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Self-affinity in the gradient percolation problem.

Alex Hansen1, G George Batrouni, Thomas Ramstad

  • 1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. Alex.Hansen@phys.ntnu.no

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 16, 2007
PubMed
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The solid-on-solid front in 2D gradient percolation exhibits self-affine scaling with a Hurst exponent of 2/3 up to a cutoff length. Beyond this, it behaves like uncorrelated noise, a crossover effect.

Area of Science:

  • Statistical Physics
  • Percolation Theory
  • Complex Systems

Background:

  • Percolation theory studies the formation of connected clusters in random systems.
  • Gradient percolation introduces spatial variation in site/bond occupation probabilities.
  • Understanding the geometric properties of infinite clusters is crucial in phase transitions.

Purpose of the Study:

  • To investigate the scaling properties of the solid-on-solid front of the infinite cluster in 2D gradient percolation.
  • To determine the nature of self-affinity and identify crossover length scales.
  • To clarify the origin of previously observed multiaffinity.

Main Methods:

  • Analysis of the solid-on-solid model in a 2D gradient percolation setting.
  • Calculation of the Hurst exponent to characterize self-affine scaling.

Related Experiment Videos

  • Identification of cutoff lengths and analysis of behavior beyond these scales.

    • Main Results:

    • The solid-on-solid front is self-affine with a Hurst exponent of 2/3 up to a cutoff length scaling as g{-4/7}.
    • Beyond the cutoff, the front position exhibits uncorrelated noise behavior.
    • The self-affine scaling is robust, persisting even after removing local front jumps.

    Conclusions:

    • The study reveals a crossover from self-affine to uncorrelated behavior in the infinite cluster front.
    • Previously reported multiaffinity is attributed to overhang dominance at small distances, a crossover effect.
    • The findings provide a detailed understanding of the geometric scaling in 2D gradient percolation systems.