B4070View full textAdd to dashboardCite

“…, γ n = β n must hold. It means that larger class particles have the priority to jump out, which precisely reproduces the n class totally asymmetric zero range process explored in [12] after reversing the labeling of the classes 1, 2, . .…”

Integrable Structure of Multispecies Zero Range Process

“…However, the formula (6.19) is often more efficient than the fusion procedure practically. It also reveals a hidden 3D structure in a class of R matrices [4] and has led to another application to the multispecies totally asymmetric simple exclusion and zero range processes when ∀ i = 1 and ∀ i = 0 [30,32]. Except for the two cases however, these R matrices do not satisfy the sum-to-unity in general 16 and we have not found an application to stochastic systems.…”

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.…”

“…We note that our weighting scheme can be extended to multiline process used to determine the steady state distribution of the totally asymmetric zero range process (TARZP) on a ring, where multiple particles can occupy the same site. This comes from the fact that the TARZP steady state distribution can also be computed using a tensor product of Kirillov-Reshetikhin crystals (under ranklevel duality) using combinatorial R-matrices with analogous connections to corner transfer matrices and the tetrahedron equation [KMO16c,KMO16d]. Thus, we expect that a similar description of σ-twisted multiline process can be defined such that the weighting is invariant under the action of the combinatorial R-matrix.…”