  The study of the properties of glass-forming liquids is difficult for many
reasons. Analytic solutions of mean field models are usually available only for
systems embedded in a space with an unphysically high number of spatial
dimensions; on the experimental and numerical side, the study of the properties
of metastable glassy states requires to thermalize the system in the
supercooled liquid phase, where the thermalization time may be extremely large.
We consider here an hard-sphere mean field model which is solvable in any
number of spatial dimensions; moreover we easily obtain thermalized
configurations even in the glass phase. We study the three dimensional version
of this model and we perform Monte Carlo simulations which mimic heating and
cooling experiments performed on ultra-stable glasses. The numerical findings
are in good agreement with the analytical results and qualitatively capture the
features of ultra-stable glasses observed in experiments.
