Universality of stochastic Laplacian growth

Abstract: We consider a stochastic Laplacian growth problem in the framework of normal random matrices. In the large $N$ limit the support of eigenvalues of random matrices is a planar domain with a sharp boundary which evolves under a change in the size of matrices. This evolution can be interpreted as a stochastic growth process. We show that the most probable growth scenario is similar to deterministic Laplacian growth. The other scenarios involve Laplacian growth in the presence of fluctuations. We use the random matrix approach to determine a probability distribution function of fluctuations. The partition function of fluctuations is shown to be universal. It does not depend on the shape of the initial domain and depends on the strength of fluctuations and the geometry of the problem only.