Largest cluster in subcritical percolation

Area of Science:

• Statistical Physics
• Complex Systems
• Network Science

Background:

• Percolation theory studies the formation of connected clusters in random networks.
• Understanding the behavior of the largest cluster is crucial for characterizing phase transitions.
• Subcritical percolation (below the critical probability) exhibits unique statistical properties.

Purpose of the Study:

• To investigate the statistical distribution of the largest cluster's size in subcritical percolation on finite lattices.
• To determine how this distribution scales with lattice size (N).
• To connect these findings to extreme value theory and renormalization group methods.

Main Methods:

• Theoretical analysis of the cumulative distribution function for the largest cluster size.
• Derivation of scaling laws for the mean and standard deviation.
• Monte Carlo simulations on large 2D square lattices (up to 30 million sites).

Main Results:

• The cumulative distribution function converges to the Fisher-Tippett (Gumbel) distribution for large N.
• The mean cluster size scales as s(*)(xi) log N, where s(*)(xi) is a crossover size.
• The standard deviation is bounded, with persistent fluctuations, and finite-size scaling is observed.

Conclusions:

• The statistical behavior of the largest cluster in subcritical percolation is well-described by extreme value theory (Gumbel distribution).
• The observed scaling laws and simulation results support a renormalization group approach.
• This study provides a robust framework for understanding large-scale cluster formation in random systems.