Density of eigenvalues and its perturbation invariance in unitary ensembles of random matrices

TL;DR

This paper investigates the eigenvalue density in unitary random matrix ensembles, deriving differential equations for the density and proving its invariance under polynomial perturbations of the weight function.

Contribution

It introduces a method to compute eigenvalue densities via differential equations and establishes their invariance under polynomial weight perturbations.

Findings

Derived differential equations for eigenvalue densities.

Calculated moments of the density directly.

Proved invariance of density under polynomial perturbations.

Abstract

We generally study the density of eigenvalues in unitary ensembles of random matrices from the recurrence coefficients with regularly varying conditions for the orthogonal polynomials. First we calculate directly the moments of the density. Then, by studying some deformation of the moments, we get a family of differential equations of first order which the densities satisfy (see Theorem 1.2), and give the densities by solving them. Further, we prove that the density is invariant after the polynomial perturbation of the weight function (see Theorem 1.5).