Dynamical transition of glasses: from exact to approximate

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.