Compact support probability distributions in random matrix theory

We consider a generalization of the fixed and bounded trace ensembles introduced by Bronk and Rosenzweig up to an arbitrary polynomial potential. In the large-$n$ limit we prove that the two are equivalent and that their eigenvalue distribution coincides with that of the canonical ensemble with measure $\exp [-n\mathrm{Tr}V(M\left)\mathrm{}\right].$ The mapping of the corresponding phase boundaries is illuminated in an explicit example. In the case of a Gaussian potential we are able to derive exact expressions for the one- and two-point correlator for finite n, having finite support.