Universality of stochastic Laplacian growth

Abstract

We consider a stochastic Laplacian growth model within the framework of normal random matrices. In the limit of large matrix size, the support of eigenvalues forms a planar domain with a sharp boundary that evolves stochastically as the matrix size increases. We show that the most probable growth scenario is similar to deterministic Laplacian growth, while alternative scenarios illustrate the impact of fluctuations. We prove that the probability distribution function of fluctuations is given by the circular unitary ensemble introduced by Dyson in 1962. The partition function of fluctuations is shown to be universal, depending solely on the fluctuation intensity and the problem’s geometry, regardless of the initial domain shape.