Random matrices and Laplacian growth

Abstract

The theory of random matrices with eigenvalues distributed in the complex plane and more general "beta-ensembles" (logarithmic gases in 2D) is reviewed. The distribution and correlations of the eigenvalues are investigated in the large N limit. It is shown that in this limit the model is mathematically equivalent to a class of diffusion-controlled growth models for viscous flows in the Hele-Shaw cell and other growth processes of Laplacian type. The analytical methods used involve the technique of boundary value problems in two dimensions and elements of the potential theory.