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

摘要

In the present paper, fixed trace beta-Hermite ensembles generalizing the fixed trace Gaussian ensembles 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 Gaussian orthogonal, unitary, and symplectic ensembles. Furthermore, for even beta, analogous to beta-Hermite ensembles, a multiple integral of the Konstevich type can be obtained.