On the "Matrix Approach" to interacting particle systems

TL;DR

This paper explores the algebraic matrix approach for interacting particle systems, extending it to time evolution for certain models, but also identifies its limitations with models like voter and contact processes.

Contribution

It provides a general recipe to derive correlation formulas and analyzes the applicability of the matrix approach to various interacting particle systems.

Findings

Matrix approach extends to time evolution for some models.

Fails for models like voter and contact processes.

Provides a framework for deriving correlation functions.

Abstract

Derrida et al. and Sch\"{u}tz and Stinchcombe gave algebraic formulas for the correlation functions of the partially asymmetric simple exclusion process. Here we give a fairly general recipe of how to get these formulas and extend them to the whole time evolution (starting from the generator of the process), for a certain class of interacting systems. We then analyze the algebraic relations obtained to show that the matrix approach does not work with some models such as the voter and the contact processes.