Fixed trace $\beta$-Hermite ensembles: Asymptotic eigenvalue density and the edge of the density

TL;DR

This paper studies fixed trace $eta$-Hermite ensembles, proving the Wigner semicircle law for all $eta$, analyzing edge behavior, and deriving explicit limits for classical ensembles, using moment methods and asymptotic analysis.

Contribution

It extends the understanding of fixed trace $eta$-Hermite ensembles by establishing the semicircle law and edge limits, including explicit results for classical cases.

Findings

Proves Wigner semicircle law for all $eta$.

Establishes edge scaling limits for fixed trace ensembles.

Provides explicit limits for fixed trace GOE, GUE, GSE.

Abstract

In the present paper, fixed trace $\beta$-Hermite ensembles generalizing the fixed trace Gaussian Hermite ensemble are considered. For all $\beta$, we prove the Wigner semicircle law for these ensembles by using two different methods: one is the moment equivalence method with the help of the matrix model for general $\beta$, the other is to use asymptotic analysis tools. At the edge of the density, we prove that the edge scaling limit for $\beta$-HE implies the same limit for fixed trace $\beta$-Hermite ensembles. Consequently, explicit limit can be given for fixed trace GOE, GUE and GSE. Furthermore, for even $\beta$, analogous to $\beta$-Hermite ensembles, a multiple integral of the Konstevich type can be obtained.