# Matrix Product representation of the stationary state of the open Zero Range Process

## TL;DR

This paper demonstrates that the stationary state of the open Zero-Range Process can be represented using a site-independent matrix product form, linking it to algebraic relations and hidden Markov chains.

## Contribution

It introduces a novel matrix product representation for the open ZRP's stationary state, which was previously known to have an inhomogeneous factorized form.

## Key findings

- Matrix product form with site-independent matrices derived from algebraic relations.
- Explicit matrix and boundary vector representations of the stationary state.
- Connection established between matrix product form and hidden Markov chains.

## 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 one-dimensional 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.