Random matrices, random polynomials and Coulomb systems

TL;DR

This paper reviews classical Coulomb gas interpretations of eigenvalues in Gaussian random matrices and introduces a new solvable two-dimensional interacting particle system related to polynomial zeros.

Contribution

It provides a comprehensive review of electrostatic analogies in random matrix theory and introduces a novel solvable model for interacting particles based on polynomial zeros.

Findings

Eigenvalue distributions interpreted as Coulomb gases

Electrostatic analogies elucidate matrix eigenvalue behavior

New solvable two-dimensional particle system based on polynomial zeros

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.