  In this paper we find spectral properties in the large $N$ limit of Dirac
operators that come from random finite noncommutative geometries. In particular
for a Gaussian potential the limiting eigenvalue spectrum is shown to be
universal regardless of the geometry and is given by the convolution of the
semicircle law with itself. For simple non-Gaussian models this convolution
property is also evident. In order to prove these results we show that a wide
class of multi-trace multimatrix models have a genus expansion.
