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A systematic analytic treatment of fluctuations in Laplacian growth is given.The growth process is regularized by a short-distance cutoff $\hbar$ preventingthe cusps production in a finite time. This regularization mechanism generatestiny inevitable fluctuations on a microscale, so that the interface dynamicsbecomes chaotic. The time evolution of fluctuations can be described by theuniversal Dyson Brownian motion, which reduces to the complex viscous Burgersequation in the hydrodynamic approximation. Because of the intrinsicinstability of the interface dynamics, tiny fluctuations of the interface on amicroscale generate universal patterns with well developed fjords and fingersin a long time asymptotic.

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