Fixed trace β-Hermite ensembles: Asymptotic eigenvalue density and the edge of the density

Abstract

In the present paper, fixed trace β-Hermite ensembles generalizing the fixed trace Gaussian ensembles are considered. For all β, 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 β, the other is to use asymptotic analysis tools. At the edge of the density, we prove that the edge scaling limit for β-HE implies the same limit for fixed trace β-Hermite ensembles. Consequently, explicit limit can be given for fixed trace Gaussian orthogonal, unitary, and symplectic ensembles. Furthermore, for even β, analogous to β-Hermite ensembles, a multiple integral of the Konstevich type can be obtained.