Gilbert's disk model (1961): place Poisson-distributed points in the plane with intensity λ;
connect two points if their distance ≤ 2r. Percolation occurs when a giant component spans the system.
Critical threshold: λ_c π r² ≈ 1.128 (η_c ≈ 0.3574 in area fraction)
Below threshold: only finite clusters. Above: an infinite spanning cluster emerges.
The dimensionless parameter η = λπr² (area fraction covered) controls the transition.
This model describes wireless network connectivity, epidemic spread, and material conductivity.