Generalized Gaussian Random Unitary Matrices Ensemble

TL;DR

This paper introduces a generalized ensemble of Hermitian matrices, analyzes the asymptotic eigenvalue density, and proves convergence to a generalized semi-circle law as matrix size increases.

Contribution

It extends the classical Wigner semi-circle law to a broader class of Hermitian matrices called the Generalized Hermitian or Chiral ensemble, with explicit density formulas and convergence results.

Findings

Eigenvalue density converges to a generalized semi-circle law for large matrices.

Laplace transform of finite-n density expressed via hypergeometric functions.

Provides asymptotic analysis of the eigenvalue distribution.

Abstract

We describe Generalized Hermitian matrices ensemble sometimes called Chiral ensemble. We give global asymptotic of the density of eigenvalues or the statistical density. We will calculate a Laplace transform of such a density for finite $n$, which will be expressed through an hypergeometric function. When the dimensional of the hermitian matrix begin large enough, we will prove that the statistical density of eigenvalues converge in the tight topology to some probability measure, which generalize the Wigner semi-circle law.