Random matrices and Laplacian growth

TL;DR

This paper reviews the theory of random matrices and beta-ensembles, revealing their equivalence to Laplacian growth models in viscous flows, and discusses analytical methods involving boundary value problems and potential theory.

Contribution

It establishes a mathematical connection between random matrix eigenvalue distributions and Laplacian growth processes, providing new analytical insights.

Findings

Eigenvalue distributions in large N limit match growth models

Equivalence between beta-ensembles and viscous flow growth processes

Analytical methods involve boundary value problems and potential theory

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.