Generalized $\beta$-Gaussian Ensemble Equilibrium measure method

TL;DR

This paper studies the eigenvalue distribution of generalized $eta$-Hermitian matrices, extending classical laws like Wigner's semi-circle law through equilibrium measure methods in the large matrix limit.

Contribution

It introduces a generalized equilibrium measure approach to analyze eigenvalue densities of $eta$-Hermitian ensembles, broadening understanding beyond classical cases.

Findings

Eigenvalue density asymptotics match generalized semi-circle law

Method applies to chiral and other $eta$-ensembles

Provides a framework for large $n$ eigenvalue analysis

Abstract

We investigate $\beta$-Generalized random Hermitian matrices ensemble sometimes called Chiral ensemble. We give global asymptotic of the density of eigenvalues or the statistical density. We investigate general method names as equilibrium measure method. When taking $n$ large limit we will see that the asymptotic density of eigenvalues generalize the Wigner semi-circle law.