Evolving fractal dimensions in iterative bicolored percolation

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.