How universal is the Wigner distribution?

We consider Gaussian ensembles of m complex N × N matrices. We identify an enhanced symmetry in the system and the resultant closed subsector, which is naturally associated with the radial sector of the theory. The density of radial eigenvalues is obtained in the large N limit. It is of the Wigner form only for m = 1. For m ⩾ 2, a new form of the density is obtained. (paper)