Harmonic measure for percolation and ising clusters including rare events

Area of Science:

• Statistical physics
• Complex systems analysis
• Fractal geometry

Background:

• Critical percolation clusters and Ising-model Fortuin-Kastelyn clusters are fundamental in understanding phase transitions and complex systems.
• The harmonic measure is a crucial tool for analyzing the geometry and probabilistic properties of these clusters.

Purpose of the Study:

• To compute the harmonic measure of critical percolation and Fortuin-Kastelyn clusters.
• To determine the multifractal D(q) spectrum, including small and negative q values.
• To compare findings with existing theoretical predictions.

Main Methods:

• Utilized a biased random-walk sampling technique.
• Achieved measurement of extremely small probabilities (down to 10^-300).

Main Results:

• Obtained the multifractal D(q) spectrum for both cluster types.
• External hull results align with Duplantier's predictions for percolation, including the -23/24 exponent for harmonic measure distribution.
• Complete hull analysis revealed a consistent probability decay exponent of -1 for both critical percolation and Ising models.

Conclusions:

• The biased random-walk method accurately quantifies harmonic measure in complex fractal systems.
• Theoretical predictions for external hulls are validated, offering insights into percolation theory.
• A universal decay behavior is identified for the complete hull across different critical systems.