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Renormalisation of 2D Cellular Automata with an Absorbing State

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Abstract

We describe a real-space renormalisation scheme for non-equilibrium probabilistic cellular automata (PCA) models, and apply it to a two-dimensional binary PCA. An exact renormalisation scheme is rare, and therefore we provide a method for computing the stationary probability distribution of states for such models with which to weight the renormalisation, effectively minimising the error in the scale transformation. While a mean-field approximation is trivial, we use the principle of maximum entropy to incorporate nearest-neighbour spin-correlations in the steady-state probability distribution. In doing so we find the fixed point of the renormalisation is modified by the steady-state approximation order.

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Acknowledgments

This work was supported by an EPSRC Doctoral Training Centre Grant (EP/G03690X/1).

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Authors and Affiliations

  1. School of Electronics and Computer Science, University of Southampton, University Road, Southampton, SO17 1BJ, UK

    Iain S. Weaver & Adam Prügel-Bennett

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  1. Iain S. Weaver
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  2. Adam Prügel-Bennett
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Correspondence to Iain S. Weaver.

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Weaver, I.S., Prügel-Bennett, A. Renormalisation of 2D Cellular Automata with an Absorbing State. J Stat Phys 159, 211–220 (2015). https://doi.org/10.1007/s10955-014-1142-4

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  • DOI: https://doi.org/10.1007/s10955-014-1142-4

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