Construction of a matrix product stationary state from solutions of finite size system

TL;DR

This paper presents a method to construct matrix product stationary states for stochastic models with finite states per site, providing conditions and verification techniques for their validity across system sizes.

Contribution

It introduces a necessary condition for finite-dimensional matrix product states and a construction method from small system solutions, applicable when N ≤ M.

Findings

Derived a necessary condition for finite M-dimensional matrix product states.

Provided a construction method from small system stationary states.

Validated the method with examples including the asymmetric exclusion process.

Abstract

Stationary states of stochastic models, which have $N$ states per site, in matrix product form are considered. First we give a necessary condition for the existence of a finite $M$-dimensional matrix product state for any ${N,M}$. Second, we give a method to construct the matrices from the stationary states of small size system when the above condition and $N\le M$ are satisfied. Third, the method by which one can check that the obtained matrices are valid for any system size is presented for the case where $M=N$ is satisfied. The application of our methods is explained using three examples: the asymmetric exclusion process, a model studied in [F. H. Jafarpour: J. Phys. A: Math. Gen. 36 (2003) 7497] and a hybrid of both of the models.