Percolation of random compact diamond-shaped systems on the square lattice

Area of Science:

• Statistical Physics
• Materials Science
• Network Theory

Background:

• Site percolation models are crucial for understanding connectivity in disordered systems.
• Diamond-shaped neighborhoods introduce complex connectivity patterns.
• Previous studies focused on monodisperse or single-sized neighborhoods.

Purpose of the Study:

• To analyze site percolation with random diamond-shaped neighborhoods of varying radii.
• To investigate the convergence of the product of average neighbors and critical threshold.
• To explore the relationship between discrete percolation models and continuum models.

Main Methods:

• Simulation of site percolation on a square lattice.
• Analysis of neighborhoods with radii uniformly chosen from a set {i, i+1, ..., m}.
• Examination of the critical percolation threshold (p_c) and average number of neighbors (z_bar).

Main Results:

• For fixed i, z_bar(i,m) * p_c(i,m) converges to a constant as m approaches infinity.
• When i=m (single diamond sizes), z(i) * p_c(i) converges to 2^d * eta_c (continuum percolation threshold).
• The model shows differences compared to systems of deposited objects with size distributions.

Conclusions:

• The study confirms expected convergence behaviors in percolation models with random neighborhood sizes.
• It highlights the limitations of direct mapping from discrete to continuum models when object sizes are distributed.
• Findings contribute to understanding connectivity in systems with complex, random structures.