Discs of radius r centered at Poisson process points. Find the percolation threshold λ_c · πr² ≈ 1.128 and watch the giant connected component emerge at the critical density.
Boolean model / Gilbert graph (1961): discs of radius r at Poisson(λ) points. Critical: λ_c = 1/(π·r²) (in 2D, λ_c·π·r²≈1.128 exactly). Equivalently: critical coverage fraction η_c=1-e^{-λ_c·πr²}≈0.677. Above λ_c: infinite connected component percolates. Connected to wireless network connectivity (Gupta-Kumar 1998), random geometric graph theory. Critical exponents same as 2D lattice percolation (universality).