2018/03/22 by Eric Bertin, Éric Bertin, Matthieu Vanicat
Mathematics · Physics and Astronomy · #Algebraic number #Asymmetric simple exclusion process #Geometry #Lattice (music) #Markov chain #Master equation #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Matrix multiplication #Physics #Product (mathematics) #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Statistical physics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech
paper · pdf · doi:10.1088/1751-8121/aac196
11 pages
arxiv created 2018/03/22 · openalex publication_date 2018/05/01 · arxiv updated 2018/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract Many one-dimensional lattice particle models with open boundaries, like the paradigmatic asymmetric simple exclusion process (ASEP), have their stationary states represented in the form of a matrix product, with matrices that do not explicitly depend on the lattice site. In contrast, the stationary state of the open 1D zero-range process (ZRP) takes an inhomogeneous factorized form, with site-dependent probability weights. We show that in spite of the absence of correlations, the stationary state of the open ZRP can also be represented in a matrix product form, where the matrices are site-independent, non-commuting and determined from algebraic relations resulting from the master equation. We recover the known distribution of the open ZRP in two different ways: first, using an explicit representation of the matrices and boundary vectors; second, from the sole knowledge of the algebraic relations satisfied by these matrices and vectors. Finally, an interpretation of the relation between the matrix product form and the inhomogeneous factorized form is proposed within the framework of hidden Markov chains.