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The random Lorentz gas (RLG) is a minimal model of transport in heterogeneous media that exhibits a continuous localization transition controlled by void space percolation. The RLG also provides a toy model of particle caging, which is known to be relevant for describing the discontinuous dynamical transition of glasses. In order to clarify the interplay between the seemingly incompatible percolation and caging descriptions of the RLG, we consider its exact mean-field solution in the infinite-dimensional d→∞ limit and perform numerics in d=2...20. We find t