A finite variant of the Toom Model

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.