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Coastlines and percolation in a model for hierarchical random deposition

Author

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  • Berx, Jonas
  • Bervoets, Evi
  • Giuraniuc, Claudiu V.
  • Indekeu, Joseph O.

Abstract

We revisit a known model in which (conducting) blocks are hierarchically and randomly deposited on a D-dimensional substrate according to a hyperbolic size law with the block size decreasing by a factor λ>1 in each subsequent generation. In the first part of the paper the number of coastal points (in D=1) or coastline segments (in D=2) is calculated, which are points or lines that separate a region at “sea level” and an elevated region. We find that this number possesses a non-universal character, implying a Euclidean geometry below a threshold value Pc of the deposition probability P, and a fractal geometry above this value. Exactly at the threshold, the geometry is logarithmic fractal. The number of coastline segments in D=2 turns out to be exactly twice the number of coastal points in D=1. We comment briefly on the surface morphology and derive a roughness exponent α. In the second part, we study the percolation probability for a current in this model and two extensions of it, in which both the scale factor and the deposition probability can take on different values between generations. We find that the percolation threshold Pc is located at exactly the same value for the deposition probability as the threshold probability of the number of coastal points. This coincidence suggests that exactly at the onset of percolation for a conducting path, the number of coastal points exhibits logarithmic fractal behaviour.

Suggested Citation

  • Berx, Jonas & Bervoets, Evi & Giuraniuc, Claudiu V. & Indekeu, Joseph O., 2021. "Coastlines and percolation in a model for hierarchical random deposition," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 574(C).
    • Handle: RePEc:eee:phsmap:v:574:y:2021:i:c:s0378437121002703
    DOI: 10.1016/j.physa.2021.125998
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    References listed on IDEAS

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    1. Amnon Aharony & Dietrich Stauffer, 2002. "Scaling In A Simple Model For Surface Growth In A Random Medium," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 13(05), pages 603-612.
    2. Indekeu, J.O & Fleerackers, G & Posazhennikova, A.I & Bervoets, E, 2000. "Exotic marginal fractals from hierarchical random deposition," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 285(1), pages 135-146.
    3. Posazhennikova, Anna I. & Indekeu, Joseph O., 2014. "Magnetic hierarchical deposition," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 414(C), pages 240-248.
    4. Indekeu, Joseph O. & Fleerackers, Gunther, 1998. "Logarithmic fractals and hierarchical deposition of debris," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 261(3), pages 294-308.
    5. Indekeu, J.O. & Giuraniuc, C.V., 2004. "Cellular automaton for bacterial towers," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 336(1), pages 14-26.
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    Cited by:

    1. Xinyu Hu & Yidian Wang & Hui Wang & Yi Shi, 2022. "Hierarchical Structure of the Central Areas of Megacities Based on the Percolation Theory—The Example of Lujiazui, Shanghai," Sustainability, MDPI, vol. 14(16), pages 1-20, August.

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