The random Lorentz gas (RLG) is a minimal model for transport in disordered media. Despite the broad relevance of the model theoretical grasp over its properties remains weak. For instance the scaling with dimension $d$ of its localization transition at the void percolation threshold is not well controlled analytically nor computationally. A recent study [Biroli et al. Phys. Rev. E 103 L030104 (2021)] of the caging behavior of the RLG motivated by the mean-field theory of glasses has uncovered physical inconsistencies in that scaling that heighten the need for guidance. Here we first extend analytical expectations for asymptotic high-d bounds on the void percolation threshold and then computationally evaluate both the threshold and its criticality in various d. In high-d systems we observe that the standard percolation physics is complemented by a dynamical slowdown of the tracer dynamics reminiscent of mean-field caging. A simple modification of the RLG is found to bring the interplay between percolation and mean-field-like caging down to d=3.
