A cellular automaton describes a process in which cells evolve
according to a set of rules. Which rule is applied to a specific cell
only depends on the states of the neighboring and the cell itself.
Considering a one-dimensional cellular automaton with finite range,
the positive rates conjecture states that under the presence of
noise the associated stationary measure must be unique. We restrict
ourselves to the case of nearest-neighbor interaction where
simulations suggest that the positive rates conjecture is true. After
discussing a simple criterion to deduce decay of correlations, we show
that the positive rates conjecture is true for almost all
nearest-neighbor cellular automatons. The main tool is comparing a
one-dimensional cellular automaton to a properly chosen
two-dimensional Ising-model. We outline a pathway to resolve the
remaining open cases and formulate a conjecture for general Ising
models with odd interaction.

This presentation is based on collaborative work with Maciej
Gluchowski from the University of Warsaw and Jacob Manaker from UCLA