A perturbed cellular automaton with two phase transitions for the ergodicity
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916825944555520 |
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| author | Marsan, Hugo Sablik, Mathieu Törmä, Ilkka |
| author_facet | Marsan, Hugo Sablik, Mathieu Törmä, Ilkka |
| contents | The positive rates conjecture states that a one-dimensional probabilistic cellular automaton (PCA) with strictly positive transition rates must be ergodic. The conjecture has been refuted by Gács, whose counterexample is a cellular automaton that is non-ergodic under uniform random noise with sufficiently small rate. For all known counterexamples, non-ergodicity has been proved under small enough rates. Conversely, all cellular automata are ergodic with sufficiently high-rate noise. No other types of phase transitions of ergodicity are known, and the behavior of known counterexamples under intermediate noise rates is unknown.
We present an example of a cellular automaton with two phase transitions. Using Gács's result as a black box, we construct a cellular automaton that is ergodic under small noise rates, non-ergodic for slightly higher rates, and again ergodic for rates close to 1. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03485 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A perturbed cellular automaton with two phase transitions for the ergodicity Marsan, Hugo Sablik, Mathieu Törmä, Ilkka Cellular Automata and Lattice Gases Dynamical Systems Probability 60K35, 60J05, 37B15, 37A50 The positive rates conjecture states that a one-dimensional probabilistic cellular automaton (PCA) with strictly positive transition rates must be ergodic. The conjecture has been refuted by Gács, whose counterexample is a cellular automaton that is non-ergodic under uniform random noise with sufficiently small rate. For all known counterexamples, non-ergodicity has been proved under small enough rates. Conversely, all cellular automata are ergodic with sufficiently high-rate noise. No other types of phase transitions of ergodicity are known, and the behavior of known counterexamples under intermediate noise rates is unknown. We present an example of a cellular automaton with two phase transitions. Using Gács's result as a black box, we construct a cellular automaton that is ergodic under small noise rates, non-ergodic for slightly higher rates, and again ergodic for rates close to 1. |
| title | A perturbed cellular automaton with two phase transitions for the ergodicity |
| topic | Cellular Automata and Lattice Gases Dynamical Systems Probability 60K35, 60J05, 37B15, 37A50 |
| url | https://arxiv.org/abs/2507.03485 |