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Stochastic Loewner evolution relates anomalous diffusion and anisotropic percolation.

Heitor F Credidio1, André A Moreira1, Hans J Herrmann1,2

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Anisotropic percolation perimeters originate from the stochastic Loewner evolution (SLE) process. This study links anomalous diffusion and fractal anisotropy, revealing new interpretations for non-Markovian processes.

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Fractal Geometry

Background:

  • Percolation theory describes phase transitions in random systems.
  • Anisotropic fractal structures exhibit direction-dependent properties.
  • Stochastic Loewner Evolution (SLE) models random curves and interfaces.

Purpose of the Study:

  • To elucidate the origin of anisotropic percolation perimeters.
  • To connect anomalous diffusion with fractal anisotropy using SLE.
  • To explore mathematical and physical interpretations of non-Markovian processes.

Main Methods:

  • Extensive numerical simulations of percolation clusters.
  • Analysis of scaling limits using the stochastic Loewner evolution (SLE) process.
  • Utilizing fractional Brownian motion to model anomalous diffusion.

Main Results:

  • Perimeters of multilayered and directed percolation clusters are SLE scaling limits of anomalous Brownian motion.
  • Superdiffusive and subdiffusive behaviors were observed for different cluster types.
  • Anisotropic traces were generated using long-range correlated time series, confirming the hypothesis.

Conclusions:

  • Anisotropic percolation perimeters are fundamentally linked to the SLE of anomalous diffusion processes.
  • The study provides novel insights into the interplay between fractal anisotropy and non-Markovian dynamics.
  • This work bridges mathematical and physical perspectives on critical phenomena and random processes.