Gradient Percolation — Invasion Front Roughness

Percolation probability p(x) varies linearly across the lattice. The invasion front traces the percolation threshold p_c≈0.593. Front width scales as ξ ~ g^{-ν/(1+ν)} with ν=4/3.

Blue = occupied. Orange line = invasion front. Width of front ~ g^{-4/7} ≈ percolation correlation length.