  One of the most actively debated issues in the study of the glass transition
is whether a mean-field description is a reasonable starting point for
understanding experimental glass formers. Although the mean-field theory of the
glass transition -- like that of other statistical systems -- is exact when the
spatial dimension $d\rightarrow\infty$, the evolution of systems properties
with $d$ may not be smooth. Finite-dimensional effects could dramatically
change what happens in physical dimensions, $d=2,3$. For standard phase
transitions finite-dimensional effects are typically captured by
renormalization group methods, but for glasses the corrections are much more
subtle and only partially understood. Here, we investigate hopping between
localized cages formed by neighboring particles in a model that allows to
cleanly isolate that effect. By bringing together results from replica theory,
cavity reconstruction, void percolation, and molecular dynamics, we obtain
insights into how hopping induces a breakdown of the Stokes--Einstein relation
and modifies the mean-field scenario in experimental systems. Although hopping
is found to supersede the dynamical glass transition, it nonetheless leaves a
sizable part of the critical regime untouched. By providing a constructive
framework for identifying and quantifying the role of hopping, we thus take an
important step towards describing dynamic facilitation in the framework of the
mean-field theory of glasses.
