KPZ Interface Roughening

The KPZ equation (Kardar, Parisi, Zhang 1986) describes the stochastic growth of an interface: ∂h/∂t = ν∇²h + (λ/2)(∇h)² + η, where h(x,t) is the height, the first term is surface tension (diffusion), the nonlinear term drives asymmetric growth, and η is white noise. The interface width W = √⟨(h−⟨h⟩)²⟩ grows as W ~ t^β with KPZ exponent β=1/3 in 1+1D (vs. β=1/4 for the linear Edwards-Wilkinson equation with λ=0). At late times W saturates at W ~ L^α with roughness exponent α=1/2. KPZ universality class includes ballistic deposition, TASEP, and random matrix GUE statistics.