Multispecies TASEP and the tetrahedron equation

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.

18 pages

References in corpus (5)

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Cited by in corpus (24)

• Multispecies TASEP and combinatorial $R$
• Exact solution to integrable open multi-species SSEP and macroscopic fluctuation theory
• Colored five-vertex models and Lascoux polynomials and atoms
• Integrable boundary conditions for multi-species ASEP
• Matrix product solution to a 2-species TASEP with open integrable boundaries
• Double Grothendieck polynomials and colored lattice models
• Combinatorial mappings of exclusion processes
• Multispecies totally asymmetric zero range process: I. Multiline process and combinatorial $R$
• Matrix product solution to multi-species ASEP with open boundaries
• Matrix product solutions to the reflection equation from three dimensional integrability
• Inhomogeneous generalization of multispecies totally asymmetric zero range process
• Limiting current distribution for a two species asymmetric exclusion process
• Matrix product formula for $U_q(A_2^{(1)})$-zero range process
• KPZ fluctuations in finite volume
• Transition probability and total crossing events in the multi-species asymmetric exclusion process
• Quantum spin chains from Onsager algebras and reflection $K$-matrices
• Multiline queues with spectral parameters
• Matrix product construction for Koornwinder polynomials and fluctuations of the current in the open ASEP
• Integrable Structure of Multispecies Zero Range Process
• Tetrahedron equation and Schur functions
• Quasi-solvable lattice models for ${\mathrm{Sp}}_{2n}$ and ${\mathrm{SO}}_{2n+1}$ Demazure atoms and characters
• Multispecies totally asymmetric zero range process: II. Hat relation and tetrahedron equation
• Exactly Solved Models and Beyond: a special issue in honour of R J Baxter's 75th birthday
• Factorization of rational six vertex model partition functions