  Glassy dynamics is intermittent, as particles suddenly jump out of the cage
formed by their neighbours, and heterogeneous, as these jumps are not uniformly
distributed across the system. Relating these features of the dynamics to the
diverse local environments explored by the particles is essential to
rationalize the relaxation process. Here we investigate this issue
characterizing the local environment of a particle with the amplitude of its
short time vibrational motion, as determined by segmenting in cages and jumps
the particle trajectories. Both simulations of supercooled liquids and
experiments on colloidal suspensions show that particles in large cages are
likely to jump after a small time-lag, and that, on average, the cage enlarges
shortly before the particle jumps. At large time-lags, the cage has essentially
a constant value, which is smaller for longer-lasting cages. Finally, we
clarify how this coupling between cage size and duration controls the average
behaviour and opens the way to a better understanding of the relaxation process
in glass--forming liquids.
