Multispecies TASEP and the tetrahedron equation

TL;DR

This paper introduces a 3D lattice model framework that unifies multispecies TASEP and demonstrates its integrability through the tetrahedron equation, providing new insights and proofs for steady state formulas.

Contribution

It develops a novel 3D integrable lattice model approach to multispecies TASEP, connecting tetrahedron equation with matrix product states and steady state analysis.

Findings

Established commutativity of transfer matrices via tetrahedron equation

Derived bilinear relations with various boundary conditions

Provided a new proof of the steady state formula for multispecies TASEP

Abstract

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 $m \times n$ 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.