Exponential ergodicity and the space-time Gibbs states of one-dimensional probabilistic cellular automata

A positive-rate probabilistic cellular automaton (PCA) defines a Gibbs specification in one more dimension, its space–time interaction, in which the future is a boundary like the past and the sides. For finite-alphabet, finite-radius, strictly positive PCA on ℤ we prove that uniform exponential ergodicity of the dynamics is equivalent to exponential weak spatial mixing of this specification in arbitrary finite volumes of ℤ² under arbitrary exterior conditions, so that the specification has exactly one DLR state, the stationary trajectory law, with exponentially decaying correlations. This answers in one dimension, under exponential ergodicity, the question of Lebowitz, Maes and Speer whether every Gibbs measure of the space–time interaction is a trajectory law of the PCA, previously known for periodic measures. In every dimension and with no rate, the relative-entropy defect of any Gibbs measure from the dynamics is a surface term; hence for an ergodic PCA the stationary trajectory is the only uniformly recurrent Gibbs measure, the translation action on the Gibbs simplex is uniquely ergodic, and all-DLR uniqueness holds exactly when every Gibbs measure is uniformly recurrent. The forward proof removes the far future boundary from Bérard's coupling from the past at an entropy cost that only one-dimensional geometry absorbs. Finite certificates for the hypothesis are exactly computable; for the noisy majority rule they give a unique space–time Gibbs state down to noise 0.485, including noise one half, and Marsan, Sablik and Törmä's fixed radius-one automaton has one space–time Gibbs state at small noise (granting structural estimates from their low-noise proof) and at large noise, and several in between.