Positive-rate PCA and IPS with stationary Bernoulli measures are rapidly forgetful
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910226119131136 |
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| author | Marcovici, Irène Taati, Siamak |
| author_facet | Marcovici, Irène Taati, Siamak |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_16904 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Positive-rate PCA and IPS with stationary Bernoulli measures are rapidly forgetful Marcovici, Irène Taati, Siamak Probability Statistical Mechanics Mathematical Physics Cellular Automata and Lattice Gases 60K35, 82C22, 82C20, 68Q80 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. |
| title | Positive-rate PCA and IPS with stationary Bernoulli measures are rapidly forgetful |
| topic | Probability Statistical Mechanics Mathematical Physics Cellular Automata and Lattice Gases 60K35, 82C22, 82C20, 68Q80 |
| url | https://arxiv.org/abs/2605.16904 |