Inefficiency of the block approximation in diploid Probabilistic Cellular Automata
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910022339919872 |
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| author | Cirillo, Emilio N. M. Vliem, Joram L. Schuricht, Dirk Spitoni, Cristian |
| author_facet | Cirillo, Emilio N. M. Vliem, Joram L. Schuricht, Dirk Spitoni, Cristian |
| contents | We study a probabilistic cellular automaton obtained as a mixture of the additive elementary rules 60 and 102. We prove that, for any finite periodic lattice and for mixing parameter $λ=1/2$, the system almost surely reaches the absorbing all-zero configuration in finitely many steps. In addition, Monte Carlo simulations indicate as well the presence of a zero-density stationary state in a finite interval around $λ=1/2$. Despite this absorbing behavior, both mean-field and block approximation schemes predict a stationary state with non-zero density. This failure, traced to the additive and mirror symmetries of the deterministic components, highlights a fundamental limitation of finite-block approximation in capturing the global dynamics of probabilistic cellular automata. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13839 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inefficiency of the block approximation in diploid Probabilistic Cellular Automata Cirillo, Emilio N. M. Vliem, Joram L. Schuricht, Dirk Spitoni, Cristian Cellular Automata and Lattice Gases Probability 60K35, 37B15 We study a probabilistic cellular automaton obtained as a mixture of the additive elementary rules 60 and 102. We prove that, for any finite periodic lattice and for mixing parameter $λ=1/2$, the system almost surely reaches the absorbing all-zero configuration in finitely many steps. In addition, Monte Carlo simulations indicate as well the presence of a zero-density stationary state in a finite interval around $λ=1/2$. Despite this absorbing behavior, both mean-field and block approximation schemes predict a stationary state with non-zero density. This failure, traced to the additive and mirror symmetries of the deterministic components, highlights a fundamental limitation of finite-block approximation in capturing the global dynamics of probabilistic cellular automata. |
| title | Inefficiency of the block approximation in diploid Probabilistic Cellular Automata |
| topic | Cellular Automata and Lattice Gases Probability 60K35, 37B15 |
| url | https://arxiv.org/abs/2602.13839 |