Phase transitions in random mixtures of elementary cellular automata

Emilio N.M. Cirillo; Francesca R. Nardi; Cristian Spitoni

doi:10.1016/j.physa.2021.125942

Phase transitions in random mixtures of elementary cellular automata

Emilio N.M. Cirillo, Francesca R. Nardi, Cristian Spitoni^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

We investigate one-dimensional probabilistic cellular automata, called Diploid Elementary Cellular Automata (DECA), obtained as random mixtures of two different elementary cellular automata rules. All the cells are updated synchronously and the probability for one cell to be 0 or 1 at time t depends only on the value of the same cell and that of its neighbors at time t−1. These very simple models show a very rich behavior strongly depending on the choice of the two elementary cellular automata that are randomly mixed together and on the parameter which governs probabilistically the mixture. In particular, we study the existence of phase transition for the whole set of possible DECA obtained by mixing the null rule which associates 0 to any possible local configuration, with any of the other 255 elementary rules. We approach the problem analytically via a mean field approximation and via the use of a rigorous approach based on the application of the Dobrushin criterion. The main feature of our approach is the possibility to describe the behavior of the whole set of considered DECA without exploiting the local properties of the individual models. The results that we find are consistent with numerical studies already published in the scientific literature and also with some rigorous results proven for some specific models.

Original language	English
Article number	125942
Pages (from-to)	1-16
Number of pages	16
Journal	Physica A: Statistical Mechanics and its Applications
Volume	573
DOIs	https://doi.org/10.1016/j.physa.2021.125942
Publication status	Published - 1 Jul 2021

Bibliographical note

Funding Information:
The authors thank R. Fernandez and N. Fatès for very useful discussions. ENMC expresses his thanks to the Mathematics Department of the Utrecht University for kind hospitality and STAR for financial support. The research of Francesca R. Nardi was partially supported by the NWO Gravitation, Italy Grant 024.002.003 –NETWORKS and by the PRIN, Italy Grant 20155PAWZB Large Scale Random Structures.

Publisher Copyright:
© 2021 The Author(s)

Funding

The authors thank R. Fernandez and N. Fatès for very useful discussions. ENMC expresses his thanks to the Mathematics Department of the Utrecht University for kind hospitality and STAR for financial support. The research of Francesca R. Nardi was partially supported by the NWO Gravitation, Italy Grant 024.002.003 –NETWORKS and by the PRIN, Italy Grant 20155PAWZB Large Scale Random Structures.

Keywords

First hitting times
Mean field
Probabilistic cellular automata
Stationary measures
Synchronization

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Cite this

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title = "Phase transitions in random mixtures of elementary cellular automata",

abstract = "We investigate one-dimensional probabilistic cellular automata, called Diploid Elementary Cellular Automata (DECA), obtained as random mixtures of two different elementary cellular automata rules. All the cells are updated synchronously and the probability for one cell to be 0 or 1 at time t depends only on the value of the same cell and that of its neighbors at time t−1. These very simple models show a very rich behavior strongly depending on the choice of the two elementary cellular automata that are randomly mixed together and on the parameter which governs probabilistically the mixture. In particular, we study the existence of phase transition for the whole set of possible DECA obtained by mixing the null rule which associates 0 to any possible local configuration, with any of the other 255 elementary rules. We approach the problem analytically via a mean field approximation and via the use of a rigorous approach based on the application of the Dobrushin criterion. The main feature of our approach is the possibility to describe the behavior of the whole set of considered DECA without exploiting the local properties of the individual models. The results that we find are consistent with numerical studies already published in the scientific literature and also with some rigorous results proven for some specific models.",

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author = "Cirillo, {Emilio N.M.} and Nardi, {Francesca R.} and Cristian Spitoni",

note = "Funding Information: The authors thank R. Fernandez and N. Fat{\`e}s for very useful discussions. ENMC expresses his thanks to the Mathematics Department of the Utrecht University for kind hospitality and STAR for financial support. The research of Francesca R. Nardi was partially supported by the NWO Gravitation, Italy Grant 024.002.003 –NETWORKS and by the PRIN, Italy Grant 20155PAWZB Large Scale Random Structures. Publisher Copyright: {\textcopyright} 2021 The Author(s)",

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N2 - We investigate one-dimensional probabilistic cellular automata, called Diploid Elementary Cellular Automata (DECA), obtained as random mixtures of two different elementary cellular automata rules. All the cells are updated synchronously and the probability for one cell to be 0 or 1 at time t depends only on the value of the same cell and that of its neighbors at time t−1. These very simple models show a very rich behavior strongly depending on the choice of the two elementary cellular automata that are randomly mixed together and on the parameter which governs probabilistically the mixture. In particular, we study the existence of phase transition for the whole set of possible DECA obtained by mixing the null rule which associates 0 to any possible local configuration, with any of the other 255 elementary rules. We approach the problem analytically via a mean field approximation and via the use of a rigorous approach based on the application of the Dobrushin criterion. The main feature of our approach is the possibility to describe the behavior of the whole set of considered DECA without exploiting the local properties of the individual models. The results that we find are consistent with numerical studies already published in the scientific literature and also with some rigorous results proven for some specific models.

AB - We investigate one-dimensional probabilistic cellular automata, called Diploid Elementary Cellular Automata (DECA), obtained as random mixtures of two different elementary cellular automata rules. All the cells are updated synchronously and the probability for one cell to be 0 or 1 at time t depends only on the value of the same cell and that of its neighbors at time t−1. These very simple models show a very rich behavior strongly depending on the choice of the two elementary cellular automata that are randomly mixed together and on the parameter which governs probabilistically the mixture. In particular, we study the existence of phase transition for the whole set of possible DECA obtained by mixing the null rule which associates 0 to any possible local configuration, with any of the other 255 elementary rules. We approach the problem analytically via a mean field approximation and via the use of a rigorous approach based on the application of the Dobrushin criterion. The main feature of our approach is the possibility to describe the behavior of the whole set of considered DECA without exploiting the local properties of the individual models. The results that we find are consistent with numerical studies already published in the scientific literature and also with some rigorous results proven for some specific models.

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Phase transitions in random mixtures of elementary cellular automata

Abstract

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Keywords

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