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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