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Fractional Brownian motion as a rough surface

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  • Cheraghalizadeh, Jafar
  • Valizadeh, Neda
  • Tizdast, Susan
  • Najafi, Morteza N.

Abstract

In this study we consider two dimensional fractional Brownian motion (fBM) as a rough surface, tuned by the Hurst exponent H, with a focus on the scaling relations. While this system fulfills partially the requirements for self-similar Gaussian surfaces, (like Gaussian height distribution), it shows deviations for the global features. A thorough examination of the Kondev’s hyperscaling relations is presented. The global features associated with level lines which define contour loop ensemble (CLE) are studied in terms of H. We show that in the thermodynamic limit the fractal dimension of loops (Df), the critical exponent of the distribution of loop length (τl), and the gyration radius (τr) vary linearly with H. The hyperscaling relations between these quantities however challenge this hypothesis. While Df, τl and τr fulfill partially the Kondev relations, the loop correlation exponent xl depends weakly on H, i.e. shows deviations from 12 which is hypothesized by Kondev to be super-universal for the Gaussian surfaces.

Suggested Citation

  • Cheraghalizadeh, Jafar & Valizadeh, Neda & Tizdast, Susan & Najafi, Morteza N., 2024. "Fractional Brownian motion as a rough surface," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 646(C).
    • Handle: RePEc:eee:phsmap:v:646:y:2024:i:c:s0378437124004163
    DOI: 10.1016/j.physa.2024.129907
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    References listed on IDEAS

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    1. Najafi, M.N. & Ghaedi, M. & Moghimi-Araghi, Saman, 2016. "Water propagation in two-dimensional petroleum reservoirs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 445(C), pages 102-111.
    2. Najafi, M.N. & Ghaedi, M., 2015. "Geometrical clusters of Darcy’s reservoir model and Ising universality class," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 427(C), pages 82-91.
    3. Jafar Cheraghalizadeh & Morteza N. Najafi & Hossein Mohammadzadeh, 2018. "Gaussian free field in the background of correlated random clusters, formed by metallic nanoparticles," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 91(5), pages 1-11, May.
    4. Rostek, S. & Schöbel, R., 2013. "A note on the use of fractional Brownian motion for financial modeling," Economic Modelling, Elsevier, vol. 30(C), pages 30-35.
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