Positive-rate PCA and IPS with stationary Bernoulli measures are rapidly forgetful

Irène Marcovici; Siamak Taati

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arXiv · 2605.16904

Positive-rate PCA and IPS with stationary Bernoulli measures are rapidly forgetful

Irène Marcovici·Siamak Taati

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Abstract

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 system as a perturbation of another system with noise. The ergodic behaviour results from a competition between the accumulation of randomness due to noise and the diffusion of randomness due to local information exchange. We show that, in two and higher dimensions, the positive-rate probabilistic cellular automata that admit stationary Bernoulli measures are algorithmically indistinguishable from those that do not.

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math.PR cond-mat.stat-mech math-ph math.MP nlin.CG

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Irène Marcovici, Siamak Taati. 2026-05-16. Positive-rate PCA and IPS with stationary Bernoulli measures are rapidly forgetful. https://arxiv.org/abs/2605.16904

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