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Phase Diagrams of Majority Voter Probabilistic Cellular Automata

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Abstract

We present a detailed numerical analysis of the phase diagrams for some majority voter probabilistic cellular automata and connect the results with theory, enabling us to prove many of the observed features.

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Acknowledgments

The research of the authors has been funded by The Alfred P. Sloan Foundation, New York.

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Authors and Affiliations

  1. College of Engineering Mathematics and Physical Sciences, University of Exeter, Harrison Building, Streatham Campus, Exeter, EX4 4QF, UK

    Piotr Słowiński

  2. Centre for Complexity Science and Mathematics Institute, University of Warwick, Zeeman Building, Coventry, CV4 7AL, UK

    Robert S. MacKay

Authors
  1. Piotr Słowiński
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Correspondence to Piotr Słowiński.

Appendix: Connections Between Percolation Operators and Contact Process

Appendix: Connections Between Percolation Operators and Contact Process

An interesting model worth studying together with the case of \(p=0\) or \(q=0\) of the NEC PCA is the contact process. The contact process with transition rates \(\omega _r^x\) for the state \(x\) on site \(r\)

$$\begin{aligned} \omega _r^x = {\left\{ \begin{array}{ll} 1 \rightarrow 0, \text{ with } \text{ rate } 1,\\ 0 \rightarrow 1, \text{ with } \text{ rate } \lambda {\sum }_{s \in \eta _r}{x_s}, \end{array}\right. } \end{aligned}$$
(22)

defined as in [12], is a continuous-time analogue of the percolation operators. By analysing the transition rates of the contact process one can observe that the contact process can be considered as a percolation operator with noise rate depending on the neighbourhood. More specifically, the state of site \(r\) can change to 1 from 0 only if there is at least one site in state 1 in its neighbourhood; this is exactly as for the case of the maximum function and the percolation operators. For both processes the all 0 state is absorbing.

Furthermore, if we compare parameters of continuous-time “well-stirred” approximations of the Stavskaya operator and the contact process in dimension one with one neighbour (\(n=2\)), we see that the parameter of the Stavskaya operator is a compactified analogue of the parameter of the contact process equation.

$$\begin{aligned} \frac{d\theta _S}{dt}&=(1-p) \theta _S(1-\theta _S)-p \theta _S,\nonumber \\ \frac{d\theta _C}{dt}&=\lambda \theta _C(1-\theta _C)-\theta _C,\end{aligned}$$
(23)
$$\begin{aligned} p&=\dfrac{1}{\lambda +1},\nonumber \\ \lambda&=0 \rightarrow p=1,\nonumber \\ \lambda&=\infty \rightarrow p=0. \end{aligned}$$
(24)

Here \(\theta _S\) and \(\theta _C\) are the proportions of 1s in the state of the Stavskaya process and in the contact process, respectively. For a formal comparison of discrete and continuous time models see [10].

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Słowiński, P., MacKay, R.S. Phase Diagrams of Majority Voter Probabilistic Cellular Automata. J Stat Phys 159, 43–61 (2015). https://doi.org/10.1007/s10955-014-1156-y

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