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

High-dimensional percolation criticality and hints of mean-field-like caging of the random Lorentz gas