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date: 22 September 2019

Abstract and Keywords

This article reviews the theory of random matrices with eigenvalues distributed in the complex plane and more general ‘beta ensembles’ (logarithmic gases in 2D). It first considers two ensembles of random matrices with complex eigenvalues: ensemble C of general complex matrices and ensemble N of normal matrices. In particular, it describes the Dyson gas picture for ensembles of matrices with general complex eigenvalues distributed on the plane. It then presents some general exact relations for correlation functions valid for any values of N and β before analysing the distribution and correlations of the eigenvalues in the large N limit. Using the technique of boundary value problems in two dimensions and elements of the potential theory, the article demonstrates that the finite-time blow-up (a cusp–like singularity) of the Laplacian growth with zero surface tension is a critical point of the normal and complex matrix models.

Keywords: random matrices, complex plane, beta ensemble, Dyson gas, correlation function, eigenvalue, N limit, Laplacian growth, matrix model

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