Continuum Percolation: Boolean Model

Boolean model: Poisson point process with intensity λ; each point gets disk of radius r.

Coverage fraction: P(covered) = 1−e^{−λπr²}

Critical intensity: η_c = λ_c πr² ≈ 1.128
(critical density λ_c ≈ 0.359/r²)

At threshold, a spanning cluster first connects opposite sides — the percolation transition.

η_c ≈ 1.128 Threshold λ_c·πr² exact is open