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Advances in Unconventional Computing pp 239–266Cite as

Percolation Transition and Related Phenomena in Terms of Grossone Infinity Computations

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Part of the book series: Emergence, Complexity and Computation ((ECC,volume 22))

Abstract

In this chapter, a number of traditional models related to the percolation theory is taken into consideration: site percolation, gradient percolation, and forest-fire model. They are studied by means of a new computational methodology that gives a possibility to work with finite, infinite, and infinitesimal quantities numerically by using a new kind of a computer—the Infinity Computer —introduced recently. It is established that in light of the new arithmetic using grossone-based numerals the phase transition point in site percolation and gradient percolation appears as a critical interval, rather than a critical point. Depending on the ‘microscope’ we use, this interval could be regarded as finite, infinite, or infinitesimal interval. By applying the new approach we show that in vicinity of the percolation threshold we have many different infinite clusters instead of one infinite cluster that appears in traditional considerations. With respect to the cellular automaton forest-fire model, two traditional versions of the model are studied: a real forest-fire model where fire catches adjacent trees in the forest in the step by step manner and a simplified version with instantaneous combustion. By applying the new approach there is observed that in both situations we deal with the same model but with different time resolutions. We show that depending on ‘microscope’ we use, the same cellular automaton forest-fire model reveals either the instantaneous forest combustion or the step by step firing.

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Notes

  1. 1.

    We are reminded that a numeral is a symbol or a group of symbols that represents a number . The difference between numerals and numbers is the same as the difference between words and the things they refer to. A number is a concept that a numeral expresses. The same number can be represented by different numerals. For example, the symbols ‘8’, ‘eight’, and ‘IIIIIIII’ are different numerals, but they all represent the same number..

  2. 2.

    Nowadays not only positive integers but also zero is frequently included in \(\mathbb {N}\). However, since zero has been invented significantly later than positive integers used for counting objects, zero is not include in \(\mathbb {N}\) in this text.

  3. 3.

    For example, if we add only one occupied site in our greed, then p increases by , and that is the smallest step along p we can distinguish in our lattice.

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Acknowledgments

This work was supported in part by grant from the Government of the Russian Federation (contract No. 14.B25.31.0023) and by the Russian Foundation for Basic Research (projects No. 13-05-12102 ofi_m, No. 13-05-01100 A, No. 15-01-06612 A).

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Authors and Affiliations

  1. Institute of Applied Physics of Russian Academy of Science, Nizhni Novgorod, Russia, Lobachevsky State University, Nizhni Novgorod, Russia

    Dmitry I. Iudin

  2. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  3. Lobachevsky State University of Nizhni Novgorod, Nizhni Novgorod, Russia

    Yaroslav D. Sergeyev

Authors
  1. Dmitry I. Iudin
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  2. Yaroslav D. Sergeyev
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Correspondence to Dmitry I. Iudin .

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  1. Unconventional Computing Centre, University of the West of England Unconventional Computing Centre, Bristol, United Kingdom

    Andrew Adamatzky

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Iudin, D.I., Sergeyev, Y.D. (2017). Percolation Transition and Related Phenomena in Terms of Grossone Infinity Computations. In: Adamatzky, A. (eds) Advances in Unconventional Computing. Emergence, Complexity and Computation, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-33924-5_11

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  • DOI: https://doi.org/10.1007/978-3-319-33924-5_11

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