Critical percolation clusters in seven dimensions and on a complete graph

Area of Science:

• Statistical Physics
• Complex Systems
• Network Science

Background:

• Percolation theory studies the formation of connected clusters in random networks.
• Understanding cluster geometry is crucial for diverse fields, from materials science to epidemiology.
• High-dimensional lattices and complete graphs represent distinct network topologies.

Purpose of the Study:

• To investigate and compare critical bond percolation on a seven-dimensional hypercubic lattice and a complete graph.
• To analyze the scaling properties and fractal dimensions of different types of clusters (whole, leaf-free, bridge-free).
• To elucidate the differences in geometric structures between high-dimensional percolation and complete graph percolation.

Main Methods:

• Numerical simulations of critical bond percolation on a seven-dimensional hypercubic lattice and a complete graph.
• Classification of occupied bonds into bridge and nonbridge types.
• Analysis of cluster size distributions, fractal dimensions, and scaling behaviors.

Main Results:

• Identical volume fractal dimension (d_f*=2/3) and exponent (τ=5/2) for cluster size distributions in both systems.
• Distinct behaviors in the fraction of nonbridge bonds (finite in 7D, vanishing in CG) and leaf-free cluster dimensions.
• The probability distribution of the largest cluster size collapses for both systems, indicating universal scaling behavior.

Conclusions:

• While some scaling properties are universal, the geometric structure of high-dimensional percolation clusters is not fully captured by complete graph models.
• The classification and analysis of bond types reveal fundamental differences in cluster connectivity and topology.
• This study highlights the importance of lattice dimensionality in determining the intricate geometric properties of random networks.