Data and scripts from: Glass-like caging with random planes

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 $d\rightarrow\infty$ analysis and is exactly solvable in finite $d$ in some regimes. By joining analysis with numerical simulations, we uncover geometrical signatures of the dynamical and jamming transitions and obtain insight into the origin of activated processes. Translating these findings into the context of standard glass formers further reveals the role played by non-convexity in the emergence of Gardner and jamming physics.