A finite variant of the Toom Model

TL;DR

This paper studies a finite version of the Toom model with closed boundaries, providing explicit formulas for steady state densities, correlation functions, and exact eigenvalues, revealing a simple yet non-product form distribution.

Contribution

It introduces a finite variant of the Toom model and derives explicit formulas for steady state properties and eigenvalues using algebraic methods.

Findings

Steady state distribution is non-product but simple.

Explicit formulas for densities and correlations.

Exact eigenvalues and multiplicities of the transition matrix.

Abstract

We present results for a finite variant of the one-dimensional Toom model with closed boundaries. We show that the steady state distribution is not of product form, but is nonetheless simple. In particular, we give explicit formulas for the densities and some nearest neighbour correlation functions. We also give exact results for eigenvalues and multiplicities of the transition matrix using the theory of ${\mathscr R}$-trivial monoids in joint work with A. Schilling, B. Steinberg and N. M. Thi\'ery.