  The random Lorentz gas (RLG) is a minimal model of transport in heterogeneous
media. It also models the dynamics of a tracer in a glassy system. These two
perspectives, however, are fundamentally inconsistent. Arrest in the former is
related to percolation, and hence continuous, while glass-like arrest is
discontinuous. In order to clarify the interplay between percolation and
glassiness in the RLG, we consider its exact solution in the
infinite-dimensional $d\rightarrow\infty$ limit, as well as numerics in
$d=2\ldots 20$. We find that the mean field solutions of the RLG and glasses
fall in the same universality class, and that instantonic corrections related
to rare cage escapes destroy the glass transition in finite dimensions. This
advance suggests that the RLG can be used as a toy model to develop a
first-principle description of hopping in structural glasses.
