  We regularize the Laplacian growth problem with zero surface tension by
introducing a short-distance cutoff $\hbar$, so that the change of the area of
domains is quantized and equals an integer multiple of the area quanta $\hbar$.
The domain can be then considered as an aggregate of tiny particles (area
quanta) obeying the Pauli exclusion principle. The statistical theory of
Laplacian growth is introduced by using Laughlin's description of the integer
quantum Hall effect. The semiclassical evolution of the aggregate is similar to
classical deterministic Laplacian growth. However, the quantization procedure
generates inevitable fluctuations at the edge of the droplet. The statistical
properties of the edge fluctuations are universal and common to that of quantum
chaotic systems, which are generally described by Dyson's circular ensembles on
symmetric unitary matrices.
