Quantized Laplacian growth, I: Statistical theory of Laplacian growth

Abstract

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.