Faster calculation of the percolation correlation length on spatial networks

Area of Science:

• Physics
• Statistical Mechanics
• Network Science

Background:

• Percolation theory is crucial for understanding phase transitions in diverse systems.
• The correlation length (ξ) at criticality is a key parameter, but its measurement in 2D systems is computationally challenging.
• Existing disjoint set algorithms accelerate percolation calculations but cannot determine ξ.

Purpose of the Study:

• To develop a novel algorithm for precisely measuring the correlation length (ξ) during percolation.
• To integrate correlation length measurement into efficient disjoint set-based percolation algorithms.
• To enable the study of critical phenomena in various spatial network topologies.

Main Methods:

• Utilizing the parallel axis theorem to track the correlation length.
• Implementing a single-sweep algorithm that leverages disjoint sets for efficiency.
• Applying the method to both lattice structures and general spatial networks.

Main Results:

• The algorithm accurately measures the correlation length (ξ) with arbitrary precision.
• It enables the calculation of ξ for the entire percolation process in a single pass.
• The method is applicable to lattices and complex spatial network topologies.

Conclusions:

• This new algorithm provides an efficient and precise tool for measuring the correlation length in percolation.
• It enhances the study of critical phenomena in 2D spatial systems and networks.
• The approach facilitates a deeper understanding of phase transitions in complex systems.