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

In the present paper, fixed trace $β$-Hermite ensembles generalizing the fixed trace Gaussian Hermite ensemble 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 GOE, GUE and GSE. Furthermore, for even $β$, analogous to $β$-Hermite ensembles, a multiple integral of the Konstevich type can be obtained.

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