Quantized Laplacian growth, II: 1D hydrodynamics of the Loewner density

A systematic analytic treatment of fluctuations in Laplacian growth is given. The growth process is regularized by a short-distance cutoff $\mathrm{\hslash }$ preventing the cusps production in a finite time. This regularization mechanism generates tiny inevitable fluctuations on a microscale, so that the interface dynamics becomes chaotic. The time evolution of fluctuations can be described by the universal Dyson Brownian motion, which reduces to the complex viscous Burgers equation in the hydrodynamic approximation. Because of the intrinsic instability of the interface dynamics, tiny fluctuations of the interface on a microscale generate universal patterns with well developed fjords and fingers in a long time asymptotic.