KPZ Interface Growth
Kardar-Parisi-Zhang equation: ∂h/∂t = ν∇²h + (λ/2)(∇h)² + η(x,t). The nonlinear λ term breaks up-down symmetry and puts the interface in the KPZ universality class with roughness exponent β=1/3, not β=1/4 (Edwards-Wilkinson, λ=0).
KPZ universality: The 1D KPZ equation (Kardar, Parisi, Zhang 1986) has exact exponents χ=1/2 (roughness), β=1/3 (growth), z=3/2 (dynamic). W(t) = ⟨(h-⟨h⟩)²⟩^{1/2} grows as t^{1/3} before saturating at L^{1/2}. The λ term creates a preferred direction of growth, relevant to bacterial colony fronts, fire spreading, paper burning. The EW class (λ=0) has β=1/4, z=2. Tracy-Widom (GUE) distribution governs height fluctuations at long times — connecting interface growth to random matrix theory.