  We introduce a family of glassy models having a parameter, playing the role
of an interaction range, that may be varied continuously to go from a system of
particles in d dimensions to a mean-field version of it. The mean-field limit
is exactly described by equations conceptually close, but different from, the
Mode-Coupling equations. We obtain these by a dynamic virial construction.
Quite surprisingly we observe that in three dimensions, the mean-field behavior
is closely followed for ranges as small as one interparticle distance, and
still qualitatively for smaller distances. For the original particle model, we
expect the present mean-field theory to become, unlike the Mode-Coupling
equations, an increasingly good approximation at higher dimensions.
