The Level Densities of Random Matrix Unitary Ensembles and their Perturbation Invariability

TL;DR

This paper investigates the level densities of various random matrix unitary ensembles using operator methods, demonstrating their limit spectral distributions and showing invariance under polynomial perturbations.

Contribution

It introduces a general operator-based approach to analyze level densities and establishes their perturbation invariability in the weak sense.

Findings

Limit spectral distributions for Gaussian, Laguerre, and Jacobi ensembles are recovered.

Level densities remain invariant under polynomial multiplicative perturbations.

Operator methods effectively analyze ensemble spectral properties.

Abstract

Using operator methods, we generally present the level densities for kinds of random matrix unitary ensembles in weak sense. As a corollary, the limit spectral distributions of random matrices from Gaussian, Laguerre and Jacobi unitary ensembles are recovered. At the same time, we study the perturbation invariability of the level densities of random matrix unitary ensembles. After the weight function associated with the 1-level correlation function is appended a polynomial multiplicative factor, the level density is invariant in the weak sense.