Positive-rate PCA and IPS with stationary Bernoulli measures are rapidly forgetful (opens in new tab)

We prove that every probabilistic cellular automaton with strictly positive transition probabilities that admits a stationary Bernoulli measure is exponentially ergodic. Moreover, the mixing time of any finite region in such a system is logarithmic in the diameter of the region. A similar result holds in continuous time for positive-rate, finite-range interacting particle systems. The proofs use entropy, and rely on a representation of the sys...