  The richness of the mean-field solution of simple glasses leaves many of its
features challenging to interpret. A minimal model that illuminates glass
physics the same way the random energy model clarifies spin glass behavior
would therefore be beneficial. Here, we propose such a real-space model that is
amenable to infinite-dimensional analysis and is exactly solvable at high and
low densities in finite dimension. By joining analysis with numerical
simulations, we uncover geometrical signatures of the dynamical and jamming
transitions and provides insight into the origin of activated processes.
Translating these findings to the context of standard glass formers further
reveals the role played by non-convexity in the emergence of Gardner physics.
