Duality with real-space renormalization and its application to bond percolation

Area of Science:

• Statistical Physics
• Complex Systems
• Materials Science

Background:

• Percolation theory studies the connectivity of random networks.
• Understanding bond-percolation thresholds is crucial for various physical phenomena.
• Inhomogeneous probabilities introduce complexity to standard percolation models.

Purpose of the Study:

• To derive the exact solution for bond-percolation thresholds on a square lattice with inhomogeneous probabilities.
• To present a more straightforward formulation compared to existing methods.
• To develop generic formulas applicable to other lattice structures.

Main Methods:

• The study employs duality analysis, a technique originating from spin-glass theory.
• Real-space renormalization is utilized as a core component of the methodology.
• The approach focuses on a direct derivation of the percolation thresholds.

Main Results:

• Exact solutions for bond-percolation thresholds on inhomogeneous square lattices were obtained.
• A novel, simplified formulation of the problem was established.
• Generic formulas were derived, extending beyond the square lattice.

Conclusions:

• The developed method provides an exact and efficient way to determine bond-percolation thresholds.
• The findings offer a valuable tool for analyzing complex systems with varying connectivity probabilities.
• The generic formulas pave the way for estimating percolation thresholds on diverse lattice types.