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Evolving fractal dimensions in iterative bicolored percolation.

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Summary
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This study introduces a novel percolation process that maintains criticality and modifies fractal dimensions across generations. This offers a new geometric mechanism for understanding scale invariance in critical systems.

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Geometric Probability

Background:

  • Criticality is often viewed as a sensitive fixed point in renormalization group theory.
  • Understanding systems that maintain scale invariance across different scales is crucial.

Purpose of the Study:

  • To introduce an iterative bicolored percolation process.
  • To demonstrate its ability to preserve criticality and alter fractal dimensions.
  • To explore the geometric mechanisms behind evolving fractal dimensions.

Main Methods:

  • Developed an iterative bicolored percolation process in two dimensions.
  • Utilized the conformal loop ensemble to derive exact fractal dimensions.
  • Performed large-scale Monte Carlo simulations for confirmation.

Main Results:

  • The process generates a hierarchy of critical generations from initial critical configurations.
  • Exact, generation-dependent fractal dimensions were derived and confirmed.
  • The evolutionary path depends on the initial state's universality class and critical structure.

Conclusions:

  • Established a general geometric mechanism for evolving fractal dimensions while preserving scale invariance.
  • Demonstrated that criticality can persist through successive coarse-graining steps.
  • Highlighted the role of initial state properties in determining critical exponents.