The random Lorentz gas (RLG) is a minimal model of both percolation and glassiness which leads to a paradox in the infinite-dimensional 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. Phys. Rev. E {2021} 103 L030104]. Here we expand on the d→∞ 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→∞ RLG. These insights help chart a path toward a complete description of finite-dimensional glasses.
