Monte Carlo investigation of the critical behavior of Stavskaya’s probabilistic cellular automaton

Mendonça, J. Ricardo G.

doi:10.1103/physreve.83.012102

Cited by 13 publications

(7 citation statements)

References 20 publications

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“…A sharp numerical estimation based on a Monte Carlo simulation provided by Mendonça (2011) gives e c ¼ 0:29450. In this reference, the critical exponents of the model are numerically studied and indicate that this phase transition belongs to the directed percolation universality class of critical behaviour.…”

Section: Theoretical Resultsmentioning

confidence: 99%

Supercritical probabilistic cellular automata: how effective is the synchronous updating?

Louis

2015

Nat Comput

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Probabilistic cellular automata generalise CA by implementing in a synchronous way an updating rule defined through a probability. A probabilistic synchronous updating scheme does it mean an efficient parallel evolution mechanism? This article deals with the question of quantifying the effectiveness of the parallel updating. A good indicator of this effectiveness is the fraction of components whose value is updated between two time steps. Two classes of parameterised models are considered. Multiple stationary distributions may occur when an infinite number of interacting components is considered (ergodicity breaks/supercritical regime). As a consequence, these models both exhibit different dynamical regimes in the corresponding case when a finite number of sites are interacting. These two classes' non trivial steady states are of different nature. One is a family of positive rates reversible PCA dynamics. The other one is the Stavskaja PCA dynamics. It exhibits an absorbing state. Thanks to numerical simulations, both these PCA dynamics are shown to behave nearly asynchronous when these phase transition phenomena occur.

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Section: Theoretical Resultsmentioning

confidence: 99%

Supercritical probabilistic cellular automata: how effective is the synchronous updating?

Louis

2015

Nat Comput

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“…The exact value of α * is not known and only theoretical lower and upper bounds or estimates from computer simulations are available. Toom [11] proved that α * ∈ (0.09, 0.323) and Mendonça [5], through computer simulations, estimated α * ≈ 0.29450 (5). We revisit the method used in [11] to improve the lower threshold for α * and show that α * > 0.11.…”

Section: Introductionmentioning

confidence: 96%

An Improved Lower Bound for the Critical Parameter of Stavskaya’s Process

Ramos¹,

S.²,

Rodríguez

et al. 2020

Bull. Aust. Math. Soc.

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We consider Stavskaya’s process, which is a two-state probabilistic cellular automaton defined on a one-dimensional lattice. The state of any vertex depends only on itself and on the state of its right-adjacent neighbour. This process was one of the first multicomponent systems with local interaction for which the existence of a kind of phase transition has been rigorously proved. However, the exact localisation of its critical value remains as an open problem. We provide a new lower bound for the critical value.

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“…By varying the ECA chosen in the mixture, the class of DECA considered in the present manuscript is indeed very rich an includes among the others: the percolation PCA studied in [1] and [16], the noisy additive PCA [11], the Stavskaya's PCA [13] and the directed animals PCA [5].…”

Section: Introductionmentioning

confidence: 99%

Phase transitions in random mixtures of elementary cellular automata

Cirillo,

Nardi,

Spitoni

2021

Preprint

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We investigate one-dimensional Probabilistic Cellular Automata, called Diploid Elementary Cellular Automata (DECA), obtained as random mixture 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 rule. 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 distinguishing trait of our result 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 coherent with numerical studies already published in the scientific literature and also with some rigorous results proven for some specific models.

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Monte Carlo investigation of the critical behavior of Stavskaya’s probabilistic cellular automaton

Cited by 13 publications

References 20 publications

Supercritical probabilistic cellular automata: how effective is the synchronous updating?

Supercritical probabilistic cellular automata: how effective is the synchronous updating?

An Improved Lower Bound for the Critical Parameter of Stavskaya’s Process

Phase transitions in random mixtures of elementary cellular automata

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