  The random Lorentz gas is a minimal model for transport in heterogeneous
media. Upon increasing the obstacle density, it exhibits a growing subdiffusive
transport regime and then a dynamical arrest. Here, we study the dimensional
dependence of the dynamical arrest, which can be mapped onto the void
percolation transition for Poisson-distributed point obstacles. We numerically
determine the arrest in dimensions d=2-6. Comparing the results with standard
mode-coupling theory reveals that the dynamical theory prediction grows
increasingly worse with $d$. In an effort to clarify the origin of this
discrepancy, we relate the dynamical arrest in the RLG to the dynamic glass
transition of the infinite-range Mari-Kurchan model glass former. Through a
mixed static and dynamical analysis, we then extract an improved dimensional
scaling form as well as a geometrical upper bound for the arrest. The results
suggest that understanding the asymptotic behavior of the random Lorentz gas
may be key to surmounting fundamental difficulties with the mode-coupling
theory of glasses.
