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Coastlines violate the Schramm–Loewner Evolution

Author

Listed:
  • Abril, Leidy M.L.
  • Oliveira, Erneson A.
  • Moreira, André A.
  • Andrade, José S.
  • Herrmann, Hans J.

Abstract

Mandelbrot’s empirical observation that the coast of Britain is fractal has been confirmed by many authors, but it can be described by the Schramm–Loewner Evolution? Since the self-affine surface of our planet has a positive Hurst exponent, one would not expect a priori any critical behavior. Here, we investigate numerically the roughness and fractal dimension of the isoheight lines of real and artificial landscapes. Using a novel algorithm to take into account overhangs, we find that the roughness exponent of isoheight lines is consistent with unity regardless of the Hurst exponent of the rough surface. Moreover, the effective fractal dimension of the iso-height lines decays linearly with the Hurst exponent of the surface. We perform several tests to verify if the complete and accessible perimeters would follow the Schramm–Loewner Evolution and find that the left passage probability test is clearly violated, implying that coastlines violate SLE.

Suggested Citation

  • Abril, Leidy M.L. & Oliveira, Erneson A. & Moreira, André A. & Andrade, José S. & Herrmann, Hans J., 2024. "Coastlines violate the Schramm–Loewner Evolution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 653(C).
    • Handle: RePEc:eee:phsmap:v:653:y:2024:i:c:s0378437124005752
    DOI: 10.1016/j.physa.2024.130066
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    References listed on IDEAS

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    1. Zhan, Dapeng, 2019. "Decomposition of Schramm–Loewner evolution along its curve," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 129-152.
    2. Vicsek, Tamás & Cserző, Miklós & Horváth, Viktor K., 1990. "Self-affine growth of bacterial colonies," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 167(2), pages 315-321.
    3. Dietler, Giovanni & Zhang, Yi-Cheng, 1992. "Fractal aspects of the Swiss landscape," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 191(1), pages 213-219.
    4. N. Posé & K. J. Schrenk & N. A. M. Araújo & H. J. Herrmann, 2018. "Schramm–Loewner evolution of the accessible perimeter of isoheight lines of correlated landscapes," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 29(01), pages 1-10, January.
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