Continuum Percolation: Random Disk Model

Boolean Disk Model: N disks of radius r are placed uniformly at random in the unit square. Two disks overlap if their center distance < 2r. Area fraction φ = Nπr² (before boundary correction). The percolation threshold φ_c ≈ 0.6763 (Quintanilla-Ziff 2007) for spanning cluster from left to right.

Universality: Despite being a continuous model, the critical exponents are identical to 2D lattice percolation (same universality class): β=5/36, ν=4/3, η=5/24. Only the threshold value depends on geometry, not the critical behavior.

Excluded area: Each pair of disk centers has excluded area π(2r)² = 4πr². The mean number of overlapping neighbors at threshold: ⟨n_c⟩ ≈ 4.512. This is the continuum analog of the site percolation threshold times coordination number.

Colors: Each connected cluster has a unique color. Red = spanning cluster (if it exists). White disks are isolated.