Unifying the percolation and mean-field description of the random Lorentz gas

The random Lorentz gas (RLG) is a minimal off-lattice model of transport in porous media. It is also a minimal model of structural glasses. The exact mean-field, infinite-dimensional limit solution of the RLG indeed predicts a discontinuous dynamical caging transition akin to that of simple liquid glass formers. The RLG, however, is also in the percolation universality class, in which the dynamical caging transition is continuous. These two descriptions are thus fundamentally contradictory. To resolve this paradox, we study the caging regime of the RLG as a function of dimension. We find that the static cage size quantitatively matches the mean-field predictions, and that the percolation transition and the (finite-dimensional echo of the) dynamical transition are clearly distinguishable in all dimensions. As dimension increases, however, the system dynamics grows increasingly controlled by the dynamical transition. In fact, the escape time from the dynamical cage grows exponentially quickly with d, thus resolving the paradox. More significantly, cage escape events are similar to hopping processes, which had mostly eluded theoretical description in standard supercooled liquids.