Universal features of cluster numbers in percolation

Area of Science:

• Statistical Physics
• Complex Systems
• Materials Science

Background:

• Percolation theory describes the formation of clusters in random systems.
• The number of clusters per site, n(p), at the critical point (p=p_c) is generally not a universal quantity.
• Properties of n(p), such as finite-size corrections and scaling behavior, can exhibit universality.

Purpose of the Study:

• To elucidate the different levels of universality for n(p) properties.
• To investigate the dependence of universality on system shape versus lattice type.
• To propose a new criterion for an absolute metric factor based on scaling function behavior.

Main Methods:

• Theoretical analysis.
• Extensive studies on 2D and 3D systems.
• High-order series analysis, Monte Carlo simulations, and exact enumeration.
• Utilizing the Sykes-Essam matching polynomial.

Main Results:

• Precise values for n(p_c) were determined for several systems.
• A clear demonstration of the singularity in the second derivative of n(p), n''(p).
• Identification and analysis of metric scale factors.
• Exact relations between lattice and matching lattice properties were derived.

Conclusions:

• Universality in percolation cluster properties is multifaceted, with some aspects depending on system shape.
• The proposed criterion for an absolute metric factor offers a novel approach.
• The study provides a comprehensive understanding of universality in n(p).