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Epidemic Automaton and the Eden Model: Various Aspects of Robustness

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Probabilistic Cellular Automata

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 27))

Abstract

The two-dimensional probabilistic cellular automaton Epidemic models the spread of an epidemic without recovering on graph. We discuss some well-known and less well-known properties of Epidemic on a finite grid and its analogous on the infinite square lattice: the Eden model. This survey is intended for non-probabilists and gives a detailed study of the robustness of a cellular automaton with respect to several sources of randomness.

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Notes

  1. 1.

    We write \(f_n=\varTheta (g_n)\) when there exist two positive numbers \(c_1,c_2\) such that, for n large enough, \(c_1g_n\le f_n\le c_2g_n\).

References

  1. Basdevant, A.L., Enriquez, N., Gerin, L., Gouéré, J.B.: The shape of large balls in highly supercritical percolation. Electron. J. Probab. 19, 1–14 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blair-Stahn, N.: First passage percolation and competition models (2010). arXiv:1005.0649

  3. Couronné, O., Enriquez, N., Gerin, L.: Construction of a short path in high dimensional first-passage percolation. Electron. Commun. Probab. 16, 22–28 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cox, J., Durrett, R.: Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9(4), 583–603 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cox, J., Kesten, H.: On the continuity of the time constant of first-passage parcolation. J. Appl. Probab. 18, 809–819 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dhar, D.: First passage percolation in many dimensions. Phys. Lett. A 130, 308–310 (1988)

    Article  MathSciNet  Google Scholar 

  7. Eden, M.: A two-dimensional growth process. In: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, vol. IV, pp. 223–239. University of California Press, Berkeley, Calif (1961)

    Google Scholar 

  8. Fatès, N., Gerin, L.: Examples of fast and slow convergence of 2d asynchronous cellular systems. J. Cell. Autom 4, 323–337 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Fatès, N., Morvan, M., Schabanel, N., Thierry, E.: Fully asynchronous behavior of double-quiescent elementary cellular automata. Theoretical Comput. Sci. 362, 1–16 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Garet, O., Marchand, R.: Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster. ESAIM Probab. Stat. 8, 169–199 (2004) (electronic)

    Google Scholar 

  11. Gerin, L.: Aspects probabilistes des automates cellulaires, et d’autres problèmes en informatique théorique. (2008). Thèse de l’Université Nancy 1

    Google Scholar 

  12. Grimmett, G.: Percolation, vol. 321, 2nd edn. Springer, Berlin (1999)

    Google Scholar 

  13. Hammersley, J.M., Welsh, D.J.A.: First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In: Proceedings of the International Research Seminar Statistical Laboratory, University of California, Berkeley, Calif pp. 61–110. Springer, New York (1965)

    Google Scholar 

  14. Janson, S., Łuczak, T., Rucinski, A.: RanDom Graphs. In: Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York (2000)

    Google Scholar 

  15. Kesten, K.: Aspects of first passage percolation. In: École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Mathematics, vol. 1180, pp. 125–264. Springer, Berlin (1986)

    Google Scholar 

  16. Marchand, R.: Strict inequalities for the time constant in first passage percolation. Ann. Appl. Probab. 12, 1001–1038 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Richardson, D.: Random growth in a tessellation. In: Proceedings of the Cambridge Philosophical Society 74, 515–528 (1973)

    Google Scholar 

  18. Szpankowski, W., Rego, V.: Yet another application of a binomial recurrence order statistics. Computing 43, 401–410 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

I am grateful to two anonymous referees for their careful reading of the first version of this article.

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Authors and Affiliations

  1. CMAP, École Polytechnique, Route de Saclay, 91128, Palaiseau Cedex, France

    Lucas Gerin

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  1. Lucas Gerin
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Correspondence to Lucas Gerin .

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Editors and Affiliations

  1. Laboratoire de Mathématiques et Applications UMR 7348, Université de Poitiers, CNRS, Poitiers, France

    Pierre-Yves Louis

  2. University of Technology, Eindhoven, Noord-Brabant, The Netherlands

    Francesca R. Nardi

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Gerin, L. (2018). Epidemic Automaton and the Eden Model: Various Aspects of Robustness. In: Louis, PY., Nardi, F. (eds) Probabilistic Cellular Automata. Emergence, Complexity and Computation, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-65558-1_12

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  • DOI: https://doi.org/10.1007/978-3-319-65558-1_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-65556-7

  • Online ISBN: 978-3-319-65558-1

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