  The random Lorentz gas (RLG) is a minimal model of both percolation and
glassiness, which leads to a paradox in the infinite-dimensional,
$d\rightarrow\infty$ limit: the localization transition is then expected to be
continuous for the former and discontinuous for the latter. As a putative
resolution, we have recently suggested that as $d$ increases the behavior of
the RLG converges to the glassy description, and that percolation physics is
recovered thanks to finite-$d$ perturbative and non-perturbative (instantonic)
corrections [Biroli et al. arXiv:2003.11179]. Here, we expand on the
$d\rightarrow\infty$ physics by considering a simpler static solution as well
as the dynamical solution of the RLG. Comparing the $1/d$ correction of this
solution with numerical results reveals that even perturbative corrections fall
out of reach of existing theoretical descriptions. Comparing the dynamical
solution with the mode-coupling theory (MCT) results further reveals that
although key quantitative features of MCT are far off the mark, it does
properly capture the discontinuous nature of the $d\rightarrow\infty$ RLG.
These insights help chart a path toward a complete description of
finite-dimensional glasses.
