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Random interfaces generated by the addition of structures of variable size.

Nicolas Pétrélis1, François Pétrélis2

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This study introduces a new class of random interfaces formed by depositing structures with heavy-tailed distributions. These interfaces exhibit universal self-affine behavior, potentially explaining correlations in phenomena like earthquakes.

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Materials Science

Background:

  • Random interfaces are fundamental in various scientific fields.
  • Understanding the statistical properties of these interfaces is crucial for modeling complex phenomena.
  • Previous models often assumed simpler distributions for deposited structures.

Purpose of the Study:

  • To investigate the statistical behavior of random interfaces generated by depositing objects with heavy-tailed width distributions.
  • To identify universal asymptotic behaviors and critical transitions in these random interfaces.
  • To elucidate the mechanism behind long-range correlations in such systems.

Main Methods:

  • Exact solvability analysis for randomly centered structure deposition.
  • Focus on heavy-tailed distributions for structure width.
  • Identification of key parameters governing asymptotic behavior: structure shape, width distribution exponent, and height-width relationship exponent.

Main Results:

  • The interface generically evolves towards a self-affine random curve.
  • Asymptotic behavior is universal, depending on three key parameters.
  • Transitions in the parameter space lead to different asymptotic behaviors, including fractional Brownian motion.
  • A new class of robust random interfaces is identified.

Conclusions:

  • The study reveals a new class of random interfaces with robust properties.
  • The identified mechanism explains long-range correlations relevant to fields like earthquake physics and energy propagation.
  • The universality of the asymptotic behavior offers a powerful framework for diverse applications.