Critical exponents for isotropically directed percolation on hierarchical lattices

Area of Science:

• Statistical Physics
• Complex Networks
• Percolation Theory

Background:

• Directed percolation is a critical phenomenon studied on various lattices.
• Hierarchical lattices possess self-similar structures and complex connectivity.
• Understanding critical exponents is crucial for characterizing phase transitions.

Purpose of the Study:

• To develop a general renormalization-group approach for calculating critical exponents in directed percolation on hierarchical lattices.
• To apply this method to analyze isotropically directed percolation, determining exponents for giant components.
• To investigate percolation on small-world hierarchical lattices, including those with long-range bonds.

Main Methods:

• Renormalization-group calculations
• Monte Carlo simulations
• Exact determination of critical exponents using a novel renormalization-group approach

Main Results:

• A general renormalization-group method was established for exact critical exponent determination.
• Two critical exponents, β_{scc} and β_{out}, were identified for isotropically directed percolation.
• The method was successfully applied to diverse hierarchical lattices, including small-world networks.

Conclusions:

• The developed renormalization-group approach offers a simpler and exact alternative to existing methods.
• The study provides a framework for analyzing critical phenomena on complex hierarchical structures.
• Explosive percolation was observed in small-world hierarchical lattices with long-range bonds.