Random matrices, random polynomials and Coulomb systems

Abstract

It is well known that the joint probability density of the eigenvalues of Gaussian ensembles of random matrices may be interpreted as a Coulomb gas. We review these classical results for hermitian and complex random matrices, with special attention devoted to electrostatic analogies. We also discuss the joint probability density of the zeros of polynomials whose coefficients are complex Gaussian variables. This leads to a new two-dimensional solvable gas of interacting particles, with non-trivial interactions between particles.Comment: 8 pages, to appear in the Proceedings of the International Conference on Strongly Coupled Coulomb Systems, Saint-Malo, 199