On the ubiquity of matrix-product states in one-dimensional stochastic processes with boundary interactions

Abstract

Recently it has been shown that the zero-energy eigenstate -- corresponding to the stationary state -- of a stochastic Hamiltonian with nearest-neighbour interaction in the bulk and single-site boundary terms, can always be written in the form of a so-called matrix-product state. We generalize this result to stochastic Hamiltonians with arbitrary, but finite, interaction range. As an application two different particle-hopping models with three-site bulk interaction are studied. For these models which can be interpreted as cellular automata for traffic flow, we present exact solutions for periodic boundary conditions and some suitably chosen boundary interactions.

LaTeX(2e), 12 pages, 2 figures

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• Facilitated Asymmetric Exclusion
• Exact solution of the zero-range process: fundamental diagram of the corresponding exclusion process
• Matrix Product States of Three Families of One-Dimensional Interacting Particle Systems
• Matrix product approach for the asymmetric random average process
• Analytical study on the criticality of the Stochastic Optimal Velocity model
• Exact matrix-product states for parallel dynamics: Open boundaries and excess mass on the ring
• Construction of a matrix product stationary state from solutions of finite size system
• Exact stationary state for a deterministic high speed traffic model with open boundaries
• Particle redistribution and slow decay of correlations in hard-core fluids on a half-driven ladder
• Exclusion Processes with Avalanches
• Stochastic Dynamics of Growing Young Diagrams and Their Limit Shapes
• A new family of exactly solvable disordered reaction-diffusion systems